Ss ee ae, ee a ee

NEW YORK UNIVERSITY
COLLEGE OF ENGINEERING
RESEARCH DIVISION
®
Department of Meteorology and Oceanography

NOTES ON THE WIND DRIVEN OCEAN CIRCULATION

QE-ET-PR4

PERST,

x
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Prepared for
OFFICE OF NAVAL RESEARCH
Contract No. Nonr-285(12)
Washington, D.C,

Given in Loving Memory of

Raymond Braislin Montgomery
Scientist, R/V Atlantis maiden voyage
2 July - 26 August, 1931

2K 2K 2 2K 2K KK

Woods Hole Oceanographic Institution

Physical Oceanographer
1940-1949
Non-Resident Staff
1950-1960
Visiting Committee
1962-1963
Corporation Member
1970-1980
KKK KK
Faculty, New York University
1940-1944
Faculty, Brown University
1949-1954
Faculty, Johns Hopkins University
1954-1961
Professor of Oceanography,
Johns Hopkins University
1961-1975

HANNA

0 0301 0043553 3

NEW YORK UNIVERSITY
COLLEGE OF ENGINEERING
RESEARCH DIVISION

Department of Meteorology and Oceanography

NOTES ON THE WIND-DRIVEN OCEAN CIRCULATION

By

Gerhard Neumann

New York University

Prepared under a contract sponsored by the
Office of Naval Research,

Washington, D.C.

Preliminary Distribution

May 1954

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xX.

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TABLE OF CONTENTS

Preface wanGdicAibsinraCichen smectite nelitente! lel Mellel et 6
IMaKHicOC bonnie 5 o 6 o GO 6 oO .0 0.6 (OO) Oo) OO
BELGIGCHeI@EUL SSEAVILEN? G6 G6 6 6 60 6 6 0 6 6 D6

The differential equation of horizontal mss
transport e e e e e ° e e e @ e e e e @ e e

The "zero layer" in the Atlantic Ocean...
A simple model of a stratified ocean... .
Notes on the influence of the wind.....
The hydrostatic mass compensation .... .
The coefficient of internal friction (r) .

The wind stress at the sea surface .....

Conclusions, and remarks on the model of the
arAvenwoceanncirc dat vOMmnren esis lets luis tlre

Outi ne NOt tuUtLUTeh WOOF cabeute: fells us) “oulau uss
AcknowMediemontiSirus: ee ween tne) te. eect cere: “ont os

References Stites collh esr teh) elm sumoncuieein ies esnreee wererare

Page

Preface and Abstract

The main important difference between the dynamics of the
atmosphere and of the oceans is concerned with the driving forces.
The atmospheric circulation may essentially be compared with a
thermodynamic engine driven by the temperature differences between
low and high latitudes on the rotating earth, and this circulation
is to some degree modified by the land and water distribution and
by oceanographical conditions at the sea surface.

In the oceans, the temperature (and salinity) differences be-
tween the equator and the poles play only a minor role as long as
they are considered as a cause of the significant water movements
in the upver layers of the sea. The ocean currents are generated
and maintained chiefly by the driving forces of the wind. The
"thermodynamic engine" (including the"halinodynamic") is relatively
insignificant in the oceans, at least in the upper layers, where
currents with measurable velocities occur. Weak thermo-haline cir-
culations, however, will dominate in the deeper strata of the
oceans beneath a layer which will be called the "layer of no motion"
for the wind induced ocean circulation in this and the following
reports.

This report deals only with the wind driven ocean circulation.
The temperature and salinity structure is considered only as far
as the density stratification is involved in the final development
of the wind driven current systems. This density stratification

is--beyond the minor influence of "density currents"--very important,

Blak

since in a stratified ocean a "zero layer" (layer of no motion)
for the wind driven circulation must develop, which may be con-
sidered as the lower boundary of the circulation, where the velo-
city of the wind driven currents vanishes. Therefore, the "zero
layer" can also be replaced by a rigid but frictionless boundary,
and the necessary resistance in order to balance the driving
forces is brought about only by the virtual internal friction.

The wind driven circulation in an ocean with a variable
depth, D, of the currents is analyzed, and it is found that the
relative change of the depth D with latitude equals the relative
change of the Coriolis parameter in the large scale planetary
circulation. This suggests that the ocean reacts to the planetary
vorticity effect in such a way that it rather tends to adjust its
level of no motion for the wind driven circulation than to dis-
place the whole gyre with relatively high current velocities to
the west. Since the depth of the layer of no motion for the wind
driven circulation is also strongly related to the vertical den-
sity stratification, the mechanism of mutual adjustments between
the current systems and the mass distribution, as it is finally
observed in the oceans, appears rather complicated.

For a quantitative study of the wind driven ocean circulation
knowledge of the wind stress at the sea surface, and of the virtual
internal friction is necessary. Both of these important forces
have been discussed in detail in this report. With regard to the
internal friction a simple expression corresponding to Guldberg-
Mohn's expression was used, and coefficients of virtual internal

friction were determined. It seems that a more general treatment

Lit

of the internal friction will not essentially change the results.

The opinions of different authors are most controversial in
discussion of the wind stress at the sea surface, especially at
wind velocities less than 10 m/sec. It is this critical range of
wind speeds which comes into consideration when dealing with clima-
tological wind data, over most parts of the sea surface. At present,
more and more evidence has been accumulated which shows that the
effective wind stress at low wind speeds is larger than it was
hitherto assumed by most of the workers on this subject. There-=-
fore, the stress formula as suggested by the author in 1948 was
used to calculate the wind stress for given wind velocities.

Some special cases of the wind driven circulation beneath the
layer of frictional influence are analyzed, such as the meridional
flow along the east coast of the United States, and the zonal flow
of the Antarctic Circum Polar Current. The outline of future
work briefly discusses the ideas which may lead to a better approach
of wind driven circulation in a real ocean, taking the true average
wind field and the boundaries into account. This approach is based
on a numerical integration of the differential equation for the
horizontal mass transport in an ocean where the wind induced cur-

rent systems have a variable depth.

iv

NOTES ON THE WIND-DRIVEN OCEAN CIRCULATION

I. Introduction

Ocean currents with the greatest velocity are observed in
the upper layer of the sea with depths varying between two hundred
and two thousand meters. This circulation is generated and main-
tained mainly by the driving force of the wind. The question is,
to what extent can the observed conditions be explained on the basis
of the hydrodynamical theories as they have been developed up to
the present time?

The problem may be divided into two parts. Part one is con-
cerned with the "gross" or general features of the oceanic circu-
lation, and part two with the details which recently have been re-
vealed in some individual branches of the general circulation by means
of refined techniques for oceanographical observations.

There is no doubt that not only the details but also the gross
features of the oceanic circulation are the result of a complicated
mechanism in which different forces or factors are involved. It is
possible that the mechanism which governs some outstanding "gross"
features, such as the "intensification" of the Gulf Stream along
the east coast of North America and the "intensification" of the
Kuro Shio along the coast of eastern Asia, is essentially the same
as the mechanism which may account for some "details" as observed
in the western North Atlantic and in other individual branches of
the general current systems of the oceans. Such details are, for
example, the series of overlapping currents "arranged somewhat
like the shingles on a roof" (Iselin, 1952), and certain "counter

currents." It seems to be the tendency of ocean currents in strati-

aL

fied water to split up into narrow streams of higher velocity
rather than to form a broader and weaker band of flow. This
"streakiness" of ocean currents is not only present within the
swift Gulf Stream and Kuro Shio, but also in other branches of
the circulation along the eastern and western boundaries of the
oceans as well as in mid-ocean regions, although the individual
"streaks" are much weaker. At present, no satisfactory explanation
for such details in the flow pattern of ocean currents has been
given.

The general circulations in the Atlantic and Pacific Ocean
are asymmetrical with respect to the equator, and this asymmetry
is related to the asymmetry of the general atmospheric circulation.
As a consequence of this a special equatorial current system is
observed, which from the dynamical point of view needs some further
consideration. Common to the oceans, including the Indian Ocean,
is the fact that the equatorial currents are relatively strong
compared with the currents of the subpolar gyre. This fact, as
well as the east-west asymmetries in the North Atlantic, in the
North Pacific and partly in the Indian Ocean cannot be explained
in a simple way by the wind field itself. For example, the Gulf
Stream and Brazil Current are almost equivalent branches of the
subtropical gyre on the Northern and Southern Hemisphere, yet the
Gulf Stream transports about ten times as much as the Brazil Cur-
rent, and the gross features of the equivalent circulation systems
on both hemispheres show striking differences.

The present report is concerned with some notes on the general

or "gross" features of the wind driven circulation. Although no

final answer to the general problem of the horizontal and verti-
cal current stratification can be given, it is hoped that these
notes may serve to stimulate further interest in the subject

along lines which will result in a resumption of the problem in

the way it was initiated by Walfried Ekman.

II. Historical review

Approximately 50 years have passed since Walfried Ekman
placed our conception of the wind driven ocean currents on a com-
pletely new basis. In his well known theory he showed how the
effect of the wind goes far beyond the generation of pure drift
currents, and how the gradient currents in the deeper layers are
maintained by the indirect action of the wind. His studies were
directed toward the calculation of the total current system for
the given wind system over the oceans. He tried to solve the
problem for the vertical and the horizontal velocity distribution.
The difficulties which rendered Ekman's general analysis very dif-
ficult have not yet been overcome. Ekman, however, succeeded in
solving essential partial problems, and most of his results of
the first paper on the subject (1905) have survived half a cen-
tury; they still belong to our basic knowledge in dynamical oceano-
graphy.

Ekman was the first to introduce eddy viscosity (virtual
friction) into the theory of ocean currents. Ina paper from 1923
he deduced some important laws which, in the case of steady motion,
the currents and particularly their curl must obey. The acceler-=
ation of the water, originally regarded as effectively eliminated

from the equations, was later found to be important as far as the

"“deep-current™ is concerned. In his revised theory of 1932, the
dynamical theory of the wind driven ocean circulation and its
problems were developed by Ekman to such a degree of completeness,
that relatively few additions have been made since that time. Prob-
ably, the only point where Ekman's model of steady state ocean
currents has been revised essentially, and where in the future
essential revisions have to be expected, is concerned with the
mechanism and the effect of internal friction and vertical density
stratification.

A new epoch of theoretical research in the field of wind
induced ocean currents started with the work by H.U. Sverdrup
(1947), H. Stommel (1948), K. Hidaka (1949) and W. Munk (1950).
These authors essentially gave up trying to evaluate the vertical
velocity distribution and confined the analysis to the horizontal
mass transport by introducing the vertically integrated mass trans-
port as the dependent variable. This was necessary in order to
avoid certain difficulties which rendered Ekman's analysis too
complicated. It was possible to examine the more general case of
a baroclinic ocean without having to specify the nature of the
vertical distributions of density and currents, although even in
this case, some simplifying assumptions had to be made. The prob-
lem was to derive the gross features of the horizontal mass trans-
port of the oceanic circulation for a given wind system. These
stimulating theories can be considered as a remarkable step in
dynamic oceanography, and the results obtained by the various models
are of great interest.

H. Stommel succeeded in 1948 in showing that due to the effect

4

of the varying Coriolis force with latitude the gyre of the
general circulation around the Sargasso Sea must be strongly
displaced toward the west in the case of a constant depth of

the wind driven circulation. The results are very strong cur-
rents along the east coasts of the continents. This "westward
intensification" of ocean currents is obvious in the Gulf Stream
and the Kuro Shio, and it seems that a satisfactory explanation
of this striking gross feature of the oceanic circulation has been
given. However, remarkable exceptions are found in the South At-
lantic and South Pacific, and Stommel's model needs further con-
sideration.

Using interesting mathematical methods, Munk and collaborators,
Hidaka and others extended the theoretical research to more natural
wind distributions and more realistic ocean boundaries.

Munk's investigation (1950) has served to emphasize the fun-
damental importance of the wind stress curl rather than of the wind
stress vector in determining the transport of ocean currents in
meridionally bounded oceans. In the case of an irrotational wind
field, such as a uniform west wind over an ocean of constant depth
bounded on its western and eastern sides, it follows that there is
no net mass transport, and that the stress is balanced by the pres-
sure distribution resulting from the piling up of water against the
lee shore. This result becomes understandable only when keeping
the two-dimensional model and its theoretical basis in mind. The
theoretical basis of the model is essentially a differential equa-
tion representing a vertically integrated version of the vorticity
equation in the case of an ocean of constant depth. It expresses

a balance between the lateral stress torque, the planetary vorticity,

D)

and the wind curl. It is evident that the vertical velocity com-
ponent must be taken into account in order to explain the steady
state circulation in a bounded ocean. Otherwise, the result that
the permanent ocean currents are only related to the rotational
component of the wind stress field over the ocean would indicate
that the currents must vanish were the wind stress field irro-
tational.

Most of the important assumptions of the previous models are
concerned with the lower boundary of the wind-induced circulation
system, or the "depth" of the currents, and with the concept of
lateral and vertical eddy viscosity, especially with the concept
of isotropic lateral mass exchange.

With the attempt to explain some of the observed detail flow
patterns in the Gulf Stream north-northwest of the Azores, in the
California current and the Norwegian current by taking non-iso-
tropic lateral mass exchange into account, the author's attention
was called to some probable relationships between the vertical
density gradient, the depth of the current, the variation of lati-
tude while the current flows in meridional direction, and the "age"
of the current, that is, the time when a wind of a certain velocity
starts to blow over an ocean originally at rest.

The varying depth of the wind generated oceanic circulation
is the first fact with which we are concerned, and which has been
neglected in the previous models even in the case of a steady
oceanic circulation. This depth depends on the density strati-
fication of the ocean, the "age" of the current, and possibly on
internal friction (lateral and vertical). At this point some

difficulties arose, so that we are facing the problems of the gross

6

features of the steady oceanic circulation again.

In the present report it will be shown that the depth of the
wind induced oceanic circulation plays a significant role in the
hydrodynamic analysis of the problem. A more realistic model of
the wind driven circulation of the North Atlantic Ocean will be
treated by numerical integration of the basic differential equa-

tions in a subsequent report.

—— ee ee

Rectangular coordinates may be assumed for a first simplified
model. Let the y-axis point northward, the x-axis eastward, and
the z-axis vertically upwards from the undisturbed sea surface which
coincides with a level surface. The boundaries of the ocean model
may be considered as vertical walls, which at first approximation
would account for the continental slope. Besides these lateral
boundaries the lower boundary of the current system has to be fixed.

It is known that in equatorial regions the layer of no motion
("zero layer") for the wind driven circulation is found in about
200-300 m of depth, and it increases with increasing latitude to
approximately 1500 m in 45°-50°N. (The depth of this "zero layer"
should not be confused with the depth of the layer of frictional
influence.) There is much evidence for this fact, and it has been
assumed by most oceanographers that there is a continuous change
of this "level of no motion" between low and high latitudes. Much
support of this point of view has been given by A. Defant (1941)
in his extensive analysis of the Atlantic circulation, where inter-
esting details of the layer of no motion have been revealed.

The "zero layer" of the oceanic circulation in the wind affected

surface and subsurface layers may be considered as the Lower_boundary

7

of the current system. It may be replaced by a rigid surface
where the velocity vanishes (no friction at this boundary). There-
fore, the "zero layer" has to be considered as the depth D of our
model of wind driven ocean currents, and D is a function of x and y.
Let f = 2m sing be the Coriolis parameter, p the pressure,
u and v the horizontal components of the current velocity vector ¢
in the x and y direction, respectively, y [em7+ er sec +] the co-
efficient of eddy viscosity in the lateral, and p the coefficient
of eddy viscosity in the vertical direction, We the Laplace oper-
ator in the x-y plane and p the density. The hydrodynamic equa-

tions of steady horizontal motion are

piv + =
ie aee Ox
v
-pfu+ pL +0:V ave = = 0,
dz° ay
the hydrostatic equation
9
= =a pls (2)

and the equation of continuity

gus gu + om = 0, (3)

The system of hydrodynamic equations (1) to (3) together with
adequate boundary conditions provides the basis for a very general
solution, that is, to find the horizontal and the vertical distri-
bution of u and v. Although this solution is of great interest,
the mathematical analysis is too complicated. Therefore the
question of the horizontal mass transport may be considered only.
If the vertically integrated mass transport M, with the components

U, V is introduced as the dependent variable,

8

)
u= f puar , ve f py dz.

=D =D
Since the integration is carried out from a depth where the total
motion is zero (u = v = w= 0), to the surface, (z = O)% it follows

from (3) that

) )
- i pu dz +¢. ih pv dz=0,
Ap) aS
_ gu Wie
div w= $0 +900, (4)

Equation (4) states that the horizontal divergence of the

and that

total mass transport equals zero, in contrast to the divergence

of the mass transport in individual levels. For example in the
layer of frictional influence (wind drift current), the divergence
may be zero in exceptional cases only.

The special form of the equation of continuity for the verti-
cally integrated mass transport (4) permits the introduction of a
"stream function", W » for the mass transport, and
dW /oy

v=-dWAx
Integration of equations (1) over the depth between the level of

U

(5)

no motion (z = -D) and the surface (z = 0) gives

ope nd
ae ay eo! ee

@)
2 =
V 2 0
yy af ca Vin O

—

* Actually z = C » where C the elevation or depression of the sea
surface with reference to,the level surface of the undisturbed sea
surface. However, since is a very small quantity it can be
neglected against D. ;

(6)

where Op/ox and dp/dy represent average values of the horizontal
pressure gradients in the whole layer between z = -D and z=0.

The two integrals in equations (6) are equal to the differences
of the components of the tangential stress at the surface and at
the depth =D. The tangential stress at the surface can be evalu-
ated from the wind at "anemometer height" by means of an empirical
formula (Neumann, 1948). This question, however, needs some further
consideration and will be discussed in more detail in section IX
of this report. The tangential stress in the depth z = =D equals
ZeYrO.

Besides the wind stress at the surface, horizontal and vertical
shearing stresses in the water have to be taken into account. The
dynamical effect of these internal stresses is evident in the vir-
tual internal friction. Since it is difficult to introduce internal
friction in a most general form, and in order to simplify the dif-
ferential equations for the mathematical analysis, it seems justi-
fiable for the purpose of the present report to assume that the
frictional forces are proportional to the total mass transport M.
The combined effect of the virtual friction in horizontal and verti-
cal direction will be considered as an "effective" internal friction.
For the whole water column between z = =D and z = O the components

of the effective internal frictional forces are assumed to be

Xie abe
Ry s+ Qe ;

where r[sec 7+] represents an effective "friction"-coefficient. A

R

CD

plausible average value of this coefficient has to be determined

later.

10

With these assumptions equations (6) can be written

f he - oe SP 38 ay + & D=0
qu ry - rgb + Bo

T._ and Ty are the components of the wind stress at the sea sur-

(8)
0

x
face in the direction of the x-axis and y-axis, respectively.

Cross differentiation of (8), and subtraction leads to

r (Sof EY) «egy «Bae ap ae)

According to equations (8)
ggugv .gangh . tz go tu gp goat , gpg) (10)

By substituting this into equation (9), it follows

Dee ee bb ieee bas Boe)

Ox dy?

8 I =a (11)

With this differential equation and adequate boundary conditions

the horizontal mass transport is uniquely determined in the ocean
under consideration. The boundary conditions for the North Atlantic
Ocean will be specified later.

In the special case where the depth D is assumed to be constant,

it follows from (11)

rv-w ts gf qv = - curl, T ,» for D = const.

This equation is identical with Munk's differential equation for
the horizontal mass transport (19503 equ. 6), if the sign of W is
changed (Munk defined V = gW /ox, was aWV Ay) and the term rV< W

11

is replaced by A,V in Munk's notation.

Before an evaluation of equation (11) several important factors
involved in the model have to be discussed and to be fixed. This
is concerned with D as a function of x and y, with the wind stress
at the sea surface and with the effective friction coefficient,

r, in the water as defined before.

IV. The "cero-layer" in the Atlantic Ocean

A successful attempt to determine the depth of the zero layer
(layer of no motion) in the Atlantic Ocean was made by A. Defant
(1941). When comparing the differences in the dynamic depth of
given pressure surfaces between adjacent oceanographic stations,
it was found that in certain levels this difference was almost
constant over a rather large depth interval. This would mean that
either the whole body of water throughout this depth interval
(sometimes with a thickness of 500-800 meters) has a constant velo-
city in the vertical direction, or that this layer is absolutely
at rest.

Defant's analysis of the North and South Atlantic Ocean has
shown that it is much more plausible to assume that this layer is
absolutely at rest, than that it is in relative motion to some
other "reference level" in the body of water. Defant assumed the
average depth of this layer to be the "level of no motion" for his
dynamical computations. This hypothetical level of no motion was
traced throughout the whole North and South Atlantic, and it was
found that it composes without any constraint to a closed layer
of varying thickness. This fact and others as pointed out by

Defant may serve to strengthen our confidence in Defant's analysis

12

of the circulation of the Atlantic. Any other assumption of the
level of no motion would lead to unreasonable and contradictory
results about the vertical structure of the Atlantic circulation.
The same hypothesis was applied to a dynamical analysis of the
Black Sea circulation by the author (1942,1943). In this case

the results could be checked against other techniques in deter-
mining the level of no motion, and the computed dynamical structure
of the Black Sea was in good agreement with the observed conditions.

Defant's results for the Atlantic Ocean are shown in figure l.
The depth of the “zero layer" (D) increases with increasing lati-
tude, both, on the Northern and on the Southern Hemisphere, In the
North Atlantic, the depth of the zero layer is more irregular than
in the South Atlantic, which has its analogy in the more compli-
cated current pattern of the North Atlantic.

Based on the chart shown in figure 1, zonal averages of the
depth of no motion (D) were computed for the North Atlantic, and
plotted against the latitude in figure 2. For the South Atlantic,
individual values were used as given along the meridian of 20°W,
since the variation of D in the east-west direction is small com-
pared with the variation of D in the meridional direction. The
values of D, according to A. Defant's chart, are indicated in figure
2 by circles, whereas the curve represents the function D = -K sino.
The constant K is different on the Northern and Southern Hemisphere,
but the variation of D with latitude follows this law closely. It
is seen that the zonal averages of the depth of the layer of no
motion in the North Atlantic, and the corresponding depths along
20°W in the South Atlantic, can be represented very closely by

D(p) = -K sing, (12)

13

ERTT

NST ; VW
—_ a = Z
aS Sly |. MkasAG
. ‘tt ; A 5
wae Sate a
=a wt SS =
=| ir ; ‘ali
f at 4
fT nm | Sm |
(lesa. fe
pace :
~

Fige 1. Depth of the "Zero Layer" in the Atlantic Ocean,
according to A. Defant (191). The figures at the
lines are to be multiplied by 100, thus denoting
the depth in meters,

14

N 60° 50° 40° 30° 20° 10° o° 10° 20° 30° 40° 50° (oe 8
fe) ———SEEEe fe)

Figure 2. Average depth, D, of the layer of no motion in the Atlantic Ocean.
The curves represent D (¥) = const. sing , and the
values according to A. Defant (191) are indicated by circles (0).

or, the relative variation of D is given by

492-2
Doy R ctn @ ,
where R is the radius of the earth. Since
1982 2
there is much evidence that in the "gross" features of the oceanic

circulation

se = £2 , (13)
and the important term

(gt - § 2)
y D oy ¢
in equation (11) drops out in the "gross" features of a model where

only zonal conditions are considered.

In the model by Stommel (1948), Hidaka (1949) and Munk (1950)

15

D was assumed to be constant. Therefore, the term responsible
for the "westward intensification", gt gy » which contains the
variation of the Coriolis parameter with latitude (planetary vor-
ticity) was not cancelled out by the variation of the depth D with
latitude (topographic vorticity, according to Ekman's terminology).

It seems unlikely that the east-west asymmetry of the gross
features of the oceanic circulation is caused by the variation of
the Coriolis parameter with latitude, but since there is an inten-
sification of ocean currents along the east coasts of North America
and Bastern Asia, this particular flow pattern requires another
explanation. This may also answer the question why an analogous
intensification of the subtropical gyre (Munk, 1950) is not found
in the Southern Hemisphere.

It seems more likely that the ocean reacts to the planetary
vorticity effect in such a way that it rather tends to adjust its

i a a a

Whether or not this adjustment is complete is a question which still
has to be answered.

It may be added that the relationship as stated in equation (13)
does not say anything about “cause and effect". The zero layer may
be the result of the current system, and probably it is to some
degree in the gross features and even in the details. However,
several other external and internal forces and factors, besides those
taken into account, are involved in the mutual relationships between
the oceanic circulation and the stratification. Some of these
factors are the large scale heat gain and loss through the sea sur-

face and evaporation, precipitation, and lateral and vertical mixing

16

processes. The first mentioned climatic factors and mixing processes
are especially responsible for some large scale density strati-
fications. With "large scale density stratification" in this sense
is meant, for example, the very weak vertical density gradient in
polar regions and the strong, significant vertical density gradient
in tropical regions, both taken as an average between the surface
and the bottom water. The vertical density stratification is an
important factor in the oceanic circulation, which, at least more
indirectly, may determine the wind driven circulation. This will

be shown by a simple model in the next two sections.

V. A_simple model of a stratified ocean

Disregard at first the differences in the salinity distribution
and assume the temperature distribution as the main factor of the
density stratification. The vertical temperature distribution as
observed in the deeper layers of the oceans is to a certain degree
usually considered as the result of the oceanic circulation or the
"spreading" of water masses (G. Wust, 1935 ime Lndish ts TOnlyapancLy,
true, as shown by H.U. Sverdrup (1939). Stationary conditions which
appear on the average "gross" features to be rather similar to the
observed conditions in the ocean may exist even in the case where
the components u, v, w, of the oceanic circulation are zero; that
is, in an ocean completely at rest.

Assume in the first step an ocean at rest. This "ocean" may
cover the whole earth, and only the sun's radiation (with zero de-
clination) may warm up the equatorial regions. The depth of the
ocean may be constant (= h meters). In this case, the resulting
temperature distribution will be symmetrical in the meridional di-

rection and equal in the zonal direction. Since u = v = w= 0 by

17

assumption,

gg. (9). 9 (2H) od (ead) ,

where IT is the temperature, and A, A A, are exchange coefficients

y?
for heat conduction. With an east-west symmetry, as in our model,
we need only to consider the resulting temperature distribution

in a meridional cross section. Stationary conditions are possible

g. (4 92). 9. (292)

Due to the sun's radiation there is a heat gain at the equator

alng

and a heat loss at the poles. Suppose that a meridional temperature
distribution is maintained at the sea surface, and let T(y) at the
sea surface be

e T
TCa= Zs +T, cos apy; (14)

where Th is the temperature of the sea surface at the pole,
(y =b), qT + 1, the temperature at the equator (y = 0).

If, for simplicity, we assume A, and A, to be constant in

u/
space, but A, > Aas a solution of the differential equation
2 2
A en 22h 2G (15)

HS cosh ath ="z)'. Te
T(y,z) = Ty, mabe Basho Gh cos SPY > (16)
where
nS Xv
2b AL

Assume that the ratio A,/A, ~ 10’, and then, with b = 90° ~ 10,000 Kn,
a= 0.497 .
Further, let T, = 25°C, Tm = O0°C, and let h = 6000 n.

18

With the assumptions made, stationary conditions of the
temperature distribution in an ocean at rest would be possible if
the temperature in different depths at different latitudes follows
equation (16). In the following table this temperature distribution
in a meridional cross section between 90° latitude and the equator

is computed on the assumption that AL/A, = 10’.

50 40 30 20 10° 0)

Oc4O; 25 Or 364 95.65) 8.9 1255 1654 2007 *25.0
OLA) Wacky Beal) sigs big Vey vale le ate) waliyacy
OnljmeOwou lees eee "B63. 4.7 Gedy "FoF 952
OnO9snOn85, 0579) 1635 2.1 269 Be0n 468) 568
OnOon Ones Oe 52 (0.09 36 1.9 cy Vlad shah)
OsO45AO G7 etOegO 0705 eOOs4 | W041 2532) 2.8
O040,0515 0.34 0.56 0.88 1224 1.62) 2.05 2.46

0 0
1 0
2 0
3 )
4 0
5 0
6 0

This temperature stratification (figure 3a) shows rough simi-

larities with the observed conditions in its very gross features.
These similarities are the strong vertical stratification in tropical
and subtropical regions with decreasing vertical gradients towards
the bottom and the homogeneity at the poles. The distribution of

the isotherms in this model (figure 3a) may lead to the impression
that the cold water from polar regions spreads out along the ocean
bottom towards equatorial regions. Even a more realistic picture
could easily be derived with the assumption of an asymmetrical ten-

perature distribution at the surface (Northern Hemisphere warmer

than the Southern).

VI. Notes on the influence of the wind

In the second step, suppose that the wind blows over this ocean.

19

ON
(©)
Depth in (Km)

n non fF w NY

|
i}
|
\

nh Od

Fig. 3a. Computed temperature distribution in an ocean at rest \cfeleett Di
of heat conduction with given sea surface conditions A,/A, = 10!),

3b. Adjustment of mass distribution (temperature) to wind induced cir-
culation with the additional assumption of geostrophic equilibrium
(schematic). The line + -+ -+ indicates the layer of no motion
for the wind induced circulation.

20

The prevailing winds are east winds between the equator and 30°,
west winds between 30° and 60°, and weaker east winds between 60°
and the pole. The primary effect of the winds acting at the sea
surface is to produce drift currents with velocities which decrease
with depth and which are essentially limited to the upper 50 m or
100 m of depth (layer of frictional influence). In our model, fig-
ure 3b, which is considered to represent the Northern Hemisphere,
the total mass transport of water in the layer of frictional in-
fluence (d) is directed toward the north between the equator and
30°N, and between 60°N and the pole, whereas the total mass trans-
port is directed towards the south between 60° and 30°. The con-
sequences are a divergent flow in equatorial regions and around 60°
and a convergent flow in the latitudes around 30° and near the pole.
The divergence (or convergence) of this mass transport is propor-
tional to the curl of the wind stress, and it can easily be computed.

The unequal divergence at different localities, which strongly
depends on details of the wind field, sets up a slope of the sea
surface, in our simplified model, which drops from 30° toward the
equator and towards 60° and rises again from 60° toward the pole as
indicated schematically in figure 3b. From this, hydrostatic pres-
sure gradients result in the water, and, as an indirect consequence
of the wind effect, gradient currents will develop in the deeper
layers (in addition to the "density gradient currents", simply called
"density currents", due to the original horizontal temperature dif-
ferences). The depth of the wind induced gradient currents depends
on the vertical stratification of the ocean water.

In what follows, we may neglect the "density currents" as they

would result from the original temperature distribution in the model.

21

To examine qualitatively the wind effect on the stratified ocean,

as shown in figure 3a, let us assume that the velocity of the wind
induced gradient currents decreases with increasing depth. In the
steady state the sea surface slopes mist have reached a certain
stationary inclination, and the mass distribution (temperature field)
must have changed in such a way that beneath the layer of frictional

influence from point to point the geostrophic equilibrium condition

bz = tany 3 eo

is nearly fulfilled, or that, approximately,

qu. gig

zZ f p oy

holds. Here, u is the zonal velocity of the gradient current in
the x-direction (indicated in figure 3b by feathered arrows), p the
density (which in our model is only a function of the temperature
at constant pressure), and y the inclination angle between the lines
of equal density and the (horizontal) level surfaces. Qualitatively,
the distribution of isotherms in the meridional cross-section under
these conditions should look like the dashed lines, as shown in
figure 3b.

When looking in the direction of the current between 60°N and
30°N the isotherms must slope down from the left hand side to the
right hand side, that is, from north to south, provided the current
decreases with increasing depth, and they must slope down from the
equator towards 30°N, since the wind induced gradient current in
this region flows from east to west. This leads to a thick warm
water accumulation around 30°N, and to a relative cooling in corres-

ponding depths in equatorial regions. Furthermore, the vertical

22

temperature gradient in certain subsurface levels between 30° and

the equator must increase, and beneath this region decrease, when
compared with the original conditions for an ocean at rest, since

the geostrophic equilibrium condition requires a tilting up of the
isotherms toward the equator in the layer of the equatorial (grad-
ient) current. The depth of this layer depends on the stratification,

and it is indicated by the line + +-+ in figure 3b.

VII. The hydrostatic mass compensation

The condition of a steady state requires that for each column
of water, the divergence of the total mass transport M from the sur-
face to the bottom or to the level of no motion equals zero. Let
My be the mass transport of the drift current (in the layer of
frictional influence), and let M, be the mass transport of the
gradient current (from the surface to a depth where the current
vanishes); then div (M, + M,) = 0, or div M, = - div M, for each
column. Only the total mass transport, integrated from the surface
to the layer of no motion, is non-divergent. But the layer of
frictional influence is a divergent layer, and this divergence must
be balanced by divergent gradient currents in deeper layers.

In the case of an ocean with constant, or almost constant
potential density from the surface to the bottom, the hydrostatic
pressure gradient, as caused by the "wind set up", should practical-
ly extend down to the bottom (as in figure 3b near 90° of latitude).
Here, a "bottom current" as assumed in Ekman's “elementary current
system" would provide the necessary anisobaric mass transport in
order to balance the anisobaric mass transport in the layer of

frictional influence. However, in a strongly stratified ocean, (as

23

in fig. 3b between 30° and 0°), a layer of no motion may develop
(with steady state conditions) in a certain depth above the bottom
by "mass-compensation" of the hydrostatic pressure differences re-
sulting from sea surface slopes. In this case, the necessary balance
for the mass transport in the layer of frictional influence has to
be provided for completely by the wind induced gradient current.

If the hydrostatic equation holds in vertical direction, the

pressure p at a depth, h, is given by

where Ce. is the elevation of the sea surface above the level sur-
face z = 0, and Po is the density of the surface.

Assume two oceanographic stations, A and B, fairly close to-
gether, for which the vertical density stratification is known, and
suppose A é is the elevation of the sea surface at station B re-
lative to station A where G. = 0. Then, the depth of the level
of no motion for the current perpendicular to the line AB, between
these stations is given at an average (constant) depth, z = -D, where

(pia = (pp 3) lp

and

aC =-+ i (pi, = Pp) Gz. (17)
Po

Let the coordinate system be oriented with the x-axis in the (zonal)

wind direction. Then, in the layer of frictional influence

24

T,
(iy Sarr (M,), = 0 (18)

x
according to Ekman.

The wind induced gradient current is a non-divergent current
only as a first approximation. As an effect of internal friction
there must be a slight deviation from the direction parallel to
the isobars. The average angle of deviation (from high to low
pressure) in the gradient layer may be denoted by a. Let u, be

the zonal component of the geostrophic current in the case of no

internal friction, then

and 0 0 P
U,. ff pu, dz f A dy Zz

Suppose that the actual total horizontal mass transport, U, in
the gradient current is deflected slightly from the direction of
U, by the angle a, as schematically indicated in figure 4. The
magnitude of a was estimated as 1°-2° in higher latitudes and

5°-6° in lower latitudes (Neumann, 1952).

2 0~«~8
Fige h. ‘The difference vector U =U, represents approximately the aniso-
baric mass transport to balafice the mass transport M, in the layer

of frictional influence.

25

=>
The difference vector U - U. represents approximately the
anisobaric mass transport M, to balance the mass transport My

in the steady state, and

= Op
M, = - + sina f oy dz. (19)

=D

With the hydrostatic equation

fe) © op
ye jy 12

and

(p, - Prd
My = é g sina ({== ae | dz (20)

for two closely spaced stations, A and B, with a distance A y apart.
From the steady state condition, div M = 0, and from (18) and
(20) we have with the assumption that between the two closely

spaced stations the variation of My and My can be neglected

0) 0
D

This equation can be solved numerically for D, the level of no
motion. The zonal wind stress, TT,» can be determined from the
wind field, and with regard to a the assumption may be made that a
in the region of the equatorial current is approximately 5°-6° and

decreases to 1.5° in 60° latitude.

26

Figure 5 shows the zonal average density distribution in a
meridional cross section through the South Atlantic Ocean. The
zonal average of the density was computed from the data as given
by G. Wust and A. Defant (1936) in the atlas of the "Meteor" work.

From Schott's (1942) wind charts, zonal average wind velo-
cities in the South Atlantic have been derived, and the wind stress,
T,» has been computed by means of an empirical formula (Neumann,
1948). More information about the wind stress is given in section
IX of this report. The zonal wind velocities Wee the wind stress
La and the estimated values for a are given in Table l.

Let us aSsume equally spaced "stations" with a meridional

distance of A y = 5°, between 5.5°S and 57.5°S. From (21) with
Awe-woo> anand O-= (po — 2) x 103, we have for an estimate

) 0)
_D -D A B

The average vertical density distribution as shown in figure

5 is tabulated for the numerical integration in table 2.

Table 1

Latitude HOM Os a5 10 1 20 2

W,, (m/sec ) O45 eesO) — AL@ Shige eas 0

T,(dyne/em*) 0.05 0.57 0.9 1.38 0.74 0

ac as 7e7 602 4.7 309 322
Latitude OBS 40 45 50 55 60°S
W, (m/sec ) Bly) 4.9 Bath Cl 8.7 7.6 4.2
T,,(dyne/em*) 0.50 1.22 2.27 3.25 2.87 2.36 9.97
a° Dig Be De Ol) eG 1S

SOOO OOO OO
SS SS SS SS SS SS

27

°(@) suotzetnoTeo yyTm pereduos (+ +— -+) quejed °y 04 S3urp109008 uoTyou ou Fo IakeT ayy jo
uqgdep eS3ereae pue Sue|09 OFZUETLV YZNOS ayy Ysnory, (%) UoTZOesS AQ TsUep TeUOTpTIew asereay

09 cs Os Gb Ov ce o¢ Gz 02 cI Ol S 0)
Se apnyiyo7

°S °3Td

28

98° 58° v8 ° €g° v8° vish 98° 48° 98° 48° 48° 88° O0SE
S8° S8° cB ° cB ° Tg° v8° 98 48° 98° 98° 48° 48° ooot
S8° v8 ° ESS 08° 08° T8° S8° S8° S8° v8 ° 98° hele 00S2
G8° €g° gZ° GL° tZ° GL° 08° 2g ° 78° cg ° €g° €g° 0002
vg° 78° 94° 89° €9° 89° vL° gZ° 62° 64° 0g° T8° OS2T
€g° Tg° eL° 79° vS° 65° 99° oZ° €Z° vL° S2° 94° OOST
Age gZ° S9° TS° €vr° 9° vS° 6S° A €9° S9- S9° OS2eT

mo. S/4e LS° Tv° Te° EES Le° vr° 9r° Lv° gr? (es7° OOOT
62° 69° 60° Ago 1sae Oc° SGe 1hg° ve? Gee 9e° (GSC 00g
gZ° €9° Ov? SAG GOebc OO elon COrLce ea. mOlLeicameyice See eae 009

€L°Ge 9S°le O€°4Ze vile 06°92 €4°9e e4°9e 64°92 6°9e SO°9e ZO°4e vO°Ze O0F
Goctc soecele Sbide. T6°9e @9°9c -Z¥°9d E29 O89e 6E°9G —ZS29c ~1479e -99°9¢ ‘00d
Ome LeuOOede vleGe ver9e beGea 9G9Ge ve°Ge OtrGe ThcSc Tretoe Sieve. eScech 20

——

——

09-55 SS-0S 0S-G¥ G¥-OF OP=St Gt-0F Ot=Se Se=0¢ Oc=Si ST-OL OT-S $-0 yadep

2 eTde,

29

For two fictitious "stations" at 42.5°S and 47.5°S, respect-

ively, the following vertical distribution of

i O
i = | f ( o. = o.. Jaz dz
ny D A B

is derived:

ea Ac, = :
m —= = = =
an = Ut) Ao Ae zo, - Az) T TA z I
x 10* x 10° P10" | 2100) nee
0) 0.40 9) 0
0.64 ? 0.32 0.64 5a
00 0.24 0.64 A

: 0.40 0.84 1.68

400 0.16 1.04 2.32
0.33 AZO LaZsl@)

600 0.17 1:37 4.73
0.34 1.54 3.08

800 On? TA y Al 7,81
0.33 MG elses)

1000 0.16 2.04 11.56
0.375 epee | Gece)

1250 0.14 2.415 WG) 51.3)
0.300 2.565 6.42

1500 0.10 2675) 23D)
0.225 2.628 17208

1750 0.08 2.940 30.63
0.137 3.085 7.70

2000 0.03 3.077 38.33
0.100 3.127 15.64

2500 Ono! SlL7. BB SY
0.025 3.189 15.95

3000 0.00 3.202 69.92

Tt
With Ges 3625 dyne/cn*, and a = 1.9°, Sing 2 565) 3 10” =
Dox 10°,

therefore D = 2500 m for the zonal component. This compares fairly
well with the depth of the layer of no motion as shown in figure 2.
Similar computations for higher latitudes (55°S) lead to values

which are a little too large, whereas the depth D computed for the

30

South East Trade wind region (10°S through 35°S) agree well with
Defant's values. The computed values D are plotted in the den-
sity section, figure 5, for comparison with the depth D in the
central parts of the South Atlantic according to figure 2.
Probably, the estimated angles a are the most inaccurate.
However, the estimate of D by equation (22) should serve only to
check the order of magnitude of the factors involved in this pos-
sible mechanism of “mass compensation" to determine the level of
no motion. Another fact which must be kept in mind is that this

calculation only considers the meridional component of the ani-

sobaric mass transport.

At this point, the dashed curve in figure 5 may at least be
mentioned briefly. It connects the maximum depth of the lines of
equal density ( @,), and is in some way related to the tilting of
the axis of the subtropical gyre. This tilting in meridional
direction toward the poles with increasing depth has already been
mentioned by A. Defant (1941), but so far no explanation has been

given for this gross feature of the oceanic circulation.

VIII. The coefficient cf _ internal friction (r)

In figure 6, i, is the vector of the geostrophic mass trans-
port, and il is the vector of the true mass transport which is de=-
flected by the angle a from the direction parallel to the isobars.
The difference vector, u, = Ml » is in some way related to the average
internal friction in the whole water column from z = 0 to z = =D.

Assume horizontal, non-accelerated motion under the influence

of internal friction. Then

aL pe = => a,
= z grad p#+#M xftReEO, (23)

31

Figure 6. Force diagram of a non-accelerated gradient
current with friction.

where R is the vector of the internal frictional force. Fora

geostrophic current without friction,

eee > =)

- 4 erad p+, xf = 0 (24)
On substituting (24) into (23), we have
> > = =

(MZ, - Mx feR. (25)

Thus, the vector of internal friction is perpendicular to the
difference vector If, = M, and only in exceptional cases is B equal
to zero.

Suppose a is known, then 8 and IR | = R can be computed. In
equation (8), for the components of the effective internal friction,

we used for a first approximation Guldberg-Mohn's expression

RS =- rU

by assuming R =-=rM. As in figure 6, let é be the vector of the

32

average pressure gradient force and C the vector of the Coriolis
force. Then from figure 6,

G sina = rM cosg (26)
and G cosa = 2a sing M + rM sing , (27)
since the components of the forces in the direction of M and per-
pendicular to it must balance. Kec i, - i is proportional to the
average "dragging force", according to Sprung.

From (26) and (27)

_ 2w sing
etna = — cosp + tanp (28)
H _ 20 sing M
tang = ctna @osina (29)
In the case where Bp ~0
~ 2 sin
ctna = ae . (30)

So far, no direct computations of r have been made in dynamic
oceanography, and it seems that the only estimates of a are those
used in this report (Neumann, 1952).

Whether (30) can be used instead of (28) to calculate r from
a, depends on the order of magnitude of —B. However, data which are
accurate enough to compute Bg are not available.

The mass transport of the Antarctic Circum Polar Current between
65°S and 45°S, is given by

Ms 1014 Cer sec +] ;

approximately, according to Sverdrup et al (1946). (This value may
be even a little larger.) The average meridional pressure gradient
G can be estimated fairly well from the "Meteor" data published by
G. Wust (1939). Using "Meteor" Stations Mt 130, 124, 60, and 10,

the following pressure values, p, in decibar are obtained, relative

to the sea surface where p = 0:

33

9) 0 0 0 0
50 51.368 51.361 51.358 51.336
100 102.772 102.740 102.730 102.682
150 154.192 154.137 154.114 154.043
200 205.626 205.552 205.513 205.425
300 308.529 308.428 308.355 308.237
400 411.481 411,361 411.252 411.110
600 617.534 617.388 617.209 617.021
800 823.783 823.616 823.377 823.137
1000 1030.228 1030.039 1029.758 1029.461
1200 1236. 869 1236.660 1236. 343 1236.000
1400 1443.703 1443.477 1443.127 1442.745
1600 1650. 728 1650.489 1650.109 1649.695
2000 2065.352 2065.092 2064.659 (2064.191)

2500 2584.70 2584.41 2583.93 ~

3000 3105.24 3104.92 3104.39 ~

decibars
Oy Any deeibers

Mt. 124

Mt. 60

From these data G ~ 1.4 x 107* dyne/em.

Mt e 10

14

With M = 10 gr

sec", D = 3000 m and a total width of 2.22 x 10° Km, it follows

per unit area of the cross section that

1

emred < 1014 er sec71

Migr sec”
Boers o¢ 8\-(0) 5.3 1023 em@

= 1.5,

and, with a ~ 1.7°, according to Table 1, for 9 = 55°,

tanB = 34 - 43.
This result can only serve to check the order of magnitude of
a, G, and M, since tans is given as a small difference between
two larger quantities which are known only approximately. There-
fore, B can not be determined from these data. However, since the

deviation of the true mass transport from the direction of the

34

geostrophic mass transport is very small, the assumption may be
justified that Bp ~ O for the purpose of estimating r. Then, from
(30) and table 1 for 9 = 55°

20 sing
etna

In the lower layers of the atmosphere, r is on the average

re-~

= BV cVl ee 1076 [sec]
approximately 100 times larger. H.U. Sverdrup and J. Holtsmark
(1917) derived for 10 North American stations average values from
50-100 individual observations. They found r between 7 x 107? and
Dah ea tome Becaun and £ between -12° and +48°,

The height of the anemometer above the ground is of great in-
fluence. Both r and 8 decrease with height, as shown by Sverdrup

and Holtsmarks:

height (meters) 25 25-49 50
r x 10° 2.20 1.99 1.56 sec
8 250 20° 17°

With increasing height, B and r decrease in the lower layer of
the atmosphere, but it seems that no fairly well established in-
formation about these friction coefficients is available for the
higher troposphere, which is comparable with the layer of wind
driven ocean circulation beneath the layer of frictional influence.
It may well be, that in the higher troposphere B-—>0, magn. a * 2°,
and magn. r * 10m 1076,

According to equation (30) etna should vary approximately
proportionally to sing, if r is considered to be constant. Since
a@ is given in table 1 by previous results, r in different lati-
tudes is shown in the following table:

35

Latitude 10° 208 JOSE kes SOSmmanOOus
a 6.2 3.9 2.7 2.1 1.7 Th 5
6 2075 Boks 243) soes4 330) 3630
With the estimated angles a, r is almost constant. This may

5p Se Alo)

indicate that in the average for the whole layer of the gradient
current, frictional conditions are rather uniform in the different
branches of the circulation system, although under special con-
ditions significant local variations of the effective frictional
forces may occur. For the numerical integration of the differential
equation (11), however, an average value
PS sg ax 1076 [sec]

will be used.
IX. The wind stress at the sea surface

Estimates of the total wind stress at the sea surface rest
largely on observations of the vertical distribution of mean wind
near the sea surface. Such wind profile measurements in the lowest
layers over the sea surface have been compared with the velocity
distribution in circular pipes, so that the interfacial stress could
be deduced from the measured wall stress in pipes. This method
Simply applies the results of experimental, aerodynamical studies,
and is, therefore, based on the assumption that such a comparison
is justified.

Regardless of the question as to how far the analogy between
the wall stress in pipes and the wind stress at the free sea surface
goes, there are still other objections against the method used ,

especially with regard to the method of observing the average verti-

cal wind profile with cup anemometers at fixed "heizhts" above the

36

undisturbed sea level (or above the highest wave crests).

Most of the observations seem to indicate that with an adia-
batic lapse rate in the lower layers above the sea surface the
observed wind profile can be approximated fairly well by a loga-
rithmic law within a certain height interval. However, in the lowest
layers, some 50 or 100 cm above the crests of the sea surface rough-
ness elements (waves), the logarithmic curve gradually steepens
more and more when approaching the "sea surface". This feature is
characteristic for the observed wind profiles over the rough sea
surface, and a careful analysis of the observations shows that the
logarithmic law for the wind profile just above the waves is a
rather questionable approximation.

The value and accuracy of wind profile measurements with cup
anemometers for evaluating the wind stress at the sea surface have
been discussed in some detail by the author (1949, 1951). There
are mainly two facts which make this method unfit for stress com-
putations:

First of all, the questions remains open as to what level has
to be chosen as the "height of the sea surface", or, in other words,
what is the "sea surface" in the case of an air-sea interface with
traveling waves? Any attempt to derive a vertical wind profile
from observed wind speeds at fixed heights above the crests, re-
quires a clarification of this question.

Secondly, wind measurements with cup anemometers at fixed
heights above the wave crests are unfit to derive true average wind
speeds over the rough sea surface. The wind at a fixed height over
the trough of a "wave" may have only 40% - 50% or less of its speed

over the crest. With steep waves, even a "lee-eddy" may develop

37,

somewhere near the trough, and the wind direction may be reversed.
Since wind profile measurements near the surface have been taken
only at wind speeds of, approximately, less than 10 m/sec, the
rather short period wave motion passing by underneath the cup ane-
mometers "speeds up" the anemometers every time a crest passes.

The inertia of the anemometers does not allow slowing down over

the troughs, and therefore, almost maximum wind speeds are observed
instead of average values. When used for stress computations, too
small values of the stress are obviously derived by this procedure.
The "critical wind speed" at 7 m/sec as discussed by W. Munk (1947)
can be explained as a result of the inaccuracy of the data obtained
by wind profile measurements (Neumann, 1951).

Another more direct method of deriving the wind stress at the
sea surface is based on observations of the steady state sea sur-
face slope in which the water is piled up due to wind ("wind set
up"). This method is the oldest, and it was first used by W. Ekman
in 1905. Since that time much more data on the "wind set up" in
different seas and lakes have accumulated, and with these data, the
method was reexamined and critically applied by the author in 1948.

The stress, T , may be expressed in terms of a resistance co=
efficient, y°, which is defined by

ay) Ae eG (31)
where p' is the density of the air, and W is the mean wind speed at
a given distance above the sea surface. In practice, this distance
is given by the “anemometer height", about 10 m. The question is

whether ‘=

is a constant, or whether it depends on the wind speed
itself. In the case of a flow over a solid surface with well de-

fined and constant roughness conditions, such as grass land or snow,

38

e may be a constant for different wind speeds. However, in the

case of the sea surface, the "roughness" is given by the composite
wave motion. Since the wave motion is a function of the wind speed,
W, (and of the state of wave development), y° will probably also be
a function of the wind speed. (In the case of stable stratified
air, y also depends on the stability of the air stratification. )

The results obtained by the author in 1948 indicate that under
normal conditions (with an adiabatic lapse rate in the lowest layers)

a varies inversely proportional to the square root of W, and

ut lem/seo1}) 1/2
W [cm/sec ]

for winds from 1 to 40 m/sec. Thus, the wind stress at the sea

y° = 9.0 x 10°3( (32)

surface (for a fully arisen sea) can be represented by
eet Ol ogous Wao) Laynevencie (33)
where W is to be inserted in centimeters per second.

A comparison of (32) with the corresponding resistance co-
efficients obtained from wind profile measurements was made in
1951, and the results are reproduced in figure 7, where more recent
cup anemometer measurements by U. Roll (1948) show again the char-
acteristic discrepancy between the 7" values as derived from "slope"
observations and from wind profile observations. By disregarding
the results based on "slope" observations at wind speeds of less
than 10 m/sec, and replacing them by the results of the inadequate
cup anemometer measurements, a "jump" of - from low to high values
appears erroneously at 10 m/sec wind velocity.

A critical review of the accuracy and a summary of the results
obtained by different methods, including wind slope observations

in wind tunnels has been recently made by R. B. Montgomery (1952).

39

© Gulf of Bothnia
winner * Baltic Sea
Gbservaticns Bae ones

e@ Lake Okeechobee

@ Baltic Sea( J.Hela)

x North Sea (Schalkwijk)

oORoll (wind profile measurements)

Fige 7. Resistance coefficient (y?2 =

z mE at the sea surface

as a function of wind velocity (W m/sec), according to

slope-observations (wind set up) and wind profile
measurements, (After G. Neumam, 1951).

He finally comes to the rather disappointing conclusion that at
no wind speed is the stress (or the resistance coefficient) known
confidently to within half its value.

At present, a new, direct approach for measuring the inter-
facial stress at the sea surface is being undertaken by L.A.E. Doe

as part of this project. It is hoped to derive the surface stress

40

from direct measurements of the eddy stress, -p[{w'u'], at a short
distance from the interface, and from wind profile measurements

at constant heights above the true wavy sea surface with a special
type of practically inertialess anemometers.

For a quantitative study of the wind driven oceanic circulation,
the exact knowledge of T (or y*) is very important. Since the
average climatological wind data used in such a study are less than
10 m/sec in most areas over the oceans, the most contraversial re-
gion with W < 10 m/sec comes into consideration in a critical way.
W. Munk (1950) in his study of the wind driven oceanic circulation
came to the conclusion that the stress values as derived from wind
profile measurements are much too small to supply the necessary
driving force of the wind at low wind speeds. The author agrees
completely with W. Munk's statement that the transports of the
large oceanic currents are probably as good an indicator of the
overall stress exerted by the winds on the ocean as any of the
measurements on which the value of 7" is now based. Similar con-
clusions have been drawn with regard to the wave generating forces
by wind (Neumann, 1950).

Recently, in a careful analysis of storm tide data on Lake
Erie, G. H. Keulegan (1952) also derived wind stresses for larger
wind speeds which are in substantial agreement with the values sug-
gested by the author (1948).

There is, at present, no evidence which really shows dis-=
agreement with the use of the empirical stress formula as proposed
by equation (31) with the variable resistance coefficient of (32).

Therefore, after a critical reexamination of the papers on this

41

subject since 1948, this formula has been applied for computing
the wind stress in table lof this report, and it is also being
used for a numerical analysis of wind driven circulation in the

North Atlantic Ocean.

re ee ES ES rn

ocean circulation

a) The differential equation (11) for the horizontal mass
transport of the wind driven oceanic circulation seems to promise
a more realistic approach to a quantitative analysis of the hori-
zontal mass transport in the upper layers of the sea. It comes
a little closer to Ekman's original formulation of the problen,
since the "topographic curl effect" was shown to be of significant
importance throughout all of his papers.

By writing equation (9) in terms of the vertically integrated
mass transport, M, with the components, U and V, it follows that

curt, w= Y QE 2($2 PP SP) 2 cuit ’ (33)

where 9p/dx and dp/dy again represent average values of the hori-
zontal pressure gradients in the whole layer between z = =D and

z=0O. With A3| = gA(B)|> + ely |> and

da . ~ Op:
(D+6) $k =p $b

az | = op 2+ gp Bm gs We |
ti (34)
= = ep Ds es pe

-D

42

On substituting (34) into (33) we have
De (ee ae Se)
2 (E BE BW) dower ,

curl, M

and with M= -
- curl, R = vf - curi,t - gd ($2 32 - $2 a2)
oe 2-3-2). 2a

Only, if D = constant
ee a
pecur iste aay) a Secure as (36)

and the equation expresses a balance between the curl of internal
stresses (friction), and the planetary vorticity and the curl of
the wind stress. The differential equation was used in this form
in the preceding papers on the wind driven circulation.

pb) The last term on the right hand side of (35) can be inter-
preted as the z-component of a vector product of two vectors,
>
a

Bree saianceeradion Sipe iri dle thelansie: between the

> >
vectors a and b , then

go 3 - ge oe = ab siné = 0

would mean that a iN} rs 5» and the lines of equal dynamic height of
the sea surface (relative to a level surface) are parallel to the
lines of equal depth D.

Similarly, the term

QB gp _ op aD _
Ox OY dy Ox

would express the requirement that the lines of equal D and the

isopyenals p are parallel. Approximately 35 years ago, Bj. Helland-

a3

Hansen and W. Ekman (1923) noticed that over large areas in the
Norwegian Sea the large scale spatial distribution of temperature
and salinity (density), followed closely the contour lines of the
pottom topography. Furthermore, when studying large scale conditions
in the oceans, they found a remarkable parallel between the iso-
therms and isohalines and the contour lines of the isobaric surfaces.
This important law became known as the "law of the parallel sole-
noids." Although this law seems very well verified in the gross
features of the large scale conditions, there are slight deviations
which are nevertheless very important.

Precise leveling along the American East Coast shows that the
mean sea level increases towards the north from Florida to Nova
Scotia (G. Dietrich, 1937). The magnitude of this slope is Tome
and thus it is of the same order as the sea surface slopes derived
from data in the Caribbean Sea (H.U. Sverdrup, et al., 1946). In
the Caribbean Sea, as in many other parts of the ocean, the slope
of the sea surface is practically compensated for by the distri-
bution of mass. However, this is not the case in the region of
the Gulf Stream along the east coast of North America. G. Dietrich
(1937) has shown that along the continental slope the distribution
of density does not indicate any rise of the sea surface, and this
is true also in the case where the sea surface topography is re-
ferred to the 2000 dbar-surface instead of the oxygen minimum layer,
which was used by Dietrich (H.U. Sverdrup et al., 1946).

With the assumption that within the layer of the wind induced

gradient current curl, R= curl ,t » then from (35)

44

or a
r¥-c[(OB rz) B-CH FH IR -0 or
The x,y-coordinate system can be oriented in such a way that
the y axis points ina direction 7) of the flow of the Gulf Stream
along the east coast of the United States. Then, the € -axis,
perpendicular to 7) , points toward the southeast, approximately.
In this case, it is found from figure 1, that ODE = -2.5 x 1073
between Cape Hatteras and Nova Scotia. With the assumption that

0D/97 = 0, or at least dD/on < AD/OE , it follows from (37) that

9S cos ase + gio E+ go) BD =o. (38)

This equation relates the (average) density distribution and the
Slope of the sea surface in the direction of the flow with the slope
of the layer of no motion perpendicular to the flow and the verti-
cally integrated mass transport of the Gulf Stream while moving
northeastward.

Suppose that there is no compensation of AYE /91n by the density
distribution p(7), and, therefore, in the direction of flow dp/07
= 0. Then gf /o7] must be positive, since V(dOf/dy) is a positive
quantity, and OD/9E <0. The order of magnitude of a6 /o7 can

be estimated with the assumptions made before, and

goog Mat sig

gp - 2.5-1073 2 Se cee ae

gp 2.5 +1079

between Cape Hatteras and Nova Scotia where 9 = 39°, and R = 6370 km
is the radius of the earth. According to H.U. Sverdrup et al.(1946)
the total mass transport of the Gulf Stream towards the NE eqals
Senta. m per sec. With a total width of the current of, approxi-
mately, 250 kn,

45

7?) BoB) 52 gO fies eat’ age y

and from (39)
06/69 = 1e15 x lo

This compares well with the observed magnitude of the slope
between Cape Hatteras and Nova Scotia (Sverdrup et al., 1946). If
this slope did not exist, and the assumptions on which (38) is based
hold, p would have to increase toward the NE in the axis of the flow.

If the term V(df/dy) would be disregarded, it is seen from
(38) that an upward slope in the direction of flow must be compen-
sated for by a decrease of the average density of the Gulf Stream
water off the continental shelf.

It should be kept in mind that equation (38) holds only in the
case of vanishing by small friction. It certainly does not apply
to the surface layers of frictional influence, and, if present,
to a bottom layer of frictional influence (Ekman's bottom current).
Similarly, the law of the parallel solenoids does not apply to
layers where frictional influences dominate.

ce) With a discussion of equation (11) for the horizontal mass

transport, the case of a simple zonal model may be considered,

ey
y © Dooy 2

and any effects of meridional boundaries are disregarded. Further,

where (see section IV)

suppose that T, = T,(y), Ge ai0s OD/dx = 0, V=0. With these

assumptions, equation (11) reduces to
amy a gpgW 2 (cox _ tx 2d)
dye) Dey Os) FE Oy DE OSes

or

46

onan) 2 eam)

D dy dy

By integration, it follows that

GNA Raina a

ag TS ’ (40)
where C is a constant, which becomes zero if U=Ofor T, = 0.
This result shows that the zonal mass transport, U, is proportional
to the zonal component of the wind stress, TY This is different
from Munk's general result where in a confined ocean of constant
depth it was found that the horizontal mass transport of permanent
ocean currents depends only on the rotational component of the wind
stress field over the ocean. Of course, in an ocean bounded by con-
tinents, the assumptions which led to (40) are in general not justi-
fied any longer. Even in the case where Ty may be assumed to be
zero, $D/@x is not necessarily zero, and V # 0, at least for some
distance along the boundaries. The whole system of horizontal mass
transport will be modified by the boundary conditions, but besides
a dependence of the mass transport on the curl of T , there will
also be a dependence on T.

Equation (40), however, will hold approximately, in the central
parts of the oceans, far away from the boundaries. Probably, it
can best be applied to the Antarctic Circum Polar Current, just where
W. Munk and E. Palmén (1951) found discrepancies with the results
of Munk's theory.

In the Antarctic Circum Polar Current between 65°S and 45°S,
according to table 1, T, * 2.0 dyne/em*, and r = 3.34 x 107° gecnl

(see page 30). Thus according to (40)
U=6x 10 er cm sec,

47

and the average total mass transport through a section between
65° and 45°S is, therefore,

3 IS SiS) oe 101+ [er sec +],

6x 10? 56 BAB 52 110)
which agrees well with the observed mass transport. According to
A.J. Clowes' (1933) computations the mass transport through the

Drake Passage is about 1.1 x nO er sec =, and it is very probable
that the transport in the central parts of the South Atlantic Ocean

is approximately 20% greater.

XI. Outline of future work

Equation (11) is being used for determining the horizontal
mass transport of the wind driven circulation in the North Atlantic
Ocean between the equator and about 55°N. The east and west bound-
aries of the ocean are the continents. They are approximated as
closely as possible by broken straight lines following the conti-
nental shelf along the 200 m isobath.

The boundary condition along the continental slopes, which may
be identified with the continents themselves, requires that the
mass transport be parallel to the boundary curves. Since an additive
constant with the stream function W makes no difference, the boundary
condition is given by My = 0 along the continents.

However, since this model is not completely enclosed by land,
more complicated conditions have to be introduced along the northern
and southern boundaries, and between the Antilles Are and Florida.
The Guiana Current along the northeast coast of South America trans-
ports an appreciable amount of water across the equator which enters
the model ocean in the southwest corner. This fact cannot be dis-

regarded. It has to be taken into account as a "source" by the

48

boundary conditions along the equator. Another "source", which
is probably very important, is the flow of water through the Straits
of Florida.

Along the northern boundary a "source" has to be considered
northeast of the Grand Banks due to the inflow of the Labrador Cur-
rent. An important "sink" between Great Britain and Iceland is
given by the transport of the North Atlantic Current which accounts
for a net balance of the water in the region covered by the model.
The total net intake of water between the equator and the northern
boundary, that is, the sum of the "sources" and "sinks", must be
zero.

For a numerical integration of the differential equation (11)
the surface of the ocean is thought to be covered by a series of
lattice points as given by the intersections of a set of rectangular
lines in the north-south and the east-west direction. The lattice
of this net consists of "interior" points and "boundary-points",.
Interior points are called such points which are surrounded by ad-
jacent points belonging to the lattice of the covered ocean area.

An interior point, therefore, is always surrounded by four adjacent
points, whereas a “boundary point" has less than four adjacent points.
The unit mesh width of the lattice was chosen as d = 222 km; how-
ever, in the western North Atlantic it may be necessary to use a
smaller unit (55.5 km) in order to get some of the details of the
Gulf Stream flow pattern. This section will be considered by a
special evaluation of the basic differential equation.

The differential equation (11) for the horizontal mass trans-

port can be written in the form

ca)

ry “w + F(x,y) gu + G(x,y) qu - H(x,y) = 0 ,

where the coefficients F, G and H have been plotted as functions of
x and y on charts. The wind stress was computed for the month of
February by means of formula (33), using the wind charts put out

by the Marine Branch of the Meteorological Office, Air Ministry,
London (1948).

If the subscripts j,k indicate an interior point, and the ad-
jacent points of the lattice are denoted by subscripts (j,k-1),
(j,k+1), (j-1,k), (j+1,k), respectively, the difference equation for
the horizontal mass transport becomes

Wied Waser tk = Woo 2 4G
p (anton * Yate * Vas )

Pye Piatt seed) « 0, , (Pistats Michit) — 5

j,k
Such a difference equation can be established for each interior
point. There are as many equations as interior points, and since
the number of interior points equals the number of unknowns, with
the boundary conditions, W is uniquely determined in the interior
of the area, provided that the determinant of the system of equations
is unequal to zero.

Basic data for the determination of the coefficients F, G and

H as functions of x and y, and the results of the numerical inte-

gration will be published in the next report.

50

Acknowledgments
The author wishes to express his indebtedness to Drs. A. J.

Abdullah, B. Haurwitz, J. E. Miller and W. J. Pierson, Jr. for
many stimulating discussions on the subject, and to the Office
of Naval Research for financial support. In preparing the typing
of this report, the assistance of Mrs. S. Wladaver is greatly

appreciated.

sl

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ie ,» 1948: Uber den Tangentialdruk des Windes und die
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54

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