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METEOROLOGY

THEORETICAL AND

APPLIED

By
K. WENDELL HEWSON, Ph.D.,

1)1 It K(. I OK, I>lKH;s|0> IMtOJKCT
lilt- M \SS4CIHJSKTTS irShl'ITIJTF. OF TI-C II NOF.OGY

and

RICHMOND W. LONGLEY, M.A.

MKTKOKOl.or.IST IN TH K M KTKOUOL>GI<: 4L
8KKVIf:iC OK r \I\4D 4

NEW YORK

JOHN WILEY &: SONS, INC.
CHAPMAN &: HALL, LTD.

BY
EDGAR WENDELL HEWSON

AND
RICHMOND WILBRRFORCE LONGLEY

This book or any part thereof must not
be reproduced in any form without
written permission of the publisher.

SECOND PRINTING, AUGUST, 1951

PRINTED IN THE UNITED STATES OF AMERICA

PREFACE

Two main considerations have prompted us to write a text on meteor-
ology. Recent books on the subject are either too elementary or too
advanced for effective use in our own and similar classes for beginners;
the need for an introductory treatment of basic meteorological theory
is particularly great. In the second place, a closer integration of fore-
casting technique with the theory on which it is founded is desirable.
The present work endeavors to meet both these needs.

The development of the subject is an outgrowth of experience gained
by one of us (Hewson) in 1938 and for several years thereafter in teach-
ing students of meteorology in the University of Toronto, and more
recently by both of us in instructing technical personnel entering the
Meteorological Service of Canada. The book is designed to appeal
to two types of readers: (1) persons studying meteorology for the first
time, either in a university, in one of the official weather services, or in a
private meteorological organization; (2) practising forecasters who
may wish to brush up on the latest developments in their field. An
elaborate knowledge of mathematics and physics is not presumed; the
development should be followed without difficulty by anyone with an
understanding of basic physical principles who has had a first course in
calculus. Even the knowledge of calculus is not essential for mast of
the second part of the book.

The two parts, the first on basic theory and the second on applications,
may be studied at the same time or consecutively. Each is to a certain
extent a self-contained unit, although cohesion is maintained by nu-
merous cross references. Thus those interested primarily in the theory
will concentrate on the first part, dipping into the applications only here
and there, whereas the student of forecasting will spend most of his
time on the second part, going back to the theory only when he wishes
to learn the assumptions. on which a given result depends. Mathe-
matical formulas used in the applied meteorology are derived rigorously
in the theoretical portion so that the student may judge for himself the
scope and degree of validity of any given equation.

The aim of the book is to give a well-rounded view of the essentials of
the science of meteorology at the present time, as shown by the inclusion
of chapters on instruments and observations, climatology, map analysis

iii

iv PREFACE

and forecasting procedure, the applications of meteorology to other
specialized fields, and the statistical analysis of meteorological data.
So far as the authors know the last-named subject is treated here for
the first time in any textbook. The problems and exercises provide a
means for the student to clarify his ideas on the topics discussed.

We are indebted to the following persons and organizations for per-
mission to reproduce a number of illustrations: Mr. W. H. Bigg, Pro-
fessor J. Bjerknes, Professor S. Chapman, Dr. W. Elsasser, Mr. J. J.
George and Eastern Airlines, Professor B. Haurwitz and McGraw-Hill
Book Co., Mr. R. K. Linsley, Jr., and the U.S. Weather Bureau, Mac-
millan and Co., London, Mr. W. E. K. Middleton, Mr. C. M. Penner,
Dr. S. Petterssen, the Royal Society of London, Sir Napier Shaw, Sir
George Simpson and the Royal Meteorological Society, Dr. G. I. Taylor,
and Professor G. T. Trewartha and McGraw-Hill Book Co. We wish
to thank Mr. H. T. Gisbome, the U.S. Forest Service, and the U.S.
Weather Bureau for the originals of various figures. We are also in-
debted to Dr. N. K. Johnson, Professor H. Landsberg, Mr. C. M. Penner,
and to the Toronto Transportation Commission.

We regret that it has been impossible under present conditions to
obtain permission for the use of certain diagrams. Full acknowledg-
ment, however, has in all cases been made at the point of insertion.

A number of other persons have provided greatly appreciated assist-
ance. Among these are Mr. J. M. Leaver who made a number of valu-
able suggestions of material which could advantageously be included
and Mr. W. E. K. Middleton who suggested several revisions. We are
also greatly indebted to Dr. D. Brunt, Professor of Meteorology in the
University of London, under whom one of us (Hewson) studied for
several years. We have drawn freely on the material covered in Pro-
fessor Brunt's lectures.

The careful typing of the manuscript by Mr. D. W. Robertson and
the accurate drafting of the figures by Mr. Frank Rayfield are much
appreciated.

E. WENDELL HEWSON
RICHMOND W. LONGLEY

Toronto
November, 1943

CONTENTS

CHAPTER PAGE

1. OBSERVATIONAL FACTS OF THE ATMOSPHERE

1. Surface Temperature Distribution 1

2. Upper Air Temperature Distribution 4

3. The General Pressure Distribution over the Earth 8

4. Wind Distribution 10

5. Composition of the Atmosphere 11

PART L THEORETICAL METEOROLOGY

2. STATICS OF THE ATMOSPHERE

6. Pressure and Temperature Units 14

7. The Equation of State for Dry Air 15

8. Geopotential 16

9. The Variation of Pressure with Altitude 18

10. Units of Atmospheric Water Vapor Content 20

11. Virtual Temperature and Height Computations 23

3. THERMODYNAMICS OF DRY AIR

12. The First Law of Thermodynamics 28

13. Adiabatic Relationships for Dry Air. Potential Temperature 30

14. The Dry Adiabatic Lapse Rate. The Stability of Dry Air 31

15. The Adiabatic Lapse Rate for Moist, Unsaturated Air .... 34

16. The Effect of Ascent and Descent on Lapse Rate and Sta-

bility 36

17. The Carnot Cycle. Heat Engine Efficiency 40

18. Entropy 43

4. THERMODYNAMICS OF MOIST AIR

19. The Clausius-Clapeyron Equation 47

20. The Saturated Adiabatic Lapse Rate. The Stability of

Saturated Air 50

21. Condensation Level. Dew-Point Changes in Adiabatic

Motion 52

22. Therrnodynamic Diagrams 54

23. The Psychrometer Equation 59

24. Wet-Bulb Temperature. Wet-Bulb Potential Temperature 60

25. Equivalent Temperature. Equivalent Potential Tempera-

ture. Rossby Diagram 63

vi CONTENTS

CHAPTER PAGE

5. RADIATION IN THE ATMOSPHERE

26. The Laws of Black-Body Radiation 67

27. The Law of Absorption 70

28. Solar Radiation 72

29. Terrestrial and Nocturnal Radiation 76

30. Radiation with Cloudy Skies 80

31. The Heat Balance in the Atmosphere 82

32. Radiative Equilibrium in the Stratosphere 85

6. ATMOSPHERIC MOTIONS UNDER BALANCED FORCES

33. The Pressure Gradient Force 88

34. The Deflecting Force of the Earth's Rotation (Coriolis Force) 90

35. The Geostrophic Wind Equation 96

36. The Thermal Wind Component 98

37. The Gradient Wind Equations 101

7. FRONTAL SURFACES

38. The Slope of Frontal Surfaces 105

39. The Pressure Trough at Fronts 108

40. Pressure Tendencies below a Frontal Surface 110

41. Frontogenesis and Front olysis. Temperature Gradient

Changes in a Wind Field 112

8. GENERAL KINEMATICS AND DYNAMICS OF AIR MOTIONS

42. The Movement of Significant Curves in the Pressure Field. 115

43. The Movement of Troughs, Wedges, and Fronts 118

44. Streamlines and Trajectories 123

45. The Equation of Continuity. Divergence and Convergence 125

46. The Isallobaric Wind 128

47. The Origin of Pressure Changes 131

48. The Movement of Upper Troughs and Wedges 132

49. Circulation and Vorticity 135

50. Rate of Change of Circulation 136

9. TURBULENCE

51. Streamline and Turbulent Motion 144

52. Turbulent Transfer of Momentum. The Wind Variation

with Height in the Frictional Layer 145

53. Turbulent Transfer of Heat 150

54. Turbulent Transfer of Matter 154

55. Evaluation of the Coefficient of Eddy Diffusivity 157

10. STATISTICAL ANALYSIS OF METEOROLOGICAL DATA

56. The Purpose of Statistics 161

57. Measures of Central Tendency. Computation of the Mean 161

CONTENTS vii

CHAPTER PAGE

58. Measures of Variability. Standard Deviation .' 100

59. The Theory of Errors 109

00. Method of Least Squares 174

01. Correlation 177

02. Harmonic Analysis 179

PART II. APPLIED MKTKOROLOGY

11. METEOROLOGICAL INSTRUMENTS AND OBSERVATIONS

03. Pressure 180

04. Temperature 190

05. Humidity 192

00. Wind 193

07. Clouds 197

08. Precipitation 204

09. Sunshine and Radiation 200

70. Visibility 207

12. THE GENERAL CIRCULATION OVER THE EARTH

71. Circulation on a Non- Rotating (ilobe 209

72. The Effect of the Earth's Rotation 211

73. The Influence of the Land Masses 213

74. Air Masses and Their Source Regions 214

13. TEMPERATURE AND HUMIDITY IN THE ATMOSPHERE

75. The Temperature of the Air at the Earth's Surface 217

70. Temperature in the Free Air 218

77. Conservative and Representative Properties 221

78. Lapse Rate and Stability 222

79. Diurnal Temperature Variations 222

80. Potential Temperature 224

81. Water Vapor Content 224

82. Dew Point 220

83. Wet-Bulb Temperature 227

84. Wet-Bulb Potential Temperature 232

85. Equivalent and Equivalent Potential Temperatures 232

80. Summary of Degree of Conservatism of Properties 235

14. STABILITY AND INSTABILITY

87. General Considerations 230

88. Conditions Required for Stability 230

89. The Stability of Moist, Unsaturated Air 237

90. Conditional Instability 237

91. Stable Type of Conditional Instability 239

viii CONTENTS

CHAPTER PAGE

92. Latent Instability 240

93. Potential Instability 242

94. The Relationship between Latent and Potential Instability 244

95. The Development of Potential and Latent Instability 244

96. Summary of Stability and Instability of Moist, Unsaturated

Air 246

15. CHARACTERISTIC PROPERTIES OP DIFFERENT AIR MASSES

97. Systems of Classification 248

98. Polar Continental (P c ) Air Masses in Winter 249

99. Polar Maritime (PM) Air Masses in Winter 251

100. Tropical Maritime (TV) Air Masses in Winter 254

101. Tropical Continental (T c ) Air Masses in Winter 256

102. Polar Continental (Pc) Air Masses in Summer 257

103. Polar Maritime (P M ) Air Masses in Summer 258

104. Tropical Maritime (T M ) Air Masses in Summer 259

105. Tropical Continental (Tc) Air Masses in Summer 260

106. The Rossby Diagram as an Aid to Classification of Air

Masses 260

107. Comparison of Air Masses 262

16. CYCLONES AND ANTICYCLONES

108. General Characteristics of a Front 264

109. Frontogenesis and Frontolysis in Deformation Fields 265

110. Frontal Zones. Frontogenesis between Air Mass Source

Regions 271

111. Air Masses and Cyclones 272

112. Life History of a Frontal Depression 272

113. Upper Air Conditions above Frontal Depressions 281

114. Other Types of Depressions 282

115. Tropical Hurricanes. Tornadoes 283

116. Comparison of Upper Air Conditions above Cyclones and

Anticyclones 285

117. The Cold Anticyclone 287

118. The Warm Anticyclone 288

119. Convergence and Divergence in Cyclones and Anticyclones 289

17. WINDS

120. Geostrophic and Gradient Winds 295

121. Thermal Winds. Isallobaric Winds 298

122. Monsoon Winds 302

123. The Effects of Friction. Diurnal Variations 302

124. Slope and Valley Winds 306

125. Land and Sea Breezes 310

CONTENTS ix

CHAPTER PAGE

18. CONDENSATION AND PRECIPITATION

126. Saturation 312

127. Condensation on Nuclei 313

128. The Formation of Rain Droplets 314

129. The Types of Rainfall 315

130. Other Types of Precipitation 319

19. FORMATION AND DISSIPATION OF FOG

131. The Effect of Evaporation 322

132. Turbulent Mixing 325

133. Adiabatic Cooling 32G

134. Non-adiabatic Cooling 326

135. The Dissipation of Fog over a Snow Surface 334

136. The Forecasting of Fog 338

20. CLOUDS

137. Frontal Clouds 343

138. Convection Clouds 345

139. Turbulence Clouds 350

140. Orographic, Billow, and Artificial Clouds 355

141. Convergence Clouds 356

21. ICING ON AIRCRAFT

142. Types of Ice Deposit 361

143. Process of Deposition 362

144. Variation of Ice Deposits with Temperature and Season. 363

145. Icing, Cloud Forms, and Stability 365

146. Ice Formation by Supercooled Raindrops 367

147. Means of Avoiding Deposition of Ice 367

148. Forecasting the Deposition of Ice 368

22. THUNDERSTORMS

149. The Potential Gradient. Electrical Charges in the Atmos-

phere 370

150. The Origin of Thunderstorm Electricity 371

151. Variations in Time and Place of Occurrence of Thunder-

storms 373

152. Thunderstorm Forecasting 377

23. CLIMATOLOGY

153. Importance of Climatology to the Meteorologist 380

154. The Factors Governing Climate 381

155. Koppen's Classification of Climate 385

156. The Tropical Rainy Regions 387

157. The Arid Regions 388

X CONTENTS

CHAPTER PAGE

158. The Warm Temperate Rainy Regions 390

159. The Cold Snowy Forest Regions 393

160. The Polar Regions 394

24. MAP ANALYSIS AND FORECASTING PROCEDURE

161. Material Available for the Forecaster 396

162. The Plotting of Data on the Surface Chart 397

163. Construction of Isobars 398

164. The Identification of Air Masses and the Location of Fronts 401

165. The Use of Upper Air Data 403

166. Isentropic Analysis 405

167. Forecasting Positions of Pressure Fields and Fronts 408

168. Forecasting Condensation Phenomena 414

169. An Example of Map Analysis 415

170. Long-Range Forecasting 427

25. METEOROLOGY APPLIED TO VARIOUS HUMAN ACTIVITIES

171. Transportation 431

172. Agriculture 433

173. Forestry 436

174. Heavy Industry 438

175. Hydrology 442

176. Public Utilities 445

177. Sports 446

178. Retail Merchandise 451

APPENDIX 453

ANSWERS TO PROBLEMS AND EXERCISES 458

INDEX 459

SYMBOLS

Symbols used in the text but not in this list are defined in that portion of the
text in which they are used.

a Absolute humidity; absorptive power; Wien's constant.

A Reciprocal of the mechanical equivalent of heat; Austausch coeffi-
cient.

A Degrees Absolute.

c Velocity of pressure system; deviation of assumed mean from mean.

c p Specific licat of dry air at constant pressure.

c'pj c pm Specific heat of aqueous vapor and of moist unsaturated air at con-
stant pressure, respectively.

c v Specific heat of dry air at constant volume.

C Circulation.

C Degrees Centigrade.

Differentiation with respect to time on an individual particle.

(it
V

Local or partial differentiation with respect to time.
at

e Partial pressure of aqueous vapor; emissive power.

e s Saturation pressure of aqueous vapor.

E Radius of the earth; black-body radiation.

/ Relative humidity.

F Degrees Fahrenheit.

ff Acceleration of gravity.

/ Intensity of radiation.

,/ Mechanical equivalent of heat.

k Absorption coefficient.

K Coefficient of eddy diffusivity.

/ 2co sin 0; generalized absorption coefficient.

L Latent heat of vaporization.

m Molecular weight of air; mass of absorbing medium.

in' Molecular weight of aqueous vapor.

M Arithmetic mean.

AT Assumed arithmetic mean.

n Distance along a normal.

p Pressure.

Q Heat.

r Coefficient of correlation; radius.

R Gas constant for dry air.

OBSERVATIONAL FACTS OF THE ATMOSPHERE [Chap. 1

Sec. 1]

SURFACE TEMPERATURE DISTRIBUTION

OBSERVATIONAL FACTS OF THE ATMOSPHERE [Chap. 1

The warmest regions of the earth are found on the land masses, and
move north or south with the sun. In July the warmest regions, at
about 30 N latitude, are in the southwestern United States, in the
Sahara Desert, and in Arabia. During January the Australian Desert
is the wannest spot on the earth's surface.

Equatorial Radius
FIG. 3. The variation of free air temperature with latitude.

10 15 20 25
Dyn Km

2. Upper Air Temperature Distribution. The temperature in the
free atmosphere decreases with height on the average at the rate of
about 6 C per km, or 3 F per 1000 ft. This is an average rate, but
decreases as great as 10 C per km are not infrequent, and at times an
increase of 5-10 C per km occurs.

Under average conditions, the decrease in temperature with height
continues until the temperature reaches about 45 C in the polar

Sec. 2\

UPPER Alii TEMPKRATUKE DISTRIBUTION

regions and about -75 C in the equatorial regions. This takes place
at a height of about 9 km at the poles and about 18 km at the equator.
At these levels the fall in temperature ceases. The temperature above
this height remains constant in the polar regions, whereas above the
equatorial regions the temperature begins to rise again. Fig. 3 il-
lustrates the variation of temperature with height up to 25 km at
various latitudes of the northern hemisphere in winter. In summer,
the temperatures are generally 5 to 10 C higher in the temperate and
polar regions.

300

200

.2*

100

200 400 600 800 1000 1200
Temperature (A)

FIG. 4. The variation of air temperature with height. (After Martyn and Pulley.)

In order to distinguish between the two layers of the atmosphere, the
lower layer where there is usually a regular drop of temperature with
height is called the troposphere and the layer above is called the strato-
sphere. The surface separating the two layers is the tropopause. Its
position is shown by the broken line in Fig. 3.

The average temperature distribution as found for the northern
hemisphere is in general duplicated, as far as can be determined from
the limited number of observations available, in the southern hemi-

OBSERVATIONAL FACTS OF THE ATMOSPHERE [Chap. 1

H
2?

Sec. 3]

GENERAL PRESSURE DISTRIBUTION

!
i.

*x
I|

if
lo

S "

8 OBSERVATIONAL FACTS OF THE ATMOSPHERE [Chap. 1

sphere. An anomaly exists, however, during the Antarctic winter.
Observations taken by expeditions to the Antarctic continent show that
the decrease of temperature with height found in the lowest layers
persisted up to a height of 12 km, the upper limit of observations,
where the temperature was 80 C. Thus the tropopause found in
the Antarctic during the summer and in the Arctic both winter and
summer disappears during the Antarctic winter.

Because of the rarity of the atmosphere above 30 km, investigations
by direct means cannot be carried out. Through various observations
on meteors, abnormal sound wave propagation, noctiluccnt clouds,
aurorae, radio beams, etc., probable values of the temperature at
heights ranging from 30 to 300 km have been obtained. The probable
variation of temperature with height in the upper atmosphere, based
on the results of these indirect investigations, is given in Fig. 4. Above
the isothermal layer noted by direct observation there appears to be
an increase in temperature to about 60 km, another decrease thereafter
until a height of 80 km is attained, arid then an increase to about
1000 Absolute at 300 km.

3. The General Pressure Distribution over the Earth. The pres-
sure of the atmosphere at a point on the earth's surface is a measure of
the weight of a column of air above that point. This pressure varies
from day to day, and from one part of the* earth to another. There
are two causes for this pressure variation at a given point. As a result
of the wind distribution, there may be an accumulation of air above a
given point on the surface, producing a rise in pressure. A depletion
of air would cause a fall in pressure. A second cause which may be in
operation is the horizontal transport of air of greater or lesser density,
producing a rise or fall of pressure.

The spatial variations in pressure may be indicated by means of
isobars. These are curves which join points of equal pressure in the
same manner that lines on a contour map join points at the same height
above sea level. The isobars are usually much more regular in their
shape than contour lines, but they too indicate " highs/' " lows/ 7
" ridges," and " troughs."

Figs. 5 and 6 give the average pressure distribution over the earth
for the months of July and January. The southern hemisphere, with
its uniform surface, shows a regular pattern. The most significant
features in this hemisphere are a trough of low pressure near the equator,
several centers of high pressure at about 25 latitude with connecting
ridges, and a high-pressure system over the south pole. There is some
displacement of these features north and south with the sun.

In the north, the areas of land and ocean interfere with the regularity

Sec. 3]

GENERAL PRESSURE DISTRIBUTION

of these main features. In winter the sub-tropical high-pressure belt
is connected to high-pressure areas over the continents, while the low-
pressure belt in the sub-polar regions is concentrated in two deep lows
in the northern Atlantic and northern Pacific. In summer the sub-
polar low-pressure belt has extended over the continents, and the high-
pressure belt of the sub-tropics is limited to two high-pressure systems
over the oceans.

_Q
D

-2

-3

12 16

Time (h)

FIG. 7. The diurnal variation of pressure, Batavia. Java.

In the vertical, the pressure decreases rapidly with height as a result
of the decreasing amounts of air above. At a height of 5 km or about
18,000 ft, one-half of the atmosphere is below the observer. Only
one-tenth of the total atmosphere; is spread over the region above
16 km, and observations of the height of the aurora prove that matter
is present at heights of 1000 km above the surface.

The pressure at any point varies irregularly, and from many causes.
These variations supply one of the major studies for the meteorologist,
for they are closely linked with the weather changes. There is, though,
a pressure variation that follows a regular cycle. This cycle has a
daily period, with two maxima and two minima. It is most in evidence

10 OBSERVATIONAL FACTS OF THE ATMOSPHERE [Chap. 1

in the equatorial regions where the lack of other pressure variations
permits it to be seen in the daily curve. In the temperate latitudes,
the passage of pressure centers causes such wide variations in the pres-
sure that the diurnal wave is apparent only during spells of little baro-
metric activity. By averaging over a long period for each hour of the
day, the variations which are due to the passage of cyclones and anti-
cyclones are eliminated, and the daily cycle becomes apparent. The
daily cycle for Batavia, Java, is given in Fig. 7. The minimum occurs
at about 16 h, local time, with a second minimum at about 04 h. The
maximum occurs at about 08 h, with the secondary maximum at 22 h.
Between 08 h and 16 h the pressure drops approximately 3.5 millibars,
then rises by almost the same amount by 22 h. The cycle at higher
latitudes has its times of maxima and minima at the same time of day,
but the amplitude of the pressure waves is less.

4. Wind Distribution. It will be shown in Chapter 6 that there is a
close relationship between the pressure gradient and wind direction
and speed. It follows that the average wind directions are related to
the average atmospheric pressure distribution. Because of this close
relationship, the average wind distributions are given by means of
arrows on the appropriate pressure maps, Figs. 5 and 6.

The winds at the earth's surface most nearly approaching constancy
are those blowing from the sub-tropical high-pressure systems toward
the trough of low at the equator. These are the northeast trades of
the northern hemisphere and the southeast trades of the southern
hemisphere. The temperate zone lying between the sub-tropical high
and the sub-polar low is often called the region of westerlies, since the
prevailing direction of the wind is west. The westerlies are more
variable in their direction than the trades, as indicated above. About
the poles the winds are variable, but tend to be easterly in direc-
tion.

The seasonal variation in pressure over the land masses produces a
corresponding seasonal variation in the wind distribution. This varia-
tion is most marked over Asia. In winter the high-pressure area gives
rise to the dry, cold, winter monsoon winds blowing from the center of
the continent over China and India. In summer the winds under the
influence of the low pressure over the continent blow in the opposite
direction, giving warm, moist, southeast to southwest winds over south-
eastern Asia. A corresponding change in winds occurs over North
America, but it is less pronounced, with variations from the general
flow occurring more frequently.

The winds found in the free atmosphere will be discussed in con-
nection with the general circulation in Chapter 12.

Sec. 5} COMPOSITION OF THE ATMOSPHERE 11

5. Composition of the Atmosphere. The atmosphere near the earth's
surface is a mixture of gases, with very little variation in the relative
proportions of its constituents. A sample of pure dry air is made up
of 78.09 per cent nitrogen and 20.95 per cent oxygen by volume. The
remainder, a little under 1 per cent, is largely argon and carbon dioxide
with smaller amounts of hydrogen and the rare gases.

Near human habitations there are variations in the amounts of some
of the constituents. But, except for one region, samples of unpolluted
air from the troposphere show variations in the amounts which are
within the limits of error of the analysis. The exceptional region is the
Antarctic continent, where the samples analyzed showed a deficiency
in oxygen, with the proportion by volume being 20.54 per cent.

The relative amounts of the different gases were constant in all
samples taken in the atmosphere up to a height of 20 km. Above that
level there have been a few samples obtained by means of three manned
balloons and some free balloons. These have shown that the con-
stancy of composition characteristic of the lower atmosphere no longer
prevails. The proportion of helium increases above 20 km, and the
proportion of oxygen decreases, although these changes are only small.
The maximum height from which samples were obtained was 31 km.

According to Dalton's law, in a mixture of gases the total pressure
exerted is the sum of the pressures of the constituents. Therefore
the decrease in amount with height will depend on the molecular weight
of the gas, the heavier gases being concentrated near the earth's surface.
In the lowest layers, turbulent eddy motion is sufficient to produce
complete mixing. Above this layer, turbulent mixing is less active, and
so the gases tend to follow Dalton's law more closely.

Assuming that the gases are distributed according to Dalton's law,
then the amount of oxygen one of the heavier gases decreases
rapidly with height. Helium and hydrogen, if present in appreciable
quantities, would, because of their small molecular weights, extend to
great heights, forming a large proportion of the upper atmosphere.
An examination of the spectrum of the aurora, which occurs at heights
varying from 70 to 1000 km, shows the presence of nitrogen and oxygen
bands and lines, but none of hydrogen and helium.

From this fact either of two inferences is possible. One possibility
is that hydrogen and helium are not present in appreciable quantities
at great heights. The alternative inference is that helium is present
in the upper atmosphere, but that the excitation potential is not great
enough to produce the emission of helium lines. Helium is being
produced constantly as a result of the erosion of igneous rocks which
contain radioactive elements. In this manner, 1.2 alpha-particles are

OBSERVATIONAL FACTS OF THE ATMOSPHERE [Chap. I

being produced per gram per second. An alpha-particle is a helium
atom charged with two units of positive electricity. During geological
time, most of these helium atoms have escaped from the upper atmos-
phere to outer space, but estimates based on geological data suggest
that, in the outermost regions, helium still forms a considerable if not

250

FIG. 8.

20 40 60 80 100
Percentage

Proportions (by mass) of the constituents of the atmosphere.
(After Chapman and Milne.)*

preponderant part of the gases present. Fig. 8 shows the composition,
by mass, of the atmosphere, under the assumptions that the separation
of gases according to Dalton's law starts at 20 km, and that helium,
but not hydrogen, is a permanent constituent of the atmosphere at
great heights. In this figure, if a horizontal line is drawn at any given
height, the lengths of the intercepts made on it by the curves give the
proportion of the different constituents. As indicated above, it is not
yet clear whether helium predominates at great heights, as indicated
in Fig. 8, or if the upper atmosphere is comprised almost exclusively
of oxygen and nitrogen.

Measurements have shown that ozoric occurs in appreciable amounts
in the height interval from 20 to 40 km. The amount of ozone at these
levels is correlated with the surface pressure distribution, being a

* In a private communication, Professor Chapman has indicated that the dia-
gram does not accurately portray his present conception of the composition of the
atmosphere.

Sec. 6} COMPOSITION OF THE ATMOSPHERE 13

maximum just to the west of a center of low pressure, and a minimum
to the west of a center of high pressure. The full significance of this
relationship is not yet apparent.

Until now the discussion has been concerned with dry air. But
there is one other constituent, water vapor, which is very important
in weather processes. Water vapor varies in amount in the atmosphere
from almost zero up to a maximum of nearly 3 per cent by mass. Be-
cause there is an upper limit to the amount of water vapor which can
occupy a given space, and this upper limit decreases with decreasing
temperature, very little water vapor is found except in the lowest
layers of the troposphere. The variation in the amount of water
vapor in the atmosphere is significant to the meteorologist since its
condensation produces clouds and rain. A knowledge of the causes
and effects of its variations is therefore essential for an understanding
of weather processes.

BIBLIOGRAPHY*

Admiralty Weather Manual, London, H. M. Stationery Office, 1938. Chapter 8.
Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapter 1.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 2

(1936).
5. Haurwitz, B., "The Physical State of the Upper Atmosphere," ./. Roy. Aatr. *SV>c.

Can., 1937, 1938.
5. " The Upper Atmosphere," Q. J. Roy. Met. Soc., 65, 303 (1939).

* The numbers which appear at the beginning of references are section numbers
and indicate that the references are applicable only to the sections referred to.
Other references apply to the chapter as a whole.

PART I. THEORETICAL METEOROLOGY

CHAPTER 2
STATICS OF THE ATMOSPHERE

6. Pressure and Temperature Units. The unit of force in the centi-
meter-gram-second (cgs) system is the dyne. This represents the
force which, applied to a mass of 1 gm, produces an acceleration of 1 cm
per sec per sec. The unit of pressure in the cgs system is, then, the
dyne per square centimeter. A much larger unit, the bar, is equal
to 10 6 dynes per sq cm. In meteorology, the pressure unit which is
generally used is the millibar, which is one-thousandth of a bar.

1 millibar (mb) = 10 3 dynes cm" 2

A pressure of 1 bar is the same as a pressure of 1000 mb. The standard
pressure of 760 mm of mercury used in physics and chemistry is 1013.2
mb. A pressure of 1 bar is nearly as great as the average sea level
pressure over the earth's surface.

The barometer, described in section 63, measures atmospheric pres-
sure in terms of the height of a column of mercury, the weight of which
is just sufficient to balance the atmospheric pressure. The relation-
ship between the millibar and the equivalent height of mercury is as
follows.

1000 mb = 750 mm of mercury = 29.53 in. of mercury

The three temperature scales in common usage in meteorology are
the Fahrenheit, Centigrade, and Absolute scales. The fixed points in
each of the three scales are given in the following table.

SCALE FREEZING POINT

BOILING POINT

DIFFERENCE

Fahrenheit (F)
Centigrade (C)
Absolute (A)

,32

273

212
100
373

180
100
100

To convert from one scale to the other, the following equations may

Sec. 7] THE EQUATION OF STATE FOR DRY AIR 15

be used.

T Q F = ? T C + 32 = - T A - 459.4
5 5

T C = - (T F - 32) = T A - 273
t/

T A = T C + 273 = jj 7' F + 255.22
9

In the English-speaking countries, the Fahrenheit scale is used on the
surface weather map to express the air temperature and dew point.
It is also used in some of these countries in measuring and plotting
upper air data. In others, the Centigrade or Absolute scale is used
for this purpose. In other countries practice varies, but the Centi-
grade scale is used widely. The Absolute scale must be used when
carrying out computations in the cgs system. In this book, it will be
assumed that the letter T refers to temperature in degrees Absolute
unless otherwise indicated.

7. The Equation of State for Dry Air. Two basic relationships of
thermodynamics are the laws of Boyle and Charles. Boyle's law
states: The pressure of a given mass of a gas at constant temperature
varies inversely as the volume. Charles' law is: The temperature
in degrees Absolute of a given mass of gas at, constant pressure varies
directly as the volume. These two laws may be combined in the
following manner. First, assume that a unit mass of gas has a pres-
sure pi, a volume v\, and a temperature T\. Let pi and v\ vary at
constant temperature TI, until a pressure p 2 and a volume v are ob-
tained. From Boyle's law it follows that

P\v\ = p 2 v and v = - (7-1)

Now let the temperature T\ change to 7 T 2, the pressure remaining con-
stant at p 2 . From Charles' law, then

v jTi v 2 7 T i

l and v = ^ (7 . 2 )

V> 2 L 2 i 2

where v 2 is the resulting volume.
Combining (7-1) and (7-2) gives

The result would have been the same had the conditions varied simul-

16 STATICS OF THE ATMOSPHERE [Chap. 2

taneously. Equation 7-3 indicates that when a unit mass of a perfect
gas is undergoing variations in pressure, volume, and temperature,

pv oc T (74)

Introducing the appropriate constant, (74) becomes

pv = T (7-5)

where R* is called the universal gas constant, and has the value
83.14 X 10 6 ergs per degree, and m is the molecular weight of the gas.
This equation may be used for any perfect gas, when its molecular
weight is inserted. It may also be used for a mixture of gases, pro-
vided that a value is substituted for in appropriate for the mixture.
Of the gases comprising the atmosphere (see section 5), only water
vapor condenses at the temperatures and pressures encountered in the
atmosphere, so that absolutely dry air may be considered as a perfect
gas. Although the constitution of the atmosphere is subject to varia-
tions, these are small, and 28.97 may be taken with sufficient accuracy
as the molecular weight of dry air. The value of the gas constant for
dry air, R, is given by

fl* 83.14 X 10 6

or, in basic cgs units

R = 2.87 X 10 6 cm 2 see" 2 deg" 1

The specific volume v of a gas is defined as the volume occupied by
unit mass of that gas, so that

v = -
P

where p represents the density of the air.

The equation of state for dry air may be given therefore in either
of the two following forms.

pv = RT (7-6)

p = R P T (7-7)

8. Geopotential. The earth is not quite a perfect sphere since its
equatorial radius is 6378 km, whereas its polar radius is 6357 km. In
practically all meteorological problems this divergence from a true

Sec. 8} GEOPOTENTIAL 17

spherical shape is of no significance, and a value of 6370 km may be
assumed.

The angular velocity w of the earth is 7.29 X 10~~ 5 radians per second.

The standard value of the acceleration of gravity g$, 980.62 cm per
sec per sec, is that at mean sea level at latitude 45. The acceleration
of gravity varies both with latitude <f> and with height above mean sea
level z. The variation is given by the following expression.

g = 980.62(1 - 0.00259 cos 20) (1 - 3.14 X 10~ 7 z) cm sec" 2

where z is in meters. Accuracy of this order is necessary only in certain
phases of meteorological work, and in most instances the variation of
g with latitude and height may be neglected, and the value 981 cm
per sec per sec assumed.

The concept of geopotential has been introduced into meteorology
in order to make allowance for the variation of g with height when
that is necessary. The geopotential ^ at a height z is defined as the
potential energy of unit mass at that height. The potential energy at
a point is, by definition, the work required to raise unit mass from
some standard level, usually mean sea level, to that point.

From the inverse square law for gravitational attraction, it follows
that at a height z above mean sea level, the acceleration of gravity g
is given by

g _ E 2

0o (E + z) 2

where E is the radius of the earth. Then, approximately

The geopotential is, therefore,

=/*

In cgs units, the unit of geopotential is

1 gm' X 1 cm X 1 cm X 1 see" 2

A more convenient unit for meteorological computations is 10 5 times
as great as the cgs unit, and is known as the dynamic meter, which,
expressed in meter-gram-second units, is

10 5 X 1 gm X ~ X - X 1 sec" 2 = 10 mgs units

18 STATICS OF THE ATMOSPHERE [Chap. 2

The geopotential in dynamic meters is, therefore, from (8-2)

(8-3)

where is given in meters per second per second and z and E arc
given in meters. The geopotential in dynamic meters is therefore only
about 2 per cent smaller than the geometric height in meters. In
practice, tables are used to convert from meters to dynamic meters.
Table I in the Appendix gives the corresponding values of z and 4> for
various latitudes.

From the foregoing, it can be seen that the dynamic meter is not a
unit of length, but a unit of energy. This fact must be kept in mind
if confusion in its use is to be avoided. The use of the dynamic meter
in evaluating the geometric height corresponding to significant points
of an acrological ascent is outlined in section 11.

9. The Variation of Pressure with Altitude. The pressure at any
height in the atmosphere is, by definition, the weight of the vertical
column of air of unit cross section which extends from that height to the
top of the atmosphere. It follows directly then that pressure decreases
with increasing height in the atmosphere. The difference in pressure
between height z and height z + dz may be found in the following
manner. Since unit cross section is being considered, the volume of
the clement of air is dz, and its mass is p dz, where p denotes the average
density in the height interval dz. Since weight is given by the product
of the mass and the acceleration of gravity, and since p decreases with
increasing z, then

dp = -gpdz (9-1)

Substituting for p from (7-7), an alternative form is obtained

This is the basic statical relationship of meteorology, and it is used
frequently in subsequent sections.

The variables are readily separated, and if T can be expressed as a
function of z, the differential equation may be integrated. Two cases
are of fundamental importance.

The first of these is the situation in which T is constant with height,
i.e., the portion of the atmosphere under consideration is an isothermal
layer. Equation 9-2 may be expressed as

dp Q

Sec. 0] VARIATION OF PRESSURE WITH ALTITUDE 19

Integration of (9-3) leads to

log = - (z z ) (94)

P = Poc * 7 ' ( *~* (9-5)

where po is the pressure at height 2 .

The second case is that in which T decreases at a constant rate
with increasing z. Then the temperature T at any height is given by

T = To - * (9-6)

TO is the temperature at z , and a is known as the lapse rate of tem-
perature.

Differentiating and rearranging (9-(>) lead to

(IT
dz = - (9-7)

Substituting (9-7) in (9-2) gives

dp q dT

= 77~ -7JT (9-8)

p /te 7 T v

Integration of (9-8) then gives

f rii \ g

--^r < 9 - 9 )

^ ^ o /

If, as a first approximation, it is assumed that the troposphere (see
section 2) is composed of dry air having a constant lapse rate of tem-
perature, then (9-9) gives the variation of pressure in the troposphere
under those conditions. If, in addition, T is assumed to be constant
with height in the stratosphere (section 2), the pressure in the strato-
sphere may be obtained by combining (9-5) and (9-9). Thus the
pressure p at any given height z in the stratosphere is given by

(m y \ (*""*<)

~~ -Y a c *<r.-,> (9-10)

where p and T are the pressure and temperature at ZQ (frequently
PQ and TO are taken at the earth's surface, so that z = 0), z t is the
height of the tropopaxise, and (T az t ) is the temperature of the
stratosphere. Fig. 9 depicts on a temperature-height diagram the
situation postulated above.

STATICS OF THE ATMOSPHERE

[Chap. 2

CP
ft)

T -oczf TO

Fia. 9. Temperature changes with height.

10. Units of Atmospheric Water Vapor Content. Up to this section,
the atmosphere has been assumed to be perfectly dry, and no allowance
has been made for the water vapor present in it. However, water
vapor plays a fundamental role in atmospheric processes and must
be considered in detail. Not only does water in the gaseous state vary
greatly in amount throughout the atmosphere, but also it appears in
the liquid and solid states. The effect on weather processes of water
in its three states and of changes from one state to another forms the
subject matter of a large proportion of meteorological investigations.

As long as water remains in the gaseous state, it acts as a perfect gas,
and its variations are subject to the relationship given in (7-5). The
molecular weight of water vapor is 18.016, and its gas constant is,
according to section 7,

' - - 83 ' 14 X 1Q6 = 4.62 X 10 6 cm 2 secT 2 deg' 1

18.016

NOTE. In this and following sections, when water vapor is referred
to specifically, the property under consideration is indicated by a
prime symbol; thus the density of water vapor is designated p'. An

Sec. 10] ATMOSPHERIC WATER VAPOR 21

exception to this rule is made in the pressure of water vapor, where e
is used instead of p in order to conform to general practice.
The density of water vapor is, according to (7-5), since v f = l/p

(lo-i)

The density of moist air is the sum of the density of water vapor and
of dry air. From Dalton's law it follows that if p is the total pressure
of the moist air, then p e is the partial pressure of dry air, and the
density of the moist air is

Now R* = m'R' = mR, so that

R' = (10-3)

and (10*1) becomes

m RT' " ' RT'

t m c e

P =~^> = *-^F< (10"*)

where

_ m_ _ 18.016
C " ^ = 28.97

= 0.622

In a mixture of dry air and water vapor, the latter is at the same tem-
perature as the former, so that T f = r l\ and with (104), (10-2) becomes

i
P

RT RT
or

The above equation shows that a given volume of moist air is lighter
than an equal volume of dry air at the same pressure and temper-
ature. This fact vshould be obvious, since water vapor is lighter than
dry air, and mixing them will reduce the density of the mixture.

If a closed vessel contains initially dry air and some liquid water,
the whole being at a constant temperature, the space is said to be sat-
urated with water vapor when steady conditions in the space above
the liquid water have been attained. Even if the dry air were not

22 STATICS OF THE ATMOSPHERE [Chap. 2

present, the amount of water vapor present at saturation would be
the same. When space is saturated in this manner, no more liquid water
evaporates, and there is an upper limit to the amount of water vapor
which may occupy the given space. The pressure exerted by water
vapor under saturation conditions will be denoted by e s . The fore-
going shows that it is not strictly correct to speak of saturated air since
space, not air, is saturated. However, that term is commonly used
and has the advantage of brevity, if not of literal accuracy. The sat-
uration vapor pressure e s is a function of temperature. The rela-
tionship between e s and T derived in section 21 is given in tabular form
in Table II in the Appendix and is shown graphically in Fig. 133,
section 132. The various aspects of saturation are discussed in greater
detail in section 126.

The amount of water vapor in the atmosphere may be expressed in
any one of four ways. The first of these is in terms of the absolute
humidity, denoted a, which is the number of grams of water vapor in
unit volume, or the density of water vapor. The absolute humidity is
then given by (104). Thus,

a = e JL (10-6)

The second measure of water vapor content is the relative humidity /,
which is defined as the ratio of the actual vapor pressure to the sat-
uration vapor pressure at the temperature of the air in question. The
relative humidity is, then

/ = " (10-7)

If the relative humidity is to be given as a percentage, as is customary
in practice, the right-hand side of (10-7) must be multiplied by 100.

The third unit is the humidity mixing ratio x, defined as the ratio of
the density of water vapor to that of dry air. Using (7-7), with p e
substituted for p t and (104), it is seen that

x = (10-8)

p - e

If the air is saturated, the saturation mixing ratio x s has the value

x s = ^ (10-9)

P e s

The last measure is the specific humidity s, which is the ratio of the
density of water vapor to that of moist air. Hence, from (104) and

Sec. 11] HEIGHT COMPUTATIONS 23

(10-5), it follows that

= JT T ( 1(MO )

p - (1 - e)e

Since p is of the order of one hundred times as great as e, for most
practical purposes the latter may be neglected in comparison with the
former, and (10-8) and (10*10) become

x = s = - (10-11)

or for saturated air

x s = s s = * (10-12)

The absolute humidity is rarely used in meteorology, whereas the rel-
ative humidity is used widely. The humidity mixing ratio is often
more convenient than the specific humidity for theoretical work, and
it will be found frequently in the following sections.

The properties of these units of moisture content and their sig-
nificance in practical meteorology are discussed in detail in section 81.
Their values arc in practice usually determined from measurements
made with the hygrometer, an instrument described in section 65.

11. Virtual Temperature and Height Computations. For most pur-
poses, the equation of state for dry air may also be used without mod-
ification for moist, unsaturated air. On some occasions, however, a
more accurate relationship is necessary. The appropriate gas con-
stant R m for moist, unsaturated air may then be used instead of the
gas constant for dry air R. R m is not constant for all amounts of water
vapor in the air, but varies with the vapor pressure e and the total
pressure p. The appropriate expression for R m may be derived in the
following manner. The density of moist, unsaturated air is given by
(10-5), so that the equation of state for moist, unsaturated air is
given by

and

R m = (11-2)

24 STATICS OF THE ATMOSPHERE [Chap. 2

Substituting in (11-2) for e/p from (10-11) leads to an alternate form

R m = ^ r (11-3)

- X(L - C)

This equation shows that R m is a function of the humidity mixing
ratio alone.

It is often more convenient to use a different value of the temper-
ature, known as the virtual temperature T v , rather than a different
value for the gas constant. The equation of state for moist, unsat-
urated air then becomes, using the same procedure as in (11-1)

= ~ (11-4)

itx L /yj 111 v

and so

T v = (11-5)

or with (10-11)

v e-x(l-t) ^^

Thus the virtual temperature of moist, unsaturated air may be defined
as the temperature at which dry air of the same pressure would have
the same density as the moist air.

Virtual temperature is used to determine the height in the atmos-
phere corresponding to various pressures as found from aerological
ascents. A number of methods of determining heights have been
developed, but that of V. Bjerknes and Sandstrom is used widely, and
will be given in its broad outlines here.

The statical equation in the form given in (9-2), with T v substituted
for T, since moist, air is being considered, may be combined with (8-2)
in the differential form to give

d& = -RT V (11-7)

</<*> = -RT v d(logp) (11-8)

On integration, (11*8) becomes

/log pi

* 2 -4> t = R I T v d(logp) (11-9)

^loe

Sec. 11]

HEIGHT COMPUTATIONS

Now on a chart having pressure on a logarithmic scale as ordinate
and temperature on a linear scale as abscissa, plot the virtual tem-
perature at each significant point of the aerological ascent against the
corresponding pressure. The resulting curve may then appear as
depicted in Fig. 10, where the required curve is LM. Consider first

^uu
?nn

G \B E

DUU

o>
o 700

ft on

D \

QOO

-10

T vc 10

FIG. 10. The computation of heights in the atmosphere.

the portion A B of the curve between 700 and 600 mb, so that p\ = 700
mb and p 2 = 600 mb. Draw an isotherm DCE from 700 to 600 mb
in such a manner that the area CEB is exactly equal to the area CD A.
The virtual temperature at (7, T vc , is with sufficient accuracy 5 C.

According to (11-9), the difference in geopotential between 600 and
700 mb is proportional to the area ABGF, and since area CEB = area
CD A, the difference in geopotential is also proportional to the area
DEGF. Equation 11-9 may then be written

(11-10)

or in this case

$600 - $700 = R X 278 log

26 STATICS OF THE ATMOSPHERE (Chap. 2

The difference in geo potential, or in geodynamic height, between any
two levels may be computed by means of (11-10). If a similar pro-
cedure is followed for other layers, the geodynamic height for the
whole column is obtained by a process of summation.

In actual practice, the temperature pressure curve is plotted. Suit-
able graduations on each standard isobar on the chart permit the rapid
determination of the corresponding virtual temperature when the
relative humidity is known, without the necessity of having recourse
to (11-5) or (11-6). The virtual temperature curve is then plotted
beside the other curve. In the above discussion, the cgs system was
used, but in practice, the dynamic meter is taken as the unit, so that
the difference in geodynamic height between two pressures is obtained.
Tables, in which the unit is 0.98 of the dynamic meter, and hence
approximately the geometric meter, give the difference in height
between the standard isobars, which are 100 mb apart, for various
values of the virtual temperature. For the bottom and top portions
of the curve which do not extend from one standard isobar to the next
a slightly modified procediire is followed. To evaluate the whole
ascent, the height differences between the top and bottom of each
layer must be added. If now log p is plotted against height, the result-
ing curve should be approximately a straight line, and the heights
corresponding to the pressures at significant points of the ascent curve
can be obtained by interpolation.

PROBLEMS AND EXERCISES

1. (a) Derive the following expression for the vertical distribution of the density
when the lapse rate of temperature is constant

'T Q - az

Assume that the air is dry and that g is constant.

(b) For what value of the temperature lapse rate is the density constant with
height?

2. Assume that the lapse rate of temperature in the troposphere is 6 C per km,
whereas the temperature at all heights in the stratosphere is equal to that at the tropo-
pause. Compute the change in surface pressure when the height of the tropopause
increases from 10 to 11 km, if initially the surface pressure is 1000 mb, and the sur-
face temperature has a constant value C. Assume that the pressure at a height
of 50 km is constant.

Why is the value for the pressure change obtained so much greater than that
actually ever observed?

3. Calculate at what height in a dry atmosphere the pressure is one-half of that
at the surface when the surface temperature is 10 C, and the lapse rate is (a) 6 C
per km, (6) zero. Assume that g is constant.

BIBLIOGRAPHY 27

BIBLIOGRAPHY

Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapter 2.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapters 1, 2.
Koschmieder, H., Dynamische M eteorologie, Leipzig, Akad. Verlag., 1933. Chapters

1,2.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 3

(1930), Chapter 6.

CHAPTER 3
THERMODYNAMICS OF DRY AIR

12. The First Law of Thermodynamics. In Chapter 2 it was implied
that air has a certain heat content, but no attempt was made to con-
sider this heat content in detail. But heat is a form of energy which
may be converted into other forms of energy. The study of the
dynamical aspect of heat is known as thermodynamics, and it has a
wide application to the problems of theoretical meteorology.

The first law of thermodynamics is a statement of the principle of
the conservation of energy, expressed in such a manner as to include
the energy of heat. Any body in a specified state has an internal
energy associated with the configuration and motion of its molecules.
It is known that thermal energy may be changed into mechanical energy,
i.e., that heat may be converted into work. The first law of thermo-
dynamics states: The heat taken in by a substance is equal to the
increase in its internal energy plus the work done by the substance.

The unit of heat in the cgs system is the caloric, which is defined as
the amount of heat necessary to raise the temperature of one gram of
water one degree Centigrade. The cgs unit of work is the? erg, defined
as a force of one dyne acting through a distance of one centimeter.
These units are connected by the relationship that 4.18 X 10 7 ergs
are equal to 1 calorie. The mechanical equivalent of heat

J = 4.18 X 10 7 erg cap 1

In meteorology it is frequently more convenient to use the reciprocal of
the mechanical equivalent of heat, denoted by A. It has the value

A = 2.39 X 10~ 8 cal erg~ l

The specific heat c of a substance may be defined as the amount of
heat dQ necessary to increase the temperature of one gram of the sub-
stance by the amount dT divided by dT. It follows that

dQ
C ~dT

For gases, there are two specific heats to be considered, depending on
whether the volume or the pressure is kept constant. The specific

Sec. 12] FIRST LAW OF THERMODYNAMICS 29

heat at constant volume is denoted as c v ; the specific heat at constant
pressure is denoted as c p . For air

c v = 0.170 cal gnT" 1 deg"" 1
c p = 0.239 cal gm" 1 deg" 1

The work done by a gas as it expands may be obtained in the following

manner. Let iSi and /S 2 denote the surface of a gas before and after

a small expansion against an external

pressure p which is constant over the

surface (Fig. 11). The pressure of the

gas under consideration is assumed to be

slightly greater than the external pressure.

Consider the element dS of the surface

and denote its displacement along the

normal as dn. The work done by the gas

as it expands is then

S(p dS) dn = pS dS dn = p dV

Therefore, the work dW done by unit

mass of the gas is FlG n The work done

dW = p dv (12-1) thr Ugh the expansi n f a gas '

Expressed in heat units, (12-1) becomes

dW = Apdv (12-2)

Consider first the case in which an amount of heat dQ\ is supplied to
unit mass of the gas, volume remaining constant. Then

dQi = c v dT (12-3)

This expression gives the increase in internal energy of the gas. If now
the volume is permitted to increase by an amount dv, the work done
is given by (12-2). When heat dQ is supplied, then, and temperature
and volume vary simultaneously, it follows that

dQ = dQi + dW = c v dT + A p dv (124)

This expression states the first law of thermodynamics in mathematical
form.

The relationship between c p and c v may be readily obtained with
the aid of (124). Differentiation of (7-6) loads to

pdv = -vdp + R dT

30 THERMODYNAMICS OF DRY AIR [Chap. 3

Substitute this expression in (124) to obtain

dQ = (c v + AK) dT - Avdp (12-5)

If the pressure is constant, dp = 0, and from (12-5) it follows that the
specific heat at constant pressure

c v + AR (12-6)

This relationship holds for a perfect gas, and it is therefore applicable
to air as long as it remains unsaturated.

13. Adiabatic Relationships for Dry Air. Potential Temperature.
If a mass of a substance such as air undergoes variations in state in
such a manner that there is no exchange of heat between the substance
and its environment, the process by which this occurs is said to be
adiabatic. When this condition is fulfilled, (12-5) becomes

c p dT = A v dp
or, with (7-6)

~r c p p

Integration of (13-1) leads to

T / ;; V

T' = (^) < 13 ' 2 >

1 o \Po/

where TO is the temperature at pressure />o and

K = - 0.288

Substituting (7-6) in (13-2) gives a second form

) (133)

and substituting (7-7) in (13-2) leads to a third form for the adiabatic
relationship

- = (-) * (134)

Po \Po/

It is frequently desirable to have some property of dry air which
is invariant during adiabatic processes. Such a property is the poten-
tial temperature 6, defined as the temperature which the air would
attain if brought adiabatically to a standard pressure, usually 1000 mb.

Sec 14] DRY ADIABATIC LAPSE RATE 31

From (13-2) it can be seen that

0.288

) (13-5)

The fact that 6 is invariant for adiabatic processes may be readily
demonstrated. Assume that initially the air at pressure PQ has a tem-
perature T . According to (13-5), its potential temperature is
given by

o T f 1000 Y ma

t/o = 7 o I I (lo-u;

If, as a result of an adiabatic process, its pressure becomes p\, then,
according to (13-2), its temperature

1\ = ? 7 o(-Y (13-7)

The potential temperature of the air is now, from (13-5),

By substituting for TI from (13-7) in (13-S), it follows that

Po Pi \ />(> /

By comparing (13-6) and (13-9), it is seen that 0o 0i- Thus, no
matter how many adiabatic processes occur, tlu^ potential temperature
of the air in question does not change.

In the atmosphere, no processes arc completely adiabatic, since there
is some mixing between a given mass of air and the surrounding air,
arid there may be loss or gain of heat by radiation. However, these
are secondary effects, and they may usually be neglected. A discussion
of the practical aspects of the use of potential temperature is given in
section 80 and after.

14. The Dry Adiabatic Lapse Rate. The Stability of Dry Air. The
rate of change of temperature with pressure of air ascending or descend-
ing adiabatically is given in (13-1). In order to find the rate of change
of temperature with height, (9-3) may be utilized in the form

where f indicates the temperature of the surrounding air, not that
of the air undergoing adiabatic vertical motion, which is denoted T.

32 THERMODYNAMICS OF DRY AIR [Chap. 3

Combining (13-1) and (9-3) in the form given above, it follows that

-f=tl

T is very nearly equal to 7 T , the difference between the two tempera-
tures being of the order of 1 or 2 per cent of their value. Their ratio
may then be taken as unity with sufficient accuracy for most purposes.
This rate of decrease of temperature with height of adiabatically
ascending dry air is known as the dry adiabatic lapse rate T, and is
given by

Aa
r - = 9.8 C km" 1 (14-2)

In most practical work the value of F is taken to be 10 C per km.

It follows from (14-2) that, on a chart having height as ordinate and
temperature as abscissa, the curve representing the variation of tem-
perature with height of dry air moving adiabatically is a straight line.
Such a line, which represents graphically the dry adiabatic lapse rate,
is known as a dry adiabat. For many purposes, however, it is more
convenient to use a chart on which temperature and pressure, rather
than temperature and height, may be plotted. A chart of this type,
having pressure on a logarithmic scale as ordinate and temperature on
a linear scale as abscissa, has already been mentioned in section 11.
With the aid of (13-2) it is possible to construct dry adiabat s on this
diagram, as shown in Fig. 12. The dry adiabats on this chart, descend-
ing from left to right, are not straight lines, but their curvature is very
small. Since it is possible to determine readily the variation of tempera-
ture with pressure during adiabatic processes by means of this chart, it
is known as an adiabatic chart. The potential temperature of a mass
of air may also be found directly from this chart. It is obtained by
proceeding from the point in the diagram representing the condition
of the air along the dry adiabat tnrough that point until standard
pressure (1000 mb) is reached. The temperature at standard pressure
is then the potential temperature.

This chart may be used in the discussion of the stability of dry air.
Let the curve ABODE in Fig. 12 represent the variation of temperature
with pressure as found by means of an aerological ascent. Consider
first a mass of air in the layer AB which has temperature TI and pres-
sure pi. Now give this air an upward displacement. As it ascends,
its temperature decreases at the dry adiabatic lapse rate, so that by
the time it reaches pressure p 2 , its temperature is TV Assuming that
the surrounding air has not been disturbed by this displacement, the

Sec. 14]

DRY ADIABAT1C LAPSE RATE

temperature of the surrounding air at p2 is T 2 . But since T 2 > T 2 ,
the displaced air is lighter than its environment, and if the displacing
force ceases to act, it continues to rise until it comes into equilibrium
with the surrounding air. Similarly, if the air has been displaced down-
ward, it will continue to subside after the force acting upon it is released.
The air (p b 7^) is then said to be in unstable equilibrium. The same
considerations apply to any mass of air in the layer ABj so that from

a
o

Pa
P,

T 2 T

T-

FKJ. 12. The relation between lapse rate and stability.

A to B the lapse rate is of the unstable type. The condition for in-
stability is that the lapse rate of the environment shall be greater than
the dry adiabatic lapse rate. Next consider a mass of air (;; 3 , T 3 ) in
the layer BC. Let this air be displaced dry adiabatically from p 3 to p 4 .
At /; 4 , the temperature of the displaced air is 7 T 4 , and that of the en-
vironment is T 4 . In this case 7 T 4 < T 4 , the displaced air is heavier
than the surrounding air, and if the displacing force is removed, the
air sinks back to its initial position. Similarly, air displaced downward
tends to return to its initial position. This air is said to be in stable
equilibrium. From B to C the lapse rate is of the stable type. The
criterion for stability is that the lapse rate of the environment shall be
less than the dry adiabatic lapse rate. Similar considerations show that
a mass of air in the layer CD, if given an upward displacement, will
neither continue to ascend nor descend to its initial position when the

34 THERMODYNAMICS OF DRY AIR [Chap. 3

displacing force is removed, but it will remain at the pressure attained
as a result of the displacement. The air is then in neutral equilibrium,
and the lapse rate is of the neutral type. The criterion for neutral
equilibrium is that the lapse rate of the environment shall be equal to
the dry adiabatic lapse rate. If the lapse rate of the environment is
denoted by a, then the criteria may be stated in the following form.

Unstable a > T
Stable a < T

Neutral a = T

15. The Adiabatic Lapse Rate for Moist, Unsaturated Air. The

humidity mixing ratio of a mass of air is given by the mass of the water
vapor per unit mass of dry air. This follows directly from the defini-
tion given in section 10. From this method of expressing the mixing
ratio, it follows that it is invariant during ascent or descent, as long as
condensation and evaporation processes do not occur and if mixing
with the surrounding atmosphere is so small as to be negligible. It
will now be shown that the lapse rate for moist, unsaturated air is a
function of the mixing ratio, but that this lapse rate is, for all practical
purposes, the same as the dry adiabat ic lapse rate.

The specific heat at constant pressure of 1 gm of moist, unsaturated
air is given by

c pm = ^^5 (15.1)

I + X

where c p is the specific heat for water vapor at constant pressure. By
using this value for the specific heat and the gas constant R rn for moist,
unsaturated air given in (11-3), (13-1) becomes

v - ( 15 '2)

T c pm p

where p is now the total pressure. Substituting for c pm and R mj (15-2)
becomes

dT _ A(l+?)eR dp

T ~ (c p + xc' p )[t - (1 - )x] p

A(l+x)eR dp ARdp

, (15-3)

bc p p

where

Sec. 16] UNSATURATED ADIABATIC LAPSE RATE 35

Now

Cp = 0.465 cal gm"" 1 deg"" 1

c p = 0.239 cal gm" 1 deg"" 1
and

= 0.622

By substituting these values in (154) and neglecting x 2 in comparison
with x, the equation becomes

Integrating (15.3) gives, then,

T /p\ m

V = (n't

* o \Po/

where

Since x is of the order of 10~ 2 , K m ~ K, and (13-2) and (15-6) may be
considered as identical. If tho moisture content of the environment,
is taken into account, (9-3) becomes

dp q

where T v is the virtual temperature of the environment at the pressure
under consideration. Combining (15-3) and (15-7) gives

_ ^L - A(J T

" dz ~ c(l + '

Since T v is very little different in value from 7 T , their ratio may be taken
as unity, as was done with T and T in section 14, and (15-8) becomes

dT Ag

__ _ __ ^ _ / 1 r Q \

dz c p (l + 0.34x) V ;

The error involved in expressing (15-8) as (15-9), or (14-1) as (14-2),
will, in practically all cases, be greater than that involved in neglecting
0.34z in comparison with 1. It is therefore permissible to state that
the adiabatic lapse rate for moist, unsaturated air T m is given by

T m = = 9.8 C km" 1 (15.10)

THERMODYNAMICS OF DRY AIR

[Chap. 3

Equations 14-2 and 15-10 are identical, and the same equation may
be used for the adiabatic lapse rate, whether the air is completely dry
or not. The only restriction imposed is that the air shall not be satu-
rated.

16. The Effect of Ascent and Descent on Lapse Rate and Stability.
The ascent and descent of air masses of broad horizontal extent is
a frequent occurrence in the atmosphere. When such vertical motions
occur, the lapse rate of the air in question usually changes. The

::::::; : :x-*;*; ^**'

"3 .:-.-.-':.::;' : - : :i;v'-.- i v- > ' : - .''^'

:.;:;;:.:.. . . ^3
:i ,' : - ' : -" fnT)

Fici. 13. Variation of lapse rate with subsidence of a layer of the atmosphere.

variation of lapse rate with height, or with pressure, may be determined
in the following manner. In Fig. 13, a layer of air is shown before and
after subsidence has taken place. At the top of the layer, initially
the pressure is p l9 and the temperature is T\, whereas at the bottom of
the layer, the pressure is p 2 , and the temperature is T 2 . The lapse
rate in the layer is ai, and the vertical thickness of the layer is h^
After subsidence has occurred, the conditions at the top of the layer
are denoted (p 3 , 7' 3 ), and those at the bottom are denoted (p 4 , T).
The top of the layer has subsided through a vertical distance h\ + h 2 ,
and the bottom through a distance A 2 + ^3- The new lapse rate
resulting from the subsidence is 3 . To find the value of 3 , it will first
be evaluated in terms of i and hi and ft 3 , and then in terms of ai, p b
p 2 , Pa, and p 4 .
Initially

- T 1 =

(16-1)

Sec. 16] VERTICAL MOTION AND STABILITY 37

and finally,

(16-2)

In addition, T\ and T 3 , T 2 and r l\ are related in the following manner.
Is- T, =T(h l +h 2 ) (16-3)

^4 - T 2 = F(A 2 + A 3 ) (164)

By subtracting (16-3) from (1(54), it follows that

(7' 4 - 7 T 3 ) - (7 ? 2 - 7'0 = F(A 3 - AO (16-5)

Substituting (16*1) and (16-2) in (10-5) and rearranging lead to

3
To the right-hand side of (10-0) add a.\ and subtract a\ , and after

rearranging further, (10-0) becomes

~ /? M (16-7)

Consider now three cases when the initial lapse rate of the layer is less
than, equal to, or greater than the dry adiabatic lapse rate.

(a) a\ < F. If contraction in the vertical accompanies the vertical
motion of the layer, h\ > A 3 , and it is soon that <* 3 < a\. This decrease
in lapse rate means that a corresponding increase in stability takes
place. If, on the other hand, vertical expansion of the; layer occurs,
hi < A 3 , and it follows that 3 > i. A decrease in stability results from
this increase in lapse rate.

(b) oti = F. When the actual lapse rate in the layer is the same as
the dry adiabatic lapse rate, vertical motion of the layer does not pro-
duce any change in lapse rate, irrespective of any expansion or con-
traction that may occur.

(c) e*i > T. A similar application of (10-7) shows that, in this case,
vertical contraction leads to an increase in lapse rate and, consequently,
a further increase in the degree of instability of the air in the layer.
Conversely, vertical expansion decreases the lapse rate, and thus de-
creases the instability.

If it is desired to evaluate the variation in lapse rate in terms of
changes in pressure, rather than in terms of changes in height, the
following modification is introduced. The relationship between the
pressure at the bottom and top of a layer having lapse rate !, accord-

THERMODYNAMICS OF DRY AIR

[Chap. 3

ing to (9-9), may be expressed as

for.

(-7 = i -

\P2/

fc T 4i fa

= ^~L " v~

Ret,

(16-8)

Expanding (pi/p%) in series results in
#i

R&I pi R c

o P2 2y

) + higher terms (16-9)

Under average lapse rate conditions, i.e., ai = 6 C km l , Ra\/g is of
the order of 0.2. If conditions in the lower troposphere only, where
Pi > 500 mb, are considered, and only layers such that p 2 PI < 200
mb are taken, then

and

A PA'

V P2/

and to a first approximation, (16-9) becomes

By substituting (16-10) in (16-8), the latter becomes

log-

Similarly,

s' P2

, p 3

= log

Q P*

(16-10)

(16-11)

(16-12)

Substituting (16-11) and (16-12) in (16-7) then gives

Z T 2log-

\ P4

(16-13)

Sec. 16} VERTICAL MOTION AND STABILITY 39

According to (13.2)

T* = (P2\

T, W

Substituting in (16-13) then gives

I/ \
Pi \
P2 lg
- 1

log .

P4 /

(16-14)

where K = 0.288. A further simplification may be introduced in (16-14)
by expanding the logarithmic terms in series. Thus

log = ( - 1 ) - U - - 1 ) + higher terms (16-15)

P'2 \P2 / \P2 /

The logarithmic term in the denominator of (16-14) may be similarly
expanded. If only thin layers are considered, such that p 2 p\ and
P-i ~ P3 < 100 mb, then only the first term need be retained, and
(16-14) becomes

2~~ l (Pi ~ P2> .1
^j - Ij

x r . /IAI\

o* = x - (T - ^^j - I (16-16)

It must be emphasized that (16-14) and (16-16) are valid only for the
lower troposphere, and for comparatively thin layers. The greater the
pressures and the thinner the layers, the greater the accuracy of these
expressions becomes. In practice these limitations are not serious, as
a knowledge of the variation of lapse rate with ascent or descent is
usually required only in the lower atmosphere, and for relatively thin
layers. The accuracy of (16-7) is greater than that of either (16-14)
or (16-16), but even the use of (16-7) involves a slight error, since the
value of F, 9.8 C per km, is only approximate, as shown by (14-1).

If an adiabatic chart, or one of the thermodynamic charts discussed
in section 22, is available, the variation of lapse rate may be determined
directly. It is only necessary to plot the pressure and temperature at
the top and bottom of the layer in its initial position, and then move
along the appropriate dry adiabats until the pressures at top and bottom
of the layer in its final position are reached. The change in lapse rate
may then be determined in a qualitative manner by inspection.

The variation of lapse rate which results from vertical displacement
is of considerable significance in meteorology. This effect is especially
noticeable in anticyclones, where the subsidence of the air often pro-
duces a marked decrease in lapse rate and a corresponding increase in

THERMODYNAMICS OF DRY AIR

[Chap. 3

stability. The significance of the foregoing results will be brought out

in greater detail in Chapter 15.

17. The Carnot Cycle. Heat Engine Efficiency. In section 12 it

was shown that whim a gas expands it does work on the environment.

The work done by a unit mass of gas as it changes from its initial state

denoted pi, vi to its final state
p 2 , v% may be evaluated graph-
ically by means of the indi-
cator diagram. In this dia-
gram, the ordinate is pressure
and the abscissa specific vol-
umes, both on linear scales.
Consider, for example, the in-
dicator diagram shown in Fig.
14. According to (12-1), the
work done, dW, when the
unit mass of gas increases
in volume by the amount
dv is p dv, which is the
stippled element of area shown

Fid. 14. Work on an indicator diagram.

in the diagram. The total work done by the gas as its state changes
from pi, i>i to p 2 , v% along the path ABC is then the summation
of such elements of area under the curve, which is the area ABCFEA.
If now the gas at p 2 , ?'2 returns to its initial state along the same path
work must be done on the gas by the environment, since in this case the
gas must be compressed as it returns. The amount of work done on
the gas in this second stage is exactly the same as that done by the gas
in the first stage, so the net amount of work done in the process is zero.
If, however, the gas returns, not by the path CBA, but by the path
CDA, the resulting amount of work is not zero. The work done on
the gas by the environment in this instance is given by the area
ADCFEA. There is then a net amount of work done on the gas equal
to the area ABC DA. This fact may be stated in another manner by
saying that the work done by the gas as it completes a cycle on the
indicator diagram in a counterclockwise manner is negative. If the
state changes from pi, v\ to p 2 , *> 2 along the path ADC and returns to
Pi, #1 along the path CBA, i.e., the cycle is carried out in a clockwise
manner, the work done by the gas is positive and equal in amount to
the area A BCD A.

The equations for adiabatic and isothermal changes of dry air in terms
of potential temperature on the indicator diagram will now be derived.

Sec. 17] THE CARNOT CYCLE 41

Equation 13-3 for adiabatic motion may be expressed as

i i

pi,i-. = P Q T=: (174)

If the ratio c p /c v be denoted 7, then, since

P p

(17-1) becomes

pv y = poK (17-2)

The temperature of air having pressure /> and specific volume v is by
definition 0, the potential temperature, so that from (7-6) it follows that

Wo = RO (17-3)

Equation 17-3 may be rearranged so as to take the form

where the constant

j =

Combining (17-2) and (174) then gives the desired form of the adia-
batic equation on the indicator diagram.

pv y = J0 y (17-5)

Each adiabat on tho indicator diagram may thus bo designated by a
value of the potential temperature.

The amount of heat Q which must be supplied to the air to permit-
it, to move* isothmnally from adiabat 0i to adiabat 2 on the indicator
diagram will next be obtained. Since T is constant, with the use of
(7-6) (12-5) becomes

dQ = -Avdp - -ART

By integrating,

r p * dv PI

Q = -ART I ~ = ART\o% 1 - (17-fi)

J Pi P 7'2

From (13-5) it follows that

MY = ??
\P 2 / i

THERMODYNAMICS OF DRY AIR

[Chap. 3

(17-7)

Substituting (17-7) in (17-6) leads to the required equation for iso-
thermal motion,

(17-8)

This equation shows that the quantity of heat which must be supplied
to the air to permit it to go isothermally from one adiabat to another

adiabat is proportional to
the Absolute temperature T
of the isothermal.

I \ ^\ Since T is constant, the

I 9\ \ equation for the isothermals

on the indicator diagram is,
from (7-0),

p = RT = Constant (17-9)

It is now possible with the
aid of (17-5) and (17-9) to
draw adiabat s and isother-
mals on the indicator dia-
gram. Two adiabats, demoted
0i and 2 , and two isother-

FICJ. 15. Carnot cycle.

mals, denoted T\ and T 2 , on such a chart are shown in Fig. 15. Con-
sider a cycle of changes around the curve abcda. The amount of heat
which must be supplied to the air to carry it from a to b isothermally
at temperature 7 7 2 i> according to (17-8),

Q 2 = c p T 2 log

(17-10)

The change from b to c is adiabatic, so no heat is supplied. From
c to d, heat must be removed of amount

Q l =

(17-11)

The final stage from d to a is adiabatic, and no heat is supplied or re-
moved.

A cycle of this type, comprising two isothermal and two adiabatic

Sec. 18] ENTROPY 43

stages is known as a Carnot cycle. It is a cycle of the reversible type.
If the process is carried out in the reverse order, counterclockwise
around adcbdj all the operations will be repeated in the inverse order
and in the opposite sense. As was shown by means of Fig. 14, the
unit mass of air does work on the environment in the original cycle
described. When the cycle is carried out in the reverse order, the
environment does work on the air.

The Carnot cycle may also be thought of in terms of a reversible
heat engine. Such an engine takes in heat Q 2 at temperature 7 T 2 and
rejects heat Q\ at temperature T\, the changes in temperature from
T 2 to TI and from TI to T 2 taking place adiabatically. The mechanical
work W done in the cycle is equal to the area abcda. Since Q 2 is the
heat taken in at To along oh, and Qi is the heat given out at TI along cd,
and as there is no heat transfer along he or rfa, the mechanical work
done must also be equal to Q 2 Qi, expressed in work units. The
efficiency of a heat engine is defined as the ratio,

Work output

Efficiency = -
Heat input

The efficiency of a reversible heat engine is, then,

Q 2 ~ Qi Qi

If (1740) and (17-11) are substituted in the above equation, it follows
that

Efficiency of a reversible engine = 1 (17-12)

-/2

Carnot/ s principle and the second law of thermodynamics are implicit
in the above treatment. Carnot \s principle states that a reversible
engine can produce? the maximum amount of mechanical work derivable
from a given quantity of heat lowered through a given range of tempera-
ture. The axiom known as the second law of thermodynamics was
stated by Clausius as follows: It is impossible for a self-acting machine,
unaided by any external agency, to convey heat from one body to
another at a higher temperature.

18. Entropy. By referring again to Fig. 15, it can be seen that the
amount of heat which must be supplied to unit mass of air at a to take
it to c along the path ahc is, according to (17-10),

Q 2 = c P 7' 2 log (18-1)

44 THERMODYNAMICS OF DRY AIR [Chap. 3

Similarly, if it goes from a to c along the path oe/c, the heat supplied is,
from (17-11),

Qi = c p r!logj (18-2)

It can be seen from this example that the amount of heat which must be
supplied to unit mass of air to change its state from pi, v\ to p 2 , v< 2 is
not a function of the initial and final states alone, but also depends on
the path by which the transformation occurs.

If (18-1) is divided by T 2 and (18-2) by TI, the equations become

Q 2 2

= c p \o (18-3)

^cplogj 2 (184)

J- 1 ^l

If the heat supplied depends on the path, (18-3) and (184) show that
the heat divided by the temperature at which it is supplied is inde-
pendent of the path. The quantity Q/T, or the more general expres-

sion / , is known as the entropy <t> and occupies a fundamental
t/ 1

position in thermodynamical theory. It is defined by the equation

d* = ^ (18-5)

In the preceding discussion, the transformation from one state to
another state was accomplished by one isothermal process and one
adiabatic process. Any type of transformation, however, may be
treated in a similar fashion by assuming the path to be made up of a
series of infinitesimal isothermals and adiabats, and the result is the
same.

From (18-3) or (184) it may be seen that the increase in entropy

from a to c is given by

c- 0a = Cplogf (18-6)

Equation 18-0 may also be derived by substituting dQ from (12-5) in
(18-5), using the differential form of (13.5), and then integrating.
Since only changes of entropy, not its absolute magnitude, are of sig-
nificance, (18-0) may be expressed

= c p log0 (18-7)

Since the change of entropy during a reversible process is a function

Sec. 18] ENTROPY 45

of the initial and final states only, it follows that during a reversible
cycle, with the air returning to its initial state, there is no change in
entropy. If, during the process, heat is lost by some irreversible
process such as conduction, the cycle is no longer reversible, and there
is a net increase in the entropy of the system. It is therefore neces-
sary to be certain that no irreversible processes occur before assuming
the constancy of the entropy during a cycle.

An expression for the entropy of dry air may be derived by substi-
tuting dQ from (12-5) in (18-5). With (7-0), it follows that

d<t>=c p ^-AR ( ^ (18-8)

In meteorology, it is frequently assumed for the purposes of computa-
tion that air at a pressure of 1000 mb and with a temperature of 100 A
has zero entropy. Integrating (18-8) on this assumption gives for the
entropy of 1 gm of dry air

(18-9)

The entropy with respect to some unspecified level of entropy may then
be stated as

= c p log T - AR log p (18- 10)

The entropy 0' of 1 gm of saturated water vapor at temperature T
is equal to the entropy of 1 gm of liquid water at the same temperature
plus the entropy necessary lo convert it into vapor. Thus

/dT L
c y + y, (18-ii)

where c is the specific heat of liquid water in contact with its saturated
vapor, and L is the latent heat of vaporization at temperature T. The
assumption that c is constant involves little error, and (18-11) may be
written

*' = c log7' + | (18-12)

By combining (18-10) and (18-12) and modifying slightly, it is seen that
the entropy <t> y of 1 gm of saturated air is given by

<t> + x s <t>' c + x s c AR

46 THERMODYNAMICS OF DRY AIR [Chap. 3

p e s represents the partial pressure of the dry air and therefore must
be used instead of p.

Equation 18-13 cannot be used without modification for the entropy
of unsaturated air, since the latent heat of vaporization L is defined as
the quantity of heat which is necessary to evaporate 1 gm of liquid
water in contact with its saturated vapor. If the entropy of 1 gm of
unsaturated water vapor at temperature T is required, the process of
evaporation must be considered as taking place at the dew-point tem-
perature Td, and then the additional entropy required to raise the
temperature from T<i to T must be computed.

PROBLEMS AND EXERCISES

1. A layer of dry air extending from 700 to 050 ml) has a lapse rate of 1 C per km.
If this layer subsides in such a manner that the pressure difference between the
bottom and top of the layer does not change during the descent, through what pres-
sure interval must the layer subside if it is to become isothermal?

2. A unit mass of dry air undergoes a reversible cycle of changes. If this cycle is
represented by a closed curve on a diagram having entropy as ordinate and temper-
ature as abscissa, show that the work done during the cycle is equal to the area
enclosed by the curve. In addition, show that if the cycle is performed in the clock-
wise direction, work is done on the environment by the air and, conversely, that if
the cycle is performed in the counterclockwise direction, work is done on the air by
the environment.

BIBLIOGRAPHY

Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapters 2, 3, 4.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapters 2, 3, 4.

Koschmieder, H., Dynamische Meteorologie, Leipzig, Akad. Verlag., 1933. Chapter 3.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 3

(1930), Chapter 6.

CHAPTER 4
THERMODYNAMICS OF MOIST AIR

19. The Clausius-Clapeyron Equation. One of the most interesting
applications of thermodynamic theory to problems associated with
meteorology is the use of the classical Clausius-Clapeyron equation to
determine the rate of change of saturation vapor pressure with temper-
ature. This equation may be derived through tiie consideration of a
specific ( 1 arnot cycle.

Consider first a liquid enclosed in a cylinder with a piston at one
end. The space between the surface of the liquid and the piston is
occupied by saturated vapor at pressure p, which is a function of the
temperature only. Keeping the temperature constant, increase the
volume by raising the piston. Some of the liquid will then evaporate
to maintain the vapor pressure constant. Thus while some liquid
remains, increasing volume leaves the pressure unchanged. It follows,
then, that the isothermal of a liquid and its vapor in equilibrium is a
line of constant pressure. In the p-v diagram shown in Fig. 16, this
process is represented by the horizontal lines AB or A f B r . When the
volume has been increased until all the liquid has evaporated, as at B,
a further increase in volume will lead to a decrease in pressure, as in a
gas. If, on the other hand, the piston is depressed so that the vol-
ume decreases, condensation of the vapor will occur. When all the
vapor has condensed, as at, A, a further compression leads to a very
great increase in pressure, as shown in the figure, because liquids have
a very low compressibility.

Fig. 16 shows that the length of the horizontal portion AB or A f B f
of the isothermal decreases with increasing temperature until, for the
isothermal indicated by a broken line, it reduces to an infinitesimal
length. This latter isothermal is known as the critical isothermal,
arid its temperature T c is known as the critical temperature. The
corresponding volume is v c , and the corresponding pressure is p c . The
state specified by p c , v c , and T c is known as the critical state. For very
high temperatures the isothermals become equilateral hyperbolas, as
the properties of the substance approach those of a perfect gas. The
dotted line and the broken line in the figure divide the area into four
sections. In one section the substance acts as a liquid, in another as

THERMODYNAMICS OF MOIST AIR

[Chap. 4

a gas, in a third as vapor, and in the fourth as a mixture of liquid
and vapor.

Now focus attention on the portion of Fig. 16 under the dotted line,
where liquid and vapor co-exist in equilibrium, and assume that the
liquid is water and the vapor is water vapor. The lower curve repre-
sents an isothermal for water and its vapor for temperature T 7 , and
the upper curve a similar isothermal for temperature T + dT. Points
A and A.' represent the condition of 1 gm of liquid water, and points B

Liquid

FUJ. 10. Pressure-volume changes in the liquid, vapor, and gaseous states of a

substance.

and B' that of 1 gm of saturated water vapor at the temperatures T
and T + dT respectively. A'C and B f D are adiabats. Take 1 gm
of water around the Carnot cycle A 1 B'DC, starting at A'. The heat
taken in along A f B f which is necessary to convert the initial 1 gm of
liquid water at A f to 1 gm of saturated water vapor at B' is the latent
heat of vaporization L + dL at temperature T + dT. Now consider
the water and its vapor as the working substance in a heat engine which
takes in heat L + dL at T + dT and rejects heat L at T 7 . Since the
process is a reversible one, the efficiency is, according to (17-12),

1 -

Sec. 19] THE CLAUSIUS-CLAPEYRON EQUATION 49

The work done in the cycle is, then, given by

(L + dL)

dW =

T \

T + dT)

A
in work units and, therefore, to the first order,

m ^~A~T ^ 19>1 ^

The work done is also equal to the area enclosed by the cycle A* B'DC
and so, with sufficient accuracy,

dW = (AB)dp

If vi and v 2 represent the specific volumes of liquid water and water
vapor respectively at T, then

dW = (v 2 - v^dp (19-2)

Equating (19-1) and (19-2) gives the Clausius-Clapeyron equation,

- ^- _

-^ - (V2 -

dp L_

dT ~~

Since the pressure is that of saturated water vapor, replace p by cv
to conform \vith previous usage. Now v 2 ^> v\, so to a first approx-
imation

de s L

dT ATv 2
Substituting from the gas equation,

e 8 v 2 = R'T

it follows that

dg __ Le 8
dT ~ AR'T 2

(194)

This equation demonstrates the scope of the second law of thermo-
dynamics, in that its use permits the development, from theoretical
consideratioas, of a formula having significance in certain aspects of
meteorology. The special form of the Clausius-Clapeyron equation
developed above will be used in the following two sections.

50 THERMODYNAMICS OF MOIST AIR [Chap. 4

20. The Saturated Adiabatic Lapse Rate. The Stability of Saturated
Air. The rate of change of temperature with height in dry air ascending
adiabatically was derived in section 14 and expressed by equation
14-2. It is also possible to derive a similar equation for saturated air
ascending adiabatically, giving the saturated adiabatic lapse rate. The
latter is smaller than the dry adiabatic lapse rate since condensation
occurs as ascent to lower pressures proceeds, and latent heat of con-
densation is released and is taken up by the air.

According to (12-5), with (12-6) and (7-6),

dQ = c p dT - ART ~-
P

This equation applies when 1 gm of dry air is considered. If saturated
air is taken, then 1 gm of dry air is associated with X H grams of water
vapor, where x s is the saturation humidity mixing ratio. The heat
added to the air owing to condensation as it ascends through a pressure
interval dp is Ldx s , the minus sign showing that Q increases as x s
decreases. Neglect the effect of the specific heat of liquid water and
water vapor, which is very small in comparison with that of the latent
heat, and it follows that (12-5) above becomes

-Ldx, = c p dT - A RT (20- 1 )

Differentiating the approximate equation 10-12 for the saturation
humidity mixing ratio gives

/de s e s \

dx s = e I --- 9 dp]

\P P /

Now, combining (19-4) and (10-3) gives

de s Le s

Substituting (20-3) in (20-2) gives

Sec. 20} SATURATED ADIABATIC LAPSE RATE 51

Substitute (204) in (20-1) and rearrange to give

where

N - Le>
N ~ AK~T

The variation of temperature with height, not with pressure, is re-
quired. So replace dp by dz with the aid of (9-3). It follows that, if
the small difference in temperature between rising and environment air
is neglected,

__dT _ gA p + N
dz c eL

Finally, by making use of (14-2), the following expression is obtained
for the saturated adiabatic lapse rate

(20-6)

C p l

Since in the atmosphere eL > c p T, it is clear that F'< F as mentioned
above. If the water vapor condenses into liquid water, even at sub-
freezing temperatures, the latent heat is given by

L = 734 - 0.517 7 cal gm" 1 (20-7)

where T is in degrees Absolute, and the saturation vapor pressure e s
is that with respect to a water surface. Values for the latter may be
obtained from Table II in the Appendix, or from (214). If the process
is one of sublimation from the vapor to the solid (ice) state, then the
latent heat of sublimation

L = G77 cal gm" 1 (20-8)

should be used, along with values of e 8 appropriate over ice, as found
from Table II in the Appendix.

Some values of the saturated adiabatic lapse rate in C per kilometer
are given in the following table, for which the values are computed with
the aid of equation 20-6. Three assumptions were made in the deri-
vation. First, the thermal capacities of the liquid water and water
vapor in the air in question were assumed to be negligible. Second,
the approximate equation 10-12 for the humidity mixing ratio was

52 THERMODYNAMICS OF MOIST AIR [Chap. 4

considered to be sufficiently accurate for use in the derivation. Finally,
the difference in temperature between rising and environment air
was neglected. Computations using more accurate equations, how-
ever, show that the error involved is slight, and that the values given
in the table are sufficiently accurate for most purposes.

SATUHATED ADIABATIC LAPSE RATE
(C per kilometer)

Condensation Sublimation
7>(mb) 1000 600 1000 600

5.9 4.9
8.6 7.9

The criteria for the stability or instability of saturated air are sim-
ilar to those for dry air, except that the saturated adiabatic replaces
the dry adiabatic lapse rate. To be absolutely rigorous, the water
vapor content of the air surrounding the particle under consideration
should be taken into account since the density of air is a function of its
moisture content. This effect is small, however, and is usually neg-
lected. The following, then, are the criteria for determining the type
of equilibrium of a given particle of air.

Stable equilibrium a < r'
Neutral equilibrium a = r'
Unstable equilibrium a > r'

*If confusion is to be avoided, the stability of saturated air should be
considered with respect to upward motion. If no liquid water is
present in the air, its temperature will increase at the dry adiabatic
lapse rate if given a downward displacement since it is no longer sat-
urated. Only if water droplets arc present to maintain saturation con-
stantly by their evaporation will air descend at the saturated adiabatic
lapse rate.

21. Condensation Level. Dew-Point Changes in Adiabatic Motion.
It was shown in section 15 that when moist, unsaturatcd air ascends
adiabatically, its temperature decreases at a rate which is very nearly
the dry adiabatic lapse rate. As ascent proceeds, the air will reach a
temperature at which the water vapor present is just sufficient to sat-
urate the air. The height in the atmosphere at which this occurs is
known as the condensation level. At this point the dry-bulb temper-
ature and the dew point coincide.

Sec. 21] CONDENSATION LEVEL 53

In sections 14, 15, and 20, expressions were found for the rate of
change of temperature with height in adiabatically ascending dry air,
unsaturated moist air, and saturated air. It is now desirable to de-
termine an expression for the rate of change of dew point with height
in adiabatically ascending, moist, unsaturated air. To do this, an
expression for the saturation vapor pressure is necessary. Such an
expression may be obtained by integrating the special form of the
Clausius-Clapeyron equation given by (194), assuming L to be con-
stant. Separating the variables gives

Lf = AR> * (21-1)

J- 8

and performing the integration leads to

When TQ = 273 A, e 80 = 6.11 mb, and so (21-2) becomes

The gas constant R r for water vapor is, according to section 10, given by
R' = 4.62 X 10 6 cm 2 sec" 2 deg" 1

and the constant value of L is taken as 595 cal per gm, that for tem-
perature 273 A. Substituting for L, A, and R f in (21-3) and rear-
ranging give the saturation vapor pressure

8 .70 2340

e s = 6.11 X 10 T mb (214)

By substituting, following Haurwitz, the actual vapor pressure e for e 8
and the dew point T d for T in (21-1), it follows that

- = 5.38 X 10 3 ^ (21-5)

e 1 d

Now differentiate the expression for the humidity mixing ratio. (lQ-,8),
giving

*_E*-*>. (2 , 6)

But the humidity mixing ratio of ascending unsaturated, moist air is
constant, provided there is no mixing between the rising air and its

54 THERMODYNAMICS OF MOIST AIR [Chap. /,

environment, and so dx = 0. It follows, then, that

^ = ^ (21-7)

e p

In addition, equation 9-3 states that for the environment

Combining (9-3), (21-7), and (21-5) leads to the following expression
for the rate of change of dew point with height in adiabatically ascending
unsaturated, moist air.

- ^ = 6.35 X 1(T 8 ^ (21-8)

The above equation is used in section 139 to determine an approx-
imate formula for the condensation level which has proved very useful
in forecasting the heights of the bases of several types of clouds.

22. Thermodynamic Diagrams. It was indicated in section 14 that
dry adiabats may be drawn on a diagram in which atmospheric
pressure on a logarithmic scale is ordinate, and temperature on a linear
scale is abscissa. It is also possible to allow for the moisture content
of the atmosphere on such a chart by introducing lines of constant sat-
uration humidity mixing ratio x s from (10*9), and saturated adiabats
from (20-6), or from a more accurate equation. In practice, true sat-
urated adiabats are not used, but rather pseudo adiabats. Pseudo
adiabatic ascent is similar to true saturated adiabatic ascent in all
respects, except that liquid water is assumed to be precipitated as soon
as it has condensed. Considering the behavior of condensation phe-
nomena such as cloud and precipitation, it can be seen that in the at-
mosphere the actual process is neither true nor pseudo adiabatic, but
somewhere intermediate between the two. Pseudo adiabats can be
computed a little more readily than saturated adiabats, since the spe-
cific heat of liquid water does not enter the calculations, but the dif-
ference between the two is so small that it is of no consequence. Since
pseudo adiabats are shown on the chart, it is generally known as the
pseudo adiabatic chart. On it, both dry and pseudo (or saturated)
.adiabatic motions may be determined with ease.

Another thermodynamic diagram which is widely used is the tephi-
gram (T-^-gram). This diagram has the entropy of dry air as
ordinate and temperature as abscissa, both on a linear scale. From
(18-7), it follows that potential temperature on a logarithmic scale is
equivalent to entropy on a linear scale. In practice, it is usual to think

Sec. 22]

THERMODYNAMIC DIAGRAMS

of the ordinate as log 6, rather than as <. The tephigram is shown in
Fig. 17, with pressure lines sloping upward to the right, pseudo adiabats
sloping upward to the left, and saturation humidity mixing ratio lines

\ \

-30 -20 -10

Temperature (C) *-

FIG. 17. The tephigram.

not quite vertical, but sloping slightly upward to the left. Dry adi-
abats are horizontal, and isotherms arc vertical. A full-size tephigram
is provided with this book in a pocket at the back. Dry adiabats and
isothermals are shown on this chart with full green lines. Pressure
lines and pseudo adiabats are shown in full orange lines, and satura-
tion humidity mixing ratio lines in broken orange lines.

Other types of thermodynamic charts have been developed from
time to time, but the pseudo adiabatic chart and the tephigram are
two of the most widely used diagrams.

56 THERMODYNAMICS OF MOIST AIR [Chap. 4

It was proved in section 17 that the work done by a mass of air in
going through a cyclic process is proportional to the area enclosed by
the curve representing the various stages of the process on a p-v dia-
gram. It can be shown that on thermodynamic charts of the types
given above, areas are also proportional or equal to work done, or
energy released. In other words, the pseudo adiabatic diagram and
the tephigram are proportional or equal area transformations of a
p-v diagram.

Consider a unit mass of air moving vertically, having density p,
temperature T } and pressure p. The surrounding air at the same level
has density p, temperature T, and pressure p. The pressure of the
moving air is always the same as that of the environment. It follows
from Archimedes 7 principle that the force acting on the rising air is,

since p = - >
v

F = a = pva = pvg pvg
and so

But

p = RpT = RpT
and therefore

It follows then that

T \ T - T
==(J ~ =(J ~ - (22>2)

Air moving a distance dz does work dW and therefore

T T T T

dW = Fdz =

Using (9-1) and (7-7) and multiplying by A to convert to thermal units
give

dW = -AR(T - ?) (22-3)

dW = -AR(T - f) rf(log p) (224)

Sec. 22}

THERMODYNAMIC DIAGRAMS

Fig. 18 shows the variations of pressure and temperature of an
ascending mass of air, plotted on a pseudo adiabatic chart. ABC rep-
resents the state of the environment, while DEF indicates the "varia-
tions of p and T experienced by a mass of air as it rises from position D
to position F. The moving air at G is surrounded by air whose con-

FIG. 18. Energy released by rising air on a T-log p diagram.

dition is given by the point G at the same pressure. HR is a later
position of GG, separated by a small interval d(\og p). To a high
degree of approximation, GGHH may be considered as a rectangle
of area (T - T) d(\og p). From (224), since d(log p) is negative,
it follows that

dW = AR X Area GGffH

Therefore, as the mass of air in question ascends from D to F, the
total amount of energy liberated is proportional to the area DBFED.

Tn a similar manner, it is possible to determine from the tephigram
the amount of energy released by an ascending mass of air. In Fig.
19, ABC represents the environment curve, while DEF gives the suc-
cessive values of <t> and T as the air ascends. The ascending air, when
at G, is surrounded by air whose state is represented by G at the same
pressure. HR is the position of GG a short time later; the lines

58 THERMODYNAMICS OF MOIST AIR [Chap. 4

GK and MLGJ are ordinates in the diagram. Equation 18-8 states that

, A dT dp

a<p = C-n - - AK

T p

FIG. 19. Energy released by rising air on a tephigram.

Now GK is the change in entropy when the pressure changes by dp
while T remains constant. It follows that

GK = -AR ^
P

Therefore

-AR(T -

= Area MGKL

= Area GGK J
= Area

since the ends of the strip, the two triangles GKH and GJH, can be
considered equal, the difference being a small quantity of the second
order. Now (22-3) states that

so that

dW = -AR(T- f)

dW = Area GGEH

Sec. 28] THE PSYCHROMETER EQUATION 59

By integration, it follows that the energy released during the ascent
from D to F is equal to the area DBFED, if expressed in heat units.

23. The Psychrometer Equation. The psychrometer, or hygrometer
as it is sometimes called, is described in section 65. For present pur-
poses, however, it is sufficient to state that the psychrometer is an
instrument for measuring the moisture content of the atmosphere. It
consists of two ordinary thermometers, the bulb of one of which is kept
moist by one of several suitable processes. If the air is saturated,
there is no evaporation from the wet bulb, and both thermometers
show the same temperature. If the air is unsaturated, however, there
will be evaporation from the wet bulb, cooling it, and the wet-bulb
thermometer then gives a lower reading than the dry-bulb one. The
theory of this cooling process is given below.

Assume that unsaturated air is moving past the wet-bulb ther-
mometer. The air, as it approaches the wet bulb, has x grams of water
vapor associated with 1 gm of dry air at temperature T. Evaporation
of water from the wet bulb saturates the air, so that the air leaving
the wet bulb comprises x sw grams of water vapor with 1 gm of dry air,
now at the temperature of the wet bulb T w . During this process,
%sw % grams of liquid water have evaporated from the wet bulb, the
evaporation occurring at the temperature of the wet bulb, presumably.
It is not obvious that evaporation occurs at T w , but it is an assumption
the validity of which must be assessed in the light of the results obtained.
The thermodynamical process may be stated mathematically in the
following form.

(c p + xc p ) (T - T w ) = L w (x sw - x) (23-1)

c' p is about twice as great as c p , but since x is of the order of magnitude
of 0.01, it is permissible to neglect xc' p in comparison with c p , as a first
approximation. Equation 23-1 then becomes

c p (T - T w ) = L w (x sw - x) (23-2)

Substituting (10-11) and (10-12) in (23-2) leads to

JW (23 .3)

where e sw is the saturation vapor pressure at T w . With this equation,
therefore, it is possible to compute the actual vapor pressure when the
values of T and T w are measured. This equation is commonly referred
to as the psychrometer equation. The consistency of the results ob-
tained in using this formula for measuring humidity is the justification
for the assumption that the evaporation takes place at the tempera-
ture T w .

60 THERMODYNAMICS OF MOIST AIR [Chap. 4

24. Wet-Bulb Temperature. Wet-Bulb Potential Temperature. A

definition of wet-bulb temperature follows from what has been given
in the previous section. Consider 1 + x grams of unsaturated air. The
mixing ratio of air saturated at the wet-bulb temperature is x sw . Evap-
orate an amount of liquid water x 8W x grams into the air, which is
sufficient to reduce the dry-bulb temperature to the initial value of the
wet-bulb temperature. The air is then saturated, and no further
evaporation can take place, even if more liquid water is introduced.
It follows then that the wet-bulb temperature may be defined as the
lowest temperature to which air may be cooled by evaporating water
into it. It is not necessary that the lowering of the temperature be
carried out in one step. Evaporate x\ x grams of water into the
air, which brings the dry-bulb temperature down to a value inter-
mediate between the initial dry- and wet-bulb temperatures. At this
point, it is still possible to cool the air to T w by evaporating x sw x\
grams of water into it, making a total evaporation of x sw x grams in
the two stages. Since the limit of cooling has not changed during the
process visualized, it follows that the wet-bulb temperature T w has
remained constant. This result, that the wet-bulb temperature of air
is unaltered by evaporating water into it, is of considerable significance
and finds many applications in both theoretical and practical aspects of
meteorology.

Normand deduced the two following propositions from the basic
assumption stated in section 23.

Proposition I. The heat content of the air is equal to the heat con-
tent of the same air, saturated at the wet-bulb temperature, minus the
heat content of the additional liquid water required to saturate it.

Proposition II. The entropy of the air is equal to the entropy of the
same air, saturated at the wet-bulb temperature, minus the entropy of
the additional liquid water required to saturate it.

Proposition I is obvious, and is indeed just a restatement of the
fundamental assumption; the second proposition is not so obvious.
The process is not reversible, and there is a gain in entropy, since heat
is lost by the air in cooling from T to T w , and the same amount of heat
is taken up by water at the temperature T w .

It is, however, possible to evaluate the gain in entropy, and so deter-
mine the order of magnitude of the increase. With (23-1), the gain in
entropy by the water is given by

See. 24] WET-BULB TEMPERATURE 61

The loss of entropy of the original moist air is

dT T

-57 = (P

The net gain in entropy of the system is

; lo g sr

\ -* w J- w

But

fTl / fjl /TT \ fT\ /77 XT' T* \ 2

log = log( I H - -) = " - H ") + higher terms

J- w \ * w / * w \ -* iy /

It follows then that

^2 1

r /? T - T\A

A0 = (c p + CpZ) H I J

I \ I w /

+ higher powers f (24-1)

Equation 24-1 shows that the net gain of entropy of the system is ap-

(T - 7 T ,A

proximately equal to \ f J times the gain of entropy of the

\ ' w /

water. This is a small fraction and may be neglected, and the process
visualized may be considered as isentropic. Proposition II may then
be assumed to be verified to a high degree of approximation.

This second proposition may be used to prove a third proposition,
also due to Normand.

Proposition III. The dry adiabat through the dry-bulb temperature,
the saturated adiabat through the wet-bulb temperature, and the
humidity mixing ratio line through the dew point intersect at a point.

First consider a mass of 1 + x s grams of air, saturated at B (Fig. 20),
which goes through the following process. This air ascends, its tem-
perature decreasing at the saturated adiabatic lapse rate until it reaches
D, where its mixing ratio is x. All water which has condensed during
the ascent, i.e., x s x grams, is shed at D. If descent next takes place,
warming will proceed at the dry adiabatic lapse rate, and upon reaching
the original pressure, the temperature of the air will be that denoted
by A in Fig. 20. It follows that

Entropy of 1 + x 8 grams of air, saturated at B = Entropy of 1 -^ x
grams of moist air at A + Entropy of x s x grams of liquid water
shed at D (24-2)

THERMODYNAMICS OF MOIST AIR

[Chap. 4

Next let A' denote the dry-bulb temperature of 1 + % grams of air
whose wet-bulb temperature is B, and whose mixing ratio is x. Propo-
sition II states that

Entropy of 1 + x s grams of air, saturated at B = Entropy of 1 + x
grams of moist air at A ' + Entropy of x s x grams of liquid water
removed at B (24-3)

A comparison of (24-2) and (24-3) shows that

Entropy at A = Entropy at A'

and therefore A and A f must coincide.

FIG. 20. Dry-bulb, wet-bulb, and dew-point temperature changes in adiabatically

ascending air.

The line CD represents the humidity mixing ratio line through D.
Since the dew point moves along the actual mixing rat io line during dry
adiabatic processes, as indicated in section 21, the point C where CD
intersects the isobar through A is the dew point of the air at A.

The above proof is not exact, since in (24-2) the liquid water was
removed at D, the temperature of the condensation level, whereas in
(24-3) the water was removed at the wet-bulb temperature. However,
the errors due to this are only slight, and Proposition III may be taken
as verified to a satisfactory degree of accuracy.

The above results show that when air undergoes adiabatic changes
in the saturated or unsaturated state, the wet-bulb temperature follows
a single saturated adiabat. It is possible, then, to define a wet-bulb
potential temperature 6 W as the wet-bulb temperature which the air
assumes when brought adiabatically to the standard pressure of 1000 mb.
The value of B w can be determined readily from a pseudo adiabatic
chart, a tephigram, or any other thermodynamic chart. Simply follow

Sec. 25} EQUIVALENT TEMPERATURE 63

the saturated adiabat through the wet-bulb temperature until the
1000-mb line is reached. The temperature at that point is B w . The
pseudo adiabats on the tephigram in the pocket at the back of this
book are labeled with their appropriate values of 6 W . The wet-bulb
potential temperature does not change during evaporation or condensa-
tion processes in the free; air, nor during adiabatic ascent and descent.
The properties of wet-bulb and wet-bulb potential temperature are
discussed in greater detail in sections 83 and 84. The use of the wet-
bulb temperature in forecasting procedures is brought out in Chapters
14, 19, and 20. Normand's third proposition has many applications in
practical meteorology, as will be seen in later sections.

25. Equivalent Temperature. Equivalent Potential Temperature.
Rossby Diagram. The wet-bull) temperature is invariant for evapora-
tion processes in the free air. There is another quantity, the equivalent
temperature 7 7 C , which has similar properties. The equivalent tempera-
ture may be defined as the temperature attained by a mass of air,
originally saturated or unsaturated, when it ascends until all its water
vapor is condensed and precipitated, and it is then brought back adia-
batically to its initial pressure. Since one dry adiabat, and one only,
is asymptotic to the saturated adiabat which the wet-bulb temperature
of the rising air follows, the equivalent temperature is a single-valued
function of the wet-bulb temperature. In certain circumstances, the
two may be used interchangeably, but the wet-bulb temperature has
certain advantages in practice which give it a much wider usefulness.

The equivalent temperature may also be defined in a slightly different
manner. Consider (23-1)

(c p + *c' p )(T - T w ) = L w (x aw - x)

If T e is the temperature of absolutely dry air which has a wet-bulb
temperature T w , then the above equation must be valid when T = T.
and x = 0, and it becomes

c P (T e - T w ) = L w x sw (25-1)

TI = TV +^ (25-2)

c p

The two definitions are not identical, and give values for the equivalent
temperature which arc slightly different. The difference is small, how-
ever, and for most purposes may be neglected.

The equivalent potential temperature B e of a mass of air may be defined
as the temperature attained when the air ascends until all water vapor

64 THERMODYNAMICS OF MOIST AIR [Chap. 4

is condensed and precipitated, and it then descends dry adiabatically
to the standard pressure 1000 mb. It may also be defined by sub-
stituting O e and 6 W for T e and T w in (25-2) to give

.=*+ (25-3)

c p

where L w and x sw now have values appropriate for O w .

The equivalent and equivalent potential temperatures of a mass of
air may be deduced from the pseudo adiabatic diagram or the tephigram
without difficulty. Proceed along the saturated or pseudo adiabat
through the wet-bulb temperature until the saturated adiabat becomes
approximately parallel to the dry adiabat s, indicating that the vapor
content of the air is so small that it can be neglected. Thereafter, pro-
ceed dry adiabatically to the initial pressure for T e or to 1000 mb for O e .

It can be seen from both definitions of O e that it is a single-valued
function of 6 W . Practical considerations, however, usually favor the
use of the wet-bulb potential temperature.

Rossby has constructed a chart in which the ordinate is potential
temperature* on a logarithmic scale, and the abscissa is humidity mixing
ratio on a linear scale. It follows from the first definition of Q e given
above (due to Rossby) that the value of O e of a mass of air is determined
by its potential temperature and its mixing ratio. Lines of constant O e
may then be drawn on a chart with these coordinates. The chart with
these coordinates, and with lines of constant equivalent potential tem-
perature on it, is known as the Rossby diagram, or the equivalent potential
temperature diagram.

Since 6 and x do not vary during dry adiabatic processes, a curve on
the diagram representing atmospheric conditions in the vertical will
not change its position as a result of dry adiabatic motion. Evapora-
tion or condensation processes will vary its position, however, since
both 6 and x are changed thereby. Stability and instability, with
respect to both dry and saturated adiabatic displacements, may be
determined. No pressure or temperature lines appear on the chart,
and the lack of these may be inconvenient at times. It is not an equal
or proportional area transformation of a p-v diagram, and therefore
it cannot be used to study energy relationships. The Rossby diagram
was intended primarily as a tool to assist in air mass analysis, and it is
most useful when employed for that purpose. It is better in distinguish-
ing between air masses than other diagrams in general use, since curves

* More accurately, partial potential temperature, the temperature attained by air
descending dry adiabatically from the partial pressure of dry air, p e, to standard
pressure, 1000 mb.

Scc.8fi] EQUIVALENT TEMPERATURE 65

representing dry cold air appear to the far left of the diagram, while
those representing warm moist air are found to the right. Differences
between intermediate air masses are brought out clearly. The use of
the Rossby diagram in classifying air masses is indicated in section 106.

PROBLEMS AND EXERCISES

1. Integrate the Clausius-Clapeyron equation when the latent heat of vapor-
ization

L = 734 -0.51Tcalgm- 1

where T is in degrees Absolute. Use the equation derived to compute the saturation
vapor pressure e s at 293 A. Also compute e s for the same temperature using (21.4),
and compare each value obtained with the observed value as given in Table II in
the Appendix.

2. Determine the equivalent temperature of a mass of air with pressure 900 mb,
temperature 273 A, and relative humidity 80 per cent by each of the two methods
outlined in section 25. When using (25.2), first assume a process of condensation,
and then repeat the computation assuming a sublimation process. Compare the
three answers, remembering that the pseudo adiabuts on the tephigram are con-
structed on the assumption that water vapor condenses into liquid water, even at
sub-freezing temperatures.

3. Show that in Refsdal's " aerogram " (ordinate T log /?; abscissa log T), the
energy liberated by a mass of air ascending through its environment, which is denser
thnn the rising jiir at each level, is proportional to the area between the environment
curve and the ascent curve.

4. Determine, using the full-scale tephigram at the back of the book, the relative
humidity of air with dry-bulb temperature 12 C, at a pressure of 880 mb, when the
wet-bull) temperature varies from 11 C to 1 C. The relative humidity should be
determined for every other degree Centigrade, as 11, 9, 7, 1 C.

Then determine the relative humidities for the same values, using the following
derivative of Regnault's empirical equation. Regnault's formula is the basis of
most hygrometric tables.

- T w }(\

c = e sw - ().0006/j(r - T w }[ I

where e - the actual vapor pressure.

e sw -- the saturation vapor pressure at the wet-bulb temperature.
p = pressure.

T = dry-bulb temperature in C.
T w = wet-bulb temperature in C.

Saturation vapor pressures are given in Table II in the Appendix.

Plot relative humidity against wet-bulb temperature for each of these two deter-
minations, and show at what relative humidity the maximum difference between the
two curves occurs.

5. (a) By using the Clausius-Clapeyron equation and expressing the latent heat
of vaporization L in the form

L = Lo + (c'p - c)(T - To)

66 THERMODYNAMICS OF MOIST AIR (Chap. 4

show that the entropy of vaporization of 1 gm of liquid water

r)logT-

(6) Use the above equation and assume that the constant C = to show that the
expression for the entropy of 1 gm of saturated air given by (18.13) may be modified
to the form

c p + x s c'p AR

- -

This expression also gives the entropy of unsaturated moist air if the actual value of
the vapor pressure e is used instead of the saturated vapor pressure e s , and if x is
used instead of x 3 .

BIBLIOGRAPHY

Brunt, D., Physical ami Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapter 4.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapters 3, 4.
Koschmieder, II., Dynamische Metcoroloyie, Leipzig, Akad. Verlag., 1933. Chap-

ter 3.

19. Cork, J. M. f Ifrnt, New York, John Wiley and Sons, 1933. p. 230.

20. Fieldstad, J. E., " Graph ische Methoden zur Ermittelung adiabatischer zu-

standsanderungen feuchter Luft," Gcofys. Ptibl., 3, No. 13, 1925.
22. Werenskiold, W., " On Equal Area Transformations of the Indicator Diagram

and a New Aerological Chart," Geofys. PubL, 12, No. 6, 1938.
22. Shaw, Sir N., Maniuil of Meteorology, Vol. 3 (1930), p. 269.

24. Normand, C. \V. B., Wet-Bulb Temperatures and the Thermodynamics of the Air,

Mem. Indian Met. Dept., 23, 1, 1921.

25. Rossby, C.-G., Thrrmodi/ nam ics A pplied to Air M /tss Analysis, Mass. Inst. Tech.

Met. Papers, 1, No. 3, 1932.

CHAPTER 5
RADIATION IN THE ATMOSPHERE

The general term radiation embraces the energy transmitted by the
whole range of electromagnetic waves from cosmic rays, having very
short wave lengths of the order of 10~ 12 cm, to long radio waves having
wave lengths of the order of 10 cm. The meteorologist usually, how-
ever, is concerned only with radiation in a relatively small band of wave
lengths, from 2 X 10~ 5 cm to 5 X 10~ 3 cm. The unit of wave length
which is commonly used in meteorology is not the centimeter, but the
micron, /z, which is equal to 10~ 4 cm. The range under present con-
sideration is therefore from 0.2 to 50 ju. This range may be divided
into two generic groups. The first includes the radiation of the sun,
which at its surface has a temperature of about 6000 A. This solar
radiation, or short-wave radiation, as it is often called, occurs in the wave
length interval from 0.2 to 4 ju, of which only the radiation from 0.4 to
0.7 /z can be detected by the human eye. The second group of wave
lengths is that emitted by the earth and its atmosphere, at temperatures
ranging from 200 to 300 A. This radiation, known as terrestrial or
long-wave radiation, extends from 4 to 50 ju.

26. The Laws of Black-Body Radiation. Before discussing the
radiative processes in the atmosphere, it is desirable to review the
several laws which of necessity form the background of any treatment
of radiation.

(a) Kirchhoff's Law. If radiation of a specified wave length p is inci-
dent at the surface of any body, then the absorptive power a^ of the body
for that wave length is defined by Kirchhoff as the ratio of the radiant
energy absorbed to the total incident radiant energy. In addition,
Kirchhoff defined a perfectly black body as a body which absorbs all the
radiation which falls on it, irrespective of whether the radiation is in
the visible spectrum or not. It follows therefore that the absorptive
power a M of such a body is unity. For all actual substances, however,
the absorptive power is a proper fraction, the value of which depends on
the nature of the substance, on its temperature, and on the wave length
of the incident radiation.

If e M dju is the radiation emitted per second between the wave lengths M
and ju + dp, from unit area of a body, e^ is known as the emissive power

RADIATION IN THE ATMOSPHERE

[Chap. 5

of the body for the wave length M- Kirchhoff's law states that the
ratio of the emissive and absorptive powers for a given temperature and
a given wave length is the same for all bodies, and is equal to the emis-
sive power of a perfectly black body. This law may also be stated as

where E^ represents the black-body radiation at wave length /*, and for
any specified temperature. When e^ is a fixed fraction of K^ for all wave
lengths, the body is referred to as a gray body, and the radiation as gray
radiation.

This law is of great importance in discussing atmospheric radiative
processes, for it indicates that if a layer of the atmosphere absorbs
radiation of a certain wave length, that layer will itself also radiate
energy of that same wave length. Conversely, if a layer is transparent
to radiation of another wave length, it will not emit radiation of that
wave length.

(b) Planck's Law. The variation of black-body radiation at known
temperatures with wave length was investigated experimentally by
Lummer and Pririgsheim, and later theoretically, using quantum princi-
ples, by Planck. The expression obtained by the latter gave values in
good agreement with the experimental values. Planck's formula states
that

5 ^- I)- 1

u.uo

* 002

8
s 001

c
o

\\273

^-^

) 10 20 30 40 5(
Wave* Length m p

(o)

7 c

\6C

00A

V i n*in s -

c
Q

01 012345

Wave Length .n/j

(b)

FIG. 21. Energy radiated by a black body at (a) 273 A and at (b) 6000 A.

where Ci and c% arc constants. The distribution of energy in the
spectrum of a black body at a temperature of 273 A is shown in Fig. 2 la,
and that at a temperature of 6000 A is shown in Fig. 216. It may be
assumed without serious error that the earth radiates as a black body
at a temperature of about 273 A, and that the sun radiates similarly
at 6000 A. Those two curves show, then, the energy radiated by the

Sec. 26} THE LAWS OF BLACK-BODY RADIATION 69

earth and the sun at their respective temperatures. The great differ-
ence between the amount of energy radiated by the sun and that radi-
ated by the earth may be seen from a comparison of the vertical scales
of the two figures. The maximum energy emitted per unit wave length,
area, and time by the sun is very nearly 5 X 10 6 times as great as that
emitted by the earth. The radiation is emitted in the two wave length
intervals mentioned in the introductory section of this chapter.

(c) Wien's Law. Before Planck derived his expression for the energy
distribution of black-body radiation, Wien had developed an equation
of the form

where the constants are the same as in Planck's equation. This expres-
sion is less accurate than that of Planck, but the difference between the
two is small, except for large values of ju or T. To determine the value
of jit for which E^ is a maximum, differentiate Wien's equation partially
with respect to ju, and set the result equal to zero, as indicated below.

^ = Ciir+e'^bvT 1 ! 1 - 1 - 5) =

a/*

The wave length n m for which E^ is a maximum is then

('2 a

Mm = gy = y 7

The constant c 2 has been evaluated as 14,385 micron degrees, so that

= a = 2877 micron degrees
5

This figure agrees well with the average of recent experimental results,
2897 micron degrees.

Wien's law thus states that the wave length for maximum black-body
radiation is inversely proportional to the temperature. Using the ex-
perimental value of a, ju m has for 273 A the value 10.6 /x, and for 6000 A
the value of 0.48 ju, in excellent agreement with the positions of the
maxima shown in Fig. 21.

(d) Stefan's Law. The total emissive power of a black body may be
obtained by integrating Planck's expression for all wave lengths from
to oo . The method of integration is given in advanced textbooks on
heat, and will not be reproduced here. When the integration has been
performed, the result is

E = I

70 RADIATION IN THE ATMOSPHERE [Chap. 5

This is Stefan's law, which states that the total black-body radiation
is directly proportional to the fourth power of the temperature. Stefan's
constant

<r = 8.14 X KT 11 cal cnT 2 muT 1 deg- 4

It follows that the areas under curves of the type shown in Fig. 21 are
proportional to the fourth powers of the respective temperatures.

Of the four laws for black-body radiation given above, KirchhofFs
law and Stefan's law are most widely used in meteorology, as will be
noted in the following sections of this chapter.

27. The Law of Absorption. Consider a beam of parallel radiation
of intensity / which falls perpendicularly on a layer of some absorbing
medium of density p and thickness dz. The mass din of the medixim
traversed by the radiation is then p dz. The amount of radiation ab-
sorbed dl is proportional to tho intensity of the beam and the mass of
the absorbing medium traversed. Thus

dl QC Jdtn

The negative sign indicates that dl represents a decrease in intensity
of the beam, rather than an increase, as it traverses the layer. Intro-
ducing a constant fc, the equation becomes

dl = -kldm (27-1)

The constant k is known as the absorption coefficient of the absorbing
substance. Integration of (27-1) gives

log- = -km (27-2)

Jo
or

/ = Ioe- km (27-3)

where 7 is the intensity of the radiation when m = 0, and 7 is the in-
tensity after traversing a layer containing mass m. The use of this
equation is sometimes facilitated if it is stated as

/ = / 1(r o- 4343 * (274)

Equations 27-2, 27-3, and 274 are mathematical statements of the
fundamental law of absorption, known as Beer's law.

The transmission by a layer is defined as the ratio of the intensity of
the radiation emerging from the layer to that incident on the layer.
The transmission coefficient T is defined as

T = (27-5)

Sec. 27]

THE LAW OF AHSOHPTION

The coefficient r is a function of k and m. If the radiation occurs in an
interval of wave lengths in which k does not vary, from (27-3) r may be
expressed as

r = e- km (27-6)

If k varies in the wave length interval, some value of ft, representing the
absorption in that interval, must be adopted.

FIG. 22. Variation of absorption with frequency in a line spectrum. (After

Elsasser.)

An equation such as (27-0) or a slightly modified form, is adequate
if the absorption is continuous throughout the spectrum. However,
if the absorption spectrum is not continuous, but is made up of a series of
discrete lines, it is much more difficult to set up an adequate trans-
mission formula. Elsasser assumed that the variation of the absorp-
tion coefficient k with frequency v (the reciprocal of the wave length)
for a series of lines in a portion of a band spectrum may be represented
in the manner shown in Fig. 22. He obtained a transmission function

TJ = 1 -

where $ is the probability integral (see section 59)

2 r x

<t>(x) = - I (T^dx
V/TT Jo

(27-7)

(27-8)

and I is the generalized absorption coefficient. Elsasser 's transmission
function may be compared with the simple form given in (27-6) for
small values of m, i.e., for thin layers. The probability integral may
be expanded in the power series

2 / -3

72 RADIATION IN THE ATMOSPHERE [Chap. 6

By neglecting terms of higher order, (27-7) becomes

/2/w
TI = 1 - J (27-9)

lr 7T

The expansion of e~ km in series gives the relation

e-" m = 1 - km + ^

2 1

By neglecting terms of higher order, (27-6) becomes

r = 1 - km (27-10)

A comparison of (27-9) and (27-10) shows that for thin layers the
absorption in a band spectrum is proportional to the square root of the
absorbing mass, whereas for a continuous spectrum, it is proportional
to the mass. The physical basis of this result may be seen from Fig. 22.
For thin layers, the absorption occurs in the sloping portions of the
curves, as at a, giving a non-linear relationship between absorption and
mass. As the thickness of the layer increases, the values given by
Elsasscr's transmission function approach those given by (27-6). For
thick layers, the regions between the linos, as at />, become effective in
absorption, and the absorption then increases directly with mass.

For most purposes, when considering band spectra, the transmission
is obtained with sufficient accuracy by substituting / for k in (27-0).

The above results are of considerable significance in the study of
radiation in the atmosphere, since both water vapor and carbon dioxide,
which absorb terrestrial radiation strongly, have line or band spectra.

28. Solar Radiation. In discussing the solar radiation which reaches
the earth and its atmosphere, it is first necessary to know the intensity
of the radiation incident at the outer limits of the atmosphere. The
intensity varies with the latitude and the time of year. For instance,
the maximum amount of radiant energy falling on a horizontal unit sur-
face just outside the atmosphere during any day of the year is about
1100 calorics. This intensity, nearly 0.8 cal per cm 2 per miri, occurs at
the poles during the summer solstices. At this time, the energy input
varies little with latitude, only decreasing to about 0.6 cal per cm 2 per
min at the equator. The variation with latitude is much more marked
at the equinoxes, the values ranging from zero at the poles to about 0.0
at the equator.

A measure of the intensity of the radiation at the outer limits of the
atmosphere is the solar constant. This is defined as the amount of energy
falling perpendicularly on unit area in unit time when the distance be-
tween earth and sun is at its mean value. The solar constant has the

Sec. 28} SOLAR RADIATION 73

value 1.94 cal per cm 2 per min. The determinations of the solar con-
stant are made at solar observatories located on mountains, where the
air is clear, and the diminution of the beam in traversing the atmosphere
is a minimum. The values obtained show a range of 2 or 3 per cent of
its mean value as given above. It is not yet clear whether these varia-
tions arc real, or if they result from inaccuracies in correcting for the
absorption and scattering of the solar beam before it reaches the instru-
ments. The question is an important one, however, for if significant
variations in the output of solar energy do occur, such variations may
be related to terrestrial weather conditions, and a study of the relation-
ship between the two might lead to improvements in long-range fore-
casting techniques.

When solar radiation enters the earth's atmosphere, it is attenuated
by three distinct processes. The first of these is through scattering by
small particles, such as the molecules of the air, and by impurities such
as dust particles. This scattering was investigated by Haylcigh, who
developed an expression of the type

T T i>& x
1 1()C

where IQ is the intensity of the incident beam, 7 is the intensity of the
beam after it has traversed a distance x, and ft is the coefficient of scatter-
ing. Rayleigh showed that is inversely proportional to the fourth
power of the wave length. Thus in the visible portion of the solar
spectrum, blue light- is scattered more; than red light-, since the former
has a shorter wave length than the latter. This fact accounts for the
blue color of the sky, since blue light is scattered more than red, and
there is therefore a preponderance of blue light- in the diffuse radiation
in the sky. This also accounts for the red color of the sun at sunrise
and sunset. The solar rays then traverse a long path through the
atmosphere, during which time the blue is scattered from the beam,
leaving a preponderance of rod in the direct rays which roach the eye.

The second process which depletes the solar beam is absorption.
This absorption is almost entirely due to water vapor in the atmosphere.
The variation of energy absorption with water vapor content of the
atmosphere is shown in Fig. 23. The abscissa in this diagram repre-
sents the number of grams of water vapor in the column of atmosphere
of cross section 1 sq cm traversed by the solar radiation. By comparing
the absorption values shown with the solar constant, it is seen that from
G to 13 per cent of the solar beam may be absorbed by water vapor in
passing through the earth's atmosphere. Ozone at high levels also
absorbs a small fraction of the solar radiation, as indicated in section 32.

Finally, the radiation from the sun may be reflected and absorbed by

RADIATION IN THE ATMOSPHERE

[Chap. 5

'02468
Water Vapor (gm cm" 2 )

FIG. 23. Absorption of solar radiation by water vapor. (After Hoelper.)

100

Reflection

Transmission

Absorption

10 20 40 6080100 400 1000

Thickness of Cloud in Meters

4000 10000

FIG. 24. The reflection, absorption, and transmission of solar radiation by clouds
of varying thicknesses.

Sec. 28} SOLAR RADIATION 75

clouds. The reflecting power, or albedo, of clouds for solar radiation
varies widely, depending mainly on the thickness of the cloud in ques-
tion, and the amount of liquid water within it. It has been found, from
theoretical considerations, that the albedo for thin, diffuse clouds may
be as small as 0.1 or 0.2 (an albedo of 0.2 means that 20 per cent of the
incident radiation is reflected), or for very thick, dense clouds it may
be higher than 0.9. For clouds of average thickness and density, the
albedo is about 0.8. The percentage of solar radiation reflected by
clouds of varying thickness containing 1.0 gm of liquid water per m 3 is
shown in Fig. 24. The maximum absorption of solar radiation in
clouds is about 7 per cent of the incident radiation, which occurs in
very thick, dense clouds. For thin, diffuse clouds the absorption is
small, being loss than 1 per cent. The variation of absorption with
thickness of clouds having the same water content as above is shown
in Fig. 24. The third curve in the figure gives the percentage trans-
mitted. These curves are for cloud droplets of radius of 5 X 10~~ 4 cm.
There is some variation in the values given in the figure for clouds
comprised of larger or smaller droplets. The nature of the relationship
for clouds made up of ice crystals has not been determined.

Part of the solar radiation reaching the earth's surface is reflected,
and the remainder is absorbed. Consider first the radiation incident on
land masses. Most of the substances comprising the surface of the
continents, such as rock and vegetation, have albedoes varying from
0.1 to 0.3. The only important exception is a snow surface, which has
an albedo of about 0.8. The radiation which is not reflected is absorbed
by the surface layer of the earth, raising its temperature, and this
heated surface in turn warms the adjacent air. If the values of the
specific heat and conductivity of the soil arc high, the increase in tem-
perature just at the surface is less than if they arc low. Moist soils have
a greater specific heat and conductivity than dry ones. For this reason
moist sand, for example, does not become as warm as dry sand on a
clear summer day. In the former case, part of the heat is used in
evaporating the moisture, and this also prevents a marked increase in
temperature.

Over the oceans, the albedo of the water surface varies with the
altitude of the sun. With the sun near the horizon, the albedo of a
smooth sea surface is about 0.4, whereas it is about 0.03 when the sun
is near the zenith. The presence of waves causes little change with
zenith sun, but produces an increase in the albedo when the sun is low.
About 70 per cent of the incident radiation is absorbed in traversing
the first meter of the ocean. The process of mixing, however, distrib-
utes this heat throughout a considerable layer of water, so that the

RADIATION IN THE ATMOSPHERE

[Chap. 5

heating of the surface waters is very small. Not all this heat is effec-
tive in raising the temperature of the water, since about 30 per cent of
the incoming energy is used in evaporating water. It has been esti-
mated that about 2 mm of water per day are evaporated from the oceans.
29. Terrestrial and Nocturnal Radiation. Water vapor is much
more effective in absorbing terrestrial radiation than in absorbing solar
radiation. Carbon dioxide, which absorbs a negligible amount of solar
radiation, is a strong absorber in several portions of the spectrum of

C0 2

H 2 O

H 2

.illli

15 20

Wave lenglh in ^i

FIG. 25.

The variation in the absorption of terrestrial radiation by water vapor
and carbon dioxide. (After Klsasser.)

terrestrial radiation. As mentioned in section 27, the absorption by
water vapor and carbon dioxide is by groups of lines and must be
treated in a special manner. The absorption spectra of these two sub-
stances from 5 to 30 /* are shown in a schematic mariner in Fig. 25. The
absorption at any given wave length is proportional to the length of the
vertical line at that wave length. From Kirchhoff s law, it follows that
the radiation at the same wave length is also proportional to the length
of the line. From the diagram it is seen that there is a strong band of
water vapor absorption and radiation from 5 to 8 ju, and another com-
mencing at 18 ju> and extending to greater wave lengths. At wave
lengths greater than 30 /x, the absorption and radiation by water vapor
are still more intense. There is a band of intense absorption and radia-
tion by carbon dioxide extending from 13 to 17 /x, and a smaller band,
not shown in the figure, centered at about 4.3 /x. A number of experi-
mental determinations of the absorption at several regions of the spec-
trum have been made, and from these measured values, Elsasser has

Sec. 89] TERRESTRIAL AND NOCTURNAL RADIATION 77

computed the following values of the generalized absorption coefficient /.

Peak of the 6 M band I = 125

Region from 8 to 13 /z I = 0.10

Region near 21 /z I = 17

Peak of the band at p > 30 I = 3000

Using these values and a technique beyond the scope of this book,
Elsasser constructed a radiation chart which permits the evaluation of
the radiation currents in the atmosphere when the vertical temperature
and water vapor distribution arc* known. In the following paragraphs
a simpler method of obtaining the approximate heat loss of the surface
of the earth or of clouds is developed.

Any given portion of the earth's surface may be assumed to radiate
as a black body at the temperature of that portion. The distribution
of energy emission with wave length at a temperature of 273 A is shown
in Fig. 2 la. It follows from what has been said in the preceding parar
graph that the atmosphere also emits radiation in certain wave lengths.
Part of the energy emitted reaches the earth's surface. Thus the net.
loss of heat at the surface is given by the difference between the upward
beam of black-body radiation and the downward beam of radiation, of
the black-body type in certain wave length intervals only. The differ-
ence between these; two beams is known as the nocturnal radiation, and
may be measured on a clear night by a radiation instrument with its
radiating element placed horizontally. The; instrument measures the
difference between the black-body radiation emitted by its element and
the radiation received on the element from the atmosphere above.

Two empirical formulas from which the nocturnal radiation may be
computed have been given. The expression developed by Angstrom
has the form

~ = A - B ICT* (29-1)

where R is the downward radiation from the atmosphere, o-T 4 is the
black-body radiation at the surface temperature 7 T , e is the vapor pres-
sure at the surface in millibars, and A, B, and 5 are constants having
the values 0.81, 0.24, and 0.052. The second equation, given by
Brunt, is

; = a + bV~e (29-2)

<fl

where a = 0.44, b = 0.080, and e is the vapor pressure in millibars.
In general, the values obtained by the use of these two formulas are in

78 RADIATION IN THE ATMOSPHERE [Chap. &

good agreement. It may be seen that these equations give valid results
for the radiation only for limited ranges of values of e by assuming that
the air at the surface is perfectly dry, i.e., e = 0. According to the ex-
pressions given, the downward radiation is then about one-half of the
black-body radiation, an improbable result in view of the dependence
of radiation by the atmosphere on water vapor.

The nocturnal radiation N is given by the expression

A' = a r 4 - n (29-3)

Substitute for R from (29-2) and (29-3) becomes

AT = a r\l -a- b\^) (294)

An expression similar to (29-4), but not involving the vapor pressure e,
may be developed by considering the absorption spectrum of water
vapor and carbon dioxide shown in Fig. 25. On the basis of the data
given in this figure, it is possible to divide the spectrum of water vapor
and carbon dioxide absorption into two main regions, as given below.

(a) Complete absorption.

By a small amount of wai er vapor from 5 t o 8 /*,
By a small amount of carbon dioxide from 13 to 17 JJL,
By a moderate amount of water vapor from 17 to 20 p,
By a small amount of water vapor for ju, > 20.

(b) Complete transmission.

By a large amount of water vapor or carbon dioxide from 8
to 13 /z.

A small amount of water vapor or carbon dioxide may be taken as the
amount normally present in a vertical column of air near the surface,
of unit cross section and a height of 50 m. A moderate amount of
water vapor is that in a similar column I km high, and a largo amount
of water vapor or carbon dioxido is that in a column extending from
the surface to the upper limit of the atmosphere. It follows that the
layer of air extending from the surface to a height of 1 km will absorb
the upward beam of black-body radial ion from the earth's surface from
5 to 8 M, and at wave lengths greater than 13 /z. From KirchhofFs
law, it follows that this layer radiates as a black body at its appropriate
temperature at these same wave lengths. It. may be assumed as a
first approximation that this downward black-body radiation from the
layer is equal in magnitude to the upward radiation from the earth at
the same wave lengths. From 8 to 13 /z there is no absorption or radi-
ation by the atmosphere, and the black-body radiation from the surface
proceeds undiminished to outer space, and as a result the surface cools.

Sec. 29}

TERRKSTRIAL AND NOCTURNAL RADIATION

Now consider Fig. 20, which shows Planck's curve for black-body radia-
tion at 273 A, which will be taken as the surface temperature: The
hatched area under the curve represents the energy lost to outer space
in the transparent band. At the other wave lengths, the surface
receives as much energy from the atmosphere as it emits, and there is
thus no net loss of energy in these wave lengths. It was assumed
above that the black-body radiation downward from the atmosphere

273A

35 40

15 20 25
Wave Length m jj

FIG. 26. Energy transmitted by mid absorbed in the atmosphere.

to the surface was emitted at the same temperature as that radiated
upward from the surface. For all absorbing wave lengths except
those from 17 to 20 /*, the absorption and radiation occur in the lowest
50 m of the 1-krn layer, and it is therefore valid to assume that there is
no significant variation in temperature in this height interval. From
17 to 20 ju, however, the downward radiation comes from a layer 1 km
in height, and the mean temperature of this layer will be, under average
conditions, about 3 C lower than the surface temperature. There is
thus a small net loss of energy in this interval. To allow for this loss
without unduly complicating the diagram, the transparent band has
been drawn as extending from 8 to 14 ju, instead of from 8 to 13 jw.

If now the ratio of the hatched area to the total area under Planck's
curve is denoted by 7, it follows that the net loss of energy, or the
nocturnal radiation, is given by 7<r7 74 , since the area under the curve
is proportional to <rT 4 according to Stefan's law. Measurement of the
areas by planimeter gives the following values of 7 for various tem-
peratures.

200
0.22

225
0.27

250
0.32

273
0.35

300
0.37

325
0.39

The variation in 7 in the range of surface temperatures from 250 to

80 RADIATION IN THE ATMOSPHKHK [Chap.fi

300 A is small, and sufficient accuracy is obtained by assuming that
7 is constant and equal to 0.35. The nocturnal radiation is then
given by

N = yaT 4 = 2.85 X HT 11 !/ 74 cal cm~ 2 min" 1 (29-5)

This equation gives results of the same order of magnitude as those
given by the empirical relationship (294).

Since clouds radiate effectively as black bodies, this expression may
also be used to determine the approximate radiational loss of heat at
the top of a cloud. When the cloud is at medium or high altitudes,
the absorbing layers arc considerably thicker than 50 m and 1 km, since
there is less water vapor present. As a result, the net loss of heat in
the absorbing bands, and especially from 17 to 20 ju, is greater. Thus,
although 7 as given above decreases with temperature, the effective
value of 7 does not vary much from 0.35, as a more detailed study of
the areas shows. Thus equation 29-5 may also be used without mod-
ification to obtain an estimate of the radiational heat loss from the
top of clouds.

It must be emphasized that equations of the types (294) and (29-5),
when applied to specific situations, give results which only roughly
approximate the true values. If more accurate values are required,
the vertical distribution of temperature and water vapor must be
known, and more refined and complicated methods involving the use
of a radiation chart, such as that developed by Elsasser, must be used.

A knowledge of the loss of heat at the surface by nocturnal radiation
is important in forecasting minimum temperatures, in forecasting
radiation fog (section 134), and for frost forecasting (section 172).

30. Radiation with Cloudy Skies. When clouds arc present, the
loss of heat from the surface by nocturnal radiation is reduced. Since
clouds act as black bodies in absorbing and emitting long- wave radia-
tion, it follows that the upward radiation in the transparent band from
the earth's surface does not escape to outer space, but is absorbed by
the base of the cloud layer. The cloud base also radiates as a black
body in all wave lengths. In the absorbing bands, the radiation is
absorbed by the air just below the cloud, and black-body radiation
from this air returns to the cloud base. There is thus no net loss of
heat in the absorbing bands. In the transparent band, the black-body
radiation at the temperature of the cloud base reaches the earth's sur-
face undiminished in intensity. The net loss of heat at the surface is
then given by the difference between black-body radiation in the trans-
parent band at the temperature T$ of the surface and that in the same

Xec. 30} RADIATION WITH CLOUDY SKIES 81

band at the temperature T B of the cloud base. Thus, with cloudy skies

AT = 7* (71 - ri) (30-1)

It will now be shown that the loss of heat at the surface is approx-
imately proportional to the height of the cloud. Denote the dif-
ference in temperature between the surface and the cloud base by
AT 1 . Thus

T B = TS - AT 7 (30-2)

Substituting (30-2) in (30-1) gives

N = 7<r|'/1 - (7\s - AT) 4 ] (30-3)

By the binomial theorem

(Tx - AT 1 ) 1 = '/I - 4T^AT' + -~ 7'|(AT) 2 ----

If suitable values such as T& = 280 A and AT 7 = 10 A arc sub-
stituted in this equation, it is seen that sufficient accuracy is obtained
by retaining only the first two terms on the right-hand side. Thus

(T s ~ AT) 4 - T\ - 4T|AT' (304)

Substituting (304) in (30-3) leads to

N = 4 T er7t*AT = 4y<rT* s (Tx - T B ) (30-5)

Thus for any specified value of TS

N T s - T B

If it is assumed that the temperature in the atmosphere decreases
linearly with height, the heat loss by nocturnal radiation is proportional
to the height of the cloud. The general validity of this result is shown
by measured values of the nocturnal radiation with clouds at various
heights, given in Fig. 134, section 134. It is also of interest to note
that for any given value of AT' the loss of energy from the surface by
radiation varies as the third power of the surface temperature when
the sky is clouded, but when it is cloudless, the loss varies as the fourth
power of the surface temperature. Dividing (29-5) by (30-5), it is
seen that the radiational heat loss for the same surface temperature is

times as great when the sky is clear as when it is clouded. If
4AT

TS = 280 A and AT = 12 A, the heat loss is about six times greater
for clear than for cloudy skies. This result is also of significance in
forecasting the occurrence of fog and frost.
In a similar manner the net loss of heat by a cloud may be computed.

82 RADIATION IN THE ATMOSPHERE [Chap. 5

The same reasoning that showed that the loss of heat by the surface is
given by (30-1) also indicates that the gain of heat at the cloud base
is equal to

If the temperature of the top of the cloud is denoted T T , the loss of
heat at the top is equal to

The net loss of heat by the cloud is then

In the lower troposphere a cloud always loses heat by radiation. To
show this, consider the limiting case where the heat lost at the top is
just equal to that gained at the base. This occurs when

TO- (T 4 T - T 4 S + T 4 B ) = (30-6)

T 4 T + T 4 S = T 4 S (30-7)

If the cloud is not very thick, it is possible to state, with sufficient
accuracy, that

n + T 4 n = 27t, (30-8)

where TM is the mean temperature of the cloud. Substituting (30-8)
in (30-7) obtains

T 4 M = %n (30-9)

The mean surface temperature at latitude 50 N is approximately
281 A. Substituting this value in (30-9), and solving for T M gives
TM equal to 236 A. This temperature is the average temperature
in the atmosphere at a height of 7 km at latitude 50 N. Thus clouds
of small or medium thickness below this height always lose heat by
radiation. For thick clouds the loss is even more pronounced, for
they extend to lower levels, and gain less heat at the base, while the
heat loss at the top is undiminishcd.

31. The Heat Balance in the Atmosphere. Since there is no evidence
of any significant increase or decrease in the mean temperature of the
earth's atmosphere over a period of years, it is evident that the radiant
energy received from the sun and that emitted by the earth and its
atmosphere to outer space must be equal. The total amount of heat
reaching the earth during a year may be computed without difficulty
by using the value of the solar constant, 1.94 cal per cm 2 per min, and

Sec. 31}

THK HEAT BALANCE IN THE ATMOSPHERE

remembering that the area intercepting solar radiation at any instant
is the cross-sectional area of the earth. A simple computation shows
that the earth receives energy from the sun at the rate of 130 X 10 22
cal per annum.

The various transformations of this energy before it is radiated
again to outer space are of interest and will be discussed in the fol-
lowing paragraphs. The discussion follows that given by Landsberg,
with minor modifications based on the work of Baur and Philipps, and
Mollcr. The unit of energy used is 10 22 cal per annum. Of the 130
units received by the earth from the sun, clouds reflect 39 and the
atmosphere scatters 12 back to outer space, the atmosphere absorbs
19, and 35 reach the earth's surface in the direct solar beam, and 25
reach it as diffuse sky radiation. There is an exchange of heat between
the earth and its atmosphere by terrestrial radiation, by condensation
and evaporation processes, and
by turbulence. Heat is ab-
sorbed at the earth's surface in
the process of evaporating y
water, the water vapor is carried
upward, and when it, condenses,
this heat is released to the
atmosphere. Then; is thus a
transfer of heat from the sur-
face to the atmosphere. Equa-
tion 53-11 shows that when the
lapse rate of temperature is of
the stable type, there is a trans-
fer of heat downward by tur-
bulence, and when the lapse
rate is of the unstable type,
there is a transfer of heat up- Latitude

ward. The average lapse rate _ ^ _, f . r , , . . i

, , i , /.o /-i F IG - 27. 1 he variation of (a) incoming and

in the troposphere is about 6 C (6) outgoing rad i ati on with latitude. (After
per km, and therefore stable, Simpson, Baur and Philipps.)

so there is on the average a

transfer of heat from the atmosphere to the surface of the earth by
turbulence.

The various amounts of heat transferred by these processes are given
in the following table. The letter (S) indicates that the radiation is
of the short-wave or solar type, and (L) shows that it is of the long-
wave or terrestrial type. The upward and downward fluxes of energy
through a surface at the outer limit of the atmosphere and through a

RADIATION IN THE ATMOSPHERE

[Chap. 5

surface at mean sea level are given in the upper portion of the table.
At the bottom the gain and loss of heat in the atmosphere between these
two surfaces are indicated. The values given in the table are only
approximate, but they indicate the order of magnitude of the several
processes involved.

TIIE HEAT BALANCE IN THE ATMOSPHERE
Unit 10 22 cal per annum

The Downward Flux
from the

The Upward Flux
from the

At the outer limit of
the atmosphere

Sun 130

Clouds (S) 39
Atmosphere (S) 12
Atmosphere (L) 65
Earth (L) 14

130

At the earth's surface

Sun (direct) (S) 35
Sun (diffuse) (S) 25
Atmosphere
By radiation (L) 125
By turbulence 5

Earth to outer space (L) 14
Earth to the atmosphere
By radiation (L) 146
By evaporation 30

190

The atmosphere

(Sains by

Loses by

The absorption of
radiation from the
Sun (S) 1?)
Karth (L) 146
Condensation 30

The emission of radiation to
The earth (L) 125
Outer space (L) 65
Turbulence 5

195

The heat, balance of the earth as a whole has been discussed above.
Investigation of the average intensities of the incoming beams of solar
radiation and outgoing beams of terrestrial radiation at various lat-
itudes have also been carried out. Curve (a) in Fig. 27 shows the
variation in intensity with latitude of the effective incoming solar
radiation, according to Simpson. In obtaining this curve, allow-
ance was made for the reflection of solar radiation from the atmos-
phere, including that from the? average amount of cloud found at each
latitude, and from the earth's surface. Curve (6) gives the variation
in intensity with latitude of the outgoing terrestrial radiation, accord-
ing to Baur and Philipps. This curve is computed on the basis of

Sec. 32} RADIATIVE EQUILIBRIUM IN THE STRATOSPHERE 85

assumed distributions of temperature and water vapor with height at
each latitude. The figure shows that from the equator to latitude 30
the incoming radiation is greater than the outgoing radiation, whereas
from 30 latitude to the pole the situation is reversed. It is evident
from this that there must be a transport of heat from equator to pole
if the mean temperature distribution over the earth is to be maintained.
The significance of this transport of heat in the general circulation of
the atmosphere is discussed in Chapter 12.

32. Radiative Equilibrium in the Stratosphere. In the troposphere
the lapse rate appears to be controlled by mixing processes which occur
with turbulent motion. An atmosphere in which the vertical tem-
perature distribution is governed entirely by mixing is said to be in
convective equilibrium. The effect of turbulent mixing in the tropo-
sphere is discussed in detail in sections 53 and 76.

It has been believed for some time that radiation plays an important
part in maintaining that isothermal, or nearly isothermal, portion of
the atmosphere known as the stratosphere. It was suggested in 1909
that radiative equilibrium, a condition in which each element of air
gains as much energy by absorption of radiation as it loses by emission,
accounted for the presence of the stratosphere. These early investi-
gations depended on the assumption that water vapor in the atmosphere
radiated and absorbed long-wave radiation as a gray body, i.e., that in
all wave lengths it emitted and absorbed a fixed fraction of the black-
body radiation at the appropriate temperature. The work of Simpson
at a later date showed that this assumption is not valid, even as a first
approximation. The wide variation of absorption with wave length
is shown in Fig. 25. A further complication in the mathematical
treatment is introduced by the fact that the radiation does not occur in
parallel beams, but is diffuse. These early investigations were based
on certain assumed vertical distributions of water vapor. For instance,
an expression for the variation of the mean vapor pressure with height
was given by Hann as

where c is the vapor pressure at the surface, and e is that at a height
h in kilometers. Later work has shown that expressions of this type
are not adequate. Progress in the investigation of radiative processes
in the stratosphere has been hampered by the lack of instruments
which are capable of providing accurate humidity measurements at the
low temperatures encountered in the stratosphere.

Studies of the ozone distribution in the atmosphere have provided
information which may prove useful in accounting for the presence of

86 RADIATION IN THE ATMOSPHERE [Chap. 5

the stratosphere. It has been shown that ultra-violet radiation from
the sun is absorbed by ozone located at heights from 10 to 40 km, with
a mean height of 22 km. It is estimated that from 4 to 6 per cent of
the incident solar radiation is absorbed by this means. It is probable
that the increase in temperature with height in the stratosphere in the
tropics, and the more nearly isothermal conditions near the pole,
shown in Fig. 3, are connected with the observed increase in ozone from
equator to pole. The precise nature of the relationship between these
two elements is not known, however.

Any complete explanation of the stratosphere must also take account
of the advection, or horizontal transport of air, which takes place in
the stratosphere. A satisfactory explanation of the stratosphere
awaits the integration of the separate effects of solar and terrestrial
radiation, ozone, and advection into a comprehensive and consistent
theory.

PROBLEMS AND EXERCISES

1. The radius of the sun is 430,000 miles, and the radius of the earth's orbit is 93
million miles. Using the value of the solar constant, compute the temperature of
the sun if it radiates its a black body.

2. According to the figures given in section 31, 39 per cent of the solar radiation
incident at the outer limits of the atmosphere is reflected back to space. Keeping
this in mind and using the value of the solar constant, compute the effective radiating
temperature of the earth and its atmosphere taken as a unit if it radiates as a black
body.

3. According to the figures given in section 31, 15 per cent of the solar radiation
incident at the outer limits of the atmosphere is on the average throughout the year
absorbed in traversing the atmosphere. Compute the average increase in tempera-
ture of the atmosphere for a 24-hour period attributable to the absorption of solar
radiation. Assume that the mean pressure at mean sea level over the earth is 1013
inb.

4. Solar radiation of intensity 1 cal cm~ 2 inin" 1 falls on the top of a cloud ex-
tending from 800 to 700 mb. Assume that 6 per cent of this radiation is absorbed
in traversing the cloud. The temperature at the top of the cloud is 5 C, that at the
base of the cloud is 10 C, and the surface temperature is 18 C. Neglecting the
specific heat of liquid water, 1 cal gm~ l deg"" 1 , in comparison with the latent heat of
condensation, approximately 591 cal gm" 1 at the temperature of the cloud, compute
the decrease in temperature of the cloud as a whole in 1 hour.

BIBLIOGRAPHY

Brunt, D., Physical ami Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapters 5, 6, 7.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapter 5.
Problems of Modern Meteorology, London, Royal Meteorological Society, 1934,

Number 7.

BIBLIOGRAPHY 87

Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 3
(1930), Chapters 4, 5.

26. Cork, J. M., Heat, New York, John Wiley and Sons, 1933. pp. 145-170.

27. 29. Klsasser, W. M., Heat Transfer by Infrared Radiation in the Atmosphere,

Harvard Met. Studies, No. 6, 1942.

28. Aldrich, L. B., " The Reflecting Power of Clouds," Smithsonian Inst., Misc. Coll.,
69, No. 10, 1919.

28. Ann. Astrophys. Obs. Smithsonian Inst., Washington, D. C., 2, 1908; 3, 1913;
4, 1922; 5, 1932.

28. Hewson, K. W., " The Reflection, Absorption and Transmission of Solar Radia-
tion by Fog arid Clouds," Q. J. Roy. Met. Soc., 69, 47-02 (1943).

28. Paranjpe, M. M., " Variations of the Solar (Constant and Their Relation to

Weather/' Q. J. Roy. Met. Soc., 64, 459-474 (1938).

29. Brunt, D., " Notes on Radiation in the Atmosphere," Q. J. Roy. Met. Soc., 58,

389-418 (1932).
29. " Emission and Absorption of Radiation in the Atmosphere," Q. J. Roy. Met. Hoc.,

68, 197-214 (1942).
31. Baur, F., and H. Philipps, " Der Warmehaushalt dor Lufthiille der Nordhalb-

kugel," Gcrl. Bcitr. Geophy*., 45, 82-132 (1935); 47, 218-223 (1936).
31. Landsberg, H., Physical Climatology, State College, Pennsylvania State College,

1941. pp. 8 M 07.
31. Moller, F., " Bemerkungen zur Warmebilunz der Atmosphare und der Erdober-

flache," Gerl. Beitr. Gcophytt., 47, 215-217 (1936).
31. Simpson, Sir G. C., Home tftiultes in Terrestrial Radiation, Mem. Roy. Met. Soc.,

2, No. 16, 1928.

Further Studies in Terrestrial Kadiafion, Mem. Roy. Met. Soc., 3, No, 21, 1928.
Distribution of Terrestrial Radiation, Mem. Roy. Met. Soc., 3, No. 23, 1929.

CHAPTER 6
ATMOSPHERIC MOTIONS UNDER BALANCED FORCES

The atmosphere is a fluid, and as such its motions may be analyzed
by means of hydrodynamics. Using hydrodynamical principles, it is
possible to set up equations of motion for a fluid on a rotating earth
which are perfectly general in their application. If a perfectly general
solution could be arrived at, it would then be possible to forecast the
weather by means of the hydrodynamical equations. The complex-
ities of the atmosphere are so great, however, that a number of simpli-
fying assumptions are necessary before these equations can be solved.
As a consequence, the equations of motion yield results of only limited
applicability. The general equations of motion will not be derived,
therefore, but attention will be focused on the simpler equations which
result when secondary effects are neglected. In particular, in this
chapter only steady motions, i.e., those not involving accelerations, of
the air in question will be considered. If no net acceleration is present,
the motion must be under balanced forces. Two of the most important
of these forces will now be considered in detail.

33. The Pressure Gradient Force. Since the atmosphere is a fluid,
the pressure in it will vary from one portion to another, but the vari-
ation is always continuous. Because of these variations in pressure
throughout the atmosphere, both in the horizontal and vertical, there
is a force acting on any given element of air.

The magnitude and direction of this force may be derived in the fol-
lowing manner. Consider a parallelepiped with the dimensions dx,
dy, and dz, as shown in Fig. 28. Assuming the atmosphere to be a
non-viscous fluid, then the pressure acts over any surface in a direction
normal to that surface. The total force, then, acting on face AEHD
is p dy dz, since pressure is force per unit area, where p represents the
average pressure over the face. Since this surface is parallel to the yz
plane, the force acts in the x direction. If dp/dx is the x component

of the pressure gradient, the pressure over the face BFCC is 7; H dx,

. ( _, dp \
is i p H dx )(

\ dx /

so the force on that face of the parallelepiped is ( p + dxjdydz,

acting in the negative x direction. The resultant force acting on the

Sec.33\ THE PRESSURE GRADIENT FORCE

volume dx dy dz in the x direction is then

p dy dz

dx ] dy dz = dx dy dz

n ~ ' dx

In meteorology, as in hydrodynamics, unit mass of fluid is considered,
so the expression must be divided by the mass in the volume element,

FIG. 28. The determination of the pressure gradient force.

which is p dx dy dz, where; p is the mean density. The force per unit
mass in the x direction is then

1 dp

p dx

(33-1)

If p increases with .r, then dp/dx is positive, and the negative sign indi-
cates that the pressure; gradient force acts in the negative x direction,
i.e., from high to low pressure. In a similar manner, it may be shown
that the pressure gradient force in the y direction is

and in the z (vertical) direction it is

p dz

(33-2)

(33-3)

The resultant of dp/dx and dp/dy is known as the pressure gradient
as the term is generally used in meteorology. If the vertical com-

90 ATMOSPHERIC MOTIONS UNDER BALANCED FORCES [Chap. 6

ponent of the pressure gradient force (33-3) is exactly balanced by the
acceleration of gravity 0, it follows that

This is the fundamental statical equation already given in (9-1). This
equation is highly accurate except when large vertical accelerations
occur, as in strong convection currents.

The horizontal components of the pressure gradient force are of
primary importance in the study of atmospheric motions, as will be
seen in later sections.

34. The Deflecting Force of the Earth's Rotation (Coriolis Force).
The pressure gradient force, which has just been considered, is a true
force. The second force is an apparent force which results from the
rotation of the earth, rather than a true force. In the following dis-
cussion, motion with respect to two frames of reference, one being the
solar system, or more simply space, and the other the rotating earth,
will be considered. The Newtonian mechanics holds without modi-
fication in a stationary frame of reference, or in one moving with a
constant velocity of translation, but not in a rotating frame of reference.
The simple form of Newton's laws applicable in space does not therefore
apply on the rotating earth, which is the frame of reference for atmos-
pheric motions. It is, however, much more convenient to use Newton's
laws in their simple form on the earth. The force under consideration
in this section, known as the deflecting force, is one of two fictitious
forces which, if included in the equations of motion of a particle on the
rotating earth, permit the use of Newton's laws in their simple form
for a fixed frame of reference.

Before proceeding with the mathematical treatment, an indication
of the general nature of this fictitious force may be helpful. The por-
tion of the earth's surface near the poles may, without serious loss of
accuracy, be considered as a plane which rotates about the polar axis
with the angular velocity of the earth. Only forces acting in the hori-
zontal will be considered, so that the force of gravity which acts ver-
tically does not have to be taken into account. It is assumed that
frictional forces are absent.

A man stationed at the north pole throws a ball horizontally. To
an observer in space the ball moves in a straight line with uniform
velocity, while the earth rotates beneath it. To the man at the pole,
however, the ball appears to curve to the right. Referring to Fig. 29,
to the observer in space the ball thrown from travels in a straight
path and reaches P after time t. When the man at throws the ball,

Sec. 34]

CORIOLIS FORCE

he is facing the point P. During the time t, however, he rotates with
the earth through an angle w, where co is the angular velocity of the
earth, so that with respect to the space frame of reference he faces the
point PI at the end of time t. He has not changed his position on the
earth, and still faces in the same direction, since P and PI represent
the same point on the earth's
surface. To him, then, the ball
has been deflected to the right,
following the curved path shown
in the figure. He naturally at-
tributes this curved motion to a
horizontal deflecting force. But
no true force has been acting
during the time t\ it only ap-
pears to the observer at the pole
that a force has been acting.
Just as the ball leaves 0, its
velocity is the same to both the
observer in space and to the
man at 0. The time scale is the
same in both frames of reference,
so that to both observers it

FIG. 29. The path of an object moving
from the north pole.

reaches point P after the time interval t. In the fixed frame of ref-
erence, the; ball has traveled to P in a straight path with a uniform
velocity in conformity Avith the Newtonian laws, since no external
forces are acting. But to the man at the ball has traversed the
longer curved path in the same time, so that to him it has undergone an
acceleration in the direction of motion, which he attributes to a force
acting in that direction. It is clear, however, that no true force has
been acting. This impression results from the operation along the direc-
tion of motion of a component of the second of the two fictitious forces
mentioned, the centrifugal force.

The magnitudes of these two fictitious forces will now be determined
mathematically. In Fig. 3()a the xy system of axes is fixed in space,
with the origin at the north pole. The x^y\ axes also have their origin
at the north pole, but this system is fixed to the earth, so that it rotates
about the polar axis with the angular velocity co of the earth. At time
t = the two sets of axes coincide, so that after time / the angle between
x and x\ and between y and y\ is co. Consider the motion of a unit
mass at P as, for example, the ball -of the previous paragraphs, or a
unit mass of air, relative to the two coordinate systems.

It may be seen from Fig. 30a that the Xi and y\ coordinates of P

92 ATMOSPHERIC MOTIONS UNDEK BALANCED FORCES [Chap. 6

in terms of fixed coordinates x and y are given by

Xi = x cos co/ + y sin to/ (34-1)

?/i = y cos to/ x sin to/ (34-2)

Now differentiate (34-1) with respect to time. In the following treat-
ment a single dot above the coordinate symbol denotes the first deriv-

KKJ. tf(). Motion with respect to rotating frames of reference.

ative with respect to time; two dots denote the second derivative with
respect to time.

x\ x cos to/ + y sin co/ to.?' sin co/ + co?/ cos co/ (34-8)

Multiplying (34-2) by co and substituting in (34-tt) lead to

TI = a" cos co/ + ?) sin GO/ + co?/] (34-4)

Similarly,

yi = y cos co/ .c sin co/ co// sin co/ cox cos to/

= ?y cos co/ .r sin co/ co.r t (34-5)

Differentiating (34-4) with respect to time gives

x { = x cos co/ + y sin co/ + co//i coj sin co/ + co?/ cos co/ (34-6)
Multiply (34-5) by co and substitute in (34-tt)- Then

Xi = x cos co/ + ;!/ sin co/ + 2co7/! + co 2 .^ (34-7)

Similarly,

y l == y cos co/ x sin co/ corfi co?/ sin co/ cox cos co/

= y cos co/ - x sin co/ r 2wr t + co 2 7/ t (34-8)

In Fig. 306 the same coordinate systems are shown, with the X and Y
components of the true forces on unit mass at I* in the fixed frame of

Sec. 34] CORIOLIS FORCK 93

reference shown as vectors by thick lines parallel to the :r and y axes.
A vector may be defined as any quantity which has both magnitude
and direction. Velocities, angular velocities, and forces, for example,
are vectors. The length of the line is proportional to the magnitude
of the quantity, and the direction of the line denotes the direction of
the quantity. Vectors may be combined according to the parallelo-
gram law. As may be seen from the figure, the components in the
rotating x\y\ system of these two components are given by

Xi = A" cos wt + Y sin wt
YI = Y cos a?/ X sin wf

Now since unit mass is being considered, X = x and Y = y in the fixed
system. Thus

Xi = x cos co/ + y sin ut (34-0)

YI = y cos wt - x sin ut (34-10)

By substituting (34-0) and (34-10) in (3-1-7) and (34-8), it follows (hat

Xi = A' i + 2C07/! + W^ (34-11)

yi = Yt- 2wx! + oA/i (34-12)

Now if

2w?/, - 2wr, = X( (34-13)

-2co>! = -20,//! = Y( (34-14)

and

co 2 .n = A'f, co 2 / yi = K{' (34-15)

equations 34- LI and 34-12 may be written

Jr, = X l +X[ + X' l f (3440)

Si = I'i + H+ ^' GW-17)

G. Coriolis in 1831 calked X\ and Y( tlic^ components of the " force
centrifuge composee " to distinguish them from those of the " forcti
centrifuge ordinaire/' X( r and Y{' . Thci former are the components of
the deflecting force and the latter the components of the centrifugal
force mentioned in the earlier general discussion.

The direction of these fictitious forces is shown in Fig. 31a. If the
velocity \\ denotes the resultant of the velocity components HI arid v^
then it can be seen from (34-13) and (34-14) that the deflecting force?
acts perpendicular to and to the right of the direction of motion of the
unit mass at P at any instant. If x\ and y\ are the components of the

ATMOSPHERIC MOTIONS UNDER BALANCED FORCES [Chap. 6

radius r\ from the center of rotation to the unit mass at P, then the
centrifugal force on P acts outward along r\ according to (34-15).

From (34-16) and (34-17) it can be seen that if these two fictitious
forces are added to the true forces present, then Newton's law that
force equals acceleration for unit mass holds in the rotating frame of
reference, the earth in this instance.

2uV,

(a)

FIG. 31. (a) The direction of action of centrifugal and deflecting forces, (b) The
component of the earth's angular velocity at latitude </>.

At the pole the centrifugal force is zero, since r\ = 0. At a distance
of, say, 100 m from the pole, it is very small, being 5.3 X 1()~~ 5
dyne. The deflecting force is, in general, much larger near the pole.
For a velocity of 20 m sec" 1 , it is 2.9 X 10" 1 dyne. Thus in the
situation depicted in Fig. 29 tho centrifugal force may be neglected
in comparison with the deflecting force.

The angular velocity of the earth about its polar axis is o>. The
effective angular velocity about any other vertical axis is readily found
since angular velocity is a vector.* The line representing the vector
coincides with the axis about which the rotation occurs; the length of
the line is proportional to the magnitude of the angular velocity; the
direction of the vector is the direction of advance of a right-hand screw
rotating In the same sense as the body. Thus in Fig. 3 lib the angular
velocity of the earth is represented vectorially by the thick line coincid-
ing with the polar axis of the earth. The angular velocity about a
vertical axis at latitude </> is then, according to the figure, co sin </>. If
this value of the angular velocity is substituted in (34-13) and (34-14),
the horizontal components of the deflecting force at latitude are

X{ = 2o> sin <t>v l (34-18)

Y{ = -2co sin 07/i (34-19)

* J. H. Jeans, Theoretical Mechanics, Boston, Ginn and Company, 1907, p. 287.

Sec. 34] COUIOLIS FORCE 95

If, as before, the velocity V denotes the resultant of the velocity com-
ponents HI and !, then t,ho magnitude of the deflecting force on unit
mass of air moving over the earth's surface at latitude with velocity
Vila

2co sin <t>Vi (34-20)

Since in meteorology the frame of reference is always the rotating earth
it is not necessary to retain the subscripts. Equations 34-18 and
34-19 show that the deflecting force at any instant always acts per-
pendicularly to the direction of motion and to the right of it. In the
southern hemisphere the magnitude of the deflecting force is the same,
but it then acts to the left of the direction of motion. It follows then
that the deflecting force tends to change the direction of motion of an
air particle but not its speed.

A more extensive analysis of the motion of a particle in three dimen-
sions shows that there is also a vertical component of the deflecting
force. However, this component is extremely small in comparison
with the force of gravity, and only rarely is it of significance. A situa-
tion in which this vertical component must be taken into account is
discussed in section 38. The horizontal components of the deflecting
force arc of the same order of magnitude as the horizontally acting true
forces iri the atmosphere, such as the pressure gradient force discussed
in the previous section, and they therefore form an essential part of any
discussion of horizontal atmospheric motions.

The second of the two fictitious forces, the centrifugal force, is zero
at the pole, and was therefore omitted from the previous discussion.
However, for </> < 90 it is of the? same order of magnitude as the deflect-
ing force. Since this force always acts perpendicularly to and outward
from the axis of rotation of the earth, as shown in Fig. 3 la, it follows
that for 90 > > there is a horizontal component of the centrif-
ugal force acting to the south in the northern hemisphere. This force
acts whether the particle under consideration is moving or not. Ac-
cordingly, a unit mass initially at rest on the earth's surface should,
if unrestrained, move toward the equator. This does not happen,
however, since the earth is not a true sphere, and the attractive force
of the earth's mass on the mass considered does not act in a direction
exactly normal to the earth's surface. As a result of this, there is a
component of gravity toward the north which exactly balances the
southward component of the centrifugal force. The horizontal com-
ponent of the centrifugal force is thus always balanced by an equal and
opposite force, and it is not necessary to consider it in the analysis of
horizontal atmospheric motions. A similar process of compensation
operates in the southern hemisphere.

96 ATMOSPHERIC MOTIONS UNDER BALANCED FORCES [Chap. 6

The centrifugal force discussed in this section must not be confused
with the one introduced in section 37. In the latter the centrifugal
force is that resulting from the circular motion of a mass of air about
a center of rotation on the earth's surface, and is independent of the
rotation of the earth. The centrifugal force in the present section
results from the rotation of a mass of air around the polar axis of the
earth, and the two are therefore entirely distinct. The shape of the
earth has no effect on the centrifugal force on a mass of air rotating
about a point on the earth's surface, and the force in this case is not, of
course, zero.

35. The Geostrophic Wind Equation. The first of the atmospheric
motions to be considered is that resulting when the pressure gradient
force just balances the deflecting force. A mass of air at rest in a pres-
sure field is acted upon only by the pressure gradient force. As soon
as it commences to move as a result of that force, however, the deflect-
ing force starts to operate, and the more rapid the motion, the greater
the corresponding deflecting force becomes. It is clear, therefore, that
air does not flow perpendicular to the isobars, from high to low pressure,
but that it is subjected to a force which continually acts to deflect it
to the right. In this case, however, there is no balance of forces and
the air undergoes an acceleration. This situation is not, therefore,
within the scope of this chapter.

The only situation in which the two forces may balance is that where
the air is moving parallel to isobars which are themselves straight and
parallel. The resulting wind is known as the yeostrophic wind. The
balance of forces may be seen with the aid of Fig. 32. The isobars are
parallel to the y axis, with pressure increasing in the positive x direc-
tion. The pressure gradient force always acts in the negative x direc-
tion in this case, irrespective of the; direction of motion of the air. The
only direction of motion which permits a balance is that shown in the
figure, the air flowing in the positive y direction. With a suitable
value of v, then, the deflecting force will exactly balance the pressure
gradient force and steady motion results. The geostrophic wind equa-
tion is, therefore,

2wsin<v=- (35-1)

p dx

and v is the geostrophic wind velocity.

This equation may be used in any situation as long as the y axis is
chosen parallel to the isobars. If the x axis is chosen parallel to the
isobars, similar reasoning shows that the equation takes the form

2cosin0u= -- (35-2)

pdy

Sec. 36\

THE GEOSTROPHIC WIND EQUATION

When the isobars are parallel to neither x nor y axis, but have some
intermediate orientation, the geostrophic wind velocity V may be
considered as the resultant of the two component velocities u and v in

Isobars

" i dp

FIG. 32. The balance of forces for geostrophic motion.

the x and y directions respectively. In addition, dp/dx and dp/dy are
then the components of the existing pressure gradient in the x and y
directions, and the geostrophic motion is defined by two following equa-
tions.

sm <pv= ----

I dp
----
p dx

2o> sin </> u =

1 dp
P dy

(35-3)

(354)

Buys Ballot's law follows directly: In the northern hemisphere an ob-
server who stands with his back to the wind will have lower pressure to
his left than to his right; in the southern hemisphere, the contrary
holds.

It is assumed in the above treatment that no frictional forces are
present, or if present, they are so small as to be negligible. This is an
accurate statement of conditions in the atmosphere at heights greater
than 2000 or 3000 ft above the surface. At lower levels, the frictional
drag of the irregularities at the earth's surface on the moving air is

98 ATMOSPHERIC MOTIONS UNDER BALANCED FORCES [Chap. 6

appreciable and leads to a decrease in velocity. The pressure gradient
force is not affected thereby, but the deflecting force is reduced. This
leads to air motion across isobars, from high to low pressure. Steady
motion results, with the deflecting and frictional forces balancing the
pressure gradient force. The flow across isobars may be noted on any
surface weather map. This question is discussed further in section 52,
and its practical significance in explaining diurnal variations of wind is
brought out in section 123.

It has not been proved from theoretical considerations that air moves
under the balanced conditions postulated above. Studies of air mo-
tions in the free atmosphere, however, have shown that masses of air
frequently do move without appreciable acceleration for great distances
over the earth's surface. This observed fact is the final justification
for the use as a first approximation of the geostrophic wind relationship
in meteorology. That the geostrophic wind is only an approximation
is demonstrated by the fact, shown in section 47, that no pressure
changes at the earth's surface are possible with simple geostrophic
motion. Non-geostrophic motions are discussed in sections 46 and 121.

Representative values of the geostrophic wind velocity, computed by
means of (35-1), are given in the following table. The density of the
air is taken to be 1.1 X 10~~ 3 gm cm~ 3 , the average density at a height
of 1 km above mean sea level.

GEOSTROPHIC WIND VELOCITY IN METERS PER SECOND

Pressure

gradient 1 nil) per 100 krn 2 ml) per 100 km 4 mb per 100 km
Latitude?*

30 12.5 25.0 50.0

45 8.8 17.7 35.3

60 7.2 14.4 28.8

A graph giving geostrophic winds corresponding to the distance be-
tween successive isobars is shown in Fig. 120, section 120. Such a
graph facilitates the determination of geostrophic winds for forecasting
purposes.

36. The Thermal Wind Component. Equation 9-1 shows that the
rate of change of pressure with height is a function of the density only,
if the small variations in the acceleration of gravity are neglected. It
can be seen from this equation that the pressure decreases more rapidly
with height in cold air than in warm air. Consider two points at the

Sec. 80] THE THERMAL WIND COMPONENT 99

earth's surface which have equal pressure, but with colder air above
one than above the other. At a height of 1 km, say, the pressure in the
colder air will therefore be less than that in the warm air. An isobar
may join the two points at the surface, but cannot join the two points
I km above these, since the pressures there are not the same. It
follows that when a horizontal temperature gradient is present in the
atmosphere, the pressure gradient and hence the geostrophic wind must
vary with height. This variation may be a change in direction, in
speed, or both. The geostrophic wind at 1 km may then be thought of
as the resultant of two components, one the surface geostrophic wind,
and the other component resulting from the horizontal temperature
gradient. This latter component is known as the thermal wind com-
ponent. The magnitude; of this component is obviously proportional
to the horizontal temperature gradient. Its direction is given by the
following rule, which follows directly from the foregoing considerations:
The thermal wind component blows around low temperature in the
same sense that the geostrophic wind blows around low pressure, keep-
ing low temperature to its left.

Isobars

u o

At the surface .X^ At 1 k.lometer

(a) (b)

FIG. 33. Variation of geostrophic wind with height.

A typical variation of geostrophic wind with height is shown in Fig. 33.
The pressure distribution and the geostrophic wind UQ at the surface are
shown in Fig. 33a. The air to the west is colder than that to the east.
Conditions at 1 km arc indicated in Fig. 336. The thermal component,
blowing from the south, is added vectorially to UQ, giving the geostro-
phic wind u at 1 km. If colder air were to the north, the geostrophic
wind would increase with height without undergoing any change in
direction. The above considerations apply equally well to any levels
in the atmosphere. If the geostrophic wind at 2 km is known, that at
3 km, or 4 km, can be found in this qualitative fashion, provided that
the approximate direction and magnitude of the horizontal temperature
gradient arc known.

100 ATMOSPHERIC MOTIONS UNDER BALANCED FORCES [Chap. 6

Equations for this variation of geostrophic wind with height may be
derived. Using (7-7), the statical equation may be expressed as

1 *E = _ J_ /30.D

p dz RT

Differentiating (36-1) with respect to x and changing the order of
differentiation give

dz \p dx) ~" dx \p dz) ~ dx \ Rf) ~ RT 2 ~dx ^ ' '
Similarly, differentiating (36-1) with respect to y gives

= 7^2 ? (36-3)

Substituting for p from (7-7) in (35-3) and (354), the equations for the
geostrophic wind components u and v, leads to

? = 7 ~ 2 (36<4)

U R 1 dp

T = ~ivi (36 - 5)

where I is the Coriolis parameter 2w sin </>. Differentiating (364) and
(30-5) with respect to z gives

(36.6)

(36-7)
Substituting (36-2) and (36-3) in (36-6) and (36-7) respectively, gives

te i*,i ,**.,.. ( 36 ' 9 )

Integration between the levels ZQ and z leads to the expressions

(36-10)
(36-11)

Sec. 37] THE GRADIENT WIND EQUATIONS 101

where U Q and VQ are the geostrophic wind components and T Q the tem-
perature at height z . Rearranging gives

T qT r* 1 dT

v = ^+'-rJ^^ < 3(M2 >

1 dT

^-T-* (36-13)

J dy

Several simplifying approximations may now be made. Since T and 7 7
are not likely to differ widely and they are in degrees Absolute, the
ratio T/TQ may be taken as unity. In addition, T may be taken as the
mean temperature of the layer from 2 to 2, and dT/dx and dT/dy may
be assumed to be constant with respect to z. The integration of the
right-hand terms of (36-12) and (30-13) is then straightforward, and
leads to

tt=M o"JST-(- *o) (38-15)

67 dz/

The right-hand terms are the thermal wind components in the y and x
directions and v and u are the geostrophic wind components at height
z, expressed in terms of the geostrophic wind components VQ and U G at
height z G and the appropriate thermal components. These equations
will be used in section 121 to derive a simple procedure for forecasting
upper winds when direct observations of the latter are not available.

37. The Gradient Wind Equations. When isobars are curved, the
motion of the air can no longer be linear, and the centrifugal force must
also be considered. The centrifugal force has the magnitude v 2 /r,
where v is the velocity of the; particle, and r the radius of curvature of
its motion; it acts in the direction of increasing r. The air motion
corresponding to a balance among the pressure gradient force, the
deflecting force, and the centrifugal force is known as the gradient wind.
The radius of curvature of the motion must be constant, if the motion is
to be steady, and so the wind must blow in circular paths.

Consider a region of low pressure, having circular isobars, as indi-

cated in Fig. 34. Since the pressure gradient force --- always acts

p dr

at right angles to the isobars, while the deflecting force and centrifugal
force always act at right angles to the direction of motion, it follows
that the only direction of motion which permits a balance of forces is

102 ATMOSPHERIC MOTIONS UNDER BALANCED FORCES [Chap. 6

that parallel to the isobars. Theoretically either clockwise or counter-
clockwise motion is possible. If clockwise motion is assumed, the
centrifugal force must balance the sum of the pressure gradient and

deflecting forces. The centrifugal
force is, however, considerably
smaller than either of the other two
under ordinary meteorological con-
ditions. Only when r is small, (r <
100 km), does the centrifugal force
approach the sum of the other
two in magnitude. It is clear then
that the hypothesis of clockwise mo-
tion around a low-pressure area
must be discarded. The alternative,
counterclockwise motion, as shown
in Fig. 34, means that the pressure
gradient force just balances the sum
of the deflecting and centrifugal
forces. No limitations of the above
kind are present, and the balanced
condition may hold for any value of
the radius of curvature. This re-
sult is confirmed by an inspection of
weather maps, which show counterclockwise winds around low-pressure
areas.

The balance of forces for the cyclonic case is given by

FIG. 34.

The balance of forces in a
cyclone.

p dr

(37-1)

where I is, as in section 36, the Coriolis parameter 2co sin <. It follows
from a comparison of (37-1) and (35-1) that the gradient wind in
cyclones is less than the geostrophic wind for the same pressure gradient.
Solving the quadratic (37-1) leads to

r dp
p dr

(37-2)

The negative sign before the radical is rejected because if it is retained,
v j when dp/dr = 0, a result having no physical significance.

For an anticyclone, similar reasoning shows that the balance of forces
may be expressed as

1 dp v 2

- -^ = to - - (37 ' 3 >

p dr r

Sec. 37] THE GRADIENT WIND EQUATIONS 103

A comparison of (37-3) and (354) shows that the gradient wind in an
anticyclone is greater than the geostrophic wind for the same pressure
gradient. The velocity must have a maximum value since the balance
of forces is destroyed if the centrifugal force becomes greater than
the deflecting force.

The solution of the quadratic (37-3) also brings out this limitation of
the possible velocity,

The positive sign before the radical is rejected in this case, since if
dp/dr = 0, v ^ 0, and the equation loses its physical significance.

The maximum possible velocity in an anticyclone occurs if the radical
in (37-4) is equal to zero, i.e., if

I^=? ' (37-5)

P dr 4 v '

It is necessary, then, since this maximum cannot be exceeded, that the
following conditions be obeyed in the anticyclone.

dp P l 2 r

i * T (37 -' 5)

Thus the pressure gradient over an anticyclone is subject to certain
restrictions, if motion under balanced forces is to occur. If the pres-
sure gradient at great distances from the center is as large, or nearly
as large, as possible, according to the criterion given in (37-0), then the
pressure gradient must become progressively smaller as r becomes less.
Pressure charts show that that condition is fulfilled, and large pressure
gradients are never noted in the central portions of anticyclones.

Representative values of the gradient wind velocity in meters per
second at latitude 45, when the pressure gradient is 1 mb per 100 km,
are given in the upper portion of the following table. The density p is
taken as 1.1 X 10~ 3 gm per cm 3 . In the last line of the table are shown
the maximum possible values of the pressure gradient for the given
values of r y computed from (37-6), using the same values of p and < as
above.

r (km) 100 300 500 1000

Gradient wind velocities (m per sec)

Cyclone 5.7 7.2 7.7 8.2
Anticyclone 11.3 9.8

Maximum pressure gradient possible

(mb per 100 km) 0.3 0.9 1.5 2.9

104 ATMOSPHERIC MOTIONS UNDER BALANCED FORCES [Chap. 6

As for the geostrophic wind equations, the gradient wind equations
apply only at heights greater than 2000 or 3000 ft above the surface
where the effects of surface friction are negligible.

In the frictional layer there is a flow of air across the isobars from
high to low pressure in both cyclones and anticyclones. The deflecting
force is nearly always much greater than the centrifugal force, and a
decrease in the former explains the observed flow of air from high to
low pressure near the surface in both types of pressure distribution.

The gradient wind also varies with height if there is a horizontal
temperature gradient in the atmosphere. The ideas given in section 36
to account for the variation of pressure gradient with height are equally
applicable here. The variation of gradient wind with height may then
be determined in the same qualitative fashion.

Strictly speaking, the radius of curvature of the moving air should
be taken. However, it is often sufficient to take the radius of curvature
of the isobar instead.

PROBLEMS AND EXERCISES

1. A small mass of air is set in motion with a velocity of 10 m per sec towards the
east. It moves only in a region where the pressure gradient is everywhere zero, and
friction is assumed negligible. Determine quantitatively the motion of the mass of
air if it is set in motion at

(a) Latitude 80 N.

(b) Latitude 40 N.
(r) Latitude 0.

2. Keeping in mind the data supplied in sections 2 and 3, how may one account for
the general westerly drift of air in middle latitudes? How account for the fact that
the velocity of these westerly winds usually increases with height?

3. Under what conditions might the pressure gradient force and centrifugal force
be so great in comparison with the deflecting force that the latter could be neglected?
Is it possible that motion under balanced forces in such circumstances ever occurs
in the atmosphere? Name one type of atmospheric motion which might be in-
cluded in this category.

4. Show that the maximum possible velocity of anticyclonic winds is twice as
great as the geostrophic wind for the same pressure gradient.

BIBLIOGRAPHY

Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapters 8, 9.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapters 6, 7.

Koschmieder, H., Dynaminche Meteorologie, Leipzig, Akad. Verlag., 1933. Chapter 6.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 4

(1931), Chapter 2.

CHAPTER 7

FRONTAL SURFACES

38. The Slope of Frontal Surfaces. It has been known for many
years that currents of air at different temperatures and with different
velocities could move side by side, the wanner overlying the cold, with-
out appreciable mixing between the two air masses. The surface of
discontinuity between two air masses is known as a frontal surface. In
the atmosphere, frontal surfaces are nearly horizontal, their average
slope being about 1 in 125. It will be shown that the slope depends on
the differences of temperature and velocity in the two air masses. In

Front x

FIG. 35. The determination of the slope of a frontal surface.

the mathematical discussion that follows, it is assumed that a true
mathematical surface separates the two currents of air. In the atmos-
phere, of course, there is actually a zone of transition of finite width.
The horizontal extent of the zone varies from 10 or 20 miles for sharp
frontal surfaces, to several hundred miles for diffuse ones. The lijie of
intersection of a frontal surface with the earth's surface is known as a
front. When the motion of a front is such that warm air replaces cold
air at a fixed point when the front passes that point, it is said to be a
warm front. Conversely, if cold air replaces warm air, it is known as a
cold front. The general characteristics of fronts are discussed in sec-
tion 108 and following.

Consider now a front lying in a line running from west to east, with
the frontal surface sloping upward to the north. The situation is illus-
trated in Fig. 35, where the x axis is positive towards the east, the y axis

105

106 FRONTAL SURFACES (Chap. 7

is positive towards the north, and the z axis is vertical. The angle
which the frontal surface makes with the horizontal xy plane is denoted
by 6. Consider the rectangle a&cd, of length dy and height dz, lying in
the yz plane. The subscript 1 denotes the denser air mass which lies
under the frontal surface and north of the front. The subscript 2 de-
notes the lighter air, extending from south of the front up to and over
the frontal surface.

Since the pressure is continuous at the frontal surface, it is immaterial
whether the point a is approached from air mass 1 or air mass 2; the
pressure attained on reaching a is the same in both. Similarly at c,
so that

In addition

~ p ai = (P6, - P,) + (p ci - P6,) =~dy + j ^dz (38-2)

dy dz

and

o <>

PC, - Pa, = (Pd 2 - Pa,) + (Pc 2 - Pd 2 ) =-+-dz + -f-dy (38-3)

dz dy

Substituting (38-2) and (38-3) in (38-1) and rearranging give

(38 .4)

dy d
Therefore

dpi

tan , * _*L_JL (38.5)

dy dpi _ dp 2

dz dz
According to (334),

^?--Wi and ^ = -ffP2

and so

dpi dp 2

tan,.!** - 2L (38-6)

Q Pi - P2

Another form of the equation for the slope is obtained by substituting

Sec. 38] THE SLOPE OF FRONTAL SURFACES 107

for dpi/dy and dp 2 /dy from (354), giving

ton, _g^* o* -put) (387)

g PI - p2

Here ^ 2 and MI represent the components of the geostrophic wind parallel
to the front. Substituting for p from (7-7) gives finally

(388)

Various values of the slope of a frontal surface at latitude 45 for
TI = 273 A and m = 10 m per sec, when T 2 varies from 278 to 288 A
and u 2 varies from 20 to 50 m per sec, are shown in the following table.

SLOPE OF A FRONTAL SURFACE
When TI = 273 A; u\ = 10 m per sec
112 (m per sec) 20 30 40 50

TTT TS" ^3~ T

TBT TTT TT8~ "88"

T5TF "SPITS 1 T78 1 32

This table shows that the smaller the 1 temperature difference and the
greater the wind component difference between the two air masses, the
greater is the slope of the frontal surface separating them. According
to (38-8), the frontal surface becomes vertical when T\ = T 2 . A more
rigorous analysis, however, taking into account the vertical com-
ponent of the deflecting force, as well as the horizontal component,
shows that the surface is not vertical for T\ = T 2 . Neglecting this
vertical component introduces a significant error only for very small
differences in temperature, and its omission in no way affects the ac-
curacy of the values given in the table above. There is possibly, then,
a small value of T 2 TI which will give a vertical frontal surface
between two air masses. Small portions of certain types of frontal
surface are sometimes vertical, but from this it is not to be assumed
that the temperature difference between the adjacent air masses is very
small. Vigorous vertical motion, which is not geostrophic, frequently
occurs near frontal surfaces, and when such vertical motion is present,
equation 38-8 cannot be applied with any degree of accuracy.

108

FRONTAL SURFACES

[Chap. 7

39. The Pressure Trough at Fronts. From the results derived in
section 38, it will now be shown that there must always be a discon-
tinuity in the isobars at a front. Since pressure is continuous at the
front, it follows that at the front

Pi = P2

and the horizontal pressure gradients parallel to the front at the frontal
surface in the two air masses are related by the equation

dpi ___ dp2
dx dx

(39-1)

The relationship between the horizontal pressure gradients normal to
the front is given by (38-6). Since the warmer and lighter air overlies
the colder and denser air, tan 6 > and also p\ > p 2 , so that

^T > IT < 39 ' 2)

dy dy

These facts are shown diagrammatically in Fig. 36, which is a two-
dimensional representation of Fig. 35. As before, the x axis may be

FIG. 36. The pressure trough at a front. (From Haurwitz, Dynamic Meteorology,

McGraw-Hill Book Co.)

Sec. 39} THE PRESSURE TROUGH AT FRONTS 109

considered as positive towards the east, and the y axis as positive
towards the north. The x and y components of the pressure gradient
are indicated in the figure, and the resultant pressure gradients are
denoted dpi/dn for air mass 1 and dp 2 /dn for air mass 2. By definition,
the isobars are perpendicular to the pressure gradient. AB therefore
represents an isobar in air mass 1 and BC an isobar in air mass 2. From
this diagram it is seen that the isobars are always V-shaped at a front,
with lower pressures within the V. The discontinuity in isobars at
fronts may be seen in Fig. 109 of section 112.

It is also possible to develop a formula for the angle between the two
arms of the V made by the isobars at a front. In Fig. 36, a represents
the angle between the pressure gradient and the x axis in air mass 1
and 7 represents that between the pressure gradient and the x axis in
air mass 2. Similarly & and 5 represent the angles between the x axis
and the portions A B and BC of the isobar in the two air masses. There-
fore

, dy _ dy

tan a = - and tan y =
dp\

dx dx

The negative sign appears because the pressure in air mass 2 increases
in the negative y direction. In addition

tan
and

a = tan f - ft J = cot
tan 7 = tan ( ~ 6 j = cot 6

Thus

dpi

cot ft = (39-3)

dpi

dx
and

cot 8 = (394)

110 FRONTAL SURFACES [Chap. 7

If X denotes the angle between the two portions of the isobar, then

cot |8 cot 51

cot X = cot (ft + 5) =

cot + cot 6

(39-5)

By substituting (39-3) and (394) in (39-5), rearranging, and remember-
ing that dpi/dx dpz/dx, it follows that the angle made by the two
portions of an isobar at a front is given by

cotX =

dgl /3P2 _ dpi\
dx \dij dy )

(39-6)

40. Pressure Tendencies below a Frontal Surface. There are several
atmospheric processes which cause variations in pressure at a given
point. One of these is the advective, or horizontal transport of air of a
different density. If Fig. 35 is referred to, it can be seen that if the
frontal surface depicted there moves northward, the pressure at a
station in the cold air will fall, for the cold air aloft is being replaced by
warm air. The pressure tendency, i.e., the pressure change at a fixed
point during a specified interval of time, usually 3 hours, resulting from
advection, may be obtained in an approximate manner in such a situa-
tion.

If no vertical motion occurs, the velocity v of the frontal surface is
the same as that of the adjacent air masses. In time A2, the frontal

surface moves a distance v&t,
i.e., the surface front initially
at A, as shown in Fig. 37, has
advanced to D. Regarding D
as fixed, the pressure at D falls,
since the air in the column of
height h, of mean density pi,
has been replaced by lighter
air of mean density p 2 . It is
assumed that the pressure in

A
FIG. 37.

The pressure tendency at a front.

the warm air at height h is constant with respect to both space and
time. The change of pressure in the time A2, or the tendency at the
surface, is then given by PA PD- According to (9-9), the pressures
at A and D are given by

PA = PB [ 1 ~ -zr

'Rat

(40-1)

Sec. 40]
and

PRESSURE TENDENCIES

111

(40-2)

where <*i and 2 represent the lapse rates in air masses 1 and 2, re-
spectively. Thus, since PR = p c , it follows that

Multiplying through leads to the form

PA PD = PD

1-11-

(40-3)

(40-4)

Expanding the terms on the right-hand side of (40-4) by the binomial
theorem and neglecting terms of order higher than the first lead to

But h = vAt tan 6, and denoting 1\\
Ap, it follows that

as AT, and PA PD as

tan AT

(40-6)

Put in the values of the constants; assume that PD = 1000 mb, that
TA^D = 273 2 , and that the slope of the warm front is 1 in 150. It
then follows that the tendency

= -0.033VAT mb per 3 h (40-7)
At

when v is in meters per second and AT is in C.

If v is in miles per hour and AT in F, (40-7) becomes

= -O.OOSvAT mb per 3 h (40-8)
At

Cold fronts slope in the opposite direction from that shown in Fig.
37 and have a slope of about 1 in 75. Thus the tendency behind a cold

112 FRONTAL SURFACES (Chap. 7

front is positive and is given approximately by

= +0.066M7 7 mb per 3 h (40-9)

/At

if the units are meters per second and C, or

^ = +0.016yA r jT mb per 3 h (40-10)

if the units are miles per hour and F.

Advection and the other factors causing pressure changes are dis-
cussed in section 47. The difference between the actual tendency as
observed at a station and that computed from one of the foregoing
equations gives the magnitude of the factors, other than advcction,
which are also present. This knowledge is frequently useful in fore-
casting.

41. Frontogenesis and Frontolysis. Temperature Gradient Changes
in a Wind Field. When new fronts and frontal surfaces develop in the
atmosphere, frontogencsis is said to occur. The reverse process of the
dissipation of fronts and frontal surfaces is known as frontolysis. Any
given front usually persists for a considerable period of time, and espe-
cially persistent ones may be followed for a week or more as they
cross the continents and oceans. As a frontal surface moves, it may
be subjected to frontogenetical or frontolytical processes, which will
cause an intensification or weakening of the contrast in properties across
the frontal surface. In forecasting, it is necessary to consider the
probable change in intensity of a front, as well as its change in position.

Since a frontal surface is not a true surface of discontinuity, the
change of a property, such as temperature, from one side of this zone
to the other is marked but not discontinuous in the mathematical sense
at any point. The frontal zone is usually narrow enough to be indi-
cated on a weather map as a line. If a contrast in temperature is taken
as delineating the position of a frontal zone, then the frontal zone is,

more specifically, a region of maximum value of

where n indi-

cates a horizontal direction normal to the isotherms. Any process

which produces an increased value of

is frontogenetical, while

frontolytical processes produce a decrease in the value of .

Any special distribution of winds which tends to bring masses of air
with differing temperatures closer together is called a deformation field.

Sec. 41]

FRONTOGENESIS AND FRONTOLYSIS

113

The simplest type of deformation field occurs when the wind velocity
varies along the normal to the isotherms. The condition for fronto-
genesis or frontolysis in such a wind field may be seen from the fol-
lowing kinematical considerations. In this section, only the question
of how frontogenesis occurs, not why it occurs, will be considered.
To determine whether frontogenesis or frontolysis is occurring, it is

The total derivative with respect to

necessary to evaluate -
at

dT
dn

time must be used since frontogenesis occurs in the moving air along a
line denoting the position of the same particles, and not at a line fixed
in space. It follows from the expression for a total derivative in
terms of partial derivatives that

'dn

Multiplying and dividing by dT/dn give the form

/ d 2 T

d 2 T

dn\

\dt dn

dn 2

dt) dT

(41-1)

Now evaluate d 2 T/dn dt. If T is invariant with respect to time

dT dT dTdn

= 1 =0

dt dt dn dt

and so

dTdn
Ihidt

Differentiating partially with respect to n gives

dndt dn\dt/ dn 2 dt

Substituting (41-2) in (41-1) then gives

dndn\dt

'dn

d^(dn\

dn\dt)

(41-3)

But dn/dt is v n , the velocity along the normal, so that (41-3) becomes

dv
dn

(414)

114 FRONTAL SURFACES [Chap. 7

The following conditions for frontogenesis and frontolysis may now
be given.

If > 0, then
dn dt

> 0. and frontogenesis occurs.

< 0, and frontolysis occurs.

The application of these results to frontogenesis and frontolysis in
various types of deformation fields is considered in detail in section 109.

PROBLEMS AND EXERCISES

1. The component of the pressure gradient normal to a front is + 4 mb per 200 km
in the cold air mass, and +4 ml) per 400 km in the warm air mass. The component
of the pressure gradient parallel to the front is +4 mb per 300 km in both air masses.
Compute the angle at the front between the two portions of an isobar.

2. A warm front lies in a north-south direction. The isobars, drawn at 4 mb
intervals, are 200 km apart along the front, and extend westward in the warm air
and northeastward in the cold air. Compute the slope of the frontal surface at the
point where the 1000 mb isobar intersects it, when the surface temperature in the
cold air is C, and that in the warm air is 7 C.

BIBLIOGRAPHY

Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapter 10.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapter 8.

Koschmieder, H., Dynamische Meteorologie, Leipzig, Akad. Verlag., 1933. Chapter 6.
41. Petterssen, S., Weather Analysis and Forecasting, Now York, McGraw-Hill Book

Co., 1940. Chapter 5.
41. Petterssen, S., "Contribution to the Theory of Frontogenesis," Geofys. PubL,

11, No. 6, 1935.

CHAPTER 8
GENERAL KINEMATICS AND DYNAMICS OF AIR MOTIONS

42. The Movement of Significant Curves in the Pressure Field. It

is frequently convenient to be able to determine the movements of
certain significant points in the pressure distribution, such as the
centers of highs and lows and of certain significant lines, such as fronts.
Such determinations may be made readily from kinematical consid-
erations, following Petterssen's method of procedure, providing cer-
tain conditions are met. It is w r ell to keep in mind that the treatment
is purely a kinematical one, and therefore that the equations developed
in this and the next section indicate only how the motion is occurring,
but not why it is occurring. The pressure field is chosen for analysis,
rather than the temperature field or the moisture field, since pressure
can be measured more accurately than any of the other meteorological
elements. C Certain effects accompanying high winds may give non-
representative values of the pressure, as indicated in section 63, but
these are of comparatively rare occurrence. The errors in correcting
from station pressure to sea level pressure at high level stations are
more serious, but if pressures at stations at lower altitudes only are
used, this source of error is avoided. The following analysis applies
not only to pressure itself but also to functions of pressure, such as the
spatial and temporal derivatives of pressure.

The pressure distribution in a horizontal plane, as at mean sea level,
is specified completely by the expression

p = p(x,y,t) (42-1)

The isobars on a synoptic weather map give this distribution at the
time when the observations were made. Thus the equation of one of
the significant curves, the isobar, is

p(x,y,t Q ) = Constant

If a curve joins points of equal pressure change in a specified interval,
i.e., of equal tendency, this curve is known as an isallobar. The ten-
dency is expressed by dp/dt, and is a function of x,y,t. The equation
of an isallobar is then

= Constant

115

116 KINEMATICS AND DYNAMICS [Chap. 8

Other curves, some of which will be discussed later, may be similarly
specified.

A formula for the velocity of a significant curve will now be developed.
The pressure itself will be discussed here, as the procedure is more readily
understood using pressures, but the argument applies equally well to
derivatives of the pressure, such as the tendency dp/dt or the pressure
gradient dp/dx. Since pressure in a horizontal plane is a function of
x,y,t as indicated in (42-1), the total differential dp, expressed in terms
of partials in the standard manner, is

The x and y axes in this equation are fixed with respect to the earth.
Then

dp _ dp dpdx dpdy

'dt = ~dt dxlTt lhj~dt

dp dp dp

= -- h Ui -- \- v\
dt dx l dy

Here dp/dt represents the pressure variation with time on a moving
particle of air, and ?/i and v\ represent the velocity components of the
particle in the x and y directions. The term dp/dt represents, then,
the pressure variation with time at any fixed point at or above the
earth's surface, i.e., it is the tendency. Orienting the axes so that the
motion is in the x direction only, the equation reduces to

* = *-* + ttl ^ (42-2)

dt dt dx ^

If the axes are moving with the pressure system, the pressure variation
with time on the moving air particle is

^ = ^ + u 2 % (42-3)

dt dt 2 dx ^ '

In this equation, dp/dt represents the pressure variation with time in
the moving system and thus represents the deepening or filling of the
system. The velocity u 2 is that relative to the axes moving with the
pressure system. The terms dp/dt and dp/dx are the same irre-
spective of whether the motion is referred to the fixed or the moving axes.
Equating (42-2) and (42-3) gives

-+<*-*>-+-

Sec. 42] MOVEMENT OF SIGNIFICANT CURVES 117

where c, being the difference between the velocity relative to the fixed
axes and that relative to the moving axes, is the velocity of the pres-
sure system relative to the axes fixed on the earth. Equation 424
cannot be solved as it stands, for there are two knowns, the tendency
and the pressure gradient, and two unknowns, the deepening or filling
term and the velocity. If only systems in which neither deepening nor
filling occurs are considered, then solving for the velocity leads to

- -
C ~ ~ dp

This gives the velocity of an isobar. Since the development has been
perfectly general, this equation applies not only to p, which is con-
stant along an isobar, but also to any function of p, f(p), the constancy
of which specifies a significant curve of the pressure field. Thus the
velocity of any significant curve of the pressure field is given by

dx
For example, an isallobar is a significant curve, being specified by

/Op) = = Constant
dt

The velocity of an isallobar is therefore, according to (42-5), given by

c ~

dxdt
The acceleration A of a significant curve is given by

(42 ' 6)

To obtain the acceleration, it is necessary to develop an expression for
the term d 2 p/6t 2 . According to (424)

8p dp dp

= -- \- c

dt dt dx

118 KINEMATICS AND DYNAMICS [Chap. 8

Since the operator 8/dt is, in terms of partials,

_
8t ~~ ~di dx

then

= - = - + C + C - , C

dt 2 U\U/ dt\dt dx/ dx\dt dx

~ dt 2 dx<lt dx 2

Substituting from (42-6), the equation becomes

Since there are two unknowns, 5 2 p/5t 2 and ^i, in this equation, then it
is possible to solve for A when 8 2 p/8t 2 = 0. The term 6 2 p/dt 2 =
when the rate of deepening or filling of the moving pressure system
is constant. With this situation, the acceleration of the system is,
from (42-7),

A = -

dp
dx

The meaning of these terms is brought out in the next section when
discussing the acceleration of a front.

The acceleration of any significant curve f(p) = constant is, there-
fore, given by

. 9 , 2

43. The Movement of Troughs, Wedges, and Fronts. The equation
for the curve which characterizes a pressure trough may be obtained
in the following manner. In order to simplify the discussion, a sym-
metrical trough, of the type shown in Fig. 38a, will be considered. The
trough line may be defined as the line joining the points on successive
cyclonically curved isobars at which the curvature is a maximum. The
broken line in Fig. 38a denotes the trough line. If the x axis is chosen

Sec.

MOVKMKNT OF FRONTS

119

perpendicular to the trough line, as shown in the figure, then the trough

line is specified by the conditions

that \ \ L

and

da-

(43-1) \

(43-2)

m , /,.., iv . ,1 , /.ji FIG. 38. Isobars at (a) a trough and

Then (43-1) is the equation of the (6) ft ^^ 8

trough line. Substituting f(p) =

dp/dx in (42-5), it follows that the velocity of a trough line in the x

direction is

<*>

dx 2

The numerator represents the x component of the isallobaric gradient
and may be evaluated from the isallobars on the weather map. The
denominator may bo evaluated from the isobars, the sign of the term
being given by (43-2).

A wedge line, shown in Fig. 386, is specified by the conditions that

and

(434)

(43-5)

The velocity of a wedge line in the x direction is, then,

c =

dxdt
d 2 p
dx 2

(43-6)

where the sign of the denominator is negative, as indicated by (43-5).

The movement of a cyclone in which no fronts are present may now
be determined. Choose two axos, x and y, not necessarily at right
angles, but intersecting near the center of the cyclone, in such a manner

120 KINEMATICS AND DYNAMICS {Chap. 8

that along the y axis dp/dxi = and along the x axis dp/dyi = 0,
where x\ and y\ are axes perpendicular to the y and x axes respectively.
The velocity c y of the y axis in a direction along the x\ axis is given,
according to (43-3), by

C v =

dx\
The velocity c x of the x axis in the y\ direction is given by

After an interval of time At, the y axis will have moved a distance
c v At in the x\ direction, and the x axis will have moved a distance
c x At in the y\ direction. The intersection of the x and y axes in their
new positions then gives the location of the center of the cyclone at
the end of the time interval At. The movement of an anticyclone may
be found in a similar manner, using (43-G).

The formula for the acceleration of a trough or wedge line is obtained
by substituting dp/dx for/(p) in (42-8).

In section 38 it was shown that, at the front separating air masses
denoted 1 and 2, p\ p 2l since pressure is continuous. Thus a front
is a significant curve of the pressure field, since PI p 2 has the con-
stant value zero along it. The function f(p) characterizing a front
is therefore given by

/(P) = Pi - P2 (43-7)

Substituting (43-7) in (42-5) gives for the velocity of a front

c = - -^ 2L (43-8)

dp\ dp 2

dx dx

The tendency dpi/dt at any given point on the front in air mass 1 is
most accurately obtained by drawing the isallobars in air mass 1, as
indicated in Fig. 39, and noting the value of the isallobar which inter-
sects the front at the chosen point. Similarly, the isallobars in air
mass 2 determine the value of dp 2 /dt in that air mass. Typical dis-

Sec. 43]

MOVEMENT OF FRONTS

121

tributions of isallobars in the vicinity of fronts are shown in Fig. 110
of section 112. The denominator of (43-8) is readily evaluated from
the distribution of isobars in the two air masses. The exact method of
procedure for evaluating the four terms of (43-8) is given in section
167, and an example is worked out in section 169.

The formula for the acceleration of a front is obtained by substi-
tuting (43.7) in (42.8). Thus

\ ___ _

dt 2

dx dt dx dt

)

dx 2 }

dpi
dx

(43-9)

This acceleration formula is made up of three groups of terms. Denot-
ing the portion of the acceleration arising from the first group as A\,
that arising from the second group as A%, and that from the third group
as A 3 , (43-9) may be expressed as

A = A l + A 2 + A 3 (43-10)

Insufficient data are given on an ordinary weather map to permit
the accurate determination of each of these groups, more particularly
the first group, but the sign of
found in the fol-
Fig. 39 shows

each may be

lowing manner.

the distribution of isobars (full

lines), isallobars (broken lines),

and the barogram (curve abc at

the bottom of the diagram) in

the vicinity of a warm front.

Curve abc is not the barogram

trace at any one station, but rep-

resents rather a composite pic-

ture of the slopes of the baro-

grams at stations along the x axis

at any given instant of time. It is this curve which cannot be ob-

tained accurately from the data given on weather maps.

Since the x axis is chosen parallel to the isobars in the warm air,
dp2/dx = 0, and the first term becomes

FIG. 39. Isobars, isallobars, and baro-
grarn at a warm front.

dt 2

(43-11)

122 KINEMATICS AND DYNAMICS [Chap. 8

In addition, the tendency in the warm air is uniform, i.e., ab is a straight
line, so that 6 2 p 2 /dt 2 = 0. In the cold air ahead of the front the
barogram trace be is curved anticyclonically, using Pctterssen's termi-
nology, so that the magnitude of the negative tendencies increases
with time, and therefore d 2 pi/dt 2 < 0. In the cold air the pressure
increases in the positive x direction, so that dpi/dx > 0. Thus
A l >0.

The portion of the acceleration A 2 resulting from the second group
is given by

dXdt dXdi > (43-12)

dpi

In the cold air the isallobars are negative, and become less negative
in the positive x direction, so that d 2 p } /dxdt > 0. In the warm air,
since the isallobars are very nearly parallel to the x axis, d 2 p 2 /dx dt ~
and may be neglected. Since the velocity c and dpi/dx are both pos-
itive, it follows that A 2 < 0.
The acceleration A$ contributed by the third group is given by

**- **' (43-13)

Since the pressure in the cold air increases in the positive x direction,
but at a diminishing rate, as shown by the configuration of the isobars,
then d 2 pi/dx 2 < 0. The isobars in the warm air are parallel to the
x axis, and therefore d 2 p 2 /dx 2 = 0. As c 2 and dpi/dx are both posi-
tive, it follows that A$ > 0.

Thus A i and ^4 3 produce positive accelerations, and A 2 produces a
negative acceleration, or a retardation. Assuming, as a first approx-
imation, that the magnitude of the acceleration contributed by each
term (43-11), (43-12), and (43-13) is the same, then the front experi-
ences a small net acceleration in the positive x direction.

The sign of the acceleration of fronts with other barograms and dis-
tributions of isobars and isallobars may be evaluated in a similar
manner. This qualitative method of analysis is used in section 112
to discuss the relative motion of the warm and cold fronts in four types
of frontal depressions.

Sec. 44}

STREAiMLINES AND TRAJECTORIES

123

44. Streamlines and Trajectories. In a field of moving particles of
air, the streamlines are defined as the curves to which, at a given instant
of time, the velocity vectors for each par-
ticle are tangential. In Fig. 40, the
curved line represents a streamline, with
the tangent at one particular point indi-
cated. The equation of the tangent line is

y = a + bx

where b represents the slope of the tangent
line. Differentiating,

dy = b dx
or the slope b of the line is

FHJ. 40. The tangential ve-
locity at a streamline and its
components.

But if V represents the tangential velocity, having components u and
v in the x and y directions respectively, then the slope b is also given by

Thus the differential equation of the streamline is

rfy = v
dx a

(44-1)

Now consider the streamlines in geostrophic motion. By multiplying
(35-3) by dx and (354) by dy and denoting the Coriolis parameter
2o? sin by /, it follows that

Iv dx = - dx
p dx

1 dp

lu dy = --- dy
p dy

Subtracting (44-3) from (44-2) leads to

(44-2)
(44-3)

(44-4)

But

124

KINEMATICS AND DYNAMICS

[Chap. 8

In section 47 it is shown that with geostrophic motion the pressure
cannot vary with time, and therefore dp/dt = 0. Thus

dp = dx
dx

j
dy

and (444) becomes

But, from (44-1)

l(vdx udy} =
P

v dx u dy

(44-5)

(44-6)

With geostrophic motion, therefore, dp = 0, and the streamlines co-
incide with the isobars.

A streamline gives an instantaneous picture of the motion of a group
of particles. A trajectory, on the other hand, gives a picture of the
motion of a single particle during a finite interval of time. The tra-
jectories of several particles of
air in a depression near the
British Isles are shown in Fig. 41.
The motions indicated occurred
during the interval from 08 h on
September 10 to 08 h on Septem-
ber 11, 1903. The small circles
connected by broken lines repre-
sent successive positions of the
center of low pressure.

Only when the motion is station-
ary do the streamlines and trajec-
tories coincide. Geostrophic mo-
tion is of necessity stationary, so
that the streamlines and trajec-
tories are the same. When a
pressure system moves as a whole,
there are usually very marked
differences between the stream-
lines and trajectories, as, for ex-
ample, in Fig. 41. This distinc-
tion between streamlines and trajectories is important in fore-
casting air motions, especially in the vicinity of pressure centers.
Unless the difference between the two is kept in mind, there is a tend-
ency to use the streamlines, instead of the probable trajectories, in
forecasting the position of a mass of air.

FIG. 41. The trajectories of air particles
near a depression, September 10-11, 1903.
(After Shaw and Lempfert.)

Sec. 45]

EQUATION OF CONTINUITY

125

45. The Equation of Continuity. Divergence and Convergence.

One of the basic concepts of hydrodynamics is that nowhere in a system
is fluid created or destroyed. This fundamental condition is expressed
in the equation of continuity. Consider the rectangular parallelepiped
shown in Fig. 42, with sides dx, dy, dz parallel to the x, y, z axes. The

yy
XH G

/^A c/

0)2 dpu
*pu* dx d X dydz t

pudydzdf - |

/ A dx B

FIG. 42. The derivation of the equation of continuity.

velocities in the x, y, z directions are u, v, w, and the density of the
fluid is p. The transport of fluid in the positive* x direction through
the face AEHD in the time dt is pit dy dz dt y whore pu is the average
value for the face, and that through the face BFGC in the same direc-

tion is ( pu -| -- dx } dy dz dt. The increase in mass in the parallel-
\ yX /

epipcd resulting from the x component of motion is then the amount
entering through face AEHD minus the amount leaving through face

BFGC, or - dx dy dz dt. Similar expressions give the increase
dx

in mass resulting from y and z components of the motion. The total
increase in mass in the parallelepiped is then

Since the fluid is neither created nor destroyed and since the mass
in a given volume increases, the density of the fluid must increase in

time dt from p to p + dt. Thus the total increase in mass in the
dt

parallelepiped may also be expressed as

dx dy dz dt
dt

(45-2)

126 KINEMATICS AND DYNAMICS [Chap. 8

Equating (45-1) and (45-2) gives the equation of continuity
dp dpu dpv dpw

(45>3)

If the fluid is incompressible, p is constant, and (45-3) reduces to

du dv dw

The equation of continuity is frequently used in this form in meteor-
ology for certain types of computations. The variations in p in the
horizontal are very small, but in the vertical there is a rapid decrease
of p with height. If a layer of small vertical extent only is considered,
however, it is permissible, as a first approximation, to assume that p is
constant throughout the layer, and therefore to use (454).

The expression

dpu dpv dpw
dx dy dz

is known as the divergence of the transport. When it is positive, there
is a decrease of mass per unit volume per unit time, according to (45-1).
When this expression is negative, it is known as the convergence, for
there is then an increase of mass per unit volume per unit time. Sim-
ilarly, for an incompressible fluid, this divergence is

du dv dw

" l~ 7 h T~
dx dy dz

which equals zero, according to (454). The expression (du/dx) +
(dv/dy) is often called the horizontal divergence, and dw/dz the ver-
tical divergence. From (454) it follows that

du dv dw

^ + ^ = ~^ (45 ' 5)

Thus, in an incompressible fluid, the horizontal divergence equals the
vertical convergence.

The method of using (45-5) for comparatively thin layers of air, in
which p may be considered constant without introducing serious error,
is given below. Assume that the air moves in the positive y direction,
i.e., northward, so that u = 0. Then (45-5) becomes

dv & w

r- = ~ (45-6)

dy dz

Sec. 45} EQUATION OF CONTINUITY

Expressing (45-6) in terms of finite differences gives

Av
Ay

Aw
Az

- 2/0

Z l ~

127

(45-7)

(45-8)

Assume that the xy plane shown in Fig. 43 coincides with the surface
of the earth. The dimensions of the parallelepiped as drawn are
xi, 2/1, and Zi. Thus y = and z = 0.
Since the base of the parallelepiped coin-
cides with the surface of the earth, W Q =
also. Equation 45-8 then becomes

= -- (v\ v )

(45-9)

The same result is, of course, obtained by

determining the net transport across the

two faces in and parallel to the xz plane

and equating this to the outflow across Fi. 43. The vertical velocity

the horizontal plane at height z v . The ^suiting from horizontal con-

,., i , ii-ii IA- vcrgence in the atmosphere,

vertical velocity at height Zi, resulting * *

from horizontal divergence or convergence, is thus obtained.

Consider the case- in which straight and parallel isobars lie in a north-
south direction at a given latitude, with pressure increasing to the
east. The geostrophic wind is then south, of magnitude given by (35-1),

v _

2w sin <t> p dx

The vertical velocity Wi at a height of z\ between latitudes fa and <
is then obtained by substituting (35-1) in (45-9). Providing that p
and dp/dx are constant with latitude, then

dp /si

sin 0o sin

2wp?/i dx \ sin fa sin

(45-10)

The distance yi over the earth's surface between latitudes fa and <fo
is given by

'h - *>) (45-11)

128 KINEMATICS AND DYNAMICS (Chap. 8

where E is the radius of the earth. Introducing (45-11) into (45-10)
leads to the form

sn <t>i sn

\ <i < /

Horizontal convergence occurs since sin < < sin fa and fa > < and
the air ascends between latitudes fa and < .

If the geostrophic wind were north, there would be horizontal di-
vergence and descending air motion. When straight and parallel
isobars extend from fa to $ , but in a direction other than the north-
south one, the vertical velocity is less than that given by (45-12), since
?/i then has a value greater than that given by (45-11).

An example of the application of (45-10) to an actual weather map
is given in the latter part of section 119. The vertical velocities re-
sulting from the variation of the gradient wind with latitude are also
discussed in section 119. The horizontal divergence and convergence
resulting from the variation in the curvature of the streamlines are
discussed in section 141.

46. The Isallobaric Wind. In the equations of motion of an air
particle given in sections 35 and 37, it was assumed that the forces
acting on a moving air particle were balanced. When the forces are
balanced in this manner, the motion of the air particle is not accelerated,
but is steady. Brunt and Douglas studied the motion of an air particle
when the forces arc unbalanced, so that the particle experiences an
acceleration. Sutcliffe's variation of Brunt and Douglas' treatment is
given below.

According to section 35, there are two forces acting on an air particle
moving in a straight line over the earth's surface. If the motion is
northward, i.e., in direction y, and parallel to isobars which are them-
selves straight and parallel, the deflecting force of the earth's rotation
acts in the positive x direction. Since pressure increases with #, the
pressure gradient force acts in the negative x direction. If these two
forces do not balance, the net force in the positive x direction is the
deflecting force minus the pressure gradient force. According to
Newton's law, force equals acceleration for unit mass. Thus the
acceleration du/dt of unit mass of air in the positive x direction equals
the net force in the same direction. These facts may be expressed in
the form of an equation as

-!,-!* (46-1)

dt p dx ^

If the motion is eastward, with the isobars lying in an east-west direc-

Sec. 46] THE ISALLOBARIC WIND 129

tion, similar considerations show that

dv 1 dv

-=-&-- / (46-2)

dt p dy v

If u and v are the x and $/ components of velocity F, rather than two
distinct velocities, then (46-1) and (46-2) together specify the motion
of unit mass of air when the forces are not balanced.

Denoting the x and y components of the gcostrophic wind velocity
as u g and v g , and substituting for the pressure gradient force terms in
(46-1) and (46-2) from (35-3) and (354), the former become

-77 = l(v ~ v g ) = In (46-3)

- = -l(u - u,) = -lut (46-4)

where Ui and Vi are the departures of the wind components u and v
from their geostrophic values u g and v g . Expressing the total dif-
ferential du in terms of partials

and so

du du du du

du = dt + dx + dy + dz
dt dx dy dz

du du du du du

= \- u \- v \- w

dt dt dx dy dz

since

dx dy _ dz

= u, - = v, and = w

dt ' dt ' dt

Since u = u g + u^ it follows that
du du g dUi du g du{ du g dui du g dUi

= 2 + + U + U-~~ +V + V +W + W

dt dt dt dx dx dy dy dz dz

(46-5)

Brunt and Douglas have given some evidence to show that under
ordinary atmospheric conditions the first term on the right-hand side
is much greater than the remaining terms on that side, so that, as a
first approximation, the latter may be neglected. Thus (46-3) and
(46-4) become

^ = b, (46-6)

= -hit (46-7)

130 KINEMATICS AND DYNAMICS [Chap. 8

By differentiating (35-4) and (35-3) partially with respect to time and
changing the order of differentiation, it follows that

dv g __ i d (dp\

et "/ptoW (46 ' 9)

Substituting (46-8) and (46-9) in (46-6) and (46-7) then leads to

The terms ( ) an( l I ~) are the x and y components of the
dx\dt/ dy\dt/

isallobaric gradient, and ui and Vi are therefore known as the com-
ponents of the isallobaric wind. It is seen from (40-10) and (46-11)
that the isallobaric wind blows at right angles to the isallobars toward
the lowest isallobars. The actual wind is obtained by adding vec-
torially the geostrophic and isallobaric winds.

Several attempts have been made to check these formulas, but with-
out any decisive results, owing to the difficulties involved. It follows
from the foregoing statements that there is horizontal divergence from
an isallobaric high and horizontal convergence in an isallobaric low.
Descending air motion therefore occurs in an isallobaric high, and
ascending air motion in an isallobaric low. The essential validity of
the results is attested by the observed fact that, in general, clear skies
accompany an isallobaric high and clouded skies accompany an isal-
lobaric low.

This isallobaric effect, however, does not always give a complete
explanation of atmospheric developments. It appears that, in certain
conditions, the terms in (46-5) which have been neglected assume im-
portance. When this occurs, the mathematics becomes very involved,
and a satisfactory method of solving the more complex equations has
not yet been devised, although attempts in this direction have been
made.

An example of an isallobaric wind obtained from the data on a
weather map is worked out in section 121, and the significance of isal-
lobaric winds in forecasting is discussed in Chapter 24.

Sec. 47] THE ORIGIN OF PRESSURE CHANGES 131

47. The Origin of Pressure Changes. If it is assumed that the at-
mosphere is in statical equilibrium, a valid assumption except near
cumulonimbus clouds where marked vertical accelerations often occur,
it is possible to determine the several factors which cause pressure
changes at any level. The pressure p at any height z may be found by
integrating (9-1) from z to . Thus

gpdz (47-1)

Differentiating partially with respect to time and assuming the accel-
eration of gravity g to be constant, it follows that the tendency at any
level z is

S* < 47 - 2)

Substitute for dp/dt from the equation of continuity 45-3, and (47'2)
becomes

dp T 00 Sdpu dpv\ 1 f 00 dpw

- = -q I (^- + jofe-,// *dz (47-3)

dt J z \dx dy/ J z dz v '

Then by carrying out the differentiation in the first term and rearranging,
and integrating the second term, (47-3) takes the form

dp r (du dtA rt dp d p \

- - = -g I P[ + )dz - g I 1^ + v :r)<fe
dt J z \dx dy/ J z \ dx dy/

(47-4)

since p = at z = <*>.

This equation shows that a pressure variation at height z can arise
from the operation of one or more of three distinct processes. The
first term represents the effect of horizontal divergence or convergence
at all heights greater than z. The second gives the effect of horizontal
advection of air of different density at heights greater than z. The
last term represents the effect on p of vertical motion at the height z.

At the surface of the earth w = 0, so that the variation of pressure
p$ with time at any fixed point at the surface, i.e., the tendency at
the surface, is

132 KINEMATICS AND DYNAMICS [Chap. 8

Thus the pressure at the surface can vary only as a result of horizontal
divergence or convergence or of advection of air of different density at
higher levels.

Although equations 47-4 and 47-5 are comparatively simple, the
determination of the actual divergence or convergence and advection in
the troposphere and stratosphere requires a quantity and accuracy of
upper air data which have not yet been attained. Valuable deductions
may be made, however, from special types of air motion which arc fre-
quently observed, such as that discussed in the next section.

When the air motion at all levels in the atmosphere is geostrophic,
the tendency at the surface must be zero. The equations 35-3 and
35-4 may be expressed in the form

1 dp
/ - y - (47-6)

Differentiating (47-6) partially with respect to y and (47-7) with re-
spect to x obtains

(47-8)
(47-9)

dy I dx dy
dpu 1 3 2 p

dx I dx dy

By substituting (47-8) and (47-9) in (47-5),

^ =
dt

Since no changes in surface pressure are possible in a geostrophic wind
field, it follows that variations in the distribution of pressure at the
surface can occur only with departures from geostrophic motion, such
as, for example, the isallobaric wind components discussed in the pre-
vious section.

48. The Movement of Upper Troughs and Wedges. Investigations
of the pressure field at high levels have shown that the isobars are
distributed in a sinusoidal pattern, the general trend of the isobars being
parallel to the parallels of latitude. The relationship between the
resulting northward and southward motions at these levels and the
surface centers of high and low pressure is discussed in section 113.
The observed east or west motions of these upper troughs and wedges

Sec. 48]

UPPER TROUGHS AND WEDGES

133

may be accounted for by horizontal divergence and convergence in
these systems.

A sinusoidal system of isobars approximating those often occurring in
the upper air is shown in Fig. 44. At rr and x 2 the isobars are curved
cyclonically. The gradient wind equation 37-2,

FIG. 44. Convergence and divergence with sinusoidal isobars.
(After J. lijcrkncs.)

gives the velocity of the air motion at these values of x. At Xi and x%
the isobars are curved anticyclonically, and the gradient wind velocity
is, according to (37-4),

_fr P7~

~2 \4'

r dp
p dr

It was shown in section 37 that, for equal values of r, p, and dp/dr,
when variations in latitude < included in the Coriolis parameter I are
neglected, vi$ > v 0t2 . Thus, as indicated in the figure, there is diver-
gence between XQ and x\, convergence between x\ and x 2 , and divergence
between x 2 and 3 . If it is assumed that advection of air of different
density and vertical motions are absent, the pressure falls, according to
(47-4), between XQ and x and between x 2 and x 3 , and rises between Xi
and x 2 . This is equivalent to a motion in the positive x direction, i.e.,
eastward, of the system of isobars.

It is implied in the above treatment that the north-south amplitude
of the isobars is not great, the variations in latitude <t> being small enough
to justify neglecting them. However, if an isobar extends over a con-
siderable range of latitude, as sometimes happens, it is no longer per-
missible to assume that I is constant. The effect of introducing the

132 KINEMATICS AND DYNAMICS [Chap. 8

Thus the pressure at the surface can vary only as a result of horizontal
divergence or convergence or of advection of air of different density at
higher levels.

When the air motion at all levels in the atmosphere is geostrophic,
the tendency at the surface must be zero. The equations 35-3 and
354 may be expressed in the form

1 dp
PV = - - (47-6)

1 dp
PU = -- t - (47-7)

Differentiating (47-l>) partially with respect to y and (47-7) with re-
spect to x obtains

dpv I 6 2 p

(47-8)

dy I dx dy
Opu 1 2 p

(47-9)

dx I dx dy

By substituting (47-8) and (47-9) in (47-5),

^>=0
dt

Sec. 48]

UPPER TROUGHS AND WEDGES

133

may be accounted for by horizontal divergence and convergence in
these systems.

A sinusoidal system of isobars approximating those often occurring in
the upper air is shown in Fig. 44. At ,r and ,r 2 the isobars are curved
cyclonically. The gradient wind equation 37-2,

.
"0,2 = ~ 9 + J T >

p ., r dp

J T >- +
\ 4 p dr

FIG. 44. Convergence and divergence with sinusoidal isobars.
(After J. B jerk nes.)

gives the velocity of the air motion at these values of x. At- x\ and x< t \
the isobars are curved anticyclonically, and the gradient wind velocity
is, according to (37-4),

r dp

It was shown in section 37 that, for equal values of r, p, and dp/dr,
when variations in latitude included in the; Coriolis parameter / are
neglected, 1^3 > # ,2- Thus, as indicated in the figure, there is diver-
gence between X G arid x\ y convergence between x\ and x 2 , and divergence*
between x% and 0*3. If it is assumed that advection of air of different
density arid vertical motions are absent, the pressure falls, according to
(47-4), between :r and jc\ and between x< 2 arid x 3 , and rises between x\
and ^2- This is equivalent to a motion in the; positive x direction, i.e.,
eastward, of the system of isobars.

It is implied in the above treatment that the north-south amplitude
of the isobars is not great, the variations in latitude <j> being small enough
to justify neglecting them. However, if an isobar extends over a con-
siderable range of latitude, as sometimes happens, it is no longer per-
missible to assume that I is constant. The effect of introducing the

134 KINEMATICS AND DYNAMICS [Chap. 8

variations in sin </> in I is to reduce the difference in velocity between
.TO and j:i, and thus the eastward velocity of the system of isobars. For
large ranges of latitude z' may equal v\, and when this happens, the
isobaric system is stationary. This condition is met when (37-2)
equals (374), or

/ o 9 . 9 r dp . / 9 > . 9 r dp

cor sin </> + * orr sur </> + = cor sin t ^/ co~r" sin" ^ -----

\ p dr \ p dr

(484)

This equation may be solved for </> and fa by a graphical method,
assuming certain values for r, p, and dp /dr. The values of </> and <i
for stationary isobars when p = 0.8 X 10"~ 3 gm per cm 3 , dp/dr = 1 mb
per 100 km, and r equals in succession 500, 1000, and 1500 km are given
in the following table. The distribution of isobars is symmetrical
about latitude 45 in each case.

CONDITIONS FOR STATIONARY ISOBARS
r (km) " <*>o (deg) t (cleg) A0 (deg)

500 12.5 31 59 28

1000 12.2 38 51 J- 13

1500 12.2 41 49 8

When the radius of curvature is small, say 500 km, the distance
#2 %o, which may be called for convenience the wave length of the
motion, will in general be small. For such isobaric systems to be sta-
tionary, a single isobar must extend through a large interval of latitude,
28 in the example in the table. Such a latitudinal variation with a
short wave length rarely, if ever, occurs. Thus systems with short
wave lengths generally move eastward. They are associated with the
traveling highs and lows of middle latitudes. The variation in latitude
required for stationary isobaric systems with greater radii of curvature
and hence of greater wave lengths is considerably less, as indicated in
the table. For example, for r = 1000 km, if A</> < 13, the system
moves eastward; if A< > 13 it moves westward. Since latitudinal
variations of the order of 13 with longer wave lengths are of frequent
occurrence, it follows that such systems tend to remain stationary, or
nearly so. It is possible that considerations of this kind account, at
least in part, for the motions of the semi-permanent centers of action
in the atmosphere, such as the Aleutian and Icelandic lows, and the
sub-tropical highs. The study of such isobaric systems may thus be
useful in long-range forecasting techniques, discussed in section 170.

Sec. 49}

CIRCULATION AND VORTICITY

135

49. Circulation and Vorticity. The concept of circulation is funda-
mental in hydrodynamics, and has a number of applications in meteor-
ology. The circulation around a closed curve is, by definition, the line
integral of velocity around the curve. Defined mathematically, it is

C = j> (udx + v dy + w dz)
If the curve is in the xy plane, (49-1) becomes

C = ' (udx + v dy)

(49-1)

(49-2)

The circulation may also be ex-
pressed in polar coordinates. The
circulation around the closed curve
shown in Fig. 45 is obtained in the
following manner. A and B arc two
points on the curve separated by
a small distance rf.s, and V is the
velocity at A, a distance r from the
instantaneous center of rotation 0.
Since a is the angle between V and
ds t the component of V along d,s* is
V cos a. Thus the circulation corresponding to (49-2) is

FIG. 45. The computation of
circulation .

C = V cos ads

(49-3)

If the instantaneous angular velocity about O is )f, then V =
In addition,

cos a =

Ydt rdd

If substitutions are made for V and cos a and the limits of integration
are changed, equation 49-3 becomes

c = /i,

But the integral

(494)

(49-5)

gives the area enclosed by any plane curve. The quantity f, which is
twice the angular velocity, is known as the vortidty. If f is constant

136 KINEMATICS AND DYNAMICS [Chap. 8

over the area, then by substituting (49-5) in (494), it is seen that
circulation = area X vorticity. Thus the circulation around a circle
of radius r is given by

C = 7rr 2 f

When vorticity is present in a fluid, the motion is called rotational.
When vorticity is absent, the motion is irrotational. The term rota-
tional as defined above does not always have the ordinary connotation
of that word. One condition in which circular fluid motion may be
irrotational is brought out in problem 4 at the end of this chapter. The
rotation of a solid is rotational motion. The earth's atmosphere as a
whole rotates as a solid, with the same angular velocity as the earth, so
that the mean vorticity of the atmosphere is 2co.

50. Rate of Change of Circulation. A knowledge of the rate of change
of circulation, and thus of vorticity, with time is necessary for the dis-
cussion of a number of meteorological problems. Kelvin's circulation
theorem, which gives the rate of change of circulation with time, may
be derived in the following manner. If (49-1) is differentiated with
respect to time

dC r fdii dv dw
= <p I djc H dy H

+ f \u 4, (dr) + v ^ (dy) +w^ (&)1 (50- 1 )
J L dt dt dt J

Since

dx dy dz

= 7/, 7 = V, = W

dt dt dt

the second term on the right-hand side may be written as

<f (u du + v dv + w dw) = \ <b d(u 2 + v 2 + w 2 ) =

since the integration is that of a total differential carried out around a
closed path. Thus Kelvin's circulation theorem

(50-2)
follows directly. By a slight modification of (46-1), (46-2), and (334),

Sec. 50} RATE OF CHANGE OF CIRCULATION 137

the equations of motion become

du 1 dp

-=lv--f+X (50-3)

at p dx

dv 1 dp

= -lu - - -* + Y (504)

at p dy

The terms X, F, Z represent external forces, such as those resulting
from friction. Equation 50-5 gives the vertical acceleration when the
gravity, pressure gradient, and external forces are not balanced. If
(50-3), (504), and (50-5) are substituted in (50-2), the latter becomes

- <j>g dz - 2co <j> sin <f>(u dy - v dx) (50-0)

Since, when there arc no changes of pressure at. a point,

dp dp dp

dx dy dZ

the first integral becomes

P
The second integral may be written

W =f(X dx + Y dy + Z dz)

where W represents the work done by the external forces.

Since the integration is around a closed path, the third integral is
zero. Thus,

dz =

The fourth integral may be interpreted in the following manner.
Consider a cyclonic circulation at latitude 0. At that latitude take
the x axis positive to the east and the y axis positive to the north and
project these axes on the equatorial plane of the earth. The pro-

138

KINEMATICS AND DYNAMICS

[Chap. 8

jected axes are denoted xi and
that

x =

dx =

The x and Xi axes are parallel, so

and u = u\

The y and y\ axes are not parallel; the relationship between them may
be seen from Fig. 4(>, which represents a cross section of the earth

through the origin of axes x and y and
through the axis of rotation of the earth.
It is apparent from this diagram that

yi = y sin <#>

and hence dy\ = dy sin and v\ = v sin </>.

The area F enclosed by the projection of
oi~ the circulation at latitude $ on the equato-
rial plane is given by

F = j>*idyi (50-7)

The rate of increase of area dF /dt is ob-

,, Ar , ru . . . ,
IIG. 46. I he projection of

axes with origin at latitude <
ori the equatorial plane of
the earth.

tained by differentiating (50-7) with respect
* fc> v / i

f ip

(it

= <f (
J

+ X

(50-8)

since dx\/dt = ui and d(dyi/dt) dv\. Dividing (50-8) into two in-
tegrals and subtracting V L dxi from the first and adding it to the second
lead to

dF r r

= J> (HI diji - vi dxi) + <t> (jri dvi +

+ vi dxi) = j>d(xivi) =

= y (

But

since the integration is around a closed path. Thus

Substituting for HI, v\, dx\, and dy\ in (50-9) leads to
dF

dF P

= <f sin (u dy v dx)
dt J

(50-9)

(50-10)

Sec. M] RATE OF CHANGE OF CIRCULATION 139

which is the fourth integral of (50-6). The latter may now be written

r -2o> (50-11)

This equation gives V. Bjerknes' variation of Kelvin's circulation
theorem, and the method of derivation gives precisely the same result
as that in which the vertical components of the deflecting force are
taken into account.

The circulation theorem as given in (50-11) has a number of applica-
tions to meteorology. For example, considering the last term only, if
convergence occurs in the air motion over an area of low pressure at
latitude <, then in the equatorial plane dF/dt < 0, and dC/dt > 0.
This agrees with the observed fact that the greater the convergence in
the lower levels of a depression, the greater the cyclonic motion of the
system becomes. In anticyclones, an increase in anticyclonic motion
accompanies an increase in divergence in the lower levels.

This question may also be discussed quantitatively. Consider a
system of air particles moving in horizontal circular paths about a
center 0. The rate of change of circulation around the circular path
of radius r when divergence or convergence occurs may be obtained in
the following manner. The circuit of particles around which the cir-
culation is taken encloses an area irr 2 . If the aroa is sufficiently small
that it may be considered as located at latitude </>, then the area F of
its projection on the equatorial plane is given by

F = 7rrsin0 (50-12)

The change in area with time as divergence or convergence occurs is ob-
tained by differentiating (50-12) with respect to time. Thus

dF dr

= 2*Tsin0- (50-13)

at at

and

dC dF

= 2w - = 47rcor sin <t>v r (50-14)

at at

where v r denotes the radial velocity dr/dt. Since the length of the
circuit is 2?rr, the mean acceleration around this circuit is, from (50-2),

W 1 dC

- (50-15)

140 KINEMATICS AND DYNAMICS [Chap. 8

Thus, by substituting in (50-15) from (50-14), it follows that

= -2wsin0t> r (50-16)

(it

The mean acceleration is thus that which results from the deflecting
force of the earth's rotation, given by (34-20), and acts perpendicular to
the radial velocity, and thus tangential to the circular motion. If conver-

I r r

gence occurs in a cyclone, the motion is inward, ?; r < 0, and > 0,

and the intensity of the cyclonic motion increases. If the radial velocity
inward at any portion of the cyclone at 40 latitude is 1 m per sec, then

TlV

= 9.4 X HP 3 cm soc~ 2
at

If the mean acceleration is constant, at the end of an hour the tangen-
tial velocity V has increased by 0.34 m per sec.

If a cyclone in which little convergence is occurring moves south-
ward, the projection on the equatorial plane of a circuit- of air particles
decreases, and the intensity of the cyclonic motion increases. Similarly
the anticyclonic circulation decreases as a high-pressure system moves
southward. The converse holds for northward motions, llossby at-
tributes the development of cyclonic curvature in southward-moving
air at high levels and of anticyclonic curvature in northward-moving
air to the influence of this term of the equation. The large-scale
cyclonic and anticyclonic eddies which have been found from studios
of the motions in isontropic surfaces are discussed briefly in section 1(H).

The rate of change of circulation resulting from the northward or
southward movement of circular cyclonic or anticyclonic sj'stems may
be readily computed, (-onsidor cyclonic motion with radius r. The
projected area on the equatorial plane is given by (50-12),

F = 7T/' 2 sin

If now the system as a whole moves northward with velocity v, r remain-
ing constant, the projected area varies with time. Differentiating
(50-12) with respect to time leads to

'| = ,rcos^ (50-17)

dt dt

Since

^ _
~dt ~~E

Sec.oO] RATE OF CHAXC.E OF CIRCULATION 141

where E is the radius of the earth, then

d( T 2co7rr 2

-.- = ~ " cos<t; (50-18)

Since the length of the circuit is '2irr, the mean acceleration around this
circuit is

d \ cor

If the system moves southward at latitude 40 with a velocity of
10 m per sec, r having the constant value 400 km, then

= ;j.5 x 1(F 3 cm sec" 2
dt

If the mean acceleration is constant, at the end of an hour the mean
tangential velocity I* has increased by 0.13 m per sec.

It must not be inferred from the foregoing that the cyclonic circula-
tion of southward-moving depressions always increases, and similarly
for the other motions discussed. It appears that the only term of those
in (50-11) which produce's an increase in horizontal circulation is
2w(dF/dt). The effect of friction, included in the M r term, is always
to decrease cither cyclonic or anticyclonic circulation. It has been

shown by Brunt that the term r cannot- produce dosed horizontal

J p

cyclonic circulations. In addition, atmospheric motions at the surface
and those at higher levels are related, as shown in section 11(5, and
developments in the upper air may offset or even reverse the changes
which might be predicted on the basis of the discussion in this section.
If the circuit of air particles is in the yz plane, then dF/dt since
F is at all times zero. In addition, this term is zero if the j, ?/, and z
coordinates of every point on the circuit are constant, and if no change;
in latitude occurs. Equation 50- 1 1 then reduces to

(50-20)
dt J p

The circulation theorem in this form is used in section 71 in discussing
the meridional circulation on a non-rotating globe. The method of

evaluating the term r - is given in that section. The process out-
J p

lined there may also be used in studying slope and valley winds (sec-
tion 124) and land and sea breezes (section 125).

142 KINEMATICS AND DYNAMICS [Chap. 8

PROBLEMS AND EXERCISES

1. A warm front extends from the northwest to the southeast. Isobars, equally
spaced along the front, run northward ahead of the front and westward behind it.
The barogram trace at a station just ahead of the front is curved cyclonically, and
that at a station just behind the front is a straight line. In the cold air, the nega-
tive isallobars run parallel to the front, and become less negative with increasing
distance from it. In the warm air, the isallobars run westward from the front.
What is the sign of the acceleration of the front in this situation?

2. Penally spaced isallobars run from west to east over a portion of the earth's
surface. The isallobar +4 mb per 3 h is found at latitude 40 N, and the isallobar
4-1 mb per 3 h is found at latitude 45 N. Assuming that this isallobaric gradient
extends from the surface to a height of 2 km, and that the average density in this
layer is 1.0 X 1(P 3 gm cm~ 3 , compute the average vertical velocity at a height of
2 km between latitudes 40 and 45.

3. A layer of air extending from the surface to a height of 2 km moves from north
to south. The velocity varies with latitude, but not with height. At a certain
latitude, the north wind has a velocity of 10 m per sec, whereas 1000 km to the south
it has a velocity of 11 m per sec. The layer has a uniform tcnif>erature of 7 C
throughout, and the mean pressure of the air in the layer is constant, being 900 rnb.
Atmospheric conditions above 2 km cause no variations in surface pressure. Com-
pute the pressure tendency in mb per 3 h at the surface under these conditions.

4. Air particles move in circular paths about a center 0, with velocities given by
V = f(r). Compute the circulation about an area bounded by two radii which
contain an angle 9 and by two arcs subtending the angle 0. Under what condition
will the circulation be zero?

5. A layer of air extending from the surface to a height h moves parallel to the x
axis with a velocity which is constant with height, but which increases at a constant
rate with x. The layer is isothermal both vertically and horizontally. If atmos-
pheric conditions above the level h cause no variations in surface pressure, show that
the pressure tendency at the surface is given by

-oh
dp ( RT \ / a u d

BIBLIOGRAPHY

Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapters 8, 15.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapters 6, 7.

Koschmieder, II., Dynamische Meteorologie, Leipzig, Akad. Verlag., 1933. Chapter 10.
44. Ibid., pp. 161-168.
42, 43. Pettcrssen, S., Weather Analysis and Forecasting, New York, McGraw-Hill

Book Co., 1940. Chapter 9.
42, 43. Petterssen, S., "Kinematical and Dynamical Properties of the Field of

Pressure with Applications to Weather Forecasting," Geofys. Publ. 10, No. 2,

1933.
44. Petterssen, S., op. cit., pp. 224-227.

BIBLIOGRAPHY 143

40. Brunt, D., and C. K. M. Douglas, The Modification of the Strophic Balance for

Changing Pressure Distribution and Its Effect on Rainfall, Mem. Roy. Met. Soc.,

3, No. 22 (1928).
46. Sutcliffe, R. C., " On Development in the Field of Barometric Pressure," Q. J.

Roy. Met. Soc., 64, 495-504 (1938).
48. Bjerknes, J., " Theorie der aussertropischen Zyklonenbildungen," Met. Z. t 59,

462-466 (1937).

CHAPTER 9
TURBULENCE

51. Streamline and Turbulent Motion. There are two main typos
of air motion, streamline and turbulent. The nature of the difference
between these types may be understood by considering the motion of
water, which may be more readily observed. Streamline motion often
occurs near the center of a slowly moving mill stream. At any instant
the streamlines of the flow are parallel straight lines. Near the banks,
however, the motion is irregular and broken into small eddies. This
latter motion is turbulent. The irregularity of the flow is more notice-
able in the rapids of a swiftly moving stream.

Turbulent motion in the atmosphere is made visible by smoke. For
instance, the smoke leaving a burning cigarette in a quiet atmosphere
ascends at first with streamline flow. There is an upward acceleration,
however, and at a certain point the stability of the flow breaks down,
and the motion becomes turbulent, as shown by the irregularity of the
smoke pattern. A similar effect may bo noted in smoke leaving a
chimney.

No clear-cut physical picture of the nature of the eddies comprising
turbulent motion can be visualized. This lack of rigid definition of
the quantity to bo studied lias seriously hampered the development of
adequate methods of analysis of turbulent flow in the atmosphere. The
gusts and lulls shown on the records of certain typos of wind- recording
instruments indicate the presence of turbulent motion iri the atmosphere
but tell little of the nature of the eddies.

An adequate theory of turbulent motion must account for the ob-
served variation of the wind with height, near the surface, whore the
latter exerts a frictional drag on the air flowing over it. It must also
account for the variation of temperature with height near the surface,
and the diffusion of matter in the atmosphere. It has boon shown that
the effect of molecular movements is far too small to account for the
observed variations in these elements.

So far, the main attempts to formulate a theory have been made on
the basis that eddy motion in the atmosphere is completely analogous
to molecular motion in a gas, the only difference being one of scale.
Evidence mounts that this approach is not adequate, but no superior

144

Sec.

TURBULENT TRANSFER OF MOMENTUM

145

method of attack on the problem has as yet been developed. Thus the
transport of momentum, heat, and matter by eddy motion in the
atmosphere is assumed to be similar to the transport of these quantities
by molecular motion in a gas. The formulas developed in the following
sections of this chapter are the same as those for molecular viscosity,
conduction of heat, etc.

52. Turbulent Transfer of Momentum. The Wind Variation with
Height in the Frictional Layer. The retardation of the air flow near
the earth's surface is assumed to be due to a type of internal friction,
known as eddy viscosity. This method of approach resulted from the
work of Osborne Reynolds, who postulated that the system of eddy
stresses caused by internal friction is a result of the deviations of the
velocity from its mean value. The; investigations of Taylor indicate
that, if the instantaneous deviations from the mean wind velocity u, the
x axis being oriented to coincide with the moan wind, are w', v ', and w'
in the #, ?/, and z directions respectively, then the mean values of u' 2 ,
v' 2 , and it/ 2 are equal. This suggests that there is equipartition of the
energy of the eddies in these three directions. Other investigations
suggest, however, that the cross-wind component is greater than the
vertical component,. It is probable that the latter is a more accurate
statement of conditions near the surface.

The frictional drag of a slowly moving air current on a more rapidly
moving current adjacent to it is assumed to be a result of the transport
of momentum from one current
to the other by means of eddies.
If a mass of air from the slowly
moving current penetrates the
rapidly moving current, conserv-
ing its initial momentum during
the process, the result is a retar-
dation of the faster current.
Consider air flowing in the x
direction with a mean velocity
it. the instantaneous deviations

FIG. 47.

Vertical transfer of momentum
through turbulence.

from u being u and w in the x

and z directions respectively. In

Fig. 47 dx dy represents an clement of the horizontal area S. If w f

is the instantaneous mean vertical velocity over the area dx dy, then the

mass of air flowing upward across area dx dy in unit time is

pw' dx dy
The amount of x momentum dl x moving upward through area dx dy in

146 TURBULENCE (Chap. 9

unit time is then given by

dl x = pw'(u + u 1 ) dx dy (52-1)

since momentum equals mass times velocity. The amount of x mo-
mentum moving upward through the area S in unit time is, therefore,

I x = I I pw'(u + u') dx dy

= u I I pw' dx dy + if pw'u 1 dx dy (52-2)

Since there is no net transfer of mass through the large area S during
unit time, it follows that the first integral on the right-hand side of
(52-2) is zero, and the equation becomes

I x = ff pw'u' dx dy (52-3)

Now consider an eddy which starts from level 2 with the mean x
component of velocity u Zo appropriate to that level. It arrives at
level z with the same x component of velocity U ZQ . The mean velocity
of the air motion at the new level z is u z , so that the instantaneous
deviation u' of the eddy motion from the mean velocity u z at that level
is UZ Q u z . The deviation u at the level z is then given with sufficient
accuracy by the expression

u' - ^ fo, - ,) - - g (z - z.) (524)

Substituting (524) in (52-3) leads to

Ix = - / / PW' ^ (z - z ) dx dy

= - ~ ff PW' (z - z ) dx dy (52-5)

The quantity w (z z ), where the bar indicates that the mean
value of the product is to be taken, is known as the coefficient of eddy
diffusivity and is denoted by K, using the notation introduced by
Taylor. The units of K in the cgs system arc cm 2 sec" 1 , which are the
same as those of the coefficient of heat conductivity K and of the coeffi-
cient of kinematic viscosity v for molecular processes. The coefficient K
is an attribute of eddy motion over a large area for a considerable
period of time, not one of eddy motion at a point at a given instant of
time. The linear dimensions of the area should be at least 10 km and

Sec. 52} TURBULENT TRANSFER OF MOMENTUM 147

the interval of time at least 5 min. The quantity pw'(z 2 ), or Kp,
was called the Austausch (exchange) coefficient A by Schmidt. The
units of A in the cgs system are gm cm" 1 sec" 1 , the same as those of the
coefficient of molecular viscosity ju. The mean value of z 2 , denoted
z z Q) was called by Prandtl the Mischungsweg (path of mixing).

The eddy transfer of x momentum upward through unit area may
then be written, according to (52-5), assuming that K and p are mean
values over the unit area,

I x = -Kp (52-6)

Now consider a disc of unit cross section and thickness dz. The amount
of x momentum entering through the bottom is I x and the amount

df x

leaving through the top is I x -\ dz. The net gain in x momentum

in the disc is then dz or, using (52-6),

(52-7)

, I/*/

The units of this expression are

_, _ 9 gm cm sec"" 2

gm cm sec ^

cm*

The expression 52-7 then represents a force per unit area, or a shear-
ing stress exerted by a moving current of air on a slower moving current
adjacent to it. This expression thus gives the magnitude of the fric-
tional forces acting in the atmosphere, which are most marked in the
surface layers.

If unit volume is considered, dz = 1, so that (52-7) becomes

(52-8)

If the atmosphere is assumed to be incompressible, so that p is con-
stant, the frictional force per unit mass is obtained by dividing by p.
Thus, from (52-8), it is seen that the frictional force per unit mass
resulting from eddy viscosity is

When the mean velocity of the air stream has the components u and #,

148 Tl'KHKLKXCE [Chap. 9

and when K is constant with height, the x and y components of the
eelely frictional force are;

K ^ and /vgi (52-10)

It will be noted that seve-ral assumptions have been made in this
derivation. In particular, it has bes'ii assumes 1 that an eelely conserves
its momeTitum jus it moves from erne* le i vcl to another. Since 1 the eddy
moves through a varying pressure 1 fie i ld as it ascends or descenels, how-
e've 1 !', it is difficult to ses 1 hew this cemelitiem can be obeyesl. Never-
theless, tlw e'xpirssiems in (52-10) give^ results which are approximately
e;e>nrct.

The gene'ral es{ua(ions e>f horizontal motion given by (4(5-1) and
(4(5-2) may now be 1 evxpande'd to include 1 the 1 e 1 fleet of frictiemal fe>rce\s
resulting from turbulemt air me>tion. The e*quatie)ns thus become*

- 1( ^ + K <lJ^ (52-11)

dr 1 dp d*v

- = -/- 7/-+KT1 (52-12)

(It p <)y dz

Uneler certain assumptiems the^se 1 esjiiafions may be solves! te> give the
variation e>f wine I with height in the* surface 1 layeTS where the 1 frictiemal
force*H are large 1 . The various steeps le^ieling to the solution are 4 toe>
lengthy and involvevl to be 1 include'd heMr. A simpler analysis give>n by
(iiildberg anel Me>hn will be* usesl instead.

When the 1 isobars are 1 straight anel parallel, balanced motion e>mirs
in the fres 1 atmetspluMr when the 1 eleiles?ting foire 1 exactly balancers the*
pressure* graelie^it fore-e 1 , as slu>\\n in ses'tiem ,S/i. If was also indicatesl
in that ses-tion that in the 1 leiwcr ljiye*rs e>f the* atme)sphe*re*, up to heights
of 2000 or MOO ft, the 1 action of frictie>nal forces causes the air te> flow
acre>ss the isobars towarel le>\ve*r pirssmr. 'The retareling e^Tes't e>f fric-
tie)ii cause's a smaller velocity than in the fres 1 atme)sphere, so that the 1
ele^lecfing fe)ire* is smalleM' than the 1 pressure graelient fe)rere*. This situa-
tion is illustrates! in Kig. 48. The* ise>bars are paralle 1 ! to the 1 y axis,
with pirssure increasing in the pe*sitive .r eliirctiem. Since the 1 velocity
r is ne>t parallel te> the 1 ise>bars, and the* elcilrcting fe>rce is pcrpcnelicular
te> the 1 elitre'tion e)f motie>n of the air, while the pirssure gradienit fe>ire
acts in a eliire*tie)ii perpendicular te> the ise)bai*s, it follenvs that the*se twe>
forces de> ne>t act in oppe>site* directions, anel therefore cannot be* bal-
ancesl. Ouleibe*rg and Malm assumes I that the frictional force is pro-
portional to the velocity, and hence is given by kV, where /; is the

TURWLKXT TRANSFER OK MOMKNTl'M

140

coefficient of friction, and acts in a direction opposite to that of the
velocity. For a balance among deflect ing, pressure gradient, and fric-
tional forces, therefore, the f Fictional force must be equal to the resultant
of the other two and oppositely directed. It follows then that the

FKJ. 18. Dcvi.'ilinri of tin 1 \\iml by friction

resultant of those two, denoted by OH in the figure , must lie* along the
wind vector I". The friefional force* AT, represented by a vector in the
diagram, is equal in magnitude to (Hi. Since* AOCH is a parallelo-
gram, BC = AO = (1 p)(c)/) V).r). Considering triangle OBC, it fol-
lows that

Thus

n + </n'- "^

(52-13)

But, according to (^

1 /)/>
p dx

where r denotes the geostrophic wind velocity. Substituting in (52-13)
then gives

r = iv g (/; 2 + / 2 r ! (52-14)

150

TUHBrLKNCE

[Chap. 9

If represents the angle which the wind vector makes with the isobars,
then

kV k , R oi^

tan 9 = = y (52-15)

A more complete analysis shows that the assumptions of Guldberg
and Mohn are only approximate. Nevertheless the equations 52-14

and 52-15 have the virtue of sim-
plicity, while not deviating too far
from accuracy. They are used in
section 123 to account for the ob-
served diurnal variation of wind in
the lower levels.

53. Turbulent Transfer of Heat.
The following treatment of the
vertical transfer of heat by eddy
motion follows that given by Brunt.
In Fig. 49, Ah represents a por-
tion of a stable environment curve
Kir,. 41). Ifclciy tnmsfor of heat. plotted on a temperature-height

diagram. The temperature of a

particle of air at A at height 2 () is 7',,, while that of a particle at B at
height z is 7 T . Since the temperature, decreases with height, it follows

7' = r - a -

(53-1)

Now consider an eddy initially at A which ascends from height 2 to
height z, its temperature decreasing at the dry adiabatic lapse rate F
during the process. It arrives at ( v at height z with the temperature
T\ given by

7'i = 7' - T(z -
By substituting (KM) in (5.V2),

(53-2)

- z ) (53-3)

The heat content of the air comprising the eddy of mass m when the
latter ascends through a horizontal surface at level z is therefore

wr n

Sec., 53} TURBULENT TttAXSFKtt OF HKAT 151

The total heat content of all the eddies ascending through unit area in
unit time is then

snicp [r - (r + ^(z - * )J

(dT\
r + JSw(-*o) (534)

where the symbol 2 signifies a process of summation

Similarly, the total heat content of all the eddies descending from
level z'j through unit area at level z in unit time is

(dT\
r + JiVu - M>)

(53-5)

The net upward transfer of heat by eddy motion, obtained by sub-
tracting (f),'*-r>) from (. r >3-4), is thrn

(Rl-fl)

As mucli mass is transported upward across the surface :is is trans-
ported downward across the same surface*, so that -/// = 2/// 7 , and the
first term of (5U-(>) is zero. The net upwanl transfer therefore becomes

(53-7)

('j f / T \
F + ~ j [S

Now consider two elements NI and *S f 2 of a large* horizontal
Over the small area S'i t-h(re is a mean upward eddying velocity w[.
The mass of air ///, passing upward through *S'i in unit time is pir[S\.
Over S 2 th(M-e is a mean downward (tidying velocity *r 2 . The mass
of air m ])assing downward through *S f 2 in unit time is then
Substituting these* values in (53-7) gives for the; upward transfer

-c p (r + ^ [Spies', (z - z ) + Spu&f 2 (a - {)] (KJ.8)

Adding the contributions of all such small elomonls it follows that the
net upward transfer of heat over the area .S* is

152 TURBULENCE [Chap. 9

where, as in the previous section, the bar denotes the mean value of the
product, in this case the mean value over the area S. The net upward
transfer through unit area is, therefore,

- r p r + [p w'(z-go)] (53-9)

But the expression w'(z z ) is the coefficient of eddy diffusivity A',
so that the net upward transfer of heat through unit area is

-Kpc p lr + } (53-10)

If the rate of decrease of temperature with height of the environment,
i.e., the lapse rate of the environment, is denoted a, so that dT/dz = ,
then the net upward transfer of heat is

-Kpc p (V - a) (53-11)

For a stable lapse rate F > a, and the turbulent transfer of heat is
downward. When F = a there is no vertical transfer of heat and when
the environment is unstable, so that F < a, the transfer is upward.
This result has numerous applications in meteorology, both theoretical
and applied, many of which will be noted in other sections of this book.
The assumptions made in obtaining this result should be noted. It
was assumed that the density was uniform, that the air particles at 2
and ZQ were normal specimens of the air at those levels, and that the
lapse rate of the environment was constant over the mixing length

Z ZQ.

From (53-10), the upward eddy transfer of heat Q through unit area
in unit time is given by the expression

Q = -Kpc,

The units of this expression are, of course, cal cm"" 2 sec"" 1 . Now con-
sider a disc of unit cross section and thickness dz. The amount of heat
entering through the bottom is Q and the amount leaving through the

top is Q + dz. The net gain of heat in the disc is then dz, or

dz dz

using (53-10),

d T __ / dT\T

5 (53-12)

Sec. 63} '

TURBULENT TRANSFER OF HEAT

153

The rate of gain of heat in the disc may also be expressed as

j dT
P dzc p -

or, with little error, by

pdzc p (53-13)

Equating (53-12) and (53-13) gives, since c p is a constant,

dT _
P dt ~ '<

Now T is a const ant, and p and A' may be assumed to be constant, so
that finally

dT d 2 T

~~~ **- o

dt dz 2

(53-15)

This equation gives the effect of turbulent transfer of heat only in
causing changes in temperature. A knowledge of the radiative transfer
of heat, is also necessary for a
complete determination of the rate
of change of temperature.

Equation 53-15 has the same form
as the equation for the conduction

of heat, by molecular motions, the j iA> / y\. 0*1 Hi -T o )

latter having the coefficient of con- I
ductivity K instead of K as in the
former. The uso of this equation
in evaluating K is outlined in sec-
tion 55.

Taylor has given a number of
interesting applicat ions of this equa-
tion to atmospheric processes. One
of these was a discussion of the
variations which occur in a mass of
air when the surface temperature
is suddenly lowered, as when in sum-
mer a current of air leaves a warm

land surface and moves over a relatively cold sea surface. The bound-
ary conditions are as follows. As shown in Fig. 50, the air leaves the
warm land surface with temperature TQ at the level 2=0 and with
temperature T = T Q az &t level z when t = 0. The lapse rate a is

Flo 60

air mass

through turbulence.

154 TURBULENCE [Chap. 9

assumed to bo initially constant. At t > and z = 0, T = TI, the
temperature of the cool sea surface. The solution of (53-15) with
these boundary conditions is*

[2 f*z/2Kt __ _ / \

1- I 6 4JB d /

W^O \2V / ^

(53-16)

The sudden change in temperature of the surface air is given by T\ 3T .
The term to the right is the probability integral discussed in section
59. Taylor .showed from (53-10) that the time t required for a change
of temperature of amount 0.1 (T\ T G ) to occur at height z was speci-
fied approximately by the equation

2\/~Kt "
or

z 2 = 4Kt (53-17)

He assumed that the change of temperature 0.1(7^ TO) was suffi-
ciently small to bo considered as just the beginning of the change, so
that (53-17) gives the approximate height to which the effect of the
surface cooling extends at the end of time t. The heights of cooling at
the end of times 1, 2, and 3 are shown in Fig. 50. The result is prac-
tically the same if the surface cooling is not, instantaneous, but gradual.
lladiational processes will tend to increase the height to which the
cooling extends during an} r given interval of time. The use of this
equation in indicating the height to which advection and radiation fog
will extend is mentioned in section 134.

54. Turbulent Transfer of Matter. The equation for the vertical
transfer of matter by eddy diffusion may be derived in a manner similar
to that used to obtain the equation of heat transfer, given in the pre-
vious section. Consider any entity in the atmosphere the mass of
which associated with unit mass of air does not vary with vertical
motion. Such an entity might be, for example, atmospheric dust,
carbon dioxide, or water vapor as long as the air remains unsaturated.
Denote the mass of the entity in unit mass of air by the symbol x-
For water vapor the s]>ecific humidity s would be substituted for x>
since the specific humidity is defined as the mass of water vapor per
unit mass of moist air.

* H. S. Carslaw, The Conduction of Heat, London, Macmilhin Co., 1921, pp. 46-47.

Sec. 54}

TURWJLEXT TRANSFER OF MATTER

155

The portion AB of the curve
shown in Fig. 51 gives the rate of
change of x with height in the in-
terval from z to z. The entity has 1
the value xo at the point A at z
height 2 and the value x at the 2o
point R at height z. It follows,
then, that

Xo = X ~ ~ (z - *o) (54-1)

Fin. 51. K<Mv transfor of matter.

An eddy of mass m ascends from
A at level z t) , conserving its initial value of xo as it rises to C. The
amount of the entity which passes upward through a horizontal surface at.
level z is wxo, or, with (54-1),

(64-2)

The total mass of the entity contained in all the eddies rising through
unit area in unit time is thus

[ x -(-

(54-3)

Similarly, denoting the mass of each downward moving eddy by m* ', the
total mass of the entity carried downward through unit area in unit
time by all descending eddies is

*') (544)

The net upward flux of the entity is then

z () ) 2m' (z z ( ') (54-5)

Since 2m = 2m', the first term is zero and the net upward flux becomes

(54-6)
The mass of air m passing upward in unit time through the small area

- ^-

156 TURBULENCE [Chap. 9

51 is pw[Si, while the mass m f passing downward in unit time through

5 2 is pru&S*. Substituting these values in (54-6) gives for the net
upward flux

(54-7)

The net upward flux over the large area S is then the sum of all such
contributions

where the bar denotes the mean value of w'(z 2 ) over the area S.
The net upward flux of the entity per unit area is thus

~ Kp te ( 54 '8)

The units of this expression are gin cm~ 2 sec"" 1 .

The assumptions made are similar to those in the previous section.
It was assumed that the density was uniform, that an eddy was a normal
sample of its environment when it started to ascend or descend, and
that d\/dz was constant over the mixing length.

The net upward flux of mass M of the entity per unit area per unit
time is given by, according to (54-8)

The amount of mass of the entity entering through the bottom of a disc
of unit cross section and thickness dz in unit time is J/ and the amount

leaving through the top is M + - dz. The net gain of mass in the

disc is then - dz or
dz

-\Kp-~\dz (54-9)

Sec. 65] COKFKICIENT OF KDDY DIFFUSIV1TY 157

The rate of gain of the entity x in the disc may also be expressed as

or, with sufficient accuracy, by

P dz d -Z (54-10)

Equating (5-1-9) and (54-10) and assuming that p and K arc constant
with height obtain

$X t .d*X /rl 1|N

= A ---, (54-11)

dt dz

This equation, along with (54-8), is useful in determining (he diffusion
of water vapor through the atmosphere by turbulent, mixing processes.
It has the same form as the equation for the diffusion of the molecules
of a gas.

55. Evaluation of the Coefficient of Eddy Diffusivity. The diurnal
variation of temperature at the earth's surface may be expressed as a
harmonic function of the time. The daily temperature period is not a
true harmonic function, however, since the maximum and minimum of
temperature do not occur 12 h apart. In general, the maximum occurs
at 14 h while the minimum does not occur at 02 h but at sunrise. As a
first approximation, however, the variation of surface temperature with
time may be expressed as

T = T + l!co* 2 ~t (55-1)

where TO is the mean temperature* for the day, B is the amplitude of
the surface temperature variation, and I) is a period of 24 h. Time; / is
taken to be zero when the temperature is a maximum, so that the
maximum temperature is 7 T + B.

When the above boundary conditions are satisfied, the solution of
(53-15) is*

T = T Q - az + Be-** cos (^ t - X*J (55-2)

where

X = ^ (65-3)

* H. S. Carslaw, The Conduction of Heat, London, Macmillan Co., 1921, pp. 47-50.

158 TURBULENCE " [Chap. 9

The az term makes allowance for the mean lapse rate during the
period. It is seen from (55-2) that the amplitude of the temperature
variation decreases exponentially with height, Avhile the maximum
temperature occurs after an increasing interval of time after the surface
maximum is attained, as shown by the increasing lag of the phase
angle Xz with height. The ratio of the amplitudes at any two heights
z\ and 22 is e~ X(2:i ~ . If the amplitude of the diurnal variation at
height z\ is B\, while the amplitude at height z 2 is B 2 , then

e-^~^ = |i (554)

The method of determining the harmonic function which best fits the
observed temperatures is given in section 62. The maximum deviation
from the mean temperature is then the amplitude B. The value of X,
and so of /, can then be determined from (554) and (55-3).

It was assumed in deriving (53-15) that K is constant. However,
it is known that K has marked daily and vertical variations. Since
observations at different heights extending over a 24-h period at least
are used in determining values of K by the above method, it must be
realized that there may be wide divergences between values computed
in this way and actual values of the eddy diffusivity at any given height
and time.

Taylor, using this method, computed values of K from hourly tem-
perature observations made during the five-year period 1890 94 on the
Eiffel Tower at heights of 123, 197, and 302 m above the surface, and at
a height of 18 m above the terrace of the Bureau Meteorologique. The
variations of K with height and during the year arc shown in Fig. 52.
It can be seen that, in general, K is greater in summer than in winter.
In addition, it increases with height in summer, but decreases with
height in winter. He found a mean value of approximately K) 5 for the
height interval 18 to 302 m for the year. Using (53-17), Taylor found
values over the ocean which were considerably less than these, of the
order of 10 3 . Presumably the lapse rates in winter and over the ocean
are smaller than over a land surface in summer so that the dimensions
and activity of the eddies are less in the former than in the latter situa-
tions.

The effects of radiative as well as of turbulent diffusion of heat are
included in the above results. It follows then that the separate con-
tribution of eddy diffusion is somewhat less than suggested by the
values given in the figure. More recent investigations, however, have
shown that, even allowing for the effects of radiation, there arc internal

PROBLEMS AND EXERCISES

159

discrepancies in the results obtained through the use of (53-15). The
presence of these discrepancies suggests that the latter equation does

T 20

I/)

*E 15

I I

197-302m

123 -302m-

18-302m

JJ
Month

FIG. 52. Annual variation of the coefficient of eddy diiTusivity. (After Taylor.)

not give a completely accurate description of eddy diffusion of heat in
the atmosphere.

PROBLEMS AND EXERCISES

1. Show th.it the effect of turbulent transfer of heat in the atmosphere is always
to smooth out irregularities in the lapse rate.

2. In winter a warm mass of air advances from a tropical sea surface over a cold
continent. If A" has the constant value 2 X K) 4 cm 2 sec" 1 , how long, in days, will it
take the surface cooling to extend to heights of (a) 1 km, (/>) 2 km, and (c) 3 km?

3. Compute the mean change in the specific humidity between the pressures 950
and 0,50 mb during a 24-h period when the variation of specific humidity with height
is constant during the period, being a decrease of 1 gin per kg of moist air per km
at 950 nib and zero at (350 mb. The coefficient of eddy difTusivity has the value
6.8 X 10 4 cm 2 sec" 1 , which is constant with height. The density of the air at 950
mb is 1.0 X 10~ 3 gm cm~ 3 .

4. At heights greater than 1 km above a given aerological station the specific
humidity varies inversely with height. At 3 km the specific humidity is 4 gm per kg
of moist air. Assuming that K is constant with height and equal to K) 5 cm 2 sec" 1 ,
compute the rate of change of specific humidity with time at 3 km. Express the
final answer in grams per kilogram of moist air per hour.

5. An isothermal winter air mass with temperature 10C has a surface pres-
sure of 1000 mb. If K at 500 mb is 10 3 cm 2 sec" 1 , compute the mean increase
in temperature in the layer of air below during a 1-h period.

160 TURBULENCE [Chap. 9

BIBLIOGRAPHY

Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapters 11, 12.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapters 10, 11.

Koschmieder, H., Dynamische Meteorologie, Leipzig, Akad. Verlag., 1933. Chapter 7.
Problems of Modern Meteorology, London, Royal Meteorological Society, 1934.

Number 5.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press, Vol. 4

(1931), Chapter 4.
Cowling, T. G., and A. White, " The Eddy Diffusivity and the Temperature of the

Lower Layers of the Atmosphere," Q. J. Roy. Met. Soc., 67, 276-283 (1941).
Lettau, H., Atmonpharische Turbulenz, Leipzig, Akad. Verlag., 1939.
Taylor, G. L, Report by, in Report on the Work Carried Out by the S. 8. "Scotia," 1913.

London, H. M. Stationery Office, 1914. Pages 48-68.
Taylor, G. I., " Eddy Motion in the Atmosphere," Phil. Trans. Roy. Soc., A215, 1

(1915).
Taylor, G. I., "Phenomena Connected with Turbulence in the Lower Atmosphere,"

Proc. Roy. Soc., A94, 137-155 (1918).
Taylor, G. I., " The Transport of Vorticity and Heat through Fluids in Turbulent

Motion," Proc. Roy. Soc., A135, 685-702 (1932).

Taylor, G. I., " Turbulence," Q. J. Roy. Met. Soc., 53, 201 -211 (1927).
Schmidt, W., Der Massenawttausch infreier Luft und verwatulte Erscheinungen, Ham-
burg, Henri Grand, 1925.

CHAPTER 10
STATISTICAL ANALYSIS OF METEOROLOGICAL DATA

56. The Purpose of Statistics. Meteorology is similar to many other
sciences since, in its study, measurements are taken of different variables.
To understand their significance, they must be condensed and analyzed.
There are many ways of doing this, some graphical, some numerical.
Thus the isobars on the synoptic weather chart show in a pictorial
manner the values of the sea level pressures at the numerous reporting
stations and their variation over the region, and thus permit the sig-
nificant features of the pressure distribution to be grasped. The
different methods whereby these data are condensed and analyzed
form a part of the field of statistics. Therefore a knowledge of some of
the elementary ideas of statistics is helpful to the meteorologist.

When the data are in numerical form, the significant features of the
series can be summarized by choosing two or three values, which are
then used to represent the series. Of these measures, two are used
most often in meteorology. The first is a measure of central tendency,
or an average, and the second is a measure of variability. The use
and the method of calculation of these will be shown in the following
sections.

57. Measures of Central Tendency. Computation of the Mean.
When the amounts of annual precipitation for a station for a number
of years are obtained, it is found that they vary from, say, 20 in. to
50 in. The selection of one number to represent all values, to be called
the annual rainfall, might be made in different ways. From one point
of view it would seem desirable to choose the number of inches of rain
that was found to recur most often in the series. This would give the
mode. Another way to select a value to represent the scries is to choose
the middle one after the numbers in the series have been arranged in
increasing order of magnitude. This gives the median. A third way
is to find the total rainfall during the period, and then divide by the
number of years comprising the period. This value is called the arith-
metic mean, or sometimes the mean. Other methods are used at times,
but the mean, median, and mode are the most commonly used measures
of central tendency.

The mode of a number of observations is that value for the variable x

161

162 STATISTICAL ANALYSIS [Chap. 10

which recurs most frequently. The table on page 163 presents a series
of observations in a convenient form called a frequency distribution.
The mean temperature for every April 13 from 1841 till 1940 for
Toronto, Ontario, was obtained, and the values classified. Two values
were found between 25.5 and 27.4 F, four bet ween 27.5 and 29.4 F, etc.
The number of values in each class is set forth in column 3. The mid-
points of the class intervals are given in column 2.

According to column 3 of the table there were more days with mean
temperatures in the class interval 37.5 to 39.4 F than in any other
interval. Hence the modal class interval is 37.5-39.4 F, or, to use the
midpoint of the interval, 38.5 F is the mode.

Since the mode is that value that occurs most frequently, it is a
typical value for the series. It is used in meteorology in speaking of
the region of the westerlies, or the northeast trade wind belt. For in
certain regions of the earth these are the modal wind directions. The
most frequent storm track is a mode of various tracks. Since 1 , though,
it presents difficulties in calculation in some series and since it fails to
be typical except for series with many observations, the mode is reserved
for particular series, such as those mentioned.

Another measure for the distribution is the median. If all the values
of the variable x are arranged in numerical order, the median is the
middle one of the scries so arranged, or with an even number of observa-
tions the average of the middle two. It thus typifies the series since half
the observations are on cither side.

Although the meaning of the median is easily understood, its de-
termination becomes more difficult when the number of observations
is large. When the observations are grouped in classes, such as in the
distribution given in the table, the assumption must be made that
the values in any class interval are arranged at regular intervals. With
this assumption, the median can IMJ computed. The median for the
distribution in the table is in the interval 39.5-41.4 F, since there are
47 of the 100 observations having values below those in that interval.
On the assumption of an even distribution of the observations, the three
additional values required to give one-half the observations will lie in the
lower 3/13 of the class interval 39.5-41.4 F. The median, then, equals

39.5 + ( T 3 3 of 2.00) = 39.5 + 0.46 = 39.96 F

The median is particularly useful in typifying a series with a small
number of observations. The mode would seldom have meaning, and
the mean value would be influenced greatly by one exceptional value.
For example, if the series is 1, 1, 3, 4, 14, the median 3 is more repre-
sentative of the numbers than the mode, 1, or the mean, 5. Such

Sec. 57}

MEASURES OF CENTRAL TENDENCY

163

COMPUTATION OF THE MEAN AND TUB STANDARD DEVIATION

Frequency distribution of daily mean temperatures for
April 13th's, for Toronto, Ontario, during the period 1841-1940*

Interval
F

(1)

Midpoint Frequency

(2)

25.5-27.4

26.5

27.5 29.4

28 5

29.5-31.4

30.5

31 5 33 4

32 5

33 5-35 4

34 5

35 5-37.4

36 5

37.6-39.4

38.4

39.5-41.4

40.5

41.5-43 4

42 5

43 5 45 4

44 5

45.5-47.4

46.5

47.5-49.4

48.5

49.5-51.4

50.5

51 5 53.4

52.5

53.5-55.4

54.5

55.5 57 4

56.5

57.5-59.4

58.5

Totals

(3)

4
4
3
6
11
17
13
9
9
7
4
5
2
1
2

_1_
100

Deviation

from

Arbitrary
Origin

d 1
(4)

-8
-7
-6
-5
-4
-3
-2
-1

(5)

-16

-28
-24
-15
-24
-33

o/t

Orr

-13

9
14
12
20
10

6
14

-187
93

fd'*
(6)

128

196

144

28
36
80
50
36
98
64
1220

94
Correction to arbitrary mean, c (in class interval units) = = 0.94.

Correction to arbitrary mean (F) = 1.88.
Mean = 42.5 - 1.88 = 40.62 F.

Z/</' 2 1220

s 2 (in class interval units) = = - = 12.20.

N 100

<r 2 (in class interval units) - s 2 - c 2 = 12.20 - 0.88 = 11.32.
<T (in class interval units) = 3.36.
a- (F) - 6.72.

* Richmond W. Longley, " The Frequency Distribution through the Year of
Abnormally High and Low Daily Mean Temperatures at Toronto," J. Roy. Astron.
Soc. Can., 36, 225-236 (1942).

164 STATISTICAL ANALYSIS [Chap. 10

series are found in the rainfall of some regions of the earth where rain-
fall is infrequent, but very heavy showers occur at irregular intervals.
The mean annual rainfall, in such a region, would not give as correct an
impression of the lack of rain as would the median value for the series.
In general, though, the median is not used to any great extent in the
field of meteorology, partly because of the difficulty of computation, and
partly because the mean is more suitable in most meteorological series.
The most commonly used average is the arithmetic mean. It is
defined as the sum of the values of the variable x divided by the num-
ber of observations, N. The arithmetic mean is expressed by the equa-
tion

_ x l + x 2 H ----- \-X N

M x =|* (57-1)

where M x is the arithmetic mean of the variable or, and 2 represents
tho process of summation.

With ungrouped data, the computation of the mean is done by adding
the individual items and dividing by the number of items. Since in
some series the number of items is large 1 , this would become burden-
some and, to avoid the excessive labor involved, the items arc fre-
quently grouped in classes according to their respective values. Thus
the mean temperatures for Toronto for the different April I3th's were
grouped in classes, which are given in the table. When this is done,
the assumption is made that every item in an interval has the value of
the midpoint of the interval. Thus it is assumed, referring to the table,
that there were 2 days, moan temperature 20.5 F, 4 with mean tem-
perature of 28.5 F, etc. To compute the mean, then, it is necessary
only to multiply each midvalue by the number of times that that value
was supposed to occur, add the products, and divide by the number of
items. In terms of tho figures given in the table, this would be repre-
sented as

_ r (2 X 20.5) + (4 X 28.5) + (4 X 30.5) + (3 X 32.5) + + etc.

JM = ----

100

One method of avoiding the necessity for lengthy computations in
the determination of the moan is by the use of an assumed mean, M '.
The assumed mean is then subtracted from each value of the series,

Sec. 57} MEASURES OF CENTRAL TENDENCY 165

resulting in a series of values of (x - A/')- The algebraic average, c,
of these is taken

^jO = *i - ^' + *2 ~ A/ 7 + + X N - M f

N N ( }

A/ 7 = Af - A/ 7

M = c + M' (57-3)

Thus the arithmetic mean is obtained by adding the mean of the devia-
tions to the assumed mean. An immediate corollary to this theorem
is that the sum of the deviations from the real mean is zero.

A second method of avoiding excessive computation with grouped
data involves the use of the class interval as a unit and then returning
to the original units after the computation is finished. In the computa-
tion of the table, the assumed mean is taken as 42.5 F. The column
headed d r gives the difference in class intervals of each class from the
assumed mean, with negative signs for those less than the assumed
mean. There are, then, 2 occurrences 8 class intervals from the
assumed mean, 4 which arc 7 class intervals away, etc. By multi-
plying these to give the figures in the column headed fd f and adding,
it is found that the total is 94 class intervals. The mean of these
values then, 94/100 = 0.94 class intervals. The mean according to
(57-3) is 0.94 class intervals when the origin is at the assumed mean,
or 1.88 F below the assumed mean of 42.5. The mean of the distribu-
tion is, then,

42.5 - 1.88 = 40.02 F

The value of the mean, 40.02 F, gives a typical figure for the whole
distribution, and it is frequently used as such. Its advantages over
other measures of central tendency arise from, first, the ease with which
its significance is understood, second, the simplicity with which it may
be computed, and third, the possibility that it can be manipulated
algebraically. Also it is affected by the value of every term of the
scries, but this is sometimes a disadvantage when an extreme value
changes the mean by an unduly large amount.

In meteorology, as in other sciences, the mean is used extensively
to typify a distribution. In climatological tables values are given for
the mean monthly temperature, the mean annual rainfall, etc. Mean

166 STATISTICAL ANALYSIS [Chap. 10

pressures are given on mean pressure maps. Mean winds are given,
too, on climatological maps. Here, though, the direction must be con-
sidered, and the mean wind is a vectorial average rather than a numerical
average. For some series the mean is not completely satisfactory.
Such a scries is that representing the cloudiness over a station. The
mean cloudiness may be computed as 0.6. This, though, could be an
average of a series in which most of the observations are either overcast
or clear. Thus the statement that the mean cloudiness is 0.6 says
nothing about the probable weather at any time during the series. In
that respect it is different from the mean value of 40.62 F for the series
of temperatures, since most of the observations of temperature lie near
the central part of the distribution, and so the mean gives a reasonable
probable value for a member of the series. In a series such as cloudi-
ness the mean is not a typical value, and when a mean of the series is
given, it must be interpreted with that understanding.

58. Measures of Variability. Standard Deviation. A measure of
central tendency, either the mean, the median, or the mode, represents
one feature of the series of observations. The purpose of each is to
represent a value around which, to a greater or less degree, the series is
grouped. But they by no means completely typify a series. For
example, the mean of the numbers 3, 5, 6, and 6 is 5. But 5 is also the
mean of the four numbers 11, 5, 2, and 2. Yet in one case the numbers
are grouped closely about the mean, and in the other the mean is close

to only one of the individual fig-
ures of the series. In order to
represent this difference between
various scries, some method of
determining the spread of the
series is desirable.

A graphical method for show-
ing the variation in the wind di-
rection and force is the wind
(b) rose, two of which arc depicted

FIG. 53. Wind roses: () at 42.5 N, in Fi & 53 ' Di ^m (a) is for
47.5 W; (b) at 7.5 N, 22.5 W. a P lacc m thc region of west-

erlies off the cast coast of New-
foundland, and diagram (b) is for a region in the northeast trade
wind belt off the west coast of Africa. The length of the line drawn in
any direction is proportional to the frequency of the wind from that
direction, while the number of feathers on the wind arrow indicates the
average force of wind in the Beaufort scale, given in section 66, in that
direction. The percentage of calms is entered inside the circle. Thus
the extent to which the wind at a given place tends to blow in one direc-

Sec. 68] MEASURES OF VARIABILITY 167

tion, or to vary its direction, may be seen by an inspection of the wind
rose.

Three numerical measures of the variability of a scries are the range,
the mean deviation, and the standard deviation. The range is merely
the difference between the largest and the smallest value of the series.
Thus it is easily understood, but has little real meaning. The state-
ment of the maximum and minimum temperatures is one of the ways
in which the range is used in meteorology and climatology. But it
is quite possible to postulate; a temperature variation for a day, such as
might occur if a sharp cold front (section 38) passed the station shortly
before midnight, in which the range failed to express satisfactorily the
variation that occurred.

Another measure of the; variability is the mean deviation. To obtain
this, the difference between the mean anei each of the observations of
the series is taken. The mean of these differences is the>n calculatcel,
the signs of the terms being disregarded. In this manner, a representa-
tive figure for the; variation of the individual items from the mean value
is obtained. The mean eleviations of the two scries in the first para-
graph of the sectie;m arc 1 and 3, mspcctively.

A more satisfactory measure of the variation from the mathematical
point of view is the standard deviation. Te> obtain this, the differences
are take3n, as in the me>an deviation, and then squared. The mean of
these squared differences is found, ami the square root of this mean is
taken. Expressed in the form of an equation, the standard deviation

2(x - M) 2

77 (584)

By squaring the; eliffere;nce^s, the algebraic inconsistency of ignoring
the signs, as is done with the mean deviation, is eliminated.

Of the two deviations, the stanelarel deviation is used more fre-
quently. However, the semiring of the differences allows a term of
the series which is extreme in its value to influence the value of a to a
disproportionate; degree. For that mison, the mean deviation is uscel
at times for scries in which an occasional extreme is likely to occur.
Such a series is that giving the; amount of annual rainfall at a station.
So in computing the variability of rainfall, the mean deviation is used
instead of the standard deviation.

In order to develop a formula with which to compute cr, square (58-1)
to give v , 2 _

168 STATISTICAL ANALYSIS [Chap. 10

From (57-1), it follows that

cr 2 = ^ - 2M 2 + Af 2

y r 2

= ^- - M* (58-2)

Thus the standard deviation may be computed by squaring, not the
deviations, but the original values, finding the mean of these, subtract-
ing the square of the mean of the series, and then taking the square
root. Thus in the series 11, 5, 2, and 2, given in the first paragraph of
this section, the value of a is found to be

As with the mean, the computation of the standard deviation is
simplified by the use of an assumed mean, M 1 .

2 2

* =

N
(s - M') + (M' - M)] 2

2 (M' - M)

N ' v N N

But from (57-2)

2(3 - M 7 )

N
Hence

M -M'

_ 2 (M ; - M) 2 + (M' - M) 2

-S (5S.3,

The first term is the mean of the squares of the deviations from the
assumed mean. From this is subtracted the square of the deviation of
the real mean from the assumed mean. The square root of the differ-
ence is then taken. By transposition in (58-3)

M') 2 2 2

- -' + c

Sec. 69} THE THEORY OF ERRORS 169

Hence the minimum value of is found when c 0, or the

mean of the squared deviations is a minimum when the deviations are
taken from the true moan.

The use of class interval units when the data are grouped is an aid
in the computation of a. Class interval units are used in the compu-
tation of <T for the distribution of the table of section 57. Column
gives the squares of the deviations, in class interval units, times the
number of observations having that squared deviation, i.e., fd' 2 . The
mean of these squared deviations, indicated by s 2 , is found by dividing
the sum, 1220, by the number of observations, 100, giving 12.20. In
addition, c 2 = 0.88. Hence, from (58.3)

a 2 = 12.20 - 0.88 = 11.32
Therefore

er (in class interval units) = 3.36

Now returning to the original units, since the class interval is 2 F, it
follows that

a = 6.72 F

The meaning of this figure, 6.72 F, may be seen by comparing it
with similar figures for other times in the year. Thus the standard
deviation for the Januaiy 13 temperatures for Toronto for the same
period is 10.07 F and for August 13 it is 4.99 F. Thus the mean
temperatures in the middle of winter varied more from year to year
than those in spring, while the mean temperatures in the middle of
summer were the most uniform.

59. The Theory of Errors. The value of 40.62 F has been found for
the mean of the mean temperatures for April 13, for Toronto. By
using the average increase of the monthly mean temperatures from
March to May it is found that at Toronto during April the mean daily
temperature increases at the rate of 0.38 F per day. It seems reason-
able to expect, then, that the mean temperature for April 14 would be
40.62 + 0.38; for April 15, 40.62 + 2 X 0.38, etc. Calculation shows
that these results are approximately correct, but the computed means
show minor variations from the value that the hypothesis would indi-
cate.

The causes of the irregularities found in the broad uniform trend in
temperature are the factors other than the increasing amount of inso-
lation received each day that contribute to the daily temperature. In
any one year there are periods during which the temperature is below

170 STATISTICAL ANALYSIS [Chap. 10

normal, followed by periods with the temperature above normal. The
variations are closely associated with the passage of cyclones and
anticyclones. Yet there is no tendency for cold spells to recur at the
same part of the month each year. Hence the variation from the normal
rise in temperature is attributed to " chance," that is, to the sum total
of a large number of causes, no one of which is predominant.

If the mean temperatures were taken for a long series of years, the
effect of these chance factors would cancel out for any particular day.
The computed mean would be the true mean for that day. For shorter
periods the chance variations give a value for the computed mean
which is not quite large enough in some instances, and too large in
others. For example, it is possible that the April 13th\sfor the period
1841 to 1940 had more days colder than normal than would be expected
if all April 13th's were considered, making the computed mean value of
40.62 F too small. How nearly correct can one assume this value to be?

If a large number of values for some variable x exists, take for con-
sideration a fraction of these values. Choose the individual values
entirely at random, that is, in such a manner that every value has an
equal chance of being selected, and so that the choice of any one value
has no influence on the choice of any other. The selection of five cards
from a well-shuffled pack would be a random sample. If t he means of
sample after sample of the original series are found, these means will be
grouped about the true mean in a standard pattern. For example,
the true mean value of a pack of 52 cards is

4(1 + 2 + + 12 + 13)
52

As the number of sample means increases, the distribution of mean
values approaches more and more closely the mathematical curve
given by the equation,

' -^

(59-1)

where N f represents the number of samples, c is the base of the Naperian
logarithms, and <r is a constant, the standard deviation of the distribu-
tion. The ordinate y gives the frequency of occurrence of a deviation x
from the true mean. Thus the frequency of occurrence of a sample of
mean value 5 or 9 from the pack is

--r*

if there have been 50 sample selections made from the pack.

Sec. 69]

THE THEORY OF ERRORS

171

This curve is frequently called the normal curve of error or the proba-
bility curve. A graph of it is given in Fig. 54. The normal curve of
error enters frequently in the field of statistics. For that reason it has
been the basis for much study by students in this subject. For a com-
plete discussion of the properties of the curve, reference should be made
to any standard text in that field.

-3CT

-2CT

-CT O +CT +2CT -+3CT

FKI. 54. The normal rurvo of error.

The following considerations show why this curve is known as the
probability curve. If N possibilities arc all equally likely, and if y of
these N realize the event A, the probability PA of the event A occurring
is defined statistically as

Thus if a die is absolutely homogeneous, the probability of 4 dots, for
example, turning up is %. In other words

7>4 = i

Thus, since as indicated above, y in (59-1) represents the frequency of
occurrence of a deviation x from the true mean, it follows that the
probability of a deviation x is given by

P* = 7^

172 STATISTICAL ANALYSIS [Chap. 10

Since the distribution of means is given by a form of the normal
curve of error, a computed mean can be expected to lie close to the
true mean. The total area under the curve (59-1) is found to be N'.
If N f t then, is taken as equal to 1, the proportion A of the total area
which lies within the limits of x and +x is given by

l /* __*L \/2 r x -

A = ^ / e dx = - / e 2 * 2 <b (59-2)

This gives the probability that an observed value lies within x of the
true value, provided that a is known. Values of A in units of x/<r arc
given in statistical tables.

One important value of this function (59-2) is found when x = a.
Then

A = 0.6827

Thus the probability of an occurrence lying within a- of the true value
is about %. For limits of 2or, the probability is about ^o, and for 3a,
about 9 %oo- The ordinates at x = 0- are indicated by the lines AB
and CD in Fig. 54.

In order to apply this discussion to the problem, it is necessary to
know the standard deviations of the distribution. By statistical
theory, it can be shown that

(59-3)

where <TM is the standard deviation of the frequency distribution of
the means, a the standard deviation of the total distribution from which
the sample is taken, and N the number of observations in the sample.
This value is sometimes called the standard error of the mean. For
example, if it is assumed that the value of a for the distribution of
temperatures for all April 13th's is the same as the value of a of the
100-yr sample studied, then

\/100

= 0.67

Thus the computed mean of 40.62 F has a probability of % of lying
within 0.67 F on one side or the other of the true mean, a probability
of l %$ of lying within 2 X 0.67 F of the true mean, and almost com-
plete assurance of lying within 3 X 0.67 F of the true mean.

Consider the means for all the days in April. The true mean values
for the daily temperatures of April should lie on a smooth curve, in-

Sec. 69} THE THEORY OF ERRORS 173

creasing from the first till the end of the month. Of the thirty computed
means, about twenty of them should lie within a value of 0.07 V of the
smooth curve, and only one or two should lie farther than 1.34 F from
this curve. Thus a line through the computed means will have irregu-
larities, with the magnitude of these variations depending on the value
of djf. To indicate that the value of a statistical magnitude is not
uniform for all similar series, and to indicate the amount of variation
expected in such series, the value of the standard error of the magni-
tude is included with the magnitude by the use of the sign . The
range thus indicated should, in two-thirds of the computations, include
the true value of the magnitude. Thus the mean temperature for
April 13 is 40.62 0.67 F.

The accuracy of a measure is sometimes given in terms of a quantity
which is 0.0745 times the standard error. This is called the probable
error. Ordinates at 0.0745(7 from the origin cut off one-half the
area under the probability curve. Hence deviations from the mean
greater than and less than the probable error are equally probable,
thus explaining the name. Since practice in this respect is not uniform,
care should be taken to ascertain which of the two, the standard error
or the probable error, is being used.

The same considerations hold for a as for M . The standard devia-
tion of the frequency distribution of different values of is

V2N
For the distribution in the table of section 57,

0.72

= 0.48

A/200

Hence

o- = 0.72 0.48 F

Further discussion of the standard errors of statistical magnitudes
and their use other than to indicate the relative accuracy of the magni-
tude can be found in the standard texts on statistics. In meteorology
and climatology, some observations have been made for a short period
only. Reliance may be placed on means, etc., from such series, provided
the computed values are approximately equal to the true values. The
degree to which this is true can be learned from the standard errors.
The standard errors are sometimes included in meteorological statistics
but an even more extensive use of these would be helpful in inter-
preting the data.

174

STATISTICAL ANALYSIS

[Chap. 10

60. Method of Least Squares. Column 1 of the table on page 175
gives a series of 25 values of the magnitude of the decrease with height
of the wet-bulb potential temperature, O w indicated by X, in the warm
sectors of depressions over western Europe. (For a description of the

30 h-

j 25

| 20

c 15
j?

g 10

1 2 3 4 o 5 6 7
Decrease in e w (C)

Fio. 55. The relation between decrease in B w with height and maximum rainfall.

warm sector of a depression see section 111.) Associated with each
observation there is given in column 3 the? maximum rainfall in milli-
meters per 12 h that occurred at any of the stations along the trajectory
of the air during the 24-h period following the upper air sounding. Fig.
55 gives the same data plotted with the decrease of W , X, as the abscissa,
and the rainfall, R, as the ordinate.

Even a casual glance at the figures or the plotted points reveals that
there is a tendency for large values of one to be associated with large
values of the other and similarly for small values. It is desirable to
express more exactly the relationship which exists between these two
quantities. This may be done geometrically by drawing a straight line
which best represents the relationship. The definition of a line of best
fit could be given in several ways. It is usually defined in such a
manner that the relation of the points to the line is the same as that
of the individual observations of a series to the mean of the series. In

Sec. 60] METHOD OF LEAST SQUARES 175

THE RELATIONSHIP BETWEEN THE DECREASE IN WET-BULB
POTENTIAL TEMPERATURE WITH HEIGHT AND SUBSEQUENT PRECIPITATION*

(1)

(2)

(3)

(4)

(5)

(6)

Decrease in

Decrease

Maximum rainfall

Ow (C)

(Unit iC)

(mm per 12 h)

X 2

0.75

1.5

2.0

1.0

2.25

1.0

2.0

1.25

3.25

169

441

273

2.25

1.75

0.0

3.5

196

112

1.0

3.75

225

2.0

3.75

225

324

270

2.5

100

121

110

4 25

289

676

442

2 25

0.25

1.5

1.25

2 5

100

199

179

2049

2179

189T

179 = 25a -f
1891 - 199a + 20496
a = -0.83 b = 1.01
y = -0.83 -f 1.01s
- -0.83 4- 4.04A r

other words, the best fitting line is the line such that the sum of the
squares of the distances from the points to the line is a minimum.

Let a number N of paired values of variables X\Y\, X 2 Y 2 , ^3^3,
X N Y N be given. It is desired to determine the best fitting straight
line,

y = a + bX (60-1)

for the values, where y indicates the ordinate of a point on the line.

* Data from E. W. Hewson, " Rainfall in Depressions," Q. J. Roy. Met. Soc.,63,
323-335 (1937).

176 STATISTICAL ANALYSIS [Chap. 10

For each value X there is a corresponding value of y on the straight line.
By definition, the sum of all the values (Y y) 2 is a minimum. If
the value of y from equation 60-1 is used, it follows that there are N
terms of the form

( Yi - a - bXi) 2 = Yl + a 2 + 6 2 Xf - 2aYi - 2bX i Y i + 2abX,

(t = 1 ... JV)

The sum is given by the expression
F(X,Y,a,b) = ZY? + Na 2

The values of a and 6 are to be chosen such that this sum is a minimum.
To satisfy this condition, the equations

da "

d6 ^
must be satisfied. Or,

2Na - 22 Yi + 2b2Xi =

Transposing

y v = M fl _u AY v

(60-2)

These equations are called the normal equations. By solving these for
a and 6, values are found which, substituted in the equation 60-1, give
the best fitting line for the paired values (X, F).

In the table column 2 gives the values of X multiplied by 4, called a',
to simplify the computations. Columns 4 and 6 are computed to aid
in the determination of the lino of least, squares. Using the totals given
in the table, the normal equations become

179 = 25a + 1996
1891 = 199a + 20496

By solving these,

a = -0.83 6 = 1.01
The equation of the line is

R = -0.83 + l.Olr

Sec. 61] CORRELATION 177

Since x is in quarter degrees, this becomes

R = -0.83 + 4.04X

This equation is plotted on Fig. 55. A glance at the figure shows that
the line follows the distribution of points fairly well.

The line of least squares, or the regression line as it is sometimes
called, can be used to obtain the approximate relationship between any
paired numbers. One use of it in economics, and to a lesser degree in
meteorology, is to determine the line of best fit of a time series. By
this line a rate of increase can be determined, and by projecting into
the future, this rate can be used for the prediction of future values.
This is done on the assumption that the trend will continue in the
future as in the past., an assumption which must bo checked by a study
of conditions, not by statistics.

61. Correlation. The straight line fitted to the data in section 60 is
the best fitting line. Yet even the best fitting line may not, be par-
ticularly good. Some measure of the closeness with which the line fits
the data should be obtained to indicate* goodness of Jit.

Such a measure of the goodness of fit was derived by Karl Pearson,
and is called the coefficient of correlation. Consider the same series
of paired variables A'lFj, -Y#Kjv as was used in section 60. The
standard deviation ay () f the dependent variable Y is a measure of the
variability of Y from its mean. A similar measure can be obtained by
considering the distances from the points to the straight line. Take
the deviation for each observation of the variable Y f from the line of
best fit, square, add, and take the square root, of the, sum. The measure
thus obtained, 6V, indicates the variability about the straight line,
y = a + bX, given in ((>() 1). If the line were through the mean MY
of the series and parallel to the .r axis, the deviations from the line would
equal the deviations from the mean and Sy <ry. This would indicate
the absence of any relationship bet ween ^Y and Y. Thus ay would be
the maximum value of SY. The line, though, was computed so that Sy
is a minimum, so generally SY < oy- A comparison of SY and ay gives
a measure of the goodness of fit of the line.

The formula developed by Pearson is

r = -J 1 - - 2 ~ (6M)

where r is the coefficient of correlation. The sign used is the same as
that for 6, the slope of the regression line.

A study of this formula brings out some significant facts concerning r.

178 STATISTICAL ANALYSIS [Chap. 10

If the data fit perfectly along the line, Sy = 0, and so r = 1. This
would be the maximum value for r. On the other hand, SY has for its
maximum value that of ay, so the minimum value of r = 0. Thus a
value for r near means that there is little relationship between the
two variables. On the other hand, a value of r greater than 0.80 or less
than 0.80 indicates that in general there is a close relationship between
the variables. Since the sign corresponds with that of the slope of the
regression line, a positive value for r indicates that large values of one
variable are associated with large values of the other, while a negative
value for r indicates that the two quantities vary in an opposite sense.

A value of r which might be considered satisfactory varies in the
different fields in which statistics is applied because of the control which
may be exercised over variables that might cause changes in the de-
pendent variable under consideration. Experiments in physics keep
constant as many factors as possible in order that some particular
relationship may be studied. The correlation, then, should be high.
In other fields the variables are less subject to control, and so smaller
values of r may indicate a probable common variability. The following
values of correlation coefficients were obtained by W. H. Dines* for
different pairs of meteorological variables.

Pressure at mean sea level and surface temperature r 0.16

Pressure at mean sea level and at 9 km r = 0.68

Temperature at 4 km and pressure at 9 km r = 0.82

Height of tropopause and temperature of tropopause r 0.68

Height of tropopause and surface temperature r = 0.30

The determination of the correlation coefficient can be made from
one of several formulas which have been developed. If the regression
line (60-1) has been determined, the value of r can be obtained by the
formula

r = 6 (61-2)

Other formulas that are useful at different times are

2 -

r - - 2 2 - (614)

and

2 aVY

- 2F 2 -

where MX, My indicale the mean values for X and Y.
* W. H. Dines, M. O. Geophys. Mem., Nos. 2 and 13.

Sec. 62] HARMONIC ANALYSIS 179

Using the data of the table given in section 60, and substituting
in (61-3),

MR = = 7.16 mm

199
M x 7.96 quarter degrees C

2049

- (7.96) 2 = 4.31 quarter degrees C

25
it follows that

= 1891 - 25 X 7.96 X 7.16 =
25 X 5.99 X 4.31

Since r is a dimensionless number, its value will be independent of the
units used in its determination, provided that the latter are used con-
sistently throughout.
The standard error for r

<*r = ~^ (61-5)

= 0.096

r = +0.72 0.096

The value of r for the 147 items from which this group of 25 was
chosen was 0.78, which is within the range of the standard error of
the computed value for r. The value of 0.78 indicates that there is a
definite and reliable relationship between the decrease in wet-bulb
potential temperature with height and the rainfall during the succeed-
ing 24 h below the warm frontal surface.

62. Harmonic Analysis. Section 60 treated the problem of fitting a
straight line to a set of points, on the assumption that the relationship
between the sets of values could best be represented by a straight line.
It is possible to extend the method in order to fit a polynomial of a
higher degree to fit the data. Thus the best fitting parabola

y = a + bx + ex 2
* A shorter method for computing r is given in the Appendix.

180

STATISTICAL ANALYSIS

[Chap. 10

may be determined, or the cubic equation may be used. These are
seldom used in meteorology, although other branches of science find
value in fitting non-linear functions to the data.

There is, though, in meteorology a large number of variables which
have periodic fluctuations. Some of these, such as temperature and
precipitation, are related to the seasons, thus showing an annual varia-
tion. Others, such as temperature, dew point, and pressure, have a

63 r

~ 54

I-

51
48

00 03 06

09 12 15
Time (h)

18 21 24

FIG. 50. Moan hourly temperutures for June at Leafield, Kngland. (After Johnson

and 1 ley wood.)

daily period. Fitting a curve to such data requires the use of the
trigonometric functions. The figures in column 2 of the accompanying
table give the values for the temperatures at a height of 1.2 m averaged
for each hour of the day for June for a period of five years at Leafield,
Kngland.* They are plotted in Fig. 50.

If t represents the time in hours after midnight and T the mean
temperature at time /, then the relationship between T and t can be
best represented by the cosine function,

T = To + a cos t -

or a similar sine function. With this function, the amplitude of the

* Data from N. K. Johnson and fJ. S. P. Hey wood, " An Investigation of the Lapse
Rate of Temperature in the Lowest Hundred Meters of the Atmosphere," M. O.
Gcophys. Mem., No. 77, 1938.

Sec. 62] HARMONIC ANALYSIS 181

THE EVALUATION OF FOURIER COEFFICIENTS

(1)

(2)

(3)

(4)

Mean June

Hour t

Temperature T

cos^-t

sin /

(F)

50.1

1.000

0.000

49.3

0.996

0.259

48.7

0.866

0.500

48 3

0.707

47 8

0.500

0.866

48.8

0.259

0.966

50 4

000

1 . 000

52 7

-0 259

0.966

54 9

-0.500

866

56 6

-0 707

707

58.0

-0.866

0.500

59.6

-0.966

259

60 5

-1.000

0.000

61.4

-0.966

-0 259

61.7

-0.866

-0 500

61.6

-0.707

61.5

-0.500

-0.866

60.8

-0 259

-0.966

59.6

000

-1.000

57.6

259

- 0.966

55 1

500

- 0.866

53 2

0.707

-0.707

0.866

-0.500

51 1

0.966

-0.259

1321 3

24Ao = 1321.3 AQ - 55.05
12,4 1 - -10.4 (1.000) - 20.6 (0.966) - 19.0 (0.866) - 16.7 (0.707)

-13.5 (0.500) - 7.1 (0.259)
= -67.15
AI = -5.60

-3.6 (0.259) - 7.0 (0.500) - 9.9 (0.707) - 13.9 (0.866) - 16.9 (0.966)

-9.2 (1.000)

-48.99

-4.08

T = 55.05 - 5.60 cos t - 4.08 sin t
12 12

182 STATISTICAL ANALYSIS [Chap. 10

variation in T is givon by a. As t increases from to 24 h, T passes
through a complete cycle, returning to its original value when t = 24.
$ is the angle of lag for the cycle. Expanding the expression above,

T = To + a cos <j> cos t + a sin </> sin t
\2i \2i

= 7' + A i cos - 1 + i sin --_ < (62-1)

where

^ l = a cos < BI = a sin

In this form, the expression gives three terms of the Fourier series

T = AQ +Ai cos + A 2 cos 20 + A 3 cos 30 +
+ #1 sin + B< 2 sin 26 + 3 sin 30 +

where 9 = - t, and /1 = 7' n .
iZ

Fourier's theorem states that, if T is a function of 6 which is finite
and has only a limited number of maxima and minima and of finite
discontinuities within the range < < 2?r, then the function may be
represented by the series, except at the points where the function is
discontinuous. The constants are evaluated by the formulas

, = X - f

27rJ ()

TdS or XT

27T

A0
T cos id dd or - 27" cos id (62-2)

7 T sin id dd or 2)7 7 sin iO

The first form is used in the case where T is a continuous function. A
complete evaluation of the constants permits any function to be fitted
perfectly by the series. If, however, the series is evaluated only for
an equal number, ///, of sine and cosine terms, then the least squares
solution for the periodic curve with 2m + 1 constants is obtained. In
the problem under discussion, the equation desired has three constants,
TO, A\, B\ y and one cosine and one sine term, according to (62-1).
The intervals are 1 h, and so A0 = Tr/12. Furthermore, t = at 24 h,

PROBLEMS AND EXERCISES 183

Then from ((52-2)

1 24
r =A =-JT

1 24 7T

^-rrcos-;

In the accompanying table, columns 3 and 4 give the values of the
cosine and sine functions by which the values of T must be multiplied.
Since, in this instance, the number of intervals is a multiple of 4, there
is a repetition of certain numbers in these columns, with changes in
sign. Here the amount of calculation is reduced by adding and sub-
tracting the appropriate values of T before multiplication. When this
is done, the following values are obtained.

TQ = ~ x 1321.3 - 55.05
24

A! = -^ X (-67.15) = -5.60

1.Z

B l = X (-48.99) = -4.08
12

The best fitting curve has the equal ion

T = 55.05 - 5.00 cos - t - 4.08 sin ~ t

\2t \i

This curve is plotted with the given data in Fig. 5(5.

By fitting a sine curve to a set of data in the above manner, signifi-
cant facts about the variation of any meteorological dement may be
determined to aid in understanding the periodicities in the atmosphere.
The amplitude of a sine curve which best fits the observed diurnal
temperature variation, along with the amplitude obtained in a similar
manner at a greater height, (see problem 3 of this chapter) may be used
in the manner outlined in section 55 to compute the value of the coeffi-
.cient of eddy diffusivity.

PROBLEMS AND KXKUC1SKS

1. The distribution of the annual precipitation for the period 1886-1938 for the
states of New York arid North Dakota is given in the following table.* Compute
the means and standard deviations for the two distributions.

* Data from Climate and Man, 1941 Yearbook of Agriculture, U. S. Department
of Agriculture, Washington, D. C.

184

STATISTICAL ANALYSIS

[Chap. 10

NEW YORK

NORTH DAKOTA

Amount of

Number of

Precipitation Occurrences

(in.)

31.00-32.99

33.00-34.99

35.00-36.99

37.00-38.99

39.00-40.99

41.00-42.99

43.00-44.99

45.00-40.99

47.00-48.99

49.00-50.99

Total

Amount of
Precipitation

(in.)

7.00- 8.99
9.00-10.99
11.00-12.99
13.00-14.99
15 00-16.99
17.00-18.99
19 00-20.99
21.00-22.99

Number of
Occurrences

1
2
2

6
11
18
11

2. The following table* gives the height of 30 meteorological stations situated on
the plateau and the eastern slope of the llocky Mountain range between latitudes
35 and 49 N, and the mean temperature for these stations for July, 1940. Com-
pute the line of least squares which gives the relationship between these two variables,
and the coefficient of correlation.

Altitude of Station
(hundreds of feet)
36
25
41
32
26
32
61
54
38
62
28
53
48
14
25

Mean Temperature
July, 1940

74.6
71.9
70.4
71.8
77.2
77.0
70.1
72.6
73.2
63.7
79.6
74.6
76.8
82.8
82.0

Altitude of Station
(hundreds of feet)

Mean Temperature
July, 1940

80.4
80.0
81.4
70.8
74.5
72.4
71.6
81.0
80.8
68.7
74.9
74.0
72.8
76.3
74.4

3. The mean values for the hourly temperatures for June for a height of 12.4 m
at Leafield, England, are given in the following table, f Fit a sine curve to the data.
Compare the values of the amplitude and the time of maximum temperature with
the results for 1.2 m analyzed in the text.

* Data from Monthly Weather Review, 68, 202-203 (1940).
t Data from N. K. Johnson and G. S. P. Heywood, loc. tit.

BIBLIOGRAPHY 185

Time Temperature Time Temperature

(0 (F) (/) (F)

01 50.5 13 59.9

02 49.9 14 60.4

03 49.4 15 60.5

04 49.0 16 60.6

05 49.2 17 60.2

06 50.0 18 59.4

07 51.9 19 57.9

08 53.7 20 56.0

09 55.2 21 54.4

10 56.5 22 53.2

11 58.0 23 52.3

12 59.0 24 51.2

4. Mean values are given for the height of the tropopause over Sault Ste. Marie,
Michigan, in different meteorological situations.*

Height

(km)

Rear of a low 10.28

Center of a low 10. 75

Front of alow 11.08

Rear of a high 12.36

Center of a high 1 1 . 55

Front of a high 10.91

Assume that these are spaced at equal distances and fit a sine curve to the points.
The significance of these data is discussed in sections 113 and 116.

5. Using the amplitudes of the sine curves fitted to the mean values of the hourly
temperatures for June at Loafield at heights of 1.2 and 12.4 m, as given in section 62
and in problem 3, compute the mean value for June of the coefficient of eddy diffusiv-
ity between these levels. The method of procedure is outlined in section 55.

BIBLIOGRAPHY

Brunt, D., Combination of Observations, London, Cambridge University Press, 1931.
Mills, F. C., Statistical Methods, New York, Henry Holt & Company, 1924.
Whittaker, E. T., and G. Robinson, The CalciUms of Observations, London, Blackie

and Sons, Ltd., 1924.
Yule, G. Udny, and M. G. Kendall, An Introduction to the Theory of Stati8tics,London,

Griffin, 1937.

* Data from C. M. Penner, " The Effects of Tropospheric and Stratospheric Ad-
vection on Pressure and Temperature Variations," Can. J. Research, A19, 1-20 (1941).

PART II. APPLIED METEOROLOGY

CHAPTER 11
METEOROLOGICAL INSTRUMENTS AND OBSERVATIONS

The raw material from which tho weather map is made and by means
of which the meteorologist makes his forecast consists of the reports
of a number of observations and of readings of instruments from each
of a large number of meteorological stations. Whenever possible
numerical values of the weather elements are obtained, but with some
of these elements, such as cloud types, the observer's judgment must
1)0 relied on. Brief descriptions of the most important instruments and
of the methods of observing the several meteorological elements, along
with the criteria used, are given in this chapter. However, complete
instructions for taking observations are not given. Explicit directions
for observers may be found in the handbooks supplied by official weather
services to their observing stations.

A meteorological instrument must fulfil demands that are not made
upon a laboratory instrument. It must continue to operate in all kinds
of weather and with little or no care, or with the care of a tinkering
amateur, which may be worse. A high degree of accuracy is sometimes
sacrificed for simplicity in order to meet these demands. Usually, ex-
tremely accurate values of the meteorological elements are not represen-
tative, as that term is defined in section 77, of the atmospheric condi-
tions over the area surrounding a station and thus have little value to a
forecaster. The pressure is a representative property and so it is meas-
ured more accurately than the other elements.

63. Pressure. The pressure at any level in the atmosphere is the force
exerted on unit area at that level by the vertical column of air extending
to the outer limits of the atmosphere. A method of determining the
pressure was discovered in 1G43 by Torricelli. A replica of his barom-
eter may be constructed in the following manner. Invert a glass tube
about 36 in. long filled with mercury into a bowl of mercury. The
mercury descends from the top of the inverted tube to a point where the
weight of the column of mercury balances the weight of a column of air

186

Sec. 63] PRESSURE 187

extending from the surface of the mercury in the bowl to the top of the
atmosphere. The mercury barometer is still the standard instrument
for measuring pressure. Improvements in design have permitted more
accurate measurements of the weight of the mercury column, but the
basic principle remains the same.

Originally the weight of the air was given in terms of the length of
the column of mercury in the barometer, in inches in the English system
of units, and in millimeters in the metric system. Because this height
doponds on the temperature of the morcury and on the value of the
acceleration of gravity, the height of the morcury as read is corrected
to standard values of T and g. An equation showing the variation of g
with latitude and height is given in section 8.

To eliminate the inconsistency of measuring a force on unit area by a
unit of length, the air pressure is now measured for meteorological
purposes in terms of a unit of 1000 dynes em"" 2 called a millibar (mb).
As indicated in section 0, the length units and the dynamic units are
related as follows.

1 in. mercury = 83.80 mb
1 mm mercury = 1.338 mb

The length of the column of mercury as road on the barometer, even if
the latter is calibrated in millibars, does not give 1 the correct pressure.
It must still be corrected for variations in T and g in order to obtain the
correct value of the atmospheric pressure.

There arc two types of barometers in general use. As the pressure
changes, the level of the mercury in the cistern at the base of the mer-
cury column, as well as that of the mercury near the top of the tube, rises
and falls. In the Fortin barometer the tube and scale are fixed, but the
level of the mercury in the cistern is adjusted for each reading of the
barometer. The tip of an ivory cone, attached to the barometer case
so that its point is just above the level of the mercury in the cistern,
coincides with the zero point of the height scale. The cistern is made
partly of leather, and the top of the 1 mercury is made to coincide; with
the ivory point by adjusting a screw which raises or lowers the base of
the leather bag, and thereby the level of the mercury in it. The height
of mercury can then be measured on the scale.

A second method of measuring the height of the column of mercury
is utilized in the Kew barometer. The construction of the cistern of the
barometer is illustrated in Fig. 57. The cistern A is of cast iron or
similar metal with a boxwood top D. This is held in place by means of
a screw C. The barometer tube E is cemented into the boxwood to keep
it steady. The metal flanges at B are provided to damp out the oscilla-

188

INSTRUMENTS AND OBSERVATIONS

[Chap. 11

tions of the mercury when the barometer is transported. The pores of
the boxwood provide a means whereby the pressure of the air in the cis-
tern and that outside are equalized.

Since the total volume of the mercury is fixed, any given drop in
pressure will cause the level in the tube to drop a fixed amount and that

in the cistern to rise by a correspond-
ing amount, depending on the cross-
sectional areas of tube and cistern.
Assume that the surface of the mer-
cury in the tube is one forty-ninth
of the surface of the mercury in
the cistern. With a fall of 1 in. in
the atmospheric pressure, the mer-
cury in the tube will fall 0.98 in.
and the mercury in the cistern will
rise 0.02 in. Thus a distance along
the tube of 0.98 in. can be treated
as 1 in. in indicating a change in
the atmospheric pressure. Knowing
the exact ratio between the two sur-
faces in any barometer, the scale
may be adjusted so that the actual

height may be read off directly.
FIG. 57 The cistern of a Kcw barom- After thc hci ht of the mcrcury
eter. (From Middleton, Meteorologi- u , , , . , ,, j

cal Instruments, University of Toronto has beon determined, the measured
Press.) value of the pressure must be cor-

rected for errors in the manufacture

of the instrument as well as for the temperature of the barometer and the
value of the acceleration of gravity at the station as described above.
The value of g is constant for any given station, and the value of the
instrument correction is constant for any given barometer. Thc only
variable which will change the value of the correction is the temperature.
A correction card accompanies each barometer, giving thc total correc-
tion corresponding to any temperature of the barometer.

An error in the reading of the barometer occurs at some stations with
high winds. The level of the mercury in the barometer fluctuates be-
cause of a pumping effect resulting from the variations of the wind
velocity. No adjustment can be made for this error. A similar error
occurs with mercury barometers on shipboard through the motions of
the ship.

Since pressure changes rapidly with altitude, the values of the station
pressure determined as described above have little significance. For

Sec. 63] PRESSURE 189

pressures to be compared, they must be corrected to a standard level.
This usually is mean sea level, although in plateaus and mountainous
regions the standard may be nearer the general level of the terrain. To
assume that there is a sea level pressure is equivalent to assuming that
there exists a column of air from sea level to the height of the top of the
atmosphere. An approximate value of the weight of an imaginary
column of air between the level of the station and sea level can be de-
termined if the height of the station and the mean temperature of the
air column are known. An approximation to the mean temperature is
found by averaging the temperature of the air at the station and the
corresponding temperature twelve hours ago. The latter is included
to provide a means of partially eliminating non-representative values for
the current temperature. These corrections to sea level are calculated
for each station with the values of the correction varying with the value
of the mean temperature of the column of air. These corrections are
added for all stations except those below mean sea level.

Another type of barometer is the
aneroid barometer, the basic parts
of which arc shown in Fig. 58. This
instrument consists of a partially
evacuated thin metal cylinder C,
whose circular sides are prevented p'

from collapsing under the pressure of FIG. 58. A simple aneroid barometer,
the atmosphere by a spring 8. When (From Middleton, Meteorological /ra-
the pressure increases, the top of d*****. University of Toronto
the cylinder approaches the bottom ress

of the cylinder which is fixed to a base plate; as the pressure decreases,
the top recedes. These motions arc communicated to the spring and
then to the lever L by means of the knife edges K and K' which pass
through posts P and P f attached to the two circular faces of the cylinder
at their center. The deflection of L is magnified by mechanical means
not shown in the figure. The cylinder is very flexible and therefore the
range of the deflection of L is governed almost entirely by the tensile
characteristics of the spring. The instrument is calibrated by com-
parison with a standard mercury barometer.

The aneroid barometer is not so accurate an instrument as the mer-
cury barometer, but it has the advantages of small size and the ease
with which it can be transported without danger of being damaged.
Since pressure is related to height in the manner specified by equations
9-5 and 9-9, the aneroid is installed in airplanes as a height-measuring
instrument. When its scale is calibrated in height units such as feet
or meters, it is known as an altimeter. An altimeter gives correct height

190

INSTRUMENTS AND OBSERVATIONS

[Chap. 11

readings only under certain standard conditions. At other times a
correction to the reading, depending on the surface pressure and the
mean temperature of the air column below the aircraft, gives sufficient

accuracy for the purposes of air naviga-
tion.

With other aneroids the pointer is
replaced by a pen which records its
movements on a revolving drum. After
the instrument has been calibrated, the
movements of the pen give a record of
the pressure changes. This instrument is
called a barograph and the pressure record
which it makes is known as a barogram.

64. Temperature. In meteorological
observations the temperature is meas-
ured on either the Fahrenheit or the
Centigrade scale. The 1 freezing and boil-
ing points on these and on the Absolute
scale, along with conversion equations,
are given in section b'. The variations
of usage for different purposes and in
different countries an; also discussed in
that section.

The temperature of the air is meas-
ured by means of an ordinary mcrcury-
in-glass thermometer, called a mercury
thermometer. Such a thermometer rap-
idly assumes a temperature which repre-
sents an equilibrium with the environ-
ment, heat being transferred to and from
the thermometer by conduction, con-
g - ' vcction, and radiation. When surround-

FIG. 50. Stevenson srropn. ing objects have approximately the same
(From Middlotoii, M rtenroli>gical temperature as that of the air, the
Instrument*, r.iivorsity of measured temperature mil be repre-
Toronto Pros*,) aont ative of the air temperature at

the height of the thermometer. A thermometer exposed in the
open would be heated by insolation during the day and would
read too high, but during the night it would be cooled by terres-
trial radiation (see section 29) and read too low. To eliminate as much
as possible such errors resulting from the radiative transfer of heat,
thermometers are housed in Stevenson screens made of wood, one of

Sec. 64] TEMPERATURE 191

which is shown in Fig. 59. Such screens vary in construction in different
countries, but they must be designed so that little or no heat can be
transferred by radiation from outside to inside, yet a free flow of air is
impeded as little as possible.

Frequently conditions occur in which the temperature varies markedly
within a short distance, or at one spot within a short interval of time.
Thus during a partly cloudy afternoon a thermometer exposed in the
open may vary as much as 5 F during a period of half an hour, or the
difference between the temperature at the top and bottom of a hill may
amount to 15-20 F during a still winter night. Thus it is useless to
attempt to determine the temperature of the air more accurately than
to the nearest degree, although for the determination of the water vapor
content as described in section 65, the readings should be takon to the
nearest tenth.

FIG. 60. Capillary on the maximum FIG. 61. Index of the minimum ther-

[thermometer. (From Middleton, mometer. (From Middleton, Meteoro-

Meteorological Instruments, University logical Instruments, University of

of Toronto Press.) Toronto Press.)

To measure the maximum temperature a special type of thermometer
is used in which there is a constriction in the capillary of the thermometer
near the bulb, as shown in Fig. 60. With rising temperature the mer-
cury is pushed past the constriction so that the thermometer gives the
correct reading. When the temperature begins to fall, the mercury
breaks at the constriction, leaving the mercury above to register the
maximum temperature. The thermometer is reset by whirling it or
swinging it while holding it at the end away from the bulb. This causes
the mercury above the capillary to return past the constriction.

When the temperature falls below 39 C, mercury freezes, and so
at lower' temperatures mercury thermometers are no longer useful. At
these temperatures organic liquids such as ethyl alcohol must be used,
but because thermometers using such liquids are less accurate than mer-
cury thermometers, they are not used in general practice except for
minimum thermometers. In the capillary of a spirit thermometer which
is used for a minimum thermometer there is a small index as shown in
Fig. 61. The index slides easily along the capillary which is kept in a

192 INSTRUMENTS AND OBSERVATIONS [Chap. 11

horizontal position. When the temperature falls the surface tension
at the free end of the liquid draws back the index as the liquid as a whole
recedes. When the temperature rises the liquid flows past the index
leaving it at the point of minimum temperature. The thermometer is
reset by tapping it gently to allow the index to slide down till it comes
again in contact with the surface of the liquid.

When two metal strips of different substances are clamped together,
the difference in the coefficients 'of expansion of the two metals causes
the combined strip to change its curvature as the temperature varies.
This principle is used in the bimetallic thermometer. In the latter a
pointer is attached to the strip which is in the form of a helical coil.
As the temperature varies, the end of this pointer swings through an arc.
This instrument is not stable enough for precision temperature measure-
ments, but is sufficiently accurate for most purposes if calibrated fre-
quently by comparison with a standard mercury thermometer. If a
pen is attached to the end of the pointer so that the former makes a trace
on a revolving drum, the instrument becomes a recording thermometer,
or thermograph. In this form the bimetallic thermometer is widely used
in meteorological work.

65. Humidity. The water vapor content of the atmosphere may be
measured in several ways. Its value at the surface of the earth is re-
ported by the dew point or by the relative humidity. These quantities
are discussed in sections 10, 21, 81, and 82. Briefly, the relative humidity
indicates how near the air is to being saturated. The dew point gives
the temperature at which the air would be saturated if it cooled so that
no change in water vapor content occurred in the cooling. A humidity-
measuring instrument is known as an hygrometer.

The dew point may be determined directly, but in meteorological
practice it is computed by using the ordinary thermometer, referred to
for purposes of differentiation as the dry-bulb thermometer, and a wet-
bulb thermometer. When the bulb of a thermometer is kept moist,
evaporation occurs. The heat required for evaporation is taken from
the air surrounding the wet bulb, and the thermometer registers a
temperature lower than that shown by the dry-bulb thermometer in the
same atmosphere. The difference is dependent on the amount of evapo-
ration which in turn is determined by the relative dryness of the air. A
discussion of the theory is given in sections 23 and 24. The basic
features of the wet- and dry-bulb hygrometer are shown in Fig. 62.
A piece of muslin covers the wet bulb, which is kept moist by the water
which travels from the container along the wick to it. To obtain satis-
factory readings of the wet-bulb temperature, it is necessary that a fresh
supply of air should flow continually past the wet bulb. One method of

Sec. 66]

WIND

193

accomplishing this is used in the sling psychrometer. The wet- and
dry-bulb thermometers are mounted on a frame which can be rotated
rapidly. In another type of psychrometer the two thermometers are
inserted in a duct or ducts through which air is drawn by means of an
electric fan, thus ventilating the
bulbs. When readings of the wet-
and dry-bulb thermometers have
been taken, the values of the dew
point and of the relative humidity
can be obtained from prepared tables.
When the temperatures are above
freezing, the wet bulb is cooled by
the evaporation of liquid water. At
these temperatures the readings
are fairly accurate. When the tem-
peratures are below freezing, the
bulb is kept damp by dipping it in
water. Since the water freezes and

FIG. 62. A simple wet- and dry-bulb
hygrometer. (From Middleton, Meteor-
ological Instruments, University of
Toronto Press.)

the rate of evaporation and the conductivity vary with the thickness
of the ice coat, errors are greater. Furthermore, for a small change in
the value of the wet-bulb temperature, there is a large change in the
value of the dew point. For these reasons the dew point obtained by
this method is not accurate for low temperatures.

The relative humidity of the air can be measured directly by an in-
strument which operates through the changes in the length of hair
caused by changes in the humidity. When the relative humidity of the
air increases, human hair increases in length slightly. These changes
are magnified so that they can be read through the movements of a
pointer along a scale or of a pen making a trace on a revolving drum.
The scales are calibrated by comparison with values obtained through
the use of a psychrometer. In spite of careful manufacture, the hair
hygrometer does not give consistently accurate readings, and these
instruments should be checked regularly by comparison with values for
the relative humidity obtained by other means. The accuracy de-
creases with decreasing temperature, the lag being so great at temper-
atures below 20 C that the instrument is practically useless.

66. Wind. There are many methods of measuring the wind direc-
tion and velocity. A brief description of only a few of these will be
given here.

An early method of indicating the velocity of the wind was that de-
vised by Admiral Beaufort in 1805. He divided the wind velocities
into thirteen classes according to their effect on objects about the sea.

,194

INSTRUMENTS AND OBSERVATIONS

[Chap. 11

This Beaufort scale has been adapted to land winds, and then the veloci-
ties determined in miles per hour corresponding to each of the classes.
The following table gives the Beaufort scale, with the descriptive name,
the corresponding velocity range, and the effect of the wind on common
objects. With the aid of the descriptive material given in the last col-
umn, an estimate of the wind velocity can be made without the use of
instruments. In comparing the values obtained by this means and
those obtained from an anemometer, it should be remembered that the
wind velocity changes rapidly with height near the surface of the earth.

BEAUFORT SCALE OF WIND VELOCITIES

Beaufort
Number

Description
of the Wind

Velocity
(mph)

Effect of the Wind

Calm

Less than 1

Smoke rises vertically.

Light air

1 to 3

Wind direction shown by smoke

drift but not by wind vanes.

Light breeze

4 to 7

Wind felt on face; leaves rustle.

Gentle breeze

8 to 12

Leaves and twigs in constant

motion; wind extends light flag.

Moderate breeze

13 to 18

Raises dust and loose paper;

small branches are moved.

Fresh breeze

19 to 24

Small trees in leaf begin to sway.

Strong breeze

25 to 31

Large branches in motion; whist-

ling heard in telegraph wires.

Moderate gale

32 to 38

Whole trees in motion; inconven-

ience felt in walking against wind.

Fresh gale

39 to 46

Breaks twigs off trees; generally

impedes progress.

Strong gale

47 to 54

Slight structural damage occurs.

Whole gale

55 to 63

Trees uprooted; considerable

structural damage occurs.

Storm

64 to 75

Accompanied by widespread damage.

Hurricane

Over 75

This is the scale used in weather reports to give the velocity of the
wind at the reporting station.

The cup anemometer is one of the most widely used instruments for
measuring the wind velocity. In this instrument hemispherical cups
are attached to the ends of horizontal arms which extend outward radi-
ally from a vertical shaft. This shaft rotates readily, and the number
of rotations during any given interval of time is counted by mechanical
means.

When the cups are exposed in the open, the wind is caught by the
inside of the cup. The force exerted causes the cups to rotate. By

Sec. 66] WIND 195

calibration as, for example, in a wind tunnel, the number of rotations
which the cups will make when a mile of wind goes by can be determined.
From this value the number of miles of wind that pass during any time
interval, and hence the velocity, can be obtained. The number of cups
on a shaft varies. A shaft with two cups is not satisfactory since the
force varies greatly, depending on the angle between the horizontal arms
and the wind at any instant. Usually three or four cups are used. The
number of rotations for a mile of wind will vary with the number of cups
and their dimensions.

It will be noticed that the wind speed determined by means of the cup
anemometer is obtained by counting the number of revolutions of the
cups in any given time interval t. The result is then an average velocity
over the interval. As t decreases the value of the velocity approaches
more nearly the instantaneous velocity. When the wind is gusty its
velocity changes rapidly; the inertia of the cups will then tend to keep
them rotating at the same speed. The anemometer then fails to register
the extreme values of the velocity. The tendency is for the readings to
be too high with gusty winds for the cups accelerate with a gust more
rapidly than they decelerate with a lull.

A second method of measuring the wind velocity makes use of the
increased air pressure in a tube when a wind blows into the open end of
the tube and the decreased pressure when a wind blows across the mouth.
The anemometer using this principle was designed by W. H. Dines.
The head of a Dines anemometer is shown in Fig. 63.

By means of the wind vane, the mouth A of a horizontal tube is kept
facing the wind. Ball bearings at C permit the vane to rotate readily.
The additional pressure in the horizontal tube is transmitted by means
of slots at B through the stationary tube D and out through F. To
insure efficient performance the fit between D and the rotating tube at
E must be close so that the pressure is transmitted undiminished through
this portion of the instrument.

A group of holes is located in the pipe at G. The suction of the wind
blowing on these holes is transferred through H to the outlet J. The
assembly is supported by means of the pipe mast M\ the cone and cylin-
der L form a covering for the pipe and connections. The rod K is
attached to the vane, passes through the center of the anemometer tube,
and actuates the direction-recording mechanism.

The tube F transmits the pressure increase from the horizontal tube
to the interior of a float in a sealed water tank. The decrease in pres-
sure at G causes a corresponding pressure decrease in the air space above
the float. Thus both the pressure increase inside the float and the
decrease in the air space above cause the float to rise. The design of

196

INSTRUMENTS AND OBSERVATIONS

(Chap. 11

the float is such that its vertical displacement is directly proportional
to the wind velocity. This displacement is recorded on a rotating drum.
The Dines anemometer thus measures the effect of the wind at each
instant and so permits the determination of the instantaneous velocity.
It is therefore more suitable for measuring the maximum velocity and
the gustiness of the wind.

FIG. 63. The head of u Dines anemometer. (From Middleton, Meteorological
Instruments, University of Toronto Press.)

The cup anemometer, as well as the Dines, can be designed to record
the wind speed on a revolving drum. The record of wind direction is
usually made at the same time.

Above the surface of the earth the direction and speed of the wind are
obtained by observing the movements of a balloon which is free to move
with the currents of air. A balloon is filled with a light gas, hydrogen
or helium, and allowed to ascend. If the balloon is filled with a given
amount of gas, the rate of ascent will be almost constant, and so the
height above the earth at any instant can be calculated, using the length

Sec. 67] CLOUDS 197

of time since the balloon was released. The path of the balloon is
followed by the observer with the help of a theodolite. At regular in-
tervals the azimuth and elevation angles are determined. These two
angles, with the height as given by the time clasped since release, de-
termine the position of the balloon. With the help of tables and a
plotting board the observer is able to calculate the average horizontal
velocity of the balloon for each time interval, and so the horizontal
velocity and direction of air in the layer through which the balloon
passed. By interpolation the velocity of the air and the corresponding
direction at given levels are found, and these are reported to the fore-
cast offices for use in analyzing the map and preparing the forecast.

The method of determining the wind velocity of the upper air with
balloons has several sources of error. The most serious of the errors is
caused by the assumption of a constant rate of ascent for the balloon.
With light rain or snow falling, the buoyant force decreases and the
balloon will not rise at the normal rate. Even if there is no precipita-
tion, the presence of vertical air currents will cause the rate of ascent to
vary and make calculations less reliable. This error can be eliminated
by following the balloon with two theodolites placed at a distance from
each other. The position of the balloon is then determined, using the
azimuth and elevation angles as measured by the two theodolites.

Free balloons are also used in meteorology to determine the temper-
ature, pressure, and moisture content of the upper air. A small light-
weight box is suspended below a large balloon filled with hydrogen or
helium. Within the box there are an aneroid barometer, a bimetallic
thermometer, and an hygrometer. These arc connected to a very small
radio transmitter in such a manner that signals giving pressure, temper-
ature, and humidity at various levels are radiated by the instrument as
it ascends. These signals are picked up and recorded by a special re-
ceiving set at the station from which the balloon was released. An in-
strument of this type is known as a radiosonde. Using the values of
pressure, temperature, and humidity received in this manner, the ob-
server is able to compute the height of the balloon at any time by the
method outlined in section 11. He then codes the values to send them
off to the different forecast centers.

67. Clouds. Clouds and cloud types are classified according to a
system introduced early in the nineteenth century. The standard for
classification is laid down by the International Meteorological Organi-
zation and is presented fully in the International Cloud Atlas. This
atlas also includes a large number of excellent plates of the different
types of clouds.

The classification of clouds is very necessary to the forecaster. The

198

INSTRUMENTS AND OBSERVATIONS

[Chap. 11

development of different types of clouds is the result of various processes
taking place in the atmosphere, as shown in Chapter 20, so a knowledge
of the cloud type aids the forecaster in determining the physical develop-
ments in an air mass. In many regions of the earth, owing to lack of
upper air observations, this is the only method whereby the forecaster
is able to learn something of the physical processes occurring in the air
above the surface layers.

Cloud types are divided into two main divisions, cumulus (having
the appearance of closely packed wool) type clouds, and stratus (layer)
type clouds. A second basis of division is in terms of the height of the
base of the cloud. The high clouds are called cirrus or have the prefix
cirro- in their names. Middle clouds are distinguished by the prefix
alto-. Low clouds have no distinguishing prefix. The prefix fracto-
is used to indicate cloud which is a part of a larger cloud that has been
broken by the wind. Nimbo- or nimbus included in the name of a cloud
indicates that it is closely associated with precipitation.

The following table gives the names, levels, and abbreviations for the
common types of clouds.

THE CLASSIFICATION OF CLOUDS

Mean

General
Catescory

Upper Level

Lower Level

Type

Abbreviation

Cirrus

High

35,(KX)

20,000

Cirrostratus

Cirrocumulus

Middle

20,000

0,500

Altocumulus
Altostratus

Ac
As

Strutocuinulus

Low

6,500

0.1

300

Stratus

Nimbostratus

Clouds with vertical
development

35,000

0.5

1,600

Cumulus
Cumulonimbus

Cu
Cb

The high clouds are seldom found below 15,000 ft, their height usu-
ally being between 20,000 and 35,000 ft. These clouds are very thin
and white, and are composed of ice crystals. Cirrostratus (Fig. 64)
is a thin veil which spreads over the sky, sometimes merely giving the
sky a milky appearance. It frequently produces a halo about the sun
or moon. Cirrus (Fig. 65) is similar, but is fibrous in nature, and is
frequently seen with hooks at the ends of individual cloud elements.

FIG. 64. Cirrostratus. (From Cloud Forms and Stoics of the Sky, U. S. Weather

Bureau.)

FIG. 65. Cirrus. (From Cloud Forms and States of the Sky, U. S. Weather Bureau.)

199

200

INSTRUMENTS AND OBSERVATIONS

[Chap. 11

It is cirrus which the fishermen describe as " mares' tails." Cirrocu-
mulus occasionally occurs and appears as a group of small rounded
bunches of white cloud at the edge of a cirrus or cirrostratus layer.

The clouds of the middle levels are usually found between the heights
of 6000 and 20,000 ft, although in a region of low pressure the tops of
these clouds may extend to a height of 23,000 ft. Altostratus (Fig. 66)
is grayish or bluish in color. When thin, it permits the sun's disc to be
seen; when thick, it hides the sun or moon, or only allows their position

FIG. 66. Altostratus.

(From Cloud Farms and States of the Sky, U. S. Weather
Bureau.)

to be determined by a bright spot on the cloud. In the vicinity of a
warm front, the base of an altostratus cloud descends to a height of
1000 to 2000 ft and is then called nimbostratus. This is a low cloud
and is usually accompanied by rain or snow. Altocumulus clouds
(Fig. 67) are formed of globular masses and are often thick. When
they become closoly packed, they may be distinguished from alto-
stratus by the irregularities on the under surface of the cloud. They
are white or gray in color. Sometimes they occur in a regular pattern
to give a " mackerel sky." One species of altocumulus is altocumulus
castellatus. In this type of cloud several tufts showing vertical de-

Sec. 67]

CLOUDS

201

velopment extend upward in the shape of small turrets from an alto-
cumulus cloud.

Low clouds may be found with their bases almost reaching the earth's
surface while in other situations the base may be as high as 10,000 ft.
Stratus is a thin gray cloud, usually occurring at a low level. Fog is

FIG. 67. Altocumulus.

(From Cloud Forms arid States of the Sky, U. S. Weather
Bureau.)

a stratus cloud at the earth's surface. Fractostratus or scud is formed
when a layer of stratus is broken by the wind or when a layer of stratus
is in the process of formation. Stratocumulus (Fig. 68) closely re-
sembles stratus since it is found in a layer, and is of a gray color. It is
variable in thickness and has light spots over its surface or is formed
of individual clouds with clear spaces between. Nimbostratus is a low

Fia. 68. Stratocumulus. (From Cloud Forms and States of the Sky, U. S. Weather

Bureau.)

FIG. 69. Cumulus of fine weather. (From Cloud Forms and States of the Sky,

U. S. Weather Bureau.)
202

FIG. 70. Heavy cumulus. (From Cloud Forms and States of the Sky, U. S. Weather

< Bureau.)

FIG. 71. Cumulonimbus. (From Cloud Forms and States of the Sky, U. S. Weather

Bureau.)
203

204

INSTRUMENTS AND OBSERVATIONS

[Chap. 11

cloud resulting from the downward extension of altostratus in the manner
indicated in the discussion of the latter.

The low clouds of the cumulus type are usually easily distinguished
from the low stratus clouds. Cumulus clouds (Figs. 69 and 70) may
be small rounded clouds of slight vertical development, called cumulus,
or fine weather clouds. These clouds sometimes develop into large
clouds, swelling both vertically and horizontally to form heavy cumulus.
Heavy cumulus clouds have definite rounded edges. When the edges
about the top begin to form a fibrous veil, it has turned into a cumu-
lonimbus cloud (Fig. 71). This formation indicates the presence of ice
crystals at the top of the cloud. When the vertical development is
inhibited by an isothermal layer, the cloud top spreads horizontally so
that it resembles an anvil.

Sometimes the type of cloud is easily determined. On other occa-
sions the cloud appears to be intermediate between two related cloud
types. In such instances careful observation is necessary to classify
the cloud properly.

Cfl)

(c)

FIG. 72. Rain gages of (a) the British Isles, (b) Canada, (c) the United States.
(From Middleton, Meteorological Instruments, University of Toronto Press.)

68. Precipitation. Rainfall is measured by means of a rain gage.
The rainfall as recorded is the average depth of water which would fall
on a level area. Three types of rain gages are illustrated in Fig. 72.
The three gages shown are standard types used in (a) the British Isles,
(6) Canada, (c) the United States. The rain gage consists, first, of a

Sec. 68] PRECIPITATION 205

large cylindrical vessel which is fixed solidly upon or just above the
earth's surface; second, a metal funnel which fits firmly over the top
of the vessel, and which has a strong beveled upper edge; third, a re-
ceiver to be placed inside the vessel and under the delivery tube of the
funnel. The rain falling into the funnel is carried through the delivery
tube into the receiver inside the vessel. Two methods of measuring the
amount of rain which has fallen are used. In Canada and Great Britain,
when the observer measures the rainfall, he removes the receiver from
the vessel and empties the contents into a glass graduate. The area of
the cross section of the glass graduate is a known proportion of the area
of the top of the cylinder. Knowing this proportion, one is able to
calculate the amount of rain per unit area, or the rainfall. Usually the
glass graduate is calibrated to read off directly the depth of the rainfall.
In the United States, on the other hand, the amount of rain is measured
by dipping a thin measuring stick into the cylinder through the hole in
the funnel, and noting the length of the graduated stick which is wetted.

In the types of rain gage described, the loss by evaporation is small,
provided the funnel is fitted closely to the vessel. The possibility of
rain splashing into or out of the funnel must be considered. The tops
of the smaller rain gages are about 1 ft above the ground level to guard
against the rain splashing in from the outside. The top of the larger
gage of the United States Weather Bureau is usually about 2.5 ft above
the ground. If the sloping sides of the funnel are several inches below
the level of the rim, there is little danger that rain which has fallen into
the gage will splash out again. Eddies will at times carry the falling
rain away from the gage and thus make the reading inaccurate. With a
careful choice of site this error can be minimized.

Several types of gages have been devised by which the rate of fall
can be determined. In the tipping-bucket gage, when a unit of rain, as
0.01 in., collects in the bucket, it tips and empties while a companion
bucket on the other side of the pivot is filling. The time at which the
bucket tips is recorded on a revolving drum. The rainfall during any
given interval of time is then readily determined. Float gages record
the height of a float which ascends as the level of water in the vessel
into which the falling rain is carried rises. When the vessel becomes
full, a device such as a siphon acts to empty the vessel in a short time,
after which the vessel again fills by the falling rain. These instruments
do not give highly accurate values, and so are not used for primary
measurement of rainfall. Furthermore they are not satisfactory for
cold climates since the instrument may be ruined if the water in it
freezes.

The accurate measurement of snowfall is more difficult than the

206 INSTRUMENTS AND OBSERVATIONS [Chap. 11

measurement of rainfall. When the fall is measured in a gage similar to
a rain gage, the eddies about its mouth do not permit a representative
amount of snow to fall into the gage. The effects of shields have been
studied, but these do not fully eliminate the errors resulting from the
eddies. Heating the gage will cause the amount measured to be smaller
than the average fall since some of the moisture evaporates when melted.

The other common method of measurement of snowfall is by measur-
ing the fall of snow at several places over a level area where there has
been no drifting, and taking an average value of the measured depths.
This necessitates the determination of the level of the old snow, a task
which is sometimes difficult. The depth of snow is converted into equiv-
alent rainfall by dividing by 10 or 12. This is only an approximation
since the density of newly fallen snow varies widely. With snow pellets
the density may be as high as 20 or 25 Ib per cu ft. Thus any method
of snowfall determination gives approximate values only.

69. Sunshine and Radiation. The measurement of neither of these
two weather elements is as yet of direct interest to the weather fore-
caster. These elements are of interest, though, to the theoretical
meteorologist and the climatologist. The theoretical meteorologist is
concerned with the heat balance of the atmosphere, and so with the
losses and gains in heat energy in the different layers of it. The clima-
tologist prefers to use the continuous record of sunshine rather than the
intermittent record of cloud observations in obtaining his value for the
average cloudiness of the day.

Fig. 73 shows a Campbell-Stokes sunshine recorder. A glass sphere
focuses the rays of the sun on a specially prepared card. When clouds
do not impede the radiation from the sun, the heat concentrated at the
focal point is sufficient to char the card. The focal point moves over
the card as the sun moves across the sky, leaving a record on the card
of the times when the sun was visible. Different slots are provided to
care for the variation of the angle of elevation of the sun during the sea-
sons. This instrument has many of the desirable features of the best
type of meteorological instrument, such as simplicity, durability, lack
of the need of adjustment, ease of understanding the results. There are
some minor possibilities of error. One of these is in the exact deter-
mination of the time of passage of a cloud over the sun, for the burn
spreads over the surrounding parts of the card. The use of the record
to determine the cloudiness for any particular day is open to criticism
since the sky can be covered with broken clouds only, and yet no sun-
shine may be recorded. Averages would tend to decrease the error
produced by this effect.

The standard instrument for the measurement of solar radiation,

Sec. 70] VISIBILITY 207

according to the International Meteorological Organization, is the
Angstrom compensating pyrheliometer. Solar radiation is received on
two thin strips of metal coated with black. Sensitive thermojunctibns
are attached to the back of these strips, but electrically insulated from
them. These are connected through a sensitive galvanometer. When
desired either strip can be shielded from the sun and an electrical current
passed through it. To make an observation one of the strips is shielded

FIG. 73. Campbell-Stokes sunshine recorder.

and sufficient current is passed through it so that the temperatures of
the two strips are the same. The energy absorbed by the unshielded
strip is then equal to the electrical energy communicated to the other.
This second can be measured, and so the first determined.

The silver disc pyrheliometer is another commonly used instrument for
measuring the sun's radiation. Sunlight is allowed to fall on a silver
disc for a stated time interval and the rise in temperature of the disc is
carefully measured. The disc is mounted so as to minimize the trans-
fer of heat to or from it by processes other than radiation. The instru-
ment is simply designed and gives very accurate results.

70. Visibility. Visibility is defined as the distance in a horizontal
direction at which objects can be distinguished. It is useful for the
airman since he uses it in deciding whether he can land safely at an
airport. To the forecaster the visibility is indicative of certain features
of the atmosphere, such as the stability.

About any observing station a number of visibility marks are chosen
and the distance of each from the observing tower determined. For
daylight observations, the marks are prominent objects on the skyline.

208 INSTRUMENTS AND OBSERVATIONS [Chap. 11

For night observations, the marks are certain lights in the vicinity.
The visibility of the lights is converted into daylight visibility by a
comparison of the visibility before and after sundown in the same
meteorological situation. The observer then makes his estimate of
the visibility by noting the farthest visibility mark which he is able
to see from the tower, and then noting whether the visibility in other
directions is about the same. The reported visibility is the average
visibility in all directions.

Care is necessary in the choice of visibility marks. They should
stand out clearly against the skyline or against their background. At
some stations there are no suitable visibility marks beyond a certain
distance. The visibility, when it is greater than the farthest visibility
mark, is then estimated by determining the clarity with which the
farthest mark stands out. With practice this method gives a reliable
measure of the visibility at the station.

BIBLIOGRAPHY

Glazebrook, Sir Richard, Dictionary of Applied Physics, Vol. Ill, London, Mac-
millan and Co., 1923.

Middleton, W. E. K., Meteorological Instruments, Second Edition, Toronto, Univer-
sity of Toronto Press, 1943.

United States Weather Bureau Circulars, Washington, D. C.

Handbooks of the Meteorological Office, London, H. M. Stationery Office.

Handbooks of the Meteorological Service of Canada, Toronto, Meteorological Office.

67. International Meteorological Committee, International Atlas of Clouds and States
of the Sky, Paris, 1932.

70. Middleton, W. E. K., Visibility in Meteorology, Second Edition, Toronto, Uni-
versity of Toronto Press, 1941.

CHAPTER 12
THE GENERAL CIRCULATION OVER THE EARTH

71. Circulation on a Non-Rotating Globe. The movement of air over
the earth's surface is affected by many different factors. It is easier to
understand the final distribution of winds and pressure if the most im-
portant factors are considered separately.

If the atmosphere on a non-rotating globe were heated by contact
with the uniformly heated surface of the globe, the only type of motion
which might occur would be local convection currents. Equilibrium
in the system is reached when the total outgoing radiation is equal to
the input of energy. Since, though, some of this heat would be lost
from the upper atmosphere, the unstable temperature distribution
would persist, and the convection currents continue to develop.

If now the sun is assumed to revolve around the non-rotating globe,
the heating is not uniform and the situation is more complex. Curve
(a) in Fig. 27 (section 31) shows the variation in intensity with latitude
of the effective incoming solar radiation. This curve thus gives the
relative amount of energy received at the different regions of the non-
rotating globe. It is clear from this diagram that the equatorial region
receives much more solar energy than the polar regions, and the air at
the equator will become warmer than that at the poles. As a result of this
heating, the air will expand and rise. Near the upper limit of the at-
mosphere the amount of air above some given height will exceed the
amount of air above the same altitude in the colder regions, and so the
pressure will be greater. A pressure gradient will be produced, and the re-
sulting pressure gradient force (see section 33) will move the heated
air at higher levels toward the colder parts of the earth. Here the
total weight of the atmosphere will increase and the weight of the air
at the equatorial part of the globe will decrease. Hence a pressure
gradient will be produced at the surface from warm to cold regions.
A complete circulation (see Fig. 74) will result, with air rising over the
region heated most strongly by the sun, moving in the upper atmosphere
to the polar regions, subsiding there, and traveling near the surface to its
starting point.

In section 49 the idea of circulation was defined. In general terms,
the circulation C about a closed curve measures the flow along that curve,

209

210

GENERAL CIRCULATION

[Chap. 12

and is obtained by multiplying the velocity V for every small part of the
curve ds by ds, and summing the products. The rate of growth of circu-
lation is, according to (50-20),

dC = _
dt ~~

or, with (7-7)

- > RTd(logp) + W

FIG. 74. Circulation on a non-rotating
globe.

In this equation t represents time,
R is the gas constant for dry air,
T the temperature in degrees Ab-
solute, p the pressure, and W the
term for the external forces. The
A B rate of change of circulation
around the closed curve ABDC
shown in Fig. 74 may now be
computed. This closed curve
may be specified as comprising two vertical lines, one above the
equator, and the other above the pole, which are joined by the two iso-
bars, p = 300 mb and p = 1000 mb. Along BD and AC

d (log p) =

and so the contributions to are zero. If T E is the mean temperature

of the column AB,

<J A

RTd(\ogp) =

Similarly if Tp is the mean temperature of CD,

f RTd(logp) = RTplog^

J D

If no external forces are acting, so that W = 0, then

Assume that the difference in mean temperature between equator and
pole is 30 C. Then the magnitude of the circulation at the end of 3 h

C = 11.2 X 10 11 cm 2 sec~ 1

Sec. 72} THE EFFECT OF THE EARTH'S ROTATION 211

The length s of the curve is approximately 2 X 10 4 km. Hence the
average velocity at the end of 3 h

__ o

y = = 5.6 m per sec
$

The accelerating force is still present, and V would continually in-
crease but for two counteracting forces. As the velocity increased, the
transport of the cold air to the equator and the heated air to the poles
would also increase, and the temperature difference could not be main-
tained. The accelerating force then decreases proportionately to the
decrease in the difference in temperature. Another force results from
the retarding effect of friction. The computations assumed external
forces zero, but such forces are actually present and the surface air
moving toward the equator would be retarded by the friction between
the air and the globe's surface. The velocity would not, then, increase
indefinitely. On a non-rotating globe these forces would finally reach
an approximate balance, with deviations being minor departures from
the general circulation.

72. The Effect of the Earth's Rotation. The preceding section dis-
cussed the circulation that would develop on the earth if the sun re-
volved about the earth rather than the earth revolving on its axis.
When the .rotation of the earth is considered, another force is exerted
on the moving particles of the atmosphere.

As explained in section 34, there is an apparent force acting on all
particles moving over the earth's surface which tends to change the
direction of motion to the right in the northern hemisphere, and to the
left in the southern hemisphere. The force has no effect on the speed
of the particles. In the discussion that follows only the northern hemi-
sphere will be considered. A similar discussion would be true for the
southern hemisphere, except that the directions mentioned would, at
times, differ.

The magnitude of the force arising from the earth's rotation, called
the deflecting or Coriolis force, is 2co sin < V, where < is the latitude, and
co is the angular velocity of the earth. If, in the circulation discussed
in section 71, the air particle starts in the upper air toward the north pole,
it is not affected by the deflecting force immediately since at the equator
the force is zero. But as the air advances northward, the deflecting
force increases as < increases, and, by the time that it has reached lati-
tude 20 or 30 N, the motion will have a marked eastward component.
At this latitude there is an accumulation of air which leads to a high-
pressure region at the earth's surface. At the surface on the southern
side of this high-pressure area the winds blow toward the equator, but

212 GENERAL CIRCULATION [Chap. 12

again they are affected by the deflecting force. Thus they become
northeast winds, rather than northerly winds, and form the northeast
trade winds shown in Figs. 5 and 6 of section 3. The high-pressure belt
in the vicinity of 30 N, often referred to as the horse latitudes, is called
the sub-tropical high.

A similar thermally produced circulation is found in the vicinity of
the poles. The air subsiding as a result of cooling in the lower layers
at the poles moves toward the equator in diverging currents. These
northerly winds are deviated to the right by the deflecting force as they
advance southward, and thus become east winds. When the air motion
is in this direction the deflecting force and the pressure gradient force
balance and the motion is steady. The air slowly becomes heated and
rises, returning aloft to the pole.

On the northern side of the sub-tropical high-pressure belt the winds
blow from west to east, giving the westerlies of the temperate zone.
_ Near the surface, the slow

movement of the air across
the isobars results in a north-
ward flow of air which trans-
ports the heat of the equato-
rial regions toward the poles.
At the northern edge of the
temperate zone it meets the
polar air moving westward.
A low-pressure trough forms
at the junction of these two
\ currents. At times the warm
moist westerly current rises

Equator A over the colder polar air, giv-

FIG. 75. Longitudinal cross section of the ij * widespread precipitation,
atmospheric circulation on the earth. lhe warm air n w continues

northward at high levels to

subside over the poles. The accumulation of air there which would result
from this transport of air in a northward direction is prevented by occa-
sional outbreaks from the polar high which carry the surplus air south-
ward into the region of westerlies, and so down to the sub-tropical high-
pressure region. A diagram of the general circulation, showing the
above features on a vertical section of one-quarter of the earth, is given
in Fig. 75. The wind distribution at the earth's surface is given in
Figs. 5 and 6, section 3.

The foregoing discussion attempts to describe and explain the circu-
lation of air that is observed upon the earth. That this attempt leaves

\\.

Sec. 73} INFLUENCE OF THE LAND MASSES 213

something to be desired is well understood by the authors. There has
been as yet no completely satisfactory exposition of certain of the mpjor
features of the general circulation. Why do the sub-tropical high-
pressure belts occur at about 30 from the equator? Precisely how does
the air which gradually moves from the equator to the poles find its way
back again to the equatorial regions? Why is the circulation from
equator to pole divided into three cells? These and other questions
still await a satisfactory answer from meteorologists.

73. The Influence of the Land Masses. On a globe with a uniform
surface the cff ect of the apparent movement of the sun north and south
with the seasons, which occurs as a result of the inclination of the earth's
axis, would not be pronounced. The heat equator, and so the equa-
torial low-pressure region, would follow the sun slowly, and the major
centers of pressure would move in the same direction. The polar high
would decrease in intensity during the summer as a result of the large
amount of heat that is received during the polar day. At the other end
of the globe's -axis, the long, cold, polar night would intensify the high-
pressure region there. This describes to a large extent the seasonal
changes which actually occur in the southern hemisphere, since there
are no large land masses there to modify appreciably the general cir-
culation.

An anomaly exists in the southern hemisphere in that the air of the
Antarctic continent contains less oxygen than the air over the remainder
of the earth (sec section 5). This fact suggests that the exchange of
air between that continent and the surrounding area is small. The sub-
polar low-pressure trough of the southern hemisphere coincides closely
with the boundary of that continent. This trough, then, would be the
outer limit to a closed polar circulation. The air that moves slowly
across the isobars from the sub-tropical high must return to tropical
latitudes without penetrating this great polar cell.

The same conditions do not exist in the northern hemisphere. It
appears likely, then, that the southward-moving outbreaks of air across
the continents, which extend almost to the pole, permit a complete mix-
ing of the air near the north pole and the air from more southerly regions.

In the northern hemisphere the proportion of land and sea is more
nearly equal, and both land and water masses extend from the tropical
regions to the Arctic. There is a difference in the heating effect of the
sun on sea and on land. Both the specific heat and the conductivity
of soil are lower than for water, and as a result the heat absorbed pro-
duces a greater increase in temperature in the surface layers of soil than
in those of water. Thus in summer the water is cool relative to the land.
The air above the water is cooled, and the high-pressure areas over the

214 GENERAL CIRCULATION [Chap. 12

oceans are intensified in this manner. Over the land the heat gained
by the air from the warm surface causes the air to expand, so that there
is a smaller mass of air over a given area, resulting in a decrease in the
surface pressure. For these reasons, the sub-tropical high-pressure belt
does not extend continuously around the earth in summer, but becomes
indistinct over the heated continents, and more extensive and pro-
nounced over the ocean areas. Meanwhile the sub-polar lows become
prominent and develop over the land areas. The low over Asia is
especially marked. The circulation about this low, moving counter-
clockwise, gives the monsoon, an extensive flow of air from the Indian
Ocean over India, and from the Pacific over China (see Fig. 5, section 3).
The low over North America is not quite so prominent; yet the mean
winds for the summer months show a definite monsoonal effect.

During the winter the difference between land and sea is reversed.
The snow surface over the northern portions of the continents reflects
most of the incident solar radiation since the albedo of snow is high
(section 28). On the other hand, the loss of heat by long- wave radia-
tion from the earth's surface is pronounced, and since the conductivity
of snow is very small, a large fall in temperature of the snow surface is
the result. The air just above is in turn cooled. (Note the crowding
of the isotherms over the land areas in winter, Fig. 2, section 1.) When
the surface water loses heat by long-wave radiation, it is replaced by
warmer waters from below, and the oceans remain relatively warm.
As a result high-pressure areas develop over the continents and the
sub-tropical highs become less extensive over the oceans. In the region
of 60 N the sub-polar low becomes prominent over the oceans, appearing
as the semi-permanent Aleutian and Icelandic lows during the winter
(see Fig. 6, section 3). The monsoon circulation becomes predominant
over Asia, but now the air motion is clockwise about an extensive high-
pressure system, and the winds blow from the land over the water,
giving cold, dry winds during the winter season.

Aloft these features are less noticeable. The continental highs and
the polar high of winter have disappeared at the 2 km level, but the sub-
tropical high may still be detected. The general trend of the isobars
is in concentric circles about the low-pressure area at the pole. During
the summer the continental lows extend to greater heights than their
winter counterparts, the anticyclones, but have disappeared by 4 km.
Above that height the polar low and the remnants of the sub-tropical
highs are the chief features of the pressure distribution.

74. Air Masses and Their Source Regions. In the center of a high-
pressure system, such as the Azores high, the air subsides slowly, and
the winds are light. The underlying surface is water having a nearly

Sec. 74] AIR MASSES AND THEIR SOURCE REGIONS 215

uniform temperature over extensive areas. As the air subsides, that
near the surface will assume a uniformity of properties as it comes in
contact with the water surface below. These properties will be trans-
ferred upward through turbulent and convective eddies. The result
is that the air throughout the anticyclone is nearly uniform in any
horizontal layer, and these properties change in all vertical columns at
about the same rate.

When, after having acquired the properties of a region, part of the
air flows to other regions, it retains for a certain period of time its former
characteristics. It can be identified, and its trajectory can be inferred
from these. This traveling anticyclone, or portion of one, is called an
air mass, and the region from which it derived its properties is called its
source region. Adjectives are added to the term air mass to describe
some of the most prominent characteristics of each. Thus the terms
equatorial, maritime, and polar are a few of those used to describe
different types.

As suggested above, the traveling air mass must be extensive, with
uniform or nearly uniform properties in the horizontal. If a true air
mass is to develop, the underlying surface must be uniform, and the air
must be stagnant over it long enough to assume the characteristics of
the source region.

The principal source regions are the areas where pronounced anti-
cyclones develop, as described in sections 3 and 73. The source region
for a tropical maritime air mass in the northern hemisphere is either the
Atlantic or Pacific Ocean, in the vicinity of latitude 30 N, where the
sub-tropical highs occur. That for a tropical continental air mass is the
Sahara Desert. An Arctic type of air develops over the Arctic and
Greenland high-pressure regions. In winter, polar continental air
masses develop in high-pressure systems over Siberia and northwest
Canada.

An area of the earth's surface may be so large and so uniform in its
properties that a body of air traveling over it becomes modified until it
takes on definite properties characteristic of the surface. It is also
described as an air mass. Such regions having sufficiently uniform
characteristics are located in the north Pacific and northeast Atlantic
oceans. Moving over these regions, a body of air may become modified
to a point where it can be described as a polar maritime air mass. The
same process takes place over polar continental regions during the
summer, producing summer polar continental air masses. A third region
of modification is on the southern edge of the sub-tropical high, where
air masses to which the term " equatorial " is applied form.

Chapter 15 describes the air masses from the several source regions

216 GENERAL CIRCULATION [Chap. 12

that affect North America and Europe. The air masses of the southern
hemisphere have not been the subject of study to the same degree as
those of the northern hemisphere. Nevertheless, similar considerations
would permit some of the conclusions about source regions and air mass
types given in this section and in Chapter 15 to be applied to the region
south of the equator.

BIBLIOGRAPHY

Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapter 19.
Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941.

Chapter 13.
Petterssen, S., Weather Analysis and Forecasting, New York, McGraw-Hill Book Co.,

1940. Sections 63-72.
Problems of Modern Meteorology, London, Royal Meteorological Society, 1934.

Number 2.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 2

(1936), Chapter 7.
Bjerknes, V., and collaborators. Physikalische Hydrodynamik, Berlin, Verlag.

Julius Springer, 1933. Page 680.
Rossby, C.-G., in Climate and Man, 1941 Yearbook of Agriculture, Washington,

U. S. Department of Agriculture, 1941. Pages 599-654.

CHAPTER 13
TEMPERATURE AND HUMIDITY IN THE ATMOSPHERE

75. The Temperature of the Air at the Earth's Surface. Any small
mass of air, referred to for convenience as a particle of air, is subject
to physical processes which tend to change its properties. It is desira-
ble to study in detail these different processes and the changes resulting
from them. The first property to be studied is the temperature of a
particle of air near the surface of the earth.

Radiation is one process which causes wide variations in the surface
air temperature. There are two ways in which radiation may act to
cause a variation in properties. During the daytime the short-wave
radiation from the sun, sometimes referred to as insolation, passes
through the earth's atmosphere with very little absorption, as indicated
in section 28, and thus without appreciably warming it. The surface
of the earth absorbs a large fraction of this radiation, and radiates it as
long-wave radiation. The incoming short-wave radiation, especially
on a clear day, is greater than the outgoing long-wave radiation, so the
temperature of the ground increases. Consequently the temperature
of the air in contact with the ground also increases. During the night,
on the other hand, there is no insolation, and when the sky is clear the
ground cools by means of long-wave radiation to outer space, as ex-
plained in section 29. The air in the lower layers which has been in
contact with the ground will therefore cool.

Insolation and nocturnal radiation may produce marked variations.
Fig. 76 shows typical variations of temperature with height, i.e., lapse
rates, during the day and night. The type of lapse rate usually found
during the daytime is shown by curve (a) of Fig. 76. There is a more
or less steady decrease in temperature with height, which on the average
amounts to 6 or 7 C per km. During a cloudless night, when the wind
velocity is not high, the surface of the ground cools by means of noc-
turnal radiation, and the temperature of the air just above also decreases.
This cooling may be sufficient to produce near the surface an isothermal
layer, i.e., a layer in which there is no variation in temperature with
height, or an inversion, a condition where temperature increases with
height. An inversion in the lower levels is shown by curve (6) of
Fig. 76. The degree of inversion which may develop in any given

217

218

TEMPERATURE AND HUMIDITY

[Chap. IS

vtt

region depends on a large number of local features and conditions.

During the night, therefore, the temperature of the surface air may show

wide variations in a given air
mass. In these ways radiation
causes the temperature of the
surface air to change.

The processes of evaporation
and condensation sometimes
cause a change in temperature.
Evaporation of water from
water or moist earth surfaces
requires heat. If the source of
this heat is the water or ground,
the evaporation will not cause
any appreciable change in the
temperature of the air. This is
the usual situation because of

FIG. 76. Temperature variations with
height of the air near the earth's surface.

the large thermal capacity of
the water or moist ground
compared with that of air. If
the supply of heat, though,
were to come from the air, cool-
Conversely, condensation supplies heat. Thus if

ing would take place,
the air cools, as, for example, by radiation, to the point where some of
the water vapor condenses, the latent heat of condensation is released.
Thereafter the rate of decrease of temperature is retarded.

At the earth's surface mixing of air of different temperatures occurs
as, for example, in coastal regions. The resulting temperature will be
a weighted average of the two original temperatures.

76. Temperature in the Free Air. Above the lowest layers the direct
influence of the earth's surface on the temperature is not present.
Insolation, which passes with little absorption through the atmosphere,
according to section 28, imparts to it a very small amount of energy.
As indicated in section 29, the water vapor in any given layer of air
absorbs long-wave radiation from the earth's surface and from other
layers. The water vapor in the layer also radiates energy, and since
under average conditions the energy radiated is always greater than
that absorbed, a decrease in temperature results.

This exchange of energy is illustrated by the diagrams of Fig. 77.
These give, considered separately and then together, the rate of heating
by the absorption of solar radiation, and the rate of cooling resulting
from terrestrial radiation processes as computed by Moller for the air

Sec. 76]

TEMPERATURE IN THE FREE AIR

219

over Lindenberg. From Fig. 77a it is seen that with clear skies the
net result is a decrease in temperature of approximately 1 C per day.
With cloudy skies, condensation of water vapor supplies heat to the
atmosphere, the effect of which is allowed for in the balance illustrated
in Fig. 776. Here, too, the net result is a gradual cooling except in the
lowest 3 km. More recent data on the absorption of terrestrial radia-
tion in the atmosphere will change the numerical values on which these

2 5

-2

-1

(C day"')

(a)

-2

-1

A! (C day" 1 )

(b)

FIG. 77. The loss and gain in energy in the free atmosphere (a) with clear skies,
(b) with cloudy skies. (After Moller.)

figures are based. Nevertheless the final result will differ only in degree
from that illustrated. The net cooling resulting from radiation in the
atmosphere is a function of the water vapor content. In the tropo-
sphere when the sky is clear, it varies from 2 to 3 C per day in air from
polar regions to about 1 C per day in air from the tropics. The effect
of insolation during the day and terrestrial radiation at night is to
produce a large diurnal variation of temperature of the earth's surface.
The air from the surface to about 1 km experiences a similar variation,
since heat is transported in the vertical by turbulence. During the
day, then, there is an increase in temperature in the lower layers and a
slight decrease at higher levels. During the night the temperature at
all levels decreases, but the cooling at upper levels is small compared
with that just above the surface of the earth. The gradual cooling by
radiation at the upper levels is in part compensated for by a transfer
of energy from the surface layers by means of rising currents of air.

In the free atmosphere evaporation takes place from water droplets.
The heat content of these is insufficient to supply the latent heat of
vaporization required. This heat comes from the surrounding air,
cooling it. The opposite process of condensation, which takes place in
the formation of clouds, heats the air by releasing the latent heat of
condensation.

In the free atmosphere the air particles do not always remain at the

220 TEMPERATURE AND HUMIDITY [Chap. 13

same level. If the change in level occurs without transfer of energy to
or from the environment, i.e., adiabatically, the ascent and descent of
air cause large changes in free air temperature. As long as the air
remains unsaturated, the variation of temperature with pressure is
given by equation 13-2

rnt I r
= 7 l

where T = the temperature of the air at pressure p.

TO the temperature of the air at initial pressure PQ.
K = AR/Cp = 0.288.

The temperature variation with ascent or descent given by a slightly
modified form of this formula is known as the dry adiabatic lapse rate, F,
and amounts to approximately 1 C per 100 m, or 5.4 F per 1000 ft, as
shown in section 14.

If the ascent of the air continues beyond the saturation point, water
vapor condenses, and the latent heat of condensation is taken up by the
air, warming it. The temperature of the ascending air then decreases
at the saturated adiabatic lapse rate, r', which is discussed in section 20.
The saturated adiabatic lapse rate is not constant, as the dry adiabatic
lapse rate is, but shows wide variations. At high temperatures, where
the water vapor content of saturated air is great, the saturated adiabatic
lapse rate is much less than the dry adiabatic, being only about 0.4 C
per 100 m. The water vapor content is small at low temperatures, and
consequently the amount of latent heat released with -ascent is small.
The saturated adiabatic lapse rate is therefore very nearly equal to the
dry adiabatic under such conditions. Actual values at various pres-
sures and temperatures may be seen in the table in section 20.

One phenomenon of nature can be explained by the difference between
the dry and the saturated adiabatic lapse rates. When moist air flows
up the side of a mountain range, the temperature drops at the dry
adiabatic lapse rate until it becomes saturated, and after that at the
saturated adiabatic. After saturation, part of the water vapor which
condenses falls as rain or snow. When the air, which has lost much of
its moisture during ascent, subsides on the other side of the mountain,
the temperature rises at the dry adiabatic lapse rate after the clouds
have evaporated. Because of the difference between the two lapse
rates, the air on the lee side of the mountain will be warmer than it was
at the same level on the windward side. The extent of the increase
will depend on the temperature and moisture content of the air initially,
the height to which it is lifted, and the fraction of liquid water precipi-

Sec. 77] CONSERVATIVE PROPERTIES 221

tated. With a lift of 8000 ft the increase will amount to about 10 F
under average conditions. This phenomenon occurs on the southern
side of the Alps, producing the Fohn winds of northern Italy, and on
the eastern side of the Canadian Rockies, producing the Chinook winds
which warm the prairie provinces. Less famous examples of the same
phenomenon are found in other regions.

Turbulent mixing in the vertical may produce large temperature
changes in an air mass. According to (53-11), the net upward flux of
heat across a horizontal unit area in unit time is

-Kpc p (T - a),

where K = a quantity denoting the intensity of the turbulent motion,

known as the coefficient of eddy diffusivity.
p = the density of the air.

c p = the specific heat of dry air at constant pressure,
a = the lapse rate of the environment.

It can be seen from this formula that the flow of heat which is due
to eddy motion is downward if a < F, and upward if a > T. It there-
fore follows that the effect of vertical mixing is always to make the
actual lapse rate more closely approach the dry adiabatic if the air is
unsaturated, or the saturated adiabatic if the air is saturated. The
only case where vertical mixing causes no change in temperature is that
in which the actual lapse rate is equal to the dry adiabatic. For all
other lapse rates the free air temperature changes with vertical mixing
processes. As long as the air remains unsaturated the changes in
water vapor content of the air mass resulting from mixing do not appre-
ciably affect the temperature.

77. Conservative and Representative Properties. In the last two
sections the physical processes of radiation, evaporation and condensa-
tion, ascent and descent, and turbulent mixing have been studied with
respect to air temperatures. In general, as a result of these processes,
the temperature changes. However, there are exceptions to this gen-
eralization. For instance, if the heat of vaporization during evapora-
tion is supplied by an outside source, the temperature remains constant.

The term conservative is applied to any property which remains con-
stant when the body is acted upon by some process. Thus the tem-
perature of an air particle is conservative with respect to evaporation
when the heat of vaporization is supplied by an outside source. In the
air the magnitude of all properties changes slowly, so conservatism in
the air is relative. If the magnitude of the property remains constant
within the range of error of the observation for a period of 12 h at the

222 TEMPERATURE AND HUMIDITY [Chap. 13

earth's surface, or from 24 to 48 h in the free atmosphere, it is said to
be conservative.

To be of value a property must be not only conservative but also
representative. A representative property is one which characterizes
an extensive region of the atmosphere adjacent to the point of observa-
tion. For example, the temperature at any level in a uniform air mass
is representative. On the other hand, the temperature at the surface
below a nocturnal inversion varies with the cloudiness and wind and so
cannot be considered a representative property.

Conservative and representative properties are useful in analyzing
and classifying various masses of air. In the succeeding sections of
this chapter the different properties of an air particle will be defined and
discussed and their conservatism or lack of it determined under the
influences mentioned above.

78. Lapse Rate and Stability. It was indicated in sections 14 and
20 that the lapse rate of an air column and its stability are closely
related. It will be shown in Chapter 15 that each air mass has a repre-
sentative lapse rate. When adiabatic ascent or descent of a layer occurs,
the lapse rate changes but not rapidly, as shown in section 16. Simi-
larly, since cooling by radiation takes place throughout the column, the
lapse rate does not change appreciably above the surface layers. The
intensity of turbulence varies inversely with the stability of the air, so
that stable lapse rates in the free air change only very slowly, since the
amount of turbulence is small. In an unstable layer the turbulence is
marked and tends to change the equilibrium to the neutral type. The
most rapid change takes place near the surface of the earth. In general,
since the lapse rate changes but slowly, it is relatively conservative, and
so is frequently used in identifying air masses.

79. Diurnal Temperature Variations. In Fig. 78, curve (a) repre-
sents the temperature variation with height in a column of air which
is nearly in neutral equilibrium, and curve (6) represents that in a column
of air which has great stability. The dotted lines represent the dry
adiabatic lapse rate. If these two curves represent the lapse rates in
two different air masses in the early morning, then after the sun rises
insolation will increase the temperature of the earth's surface and hence
that of the air just above. When the lapse rate becomes greater than
the dry adiabatic, the air becomes unstable (section 14) and vertical
currents develop which readjust the temperature distribution. Thus
as long as the surface temperature is increasing, the lower layers will
have a lapse rate close to the dry adiabatic.

Assume that the surface temperature increases by the same amount
for the two columns. The limiting dry adiabatic line will intersect the

Sec. 79]

DIURNAL TEMPERATURE VARIATIONS

223

original curve at a higher level for the unstable column than for the
stable column. A thicker layer of air must then be heated in the un-
stable column for the same increase of surface temperature, requiring
the input of more energy. Or, if equal amounts of energy are available,
the surface temperature will rise farther in the stable column than in
the unstable column, but the increase will extend to a greater height in

FIG. 78. Temperature increase through insolation.

the unstable column of air. An additional increase of the surface
temperature, after the early morning rise, requires the heating of a thick
layer and so the input of a large amount of heat. Therefore the maxi-
mum temperature for the day varies little from day to day in the same
air mass when the sky is clear, but differs from one air mass to another.
The maximum temperature is then a representative and conservative
property.

The cooling at night is decreased by a cloud cover, and the daily
range of temperature is consequently small. But even with clear skies,
the minimum temperature varies with the location. A thermometer in
a hollow into which cold air from nearby slopes drains readily, and in
which the wind cannot keep the air stirred, will register a lower minimum
temperature than one in a more exposed location. Thus the minimum
temperature and so the range of temperature are neither representative
nor conservative properties of an air mass, but they vary with local
conditions.

224 TEMPERATURE AND HUMIDITY [Chap. IS

80. Potential Temperature. The potential temperature of a mass
of air is defined as the temperature which that air would attain if
brought dry adiabatically to a standard pressure, usually 1000 mb.
By referring to equation 13-5, it is seen that the temperature correspond-
ing to the standard pressure 1000 mb, i.e., the potential temperature 0,
is given by

6 =

The equation shows that the potential temperature of a given mass
of air is a function of its temperature and pressure. It is therefore not
conservative with respect to radiation, evaporation (except when the
heat of vaporization comes from an external source) and condensation,
and turbulent mixing (heat) processes, since the temperature of the
air is changed by all these. But dry adiabatic ascent and descent are
reversible processes, and hence it follows that, as long as there is neither
evaporation nor condensation, the temperature of a specified air particle
brought adiabatically to 1000 mb will always be the same, no matter
how much or how often the air ascends or descends before reaching that
pressure. If that condition is fulfilled, therefore, ascent and descent
produce no changes in the potential temperature (see section 13).
Any change in water vapor content resulting from turbulent mixing will
produce no significant variation in the potential temperature.

81. Water Vapor Content. There are four methods by which the
amount of water vapor in the atmosphere may be expressed.

Absolute humidity, a, is defined as the mass of water vapor per unit
volume of air. Its value is, according to (10-0),

a = RT gm P01 Cm3

where = 0.622.

e = the vapor pressure in dynes per cm 2 .

If e is expressed in millibars, then

a = 217 ^gm perm 3 (81- 1)

The abolute humidity is not at all conservative, since all the processes
listed in section 77 may produce large variations in either e or T. This
measure of atmospheric humidity has therefore a limited use in meteor-
ology.

Sec. 81] WATER VAPOR CONTENT 225

Specific humidity, s, may be defined as the mass of water vapor per
unit mass of moist air. From equation 10-10, it is seen that

e
s - - - gm per gm

p - (1 - c)e
or

Since the specific humidity is a ratio between two masses, neither
of which is changed by a change in temperature, it is conservative for
those processes involving a variation in temperature. It is not, though,
conservative for those changes involving a change in water vapor con-
tent, i.e., for evaporation and condensation and turbulent mixing
processes. Specific humidity thus finds a wide use in meteorological
analysis.

Humidity mixing ratio, x, is the mass of water vapor per unit mass
of dry air. It can be expressed by the equation 10-8

e
x = e - gm per gm

p e

e
x = 022 --- gm per kg

p e

The numerical difference between the values of the specific humidity
and mixing ratio for any given particle of air is very small, and in
practice the two are used interchangeably. The values of these ele-
ments are given with sufficient accuracy by the formula 10-11

x = 5 = c - gm per gm
P

It follows from the foregoing, of course, that the mixing ratio has
the same degree of conservatism and the same range of use as the specific
humidity.

Relative humidity, /, is defined as the ratio of the actual vapor pres-
sure to the saturation vapor pressure at the same temperature. Its
value is given by the equation 10-7,

226 TEMPERATURE AND HUMIDITY [Chap. 13

where e 8 is the saturation vapor pressure at the given temperature. If
the relative humidity is expressed as a percentage, the equation takes
the form

/=100-

By combining (10-7), (10-11), and (10-12), it is seen that the relative
humidity can also be expressed as the ratio of the actual specific humid-
ity to the specific humidity of air saturated at the same temperature.
The same relation holds if the mixing ratio is used instead of the specific
humidity. Thus, the relative humidity may also be computed from
the equation

/ = 100 - = 100 - (81-2)

s s x 8

where s 8 and x a indicate the values required for saturation.

Since the saturation vapor pressure is a function of temperature as
shown by (21-4), and the actual vapor pressure may undergo changes
as a result of evaporation, and of turbulent mixing processes, it follows
that the relative humidity varies with all the processes considered in
section 77. It is therefore not a conservative element, but nevertheless
it has a wide range of use in many phasos of meteorological theory and
practice.

82. Dew Point. The dew point of a particle of air is the temperature
to which the air must be cooled at constant pressure and constant water
vapor content in order to become sat united. KithcT the saturation
vapor pressure or the amount of water vapor per unit volume required
for saturation, i.e., the saturation value of the absolute humidity, is a
single-valued function of the temperature as given by the equation 21-4

H.573-*"

f, = 6.11 X 10 7 mb

or by (81-1). Therefore the dew point is a single-valued function of
the amount of water vapor per unit volume. For that reason, then, any
process changing the amount of water vapor per unit volume changes
the dew point. Thus the dew point of a particle of air is not conservative
for evaporation and condensation, nor for turbulent mixing with air of
a different water vapor content. It is, though, conservative for radia-
tional heating or cooling to the point of saturation, or for turbulent
mixing when the mixing masses of air differ only in temperature.

Changes of pressure cause variations in the vapor content of a given
volume since the volume varies inversely with the pressure at constant

Sec. 83} WET-BULB TEMPERATURE 227

temperature, as shown by (7-6). Thus the dew point also changes.
With ascent the mass of water vapor per unit volume decreases, lowering
the dew point. This change with ascent is small, being about one-sixth
the change of temperature under dry adiabatic conditions. The varia-
tion of dew point with height in adiabatically ascending air is given
by (21-8).

Several types of thermodynamic charts used in the analysis of upper
air data are described in section 22. One of the most widely used of
these is the tephigram. This chart is shown in Fig. 17, section 22,
and a full-scale copy is provided at the back of the book. In the tophi-
gram the abscissa is temperature on a linear scale and the ordinal c is
entropy on a linear scale, which is equivalent, according to (18-7), to
potential temperature on a logarithmic scale. The path of a particle
of air moving dry adiabatically is then given by a horizontal line of the
diagram. Lines of equal pressure slope upward to the right, and lines
of equal saturation mixing ratio are dotted lines which are almost paral-
lel to the isotherms but slope upward to the left. A fifth set of linos
slopes upward to the left and gives the saturated adiabatic lapse rates.

On the tephigram the dew point of a particle of air can bo found by
noting the intersection of the constant -pressure line and the humidity
mixing ratio line corresponding to the moisture content of the air. Tho
temperature at this point of intersection is the dew point. For example,
from the full-scale tephigram it is seen that the dew point of air with
temperature 5 C, pressure 800 inb, and mixing ratio 2.5 gm is 9 C.
There is no change in the moisture content of air rising dry adiabatically,
and so the dew point will always lie on the same mixing ratio line. For
this reason those saturation mixing ratio linos are frequently called
dew-point linos. When the air has boon lifted until the temperature
and the dew point coincide, saturation is attained. The level at which
this occurs is called the condensation level, or the lifting condensation
level to distinguish it from another type of condensation level described
later.

83. Wet-Bulb Temperature. The wet-bulb temperature of a given
mass of air may be defined as the lowest temperature to which that
air may be cooled by evaporating water into it. In practice it is meas-
ured directly by means of a ventilated wet-bulb thermometer as indi-
cated in section 65. It is denoted by the letter T w .

It has been shown by Normand that, to a high degree of approxima-
tion, the dry adiabat through the dry-bulb temperature, the saturated
adiabat through the wet-bulb temperature, and the saturation mixing
ratio line through the dew-point temperature all meet at a point. The
proof of this proposition is given in section 24. This relationship is

228

TEMPERATURE AND HUMIDITY

[Chap. 18

shown on the tephigram given in Fig. 79 in which A represents the
dry-bulb, D represents the wet-bulb, and B represents the dew-point
temperature. The three lines intersect at C, the condensation level.
Thus if any two of these quantities and the pressure are known, the
other two may be obtained directly with the aid of an adiabatic chart
such as the tephigram.

FIG. 79. Determination of the wet-bulb temperature.

This relationship is useful in many ways. For instance, if the tem-
perature, pressure, and relative humidity of a mass of air are known,
the wet-bulb temperature may be obtained readily from the tephigram.
The saturation mixing ratio for the air is given by the saturation mixing
ratio line through the dry-bulb temperature A on the tephigram.
According to (81-2), the actual mixing ratio is found with sufficient
accuracy by multiplying the saturation mixing ratio by the relative
humidity expressed as a fraction. Proceed horizontally along the dry
adiabat from A until the mixing ratio line corresponding to the actual
mixing ratio is reached at C. Follow the saturated adiabat through
tliis point until it meets the pressure line through A at D. The tem-
perature at D is then the wet-bulb temperature of the air. For example,
if the particle of air has pressure 950 mb, temperature 3 C, and relative
humidity 60 per cent, the dew point is 4 C, and the wet-bulb tem-
perature is C. Because of the approximations in Normand's treat-
ment, the value found in this manner is not exactly the same as the

Sec. 83]

WET-BULB TEMPERATURE

229

actual reading of a wet-bulb thermometer in the air in question. How-
ever, the difference is so small that it may be considered negligible fgr
all practical purposes.

The wet-bulb temperature changes
if radiational cooling occurs. It
follows from Fig. 80 that if the dry-
bulb temperature decreases through
radiational cooling from A to E, f
the condensation level will descend *
from B to F. Remembering the
relationship between dry-bulb, wet-
bulb, and dew-point temperatures as
given by Nonnand, it can be seen
that the wet-bulb temperature will
decrease from C to G.

The relationship between the de-
crease in dry-bulb temperature with

radiational cooling and the corresponding decrease in wet-bulb tempera-
ture may be shown in the following manner.

In Fig. 80, EFD and ABD may be taken, with sufficient accuracy
for present purposes, as similar triangles. Then

FIG.

T *-

80. Change of wet-bulb tem-
perature through radiation.

FT) _
BD " AD

Triangles FDG and BDC are also similar, so that

*-R -

BD ~ CD

Thus

GD ED , CD - GD
- andso ^~

AD - ED
AD

or finally

~CD ^ AD
If a represents the angle BAD t then

sec a

(T Wl - T dl ) sec a (Ti - T dl ) sec a
where the subscript 1 refers to initial values, and the subscript 2 refers

230

TEMPERATURE AND HUMIDITY

[Chap. 13

to values after the radiational cooling has taken place.

T T ,
j. * j. (i

fji fw ___ i i / ni rjj \

* v>i * W 2 m rii \ 1 * 2/

1\ ~ l dl

Thus

(83-1)

The fraction (T Wl - T dl )/(T { - 7', A ) is a constant for any given
situation. The value of this fraction for surface pressures is approxi-
mately 0.4 at 20 ( !, 0.0 at ( !, and 0.9 at -20 C.

It follows from the definition given at the first of this section that the
wet-bulb temperature is invariant for processes of evaporation and

condensation. For if the evaporat-
ing is done in two or three stages, as
suggested in section 24, it can be
seen that the wet-bulb temperature
does not change during the process.
The air can still be cooled only to
the same temperature, so the wet
bulb must remain the same.

I'sing a method similar to that
above, tho change of the dew point

during evaporation can be related to

[ ~~"~" the change in temperature. Hefer-

>f dew punt with rillK t() F jj,. si, the initial tempera-
ture, condensation level, wet -bulb

h.i. 81. Cluin

ttvutmnition.

temperature, and dew point are denoted by ,1, It, (\ and /) respectively.
After the process of evaporation postulated has occurred, they are

denoted by K, l<\ (\
before*, it follows that

and so

(f n\spectively. Tsing similar triangles as

(ID
CD

EC
AC

AK
AC

If indicates angle RAD, then the increase in dew point resulting from
the evaporation is given by

(7 T /i - r (/| ) sec a (1\ - T 2 ) sec a

Wl - T dl ) sec

sec a

(83-2)

Sec. 83]

WET-BULB TEMPERATURE

231

The fraction is constant for any one situation. Ite value for surface
pressures is approximately 0.0 at 20 (', 1.5 at C\ and t> at -21) C\

Normand's treatment brings out the fact that the wet-bulb temper-
ature varies at the saturated adiabatic lajwe rate as air ascends or
descends. Thus if air originally at a pressure of KHH) mb ascends to
the condensation level, the wet-bulb temperature, as indicated in Fig. 79,
will decrease from D to C. At C the dry-bulb, wet-bulb, and dew-
point temperatures coincide, and with further ascent all three move*
along the same saturated adiabat, CK. For example, if air with
p = 800 mb, T = 10 (\ T {t , = 4 C\ T d = -2T ascends, it can be
seen from the tephigram that the condensation level is reached at

i
I

Fin. S2. Change of wot -hull i tem-
lirrature with change in moisture.

l <f in. 8,'J. ClmngpM in ilry-hiilh, wet-
hull >, and (icw-|M)int t<Mn|M'rature.M
with turbulent mixing.

a pressure of (>70 mb, and if ascent continues to a level where, the pres-
sure is 500 mb, the three element* are identical at this level, being
li).5 C. It is therefore* obvious that \vet-bulb temperature is not, a
conservative property in processes which involve* ascent and descent of
the air.

The result of temperature changes caused by turbulent mixing of
different air masses is exactly the same as that caused by radiational
heating and cooling, provided the actual moisture content does not
change with the mixing. The corresponding changes in wet-bulb tem-
perature are therefore also the same and may bc sieri from Fig. 80. Jf
there is a change in water vapor content with the turbulent mixing,
but no change in temperature, the type of variation is shown in Fig. 82.
A decrease in mixing ratio was the result of turbulent mixing, the dew
point decreasing from B\ to BZ, and the wet-bulb temperature decreas-
ing from DI to D 2 .

232 TEMPERATURE AND HUMIDITY [Chap. 13

Turbulent mixing usually, however, results in changes in both tem-
perature and moisture content. In Fig. 83 the dry-bulb temperature
increases from AI to A 2j and the dew point increases from B\ to 2 -
Under these conditions, the wet-bulb temperature increases from DI
to D 2 .

Wet-bulb temperatures may be used very conveniently in conjunc-
tion with dry-bulb temperatures in the analysis of upper air data, as on
the tephigram. The advantages of the use of this element are discussed
in the next chapter.

84. Wet-Bulb Potential Temperature. The wet-bulb potential
temperature of an air mass, denoted by O w , is defined as the wet-bulb
temperature of that air when brought adiabatically to a standard pres-
sure, usually 1000 mb (see section 24).

It was indicated in the previous section that the wet-bulb tempera-
ture of ascending or descending air changes at the saturated adiabatic
lapse rate. The wet-bulb potential temperature is therefore obtained
by noting the temperature at the point of intersection of the saturated
adiabat through the wet-bulb temperature with the 1000-mb line. The
various saturated adiabat s on a tephigram are frequently specified by
their corresponding wet-bulb potential temperatures, in the manner
shown on the full-scale tephigram. For example, it may be found with
the aid of the tephigram that air with p = 700 mb, T = 3 C, and
/ = 44 per cent has a wet-bulb potential temperature of 14 (\

It follows by definition that wet-bulb potential temperature is con-
servative for processes of ascent and descent, even if condensation
occurs during ascent. Like the wet-bulb temperature, it is also conserva-
tive for evaporation and condensation. It is not conservative for
processes of radiation and transfer of heat and water vapor by turbulent
mixing. These latter, however, are usually of secondary importance
in the free air above the region of influence of surface friction. Thus
wet-bulb potential temperature is one of the most conservative elements
which may be used in the analysis of upper air data.

85. Equivalent and Equivalent Potential Temperatures. According
to section 25, the equivalent temperature T is the temperature of
absolutely dry air whose wet-bulb temperature is T w .

There are two methods of defining the equivalent temperature. The
first defines it as the temperature attained if all the water vapor in the
air is condensed, and the latent heat of condensation so released is
added to the air, the whole process being carried out at constant pres-
sure. The equivalent temperature is then given by the equation 25-2.

T = T

Sec. 85} EQUIVALENT TEMPERATURE 233

where L w = the latent heat of condensation at temperature T w .
x sw = the saturation mixing ratio at T w .

It can be seen that the second term on the right-hand side of the
equation represents the increase in temperature resulting from the
release of the latent heat of condensation.

It may also be defined as the temperature attained by a mass of air
which ascends until all the moisture in the air condenses and is precipi-
tated, and which then descends dry adiabatically to the original pres-
sure. Thus in Fig. 84, air with dry-bulb temperature A and wet-bulb

T -

Fici. 84. Equivalent, equivalent potential, and wet-bulb potential temperatures.

temperature D ascends to the condensation level C. The air continues
to ascend to F, the moisture condensing and being precipitated during
the ascent. All the water vapor has been removed by the time the
air reaches F, and it then descends dry adiabatically to the original
pressure at E. The temperature at this latter point is then the equiv-
alent temperature.

These two methods of definition give values of T e which are ordinarily
very nearly the same. For example, consider air at 900 mb pressure,
with dry-bulb temperature 4 C, and relative humidity 80 per cent.
Substituting appropriate values in the equation 25-2 gives

T e = 15| C approximately

By using the other definition, and with the aid of the tephigram, the
value obtained is

T e = 16 C approximately

Different values are obtained by the two definitions because in one the
removal of the moisture is assumed to be carried out at constant pres-

234 TEMPERATURE AND HUMIDITY [Chap. 13

sure, whereas in the other the pressure arid temperature vary greatly
during the process.

Jt can be seen from Fig. 84 that the saturated adiabat through the
wet-bulb temperature is asymptotic to the dry adiabat through the
equivalent temperature, i.e., the equivalent temperature is a single-
valued function of the wet-bulb temperature. It therefore follows
that the equivalent temperature 1 is conservative or non-conservative in
exactly the same sense as the wet -bulb temperature is. Thus it is con-
servative for evaporation and condensation, but not for the other
processes discussed in section 77.

The; equivalent potential temperature. O c (section 25) erf an air mass may
be defined as the equivalent temperature of that, air if brought adiabati-
cally to a standard pressure, usually 1000 nib.

There are several ways in which this property may be defined. The
first defines it as the temperature; attained if the; air is brought dry
adiabatically to a pressure of 1000 mb, then all the; water vapor in the
air is condensed, and the latent heat released in the process is added to
the air, raising its temperature. From equation 13-5 it can be seen that
the equivalent potential tempe,rature may also be given by the equa-
lion

/1000V
Be = T e ( - )
\ P /

The value of O e obtained will depend on which value of T e is used. The
values of 9 e obtained by using the* two definitions of T e arc very nearly

A value of 6 e may readily be; found with the aid of the tephigram.
As shown in Fig. 84, it is obtained by descending dry adiabatically from
K to the* 1000-mb line at (}. In the example given, using T G = 16C,
the value of e is 25(-.

It will be remembered that the value of the wet-bulb potential temper-
ature may be obtained by noting the temperature at the point of inter-
section of the saturatcel adiabat through the we^t-bulb temperature
with the 1000-mb line, indicated by // in Fig. 84. Thus, since FH
and FG are asymptotic, O e is a single-valued function e>f 0,,,, and 6 e
exhibits exactly the same degree of conservatism as d w . 9 C is therefore
conservative for proce\sses of ascent and descent, and evaporation anel
condensation, but not for radiation and turbulent mixing.

In actual practice Q w is more convenient to use than 0,.. The former
may be obtained from any adiabat ic chart, whereas O f cannot be found
at high temperatures if the pressure lines on the chart do not extend
to values lower than 400 or 500 mb. Furthermore, W , values may be

BIBLIOGRAPHY

235

Obtained directly if the wet-bulb curve is plotted on a tephigram on
which the saturated adiabats arc appropriately labeled.

86. Summary of Degree of Conservatism of Properties. The results
of the discussion of the comparative conservatism of properties outlined
in the previous sections are summarized in the following table.

DKCIIKK OK CONSERVATISM OF V \KIOUS PKOPNKTIKS

^xConscrvative for

^X. Processes

evapora-
tion and

Ascent

Turbulent Mixing

^X. of

Radiation

Condensa-

and

^x.

tion

Descent

Heat

Water

Property ^X.

Vapor

Dry-hull) temperature

Yes 1

Absolute humidity

Specific humidity

Yes 1

Mixing ratio

Yes 1

Relative humidity

Potential temperature

Yes 1

Dew-point temperature

Yes 1

Wet-bulh temperature

Yes

Wet-bulb potential

Yes

temperature

.Equivalent tempera-

Yes

ture

Equivalent potential

Yes

temperature

1 As long as condensation does not occur.

PROBLEMS AND EXERCISES

1. A particle of air has a mixing ratio of 3 gin per kg. By moans of the tephi-
gram, determine the difference between T and T e . How much variation is there
in this value for different original temperatures and pressures? Repeat the process
for 1.5 gin per kg. How can the equivalent temperature of a particle of air, T ----
27 C, p = 950 nib, x = 15 gm per kg, be evaluated using the results obtained?

2. A mass of air with temperature T becomes stationary over a water surface of
temperature r l\. What changes will take place in T and T w before equilibrium is
reached? Consider this situation both when 1\<T W and when T W <T\ <T.

3. Which of the properties of the surface air mentioned in the summary are con-
servative as air moves over a dry surface which has a temperature higher than that of
the air?

BIBLIOGRAPHY

Byers, H. R., tiynoplic and Aeronautical Meteorology, New York, McGraw-Hill Book

Co., 1937. Chapter 1.
Petterssen, S., Weather Analysis and Forecasting, New York, McGraw-Hill Book Co.,

1940. Chapter 1.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 2

(1936), Chapters 4, 5.

CHAPTER 14
STABILITY AND INSTABILITY

87. General Considerations. The conditions for stability of air are
the same as those for a solid body. To illustrate these conditions, con-
sider the result if a force is applied to a body. If the body is in stable
equilibrium, it will return to its original position when the force is
released; if in unstable equilibrium, it will move farther away from its
original position; and if in neutral equilibrium, it will not change its
position after the force is released. Thus a cube resting on a horizontal
surface is stable, a cone balanced on its point is unstable, and a sphere
on a horizontal surface is in neutral equilibrium. The complexity
arising when the stability of air is considered is due partly to the com-
pressibility of the gas, and partly to the variable amount of water
vapor present in the atmosphere.

There are several assumptions involved in the treatment which is
to be given in the following sections. One of the most important of
these is the assumption that an ascending or descending mass of air
may be thermally segregated from the surrounding atmosphere, so that
there is no transfer of heat from one to the other. This assumption
represents only a rough approximation to what happens in the atmos-
phere. Actually, of course, the motion is not truly adiabatic, since
there must be some mixing between the ascending air and that air
constituting the environment. It can therefore be seen that the tem-
perature of vertically moving air does not change at exactly the dry or
saturated adiabatic lapse rate. Furthermore, it follows that this mixing
also produces changes in the actual lapse rate of the environment. No
suitable method of allowing for this mixing has yet been developed, and
so its effect cannot be taken into consideration directly. However,
the error introduced by neglecting this factor is not usually great.

88. Conditions Required for Stability. The conditions for stable,
unstable, or neutral equilibrium were found for dry air in section 14, for
unsaturated moist air in section 15, and for saturated air in section 20.
The results can be expressed briefly as follows. If the observed lapse
rate is less than the adiabatic lapse rate, the air is stable; if greater, the
air is unstable; and if equal to the adiabatic lapse rate, the air is neutral.
For dry and for moist unsaturated air until saturation is reached, the

236

Sec. 90] CONDITIONAL INSTABILITY 237

dry adiabatic lapse rate is to be used. For saturated air. the saturated
adiabatic lapse rate is to be used. The conditions previously derived
can be summarized as follows.

FOR DRY AIR, OR
MOIST, UNSATURATKD AIR FOR SATURATED Aiu

Stable equilibrium a < T a < F 7

Unstable equilibrium a > F a > F 7

Neutral equilibrium a = F a = F'

89. The Stability of Moist, Unsaturated Air. When the possibility
of saturation occurring in unsaturated air is considered, the question of
its stability is much more complicated than for either dry or saturated
air because the stability may depend not only on the vertical tempera-
ture distribution, but also on the vertical water vapor distribution.
When a particle of moist but unsaturated air is lifted in the atmosphere,
its lapse rate changes at the point of saturation. Before saturation
the temperature falls at the dry adiabatic lapse rate, and after satura-
tion the fall is at the saturated adiabatic lapse rate. Thus it is possible,
when the lapse rate of the environment lies between the two adiabatic
lapse rates, that a rising particle of air will be cooler than the environ-
ment for the first part of the ascent, but warmer than the environment
at a higher point. Thus air may be stable if it undergoes only a small
vertical displacement, but may become unstable for large vertical
displacements.

This suggests three main headings under which the stability of un-
saturated air may be discussed:

(a) a < I 1 ', absolute stability.

(6) a > F, absolute instability.

Since the least decrease of temperature with height in a particle
rising adiabatically is F 7 , then when a < F 7 the particle will never be
warmer than the environment, and the air can never become unstable.
Similarly, when a > F, the rising particle will always be warmer than
its environment, and the air will be absolutely unstable. When the
lapse rate is greater than the saturated adiabatic but less than the dry
adiabatic, the stability is said to be of the conditional type, for the
stability is conditional on the water vapor content. This type of
stability will be discussed in considerable detail in the following sections.

90. Conditional Instability. Upper air observations plotted on a
tephigram are shown in Fig. 85. Dry-bulb temperatures are shown by
the curve AHB, and corresponding wet-bulb temperatures are indi-

238

STABILITY AND INSTABILITY

[Chap. 14

cated by the broken curve CD. Consider now a particle of air at the
surface, with dry bulb A and wet bulb C, and assume that it is given an
upward displacement. The dry-bulb temperature decreases at the
dry adiabatic rate, and the wet-bulb temperature decreases at the
saturated adiabatic rate. The air becomes saturated on reaching E,
and its temperature drops at the saturated adiabatic rate with further
ascent. At point F the temperature of the ascending air is the same

8 1

FIG. 85. Conditional instability.

as that of the environment. In the ascent from the surface to F, the
rising air has, at each level, been colder than the surrounding air and
therefore denser. It is clear, then, that energy from some outside
source must have been supplied to move the air from the surface to F.
The construction of the tephigram and of adiabatic charts in general
is such that the amount of this energy is proportional to the area en-
closed on the lower side by the line (AEF in Fig. 85) traced out by the
rising air and on the upper side by the line (AKF in the figure) denoting
the environment (see section 22). This area is shown by vertical
hatching in Fig. 85. As the particle of air in question ascends to levels
above F, it will be warmer and therefore less dense than the surrounding
air. The ascent along the saturated adiabat will therefore continue
until the temperature of rising air and environment become equal again,
as at G. In this case, the energy necessary to move the mass of air
from F to G is supplied by the environment itself and is proportional
to the area FHGL, indicated by horizontal hatching in the figure.

Thus it can be seen that if the vertical displacement of the air by
some outside agency does not carry it to a height as great as F, the air

Sec. 91] STABLE TYPE OF CONDITIONAL INSTABILITY

239

will tend to sink back to its former position as soon as the displacing
force is removed. In other words, the air is stable for small displace-
ments. On the other hand, if the air is carried only a short distance
above F by outside forces, it then ascends to greater heights by means
of the energy supplied by the environment. Here the air is unstable
for displacements to heights above F, i.e., for large displacements.
Summarizing, it may be said that the air at the surface is stable for
small displacements, but unstable for large displacements. This is a
typical example of conditional instability.

FIG. 86. Stable type of conditional instability.

91. Stable Type of Conditional Instability. If one defines conditional
instability by saying that it exists if the actual lapse rate is intermediate
in value between the dry and saturated adiabatic rates, there is one
case included in this category in which instability cannot possibly
occur. The air in such a situation cannot therefore be considered in
any real sense to be conditionally unstable, since it is absolutely stable
for all displacements. It has been called by Normand the stable type
of conditional instability, although in rigorous terminology the adjec-
tive " conditional " would not be used at all. An example of this type
is given in Fig. 86. The curve ABC shows the dry-bulb temperatures
and the broken curve DEF gives the corresponding wet-bulb tempera-
tures. Now consider that a parcel of the surface air is given an upward
displacement. In the ascending air the dry-bulb temperature decreases
at the dry adiabatic rate, and the wet-bulb at the saturated adiabatic
rate, until the condensation level K is reached. Thereafter both de-
crease at the saturated adiabatic lapse rate. It can be seen from the

240

STABILITY AND INSTABILITY

[Chap. 14

figure that the temperature of the rising surface air is less than that of
the environment at all levels. The surface air is therefore stable for
all displacements. If the air at each level is considered in the same
manner, it can be seen that the air in the whole column is stable for all
displacements, both large and small. Inspection of the curves shows
that the air with dry-bulb temperature G and wet-bulb temperature E
comes nearest to actual instability. The air which most closely ap-
proaches instability in this manner is always that which in the lower
levels has the maximum wet-bulb potential temperature. It can
readily be shown that the air with temperature G has the maximum
value of Q w .

The criterion for deciding whether the stable type of conditional
instability is present or not is as follows : If no saturated adiabat through
the wet-bulb curve cuts the environment curve at a higher level, the
air is stable, even if a > F'.

<D
O

FIG. 87. Latent instability.

92. Latent Instability. Another type of conditional instability is
known as latent instability. This type is illustrated in Fig. 87. The
dry-bulb temperatures are represented by ABC, and the wet-bulb
temperatures by DEFG. It can be seen that the surface air, with dry
bulb A and wet bulb Z>, is stable, since the saturated adiabat through D
does not cut the environment 'curve at a higher level, and ascending air
will therefore always be cooler and denser than the surrounding air.
But consider air with dry bulb H and wet bulb E. If this air is dis-
placed to a height above K, in the manner outlined in section 90, it will
then be wanner and less dense than the surrounding air, and will con-

Sec. 92\

LATENT INSTABILITY

241

tinue to rise until its density becomes equal to that of the environment
at L. The energy which must be supplied from external sources is
proportional to area HJK, while the amount of energy derived from
the environment is proportional to area KBL, as shown in section 22.
Latent instability has been subdivided into two types:

(a) Real latent instability, area KBL > area H JK.

(b) Pseudo latent instability, area KBL < area HJK.

Where the area KBL is much greater than the area H JK, strong ascend-
ing currents will result from only a small input of energy from external
sources. Thunderstorms frequently accompany such strong ascending

o
o

Fia. 88. Layer having latent instability.

currents. However, in pseudo latent instability, the initial input of
energy required to bring the air to the point K is great. Such large
amounts of energy from outside sources are rarely available, so that the
air is, in effect, stable, and ascending currents rarely occur.

The presence or absence of latent instability is readily determined
from the dry-bulb and wet-bulb curves plotted on a tephigram. Such
curves are denoted by ACH and KGJ in Fig. 88. The saturated
adiabat CGE tangential to the dry-bulb curve at C gives the layer in
which latent instability exists. This layer, as can be seen from the
figure, is that which has wet-bulb temperatures given by the curve
EFG, E and G being the points of intersection of the tangential saturated
adiabat and the wet-bulb temperature curve below C. Thus particles
of air in the layer MN between the pressures corresponding to points
E and G are said to have latent instability. A paradoxical situation

242 STABILITY AND INSTABILITY [Chap. 14

exists in the portion ML of the layer M N, for particles of this layer are
absolutely stable for small displacements according to the criterion of
section 89; yet for large displacements the particles will realize latent
instability. The saturated adiabat DBF tangential to the wet-bulb
curve at F shows the layer which has liability, for it is in the layer BD
that energy will be released to force the rising particles to ascend further.

The determination of the boundary between the layer showing real
latent instability and that showing pscudo latent instability can be
made only by trial and error. It is seldom important, though, for the
greatest value to be derived from an analysis of the data comes from
determining the presence or absence of real latent instability in the
column, and so the probability of vertical currents.

93. Potential Instability. In the previous sections, one main type
of stability has been considered, that of a particle in an undisturbed
environment. Another main type, the stability of a particle of air
immersed in a layer of air which moves as a whole, will now be examined.
A particle of air might be described as a mass of air whose three dimen-
sions are of the same order of magnitude. In the atmosphere, the
dimensions of a particle are usually small, ranging up to lengths of
perhaps two miles. In the case of a layer of air, however, the vertical
extent of the mass is very much less than the horizontal extent. For
example, a layer might be half a mile in thickness but might extend for a
thousand miles in the horizontal. In the following treatment, the layer
of air does not ascend or descend through an undisturbed environment,
but the whole air mass, or a large section of it, ascends or descends.
The air above and below the layer therefore undergoes the same vertical
motion as the layer itself.

Consider a layer of air between 800 and 900 mb, whose dry-bulb
temperature varies from A to B, and whose wet-bulb temperature varies
from C to Dj as shown in Fig. 89. There is, of course, air both above
and below this layer, but in order to simplify the diagram its lapse rate
is not shown. The air is unsaturated, and since the lapse rate is less
than the dry adiabatic the air in this layer is stable. Now let the
whole air mass ascend until the air which originally extended from
900 to 800 mb (layer AB) is saturated. It can be seen from the
figure that the air in this example becomes saturated after an ascent of
200 mb. But the lapse rate is greater than the saturated adiabatic,
and since the air is saturated, the air in that layer will be absolutely
unstable a-s indicated in section 88. It is, however, not usual for the
air at each level in the layer to become saturated after an ascent through
the same pressure interval, such as 200 mb, as suggested in Fig. 89.
The relative humidity at the base of the layer is often considerably

Sec. 93]

POTENTIAL INSTABILITY

243

greater than that at the top, with the result that the air at the base
becomes saturated before that at the top. Thereafter, with further
ascent, the air in the lower portion of the layer cools at the saturated
adiabatic rate, while the drier air near the top cools at the greater dry
adiabatic rate. Hence the difference in temperature between bottom
and top will become greater, and so instability will develop. The
breakdown of the unstable layer will, in general, be accompanied by

en
o

FIG. 89. Potential instability.

strong vertical air movements in the layer and in the air just above and
below it. It can readily be seen from a study of several cases that if
the lapse rate of wet-bulb temperature in a given layer is greater than
the saturated adiabatic lapse rate, that layer will eventually become
unstable if it is lifted far enough. Thus a layer may be said to have
potential instability if the lapse rate of wet-bulb temperature in that
layer is greater than the saturated adiabatic lapse rate. This type is
sometimes referred to as convective instability.

Potential instability may also be realized by evaporation. If a
potentially unstable layer lies under a warm frontal surface, then the
evaporation from the falling rain drops will saturate the layer, cooling
the air at each level in the layer to the appropriate wet-bulb temper-
ature which will remain constant in the process. The lapse rate will
then be greater than the saturated adiabatic and so the layer will be
unstable.

As an example, consider the meteorological ascent in which the fol-
lowing values of pressure, temperature, and humidity were recorded.

244 STABILITY AND INSTABILITY [Chap. 14

Pressure 1000 900 800 700 600 mb

Temperature 15 12 6 2 -3 C

Relative humidity 62 51 58 11 15 per cent

It will be seen after plotting the dry- and wet-bulb temperatures on a
tephigram that there is potential instability in the layer from 700 to
800 mb. The air at 800 mb will become saturated after an ascent of
90 mb, while that at 700 mb must ascend 250 mb before reaching
saturation. The realization of the potential instability will then com-
mence in the lower part of the layer after an ascent of over 90 mb, but
the potential instability of the whole layer will not be realized until
it has ascended 250 mb.

94. The Relationship between Latent and Potential Instability. For
latent instability to be present, a saturated adiabat through a point
on the wet-bulb curve must intersect the dry-bulb curve at a higher
level. When such a situation exists, it is necessary that the slope of
the wet-bulb curve be greater than that of the saturated adiabat.
Hence potential instability must be present in part of the air column.

The reverse situation need not be true. The wet-bulb curve may
have, at times, a slope greater than that of the saturated adiabat, even
though the air column itself is stable. So the presence of potential
instability does not mean that latent instability is also necessarily
present, although it often is.

These statements can readily be verified if a number of examples are
plotted on the tephigram. For instance in the latter part of section 93
there is a numerical example in which potential instability is present,
but without latent instability.

95. The Development of Potential and Latent Instability. Using
again the criterion for potential instability, that the lapse rate of the
wet-bulb curve must be greater than the saturated adiabatic lapse rate,
it follows that any process raising the wet-bulb temperature in a lower
layer while lowering it in a higher layer favors the development of
potential instability. Any process raising the wet-bulb temperature
in a lower layer while lowering the dry-bulb temperature at higher levels
favors the development of latent instability.

It follows from section 83 that the wet-bulb temperature at lower
levels rises under the following circumstances.

(a) It rises when heat is added to the air near the surface on a sunny
afternoon through radiation and turbulent mixing.

(6) It always rises in the lower layers of an air mass which passes
from a land surface to a water surface whose temperature is greater
than the dry-bulb temperature of the air. If the temperature of the
water surface lies between the dry- and wet-bulb temperatures of the

Sec. 95] DEVELOPMENT OF INSTABILITY 245

air, the wet-bulb temperature of the air near the surface first decreases,
and then increases, becoming the same as that of the water surface when
saturation is reached. The foregoing statement holds only if the dry-
bulb temperature decreases to that of the water surface very rapidly,
and the process of evaporation from the surface which produces satura-
tion proceeds more slowly. The wet-bulb temperature is always higher
when the final, steady state is reached. If the surface temperature of
the water is less than the wet-bulb temperature of the air, the latter
always undergoes a decrease.

(c) There is an increase in wet-bulb temperature in the lower layers
of air moving over warm, moist soil, in exactly the same manner as if
the air were moving over a water surface.

(d) If there is turbulent mixing of the air mass in question with a
moister air mass of the same temperature there will be an increase in
wet-bulb temperature of the former.

The dry- and wet-bulb temperatures of air at upper levels are lowered
by means of advection of cold, dry air, usually from northern latitudes.
Thus it can be said that, in general, a southerly wind at lower levels
with a northerly one at upper levels favors the development of poten-
tial and latent instability, while reverse winds inhibit the development of
these types of instability. Even in the situation where the velocity
of a southward-moving current of air increases with altitude, potential
and latent instability may result, since such a wind field will increase
the lapse rate of an air column.

A second means of development of latent and potential instability is
through the action of radiational cooling. The top of a cloud layer
will lose the maximum amount of heat during the night, as shown in
section 30, lowering both the dry- and wet-bulb temperatures, which at
this point will be equal. This occurs frequently above the cloud bank
above a warm front, giving rise to vertical currents and thunderstorms
in the overrunning air at night.

There is another process which develops and increases latent instab-
ility, but not potential instability. This process is the ascent of a
whole air mass, as in flowing over an extensive mountain range. Refer-
ring to Fig. 90, the dry-bulb temperature before ascent is given by ABC,
while the corresponding wet-bulb curve is denoted by DEF. No
saturated adiabat through the wet-bulb curve cuts the environment
curve at a higher level, so that there is no latent instability present at
this stage. Now assume that the whole column has ascended 100 mb,
the dry-bulb temperature of the air originally at 1000 mb decreasing
from A to G, and the corresponding wet-bulb temperature decreasing
from D to K^ and similarly with the air at higher levels. The dry-bulb

246 STABILITY AND INSTABILITY [Chap. 14

curve after the ascent is given by the curve GHJ and the corresponding
wet-bulb curve by KLM. It can be seen that there is now real latent
instability present. A necessary condition for this latent instability to
develop is the initial presence of potential instability in the air column.
It can be seen from the figure that the potential instability has not yet
been realized by the ascent of the whole column through a pressure
interval of 100 mb. A further ascent of about 30 mb is necessary

FIG. 90. Development of latent instability where potential instability is present.

before the realization of this potential instability commences. Thus,
merely through ascent, latent instability in the column has developed.
Bodily ascent of an air mass occurs chiefly along mountain slopes and
frontal surfaces, and therefore latent instability resulting from ascent
develops most frequently in such air currents.

96. Summary of Stability and Instability of Moist, Unsaturated Air.
The classifications dealing with moist, unsaturated air, which have been
treated in the last few sections, can be summarized as follows.

(a) Absolute stability, a < T'.

(b) Absolute instability, a > T.

(1) Stable type of conditional instability: No saturated adiabat
through the wet-bulb curve intersects the environment curve
at a higher point.

BIBLIOGRAPHY 247

(2) Latent instability : A saturated adiabat through the wet-bulb
curve intersects the environment curve at a higher point.
(i) Pseudo latent instability: The energy released by the

moving particle is less than the energy necessary to

move it to a point of instability.

(ii) Real latent instability: The energy released by the mov-
ing particle is greater than the energy necessary to move
it to a point of instability.

(d) Potential instability: The lapse rate of the wet-bulb temperature
is greater than the saturated adiabatic lapse rate.

PROBLEMS AND EXERCISES

The following tables give the values of the pressure p, the temperature T, and the
mixing ratio x, for different meteorological ascents. After plotting the data on the
tephigram, determine the tyi>es of instability present and the layers in which each is
present.

mb
C
gm per kg

1. p

2. p
T

3. p
T

987
21
15.2

1011
16
8.1

979
19
13.7

936
21
15.6

955
12
7.0

921
18
13.0

863
16
13.2

909
7
7.0

805
11
9.8

825
14
12.3

864
5
6.0

684
4
5.3

741
10
9.8

834
9
5.7

570
4

2.3

680
6
7.1

755

6
3.8

457
-16
1.2

591
-1
6.0

719
6
4.2

C
gm

533 514
-6 -5
4.7 3.7

632 598
1
2.6 2.2

per kg

473
-8
2.2

mb
C
gm]

408
-16
1.4

per kg

4. Plot on a tephigram the dry- and wet-bulb curves for the data below. If the
layer is lifted 50 mb, plot the resulting dry- and wet-bulb curves. What changes in
stability have taken place as a result of the lifting?

p 1020 961 905 854 757 mb
T 15.8 15.7 12.9 10.7 7.0 C
x 9.9 9.6 8.3 6.9 4.0 gm per kg

BIBLIOGRAPHY

Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book

Co., 1937. Chapter 2.
Petterssen, S., Weather Analysis and Forecasting, New York, McGraw-Hill Book Co.,

1940. Sections 27-54.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 3

(1930), Chapter 7.
Normand, C. W. B., " On Instability from Water Vapor," Q. J. Roy. Met. Soc., 64,

47-36 (1938).

CHAPTER 15

CHARACTERISTIC PROPERTIES OF DIFFERENT AIR MASSES

97. Systems of Classification. As indicated in section 74, air which
remains for several days over a uniform surface acquires certain char-
acteristics typical of that region. An understanding of these character-
istics is useful since it helps explain the weather associated with the air
mass at the source region and as it leaves that region.

Air masses are identified by two different features of their source
region. The first distinction is made on the basis of the latitude of the
region. Thus one speaks of equatorial, tropical, polar, and Arctic air.
Equatorial air, denoted E, is very warm, moist, and unstable. Its
source region lies between the two trade wind belts. Tropical air, T,
has its source region in the sub-tropical high-pressure belts. In general
it is not as moist in the upper layers as equatorial air, but it is very
warm. Polar air, P, in spite of its name, has its source region in lati-
tudes from 45 to 70. The term Arctic, A, is applied to air which
comes from the polar high. It is very cold and stable in the winter,
with the temperatures increasing somewhat in the summer. In this
chapter equatorial and Arctic air will not be discussed. Equatorial
air is very similar to moist tropical air; Arctic air sometimes appears
over northern Europe, but rarely over northern Canada. In general the
differences between Arctic air and polar air are relatively unimportant.

The second differentiating feature is the type of underlying surface.
The two kinds of surface are maritime, M, and continental, C. A
third distinction is sometimes made by comparing the surface tem-
perature of the air mass after it leaves its source region and that of the
underlying surface. Thus W indicates that the air is warmer than the
surface, and so is becoming more stable, while K indicates that the air
is colder than the underlying surface, with instability developing. For
example, PM indicates polar maritime air, and A/ TV indicates tropical
maritime air that is moving over a relatively cold surface.

Another method of classification uses the same latitudinal differentia-
tion of tropical, polar, etc., but it uses more specific terms to indicate
the source regions. For example, the polar maritime air masses are
divided into polar Pacific and polar Atlantic, indicated by the letters
PP and PA. Modified air masses are indicated by the letter N, as #Pp.

248

Sec. 98] WINTER POLAR CONTINENTAL AIR 249

Similarly distinctions are made in the other types of air masses. In
general the method of classification based on latitude and underlying
surface will be used in this text although significant differences between
similar source regions will be noted.

98. Polar Continental (P c ) Air Masses in Winter. The main sources
of PC air in winter are the polar anticyclones over northwest Canada
and over Russia. The surface temperatures are very low in these
regions. The subsiding air in these anticyclones tends to dissipate any
cloud present, and the resulting clear skies permit rapid cooling of the
snow surface by radiation to outer space. Large inversions, often as
great as 20 C, develop over these cold snow surfaces. The cooling in
the layers just above the surface is often sufficient to cause the deposi-
tion of frost on the snow surface. In this manner the moisture content
of the surface air decreases. Turbulent mixing of this dry air with
moister air at higher levels, although a slow process in such circum-
stances, ultimately leads to an appreciable drying of the whole air mass.
At the outer edges of the anticyclone, where the wind velocities and
therefore mechanical turbulent mixing arc greater, inversions and iso-
thermal Ia3'crs are not so frequently found. Mean dry- and wet-bulb
temperature conditions in P c air during the winter at Ellendale, North
Dakota, are shown in Fig. 91. This figure also gives the mean free air
temperatures above Fort Smith, Northwest Territories, Canada, for
the month of January, 1937. This curve represents the average of all
ascents, irrespective of air mass type. Nevertheless most of the ascents
arc in P c air, and other types of air mass, if present, would have become
modified greatly by the time they reached Fort Smith. It will be noted
that there is no major difference in the two temperature curves, except
for the lower temperatures and greater inversion found near the surface
at Fort Smith. An inversion of about 6 C at Ellendale and of about
8 C at Fort Smith from the surface to 800 mb can be noted from the
diagram. The lapse rate from 800 to 600 mb is not large.

It can thus be seen that the air is absolutely stable. The moisture
content of the air is very low, the humidity mixing ratio varying from
about 0.3 to 0.6 gm per kg. The relative humidity in these air masses
averages about 80 per cent in the lower layers. The depression of the
wet bulb is very small at these low temperatures, because the maximum
possible moisture content of the air is so small. Conditions are very
similar over northern Russia. Mean temperatures at the surface are
about the same, but at higher levels, say about 600 mb, temperatures
are lower and lapse rates steeper. The moisture content of the air is
about the same at both source regions.
Observations in Antarctica indicate that cooling at the upper levels

250

PROPP:HTIES OF AIR MASSES

[Chap. 15

takes place slowly by radiation leading to extremely low temperatures
of the order of 80 C at a height of 12 km during the winter night.
For comparison purposes average winter temperatures for Antarctica
are given in Fig. 91.

O*
O

-50 -40 -30 -20

T(C)

FIG. 91. Temperature curves in typical winter P c air masses.

As a result of the stability, marked turbulent air motion and clouds
are absent in the source region. The visibility is good, since there are
few sources of impurities. However, the relative humidity is high so
that at the edge of an anticyclone clouds form with the mechanical
turbulence, and snow flurries sometimes occur.

Sec. 99] WINTER POLAR MARITIME AIR 251

When a winter anticyclone becomes stationary to the east of the
Rocky Mountains, PC air may extend as far south as Texas with little
modification. The surface layers become warmer through insolation,
but subsidence and horizontal divergence of air cause the inversion to
become more pronounced, as shown in section 16. If, for example, the
air at Ellendale, shown in Fig. 91, were to descend adiabatically so that
the air originally at 800 mb subsided 50 mb, and that at 700 mb sub-
sided 100 mb, the layer in which initially the temperature decreased
slightly with height would now have an inversion of 4 C. Flying
conditions in general arc good, with few clouds. With such stable
lapse rates, turbulence is slight, the coefficient of eddy diffusivity K is
small, so that according to (54-8), the vertical transport of atmospheric
impurities such as smoke is inhibited. Thus near industrial regions the
products of combustion are concentrated in the surface layers as a
result of the inversion, causing poor visibility.

A second important modifying process for winter polar continental
air over North America is the movement of the air over the Great Lakes.
The lakes present a surface of open water with a temperature greater
than C, and thus are a source of heat and water vapor. The temper-
ature of the air crossing the lakes increases, sometimes as much as
15 F, and the surface mixing ratio often increases from 0.5 gm to 2.5 gm
per kg. These modifications generally extend to a height of only 2 km,
but they are sufficient to produce conditional instability of the latent
type in the air mass. Cumulus clouds are frequent over the lakes and
on the lee shore where the land surface produces mechanical turbulence
in the air stream to set off the instability. These give frequent snow
flurries, producing " snow belts " to the lee of the lakes. The water
vapor that does not condense here is carried to the Appalachian range
where orographic lifting causes condensation of at least some of the
remainder.

99. Polar Maritime (Pu ) Air Masses in Winter. Since there are no
high-pressure regions over the northern oceans in winter, there are
no true source regions for polar maritime air. Outbreaks of polar
continental air from Siberia or Alaska move southward to the west of
the Aleutian low and are carried in the circulation about that low into
a westerly current to the coasts of British Columbia, Washington, and
Oregon. A similar situation exists for Europe, with the PC air moving
from the North American continent or Greenland, around the Icelandic
low to western Europe. In both cases the ocean surface is highly uni-
form in temperature and the path over the ocean is long enough for the
air to acquire definite characteristics.

The PC air which moves out over the ocean is cold and stable. Ad-

252

PROPERTIES OF AIR MASSES

[Chap. 15

vancing over the warmer water, there is a rapid increase in the moisture
content and temperature of the air in the lower layers. This is similar
to the modification of PC air moving over the Great Lakes, but the
time spent over the water surface is much longer, and so modifications
extend to greater heights, such as 3 to 4 km. Fig. 92 illustrates polar
Pacific air just as it leaves its source region and reaches Seattle, Wash-
ington. The surface temperature is 30 C warmer than in PC air in
winter, and the moisture content at the surface is about 5 gm per kg.

<D
O

-20

FIG. 92. Temperature curves for typical fresh and modified winter P P air masses.

Potential instability and conditional instability of the pseudo latent
type are present in the lowest 200 mb. The condensation level is low,
so that low clouds of the cumulus type develop frequently.

When polar Pacific, Pp } air flows over the mountains, the potential
instability present is often realized, and the pseudo latent instability
develops into real latent instability through orographic lifting in the
manner outlined in section 95. Heavy cumulus and cumulonimbus
clouds develop on the windward side of the ranges, giving heavy rains
and thunderstorms.

When polar Pacific air crosses the Rockies it undergoes a number
of modifications. There is a marked increase in temperature, and a
decrease in moisture content. These changes can be seen in the dry-
and wet-bulb temperature curves for modified polar Pacific air over
Ellendale, shown in Fig. 92. These upper air observations were made

Sec. 99] WINTER POLAR MARITIME AIR 253

in the early morning hours, and nocturnal radiation accounts for the
inversion in the lowest 100 mb. The lapse rate in that layer is much
greater during the day. The increase in temperature from Seattle to
Ellendale is partly due to the fact that the temperature of the air
ascending the western slope of the Rockies decreases during the latter
part of the ascent at the saturated adiabatic rate, with the condensed
moisture being precipitated, and then descends the eastern slope at the

o>
o

-40

FKJ. 93. Mean temperature curves for Bismarck and Seattle, January, 1941.

dry adiabatic lapse rate. That does not account for all the difference
between the two curves, and other factors such as subsidence and turbu-
lent mixing must be in operation. Farther east, these modified polar
Pacific air masses are warm, dry, and stable.

The previous paragraphs have described the characteristics of polar
maritime air from the Pacific Ocean. Over western Europe the lapse
rate and moisture conditions in polar maritime air from the Atlantic
are practically identical with those at Seattle. The modifications in
this air will not be the same as those in PP air over North America
because of the absence of a mountain barrier, such as the Rocky Moun-
tains. Over Europe polar maritime air tends to be modified by cooling
in the lower layers, with a consequent increase in stability.

When a low-pressure area is situated south of Newfoundland, the
easterly current north of the low brings air from the Atlantic over

254 PROPERTIES OF AIR MASSES [Chap. 15

Quebec, and at times as far west as New England. This polar Atlantic,
PA, air was previously polar continental air, but had a short trajectory
over the northern Atlantic. Since the time over the ocean is shorter
than that for polar Pacific air on the other side of the continent, the
modifications do not extend above 1 to 2 km. In this lower layer the
relative humidity is high and conditional instability is present. Turbu-
lence clouds of the stratocumulus type are frequent with light pre-
cipitation. Above this low layer of clouds the sky is cloudless.

The similarity of PC and PM air aloft is shown in Fig. 93. One pair
of curves gives the mean dry-bulb and wet-bulb temperatures over
Bismarck, North Dakota, for January, 1941, and the other pair gives
the same data for Seattle. No attempt was made to separate ascents
into groups of those taken cither in P c or PM types of air masses, but the
resemblance of the Bismarck curve to typical PC air given in Fig. 91
suggests that there was very little variation in air mass type over this
station for this month. The average for Seattle was a few degrees
warmer and slightly more stable in the lower layers during this month
than for average polar Pacific conditions. It is readily seen that above
700 mb the average conditions were almost identical.

100. Tropical Maritime (T M ) Air Masses in Winter. Tropical mari-
time air masses have their source regions in the large sub-tropical anti-
cyclones of the Atlantic and Pacific oceans at 30 N latitude. The light
winds, uniform underlying surface, and subsidence permit the develop-
ment of a uniform air mass. Air from both these source regions influ-
ences the weather over the North American continent

The tropical Pacific air which enters North America by way of the
southern Pacific coast of the United States is very similar to the tropical
Atlantic air from the Atlantic anticyclone over southwest Europe.
Fig. 94 gives mean values for tropical Pacific air at San Diego, Cali-
fornia, and for tropical Atlantic air at Trappes, Dijon, Lyon, and
Chateauroux in France. The temperature shown by the latter is less
by about 10 C, since France is farther north than San Diego, and the
relative humidity is correspondingly greater, but the stability condi-
tions are about the same in both regions.

The air in general is warm throughout, with moisture decreasing
gradually with height. There is some conditional instability of the
stable type, but little potential instability. The absence of real latent
instability is probably due to subsidence in the layers aloft. Since
this air moves into California and France from more southerly regions,
inversions resulting from the advance of the air over a cooler surface
and from subsidence are sometimes found, with accompanying stratus
type clouds.

Sec. 100} ~ WINTER TROPICAL MARITIME AIR 255

Tropical maritime air from the Atlantic region is called tropical
Gulf (70), or tropical Atlantic (T A ) air. T G air comes from the Carib-
bean or Gulf of Mexico region, while TA air comes from a region farther
north in the vicinity of the Sargasso Sea. In practice it is found that
they are indistinguishable and in the following discussion the symbol
T G will be used to denote both. At times the tropical Gulf air found
over the southeastern states is just P c air which has left the continent,
become rapidly modified over the warm waters, and has returned to
the continent.

i i

-20 -10 10 20

T(q -

FIG. 94. Temperature curves for typical winter T M air in California and France.

The mean values of winter temperatures and wet-bulb temperatures
in T G air over Groesbeck, Texas, are given in Fig. 95. The most out-
standing features are the very moist layer in the lowest kilometer, with
a relative humidity of 90 per cent, and the dry layer, with the relative
humidity less than 40 per cent, above. In the lower layer there is con-
ditional instability of the stable type and slight potential instability.
With strong winds, the mechanical turbulence frequently produces a
layer of stratocumulus cloud at night, but without precipitation. The
insolation during the day causes this layer to dissipate.

The extremely dry layer aloft tends to make the air more stable,
although the small amount of moisture at these levels causes marked
potential instability in the whole air mass. Since, though, this poten-
tial instability is released only after the ascent of the layer through a

256

PROPERTIES OF AIR MASSES

[Chap. Id

pressure interval of 200 mb, it is not likely to be realized except with
ascent at a frontal surface.

The warm dry layer aloft is of a type known as Superior air, desig-
nated by S. This air has no known surface source region, but is thought
to come from the upper portions of the tropical anticyclones. Air with
similar properties is sometimes found above a layer at P c or PM air.
Its extreme dryness is explained then by subsidence. At times, particu-
larly in summer, it appears at the surface of the earth, giving extremely
hot and dry weather.

CP
O

Fia. 95. Temperature curves for typical winter TG air masses.

Modifications occur as the TQ air moves northward over the conti-
nent. These are especially marked by the time the air reaches the
northeast portion. The observations over Boston show that near the
surface the temperature and moisture content values are lower than at
Groesbeck. On the other hand, the moisture content at higher levels
is greater. This suggests that T G air which reaches the northeast coast
may not be from exactly the same source region as that over Groesbeck.
It may be air which has had a more recent polar origin, in which mois-
ture has been carried up to higher levels by turbulence. An alternative
explanation is that the air has become thoroughly mixed during its
northward movement to the vicinity of Boston, producing a decrease
in moisture content at lower levels and an increase at higher levels. As
modified T o air proceeds northeastward over the cold ocean surface to
the vicinity of Newfoundland, dense fogs frequently develop. These
affect the coastal regions of the eastern provinces of Canada.

101. Tropical Continental (T c ) Air Masses in Winter. There is
only one source of tropical continental air in the northern hemisphere,
that being over north Africa. There are not sufficient upper air obser-

Sec. 102}

SUMMER POLAR CONTINENTAL AIR

257

vations in this air mass to be able to obtain the average conditions of
temperature and moisture content. Some general conclusions may be
reached, however, from surface data. The air is warm, stable, and very
dry. The cooling by nocturnal radiation is very great, and large in-
versions develop in the early morning hours.

102. Polar Continental (Pc) Air Masses in Summer. Polar conti-
nental air masses are quite different in summer from those in winter.
There is no snow cover over the source region, and the heating of the
ground by radiation from the sun produces a greater lapse rate in the
lower levels. As indicated in a previous section, there are no well-
marked anticyclones over northwest Canada or Russia in summer, so

-10

FIG. 96. Temperature curves for typical summer P c air masses.

that the summer PC air masses do not form in the same manner as the
winter ones do. Average upper air conditions in PC air masses in
summer over Ellendale are shown in Fig. 96. It can be seen from the
diagram that the moisture content, especially in the lower levels, is
small. The air near the surface is far from saturation, the relative
humidity being only about 45 per cent. This lack of moisture in the
lower levels permits large diurnal temperature variations, sometimes as
great as 15 C. The stability is to be classified as the stable type of
conditional instability. The figure shows that on the average there
is no potential instability in summer PC air at the time of observation.
These curves represent average conditions during the early morning
hours only. The situation is different during the day, however, when
radiation from the sun heats the earth's surface. The dry-bulb and
wet-bulb temperatures of the surface air rise, frequently producing

258

PROPERTIES OF AIR MASSES

[Chap. 15

latent and potential instability. The condensation level of the surface
air is high, and only a few high clouds of the convcctive type form. If
the moisture content of the lower air increases, convective processes
are more pronounced, and thunderstorms may occur. Air to the east
of Ellendale is usually colder because of the influence of the cool waters
of Hudson Bay. Near the Gulf of Mexico summer P c air is warmer
and more moist at all levels than that at Ellendale.

Summer PC air in western Europe is slightly cooler and much more
moist than that at Ellendale. This suggests that PC air reaching
western Europe has generally had a recent oceanic trajectory. The
temperature in PC air over eastern Europe is about the same as that
over western Europe, but the moisture content of the air is slightly less.

-10

T(q-

FIG. 97. Temperature curves for typical summer Pp air masses.

103. Polar Maritime (P M ) Air Masses in Summer. In summer the
comparatively cold ocean surfaces increase the intensity of the sub-
tropical high-pressure systems. In the Pacific the ridge of high extends
north to the Aleutians. The resulting pressure gradient between this
high and the continental low brings air from Alaska over the Pacific
and to the west coast of North America in a northwest current. This
air becomes modified sufficiently in its trajectory over the Pacific to
acquire typical properties and is called polar Pacific air.

Fig. 97 gives average conditions in this type of air mass over Seattle.
Turbulence in the moderate winds produces a lapse rate close to the
dry adiabatic in the lowest kilometer. Thus conditional instability of
the stable type is found here, and a layer of stratocumulus frequently

Sec. 104] SUMMER TROPICAL MARITIME AIR 259

forms during the night at the top of this layer. Above this layer the
lapse rate is close to the saturated adiabatic, but there is little potential
instability. The temperature of the air, since it has come directly from
the north over relatively colder water, is lower than that of PC air. In
spite of its southerly trajectory, it frequently becomes more stable as it
passes over the cold coastal waters. The condensation level lowers and
fogs develop. As PM air passes over the mountains, it acquires heat
in the lower layers and rapidly becomes modified so that it is indis-
tinguishable from PC air.

Over western Europe summer polar Atlantic air is slightly cooler
and more moist than PM air observed at Seattle.

Along the east coast of North America a return circulation occasion-
ally develops, bringing over the continent air which has passed over
the north Atlantic. This circulation develops most frequently in spring
and early summer. The waters of this region are cold for their latitude,
and so the polar Atlantic air is moist with low temperatures. Low
turbulence clouds form in this current, but above this layer the air is
clear and flying conditions are good.

104. Tropical Maritime (T M ) Air Masses in Summer. As noted in
the discussion of PM air in summer, the air trajectory at the eastern
boundaries of the oceans is from the north, and for the same reason the
return circulation gives southerly winds at the western boundaries.
For this reason the tropical maritime air reaching California and south-
west Europe is dry, stable, and relatively cool, whereas the tropical
maritime air along the southeast coast of the United States is moist and
unstable. Fig. 98 gives average conditions in this air mass at Miami,
Florida.

It will be readily noted that the relative humidity is high through-
out, with temperatures also high. The lapse rate is of the condi-
tionally unstable type, with real latent and potential instability. A
small lift of 50 mb is sufficient to release the potential instability; or
even the diurnal heating over the warm land will set off strong convective
currents. Thunderstorms are, then, frequent in this air mass. This T G
air is more nearly of the equatorial type than of the tropical type. It
is practically identical in properties with air over Batavia, Java, at
6 S latitude.

In the spring and early summer, when as sometimes happens TQ air
has moved northward and over the cold waters of the Great Lakes, it
is cooled in the lower layers. Because of the abundant moisture
present, fogs frequently develop in this air at the north shore of the lakes.
The modifications along the Atlantic coast are very marked. Moving
northeastward parallel to the east coast of the United States, it leaves

260

PROPERTIES OF AIR MASSES

[Chap. IS

the Gulf Stream to move over the cold waters of the Labrador Current.
Rapid cooling in the lower layers causes dense fogs over the Newfound-
land Banks and adjacent waters during this season.

-10

FIG. 98. Temperature curves for typical summer T G air

masses.

105. Tropical Continental (T c ) Air Masses in Summer. This type
of air mass develops over north Africa and west Asia. The air is so
dry, and subsidence is so prevalent, that condensation forms do not
develop, despite the steep lapse rate which occurs during the daytime.
If it moves northward or northeastward, it takes up considerable mois-
ture in the lower portion of the air mass, and convective clouds and
showers, and sometimes thunderstorms, develop over southeastern
Europe and southwestern Asia.

No tropical continental air is found over North America in summer
since there is no source region on this continent.

106. The Rossby Diagram as an Aid to Classification of Air Masses.
The various types of air masses may be conveniently identified by
means of the Rossby diagram. In this diagram, discussed briefly in
section 25, the ordinate is partial potential temperature on a logarithmic
scale, and the abscissa is the mixing ratio on a linear scale. Partial
potential temperature is obtained in the customary manner, but the
partial pressure of the air, p e } is used instead of the ordinary pres-
sure p. Partial potential temperature has very nearly the same value
as ordinary potential temperature, and in practice, the two may be
used interchangeably. A Rossby diagram is shown in Fig. 99. The

Sec. 106]

THE ROSSBY DIAGRAM

261

lines sloping downward from left to right are lines of constant equiva-
lent potential temperature. Pressure and temperature lines are some-
times shown on the diagram, but their usefulness is limited, since they
are applicable only if the air is saturated. Average winter conditions
in PC air at Ellendale (Fig. 91), in Pp air at Seattle (Fig. 92), and in
T G air at Groesbeck (Fig. 95) are shown in Fig. 99. The different types
of air masses may be readily distinguished. If the actual lapse rate is
equal to the dry adiabatic, and if the moisture content is the same at all

320

250

10 12

2468
x (gm kgm" 1 )

FIG. 99. Typical winter air mass curves on a Ilossby diagram.

levels, the ascent curve becomes a single point. An ascending or de-
scending particle of air will be represented by a single point on the
diagram as long as processes of evaporation and condensation do not
occur. The equivalent potential temperature at each level may be
readily determined, since lines of constant equivalent potential tem-
perature are drawn on the diagram. The presence or absence of po-
tential instability in a layer may therefore be determined directly,
since it is present if the equivalent potential temperature decreases
with height in that layer. It can be seen that the TQ air has potential
instability, but the PC has not.

The chief use of the Rossby diagram is in identifying air masses.
However, the tephigram, with both dry- and wet-bulb temperatures

262

PROPERTIES OF AIR MASSES

[Chap. Id

plotted on it, may also be used for this purpose. The lack of pressure
coordinates on the Rossby diagram is inconvenient, and the tephigram
is much better for the study of stability.

107. Comparison of Air Masses. The following table presents a
comparison of the typical air masses of North America at their source
regions. These are for average conditions, from which, of course,
variations will occur in particular cases.

AIR MASS TYPES OF NORTH AMERICA

Surface

Season

Type of
Air Mass

Temper-
ature

Stability

Cloud
Forms

Precipita-
tion

Flying
Conditions

-26C

Very stable

None

Smooth

-1C

Conditional

Stratocu-

Snow

Turbulence

modified over

instability

mulus

flurries

in lowest

Great Lakes

in lowest

with

layers;

2km

cumulus

moderate

rime icing

Win-

Potential and

Cumulus

Showers

Severe

ter

pseudo latent

and cumu-

and

turbu-

instability

lonimbus

thunder-

lence;

over

storms

severe

mountains

over

icing over

mountains

moun-

tains

T P

18 C

Stable

Some

None

Moderate

stratus

visibility

19 C

Potential in-

Stratocu-

Drizzle

Visibility

stability

mulus at

poor at

realized with

night

times

ascent at

fronts

15-25 C

Real latent

High level

None

Moderately

during the

cumulus

turbulent

day ; stable

during

at night

day

Sum-

17 C

Stable type of

Stratocu-

None

Smooth;

mer

conditional

mulus

poor visi-

instability at

with low

bility

surface

tops at

night

24 C

Real latent

Heavy cu-

Thunder-

Extremely

and potential

mulus and

storms

turbu-

instability

cumu-

lent

lonimbus

BIBLIOGRAPHY 263

PROBLEMS AND EXERCISES

1. The mean conditions in winter P c air at Boston, Massachusetts, modified as it
has moved eastward, are as follows.

Height 0.0 1.0 2.0 3.0 4.0 km

T -6.3 -14.3 -18.0 -23.0 -29.0 C

x 0.9 0.6 0.5 0.3 0.2 gm per kg

As this air moves out over the waters of the Atlantic, which are at a temperature of
2 C, what changes will take place? To what height will the turbulent currents ex-
tend? What clouds and weather can be expected when P c air in winter moves off
the continent over the warmer ocean?

BIBLIOGRAPHY

Admiralty Weather Manual, London, H. M. Stationery Office, 1938. Chapter 20.
Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book

Co., 1937. Chapter 7.
Byers, H. R., and V. P. Starr, " Circulation of the Atmosphere at High Latitudes

during Winter," Monthly Weather Review, Supplement 47, Washington, D. C.,

1941.
Chang- Wang Tu, " Chinese Air Mass Properties," Q. J. Roy. Met. Soc., 65, 33-51

(1939).
Namais, J. et al., An Introduction to the Study of Air Mass Analysis, Fifth Edition,

Milton, Mass., American Meteorological Society, 1940.
Schinze, G., " Tropospharische Luftmassen und vertikalen Temperaturgradient,"

Archiv deut. Seewarte, 52, No. 1, 1932.
Showalter, A. K., " Further Studies in American Air Mass Properties," Monthly

Weather Review, 67, 204-218 (1939).

Willet, H. C., "American Air Mass Properties," Mass. Inst. of Tech., Papers in Physi-
cal Oceanography and Meteorology, 2, No. 2, Cambridge, Mass., 1934.

CHAPTER 16

CYCLONES AND ANTICYCLONES

108. General Characteristics of a Front. One characteristic of an
air mass is its uniformity. Properties of the air in an air mass change
little if any in the horizontal direction. When, though, two air masses
lie side by side there is a rapid change in properties from one air mass
to the other. Fig. 100 illustrates this in terms of the isotherms when
one air mass is warmer than the other. This region of discontinuity
between the air masses is called a frontal surface or a front. The term
frontal surface is used for the surface of discontinuity between the two
air masses when it is desirable to make a distinction between the dis-
continuity at the earth's surface
and that above it. The term front
refers then to the line of discon-
tinuity at the ground.

As a result of turbulent mixing
between the air masses, there is no
discontinuity in the mathematical
sense across the frontal surface, but
rather a narrow zone of transition.
The limits of this zone are indicated
by AD and EC in Fig. 100. The
discontinuity is usually one of temperature, but at times the contrast
in moisture content across the surface is as marked as that of tempera-
ture.

When two liquids of unequal density, such as oil and water, are put
into the same vessel, equilibrium is reached when the lighter liquid
rests above the heavier. In the atmosphere there is a tendency for the
warmer, and therefore lighter, air to lie above the colder and heavier
air. But as a result of the rotation of the earth, the equilibrium condi-
tion is reached when the frontal surface intersects the ground at a small
angle. As shown in equation 38-8,

FIG. 100. Isotherms across a frontal
zone.

tan0 =

2co sin <

where 6 is the angle between the frontal surface and the ground, as-

264

Sec. 109] FRONTOGENESIS AND FRONTOLYSIS 265

sumed horizontal, w the angular velocity of the earth, < the latitude, g
the acceleration of gravity, T\, HI represent the temperature and the
component parallel to the front of the geostrophic wind in the cold air,
and T 2 , u 2 the corresponding quantities in the warm air. The angle 6
increases with increasing difference in velocity, and also increases with
decreasing difference in temperature across the frontal surface.

Since a front is a boundary between two air masses, it must move
when the air masses move. Thus the successive positions of the front
may be determined from the velocity components normal to the front
of the air on the two sides of the front. Some mixing does, at times,
take place to decrease the sharpness of the discontinuity, although in
other situations the front may become more distinct. The wind distri-
bution in the vicinity of a front is closely related to the changes both
in the position and the sharpness of the front.

109. Frontogenesis and Frontolysis in Deformation Fields. As men-
tioned above, there is not a mathematical discontinuity at a front, but
a zone of transition. This zone of transition is a region in which the
temperature changes rapidly with distance, or, in other words, where
the temperature gradient is large. Such a zone does not keep the
same width at all times. Mixing will change the temperature gradient,
as will a difference in the directions of the winds. When this zone of
transition is becoming sharper, frontogencsis is said to be taking place;
the reverse situation is called frontolysis. These processes are closely
associated with the distribution of the wind.

If n represents the direction of a horizontal line perpendicular to the
isotherms, the temperature gradient is given by the rate of change of
temperature with n, i.e., dT/dn. Near a front |d7Vdn| is a maximum,
the absolute value being used since n may be in the direction of rising
or of falling temperatures. For frontogenesis, |d7 7 /dn| must increase
with time, and for frontolysis it must decrease with time. As given in
equation 414

d_
dt

where v n represents the velocity in the direction of n. For any particle
of air, T is assumed to be invariant with time. Therefore frontogenesis
occurs when the velocity decreases along the normal and frontolysis
occurs when the velocity increases along the normal.

It may be shown that if the velocity is given by a linear function of
the space coordinates, it can be compounded of one or more of six
primary fields of motion. These are

266

CYCLONES AND ANTICYCLONES

[Chap. 16

(a)

(c)

(d)

Axis" of cfilatationT*'

- IH

HHf

... . o

t t t to

IZ 1 1 1 tg

i: tttri

(a) Translation.

(b) Rotation.

(d) Divergence.

(e) Dilatation.
(/) Contraction.

These fields of motion are illus-
trated in Fig. 101.

When the velocity field is one

s of pure translation or of pure

' rotation, there is no tendency

for particles of air to approach

*** ^^ or to separate from one another.

Hence there will be no tendency

for either f rontogenesis or fron-

tolysis.

Frontogenesis in a conver-
gence field is illustrated in Fig.
\\\\ 102. TI, T 2 , etc., are isotherms,

H H
t t t t

and the line AB is a normal to
these. Thin arrows indicate the
velocity. The components of
tilt velocity along the normal AB
} | | | are indicated by thick arrows.

A A A

(e)
Fia. 101.

(0
Primary fields of motion.

As can be seen from the dia-
gram, the velocity components

-T,

\ /

from B to A decrease in .value,

become negative, and then have in-
creasing negative values. Then, ac-
cording to (414), f rontogenesis takes
place in such a wind field. Since

the wind field is symmetrical with

respect to the central point, the ^-

orientation of the isotherms does

not alter this conclusion.

With a divergence field, the method
of analysis is similar. The velocity
components then increase along the
normal to the isotherms, and hence F IO j 2. Frontogenesis with con-
there is frontolysis. vergence.

Sec. 109]

FRONTOGENESIS AND FRONTOLYSIS

267

Frontolysis in a dilatation field may be understood by reference to
Fig. 103. The velocities are parallel to the axis of dilatation, and are
indicated by thin arrows. The isotherms are indicated by TI, 7 T 2, etc.;
AB is normal to them. The thick arrows along the direction AB
indicate the velocity components along the normal. In the situation
illustrated, the velocity from B to A increases from large negative
values to small negative ones, then to small positive, and finally to

FIG. 103. Frontolysis with dilatation.

large positive values. Thus dv n /dn is positive and, according to (414),
frontolysis is taking place. If, though, the isotherms are parallel to
the axis of dilatation, the velocity along the normal is zero, and so neither
frontogenesis nor frontolysis occurs.

Since the particles on the line CD, from which the particles are
diverging, are not moving, the point of intersection of an isotherm
with CD is fixed. On either side the particles on the isotherm are
moving away from CD. Hence the direction of the isotherm is chang-
ing in such a manner that the angle of intersection with CD increases,
approaching 90 as a limit. Frontolysis is then no longer taking place.

A similar discussion of a contraction field shows that contraction
causes frontogenesis unless the isotherms are parallel to the axis of
contraction. With this exception the velocity field changes the direc-
tion of the isotherms, causing the angle between the isotherms and the
axis of contraction to increase. This change of direction increases the
tendency to frontogenesis.

An example of a specific contraction field in the earth's atmosphere
will now be given. Consider the situation in which straight parallel

268

CYCLONES AND ANTICYCLONES

[Chap. 16

isobars run in a north-south direc-
tion, as illustrated in Fig. 104. Here
x is positive to the east, and y is
positive to the north. The geostro-
phic wind is south, of magnitude
given by (35-1),

1 dp

v =

sin p dx

FIG. 104. Frontogencsis in air with a
northward component of motion.

The isotherms are also straight and
parallel, and make an angle a with

the isobars, as indicated in the figure. The velocity component in the

direction n normal to the isotherms is then

Thus

v n = v sin a

sin a dp
2co sin p dx

(109-1)

Assuming that p is constant, only the latitude varies with n. Differ-
entiating (109-1) partially with respect to n gives

sin a dp d .
. ( g i n (

dx dn

sin a cos <

Substituting (109-2) in (41-4) leads to

2wp sin 2 dx dn

(109 . 2)

d!
dt

dT_

sin a cos dp d0

2wp sin dx dn

-r- >0

since both dp/dx and d<t>/dn > 0. The temperature gradient increases
with time, so that the wind field is of the frontogenetical type.

It must not be inferred from this that frontogenesis always occurs in
northward-moving air when the isotherms are as shown, although it
frequently does. Variations in the pressure gradient or in the density

may be sufficient to reverse the sign of

dT
dn

Furthermore, non-

geostrophic components, of the type discussed in section 46, may be so
large that it is no longer valid to use the geostrophic wind, even as a
first approximation.

A similar process of reasoning shows that if the geostrophic wind is
north, rather than south, the wind field is of the dilatational type, and
hence produces frontolysis.

Sec. 109]

FRONTOGENESIS AND FRONTOLYSIS

269

When contraction and dilatation velocity fields are superimposed, the
result is as illustrated in Fig. 105. AB is the axis of dilatation and CD
the axis of contraction. Thin arrows give the velocity components and
thick arrows the resultant velocities. The streamlines are given by
the smooth curves. These streamlines are similar to the isobars near

FIG. 105. Resultant of contraction and dilatation velocity components.

a col as given on a weather map, where high-pressure areas are situated
near the points F and H and lows near E and G. Because of the direc-
tion of movement of the particles, AB is sometimes known as the axis
of outflow, and CD the axis of inflow. A velocity field which is com-
pounded of a dilatation and a contraction field is known as a deformation
field.

In a deformation field, if the isotherms are parallel to the axis of
contraction, no frontogenesis results from the contraction component,
but frontolysis results from the dilatation component. Similarly, if the
isotherms are parallel to the dilatation axis, frontogenesis results along
the axis of dilatation. For isotherms in intermediate directions, the
result depends upon the relative strengths of the frontogcnetical and
frontolytical tendencies. If the angle which the isotherms make with
the axis of dilatation is less than 45, frontogenesis will occur, but if it
is greater than 45, frontolysis will result.

As shown above, both contraction and dilatation tend to rotate the
isotherms into a position parallel to the axis of dilatation. This rotation
would result in frontogenesis along the axis of dilatation. In actual
situations, though, the structure of the pressure system will rarely remain
constant long enough for this to occur.

270

CYCLONES AND ANTICYCLONES

[Chap. 16

A deformation field may be compounded with any one of the other
types of velocity fields, although not all possible combinations are
found on actual weather maps. Fig. 106 gives one resultant field that
is frequently found on synoptic charts. It is a combination of a de-
formation field and a rotation field. In this situation the axes of inflow
and outflow CD and AB are not perpendicular to each other, and the
axes of contraction and dilatation are not readily obtainable. The
axes of contraction and dilatation may be defined as follows. In a
wind velocity field consider a flexible circular ring attached to the
particles of air at any given moment. After an interval of time, the
circle will be deformed until it has approximately the shape of an ellipse.

FIG. 106. The determination of the axes of contraction and dilatation.

The major axis, along which the particles are spreading apart most
rapidly, is the axis of dilatation. The minor axis, along which the
particles are approaching one another most rapidly, is the axis of
contraction.

These axes may be found in Fig. 106 most readily by the following
steps. Bisect the angles AOC and BOG by the lines EOF and GOH.
Bisect the angles EOG and FOG by the lines MON and KOL. The line
MONy nearer the axis of outflow, is the axis of dilatation and the line
KOL is the axis of contraction. Frontogenesis, when it occurs, will be
along the axis of dilatation. The rule, as determined above, that
frontogenesis will occur when the isotherms make an angle of 45 or
less with the axis of dilatation is still useful, although the angle varies
slightly depending upon the extent of convergence or divergence.

#*c. 110] FRONTAL ZONES 271

Contraction and convergence are the two types of wind fields which
lead to frontogenesis. Contraction is found in the region of cols, and
convergence is prominent in the regions of low pressure. Tor that
reason, temperature gradients near either of these two types of pressure
distributions should be watched closely for the possible development of
fronts. Also, since high-pressure areas are regions of divergence, fronts
will dissipate near the center of an anticyclone or across a ridge.

110. Frontal Zones. Frontogenesis between Air Mass Source
Regions. Frontogenesis is especially likely to occur near the coastline
of a continent. If the air moves parallel to the coast for a long period,
so that one portion of the air mass has an extended continental trajec-
tory, while adjacent air has a long oceanic trajectory, even a small
amount of deformation or convergence may produce frontogenesis.
This would occur, for example, if the isobars over the east coastal waters
of the United States were parallel to the coast line. Rapid transforma-
tion of the air over the ocean would lead to the formation of a front
along the boundary of the Gulf Stream. This is a frequent cause of
frontogenesis, and such a development must be allowed for in forecasting.

The main sources of the temperature differences which lead to fronto-
genesis are:

(a) The north-south temperature gradient.

(6) The contrasts between air masses over continents and those over
oceans.

Deformation fields usually produce the fronts, aided occasionally by
convergence. The zone of frontogenesis nearly coincides with the region
of maximum temperature gradient of the underlying surface. In gen-
eral, it may be said that frontogenesis occurs most readily near the east
coasts of northern continents.

In winter there are five main regions where frontogenesis occurs.
There is one zone near the east coast of North America, and another
near the east coast of Asia. There is another zone in the north Pacific,
nearer to North America. This is a result of the division of the north
Pacific sub-tropical high into two cells. The deformation field between
the two cells leads to frequent frontogenesis in this region. Another
zone of frontogenesis lies to the east of Greenland, in the general region
of the Icelandic low. The last main zone is found extending in an
east-west direction along the Mediterranean.

In summer there are only three main zones of frontogenesis. The
temperature contrast between air of continental origin and that of
oceanic origin is small in summer, especially in middle latitudes. The
contrasts between cool Arctic air and warm continental air are larger,
and therefore the summer frontal zones are found at higher latitudes*

272 CYCLONES AND ANTICYCLONES [Chap. 16

One zone extends across Canada, another across northern Europe, and
the third extends from northeast Russia to Alaska.

111. Air Masses and Cyclones. The first attempt to study physical
processes in a depression from the air mass point of view was made by

Shaw and Lempfert in 1906. They
-E studied the trajectories of air in cy-
clones with the aid of one- and two-
"E hourly weather maps. Some of the

W-
COLD

Rainfall from
displaced air

of pressure v results of this investigation are shown

*~ ^ * n ^- ^ taken from one of Shaw's

Rainfall from
converging air

in Fig. 107, taken from one of Shaw's
later papers. The interaction of the

WARM \ warm and cold air masses is indicated

clearly. The similarity with later
models of a depression in its early

FIG. 107. Motion of air near a Htftgqs M developed by the Nor-
center of low pressure. (After Shaw.) . , ,.,

wegians, may be readily seen.

Helmholtz, late in the nineteenth century, .suggested that two cur-
rents with different properties could flow side by side, separated by a
surface of discontinuity. A number of meteorologists, such as Bigelow,
Hanzlik, Von Ficker, and Shaw and Lempfert suggested early in the
present century that cyclones tend to form at surfaces between warm
and cold air. These earlier ideas were developed by Norwegian meteor-
ologists, notably V. and J. Bjerknes. The former studied the question
from the mathematical standpoint, considering a depression as a wave
on a surface of discontinuity. J. Bjerknes, the son, investigated physi-
cal processes at fronts, and applied his findings to map analysis and
forecasting.

112. Life History of a Frontal Depression. The development of a
depression forming at a front, as visualized by Bjerknes, is shown in
Fig. 108. At first there is a front separating cold air, denoted 1, mov-
ing westward or slowly eastward, from warm air, denoted 2, moving
more rapidly eastward, as indicated in Fig. 108a. Then a bulge appears
in the front, the warm air advancing from the south and the cold air
retreating to the north, as shown in Fig 1086. This bulge is similar to
a wave, but the exact nature of the generating cause for the wave along
the front is not known. This stage in the development is preceded and
accompanied by more rapidly falling pressures in the neighborhood of
the wave than elsewhere, and by increasing cloudiness caused by ascent
of the warm air over the cold.

The kinematics of this northward advance of warm air mass 2 may
be discussed in the following manner. Draw an axis y positive to the
north through point B on the stationary front, as shown in Fig. 108a.

Sec. US] LIFE HISTORY OF A FRONTAL DEPRESSION

273

Since the front is stationary, the velocity c of the front is zero, and the
acceleration A of the front becomes, according to (43-9),

dpi

(112-1)

Worm

pold

(a)

Center of Low
Pressure

(d)
FIG. 108. Life history of a cyclone.

where the subscripts refer to air masses 1 and 2. According to (39-2),

~dy " Hy >

Now if the falling pressure tendencies are becoming greater more
rapidly with time in cold air mass 1 than in warm air mass 2, it follows
that

The front then has a positive acceleration northward, as shown by
(112-1) and the bulge develops, as indicated in Fig. 1086.

The bulge becomes more marked, with the overrunning causing
precipitation, and a low-pressure area forms at the point where the
warm air has advanced the greatest distance into the cold. This stage
is illustrated in Fig. 108c, the hatching indicating the extent of the
precipitation area. Now consider a point D on the front farther to the
west, where the front is still stationary, as shown in Fig. 1086. Here
the falling tendencies usually increase less rapidly with time in air
mass 1 than in air mass 2, or in the former the pressure may even be

274 CYCLONES AND ANTICYCLONES [Chap. 16

rising. Under such conditions

_
dt 2 dt 2

and, according to (112-1), this portion of the front has a southward
acceleration. The advance of this section of the front to the south is
shown in Fig. 108c.

The portion of the depression having warm air at the surface is
called the warm sector. The air to the west of the center of low pres-
sure that has formed advances southward and southeastward, as indi-
cated above, replacing the warm air at the ground. This section of the
front is called a cold front. On the other side of the low the warm air
replaces the cold, and this portion of the front is known as a warm
front. The isobars in the warm sector are usually nearly straight.
This stage is accompanied by falling tendencies ahead of the warm
front, becoming steady in the warm sector, and rising back of the cold
front. The wind veers, i.e., its direction changes in a clockwise manner
at each front. The temperatures in the warm sector are fairly uniform
with contrasting temperatures across the fronts.

The wave sometimes moves rapidly along the original front during
stages (6) and (c), never developing beyond stage (c). The rate of
movement is dependent on the velocities of the air on either side of the
front. As the wave travels eastward, the area just north of the front
will have a short spell of cloudy weather with precipitation, accom-
panied by rapidly changing winds. This development is to be expected
if falling tendencies occur to the east of the crest of the wave, and to
the west of the crest the pressure is rising.

The other possible development is a deepening of the center of low
pressure, accompanied by an increase in the amplitude of the wave,
the movement of the system in the direction of the isobars in the warm
sector, and the overtaking of the warm front by the cold. This stage
is shown in Fig. 108d. The fronts now form one line at the surface,
known as an occlusion. As the process of occlusion occurs, the cold
and dense air pushes underneath the warm air, the cold front advancing
at the same rate as the wind in the cold air behind the front. On the
other hand, at the warm front the warm air overrides the cold air.
Hence the warm air does not displace the cold air with the speed of the
air in the warm sector but at a speed which varies with different situa-
tions but averages about 60 per cent of that of the warm sector air.
Since, then, with the same pressure gradient, the cold front moves
more rapidly than the warm front, it overtakes the warm front, driving
aloft the warm air that is trapped between the two.

Sec. 112} LIFE HISTORY OF A FRONTAL DEPRESSION

60'

40'

10'

O20 |Ol6 IOI2.

40 s

20"

10"

FIG. 109. Weather map showing a frontal depression near western Europe. (From
Daily Weather Report of the Meteorological Office, London.)

Fig. 109 gives an example of a wave which has developed near the
eoast of western Europe to about the stage (c) of Fig. 108. The only

276

CYCLONES AND ANTICYCLONES

[Chap. 16

information plotted for each reporting station is the wind, the amount
of sky covered, and the temperature. The sky cover is indicated by
the amount of shading within the circle representing the station. The
wind direction is given by the shaft from the station circle, the wind
blowing along this line toward the station. The force of the wind is
indicated by the number of feathers on the shaft, each feather repre-
senting two points of the Beaufort wind scale described in section 66.
The full lines give the sea level isobars, with pressures as indicated.

FIG. 110. Types of isobars, isallobars, and barogram curves about low-pressure areas.

(After Petterssen.)

The warm front is given by a curve with semicircular black areas on it.
Triangular black areas mark the cold front, and alternate triangles and
semicircles designate an occlusion. An example of an occlusion is
found in the upper right-hand portion of the diagram.

The warm sector is located in the low situated south of Ireland.
Note the change of wind across both the warm and cold fronts, as well
as the difference in temperature. This system never reached stage (d)
of Fig. 108, as frontolysis occurred shortly after the time of the map in
Fig. 109. Near Iceland there is a low-pressure area which is now filling,
and out of which extends an occlusion. Note the wind shift across the
occlusion, as well as the difference in temperature. During the occlu-
sion process, the low at the end of the occlusion deepens and moves

Sec. 112} LIFE HISTORY OF A FRONTAL DEPRESSION

277

much more slowly. When the occlusion process is complete, the warm
air has been pushed aloft, and at the surface there is a core of cold air.
The center fills, the occlusion is subjected to frontolysis and finally
disappears.

If the method of analysis adopted by Petterssen is used, the rate of
occlusion in several types of frontal depressions may be discussed qual-
itatively with the aid of the formula for the acceleration of a front
given in section 43. It was shown in that section that the acceleration
A of a front is made up of three components, A\, A 2l and A 3 . Accord-
ing to (43-10)

A = AI + Ao + A 3

The nature of each component is indicated in section 43. For any given
distribution of isobars, isallobars (lines of equal pressure tendency),
and any type of barogram curve, the sign of each of these components
in the x direction may be determined by the method indicated in sec-
tion 43. Four types of frontal depressions, differentiated by the distri-
bution and curvature of isobars, isallobars, and barogram curves in
each, are shown in Fig. 110. The full lines represent isobars and
fronts, broken lines represent isallobars, and the curve at the base of
oach diagram represents the barogram curve. Applying the procedure
of section 43 to each, the results given in the table are obtained. Al-

SICNS OF THE VARIOUS ACCELERATION COMPONENTS
FOH WARM AND COLD FRONTS

Type of
Depression

Front

Components of Acceleration

A z

A t

Warm

(")

/< ii /At*
C()ld (-At/

Warm

(b)

Cold

/^N

Warm

( C )

Cold

(d)

"" { Ml

{;";

+
+

though the magnitudes of the three components will vary from one
depression to the next, to facilitate the following discussion it will be
assumed that each component has the same magnitude, the only differ-
ence being one of sign.

278 CYCLONES AND ANTICYCLONES [Chap. 16

It can be seen from the table that in type (a) depression the warm
front has a small net acceleration to the east. Near the center of low
pressure, at e, the cold front has a small net retardation, and farther
south, at /, it has a large net retardation. It follows, then, that this
type will not tend to occlude, but rather to move as a stable wave.
Systems of this kind frequently travel for hundreds of miles as a stable
wave. The system shown in Fig. 109 appears to be of this type. As
mentioned before, it did not occlude but dissipated as a result of f rontol-
ysis.

In type (6) the warm front is strongly retarded at all points, while
the cold front has a net acceleration towards the east. A cyclone of
this type is therefore likely to occlude rapidly.

There is a small net eastward acceleration of both fronts in type (c).
With this configuration, then, the system may move as a stable wave,
or if it occludes, the rate of occlusion will not be rapid. It is inter-
mediate between types (a) and (6).

.Warm

Colder T Cold Cold /^Colder

(a) (b)

FIG. 111. (a) Cold and (b) warm front typ(\s of occlusions.

An examination of the table shows that type (r/) occludes rapidly
near the center of low pressure at (j and /, but only very slowly, if at
all, farther south, near h undj. The result is that the northern portion
of the warm sector occludes rapidly in the early stages of development.
The process of occlusion then ceases, leaving an occlusion extending
northwestward from the tip of the warm sector to the center of low
pressure. This type of depression is of frequent occurrence. If a
fresh supply of warm moist, air flows into the warm sector, providing a
new source of energy for the system, a secondary low may form at. the
junction of the warm and cold fronts. When this happens, the original
low becomes less prominent with the development of the second depres-
sion.

Two types of occlusions occur, and are illustrated in Fig. 111. Origi-
nally the portions of air on the two sides of the occlusion were from the
same air mass and therefore nearly identical in properties. But in
moving over different underlying surfaces and in other ways, slight
differences in properties across the occlusion develop. When the air
behind the occlusion is colder, the occlusion is of the cold front type,
illustrated by Fig. Ilia. When the air behind is warmer, the occlusion

Sec. m\ LIFE HISTORY OF A FRONTAL DEPRESSION 279

is of the warm front type, illustrated by Fig. 1116. The difference in
temperature is seldom as large as that between one of the cold air
masses and the warm air mags. Hence, according to (388), the slope
of the occlusion surface below the trough of warm air is steeper than the
slope of either the cold or the warm frontal surface. At times, with a
warm front type occlusion, the passage of the upper trough of warm air
can be determined at a station by means of a change in the rate of fall
of the barometer or by an increase in the precipitation arising from the
lifting of the warm air at the upper cold front at this point.

With some fronts several waves will develop, one after another.
The average length of time between the development of successive
waves is about 24 h. When a front has become stationary the passage
of a family of these waves may cause the front to shift back and forth
several times across a station which lies near the front.

There may be secondary cold fronts behind the main cold front.
These sometimes occur in rapidly moving polar air as it flows south-
ward over a warmer land surface. Those* secondaries continue as long
as the wind velocity remains high. The distance between secondary
cold fronts in any given depression is nearly equal, but this distance
shows wide variations in different depressions. They may be any whore
from 50 to 400 mi apart, and moving with velocities from 25 to 40 mph.
A continuous succession of these secondaries may pass, some being
marked and others very woak. Thoir passage is usually characterized
by gusty, shifting winds, and a short- period of rain, with hail and
thunder on occasion. It is oft on impossible to determine the positions
of these secondary cold fronts from ordinary synoptic weather maps,
and autographic rocords must be used in conjunction with the maps
if their exact location is desired. It is obvious, therefore, that discre-
tion must bo used in plotting secondary fronts on the map. Only the
most marked of these* can safely be plotted unless sufficient time ami
data are available* to pormit exhaustive analysis.

Seclusion of the warm sector air, instead of occlusion, sometimes
occurs under certain orographic conditions. The process of seclusion
is shown in Fig. 112. The motion of the lower portion of the warm
front is retarded, as indicated in Fig 112a, by high hills or a range of
mountains. The cold front overtakes the warm front first at this
point, leaving an island of warm air secluded farther to the north, as
shown in Fig. 1126. This development sometimes occurs on the west
coast of Norway, where the coastal range juts westward in the southern
part of the country, retarding advancing warm fronts at this point.

The changes in the weather elements that occur with the approach
and passage of a front vary widely. The variations result from differ-

280

CYCLONES AND ANTICYCLONES

[Chap. 16

ences in the air masses, in the rates
of motion of the fronts, in the
amount of overrunning, in the rate
of deepening, and other causes. The
following table gives the changes
normally expected in the vicinity of
cold and warm fronts. Yet, owing
to the differences between individual
fronts, each of these changes should
be taken as the average one only, and not as invariably accompanying
every front. The changes which occur at an occlusion are not as
clearly defined as those taking place when a warm or a cold front passes.

USUAL CHANGES ACCOMPANYING THE PASSAGE OF FRONTS

The development
seclusion.

TV IIP

.type
of
Front

Property

Ahead of Front

At Frontal
Passage

After Frontal
Passage

Pressure

Steady fall

Fall ceases

Not much change;

perhaps slow fall

Temperature

Slow rise

Rise ceases

Not much change

Humidity

Gradual rise

Rise ceases

Not much change

Wind

Back, and increase

Veer and decrease

Not much change

Cloud

Ci, Cs, As, Ns

Low Ns and Fs

Perhaps St or Sc

Warm

in succession

Weather

Steady

Precipitation

Fair or drizzle,

precipitation

decreases

perhaps inter-

mittent

Visibility

Hood, except in rain,

Poor; sometimes

Poor; mist or fog

decreasing with

mist or fog

often persist

approach of front

Pressure

Fall, usually slow

Sudden rise

Rise continues

more slowly

Temperature

Not much change

Sudden fall

Pall continues

slowly

Humidity

Not much change

Sudden decrease

Low

Wind

Slight backing,

Sudden veer,

tfot much change;

increasing

perhaps squall

further veering

Cold

perhaps

Cloud

Perhaps Cu or Cb

Cu and Cb,

Sometimes As,

perhaps Ns

then Cu

Weather

Possibly rain

Heavy rains, some-

Showers

times thunder and

hail

Visibility

Poor

Sudden improve-

Good

ment

Sec. 113]

UPPER AIR CONDITIONS

281

A discussion of the types of clouds normally associated with a frontal
depression will be given in section 137. The occurrence of fog in the
vicinity of fronts is discussed in section 131 and following, and the
extent of the areas of frontal precipitation is indicated in section 129.

113. Upper Air Conditions above Frontal Depressions. In the early
stages of a depression, when the warm sector is broad, closed isobars
occur at the surface, but do not usually extend to heights as great as
3 km. As the depression intensifies, however, closed isobars extend
upward to greater heights, and in the later stages even the isobars at
7 or 8 km frequently have cyclonic curvature, as shown by the motion
of cirrus clouds near a depression.

FIG. 113. Variation (a) of pressure at various heights in kilometers and of the

height of the tropopause, and (b) of temperature at the same heights and at the

tropopause with the passage of high- and low-pressure areas. (After Penner.)

Since the air on the west side of a low is colder than that in the warm
sector, the rate of decrease of pressure with height is greater in the
former than in the latter. The center of lowest pressure at higher
levels is found, then, on the west side of the surface depression. This
deduction is substantiated by the movement of cirrus clouds, which
have a cyclonic trajectory about a point to the west of the center at
the earth's surface.

Further support for this deduction is found in the data illustrated in
Fig. 113. In Fig. 113a curves are given showing the variation in the
mean pressure p z at the level z in kilometers for the areas at the east or
front (F) side of a low, at the west, or rear (R) side, of a high, at the
center (C) of a high, at the east side of a high, at the west side of a low,

282 CYCLONES AND ANTICYCLONES [Chap. 16

and at the center of a low. Also included is the variation in the
height of the tropopause, H, for the same areas. Fig. 1136 gives the
corresponding variation in mean values for the temperature T g . These
curves were computed from the radiosonde reports from Sault Ste.
Marie, Michigan.

It will be noticed that at 4 and 8 km and at the tropopause (about
10 km) the lowest pressure is at the rear of a low. The tropopause
is highest and coldest at the rear of a high. As the diagram shows, a
wave on the tropopause is associated with the surface low-pressure area.

Under average conditions the tropopause is high and cold over the
equator, whereas at the poles it is low and relatively warm, as shown in
Fig. 3, section 2. Thus the conditions at the tropopause over the
rear of a high, i.e., over the warm frontal surface, correspond to those
occurring on the average at the tropopause a few degrees of latitude to
the south, but over the rear of a low, i.e., over the cold front, the con-
ditions are similar to those a few degrees to the north. These considera-
tions suggest the presence of a horizontal north-south wave motion of
the tropopause, which in turn produces a vertical wave motion of the
latter. The cyclonic motion in the cold and warm air in the tropo-
sphere thus extends above the tropopause into the stratosphere, as
manifested by a movement north and south of the tropopause in the
vicinity of a surface low-pressure area.

The exact nature of the relationship between the wave motions of
the tropopause and surface low-pressure areas is not known. Some
meteorologists believe that the upper wave is induced by the variations
near the surface. Others hold the opposite view, that tho decrease in
pressure in the stratosphere results in a corresponding decrease in
pressure below, at the surface of the earth. The truth probably lies
somewhere between these two opinions. More study of individual
depressions is needed before these questions can be answered with
assurance. However, it is probable that no single picture is applicable
to all depressions.

114. Other Types of Depressions. All depressions do not have the
structure outlined in the previous sections. Because of the unequal
heating over the earth's surface, there is a tendency for thermal depres-
sions to develop over warmer regions. Of course influences other than
surface heating are significant, such as upper air conditions, but some-
times this factor is predominant. The summer monsoon low of south
Asia (Fig. 5, section 3) is the most notable example of this type. The
winds flow in the usual manner about the center of low pressure, but
extensive cloud systems are not always present. On the contrary, long
periods of clear weather often occur in .regions where there are no large

Sec. 115] TROPICAL HURRICANES. TORNADOES 283

ranges of hills or mountains. It cannot be said, therefore, that cyclones
always bring disturbed weather conditions.

The development of some cyclones may be due to instability withia
the air mass. These are called instability depressions. They sometimes
develop in polar air masses as they move over warmer portions of
the earth's surface. A depression of this type sometimes forms in
winter and early spring in an extensive mass of polar continental air
which becomes stagnant over the Great Lakes and acquires heat and
thus instability from the underlying warm water. Instability is pro-
nounced under such conditions. These depressions initially have no
discernible frontal structure, although fronts may develop in them
later.

Orographic depressions sometimes form in the lee of mountain ranges.
* There is a slight tendency for compression of the air to the windward
to occur, producing a ridge of high pressure, and for rarefaction on the
lee side, producing a shallow depression there. Orographic cloud and
rain may occur over the windward slope of the mountain, while
descending motion of the air to the lee may prevent the formation of
cloud in the shallow depression. These depressions will, at times, move
from the region of their formation and into the general west to east
track of extra-tropical cyclones.

115. Tropical Hurricanes. Tornadoes. Tropical hurricanes, as their
name suggests, are violent storms developing over the tropical portions
of the oceans. These storms do not occur near the equator, since the
deflecting force of the earth's rotation (section 34) is not sufficiently
strong there to produce cyclonic motion. They develop only at lati-
tudes greater than 5 or 6.

As these storms form over the ocean in tropical latitudes where there
arc few weather reports from ships, there is some doubt as to the manner
of their formation. Two main suggestions have been put forward.
The Norwegian meteorologists and some others associate their develop-
ment with an " inter-tropic front." They believe that this inter-tropic
front lies between the trade wind systems of the northern and southern
hemispheres. The trade wind systems move northward and southward
with the seasons. The maximum northward displacement occurs during
the months of August, September, and October. Also during these
months the difference in temperature across the front between the air
of the northern hemisphere and that of the southern hemisphere is at a
maximum. For these reasons, the maximum number of hurricanes
occurs during these months. Other meteorologists believe that tropical
hurricanes develop whenever high temperature and moisture content in
the surface air coincide with suitable upper air conditions. This may

284 CYCLONES AND ANTICYCLONES [Chap. 16

occur when fresh polar continental air has moved from the continent
and has become rapidly modified in the lower layers, producing insta-
bility of the real latent type.

There are a number of characteristic features of a tropical hurricane.
One of the most striking of these is the presence of what is known as the
eye of the storm. In the eye there are higher temperatures and lower
humidities than in the outer portions of the storm region, few clouds,
nearly calm conditions, and a pressure minimum. The diameter of the
eye of the storm varies from 5 to 50 mi. The foregoing facts suggest
that there is subsiding air in the central portion of the storm. Air may
be brought down from above as a result of the extremely low pressure.
In the area outside the oye there arc very high winds and heavy rain-
fall.

It is generally thought that the latent heat released by condensation
provides the energy of tho storm. It is difficult to understand how the
air ascending from below is removed at upper levels. Several hypoth-
eses have been put forward, but no out i rely satisfactory explanation
has been developed as yet.

Fortunately only a small number of tropical hurricanes reach con-
tinental areas. The usual path of tropical hurricanes in the Atlantic
is shown in Fig. 114. The storm develops to the south of the Atlantic
anticyclone and moves westward. Generally its path is approximately
parallel to the isobars of the sub-tropical high. It acquires a north-
westward component, then moves northward and finally northeast-
ward. The storm is said to recurve when it acquires an eastward com-
ponent of motion. The storm decreases in intensity as it moves to
higher latitudes, and it may continue thereafter with all the character-
istics of an ordinary depression of middle latitudes. However, if the
hurricane does not recurve, it may move along the eastern coast of the
continent, doing great damage. If the hurricane moves inland, its
energy is rapidly dissipated, but it may give very heavy rainfall even in
its later stages.

The region of the western Atlantic from the West Indies north to
Bermuda is frequently visited by these storms. They are found, too,
in the western Pacific along the China coast, where they are called
typhoons. Relatively few occur in the southern hemisphere.

A tornado has many features of a tropical hurricane but is of much
smaller dimensions. It is a violent whirlwind several hundred yards in
diameter, moving rapidly and destroying everything in its path. They
occur most frequently in the United States and Australia. They
develop in air of great instability and are frequently associated with
cold fronts at the leading edge of an outbreak of fresh polar continental

Sec. 116}

CYCLONES AND ANTICYCLONES

285

air in the spring and early summer. They are usually accompanied by
thunderstorms and heavy rain, and at times hailstorms.

The tornado is intermediate in size between phenomena such as dust
devils on the one hand, and tropical hurricanes on the other. It is

FIG. 114. The path of a tropical hurricane over the western Atlantic.

essentially a cyclone with a center of low pressure (sec problem 3,
chapter 6), and winds moving as specified by Buys Ballot's law. The
horizontal dimensions of a tornado are too small for it to be shown by
isobars on a weather map.

116. Comparison of Upper Air Conditions above Cyclones and Anti-
cyclones. It might bo thought that convergence of air in the lower
levels would lead to high pressure. Just the reverse is true, however,
for there is divergence from anticyclones, which are high-pressure
systems. The winds move in a clockwise manner around the center of
high pressure in such a manner that the deflecting force just balances

286

CYCLONES AND ANTICYCLONES

[Chap. 16

the sum of the centrifugal force and that arising from the pressure
gradient (section 37). There is descending motion of the air in the
anticyclone, as indicated by the divergence at the surface. A common
misconception, widely held, is that the air at upper levels above an
anticyclone is colder than that at the same levels above a cyclone.
Fig. 115 shows that the reverse is true. The two curves give mean
values of the temperature distributions with height in cyclones and

14 1-

60 -50 -40 -30 -20 -10

FIG. 115. Mean temperature curves in cyclones and anticyclones over England.

(After Dines.)

anticyclones. A similar comparison can bo mado between the tem-
peratures in anticyclones and cyclones by noting the temperatures for
the center of a high and the center of a low as given in Fig. 1136, sec-
tion 113. These diagrams show two significant features. The first is
that, at nearly all levels of the troposphere, air in anticyclones is warmer
than that in cyclones. In the lowest 2 km the evidence is conflicting.
This difference is probably due to the relative development of the
cyclones. When the cyclone is occluded, the surface air at the center
of the cyclone will be colder than in a cyclone which has an open warm
sector. Thus in the lowest levels the mean temperature depends on
whether the majority of observations are taken before or after occlusion.
The second point of interest is that the stratosphere is higher and
colder above anticyclones than above cyclones.

Sec. 117} THE COLD ANTICYCLONE 287

The pressure at the base of an air column is proportional to the
density of the air above, and so is less with air columns with high tem-
peratures. The weight of the air in the troposphere over a center of
high pressure is, then, less than that over a center of low pressure.
The higher pressure at the surface must, for this reason, be the result
of heavier and therefore colder air in the stratosphere. A statistical
analysis by W. H. Dines reveals the close relationship among these
variables. Between the pressure at the surface and the pressure at
9 km, the coefficient of correlation is 0.08, Between the pressure at
9 km and the temperature at 4 km, the coefficient of correlation is 0.82.
Between the pressure at 9 km and the mean temperatures from the
surface to 9 km, it is 0.95. Between the pressure at 9 km and the
height of the tropopause, it is 0.84.

The warm air found in the troposphere above the center of high
pressure may be explained by the subsidence that accompanies the
divergence.

The low temperatures in the stratosphere may be explained as a
result of the wave motion of the tropopause discussed in section 113.
The horizontal movement in the stratosphere which accompanies the
surface pressure variation carries northward in the region above the
anticyclone the colder air of the southern stratosphere. At the same
time there is a southward movement of the warm air of the northern
stratosphere above the center of low pressure. Further evidence to
support this explanation is found in the distribution of ozone mentioned
briefly in section 5. In the normal distribution the ozone, which is
found chiefly in the region from 20 to 40 km, increases with latitude.
But near a depression the ozone content is greater than the normal for
the latitude, while near an anticyclone the amount is less than normal.

117. The Cold Anticyclone. Variations from the mean temperature
values discussed in the previous section are found in individual anti-
cyclones. Certain high-pressure systems exist, not because of cold air
in the stratosphere, but because of a shallow layer of very cold air near
the surface of the earth. These are called cold anticyclones. Semi-
permanent anticyclones of this type are found over Greenland and over
the Antarctic continent. Rapid subsidence occurs with air diverging
horizontally from the ice caps. The cold air is shallow, though, and
warm air moves in above the cold dome.

A similar situation exists over Siberia and over northwest Canada
during the winter. The underlying snow surface, being very cold as a
result of long-wave radiation, cools the adjacent air. The air is cold
only in the lowest few kilometers, and warm air lies above it. Fig. 116
illustrates a west-east cross section of an anticyclone of this type over

288

CYCLONES AND ANTICYCLONES

[Chap. 16

North America. The exact determination of the upper boundary of
the cold dome is not easy, owing to the mixing that takes place. The
eastern boundary is a steep cold frontal surface, while to the west the
warm PP air ascends the warm frontal surface.

In North America these air masses are frequently so shallow that
they are prevented from extending to the west coast of the continent
by the Rocky Mountain barrier. They then follow a southeasterly
path changing to easterly as they leave the continent and move over

10C

FIG. 116. A vertical cross section through a cold anticyclone.

the Atlantic Ocean. In Europe there is no extensive mountain barrier
and the cold Siberian anticyclone is free to move to the west. Thus
western Europe in the winter is more often under the influence of a
continental type of climate than the Pacific coast of North America.

As a cold anticyclone becomes heated during its advance to lower
latitudes, the pressure at times decreases, and such a weakened cold
dome frequently collapses as a center of low pressure moves toward it.
Another change sometimes occurs. If the cold anticyclone becomes
stationary in middle latitudes, the pressure above the cold dome fre-
quently increases as a result of advection of cold air at that level.
Thus the decrease of pressure resulting from the increase in tempera-
ture at the surface is compensated for by an increase in pressure aloft,
and the cold anticyclone is transformed into a warm anticyclone.

118. The Warm Anticyclone. Warm anticyclones, such as those
comprising the sub-tropical high-pressure belts, have high surface pres-
sures in spite of their high surface temperatures. These anticyclones
extend, then, to great heights. This type of anticyclone is also found
in the temperate zone. If one of the sub-tropical high-pressure cells
divides, one of the component parts of the system may penetrate the
region of westerlies and move in the general westerly circulation. Also,
as indicated in the previous section, at times a cold anticyclone which
has become stagnant in middle latitudes is transformed into a warm
anticyclone.

Sec. 119] CONVERGENCE AND DIVERGENCE 289

The warm anticyclone is more stable than a cold anticyclone. A
cyclone moving eastward toward one of the former will become stagnant
on the western boundary of the anticyclone or will move in the anti-
cyclonic circulation about the center of high pressure.

As shown by the variation of pressure at 8 km indicated in Fig. 113a,
section 113, the wedge of high pressure at that level is found above the
rear of the surface high-pressure system. Above the center of high
pressure at the surface the clouds at the cirrus level move from the
northwest along the eastern edge of the ridge of high at that level. In
a warm anticyclone the cirrus clouds sometimes disappear above the
high-pressure system at the surface. This fact suggests that the sub-
sidence in the anticyclone extends up to the cirrus level.

119. Convergence and Divergence in Cyclones and Anticyclones.
Some of the weather occurring in cyclones and anticyclones is the result
of convergence and divergence in these systems. In all portions of a
cyclone there is horizontal convergence in the frictional layer near the
surface, which accounts in part for the cloudiness which occurs in low-
pressure areas. Similarly, horizontal divergence in the frictional layer
accounts in part for the clear skies associated in general with anti-
cylones. However, frictional inflow and outflow do not explain the
variations in weather from one section of a cyclone or anticyclone to
another.

Convergence and divergence resulting from the variation of the
gradient wind with latitude account for many of these variations in
weather. The gradient wind velocity v in a cyclone with circular
isobars is, according to (37-2),

/ 9
* /w

wr sin </> + * /cor sin d> -\

^' p dr

since the Coriolis parameter I = 2co sin <. In this equation r represents
the radius of curvature of the motion. Consider now a cyclone centered
at latitude <. The gradient wind VQ at $o and v\ at <t>\ may be com-
puted, if r, p, and dp/dr are known. With the horizontal convergence
or divergence found in this manner, the resulting average vertical
velocity wi at height z\ between </> and 0i is obtained by substituting
for Vi v 0) yi, and Zi in (45-9), where y\ is the arc of a circle of radius r
extending from < to <t>\. Remember that the length of the chord
subtended by angle 6 is

2r sin -

290 CYCLONES AND ANTICYCLONES [Chap. 16

and it may readily be shown that the length of the arc between </> and 0i

360r

where E is the radius of the earth.

r rhe following example shows the order of magnitude of the vertical
velocities at a height of 1 km above the surface at varying distances
from the center of low pressure, when the latter is at 45 N latitude,

yclone

-2

'0 400 800

Radius (km)

1200

FIG. 117. Variations of pressure gradient with distance from the centers of cyclones

and anticyclones.

when 0o = 42.5, 0! = 47.5, p = l.l X 10~ 3 gm per cm 3 , and the
variation of the pressure gradient dp/Or with r is as given by the curve
marked cyclone in Fig. 117. The positive direct-ion of r is taken as
outward from the center. The results of the computation arc given in
the upper part of the following table. The positive velocities indicate

VERTICAL VELOCITIES AT 1 KM RESULTING FROM
THE VARIATION OF THE (JUADIKNT WIND VELOCITY WITH LATITUDE

Type of
Pressure
System

Radius
(km)

*'o
(in per ser)

Z'l

(in per sec)

Vertical Velocity wi
(cm per sec)

TCast of
Center

West of
Center

Cyclone

400
800
1200

23 2
15.4
8 6

22 2
14 5

-fO Hi
+0.16
+0 11

-0.16
-0.16
~0.ll

Anticyclone

400
800
1200

1 9
10
22 8

1 8
9 5
19.9

-0 02
-0 19
-0 52

+0.02
+0.19
+0 52

Sec. 119] CONVERGENCE AND DIVERGENCE 291

that the air is ascending, while the negative ones show that it is de-
scending. The vertical velocities at 2 km will be twice as great as
those at 1 km, shown in the table, provided that the field of motion
extends to 2 km.

A similar computation for an anticyclone may be carried out by using
the equation 374 for the gradient wind in the anticyclonic case

/ 2 2 . 9 r
v = rw sin \\irur sin <t>

\ p

and the appropriate values for dp/dr as given by the curve designated
anticyclone in Fig. 117. In deriving this equation, the positive direc-
tion for r was taken as inward toward the center of high pressure, so
that the negative values of dp/dr shown in Fig. 117 are positive when
substituted in the equation. The vertical velocities obtained in this
manner are given in the lower portion of the table.

The distributions of dp/dr with r shown in the figure arc representa-
tive of those in many cyclones and anticyclones, so that it is permissible
to make a number of generalizations on the basis of the values shown
in the table. The vertical velocities are always small except at the
eastern and western outer limits of an anticyclone. Here the vertical
velocities of about 0.5 cm per sec at 1 km and 1 cm per sec at 2 km are
sufficient to account for, at least in part, the cloudiness and light pre-
cipitation often observed near the western edge of an anticyclone, and
the clear skies near the eastern edge. The values of vertical velocities
given in the table are the maxima to be expected at 1 km from this cause
in cyclones and anticyclones, since the variation of $ and so of sin for
any given length of arc is a maximum to the east and west of the center.
Near the north-south axis of a cyclone or anticyclone, the vertical
velocities are much smaller than indicated in the table, since the varia-
tion of <#> along the arc is slight, and they become zero at the axis where
the variation of <t> along the arc is zero. The weather occurring at the
northern and southern edges of an anticyclone must thus be accounted
for in other ways, perhaps by variations in the curvature of the air
motion, as outlined in section 141.

The distribution of vertical velocities in a frontal depression resulting
from the variation of the gradient wind with latitude and pressure
gradient is shown in Fig. 118a. The circular lines are streamlines
of the air motion at distances of 400, 800, and 1200 km from the center.
Since the wind in the warm sector is approximately west when the
latter is in the position shown, the vertical velocity is very nearly zero.
If the warm sector is oriented so that the air has a considerable north-

292

CYCLONES AND ANTICYCLONES

[Chap. 16

ward component of motion, horizontal convergence and ascending
motion occur, the approximate magnitude of which may be computed
with the aid of (45-10). Fig. 1186 gives the distribution of vertical
velocities in an anticyclone, based on the figures given in the above
table.

(a)

(b)

Fio. 118.

Vertical velocities resulting from latitudinal convergence and divergence
in (a) cyclones and (b) anticyclones.

According to (47-4), ascending motion at any level results in increas-
ing pressure at that level, whereas descending motion results in de-
creasing pressure. On the basis of the above results alone, then, the
pressure at a height of 1 km should rise ahead of a cyclone and fall
behind it. The reverse actually happens, however. This apparent
anomaly is resolved because the pressure at any level also depends
on the convergence or divergence and advcction at higher levels, as
shown in section 47. The effect of vertical velocity is, in this case,
ovcrcompensated by the convergence and divergence at greater heights
in the warm sector air, of the type discussed in section 48, and by advec-
tion. Similar considerations apply to the pressure variations ahead of
and behind an anticyclone.

The anticyclone that dominated the eastern portion of Canada and
the United States at 01.30 h EST on November 5, 1942, is shown in
Fig. 119. Since the isobars on the western edge of the system are
practically straight, the variation of the geostrophic wind with latitude
may be used in determining the horizontal convergence. By using
(45-10), the average velocity of ascent at 2 km between latitudes 40
and 55 is found to be 0.5 cm per sec. The average velocity of ascent
of the warm sector air in an occluding frontal depression, which pro-
duces thick cloud systems and moderate to heavy precipitation, is
about 4 cm per sec. Thus the computed vertical velocity of 0.5 cm per

PROBLEMS AND EXERCISES

293

sec is sufficient to account for the cloud development and light precipita-
tion that occurred at the western edge of the anticyclone, as shown in
Fig. 119.

100

50'

35'

85<

FIG. 119. The weather map for 01.30h, November 5, 1942, illustrating the hori-
zontal convergence in air with a northward component of motion.

PROBLEMS AND EXERCISES

1. An airport is in the cold air 250 mi east of a warm front lying in a north-south
direction. The front is moving eastward at 15 mph. A plane is 300 mi west of the
airport and is flying at 3000 ft alx>ve the surface eastward at 100 mph. Assuming
that the warm frontal surface has a sloi>e of /^oo at what distance from the station
will the plane pass through the frontal surface?

2. If a cold front passed a station at 10.00 h, moving at the rate of 25 mph, at
what time would the winds at 2000, 5000, and 10,000 ft veer with the frontal passage
if the frontal surface is assumed to have a slope of Jo?

294 CYCLONES AND ANTICYCLONES [Chap. 16

3. A warm front, slope Kso* li es 400 mi from a station at 08.30 h and is moving
toward the station at a speed of 20 mph. At what times will the wind at 4000 and
8000 ft veer as a result of the passage of the frontal surface?

4. The warm sector air of a depression overruns the cold air ahead. The warm
air at the surface has a temperature of 67 F and a dew point of 55 F. If the front
has a slope of Joo at what distance ahead of the surface front will the air become
saturated?

BIBLIOGRAPHY

Admiralty Weather Manual, London, H. M. Stationery Office, 1938. Chapters 18, 19.
Brunt, D., Physical and Di/namical Meteorology, London, Cambridge University

Press, 1939. Chapters 17, 18.
Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book

Co., 1937. Chapters 5, 12.

Haurwitz, B., Dynamic Meteorology, New York, McGraw-Hill Book Co., 1941. Chap-
ter 15.
Petterssen, S., Weather Analysis and Forecasting, New York, McCi raw-Hill Book Co.,

1940. Chapters 4, 5, 6.
Problems of Modern Meteorology, London, Royal Meteorological Society, 1934.

Numbers 1, 9.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 2

(1936), Chapter 8.
109. Bergeron, T., " Uber die dreimensional verknupfende Wetteranalyse," Geofys.

Publ., 5, No. 6, Oslo, 1928.
109. Petterssen, S., " Contribution to the Theory of Frontogenesis," Geofys. Publ.,

11, No. 6, Oslo, 1934.

111. Shaw, W. N., and R. G. K. Lernpfert, The Life History of Surf ace Air Currents,
London, Meteorological Office, M. O. 174, 1906.

112. Bjerknes, J., " On the Structure of Moving Cyclones," Geofys. Publ., 1, No. 2,
Oslo, 1918.

112. Bjerknes, J., "Practical Examples of Polar Front Analysis over the British

Isles in 1925-26," Geophys. Mem., No. 50, London, 1930.
112. Douglas, C. K. M., " Some Aspects of Surfaces of Discontinuity," Q. J. Roy.

Met. Soc., 55, 123-147 (1929).

112. Gold, E., " Fronts and Occlusions," Q. J. Roij. Met. Soc., 61, 107-157 (1935).

113. Bjerknes, J., " Exploration de quelques perturbations atmospherique a Paide
de Bondages rapproche"s dans le temps," Geofys. Publ., 9, No. 9, Oslo, 1932.

113. Bjerknes, J., " Investigations of Selected European Cyclones by Means of Serial

Ascents, Case III," Geofys. Publ., 11, No. 4, Oslo, 1934.
113. Bjerknes, J., and E. Palmen, " Investigations of Selected European Cyclones

by Means of Serial Ascents, Case IV," Geofys. Publ., 12, No. 2, Oslo, 1937.
113, 116. Penner, C. M., " The Effects of Tropospheric and Stratospheric Advection

of Pressure and Temperature Variations, " Can.J. Res., A19, 1, 1941.
115. Mitchell, C. L., " West Indian Hurricanes and Other Tropical Cyclones of the

North Atlantic Ocean," Monthly Weather Review, Supplement 24, Washington,

D. C., 1924.
115. Tannehill, I. R., Hurricanes, Princeton, Princeton University Press, 1942.

CHAPTER 17
WINDS

120. Geostrophic and Gradient Winds. When a hole is made near
the bottom of a cask full of water, a force acts upon the liquid inside
the cask to cause the water to flow out the hole. This force arises from
the difference in pressure on the two sides of the hole, that on the
inside resulting from the weight of the water and the air above the
water, and that on the outside resulting from the weight of the air
only. Since the pressure inside is greater, the water is forced out. In
a similar manner the variations in the pressure of the air at different
points in a horizontal plane in the earth's atmosphere give rise to forces
which tend to make the particles of air move from regions of high
pressure to regions of low pressure. This force is called the pressure
gradient force, and always acts normal to the isobars toward low pres-
sures. As shown in section 33, its value on a unit mass of air is

p dn

where n is measured in the direction along the normal to the isobars.

Another force acts on any particle moving relative to the surface of
the earth. It is caused by the rotation of the earth and is called the
Coriolis force, or the deflecting force of the earth's rotation. Its value
for unit mass, as shown by (34-20), is

2co sin <t> V

In this expression V is the total velocity. The deflecting force acts
normal to the velocity, to the right in the northern hemisphere, and to
the left in the southern hemisphere. Since this force is normal to the
velocity it affects the direction of the air motion but not its speed.

When the pressure gradient force exactly balances the deflecting
force on a mass of air, the motion is said to be geostrophic. This situa-
tion is illustrated in Fig. 32, section 35, with the pressure gradient force
acting toward low pressure being just equal to the deflecting force
acting in the opposite direction. It follows that, since the pressure
gradient force acts in a direction normal to the isobars, the motion of
the air, or the wind, must be parallel to the isobars when the motion is

295

296

WINDS

[Chap. 17

geostrophic. Such winds are known as geostrophic winds. Numerous
investigations have shown that, above the heights where the surface
friction exerts an influence, winds do blow parallel to the isobars most
of the time and therefore these two forces are usually balanced. If
these two forces are equated, it follows that the geostrophic wind
velocity

2cop sin dn

(120-1)

If then on a weather chart the isobars are drawn for fixed intervals of
pressure, the geostrophic wind velocity for a given latitude is inversely
proportional to the distance between the isobars. In any given situa-
tion V g may be computed by substitution of the proper values in (120-1).
Values of V g obtained in this manner are given in the table in section
35. The necessity of doing this calculation each time may be elimi-
nated by the use of a scale such as the one shown in Fig. 120. This scale

050100 200 300 500 750 1000

Distance between consecutive 4mb isobars (miles)

FIG. 120. Geostrophic wind scale for 4-mb isobar intervals.

was computed for isobars drawn at intervals of 4 mb and for p = 1.163 X
10~ 3 gm cm~ 3 , which is the average density at 2000 ft. The abscissa
is the distance in miles between two consecutive 4-mb isobars and
the ordinate is latitude. The curves are lines of equal geostrophic
wind velocity in mph. The geostrophic wind for any given spacing
of isobars and at any latitude from 25 to 75 may thus be obtained
directly. For example, if the isobars arc 200 mi apart the geostrophic
wind is 20 mph at 50 latitude and 32 mph at 30 latitude. The scale
may readily be adjusted for use with isobars drawn at intervals other
than 4 mb. Thus with 2-mb isobars the distance between every second
isobar may be taken.

When the scale of distances for the geostrophic wind scale is made
to correspond to the scale of distances on the weather map, the value
of the geostrophic wind velocity may be determined directly. Measure
the distance between consecutive isobars and apply this to the geo-

Sec. 120} GEOSTROPHIC AND GRADIENT WINDS - 297

strophic wind scale at the appropriate latitude, reading off the answer.
Allowance must be made for the type of map projection used, since the
distance between two points on a map may not be the same as the true
distance between the points. For instance, in a conformal conic pro-
jection having standard parallels at 30 and 60, measured distances
are true distances only at these two latitudes, although the error at
intermediate latitudes is not large. As the equator or the pole is
approached, however, the error becomes progressively greater.

The scale may be used not only to determine the geostrophic wind
velocity but also the component of the velocity in any direction. Thus
if a front intersects the isobars but not at right angles, the geostrophic
wind has one component parallel to the front and one normal to the
front. The latter, useful in determining the velocity of the front, is
obtained by measuring the distance between the intersections of con-
secutive isobars with the front and using that distance to determine
the desired component of the geostrophic wind.

When the direction of motion of the particle of air is not straight
but curved, the centrifugal force acts upon the air. The magnitude of
this force is F 2 /r, where r is the radius of curvature of the path, and is
directed outward along the; radius of curvature. When the motion
is such that there is a balance among the pressure gradient force, the
deflecting force, and the centrifugal force, the wind is known as the
gradient wind. As shown in section 37, when the isobars are cycloni-
cally curved the nature of the balance among the three forces is differ-
ent from that when the isobars are anticyclonically curved. The equa-
tions for the gradient wind, as given by (37-2) and (374), are

for cyclonic curvature, and

"=2

IW_rdp
\ 4 p dr

for anticyclonic curvature, where I is the Coriolis parameter 2co sin <f>.
When the curvature is cyclonic the gradient wind is less than the
corresponding geostrophic wind; for anticylonic curvature the gradient
wind is larger. The difference increases as r decreases. Thus in a
tropical hurricane at latitude 30 and at a distance of 200 km from the
center of rotation the gradient wind is 21 m per sec when the geostrophic
wind is 50 m per sec for the same pressure gradient.

298 WINDS [Chap. 17

121. Thermal Winds. Isallobaric Winds. The geostrophic and
gradient winds are closely related to the magnitude and direction of
the pressure gradient. As shown in section 36, there is a tendency for
low-pressure areas to form aloft above cold regions, and high-pressure
areas to form above warm regions. These changes in the pressure
gradient with height result in corresponding changes in the geostrophic
wind with height. The vector difference between the geostrophic winds
at two levels is called the thermal wind in that height interval. The
thermal wind blows about an area of low temperature in the same direc-
tion that the geostrophic wind blows about an area of low pressure.
The additional component of equations 36-14 and 36-15 which is due
to the horizontal thermal gradient between the levels z and z is

n'TT

V t - - (, - 20 ) (121-1)

where dT/dn is the rate of change of temperature in the direction n,
normal to the isotherms, and T the mean temperature of the layer.
The geostrophic wind, (120-1), is given by

v _!

V ~ lp On'

where n 1 represents the direction of the pressure gradient. If the dis-
tance between isotherms is equal to the distance between isobars, it
follows that

(121-2)
V g T Ap U ;

where finite differences are substituted for differentials. The right-
hand side of (121-2) can be evaluated approximately. Assume that
p = 1.2 X 1(T 3 gm cm" 3 , A7 7 = 5 F, z - z () = 1000 ft, T = 273 A,
Ap = 4 mb. After converting these to cgs units where necessary, it fol-
lows that

V t = 0.09147.,

where V g t indicates the wind velocity obtained from the geostrophic
wind scale, using the spacing of the isotherms drawn at intervals of
5 F. With sufficient accuracy for forecasting purposes, the constant
may be remembered as one-tenth for every 1000-ft interval for which
the wind is to be calculated. Thus if the height interval is 5000 ft, the
thermal component is one-half of the value read from the geostrophic
wind scale.
The method of using the thermal wind is illustrated in Fig. 121.

Sec.

THERMAL AND ISALLOBARIC WINDS

299

Isobars at level z are indicated by p, p + 1, p + 2, and the geostrophic
wind at the point P is VgQ. Mean isotherms for the layer z to z are
indicated by T, T + 1, T + 2. The thermal wind, calculated by
(121-1) or by (121-2), is indicated by V t and is drawn as a vector from
the point P parallel to the isotherms with low temperatures on the left.
The resultant of V gG and Vt, V QJ is the geostrophic wind for level z.

FIG. 121. The application of the thermal wind.

The behavior of the thermal wind may be described by four rules.

(a) If low temperatures are associated with low pressures, the wind
increases with height.

(6) If low temperatures are associated with high pressures, the wind
decreases with height.

(d) If the wind blows from low temperatures to high, it will back
with height.

The thermal gradient from pole to equator is a permanent feature
of the troposphere. The resultant thermal wind component causes the
winds in the temperate latitudes to acquire a westerly component with
increasing height, and finally produces the west to east flow found in
the upper troposphere and the lower stratosphere.

A second cause for a variation in the winds of the free atmosphere is
the changing pressure distribution. As shown in section 46, in a region
of falling pressure there is a component of the wind toward the center
of the isallobaric low, and in a region of rising pressure there is a com-
ponent away from the isallobaric high. The isallobaric wind, as it
is named, differs fundamentally from the thermal wind. The thermal
wind is a component added to or subtracted from the geostrophic wind
at one level to obtain the geostrophic wind at another level. The isallo-
baric wind is a correction to the geostrophic wind at any given level
where there is no longer a balance of forces.

The value of the isallobaric wind, as obtained from equations 46-10

300 WINDS [Chap. 17

and 46-11 is, dropping the negative sign which indicates only the direc-
tion of the motion,

1 a " (121-3)

I 2 p dn \dt

where the normal n is taken with respect to the isallobars, and where
the direction of the isallobaric wind is along the normal toward an
isallobaric low. The tendencies plotted on the weather map may be
used to obtain the values of dp/dt and so to draw the isallobars. Di-
viding (121-3) by (120-1) obtains

Vgi ZAp t

where V g i is the value obtained from the geostrophic wind scale using
the spacing of isallobars. If dp/dt is in millibars per 3h, Ap is in milli-
bars, t is the time in seconds for which tendencies are computed, and
if again finite differences are substituted for differentials, the right-hand
side may readily be evaluated. Thus if isobars are drawn for 4-mb
intervals, and isallobars for intervals of 1-mb change in 3 h, then

In practice the isallobaric wind cannot be obtained as accurately as
the geostrophic wind. Because of this, a sufficient degree of accuracy
in middle latitudes is obtained if the constant of proportionality is taken
as one-quarter, which is the value at 40 latitude with 4-mb isobars and
1-mb isallobars.

The diagram in Fig. 122, a weather map for a portion of the southern
United States for March 12, 1941, at 07.30 h, EST, shows the evaluation
and use of the isallobaric wind. The method of plotting and drawing
up the map is given in section 162. The broken lines are the isallobars
and the full lines are the isobars. Consider the area about point A first,
which is j ust to the sout h of the center of an isallobaric high. The isallo-
baric gradient about the center is nearly symmetrical with the mean
value of 1 mb per 3 h per 80 mi. A distance of 80 mi at latitude 31
corresponds to a velocity V i of 75 mph, as can be seen from the geo-
strophic wind scale. Substituting this value in (121-4) the isallobaric
wind is found to be 23 mph. Since the isallobaric wind blows outward
from the isallobaric high, and since the point A lies to the south of the
center of the high, the isallobaric wind is nearly north, as shown by the
vector AP. The geostrophic wind, 46 mph, is represented by the vector
AQ. The resultant is given by the vector AR. Since AR has a com-

Sec. 121]

THERMAL AND ISALLOBARIC WINDS

301

ponent of 23 mph normal to the front and directed nearly southward,
the front at this point should move southward with a velocity of 23 mph.
If the isallobaric effect is neglected, the gcostrophic wind alone would
indicate a stationary front, or perhaps even a slight northward move-
ment of the front. The map succeeding this one showed that the front

100*

Fio. 122. The weather map for 07.30 h, March 12, 1941, showing isallobaric wind

components.

had moved southward, as would be forecast on the basis of the foregoing
analysis. The essential validity of this result is further confirmed by the
north to north northeast winds just to the north of the front which are
force 6 on the Beaufort scale. This figure corresponds to a velocity of
about 25 mph, which should also give the velocity of the front.

A similar analysis may be made using the isallobaric low at B. Since
the isallobaric wind blows into the isallobaric low, the isallobaric com-

302 WINDS [Chap. 17

ponent at C, just to the south of the front, is represented by the vector
CB, and the isallobaric wind is 33 mph. The geostrophic wind at C,
22 mph, is represented by the vector CD. The vector CE represents
the resultant. The component of this vector normal to the front is
about 17 mph. If the front in the region A to C moves with a constant
velocity, then the cold air is undercutting the warm air and causing some
ascent of the latter.

122. Monsoon Winds. As was described in section 3, the distribu-
tion of land and sea produces a deviation from the general distribution of
pressure which would occur on a globe with a uniform surface. This
deviation is not constant but varies from winter to summer. In winter
high-pressure areas develop over the land masses, and the low-pressure
areas over the ocean deepen. The situation is reversed in the summer
season, with the high-pressure areas over the ocean becoming more
intense and low-pressure areas developing over the land.

These pressure variations are reflected in the winds (see Figs. 5 and 6,
section 3). During the winter the winds blow anticyclonically about
and outward from the continental high-pressure areas. With the
development of the summer continental low during the spring months,
the winds reverse, blowing cyclonically from the ocean toward the land
areas.

These circulations are most pronounced over the eastern half of the
Eurasian continent where they are known as the monsoons. Similar
circulations, but on a smaller scale, are discernible over North America,
Africa, and Australia. They can be observed also around the shores of
large inland bodies of water, such as the Caspian Sea and Lake Superior,
if the mean winds arc used in order to eliminate variations produced by
passing highs and lows.

It will be noted that the direction of the monsoon wind differs in
different coastal regions. Thus in China the winter monsoon is north-
west, but in India it is northeast, with a similar difference in the summer
monsoon.

The monsoon wind is a striking feature of the general circulation. Yet
it is a result of the general pressure distribution, and as such its velocity
is determined by the pressure gradient existing at the time.

123. The Effects of Friction. Diurnal Variations. The winds which
have been discussed in the last three sections are winds which exist in
the free atmosphere. In the following three sections, winds which occur
near the surface of the earth and are affected by local influences are
treated.

The friction between the moving air and the surface of the earth
causes the winds to deviate from the geostrophic wind value and to blow

Sec. 123] THE EFFECTS OF FRICTION 303

across the isobars toward the low-pressure region. The effect of friction
on the surface wind is discussed in section 52. There, under the simpli-
fying assumption that the frictional force acts in a direction opposite to
the wind and is proportional to the velocity, it is shown, according to
(52-14) and (52-15), that the velocity

and that

tan B = -

In these formulas, k is the constant of proportionality, or the coefficient
of friction, and 6 is the angle at which the wind blows across the isobars,
as shown in Fig. 48, section 52.

It will be noted that, with decreasing values of k, the velocity increases
and the angle between the wind and the isobars decreases. Thus, over
the sea where the frictional forces are small, the actual wind is much the
same as the geostrophic wind in both magnitude and direction. For
this reason reliable wind reports from ships and from island stations may
be used to determine the direction and the spacing of the isobars, pro-
vided that isallobaric effects arc absent. The isobars should be placed
so that the surface wind is about 70 per cent- of the geostrophic wind, and
blowing at an angle of about 20 across the isobars.

Over the land areas the frictional force varies, depending on the
underlying surface. Over plains the effect of friction is only slightly
greater than over the sea. Hence the winds in such areas, when the
velocity is greater than 12 mph, may be used with confidence in drawing
the isobars. Over mountain areas the friction is great and surface winds
give little help in drawing up a weather map unless the effect of the local
topography on the wind at a given station is known.

The coefficient of friction is dependent not only on the nature of the
underlying surface but also on the stability of the air. When the air is
in neutral equilibrium or has little stability, little energy is needed to
lift a particle over an obstacle at the surface of the earth. When,
though, the air is very stable, as in an inversion, more energy is needed to
lift a particle against the stabilizing factors of the environment. It
follows that k, the coefficient of friction, is large with stable air, and small
with unstable air.

The dependence of k on the stability of the air is shown in the diurnal
variation of the wind at the surface of the earth. Curve (a) of Fig. 123

304

WINDS

[C/iap. 17

gives the mean values of the hourly wind velocities at Winnipeg, Mani-
toba, for the month of July, 1942, and curve (b) gives the values for an
individual day, July 26, 1942, during which the wind remained constantly
in the sector west to northwest. It is seen that the wind speed decreases
until about 06 h, at which time the stability of the air caused by noctur-
nal radiation (see section 29) is a maximum. Thereafter the wind
increases until 14-15 h, during which period the insolation from the sun
exceeds the radiation from the earth, and as a result the air becomes

20 r-

12
Time (h)

FIG. 123. (a) Mean hourly values of the wind velocity at Winnipeg, July, 1942;
(6) wind velocities for Winnipeg for July 26, 1942.

increasingly less stable. After 15 h the increasing stability is accom-
panied by decreasing winds. Curve (6) shows that the diurnal varia-
tion is more pronounced on some individual days than it is when values
are averaged over a number of days. This is to be expected since varia-
tions in the stability of the air occur from day to day. Thus on a day
with a thick overcast sky, radiation from the sun has little effect at the
surface of the earth, and so there is little increase in the wind velocity
during the day. On the other hand, with a clear sunny day the marked
changes which take place in the stability of the air are reflected in the
changes of the surface wind.

The diurnal variations of the wind are superimposed on the other vari-
ations. Thus in forecasting winds, the changes in the pressure gradient
and the time of day must be considered together. If the pressure grad-
ient near a given station is decreasing during the morning, the diurnal
increase may counteract the smaller value of the geostrophic wind. If,

Sec. 123}

THE EFFECTS OF FRICTION

305

instead, the pressure gradient is increasing, then the effect will be to
superimpose the one increase upon the other.

Observations have shown that the friction between the earth's surface
and the moving air affects the wind velocity in the layer from the surface
to a height of about 1 km. Because of the stability such as occurs with

10 r-

4 8 12 16 20 24
T.mc (h)

FIG. 124. The diurnal wind variation at the Eiffel Tower, Paris.

an inversion, the turbulent air motion is restricted to a thin layer near
the earth's surface. Hence the decrease of winds which takes place
during the early morning occurs only in the lower 300 m, while above this
level the winds tend to be close to the geostrophic value. But during
the time of maximum turbulence, the effect of friction is distributed over
a thicker layer of 500 to 1000 m. This results in a wind variation in the
upper levels of the frictional layer with a maximum at these levels dur-
ing the night and a minimum in the early afternoon associated with the
maximum that occurs at the ground at that time. At an intermediate
level there is a tendency for two maxima, one in the early morning when
the turbulence does not extend to that level, and one in the afternoon
when the maximum at the ground extends to that height. These facts
are brought out in Fig. 124, which gives the wind speeds for the bottom
and top of the Eiffel tower (height about 300 m) in Paris. Observa-
tions have shown that these variations in the wind velocity are associated
with backing or veering of the wind as k varies, as would be indicated by
equations 52-14 and 52-15. Thus the winds at the surface back and

306 WINDS [Chap. 17

decrease during the night and veer and increase during the day. At the
top of the frictional layer the winds back slightly as they decrease during
the day and veer and increase during the night.

124. Slope and Valley Winds. A second type of wind that is asso-
ciated with local topography is known as the slope wind. Nocturnal
radiation on the side of a hill cools the air just above the slope, while the
air at the same level at a considerable height above the ground farther
down the hill experiences but little cooling. The air just above the slope
is thus denser than its environment and subsides along the slope. The
reverse process takes place during the day if the slope is strongly heated
by the sun. The air in contact with the side of the hill becomes lighter
than the environment and rises, resulting in a flow of air up the slope.

A large-scale example* of this type of wind is found along the coasts
of Greenland. The air in contact with the ice cap that covers that con-
tinent is cooled and a high-pressure area forms over the region. The
pressure gradient force combined with the force of gravity gives rise to
currents of air descending to the* sea through the fjords and ravines along
the coast. This current is only shallow but its velocity is often high,
sometimes reaching gale forco. The return current appears as n lighter
on-shore wind at higher levels. Winds of this type are also known as
katabatic winds. When the air flow is up a slope the wind is known as an
anabatic wind.

A similar wind occurs in mountain districts. Consider a mountain
river valley with steep sides. Here there ure two slopes to be* considered.
At night the air just above the valley sides cools and sinks in the manner
outlined above. This air subsides to the lowest portion of the valley.
But the valley itself forms a slope, as shown by the flow of water in the
river. The denser air from the valley sides therefore flows downstream
under the influence of gravity just as the water comprising the river
does. During the day the valley sides are heated by insolation, the air
river these sides rises, and there is also a How of air upstream along the
length of the valley.

These air motions may also be discussed in terms of the pressure
gradient force. Consider two points at the bottom of the valley, the
elevation of the first above m s 1 being 1500 ft and the second downstream
being 1200 ft above m s 1. At night the* density of the air just above the
1500-ft level is greater at the first point because of radiational cooling
than at the second since, at the latter, air 300 ft above the valley bottom
is being considered and there is little radiational cooling at that height
above the surface. Since the weight of the air above a given level
governs the pressure at that level, according to the statical equation 9-1,
it follows that the pressure at the first point is greater than that at the

Sec. 124} SLOPE AND VALLEY WINDS 307

second. There is thus a pressure gradient directed up the valley and
therefore a pressure gradient force acting down the valley, as shown by
(33-1), giving rise to the downstream How of air. It also follows that
the mean density of the air in the height interval 1500 to 3000 ft, for
instance, is greater over the first point than over the second, so that, as
shown in section 30, the pressure above the first point at 3000 ft is less
than that above the second point at the same height. At this and adja-
cent levels, therefore, the pressure gradient and thus the wind are oppo-
site in direction to those at 1500 ft. There is thus a circulation in a
vertical plane, changes in which may be computed with (50-20), using
the method given in section 71. The up-valloy flow r extends through
a considerable height interval, and the velocities are therefore small.
The effect of friction must be considered, even if only in a qualita-
tive fashion. When the velocity increases to the point- that the retarding

effect of friction W just equals the value of the term V in (50-20)

J p

then the expression dC/dt becomes zero and steady conditions are at-
tained. During the day the pressure gradient force acts up the valley
and the wind is upstream. In the foregoing it was assumed that the
mean pressure gradient along the valley for the day was zero. If the
general pressure distribution over the mountain region is such that there
is a component of the pressure gradient along the valley, this diurnal
variation is superimposed on it, leading to a local diurnal variation in
the magnitude of the pressure gradient. Thus the wind may not be
down river at night, but up river with a smaller velocity than during
the day.

If the valley sides are steep so that the flow of air is generally either up
or down river, the deflecting force of the earth's rotation need not be
taken into account because it always acts perpendicularly to the direc-
tion of motion and is not strong enough to deflect the air up and over the
sides of the valley to any significant extent. If mountains extending
above the general level of the terrain are situated near the valley, large-
scale eddies formed in the lee of these may further complicate the air
motions. The air motions in valleys may be very complex, and the
effects mentioned above are best seen from mean values of the wind
rather than individual values.

A detailed study of the winds in the Columbia River Valley in southern
British Columbia near the city of Trail has been made. The results of
a large number of pilot balloon observations made at Columbia Gardens,
which is near the center of the valley and about 4 mi north of the inter-
national boundary, are shown in Figs. 125 and 126. The mean wind
values given in the figures are based on approximately one thousand

308

WINDS

[Chap. 17

pilot balloon observations made at successive 2-h intervals during por-
tions of the summers of 1938, 1939, and 1940. At Columbia Gardens the
valley runs in a north-south direction, descending to the south. In
Fig. 125 are shown the mean north- and south-wind components through-
out the day from the valley floor at a height of about 1500 ft above m s 1
to a height of about 5000 ft above m s 1. The curves on the figure are
inclines of velocity, expressed in miles per hour. Mean north compo-
nents are indicated by hatched areas and mean south components by
unhatched areas. Since the sides of the valley extend up to heights of

5000

4000

53000

2000

II 13

Time (h)

Fio. 125. The diurnal variation of the north-south wind components in the Colum-
bia River valley near Trail, B.C.

4000 to 4500 ft above m s 1, it can be seen that the prevailing wind above
the valley has a south component. In the valley itself the mean wind is
down valley until 10 h, although at higher levels the north wind ceases
somewhat earlier than this. From 10 to 17 h the wind is up the valley.
By 17 h the sun is no longer heating the valley floor and radiational cool-
ing has commenced, as indicated by the beginning of a down-valley flow
at that time. As cool air descends from the valley sides causing an
increase of density at greater and greater heights, the wind gradually
shifts from up to down valley until by midnight all the air in the valley
is moving down valley.

The cast-west or cross-valley components are shown in Fig. 126. As
in the previous figure the curves give velocities in mph. Hatched areas
indicate an east component of wind and unhatched areas a west com-
ponent. Since it was desired to study the effects of the sun's heating,
only velocity components of 5 mph or less were included. The inclusion
of a number of components of 25 or 30 mph arising from other causes
would tend to mask the effect of insolation. It can be seen that above

.Sec. 124]

SLOPE AND VALLEY WINDS

309

the valley the prevailing wind has a west component. In the valley
itself during the night the wind generally has an east component. This
may be due to the deflecting force acting on the air as it moves down
valley and giving it a small westward component of motion, or it may be
a manifestation of a large-scale eddy motion in the vicinity, since there
is an opposite current above.

As the sun rises it heats the west side of the valley but not the east
side. This unequal heating sets up a pressure gradient across the valley
and thus a pressure gradient force acting from east to west. This pres-

5000 p

2000

It 13
Time (h)

Fi(i. 126. The diurnal variation of the east-west wind components in the Columbia
River valley near Trail, H.C'.

sure gradient force* reinforces the oast component of wind which prevails
during the night so that from 00 to OS h thore is a maximum of 0.8 mph
at 4000 ft. This reinforcement of I ho east component probably starts
at about 05 h when the insolation roaches the upper portion of the west
side of the valley. The west component which appears at 3300 ft at
05 h may be regarded as a part of tho return circulation of the air flow
above. Near the surface the oast component continues until noon. It
is interesting to note that the direction of the cross-valley component
changes as the sun passes tho zenith and commences to heat the east
slope of the valley more strongly than the west side. The west com-
ponent continues strongly until 21 h whon the east component charac-
teristic of the night commences at intermediate levels. It would appear
that the greater heating of the east slope during the afternoon causes a
slight difference in temperature to persist during the night so that a small
west component continues noar the surface until 05 h.

The wind pattern on individual days may vary widely from that shown
above. In the small but steep side valleys extending from the main.

310

WINDS

[Chap. 17

valley, however, the air flow at the surface is often much more regular.
Surface wind observations for the month of October, 1938, in a small side
valley extending upward to the east from the main valley near Columbia
Gardens are summarized in Fig. 127. Wind directions in the sector
northeast through east to southeast were considered as down-valley
winds, while those in the opposite sector, southwest, west, to northwest
were up-valley winds. The number of occurrences of the wind in these
two sectors at each hour of the thirty-one days of the month is shown in
the figure. The regularity of the down-valley motion at night and the
up-valley motion during the day is striking. For example, the air flow
at 19 h was down valley on thirty days.

02 06

10 14
Time (h)

FIG. 1 27. Number of up-valley and down-valley winds in a side valley of the
Columbia in October, 1938.

The strength and regularity of such katabatic and anabatic winds
varies with the locality. A knowledge of the winds of this type in the
vicinity is important in understanding local variations in the distribu-
tion of ground fog. This factor is discussed further in section 134.

125. Land and Sea Breezes. The unequal heating of land and water
by the sun's rays causes one other type of local wind. During a day
when the insolation is large, in a coastal area the land surface becomes
warmer than the adjacent ocean. The heating of the surface air over
the land results in the development of a pressure gradient and hence of a
pressure gradient force acting normal to the coast line. This produces a
wind blowing from the sea to the land. At higher levels the direction of
the pressure gradient is reversed and the wind blows from land to sea.
There is thus a circulation of air the magnitude of which may be com-
puted with the aid of (50-20), using the method outlined in section 71.
As the velocity increases during the day the deflecting force acting on
the air becomes greater, so that by late afternoon the wind may be blow-
ing nearly parallel to the coast line. During the night the radiational
cooling reverses the situation, and the surface current now blows from

BIBLIOGRAPHY 311

land to sea. In this manner the land and sea breezes, which are well
known to inhabitants of coastal regions, are produced.

In the tropics these winds blow with great regularity. In the tem-
perate zone the effect is often not strong enough to do more than modify
slightly the gradient wind resulting from the general pressure distribu-
tion. If the general pressure distribution is weak, the sea breeze will
develop during the late morning, dying down an hour or two before
sundown. The night breeze off the land is generally light. These wind
systems are limited in extent, reaching only a few miles inland. They
are also surface phenomena, limited to the lower levels of the atmos-
phere. Above the surface layers the air motion is under the influence
of the general circulation in the region.

PROBLEMS AND EXERCISES

1. At 2000 ft the wind is 15 mph from the northwest. At 10,000 ft the wind is
39 mph west southwest. If these are the geostrophic winds at these two levels,
what is the direction of the mean isotherms for the layer from 2000 to 10,000 ft, and
what is the temperature gradient? Assume latitude 45 and the mean temperature
of the layer 273 A.

2. The isobars around the point of a trough line have a radius of curvature of 250
km. What is the value of the gradient wind if the geostrophic wind for the same
pressure gradient is 20 m per sec, the latitude being 45?

3. On a summer day the mean temperature of a vertical column of air over a land
surface at a distance of 5 km from an adjacent coast line is 24 C. The column
extends from 1000 to 900 mb. The mean temperature of a similar column of air
over the sea surface at a distance of 5 km from the coast is 22 C. The height of
both columns may be assumed to be 1 km. If the effect of friction is neglected,
compute the mean velocity of the circulation around this circuit 1 h after this temper-
ature difference was set up, assuming that the latter remains constant during the
1-h period.

BIBLIOGRAPHY

Admiralty Weather Manual, London, H. M. Stationery Office, 1938. Chapter 8.
Brunt, D., Physical and Dynamical Meteorology, London, Cambridge University

Press, 1939. Chapter 13.
Petterssen, S., Weather Analysis and Forecasting, New York, McGraw-Hill Book

Co., 1940. Chapter 4.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 2

(1936), Chapter 6.
124. Heywood, G. S. P., " Katabatic Winds in a Valley,'* Q. J. Roy. Met. Soc., 59,

47-57 (1933).

124. Wagner, A., " Neue Theorie des Berg- und Talwindps," Met. Z., 49, 329-341
(1932).

125. Sutcliffe, R. C., " The Sea Breeze at Felixstowe, A Statistical Investigation of
Pilot-Bailoon Ascents up to 5,500 Feet," Q. J. Roy. Met. Soc., 63, 137 (1937).

CHAPTER 18

CONDENSATION AND PRECIPITATION

126. Saturation. The saturation pressure of water vapor is defined
as the maximum pressure which water vapor can exert when in contact
with a plane surface of pure water. In the atmosphere the water sur-
faces under consideration are not usually plane, but spherical. In
addition, the water is not pure.

First to be considered is the nature of the surface of the liquid. Atmos-
pheric condensation takes place chiefly in the form of water droplets.
Owing to the surface tension of a curved water surface, the saturation
pressure e' a with respect to a water droplet is larger than the saturation
pressure e a with respect to a plane water surface which appears in the
expression 10-7 for the relative humidity. It was shown by Kelvin
that the logarithm of the ratio e' 8 /c a is inversely proportional to the
radius of the droplet. In order for small droplets to form directly in the
atmosphere, it would therefore be necessary that the vapor pressure
increase considerably over the saturation value c s with respect to a plane
surface. The state in which the actual vapor pressure is greater than e s
is called supersaturation. Small supersaturation scorns to occur in the
atmosphere sometimes but never on a scale which would permit the
direct formation of water droplets.

The saturation relative humidities for water droplets of various
diameters are given in the following table.

VARIATION OF SATURATION RELATIVE HUMIDITY WITH DROPLET SIZE

Diameter (cm) 10~ 6 10~ 5 1CT 4 10~ 3

Relative humidity (per cent) 1 26 102 . 4 100 . 23 100 . 02

The second factor in causing the vapor pressure to vary is the material
dissolved in the water. When there is a salt or an acid dissolved in the
water, the saturation vapor pressure is reduced by a factor (1 kc)
where k is a constant depending on the nature of the solute and c is the
molecular concentration of the solution. The relative humidities for
water vapor over a plane surface of sulphuric acid of various concentra-
tions, when the temperature is C, are shown belovy.

312

Sec. 127} CONDENSATION ON NUCLEI 313

SATURATION RELATIVE HUMIDITY OVER VARIOUS CONCENTRATIONS
OF H 2 SO 4 AT C

Concentration (per cent) 10 20 30 40 60 60 70

Relative humidity (per cent) 95.5 87.6 74.8 55.6 33.8 15.0 3.4

Over a saturated solution of NaCl at 10 C the relative humidity is
22 per cent less than that over pure water. It is not certain if these
relationships apply to small droplets, but they certainly do to large ones.
It can be seen, therefore, that condensation would occur on large drops
of a saturated solution of a salt with a relative humidity less than 100
per cent. This effect may be of fundamental importance in the early
stages of the development of a droplet, especially if condensation com-
mences on a small grain of the salt which acts as a nucleus.

A study of the effect of electrical charges on the saturation vapor
pressure over small drops indicates that it would be significant only if
the number of charges were very great. However, there is seldom more
than one electrical charge on a nucleus of condensation, so that this
effect is negligible in the early stages.

These various effects counteract one another. The final result in any
part of the atmosphere; is indeterminate. Yet the variation in the
relative humidity in vertically ascending air is so great that for practical
purposes it can be assumed that condensation begins when the relative
humidity is 100 per cent, in other words when the air is saturated with
respect to a plane surface of pure water.

127. Condensation on Nuclei. It was shown in the first table in the
previous section that the saturation relative humidity over a droplet of
pure water of diameter 10~ G cm is 126 per cent. But since relative
humidities as great as this never occur in the atmosphere, condensation
can commence only if there are already particles present, known as
condensation nuclei, of diameter greater than 10~" 6 cm. Since super-
saturation of only slightly more than 2.4 per cent will cause condensa-
tion on particles 10~~ 5 cm in diameter, it appears that the condensation
nuclei must be at least as large as this. Water molecules themselves
cannot act as nuclei, as their diameters are of the order of 4 X 10"" 8 cm,
and 5000 of them would have to come together to form a drop even 10~~ 6
cm in diameter. It is difficult to visualize any process by which a
number of molecules would coalesce to form a water droplet.

By means of an instrument devised by Aitken, the number of nuclei
in a given volume of air can be obtained. The Aitken counter and others
of a similar type operate by creating a very high degree of supersatura-
tion in a chamber containing the sample of air to be analyzed. Such
supersaturation causes droplets to form about particles in the air which,

314 CONDENSATION AND PRECIPITATION [Chap. 18

because of their small size, would not act as nuclei in the free atmosphere.
All the nuclei counted by such instruments are therefore not involved in
ordinary condensation processes. There is a wide range in the number
of nuclei present in unit volumes of air in different localities. Some
representative values are given in the following table.

AVERAGE NUMBER OP NUCLEI PER CM S

Locality Average Number
City 147,000

Country (inland) 9,500

Ocean 940

Mountain (2 km) 950

The most important nuclei arc composed of water-soluble salts. It is
not yet certain if ordinary dust particles act as nuclei of condensation,
although it is believed that they may function as nuclei for sublimation
in the process of formation of ice crystals. One of the most common
types of hygroscopic nuclei is composed of NaCl, carried into the atmos-
phere by the evaporation of ocean spray. Sulphuric acid particles in
the air, formed through the oxidation of sulphur dioxide by the action of
sunlight and union with water vapor, may act as nuclei for condensation.
Other products of combustion found near industrial areas may also act
as nuclei. Since condensation occurred long before the present indus-
trial era, however, it is obvious that other substances must also act as
nuclei. Nitrous acid, HNO2, is formed during natural processes, such
as lightning discharges, and particles composed largely of this acid may
therefore be present in the atmosphere to act as nuclei.

Condensation continues on the small droplet until the salt nucleus is
dissolved. Thereafter the condensation reduces the concentration of
the solution. An equilibrium situation will be reached with a large
number of small droplets. Thus in country fogs the average droplet has
a radius of from 4 X 10~ 4 to 3 X 10~~ 3 cm. In clouds the radius of
droplets may be as great as 10~ 2 cm. Since the average radius of rain-
drops under different conditions varies from 10~ 2 to 0.2 cm, some process
or processes must be in operation to cause the droplets found in clouds
and fogs to develop into drops large enough to fall as rain.

128. The Formation of Rain Droplets. According to a theory pro-
posed by Bergeron, the most common cause of the formation of rain
from cloud droplets is the difference in vapor pressure over water and
over ice at the same temperature. This difference is shown in the upper
portion of Fig. 139, section 135. When the air is above the freezing
level, the water droplets become supercooled. At the same time some

Sec. 129} THE TYPES OF RAINFALL 315

ice crystals will form by sublimation on ice crystal nuclei. With both
ice and water present, there will be a vapor pressure gradient from the
ice crystal to the water droplet. Evaporation will take place from the
droplet, with further sublimation on the ice crystal. This will continue
until the ice crystal becomes large enough to sink. It will continue to
increase in size as it falls, by coalescence with water droplets with which
it collides, by further sublimation while it remains in the freezing zone,
and by condensation after it falls below the freezing level and melts.
Researches into the temperature distribution in the upper air when rain
is falling, carried out by means of airplane ascents, have tended to
confirm this theory, for in general the tops of the clouds have extended
above the freezing level.

Since, though, light rain has been observed from clouds at tempera-
tures above the freezing point, at times raindrops must form from cloud
droplets by some other process. Bergeron further suggests that the
difference in temperature between different drops can explain the increase
in size. The vapor pressure increases with temperature. Hence when
two drops of different temperatures are adjacent to one another, equilib-
rium conditions for the drop with the higher temperature will give
supersaturation for the colder drop. Condensation will take place on
the cold drop, with evaporation from the warmer drop. With tempera-
tures near freezing this difference in vapor pressure is small for a small
difference in temperature. But when the temperature is above 12 C, a
difference of 0.5 C between droplets will lead to a difference in vapor
pressure of 0.5 mb or more. This difference is greater than the maximum
difference in vapor pressure between ice and water, and so would readily
explain the increase in the size of the droplets. This difference in
temperature between adjacent droplets could be caused by turbulent
motion in the cloud which might bring droplets of different temperatures
into proximity. This difference may also be explained at times by the
cooling of the drops at the top of the cloud by long-wave radiation, in
the manner indicated in section 30. After the drop has increased in
size sufficiently to begin to fall, it will continue to increase in size through
collision with other cloud droplets.

129. The Types of Rainfall. Precipitation is associated with clouds,
and the different types of clouds produce different types of precipitation.
The types of clouds and the causes of their formation are discussed in
Chapter 20. In general clouds are formed by the cooling of the air
below the saturation point. But precipitation of any considerable
quantity falls from only those clouds which are produced through the
lifting of a mass of air.
One region of widespread lifting of air is in the neighborhood of a

316 CONDENSATION AND PRECIPITATION [Chap. 18

frontal depression. Above the warm frontal surface of a depression the
lifting is extensive and gradual and leads to frontal rainfall. The area
of precipitation, illustrated in Fig. 108, section 112, may extend 200 or
300 mi ahead of the surface front. The precipitation ahead is usually
nearly continuous until the warm front passes. The amount of warm
front precipitation shows wide variations from one depression to another,
however. There appears to be a correlation between the degree of
potential instability in the warm sector air before its ascent and the
subsequent rainfall. The analysis in section 61 of the data given in
section 60 shows a coefficient of correlation of 0.72 0.096. With a
larger number of observations, the figure is even higher, 0.78. If there
is little or no potential instability, the rainfall will be light, while if the
potential instability is great, the precipitation may be heavy. Of course
in the latter case, if there is insufficient ascent to permit the realization
of the potential instability, the rainfall will not be great. The turbu-
lence accompanying the realization of the potential instability may
permit a more rapid ascent of the warm sector air, or it may produce
larger raindrops. In either case the result will be the occurrence of
heavier rainfall than if the ascending air were stable at all levels.

There is a small diurnal variation in both the amount and in the width
of the band of warm front rainfall, the maximum being reached during
the early morning. As suggested by R. V. Dexter, this may be
accounted for by radiational heating of the warm sector air near the
surface during the day, which increases its wet-bulb temperature and
hence its potential instability. This air then ascends over the warm
frontal surface and by the early morning sufficient ascent has occurred
to cause saturation in the layer and the realization of the potential
instability. The air which ascends during the day has been cooled at
the surface by radiation during the preceding night, and so its potential
instability is decreased.

The ascent of such large masses of air is a dynamical as well as a
thermodynamical problem, and the part played by each factor is not
clearly understood. For example, it is not clear why the air should con-
verge and rise on the warm side of the front, either on theoretical
grounds or from a study of surface winds. The dynamical aspect is
often marked in the early stages of development of a depression. Exten-
sive rainfall may occur when the bulge in the front is just developing,
well before the depression has commenced to occlude, at the stage illus-
trated by Fig. 1086, section 112.

Precipitation also occurs at the cold front, but it does not last as long
as that ahead of the warm front, nor is it as steady. It is generally of a
showery nature, and in amount it may be either light or heavy. It was

Sec. 129\

THE TYPES OF RAINFALL

317

shown in section 95 that latent instability may develop or increase
when a mass of air is lifted bodily, as at a cold front. The precipitation
may, on occasion, extend as far as 100 mi ahead of the cold front. No
entirely satisfactory explanation has yet been advanced to account for
the presence of ascending air currents so far ahead of the cold front.
There is a diurnal maximum in the amount of cold front rainfall. This
maximum is found in the afternoon, when insolation renders the air less
stable and so makes instability showers at the cold front more probable.
Occlusion rainfall shows no well-defined characteristics. Generally
speaking, if the occlusion is of the warm front type, the rainfall will
resemble warm front rainfall in some respects, and similarly with cold
front type occlusions. Heavy rainfall frequently accompanies either
type.

O>
O

-20

-10

FIG. 128. A lapse rate favorable for convectional rain.

Depressions of the middle latitudes are more intense and also more
frequent in winter than in summer, so that there is a maximum of precip-
itation of the frontal type during the winter.

Convectional rain is of frequent occurrence. This type is found when
strong convection currents develop in an air mass. Individual masses of
air may ascend until they reach their condensation level; if the lapse
rate conditions are favorable for further strong ascent to greater heights,
convectional precipitation of the showery type usually results, especially
if the top of the cloud extends above the freezing level. Upper air con-
ditions favorable to the development of convectional precipitation are
shown on the tephigram in Fig. 128. Since the lapse rate near the sur-

318

CONDENSATION AND PRECIPITATION

[Chap. 18

face is greater than the dry adiabatic, any slight perturbation will start
an upward motion in that portion of the air which is subjected to the
perturbation. If the dry-bulb temperature of the surface air is A, and
the corresponding wet-bulb temperature is B, the portion of this air
which ascends will become saturated at 0, and thereafter its tempera-
ture will decrease at the saturated adiabatic lapse rate with further
ascent. In the example shown, the air will rise to a height at which the

pressure is about 550 mb, denoted
by D. The area ACDE is a posi-
tive one, and represents the work
done by the environment on the
mass of air in question. This is,
therefore, a case of absolute in-
stability and latent instability of
the real type (section 96). Strong
ascent to considerable heights will
occur in such circumstances, and
convectional or instability showers
are almost certain to develop.
Instability of this type is likely
to occur under conditions which

FIG. 129.

A lapse rate unfavorable for
convectional rain.

have been outlined in section 95.
It will be remembered that such
instability tended to develop on warm, sunny afternoons in summer,
and in cold air masses moving over warm, moist soil and over warm
ocean surfaces.

Showers do not necessarily occur, however, when absolute or real
latent instability is present. Fig. 129 is an illustration of absolute
instability, but where convectional precipitation is not to be expected.
The ascent beyond the condensation level in such a situation will not be
sufficient to produce precipitation, although cumulus clouds will form.
It has been observed that rain does not usually start to fall until the
rising currents extend above the freezing level. The appearance of a
swelling cumulus cloud, having an anvil-shaped top with mantle, indi-
cates the formation of ice crystals and suggests that rain is about to start.

Another kind of precipitation which is of importance is orographic
rainfall. Such precipitation occurs when air ascends in flowing over a
range of hills or mountains. Orographic rainfall occurs frequently over
the western slope of the Rockies, when moist air from the Pacific moves
eastward over the continent. It can be seen that orographic rainfall
has many points of similarity with warm front rainfall. It may be
steady, or it may be of the showery type, depending on lapse rate con-

Sec. 180] OTHER TYPES OF PRECIPITATION 319

ditions. Although there may be no latent instability present as the air
flows over the ocean, marked latent instability may develop through the
ascent along the mountain slope, and heavy showers may occur.
If the lapse rate before ascent is such that no latent instability will
develop, even after ascent, the precipitation will be steady in character.
As explained in sections 119 and 141, horizontal convergence of air
produces vertical currents in regions near cyclones and anticyclones.
These currents are a contributing factor or at times the major cause of
precipitation in some areas. When convergence is the predominating
factor the rainfall is of the convergence type. The horizontal convergence
resulting from isallobaric and frictional inflow of air across the isobars
will give precipitation in some warm sectors. Also the latitudinal con-
vergence on the west side of a ridge of high pressure may produce rain
over a wide area. At times, although not always, the frontogenetical
field associated with this convergence is sufficiently strong to cause the
development of a well-marked front. Usually the ascent with con-
vergence is slow and the rainfall is light.

130. Other Types of Precipitation. Snow occurs under the same
general meteorological conditions as rain, except, of course, that tho
transition from the vapor to the solid state occurs at temperatures
lower than C. Sublimation of water vapor is believed to take place
on special non-hygroscopic types of nuclei, generally referred to as ice
crystal nuclei. The ice crystals increase in size as sublimation proceeds
and start to fall slowly through the air. These ice crystals are found
in an unlimited variety of forms and are completely symmetrical about
a hexagonal base, as shown by numerous microscopic examinations of
such crystals. Large snowflakes are aggregates of crystals, usually
moist, and are usually found when the temperature is only slightly below
the freezing point. At lower temperatures the crystals do not combine
in this manner, and such large snowflakes do not develop. The name
sleet has been given to two entirely different types of snow or ice struc-
ture. In Great Britain, sleet refers to the transitional type between rain
and snow, and it may be either a mixture of rain and snow, or partially
melted snow. It occurs only when the temperature is very near the
freezing point, usually just above it. In North America the name sleet
is given to pellets of ice which are initially small raindrops but which
freeze during their descent. This latter type of precipitation occurs
when rain falls from overrunning warm air which is at a temperature
above C into colder air below, which has a temperature lower than
C. If the water drops do not freeze during descent but become super-
cooled they will freeze upon striking the ground to form glaze.

The formation <rf hail is associated with the development of strong

320 CONDENSATION AND PRECIPITATION [Chap. 18

convection currents. In order that the hailstone may grow to any con-
siderable size, the ascending current must be sufficiently strong to pre-
vent it from falling too rapidly to the ground. A current of such magni-
tude is to be found only in the process of realization of marked latent
instability. It will be well to discuss the process of formation of hail-
stones in some detail. An ice crystal may form at high levels, and when
it becomes heavy enough it will fall, increasing in size as it descends, and
especially so if it falls through a cloud of water droplets. If there is
sufficient upward motion of the air to keep the growing ice crystal from
falling rapidly, it may develop to large proportions, and fall to the
ground as a hailstone. If the greater part of its growth has occurred at
temperatures well below the freezing point, it will be composed chiefly of
snow which is loosely packed and often referred to as white ice. Such a
particle is known as a snow pellet. A pellet of this type may not fall to
the surface, however, but may be carried upward from lower levels by a
rising current of air. While at temperatures near the freezing point
a layer of clear ice may form on the core of white ice through collisions
with supercooled water drops. As it ascends to greater heights again,
white ice will form over the layer of clear ice. If the particle makes a
number of vertical excursions in this manner, alternate layers of white
and clear ice will form, until it finally drops to the ground as hail. Not
all hailstones have such a structure, however. Many grow to large
dimensions without undergoing a number of vertical excursions, and fall
to the surface as a lump of clear ice.

The formation of large hailstones requires ascending currents strong
enough to support the hailstone until it can grow to large dimensions. It
is known that vertical velocities as great as 3000 ft per min are not
uncommon in well-developed thunderstorms, and vertical velocities of
even 5000 ft per min have been reported under extremely unstable con-
ditions. Clouds which are extensive in the vertical are necessary for
hailstone development, and will form when the air is very unstable.
Hailstones as great as 4 in. in diameter and weighing nearly 2 Ib have
been found. Studies of the aerodynamics of hailstones suggest that the
maximum weight of hailstones is about 1} Ib.

The distribution of hail over the United States is shown in Fig. 163,
section 151.

Drizzle is a form of precipitation in which the droplets are small, and
fall very slowly to the ground. Because of the large number of the drop-
lets per unit volume, drizzle tends to reduce the visibility more than
ordinary raindrops. When it occurs, it generally falls from stratus and
stratocumulus clouds.

BIBLIOGRAPHY 321

BIBLIOGRAPHY

Problems of Modern Meteorology, London, Royal Meteorological Society, 1934.

Numbers 13, 14.
Bergeron, T., On the Physics of Clouds, Memo. Met. Assoc., Intern. Union for Geodesy

and Geophysics, Lisbon, 1933.
Findeisen, W., " Die Kolloidmeteorologische Vorgange bei der Neiderschlagsbildung,"

Met. Z., 55, 121-133 (1938).
Simpson, Sir G. C., " On the Formation of Clouds and Rain," Q. J. Roy. Met. Soc.,

67, 99-133 (1941).
Stickley, A. R., " An Evaluation of the Bergeron-Findeisen Precipitation Theory,"

Monthly Weather Review, 68, 272-280 (1940).
Wright, H. L., "Atmospheric Opacity: a Study of Visibility Observations in the

British Isles," Q. J. Roy. Met. Soc., 65, 411-439 (1939).

CHAPTER 19
FORMATION AND DISSIPATION OF FOG

131. The Effect of Evaporation. Fog is defined as a cloud which
envelops the observer and reduces the horizontal range of visibility to
less than 1 km. It is formed through the condensation of water vapor
from saturated air. At least 0.5 gm of liquid water per kg of air must
be present in the atmosphere before the visibility is reduced sufficiently
to permit the classification of the condition as fog. With dense fog the
amount of condensed water may amount to as much as 5.0 gm per kg.

There are four ways in which air near the surface of the earth may
become saturated:

(a) Evaporation of water into the air.

(6) Turbulent mixing.

(d) Non-adiabatic cooling.

These four processes will be studied in some detail.

Evaporation of water is a frequent cause of fogs, especially of frontal
fogs. There are three separate cases to be considered, depending on
whether the temperature of the air in contact with the water is less than,
equal to, or greater than that of the water. The source of the heat which
is required for the evaporation of the water must also be considered in
assessing the probability of fog formation. The following list of symbols
will be used in the ensuing discussion.

T = the temperature of the air.
TI = the temperature of the liquid water.

e a i = the saturation vapor pressure at the temperature of the water.
e 8 = the saturation vapor pressure of the air.
e = the actual vapor pressure of the air.
T w the wet-bulb temperature of the air.

e 8W = the saturation vapor pressure at the wet-bulb temperature of
the air.

(a) Heat of Vaporization Supplied by the Body of Water. This situa-
tion will exist most frequently at the surface of the earth over a large
body of water, or over a moist land surface. During the evaporation,

322

Sec. 181] THE EFFECT OF EVAPORATION 323

the vapor pressure e increases, but the temperature of the air will be
considered constant.

First, consider the result if T > TI. In this case, equilibrium will
be reached when e = e a i. Then e < e 8 , and so saturation is not reached,
and thus no fog forms.

Second, if T = TI, equilibrium will be reached when e = e 9 i = e a .
Hence saturation will be reached, but no condensation occurs and thus
no fog forms.

The third condition is specified by T < TI. Equilibrium with respect
to the water surface would be reached when e = e a i. But this would
require supersaturation of the air at temperature T, since e would then
be greater than e a and so condensation will occur if suitable nuclei are
present. If the difference in temperature is great, the condensation
may be sufficient to produce a fog. The vapor will condense immedi-
ately above the water surface, giving the appearance of steam leaving
the surface of hot water before it boils. I lence fogs of this type are often
called steam fogs.

When this situation exists the underlying warm surface will supply
heat as well as moisture to the adjacent air. This addition of heat will
induce instability in the air and hence will cause rising currents. These
ascending currents will carry the fog upward where it will evaporate in
drier air aloft, thus causing the dissipation of the fog. Such fogs are
frequent over the open waters in the Arctic in the winter season, where
the difference in temperature between air and water is great. Because
of their appearance they have become known as Arctic sea smoke. If
an inversion is present aloft to dampen the vertical currents, these fogs
will persist in the surface layers, rather than being carried aloft. Steam
fogs are also found frequently in temperate latitudes over lakes in the
autumn, when these bodies of water are still warm while the air is cool.

(6) Heat of Vaporization Supplied by the Air. This situation will
occur in the free atmosphere, the evaporation taking place from falling
rain. The heat is supplied by the air since the heat content of the falling
drop is too small to permit any extensive evaporation. With the heat
being derived from the air, the temperature T of the latter will fall and
the vapor pressure e will rise, but, as was shown in sections 24 and 83,
T w will remain constant.

When T w > TI, equilibrium will be reached when e = e^. Then
e < e 9W , and saturation will not occur. When T w = TI, equilibrium
conditions will prevail when e = e aw = e a i. In this situation, saturation
will exist, but no condensation occurs, and thus no fog forms.

If T w < TI, equilibrium over the water surface would exist only ft
e = e 8l . This would require a decrease of the air temperature to T Wi

324

FORMATION AND DISSIPATION OF FOG

[Chap. 19

so that e > e 8W and the air would be supersaturated. The additional
moisture would then condense to form cloud or fog. This condition
arises when rain falls from a layer of warm air lying over a layer of cold
air at a warm front. If the wet-bulb temperature of the surface air is
less than the air temperature of the warm air aloft as shown in Fig. 130a,

Stratus

(a) (b)

FIG. 130. Lapse rate conditions above and below a frontal surface with which
(a) prefrontal fog will form, (b) stratus will form.

condensation will take place at the surface, giving a frontal fog. If the
wet-bulb temperature of the air in the lower layers is greater than the
temperature of the overrunning warm air, as indicated in Fig. 1306,
then the cloud will build down from the frontal surface to the level where
the temperature of the raindrops equals the wet-bulb temperature of
the air, forming a layer of stratus cloud. The likelihood of frontal fog
is readily determined if upper air observations, as shown in Fig. 130, are
available. However, in the absence of such observations, upper air
data in the warm sector or even surface data may be of use. Consider
the case of fog just ahead of a warm front. Assume that a sharp front
divides the warm sector air with a surface temperature at 50 F from the
cold air ahead which has a temperature at the surface of 40 F. The
surface air in both masses is saturated. The warm air flows up over the
cold air, its temperature in this case decreasing at the saturated adia-
batic lapse rate. It can be seen that fog may occur until the lower
portion of the warm sector air has ascended to a height where its tem-
perature has become less than 40 F. Saturated air at 1000 mb with a
temperature of 50 F ascends to 880 mb before its temperature decreases
to 40 F, as can be seen from the tephigram. Under average conditions
this represents an ascent of about 3400 ft. If the slope of the warm
frontal surface is 1 in 200, the fog may extend as far as 130 mi ahead of
the warm front. Low stratus cloud, formed as indicated in Fig. 1306,
may extend ahead of the band of fog, as shown in> Fig. 131. In this

Sec. 182]

TURBULENT MIXING

325

manner, a rough approximation to the horizontal extent of the fog may
be obtained. Since a considerable difference in temperature between

-130 miles-

FIG. 131. The computation of the distribution of fog and stratus cloud under a

warm frontal surface.

water and air is necessary before sufficient condensation to produce fog
will occur, the actual width of the fog belt will be less than 130 mi.

132. Turbulent Mixing. Turbulent mixing in the vertical plays only
a minor part in the formation of fog, perhaps carrying the effects of
evaporation from warm rain down to the
surface. Mixing usually inhibits the forma-
tion of fog. Before mixing takes place, the
humidity mixing ratio will ordinarily decrease
with height, as shown by the curve (a) in Fig.
132. The effect of vertical turbulent transfer
of water vapor is, according to (54-8), to
make the distribution of mixing ratio more
nearly constant with height, as indicated by
curve (6), Fig. 132. There is a tendency for
condensation at higher levels, since the moist-
ure content increases there, whereas there is
less likelihood of condensation at lower levels,
where the moisture content decreases. Tur-
bulent mixing between the surface layer
and the layer above promotes, then, the development of low stratus
cloud, but retards the formation of fog.

Mixing between air of adjacent air masses, such as occurs at a front,
will at times increase the relative humidity of the colder air. The equa-
tion for the increase of saturation vapor pressure with temperature
(214) is not linear, as shown by Fig. 133. Mixing will give a mean value
in temperature and in moisture content. Thus it would be possible for
two air masses, both with relative humidity near 100 per cent, to mix
and give supersaturated air. Yet the amount of water vapor that will

Fio. 132. The vertical dis-
tribution of moisture (a)
before, (6) after vertical
mixing.

326

FORMATION AND DISSIPATION OF FOG

[Chap. 19

condense is never great enough to form a fog. Thus horizontal mixing
alone will not cause the formation of a fog. Yet the process will aid in
the saturation of air, and so will assist in the formation of fog when some
other fog-forming process, such as radiation, is present.

550

^ 40

u 30

10
</>

-30 -20 -10 10 20 30 40
T(C)

Pia. 133. The variation of saturation vapor pressure with temperature.

133. Adiabatic Cooling. Adiabatic cooling is the cause of upslope
fog, which forms when the air is forced to rise above its condensation
level as it ascends the side of a hill or mountain. Such fogs will form
only if the air is stable, if the amount of turbulence is small, and if the
relative humidity is high. Fog will not develop if there is potential
instability in the air which would be released by the ascent. If there is
sufficient ascent, the turbulent mixing resulting from the release of the
potential instability will prevent the formation of fog.

134. Non-adiabatic Cooling. Two types of fog, radiation and advec-
tion fog, result from non-a diabatic cooling of surface air.

(a) Radiation or Ground Fog. The formation of radiation fog is
favored by a number of conditions:

(1) Clear sky.

(2) High relative humidity.

(3) Increasing moisture content with height.

(4) Slight turbulence, but not completely calm.

(5) Not too stable stratification.

Sec. 134]

NON-ADIABATIC COOLING

327

Long-wave radiation to outer space proceeds most rapidly when there
are no clouds in the sky, as indicated in sections 29 and 30. Radiation
fogs are, therefore, most likely to occur on cloudless nights. It was
shown in section 30 that there is a relationship between the amount of
long-wave radiation to outer space and the height of cloud. This rela-
tionship is that the net nocturnal radiation from the ground is roughly
proportional to the height of the cloud. Actual average values as deter-
mined by Asklof are given in Fig. 134. The type of cloud present at
each level at the time at which the observations were made is indicated
-j |

o
5 4

en
o

/NS,ST,orSC

if)

ft>

005 010 015

Net Radiation (cal cm" 2 min"')

020

FIG. 134. Nocturnal radiation with cloudy and clear skies. (After Asklof.)

in the figure. When the sky is covered with high cloud, the net loss of
heat from the ground is almost as great as when the sky is clear. On the
other hand, when the sky is covered with low cloud the net loss of heat
from the ground is only about one-seventh as great as the loss when the
sky is cloudless. It follows, therefore, that the fall in temperature at
night when the sky is overcast with high cloud will be nearly as great as
if conditions were cloudless, but will be only about one-seventh of this
amount if the sky is overcast with low cloud.

It can be readily seen that high relative humidity favors the develop-
ment of radiation fog. Under such conditions, the dew-point temperar
ture is nearly as great as the dry-bulb temperature, and cooling of only
two or three degrees may be sufficient to produce saturation. If the
relative humidity is low, a much greater decrease in temperature, say
ten degrees, will be necessary before the air becomes saturated. If it
is assumed that a further decrease in temperature of three degrees is

328

FORMATION AND DISSIPATION OF FOG

[Chap. 19

required for sufficient condensation to produce a fog, a total cooling of
six degrees with the high relative humidity, and thirteen degrees with
the dry air will be necessary for fog formation. Thus if the temperature
drop during the night is nine degrees, fog will form in the one situation,
but not in the other.

Increasing moisture content with height assists in the formation of
radiation fog. As indicated previously, turbulent mixing always acts
in such a manner as to make the moisture distribution more nearly uni-
form vertically in the layer in which the mixing occurs (see Fig. 132,
section 132). Thus turbulent mixing tends to increase the moisture
content of air near the surface if the moisture content originally increased
with height. Such a moisture distribution therefore facilitates the
development of radiation fog. By the same reasoning a decrease in
moisture with height retards the formation of radiation fog.

4 r-

04
Time (h)

FIG. 136. The height of cooling by molecular conduction. (After Taylor.)

A slight amount of turbulence is necessary if radiation fog is to form.
The height to which cooling will extend upwards from the surface as the
latter cools during the night, if only the molecular conductivity of the
air is taken into account, has been computed. These computations
show, as indicated in Fig. 135, that 8 h after the ground temperature
starts to fall the cooling of the air has extended up to a height of about
4 ft only. Molecular conductivity alone, therefore, will produce only
very shallow ground fogs. Since it is known that cooling extends to
much greater heights than this, it is clear that some heat-diffusing
agency, other than conductivity, must be in operation. Radiational
cooling of the air itself doubtless plays a part in this cooling process, but
turbulent mixing is probably the main agent in diffusing heat. Accord-
ing to (53-17), the rate at which the cooling extends upward from the

Sec. 134] NON-ADIABATIC COOLING 329

surface by turbulent transfer of heat is given by

z 2 =

where z is the height to which the cooling extends during time t, and K
represents the coefficient of eddy diffusivity. The interval of time t
commences when the surface temperature begins to fall. The coeffi-
cient K is of the order of 10 3 cm 2 per sec on a clear night when the wind
is light. Thus with this value of /C, and assuming that the temperature
decrease at the ground commences at 20 h, the cooling extends to a
height of about 75 in by midnight. Thus a certain amount of turbulence
is necessary if a fog bank of any considerable depth is to form. The
turbulence must not be too great, however, or the water droplets that
form near the surface will be dispersed by the turbulence, preventing
the development of radiation fog. The degree of turbulence accompany-
ing a 2- to 8-mph breeze seems to be the most favorable for the develop-
ment of this type of fog.

It was indicated in sections 53 and 76 that the eddy transfer of heat is
downward when the lapse rate in the air is less than the dry adiabatic.
The greater the stability of the air, the greater will be the downward
transport of heat for a given value of K. Thus the effect of the cooling
of the earth's surface by radiation to outer space, in producing a surface
inversion will be retarded by the downward turbulent transport of heat
from higher levels. If the inversion becomes very marked, the turbulent
transfer of heat from air above to that near the surface may be sufficient
to prevent the formation of fog under certain conditions.

Radiation fogs occur most frequently in air of maritime origin after
it has become stagnant over a cold continent. Accurate forecasting of
such fogs is most essential, and useful methods to achieve this have been
developed. Taylor constructed a chart for predicting radiation fogs
which has been found very helpful. lie took all cases of radiation fog
at Kew Observatory, in England, which occurred during a five-year
period, and studied the preceding temperature and humidity conditions.
He found, for instance, that if the depression of the wet-bulb tempera-
ture below a given dry-bulb temperature exceeded a certain critical
value as shown by the 20-h observations, radiation fog very rarely
formed subsequently. These critical values were plotted in a manner
similar to that indicated in Fig. 136. Thus in forecasting radiation fog
at Kew, the dry-bulb temperature at 20 h would be plotted against the
accompanying depression of the wet-bulb temperature. If the point
so determined lies above the line sloping upward to the right, no radia-
tion fog is to be anticipated; if the point lies below this line, fog may
occur, but not necessarily so. In the latter case, other criteria must be

330

FORMATION AND DISSIPATION OF FOG

[Chap. 19

used in judging whether or not radiation fog will result. Another type
of chart was constructed by George for ground fogs at Chattanooga,
Tennessee. This chart, shown in Fig. 137, was constructed from obser-

16 i-

Q- 8

Dry-bulb temperature ( F)

FIG. 136. The critical value of wet-bulb depression for various air temperatures
at 20 h, for radiation fog formation at Kew, England. (After Taylor.)

-c 4
o>

o 6

I 10
5 12

|l4

18
21

v \

BhX

\ ;

JOmol^X

\ X

Qm^

\20 mph

\ N

)mpA

\ \

2 24 02 04 ( 06
Time visibility reaches gmi (h)

FIG. 137. Radiation fog formation at Chattanooga, Tenn. (After George.)

vations of the gradient wind and the time of formation of the fog during
the night for the months of March, April, and May. Similar charts
were constructed based on observations made during the other seasons

Sec. 184] NON-ADIABATIC COOLING 331

of the year. This graph could be used at Chattanooga to assist in fore-
casting. The dew-point depression at 19 h is noted. Then along the
horizontal line corresponding to that value, interpolation would be
made for the computed gradient wind. Moving vertically, the expected
time of occurrence of ground fog could be read from the scale on the
horizontal axis. But in the use of these or any similar charts, all con-
tributing factors must be kept in mind, including cloudiness, turbulence,
and the vertical distribution of temperature and water vapor. The
exact position of the limiting lines in such diagrams may be different for
different localities, and it must be determined for each area for which fog
forecasting is undertaken. There are other factors which must be kept
in mind when forecasting fog of any description and which must be
taken into account in the forecasting of radiation fog.

The first of these is the necessity of an adequate supply of conden-
sation nuclei. Such a supply is nearly always present, but the number
and type of nuclei will in part determine the density of the fog. Visi-
bility observations provide, on occasion, an indication of the quantity of
nuclei present. If the visibility is reduced to 4 to 6 mi by haze, not
dust, it is probable that there are enough nuclei available to produce a
dense fog when other conditions are appropriate. However, radiation
fogs do sometimes form when the visibility is initially greater than 6 mi,
and too much reliance should not be placed on visibility observations in
fog forecasting.

If the forecasting is for fog over a comparatively small area, such as an
airport, the topography of the surrounding country must be taken into
account. Air cooled by coming in contact with sloping ground which is
being cooled by nocturnal radiation will tend to flow down the slope,
even if the incline is only very small. In this manner cold air tends to
flow into declivities, and if the cooling is sufficient, radiation fog may
form there. Fog is not so likely to occur on the slope itself. The con-
ditions under which fog forms at a given locality depend so much on the
topography that an individual stu dy of conditions at each place at which
fog forecasting is to be carried on extensively must be made if the
maximum accuracy is to be attained.

A third factor which must be kept in mind is the possibility of a
decrease in dew-point temperature, perhaps 2 or 3 C during the night,
resulting from the deposition of dew. Dew, instead of fog, does fre-
quently form under certain conditions, and especially when the air is
calm and there is practically no turbulence. The dew-point tempera-
ture decreases when fog forms and increases again as it evaporates after
being warmed by insolation.

Also to be considered is the depression below the dew point necessary

332 FORMATION AND DISSIPATION OF FOG [Chap. 19

to provide sufficient condensed moisture to form a fog. As mentioned
in section 131, 0.5 gm of liquid water per kg of air is necessary in order
to reduce the visibility sufficiently for fog. When tha amount of liquid
water present is not so great as this, the atmospheric condition is known
as mist. Since the curve of saturation vapor pressure with temperature
is not linear, as shown by (214), the amount of moisture which would
condense for one-degree drop in temperature varies. For saturated air
with a temperature of 30 C, the amount of cooling required to form fog
is less than % C. For a temperature of 10 C, the corresponding cool-
ing is 1 C; for 10 C, it is 3 C. Hence the amount of cooling neces-
sary for fog formation is dependent on the temperature of the air as well
as the dew-point depression.

The cloud cover during the day and the following night must be
considered, too, in estimating the possibility of fog formation. If there
has been an overcast during the day, the heating effect of the sun has
been small. A clear sky during the night will then permit the air to
cool rapidly to the dew point and so lead to the formation of fog. Such
fogs are found at the rear of cold fronts. The surface air has acquired a
high humidity in passing over a warm surface which has become moist
through showers and rain. Clear skies during the night will give rise at
times to a mixing-radiation type of fog.

The presence or absence of a surface snow cover also determines, in
part, whether or not radiation fog will develop. The influence of a snow
cover will be discussed in the next section.

(b) Advection Fog. Advection fog is another type which results from
the non-adiabatic cooling of surface air. The development of this type
of fog is favored by the following conditions:

(1) High relative humidity initially.

(2) Increasing moisture content with height initially.

(3) Medium wind velocity.

(4) Stable stratification initially.

(5) Large temperature difference between air and underlying surface,

the former being the warmer.

Items (1), (2), and (4) promote the formation of advection fog in
the same manner as they promote the development of radiation fog.
Advection fog will develop more readily in warm air as it moves over a
colder surface when the relative humidity is high than when it is low.
With regard to item (3), it can be seen that strong winds, with the
accompanying marked turbulence and vertical mixing, tend to prevent
the development of fog. If the wind is low, on the other hand, the warm
air does not advance rapidly enough over the underlying surface, which

Sec. 184}

NON-ADIABATIC COOLING

333

must have a marked horizontal temperature gradient to maintain the
fog. For instance, fog may form in air flowing from a warm land surface
over a cooler ocean surface. If the water does not become progressively
cooler, however, with increasing distance from the shore, fog will form
in the air when it first reaches the water surface, but as it blows farther
out to sea, the fog will be dissipated by turbulent mixing. Thus a
medium wind velocity is essential for the development of extensive
advection fog. A large difference between the temperature of the air
and that of the underlying surface is necessary, of course, to ensure
sufficient cooling to produce a fog. Items (3) and (5) are the most
important ones in the production of advection-type fogs.

ou
0.40

^ 30

u
u

5 10

o
4
3

2
1

(b)/

/ ,

2345
Wind force (Beaufort Scale)

FIG. 138. The variation in (a) the number of occurrences of fog, and (6) the differ-
ence between the air and the sea temperature during fog with different wind velocities
on the Grand Banks of Newfoundland. (After Taylor.)

G. I. Taylor has investigated the conditions under which advection
fogs form over the Grand Banks of Newfoundland. Large horizontal
gradients in the sea surface temperature occurring in this region, result-
ing from the proximity of cold water from the Arctic seas and the warm
water of the Gulf Stream, make this one of the foggiest areas to be found
on the earth. Some of his results are shown in Fig. 138. Curve (a) of
the diagram indicates the frequency of fog formation with various wind
velocities. Fogs formed most frequently when the wind was force 3 on
the Beaufort scale (8 to 11 mph). Curve (6) of the diagram shows how
various differences* between air and water temperatures are related to

334 FORMATION AND DISSIPATION OF FOG [Chap. 19

wind force. It can be seen that the great majority of fogs, about 72
per cent, occur when the air is approximately 1 F warmer than the sea
surface and the wind is force 2, 3, or 4. Fogs rarely form when the
temperature difference is great. Such large differences occur only when
the air is moving rapidly over progressively cooler water, and the turbu-
lent motion accompanying such winds prevents the development of fog.
Low stratus clouds may form under such conditions. The vertical
extent of advection fogs may be computed from (5347) in the same
manner as for radiation fogs.

There are several types of advection fogs. Tropical air fogs occur in
tropical air as it moves gradually northward over the ocean surface
which has, of course, a latitudinal temperature gradient. Fogs of the
land and sea breeze type occur near sea coasts. These tend to form over
the land adjoining the coast in winter, because the land mass is cooler
than the ocean at that season. The temperature gradient is oppositely
directed in summer and the fogs form over the cooler ocean.

In closing this section, it must be added that over a land surface
advection fogs are usually difficult to distinguish from radiation fogs,
and the characteristics of the former may best be determined from a
study of fog formation over the ocean. Radiation fogs do not form over
the ocean, of course, because of the very small diurnal variation in the
temperature of the water.

135. The Dissipation of Fog over a Snow Surface. The processes
taking place in a fog consisting of water droplets over a snow surface will
be considered, following Petterssen, under two general headings: first,
when the air temperature is below the freezing point, and second, when
the air temperature is above the freezing point.

(a) T < C. Under these conditions, fog does not readily develop
over a snow surface, owing to the lowering of the saturation vapor pres-
sure over ice. The amount of this lowering may be seen from the upper
portion of Fig. 139 in which Ae represents the difference in saturation
vapor pressure over water and over ice. The maximum difference
occurs at a temperature of about 12 C. If the relative humidity of
the air at such a temperature is sufficiently high, the vapor pressure in
the air will be greater than the saturation vapor pressure over ice or
snow, and sublimation on the snow surface will result. This will pre-
vent the air from becoming saturated with respect to a water surface,
and only under exceptional circumstances will a water droplet fog result.
For example, at 10 C the saturation vapor pressure over water is
2.87 mb, while over ice it is 2.60 mb. Thus if air at - 10 C has a rela-
tive humidity of 91 per cent, the actual vapor pressure of the air is
exactly equal to the saturation vapor pressure over ice. If the snow sur-

Sec. 136]

DISSIPATION OF FOG OVER SNOW

335

face and the air above it cool to a temperature lower than 10 C,
sublimation on the snow surface will start, preventing saturation -with
respect to water and rendering the development of a water droplet fog
unlikely. The lower portion of Fig. 139 gives the limiting values of
relative humidity at which sublimation will start if further cooling
occurs. Not only will fog not form under such conditions, but also water
fogs transported from other regions will be dissipated if brought over

04r-

501

5
MOO

-30 -20 -10

T(C)

I 70

_0
ft)

* 60

FIG. 139. The difference between saturation vapor pressure over water and over ice,
and the relative humidity with respect to water for saturation over ice.

such snow surfaces. When the air temperature is only slightly below the
freezing point, the difference in saturation vapor pressures is small, and
if the cooling is rapid, fog may develop. However, sublimation on the
snow surface will reduce the amount of liquid water, for the above-men-
tioned reasons, and the fog will dissipate unless the cooling proceeds
steadily, or unless sufficient water vapor is brought down from the air
above by turbulent transfer. If the air cools to a temperature of 10
to 15 C, Ae, increases to its maximum, and fogs are much less likely
to form. Those which are transported over the snow surface from other
regions will dissipate rapidly.

The diurnal range over snow, at low temperatures, is small and noc-
turnal radiation alone rarely produces fog. Advection is the main cause

336

FORMATION AND DISSIPATION OF FOG

[Chap. 19

of the water fogs frequently found over the cold continental areas in
winter. Most of such fogs are very diffuse, and do not reduce the visi-
bility to any great extent. This difference in saturation vapor pressure
accounts, to a large extent, for the fact that water fogs are comparatively
rare over the cold snow-covered portions of the continents, even when
moist polar maritime air flows over them. This is in marked contrast to
the behavior of similar air when it flows over a surface which is not snow
covered, such as an ocean, which may be only five or ten degrees cooler
than the air. Dense fogs develop in such circumstances, and may per-
sist for considerable periods.

(6) T > Q C. If the air temperature is above the freezing point, it
will be taken for granted that the snow is melting. The air in contact
with the melting snow will have a temperature of approximately C,
and a relative humidity of 100 per cent. The saturation vapor pressure
is thus 6. 1 1 mb just at the snow surface. Now assume that the air at 2 m
above the surface has a temperature of 6 C and a relative humidity of

J 80

| 90
_o

100

4 6

T(C)

FIG. 140. The relative humidity for condensation on a melting snow surface.

100 per cent. The vapor pressure is then 9.35 mb. There is thus a
vapor pressure gradient of approximately 1.6 mb per m, and part of the
water vapor of the air will condense rapidly on the melting snow surface.
This condensation will continue until the vapor pressure at 2 m decreases
to 6.11 mb, i.e., until the relative humidity falls to 66 per cent. If the
relative humidity is initially less than 66 per cent, water from the melt-
ing snow surface will evaporate into the air until the relative humidity
reaches 66 per cent, which is the equilibrium value. It follows, there-
fore, that condensation occurs on the snow if the relative humidity is
greater than a limiting value, in this case 66 per cent. The magnitude
of this . limit depends on the air temperature some distance above the

Sec. 186]

DISSIPATION OF FOG OVER SNOW

337

snow surface. The relative humidity of the air above the ground is
usually greater than this limit, and therefore condensation of water
vapor on the melting snow surface occurring in the spring after the air
temperature has risen is the usual development. The limiting values of
relative humidity for various temperatures of the overlying air are indi-
cated in Fig. 140. It can be seen from the foregoing discussion that the
greater the temperature at some distance above the ground, the more
readily fog dissipates.

16-

8-|:

5 -10 -20 -30 -40

T (C)

-50

Fia. 141. Winter fog frequency on the lowlands of Siberia. (After Petterssen.)

Thus fog is formed and maintained to any large extent over a snow sur-
face only when the temperature is slightly above or slightly below the
freezing point. Fogs over snow surfaces are most prevalent during the
spring of the year.

Sublimation on sublimation nuclei is the normal occurrence at tem-
peratures from 20 to 50 C. Such sublimation processes produce
ice crystal fogs. Since both the fog and snow surface are of an ice struc-
ture, there is no difference in saturation vapor pressure over the fog
particles and the snow surface. There is, therefore, no tendency for
such ice crystal fogs to dissipate, and they may persist for long periods.

The frequency of fog formation at various temperatures over ten
meteorological stations in the lowlands of Siberia (lying to the east of
the Ural Mountains) is shown in Fig. 141. The ordinate in the diagram
is fog frequency in per cent, found from the expression 100 (//F), where
/represents the frequency of the development of fog in each temperature
interval, and F represents the frequency of occurrence of air tempera-

338 FORMATION AND DISSIPATION OF FOG (Chap. 19

tures within each temperature interval. It can be seen that the fog
frequency decreases rapidly as the air temperature falls below C.
This decrease continues until about 20 C, when a secondary maximum
in the distribution occurs. This irregularity in the curve may be due to
the fact that both water droplet and ice crystal fogs may occur in that
temperature interval. There is a further slight decrease, and then an
increase as the temperature drops to 50 C. A decrease at even lower
temperatures is to be expected, because of the very small amount of
water vapor available for sublimation processes at such low
temperatures.

136. The Forecasting of Fog. According to the previous analysis,
fogs can be classified according to their method of formation into steam
fogs, pre-warm frontal fogs, upslope fogs, radiation fogs, and advection
fogs. In general, a fog is frequently the result of two or more of these
processes acting at once. For instance, fog will form at a coastal air-
port with an easterly wind blowing off the adjacent water, the easterly
wind being associated with the pressure gradient under a warm front
lying to the south of the airport. Such a fog is due to advection and
pre-warm frontal conditions. Hence, although the causes for fog may
be classified, any particular fog may be the result of several contributing
factors.

The occurrence of fog is very variable, even in a small district. A
widespread advection fog may fail to blanket some region owing to adia-
batic heating. Hence satisfactory forecasting of fog demands the study
of its occurrence at individual localities. Such studies have been made
for a number of airports, and the conclusions are helpful in understand-
ing the general causes of fog, as well as local peculiarities. The general
conclusions are also helpful in understanding the occurrences in other
localities. But a knowledge of the local variations at aA airport will be
of value only in forecasting for that particular airport. Such a study
should include investigations of the general meteorological situation, the
cloud cover, the gradient and surface winds, the dew-point depression,
the temperature, the nocturnal cooling, the possible sources of moisture,
and the general topographic features of the neighborhood. One airport
may have fog chiefly as the result of the adiabatic cooling which occurs
when the air from the adjacent river valley is carried upward to the air-
port, as the wind increases in the early morning, after being cooled all
night by nocturnal radiation. Another airport may be in the bottom
of a valley where light winds from one direction will continue to carry
fresh air into the valley and so give no opportunity for nocturnal cooling
to reduce the temperature of the air in the bottom of the valley, while
higher winds from another direction will leave a calrii at the valley floor

Sec. 186}

THE FORECASTING OF FOG

339

and so allow cooling to take place in the stagnant air. Studies of the
causes contributing to fog at an airport, with the extent of the contribu-
tion of each, will aid in the forecasting for that locality.

Emery Wftlkcr LU. K.

FIG. 142. The distribution of fog over the British Isles. (From Bilham, The
Climate of the British Isles, Macmillan and Co.)

An examination of the distribution of fog over the British Isles, shown
in Fig. 142, brings'out a number of significant points for the forecaster.

340 FORMATION AND DISSIPATION OF FOG [Chap. 19

The most striking feature of this chart is the location of the two maxima
of fog frequency, the larger one extending northwestward from London
over the Midlands and beyond, and the other over the Clyde Basin.
Both of these regions are highly industrialized, and the correlation
between fog and atmospheric pollution by processes of combustion is
striking. It is probable that these impurities not only increase the
obscurity themselves but also provide a plentiful supply of nuclei for
condensation. The location of these areas suggests that radiational
cooling is the predominant cause of fog formation over the British Isles,
and that advection fog is of secondary importance. The lack of advec-
tion fog over the west coasts of the two main islands is especially surpris-
ing, but it is probably due to the strong turbulent mixing accompanying
the unstable motion in the air coming from the Atlantic. The high inci-
dence of fog at the northeast coast of England is to be attributed to
advectional transport of moist, stable air from the North Sea. Little
upslope fog is shown in the figure since only data from stations at levels
lower than 500 ft above m s 1 were used in constructing the diagram.

Fig. 143 gives the distribution of fog over the United States. Maxima
are found over the western coastal regions, over the western slopes of the
Appalachians, off the New England coast, and over the Great Lakes
district. These maxima arise through different causes. The time for
maximum frequency of fog over the west coast is during the summer
months. Off California at this time, there is an upswclling of cool
water along the coast which cools the surface air whereas an inversion at
higher altitudes prevents mixing with dry subsiding air aloft. Farther
to the north the fogs develop in currents of tropical air cooled on their
trajectory northward. East of the Mississippi, fogs are most prevalent
when in the winter a current of moist air from the Gulf of Mexico is
carried north-northeastward and becomes stable as it moves over the
cold surf ace. The region of maximum frequency is found where adia-
batic cooling by lifting on the slope of the Appalachians aids advec-
tion in the formation of fog. Fogs over the Great Lakes during the
spring months are associated with the stabilizing effect of the cool waters
of the lakes, and in the autumn consist of radiation and steam fogs. Off
Nantucket fog is most prevalent during the early summer. The area
around Nantucket is part of the region of maximum fog occurrence ex-
tending from New England eastward to the Grand Banks of Newfound-
land shown in Fig. 166, section 154. Much of this fog occurs in tropi-
cal Atlantic air which is carried over the cool waters of the Labrador
current. Some fog near the coast lines occurs in the air leaving the
continent during the summer. This air is cooled as it flows over the adja-
cent water,

Sec. 136]

THE FORECASTING OF FOG

341

8 !

52 co

342 FORMATION AND DISSIPATION OF FOG [Chap. 19

Fog, thunderstorms, and icing form the triad of major dangers to air
travel. Fog is included in this category partly because it may blanket a
wide area and thus increase the difficulties of air navigation, and partly
because it may cover a particular landing field so completely and so
rapidly without a great deal of warning. Hence the forecaster must be
continually on the alert to detect the earliest signs of its occurrence.

PROBLEMS AND EXERCISES

1. An air mass had the following properties just as it left a continental area and
was about to move over an ocean surface.

p 1000 950 900 800 mb

T 15 16 14.5 13 C

/ 93 75 70 70 percent

The temperature of the water surface near the coast was 12 C, and decreased at the
rate of 1 C per 100 mi in a direction normal to the coast line. Indicate what factors
in this situation promote the formation of advection fog, and those which tend to
prevent it.

2. There is a spread of 6 F between temperature and dew point at 18 h near the
center of a high-pressure area, with clear skies expected for the night. The temper-
ature is (a) 72 F in August, (b) 35 F in October, (c) 14 F in January. What
factors make ground fog more probable in one situation than in another? What else
would it be advantageous to know, in order to evaluate the probability of the occur-
rence of ground fog, and under what conditions should it be forecast for the coming
night?

BIBLIOGRAPHY

Admiralty Weather Manual, London, H. M. Stationery Office, 1938. Chapter 9.

Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book
Co., 1937. Chapter 11.

Petterssen, S., Weather Analysis and Forecasting, New York, McGraw-Hill Book Co.,
1940. Chapter 2.

Taylor, G. F., Aeronautical Meteorology, New York, Pitman Publishing Corpora-
tion, 1938. Chapter 14.

George, J. J., Fog, its Causes and Forecasting with Particular Reference to the Airports
at Chattanooga, Tenn., Camden, N. J., Richmond, Va., Louisville, Ky., and San
Antonio, Tex., New York, Met. Dept., Eastern Air Lines, Inc., 1939.

George, J. J., " Fog, its Causes and Forecasting with Special Reference to Eastern
and Southern United States," Bui. Am. Met. Soc., 21, 135-148; 261-269;
285-291 (1940.)

Petterssen, S., " Some Aspects of Formation and Dissipation of Fog," Geofys. PubL,
12, No. 10, 1939.

Taylor, G. I., " The Formation of Fog and Mist," Q. J. Roy. Met. Soc., 43, 241-268
(1917.)

CHAPTER 20
CLOUDS

137* Frontal Clouds. When the air in the vicinity of a depression,
as, for example, that in the warm sector, ascends above its condensation
level, condensation of water vapor commences and clouds form. Owing
to variations in the stability and rate of ascent in different parts of a
frontal depression, different types of clouds develop throughout the region.
Some of the various cloud types encountered are indicated in Fig. 144.
Diagram (6) shows a frontal depression as seen on a weather chart, with
the frontal system beginning to occlude. AB and DE indicate the lines
along which the vertical cross sections given in diagrams (a) and (c)
are taken. Diagram (c) gives a cross section along a line which inter-
sects both the cold and warm fronts and passes through the warm sector.
Diagram (a) gives a cross section along a line which intersects the
occlusion. As illustrated the occlusion is of the warm front type.

Warm front clouds are, in general, of the stratus type. When a warm
front approaches a station, cirrus cloud is first noted, and then cirro-
stratus. With the nearer approach of the front, altostratus is next
observed, with its base at medium heights, and this merges into low
nimbostratus as the front itself passes. The cloud systems become
more complicated if there is either actual or potential instability in the
adjacent air masses. If there is potential instability in the warm air,
ascent of the air may lead to its realization, and to the formation of
cumuloform clouds embedded in the nimbostratus or altostratus. In
such a situation, cumulus or cumulonimbus clouds may extend through
and tower above the altostratus cloud as shown in Fig. 144c. If the slope
of the frontal surface can be estimated, it will be possible to determine
where the potential instability will be realized, and the cumulus type
clouds form. If there is actual instability in the cold air mass, there will
also be cumuloform clouds just below the warm frontal surface. It can
thus be seen that stability conditions in the air masses must be considered
when forecasting the types of warm frontal clouds that will develop.

Underneath the warm frontal surface the clouds will build downward
in the cold air as explained in section 131. Near the front these clouds
may extend to the surface and give a pre-warm front fog. Frequently
too the precipitation falling through the cold air evaporates and increases

343

344

CLOUDS

[Chap. 20

the relative humidity in the lower layers. Turbulence will then cause
a layer of fractostratus or stratocumulus cloud to form which, near the
front, merges with the nimbostratus cloud.

Fia. 144. Typical clouds in a frontal depression of the type shown in (6) ; (a) cross
section through the occlusion, (c) cross section through the warm sector.

Because of the friction between the earth and the surface layers, the
cold air behind a cold front moves more rapidly at a height of 3000 to
5000 ft than it does at the surface. The cold frontal surface, which is the
leading edge of the cold air mass, tends to have a steep slope near the
ground and with some fronts the frontal surface up to 5000 ft overhangs
the front at the ground (see Fig. 192, section 177). The overrunning
nose then traps warm air underneath it. With either a steep front or an

Sec. 188} CONVECTION CLOUDS 345

overrunning nose, the forward movement of the cold air gives to the
warm air ahead of the cold front a vertical velocity which is much
greater than that along the warm frontal surface. In addition, when cold
air overlies warm air as suggested above, the disposition of air is unstable,
and violent overturning takes place. The cold nose therefore forms,
breaks down, reforms, and so on.

Cold front clouds are, then, predominantly of the cumulus type, and
occur most frequently near the front. These clouds may be of great ver-
tical extent if there has been marked potential instability in the warm
air, and if the slope of the cold frontal surface is steep and its movement
rapid. Line squall clouds of the cumulonimbus type occur under con-
ditions of extreme instability. With some fronts subsidence occurs in
the warm air above the cold front. In this situation the development of
vertical currents producing cumulus and cumulonimbus clouds at the
cold front is inhibited. The clouds then are of small vertical extent, or
in situations with marked subsidence they may be entirely absent.

The type of cloud which forms in the cold air back of the cold front
depends upon the characteristics of the fresh cold air. In moving over
the surface warmed by the air of the warm sector and moistened by
showers, the air behind the cold front quite frequently develops insta-
bility. Then cumulus of fine weather or turbulence stratocumulus
forms. With strong outbreaks of polar continental air this instability is
often sufficient to cause a showery type of precipitation. When, on the
other hand, the air behind the cold front is already modified to some
degree it remains stable, and no clouds develop. With a cold front of
this type there is rapid clearing after the frontal passage.

Clouds associated with an occlusion may be of either the stratus or
the cumulus type, depending on whether the occlusion is of the warm or
the cold front type. Since the air has already been lifted some distance
from the ground, any latent instability initially present has usually been
released. Strong vertical currents are then unlikely except at the upper
cold front of a warm front type occlusion where the cool air behind the
occlusion ascends over the cold air ahead of the occlusion.

138. Convection Clouds. Convection clouds are of the cumulus type.
Such clouds develop under various conditions, one of the most important
of which is the heating of the air near the earth's surface on a sunny day.
The forecasting of cumulus clouds which develop as a result of surface
heating will now be discussed in detail, following a method developed
by Poulter.

It is possible to forecast the approximate time and height at which
cumulus cloud will form, and also the amount of cloud to be expected.
The method of procedure for forecasting the height of the base of the

346

CLOUDS

[Chap.

cloud is illustrated by Fig. 145. The dew-point temperature as observed
in the early morning is represented by A. It is assumed that the dew-
point temperature undergoes no significant increase as the surface air

warms during the day, and by
the time that the dry-bulb
temperature at the surface
has increased to B, the base
of the cumulus cloud will be
at the condensation level D.
As the temperature rises, the
height of the base of the cloud
will increase, and the maxi-
mum height of the base may
be forecast by forecasting
the maximum temperature
which the surface air will
attain during the afternoon.
In the figure, C represents the

FIG. 145. The variation of the condensation
level through insolation.

maximum temperature expected, and E gives the corresponding maxi-
mum height of the cloud base. This method of cloud forecasting is
chiefly of use in very homogeneous air masses, such as anticyclones.
Very erroneous forecasts will result unless great care is exercised in
choosing only homogeneous air masses. The cloud amount which will
occur can be estimated by noting the relative humidity of the air at the
level at which cloud is expected to form. If the air is near saturation at
that level, the cloud droplets will evaporate slowly, and the amount of
cloud cover will be large. On the other hand, if the relative humidity
in the surrounding air is low, the cloud droplets will evaporate rapidly,
and only a few tenths of cloud will develop. A linear relationship
would suggest that if the relative humidity at the level where cloud is
expected to form is 40 per cent, there will be YIQ of cloud; if 70 per cent,
Ko of cloud and so on. This rule holds only for relative humidities above
30 per cent.

The manner in which this method can be used is best shown by means
of the example given in Fig. 146. ABDF gives the dry-bulb curve of
an early morning radiosonde ascent, GH the wet-bulb curve, and C the
dew point at the surface. The relative humidities and mixing ratios are
plotted on the curve. The average mixing ratio in the lower layers is
approximately 7.5 gm. The intersection of the 7.5 gm mixing ratio line
with the environment is at Z>, at a level of 815 mb. This level is called
the convective condensation level. It is the level at which the surface air
will become saturated by adiabatic ascent and cooling, when the energy

Sec. 138} CONVECTION CLOUDS 347

necessary to carry the air to that level is obtained by surface heating.
Dry adiabatic descent from D gives the point of intersection with the
surface pressure line at E. The temperature at E gives the surface
temperature which must be attained during the course of the day in
order that convective currents may rise to the level D and form clouds.

<D
0>
O

-10 10 20

T(C)

FIG. 146. Forecasting cumulus clouds.

In Fig. 146 the surface heating required is 9 C. But since the area
between the ascent curve ABD and the adiabat DE is small, and near
the level D there is a small layer of absolutely unstable air, it would be
expected that the required heating would take place rapidly, and that
the clouds would develop early in the morning. Further increase above
the temperature at E by insolation would be unlikely since any increase
in temperature above E would be accompanied by a corresponding
increase in temperature throughout the turbulence layer. Another
method of determining whether to expect a certain increase of tempera-
ture during the day is to observe the maximum temperature of the
previous day in the same air mass. Since the maximum temperature is
conservative, this will give a reliable approximation. The moist adiabat
through D in the diagram intersects the environment curve again at the
isothermal layer at F, 660 mb, at which point the air is again in equilib-
rium with the environment. The clouds of vertical development would

348

CLOUDS

[Chap. 20

be expected to extend from D to F, and since the relative humidity at D
is 70 per cent, a cloud cover of six- to eight-tenths should be forecast.
Since the clouds extend above the freezing level, some showers might
occur below the base of the cloud. The drops would probably evaporate
in the thick layer of dry air below the cloud base, and so it is doubtful
if showers would be observed at the ground.

<D
O

FIG. 147. Conditions in which cumulus clouds may dissipate and then reform during

the day.

Another situation which sometimes occurs is illustrated in Fig. 147.
It can be seen that if the dry-bulb temperature remains between B and
C, no cumulus cloud will form. In such a case, cloud may form earlier
in the morning; it may then dissipate as the surface temperature rises,
and then reform at higher levels, above D, in the afternoon.

There are three main types of cumulus clouds which develop in this
manner.

(a) Cumulus humilis. This type of cloud is the ordinary fair weather
cumulus which is observed so frequently on summer afternoons. The
clouds are not of great vertical extent, as can be seen from the sketch
shown in Fig. 148a or in the photograph in Fig. 69, section 67. A
typical lapse rate distribution which gives such cumulus humilis clouds
is shown in Fig. 148&. The clouds cannot extend to any great height
because of the lapse rate above the condensation level.

(b) Cumulus congestus. Cumulus congestus cloud may extend to
considerable heights, and appears to be swelling, with protuberances on
the cloud at the sides and top. Fig. 149a indicates roughly the appear-
ance of a typical cumulus congestus cloud. A photograph of this type
of cloud is shown in Fig. 70, section 67. The cloud may tower to con-

Sec. 138}

CONVECTION CLOUDS

349

siderable heights locally. In Fig. 1496, the temperature in the cloud is
greater than C at all levels, and therefore no ice crystals will form.
As the cloud extends upward, either of two things happens, depending on
circumstances. It may reach a region where the wind increases or

(a)

T-
(b)

Fio. 148.

(a) Cumulus humilis, and (6) lapse rate conditions favorable for its
development.

(o)

(b)

FIG. 149.

(a) Cumulus congestus, and (6) lapse rate conditions favorable for its
development.

changes direction rapidly with height, and the top of the towering cloud
will be dissipated by mixing, and its vertical growth halted. If some
such process does not take place, the top of the cloud will extend upward
until it reaches equilibrium with its environment. If there is an iso-
thermal or inversion layer at this level, the upward motion of the air will
cease abruptly, and the cloud will spread laterally, giving a typical anvil
type of cumulus clqjid.

350

CLOUDS

[Chap. 20

(c) Cumulonimbus. Cloud of the cumulonimbus type forms when the
upper portion of the cloud penetrates into the region where ice crystals
form. This region is at high levels during the warm portion of the year,
when convection processes are most active. Marked instability must be
present before cumulonimbus clouds will develop. A cumulonimbus

(a)

(b)

FIG. 160. (a) Cumulonimbus, and (b) lapse rate conditions favorable for its

development.

cloud, with anvil and mantle, the latter composed of ice crystals, is
sketched in Fig. 150a. Fig. 71, section 67, is a photograph of a cloud
of this type. The appearance of the mantle heralds the beginning of
precipitation from the cloud. Violent convection currents occur with
cumulonimbus cloud, and, in addition, thunderstorms and hailstorms
frequently develop. Typical upper air conditions are shown in Fig. 1506,
with the upper portion of the cloud at temperatures well below C.
Cloud droplets do not usually freeze as soon as they reach a temperature
of C. The temperature often decreases to 5 or 10 C before ice
crystals begin to form.

139. Turbulence Clouds. Turbulent air motion is found at any level
in the atmosphere but occurs with the greatest frequency and intensity
near the surface in the f rictional layer, which extends to a height of about
2000 ft on the average. It was shown in section 54 that in a turbulent
layer the upward flux of any property x which does not change owing to
vertical motion is given by the formula

(54-8)

Sec. 1S9\

TURBULENCE CLOUDS

351

where K, the coefficient of eddy diffusivity, is a constant in any given
situation. When equilibrium is reached, so that there is no transfer
upward or downward, d\/dz must equal zero, or the property x must be
uniform with height. Hence in the turbulent layer the specific humidity
s or the mixing ratio x must become uniform with height. Similarly,
equilibrium in the vertical temperature distribution is attained when the
dry adiabatic lapse rate is reached, according to section 53. Fig. 151
illustrates the distribution of moisture, temperature, and relative

lopjof
fnctional
[ayir

*s^ \

Mixing ratio
(a)

Temperature

(b)

50 100

Relative humidity (%)
(c)

FIG. 151.

The distribution of (a) moisture, (b) temperature, (c) relative humidity
in and just above the turbulence layer.

humidity after equilibrium has been reached. If the layer is thick
enough for the relative humidity to become 100 per cent, a layer of cloud
will form at the level at which this occurs, extending to the top of the
turbulent layer. Since during the mixing that has taken place heat has
been transported from the top of the turbulent layer downward, as shown
by (53-11), but has left unchanged the temperature above the layer, an
inversion will frequently be found above the cloud top.

The height of the cloud base is given by the lifting condensation level
of the air at the surface after complete mixing has occurred. According
to (21-8), the rate of decrease of the dew point Td in adiabatically ascend-
ing air is given by the equation

C cm

since T ~ T. But by (14-2) the corresponding decrease in temperature is

r
- = 9.8 X 10~ 5 C cm- 1

Consider a mass of air with temperature T and dew point T* at the

352 CLOUDS [Chap. 20

surface ascending to the lifting conden-
sation level LC, where T = Td. The
rates of change of temperature and dew
point with height, given by the above
equations, are shown in Fig. 152. It
follows directly, then, that

T = TQ+LC (139-1)
dz

Tdo

T and

FIG. 152. The decrease of tern- jm

perature and dew point with Td = Td H ~ L c (139-2)

adiabatic ascent. dz

Since T = T d , (139-1) and (139-2) may be equated to give

Substituting from (14-2) and (21-8) in (139-3) leads to

To - T d , = (9.8 X HT 5 - 6.35 X l<T*j\L c
or

L c = - ^^ - =, (1394)

9.8 X 10- 5 - 6.35 X 10~* -^

By assuming values for T and 7^, the fraction may be evaluated.
Assuming T = 280 A, T d = 270 A

L c = 1.23 X 10 4 (r - T do ) cm
or, with sufficient accuracy for evaluation purposes,

L c = 125(T - T do ) m (139-5)

When the temperature and dew point are measured in degrees
Fahrenheit,

L c = 225 (T ~ T<* ) ft (139-6)

When the moisture content is indicated by the relative humidity /, an
approximate relationship is given by

L c = 70(100 - /) ft (139-7)

Sec. 139} TURBULENCE CLOUDS 353

Thus, if the temperature is 59 F, the dew point 51 F, the relative humi-
dity is found to be 75 per cent. From (139-6) L c = 1700 ft; from
(139-7), L c = 1750ft.

There are two conditions essential for the development of turbulence
cloud:

(a) The relative humidity in the air near the surface must be great
enough to produce a low condensation level.

(6) The turbulence must extend to this condensation level.

If the lapse rate in the surface layer approaches the dry adiabatic,
and the mixing ratio is constant with height in the same layer, the follow-
ing method may be used to assist in determining if turbulence cloud will
form. First, it is necessary to estimate the height to which turbulence
will extend during the period for which the forecast is intended. Some
general rules may be given which will be of assistance in determining
this height. There are five general considerations which must be kept
in mind:

(a) The time of day.

(&) The season of the year.

(d) The stability conditions in the lower levels.

(e) The nature of the underlying surface.

During tjie night turbulence usually extends only about 500 ft above
the surface, and often not that high. The maximum height is reached
during the afternoon, when marked turbulence is frequently found at
1500 or 2000 ft above the surface. The turbulence layer is, in general,
shallower in winter than in summer. The intensity and vertical extent
of turbulence are directly related to the wind velocity, increasing as the
velocity increases. Vertical motion occurs in the atmosphere more
readily when the air. is unstable, or nearly so, than when it is stable.
Turbulence therefore extends to greater heights when the air is unstable.
Finally, the topographical features of the underlying surface exert a
large influence on the vertical extent of the turbulence layer. Over an
extensive plain, or over the ocean, the turbulent layer is usually com-
paratively shallow. Over mountainous country, on the other hand,
turbulence may extend to great heights, especially if the wind is strong.
Thus the motion of air flowing rapidly over a mountainous region on a
sunny afternoon in summer will be very turbulent. Under such con-
ditions it is not unusual for the turbulence layer to be 5000 ft in thick-
ness. Of course, it will be realized that these five factors are not inde-
pendent of each other, but each should be kept in mind when estimating
the vertical extent af the turbulent layer.

354 CLOUDS [Chap. 20

Once the height to which turbulence will extend has been estimated,
the lifting condensation level should be computed from (139-6) or
(139-7). When the condensation level is at 1500 ft, but turbulence
extends only to 1000 ft, no turbulence cloud will form. If the turbulent
layer extends to 2000 ft, however, turbulence cloud will form in the height
interval from 1500 to 2000 ft.

The foregoing procedure must be modified somewhat if the actual
lapse rate is not approximately the dry adiabatic or the mixing ratio
is not constant with height. The temperature and moisture distribution
after sufficient mixing to produce an equilibrium condition has occurred
may be estimated in the following manner. The dry adiabatic lapse
rate is 5.4 F per 1000 ft. After mixing, the difference in temperature
between the bottom and top of a turbulent layer extending to 2000 ft
will therefore be 11 F. If, during the morning before turbulence has
developed, the surface temperature is 55 F, while that at 2000 ft is
50 F, with the intermediate lapse rate nearly constant, the effect of
the vertical transfer of heat by mixing as the turbulence increases during
the morning will be to increase the surface temperature to 58 F and
decrease the temperature at 2000 ft to 47 F, so that the difference
becomes 11 F. If the mixing ratio varies uniformly from 8.3 gm at the
surface to 7.7 gm at 2000 ft, after equilibrium conditions are attained
the mixing ratio in the column has the constant value 8.0 gm, which is the
average for the column and which corresponds to a dew point of 51 F
at the surface. Substituting T = 58 F and T do = 51 F in (139-6)
gives the condensation level to be expected after the mixing occurs.

Another condition must also be kept in mind. If turbulence cloud
is to form, the lapse rate above the condensation level should not be
greater than the saturated adiabatic. If such unstable lapse rate con-
ditions extend to any considerable height, the saturated air will be
unstable, and towering cumulus clouds will develop. The lapse rate
above the condensation level is usually considerably less than the
saturated adiabatic, and indeed, when the cloud occurs as an extensive
and persistent layer, there is nearly always a strong inversion of tem-
perature just above it. Such an inversion is often maintained, and even
intensified, through cooling of the upper portion of the cloud by means of
long-wave radiation. The absorption of insolation by a cloud of this
type is small, and may be neglected, as indicated in section 28.

Turbulence cloud sometimes appears as fractostratus, or as a thick
layer of stratus or stratocumulus. Cloud of this type develops fre-
quently in tropical maritime air when it arrives in temperate latitudes,
owing to the surface cooling and the consequent increase in relative
humidity. If the amount of turbulence is only small, fog may form

Sec. 140] GEOGRAPHIC CLOUDS 355

instead of turbulence cloud. Surface cooling in continental anticy-
clonic areas in winter proceeds rapidly, and this process increases .the
relative humidity of the air near the ground. Turbulence cloud fre-
quently develops under such conditions, especially in England, where
most of the surface is not snow covered. A snow cover, at temperatures
below the freezing point, tends to reduce the relative humidity because
of the difference in saturation vapor pressure over water and over ice, as
pointed out in section 135.

140. Orographic, Billow, and Artificial Clouds. Clouds are produced
when air ascends over a mountain range in much the same manner as
warm front clouds develop. The chief differences between the two types
arise from the fact that orographic cloud is of necessity stationary, or
very nearly so, while the warm front cloud system moves with the front.
Cloud which is purely orographic in character is of the stratus type,
with a flat base. Its vertical thickness may vary considerably, depend-
ing on the height of the obstruction over which the air is flowing. Adia-
batic warming of the air as it descends on the lee side of the mountain
causes rapid dissipation of the cloud.

It has been shown previously that, if an air mass ascends, latent
instability may develop within the air. If latent instability develops,
or is intensified through ascent, and is then realized, clouds of the con-
vection type will develop in addition to the essentially orographic cloud
of the stratus type.

If there is a layer of moist but unsaturated air at a medium height
over the lower ground, orographic ascent may produce condensation in
this layer, and descent on the lee of the mountain will result in evapora-
tion of the cloud droplets. In this manner, a stationary cloud may form
at some height above the top of the mountain or hill. Clouds of this
kind are of the orographic type, and are known as lenticular clouds.
These clouds may form in the vertical currents which develop over the
slope of a mountain heated by the sun, as described in section 124. The
banner clouds which are seen about high mountain peaks are of this type.

Billow clouds form at the surface between two layers of air. Across
such a surface there frequently is found a change in the direction or speed
of the wind. This wind shear gives rise to vertical motions of the air
particles. If the lower layer of air is near saturation, the air particles
become saturated at the highest point in their path and form a cloud.
Such clouds form in a regular pattern across the sky.

Another type of cloud is observed frequently behind aircraft flying in
air at a low temperature. The clouds are formed by the passage of the
airplane, but their cause is not fully understood. Sometimes they
develop through the'adiabatic cooling arising from the decrease in pres-

356

CLOUDS

[Chap. 20

sure back of the wing tips. More frequently they are observed back of
the engine exhaust and are caused by the additional moisture or con-
densation nuclei emitted by the exhaust. They always form in a layer
of air of high relative humidity. Sometimes such condensation trails
persist for considerable periods of time while at other times they dissi-
pate rapidly.

141. Convergence Clouds. When horizontal convergence occurs
near the surface, there is a corresponding ascent of the air, as shown in
section 45. Similarly, descending motion accompanies horizontal
divergence. The vertical air motions associated with convergence and
divergence resulting from the variation of the gradient wind with latitude
around cyclones and anticyclones were discussed in section 119. It was
shown there that the computed velocity of ascent is great enough to
account for the development of cloud, and even of light precipitation.

The horizontal convergence and divergence arising from variations in
the radius of curvature of the air motions and the effect of the resulting

vertical motion on cloud formation
and dissipation will now be dis-
cussed. In order to simplify the
computations the gradient wind
where r = oo, i.e., the geostrophic
wind, is compared with the gra-
dient wind for the same pressure
gradient and for various finite
values of r. Consider, for example,
the distribution of isobars shown in

Horizontal convergence re- Fig. 153. BDF and ACE represent
suiting from the variation in the curva- para n e l isobars, straight from B to

ture of air motion. ^ , - A n i~ j i

D and from A to C, but develop-
ing curvature between CD and EF y the mean radius of curvature
at the latter being r. The velocity of inflow at CD is v$ ; the velocity of
outflow at EF is v\. According to (45-9), the vertical velocity wi,
resulting from horizontal convergence or divergence, is

w l = ~(vi - VQ)

where now y\ represents the distance between CD and EF along the
curved path and z\ represents the height at which the vertical velocity
is computed.

Assume that the streamlines and isobars coincide, an approximation
giving sufficient accuracy for present purposes. This is equivalent to
assuming that at each instant there is a balance l femong the pressure

FIG. 153.

Sec. 141] CONVERGENCE CLOUDS 357

gradient force, the deflecting force, and the centrifugal force. In the
situation shown in the figure, since the pressure gradient force is con-
stant, it follows that the centrifugal force must increase and the deflect-
ing force must decrease in such a manner that their sum remains con-
stant. Thus

h 2w sin <t> v = Constant (141-1)

By differentiating with respect to time, it follows that

2v dv v 2 dr . dv

T - -3 -- + 2w sin < -- = (141-2)

r dt r 2 dt dt

neglecting the variations in the latitude. Rearranging, the acceleration
of the motion

* = * (141-3)

dt 2r(v + rw sin <f>) dt

Since dr/dt < in the figure, and all the other terms on the right-hand
side are positive, dv/dt < 0, and there is horizontal convergence and
ascending motion along the path y\. From (35-1) and (37-2)

_ 1 dP
lp dn

and

where dp/dn = dp/dr. Substituting in (45-9) and dropping the sub-
scripts, it follows that

W= ~~ y\ 2~ T ~\4' ' p dr Ipdn,

The values for w obtained by substituting various values of the radius
and the pressure gradient in (141-4) are given in the upper portion of the
following table. The latitude is assumed to be constant with the value
47.5, and the density p has the value 1.1 X 10~~ 3 gm cnT" 3 . The height
z = 1 km and the distance y = 600 km. In the situation illustrated in
Fig. 153 the air ascends, i.e., the vertical velocities are positive. If the
wind entering the volume is the gradient wind, while that leaving is the
geostrophic, the magnitude of the vertical velocity is the same, but the
sign is opposite, i.e.; negative, and the air descends. The distribution of

358

CLOUDS

[Chap. 20

vertical motion in the vicinity of a trough of low pressure, based on these
figures, is shown in Fig. 154.

VERTICAL VELOCITIES IN CENTIMETERS PER SECOND

Radius (km) 200 400 800 1200

Pressure
Gradient

Cyclonic curvature

4mb per 100 km 2.8 2.0 1.3 1.0

2mb per 100 km 1.0 0.7 0.4 0.3

1mb per 100 km 0.3 0.2 0.1 0.1

Anticyclonic curvature
2mb per 100 km ... ... 1.0 0.5

1mb per 100 km ... 0.5 0.2 0.1

The table shows that the maximum vertical motion occurs when the
pressure gradient is large and the radius of curvature small. These
conditions are found near the center of a well-developed occluding
depression. The maximum vertical velocity shown in the table is 2.8
cm per sec, which is the same order of magnitude as the rate of ascent of

FIG. 154. Vertical currents at a
trough line.

FIG. 155. Vertical currents at a
wedge line.

air over a warm frontal surface in a rapidly occluding depression. In
the latter, the vertical velocities usually range from 3 to 5 cm per sec.
A rate of ascent of 2.8 cm per sec is therefore sufficient to produce dense
cloud and medium precipitation. A trough accompanied by cloud and
precipitation is frequently found in the rear portion of an occluding
depression. There has been a tendency to place a back-bent occlusion,
i.e., an occlusion extending outward from the center of low pressure and
to the rear of it, in such a trough. No sound physical reason has been
put forward to account for such an outward growth of an occlusion, and
the foregoing analysis explains the weather of a frbntal nature which

Sec. 141] CONVERGENCE CLOUDS 359

occurs in a trough near the center of a low without invoking any surface
of discontinuity.

As shown in Fig. 154, there is a region of subsiding air just ahead of
the trough line, in which the cloud would tend to decrease, the precipi-
tation to cease, and the temperature to rise. Thus as the trough line
approaches a station the cloud becomes less dense, the precipitation
diminishes or stops, and the temperature rises. Then just after the
trough line passes, the clouds become thicker, the precipitation starts
again, and the temperature falls, which is precisely the sequence of
weather to be expected with a cold front type of occlusion. The presence
of an area of subsiding air and one of ascending air in such close prox-
imity, both extending nearly parallel to the trough line, thus accentuates
the impression that a cold front type of occlusion has passed, although in
fact none has. The wind speed across the trough line is frequently
greater than the speed of the trough itself in such situations, which is a
further indication that no front lies in the trough. Even with smaller
pressure gradients and larger radii of curvature, the vertical velocities
shown in the upper portion of the table are sufficient to account for the
formation of cloud near the outer limits of a trough.

The vertical motions associated with a wedge of high pressure may be
computed from an equation similar to (1414). By using the same
values of $, p, etc., as before, the vertical velocities calculated are shown
in the lower portion of the table. The values for anticyclonic curvature
are greater than those for cyclonic curvature for the same values of r
and dp/dn, but owing to the upper limit on the pressure gradient in an
anticyclone, discussed in section 37, the maximum vertical velocities
attained are not so great as for cyclonic curvature.

The distribution of vertical velocities near a wedge line is shown in
Fig. 155. There is horizontal convergence and ascending motion ahead
of the wedge line, horizontal divergence and subsidence behind it, neither
of which is as great as near a trough. Thus the cloud near a wedge tends
to be concentrated just ahead of it when the latter is in the position
shown in Fig. 155. Convergence and divergence of this type frequently
account for the cloud distribution near the portion of an anticyclone
where the curvature of the isobars changes rapidly.

The cloud resulting from isallobaric convergence, discussed in sec-
tions 46 and 121, belongs in the same category.

Convergence cloud is thus associated with horizontal convergence
arising from variations in latitude and curvature, from isallobaric devia-
tions from the geostrophic wind, and from frictional inflow during the
motion of the air. Horizontal convergence arising from more obscure
causes also occurs. *

360 CLOUDS [Chap. 20

In general, convergence cloud is of the stratus type. If marked latent
or potential instability is present, however, the vertical motion accom-
panying the horizontal convergence may be sufficient to realize the
instability, resulting in the development of clouds of the cumulus type
as well.

PROBLEMS AND EXERCISES

1. The following are values of temperature and mixing ratio obtained during a
radiosonde ascent.

p 994 960 823 768 692 611 540 517 429 mb
T 26 29 18 14 10 4 -2 -2 -11 C

x 16.4 17.9 10.4 8.6 4.0 2.4 1.1 1.6 1.0 gm per kg

What cloud development is probable during the following day if it is expected
that the surface temperature will rise to 93 F? What clouds and weather are to be
anticipated if a cold front passes during the day?

2. Consider the same questions as in Problem 1 for the following ascent if the
maximum temperature expected is about 83 F.

p989 982 954 854 824 750 713 653 582 470 406 mb
T 15 19 20 13 14 11 9 6 -1 -9 -18 C
x 10.1 13.5 7.0 5.4 2.9 8.3 7.8 4.1 2.6 1.1 0.7 gmperkg

3. The following table gives the values of the temperature and mixing ratio for an
air column during the night.

p 950 861 774 710 623 588 423 mb

T 18 14 7 3-4 -4 -19 C

x 8.6 7.6 5.0 4.2 2.7 1.2 0.8 gmperkg

If the maximum temperature expected at the surface during the following day is
75 F, what cloud forecast should be made? If this were an ascent in the warm sec-
tor of a depression, what clouds would be expected as this air ascends the warm
frontal surface?

BIBLIOGRAPHY

Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book

Co., 1937. Chapter 5.
Petterssen, S., Weather Analysis and Forecasting, New York, McGraw-Hill Book Co.,

1940. Chapters 2, 6.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 3

(1930).
Douglas, C. K. M., " Physical Processes of Cloud Formation," Q. J. Roy. Met. Soc.,

60, 333-341 (1934).

138. Poulter, R. M., " Cloud Forecasting The Daily Use of the Tephigram,"
Q. J. Roy. Met. Soc., 64, 277-292 (1938).

139. Krick, I. P., " Forecasting the Dissipation of Fog and Stratus Cloud," J. Aero.
Sc., 4, 366-371 (1937).

CHAPTER 21
ICING ON AIRCRAFT

142. Types of Ice Deposit. There are four types of ice which
form on aircraft in flight :
(a) Carburetor ice.
(6) Hoar frost.

(d) Clear ice.

Carburetor ice is a type that occurs in the carburetor of the engine,
arising from the condensation of part of the moisture present in the air
when the latter expands rapidly. It occurs with outside temperatures
as high as 70 F, and no method of forecasting its occurrence has as
yet been developed. The elimination of icing of this kind is a problem
for the aeronautical engineer rather than for the forecaster.

Hoar frost is the product of the sublimation of water vapor upon the
surface of the aircraft. It is similar to frost that is deposited on the
ground and occurs when the surface of the plane is colder than the dew
point of the air. Hoar frost develops on a plane when it flies from cold
air to warm. As the plane takes on the temperature of the new environ-
ment the frost evaporates. It is not, then, a serious hazard to the plane
when it is in flight. Yet if a plane has been left outside during a clear
frosty night, the radiational cooling of the surface of the plane may be
sufficient to cause a deposit of frost. If the pilot attempts to take off
before this is removed, the change in the aerodynamic properties of the
wings may be so great that the plane is difficult to control, and the
process of taking off dangerous. The development of hoar frost in this
manner is not the concern of the meteorologist, however. Also, a coat-
ing of frost on the windshield sometimes develops when a plane descends
rapidly from a high altitude for a landing, and interferes with the vision
of the pilot.

Rime ice is white, opaque, and of a granular structure consisting of
small ice pellets. This type of ice forms principally on the leading
edges of the aircraft. It is seldom dangerous, since it does not alter
the aerodynamic properties of the wings to a serious extent. A typical
formation of rime ice is shown in Fig. 156a. Furthermore, because of
its granular structure, it is frequently dislodged by vibration.

361

362 ICING ON AIRCRAFT [Chap. 21

Clear ice is smooth and glassy in appearance and is dislodged only
with difficulty. It forms near the leading edge of the airfoil or strut,
usually in a mushroom shape as in Fig. 1566. This is the most danger-
ous type since it alters the aerodynamic properties of the wings, and
the extra weight added to the plane may produce a serious loss of lift.

(a) (b)

FIG. 156. The deposition of (a) rime ice, (6) clear ice on the wing of an airplane.

The last two types of icing are serious hazards to the aircraft, and
the remainder of the chapter will be devoted to a discussion of the phys-
ical processes in operation and the meteorological conditions present
when such ice forms.

143. Process of Deposition. It was indicated in section 128 that
water droplets exist in the free atmosphere at temperatures below C.
These supercooled water droplets have been observed by Wegcner in
Greenland at temperatures of 35 C and by Haines in Little America
at 26, 30, and 44 C. The extent of the supercooling can be
accounted for by one of or all the following factors : the surface tension
of the drop, the fact that the drop is not of pure water but is composed
of a salt in solution, or the effect of the electrical charges on the drop.
It has not been decided as yet which one of these factors is fundamental
in permitting such a degree of supercooling in tha atmosphere.

When a supercooled water droplet begins to freeze, its temperature
rises instantaneously to C, the heat necessary to raise the tempera-
ture being obtained by the release of the latent heat of fusion. If the
supercooled water droplets are at a temperature of T C, then/, the
fraction of water frozen, is given by

80/=T or /=!

The latent heat is taken as 80 cal per gm. It can be seen that the frac-
tion of water frozen is directly proportional to the degree of super-
cooling. Thus if T = 8C, 10 per cent will freeze immediately.
Since the remainder of the drop is at a temperature of C, evaporation
can take place into the cool air stream, or heat can pass by conduction
from the drop to the surrounding air or the plane. The heat of fusion
is thus dissipated by any of or all these methods and the remainder of
the drop will freeze.

Sec. 144] VARIATION WITH TEMPERATURE AND SEASON

363

The manner of freezing depends on the size and number of the drops.
When the droplets are small so that each one freezes as a unit almost
immediately on coming in contact with the plane, the ice which is
deposited has little cohesion and there are small air pockets embedded
in it. This ice, of the rime type, forms in a uniform layer along the
leading edge of the wing so that it does not materially reduce the lift.
Since it is made up of a large number of small crystals, it can be dis-
lodged easily by ice-removing devices on the wings and other parts of
the plane. For both these reasons rime ice does not endanger the air-
craft.

When the drops which the plane strikes are large, considerable water
is left after the initial freezing, and this is carried backward in the wind-
stream over the edge of the wings. The ice that is formed as this water
freezes extends over a considerable area of the plane as a solid and
tenacious mass. If, before the drops are completely frozen, other drops
come in contact with them, the mass of ice grows as a unit and becomes
even more difficult to dislodge from the wings of the plane. This is the
method of formation of clear ice. Because of its adhesive and cohesive
properties, its rapid rate of QQ
accumulation, and the
change which it causes in
the shape of the wing, it is
a major hazard to aviation.

144. Variation of Ice De-
posits with Temperature
and Season. The amount
of ice formation varies
greatly with temperature.
In general, it may be said
that the greatest deposits
of ice occur at tempera-
tures just below the freezing
point. At lower tempera-
tures the rate of accretion FIG. 157. Variation of icing with temperature
of ice is much less. Data in the British Isles: clear areas, flights without
on the dependence of icing icin & sin 8 le hatching, light icing; double hatch-
on temperature over the ing, heavy icing. (After Bi gg .)

British Isles are available and are shown in Fig. 157. The total
height of each column is proportional to the total number of
occasions when the temperature was within the limits shown and ice
formation might be expected. The unhatched portions of the columns
in the diagram reprpsent the number of cases without icing, whereas the

28 23 18 13 8 3 -2
Temperature (F)

364

ICING ON AIRCRAFT

[Chap.

hatched areas of the columns indicate the number of cases of icing of
all types in each temperature interval. Double hatching indicates the
number of occurrences in which icing was heavy. For example, out
of a total of 94 flights when the temperature was between 28 and 23 F,
there was no ice formation during 16 and some icing occurred during the
remaining 78 flights. Of the latter there were 14 cases of heavy icing.
In addition to the values shown, there were five cases of icing at temper-
atures below 2 F, but no cases of heavy icing at temperatures below
this. This diagram shows that the maximum icing of both heavy and
all types occurs in the temperature interval 28 23 F. As can be
computed from the values given in the chart, 82 per cent of all icing
occurs in the internal 32 18 F, and 79 per cent of the cases of heavy
icing occurs in this interval.

Similar data, although less in amount, have been obtained in the
United States, and the analysis of these indicates that icing occurs
under the same general conditions. The results of the analysis of 66
observations of icing are given in the table in which it can be seen that

OCCURRENCE OF CLEAR AND RIME ICING

Temperature range (F)
Mean temperature (F)
Percentage at temperatures above 18 F

Type of Ice

Clear Rime

35 to -7 29 to -20

20 12

78 36

rime ice occurs at lower temperatures than clear ice. This is to be ex-
pected, as rime results from the freezing on impact of small supercooled
water droplets, which are prevalent at lower temperatures. The large
droplets which give clear ice are found at higher temperatures.

0>
J3

1 ]_ I 1

J I

I L J

J F

MAMJJASOND
Month

Fio. 158. The annual distribution of icing in the United States.

The seasonal variation of icing on aircraft over the United States is
indicated in Fig. 158. The great majority of ice deposits occurred
during the late autumn, winter, and spring months^ These variations

Sec. 146} ICING, CLOUD FORMS, AND STABILITY 365

are, of course, largely due to the variation in the height of the freezing
level during the year. In winter the level is low, and aircraft often
encounter sub-freezing temperatures. In summer the freezing level is
high, and only under exceptional conditions will aircraft be flying at
such levels. Conditions over the British Isles are essentially the same
as those shown in Fig. 158.

145. Icing, Cloud Forms, and Stability. The deposition of ice occurs
where there are supercooled water droplets, and the larger the amount
of liquid water in a given volume, the greater the icing hazard. When
saturated air at 5 C and 800 mb cools to 10 C, 1 gm of water vapor
per kg of dry air condenses. If the air had been saturated at 5 C, and
cooled to C, the mass of water vapor condensed would have been 2
gm per kg. Thus the amount of water vapor which is condensed is
dependent on the dew point of the air before condensation commenced
as well as on the degree of cooling below that dew point.

Another factor to be considered is the length of time the cloud par-
ticles have been above the freezing level. As described in section 128,
condensation occurs in the form of water droplets when the temperature
is above freezing, and in the form of ice crystals or water droplets or
both when the temperature is below freezing. When both ice crystals
and water droplets exist at temperatures below C, the state is un-
stable, as indicated in section 128, and the ice crystals grow at the ex-
pense of the water droplets. It follows that the longer the cloud par-
ticles stay above the freezing level the less the icing hazard becomes
since an increasing proportion of the condensed moisture will be in the
form of ice crystals. Thus the most serious icing in clouds occurs when
the latter form in air which is initially warm and moist, and when the
clouds have recently developed or have recently ascended above the
freezing level.

In clouds of the cumulus type, icing is both probable and dangerous.
The clouds are formed in vertical currents which carry air from near the
surface of the earth to great heights. The amount of cooling below the
dew point is considerable; the initial temperature and moisture content
of the air before ascent are usually comparatively high ; the time inter-
val during which the cloud has existed above the freezing level is short;
and the turbulence produces large water droplets. All these features
aid in explaining the relatively high frequency of occurrence of heavy
icing on aircraft in this type of cloud.

Altostratus and nimbostratus clouds are formed in the warm, moist
overrunning air at a warm front. But the rate of ascent is small com-
pared with that in a cumulus cloud. If then the cloud is far above the
freezing level, more of the condensed moisture will be in the form of ice

366 ICING ON AIRCRAFT [Chap. 21

crystals than of liquid water. A long flight in the cloud will lead to a
considerable deposit of rime ice when the temperatures are below 8 C
(18 F). The portions of the cloud at temperatures to 8 C will
have been above the freezing level for only a short time, and sufficient
liquid water will be present to lead to clear icing, which will sometimes
be heavy. If actual or latent instability is present, a cumulus cloud
with heavy icing in its top portion may develop in the nimbo-
stratus.

Turbulence stratus and stratocumulus clouds may persist for a con-
siderable time. Since, however, the cloud droplets are continually
evaporating and forming in the vertical currents just below the cloud
base, water droplets will persist in these clouds. In stratus the droplets
are usually small and rime ice will be deposited. When, as sometimes
happens, the moisture content of the clouds is sufficient to give freezing
drizzle, the accumulation of ice may be large if the aircraft persists in
the cloud for any length of time. Since, though, stratus cloud is usually
thin, flying can be done above the cloud and thus the hazard is elimi-
nated.

Icing in stratocumulus is similar to that in stratus except that the
turbulent eddies are stronger, and the drops are larger. In the strato-
cumulus clouds that form through the instability that develops in polar
air behind a cold front, the amount of moisture may be sufficient to give
a clear ice deposit at a moderate rate if the temperatures are above
8 C. In other situations, or with temperatures below this, the icing
that occurs will be of the rime type.

The temperature at the altocumulus level is usually low enough that
any icing that takes place there will be light and of the rime type. Ob-
servations in general have tended to confirm this deduction. Heavy
clear icing in altocumulus occasionally occurs and may be due to the
presence of strong turbulent eddies arising from the instability at that
level which produces altocumulus congestus clouds.

Clouds at the cirrus level are formed of ice crystals and so do not lead
to ice deposits.

In the foregoing discussion the clouds which give serious icing are
distinguished from those that give light icing largely on the basis of the
stability of the air. Thus rapid icing is found in cumulus, cumulonim-
bus, cumulus embedded in nimbostratus, and in stratocumulus behind
a cold front. Stratus type clouds do not in general lead to rapid ice
accretion. Investigations have confirmed this conclusion to a certain
extent, the stability being evaluated in terms of the saturated adiabatic
lapse rate. Further investigations of the relationship between insta-
bility and the rate of ice accretion are needed. *

Sec. 147] AVOIDING DEPOSITION OF ICE 367

146. Ice Formation by Supercooled Raindrops. The formation of
ice on an aircraft as it flies through a cloud of supercooled droplets has
been discussed in the previous section. Very serious ice accretion may
also occur during flight beneath cloud, if the aircraft encounters super-
cooled raindrops as they are falling to earth. Such icing is likely to be
especially serious beneath frontal and orographic cloud, because of the
great extent of the cloud, both horizon-
tal and vertical, and the difficulty in ^ ^ Frontal

reaching a region free from supercooled
rain. Rapid deposition of ice fre-
quently occurs under such circum-
stances, and the airplane may be forced
to land unless there is a layer below in Fjo 15Q ^ gion of
which the temperature is greater than a warm frontal surface>

C in which it can fly.

At a warm front, the situation is serious if rain falls from warm front
cloud through cold air below, the temperature of which is below the
freezing point. The region of maximum danger below the warm frontal
surface is indicated in Fig. 159.

At a cold front, conditions may be similar, but there are significant
differences. Because of the fact that the slope of a cold front is usually
greater than that of a warm front, and the cloud system associated with
the former is often discontinuous, the area in which icing conditions
occur is not so extensive in the horizontal as near a warm front. Since
the amount of ice accretion is in part dependent on the length of time
that the aircraft remains in the region where ice accretion occurs, it is
clear that ice deposition will, in general, be less serious at a cold front
than at a warm one. On the other hand, a slowly moving cold front
often exhibits icing conditions similar to those at a warm front. In
addition, cumulonimbus clouds are frequently associated with cold
fronts, and icing in these is likely to be serious.

At occlusions serious icing may occur. The situation is similar to
that at a warm front, for aloft there is a trough of warm air from which
precipitation falls. If the temperature of the warm air is greater than
C, and that of the cold air masses below is less than C, a wide area
exists in which icing will occur rapidly.

Icing may also occur on an aircraft flying under a precipitating cloud
of the cumulus type, such as cumulonimbus. However, such clouds
are usually isolated, and it is possible on most occasions to avoid the
regions where supercooled rain may be falling.

147. Means of Avoiding Deposition of Ice. The deposition of ice
will be prevented if aircraft avoid ice-forming clouds and regions where

368 ICING ON AIRCRAFT [Chap. 21

supercooled rain is falling. If ice begins to form on the plane, then the
pilot must immediately estimate its rate of accretion, and leave the
region, usually by ascent or descent, if the icing is going to be too heavy
for continued navigation. Rules for avoiding serious icing can be
summarized as follows:

(a) Avoid, at all times, flying in air cooler than C into which rain
is falling from above. If the plane cannot fly below the freezing level,
it may be able to turn back out of the rain and climb above the region
of serious icing. Otherwise the plane should land.

(&) Avoid, at all times, flying in cumulus and cumulonimbus clouds
above the freezing level. Since these are usually isolated, it is usually
possible to avoid these clouds.

(c) Avoid flying in nimbostratus and altostratus clouds when the
temperature is between and 8 C (32 and 18 F). If the air is un-
stable, or likely to become so, avoid these clouds in the layer between
and 14 C (32 and 7 F). Since these layers normally are about
4000 ft thick for stable air, and 8000 ft thick for unstable air, it is fre-
quently possible to choose between flying above or below them to avoid
the region of serious icing.

(d) Avoid extensive flying in stratus, stratocumulus, and altocu-
mulus clouds in the temperature range to 8 C (32 to 18 F). Since
these clouds are usually thin, it is possible to get out of the cloud either
by ascending or descending.

148. Forecasting the Deposition of Ice. The forecasting problem is
considerably simplified if upper air temperatures are available. Deter-
mine the heights at which temperatures of and 14 C (32 and 7 F)
occur. If clouds are expected in the layer between, then icing is a
possibility. If cumulonimbus clouds are expected, anticipate ice accre-
tion at any level in the cloud. If no upper air temperatures are avail-
able, the height of the bottom and top of the region of serious icing can
be obtained by approximate methods. Knowing the surface temper-
atures, a rough estimate of these levels may be obtained by assuming
an average lapse rate of 6 C per km or 1 F per 300 ft. If there is a
surface inversion, allowance must be made for this. If the inversion
is due to a frontal surface, then the heights of the top and bottom of the
region of serious ice accretion above the frontal surface may be calcu-
lated by assuming adiabatic ascent of the surface air in the warm sector.
The height of the freezing level, or of the freezing levels when two air
masses are present, should be available for pilots.

Since instability increases the danger of icing, an evaluation of this
should be made. In lieu of upper air ascents, it is sometimes necessary
to deduce its presence or absence by means of the types of precipitation
that are reported at the observing stations.

BIBLIOGRAPHY
PROBLEMS AND EXERCISES

369

1. Weather reports before, at, and after the passage of a cold front gave the in-
formation found in the table.

Time

14.30

15.30

15.45

16.30

17.10

17.30

[Type

Sky

Low cloud | Height

2000

2500

obscured

1500

2000

(.Amount

tenths

Type of middle cloud

Total cloud amount

tenths

Temperature

Dew point

w j {Direction

SSW

WSW

Wind < a ,
[Speed

25G 1

25G

35G to 62

30G

35G

37G

mph

Visibility

Weather

Haze

Heavy sleet

Mod. sleet

Haze

1 Gusty.

From this information, what deductions may be made concerning the levels un-
favorable for flying in the vicinity of the front?

2. The following values were obtained by an upper air sounding through a quasi-
stationary front.

P
T

1000
2

900

-1.5

42
850

800

93
700
-6

130
600
-12

100's of ft
mb
C

The relative humidity throughout was over 70 per cent. Precipitation was
falling at the station, and in the vicinity rolling hills reached an altitude of 600 ft.
What routes are possible for a pilot who wishes to fly normal to the front from a point
200 mi on one side of the station to 200 mi on the other side?

BIBLIOGRAPHY

Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book
Co., 1937. Chapter 13.

Bigg, W. H., Ice Formation in Clouds in Great Britain, Prof. Notes, No. 81, London,
Meteorological Office, 1937.

Jordanoff, H., Safety in Flight, New York, Funk and Wagnalls, 1941.

Lacey, J. K., "A Study of Meteorological Factors Affecting the Formation of Ice on
Airplanes," Bui. Am. Met. Soc., 21, 357-367 (1940).

McBrien, R. L., " Study of Icing Problems of Transport Aircraft," Canadian Avia-
tion, 14, 21 (1941).

Minser, E. J., " Studies of Synoptic Free-Air Conditions for Icing on Aircraft,"
But. Am. Met. Soc., 19, 111-122 (1938).

Simpson, Sir G. C., Ice Accretion on Aircraft, Prof. Notes, No. 82, London, Meteor-
ological Office, 1937.

Sutcliffe, R. C., Meteorology for Aviators, London, H. M. Stationery Office, 1940.
Pages 138-143.

Taylor, G. F., Aeronautical Meteorology, New York, Pitman Publishing Corporation,
1938. Pages 313-318.

CHAPTER 22
THUNDERSTORMS

149. The Potential Gradient. Electrical Charges in the Atmosphere.

In fine weather there exists a difference in electrical potential between
the earth and the atmosphere, or a vertical potential gradient. This
potential gradient was first measured by Franklin by means of kites
and vertical rods. It is still the object of innumerable measurements.
During fine weather the gradient is positive upward with the earth
acting as a negatively charged conductor.

The average value of the potential gradient over the ocean and over
level land areas is about 100 volts per m. Individual values vary widely
from this, however. During thunderstorms it reaches at times the
value of 10,000 volts per m. Over the sea, where local factors are not
significant, there is a diurnal variation, but the time of maximum is the
same for all parts of the earth, being about 17.30 h GMT. There seems
to be a close correspondence between the variation of the potential
gradient and of the total number of thunderstorms that are occurring
over the surface of the earth. This suggests that each thunderstorm
adds a positive charge to the upper atmosphere which is carried
rapidly over the earth.

The presence of clouds overhead changes the value and frequently
the sign of the potential gradient. With the arrival of warm frontal
clouds the gradient generally changes to negative. The gradient in
the vicinity of a thunder cloud fluctuates rapidly, but it is generally
negative below the edges of the cloud and strongly positive under the
center of the cloud. With shower clouds the gradient is usually nega-
tive.

Measurements of the charges within thunder clouds were made by
Simpson and Scrase between 1935 and 1939. This was done by attach-
ing to a balloon an instrument, which they called an alti-electrograph,
and permitting it to rise through the cloud to be studied. The average
distribution of charge found in different parts of the thunder clouds in-
vestigated is illustrated in Fig. 160. At the top of the cloud, in the
portion extending to heights greater than the level of the 10 C iso-
therm, the charge was found to be positive. In the middle portion of
the cloud, between the 10 C and C isotherips, the charge was

370

Sec. 150} THE ORIGIN OF THUNDERSTORM ELECTRICITY 371

negative. In the base of the cloud a center of strong positive charge
was at times measured, below which heavy rain occurred at the surface
of the earth.

Measurements of the charge on raindrops falling to the earth have
shown that these are electrically charged. The investigations indicated
that during any one storm some drops carry negative charges, while
others carry positive charges. Nevertheless the average charge carried
to the earth in rain from thunder clouds is positive, that from shower
clouds is negative, and that from frontal rain is positive.

Neg _ -
Ram

FIG. 160. The average distribution of charge in a thundercloud, according to

Simpson.

During dust storms and snow storms there is a separation of electrical
charges. The air becomes positively charged, while the dust or snow
particles become negatively charged. The gradient developed during
these storms is sometimes very great.

150. The Origin of Thunderstorm Electricity. Two main theories are
current to account for the development of the great electrical fields
which accompany thunderstorms.

(a) The Breaking-Drop and I mpact-of -Ice-Particles Theory. In 1892
Lenard showed that pure water splashing against a solid object causes
electrification, the water becoming positively charged. In 1908 Simp-
son showed that the same separation of charge takes place when water
drops are broken up by a rapidly moving stream of air. Using this fact
as a basis, Simpson developed a theory to account for thunderstorm
electricity. He had found that the separation of charge occurred when
the water droplets were broken by a stream of air moving at a speed

372 THUNDERSTORMS [Chap. 22

greater than 8 m per sec. Vertical currents with velocities greater than
this occur in the lower front portion of a thunder cloud, producing a
separation of charge in that region. The raindrops there become posi-
tively charged, and the negative charges become attached to the cloud
particles and are carried by the turbulent currents through the cloud.
With the discovery of the positively charged region in the top of the
cloud, Simpson modified his theory. At the top of the cloud, which is
usually at a temperature lower than 10 C, the ice particles are in
turbulent motion. Collisions between them cause a separation of
charges, with the ice crystals becoming negatively charged and the
atmosphere positively charged. The ice particles increase in size and
descend, leaving a positive charge at the top of the cloud, while at lower
levels the preponderance of negatively charged ice crystals gives a nega-
tive charge to that part of the cloud. When the ice crystals fall below
the freezing level, they become raindrops. The strong vertical currents
cause most of these to break, during which process they are stripped of
their negative charges and finally take on a positive charge. The drops
which are not subjected to sufficiently strong vertical currents to be
broken up carry their charges to the ground. By this theory both the
distribution of charge in clouds and the differing charges on drops reach-
ing the surface are explained. The theory also explains the charge on
rain in showers, where the vertical currents are not as strong as in
thunderstorms.

(fe) Influence Theory. According to this theory, put forward by
C. T. R. Wilson, the charges are separated by the influence of electrical
fields,hence the name. Wilson assumes the ordinary fine-weatherpositive
potential gradient and considers how a separation of charge may occur.
He takes the case of raindrops falling through a cloud when such a posi-
tive field exists as is illustrated in Fig. 161. The positive charges aloft
and the negative charges at the surface are used to denote the positive
potential gradient of the earth's field. A drop falling in such a field
will have negative charges induced at the top of the drop and positive
ones at the bottom. Negative ions in the air will be attracted to the
lower portion of the drop while positive ones will be attracted to the
upper part of it. But because the drop is falling, negative ions will be
readily drawn to the bottom and will neutralize the induced positive
charge there. On the other hand, positive ions will have difficulty in
overtaking the falling drop, especially in view of their small mobility,
and there will be little neutralization of charge at the top of the drop.
This will leave a net negative charge on such drops and a net positive
charge on the air above from which negative ions have been removed.
In this manner, according to Wilson, the separation of charge is effected.

Sec. 161} VARIATIONS IN TIME AND PLACE 373

As originally proposed, this theory explained the positive charge at the
top of the thunder cloud and the negative charge over most of the base,
which were known to exist when the theory was put forward. If the
possibility of the separation of charges by impacts of ice crystals is
accepted, it can be modified to explain the positive charge at the base.
Below the freezing level, i.e., below the strong negatively charged layer
from C to 10 C, the ice crystals melt and fall as raindrops. Each

FIG. 161. The development of an electrical charge on a raindrop falling in an
electrical field, according to Wilson.

drop is now falling through a region which has negative charges above
and positive charges at the earth's surface and will then acquire a posi-
tive charge through the influence of the electrical field. The influence
theory, however, fails to bring out the close connection between strong
electrical fields and vertical currents since the potential gradient is pres-
ent during all types of rain.

Neither theory satisfactorily explains the preponderance of positive
charge found in warm frontal rain with stable conditions. A fully satis-
factory theory for the development of atmospheric electricity has not
yet been proposed.

151. Variations in Time and Place of Occurrence of Thunderstorms.
Thunderstorms are associated with strong vertical currents and so are
most frequently found in regions of great instability. The vertical dis-
tribution of temperature and moisture which leads to thunderstorms
is the same as that which leads to the development of cumulonimbus
clouds, discussed in section 138.

Thunderstorms are most frequent over the moist, humid regions of the
equatorial low-pressure belt. They are comparatively rare in the sub-
tropical high-pressure belt except at the western edge of a cell where
the southerly winds bring moist air from the equatorial regions. Over
the temperate zon$ there is an annual variation, with the number of

374

THUNDERSTORMS

[Chap. 22

.52

02 2

"8 8

+3 <+H

> Q)

Sec. 161]

VARIATIONS IN TIME AND PLACE

375

2 5 w "> e
< a i ( i u

^ ~ CSJ K> >

376 THUNDERSTORMS [Chap. 22

storms decreasing with increasing latitude. Very few occur in the polar
regions.

In the temperate zone the time of most frequent occurrence of thunder-
storms is during the months of June, July, and August. Very few
thunderstorms occur over the ocean, but the time of maximum frequency
of those reported is during the winter. There is a similar difference
in the time of occurrence of maximum thunderstorm activity during the
day over land and over ocean surfaces. Over the land most of the
storms occur during the period 14 to 18 h, whereas over the ocean the
maximum is from 24 to 04 h. This contrast is explained by the effects
of surface heating. Over the land the time of maximum heating is dur-
ing the afternoon in summer. The surface of the sea has a relatively
constant temperature. Hence it is warmest with respect to the air
moving off the land when that air is coldest, that is during the winter
and during the night. The maximum instability over the ocean de-
velops then, and with it the maximum of thunderstorm activity.

The effect of a water surface in decreasing instability must be kept in
mind when forecasting for some land stations. When the trajectory of
the air is over an extensive water surface during the summer, the in-
stability that may be present originally will frequently decrease, leading
to a cessation of thunderstorm activity as the air moves again over the
land. The average distribution of thunderstorms over the United
States is shown in Fig. 162. The region of maximum occurrence is over
western Florida where the moist air from the Gulf develops instability
through surface heating. A second maximum is found over New
Mexico and Colorado. Fig. 163 gives the average distribution of
hailstorms over the United States. Although there is some similarity
in the distributions of hail- and thunderstorms, the storms near the
Gulf are seldom accompanied by hail. The greatest proportion of
storms with hail is found over the plains where on the average one
storm in ten is accompanied by hail.

The effect of insolation in causing thunderstorms is 'well illustrated
by the distribution of thunderstorms over the British Isles, shown in
Fig. 164. During the summer months the prevailing winds are westerly.
As the air leaves the ocean and advances over the land surface, which
has been heated by insolation during the day, the lower layers in turn
become heated. When the air at higher levels is cold, while that near
the surface is particularly moist, this surface heating may be sufficient
to produce marked real latent instability by the time the air reaches
the Midlands, resulting in the observed maximum of thunderstorms
there. This analysis is confirmed by the fact that the area of maximum
thunderstorm activity coincides approximately with'the region of maxi-
mum temperature during the summer months.

Sec. 152}

THUNDERSTORM FORECASTING

377

152. Thunderstorm Forecasting. The forecasting of thunderstorms
has its own peculiar difficulties. Thunderstorms are local phenomena,

FIG. 164. The distribution of thunderstorms over the British Isles. (From Bilham,
The Climate of the British Isles, Macmillan and Co.)

with diameters of 8 to 12 mi. Even in a district where thunderstorm
activity is pronounced, there will be sections in which none occur. This
difference is probartfy a result of the local topography, since a slight

378 THUNDERSTORMS [Chap. 22

orographic lift may be necessary to release the instability present.
After a center of convective activity forms, it tends to persist and to
move off in the direction of motion of the current of air at 4000 to 6000 ft.
But since these storms are local, it is not possible to forecast their occur-
rence at a given station with assurance, although they can be expected
in a given district.

There are several types of thunderstorms which must be kept in mind
when forecasting.

Air mass thunderstorms may develop in several ways. Heating of the
air near the surface is the most frequent cause of this type of storm,
although advection of cold air in the middle layers is also occasionally
a contributing factor. Insolational heating of the earth's surface will
produce a steep lapse rate in the lower layers. If, in addition, the mois-
ture content of the air near the surface is high, as in tropical maritime
air masses, thunderstorms are likely to develop, especially when upper
air conditions are such that a large positive area extending to great
heights is shown on the tephigram. The development of these areas
is discussed in section 138 in connection with the development of con-
vection clouds. Air mass thunderstorms may also occur as a result of
the heating in the lower levels of a cold air mass, such as a polar con-
tinental air mass, as it moves over a warmer surface. Very steep lapse
rates will develop near the surface in this manner.

Possible changes in the upper air should be considered. If the air
is becoming colder because of a wind shear at higher levels, or if the air
near the surface is becoming more moist, instability may develop where
it was previously absent. On the surface pressure map the region of
thunderstorm activity frequently coincides with a weak low-pressure
area, although the latter may be missed because of the weak pressure
gradient usually existing over the continent during the summer.

There are several local indications that thunderstorms will develop.
The occurrence of altocumulus castellatus, that is, altocumulus with
turrets of cloud associated with it, during the early morning is an
indication of instability in the upper levels which may increase as the
surface layers are heated. Later in the day towering cumulonimbus
clouds with warm, moist conditions at the surface suggest that thunder-
storms will occur.

Frontal thunderstorms are of frequent occurrence. Latent instability
in the warm air mass is developed or increased by lifting of the air mass
in the manner outlined in section 95, and this increase in latent insta-
bility makes thunderstorm development more probable. Perturba-
tions of the air motion, arising from the vertical motions sometimes
occurring near a front, may also start the process 6f realization of the

BIBLIOGRAPHY 379

latent instability. Thus thunderstorms frequently occur 100 mi, or so
ahead of a cold front. Most frontal thunderstorms over North America
occur in warm, moist tropical Gulf air. The base of a cold front thunder
cloud is usually near the surface of the earth. On the other hand, warm
front thunderstorms frequently occur 200 to 300 mi ahead of the surface
position of the front at a high level at the point where the instability is
released. Summer thunderstorms are very frequently associated with
old occluded fronts which have taken on the characteristics of active
cold fronts. Diurnal heating of the surface air tends to intensify frontal
thunderstorms, especially those of the cold front type. The time of
maximum occurrence of these is during the afternoon.

Orographic thunderstorms have many characteristics similar to those
of warm frontal thunderstorms. The ascending motion of the air in-
creases the latent instability, and the accompanying turbulence near
the surface may be sufficient to start the process of release of the insta-
bility. An interesting feature of some orographic thunderstorms is
their tendency to remain almost stationary. Mountain slopes which
are nearly normal to the direction of the sun's rays may be very strongly
heated, and intense thunderstorms often develop over such slopes.

BIBLIOGRAPHY

Admiralty Weather Manual, London, H. M. Stationery Office, 1938. Chapter 12.
Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book

Co., 1937. Chapter 12.
Problems of Modern Meteorology, London, Royal Meteorological Society, 1934.

Number 3.
Shaw, Sir N., Manual of Meteorology, London, Cambridge University Press. Vol. 2

(1936), Chapter 2; Vol. 3 (1930), Chapter 9.
Brancato, G. N., The Meteorological Behavior and Characteristics of Thunderstorms^

Washington, D. C., U.S. Department of Commerce, 1942.
Byers, H. R., Non-Frontal Thunderstorms, University of Chicago, Inst. of Met., Misc.

Reports, No. 3, 1942.

" Discussion of Thunderstorm Problems," Q. J. Roy. Met. Soc., 67, 327-361 (1941).
McEacheron, K. B., and K. G. Patrick, Playing with Lightning^ New York, Ran-
dom House, 1940.
Simpson, Sir G. C., " The Electricity of Clouds and Rain," Q. J. Roy. Met. Soc., 68,

1-34 (1942).

CHAPTER 23
CLIMATOLOGY

153. Importance of Climatology to the Meteorologist. Meteorology,
as a science, deals with the daily weather, and more particularly with
the variations that occur, and the influences that cause these variations.
Climatology, too, is a study of the weather, but from the standpoint
of assessing and explaining the average conditions. For example, the
climatologist is interested in the statement that the average wind at a
station is westerly. The meteorologist is concerned with understanding
what conditions will produce, for example, easterly or southerly winds
at the same station, and why. The forecaster's aim is to specify what
changes are to be expected in the weather in the future. The work of
the climatologist is to determine the average weather conditions which
have occurred in the past.

To a large extent the statistical facts presented by the climatologist
are not vital to the forecaster. Yet an understanding of some of the
climatologist's findings is valuable in assessing the weather changes that
may be expected.

Very frequently a forecaster remains in one district for a considerable
period and learns the variations in the weather in that district only. If
he moves to another district, it requires from one to six months for him
to learn the broad trends and local anomalies of the weather, a thorough
knowledge of which is necessary for accurate forecasting. If the cli-
mates of the two regions are similar, the minimum period only is re-
quired; if the climates are markedly dissimilar, so that his past experience
no longer aids him when forecasting abnormal atmospheric conditions,
the full period of six months is necessary to obtain a grasp of the clima-
tology of the region. A study of climatological data giving, for example,
average maximum and minimum temperatures and average limits of
rainfall during the months of the year, is helpful. A study of climatology
also aids in understanding the problems of the forecaster in other regions
so that an exchange of ideas can take place more freely.

Even in the region in which the forecaster is primarily interested, a
knowledge of climatic types is desirable. No region is so small that
variations in the climate cannot be observecl. Thus the climate in one
part of the region may be continental in character; another part may be

380

Sec. 154} THE FACTORS GOVERNING CLIMATE 381

under the moderating influence of lakes; another part may be on the
lee side of hills where descending air is relatively d,ry and rain is less
frequent. An understanding of the general significance of these factors
will aid the forecaster in modifying a forecast to suit each portion of the
region. The forecaster may learn this by experience as he analyzes
the maps day after day. A more objective analysis by a statistical
summary of the weather for the different parts of the forecast region
would correct some of the ideas that the forecaster may have. It would
also aid a new forecaster coming into the district in distinguishing the
different climatic regions of the country. For example, studies of a
climatological character of the conditions which precede the formation
of radiation fog at various airports, of the types shown in Figs. 136 and
137 of section 134, are an aid to both the experienced and the inex-
perienced forecaster.

154. The Factors Governing Climate. The elements which, taken
together, vary over the earth's surface and constitute climate, when
they vary in any one place, produce weather. These are temperature,
precipitation, clouds, moisture content of the air, and wind. Pressure
varies, but changes in pressure are noted only because of corresponding
changes in one of the other weather elements.

The distribution of temperature over the earth is shown in Figs. 1
and 2, section 1, and the winds of the earth in Figs. 5 and 6 of section 3.
Fig. 165 gives the world distribution of precipitation. The variation
of fog for June, July, and August over the north Atlantic is shown in
Fig. 166. Fogs in this region are of the advection type, as shown by the
fact that the area of maximum fog frequency, off the coast of Newfound-
land, coincides with the area of maximum temperature gradient of the
ocean surface. Several other climatic maps are given in other sections.
Thus, the distributions of fog over the British Isles and the United States
are shown in Figs. 142 and 143, section 136; of thunderstorms in
Figs. 162 and 164, section 151; and of hail over the United States in
Fig. 163, section 151.

Two regions have differences in climate because of differences in one or
more of the following:

(a) Latitude.
(6) Altitude.

(d) Relation to the general pressure distribution.

(e) Prevailing winds.
(/) Nearness to ocean.

(g) Relation to ocean currents.

382

CLIMATOLOGY

[Chap. S3

Sec. 164}

THE FACTORS GOVERNING CLIMATE

383

384 CLIMATOLOGY [Chap. %S

These climatic controls are not independent. Thus the general pressure
distribution varies with latitude and, since the former has a controlling
influence on the prevailing winds and the ocean currents, these also
vary with latitude.

The weather, and particularly the temperature, varies markedly with
latitude. This variation arises because the distance from the equator
determines the amount of heat received directly from the sun, as shown
in Fig. 27, section 31. The tropical regions, on which the sun shines
nearly vertically throughout the year, are the warmest portions of the
earth, and the polar regions, which receive considerable insolation
during the summer, but none during the winter, as indicated in sec-
tion 28, are the coldest. In order to be able to assess the other factors
governing temperature, climatologists at times eliminate the latitudinal
effect by drawing charts of temperature anomalies, i.e., maps showing
the variation from the average for that latitude.

The temperature in the atmosphere decreases with altitude at a
fairly regular rate, so that the mean temperature at any given point at
the earth's surface is in part determined by the altitude of that point
(see problem 2, Chapter 10). The altitude and the proximity to moun-
tains also help determine the amount and type of precipitation. On the
windward side of the mountain the air rises, giving orographic clouds
and precipitation. On the lee side and sometimes on plateaus and in
valleys the air is dry and little rain falls. Regions, such as these, which
have less precipitation because of a mountain to the windward, are said
to be in the mountain's rain shadow.

As described in Chapters 12 and 15, the general pressure distribution
determines the characteristics of air masses and the frequency of their
occurrence at any location. Hence regions differently located with
respect to these will have differences in weather. For example, Florida,
on the western edge of the Azores high, with southerly winds, has an
annual rainfall of 52 in. ; Rio de Oro, on the African coast with the same
latitude but on the eastern edge of the Azores high, v/here northerly
winds prevail, has an annual rainfall of less than 10 in. Since the pre-
vailing winds are closely associated with the general pressure distribu-
tion, the effects of each on climate cannot be separated.

The ocean currents are an effective means of transferring heat from
equator to pole and so of aiding in the maintenance of the terrestrial
heat balance. Their influence is not spread uniformly and in some
regions an adjacent current of warm water from the equatorial regions
gives mild temperatures and abundant moisture. Other regions of the
same latitude have a raw, cold climate because of the proximity of a
current from the Arctic. The variations over thef ocean areas of the

Sec. 166\ KOPPEN'S CLASSIFICATION OF CLIMATE 385

mean isotherms, as shown in Figs. 1 and 2, section 1, bring out clearly
the effects of the ocean currents.

The sea exerts a moderating influence on the weather of the land
areas about its borders. The summers are not so hot, and the winters
not so cold as they are farther inland. Also the air is moist, so precipi-
tation is usually abundant and uniform throughout the year, and
droughts do not occur. With increasing distance from the sea these in-
fluences become less effective and a continental regime of less uniform
precipitation and greater variations in temperature is in control. The
extent to which the influence of the ocean reaches inland varies with
the topographic features and the direction of the prevailing wind. In
Europe, Sweden, lying to the east of the Scandinavian mountain range,
is continental in climate, but farther south the influence of the Atlantic
extends inland over the plain of northwest Europe as far east as Poland.

155. Koppen's Classification of Climate. Several different classifi-
cations of the climates of the world have been devised. The classifi-
cation of Koppcn, first published in 1918, has been widely accepted by
climatologists, and will be described here. It has a numerical method
of dividing the climatic zones, using combinations of letters to distin-
guish different regions. Each letter is rigorously defined, and these
definitions are so well known by climatologists that the climatic types
are often given by letters, rather than by more descriptive terms.

Koppcn's classification divides the world into the following five major
climatic divisions.

A climates. The moist tropical climates. The temperature of the
coldest month is greater than 18 C (64.4 F). This is the climate of
the hot, rainy, equatorial belt. Its two chief divisions are:

(a) Af, signifying regions where there is no dry season, the driest
month having at least 6 cm (2.4 in.) of precipitation.

(6) Aw, denoting regions where there is a distinct dry season, one
month with precipitation less than 6 cm (2.4 in.).

B climates. The arid climates. In these regions the evaporation
exceeds the precipitation. Although the principal arid climates are
found in the sub-tropics, this type also occurs farther north in certain
parts of the world. There are two subdivisions of this type:

(a) BW, or the desert climates.

(b) BS, or the climate of the steppes.

A further subdivision distinguishes the sub-tropical deserts (A), with
an average annual temperature of 18 C (64.4 F) or more, from the

386 CLIMATOLOGY (Chap. S3

middle latitude deserts (&), with an average annual temperature of less
than 18 C (64.4 F).

C climates. The warm temperate rainy climates. The average tem-
perature of the coldest month is less than 18 C (64.4 F) and greater
than -3 C (26.6 F) ; the average temperature of the warmest month is
over 10 C (50 F). Three distinct types are recognized, which are:

(a) C/, with at least 3 cm (1.2 in.) of precipitation in the driest month
and the difference between the wettest and driest month less than for
Cw and Cs.

(b) Cw, with a winter dry season, and at least ten times as much pre-
cipitation in the wettest month of summer as in the driest month of
winter.

(c) Cs, with a summer dry season, and at least three times as much
rain in the wettest month of winter as in the driest month of summer,
and in addition the driest month of summer having less than 3 cm
(1.2 in.) of precipitation.

D climates. The cold temperate dimates. The average temperature
of the warmest month is above 10 C (50 F) and that of the coldest
month is below -3 C (26.6 F). The two chief divisions are Df and
Dw where / and w are defined as in C climates.

E climates. The polar dimates. The average temperature of the
warmest month is under 10 C (50 F). There are two subdivisions:

(a) ET (tundra), with an average temperature of the warmest month
above C.

(b) EF (frost), with no month with a temperature above C.

To subdivide more closely, Koppen used a third set of letters defined
as follows:

a (hot summer) average temperature of the warmest month over
22 C (71.6 F).

6 (cool summer) average temperature of the warmest month under
22 C (71.6 F).

c (cool, short summer) less than 4 months over 10 C (50 F).

d average temperature of the coldest month below -38 C (-36.4F).

g (Ganges) hottest month before the summer solstice.

i (isothermal) range of temperature between the warmest and cold-
est month less than 5 C (9 F).

fc' temperature of warmest month under 18 C (64.4 F).

m (monsoon) short dry season, but sufficient moisture to give wet
ground throughout the year. c

Sec. 166} THE TROPICAL RAINY REGIONS 387

n (Nebel, fog) frequent fog.
w' rainfall maximum in autumn.
w" two distinct rainfall maxima.

The main features of the geographical distribution of the different
climatic types as defined by Koppen are given in Fig. 167.

156. The Tropical Rainy Regions. The regions with this type of
climate are those without a winter. They lie in the region of the equa-
torial low-pressure belt, generally within 20 of the equator. Along the
eastern boundaries of the continents the moist, unstable, equatorial air
is carried poleward around the sub-tropical high-pressure systems over
the oceans and the adjacent land areas, and so along these coasts the
tropical rainy climates extend farther northward.

The temperature in these regions remains high throughout the year.
The maximum temperature is not so high as it is in the sub-tropical
high-pressure belts and even farther north, but the average temperature
is greater. The region is characterized by uniformity of temperature
accompanied by high relative humidity which results in a minimum of
radiative cooling at night near the surface.

Precipitation is generally abundant, although on the borders it de-
creases, approaching the precipitation regime of the dry climates. In
these border regions there is a definite dry season followed by abundant
moisture during the time of high sun, i.e., the summer season.

The air has real latent instability and the strong insolation assists in
the release of this. Hence precipitation s usually in the form of thunder-
showers. The sky is clear during the early morning, but cumulus
clouds form with the heating of the surface layers by the sun during the
morning and these frequently develop into cumulonimbus by mid-after-
noon. Gusty winds and heavy thunder accompany the rain showers.
These clouds dissipate during the evening. Although this is the usual
type of precipitation, weak low-pressure areas sometimes develop and
persist, giving periods of dull, gray skies and steady rain.

In some regions a belt of calm exists between the northeast and south-
east trade winds. In other regions this belt becomes very narrow or
disappears, and the winds observed are the constant trade winds.
Nevertheless the winds, although steady, are not strong. The con-
stancy of the trades is shown by the wind rose in Fig. 536, section 58.
Along the sea coast the sea breeze is a regular occurrence. Fogs rarely
occur in this climatic region.

Figs. 168 and 169 illustrate the variation of the temperature and pre-
cipitation in the moist, humid regions. The solid line shows the varia-
tion of temperaturG and the dotted line that of precipitation. (This

388

CLIMATOLOGY

[Chap. 23

practice is followed in other diagrams of this type.) Belem, Brazil, at
the mouth of the Amazon in the Af regions, has a very uniform tem-
perature regime and no dry season, although the amount of rain is not
large in September, October, and November. Darwin (Fig. 169) on
the north coast of Australia, with a dry season in the months of May to
September, lies in the Aw climatic zone.

-40

\ %
\
\.

1 1

**-../
1 I

J A J
Month

10 -2

^
\

\
\

i.../

A J

Month

O o

FIG. 168. The annual variation of
temperature and precipitation at Belem,
Brazil (1S, 48 W).

FIG. 169. The annual variation of
temperature and precipitation at Dar-
win, Australia (12 S, 131 E).

157. The Arid Regions. The sub-tropical arid climates, BSh and
BWh, are found in the regions of the sub-tropical high-pressure belts,
which extend from about 15 to 30 latitude. In the western hemisphere
the land areas at these latitudes are small, and the most extensive regions
with this climate are found in Africa, Asia, and Australia.

On the eastern boundaries of the continents the trade winds of the
general circulation carry moisture inland, so the arid regions do not
extend to these coasts. Along the western boundaries the winds are off
shore and directed toward the equator around the cells of the sub-tropical
high. For this reason the western shores tend to be dry. Thus in South
America the arid regions include the coastal strip on the west from lati-
tude 35 S to 5 S. On the east the arid coastal regions extend only a,
short distance at 10 N, and a somewhat greater distance in the region
of the westerlies, i.e., they are middle latitude deserts, at 40 S to 50 S.

Middle latitude arid regions BWk and BSk are located along the east-
ern slopes of the Rocky Mountains, lying in their rain shadow, and in
central Asia, far from any source of moisture.

Since these arid regions extend over such a wide range of latitude, no
general statement can be made about their average temperature. The
clear, dry air with few clouds permits rapid changes in surface temper-
ature through the action of both solar and terrestrial radiation. Both
the daily and the annual ranges of temperature are large. Thus in the

Sec. 157}

THE ARID REGIONS

389

desert the temperature during the day may reach 85 F, then fall to
40 F by sunrise the following morning.

In the sub-tropics the air is dry and, in spite of the high surface tem-
peratures, relatively stable. This stability is caused, in the manner out-
lined in section 16, by the subsidence that occurs in the upper atmos-
phere. Turbulent eddy motion develops, but the influence of this does
not generally extend to the condensation level, and clouds are few. Less
cloudiness occurs in these regions than in any other part of the world.

Precipitation is infrequent and variable. That which falls is usually
in the form of thundershowers in unstable air that has invaded the area.
In the hot desert regions a heavy shower may occur after several dry
years. Because of the irregularity of occurrence of these showers they
provide little aid to vegetation.

15
10
5

.L J

1 J

J A J
Month

-40

FIG. 170. The annual variation of

temperature and precipitation at Cairo,

Egypt (30 N, 31 E).

en 5 -

Precipitation (in)

L. .A

J A J
Month

-40

FIG. 171. The annual variation of
temperature and precipitation at Trip-
oli, Libya (32 N, 13 E).

The variation of temperature and precipitation at a typical station
in the BW climatic zone is shown in Fig. 170. This gives the value of
these elements for Cairo, Egypt. As will be observed, the amount of
precipitation is small, and it occurs at irregular intervals.

On the edges of the regions of B type climate the precipitation is
slightly more regular. On the equatorial boundary of the sub-tropical
deserts the precipitation occurs during the season when the sun is high,
and on the side near middle latitudes it occurs when the sun is low, as
the rains of the temperate regions extend to lower latitudes during
that season.

Tripoli, Libya (Fig. 171), is an example of a station lying in the BS
climatic zone on the poleward side of the desert region. The precipi-
tation falls at the time of low sun, when the storms of the westerlies move
farther south and nearer the station.

Since strong pressure gradients are not found in this region, the winds

390 CLIMATOLOGY [Chap. S3

are light. Nevertheless the instability that develops in the lower layers
produces at times strong, gusty, surface winds accompanied by dust
storms. Land and sea breezes are common along the coasts. Fogs are
absent except along the west coastal regions where the stability and
moisture content of the air increase over the relatively cool water.

The arid climates of the middle latitudes are a result of topographic
features. They are found in Patagonia, in central Asia, and on the
plateaus and eastern slopes of the Rockies. The climates of this type
in South and North America are a result of the extensive mountain ranges
to the westward which cause the westerly winds of these latitudes to lose
a large part of their moisture as they ascend the western slopes and be-
come drying winds after they have descended the eastern slopes. The
district in Asia is inland and the winds which enter the continent have
lost most of their moisture by the time they have reached the interior.

The temperature varies widely with latitude. As in the regions of
the sub-tropical highs, the lack of clouds permits marked radiational
heating and cooling near the surface. Hence the temperatures are more
extreme, and the range is greater than at other places the same distance
from the equator.

These areas are not without precipitation. The rains and cloud forms
of the winter season are associated with low-pressure areas that move
across these regions; but these cyclones are not well developed, and the
amounts of precipitation are small. Instability developing in the cold
air behind a cold front provides another source of winter precipitation
in the form of the " blizzard " storms of the North American plains.
Summer precipitation occurs in the form of thundershowers when the
diurnal heating in the polar continental air is sufficient to release latent
instability and form cumulonimbus clouds.

Turbulence in the mid-afternoon along with the pressure gradient
which develops around the Asiatic low-pressure area produce, in the
manner described in section 123, gusty, strong summer winds over the
deserts in Asia. The winds about the winter anticyclone tend to be
more moderate. In North America the winds are strong in the air
current flowing down the eastern slopes of the Rockies as well as in the
blizzards back of a traveling low-pressure system.

158. The Warm Temperate Rainy Regions. The limits set in the
definition of the warm temperate climate restrict it to the temperate zone
between the arid regions and the region of a snow cover. Some of the
equatorial plateaus of Africa fall under this type by definition, since
their coldest month has a mean temperature of less than 18 C, but their
climate corresponds with that of the tropical rainy regions.

The normal region for this type of climate is between 30 and 40

Sec. 168}

THE WARM TEMPERATE RAINY REGIONS

391

-40

~^\

TV.

/'*.
I-.' 1

J A J
Month

2?
a.

latitude, but topographic features and ocean currents are responsible
for a much wider distribution than that. Along the western shores of
the continents the moderating influence of the warm ocean currents
permits this type to extend northward to Alaska, and to the Arctic Circle
in Norway as well as to include most of western Europe. In Asia, parts
of northern India and of southeast China are of this climatic type.

One distinct division of the warm temperate climates is the Mediter-
ranean or Cs climate. This, by definition, has a dry summer and a moist
winter. An example of this type of climate is shown in Fig. 172. In
Lisbon, Portugal, there is no absolutely
dry month, but the maximum pre-
cipitation occurs during the winter.
The mean temperature never falls
below 40 F. During the summer
season the regions having this type
are under the influence of the sub-
tropical high-pressure belt and enjoy
clear, warm days with cooling at
night. During the winter the dis-
placement toward the equator of the
westerlies of middle latitudes brings
the region under the influence of the
traveling cyclones of the middle lati-
tudes which give spells of variable weather and the winter rains.
This type of climate is found around the Mediterreanean, at the
southern coasts of Australia, in central Chile, and along the western coast
of the United States.

In summer these areas have a climate similar to that of the sub-
tropical deserts. Clouds are few and rain is infrequent. As winter
approaches, the influence of the low-pressure areas of the temperate
zones becomes more prominent and the average cloud amount increases.
The centers of cyclonic motion do not extend to these areas, however,
and so the clouds are neither so thick nor so persistent as in regions
where the ascending currents are more pronounced. The rain that falls
is more of a showery type even when these regions are under the influence
of the cyclones and the clouds tend to break up rapidly.

With the development of the winter cyclones and anticyclones, the
winds increase in strength and veer with the passage of fronts. In
summer the winds are lighter, but along the coasts sea breezes occur
regularly. Fogs along the western shores are frequent and result from
the same causes as those which produce fog along the western coasts of
the arid regions.

FIG. 172. The annual variation of
temperature and precipitation at
Lisbon, Portugal (39 N, 9 W).

392

CLIMATOLOGY

[Chap. 23

In the other major division of the warm temperate climates, classified
as C/, the precipitation is distributed more uniformly throughout the
year. One district is in the eastern United States extending north to
the fortieth parallel. Summer rains are caused by the monsoon winds
off the Gulf of Mexico. Another region with this type of climate is in
western Europe north of the Mediterranean coast lands, including the
British Isles, France, Germany, the northern Balkans, and the coast of
Norway to the Arctic Circle. The westerly flow of moist air from the
warm Gulf Stream accounts for the precipitation and moderate temper-
atures for this region. A third dis-
trict is found along the China coast,
the monsoon off the Pacific providing
the summer rains. Inland the winter
precipitation is light and the region is
classified as Cw.

An example from a fourth region is
shown in Fig. 173. New Zealand and
parts of Southern Australia lie in this
climatic zone.

The temperatures of these regions
vary according to the latitude, but the
ocean control and the high relative
humidity modify the temperature so

c:40

-40

~^\

1 1

* %

1 1

J A J
Month

10 I

FIG. 173. The annual variation of

temperature and precipitation at

Wellington, New Zealand (41 S,

175 E).

that the diurnal and annual ranges are not so great as in other dis-
tricts at the same distance from the equator. All regions experience
cold periods during the winter when outbreaks of polar continental air
from its source regions occur, and of course they are subject to the rapid
changes in temperature associated with frontal passages.

Precipitation in winter occurs with frontal low-pressure systems.
The warm sector air of these depressions is usually of maritime origin,
and so precipitation is abundant. In the summer the air from the oceans
becomes unstable in moving over the heated land and releases its mois-
ture in the form of showers and thunderstorms. Winter is the time of
cloudiness, the typical clouds being altostratus, nimbostratus, and
stratus. In the summer, even though the amount of precipitation may
be greater, the cloudiness is less, the clouds being more frequently of the
cumulus type. In the winter the maritime air moving over the cold
land surface becomes stable, and fogs of the advection type are fre-
quently widespread. In summer the most common type of fog is that
caused by radiation, but these fogs dissipate with the solar heating of
the morning.

Besides being affected by the frontal depressions of the winter season,

Sec. 169] THE COLD SNOWY FOREST REGIONS 393

the coastal regions of China and the southeastern United States are
close to the path of the tropical hurricanes. Although these storms
occur relatively infrequently, the high winds associated with them cause
extensive damage along their path. The tornadoes of the United States
also occur in the region which belongs to this climatic type.

159. The Cold Snowy Forest Regions. The cool, temperate climates
lie to the northward of the warm temperate zone in North America and
Eurasia. No continent of the southern hemisphere extends far enough
to the south to have regions with a D type climate. By definition, there
must be one month with an average temperature below freezing, and so
snow will lie on the ground for at least a short period. The northern
boundary coincides with the northern boundary of forest. In North
America it extends from the fortieth parallel to, but does not include,
the coastal regions of the Arctic and neighboring waters, except for the
coastal region along the Pacific and an arid region in the rain shadow of
the Rockies. In Eurasia, it includes Russia in Europe, and Asia north
of the fortieth parallel, except for the coastal areas along the Arctic and
the arid regions in the center of the continent.

In general the regions with this type of climate are continental in
character. They include the east coasts of the continents, but, since
the prevailing wind is westerly, these coasts are not under the influence
of the adjacent ocean as much as the regions along the west coast.
Hence the former have a climate which resembles the continental regime.
With the radiational cooling that takes place during the long nights
from the snow cover, the winter temperatures are low. It is in these
areas that the polar continental air in winter derives its characteristics
of low temperature and extreme stability. During the summer the
radiation from the sun is large and, lacking the moderating influence
of the ocean, the mid-day temperatures are high. Thus the annual
range of temperature is large. The temperature changes rapidly as the
wind shifts with the passage of the fronts associated with the traveling
cyclones. Thus these regions experience successive warm and cold
waves.

The annual precipitation over most of these areas is not great, more
falling on some of the arid regions than over a large part of the cold
temperate regions. But the evaporation is relatively much less, and
moisture is sufficient for vegetation. During the winter the average
position of the polar front lies to the south, and the precipitation falls
from overrunning air. In the summer the front lies along the northern
half of the region, and although some of the precipitation is caused by
frontal phenomena, more falls in convective showers. The maximum
for the year is during the early summer, at which time the growing crops

394

CLIMATOLOGY

[Chap.

get the most benefit from the rain. In general the winter with its
frontal clouds and low ceilings has the maximum cloudiness. But in
the center of Siberia, under the influence of the Eurasian anticyclone,
the average cloudiness is less in winter than in summer.

Leningrad, in European U.S.S.R. (Fig. 174), and Olekminsk, Siberia
(Fig. 175), illustrate the regime of the cold, snowy climates. At these
stations the continental influence is seen in the wide variation of tem-
perature between summer and winter, and the tendency to a summer
maximum of precipitation.

80
40

-40

CP o &

Precipitation (in)

1 ["""

i r

J A J
Month

80
40

-40

en 5 ^

Precipitation (in)

L-

TN^

J A J
Month

FIG. 174. The annual variation of
temperature and precipitation at Lenin-
grad, U.S.S.R. (60 N, 30 E).

FIG. 175. The annual variation of
temperature and precipitation at Olek-
minsk, U.S.S.R. (60 N, 120 E).

Most of the usual storm tracks of the extra-tropical cyclones cross
the cold temperate regions, although they may lie slightly to the south
of them during the winter months. These cyclones pass with some
regularity, the average period during the winter months being about
three days and during the summer about six days. They give rise to
fresh to strong winds with their passage, but the winds are not so strong
as those occurring about these lows after they move off the east coast
over the water areas.

160. The Polar Regions. The polar regions include the ice-covered
continents of Greenland and Antarctica, and also the coastal regions of
North America and Eurasia, bordering along the Arctic Ocean. Radia-
tional cooling during the long winter night causes extreme cold. Over
the ice caps the temperature does not rise much above freezing even in
summer, but in the other regions the temperature rises rapidly during
the short summer, the maximum being greater than 70 F along the
Canadian Arctic coast. An example of the temperature and precipi-
tation variation is given in Fig. 176, showing the average values of these
for Godthaab, Greenland.

BIBLIOGRAPHY

395

:40

-40

i^.

1 J
"c"

in c

IU o

c .

"""..

"\ T

\ \

J A
M

J
onth

The moisture content of the air is low, even during the summer, and
precipitation is light. Over the ice caps of Greenland and Antarctica
the cooling above the cold surface causes a semi-permanent anticyclone
to form and little precipitation falls from the air which is usually sub-
siding. During the winter season the continental high-pressure systems
produce a similar condition over the Arctic coastal regions, but during
the summer weak low-pressure systems
give some light precipitation. Some
stratus clouds occur even in winter,
but summer is the time of maximum
cloudiness.

In general the winds are light, al-
though the katabatic winds off the ice
caps are strong and steady. In addi-
tion, the Icelandic low is adjacent to
regions with this type of climate, and
the strong pressure gradients about
this system cause high winds and gales FIG. 176. The annual variation of
along some of the coasts in the vicinity, temperature and precipitation at
Inland fogs are rare but along the coastal Godthaab, Greenland (64 N, 52 W).
regions fogs are frequent in summer.

The climates of the world can be classified by types to aid in under-
standing and remembering them, yet borderlines are indefinite, since
there are no discontinuities in climate except those produced by oro-
graphic features. There is, in each climatic zone, a gradual change
from place to place so that the climate of each blends smoothly with the
climate of the next zone at the boundary. This fact must not be for-
gotten in any study of climates.

BIBLIOGRAPHY

Trewartha, G. T., An Introduction to Weather and Climate, New York, McGraw-Hill
Book Co., Second Edition, 1943.

Kendrew, W. B., The Climates of the Continents, Oxford, Clarendon Press, 1937.

Conrad, V., Fundamentals of Physical Climatology, Milton, Mass., Harvard Uni-
versity, Blue Hill Meteorological Observatory, 1942.

Bilham, E. G., The Climate of the British Isles, London, Macmillan and Co., 1938.

Koppen, W., and R. Geiger, Handbuch der Klimatologie, Berlin, Verlag von Gebrtider
Borntraeger, 1936.

Brooks, C. E. P., Climate, Second Edition, London, Benn, 1932.

Landsberg, H., Physical Climatology, State College, Pennsylvania State College, 1941.

Climate and Man, 1941 Yearbook of Agriculture, Washington, U.S. Department of
Agriculture, 1941.

CHAPTER 24
MAP ANALYSIS AND FORECASTING PROCEDURE

161. Material Available for the Forecaster. The previous chapters
have described the elements that make up the weather and discussed
the causes for their variations. This chapter will attempt to show how
the forecaster applies these facts when working with the synoptic weather
chart. Yet no study of books will take the place, in the development of
a forecaster, of training and experience with actual weather maps.

A large amount of data obtained from observing the weather elements
is available to the forecaster as he prepares his forecasts. This material
is collected at central points by telegraph, telephone, and radio, and it
is then distributed by teletype to the different forecast centers where
it can be used by the meteorologist. He then selects for analysis only
that part of the data which is pertinent to the particular forecast which
he must make.

The most significant material is contained in the synoptic weather
reports, which give a fairly complete description of the current weather
with some information about the past weather. The information is put
into code form to save time in transmission. The code changes from
time to time and a description of the current code is always available
at the weather offices, so it will not be described here. The report usu-
ally contains the following data:

(a) Sea level pressure.
(6) Temperature.

(d) Barometric tendency, i.e., the amount of rise or fall of pressure
during the three-hour period preceding the time of observation, along
with the type of change.

(e) The total cloud amount.

(/) Types of cloud present in the sky.

(g) The height of the cloud base.

(h) Visibility.

(i) Present, and sometimes past, weather.

(j) Wind direction and speed.

Other supplementary information, such as the amount of precipitation,
the temperature of the sea, etc., is sometimes included. These synoptic

396

Sec. 162} PLOTTING OF DATA ON THE SURFACE CHART 397

observations are taken four times a day at stated times, which are
approximately uniform over the globe, at fixed stations on land, and
on a large number of ships. They are sent to the forecast offices where
they are plotted on a map, which is called the synoptic weather chart,
ready to be analyzed by the meteorologist.

At airports the current weather is very important, and so more fre-
quent observations are taken and relayed to neighboring airports and
forecast centers where the information is of value. These " airways
sequences," as they are called in North America, include a large part of
the same type of information that is given in the synoptic weather report.
These observations are usually taken every hour with special observa-
tions taken when the weather is changing rapidly.

At a number of stations a small balloon is released at regular inter-
vals, and its path followed by a theodolite. Using the data in the
manner described in section 66, the wind speed and direction are com-
puted for different levels of the atmosphere, and this information passed
by teletype to the forecast centers. Here the values are plotted on
maps, a different one for each level, or sometimes by means of colored
inks on the synoptic weather chart.

In a fourth set of data available for the forecaster are the values of the
temperature and moisture content at the different levels in an air column
above a small number of widely distributed stations. As indicated in
section 66, a light-weight set of instruments called a radiosonde, contain-
ing a radio sending set, a bimetallic thermometer, an aneroid barometer,
and an hygrometer, is released and carried aloft by means of a balloon.
The transmitter sends back to the receiving set at the station the values
for the pressure, temperature, and relative humidity. A report of these
is sent to various forecast centers interested in the data. Here the in-
formation is plotted on adiabatic charts, such as the tephigram included
at the back of this book, or evaluated by other means.

162. The Plotting of Data on the Surface Chart. In order to be able
to include on a map all the data of the synoptic weather reports a system
of shorthand has been devised. Some data are entered by figures.
These include the sea level pressure, in millibars with the hundreds'
digits missing, barometric tendency amount in tenths of millibars,
temperature, dew point, amount of precipitation, cloud height, and
visibility. These last two are reported and entered according to a
standard scale which is accepted for general use by international agree-
ment. The wind direction is given by a shaft drawn from the station
circle, the wind blowing along the shaft toward the circle, and the force
is given by the number of feathers drawn from the end of the shaft, each
feather representing two units and a half feather representing one unit

398 ANALYSIS AND FORECASTING [Chap. 24

on the Beaufort scale given in section 66. The remaining data are given
by means of symbols.

Because of the lack of space on the reduced maps given in the text,
the symbols for the different cloud types are not included on these maps.
For that reason and since they are available with complete definitions
at any forecast office, they will not be described. The total cloud
amount is indicated by the amount of shading inside the station circle,
with four vertical lines indicating an overcast sky. A sky obscured is
denoted by three horizontal lines. The pressure tendency characteristic
is given by means of one of a set of symbols, each of which resembles
the shape of a particular barograph curve. Thus the symbol \ means
that the pressure has been falling and is beginning to rise. There are
96 different symbols for the different types of weather which are defined
by the international code, except the weather types defined as clear,
partly cloudy, cloudy, and overcast. These are built up from a number
of basic symbols. The most frequently used symbols with their mean-
ings are :

Rain Smoke <^>

Snow * Shower V

Drizzle > Thunder J<J

Fog s

Various combinations of these and other symbols are used to differen-
tiate the types of weather.

Each of these weather elements has a definite position about the
station circle, as specified by the international plotting model. These
positions are shown in (a) of Fig. 177. Diagram (b) shows a com-
pletely plotted station. The information given in diagram (6) is as
follows: sky overcast with breaks in the overcast of stratocumulus cloud
at 2000 to 3000 ft and cirrus cloud visible through the breaks; wind
ENE 7-12 mph; pressure 1019.0 mb, which has risen 1 mb and then
steadied during the past three hours; temperature 6 F; dew point
12 F; visibility between 12^ and 31 mi; weather, light continuous
snow which began 1-2 h ago, the total precipitation being 0.05 in. in
rain equivalent during the past six hours.

163. Construction of Isobars. When the meteorologist has a fully
plotted map for analysis, he begins drawing the isobars, i.e., the lines
of equal pressure, and the fronts. Since these two sets of lines are inter-
dependent, the choice of drawing fronts or isobars first is a matter of
personal preference with the forecaster.

The isobars are drawn for certain given pressures at regular intervals.
The intervals and the pressures are determined bjr convention within

Sec. 163]

CONSTRUCTION OF ISOBARS

399

each forecasting service, the interval usually being 2, 3, or 4 mb depend-
ing upon the scale of the base map. In the diagrams given here the iso-
bars are drawn for every 4 mb, starting from 1000 mb. If the reported
pressures were accurate, the drawing of the isobars would be merely a

Form of High Cloud

Air Temperature ^H
_ ^

\ TT C M

Visibility, y w

Present p^ T
Weather Us 'S

Dew / T lll

Point

Sea Temperature^
Form of Low cloud

The Circle denotes the
Position of the Station

Form of Medium Cloud
State of the Ground

(Eh-
PPP
ppa-

W f ml Par * ^ ww re ^ errm 8
W 4!['v *o last hour but not
(RR)wo time of observation

t Sea Level Barometric Pressure
Pressure Tendency Characf eristic

l Of Predominating
J Lower Cloud

(a)

-6

8 **
-12

*2
05

(b)
FIG. 177. The international station plotting model.

matter of drawing a smooth curve through points obtained by interpo-
lation between the given pressures. Errors arise, though, at the ob-
serving station because of errors in observation or calculation, because
the observations are not made at the specified time, or because the
method of reducing the pressure to sea level may not give accurate re-
sults. Sometimes the pressures recorded on the map are wrong because
of errors in transmission of the data from the observer to the plotted
map. Errors in observation, calculation, and transmission must be
expected occasionally. If the pressure for six hours ago is known, this
error can be detected at times by computing from the tendencies on the

400 ANALYSIS AND FORECASTING [Chap. $4

two maps an approximate value for the present pressure. Sometimes
with ships at sea the error is not in the transmitted pressure but the
transmitted position of the ship. The pressure and wind will sometimes
check with those to be anticipated at a position 5 or 10 of latitude or
longitude from the position as given. In that case the ship should be
considered as reporting from this alternative position. Yet ship pres-
sures are not so accurate as those for land stations since the motion of
the ship does not permit the level of the mercury to assume a steady
position, and so they cannot be relied upon as fully.

The correction to sea level is based on the assumption that a column of
air of a specified temperature distribution exists from the level of the
station to mean sea level. The temperature distribution is calculated
from the present temperature and the temperature twelve hours ago.
When the station is not over 1000 ft above sea level, the errors in this
correction are small. For higher altitudes the correction to be added
is large, being about 100 mb for 3000 ft, so that if the temperatures used
happen to be non-representative of those for the air mass as a whole, the
correction to sea level pressure will be inaccurate. Sometimes an error
arising from this cause may be detected by noting if the temperature
of the reporting station is lower or higher than representative tempera-
tures for the district. When a front lies along a mountain ridge the
correction to sea level will often produce a fictitious pressure gradient
along the front, since in the cold air the corrections to sea level will be
larger than those in the warm air. This pressure gradient does not exist
at the surface of the earth and cannot be used in computing the gradient
wind and the motion of fronts.

Because of the errors in the plotted pressures, the isobars based on
them will be irregular. It is reasonable, however, to assume that in the
free air the isobars are simple curves except at fronts, so that, except
when the pressure gradient is weak, irregularities in an isobar should be
eliminated, unless the adjacent isobars show a similar irregularity. In
weak pressure systems, an intermediate isobar will sometimes show
whether an irregularity is real or fictitious. The isobars should be
drawn so that the gradient is approximately uniform or so that there is
a regular decrease in distance between adjacent isobars as one goes from
a high-preasure region to a low-pressure region. Along a front the iso-
bars should be drawn with care, and to bring out the trough that exists
at a front, as shown in section 39. If possible they should be drawn to
show whether the isobars have a cyclonic or anticyclonic curvature at
the front, especially in the vicinity of a frontal depression, since this in
part determines the rate of occlusion (see section 112).

The wind direction and speed are very helpful in drawing the isobars,

Sec. 164} IDENTIFICATION OF AIR MASSES 401

particularly in the analysis of ocean weather maps, if the wind force is
strong enough to be representative of the pressure gradient. Over the
ocean the winds should blow across the isobars toward low pressure at
an angle of about 20, and the distance between isobars should corre-
spond to a geostrophic wind one-quarter or one-third higher than the re-
ported wind. Over land the angle at which the wind blows across the
isobars should be about 30 or 40, and the gradient should correspond to
a geostrophic wind half as strong again as the surface wind.

If a complete map of the pressure distribution over the whole world
were drawn, each isobar would be a closed curve. Since the maps drawn
are for only a portion of the globe, parts of some of these closed curves
extend beyond the map. Remembering this, and also the features of
the general distribution of pressure over the world, as shown in Figs. 5
and 6 of section 3, it is possible to extend the isobars some distance be-
yond the recorded pressures with some justification. It is sometimes
helpful to remember that when going along an isobar, high pressures
remain always on one side and low on the other.

164. The Identification of Air Masses and the Location of Fronts.
The identification of air masses from an individual surface map is very
difficult and is almost impossible unless something about the usual
sequence of weather expected over the district is known. In this latter
case an estimate of the probable sequence of developments that pre-
ceded the current map may be made, and from that a reasonable esti-
mate of the types of air masses to be expected in the different sections
obtained. Upper air data arc more conservative and more representa-
tive and are thus very helpful. By plotting the data on a Rossby dia-
gram or a tephigram, and comparing the curve with the typical curves
for the different types of air masses as given, for example, in Chapter 15,
the types of air masses present may be ascertained with reasonable
accuracy. Even when the previous history of atmospheric develop-
ments is available, it is desirable to check by means of the upper air
ascents. The best method is to combine the study of the upper air
ascents with a review of the paths the air has taken during the past 48
to 96 h. Knowing the trajectory, the source region and the amount of
modification that has taken place are readily determined.

Similarly the location of any given front is best determined from the
history of the motion of the front and a comparison of the upper air
ascents to locate the regions where a rapid change of temperature or
moisture in the horizontal is present. After the analysis of a map is
complete, the forecast position of the fronts at the time of the next map
is usually indicated. When drawing up the latter, this extrapolated
position from the previous map is useful as a starting point for locating

402 ANALYSIS AND FORECASTING (Chap. 24

the actual position of the front. A study of the previous analysis tells
the forecaster the extent of the contrast across the front and which
weather elements reveal the position of the front most clearly. A search
for the same contrasts will aid in the exact location of the new position
of the front.

Although a front is a line across which the temperature changes
rapidly, or more rapidly than in other regions, the surface temperatures
do not always bear out this generalization. Temperature is not a con-
servative property, and at times the position cannot be located by means
of temperature. The second table in section 112 gives some changes that
should take place with a frontal passage, but only rarely are all these
well marked. Because of the varying weight that may be given to
different weather elements, two forecasters analyzing the same map
may not agree on the precise position of a front, but usually the differ-
ence between the two positions will not be large. One of the most re-
liable indicators is the wind shift that occurs with a frontal passage.
Another indicator which is reliable with many fronts is the compara-
tive stability of the two air masses as indicated by cloud types, strength
of winds, and types of precipitation. The line of zero pressure change
cannot be considered the front, but a comparison of the amounts of
pressure change often gives a good indication of the frontal position,
provided the analyst remembers that stations which the front passed
during the past 3 h will not show a representative tendency for that side
of the front. During the summer the dew point is quite reliable since
it is conservative for radiational heating. With temperatures below
freezing the evaluation of the dew point is less accurate and so it cannot
be used with the same degree of assurance. In regions where the ob-
serving stations are few and upper air data not available, frontal pas-
sages may sometimes be determined by means of the temperature
changes during the past 24 h. If the comparison is attempted for a
shorter period, the diurnal temperature variation must be allowed for.
This is large in the northland where such a method is usually needed;
it is difficult to estimate accurately, and for this reason the use of the
24-h period is preferred. All these changes must be considered, with
attempts to explain them by causes other than the passage of the front,
before the position of the front is finally decided.

Considerable assistance in locating fronts and determining their speeds
may be obtained from the special airways reports. These give the
values of the weather elements at more frequent intervals than are given
by the synoptic weather reports. Since the position of the front is
known from the last synoptic chart, the weather changes at those sta-
tions in the vicinity of the front can be noted from kour to hour. The

Sec. 166} THE USE OF UPPER AIR DATA 403

time of the frontal passage can often be determined quite -accurately,
and, knowing the time of the passage at two stations and the distance
between them, the speed of the front can be computed. By studying
the reports at the various stations, those changes which occur before,
at, and after the passage of the front can be distinguished.

Besides finding the location of those fronts which were present on the
past map, it is also necessary to investigate the possibility of fronto-
genesis in other regions of the map. As shown in section 109, the areas
where cols and troughs are present are those in which frontogenesis is
most probable, although fronts may form in other regions. The char-
acteristics of the developing front will be the same as those discussed
previously except that the indications are less well defined. If the front
that appears to be developing will not affect the forecast for the region
under consideration, the meteorologist at times will leave it for further
observation before indicating its position on the weather chart.

165. The Use of Upper Air Data. The wind speeds and directions
as obtained by means of pilot balloons have several different uses. At
2000 ft above the surface of the earth, the winds are not affected greatly
by surface friction. These give approximately the gradient wind, or
the resultant of the gradient and the isallobaric winds. If there is no
isallobaric component these winds will be parallel to the isobars and
will indicate the pressure gradient at the station. Thus, except in regions
where the tendencies vary and therefore marked isallobaric compo-
nents are present, the velocity at this level is helpful in drawing the
isobars. In the neighborhood of a front a shift in the wind at 2000 ft
above the surface is more significant than a shift in the surface wind,
since the former wind is more representative of general conditions in
the air mass than the latter. Furthermore, the motion of a warm front
may be determined approximately by noting the component of wind
speed at 2000 ft normal to the front in the cold air ahead of it; similarly
the speed of the cold front is given by the component of motion normal
to it in the cold air behind the front. Vertical motions are frequently
too large in the warm sector to permit the use of pilot balloon data for
this purpose in that portion of a depression. A further use of the wind
reports can be found in certain instances. Usually near a warm front
the cloudiness is too great to get satisfactory pilot balloon reports. Yet
sometimes the balloon can be followed as it ascends through the frontal
surface, and the height of the latter is then found by means of the wind
shift. Having this information and also the position of the front at the
ground, the slope of the front may be obtained with reasonable accuracy.

The pilot balloon reports assist the forecaster in telling the pilot of
an aircraft the wiud to be expected at every level. With the wind re-

404 ANALYSIS AND FORECASTING [Chap. 24

ports available, the forecaster still has the task of forecasting the changes.
With the wind reports missing, he must estimate the direction and
spacing of the mean isotherms for the air above the surface and from
these and the surface geostrophic wind compute, in the manner indi-
cated in section 121, the value of the geostrophic wind at each level, and,
using these values, forecast the changes expected.

Further information about conditions in the upper air comes to the
forecaster through the radiosonde reports. As described above in sec-
tion 164, the plotted reports assist the forecaster to determine the air
mass type present in the different regions of the weather map. Also
the plotted data are easily analyzed to give information about the sta-
bility of the air, as shown in Chapter 14. With this information avail-
able, the forecaster is able to estimate the possibility of the development
of cumulus cloud, and its amount and height. When used in conjunc-
tion with the weather map the likelihood of the development of instabil-
ity in the air as it passes over a warm surface can be assessed. For
example, the meteorologist can decide if instability stratocumulus is
likely to develop in a fresh outbreak of polar continental air.

In the neighborhood of a front, a comparison between representative
values of temperature and moisture content in the two air masses may
be made by observing the differences at the various levels between
ascents made on both sides of the front, or between ascents made at the
same station before and after the frontal passage. In this manner the
analysis of the map from the surface data may be assisted and verified.
This comparison may be made, using the data as plotted on a tephigram
or on any of the other adiabatic charts. Another method, which per-
mits the comparison of several ascents at once, is through the use of a
cross section of the atmosphere. A cross section is taken in a vertical
plane which intersects the earth's surface in a line near which radio-
sonde ascents are made. The values of the weather elements as deter-
mined by these ascents are plotted on the cross section. An example
is given in Fig. 183, section 169. The vertical scale is greatly expanded
in comparison with the horizontal scale. For each aerological station
the temperature, the potential temperature, the relative humidity, and
the mixing ratio are plotted for various chosen levels in the ascent. By
comparing the values of these elements from station to station it is
possible to decide whether or not frontal surfaces lie between the several
stations. The actual intersections of the ascents with the frontal sur-
faces are usually manifested by a change in the lapse rate, frequently to
an inversion or an isothermal layer, accompanied by an increase in the
mixing ratio. The fronts are drawn on the cross sections by broad lines,
blue for a cold and red for a warm front, to indicate the mixing zone.

See. 166} ISENTROPIC ANALYSIS 405

The lines of equal potential temperature, or sometimes temperature,
and of equal mixing ratio are drawn by colored lines. The heights of
these lines will change at the frontal zones since these zones are regions
of discontinuity for temperature and mixing ratio. By using the heights
of the intersection of the frontal surface over two stations which are on
a line perpendicular to the front, or the intersection at one aerological
station and at the ground, the slope of the front may be determined.

When evaluating radiosonde data, inversions other than at fronts will
be found frequently. During the night a ground inversion is to be
expected. Subsidence in a stable cold air mass often causes an inversion,
in the manner indicated in section 16, and another is sometimes found
at the top of a cloud layer, the inversion arising from the turbulence in
the cloud and the radiation from the cloud top. These last two types
will be dry inversions, i.e., inversions in which the moisture content
decreases with height. In studying radiosonde ascents it will be
observed that the relative humidity sometimes does not go above 85 or
90 per cent in regions where cloud should be expected or is reported.
This is partly explained through the failure of the instrument to register
correctly because of a lag in the hygrometer. It is also possible that
condensation has occurred at relative humidities less than 100 per cent
owing to the lowering of the saturation vapor pressure over salt crystals
and other hygroscopic nuclei (see sections 126 and 127).

Upper air pressure charts may also be used in analyzing radiosonde
data. With a close network of aerological stations, isobars may be
drawn at specified levels, such as 5000, 10,000, and 20,000 ft. If
temperature and moisture observations are included, the motions of
fronts and pressure systems at the several levels may be followed. A
slight modification of this procedure has been extensively used in
Germany, where contour lines of certain fixed isobaric surfaces have been
drawn. The results obtained by the two methods will, of course, be
essentially the same.

166. Isentropic Analysis. Another system of upper air analysis has
been developed in the United States. The entropy of dry air is constant
during adiabatic motions, since heat is neither added to nor taken from
the air. Thus, in the absence of non-adiabatic processes, dry air always
moves with constant entropy, i.e., along an isentropic surface. The
study of such isentropic motion is the subject matter of isentropic
analysis.

Isentropic surfaces are also surfaces of constant potential temperature,
as shown by (18-7), and in practice the surfaces are labeled by their
potential temperatures rather than their entropy. The isentropic sur-
faces chosen for siudy must be far enough above the ground to be

406 ANALYSIS AND FORECASTING [Chap. 24

reasonably free from the operation of non-adiabatic processes such as
turbulent mixing and radiational heating and cooling, yet not so high
that they are at heights where the moisture content of the air is very
small and humidity measurements inaccurate. The potential tempera-
tures of the three standard isentropic surfaces used in the United States
during the various months of the year are given in the following table.

ISENTROPIC SURFACES IN USE THROUGHOUT THE YEAR

Potential Temperature (A)

Months First Surface Second Surface Third Surface

January-February 290 296 302

March-April 296 302 308

May- June 302 308 314

July-August 308 314 320

September-October 302 308 314

November-December 296 302 308

Either of two groups of data may be plotted on the isentropic surface,
depending on the personal preference of the analyst. Pressure, mixing
ratio, and saturation mixing ratio may be included, or alternatively,
pressure and condensation pressure, i.e., the pressure at which saturation
is attained if the air ascends adiabatically, may be used. In cither case
isobars at 50-mb intervals are drawn for the isentropic surface. The
moisture content of the air may be represented by lines of constant mix-
ing ratio, or alternatively by isobars of condensation pressure, again at
50-mb intervals.

If the isentropic surface moves much less rapidly than the air itself,
it is possible to determine whether vertical motion is occurring, as long
as condensation has not commenced. Thus, if the wind blows normal
to the isobars in the isentropic surface and toward low pressures, the air
is ascending; if it blows toward high pressure, the air is descending.
However, since the isentropic surface may be almost stationary, or may
move with the speed of the wind, or at any intermediate speed, this
criterion is not a satisfactory one. A more useful means of determining
vertical motions is through an examination of the spacing between mix-
ing ratio and corresponding saturation mixing ratio lines, or isobars and
corresponding condensation isobars for successive charts. The method
is illustrated in Fig. 178. The isobars are shown by full lines and the
condensation isobars by broken lines. The distribution of isobars and
condensation isobars on the preceding isentropic chart is shown in
diagram (a). If the spacing of the two sets of lines has changed in the
interval between charts in such a manner that at the time of the con-
temporary chart the distribution is as shown in diagram (6), i.e., isobars

Sec. 166] ISENTROPIC ANALYSIS 407

and corresponding condensation isobars closer together, then ascending
motion has occurred between the two maps. If the ascent continues,
cloud and perhaps precipitation will develop. If the change is as shown
in diagram (c), subsidence of the air is occurring, and cloud at adjacent
levels where the air is saturated will tend to dissipate.

- ----- -550mb

600mb ---- -

- ------ 600mb

650mb - -- --

- ----- -650mb

700mb - --- -

(a)

- ------ 550mb

600mb ^r^r^r^: 600mb 600mb^ ^ 550mb

650mb -- z -- _ n . 650mb :^LZJI_^ 600mb

-- "* - Umb

_ - ---- 650mb

700mb -- --- 700mb - ----

(b) (c)

FIG. 178. Isobars and condensation pressure isobars in an isen tropic surface.

Considerable evidence has been produced by Rossby and others that
there is large-scale mixing, known as lateral mixing, along isentropic
surfaces. The lateral transport of an entity, such as w r ater vapor, by
this means appears to be at times much greater than that resulting from
small-scale eddy motion of the type discussed in Chapter 9. Large
tongues of moist air, others of dry air, may often be noted on the isen-
tropic chart. The diffusion of moisture from such moist tongues into
the surrounding drier air is frequently rapid.

In general, the isentropic surfaces near a frontal surface lie parallel
to the latter and close together, as indicated by cross sections. If,
however, the isentropic surfaces intersect the frontal surface, the lateral
mixing between the two air masses may be sufficient to cause the frontal
surface to dissipate. Frontolysis in the free atmosphere may sometimes*
be forecast on this basis.

Isentropic chartsusually show the presence of a moist tongue of air

408 ANALYSIS AND FORECASTING [Chap. 24

advancing from the southwest in the warm sector of a frontal depression.
On some occasions this moist tongue splits into two portions, one with a
motion having cyclonic; curvature, the other with aiiticycloiiic curvature.
When this happens a secondary center of low pressure 1 often develops at
the tip of the warm sector.

Three non-adiabatic processes tend to destroy the conservatism of
iscntropic surfaces. The first of these is cooling of the free atmosphere
by long-wave radiation, which produces a decrease in temperature, and
so of the potential temperature, of the order of 1 to 2 C per day. Con-
vection, including vertical mixing, also produces changes in potential
temperature. The third and most serious cause of variation is the proc-
ess of evaporation or condensation. The cooling resulting from the
evaporation of precipitation falling from overrunning air may be great,
as when the relative humidity in the air through which the precipitation
falls is initially low. As soon as condensation has started in the ascend-
ing warm sector air of a depression, the isentropic surface 110 longer
coincides with the same layer of air particles and the method breaks
down.

Several deficiencies have prevented the widest possible utilization
of this method of upper air analysis. Tn the first place 1 , little is known
of just how and why isentropic surfaces move under various specified
conditions. If the motions of the isentropic surfaces themselves could
be forecast with reasonable accuracy, the value of the method would be
greatly enhanced. Secondly, an adequate statement has not as yet
appeared of the exact manner in which isentropic analysis can be used
in conjunction with the surface analysis to augment the information
provided by the latter. In order to realize* the* full potentialities of the
method, it appears to be necessary to work with isentropic charts for a
e'onsie Arabic period of time. Many me*te*ore)logists elo not have the tinier
available for this and so have been unable to utilize the method. This
is in contrast with the method of frontal analysis, the advantages and
limitations of which can be comprehe^nelcd in a comparatively short
perioei of time.

167. Forecasting Positions of Pressure Fields and Fronts. After
the synoptic weather chart has been analyzed, with the fronts and ise>-
bars entereei on the chart, it is ready to be useei fe>r forecasting. First
to be forecast are the movements erf the centers of high and low pressure
and of the fronts, since the motions of these govern to a large extent
the changes in the weather throughout the forecast region.

One of the easiest methods for forecasting the behavior of fronts and
pressure centers is to extrapolate* their positions in the elirection anel at
the speeel with which they have bee>n moving during the perioei since the
last map. In order to do this readily, the previous positions of the signi-

Sec. 167} FORECASTING PRESSURE FIELDS AND FRONTS 409

ficant points and lines are often entered upon the current weather map
by colored dots or lines. Very little justification can be given for this
method of forecasting, except that the processes which were active in
the immediate past may continue to operate in the same manner in tho
future. This is often true, and the method can give reasonable first
approximations for the movement of fronts and pressure systems.
('arc must be- taken, though, to see that no significant changes have
taken place or are taking place to make the forecast erroneous. Thus
rapid deepening of a low may be occurring, indicating that occlusion of
the fronts is imminent; or an active low-pressure area may be advancing
toward one quadrant of a large anticyclone; or a long trough may have
developed along the front; or some* other development may have
occurred, any one 1 of which indicates that the system will not continue
to move with the previous velocity.

The kinematical method of forecasting the motion of fronts, centers,
etc., as given by Petterssen, was described in a general manner in sections
42 and 43. This treatment is mathematical in derivation and use, the
movement being obtained by substituting into a formula values of cer-
tain functions of the 1 pressure, such as the pressure gradient and ten-
dency, at different points on the map. The formulas are in terms of
partial derivatives. To use them, it is necessary to replace the deriva-
tives by finite differences. An example 1 of this computation will be
given, using the formula for the velocity of a wedge. The, formula for
the velocity c normal to the wedge line is, according to (43.6),

dxdt

dx*

In this formula, x is taken in the direction perpendicular to the wedge
line.

Fig. 179 shows a wedge, with All denoting the line of the wedge arid
CD perpendicular to the line AB. By using finite differences in the
above formula,

c =

(*p\

AX VAX/
AT

JLYM
AX \AX/

410

ANALYSIS AND FORECASTING

[Chap. 24

where T represents the value of the tendency. Now AT/ Ax may be
evaluated from the map for any length Ax by taking values of the
tendency at points Ax apart. Since these points should be centered on
the line AB of which the velocity is to be determined, they should each
be a distance %Ax on either side of AB. The value of (l/Ax)(Ap/Ax)

FIG. 179. The velocity of a wedge by the kinematical rules.

may be obtained by finding the values of Ap/Ax for two lengths which
extend for a distance of Ax in both the positive and negative x directions
from the line AB, subtracting the values obtained, and dividing by Ax.
In the diagram, let Ax = OP, and let OQ = OP be measured on the other
side of on the line CD. Let X and Y be the midpoints of OP and OQ
respectively. Then

AT T x - T Y

Ax Ax

J_/M = JL/;

Ax \AxJ Ax \

PP - Po ___ Po - PQ
Ax Ax

(PP ~ Po) ~ (Po- PQ)
(A*) 2

L(T X - TV)
PP - Po) - (Po- PQ)
This can be put in the more general form

Hence, if Ax = OP = L

c =

(PL - Po) - (Po - P-L)

(167-1)

Sec. 167} FORECASTING PRESSURE FIELDS AND FRONTS 411

Since the tendencies are given on the map for 3-h intervals, this for-
mula gives the velocity of the wedge in miles per 3 h if L is expressed in
miles. The displacement of the wedge line in miles dining a 6-h
period, for example, is obtained by doubling the answer obtained from
(167-1).

Tare must be exercised in selecting a suitable value for the length L.
It is advantageous to draw tendency and pressure profiles along the x
axis to aid in this selection. The maximum accuracy is attained if L is
chosen so that at distances of 3 L and J ^L from the wedge line, the
tendency profile is nearly a straight line having a maximum slope. The
slope of the pressure profiles should be regular and representative at
distances of L and L from the wedge line.

In a similar manner the velocities of troughs, highs, and frontless lows
may be determined as indicated in section 13.

Referring to section 42, it is seen that an absence of deepening or fill-
ing of the pressure system was assumed in deriving the formula for the
velocity of an isobar. Similarly, the velocity formula when applied to a
wedge is strictly accurate; only when the x component of the pressure
gradient in the moving system is invariant with time. The formula
should be applied only to those wedges in which this condition is fulfilled.

A similar method is applicable to the motion of a front. Thus, using
(43-8),

_ L(T a ~ 7V)

where the distances L and L are now measured from the front, and T a
and T r are tendencies just in advance and to the rear of the front.
These latter are best obtained by drawing the isallobars and noting the
tendencies in the two air masses just at the front. The formula should
be applied only when deepening or filling of the pressure system is absent
or is uniform on both sides of the front.

By using the formulas one assumes that the tendency profile of the
moving pressure system will continue into the future. Thus again the
method is to extrapolate the effects of the forces already present into
the future, hut the data used here an; nearer the current situation, being
pressure changes during the last 3 h, and so are more reliable for fore-
casting purposes. These formulas, then, will give a forecast which is
more accurate than one obtained by the former method in a situation in
which changes in the pressure field have taken place recently but are
now no longer occurring. The forecasts of the motion from these formu-
las are most accurate for periods of 6 or 12 h in the future. Owing to
the frequent development of accelerations, both positive and negative
only rarely will they predict displacements accurately for a full day in

412 ANALYSIS AND FORECASTING [Chap. 24

advance. Acceleration formulas have been developed in section 43 for
the significant points and lines of the pressure field, but the evaluation of
these is very complicated and the results seldom justify the time spent.

A third procedure uses the pressure field of the current map and is
simply applied. It is based on the fact that a frontal surface is the
edge of an air mass, and so must keep ahead of the air that is moving
behind it. Since there can be no overrunning of the cold air, a cold front
must move with the speed of the winds behind it. Friction retards the
surface wind, but a representative value is given by the wind at 2000 ft
above the surface. After the component of wind at this height normal
to the front has been expressed in miles per hour, the motion of the front
in miles during a 6-h period, for example, is obtained by multiplying the
above velocity component by the time interval, 6 h.

Another value of the speed of the front is found from the component
of the geostrophic wind velocity normal to the front in the cold air,
adjusted for the curvature of the isobars. The cold front can be assumed
to move at a speed of 90 to 100 per cent of this adjusted value. Along
the warm front the warm air may override the colder air underneath.
Hence the warm front does not move with the speed of the warm air
winds, but with the speed at which the cold air under the front advances
in a direction normal to the front. This speed can be obtained approxi-
mately by noting the component of the velocity normal to the front of
either the 2000 ft or the gradient wind in the cold air. The computation
of the speed of an occlusion would depend on whether it was of the cold
or warm front type. The method of computation for the velocity of a
cold front would apply for a cold front type occlusion, and that of a
warm front for a warm front type occlusion. Frequently, though, an
occlusion lying along an extended trough moves very slowly and dissi-
pates while a new pressure center forms at the junction of the cold and
warm fronts.

The method as described in the last paragraph assumes that there is
little or no change occurring in the pressure field. If there is a large
isallobaric gradient, then the velocity will have to be obtained from the
gradient wind compounded with the isallobaric wind in the manner
explained in section 121. The observed 2000-ft wind already gives
the adjusted value.

This last method is a valid one for obtaining the instantaneous
velocity. It uses the current pressure gradient, adjusting it for the
isallobaric system. It utilizes the present motion to determine the
future displacement and does not allow for any changes in the pressure
field. This method cannot be used for pressure centers, troughs, and
wedges.

Sec. 167] FORECASTING PRESSURE FIELDS AND FRONTS 413

There is frequently very little difference among the values obtained
by the three methods of forecasting displacement, and an average of the
distances obtained from two of them gives a fairly reliable forecast.
When there is a marked difference, then the value obtained by either of
the last two methods is more reliable than that obtained by the first
method, since they both take into account recent changes that have taken
place in the pressure distribution and the effects of the isallobaric field.

Besides the methods outlined above for forecasting displacement,
there are rules, some empirical and others that can be explained by
reference to the tendencies, which tell the forecaster the direction in
which pressure centers are likely to move. A frontal depression will
move slightly to the left of the direction of the isobars in the warm sector
until it begins to occlude. After occlusion, it tends to become stationary
and to fill. A center of low pressure tends to move in the direction of the
most rapidly falling tendencies, and a center of high pressure in the
direction of the most rapidly rising tendencies. Near the periphery of a
largo semi-permanent warm anticyclone a low-pressure center tends to
move in the circulation around the anticyclone.

Forecasting involves not only the prediction of the motion of the
pressure centers and fronts but also of the increase or decrease in their
intensity. With fronts it is necessary to decide whether frontogenesis
or frontolysis is occurring. This can be determined by a study of the
existing wind field, as shown in section 109.

For pressure systems, the variation in intensity is most readily deter-
mined by noting the changes occurring in the pressure at the center of
the system. The deepening or filling that is occurring at the time of the
map is indicated by the tendencies at the centers of the highs and lows.
If it is assumed that these tendencies will continue, a high will become
stronger if the tendency at the center is positive and will become weaker
if the tendency at the center is negative. In a similar manner the fill-
ing or deepening of a cyclone can be determined. In a frontal depres-
sion the deepening or filling is determined by the tendencies in the warm
sector. Allowance should be made when using these rules for the small
pressure rise at certain periods of the day and the small fall at other
periods according to the diurnal pressure wave mentioned in section 3.

Other empirical rules are useful in determining the probability of
deepening or filling. Thus an occluding center deepens, and when the
occlusion process is complete, the center begins to fill. When a secon-
dary low forms, the primary low begins to fill. When in winter a low-
pressure system moves off the land and over the ocean, the low will tend
to deepen and become more active.

As shown in sectipn 50, horizontal convergence in the wind field is

414 ANALYSIS AND FORECASTING [Chap. 24

associated with increasing cyclonic circulation around a low and decreas-
ing anticyclonic circulation around a high; with divergence the converse
holds. Other things being equal, the cyclonic motion increases as a low
moves southward and decreases as it moves northward; the converse is
true for a high.

Many of the foregoing methods of forecasting assume that the atmos-
pheric processes in operation at the time of the map are going to con-
tinue in the future. These conditions usually do persist for a period, but
they gradually change. To extrapolate beyond the period during which
the conditions as outlined on the map will persist cannot be justified.
Over land reliable forecasts of frontal positions are limited to 12 to 24 h;
over the sea the period may be extended to 24 to 48 h. With experi-
ence a meteorologist learns to anticipate developments in the weather with
certain pressure configurations and learns to extend his forecast beyond
the periods mentioned. Thus, as a front moves off the east coast of
North America in winter, at times a center of low pressure develops on it
and then moves northeastward along the coastal regions. This results
from the inflow of warm air from over the Gulf Stream into the system,
providing a fresh supply of energy for the circulation. But there may
be no evidence of this development in the configurations of the isobars
and the isallobars from which many of the above rules and methods were
derived. With experience the forecaster is thus able to suggest the
probable weather for three days, but the forecast for the third day is not
too reliable. Methods of forecasting for a period longer than three days
will be discussed in section 170.

168. Forecasting Condensation Phenomena. The different types of
clouds are discussed in Chapter 20, and of precipitation in sections 129
and 130. The forecasting of clouds and precipitation is, of course,
based on the determination of the circumstances in which these are
likely to occur. These circumstances can be determined when the
positions of the fronts and centers of high and low pressure have been
forecast.

There is a diurnal variation in frontal precipitation, consisting of
an increase in the intensity and area of occurrence of warm frontal pre-
cipitation during the night and a decrease during the day; at a cold
front showers are more widely distributed during the day. As men-
tioned in section 129, the amount of frontal precipitation depends to a
large extent on the amount of potential instability in the warm sector
air. By a determination of the changes in the latter, it is possible to fore-
cast the subsequent variations in the frontal precipitation. Variations in
the amount of cloudiness and precipitation will also arise through a
change in the intensity of the circulation about the low-pressure area.

Sec. 169] AN EXAMPLE OF MAP ANALYSIS 415

Modifications in the air masses too will change the extent and inten-
sity of the precipitation. If the air has a trajectory over a warm
water area rather than over land, there will be an increase in the amount
of moisture in the air mass and also a change in the stability. As a
result of both these changes a difference in the cloudiness and precipi-
tation is to be expected. Variations in the air trajectory over moun-
tainous country will result in variations in the amount and extent of
orographic cloud and precipitation.

169. An Example of Map Analysis. To learn to analyze a weather
map satisfactorily, it is necessary to work with a number of maps under
the supervision of an experienced analyst. Yet it is desirable to show
how the foregoing discussion can be applied to an actual weather
situation.

Certain symbols have been recommended by the International
Meteorological Organization for indicating the details of the analysis.
These are given in the accompanying table.

SYMBOLS TO INDICATE THE RESULTS OF THE ANALYSIS

On working chart On printed map

Surface cold front Continuous blue line fTTTTTTTT

Surface warm front Continuous red line WWWWWWW 1 ^

Surface occlusion Continuous purple line FHT^P r ^'^FHr^P

Upper cold front Broken blue line r v v v v w^rv
Upper warm front Broken red line
Upper occlusion Broken purple line
Stationary front Alternate red and blue line

Areas of continuous

precipitation Continuous green area Hatched area

Area of fog Continuous yellow area Fog symbols

Showers Green triangles

Figs. 180, 184, and 185 give the weather maps for a small section of
North America for 19.30 h, November 17; 07.30 h, November 18; and
19.30 h, November 18, 1941. All times mentioned throughout the
discussion are Eastern Standard Time. Because of the difficulty of
giving the names of the reporting stations, a station will be referred to
by its coordinates of latitude and longitude when necessary, as will areas
to which it is desirable to refer. The only data plotted on the maps are :
wind, pressure, tendency, temperature, dew point, and present weather.
For a satisfactory analysis, information about cloud types and heights is
necessary, but this information has been omitted because of lack of space
on the printed map.

Previous to the evening of November 17, a ridge of high pressure had
been centered along the Alleghenies. In addition, a center of low

416

ANALYSIS AND FORECASTING

[Chap. 24

pressure had been located at about (42, 107) with a cold front to the
west moving rapidly southeastward and an almost stationary warm front
extending to the southeast. By 19.30 h on November 17, occlusion
had occurred, with the junction of the warm and cold fronts at (41, 106).
The warm front extended to the eastward, and frontogenesis had taken
place in the southerly current of returning polar continental air from
(40, 103) to (46, 90). South of this front the polar continental air was
much more modified than the air lying to the north of it.

35'

105

100'

Fia. 180. Weather map for 19.30 h, November 17, 1941.

The placement of the front as shown in Fig. 180 can be explained in
terms of winds, temperatures, and dew points. South of the front from
longitude 88 to 100 the dew point of the air ranges from about 48 to
55, whereas north of the front, the dew point decreases from 41 in
the neighborhood of the front to 26 farther north where no precipita-
tion has occurred. West of longitude 100 the air has had a trajectory
over the dry lands west of the Gulf of Mexico, and the dew points are not

Sec. 169} AN EXAMPLE OF MAP ANALYSIS 417

so high. Yet there is a difference in temperature across the front of
about 20 in these longitudes decreasing to about 8 where the front has
recently formed. Near the western portion of the front there is a
decided difference in wind direction from southerly on the south side of
the front to northerly on the north side of the front. In the region south
of Lake Superior the wind shift is only from easterly to southerly across
the front.

As is usual with a stationary front, there is little difference in tendency
across the front. The motion of the air in the cold air mass governs the
motion of the front, as mentioned in section 167. To the west the
gradient wind near the front in the cold air is weak, but becomes stronger
south of Lake Superior. However, in the latter region there is a weak
isallobaric gradient directed northward, and therefore a small isallobaric
wind component to the south. The resultant of this and the east south-
east gradient wind is an east wind parallel to the front. Thus the front
is almost stationary, with the warm air overrunning the cold in this
region. In general there are rising tendencies along the front. But at
(45, 95) the tendencies are falling, suggesting that a wave may be form-
ing in the vicinity.

South of the front the weather is clear. North of the front in the cold
air there is considerable cloudiness and precipitation, caused partly by
the overrunning and partly by the slight orographic lift in the easterly
current. Some clearing is occurring in the ridge of cold air at (42,
103).

There are not enough data given in the vicinity of the occlusion on
which to base a forecast, although in general it would be expected that
an old occlusion would remain almost stationary and dissipate. In the
vicinity of (42, 98) the difference in the tendencies on the two sides of the
front is not large. Thus, according to the kinematical rules, the motion
of the front is small, a deduction that agrees well with that given by the
previous motion and the geostrophic wind criterion. The wave at
(45, 95) has not developed sufficiently to permit a forecast of the motion,
but that development will lead to more extensive overrunning and so
more precipitation and cloudiness to the north. At longitude 90, a
computation by the kinematical rules can be made. The line AOR is
normal to the front at O and is considered positive to the south. The
unit distance L is OA = OR. The tendencies at the front are 0.4 mb per
3 h in the warm air and 0.6 mb per 3 h in the cold air. Substituting into
(167-2) gives

0.4 - 0.6

(1011 - 1009) - (1009 - 1013)

f} A

418 ANALYSIS AND FORECASTING [Chap. 24

This gives the motion in 3 h. The movement in 12 h is south % Q of OA.
The displacement is small, and the front can be assumed to be stationary.

With the development of the wave, the gradient wind will increase
slightly, but very little motion of the front is to be expected in this situa-
tion for the coming 12-h period. During the succeeding 12-h period the
wave may move along the front. If that occurs, the motion of the front
will be irregular with cloudy weather and precipitation north of the
front. West of the moving center of low pressure the front will move
south as a cold front. The other possibility is that the wave will remain
almost stationary and occlude. In that event, increased overrunning
will continue to give precipitation to the north, but the gradient wind
will increase sufficiently to cause the front to move northward.

At 23.00 h, November 17, information on the upper air conditions
was obtained at a number of stations by means of radiosondes. The
ascents which are of interest in the present analysis are those taken at
Sault Ste. Marie (46, 85), St. Paul (45, 93), Bismarck (47, 101), and
Great Falls (48, 111). Great Falls is not on the part of the map given
in Fig. 180 but is slightly to the west of the western boundary. The
figure shows that Sault Ste. Marie is in the zone of f rontogenesis, St. Paul
is in the warm sector, and Bismarck and Great Falls in the cold air,
although Bismarck is a short distance from the front and to the north-
west of the developing wave. The upper air ascents for St. Paul and
Bismarck are shown plotted on a tephigram in Fig. 181, and for Sault
Ste. Marie and Great Falls in Fig. 182. The ascent at St. Paul is entirely
in warm air, but the graph shows that the air is not completely uniform,
having an isothermal layer at 800 mb as well as a nocturnal inversion
near the surface. This lack of uniformity would be expected in polar
continental air which has become modified to varying degrees depending
on the extent of the southward excursion of the air. The air at Great
Falls is almost saturated, as is that in the lower layers at Bismarck.
The air at both stations is lying to the north of the front and is being
carried westward and so being lifted because of the topography of the
country. The air in the warm sector contains more water vapor, but
the relative humidity is lower than in the cold air, and it will require a
lift of 2000 or 3000 ft to saturate it. At Sault Ste. Marie the non-
uniformity of the warm air is again apparent, since the developing frontal
surface intersects the ascent where the increase of moisture and tempera-
ture occurs, from 950 to 900 mb, and the air above this level shows an
inversion and an isothermal layer. The frontal surface is not delineated
sharply in this region since the front is still developing.

The ascents at different aerological stations may be compared by
means of their graphs on tephigrams. Another method, as described in

Sec. 169}

AN EXAMPLE OF MAP ANALYSIS

419

O>
O

-20

-10

FIG. 181. Upper air ascent curves for St. Paul and Bismarck, 23.00 h,
November 17, 1941.

0>
o

-20

-10

T (CJ-

FIG. 182. Upper air ascent curves for Sault Ste. Marie and Great Falls, 23.00 h,
November 17, 1941.

420

ANALYSIS AND FORECASTING

[Chap. 24

section 165, is to take a cross section of the atmosphere. Fig. 183 shows
a cross section with data from these four stations plotted. The data
included are temperature, potential temperature, relative humidity,
and mixing ratio. By comparing the values of the potential temperature
and the mixing ratio for the ascent at St. Paul with those at Bismarck
and Sault Ste. Marie it is seen that there is only a small difference near
the upper portions of the ascents, but near the surface the differences

Bismarck

Sf Paul

Saulf

1500

FIG. 183. Cross section for 23.00 h, November 17, 1941.

are marked. Similarly, comparing the ascent at Bismarck with that at
Great Falls, it is apparent that at higher levels, above 2 km, the values
are not similar, but near the surface the variation is less, although even
here there is a difference, showing that the air mass is not quite uniform.
On the basis of these facts, the cold frontal surface has been drawn to
intersect the ground between St. Paul and Bismarck, and the Bismarck
ascent at about 2 km. The warm frontal surface intersects the ground
to the east of St. Paul and the Sault Ste. Marie ascent at about 1 km.
Lines of equal potential temperature for the values of 10, 15, 20, C
and for values of the mixing ratio of 2, 4, 6, gm per kg have been
drawn on the chart. Since the fronts are lines of discontinuity for these
variables, on the chart the lines for the same value of the variable meet
the frontal surface at different points of the two sides of the front. By
using the position of the cold frontal surface at the ground and the point
of intersection at Bismarck, the slope of the cold front is found to be Kso
The surface position of the warm front near Sault Ste. Marie is too
indefinite to use for determining the s ope of the fr(mt.

Sec. 169}

AN EXAMPLE OF MAP ANALYSIS

421

Fig. 184 shows the weather map for 07.30 h, November 18. In
general there has been little change in the surface pressure distribution.
The occlusion has moved slowly to the northeast, passing stations at
(40, 105), (41, 105), and (43, 106), as shown by the strong rising pres-
sures, by the shift in the winds from those with easterly to those with
westerly components, and by the drop in temperature and dew point.

105'

FIG. 184. Weather map for 07.30 h, November 18, 1941.

The temperature has decreased at other stations owing to the diurnal
variation, but, in general, the drop at these stations which the front has
passed has been more than that to the east of the front. The center of
low at the tip of the occlusion has filled about 5 mb, and with the filling
the weather has improved. Near the junction of the warm and cold
fronts the pressure is rising slowly, as indicated by the rising pressure at
(39, 102). Although some of this filling is attributable to the diurnal
pressure variation, agood deal of it is a result of the general filling of the

422 ANALYSIS AND FORECASTING [Chap. 24

low-pressure system. The cold front has moved to the southeast,
although the rate of motion cannot be determined by the data plotted
on the map. The warm front has apparently passed no stations.
The developing low, presaged by the falling pressures along the front
on the previous map, is shown by the 1000-mb pressure of (44, 98). The
temperature at the reporting station continues to be representative of
that of the cold air, but the wind has shifted from north to southeast.
As will be seen when the hourly weather reports for (46, 95) are dis-
cussed, the wind shifts when the frontal zone reaches the station, and
thus before the warm air actually arrives. A bulge has now developed
on the front to form a wave. To the east there has been little change,
although the wind shift at (46, 95) of 90 with a rise in temperature
against the diurnal variation suggests that the frontal zone is approach-
ing that station. Frontogenesis has occurred farther to the east, as
seen by the difference in cloudiness, wind, and temperature between the
stations (46, 85) and (45, 85).

The most important changes in the frontal positions will be related
to the developing wave. The occlusion will continue to dissipate with
the filling of the low-pressure area, and the warm front will not change
its position much since the geostrophic wind normal to it is light and
marked isallobaric gradients are absent. With the falling pressures on
all sides of the newly developed low-pressure system, the cyclone should
deepen and occlude during the succeeding 12-h period. With the
occluding and deepening, the wave will tend to take its own path rather
than move along the front. The motion to be expected is in the direc-
tion of the most rapidly falling tendencies around (48, 97). With the
deepening that is occurring, the fronts, and especially the cold one, will
accelerate. Hence any values obtained for the forecast displacements
of these will be approximations only. By using the kinematical rules
for the velocity of the warm front, the computed position of the latter
at the end of 12 h is found to be at the forty-seventh parallel in the
vicinity of Lake Superior, swinging northward to the forty-ninth parallel
at longitude 97. But with the eastward movement of the cold front
to the south of the low, the western portion of the warm front will be
occluded.

With the development of the wave, the precipitation to the north of
the front will become more extensive during the succeeding period. The
cold front south of the center will become more active and the lift at
the frontal surface will probably release the potential instability which
is present in the air over St. Paul and which will increase through the
diurnal heating in the surface layers. Thus light to moderate showers
should be expected with the cold frontal passage. With the movement

Sec. 169]

AN EXAMPLE OF MAP ANALYSIS

423

of the low-pressure system to the region about (49, 97), the circulation
over the Canadian prairie provinces will be northeasterly. This direc-
tion of wind carries the air over higher ground, and so the orographic
drizzle and cloud will continue. Southwest of the center the air motion
will have a north to northwest direction and will no longer be subjected
to an orographic lifting process. Clearing is expected in this region.

??A

6 ft / '

3 3*wy&

Fia. 185. Weather map for 19.30 h, November 18, 1941.

Fig. 185 shows the weather map for 19.30 h, November 18. Signifi-
cant changes have taken place in the frontal construction. The occlu-
sion near (41, 104) has now dissipated, and a weak low-pressure system
is centered at (38, 104) and is filling, as shown by the tendency at (38,
100). The low which on the 07.30-h chart was at (44, 98) has moved
northeastward to (48, 95), as was indicated by the tendencies.

The position of tie front is approximately as suggested by the analy-

424 ANALYSIS AND FORECASTING [Chap. 24

sis of the 07.30-h map. In the southern portion, in the vicinity of
(35, 103), the stronger rising tendencies behind the front than ahead of
it help in placing it. Behind the front the winds are southwest or west.
To the east of the front the winds are south, corresponding to the direc-
tion of the isobars. The dew points to the east of the front are 47 to
58, whereas in the drier air to the west the dew points are 44 near the
front and decrease with increasing distance from the front. A similar
but more marked difference exists in temperature. Along the front
lying in the trough between the two low-pressure areas, the difference in
temperature and dew point across the front is ten degrees or more, with
the values of these elements in the air behind the front decreasing rapidly
with distance from the front. This situation is quite common. The
warm air in the warm sector is usually nearly uniform in temperature
and dew point. The main change in these elements occurs in the cold
air under the warm frontal surface and behind the cold front. Owing
to the small velocity of the front at (41, 97), the tendency contrast is
small across it, but in the vicinity the thunderstorm reported and the
distant lightning are both evidence of the proximity of the cold front.
In the vicinity of Lake Superior the warm front did not move quite so
far as was forecast. The rapid increase at (46, 85) of temperature and
dew point as compared with that at (47, 88) shows that the front has
passed the former but not the latter. Although there has been some
increase in temperature in the vicinity of the low at (48, 95), at no
point is there evidence of warm sector air. The increase in temperature
can be explained by the heating produced by~thc precipitation falling
from the trough of warm air aloft. As yet there is little difference in
temperature across the occlusion, which in this situation is placed on the
basis of the wind shift from easterly to southwesterly, and the rapidly
rising tendencies to the west and southwest of the front. During the
12 h between the two maps, the pressure at the center of the low-
pressure system decreased, with the lowest pressure at 13.30 h being
999.0 mb. Since that time the pressure has risen, and the center is
filling, as shown by the mean tendencies in the region. This indicates
that the occluding process is complete, and from now on the occlusion
will dissipate.

The next problem is to forecast the motion of the fronts. In the
southern portion at longitude 103, the geostrophic wind criterion gives
the front a velocity of 40 mph. But the filling of the low during the
succeeding 12 h will cause this value to decrease during that period. As
a result of the very weak trough in the isobars at the front, the denomina-
tor of the fraction in (167-2) giving the motion by the kinematical
criterion is small. Any inaccuracy in the numerat&r will be magnified

Sec. 169] AN EXAMPLE OF MAP ANALYSIS 425

when the distance the front is expected to move is computed. . Further-
more, the tendencies as recorded west of the front are very irregular, so it
is difficult to draw isallobars. A computation on the thirty-sixth paral-
lel places the front in 12 h at the ninety-seventh meridian, but this
should be checked by the previous motion, or by a more accurate deter-
mination of the tendency field west of the front. Along the cold front
between the two lows there is no geostrophic wind normal to the front,
but in the cold air the pressures are rising more rapidly than in the warm
air. The trough with the cold front will then move slowly eastward dur-
ing the succeeding 12-h period. A motion of about 100 mi for 12 h might
be expected. This agrees with the past motion south of the center of low
pressure on the previous map. This center should move slowly toward
the region of most rapidly falling pressures, that is to the northeast, and
fill slowly. With the rapidly falling pressures to the north, the previous
motion will not be reliable for a forecast of the displacement of the warm
front. Moving it at 70 per cent of the geostrophic wind of 26 mph
would put it at latitude 49 in the Lake Superior region. The same fore-
cast position is reached by the use of the kinematical rules at longitude
87. It would be expected then that at the end of 12 h the two fronts
will meet in a center of low pressure at about (49, 92), with a short occlu-
sion to the northwest, and the cold front extending southerly and then
south westerly in a weak trough.

The most extensive area of precipitation will extend to the north of
the warm front, where overrunning is supplemented by orographic lift
from the lakes region to the ridge to the north. The precipitation about
the occlusion will become less pronounced, and to the west neither oro-
graphic lifting nor instability should cause much precipitation. Some
local showers and thundershowers are to be expected with the cold frontal
passage to the south. Data beyond the edge of the map will tell whether
the current of air from the south has a trajectory over the Gulf of Mexico.
If its moisture content is increasing as it flows over the Gulf, then con-
vergence in the south-north current may give rise to cloud east of the
frontal region.

Further data about the position and motion of fronts can be obtained
from the hourly weather reports. Some of the information contained in
the hourly reports from 01.30 to 21.30 h on November 18 from the station
at (46, 95) is given in Fig. 186. The values for the pressure, tempera-
ture, and dew point are plotted, the latter two being coincident until
13.30 h. Underneath are given the wind directions and speeds in miles
per hour, the height of the low cloud in hundreds of feet, and the visi-
bility in miles. The low cloud layer formed an overcast except when
noted. On the ch*rt for 19.30 h of November 17, this station was in the

426

ANALYSIS AND FORECASTING

[Chap. 24

(Jo) 1

(qiu) d

Sec. 170] LONG-RANGE FORECASTING 427

cold air under the warm frontal surface, and was still there at the begin-
ning of the period illustrated. This is seen by the wind direction, the
fog which was present north of the warm front, and the temperature.
The drizzle later in the period was an indication of the approach of the
front. The first sign that the station was in the frontal zone appeared
at 09.30 h, at which time the wind shifted to the southeast. The tem-
perature then began to rise more rapidly, but did not reach the tempera-
ture of the warm air until 12.30 h, when it became 52. At this time the
wind was south southwest, and the visibility had improved, showing that
the drizzle was ceasing. The latter stopped and an increase in the ceiling
took place during the next few hours. For 3 h, from 09.30 till 12.30 h,
the station was in the frontal zone. Since the front was moving with a
speed of about 5 mph, this indicates a frontal zone of 15 mi, which is not
an exceptional width. The pressure decreased while the front was
approaching, became steady with the arrival of the frontal zone, and
then decreased again in the warm sector as the center of low approached
the station. The center passed close to this station since the minimum
pressure recorded of 999.7 mb was close to the minimum, 999.0 mb, at the
center of the system.

The passage of the cold front was more sudden. Between 15.30 and
16.30 h the pressure began to rise, the temperature dropped 4 with an
equal drop in the dew point, the wind veered through 45, the ceiling
and visibility decreased for one observation, but both improved after
the frontal passage. After the cold air advanced past the station, the
pressure continued to rise, and the temperature and dew point to drop
until the end of the period illustrated. All these variations are of the
type to be expected in a situation of this kind.

170. Long-Range Forecasting. A complete discussion of the pro-
cedure of long-range forecasting is beyond the scope of this book. A
short description only of the methods by which this problem is being
attacked will be given.

One method of attempting to solve the problem is by a study of
possible variations of the energy output of the sun (see section 28).
Since the sun is the source of energy for the earth, any variation in its
output should have an effect on terrestrial weather. The atmosphere
absorbs and scatters some of this solar energy, and so the best place for
its measurement is on high mountains where there is less atmosphere and
also less atmospheric pollution above the observers, and very few clouds.
The Smithsonian Institution of Washington maintains observatories
taking measurements of the energy from the sun at Mount Wilson, Cali-
fornia, and in the Andes, and at a number of other stations. These
measurements suggest that there are variations amounting to about

428 ANALYSIS AND FORECASTING [Chap. 24

2 per cent of the total energy. It has been contended that these varia-
tions are not random but cyclic. The correction for the absorption
and scattering by the atmosphere is on the average about 25 per
cent of the total energy received at the outer limits of the atmosphere,
and it is not yet clear if these variations mentioned above represent real
variations in the energy output of the sun, or if they result from an
incorrect allowance for the effect of the atmosphere. It has been further
claimed that a correlation exists between the energy output of the sun
and the world weather. Thus, for example, the eleven-year cycle of the
sun's energy output is reflected in an eleven-year weather cycle on the
earth. Some attempts to forecast for a period of a month in advance
have been made, but as yet the method has not proven its value.
* Another method of forecasting is the result of the study of a large
number of meteorological series. By means of computing correlation
coefficients (see section 61), meteorologists have been able to show that
variations in the values of some series have been followed by correspond-
ing variations in other series. Simultaneous variations are not uncom-
mon. Thus if the pressure in Iceland falls, the pressure gradient over
the North Atlantic increases, the transport of air to western Europe
from the southwest is greater, and the temperature there rises. Sub-
sequent variations are more rare. Yet it has been found by Walker
that the summer rainfalls of India, Australia, and South Africa are
related to the distribution of pressure of the previous season in the South
Pacific and Indian Oceans. By using a combination of the values of
several variables, a formula has been obtained by which to calculate the
summer rainfall in these districts. The value of the latter has a corre-
lation of from 0.7 to 0.82 with the actual rainfall. This formula gives a
fairly reliable, although not perfect, estimate of the rainfall for the
following six months.

Similar attempts have been made to find series, especially for the
temperate zone of the Northern Hemisphere, which can be used to pre-
dict future values of the weather elements, but so far without success.

Baur has developed a method of forecasting for Germany for ten-day
periods. His forecasts are based on observations in the stratosphere and
high troposphere. According to Baur and his associates, the changes in
pressure at the earth's surface are controlled by atmospheric movements
at levels of 4 km and higher, the process whereby this occurs being known
as " steering." This group of meteorologists has shown .that the 24-h
isallobaric highs and lows at the surface follow the direction of the wind
in the upper atmosphere. Since the meteorological conditions at and
above 4 km change only slowly, the air motion at these levels can be pre-
dicted with a degree of accuracy and thus also the movement of isallo-

BIBLIOGRAPHY 429

baric centers at the surface. Knowing the current pressure system,
Baur is therefore able to use the future position of the isallobaric systems
to draw a forecast map of the pressure distribution. From this map he
makes a forecast of the weather in the different districts. His forecasts
have been highly regarded in Germany.

A method of forecasting for a period of five days has been developed by
Rossby and collaborators. Rossby has shown that the positions of the
semi-permanent centers of high and low pressure are related to the
strength of the circulation in the free atmosphere in the region of wester-
lies. The changes in this circulation are less variable than those in the
values of the weather elements at the surface. Thus by forecasting the
value of the circulation, Rossby is able to forecast the positions of the
pressure centers and so the general weather distribution. By this
method forecasts for the subsequent five days are being made for the
United States at the present time.

Because of the value that accurate long-range forecasts would have to
various industries, many more or less reliable forecasts have been issued.
Some of these methods operate in the realm of folk lore, using the posi-
tion of the moon, the thickness of the fur on wild animals, the sunshine
on Candlemas Day, etc. A few of these simple rules for forecasting for
the immediate future have stood investigation.

Other forecasts are made for periods of a week to a month by meteor-
ologists in which they combine changes arising from the annual variation
of the weather with the average weather variations accompanying the
passage of high- and low-pressure areas. As such they are valuable since
they instruct those interested in the normal changes that can be expected.
Yet it has not been proven that such methods are capable of forecasting
the abnormal variations which comprise the most interesting part of
the weather.

BIBLIOGRAPHY

Admiralty Weather Manual, London, H. M. Stationery Office, 1938. Chapters 17,
21.

Byers, H. R., Synoptic and Aeronautical Meteorology, New York, McGraw-Hill Book
Co., 1937. Chapters 8, 9.

Problems of Modern Meteorology, London, Royal Meteorological Society, 1934. Num-
bers 4, 6.

Weightman, R. H., Forecasting from Synoptic Weather Charts, U.S. Department of
Agriculture, Misc. Publications, 236. Washington, D. C., 1936.

The Weather Map, Third Edition, London, H. M. Stationery Office, 1939.

Starr, V. P., Basic Principles of Weather Forecasting. New York, Harper & Bros.,
1942.

170. Baur, F., " Die Bedeutung der Stratosphare fur die Grosswetterlage," Met. Z.,
53, 237-247 (1936).

430 ANALYSIS AND FORECASTING [Chap. 24

170. Walker, Sir G. T., "Ten-Day Forecasting as Developed by Franz Baur," Q. /.

Roy. Met. Soc., 63, 471-475 (1937).
170. Walker, Sir G. T., " Seasonal Weather and Its Prediction," Smithsonian Report

for 1935, Washington, D. C. Pages 117-138.
170. Weightman, R. H., " Preliminary Studies in Seasonal Weather Forecasting,"

Monthly Weather Review, Supplement No. 45, 1941.
170. Willett, H. C., Report of the Five-Day Forecasting Procedure, Mass. Inst. of Tech.

Papers in Physical Oceanography and Meteorology, 9, No. 1, Cambridge, Mass.

1941.

CHAPTER 25
METEOROLOGY APPLIED TO VARIOUS HUMAN ACTIVITIES

Meteorology owes a great deal to the demand put upon its facilities
by the development of aircraft, as well as to the additional information
that the aircraft has provided for the study of the free atmosphere by
the meteorologist. In return the meteorologist has given a great deal
of his time and thought to the special problems of air transportation.

But climate and weather affect the other occupations of mankind.
Climatic factors help determine, for example, the location of certain
types of hospitals, the choice of agricultural crops, and the strength of a
steel bridge. The information necessary for the solution of these prob-
lems is obtained from a study of climatological data, and the short-
range forecast provides little assistance. Yet persons engaged in differ-
ent occupations can use the information available from the forecaster to
help do more efficiently the task at hand. To serve such persons best,
it is necessary for the forecaster to understand the problem which he is
helping to solve. He can thus give more readily the information desired.

Engineers to whom some of these forecasts will go are accustomed to
accuracy in their data. Care must be taken to make certain that they
understand that, at present, forecasts cannot be relied upon completely
and that a margin of safety must be left when using the meteorologist's
forecast. On the other hand, the forecaster should learn to give a state-
ment which is more specific than the general forecast of " fair
and colder," and one which is useful in planning the activities of the next
few days.

171. Transportation. Weather forecasts have for years played an
important part in maintaining the safety of ocean travel. The major
point of interest to the navigator is the strength of the wind. Moderate
gales may cause difficulty in keeping the course as well as damage to the
ship. For the benefit of ships at sea the weather services issue storm
warnings which tell the position and direction of motion of storms, with
the result that a ship is sometimes able to change its course in order to
avoid the worst part of the storm area. Freezing rain or drizzle hinders
the work and movements about the ship, and the presence of fog fre-
quently necessitates a decrease in the speed of the ship. These are the
weather factors which the officers of the ship wish to have the meteorolo-
gist forecast.

431

432 METEOROLOGY IN HUMAN ACTIVITIES (Chap. 26

Water travel on lakes and coastal waters is affected by the same
weather elements as ocean travel. The direction of the wind and the
times for wind shifts have more significance for ships plying inland water-
ways than for those over the ocean, however. Thus, for example, when
ice is floating about, the direction of the wind may determine the
availability of a particular harbor.

With all transportation systems that handle freight, the extremes of
temperature have significance at times. These tell the responsible
authorities the possibility of danger to the cargo through exposure to
temperatures which would injure it. This is of particular significance
with respect to freezing temperatures during periods of the transport of
fruit, although other commodities as well are affected by temperature
extremes.

Poor visibility caused by fog or precipitation lead^ to slowing down of
railroad transportation. In this case, though, it is not greatly bene-
ficial to the engineer to have this information beforehand. The railroad
management, however, will be able to place its equipment better for
keeping the tracks clear if it has advance information of the approach
of a severe storm or of a high wind which would drift snow into the cuts
along the track. Also if it knew of the approach of flood conditions along
the route, plans could be made to repair any possible damage.

The meteorologist can assist the highway engineer, too, in planning the
removal of snow. A previous knowledge of the approach of a snow-
storm will provide the manager with an opportunity to assign his equip-
ment to the best advantage for handling the clearing of the roads. Not
only is he interested in the amount of snow but also the type. A heavy,
wet snow is more difficult to handle than a dry, powdery snow. The
wind direction during and after the snowfall is of interest since this will
inform the engineer as to the routes along which drifting will be serious
and those along which it will be light. Snow removal is necessary dur-
ing and after periods of drifting snow as well as after periods of snowfall,
since such drifting snow will fill up the cuts in the snow banks which have
been made to permit road travel. Hence the engineer continues to be
interested in the direction of wind and in any shift of it.

The occurrence of freezing rain on roads slows traffic and causes many
accidents. The greatest danger occurs on hills. A notice of the
approach of a sleet storm, even a couple of hours in advance, provides
the highway engineer with an opportunity to send out trucks with gravel
to these hills and so have the equipment ready to keep the danger at a
minimum.

Ice storms affect the electrical contacts made by a street car with an
overhead wire. Considerable time may be savec^ and inconvenience

Sec. 172] AGRICULTURE 433

avoided if the cars are fitted out to cope with this difficulty before they
leave the car barns. Hence the management of such transportation
systems can make use of a forewarning of the approach of these storms.

172. Agriculture. Agricultural activities are affected greatly by the
weather and the climate. The distribution of crops is determined by
the length of the growing season, the amount of precipitation, etc.
Average figures for these in the different districts of the country may be
determined from climatological data. If it were possible to forecast the
values of these for a particular year, agriculturalists would be able to
adapt the crops to the weather expected. This is done in India where the
monsoon rainfall is forecast at the beginning of the season, and an
adjustment of crops is made on the basis of the forecast. This is not
yet possible over most of the world.

Farmers have learned through experience to make a fairly accurate
forecast covering the succeeding 12 to 24 h from the appearance of the
sky and the direction of the wind. A forecast covering the succeeding
12 h is definitely helpful to them in planning their work, for some of it
requires two fine days to complete satisfactorily. Drying of hay and of
fruit is done much more easily if no rain falls during the drying period.
A farmer finds a forecast useful when planning to plow clay soil, since the
latter must be neither too dry nor too damp.

One special type of service which is saving the farmers time and
money is the forecasting of minimum temperatures and frost. When
the temperature is expected to fall below freezing, fruit growers are able
to save their fruit by lighting smudge fires and keeping the temperature
from falling so low in the orchards. Cranberry growers flood their bogs
to save the cranberries from frost. These operations would be too
expensive if no forecasts were available and they had to be carried out
every night. Sheep breeders, too, arc interested in knowing the mini-
mum temperatures during the lambing and shearing seasons.

A common method of forecasting minimum temperatures is through a
statistical analysis. Previous records are taken for the vicinity. The
relative humidity or another function of the moisture content of the
evening is plotted against the difference between the dew point of the
evening and the succeeding minimum temperature. If the points fall
along a straight line, as frequently happens, the line of least squares is
determined for the data in the manner shown in section 60. By using
the equation of the line and the relative humidity or the dew point of
the evening, a reliable forecast for the minimum temperature of the night
can be made. For example, the equation determined for San Diego,
California, is

. y = -6.6 + 0.62(r - T d )

434

METEOROLOGY IN HUMAN ACTIVITIES

[Chap,

BURNING INDEX METER

f-or use on the forests of the Nor t /tern flocSty Mounte/ns

Calendar Data

Aunua+ 21 to 31

ive Hunnic

no hyorothcn
rrtum for fhe
4, 5, or 6/D m. rrte

n 16 n 2O

n II n 15

6 ii 10 ii

n 6 n IO

n 1 n 5

ill

1 1 H 15 n

Jtjly 26 n 31

n 21 n 25

r.i

16 20 ii

n II n 2O

n 1 n IO

25% +

21 ,. 25 i>

Junft 16 M 3O

15 -to 24

26 n 3O n

n 1 n 15

IO/<? 14

1 n IO Orr+nh*r

May 16 ii 31

Under 1

1 1 H 20 n

ii 1 ii 15

21 31 n

How to use this meter,
f/rst, set fhe s//</e so that the
tf/'fh the current ctete. Second, on r
for current fue/ mo/'sture &nc/ on the
f/'nd the Burn /'no /nctex or *& '/"

Season,
7~he net effect of four factor
ch&nges tv/'fh the c&/eno'&r atete br&
C/J hours of sunsh/ne, f2) /'ntenst*

f4J cumu/&t/re c/ry/ng or tret ft no of

pre/oer hum/c//ty /'s on //ne
-ererse s/'afe /'n the co/umn
//'ne for concurrent if /no*,

~s of forest /nf/<9mm<9b/'//ty

y of so/ar r0o > 'et-/ort.
7, cur/not or cureo*, &ncf
heavy fue/s <sspec/*//y
*rth s/opes.

9ss or woe& types &no*

nmon wherever there /'3
content /& tf>e most

Humidity,-

they &re /nsuff/c/ent for pure gr<.
&re too gre&t for o'ense green ttt

Half-inch stick*.-
Thcse represent & fue/ cor
&ny tree Growth. The/r mo/sttsre
o t e f yenc/&>/e 3t'no/e /nc/e# of the

FIG. 187. Burning index meter, side A, set for relative humidity 20 per cent on
June 25. (After Gisborne.)

Sec. 173]

FORESTRY

435

lIP-

**<&

-in

? /x <9

13
\4

Wind
Velocity

20 t~ee.t
Gooise open
/ere/ around

Expt
sh&<.
sum

\\*

>sw

vet
&<s/
18

eh stick moistures

oors orovno 1 , t/n&er st&nc/0r&
asurccf t hour or more before

T& of ore?r~> c/ry weit'ori'f'
1 9.5 8.5 7.5 6.5 5^ 4.5

or / \/ \/ \

12 10 9 A 7 6 5 4

3.4

or
/ess

2.5 or /ess

Pfl

S3 5

?*>

\ 6

,34

3fl

75 7

6ft *-*" B

etc 9

1 1

II ^ 12

13 or- 14

ftt

15 16 17

Pfl

18 19 2O

21 to 24

25 or more

BURNING

Significance.-
Burr>/no /ncfe* expresses t
/to wh/ch & forest /s ejrposco*. // J
mer/c&/ sce/e for report/no the ft
or & se&son. x/ f of 3&y 5O w
spre&c/ of f/re /r> c/'ffererrt fu&
he&t of f/re /n ? forge, S0y 6S0*
W/// d/fnosf me/t G/t/sn/num, out **
f ejrpr-csses tntens/ty of cone
types ac/j&cenl' to the st&t/on
>ts o'er/reef here /t /s not expec

o*&y 0s c? wno/e /t /s the f/rst.

INDEX^'BI"

he resc//t of serer*/ con</tt/orts

'// me&n & c/'fferent r&fe of
/ types, s'ust &s & cerf&/r>
C, **/// turn /c&cf to & /f'%u/cf,
/// /e&re /ror> st/// />&//. 7~he
f/tt'ons &ffect/'no; &// ft/e/
where JBf /s me&surect.
teaf to represent & p&rf/c-

4><st not tot&/ o&s/s for*
*" cf^noef

FIG. 188. Burning index meter, side B\ for a velocity of 10 mph and fuel
moisture 8 pet cent the meter reads 40. (After Gisborne.)

436 METEOROLOGY IN HUMAN ACTIVITIES [Chap. 25

where y gives the excess of the minimum temperature over the evening
dew point in F and values of T and T d are those of the evening. The
equation varies with different localities. In some instances, the rela-
tionship is best expressed by a parabolic equation connecting y and /,
the evening relative humidity.

In general the wind velocity is not a major factor with agriculturalists
when making their plans. They are interested in this element, though,
during the seasons when spraying their orchards or field crops is neces-
sary. With high winds a great deal of the spray is wasted since it is
carried away by the wind. If rain is expected the following day, they
may feel that immediate spraying is worth while since some of the fungi
and insects will be destroyed. If, on the other hand, the following day
is going to be fine and with light winds, the farmer will often prefer to
postpone his spraying for a day till better weather has come. In such
situations the agriculturalist will desire an accurate forecast of winds.

In these and in many other ways the farmer can make use of the fore-
caster's services to plan his work more efficiently.

173. Forestry. Forest fires are the greatest hazard to the conserva-
tion of forests. Since the possibility of fires starting and the rate at
which a fire will spread are dependent upon atmospheric conditions,
those in charge of forest conservation are vitally interested in various
weather elements and forecasts of these elements.

Six factors are significant when assessing the danger of fires starting
and spreading:

(a) The season of the year.

(6) The wind velocity.

(d) The fuel moisture content.

(e) The visibility.

(/) The recent occurrence of lightning.

Although the visibility does not affect the starting or the actual
spreading of the fires, it can hinder lookout men from observing a fire
readily, thus giving the fire an opportunity to break away. Two
factors, season and fuel moisture content, are not weather elements,
although the moisture in the underbrush is closely related to the past
weather of the region. Gisborne has combined these factors in such a
manner that the relative danger of fire in the northern Rocky Mountain
region of Montana is assessed. The computation is carried out by
means of two meters, the burning index meter and the fire danger meter.
The two sides of the burning index meter are shown in Figs. 187 and 188.
By sliding the movable card, the relative humidity fqr the day is placed

Sec. 173}

FORESTRY

437

on a line with the current date. As illustrated in Fig. 187, it is adjusted
for a relative humidity of 20 per cent on June 25. Then, reversing the
card, one reads off the burning index corresponding to the current wind
velocity and fuel moisture. As shown in Fig. 188, for 8 per cent fuel
moisture and a wind velocity of 10 mph, the burning index is 40. With
the fire danger meter (not shown) this value is adjusted slightly for
visibility and recent lightning. The forest executive uses this fire

FOREST

A BL .
NtlMBERot MEN

$5O /SO &=
/I/// force

trte +
r&ew's
w/'+rt /

pro6at
cfortget

$ .

% 70

-t*7

<H 70
* 60 '

^ 60

I so'

t23

\ .

* 30'

& 10-

j
:

n mm

agr-*

? /<

? 2

3(
-IRE

4
04

5
QNC

6
>ER

7
CL

o e

o *
s

112

FIG. 189. The dependence of organization size on fire danger. (After Gisborne.)

danger rating in conjunction with the chart shown in Fig. 189 to deter-
mine the number of men required for a proper protection of the district.
In the example only 15 per cent of his organization is necessary to be on
guard against fires.

It must be emphasized that the weights given to the several meteoro-
logical elements will vary from one region to another. Thus the burning
index meter shown in Figs. 187 and 188 may be used only in the northern
Rocky Mountain area. The use of a meter of this type is permissible in
another region only after investigations of the relative influence of the
several factors in that region have been made and an appropriate meter
constructed.

438 METEOROLOGY IN HUMAN ACTIVITIES [Chap. 25

A somewhat similar method of determining the forest fire danger for
various parts of Canada has been developed by Wright and Beall. This
system is based on measurements of rainfall, evaporation, relative
humidity, and wind velocity. On the basis of statistical analyses of
these elements, tables of a fire hazard index for various periods of the fire
danger season and for various types of timber stands have been con-
structed. In applying this method to assess the fire danger, daily values
of these four elements are plotted on a fire hazard chart, and with the aid
of the tables the current hazard index is determined.

The forest ranger not only uses the meteorological elements them-
selves, but he may also use forecasts of these. The forecast values of
these weather elements may be used in place of the current values in
assessing the expected fire danger. In this regard, the forest rangers
have found it necessary to adjust the wind speeds as forecast on the
Beaufort scale to allow for the decrease of wind in the forest where their
own observations are made.

Forest executives use the forecasts also in planning for the fighting
of fires. The defense that they plan must be adjusted to allow for any
increase in wind speed or any change in wind direction. For instance,
they may be able to use even a brief shift in the wind, if warned of its
coming, to prepare a back-fire to stop the main conflagration.

174. Heavy Industry. There are a number of ways in which weather
forecasts may be used to facilitate the operation of heavy industries.
In this section, however, attention will bo focused on one particular
phase to which meteorological principles have been applied.

One of the problems arising from the operation of heavy industry is
the atmospheric pollution which frequently results from the various proc-
esses employed in the plant operations. Sulphur dioxide, for instance,
is frequently produced, and if released to the atmosphere, may damage
vegetation in the vicinity. Damage to crops in the state of Washington
occurred, for example, through the action of sulphur dioxide fumes
emitted by the smelter of the Consolidated Mining and Smelting Com-
pany of Canada, Limited, located in the Columbia River valley at
Trail, British Columbia, which is about seven miles north of the inter-
national boundary. The fixing of compensation for injury involved a
good deal of international litigation until, under the authority of an
Arbitral Tribunal set up by the governments of the United States and
Canada in 1935, a control regime based largely on meteorological criteria
was established in 1941.

The development of this regime was possible only after a compre-
hensive investigation of meteorological conditions in the Columbia River
valley near the international boundary, some of the results of which are

Sec. 174] HEAVY INDUSTRY 439

given in section 124. The regime adopted is based largely on .the wind
direction and speed and on the atmospheric turbulence in the valley.
The importance of the former in preventing the encroachment of smelter
fumes on the territory of the state of Washington will be obvious. It is
shown in section 54 that matter is diffused throughout the atmosphere
by turbulence. Thus equation 54*11 gives the rate of diffusion of an
entity x, in this case sulphur dioxide expressed as the number of parts of
sulphur dioxide per million parts of air. Owing to the difficulty of
obtaining mean values of the coefficient of eddy diffusivity for a period
of half an hour (the minimum period required for the method given in
section 55 is 24 h), a direct method of measuring the turbulence was
developed.

For this purpose a special instrument called a bridled-cup turbulence
integrator was designed by G. C. Gill. This consists essentially of an
anemometer with 22 cups. The vertical shaft is bridled to prevent con-
tinuous rotation such as occurs in the ordinary cup anemometer described
in section 66, and is so constrained that its angular displacement from
the zero position for no wind is directly proportional to the wind velocity.
When the air motion is turbulent the angular displacement of the ane-
mometer increases as the gusts grow in intensity and decreases as the
gusts die away. Each change of the velocity by 2 mph is recorded by
electrical means on a rotating drum. The degree of gustiness or turbu-
lence during a half-hour period, for example, is then obtained by count-
ing the number of deflections recorded during that period. The number
of deflections per half hour varies from zero when the atmosphere is very
stable, as during the early morning hours, to 2000 or 3000 during a hot
summer afternoon. The diffusion of the gas is closely related to the
turbulence determined in this manner.

The maximum permissible sulphur emission depends on the time of the
year since the possibility of damage is, of course, much greater during
the growing than during the non-growing season. Allowance must also
be made for the fact that crops are more susceptible to injury by sulphur
dioxide during the day when exposed to sunlight than at night. The
period of maximum danger is just after sunrise, which also coincides with
the time when the concentrations of sulphur dioxide tend to be greatest.

The maximum sulphur emission in tons of sulphur per hour (one ton
of sulphur corresponds very nearly to two tons of sulphur dioxide) per-
mitted by the control regime laid down by the Tribunal is given in the
table. Favorable winds are those from the north, east, south, and
southwest, and intermediate directions, provided that they have a speed
of 5 mph or more and have persisted for 30 min at the point of observa-
tion. All other winds are unfavorable. Most of the other provisions

440

METEOROLOGY IN HUMAN ACTIVITIES

[Chap. 25

of the regime are non-meteorological in character and will not be dis-
cussed here.

MAXIMUM PERMISSIBLE SULPHUR EMISSION
(Tons of sulphur per hour)

Time of Day

Season

Turbulence (Deflections per Half Hour)

0-74

75-149

150-349

350 and above

Wind

Un-
fav.

Fav.

Un-

fav.

Fav.

Un-
fav.

Fav.

Either
unfav. or fav.

Midnight to 3 A.M.

Growing
Non-growing

2
2

6
8

6
6

9
11

9
9

11
11

3 A.M. to
3 h after sunrise

Growing
Non-growing

4
4

4
6

4
4

6
6

3 h after sunrise to 3 h
before sunset

Growing
Non-growing

2
2

6
8

6
6

9
11

9
9

11
11

3 h before sunset to
sunset

Growing
Non-growing

2
2

5
5

7
9

7
7

9
9

Sunset to midnight

Growing
Non-growing

3
3

7
9

6
6

9
11

9
9

11
11

Few heavy industries have the flexibility of operation necessary to
follow such a regime while maintaining full production. At Trail the
plant installations were enlarged to permit such flexibility. The por-
tion of the sulphur dioxide which cannot be discharged to the atmosphere
during periods of poor diffusion is used in the manufacture of elemental
sulphur and sulphuric acid.

Fig. 190 is a photograph of the smelter at Trail, taken from a point
about a mile downstream from the smelter at 21 h on June 9, 1940.
Although the sun had set, the smoke can still be seen above the stacks.
The diffusion of the smoke at this time was slow. The bridled-cup
turbulence integrator showed 65 deflections per half hour during the
preceding hour and 10 deflections per half hour during the following hour.
The wind at the stack was south 2 mph. The rate of emission of sulphur
dioxide was low, having been 2.5 tons of sulphur per hour from 17 to
19 h; it was then increased to 3.5 tons per hour at 19 h and to 4.5 tons
per hour at 21 h. The latter rate of emission is slightly greater than

Sec. 174]

HEAVY INDUSTRY

441

442 METEOROLOGY IN HUMAN ACTIVITIES [Chap. 25

the maximum permitted at a later date by the ruling of the Tribunal.
According to the table the maximum from sunset to midnight during the
growing season when the turbulence is low and the wind unfavorable is
3 tons of sulphur per hour.

The above regime is not based on forecast data but on current obser-
vations. The difficulties of forecasting winds, and especially light winds,
and also turbulence in a valley located in such mountainous terrain are
so great that it has not yet been possible to base a control regime on fore-
cast weather developments. It should be possible, however, to prepare
adequate forecasts of wind and turbulence for industries located where
local topography docs not introduce so many complications. Even near
industrial areas which are so located that international complications will
not arise, the control of emission of gases on the basis of wind direction
may be helpful in preventing the fumigation of crops and centers of
population lying in various directions with respect to the offending plant.

175. Hydrology. Hydrology and meteorology are closely related
sciences, since they arc concerned with two separate but complementary
phases of the same cycle. The study of the processes by which water
vapor in the atmosphere condenses, the resulting water droplets form
clouds, and then fall to the ground as rain is basic in meteorology. The
hydrologist takes up the water cycle at the point where the meteorologist
stops and investigates what happens to precipitation before it evaporates
and returns to the field of meteorology. The hydrologist's studies of
the flow of water over and through the ground and in natural channels
have numerous applications in the fields of irrigation, water supply,
hydroelectric power, soil conservation, flood control, and river forecast-
ing. The relationship between river or flood forecasting and meteorol-
ogy is so close that it will be considered briefly here.

The river forecaster makes use of observations of a number of the
meteorological elements such as precipitation, temperature, wind,
evaporation, and humidity. Of these precipitation is the most import-
ant since it is used directly in determining the run-off, which is the por-
tion of the precipitation which reaches the stream channels. The
remainder of the precipitation which is not accounted for by the run-off
is known as the loss and is determined by subtracting observed run-off
from average rainfall. The loss occurs in water which is retained in the
top soil and in that which percolates to lower layers of the soil; part of
the loss is water which is intercepted by leaves of trees, etc., and the
remainder is precipitation caught in ditches, puddles, etc. The run-off
occurs either at the surface or by means of subterranean flow.

The problem of flood forecasting may be considered as made up of
two parts. The first part is the prediction of the,, rate of outflow of

Sec. 176}

HYDROLOGY

443

I 2 3
Run-off (in)

water from the headwater basins within the river district. The second
is the problem of translating the flow from the headwaters to the lower
limits of the river districts.

In headwater forecasting it is necessary to predict both the probable
volume of run-off and the time distribution of this run-off at the out-
flow station. One method of estimating
the former is by the use of a chart of the
type shown in Fig. 191 for a given basin.
The run-off for any given value of the
average rainfall is read off directly for any
type of antecedent conditions. By the lat-
ter is meant the combined effect of rain,
snow, temperature, humidity, wind, and
other elements upon the run-off character-
istics of a basin during the period just be-
fore a storm. In practice it is impossible
to compute the effects of all these, and
the flood forecaster estimates the antece-
dent conditions as very dry, dry, etc. Thus
when the average rainfall for the basin to FlG . 191 . The re i a tion be-
which Fig. 191 applies is 6 in., the run-off tween precipitation and run-off
will be 4 in. when the antecedent condi- f r varying antecedent condi-
tions are very wet, but only 1.5 in. if they tions - ( From Linsley, River
j rm a c j.u 4. Forecasting Methods. U. S.

are very dry. Ihe flow of the stream at w j ther ^^

the beginning of the rain may also be

used to determine the antecedent conditions. The net peak flow
resulting from a storm of specified duration is a function of run-off
and is determined in a similar manner.

The rate of outflow at one station along the river is determined by
the inflow at the next station upstream, plus the measured inflow from
tributary streams along the reach of the river between the two stations,,
plus the unmeasured local inflow from other sources along the reach.
The process of translating the flow from the headwaters to the lower
reaches of the stream consists of combining the inflow from all three
sources and distributing the total with respect to time at the downstream
station. The most commonly used method of predicting the outflow
at a lower station is based on a study of crest stages (heights) of numer-
ous floods at two stations. Crest stages at the upstream station are
plotted against corresponding crests at the downstream station, and a
curve of best fit is drawn through the points. The flood forecaster,
knowing the actual or predicted crest upstream, may then estimate the
crest downstream. % The time required for the crest to traverse any given

444 METEOROLOGY IN HUMAN ACTIVITIES [Chap. 25

reach, in addition to any backwater effects, must also be estimated.

The relationship between snow cover and the occurrence of floods
varies. If a thin layer of snow melts during a severe storm, the flow of
water may be augmented. On the other hand, a heavy blanket of snow
may absorb a large proportion of the rainfall and thus reduce the
flood hazard.

If snow or ice is to melt, the necessary latent heat of fusion, 80 cal per
gm, must be supplied by some external source. The two most effective
sources are the condensation of water vapor on the snow surface and
solar radiation. For example, if there is an inversion in saturated air
lying just above a snow surface at the freezing point, a vertical vapor
pressure gradient will be present. Thus, if the temperature at 2 m
above the surface is 4 C, the saturation vapor pressure at that level is
8.13 mb while that at the surface is 6.11 mb; the vapor pressure gradient
is then 1 mb per m, and rapid condensation on the snow surface occurs.
Since the latent heat of condensation, 595 cal per gm at C, is about
7.5 times greater than the latent heat of fusion of ice and must be taken
up by the ice, melting it, the condensation of 1 gm of water vapor
results in the production of 8.5 gm of liquid water at the surface. Rapid
melting occurs when the supply of saturated air is maintained either
by advection or by turbulent mixing with moist air at higher levels and
may be sufficient to increase the flood hazard.

The effect of solar radiation may also be estimated. The albedo of
fresh clean snow is high, about 0.8 as indicated in section 28. If the
solar radiation falling on a snow surface is 0.8 cal cm~~ 2 min" 1 , only
20 per cent of this, 0.16 cal cm"" 2 min^ 1 , will be absorbed, the remainder
being reflected. Under such conditions 1 gm of ice will be melted in
8 h if its temperature is initially C. If the snow is dirty, its albedo
is lower and less time is required to melt 1 gm.

The melting effect of warm air is not great since the specific heat of
air is small, 0.24 cal gm" 1 deg"" 1 . Thus the height of the column of air
of 1 sq cm cross section which must cool from 10 to C in order to
provide sufficient heat to melt 1 gm of ice is approximately 260 m.

The estimation of the daily run-off resulting from the melting of snow
requires complete data on wind, temperature, humidity, and cloudiness
from representative stations within the basin. Empirical relations
based on the correlation between observed run-off and certain combi-
nations of meteorological factors during past years may also be used. A
prediction of the seasonal run-off requires a somewhat different tech-
nique. The simplest procedure is to measure the deviation from
normal of the water content of the snow field at the beginning of the
season and then to forecast a corresponding deviatioR in the run-off.

Sec. 176] PUBLIC UTILITIES 445

At present, flood forecasting is done almost entirely on the basis of
contemporary observations, including those of some of the weather
elements. The warning of approaching flood conditions could be pro-
vided much earlier if forecast quantities were substituted in the equa-
tions and charts rather than observed ones. For example, since the
rainfall in frontal depressions is closely related to the potential insta-
bility in warm sector air as shown in sections 60, 61 , and 129, it should be
possible to devise a chart similar to that shown in Fig. 191, but with the
amount of decrease of wet-bulb potential temperature with height, i.e.,
the degree of potential instability, as ordinate instead of average rain-
fall. In this manner the probable run-off could be predicted even before
the precipitation started. Other forecast procedures might be similarly
adopted.

176. Public Utilities. Companies which supply power to the general
public can make use of several features of the weather forecast. For
instance they, along with telephone companies, telegraph companies, etc.,
have wires strung from pole to pole throughout the continent. These
wires are sometimes damaged by lightning and by sleet storms. Light-
ning with its excessive voltage may injure the wire as it uses it while
seeking a path to the ground, or it may enter through the power lines
into one of the buildings and injure some of the equipment there. The
power companies prepare for lightning before it arrives by installing
protectors about its lines and its buildings. Yet the possible damage to
the lines makes it desirable for those who control the power distribution
to know the regions in which storms are occurring. They are able to
do more to protect their lines during a sleet storm than they can during a
lightning storm. The freezing rain and drizzle cause a ring of ice to
form about the lines. This becomes so heavy in severe storms that it
breaks the lines, particularly if strong winds accompany the storm.
By routing extra power through the lines the engineers in charge are able
to heat the wires sufficiently to prevent the layer of ice from forming.
Being warned of the approach of such a storm a few hours in advance,
they can plan their distribution to allow for this extra load on some
of their lines. Thus the maintenance divisions of such public utilities
are interested in advance knowledge of the occurrence of these
storms.

With an accurate forecast of the month's rainfall at the beginning
of the month, the hydroelectric power companies could plan the use of
the available power more efficiently. Without it, they have to keep
water in reserve to allow for possible dry periods. Although the short
term forecasts do not provide them with all the desired data, even the
advance informatics that these forecasts can give about the amount of

446 METEOROLOGY IN HUMAN ACTIVITIES [Chap. 25

rainfall during the next few days is a help to them in determining the
amount of water that they may use with safety.

The power companies with the aid of the forecast offices can help in
decreasing the damage caused by floods. These companies always
have a reserve supply of water dammed up in their lakes along the rivers.
When a big storm is expected, they may release beforehand a part of
this supply. Thus the run-off of the storm must first fill the partially
emptied basins before the overflow starts down the river as a flood. In
this manner the power companies can spread the abnormal supply of
water over a longer period of time and lower the crest of the flood water.
This is possible because of advance information that the forecast office
supplies to the companies.

Companies* which supply gas for heating and lighting can also use fore-
oast information. The demand for gas varies with the temperature, and
to a certain extent with the brightness of the day. It is necessary to
have enough gas on hand to meet the demand. With some companies
the source of supply is a long distance from the city. These will have a
reserve supply to care for slight increases in the requirements, but which
may not be adequate to meet a demand that continues for a long period.
Since an interval of time must elapse between the time an extra supply
is put into the pipes at the source and the time it reaches the city, it is
necessary for the company to foresee the additional load in time to per-
mit the gas to reach the city. By learning from the forecaster the varia-
tion in temperature that is expected, the distribution manager can adjust
his supply of gas to care for the increased demand if the temperature
drops. The demand for electricity varies in a similar manner, but there
is no lag between the supply of more power at the generators and its
arrival at the city. If, though, the demand for more power can be
answered only by the use of one of the auxiliary generators which these
electric power companies usually have in reserve, then an interval of
time elapses between the demand and the provision of the additional
supply. Forewarning in terms of a forecast of temperature changes is
helpful to such companies as well as to those which supply gas.

177. Sports. Suitable weather is necessary for the various types of
outdoor sporting events. A large number are private events, and the
participants can learn from the general forecast whether the weather
will be suitable in time to arrange a postponement if necessary. For
some of the larger tournaments, such as football games, there is no
change in plans, even if unsatisfactory weather is probable. Baseball
games may be postponed because of rain, and so the managers of these
games can make use of short-period forecasts to help decide whether to
change their plans. <

Sec. 177} SPORTS 447

With the increase in popularity of skiing during the past few years,
a new demand has been made on weather forecasters. Skiing trips
usually last for two or three days, but will be spoiled if the weather is
going to be unsatisfactory. The information that skiers desire includes
the type of snow surface, the amount of new snow that is expected and
its type, and the wind velocity. The facts about the present snow sur-
face may be gathered from observers at the ski resorts. Modifications
in the surface will take place under different conditions of cloud and
temperature. In addition, the weather forecaster must adjust his fore-
cast for the different districts and altitudes in the region covered, for
orographic effects will vary over the region, and the temperature will
change with altitude. These will make a difference in the type of the
snow that will fall on the different parts of the district. The U.S.
Weather Bureau has met the demand for this type of forecast in New
England by broadcasting special forecasts regularly during the ski-
ing season.

The first successful gliding in the ascending air stream over a hill slope
was achieved by Lilienthal in Germany during the last two decades of
the nineteenth century. Annual gliding competitions were started over
the Wasserkuppe in the Rhon mountains in Germany in 1920. There,
from 1926 to 1929, the art of soaring flight developed with the discovery
by Nehring, Kronfeld, Hirth, and others that flights to considerable
heights and for great distances could be made by utilizing convection
currents in an unstable air mass or at a cold front. Since that time the
sport of soaring has spread to other countries and the number of experi-
enced sailplane pilots now runs into the thousands.

The sailplane pilot is interested in four aspects of the weather, the
wind speed and direction, the degree of instability of an air mass, the
proximity of cold fronts, and, in certain mountainous localities, the
presence of standing waves.

The wind direction governs to a large extent the direction in which a
flight may be made. Although a sailplane may fly with a cross-wind
component of motion, the greatest distances are usually traveled when
the plane advances in the general direction of the wind. The wind speed
determines in part the distance which can be covered. The most satis-
factory speed is one from 20 to 30 mph. This is great enough to carry
the plane along rapidly in the air stream, but not so great that convection
currents are disrupted and the source of lift made too irregular in its
operation.

The convection currents, or " thermals," as they are called by sail-
plane pilots, are of two types. Marked convection currents, known as
dry thermals, sometimes occur even with a cloudless sky. The rate of

448

METEOROLOGY IN HUMAN ACTIVITIES

[Chap. 26

ascent of air in these may be great enough to carry a sailplane to a height
of 3000 or 4000 ft. The horizontal extent of such thermals is not large,
and the pilot must ascend in spirals of such small radii that his craft
always remains within the region of ascending air. The surface air
must be very dry if heights as great as those mentioned above are to be
attained. Lengthy flights under such conditions are difficult and require
a high degree of skill on the part of the pilot. Soaring is easier when the
positions of thermals are indicated by cumulus clouds topping them.
The experienced pilot may guide his craft into a cloud, and, flying blind
on instruments in tight circles, may gain additional height in this way.
Some sailplanes are very strongly constructed, having a safety factor
greater than that of a fighter plane. With one of these the experienced
pilot may ascend into swelling cumulus or cumulonimbus. The vertical

2 3

Distance (mi)

30 40 50
T(F)

FIG. 192. Typical structure of the nose of a cold front.

currents in such clouds are extreme, sometimes exceeding 60 mph, so
that heights of 4 or 5 mi have been reached in a comparatively short
period of time. Such flights may be dangerous, however, owing to the
likelihood of severe icing, and the possibility of the plane becoming
damaged or getting out of control. One pilot, for example, ascended to
16,000 ft in a cumulonimbus cloud, at which height his sailplane was
torn to pieces by the violent air motion. Supported by his parachute the
pilot was carried through four vertical circuits in the cloud before reach-
ing the earth.

It has been found that the ascending currents just ahead of the nose
of a well-marked cold front are sufficiently strong to carry a sailplane up
to considerable heights. The structure of a typical marked cold front,
based on the reports of sailplane pilots, is shown in Fig. 192. Several
features call for comment. At about 2000 ft a low revolving cylindrical-
shaped cloud advances and marks the location of \yhat is often known

Sec. 177] SPORTS 449

as the line squall. This type of low cloud at a cold front is well known
to sailplane pilots, and is sometimes referred to by them as an " air-
roller." Extending upward and about 2 or 3 mi ahead of the front at the
surface is the nose of the system. The air just below this nose is ex-
tremely unstable owing to the very high lapse rate in the frontal zone
there. This instability is shown by the lapse rate (see the right-hand
section of the diagram) in the two air masses at the position indicated
by the vertical broken line at 3 mi. By referring to the criteria given
in Chapter 14, it is seen that the warm air in the lower frontal zone and
that just underneath it is absolutely unstable. The lapse rate in the
frontal zone may be twice as great as the dry adiabatic, and since the
air is saturated the resulting violent ascent will lead to the development
of swelling cumulus or cumulonimbus cloud. The inversion at the
upper frontal zone will sometimes prevent the vertical extension of the
cloud above that level. This system is unstable, the cold nose breaking
down, then reforming, breaking down again, and so on. The phenome-
non as a whole is to be accounted for by the retardation of the air flow in
the lower layers by surface friction.

As the line squall approaches, the sailplane is launched, and, keeping
just ahead of the cold nose, the pilot guides his plane in the ascending air.
The vertical currents are usually strong enough to permit him to ascend
to the tip of the nose, at about 6000 ft, where the vertical velocity of the
air is of the order of 2 ft per sec, just sufficient to permit a well-designed
sailplane to fly horizontally. Having attained this altitude it is often
possible to fly for long distances parallel to the front. A difficulty
arises when a portion of the front where the cold nose has broken down
is reached. It is then necessary to glide across the ruptured portion of
the front until a section is reached where the nose is well developed, and
ascending currents are again in evidence.

Under certain conditions standing waves occur in the air on the lee of
a range of hills or mountains. The rate of ascent of the air in parts of
these systems is sometimes high enough to carry a sailplane up to great
heights. Such currents were first utilized for soaring in the Sudetes
Mountains of southeastern Germany, where the stationary bank of
cirrus which marks the upper portion of the wave system is known as the
Moazagotl.

These standing waves frequently develop when air, after crossing
a mountain ridge, flows down the lee slope to give the warm Eohn or
Chinook winds. The structure of the wave system is depicted in
Fig. 193. If the amplitude of the waves near the surface is large,
large-scale eddies form. These eddies are known as " rotors " and are
sometimes capped by cumuloform clouds. At the cirrus level the sta-

450

METEOROLOGY IN HUMAN ACTIVITIES

[Chap. 25

tionary Moazagotl occurs. The plus signs in the diagram indicate the
regions of most rapidly ascending motion. The Moazagotl reaches its
maximum development in the colder months, and especially in the
autumn. It usually occurs when the air is conditionally unstable and
the wind speed exceeds certain critical values.

30000

_25000

-20000

1)15000

= 10000

5000

10 15

Distance (mi)

6900m /600m\
>j(419m)>r (321mb) V

/ I /*4m/sec

fJ 2700m
/ (724mb)

j,$tart

340m (967mb) |25

Fia. 193. The structure of standing waves on the lee of a range of mountains.

(After Kuttner.)

The world's altitude record for sailplanes was broken in November,
1938, by Ziller, who reached a height of over 28,000 ft flying in favorable
Moazagotl conditions. The baro-
gram for the flight is shown in Fig.
194.

Similar systems of standing waves
are known to occur in other parts
of the world, including North Amer-
ica, although these have not yet
been utilized for soaring. For in-
stance, a preliminary investigation
by W. L. Godson indicates that the
cloud formation known as the "Chi-
nook arch " which occurs in south-
ern Alberta just to the east of the
Rockies is of the Moazagotl type. Studies of standing waves in other
localities are necessary before the salient meteorological conditions ac-
companying the development of such systems can be specified.

It will be obvious from the foregoing that forecasts of wind direction
and speed, stability conditions, and the approach of cold fronts will aid
the sailplane pilot in planning his flights. <

FIG. 194. The barogram obtained
during the record-breaking sailplane
flight by Ziller (November 21, 1938) in a
system of standing waves near the
Sudetes Mountains.

Sec. 178] RETAIL MERCHANDISE 451

178. Retail Merchandise. In several ways the retail trade can
advantageously consider the expected weather in making its plans.
First, it can adjust its advertising to fit in with the current weather.
Advertising has as one of its purposes the creating of a demand for an
article. If the weather helps to suggest the use of the article, part of the
task of the advertising is done. If, on the other hand, the weather is
such that the article will not be useful for the next few days, then the
advertising is wasted on an unresponsive public. Thus bathing suits
should be advertised during the first warm spell of spring, and an adver-
tisement for rubber clothing finds a ready response on a rainy day.
Since sufficiently accurate forecasts for this purpose are not available
for more than a few days in advance, it is necessary for the advertising
staff to reserve final decision on the advertising to be run until the fore-
cast is available.

The retail stores must also plan their stocks to answer the demand
which the changes in the weather will make. For example, the supply
of rubbers, etc., must be readily available on wet days, when there will
be a big demand. There are many other examples of such spasmodic
demands created by the current weather. Coal dealers find that a cold
spell will be accompanied by a large number of orders for coal. Simi-
larly, a hot week end will cause a run on the supply of ice cream and soft
drinks. These demands can be anticipated by the retail trade by mak-
ing use of the forecast facilities of the weather office.

A satisfactory forecasting service is of value to the management of a
large department store in making the most efficient use of the help.
The number of customers varies with the weather. With inclement
weather those departments selling garments, such as raincoats, to pro-
tect the public from the weather are busy, but other departments are
slack. Also on a fine day following a stormy period the business is brisk.
Managers have several methods of making adjustments for these fluctua-
tions. They usually have occasional help whom they can call in during
rush periods. Also in the store the staff may be shifted to allow for
variations in business, with the stock room being able to absorb a num-
ber during slack hours. Although some rearrangements can be made on
short notice, the management can supervise the arrangements more
efficiently if it has the warning that a weather forecast gives. A fore-
cast for at least one day in advance is desirable, but even a few hours'
notice will at times assist the management in revising its plans. The
general public forecast satisfies the needs of the management to a certain
extent. Nevertheless the weather service, if it is to serve the trade or
other types of business well, must know to what use the forecasts are
put and what information is of value. Only thus can the forecaster

452 METEOROLOGY IN HUMAN ACTIVITIES [Chap. 25

give satisfactory service to those who wish to know what the weather is
going to be. .

BIBLIOGRAPHY

Pagliuca, S., " Some Aspects of Industrial Weather Forecasting," Bui. Am. Met. Soc.,
20, 287-293 (1939).

172. Smith, J. W., " Predicting Minimum Temperatures from Hygrometeric Data,"
Monthly Weather Review, Supplement No. 16, 1920.

173. Gisborne. H. T., Measuring Fire Weather and Forest Inflammability, U.S. De-
partment of Agriculture Circular 398, Washington, D. C., 1936.

173. Wright, J. G., and H. W. Beall, Preliminary Improved Forest-Fire Hazard
Tables for Eastern Canada, Department of Mines and Resources, Forest Fire
Research Note 5, Ottawa, Ont., 1940.

174. Trail Smelter Arbitration between the United States and Canada under convention
of April 15, 193$. Decision of the Tribunal reported March 11, 1941. Wash-
ington, D.C., U.S. Government Printing Office, Department of State Publica-
tion 1649, Arbitration Series 8, 1941.

174. Dean, R. S., and R. E. Swain, Report Submitted to the Trail Smelter Arbitral
Tribunal, U. S. Bureau of Mines, Bulletin 453, Washington, D. C., 1943.

175. Linsley, R. K., Jr., River Forecasting Methods, U.S. Department of Commerce,
Weather Bureau, 1942.

176. Gilkeson, W.R., " The Use of Weather Forecasts in Electric Power Operation,"
Bui. Am. Met. Soc., 20, 338-340 (1939).

176. Smith, C. P., " The Use of Meteorology in Natural Gas Dispatching," Bui.
Am. Met. Soc., 20, 390-395 (1939).

177. Barringer, L. B., Flight without Power, New York, Pitman Publishing Corpora-
tion, 1940.

177. Hirth, W., The Art of Soaring Flight, Translated by Naomi Heron-Maxwell.
Stuttgart, Hirth, 1938.

APPENDIX

TABLE I

HEIGHTS IN DYNAMIC METERS CORRESPONDING TO GIVEN HEIGHTS IN GEOMETRIC

METERS

Latitude

Height

1000

978.1

978.2

978.7

979.3

980.2

981.1

981.9

982.6

983.0

2000

1,956

1,957

1,959

1,960

1,962

1,964

1,965

1,966

3000

2,934

2,935

2,936

2,938

2,941

2,943

2,946

2,948

2,949

4000

3,910

3,912

3,915

3,918

3,922

3,925

3,928

3,930

5000

4,886

4,887

4,889

4,893

4,897

4,901

4,905

4,909

4,911

6000

5,863

5,866

5,870

5,875

5,880

5,885

5,889

5,892

7000

6,839

6,843

6,847

6,853

6,860

6,865

6,870

6,873

8000

7,815

7,816

7,820

7,825

7,832

7,838

7,845

7,850

7,854

9000

8,791

8,792

8,796

8,802

8,810

8,818

8,825

8,831

8,835

10000

9,766

9,767

9,772

9,778

9,787

9,796

9,804

9,811

9,815

453

454 APPENDIX

TABLE II
SATURATION VAPOR PRESSURE OVER WATER (MB)

T(A)

320

106.1

111.6

117.4

123.4 129.6

136.1

142.9

150.0

157.4

165.1

310

62.76

66.26

69.92

73.77

77.79

82.00

86.40

91.01

95.84

100.9

300

35.65

37.80

40.06

42.43

44.93

47.55

50.31

53.20

56.23

59.42

290

19.38

20.64

21.97

23.38

24.87

26.44

28.09

29.84

31.68

33.61

280

10.02

10.73

11.48

12.28

13.13

14.02

14.98

15.98

17.05

18.18

270

4.90

5.28

5.68

6.11

6.57

7.06

7.58

8.13

8.72

9.35

260

2.26

2.44

2.65

2.87

3.10

3.35

3.62

3.91

4.22

4.55

250

0.967

1.06

1.15

1.26

1.37

1.49

1.62

1.76

1.92

2.08

240

0.383

0.421

0.464

0.510

0.560

0.615

0.674

0.739

0.809

0.885

230

0.138

0.153

0.170

0.189

0.210

0.233

0.257

0.285

0.314

0.347

220

0.063

0.071

0.080

0.089

0.099

0.111

0.124

SATURATION VAPOR

PRESSURE OVER ICE

(MB)

T(A)

270

4.76

5.17

5.62

6.11

260

1.99

2.18

2.38

2.60

2.84

3.10

3.38

3.69

4.02

4.37

250

0.773

0.853

0.940

1.035

1.14

1.25

1.38

1.51

1.66

1.82

240

0.278

0.309

0.343

0.381

0.423

0.468

0.519

0.573

0.635

0.701

230

0.091

0.102

0.115

0.129

0.144

0.160

0.180

0.201

0.224

0.250

220 0.039 0.044 0.050 0.057 0.064 0.072 0.081

Data from: L. Holborn, K. Scheel, and F. Henning, TFarmeta6eZ/erc,Braunschweig,
Vieweg and Sohn, 1919; L. P. Harrison, Monthly Weather Review, 62, 247-248 (1934);
E. W. Washburn, Monthly Weather Review, 52, 488-490 (1924).

APPENDIX 455

TABLE III
VALUES OF CONSTANTS

a = 2897 micron deg. m = 28.97

A = 2.39 X 10~ 8 cal erg- 1 m' = 18.016

c p = 0.239 cal gm" 1 deg- 1 R = 2.87 X 10 6 erg deg -1

c v = 0.171 cal gm- 1 deg- 1 R* = 83.14 X 10 6 erg deg - 1

c r p = 0.465 cal gm- 1 deg- 1 R' = 4.62 X 10 6 erg deg- 1

g = 981 era sec" 2 <r = 8.14 X 10~ n cal cm" 2 mitf" 1 deg' 4

o> = 7.29 X 10~ 6 sec- 1

J = 4.185 X 10 7 erg cal' 1 T = 1.402

= 0.622

Polar radius of the earth = 6,356 km
Equatorial radius of the earth = 6,378 km
1 ft = 0.3048 m 1 m = 3.281 ft
1 mi = 1.609 km 1 km = 0.6214 mi
1 m per sec = 2.237 mph

e = 2.718 logio e = 0.4343

456

APPENDIX

ALTERNATIVE COMPUTATION OF CORRELATION COEFFICIENTS

When the number of paired variables to be correlated is large, the method of
computation outlined in section 61 becomes laborious. If, however, the data are
grouped according to class intervals and an assumed mean is used, the correlation
coefficient may be calculated much more rapidly. The modified form of equation
61.3 to be used is

where the meanings of the symbols are the same as in Chapter 10.

Each of the two sets of variables is analyzed in the manner shown in the table of
section 57. A convenient arrangement for doing this is shown below, the data used
being those given in the table of section 60.

^\tf
*^\

0-4

5-9

10-14

15-19

20-24

25-29

d' x

fxd'x

d'xd'n

3.75-4.25

3.0 -3.5

2.25-2.75

1.5 -2.0

0.75-1.25

-1

-6

0.0 -0.5

-2

-4

25
25

-1

-11

-1

f*d%

d'nd'x

r =

N -| = - 36

-t

~25

-0.04

- (-0.04) 2 = 1.28

& - NC'XCR 31 - 25(0.36 X -0.04)
25(1.44X1.28)

<0.68

APPENDIX 457

The evaluation of the d f x d' R term for d r x = 3 (top row of the chart) is carried out
in the following manner.

3[1(-1) + 1(2) + 1(4)] - 15

and similarly for the remaining class intervals. Since 2fx = 2/# = N and

2d' R d' Xy a useful check on the accuracy of the arithmetic is obtained by evaluating

these expressions for both sets of variables.

The value 0.68 obtained by this method is less accurate than the value 0.72 com-
puted in section 61. If narrower class intervals are used, however, the method will
give a correlation coefficient much closer to the true value. The above system of
calculation is especially useful when the number of paired variables is large.

ANSWERS TO PROBLEMS AND EXERCISES
Chap. 2. 1. (6) 34.2 C per km. 2. +200 mb. 3. (a) 5.4 km (6) 5.7 km.
Chap. 3. 1. 110 mb.
Chap. 5. 1. 5780 A. 2. 247 A. 3. 0.4 C. 4. 0.09 C.

Chap. 6. 1. (a) Speed 10 m per sec in a circular path of radius 69.7 km. (6) Speed
10 m per sec in a circular path of radius 107 km. (c) Speed 10 m per sec eastward
along the equator.

Chap. 7. 1. 160.5. 2.

Chap. 8. 1. Negative. 2. 0.35 cm per sec. 3. -2.33 mb per 3 h.

Chap. 9. 2. (a) 1.4 days (6) 5.8 days (c) 13 days. 3. 0.2 gm per kg. 4. 0.032
gm per kg per h. 5. 4.6 X 10~ 4 C.

Chap. 10. 1. New York M = 39.5 0.5 in.; a- = 3.52 =fc 0.34 in.; North
Dakota M = 17.0 =b 0.4 in.; <r = 2.94 0.28 in. 2. If y represents mean tem-
perature, x altitude in hundreds of feet: y = 80.38 0.148z; r = 0.51 0.13.

3. T - 54.93 - 4.06 cos ^- t 4.14 sin ~ t. 4. If d represents a unit distance

12 12

7T 7T

and the origin is taken at the rear of a low, H = 11.15 0.86 cos d 0.17 sin 7- d.

3 3

5. 1.4 X 10 3 cm 2 sec- 1 .

Note. In dealing with problems which are related to the weather map and to
forecasting, extreme accuracy is unnecessary since the calculations are often based
on assumptions which in themselves are not precise, and also only approximate values
are needed in the results.

Chap. 13. 1. 8 C; none; add ^/ of 8 to 27 C.

Chap. 16. 1. 110 mi. 2. 10.50 h; 11.50 h; 13.50 h. 3. 22.50 h; 17.10 h.

4. 100 mi.

Chap. 17. 1. Northeast-southwest; 5^ F per 100 mi. 2. 13m per sec. 3. 9.9m
per sec.

458

INDEX

Absorption, coefficient, generalized, 70,

in a spectrum, 71
law of, 70-72

of long-wave radiation, 76, 78, 79
of solar radiation, 73-74, 218, 219
power of, 67

Acceleration, of a front, 121, 277
of gravity, 17

of significant curves, 117-118
Adiabatic changes on an indicator dia-
gram, 41

Adiabatic processes, definition, 30
effect of, 220, 222-228, 232-235
in dry air, 30-32
in moist air, 34-35
pseudo, 54
Adiabats, dry, 32
Advection, development of instability

by, 245
fog, 332-334

pressure changes through, 131
Aerogram, Refsdal's, 65
Agriculture, relation of meteorology to,

433, 436

Air, adiabatic relationships for, 30-35
density of moist, 21
effect of ascent of, 36-39, 220, 224-227,

231-234, 242
entropy of, 45, 66
equation of state for dry, 15-16
gas constant for, 16, 23
molecular weight of, 16
specific heat of, 28-30, 34
stability of, 32-33, 52, 236-247
upper, composition of, 11
conditions above cyclones and anti-
cyclones, 281-282, 285-287
data, use of, 403-405, 418
pressure distribution in, 9, 132-134,

214

temperature of, 4
Air mass thunderstorms, 252, 258, 259,

260, 378

Air masses, 248-262
classification of, 215, 248
comparison of North American, 262

Air masses, cooling by turbulence, 153-
154

curves of typical, on a Rossby dia-
gram, 261

definition, 215

equatorial, 248

identification of, 401-403

polar, 215, 249-254, 257-259, 262

production of, 215-216

source regions of, 215

Superior, 255, 256

tropical, 215, 248, 254-257, 259-260,

262

Air-roller, 449
Airways sequences, 397

use of, 402-403, 425-427
Aitken, J., 313
Aitken counter, 313
Albedo, 75
Aleutian low, 214
Altimeter, 189
Analysis, harmonic, 179-183

isentropic, 405-408
Anemometers, cup, 194

Dines, 195-196
Angstrom, A., 77
Angstrom compensating pyrheliometer,

207
Antarctic, air temperatures in, 250

high-pressure cell, 213
Anticyclone, Antarctic, 213

cold, 287-288

continental, 214, 249

convergence and divergence in, 289-
293

deepening and filling of, 413

maximum pressure gradient about, 103

polar, 8, 212

pressure gradient in, 290

subsidence in, 39

sub-tropical, 8, 211
effect of, 254

upper air conditions above, 281, 285-
287

velocity of, 120, 408-414

warm, 288-289

winds about, 103

459

460

INDEX

Arctic air masses, 248

source region of, 215
Arctic sea smoke, 323
Arithmetic mean, 163-166

standard error of, 172
Ascent of air, effect of, 36-39, 220, 224-

227, 231-234, 242
Asklof, S., 327

Atlantic Ocean, effect of, on air masses,
251, 253

fog over north, 383
Atmosphere, composition of, 11

loss or gain of heat of, 83-84

pollution of, by industry, 438-442
Austausch coefficient, 147

Barograph, 190
Barometer, aneroid, 189

corrections for, 187-189

Fortin, 187

Kew, 187

Baur, F., 83, 84, 428
Beall, H. W., 438
Beaufort, Sir F., 193
Beaufort scale of wind velocities, 194
Beer's law, 70
Bergeron, T., 314
Bilham, E. G., 339, 377
Bjerknes, J., 133, 272
Bjerknes, V., 24, 139, 272
Black-body radiation, 67
Boyle's law, 15
Brunt, D., 77, 128, 129, 141
British Isles, fog over, 339

thunderstorms over, 377
Buys Ballot's law, 97

Calorie, 28

Campbell-Stokes sunshine recorder, 206,

207
Carbon dioxide, absorption of radiation

by, 76

Carburetor ice, 361
Carnot cycle, 42-43, 48
Carnot's principle, 43
Carslaw, H. S., 154, 157
Centrifugal force, 93-94, 297
Chapman, S., 12
Charles' law, 15
Charts, aerological, pseudo adiabatic, 54

evaluation of work on, 56-57
Rossby diagram, 64, 260-262
tephigram, 54-55
use of, 227-234, 404-405, 418-420
synoptic weather, 396-397
upper air, 54-55

Charts, use of, 227-234, 404-405, 418-420
Chinook arch, 450
Chinook winds, 221
Circulation, 135-141

change with convergence and diver-
gence, 139-140

change with north-south motion, 140

over the earth, 209-216

rate of change of, 136-141

zonal, development of, 209-210
Circulation theorem, Kelvin's, 136-139

V. Bjerknes' variation of, 139, 210
Clausius-Clapeyron equation, 47-49
Climate, factors governing, 381-385

Koppen's classification of, 385-387

regions, arid, 388

cold snowy forest, 393
Mediterranean, 391
polar, 394
tropical rainy, 387
warm temperate rainy, 390
Climatology, 380-395

definition of, 380
Clouds, 343-360

albedo of, 75

altocumulus, 200, 201

altostratus, 200

artificial, 355

banner, 355

billow, 355

cirrocumulus, 200

cirrostratus, 198, 199

cirrus, 198, 199

comparison in North American air
masses, 262

convection, 345-350

convergence, 356-360

cumulonimbus, 203, 204

cumulus, 202-204

development of, 345-350
forecasting of, 345-350
over a water surface, 251
vertical motion in, 241, 320, 448

description of, 197-204

effect in preventing heat loss, 81, 327

effect on solar radiation, 74-75

forecasting of, 414

frontal, 343-345

height of, 198

ice accretion in, 365-366

lenticular, 355

loss of heat by radiation from, 82

Moazagotl, 449-450

names of, 198

nimbostratus, 200

nocturnal radiation with, 80-82, 327

INDEX

461

Clouds, orographic, 355

size of droplets, 314

stratocumulus, 201-202

stratus, 201, 324

turbulence, 350-355

types of, 197-204
Color of the sky, 73
Cols, frontogenesis in, 271
Condensation, 312-314

effect of, 83, 218-219, 224-227, 230-
234

latent heat of, 51
Condensation level, 52-53, 227, 346, 352

convective, 346

lifting, 227, 352

Condensation nuclei, 313-314, 331
Condensation trails, 356
Conservative properties, definition, 221

summary of, 235
Continuity, equation of, 125-126
Contraction, axis of, 267, 270

compounded with dilatation, 269

effect on fronts, 267

field, example of, 268
Convection, clouds, 345-350

currents, 447-449

rain, 317-318

Convergence, causing vertical motion,
127, 290-293, 358

change of circulation with, 139-140

clouds, 356-360

effect on fronts, 266

equation of, 126-128

in cyclones and anticyclones, 289-293

in south-north motion, 127-128

pressure change through, 131

rainfall, 319

vertical velocity through, 121

with upper-level isobars, 133
Coriolis, G., 93
Coriolis force, 90-96, 295

effect of, 211-213
Correlation, coefficient of, 177-179

standard error, 179

use in forecasting, 428
Critical state, 47
Cross section, definition, 404

use of, 404-405, 420
Cycle, Carnot, 42-43, 48
Cyclone, 264-286

convergence and divergence in, 289-
293

deepening and filling of, 413

development of, 272-283

frontogenesis in, 271

pressure gradient Li, 290

Cyclone, upper air conditions above,

281-282, 285-287
velocity of, 120, 289, 40&-414
winds about, 102

Dalton'slaw, 11, 21
Dean, R. S., 441
Deformation field, 112, 269
Density, of dry air, 16

of moist air, 21

of water vapor, 21
Depressions, types of, 282-285
Descent of air, effect of, 36-40, 220,

224-227, 231-234
Dew point, calculation of, 193

changes in, 226-227, 230-231

definition, 226

rate of change with height, 53-54, 352

relation to radiation fog, 329-332
Dexter, R. V., 316
Dilatation, axis of, 267, 270

compounded with contraction, 269

effect on fronts, 267
Dines, W. H., 178, 195, 286, 287
Dines anemometer, 195-196
Divergence, causing vertical motion,
29%293

change of circulation with, 139-140

effect on fronts, 266

equation of, 126

in cyclones and anticyclones, 289-293

pressure change through, 131

vertical velocity through, 127

with upper level isobars, 133
Douglas, C. K. M., 128, 129
Drizzle, formation of, 320
Droplets, size of water, 314
Dynamic meter, 17, 26

Earth, angular velocity of, 17
deflecting force of rotation of, 90-96,

211-213, 295

loss of heat by radiation from, 79, 80
oblateness of, 95
radius of, 16
surface of, loss and gain of heat of,

83-84
Eddy diffusivity, coefficient of, 146, 157-

159

evaluation of, 157
Efficiency of heat engine, 43
Elsasser, W. M., 71, 76, 80
Electrical charges, in thunderclouds, 371-

373

on raindrops, 371
Emissive power, 67

462

INDEX

Energy, changes of, 28, 238

types of, 28
Entropy, 43-46, 66

surfaces of equal, 405-408
Equilibrium, convective, 85

radiative, 85
Error, normal curve of, 170-171

probable, 173

standard, 172, 173, 179

theory of, 169-173

Evaporation, development of instability
by, 243-245

effect of, 83, 218-219, 224-227, 230-
234

fog produced by, 322-324

saturation produced by, 322-324
Eye of a tropical hurricane, 284

Fields of motion, primary, 266
Flood forecasting, 445 ^

Flying conditions, comparison in North

American air masses, 262
Fog, 322-341

advection, 332-334

definition, 322

forecasting, 330-332, 338-340
charts for, 330-331

formation and dissipation of, 322-341

frontal, 324

ground, see Radiation

ice crystal, 337

in polar maritime air, 259

local factors in producing, 338-340

mixing-radiation, 332

off Newfoundland, 256, 259, 333

over British Isles, 339

over north Atlantic, 383

over United States, 340-341

radiation, 326-332

steam, 323

tropical air, 334

upslope, 326
Fohn winds, 221, 449
Force, centrifugal, 93-94, 297

Coriolis, 90-96

deflecting, of earth's rotation, 90-96
effect of, 211-213
vertical component of, 95

frictional, 148

pressure gradient, 88-90, 295
Forecasting, clouds, 414
cumulus, 345-350

deepening and filling of highs and lows,
413

floods, 445

fog, 330-332, 338-340

Forecasting, ice accretion, 368
long-range, 427-429
minimum temperatures, 433
motion of a front, 417, 422, 424
precipitation, 414
thunderstorms, 377
Forestry, relation of meteorology to,

434-438

Fortin barometer, 187
Fourier coefficients, 181
Fourier series, 182
Fourier's theorem, 182
Franklin, B., 370
Friction, coefficient of, 148-150, 303

effect of, 148-149, 211, 302-305
Front, 264-281

acceleration of, 121, 277
changes in weather at, 280, 427
cold, characteristics of, 274
clouds at, 344-345
definition of, 105

development of tornadoes at, 284
instability at, 448
nose of, 345, 448
occlusion type of, 278, 279
secondary, 279
slope of, 106-107, 345
structure of, 448

computation of movement of, 417
cross section of a cold, 448
definition of, 105, 264
development of, 265-272
forecasting motion of, 417, 422, 424
ice accretion at, 367
intertropic, 283
isobars at, 108-110

location of, 401-403, 415-416, 421-427
movement of, 265-271, 300, 403, 408-

414

pressure trough at, 108-110
symbols to indicate, 415
velocity of, 120
warm, characteristics of, 274
clouds at, 343-344
definition, 105

Frontal surface, 105-114, 264
pressure tendency below, 110-112
slope of, 106-107, 264-265, 345
wave on, 273, 274, 417, 418, 422-424
Frontogenesis, 112-114, 265-271

conditions for, 113-114
Frontolysis, 112-114, 265-271
conditions for, 113-114

Gages, float, 205
rain, 205

INDEX

463

Gas constant, for dry air, 16

for moist air, 23

for water vapor, 20

universal, 16
Geopotential, lft-18, 25
George, J. J., 330
Gisbourne, H. T., 434, 435, 436
Gill, G. C., 439
Glaze, formation of, 319
Gliding, 447-450
Godson, W. L., 450
Gradient, potential, 370
Gradient wind, 101-104, 297

causing convergence, 289
Gravity, acceleration of, 17
Gray-body radiation, 68
Great Lakes, effect of, 251, 262
Guldberg, C. M., 148, 150
Gusts, measurement of, 195-196

Hail, formation of, 319-320

over United States, 375-376
Hann, J., 85
Hanzlik, S., 272
Haurwitz, B., 53, 108
Heat, latent, of condensation, 51
of sublimation, 51

loss by radiation, 79-84, 327

mechanical equivalent of, 28

specific, 28

turbulent transfer of, 150-154
Heat balance of the atmosphere, 82-84
Heat engine, 43
Height, computation of, 24

of clouds, 198
Helium in Atmosphere, 11
Helrnholtz, II., 272
Heywood, G. S. P., 180, 184
Hirth, W., 447
Hoar frost, 361
Hoelper, O., 74
Humidity, absolute, 22, 224, 226

measurement of, 192-193

mixing ratio, 22, 225, 325

relative, 22, 192-193, 225, 351

specific, 22, 225
Hurricanes, tropical, 283-285
Hydrology, relation of meteorology to,

442-445
Hygrometer, 192

equation for, 59

Ice accretion on aircraft, 361-368
at fronts, 367
forecasting of, 368
in clouds and rain, 65-367

Ice accretion on aircraft, rules for avoid-
ing, '368

seasonal variation of, 364-365

temperatures suitable for, 363-364

types of, 361-362, 364, 366, 368
Ice crystal fog, 337
Icelandic low, 214
Indicator diagram, 40-42

changes on, 48

Industry, atmospheric pollution by, 438-
442

relation of meteorology to, 438-442
Inflow, axis of, 269
Instability, absolute, 237

at a cold front, 448

changes with motion of a layer, 37

conditional, 237-242
stable type of, 239-240
subdivisions of, 246-247

convective, 243

depression, 283

development of, 244-246

in dry air, 33

in thunderstorms, 373-379

latent, 240-242

development of, 244-246
pseudo, 241
subdivisions of, 241, 247

potential, 242-246, 316

rainfall with, 316-318

types of, 246-247
Inversion, 217

dry, 405
Isallobar, 115

at frontal depressions, 276

velocity of, 117
Isentropic analysis, 405-408
Isobars, at fronts, 108-110

conditions for stationary upper level,
134

construction of, 398-401

definition, 8

flow of air across, 300, 303

relation to geostrophic wind, 296-297

velocity of, 117

Isothermal, changes on indicator dia-
gram, 42

layer, 217
Isotherms, for earth's surface, 1-3

movement of, 267, 269

near frontal surface, 264

Jeans, J. H., 94
Johnson, N. K, 180, 184

Kelvin, Lord, 312

464

INDEX

Kelvin's circulation theorem, 136-139

Kew barometer, 187

KirchhofFs law, 67

Koppen's classification of climate, 385-

387

Koppen, W., 385, 386
Kronfeld, R., 447
Kuttner, J., 450

Land and sea breezes, 310-311

Land masses, effect of, 213-214, 271, 302

Land surface, albedo of, 75

Landsberg, H., 83

Lapse rate, adiabatic for moist air, 35

average, 4

changes of, 36-39, 222

dry adiabatic, 31, 32, 220

in free atmosphere, 217

saturated adiabatic, 50-52, 220
Latitude, variation of radiation with,

83-84

Least squares, method of, 174-177
Lempfert, R. G. K, 124, 272
Lenard, P., 371
Liability, 242
Lilienthal, O., 447
Line of best fit, 174
Line of least squares, 174-177
Line spectrum, absorption in, 71
Linsley, R. K., Jr., 443
Low, equatorial, 8, 209

secondary, 278, 413

sub-polar, 9, 212
Lummer, O., 68

Martyn, D. F., 5

Matter, turbulent transfer of, 154-157

Mean, arithmetic, 163-166

standard error of, 172
Mean deviation, 167
Median, 162

Mediterranean climatic region, 391
Meter, dynamic, 17, 26
Middleton, W. E. K, 188, 189, 190, 191,

193, 196, 204
Millibar, 14, 187

relation to other units, 14
Milne, E. A., 12
Mischungsweg, 147
Mist, 332
Mixing, turbulent, effect of, 218, 221-

222, 224-227, 231-234
Mixing ratio, 22, 225

change in, 225, 325

on the Rossby diagram, 260
Moazagotl clouds, 449-450

Mode, 161

Mohn, H., 148

Moisture content, use in forecasting,

338-340, 433

vertical distribution of, 350
Molecular conduction, extent of cooling

by, 328
Molecular weight, for dry air, 16

for water vapor, 20
Moller, F., 83, 218, 219
Momentum, turbulent transfer of, 145-

150

Monsoon, 10, 214, 302
Motion, primary fields of, 266
turbulent, 144

vertical, in atmosphere, 448-450
in cumulus clouds, 241, 320, 448
on an isentropic surface, 406
pressure changes through, 131
through convergence and diver-
gence, 127, 290-293, 358
Movement, of a front, 265-271, 300, 403,

408-414
of a thunderstorm, 378

Nehring, J., 447

Newfoundland, fogs off, 256, 259, 333

Normand, C. W. B., 60, 61, 227, 229,

231, 239

Normand's propositions, 60-61
Nuclei, condensation, 313-314

Occlusion, clouds at, 343-345

definition, 274

development of, 274, 276-277

rate of, 276^-277

slope of, 278, 279

types of, 278, 279
Outflow, axis of, 269
Oxygen, distribution of, 11
Ozone, distribution of, 12, 86, 287

Pacific Ocean, effect of, 332

Pearson, K., 177

Penner, C. M., 185, 281

Petterssen, S., 115, 122, 334, 337, 409

Philipps, H., 83, 84

Pilot balloons, 196-197

use of, 403-404
Planck's law, 68

Plotting on synoptic charts, 397-399
Poulter, R. M., 345
Prandtl, L., 147

Precipitation, comparison in North
American air masses, 262

forecasting of, 4f 4

INDEX

465

Precipitation, measurement of, 204r-206

near a developing wave, 274

types of, 319-320

world distribution of, 382
Pressure, changes in, 131-132

correction to sea level, 400

critical, 47

distribution at the earth's surface, 6-18

distribution in the upper air, 132-134,
214, 281, 287

diurnal variation of, 9

errors in reported, 400

measurement of, 186-190

tendencies, 110-112, 274

trough at fronts, 108-110

units, 14

variation with height, 9, 18-19, 299
Pressure gradient, force, 8&-90, 295

in cyclones and anticyclones, 103, 290
Probability, curve, 171

definition, 171
Pringsheim, E., 68
Psychrometer, 193

equation, 59
Public utilities, relation of meteorology

to, 445-446
Pulley, O. O., 5
Pyrheliometers, types of, 207

Radiation, 67-87

absorption of, 73-76, 78, 79, 218-219

black-body, 67

development of instability by, 245

effects of, 217-219, 224-227, 229, 232-
234, 310-311

fog, 326-332

gray-body, 68

in the stratosphere, 85-86

laws of, 67-69

long-wave, 67

absorption of, 76, 78, 79

loss of heat by, 327

nocturnal, 77-82, 217, 327

short-wave, 67

solar, 67, 72-75, 83-84, 218-219, 427
scattering of, 73

terrestrial, 67, 7&-S4, 218-219

types of, 67

with clouds, 80
Radiosonde, 197, 397

reports, use of, 404, 418-420
Radius of earth, 16
Rain, equivalent of snowfall, 206

gage, types of, 204-205

ice accretion in, 367

shadow, 384

Raindrops, electrical charges on, 371

formation of, 314-315

size of, 314
Rainfall, measurement of, 204

relation to run-off, 443

types of, 315-319

Range as a measure of variability, 167
Rayleigh, Lord, 73
Refsdal's aerogram, 65
Regnault's equation, 65
Regression line, 177
Reports, weather, 396, 397
Representative property, 222
Retail merchandise, relation of meteor-
ology to, 451
Reynolds, O., 145
Rime ice, 361, 364, 366, 368
Rocky Mountains, effect of, 252
Rossby, C.-G., 64, 140, 260, 261, 407,

429

Rossby diagram, 64, 260-262
Rotation as a field of motion, 266
Rotation of earth, effect of, 211-213

Sandstrom, J. W., 24
Saturation, 312-313

definition, 21

methods of producing, 322
Saturation vapor pressure, 312-313, 326

over ice, 335

variation of, 47-49
Scale for geostrophic wind, 296

use with isallobaric wind, 300

use with thermal wind, 299
Schmidt, W., 147
Scrase, F. J., 370
Seasons, effects of, 213
Seclusion, 279-280
Sector, warm, 274
Shaw, Sir N., 124, 272
Simpson, Sir G. C., 83, 84, 85, 370, 371
Sine curve, fitting of, 179-183
Skiing, 447
Sky, color of, 73
Sleet, formation of, 319
Slope and valley winds, 306-310
Slope, of a frontal surface, 106-107, 264-
265,345

of an occlusion, 278, 279
Smithsonian Institution, 427
Snow, belts, 251

effect on fogs, 334-338

formation of, 319

melting of, 444

pellets, 320
Snowfall, measurement of, 205-206

466

INDEX

Soaring, 447-450

Solar constant, 72, 427

Solar radiation, 72-75, 83-84, 427

absorption of, 73-74, 218-219

variation of, 83-84, 427

wave length of, 67
Source regions, of air masses, 215

frontogenesis between, 271
Spectrum, absorption in, 71
Sports, relation of meteorology to, 446-

450
Stability of air, 32-33, 52, 236-247

absolute, 237

changes in, 37, 222

classifications of, 237

comparison in North American air
masses, 262

conditions required for, 236
Standard deviation, 167-169

computation of, 163

standard error of, 173
Statistics, 161-185
Steering, 428
Stefan's constant, 70
Stefan's law, 69
Stevenson screen, 190
Stratosphere, 5

radiation in, 85-86

variation of pressure in, 19, 281

variation of temperature in, 287
Streamline, 123-124
Sublimation, latent heat of, 51
Subsidence, effect of, 36-40, 220, 224-

227, 231-234
Sunshine, measurement of, 206-207

recorder, Campbell-Stokes, 206
Surface, frontal, 105-114, 264-265, 345
pressure tendency below, 110-112
wave on, 273, 274, 417, 418, 422-424

temperature, 1-3, 190-191, 217-218

wind, 6-7, 10, 148-149, 302-305

measurement of, 194-195
Sutcliffe, R. C., 128
Swain, R. E., 441

Taylor, G. L, 145, 146, 153, 154, 158,329,
330, 333

Temperature, 217-235
changes at fronts, 273-274
changes in free air, 218-221, 351
changes in surface, 213-214, 217-218
comparison in North American air

masses, 262
critical, 47

decrease over a cold surface, 153-154
diurnal variation of, 157, 222-223

Temperature, equivalent, 63, 232-234

equivalent potential, 63, 234
diagram, 64

gradient, at a front, 265

change with frontogenesis, 265

maximum, 223, 347

measurement of, 190-192

minimum, 223
forecasting, 433

potential, 30, 224
changes in, 224
equivalent, 63, 232-234

on the Rossby diagram, 260
partial, 64

surfaces of equal, 405-408
wet-bulb, 62, 232

scales, 14

suitable for ice accretion, 363-364

surface, 1-3, 217-218

variations in surface, 190-191

variation in the free atmosphere, 4,
281, 286-287

vertical distribution of, 4, 351

virtual, 23-24

wet-bulb, 60-62, 227-232, 238-244

wet-bulb potential, 62, 232
Tendency, pressure, 110-112, 274
Tephigram, 54-55

evaluation of work on, 58

use of, 227-234
Thermals, 447
Thermodynamics, first law of, 28

second law of, 43
Thermograph, 192
Thermometers, types of, 190-193
Thunderstorms, 370-379

air mass, 252, 258, 259, 260, 378

diurnal variation of, 373

frontal, 378

instability in, 373-379

movement of, 378

types of, 378-379
Tornadoes, 284
Torricelli, 186
Trade winds, 10, 212

migration of, 283

Trajectory of air particles, 124, 272
Translation, effect of, 266
Transmission coefficient, 70
Transportation, relation of meteorology

to, 431-433

Trewartha, G. T., 2, 3, 6, 7, 386
Tropopause, 5

height of, 185, 281-282, 287

in Antarctic, 8
Troposphere, 5 *

INDEX

467

Troposphere, pressure in, 19
Troughs, movement of upper, 132-134

velocity of, 119, 408-414

weather phenomena at, 359
Turbulence, 144-159

clouds, 350-355

development of instability by, 245

effect of, 83, 325

measurement of, 439

relation to fog, 328-329, 332-333

saturation produced by, 322, 325
Typhoons, 284

United States, fog over, 340-341

hail over, 375-376

thunderstorms over, 374, 376
Units, of pressure, 14

of temperature, 14

of water vapor, 20, 224-226

Valley winds, 306-310
Vapor pressure, over ice, 335
saturation, 47, 312-313
rate of change of, 49
variation of, 326
Vaporization, entropy of, 66
Variability, measures of, 166-169
Velocity, angular, of the earth, 17
effect of friction on, 148-149
of significant curves, 117-120, 408-414
vertical, through convergence, 127,

290-293, 358

Visibility, comparison in North Ameri-
can air masses, 262
measurement of, 207-208
Volume, critical, 47
Von Ficker, H., 272
Vorticity, 135-136

Walker, Sir G., 428

Warm front, characteristics of, 274

clouds at, 343-344

definition, 105

weather changes with passage of, 280,

427
Water surface, albedo of, 75

modifying effect of, 251-252, 256, 259
Water vapor, absorption of radiation by,
74,76

amount in atmosphere, 13

density of, 21

entropy of, 45

gas constant for, 20

molecular weight of, 20

units of, 20, 224-226

Wave, length of radiation, 67
on a frontal surface, 273-274, 417-418,

422-424

standing, in the atmosphere, 449-450
Weather, at a trough line, 359
changes with frontal passage, 280, 427
chart, synoptic, 397
code for plotting, 398
effect of, on agriculture, 433, 436
on forestry, 434-438
on heavy industry, 438-442
on hydrology, 442-445
on public utilities, 445-446
on retail merchandizing, 451
on sports, 446-450
on transportation, 431-433
reports, synoptic, 396
Wedge, movement of upper, 132-134

velocity of, 119, 408-414
Westerlies, 10, 212

Wet-bulb potential temperature, 62, 232
Wet-bulb temperature, 60-62, 227-232

use of with instability, 238-244
Wet-bulb thermometer, 60, 192
Wien's law, 69
Wilson, C. T. R., 372
Wind, 295-311
anabatic, 306-310
changes at fronts, 274
Chinook, 221

distribution over the earth, 6-7, 10
diurnal variation of, 302-305
effect of friction on, 302-305
effect of in frontogenesis and frontoly-

sis, 265-271

effect of land areas on, 302
Fohn, 221

geostrophic, 96-99, 295-297
equation, 96-98
field, pressure change in, 132
use of, 412

variation with height, 99
velocity, values of, 98
gradient, 101-104, 297

causing convergence, 289
isallobaric, 128-130, 29&-302
katabatic, 306-310
land and sea, 310-311
measurement of, 193-197
monsoon, 10, 214, 302
on a non-rotating globe, 211
on a rotating globe, 212
rose, 166
slope, 306-310

surface, 6-7, 10, 148-149, 302-305
measurement of, 194r-195

468

INDEX

Wind, thermal, 98-101, 298^299
trade, 10, 212, 283
upper, 196-197

use in drawing isobars, 400-401
valley, 306-310

velocities, Beaufort scale of, 194
westerlies, 10, 212

Work, on an indicator diagram, 40
on a pseudo adiabatic chart, 56-57
on a tephigram, 58

Wright, J. G., 438

Ziller, E., 450

Zones, frontal, 264, 271

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