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Full text of "Application of similarity theory to forecasting the mixed-layer depth of the ocean."

N PS ARCHIVE 
1964 
MCDONNELL, J. 






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APPLICATION OF SIMILARITY THEORY 
TO FORECASTING THE MIXED-LAYER 

DEPTH OF THE OCEAN 



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john r. McDonnell 



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LIBRARY 

U.S. NAVAL POSTGRADUATE SCHOOL 

MONTEREY, CALIFORNIA 



APPLICATION OF SIMILARITY THEORY TO 
FORECASTING THE MIXED -LAYER DEPTH OF THE OCEAN 

jf <jf a/- % <$g 

John R. McDonnell 



APPLICATION OF SIMILARITY THEORY TO 
FORECASTING THE MIXED -LAYEF 3EPTH OF THE OCEAN 



by 
John R. McDonnell 
Lieutenant, United States Navy 



Submitted in partial fulfillment of 
the requirements for the degree of 

MASTER OF SCIENCE 



United States Naval Postgraduate School 
Monterey, California 

19 6 4 



ffp L/5RAJ5Y 

VS NAVAL POSTGRADUATE SCHOOL 
MONTEREY. CALIFORNIA 

APPLICATION OF SIMILARITY THEORY TO 
FORECASTING THE MIXED-LAYER DEPTH OF THE OCEAN 

by 
John R. McDonnell 

This work Is accepted as fulfilling 

the thesis requirements for the degree of 

MASTER OF SCIENCE 

from the 

United States Naval Postgraduate School 



ABSTRACT 
The thermal structure of the ocean, especially the uppermost mixed 
layer, greatly affects sonar ranges. In this paper, similarity theory 
is applied to the problem of forecasting the depth of the mixed layer 
during the warm season, assuming the controlling processes are secular, 
non-advective , and non-divergent. The resulting forecast method consists 
mainly of two equations. Parameters used are wind, coriolis effect, the 
coefficient of thermal expansion and a measure of the excess heat within 
the mixed layer. The constants in the equations were determined using 
data from OWS Papa (50N, 145W). The forecast method treats both seasonal 
and transitional thermoc lines . The method was tested with data from OWS 
Pap* and OWS November (30N, 140W). The tests apparently indicate wide 
applicability of this forecast method and thus tend to corroborate the 
proposal by Kitaigorodsky that the mixed- layer depth is a function of a 
universal coefficient. 



ii 



TABLE OF CONTENTS 

Section Title Page 

1. Introduction 1 

2. Description of Similarity Theory 5 

3. Variables and Processes that Affect the Mixed-Layer 

Depth During the Warm Season 7 

4. Selection of Parameters to Represent Controlling 
Processes 8 

5. Practical Application of Similarity Theory 13 

6. Application and Test of Equations 22 

7. Conclusions 29 
Bibliography 30 

Appendix 

I. Basic Procedures for Application of JT Theorem 32 

II. Relationship between Variance of the MLD and the Strength 

of the The rmoc line 36 

v 

III. Methods Used for Determining Values of Parameters 38 

IV. Forecasting Method Proposed by Kitaigorodsky 43 



iii 



LIST OF ILLUSTRATIONS 
Figure Page 

1. Typical Thermal Structure of the Upper Layer of the 

Ocean During the Warm Season at OWS Papa 3 

2. Two Possible Thermal Structures with Same Excess Heat 
Present in Warm Layer 11 

3. P versus N Based on Monthly Cliraato logical Data for the 
Seasonal Thermocline at OWS Papa 15 

4. P versus N for OWS Papa Data (Transitional Thermocline) 

with Least- Squares Best- Fit Line 17 

5. P versus N for OWS Papa Data (Seasonal Thermocline) 

with Least Squares Best Fit Line 20 

6. P versus N for OWS November Data Compared to OWS Papa 

P versus N Best Fit-Line (Seasonal Thermocline) 26 

7. Variance of MLD about the Mean versus Temperature 

Change Through the Uppermost Thermocline 37 

8. Representation of Area T Used in Determining Q™ 39 

9. Representation of Areag Used in Determining Q s 41 

10. Values of P versus N Computed Using Equations 

Developed by Kitaigorodsky 45 



iv 



LIST OF TABLKS 
Table Page 

1. Data Used to Determine Values of P and N for Mixed- 
Layer Depths Associated with Transitional Thermoclines 18 

2. Data Used to Determine Values of P and N for Mixed- 
Layer Depths Associated with Seasonal Thermoclines 21 

3. Forecast Verifications 23 

4. Statistical Results of Forecasts 24 

5. Data Used to Determine Values of P and N for Mixed - 
Layer Depths Associated with Seasonal Thermoclines 

at OWS November 27 

6. Values for the Coefficient of Thermal Expansion 

Based on a Salinity of 32,5°/oo 42 



LIST OF SYMBOLS AND ABBREVIATIONS 

/3 coefficient of thermal expansion 

MLD mixed layer depth 

Q Excess heat present in upper layer of water 

Q F Net heat flow to and from upper layer of water 

Q T Excess heat present in upper layer of water associated with a 
transitional thermocline 

Qq Excess heat present in upper layer of water associated with a 
seasonal thermocline 

TS Temperature at the surface of the ocean 

W wind speed or a representative wind speed 

f coriolis parameter 

4 
A. modified coriolis parameter (f x 10 ) 

to angular velocity of earth 

latitude 



vi 



1. Introduction. 

The detection of enemy submarines by sonar is one of the major un- 
solved problems of this decade. Despite recent technological advances 
in many fields of science, the nuclear submarine is still practically 
invulnerable. Even though sonar technology, in particular, has advanced 
rapidly, we are still unable to position our ASW forces or their sonars 
for optimum performance. This is largely because sonar ranges are either 
enhanced or reduced by refraction of sound energy in the surface layers 
and we are not able to forecast accurately the thermal structure which 
greatly determines refraction. 

Many authors have devised systems for forecasting the thermal struc- 
ture of the upper layer of the ocean based on either dynamical analysis 
or empirical relationships. The equations resulting from dynamical 
analysis are either too complicated, if all the physical processes are 
considered; or are impractical, if many simplifying assumptions are made. 
Empirical relations have been determined for certain locations and for 
limited time periods but do not appear to be valid universally. 

Another possible method for developing a forecasting system uses 
similarity theory. Kitaigorodsky CO investigated the application of 
similarity theory to the ocean thermal-structure -forecasting problem. 
Although his results have certain drawbacks, as noted in appendix IV, 
still he has shown the applicability of the method. Consequently, the 
method used by Kitaigorodsky is also applied by this author, with some 
modifications of parameters, in an effort to develop a more practical 
result. The form of P, a dimensionless coefficient inherent in the 
application of similarity theory, is determined by use of data from 



Ocean Weather Ship Papa (50N 145W). 

Figure 1 depicts a typical ocean thermal structure in the wartn sea- 
son at OWS Papa. Characteristically , there is a quasi-isothermal layer 
that extends from the surface to the upper boundary of a negative tem- 
perature gradient. 

The depth to which the quasi- isothermal layer extends is usually 
referred to as the mixed- layer depth. In this paper the mixed- layer 
depth (MLD) is defined as that depth at which the temperature of the 
water first becomes 1C less than that of the water at the surface. 

A transitional layer of negative temperature gradient between layers 
of relatively small temperature variation is referred to as a thermocline 
Four main types of thermocline can be classified, primarily according to 
degree of permanence: diurnal, transitional, seasonal, and permanent. 
Diurnal thermocline s it results from a net heat gain during 
the day, a small thermocline (AT < 1C, see fig. 1) being formed 
close to the surface by late afternoon. With a net heat loss at 
night, the thermocline will be destroyed by morning. 

Transitional thermocline: a moderately large thermocline 
(AT > 1C) is formed when diurnal heat input exceeds losses. After 
a few days or weeks it joins with the seasonal thermocline as the 
added heat diffuses downward. 

Seasonal thermocline: the transition zone which lies between 
the surface waters warmed during the summer and the colder water 
below. At OWS Papa the seasonal thermocline depth is about 20 
meters in the late summer, and about 50 meters in the spring and 
fall. It is not present in the winter. 



TEMPERATURE CO 



UJ 

2 



a. 

UJ 

o 



Diurnal 
Thermocline 

Transitional 
Thermocline 




- Seasonal 
Thermocline 



80 u - 



Figure 1 

Typical Thermal Structure of the Upper Layer of 
the Ocean During the Warm Season at OWS Papa 



l .<.. . 



Permanent thermocline; in many localities a deep therraocline 
that is relatively persistent in depth exists throughout the year. 
Below a permanent thermocline the temperature gradient (usually 
negative) is very small all the way to the bottom. 
The entire thermal structure of the ocean, though of interest, is 
not immediately as important to sound propagation with present sonars as 
the depth of the uppermost mixed- layer of the ocean. As far as this 
mixed layer is concerned, there are two basic seasons at OWS Papa. These 
are the warm season, when mixing of the upper layer is mainly due to the 
wind, ind the cool season, when mixing is mainly due to thermoh aline con- 
vection. Thus the physical processes during the two seasons are quite 
different and are usually treated separately. 

As in the paper by Kitaigorodsky, similarity theory will be applied 
to develop a method for forecasting the mixed- layer depths associated 
with the transitional and seasonal thermoc lines during the warm season 
at OWS Papa. Henceforth, diurnal therraociines will be ignored and the 
terms "MLD" and "depth of the thermocline" will be synonymous and will 
refer only to mixed-layer depths associated with either transitional or 
seasonal therraociines. Only those changes in the MLD which are secular, 
non-advective, and non-divergent will be considered. 



2. Description of Similarity Theory. 

There are three basic steps in the application of similarity 
theory, 

1) Determine the physical processes that control the physical 
phenomenon of interest. 

2) Select parameters that accurately represent the controlling 
physical processes. 

3) Apply the -ff theorem Q2j to the chosen parameters. 

The TT theorem is a method for determining a dimensionaiiy-correct 

relationship for a given set of parameters. A short description given 

by Binder Z^3 follows. 

Let A^A2A3....A n be n physical quantities which are involved in 
some physical phenomenon. Examples of these physical quantities 
are velocity, viscosity, and density. Let m be the number of all 
the primary or fundamental units (such as length, mass, and time) 
involved in this group of physical quantities. The physical equa- 
tion, or the functional relation between these quantities, can 
be written as 

j (A, A,. A 3 A m ) = 

The IT theorem states that the foregoing relation can be written as 

where each If is an independent dimensionless product of some of 
the A's. 

Thus, if there are n physical variables in a particular problem, and m 
fundamental units, then the physical relationship can be expressed in a 
form involving (n-m) dimensionless ratios. An elaborate and formal 
proof of the theorem is given by Buckingham |_2j. A simplified step-by- 
step procedure and an example are given in appendix I. 

As can be seen in appendix I, to equate these (n-m) ratios, the TT 
theorem introduces a dimensionless coefficient (P) which serves to make 

5 



the dimensionless ratios numerically equivalent. A practical example 
for the necessity of P is given in the second paragraph of section 5. 
The form of P is normally determined from actual test data. P can be a 
constant or it can be a function of some or all of the parameters used 
to determine the ratios. Buckingham [_2J suggests several ways to deter- 
mine the form of P. 

One method is to let P be a function of all of the parameters of 
the process with the exception of the one for which the relationship is 
being developed. For ease of reference, this combination of parameters 
is called N. The 1f theorem is entered with the chosen parameters and 
an expression for the dimensionless ratio N is obtained. Pairs of 
values of p and N, determined using observed data, are plotted together. 
Then the form of P(N) can be determined from this plot. 

Once the dimensionless ratios and the form of p have been deter- 
mined, then the results can be used to forecast the mixed- layer depth. 



3. Variables and Processes that Affect the Mixed-Layer Depth During 

the Warm Season. 

There is general agreement among many researchers that during the 
warming season the mixed- layer depth varies with: 

1) wind (Geary C4] , Tabata [5], Mazeika [S~] f Munk and Anderson [7]); 

2) the heat flow between upper and lower layers (Mazeika [6J > 
Rossby and Montgomery QsQ , Kitaigorodsky CQ)! 

3) coriolis effect (Rossby and Montgomery CC » Kitaigorodsky Ckl)* 

4) divergence in the upper layers of the ocean (Mazeika [b] , Tabata 
and Giovando |V]); 

5) internal wave action (Mazeika M , Tully [lOJ , Tabata and Gio- 
vando K]); 

6) advection (Mazeika C€]» Tully [j-Q] , Tabata and Giovando C?] ). 
Only the first three of these processes will be considered in developing 
a forecast method. The remainder of the influences contribute to the 
scatter of the results. 



4, Selection of Parameters to Represent Controlling Processes. 

An aid in determining representative parameters is to write the 
relevant dynamical equations describing the processes and to examine 
them term by term. 

The equations of motion in the xy -plane, assuming an unaccelerated 
flow in an unbounded and horizontally- homogeneous ocean, reduce to: 

O « f -\r-i- /o d s 

(1) 



Q = —j-M. -+- 



Oil 



/° a ^ 



*# 



where u and v are the components of velocity, ~TL V ar *d "fl are the com- 
ponents of stress on the horizontal plane, and + is the coriolis param- 
eter. At the interface, the stress is given by: 

-fV^W 1 (2) 

where ~\ is the stress on the water surface, W is the mean wind speed and 

o is a coefficient that is a function of wind speed and the height at 
which the wind is measured. This indicates that the water motions, in- 
cluding the eddies that carry both heat and momentum, depend upon the 
surface wind speed. Thus a measure of wind speed that is representative 
of the stress effects should be introduced as a parameter in the applica- 
tion of the IT theorem. However, the depth to which wind effects reach 
depends also on the coriolis parameter as shown theoretically by 
Ekman \JL1J. Thus, the coriolis parameter must also be considered in 
application of IT theory. 

The maintenance of a thermal structure involves heat flow. A common 
equation for temperature changes due to heat flow is: 

8 



where J° is the density of water, c is the specific heat of water, T is 
the temperature of the water, and K is the eddy-conductivity coefficient 
for heat. This indicates that the local rate of change of temperature 
is a function of K as well as of the differences in temperature between 
layers. But K depends upon the field of motion and the stability. Param- 
eters to represent motion have already been chosen in the previous para- 
graph. Thus, to complete the representation of the variables in the heat 
flow process, the only additional parameters that need be introduced are 
those representative of stability. 

Thus it appears that, for the conditions and assumptions given in 
section 3, the IT theorem should be entered with parameters that are 
representative of: 

1) wind stress; 

2) coriolis effect; 

3) stability. 

WIND STRESS. Here it is necessary to find a value of wind speed 
which is representative of the influence of wind on the MLD over a period 
of time. The wind enters into the mixed-layer-depth problem in the form 
of stress, and the stress coefficient ( o) is an increasing function of 
wind speed. Thus equations (I) and (2) show that higher winds should be 
weighted more heavily in estimating the influence of wind stress upon 
the MLD. 

One therefore should avoid a linearly- averaged wind, The author 
has devised an empirical method which qualitatively and objectively 



weights the stronger winds in order to obtain a suitably representative 
wind parameter. The parameter is called the "representative maximum 
wind" and is defined as: 

the average of the five highest winds of the eight usually 

reported during a chosen 24-hour period. The chosen period is 

that one having the highest winds in the interval from 72 hours 

to 12 hours prior to observation time. 
The step-by-step procedure for determining the "representative maximum 
wind" is given in appendix III. 

CORIOLIS EFFECT. The standard equation for the coriolis parameter 
times 10 will be used to represent coriolis effects: 



_n_ =x -f x io + = a lj sin <p x io 



t 



The constant, 1x10 , was introduced arbitrarily for convenience. 

STABILITY. The density gradient, and thus the stability, can be 
represented by a combination of the coefficient of thermal expansion 
( /3 ) and the temperature difference between layers; the following two 
parameters have been chosen to represent stability: 

1) the coefficient of thermal expansion (/3 ) ; 

2) the excess heat (Q) in the mixed layer over that in lower layers. 
Values of /3 used in this paper were interpolated from values given 

by Sverdrup \\2\ assuming a constant salinity of 32.5 % . These values 
are tabulated in appendix III. Constant salinity was assumed at OWS 
Papa during the warm season as in Tabata £fQ. 

The step-by-step methods used to determine Q are given in appendix 
III. One difficulty with the use of Q as a measure of stability is 
illustrated in figure 2. This shows that thermal structures of obviously 

10 



T— > 



1 




P 



T^ 




Q, =Q 



Figure 2 

Two Possible Thermal Structures with 
Same Excess Heat Present in Warm Layer 



11 



different .stability can prevail with the same Q. However, the combina- 
tion of Q, A , and the MLD give a good approximation of the stability 
of the mixed layer. (The MLD enters the forecast model in the form of 
P(N) since P(N) is initially determined from raw data that includes 
MLDs . ) 

Thus, besides the MLD itself, four parameters have been chosen for 
entry into the n theorem: 

1) the representative maximum wind (W); 

2) the coriolis parameter (-f); 

3) the coefficient of thermal expansion (/3); 

4) the excess heat in the upper layer (Q). 



12 



5. Practical Application of Similarity Theory. 

In section £ , f our parameters were chosen as representative of the 
processes effective in forming and maintaining a mixec layer at OWS Papa 
in the warm season. When the 1T theorem is applied to these four param- 
eters the results are: 



Q/3-rt- (3) 






Both of these equations are derived in the example in appendix I. 

Because the relationship between ml I) and the other parameters is 
complex, the coefficient P(N) is not in general a constant and must be 
found as a function of some of the parameters, as in appendix I. As an 
example of the ambiguity which must be eliminated by use of P, consider 
again the parameter Q; as shown in fig. 2, two quite different thermal 
structures may have the same excess heat Q. Thus a value of Q alone 
does not uniquely specify the thermal structure. Rather the structure 
is a result of the interaction of all the controlling parameters. That 
is, a wide range of possible mixed-layer depths exists for each value of 
a particular controlling parameter, whereas only one MLD exists for each 
combination of parameters. To specify the form of P(N), which is a 
function of all the prevailing parameters, several pairs of values of 
P and N are determined from experimental data and plotted together. 
Then the form of P(N) is given by this plot. 

If , in the application of similarity theory, parameters truly repre- 
sentative of the controlling processes are chosen, then the plot of P 

13 



versus N should have little scatter. If the parameters used are not 
truly representative, then large scatter results. 

To test equations (3) and (4) they were used to obtain paired values 
of P and N; these values were based on mean monthly data at OWS Papa 
tabulated by Tabata and Giovando [_9J. Values of the parameters, W, ft , 
_TL , and Q s (where the subscript S refers to seasonal thermoclines) were 
determined in accordance with Appendix III. The paired values of P and 
N are plotted in figure 3. The points have very little scatter; and, in 
fact, a straight line gives a very good least-squares fit. However, the 
use of figure 3 is limited because it represents a climatological MLD, 
whereas the MLD for a particular day can vary considerably from the clima- 
tological MLD. Figure 3 is important in that the small scatter of the 
paired values apparently indicates that the proper parameters have been 
used in equations (3) and (4). 

In order to make equations (3) and (4) useful in short range predic- 
tion of the MLD, P(N) was determined using daily data from OWS Papa. The 
data were from June to October during the years 1958 through 1962. In 
order to filter out internal wave effects, data were used only from those 
days for which six or more bathythermographs were available; the MLD 
used is a mean of the six (or more) bathythermographs. This requirement 
limited the available data such that only 22 pairs of P and N are avail- 
able for transitional MLDs and 29 pairs for seasonal MLDs. Although the 
number of paired values appears small, they represent over 200 BTs. 

Pairs of values of P and N were first obtained for MLDs associated 
with transitional thermoclines with values of the parameters W, /3 , 
Si, and Q T determined as stated in Appendix III. Note that for a 

14 



16 



14 



12 



10 



x 8 




•', »■ 



Number by each point indicates 
month of year point represents, 



NX 10 



Figure 3 



8 



P versus N Based on Monthly Climatological Data 
for the Seasonal The rmoc line at, OWS Papa 



15 



transitional thermocline the general parameter Q in equations (3) and 
(4) is represented by the more precise parameter Q^.. Figure 4 is a plot 
of pairs of P and N resulting from data listed in table 1. The least - 
squares best-fit straight line has the equation: 

P = -.a b"xio~^ + z.^i N 
Substituting this into equation (3) gives: 



MLD-.^XI0- + (^+^ (-£) 



(5) 



Thus equation (5) can be used to forecast a transitional MLD without 
determining a value of P. 

It is interesting to note the similarity between equation (5) and 
the equation derived by Rossby and Montgomery [_8J : 



MLD— V**+K a -$> (6> 



W z wind speed measured at the sea surface. 

= latitude 
Based on 39 observations, they arrived at a value for K 2 of 2.38 per 
second. It was concluded that K^ was negligible. The MLD data for these 
39 observations were obtained by interpolating between readings of tem- 
perature and salinity which are 10-25 meters apart. The average MLD for 
the 39 observations was 32 meters, and they considered a five-meter aver- 
age error as a plausible assumption. Since the data used by Rossby and 
Montgomery appeared to be of poor quality compared to the data available 
at OWS Papa, it is suggested that equation (5) is more applicable than 
(6) for forecasting the transitional thermocline. Tests in section 6 

16 



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V 








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8 












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7 














1 




6 












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x S 

0. 




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•/ 


• / 




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1 




4 




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/• 






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/ • 










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NX1CT- 

Figure 4 
P versus N for OWS Papa Data (Transitional Thermocline) 
with Least-Squares Best-Fit Line 

17 



TABLE 1 

DATA USED TO DETERMINE VALUES OF P AND N 
FOR MIXED-LAYER DEPTHS ASSOCIATED WITH TRANSITIONAL THERMOCLINES 

W Qt MLD TS 



DATE 


(KNOTS) 


*■ 
(Kg cal/cnO 


(METERS) 


(°C) 


PxlO 4 


NxlO 4 


61758 


27.4 


7.21 


26.4 


10.3 


2.0 


0.96 


61858 


27.4 


6.39 


23.4 


10.3 


1.6 


0.85 


62058 


27.8 


8.06 


28.5 


10.4 


2.4 


1.10 


71758 


20.0 


3.78 


20.4 


13.2 


1.8 


0.80 


81658 


25.6 


18.29 


34.1 


12.5 


8.6 


2.95 


71062 


20.0 


9.00 


27.2 


10.7 


4.1 


1.3 


71262 


21.0 


7.25 


30.0 


10.8 


5.0 


1.71 


8066l" 


19.6 


13.50 


23.9 


13.5 


8.0 


3.00 


90661 


26.6 


21.40 


34,8 


13.3 


9,7 


3.40 


91461 


30.6 


24.00 


39,7 


13.4 


9.4 


3.33 


62362 


12.3, 


2.98 


13.9 


10.5 


2,3 


0.92 


62762 


24.0 


5.86 


27.3 


9.5 


2.1 


0.86 


63062 


24.0 


4.01 


27.4 


10.0 


1.5 


6.10 


70262 


20.0 


6.32 


32.4 


10.1 


4.1 


1.15 


70662 


25.0 


9.63 


30.3 


10.4 


3.7 


1.40 


70762 


25.0 


10.11 


32.4 


10.3 


4.2 


1.47 


71462 


20. 


5.38 


27.6 


11.0 


3.1 


1.04 


71662 


20.0 


5.87 


28.0 


11.2 


3.5 


1.13 


72062 


17.2 


4.86 


24.3 


11.6 


3.4 


1.12 


72662 


13.2 


1.10 


14.1 


13.0 


0.8 


0.35 


72862 


17.0 


1.83 


19.8 


13.0 


1.2 


0.46 


73062 


22.8 


5.34 


23.0 


12.7 


2.1 


0.96 



18 



Indicate this to be true and also show that the terms of equation (5) 
are all of the same order of magnitude, 

Although equation (5) apparently is applicable for MLDs associated 
with the transitional thermoelinej, tests indicated that (5) was no 
longer applicable when the transitional MLD joins the seasonal MLD,. In 
this case, a slightly different approach is necessary, apparently due to 
the increased stability associated with a seasonal type thermocline. To 
take this increased stability into consideration Q s (defined in A>pendix 
III) is used in equations (3) and (4) instead of Q T . 

The general procedure for finding values of P and N for the sea- 
sonal MLD was the same as for values associated with the transitional 
MLD with the exception that Q<, was used instead of Q_,, Pairs of values 
of P and N for seasonal MLDs, based on data listed in table 2, are 
plotted in figure 5. The least-squares best-fit line for these points 

was: 

p = -6.1 X lo~ 4 -h3. 8=1 N 

Substituting into equation (3) gives: 



M LD=-i.lxio"^ Q/3 ][jij + 3 - 8 Hit:) (7) 



Equation (7) is applicable to MLDs associated with the seasonal therao- 
c 1 ine . 

An objective method has been devised to determine whether a trans i- 

-4 
tional or seasonal condition exists- When N < 3o5 x 10 , the therm©- 

-4 

cline is defined as transitional . When N > 3.5 x 10 „ a seasonal 

thermocline is said to exist. This is not a rigid rule. Obviously, even 

when N < 3.5 x 10 s if the forecast MLD using equation (5) is deeper 

than a prevailing seasonal thermae 1 ine , Chen equation (7) should be used 

rather than (5). 

L9 



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in 



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z 



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20 



TABLE 2 

DATA USED TO DETERMINE VALUES OF P AND N FOR MIXED-LAYER DEPTHS 
ASSOCIATED WITH SEASONAL THERMOCLINES 





W 


QS 2 
(Kg cal/cnO 


MLD 


TS 






DATE 


(KNOTS) 


(METERS) 


c°c) 


PxlO 4 


Nx 10' 


62762 


24.0 


28.3 


27.3 


9.5 


10.3 


4, 


91958 


50.0 


44.1 


49.8 


LI. 6 


7.6 


3.5 - 


61858 


27.4 


22.5 


23.4 


10.3 


5.6 


3.0 


91161 


27.2 


43,2 


40.5 


13.3 


21c 8 


6.7 


82761 


21 = 4 


40.6 


32.2 


14.5 


27.7 


8.5 


82361 


21.0 


41.4 


31.3 


13,8 


27.7 


8.6 


80661 


19.6 


35.7 


23.9 


13.5 


21.0 


7.9 


72062 


17.2 


28.5 


24.3 


11,6 


20.2 


6.6 


90661 


26.6 


39.7 


34.8 


13,3 


18,0 


6.3 


91461 


27.2 


40,6 


39.7 


13.4 


20.1 


6,3 


73062 


23.0 


31.8 


22.8 


12.; 


12.3 


5.7 


90961 


27.2 


40.7 


37.6 


13.4 


19.1 


6.3 


71462 


20.0 


31.0 


27.6 


11.0 


18.0 


6.0 


71662 


20.0 


27.5 


28.0 


LI. 2 


16.2 


5.3 


91461 


30.6 


39.5 


39.7 


13.4 


15.5 


5.5 


81261 


28.4 


32.5 


30.3 


13.2 


11.3 


4.9 


71262 


21.0 


29.6 


30.0 


10.7 


16.6 


5.3 


71062 


20.0 


27.5 


27.2 


10.8 


15.4 


5.2 


70262 


20.0 


26.2 


32.4 


10.1 


16.9 


4,8 


80861 


28.0 


29.6 


26.8 


13.3 


9.3 


4.5 


81461 


32.0 


37.4 


34.7 


12.8 


11.4 


4.8 


63062 


24.0 


29.1 


27.4 


10.0 


11.0 


4.4 


70662 


25.0 


30.5 


30.3 


10.4 


11.7 


4.5 


70762 


25.0 


27.6 


32.4 


10,3 


11.4 


4.0 ' 


61758 


27.4 


29.5 


26.4 


10.4 


8.2 


3.9 


61958 


27.4 


29.8 


25,5 


10.4 


8.0 


4.0 ' 


62058 


27.8 


29.3 


28.5 


10.3 


8.6 


3.8 


81658 


25.6 


39.8 


34.1 


12.5 


18.6 


6.4 


91858 


50. 


43.7 


49.4 


11.5 


7.4 


3.4 



21 



6. Application and Test of Equations. 

Equations (5) and (7) can be used to forecast MLD over any length 
of time for which the parameters can be accurately predicted. Thirteen 
tests of the equations were made using independent data for forecast 
periods of from one to four days. Five of the tests were based on data 
from OWS Papa (50N, 145W) and eight on data from OWS November (30N, 140W). 
Three of the tests were for transitional thermoclines and ten were for 
seasonal thermoclines. The tests were made using a mean MLD computed 
from four to six BTs from each day as a representative MLD for that day. 
Results are shown in table 3. 

The verification of a forecast is made difficult by the fact that 
the MLD varies randomly. One objective way to measure verification suc- 
cess is to determine if the forecast value lies within the normal range 
of variability of the MLD as measured by the standard deviations of the 
individual values. In table 3 the 13 values listed under "verified MLD" 
actually represent 63 BTs. The standard deviation with respect to the 
daily mean for these 63 bathythermographs was 3.47 meters. The percent- 
age of forecasts that fell within one and two standard deviations of the 
individual MLDs is given in table 4. The root-mean square and the alge- 
braic mean of the forecast differences are also given in table 4„ 

The algebraic mean of the forecast was computed to determine if the 
forecasts were biased. Forecasts using the equation proposed by Rtssby 
and Montgomery were biased positively whereas those based on equations 
(5) and (7) were less biased and negative. This suggests that the nega- 
tive term neglected by Rossby and Montgomery (see section 5) should be 
included as in equations (5) and (7). 

22 



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23 



TABLE 4 
STATISTICAL RESULTS OF FORECASTS 





FORECASTS BY 

EQUATIONS 

(5) AND (7) 


FORECASTS BY 
PERSISTENCE 


FORECASTS BY 

ROSSBY AND 

MONTGOMERY 

EQUATION 


X FORECASTS 
WITHIN 1 STANDARD 
DEVIATION 


69 


46 


15 


% FORECASTS 
WITHIN 2 STANDARD 
DEVIATIONS 


92 


61.5 


46 


ROOT MEAN SQUARE 
OF DIFFERENCES 
(METERS) 


4.3 


6.3 


11. 1 


ALGEBRAIC MEAN 
OF DIFFERENCES 
(METERS) 


-2.9 -1.06 

1 


8.1 





24 



To test how the function P(N) at OWS November compares with that 
found at OWS Papa, eight paired values of P and N at OWS November were 
computed. The BTs upon which the verifications were based in table 3 
were also used to obtain these paired values in the same manner as for 
OWS Papa (section 5). Figure 6 is a plot of paired values of P and N 
for the data listed in table 5; the straight line represents P(N) for 
the constants derived from data at OWS Papa. The function P(N) appears 
to be very nearly the same at both OWS Papa and OWS November. 

Two examples of the application of equations (5) and (7) to fore- 
casting follow. 

Example 1. Forecast to be made for 23 August 1957 based on BTs 

taken on 19 August at OWS Papa. 

1. From the BTs taken on 19 August 1957, Q~ was determined to be 

2 
1.74 Kg cal/cm by methods described in Appendix III. 

2. Using the August climatological data computed by Tabata and 
Giovando [V] , the average net heat transfer downward across the air- 

sea boundary at OWS Papa is approximately +0.1 Kg cal/cnr per day. 

2 
Using this information, 0.4 Kg cal/cm were added to Q„ during the 

four-day interval up to the verification day giving? 

Q T = 1.74 + 0.4 = 2.14 Kg cal/cm 2 

(Where enough current meteorological data are available, the methods 

for computing the net heat transfer proposed by Laevastu [.13J could 

be applied instead of climatology.) 

3. Values of /3 , Sl , and W were determined by the methods described 
in Appendix III. 

4. Equation (4) was entered and N was found to be less than 

25 



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00 



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c 
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H 



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3 

> «-» 
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3 


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CD 


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u 




ta 





55 




«M 


w 




z 


3 
w 




CO 


p 




3 


01 




0) 


> 




I- 






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Oh 




> 


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Oh 
CO 

5 
O 


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V 

u 
flj 

a 
B 



CJ 



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O 

CM 

.Olxd 



26 



TABLE 5 

DATA USED TO DETERMINE VALUES OF P AND N FOR MIXED LAYER DEPTHS 
ASSOCIATED WITH SEASONAL THERMOCLINES AT OWS NOVEMBER 



DATE MLD W Qs P x 10 4 N x 10 4 

(METERS) (KTS) (Kg cal/cra Z ) 

61657 27.9 13.2 32.9 26.8 7,5 

70757 43.0 17.1 31.6 13.0 4.7 

71057 37.5 19.3 22.7 11.5 4.3 

72157 44.5 23.5 25.2 24.9 6.4 

72757 49.0 17.8 28.3 28.5 9.7 

82558 34.7 15.1 39.6 34.2 10.5 

73159 43.5 18.5 33.0 23.1 6.9 

70159 32.0 18.0 18.2 9.9 3.9 



27 



3.5 x 10" . Consequently, equation (5) was used, giving a forecast 
MLD of 16.1 meters. 

5. The seasonal MLD was at 32 meters on 19 August 1957, According 
to the above forecast, the wind was not strong enough to cause the 
transitional thermocline to merge with the seasonal thermocline. 
Thus persistence is used to forecast the depth of this seasonal 
thermocline. 

6. The forecast thermal structure for 23 August 1957 consists of a 
mixed layer down to a transitional thermocline at 16.1 meters with 
a seasonal thermocline below it at 32 meters. The verification 
showed a transitional thermocline at 18.3 meters and a seasonal 
thermocline at 30.5 meters. 

Example 2. For the forecast MLD of 1 August 1959 at OWS Papa, steps 
1, 2 and 3 as done in example 1 resulted in N >3.5 x 10" . Conse- 
quently, the procedure for a seasonal thermocline was folio' ed. 
That is, Qc was determined and equation (7) was used to forecast 
the MLD. The forecast MLD was 35 meters and 39 meters verified. 



28 



7. Conclusions. 

The agreement between forecast and observed MLDs tends to verify 
the findings of Kitaigorodsky that similarity theory is useful in this 
forecasting problem. Equations (5) and (7) were successful in forecast- 
ing both large and small changes in the MLD, in the positive as well as 
in the negative sense. The concept of a universal function P(N), as pro- 
posed by Kitaigorodsky, is strengthened. 

Equations (5) and (7) appear to be useful in forecasting the mixed 
layer depth to a reasonable degree of accuracy. Some of the deviations 
can be attributed to the effects of divergence, advection and internal 
waves which were not evaluated in this study. 

Future rest arch could well be applied to determining more paired 
values of P and N for other ocean locations. This would not only help 
to fix the constants of equations (5) and (7) better, but would help to 
demonstrate further the universality of P(N), if it exists. 



29 



BIBLIOGRAPHY 

1. Kitaigorodsky, S, A. On the computation of the thickness of the 
wind-mixing layer in the ocean. Academy of Sciences, USSR, Geo- 
physics Series, no. 3, March 1960, 

2. Buckingham, E. On physically similar systems; illustrations of the 
use of dimensional equations. Physical Review, vol. 4, 1914, pp 
345-376, 

3. Binder, R. C. Fluid mechanics, Prentice Hall Inc. 1943. 

4. Geary, J, E. The effect of wind upon the mixed layer depth. M.S. 
thesis, U. S. Naval Postgraduate School, Monterey, Calif., 1961. 

5. Tabata, S. Temporal changes of salinity, temperature, and dis- 
solved oxygen content of the water at Station "P" in the northeast 
Pacific Ocean, and some of their determining factors. Journal 
Fisheries Research Board of Canada, vol. 18, no. 6, 1961. 

6. U. S. Navy Jydrographic Office. Prediction of the thermocline depth 
by Mazeika, P. A. TR 104, June 1960. 

7. Munk, W. H. and E. R. Anderson. Notes on a theory of the thermo- 
cline. Journal of Marine Research, vol. 7, no. 3, pp 276-295, 1948, 

8. Rossby, C. G. and R. B. Montgomery. The layer of frictional influ- 
ence in wind and icean currents. Massachusetts Institute of Tech- 
nology and Wood*; Hole Oceanographic Institution, Papers in Physical 
Oceanography and Meteorology, vol. Ill, no. 3. 

9. Pacific Oceanographic Group, Nanaimo, B. C. The seasonal thermo- 
cline at OWS Papa during 1956 through 1959 by Tabata, S. and L. F. 
Giovando. Fisheries Research Board of Canada, no. 157, April 23, 
1963. 

10. Pacific Oceanographic Group, Nanaimo, B. C. Oceanographic domains 
and assessment of structure in the North Pacific Ocean by Tully, 
J. P. Fisheries Research Board of Canada, File N6-13(4), 16 Sept. 
1963. 

11. Ekman, V. W. On the influence of the earth's rotation on ocean 
currents. Ark. f . Mat, Astr. och Fysik. K. Sv. Vet, Ak. , Stock- 
holm, 1905-06, vol. 2, no. 11, 1905. 

12. Sverdrup, H. V. The oceans. Prentice Hall Inc., 1942. 

13. Laevastu, T. Factors affecting the temperature of the surface layer 
of the sea. Societas Scientiarum Fennica Commentationes Physico — 
Mathematicae XXVI, 1960. 



30 



14. Monin, A, S. and A. Mo Obukhov. Basic laws of turbulent mixing in 
a ground layer of the atmosphere. Transactions of the Geophysical 
Institute, Academy of Sciences, USSR, no. 2 (151), 1954. 

15. . Oceanic observations of the Pacific: 1955. The Norpac 

Data prepared by the Norpac Committee, University of California 
Press and University of Tokyo Press, Berkeley and Tokyo, 1960. 

16. . Manuscript Report Series, Oceanographic data records, 

Ocean Weather Station "P", Fisheries Research Board of Canada, 
1956-1962. 

17. Budyko, M. I. The heat balance of the earth's surface. Translation 
by Office of Technical Services, U. S. Department of Commerce, 
Washington 25, D. C, 1958. 



31 



APPENDIX I 
BASIC PROCEDURES FOR APPLICATION OF TT THEOREM 
Let n represent the number of parameters (A, A9....A ) chosen to be 
representative of the processes involved in a particular physical phe- 
nomenon. Let m represent the number of fundamental dimensions (length, 
time, mass etc.) present in all of the parameters. Then follow these 
steps. 

1. Determine the number of " TT dimension less ratios" required to 
incorporate all the chosen parameters (given by n-m). 

2. Combine m + 1 parameters into an equation of the following form: 

where (A, A^^.A ) must together include all of the fundamental 
dimensions. 

3. Substitute the fundamental dimensions into equation (8) and 
equate like powers on each of the fundamental dimensions to zero, 



a* a* .A 4., = L M°T t 

A. A 3 - rvw+l 



4. Solve the resulting m equations with m unknowns to obtain x. , 
yi and z- . 

5. Substitute the values of x.,y ,z. into the equation for TT\ to 
obtain the first dimensionless ratio. 

6. Repeat steps 2 through 5 after changing A to another parameter 

m+1 

not previously used for any A and solve for TT 2 . Repeat as neces- 
sary until Tf. , TT , ....TT ratios are obtained. 
7 1' 2' n-m 

7. The ratios represented by TT,, TT-, ....TT are all dimensionless 

12* n-m 

32 



and can be equated one to the other to obtain a functional relation- 
ship that is dimensionally correct. The ratios are then made numer- 
ically equivalent by use of a dimensionless coefficient P. 
An example illustrates the application of the foregoing step by step 

procedure. 

Parameters Dimensions 

Q (total heat present) LT 

W (wind speed) L/t 

XL (coriolis effect) l/t 

/3 (coefficient of thermal expansion) 1/T 
H (mixed-layer depth or ML.D) L 

n = 5 m = 3 

STEP 1 n-m = 2 " If ratios" will be required 

STEP 2 11^ = Q w SL. ft and %= Q. *wV H 

To find the powers (x,y,z) which make the above products dimension- 
less: 



STEP 3 



For Til : (Lrf' (L/t/'(l/-tf (1/T) = L_° T° t° 



STEP 4 (L) rh*"fr\ ~ ° 

(T) ^.-1=0 



i 



(t) 



W " % = ° 



*-, =1 



33 



= 1 



STEP 5 



\}\ = q' w~'jn' /3 



or 
Q. -TL/3 



ir t - 

STEP 6 For If: (i_ T f ( L /t)^ ( l/t f ( L-) = L°T°t° 



(L) 



^2. + ^- + 1 = O 



(T) ^ = o 



Ct) 



'^1 -fr 



-^ = o 






fc 



= 1 



T, = q°w"'-TL H 



ir - -^ H 



STEP 7 Since both Tf^ and TT 2 are dimension less , either can be 
inverted in order to give a relation that agrees with what is observed 
physically. In this case it is known that the MLD is an increasing func 
tion of the wind, and u^ and TT2 ma y De combined into? 

-m-p-^- 

where P is a dimension less coefficient whose form must be determined 
experimentally. Substituting for ff an< * "9 an< * so * vin 8 f° r tne mixed 
layer depth gives; 



" L D = P( Q ^) 



34 



In his paper on TT theory, Buckingham [2] gave several methods to 
determine the form of p. Of these, the same method used by Kifcai*, 
sky £l] will be given as an example here. Since P may be considered a 
function of all the original parameters » they can be used to determine P 
with experimental data. If P does not turn out to be constant, it can 
be made a function of certain of the parameters and the form of that 
function can be found from the data. Thus P is made a function of N, 
which, in this example h is made to depend upon all the parameters except 
MLD, the quantity to be predicted. Use of tT theory, as before, yields 
a dimensionless ratio of these parameters ° 

1M W 

Entering the equations for both P and N with the same data and plot- 
ting the corresponding pairs gives P(N) in graphical form. 



35 



APPENDIX II 

RELATIONSHIP BETWEEN VARIANCE OF THE MLB 
AND THE STRENGTH OF' THE THERMOCLINE 

Mazeika jjQ used 356 groups of BT data t© determine that the MLD 

at OWS Echo had fluctuations with a mean amplitude of 17,85 feet and 
standard deviation ©f 10.55 feet about the mean. He attributed these 
fluctuations to internal waves. 

On the basis of 100 BT observations from OWS Papa for the months of 
September and October of the years 1958 through I960* the author deter- 
mined a system for estimating the amount of fluctuation of the MLD that 
might be expected at verification time. The temperature difference be- 
tween the sea surface and the temperature of the near- isothermal water 
just below the thermocline was used as a measure ©f the strength of the 
thermocline. The variance of each observation was computed from a five- 
day running mean. The average variance and the average AT for each two- 
meter increment of misted layer depth was then computed and plotted as 
pairs in figure 7. Figure 7 can be used to estimate the variance that 
can be expected at verification time. 

In order to smooth out the fluctuations of the MLD when an accurate 
verification Is desired 9 it is apparently best to take a series of BT 
observations over a period of at least 12 hours and then use the average 
of all the BTs to verify the forecast. 



<0 



»- 

<3 



u 

3 
60 



10 



a 

o 

0) 
V* 

4J Q) 

ra C 

M -H 
Q) f-l * 

a u 

8 2 
u e 

H U 

a, 

3 H 

(Q 

a> to 

*z 

x: a 



3 ^ 

O 00 

•n 3 
ca o 

u 

a a 



at 

u 

s 

J-l 

> 



O 



o 



O 

CM 



(gSJd}dtN) ODUeueA 



37 



APPENDIX 111 
METHODS USED FOR DETERMINING VALUES OF PARAMETERS 
WIND (W) 

1. Out of the 72-hour period preceding verification time, but not 
including the 12 hours immediately preceding verification time, 
determine the 24-hour period during which the highest average winds 
prevailed or are forecasted to occur. 

2. Fro® the eight wind reports given for that 24- hour period, 
choose the five highest speeds . The average of these five is the 
"representative maximum wind." 

EXCESS HEAT IN UPPERMOST LAYER (Q) 

When finding values of P and N 9 the computation of Q was based on 
the average BT for each particular 12-hour period. When using equations 
(5) and (7) to forecast an MLD, the most recent BTs available should be 
used to compute Q. In either case the following steps are applicable in 
determining the excess heat in the uppermost layer. 

1. Assume a transitional warm layer exists. For this situation 

compute Q™ as follows; 

a) Determine "AREA^'s, depicted in figure 8. The dashed line 
is a vertical drawn from the point of maximum curvature of the 
BT trace (roughly the bottom of the thermocline) to the surface. 

b) Or & s then given by 

-i 



Q T ^CpAREA X 10" (Kgc^l/c^) 



For OWS Papa in summer />c s 0.975 for salinity 32.5 °, oo , 
assumed to be constant (pc n Ira -JL®JL_ and AREA T in M°C). 

38 




Figure 8 



reseiitati©© of'Area* 
d in Deterraiaing 'Q_, 



-4 
2. If N > 3.5 x 10 using Q~> then a seasonal- type thertaocline is 

assumed to exist. The excess heat for a seasonal therrnocline (Qe) 

is found by the following objective method. 

a) Determine "AREAe'\ depicted in figure 9(d). The dotted 
lines, figure 9(a) 8 is a vertical drawn from the intersection of 
the BT trace asad 200 meters. HH is determined by equalizing 
areas A and B as in figure 9(b). Then 

AREAg r C - B 

b) , . 

Q s =/=>c p AREA X 10 (Kgc^l/cm^j 



40 



T > 



T > 



Q 2 



200- 





/ Trace 


« 



200 - 




(a) 



<b) 



T > 



T > 





(c) ,' (d) 

: Figure 9 

Representation ©f Area g Used In Determining Q« 



41 



TABLE 6 

VALUES FOR THE COEFFICIENT OF THERMAL 
EXPANSION BASED ON A SALINITY OF 32.5°/oo 



Temperature (°C) 


(- <10 6 (1/°C) 


> 


14.0 


205 


14.0 


- 


13.6 


200 


13.5 


- 


13.1 


195 


13.0 


- 


12.6 


190 


12.5 


- 


12.1 


186 


12.0 


- 


11.6 


182 


11.5 


- 


11.1 


178 


11.0 


- 


10.6 


174 


10.5 


- 


10.1 


168 


10.0 


- 


9.6 


163 


9.5 


- 


9.1 


158 


9.0 


- 


8.6 


152 


8.5 


- 


8.1 


147 


8.0 


- 


7.6 


141 


7.5 


- 


7.1 


135.5 


7.0 


- 


6.0 


124.5 


<6 


.0 


113 



42 



APPENDIX IV 
FORECASTING METHOD PROPOSED BY KtTAIGORODSKY 

Kitaigorodsky [ij , using similarity theory as described by Monin and 
Qbukhov [141 , developed an equation for forecasting the MLD. In develop- 
ing the equation Kitaigorodsky assumed that thermal convection was neg- 
ligible and that the vertical gradient of salinity was zero. This limits 
the use of the equation to the warm season when a stable layer exists. 
The equation does incorporate parameters that represent all the processes 
and variables that were considered in section 3 to cause or affect the 
MLD at OWS Papa. 

Equation developed by Kitaigorodsky using similarity theory; 

MLD=P- 



where i T a s a parameter proportional to the tangentia 
stress of the wind on the sea surface 

Qp - rate of heat flow from the atmosphere (LT/t), 

2 

g s acceleration of gravity (L/t ); 

/3 s coefficient of thermal expansion of sea water (l/T) 

P - a dimensionless coefficient that is a function of 
Na both N and P are determined empirically; 



(Mi is dimensionless) 



i\_ s coriolis parameter (1/t). 
The value of P in equation (8) is given by a plot of 14 correspond- 
ing values of P and N, obtained by Kitaigorodsky from data from the 

43 



NORPAG ATLAS [l5] . The large areal extent which the data represents 
makes the results widely applicable throughout the ocean during the warn 
season. 

The present author, using data from OWS Papa jj.6] has determined 11 
more pairs of values of P and N. Each of these 11 pairs represents from 
four to six BTs averaged together in an attempt to smooth out internal 
wave effects. The paired values of P and N given by both Kitaigorodsky 
and the present author are plotted on figure 10. 

In calculating P and N, it appears on the basis of the reference [l 7] 
that Kitaigorodsky used an average (climatological) heat flow to find Qp. 
The present author also used an average heat flow, the average heat flow 
since the time that the upper layer of the ocean was last isothermal. 
This average heat flow was arrived at in three steps. 

1. Calculate the heat added in the surface layer since warming 
began. The method described for Q_ in A>pendix III was used for 
this calculation. 

2. Determine the length of time elapsed since warming began. 

3. Divide value determined in 1 above by the value determined in 
2 above. 

Despite the different locations of the NORPAC and OWS Papa data and 
despite the differences in Qp, both sets of the points P versus N appear 
to fit the same curve (in figure 10). Still both methods for obtaining 
Q„ seem weak. In section 4, Qp is replaced by another parameter to 
represent stability in order to eliminate that weakness. 

Another problem in applying the method proposed by Kitaigorodsky 

dp 
lies in the steepness of the slope, ^, for the small values of N. 

44 







1 




1 


© 


>i ■ 
m 

© wt 

w i 

® © 




' 




; .. 


- 














computed 
computed 


' 


• 




• 


© 

© 


3 3 

> > 










- 


II II 
X O 




• 


X 


X 
X 

0* 


© 




X 




X 
X 


^0 

© 


X 


• 


1 


1 


1 — 1 


I 


1 





CU 



. in 



*- z 



- 10 



<*> . 



CM 



Va 



s of P versus H 



ted Using Equations Developed by Kltaigorodsky 
45 , : ■■,•■'' 



To avoid these problems, the author derived by similarity methods 
an equation with more readily-available parameters and a more stable 
P(N). 



46 



thesM1835 

Application of similarity theory to fore 




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