A study of the time-dependent wind-driven ocean circulation

OFFICE OF NAVAL RESEARCH aie

Contract N7onr -35801
Ti Op 1.

NR -041 -032

A STUDY OF THE TIME-DEPENDENT
WIND -DRIVEN OCEAN CIRCULATION
by

G. Veronis and G. W. Morgan

|e
IEE GRADUATE DIVISION OF APPLIED MATHEMATICS
| VY BROWN UNIVERSITY

1753
| PROVIDENCE, R. I.

December, 1953
All-101/110

Given in Loving Memory of

Raymond Braislin Montgomery
Scientist, R/V Atlantis maiden voyage
2 July - 26 August, 193]
KKK IK KK
Woods Hole Oceanographic Institution
Physical Oceanographer
1940-1949
Non-Resident Statf
1950-1960
Visiting Committee
1962-1963
Corporation Member
1970- 1980
RK KKK
Faculty, New York University
1940-1944
Faculty, Brown University
1949_ 1954
Faculty, Johns Hopkins University

1954-196]
Professor of Oceanography,
Johns Hopkins University
1961-1975

IIA

iii

0 0301 0035129 8

Al1-101

CKNOWLEDGEMENT

The authors wish to express their ap-
preciation to Mr. Henry Stommel for calling
to their attention the need for an investiga-
tion of the problem herein presented,

Grateful acknowledgement is made to
Mrs. Marion Porritt and Miss Ezoura Dias
for the typing, Miss Mary Melikian for the
figures, and Miss Nancy Bowers for the Mimeo-

graphing.

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Al1-101 dl

A Study of the Time-Dependent
Wind-Driven Ocean Cilecinlate aera
by

Ge Wencoutue and G. W. Meneame

Abstract. This investigation is concerned with the

large-scale wind-driven motions of the ocean and their responses
to a time variation in the wind, Starting from the equations

of motion for an inhomogeneous fluid, a detailed formulation of
the problem is presented, including the listing and discussion
of the assumptions and simplifications necessary to reduce the
general mathematical model to one which may be successfully
attacked analytically.

Since the real ocean is baroclinic, the problem is
formulated to include a non-uniform density distribution. Two
special cases are considered.

(i) An ocean consisting of two superposed layers of con-
stant density is assumed and the equations are integrated over
each layer to simplify the analysis. Attempts at an analytical

solution for this case were unsuccessfule

The results presented in this paper were obtained in the
course of research conducted under Contract N7onr-35801,

Research Assistant, Graduate Division of Applied Mathematics,
Brown University, Providence, R. I.

3 Associate Professor of Applied Mathematics, Brown University,
Providence, R.. I.

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A1l1-101 2

(ii) A more general density distribution is then assumed,
but a more restrictive assumption is made concerning the verti-
cal variation of velocity. In particular, it is assumed that
there exists a (variable) depth below which the velocities are
negligiples As a result of this assumption, a direct relvacion
is found between the thermocline and the free surface. The
equations are integrated from this depth up to the free surface,
The linearized equations are then subjected to an analytical
treatment consisting of a perturbation expansion in terms of a
parameter which is proportional to the frequency of the wind
variations The resulting equations are solved by boundary
layer technique.

Results are derived for the response of the mass trans-
port to slowly varying winds, and the effect of the wind on the
intensified stream near the western boundary is discussed in
details

The two-layer steady problem is also solved and the

steady position of the thermocline is determined.

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le Introduction. Much of the investigation, both theo-

retical and observational in the field of oceanography has center~
ed around the dynamics of ocean currents = including the mass
transport of the Gulf Stream and the Kuroshio Current, and the
general oceanic circulation. Recently interest has developed
regarding the response of the thermocline (the region of sharp
vertical gradient of density) to a time-varying wind,

Since the time of Ekman's first paper iatala a large
number of papers have appeared in some of the geophysical jour-
nals dealing with various aspects of ocean currents. However,
analytical investigations of the problem of general oceanic
circulation have met with success only in recent years. In the
past decade various interesting and meaningful mathematical
models have been suggested by numerous investigatorss Sverdrup
[2] and Reid [3] proposed a fairly simple model which seems to
give very good qualitative results for a region with only one
north-south boundary. Stomnel [4] considered two linearized
models with a simplified viscous terme His very important con-=
tribution to the overall problem is based on the difference
between the results obtained with the two models. In one case,
the Coriolis term was constant and the resulting streamline
pattern is identical with the one in a model with no rotation.
In the second case, the Coriolis term varied linearly with
latitude and westward intensification resulted = a factor which

* Numbers in square brackets refer to the bibliography at the
end of the papers

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Al1-101

was not present in the previous casee Since Stommel's paper

all problems dealing with general circulation contain a varying
Coriolis parameter. Munk [5] refined all the previous work and
included the general viscous terms in the equations of motion,
He solved the problem of a steady wind blowing over an enclosed
ocean, taking account of many of the salient features which are
present in the real ocean, Munk's work was extended by Munk
and Carrier [6] to include oceans of various geometrical shapes,
vize, triangular and semi-circular. It was further extended

by Munk,Groves, and Carrier [7] to include the non-linear terms
by means of a perturbation procedures

Along with the American publications, a number of papers
have appeared in Japan. Notable among the Japanese authors is
Hidaka, who published a series of articles covering many of the
interesting phenomena of oceanographic problems, Among his con-
tributions are a series of three papers on drift currents in an
enclosed ocean [12], [13], [14], and a contribution concerning
the neglect of the non-linear terms in the solution of problems
in dynamic oceanography [15].

Practically all of the work done so far in ocean current
problems has been confined to motions which are independent of
time. Hach publication has treated some aspect of the general
problem of oceanic cireulation, This problem essentially con-
sists of finding the dynamic pattern which results from a given
distribution of winds acting on the ocean surface,

The complete problem contains a large number of features,

such as large-scale oceanic circulation, surface waves,upwelling,

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AILLSTOAL B

ete. To find all such motions one would have to take into ac-
count the effects of the wind, density and temperature distribu-
tion, the topography of the ocean bed and possibly even such
features as salinity. Needless to say, a mathematical analysis
including all these features is impossible. It is therefore
necessary to decide what particular aspects of the problem one
wishes to studye In this paper we shall confine our attention
to large-scale wind-driven motions in the oceans and their re-~
sponses to a prescribed time variation in the wind. In the
Atlantic Ocean, such large-scale motions must include the Gulf
Stream and its counter-currents, the Sargasso Sea, etc,

The time-dependent problem has also been considered by
Ichiye [16]. We shall discuss his work later in the report.

It has been generally agreed upon by oceanographers
that the type of phenomena we wish to consider can be adequately
described by the dynamics of the problem alone, the temperature
effects being included by way of an assumed semi-empirical den-
sity distribution, At the Woods Hole Oceamgraphic Institute,
experiments with a model parabolic ocean basin verify the above
conjecture, Hence, in the subsequent analysis, we shall neglect
direct temperature dependency in the treatment of the problem
and shall include only the effects of wind and gravitation.

A large part of our report is concerned with the formu-
lation of the problem and the assumptions made to reduce the
general problem to one which can be attacked mathematically, In
the past a discussion of such assumptions has often been vague.

It was felt therefore that an explicit and detailed analysis of

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the simplifications involved in the formulation of the problem
might be welcomed by workers in this field and that it might
help to clear up any existing misconceptions concerning the

validity of some of the assumptions,

2o Discussion of Results. At this point we shall discuss,
without resorting to mathematical detail, the basic assumptions,
the results, and the conclusions of the present investigation.
In this manner we hope to convey a more integrated picture of
the physics of the problems

Mathematically, the motion which we want to study can
be defined by the Navier-Stokes equations of motion with the
viscous terms replaced by terms arising from a macroscopic vis-
cosity, vize, an eddy viscosity, The complete non-linear equa-
tions are too difficult to solve, however, so that we are forced
to make a number of simplifying assumptions which we shall list

below.

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be inhomogeneous.

2. The equations on a rotating sphere are approximated by
equations in a rectangular Cartesian system. The effect of the
sphericity of the earth is retained by allowing the Coriolis
parameter to depend on the latitude. Since we shall consider a
rectangular ocean in the Cartesian system, a few remarks must
be made concerning the region of the sphere onto which the rec-
tangle is mapped. The constant east-west distance of the rec-

tangle is preserved in the mapping of the rectangle onto the

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All=-101 7

sphere. Such a mapping is not conformal since angles between
lines are not preserved. The region under consideration must
be well removed from the north pole.

3. The vertical acceleration terms and the viscous terms
are neglected in the equation of vertical motion so that, in
efieet, hydrostatie pressure is assumed, i.e... p = g("paz, where
n is the free surface height and p = 0 at Z = The, density
p may, Of Course, be a function of the space coordanateiss ln
Appendix 3 it is shown that for the problem which is independ-
ent of time, the hydrostatic pressure assumption is necessary
only in the depths where there is no motion if one desires a
solution for the components of the mass transport only. If it
is necessary to find the shape of the free surface, however, or
if the non-steady problem is considered, this assumption or some
analogous one must be made.

4. As stated in the introduction, the thermodynamic effects
are accounted for only empirically by stipulating a density dis-
tribution, We assume p = plz = T(x,y,t)] where the function p
of the variable (z - T) can be prescribed to fit observational
datas This functional form for p makes the curves of constant
density parallel.

5. The equations of motion are integrated over the verti-
cal coordinate, Ze

In order to perform this integration it is necessary
that we specify the density distribution since p appears in
some of the integrands. We consider two caseSe

(i) The surface z = T separates two layers of constant

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Al1-101 3

density. The equations of motion in each layer are then inte-
grated over the depths of the respective layers and the non-
linear terms are neglected. We also neglect shear forces at
the bottom of the lower layer and at the interface. No assump-
tion is made concerning the vertical distribution of velocity’,
but instead, we hope to solve for the integrated velocities
(i.e., the transports) in each layer, This case is referred to
as the two-layer problem. Unfortunately, it is much too diffi-
cult to handle analytically, and consequently we must consider
a second problem,

(ii) The manner of performing the integration in this case
will lead to a considerably simplified problem which allows us
to stipulate a more general density distribution than that in
(i). The density is specified as a continuous function of depth
and the ocean is divided into three layers, A layer of constant
density, po, lies above the surface z = T(x,y,t). From z =T
down to z = T - d (d is constant) the density increases linearly
with depth from p, to the value p_jy.- Below z = T-d, the den-
sity has the constant value, Ppt

We assume that there is a depth z = = h(x,y,t) below
which the velocities may be considered negligible (in some
suitably defined sense). The pressure gradients will then also
be negligible below z = - he As a consequence of this assumption
and the previous assumption of hydrostatic pressure, a relation-

ship exists between the surface z = T and the free surface

* | Compare this with case (ii),

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Al1-101 5)

Z= 1, vize, T==- pp n ~C (where Ap =p_), -pP, and T = -C
when n= 0). Thus, if the velocities are negligible in the
depths of the ocean, the thermocline must respond immediately

to a change in the shape of the free surface in order to main=
tain negligible pressure gradients at these depths.

The three assumptions, (a) hydrostatic pressure,

(b) negligible velocities in the ocean depths, and (c) con-
stant density below the thermocline, are crucial for the present
ease. It is, of course, possible that any one or a combination
of these three assumptions may be incorrect. If this be the
case, then the thermocline need not respond to the free surface
immediately, The frequency of the wind variation which we shall
consider later in our development will be small so that assump-
tions (a) and (b) seem plausible, Thus the only motion exist-
ing below the thermocline is caused by vertical shear and this
motion decays exponentially with increasing depth according to
Ekman [1].

The equations of motion are then integrated from the
depth) z ==! h to the free surface zo =n. This problem wilaivbe
called the one-layer problem because of the single integrations
The depth, z = - h, does not appear explicitly in the integrated
equationse

lin) both ealsejs;,, the eriecy of the wind as\represented
by the shear stress at the ocean surface and appears in the
evaluation of the vertical viscous terms at the upper limit of
integration (free surface).

An additional difference between the two problems is

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Al1=-101 10

that the two-layer problem specifically restricts the fluid of
the top layer to remain in the top layer and the fluid in the
lower layer to remain in the lower layere The one=-layer prob
lem has no such restriction and an interchange of fluid may
result. However, because of the integration we have no inform-
ation concerning this vertical motion.

6. The non-linear terms in the equations of horizontal
motion are neglected. A plausibility argument for this assump-=
tion, based on the results of 7a. is presented in Appendix 2,
However, our results must now be considered tentative, since
the case presented in the appendix for the neglect of the non-
linear terms is a plausibility argument and not a justificatiom
The primary motive for neglecting the non-linear terms is our
inability to cope with them analyticallye

(othe Coriolis parameter) as iaineariZede) | invetinect,
this is comparable to linearizing the sine of an angle when the

angle varies between 15° and 60°,

With the above assumptions and simolifications we are
in a position to attempt a solution of the non-steady problem.
The ocean is chosen to be rectangular with vertical walls as
boundaries on the east and west. Because of the presence of
viscosity, the boundary conditions on these walls are that the
velocities vanish The boundaries on the north and south are
water boundaries.

The wind=stress is written as

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A11-101 IE

where W!, 1 ',w, and n are constants and t, (Fig. 1) is the
east-west component of the stress, The above form for the wind-
stress may be considered as the general term of a Fourier series
expansion so that the wind-stress may be generalized for the
linear problem. However, for our numerical example, we have
chosen wto give a period of one year and nas 2/s where s is
the north-south length of the ocean (0 < y < s). The wind-

stress component t.. is assumed identically zeroe Since the wind-

V
stress is prescribed in such a manner that its y derivative
vanishes at y = 0,s, it appears reasonable to demand that these
boundaries be streamlines and that the normal derivatives of

the velocities vanish there.

The one=layer problem is solved by the following proce-
dures The equations are non=dimensionalized. The non-dimen-=
sional velocities and free surface height are expanded in per-
turbation series with the non-dimensional time parameter as the
perturbation parameter. Each resulting set of equations is
then solved by application of the boundary layer techniques

The conditions for the validity of the expansion restrict
the time variation to a maximum frequency of seasonal oscilla-
tions in the numerical example, yearly frequency is assumed
and the perturbation terms of second-order and higher are
neglected. The error involved in neglecting the second-order
term as compared to the zero-order term is about 5%, and it is
about 20% as compared to the first-order term. The remaining
physical parameters are given values which correspond roughly to

those of the North Atlantic Ocean,

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All-101 2

The following discussion will be based on the non-=
dimensional quantities defined in the body of the reporte When-
ever dimensional quantities are mentioned, we shall include the
dimensions,

The graph of the north-south component, V, of the mass
transport vs. the east-west coordinate x! near x! = 0, the
western shore, is shown in Fig. 2 for the value y* = 0625, iwe,,
where the Gulf Stream is most pronounced. The Gulf Stream re-
gion is the region of large positive Ve The region of negative
Vv adjacent to the Gulf Stream corresponds to the offshore
counter-current,

The Gulf Stream responds to the wind in such a manner
that the mass transport and the wind are in phase whenever the
latter takes on its maximum or minimum value, At all other
times the mass tramsport lags behind the wind with the zreatest
lag occurring when the wind reaches its steady position’. At
this time the mass transport is about 9 days away from its
steady values ihe Length of this amverval ds Gy) Tane days.
is independent of the frequency for slowly varying winds.

The wind (see Fig. 1) and the mass transport attain
their maximum values at t = 7/2, The mass transport now has a
magnitude of (1 + ['/W') times its steady value. Thus, within
the accuracy of the present method of solution, the time at
which maximum transport occurs and the magnitude of the maximum

* We shall refer to the "steady position" whenever the time-
dependent contribution of the wind is zero.

** j,e., the value due to its response to a steady wind

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All1-101 13

transport are independent of the frequency. The magnitude of
the out-of-phase effect (the second term in the perturbation

series) which is largest when the wind has its steady value,

is proportional to the frequency.

The time variation of the wind affects the Gulf Stream
only by changing the mass transport through the Stream It
does not change the Stream's position.

As can be seen from Fig. 2, the relative importance of
the out-of-phase effect is greatest in the counter-current.

Figure 3 is a graph of the north-south mass transport
component near the eastern boundary of the rectangular ocean
at the latitude y! = 0.25, The accompanying out-of-phase effect
is shown at its maximum in the figure. V is negative on the
eastern coast, i,e., the mass transport is toward the southe

Figures 4, 5, and 6 show the contour lines of the free
surface in the southern half of the ocean for various times.
With the values of the contour lines multiplied by -200 the
three figures represent the contour lines of the thermocline.
Qualitatively, the results agree fairly well with observation
though some of the natural features are missing. It seems
likely, however, that most missing features result from local
effects which we have not taken into account,

Because of the lengthy computations involved, we have
calculated numerical results for only one set of values of the
parameters. It can be seen from the analytical results that if
the average depth of the top layer be changed, the values for

the deflection of the free surface and the out-of-phase

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velocities will changes Specifically, if the depth is decreased,
the free surface deflection is increased and all out-of-phase
Quantities are increased.

The above results appear to invalidate the solution of
the problem as obtained by Ichiye biG Ichiye neglected the
contribution of the non-steady term in the integrated continuity
equation. However, with the values of the parameters used in
Section 4, the magnitude of this term in the interior of the
ocean is as much as ten times that of the remaining non-steady
terms which were retained in Ichiye's analysis.

We have computed the mass transport through the Gulf
Stream for the one-layer steady problem. With the given wind
distribution our result is 26.6 x 10° metric tons per seconds
This value is about three-fourths of Munk's value [5] and about
one-third of the observed value. Munk used an empirical east-
west wind distribution,

The two-layer steady problem is solved in Section 5
where it is shown that the mass transport streamline pattern is
the same as in the one-layer problem This is to be expected
since, for the steady case, the same assumptions are made regard-
ing negligible velocities below the thermocline. Thus, the
height of the thermocline is shown to be proportional** to the
free surface deflections Since the free surface height is deter-
mined largely by the thickness of the top layer, the thermocline
* In [il6] the term corresponding to W! in the present paper

was assumed to be identically zeroes i.ée.e, the wind had a
zero mean value.

*x* The factor of proportionality is the reciprocal of the
density difference,

maf at ‘Wiad aul’ 2 7) et ult ny a
aimee osiya tin. i uf %, ae i rier 4

la mn i Taunton |
‘ (357) Tt f ( y

aS

ot, | be pabne ye, on 4 ta Liven oF
*_ and bed yelwen ‘ igi ite |
: a sank ; i ) i atl ai a
i, ; 7 ia ayant ; oe

i el Nich BAA

Np hay evs ti ’

re PI ety a Wau co inp

Es Ee eh Ti
# ees ihe ‘a
aes

wan} Sat a taro Sk mm Put Peiuy vir i, 4 i

fc Hy it of ty a pie

AY if we a ant j i ay Oy iby \ Hi hp ity 7

Cheer VME Shel wen a

A11-101 15)

variation depends on the choice of the two parameters, density
difference and thickness of top layere

By varying the two parameters we can get good qualita-
tive agreement with observations of the shape of the thermocline.
In Fig. 9, a cross-section of the computed thermocline is shown
for four pairs of values of the parameters. Because of the
rather vague definition of the actual thermocline, we cannot
state specifically the extent of quantitative agreement between
our computed results and the observed values. Consider,
however, the curve in Fig. 9 with a depth of the top layer of
200 meters and a density difference of 0.0025. For that curve
the results disagree by a factor of three when compared to some
of the measurements of the thermocline off Chesapeake Bay [10],

The two-layer non-steady problem constitutes an attempt
to drop the assumption made in the one-layer problem that the
velocities vanish at some great depth As a consequence the
problem becomes much more complicated and it is necessary to
introduce some other simplifying assumptions, vize, to neglect
the shear forces at the bottom and at the thermocline. This
may have far-reaching effects, These simplifications notwith-
standing, we were unable to obtain a solution. A brief descrip-
tion of our attempts at such a solution follows,

First, the equations are non-dimensionalized as in the
one-layer case. The integrated continuity equation for the top
layer now contains the time derivative of the magnitude of the
deviation of the thermocline from its equilibrium positions

Since this term is very large, the perturbation method used in

Tye hes aatinnnia chit tes
chic: mi peat

std

Pagers Ceomcatts a oe vine:
, = ‘
. Bit Tec “Paha as

is : cals a v
OA UMASS BS Yee
1 r he Le is
ERPS) BE OES f
To 2eyel aod «
PVG: RAS 1h,

erin Od. Bancine

al Ol) vat Ors

Fone SAO Gee aes =
a PROM DORN: 28) Ge Pre opm

aT Vepas mien us Pr bust | bens vant

ada yan
iy veuyS &

i er ae Lies, bea cae ? Ny
HAT « SATA Sete ae bbs ‘ng i oy,

vhitgke ncaa ascii wae

ogi AOWA y at age A. hh baru (oD ey a ; a a 4 sale j
he yout hati te, fo > nanan ita a
aid Ms qe. bea bif sl hua wien [

‘ $ a “31 ry
a a) 2 a da abi

mi hie

b Ne ten s ot het epee saat

the one-layer problem is restricted to a range of frequency
values corresponding to less than one oscillation every hundred
yearse Since these results are not physically interesting no
numerical results were computede

A second method of attack is then attempted. The wind-
stress term is first divided into its steady and non-steady parts
and the two problems are treated separately without resorting
to a perturbation in the time parameter. This method had been
attempted for the one-layer problem with no success, In the
present case, however, it was hoped that the new parameter in-=
volving the density difference could be used to advantage, Un-
fortunately, an analytic solution still appears to be quite
hopelesse

The one interesting fact which seems to emerge from the
attempts at the solution of our idealized, two-layer, non-steady
problem concerns the magnitude of the lower layer transporte We
music Mecalili that, in the case treated. the solution is mestricted
to the frequency range for which the thermocline responds to the
variation of the top surface in a quasi-steady manners; ee as
a result of any change in the free surface, the thermocline
assumes the same shape as it would for a steady problem with the
given free surface, except for a small out-of-phase correction.
In this case, the mass transport in the lower layer, excluding
whatever transport may be caused by shear at the interface, is
of the same order of magnitude as that portion of the transport
in the upper layer which is out of phase with the wind. Fora

higher frequency this result does not necessarily hold truée

srg Beit cine sedaect ae
on adele iG ifaotinws don. ys abla.

fan Day "ae sbaitenecitos au ti le ete a6 etn lees
Sta5g Yhevte wistut Le ibe e. |
abtvrone ‘ wig a iy yiotows 108:
fe oc Rat Beit ri ‘oni abe onaretg “out ont at ¥
wie ah eaaine i eit wm | | | . |
«itt apt omsrty es an < wat ase - nf ey ce wkerevet sone 8
"healt woe tievtie oe, Breil ek Si God wae we kh |
‘ ai ty ey ee liana t i
Sau nett eget i$: separ oat a
ihasds wf lcait ee A: ; 1 si har es te
Sy ad ayer ew genet be ite
hadsinscey a2 ni), fi Lite mae Beit ont ee.
ems of aphoue ~ ond Loowetsiets erty. He re
Be »ett od HCAS an tions a Oh (8 ° ig it oat pd ony! i, fe
dack FT tag resahtt ty Pricer aye a. ti ae at, sadede ved My
mi aitiv wl ooat phim uM ot 8 fi i Ge papi sonnet oe: )
ehOI Sues a Se phere bhimiteeg), IL | ha» * mnt ‘dagen eae 9
Rethiidne ,ravel sent eis rh gook an ee sid ett eine a a
51 rent Fsatnt ot Pex ~ i ita. ‘s ‘boawias ‘ody wet aeggnnei due
eyo

Hi

<i +n ean )

: ett alow ¥Ti4, wadioue daw yaoi

All-101 L7

A final word should be said about the lack of dquantita-
tive agreement between our computed results and observation.
The factor of three is not surprising when one considers the
very idealized model which we have assumed. A number of more
realistic assumptions may certainly affect our quantitative
results by such a factor. The inelwsion of the non-linear terms,
a better representation of the wind effects on the water, a
more natural topography, and a non-constant eddy viscosity may
well alter the quantitative results and bring them into closer

agreement with reality.

3. Formulation of the Problem. It is our aim to derive
expressions for the velocity and the pressure satisfying the

three equations of motion on a rotating sphere
== +4 ° ya + 20xd + Ox(Qxr) = - : Vp +E + au * AV)

the continuity equation

70a 20

and the boundary condition that 4 = 0 on a land-water boundary.
Here, x
q@ = (u,v,w) denotes the velocity vector relative to a

ae coordinate system rotating with the sphere,

@ denotes the angular velocity vector representing the
earth's rotation,

p denotes the pressure,

p denotes the density,

heal

denotes the external forces per unit mass (in our case,
gravitation),

* U,V,W are spherical components of velocity along the direc-=
tions) of the nadius, vine meridvans, and the paraliiels of
latitude respectively.

esate. HAN BE anycsee s
: ee Ped A tis: Me eso
Mem VeLienees Yoke feerco

L sD | a
@ a s O8 of au ty,’ ie
“ear warts, weitt

,
7 ars Pay a! Wig) Debate we
.
oe tort Ah eee
ray '
hy eee " Vie ?
sehuleynt ‘tohiae tek) ie
i) ae Ke ay TAS tobiog ne
p CS ult ne ,
( a a yd digi acti he
: ‘i ‘\ fea a IE xvid ‘a
i i al #
ui \ : “Ae ns
Pk! Siw fe mh) eae tne aL
ra } )

a AHMAD BE Rei ae vo ay ‘yy ike,
ac a he will Lanenc oad | bs gid

Sav, Lee a De ae Ur q wer

ALI=LOL 18
CW UN ;V)4 represents the eddy viscosity term (discussed
~ pelow).
Let us consider the expression for the eddy viscosity
term in a rectangular coordinate system, this being the system
in which we shall later write our eduationsSe

We define the operator (V ° A;V) as follows:

° ma) fe) ) =

where A,, Ap, A,, may depend on the space coordinates. These

5
three quantities (the coefficients of the lateral and vertical
eddy viscosity) have been measured and are known to vary through-
out the oceane The definition of the viscous coefficients and
our knowledge of their magnitudes, however, are rather vaguee
In view of this, and because of subsequent analytical simplifi-
cations, we assume that the lateral kinematic eddy viscosity
coefficients are constant and equal, so that
a(V ° Ay) = (Ce +4 SA
p axe ay” ?

a5)
where A is now a kinematic eddy viscosity and is constant. No
simplification will be made concerning Aye

Our continuity equation is valid for an incompressible
fFiluid., In the steady problem the density may be more general
and we have simply V* (pa) = 0. In the non-steady problem, the
assumption of incompressibility is imposed but the fluid may be
homogeneous.

We shall want to make use of [7] regarding the effect

of the non-linear terms. Because the results in [7] are discussed

in terms of rectangular coordinates and because the use of

| Bepawens ay ater eget on PY yee aii Se
yi Salida le hee iat wat Ries Lae

tee) OCP RRR ale. hace’

5 P Hr eet Pit) i ASS

STEN TNE BY ELEN Te oe

i

ed

ae Me a

Beh DNS eR Ce yon} bs:

‘saan lbs

¢

i
fy

bp it a )

wt ug a 7 PEGI i

A11-101 19
rectangular coordinates considerably simplifies the analysis,

we shall first transform the equations of motion from spherical
to rectangular coordinates in such a manner that the equilibrium
free surface which establishes itself in the spherical system

as a result of gravity and centripetal acceleration corresponds
to the x-y plane of the rectangular system. The apparent gravi-
tational force, iee., the force which is the resultant of true
gravity and centripetal acceleration, acts in a direction normal
to this equilibrium surfaces

In Appendix 1, it is shown that our original equations

reduce to
ou! 1 Ou! + Out fl Gan (oye, HAI .
ee 2 gy eee ey Eo 20y Ey ce OI VeA.V ;
ae ax as Q sin(=) 5 arog A, du Gy)
av' 4 yf! ov' 4 yt Ov" + 20u! sin(Z) == 1 oP ayvean.v)yt
at 0x ay R p ay #( A,V)v (2)
= £ QP =
sag (3)
GUY OME Our = 10 (4)
Ox y QZ
where
x,u' denote the east-west coordinate and velocity
respectively (x is positive eastward).
y,v' denote the north-south coordinate and velocity
respectively (y is positive northward),
z,w' denote the vertical coordinate and velocity
respectively (z is positive upward),
R is the mean radius of the earth,
g is the apparent gravitational acceleration on
the earth's surface,
2gsin(2) is the radial component of the angular velocity
R vector of the earth.

is

pRB ta Garey e
Lihrinsige
PA) Se aee
NAIA 14
elrened wt as

whens ier,

é r Wy "
wa. TO Oi

Larhr Loney

: Wuyi
Gay v, 44
f i
vé,3

i
v .
- linn
. ‘ pv
-

Witetiov of
wet 47
fee OCs AY

>

BOLE Tit

fed eatan 4 ain
oe mi

did ce
‘A bean oe
“4 te
a (
eres |
Lie ae |
; 4
s: ,
bs
i
. Le

>

7 sie

te aay ‘eh
mh ae ak

ae

r wi ive
mutta

Al1-LOl 20

The rectangular coordinate system is oriented with the
origin in the southwest corner of the ocean and with the equili-
brium surface in the x-y plane.

A number of assumptions were made in the reduction of
the four equations valid on a spherical earth to the four edua-
tions given above. These assumptions are listed here for the
convenience of the reader who does not wish to go through the
detail in Appendix l.

(1) In the radial component of the equations of motion,
the acceleration terms and the viscous terms are
neglected in comparison to g, the gravitational accel-
erations In essence, we assume hydrostatic pressures

(2) <All terms involving radial velocity are neglected in
the remaining two equations of motion on the supposi-
tion that the radial velocity is very small compared
to the lateral velocities.

(3) The variation of the radial distance, r, over the
depth of the ocean is neglected and we write rv R,
the mean radius of the earth.

(Actually, the radial distance varies by about 1/1000
Oi ales woweil Lerma, )

(4) Terms which are divided by R are neglected in compar-
ison with all other terms.

(5) The region considered must not lie close to the north

pole since some terms which have been neglected

* In Appendix 3, this assumption is discussed in more details

iu ile
ii N ie

ela nls » fixity et ace gps WP DBE aA Lapras on wat:
Ne whe sink id iswoe ent te ‘sone: ber ivouee oe ae
aCe ee ) ct ia «2 fe wt wid teh Baia mth

‘hs Hevnehanrah fe: eis ne Sis “ynane!! apt nate AeA 38 “etc 4 \
a Wet pa ot aeine Facibomsare Pe Cem mY ait Liaw ies |
WR HOY wind bate! pte Kost ames saedT aevede| ti

emt deen? oy at dete toe aber ory Yobiak.adgd ie aa

PPI Ace haeiectle ll BERR, Sie

_tdison % sho ginny Rea “hy She Rouse he EOE Nano ee
i ore euro uoey « ‘cd hoi uieles ab thee Pian ie: abe
bie Lboosdndivnrts ‘fa ae 42) WORM ONION hE es seFsen:

: ra Parl Aaah » shoe mite en OS peak ae -, SL OY 24 fd

wn’ Red yerg ux enti yh ool oy

Lekber: saay
Lemeegaive aly ae ate (Sia MOL Ae He A Riel. en
| Desi asptito 2 ae ane Be Nee DabGtht, wrth ee dees

i.
5
ah Dy ae aS poe a De j 5
‘ 4 Me ieee, We BR i Bo PR ei
f
- i x = Fe , Fi
aes vats sw ee: yO ets hee ZEISS RO Seer ee rae
7) Y = ea ¥ y eae

~~. | if e ga exis Ay he becnataen St ‘iste. alr Te. aie
Fy . 5 " ie ee LO Sh 19S Te tieunealet ane

" x Bro aw itiFed ja Bh * i “ ib AAD ne) it a ; et % tale Ke tie 3
Ravi ; ; rat ee ie

{ | ;
ry ‘ ‘
hi, om MPA MALO Tans Crip My fon ces
mt 7 i 7 : i « *
ROR TL Cota whe wets Fe ye babi L i Wey Mull Aeele

wiih ae ek CR Tee

ut i? q i ¥ } } ‘ : y "
alga HAY. O7 eRelG and sal vu ant Beso} (th oe Fit, a ae

Al1-101 ll

previously become infinite at the pole. In our prob-
lem the ocean is confined to a region lying southof
latitude 70°,

(6) An appropriate interpretation of the results as applied
to the spherical earth must be made, keeping in mind
that the boundaries have been distorted. If we con-
sider a rectangular ocean in the plane, the appropri-
ate mapping onto the sphere would preserve the con-
stant east-west length. Such a mapping is not conform-
al since angles are not preserved. (In the caseof a
Mercator projection, on the other hand, angles are

preserved, but the east-west distance is distorted. )

Let us consider the simplified equation of vertical
motion (3). In integrated form, this equation is
uy)
i = 2h paz (ea)
Zz
where n measures the deflection of the free surface from its
equilibrium position and the scale of p is chosen in such a
manner that p =O m g=ne Now, the density is a function of
temperature and salinity. In our treatment of the problem, how=
ever, we wish to avoid the analytical difficulties introduced
by ineluding, explicitly, the enemey equation and) an equation of
state. We propose instead to account for the thermodynamics of
the problem empirically by prescribing a density distribution
which roughly conforms to observation’. In particular, we
* In Appendix 3 it is shown that a specification of the density

distribution and the assumption of hydrostatic pressure are
not necessary for the steady problem.

‘a eae iets AL Od mat in
ace Batt fa iyo fy at: BY Ait HE

wit

ne >

Fay i be ay Waa

V bekiaes ex os dkual ot, shhh oy et
: hikes aie yond A airgh pattie

ie

) he = w i rla®
aon ae "ht bet Ova Tass

ATAGOTY IE RE. A ws
atte” ‘eae: et aie M ines fe ele X wr :
sent) Rigen eats a | bi Pasig (iseee)
£

Peeoowxiteh Bore lnol TiAl tne

a " Ls
ele CG iy

ier se a

AV =O 22

choose p = p[z - T(x,y,t)], where the function p of the variable
(z = T) can be prescribed to fit observational datas We observe
that this functional form for p makes the curves of constant
density parallel to each other,

A complete analysis for the unknown quantities as func-
tions of the four independent variables x,y,z,t is exceedingly
difficult and we are forced to eliminate one variable by inte-
grating our equations over the vertical coordinate, z, and then
solving for suitably defined integrated quantitiese In so doing,
we lose information concerning the dependence of the unknowns
on z Since we are primarily concerned with general oceanic
circulation and mass transport, however, and since the integra-
tion leads to a considerable reduction in diffculty, the advan-
tages gained more than balance the loss of information involved.

Actually we cannot afford a complete loss of information
concerning the vertical devendence of velocity, This will become
apparent shortly.

The general density distribution must be specialized in
order to permit integration of the equations over the vertical
coordinate. Two cases will be considered.

First, let T be a surface which separates two layers) of

constant density so that

plz = Wx yt) l= pa Or zh Gey)
and
plz = T(x,y,t)] = pp for z <T(x,y,t)a
For this problem it is convenient to choose the coordi-

nate system with the xy-planes parallel to the undisturbed

tory

a as ' bi ine
tgs, fe dg a ti i i hie
hapa (ite

F muito a alt at

oy aa eee ” vile BOT ny

Nn i

es (dha sai) ey: ry bey

hy ‘ Fi
Dy r Wee yah

WME DITA WNW date ie Yylein i

y! ce :
b EDS ate te fix

mig

dap eal ‘g pa , My
Tn Ne her)
j :

4
i hi A Ra ery
Peal a na

doageara
\ Aiea by

avs

Cath Se

MN Mee ah;

ATA LON 23

equilibrium surface and with the plane z = O at the bottom of
the ocean, the bottom being assumed plane in this problem A
layer of constant density Po extends from the bottom of the
ocean to the height z = D5 + No where the constant Do is the
average height of the lower layer and No is the height of the
disturbed surface of this layer measured from the plane z = Do.
A layer of constant density Py extends from the height

Z = D5 + No to the free surface z = Dy + 11, where D, is the
distance from z = O of the undisturbed equilibrium surface of
the upper layer and 1, is the height of the disturbed free sur-
face of the upper layer measured from z = Dj.

Then equation (3.a) becomes

Pi = gp, 0m + Dz - Zz) for the upper layer
(3.b)
Ron = g PLO te Dap by ca D5) * EPalia * By = z) for the lower layer

(3.¢)
If we denote all quantities in the upper and lower layers
by subscripts 1 and 2, respectively, the equations (1), (2) and
(4), with expressions (3b) and (3.c) substituted for the pres-

sume in thie upper and) Hower Wayers., melsvectavely. pecome

i}

t
du 6u 6u y On t

2 Fe rranael Pipa is Lue eyelid eI WO MAW ye C5)
pee ban ey eas Oe a a

! ! !
his av av y i

1 Hy eae Heal ein) aoe et PAA hy, CG)
ae + Uy ae Ve aa + eqns sin(s) g By : il

au. avi ews
a eels i ee 2G @)

Ox Oy OZ

Mer 7. | a ae MG Tee, See
7 ihe aidved, He, we C4 uf rac ii tt im bee youl tane,

7 8 panier pa oa ul ‘ui iis a
+ a o s' asia “aghth we fy becom nian)
a oe
mihi pit f Iptaat F oe , x i OB
%o verbs Gy Pasig be Ts asi Ais +2

7 Bb ph ees aie ics
b: a

A 7 iter gn? Boesapiin ih BRS ho. ata hae

ak,
nt

‘
a i

K4tOys f as prees” (eies RM FAY) Mey ota
eget wid od i bodied f eat

‘waged Cae oe yy oe iY.

tt a ad

\ ; | . G
ce) Ese. “ ay wis Loading’

ne, * ata rg ae Vet le, minke tit

Al1=101 ol

!
—— Ee bd meee ap eee = t i y =
AE Dee Mp a 2Qv, sin (3)
2 ann L(Ve
3 ely ——S be) ee A.V)u (8)
pic 1 2
Py
t ! 1
Ov 1 OV OV
ae 2 Ugeecil? Ba
At Daas th VO aT + 20u, sinGs) =
dn On
= 2 cal 1 e '
g[b a os Gl aa ] aes A,V)V5 (9)
1 ! !
Au av aw
ax | Oy | ee u ue

where a = py/po, b 4P5 = pp /po =Ae/p,*

The problem defined by equations (5) = (10) with appro-
priate boundary conditions is quite general in that no assumption
has been made concerning the vertical distribution of velocity.
As we shall see later, when the equations are integrated over gz
and linearized, the simplified problem is still too difficult
to solvee For this reason we formulate a second problem which
allows a more general density distribution but which is more
restricted in other respectSe

In this problem we retain, for the time being,the gen-

eral form p = plz - T(x,y,t)]. Then the pressure terms in
equations (1) and (2) are*
ul
foe = & OP g a 3 Oil
pice 5) ox een ee tee
7
ace =e O28 aa aS SO, (11. b)
POY |. Pilg (Oy pi ey iS

Hor) the present problem the plane z = O lies on the undisturb-=
ed equilibrium free surface.

De oaeekeo.ey Yo! nnhdint io. |

& eve” nas hee any ee a ft

Shine Th oor Tite 2

BST thy wa id wre Paces A Hoe

ores ef hot rly oR AD LENO

UHL BGT NS

bay ee

ye FY aoa ey
yn fa rat Te pan

eh
A} ig
, jr 4
ah oe a
ue 4

Lee bt aA awit OC = & poe HEN Tea ca gO

y ey STi

a

MILL sWOUL 25)

where pg = pln - T(x,y,t) ], the density at the free surface.

If these terms be substituted into (1) and (2), we have

Ou y + ut out ab fy I Oi vt sin Z =
at Be A age a (?
g I" dp Z Ot af} Vv A;V)u!
- = as Of aw SS at 2 U. (2)
ie Gx i Boe ae 5 1
J
ov! 1 Ov! i dv! AO ae iva
oa lh Sa + V we ob AW sin SS a
= ut a zu (=)
a E
SS tz 2 elo CV" BAW) vale (13)
Pia ek Ree OG ‘

As stated previously, the problem will be simplified by
integrating the equations over the vertical coordinate, Ze
Let us first consider the problem defined by the equa-
tions (4), (12), (13). We assume that there is a depth
z = = h(x,y,t) below which the velocities may be considered
negligible™ (in some suitably defined sense), and we integrate
from Z = = h up to the free surface, The depth z = - h(x,y,t)
May, Of Course, vary from point to point im the oceans since
the velocities are negligibly small below z = - h, the horizontal
pressure gradients must also be negligibly small and we may
therefore write
Ps oF (14)
=-h p ieee
We must now specialize the general form of the density
distribution because an integration involving p will actually

* This assumption is the fundamental difference between the
two problems considered.

a re Ms ae

are, RS

a Bo kB Sse aid, | on

ti yt aki drage, |
oe ald We ies Ki 4
i a pen ry ei femil> :
| 1 Sewieshannn), iyo Ve * eed
' rc. nt wis fer a Cage tin

om ELAR it re oe! uf rm ve

a
MW op: he

i r
r, oe
a
pune
‘aan ny
an bi
i
4 }
rn |

! a 1 | so 7, ’ 4 sip
<A Rint tet) Ta 4a |
“ita ie i H i @ ay he aw t ey

Malt Wii Ms ee aie
feo

eat

NWS LOL 26

have to be carried out,

Define p = p[z - T(x,y,t)] in such a way that

OS Pane constant OW Gj SB > E

a2)
{]

cl = Zips for 2 Sz wid (erdiiconstante) (15)

= (Cheloor i od S| ¢

OD
I

Ph

With this definition the density is a continuous function of
depth and the ocean is divided into three distinct layers. A
layer of constant density, P>5, lies above a region in which the
density increases linearly with depth from p, to the value p_).
Finally, at the bottom, there is a layer of constant denen iy
Pye This prescribed distribution agrees well with the observed
density distribution.

If p, as given by (15), be substituted into equation
(14), we find that*

OD Conn | Mol eile
pe) AD. OR” y Ap ay vue)

where Ap = ay Se ree

If we integrate equations (16), we obtain
T=-—— 7 -C Gu)

where z = ~ C is the constant depth of T when n = 0. Physically,
Z=-= © is an average depth of the top layer or the depth of 2
when the ocean surface is undisturbed (i.ee., in the absence of

winds). These two quantities are, of course, identical.

* The alsebraic manipulation is given in Anpendix (a).
oO C oo p

pen ren ry eb

Vigo
oy Mian
a Nee

ree

4” aD i Bi: ne

Min yeet nner rege ais

Phy oa gga,
Ne Bye HM " a,

A11=10O1 27

let us next integrate equations (12) and (13) from

*
Z=- htogz=ne The pressure terms become

(a p
at 9p Seo On es a=h on 18;
On f AES dz gD ae g Kp n aS ( a)
Ia)
(fm)
= al ap 6 3 5 gD on - ¢g aah 3n (Caley b)
J-h ee oy ae a

where D =C + d/2, and the complete equations are

\

OU 45 Nit Ome aa 5) we Onl ag = 20 gia ©)
at ax | ay R
-h Veh
= Tk
= - gp 208 - g Fab , OUP saat + (a, SH (19)
Ox Ap @)2' Oz -h

(pea 7 me
OV + 6 u! a dz + Fl v! ee dz + 2Q0U sin (2)
-h

Ot xe a
JU =-h
= - gD se - Dual ene + AAV + (A, _ aval” (20)
Bo -h
where pe iS a res
U = pu'dz, V = pv'dz,
J-h -h

p is a constant,average density,

and NC2y¥ 42 6) as A(X, o1,t) - MCx,y,-h,t).
-h

The non-linear terms, u'(x,y,n,t)6n/9t, etc., from

* See Appendix 4(b) for the details.

+s Since the vascous terms) are, in any case, only ‘approximacitons
to the actual shear stresses, we have made the further approx-

imation n
( i} ee CIN Oulyg, ~ | 6) Gi) = 4b Oy <i
es ou. ee (A Uu zh Ou
Jia, 2 Oz 3 2 p va az) 3) Ca) 33 Oz lane

nae cer). bas (st) botiieisaaa |
ei, *paic nent anton: erie bars ;

PRAT MATION | Uy “Binh SOLU), cet pint oye
Siw ‘derlese aut man: ee Bite hon ts,

a LS x

P ; ess - ir Hy
brik at te eo 7 Fatal mi; oe or ne

Al1-LOL 23

the interchange of integrals and derivatives of the velocity
terms have been neglected, We have defined U and V as mass
transport components rather than as volume transport components
(by simply including an average density in the definition) be-
cause we want to compare some of our quantitative results with
observations and with the results of Munk, both of which are
given in terms of mass transport.

The terms A duly t and A avy" must give the wind-
2 Oa oh Ser | an

stress terms since they represent the shear stress evaluated at
the upper limits (the shear stress terms at z = = h are negli-
gible since -h was chosen as the depth where the motion becomes

negligible). Thus

i

A, Out|° =, = x component of wind stress
3 0z -h *
i
A iow = T. = ¥ COMponient OF wind) siGrelsis,

3) Gals hy y

In the equation of continuity we shall want to make use
of the kinematic free surface condition [9]
4 [z= 1G) l= 0 2G 4 = io
When expanded, this equation reads

wit{T = =i eal ga + yit|l g4
x Vi

where w'| ete. denotes the value of We GeN/omo Ge) 2 B= ac

Integration of the continuity equation (4) yields

ou fi oii ona! Wye ea in ae ae ah We
x Oy p | Ox P | dy oak | :

A LY) BAR Sn. Sove Siw fi td Thi

Be tephe be hay Fey ts wri)

guia! Sieben van ga ‘ab pane
Lon Cagtsnnd pst: me WE ANS Len ret ahh

ff | hae whhwve's wees $e tapid th : hey aha RG sitio Day

*

. wa, tii (ye ee if aw )

“slit ont: ayy vai fa

a

*. biet cue te ote
Ze r aba om as 7
| | gm ina Hou “a a

* r= hed PVH Meh" Bt oe a Te a

5 gant) ay Re mee m i EM Biase, ik id i (ae

Al1-1OL 29)

where w! | is negligible by definition of h(x,y,t). Substitut-

ing the free surface condition, we have

6 Lay 2 | fie . 91)
ae OC (

Equations (19) and (20) are now further simplified by
neglecting the non-linear terms. The reader is referred to
Appendix 2 for a detailed plausibility argument concerning this
step’.

Two final simplifications will be made in equations (19)
and (20). The Coriolis parameter 22 sin(Z) will be linearized
by writing 2 2 sin() = By where B = 20/R.

In addition, if the velocities are found in some manner,
then the free surface shape can be obtained by integrating the
equations (19) and (20) (neglecting the integrals of the non-
linear terms) with respect to x and y respectively. This yields

(eb + & <a a) =) x

where X denotes a known function. The solution of this quad-

ratic equation in yn is

* It must be emphasized that the argument presented in Appendix
2 is one of plausibility and not one of justification. In
view of the desirability of obtaining an analytic solution
we neglect the non-linear terms in the hope that the results
will agree qualitatively with observation and will so furnish
a mathematical description of the ocean circulation.

SUE A eli

| hy. ‘ve paterinres Save pers ae

ra as # Hie

mi pai T. ce aN a vi

ba Bi: ed ait re ch oF My i i ae i ‘ ‘h
avi : u he EY ’ ae if , r) tas vg , m io. Le,
BRP isons aoe rae
vii Bas, Phil ig a eis te i ros
a ‘e hes th ip cr) Le bee Z a 5 | itis, pervs +4 , bak ate Fat
S94, yaar) F Beas, Py

Hin
i)

eal Ge Maines

Aaah ant 30

But Lies erp
a betas le Aare atthe Cam
Ap gp p- Ap ep p-
Tat 20
P BP p
Hence
oe ak
epD

provided the above inequality holds. It will be shown in Sec-
tion 5 that the values of the constants which are appropriate
to our problem satisfy this condition.

Hence, the final equations take the form

au = V So D ane + AAU +7 22
ae SB: zs (22)
av a dnp S

Ca ee (23)
OU . OW = 2 Bae

QU ON ee oy 8 - 5)
Ox i Oy ot ven)

The boundary conditions are 0) = Vi =)Oyon a dland-water

boundary. The wind-stress is prescribed to be

So (Wo 1 gf i
T (it I caine) icos may. ua 0)

where W', I'' represent the magnitude of the mean wind«stress
and the amplitude of the time variation of the
wind=stress, respectively,

my) is the frequency of the wind variation,

n is the wave number associated with the wind dis-=
tribution.

. At ;
Ay ae) pve ie)
ee a a Meee chetate opeehlil

oP mNiiNetie ce Ox F

j 4 *y

Al1=-1O1 hal

One can consider the above form for the wind as a typi-
cal term in a Fourier series for a more general wind distribu-
tion. The numerical results in this report are based on a value
of w corresponding to a period of one year and n is set equal
to 2n/s where s is the north-south length of the ocean.

The problem defined by equations (22), (23), (24) to-
gether with the boundary conditions and the wind-stress term
will be referred to as the one-layer problem or Problem 1;

("one layer" because the integration over z is carried out over
the entire depth).

For the second problem in which the density stratifica-
; tion is specified as two constant density layers, we have equa-
tions (5) - (10). Each equation will be integrated over the
vertical coordinate, z, with (5) - (7) integrated over the top
layer, i.e., from z = Do +5 to 2 = Dy +7, and (8) - (10)
WIONGSSIENUEI ONASID rolls’ eyes Ie wisie anes | aicOm 4 = 0) UO B= Do + Noe

As in problem 1, the non-linear terms, Be On/Ot etc.,
resulting from the interchange of differentiation and integra-
tion,are neglected. The viscous terms are integrated in the
same manner and the Coriolis parameter is again linearized.

Then the integrated forms of (5) - (10) are

aU
Mey. 4 e( Dy = Daye iy Sn ean ee (25)
34 il 1 2 1 20 Re i ba ae 2>c
av = On, Pp —
= + By, + g(D, = Do + 1y = 1d) wail = AAV, + Tiy ~ Toy (26)
Our, Pav
1 ED el NS %
ie ii aan TOOR (P49y - 2 PoNd) (27)

A i. ve poe atte: wl ob v whe
BCL ee le in tae | ey, 1 ‘s
omlitiny mt! oy Aeitil” Mee tow oF a
avon tee a rs bits Bian wee) .

vahiaw96, ‘pet i te Ptah “ J ‘bl ; : ‘

wnt we WBE) 48) exis
anes oor: Went Une oO ‘belt
ws yi et" Hite Bee spans
“vo Done be hen, . ie ae

m4

angi tovige ahs a Baie cae
mane pret ween t cd tncvatl &

i “pry ‘mate ies ‘Bt Pecktey dict

where

We specify ee to take the same form as T

u g(D, aii Np) gql bane Ts P11, J=AdVo+t, 7
ou av
a 8 = = & (Qo)
Ox ay Che
p2a+ ty ean
= : i Al
Uae = | piu,d2, Wo = | p,vi92,
eDp+ "1p Do+ Np
pPotNs ine:
— ! — 1
U., = PrUnlZ y Wo = Piao
YO Uo

are the x and y components, respectively of the wind-

stress on the free surface

are the x and y components, respectively,of the shear
stress between the lower layer and the upper layer at

the interface,

are the x and y components, respectively, of the shear
stress between water in the lower layer and the ocean

bottome

x

in Problem le

The remaining shear stress terms are assumed to be negligible.

The boundary conditions are U1 = Vi = Up = We = Oona land-

water boundary, i.¢., vanishing mass transport in each layere

These conditions are much more restrictive than the boundary

conditions of the one-layer problem since there can be no verti-~

cal interchange of transport across the interface at the bound-

aries.

Equations (25) - (30), together with the boundary condi-

tions and the wind-stress, constitute Problem 2, or the two-=

Myatt ir :#} 6 aa

'
is
*

AR

va
i

Pa te Wake! cite!

it View gi? ry bs ehh?
i: an shea Aa

vy

ia eit wat Re; ‘uh ae Bd eee
; ft ene Bia ref 9

Bia W: 0 a ie ‘
{ :
si tehound feng rh Hoogone’
‘i ‘ 4 .
; “ho Mtb uit vagy: Biwi

>

ae i aioe bod an ri ih “it er md ts

Catal te paint hus

ae ¥ ‘ato. inna bagd 4 | ik iat
Abi ag i ae ay
4) ny 1G ¢ a MEAD | vi aH Cab

|
Hert
valde

ey th :

nue }
Bee ky
ht ne

ae ar mM) vi ae : Patty.
vi Wer) eae aC Dae) ie tl Gat

Vor ee
a A

NT TOL 33

layer problem (the vertical integration being carried out in
two steps).

It may seem to the reader at this point that, since we
have integrated the equations of motion over the vertical coor-
dinate z in both problems, there is nothing to be gained by
considering Problem 2 in which the density distribution is more
specialized thaw that of Problem le Because of the importance
of this point, we shall discuss the significance of the two
problems in more details

Needless to say, the problem of greatest interest in-
cludes the more general uensity distribution of Problem 1, the
four independent coordinates x,y,z,t, and the full non-linear
equations. The wind-stress components appear as the values of
the vertical shear at the free surface z = (x,y,t). The solu-
tion of this problem would, of course, include complete inform-
ation concerning the dependence of the motion on Zz Being
unable to attack this problem, we are forced to integrate the
equations over z and to content ourselves with a solution for
the transport componentse

At first this integration over the vertical coordinate,
Z, appears to have only one shortcoming, vize, a loss of inform-
ation concerning the vertical distribution of velocity. We
cannot, however, completely afford such a loss of informa tion
in the formulation of the "transport" problem and some recourse
to field evidence is necessary. Unfortunately, however, accur-
ate observational data are extremely difficult to obtain. In

particular, it is generally held that the motion in the deep

aa hares Donnan par bk rent weedy

ot whine va oki

trai naadoine ate age isis |

vil Ronin, i ar ante ee

m) | ao oe apart nai 2h Ae te) 3h: de a ;
pcind-vorss, chet” , Mle dresicee i he

et al i tho e a it

VO GTA a Ge nen

7 . ie

ore: Feasts

geonases Wont 2

eee at ky Het.

‘BMS kerwo'tre:
Riau ae in
iA “en IO etavian
lant ; grre oy

“| ohn wf! md

Al1-101 34

layers of the oceans is negligible, but no definite conelusions
have been established to this effect. It is because of this
uncertainty that we consider the two separate problems, 1 and

2. If the motion of deep water is really negligible, the pres-
sure gradient in deep water is also negligible and the assump-=
tions of Problem 1 are justified with the result that the thermo-
cline responds instantaneously to a change in the free surface
height provided the hydrostatic pressure assumption is also
valid. Consequently, the only motion existing in the layer
below the bottom of the thermocline is that due to the shear
force exerted by the water at the depth zg = T - d onto the water
below it. Vertical shear will extend the motion to lower depths
but the velocities will decay exponentially in the vertical
direction [1] until they become negligible.

If the motion of deep water is not negligible, then we
must consider Problem 2 where no such assumption is made, In
that case, the thermocline does not necessarily respond imme-
diiawelly sional echanze an ‘they irce (surmace and. iiconsiequicmt inyaura!
pressure gradient may result. Since the fluid in the bottom
layer is homogeneous and since the wave length of the thermo-
eline is large compared to the depth of the lower layer, a
velocity with uniform vertical profile is set up, (hydrostatic
pressure being again assumed). The shear stress, Toy 9 exerted
by the water of the upper layer onto the surface of the lower
layer also causes a velocity in the lower layer. This velocity

is not uniform vertically. The problem including the effect of

7 7 - : ' _ TARE, aa 5: he v=
aT i he | ' ‘i ah : Tie ey ita) Fry ey : ae 4

: ) f

’ J ‘

ie tai bmn ananrs mali! \esin! sett: ees wen gn
eins “a aha <4 Eb i et |
five a aus “0 Ct ec

miberdy- welt cinta won eit

pone wet mas hehidion put

SS yee See). WY HE eased ca Be Ws ie i
EPA BEG SY Omire iy ae
eae Mi cl ering

Ri ED gt 8k

PSFK, ats Gs His be bey te ms Wis!

> ; 2 a b . y ; . i ‘ { 4
Hin! yt ho A aay RA RE AMY a NSE) PRS a f, e bs ;

at

120) ; u i

FE al '¢ nt

a. 1 ae vont BEE re (ntieg. a eos
‘

; i as "7
. yi bie At pyrene: De aay
4 , 4 ‘ is Tew
7 iv, bet Bag Lars aise Pieri ;
1 : } yr y r th i
ee ; sre ‘ ' oy
bial eR Ly Mal Le ie ‘ f ms i Me Fras ta sh Re SMT resto Wie,
. hd é fat the. ) 4 ‘ Teen tay ite Sula
, : , fe

7 ; if eh ee 5 ci ’ a Wey ee A
Pt ok A ed 8 NE ae ee aS See | Aare ees bi gue Fe

need ae

nike Fg
. 4

a)

7 ‘

ay

7

it “ Lite

Mle MOI 35

G and, in addition, the stress of the ocean bottom on the

Dre
lower layer, is so complex that an analytic solution is out of
the question. We therefore assume that the effects of these
shear stresses on the velocity in the lower layer are negligible
when compared to the velocity resulting from the variation of
the thermocline.

If the two problems were now solved and the results
compared with available observational data, it might be possible
to determine whether or not sensible deep-water motion exists.
As we shall see in Sec. 5, however, Problem 2 cannot be solved

by the methods used in the present paper, and numerical methods

of solution may have to be employed.

4, Solution to Problem 1, The solution to Problem 1
will be carried out by means of a boundary layer technique. For
the convenience of the reader who is not familiar with this
technique and whowishes to follow the details of the present
section, a discussion of boundary layer analysis is presented
alfal,| JNoyerchatclaln-e oy

The solution of differential equations by boundary layer
analysis can be carried out most conveniently if the equations

are first put into non-dimensional form. Let the rectangular

ocean have dimensions
OK 255 O<7 <s CPigo Dc

Choose as a reference length the north-south dimension,

s, and define dimensionless coordinates x', y' by

. ‘ sae ivy, i LW ) ee ee Aa

| a AWS as Ot

a dl fon ‘sic 40 WHER Sek ce BRD G 7

he She es: wohanton wih dd ina. the

‘por Waist ays yitaete aad ee
ore wie ‘ais, vs deat ‘ual
Gy a8 bin ate Basins th

a wn)

ary asin OS Diels Bows vas
Py valde ot ate, in 94 as yi Lists

abate 02 top re aiticarrsg iyi? ied theamass. He

Spevtes oa HP eopiiy 5 > et
“Wheaton Sadan ‘s Wes yA

gr Ae) ts a a ei, Ht nen

és % ees ‘ Meta ra Ra le es -

ay rth nes ww Ww we pl TK dah va is aa i
TAsuwAy ‘aa he we shat) wea
Boas oe 1) ath Neer AAU SNE ~ AY

Sool ot Mra Tk a ei We oe

AVERUR TOS Chemin UiLaaee Bo te ee Onesie

BLA ator hoy Blah te)

. m

; ; {\ Ce y ph y

pobar ce th Pipe mph | i.e

wo) 13 a

AV 1 =LOL 36
yo = syle (ie = Sate

Then the east-west and north-south dimensions of the ocean in

non-dimensional coordinates will be
me
O52 Seats On yi ie

We shall assume that the ocean is bounded by land on
x' = O,r and by water on y' = 0,1.

Now differentiate equation (3.23) with respect to x and
equation (3.22) with respect to y and subtract. Substituting

for the prescribed wind-stress, t,, we then have

0 (ai ot OU 4, Gi ¢ aM 2 au
at ay ay Gees ay) wy a AACS a
- [nW! + WM! sin wt]sin ny. (a)
Introducing y
aN eS 3 yl GE = WE
and defining
!
mW Wl | neue ae eee
Wee W
equation (1) becomes
w 9 av au ell Ov =
eee (HONE Gi. ON BONE cs SONICS
ca yo) OF. “ol oot
eee, oo iam ie eo iil j
ie? !
BS Bpe ax tay t@ ax 'ay fe) yt?

=W i = @ sim ailsim msy! @2))

or

J }
if. i
ie Oe an
: ‘ ike fare ea ey ) ea i Te AN robe es
i Mesne ote TS Sati wach ly, tatoiaitnry: Dem
| | dt |
pi > Ve 3):
t u a : ; i at AVP
ee fest YO Delmar RL Nee oe
MS = “ary , a - . q : / My ke

Ht. Peed Lea0w, adorn,

ey t a ba et
Vey oi-8 wis

Gere
: i th
ro
oa
:
2
i
7
oth
ita
Rive
h -
|
i
i -
4

iN) Lo LOal 37

) 7 7 1 ay 7 ee
oe a

De oie meee ON eT
Ws? axt3 ax‘ ay axt@ay! «gy 13

= (al ee Silene] satin inane! (3)

Now, since the term (1 + a sin t)sin nsy! is of order
unity’, and since this term represents the ‘inal inten generates
the velocities, it is aopropriate to choose a dimensionless
velocity which will also be of order unity. Hence we select a
non-dimensional term containing the velocity which is presumably
of order one. The term suggested by an inspection of (3) is

-BV/W and we therefore put
yo emt iy = BY
W W

We shall drop the primes from the x' and y! coordinates
and work in the non-dimensional system henceforth. With the

definitions, € = A/3s2 and 5 = w/Bg equation (3) becomes

Be Ul ty lU VE eel cee i a cee ae :

= (i 4 @ sin a) sin nsy, (4)

where Vy = 0V/éx, (Vx - Uy), = 0°V/8x0T ~ 9°U/dyee: , ete.

If we non=dimensionalize the momentum equations (3,22)
and (3.23) and the continuity equation (3.24) by means of the
above definitions, we must introduce a new parameter 9 and a

variable H defined by
= 422

a = aed a OTS e

B-s3 W

* As will be seen later, we shall choose a to be 0.2.

1)

any

Ricuatand "i al Yay he
\shtereciny AOL | iat
parent pot Me co

ae

b fu
We hE

iL, Om 38

The equations become

nsd au - nsy V+e@ = acoA (il eb cilia Gaoos inchy (5)
T x

nsd OV + nsy U+6 ols ns ¢ AV (6)
2} OT oy
and (3.8) becomes
OU + OV = 16) moist ° (7)
Cbs GY at

Attempts to solve equations (5) to (8) in closed form
were unsuccessful.e We therefore resorted to seeking solutions

by a perturbation expansion in the parameter 6.

Let 5
Uw = Us + dUz + Uy BOAO
Tey. = Si. = Oey. 4
a XO L 2 ke
H = H~ + 8H. + 8°H +
a O alt 5) ees e

Our formal procedure is to regard the coefficients U,, JU, etCe y
as coefficients in a power series in 6.
Let us substitute the expansions into equations (4),

(prey) and(7). We have
Woe ae OVix “OOO Voy = iy = oC

ey LU. a OUsis. + ees ee a OUT cco

Wee tt 8V> eon Sr Sy

ak
oxxx 7 OVI xxx ee

O
oF Voxyy ote Vi xyy + eve = Vox xy - BU xy — eee
- = fe) —~ eee - j 7

Wane Viyyy 7 (1 + a sin t)sin nsy (8)

ap al) Be |

word, booty, sity

i

j

Woh oy tit

y

“See

ry ‘
fit h tel. ur
4 an Cc * r
Ore

2 es crept ead i ing aly 4 4 be
; i ee CSRS ON

MILT SwOa 39

OU OU,

nsé[—2 +6 —= +... ]=- nsy[V, + OW sais)
OT OT
0H 0H
aee|O) peenelll ae = U + 67 + ene
+ Sls OS ae ] = nseA[U, 1 ]
- (1 + a sint )cos nsy (9)
O orl
nsd(|——= + 6 ——= + ee. | + U. +8U5 + aco
ae (olan ] nsyl 0 e 1
0H 0H
fe) i fe
Fl fo) age. oe ] = nseA [V, + dV, + ove] (10)
au aU av av aH
© at fs See a eee fee a: oy eres eo Pea Bes gy Ip
Ox Q Oy oy at OT (Gia)

If we regroup each of these equations so as to combine
the coefficients of each power of 6 4 we have, upon retaining

terms in 8° and & only:

ote: 5 Veal © Vows ELV Gx ‘ Voxyy ee Voxxy a Cosa

ee

+ (1 + a sin t)sin nsy ee one =Usy, + ylU,, + Viy!

= Wa = Clee. oo U ee bs eaes0 Caz)

ina Ua xy Pell

0H
{-nsyv, + 9 = — IMs se (il a> & sulin 4 eos ney f

aU 0H
+ Fale] OES Galen eee aS ONG Oa iolba = © Ci
{ At nS Re . ee)

0H 6V 0H
nsyU, + 9 —2 - nseAV, >+ e =0 . msl. + © —al=nseAVa sas = 0
: Oy : at 1 oy )
L Se Gi

COUa) CUR { u, WV, 0Ho
aoe + [ie * a FR fo) $000 = O Ca)

Setting each of the coefficients of 8 equal to zero we

have as the zero order equations for (12) and (15)

j

¥ (i) We oN hie ion Bb ec (i } } si yi
isd bY ne me i iy My ae SPR Wi fi aya

; a ;
Ly “fi 1 Dagraniat

my si

Len

j
it

a ee
te) vi

ir Hii ts Tee, Py

i t
ithe | ated ie ,
Cia

Al1-101 LO

BI oscar 3g Voxyy = Voxxy = Voyyy ]-Vj=01 Dio} Sali 45) Sibi inlay (CLS)

U ae W = 0 (17)

UW SO. Om sO, ae = fo (18)

With the particular wind distribution prescribed we will

also be able to satisfy the additional boundary conditions
aU,
V Sata O on Ww = O51 (18. a)

We shall proceed to solve equations (16), (17) together
with the boundary conditions (18), (18.a) for the velocities
Us and Vos

Detine a sitream fLunciron

. 8 eek)
Vo= fe, U=- x (19)

Somuaat (Ly) as) catistied adentically. — Then (6) can sbelwiriannen

ceAAy -p, = (1 + a sin t)sin nsy (20)
oyna)
where AA( ) is the biharmonic operator 4) D +
ax 6x-8y-
iy x x-dy
0 )
oy

Equation (36) is similar to the one solved by Munk [5]
and Munk and Carrier [6]. In the present case, however, the
non-dimensional time, tT, appears as a parameter, so that our
problem corresponds to a quasi-steady probleme

Equation (20) together with the boundary conditions

My y
a
y
‘ ‘
ari 5
i .

ian men ha " bbe

7 ‘eh

‘a ae

me)

Al1-101 WL

baw = oO One TOR

We W = © Cay = 0,0 (20. a)

can be solved for wv by applying the boundary-layer technique”

GOmieiae bo wMndancitels 5c — One Lhe solution mais

y=

Uy

j -1/3
GU ace Sivan) Salita! ny 4 =x + Tf = el/3 i B/S} (Cs-19))2
3 x Vze2/3
alae -r)cos( 5 )) et
Al/3 V3
x VRE ~ i
= (4/3e 13 - —£_) sin( 2) ie 2 \
V3 J} (21)
From (19) U, and V, are found to be
l 1/3 (Gene
= = ns(1 +a sin t)cos ees +rem—€ = el/36
13 73 1/3 at
Mite --s)costavee 4a = aires 38 )] aor me
2 /3 D |
22)
i) Gene 2
= (L 2 @Gilla wea may Ss 1 = 6
u “1/3
-1/3 -1/3 Wf) 288
+ Koa = ) ioc ae = V3) sin(SB2 ye a 7
3

The zero-order equations derived from (13) and (14) are

The problem defined by equations (20), (20.a) is solved in

detail in Appendix 5 by means of the boundary layer tecnnique.
The method used in the remainder of this paper is described
in detail in that sections Munk and Carrier [6] used this
method for solving the steady problem in a triangular ocean.

AERO Ser Teta. Ors
ie AM ite! ; ye Leg
on Ro TORU lie ee, ap

pasty . wt 4 i t
Ret Who Me.

PAY

1

, De NAN
in:

Oe eh ‘eee 8, poy i
iy) OM itaee wa terns He:
pears Hes ky Coto te.

} 2

nese Tea WEE td

Al1l-1LO1 42

OH

See SING GP WS SAUG Sl sP sint)cos nsy (24)

CH
Oy

- nsyU, + nseAV,. (25)

Solving for H,, we have, (neglecting terms of order e),

OH, = (1 + a sint)(cos nsy + nsy sin nsy)( - x +97 - -l/3)
-1/3
dp (CL 2b ies Sala Ga) iach Gialiny falc j 3, (ere
-1/3
a8 ice 3 = r)cos( se ae
-1/3
-1/3 Bese

1/3 : e
TGV Ole - —)sin(* Jy 2
V3 a : es

First-Order Solution
From equations (12) and (15) the terms of first order

in 8 are found to be

e[V U Se) See care yH Jt (27)

lxxx 7 Vascyy mi Bear 5 knny 1 Ox oy

Us 7 ae. =o Hee (23)

The boundary conditions are again U, = Wa = © Ola 2c = Opies

In (27) and (28) the right sides of the equations pro-
vide the driving term as did (1 + a sin t)sin nsy in the zero-
order equation. We shall proceed with the solution by means of
the boundary layer technique.

For the interior solution we assume that the functions

are smooth and hence that the derivatives are of the same order

i \ i 5 4 av ig
Dery Ms Avi ae | AW
\ ya ' : Aik vi ta de ea : " Deo
) y ; i Mii x en, uD WALES en! ‘i ; Ty ee) Path Ore te
ay ie ies / i ae sieeitaey plan Nii } : nt ; iy
thi ; i ANE aa oud NA be : b A as i
i) : hou } t i» : } : of ; ih)
. Sa ‘ , : ‘ xe pm ane Sh ee F a ea i
} hi A wat SAA tae F . ] ; oh
2 4 F

j
; “
Nes

Whe, Bet nee EK Mey ae

)

co: eto. TeiwIriet Yhitoative.

Ob har NOt Fy.

>
ee
+
?
cs
=
SE

: Y; fhe Ka ea aa hate ak a in ie) ee Ae

. . ae. (ia ot i Dis 2 eres “ag

© ‘ t, - eh Shc Rg fs :
ener ons MS Gar aah Tie Ht

SRAG PS oh tT ari cha nigeree 5 A eS Sy

Wasbie epee Ota io: csc OUR VP eh

ANUS WOAL 4.3

of magnitude as the functions themselves, The terms multiplied
by € may therefore be neglected.

Let us rewrite equations (22), (23), and (26) as the
sum of two parts - one part, with subscript i, having the same
order of magnitude throughout the domain (the "interior solution"):
the second part, with subscript b, sensibly large near the bound-
ary and negligibly small in the interior, (the "boundary layer

contribution")

1/
Oo, == ns(1 +a sint)cos nsy(- x +r-€ 3

va

13 Rea) £3 +

Usp = 7 ns(1 +asin t)cos nsy “V3
=1/3 i 3 4 1/3 Be Za
Elite’ *-n cos 2 4G@Bic a een gene i
2 73 2
Vos = = (Gl ia) sine) samaenasy,
(O)aL -1/3
Glen [cos(= Be ) +
We Siti Suki aa) Sia msi Ke 2
-1/
“1/3 . Bes sails
+ (Zee ——-- = V3) sin(——5—— )le =
V3
Ctln = (1 + a sin t)(cos nsy+nsy sin nsy)(- x +7r - 212)
3 -1/3
ch = Gh) =e ergabey 45 ish) Sakin nay 7 60/3949

aa eG eS
+((e/3-r)cos(2¥3e VC 2 a yen ee a. :
2 V3 é

Ryle uae Wiel ! PON ROU Te Un gy Ny
Oe | Dae.”

CEG STS) Av Tee EP

Phil We arik | A dN wo he

4 eee!

me T a ALi naw at

ead ‘BR, er oie tes)
ey. we iinet ot ayes
be fincgaeots oni La nol ssi 7
| pret oat ‘aye, muvee x |
yang: Wren bet wer” pda)

f j bee

MaSUen. Ly

We expect the boundary layer thickness to have the same
order of magnitude in the higher order solutions as in the zero-
Order solution, Wwvalza. e 1/3, Thus, in order to find the first-
order interior solution, we neglect all the terms with subscript

b since they are negligible in the interior, Thus immediately,

Vigo the interior portion of V,, is known and is (from (27))
Moa nee EViotee $ Uoiy = yHoil,
= oa -x +r- eY3)tcos nsy + (nsy-@n*s“) sin nsy]

(29)
From (28) and (29) the interior portion of U,, U,,, can

be computed directly, giving

cos 1/3 i
UG =) Se Saeciaee i= “ye fee oe il lOnsy sila iashy +
+ (n@s*y" + en>s3 + 2)cos nsy }] + C1 (yy)

20

where C,(y,7) is arbitrary and must be evaluated by applying
the boundary conditions to the complete solution, i.e., inter-
ior solution plus boundary layer contribution.

Before proceeding with the boundary layer analysis we
can simplify equation (27) to some extent. Near x =r,

OX

250, 1 SOGe athe OG), ala Se "OU. Wes dn

each case we are justified in using only the contribution of the

2/3) 213. ao

T= Oe 3), Uy, ae?) ena omto(e/3), Near

Vj, term provided on S Sil eing) 67 As will be shown
later, when the appropriate dimensional constants are substi-

tuted, the error involved in neglecting the other terms is

amo ise er lh Wee yams. oe nie
iim ii. bs ay 2 nie 5 ety

fat on “int me a
(entaeteent ot saat

Agee me ee Sain : a ig

a i a cs an a At ha th ai

at ey Avs, sof de etd ay) is meri )

ALISON 5

extremely small, Thus for all practical purposes, equation

(27), near the boundaries can be written

e[v 5 W

lbpxxx Ino onan Opec ss Th byyy ]
ie =-1
-1/3,(x-r)e

-1/3
ar - ae jens = +

=", WOE GE alia Da

+

a snes )Je a 6 (31)

Near x = 0, the inhomogeneous contribution which contains the

-1
term git-re can be neglected since its effect is felt only

near the eastern boundary, ise, Bee ies SHU abiLenollye 5 lalevelic

x =r, the terms multiplied by e ~~5~ ~~ can be neglected, Thus

HOG eUMe Ke sion mean a) = 0),

ee os Vi pxyy 3 Ui pxxy a Ui byyy Ive Vib
= A -1/3
= 6 OOS TG Salad Msyy ilies 21) aye cose) fe
22/3 By arn (3)
ios 1s Ve Son mea 3
4 ————sin( 5 dle
V3

Now suppose the x coordinate is stretched by substitut—

aA exe = eke (k > 0). Then (32) becomes

leer , ek _ eee ee ue
i LbEE Mbeyy IbEEy ~ © lbyyy ~ 1b

= 6 COS « Sil tasy7iL(Ges

k=-1/3
Fe

. Be Deeaie
" meme ei a3 : dle 2 ,

Ty AP
Wy
Ve

ee Besa \

ede aba) 2

\s ee) Uo)

titer ta J:

i UMN Be ; 4 ;

A11-101 We

The term of highest order derivative in & is matched
with the remaining largest term in the equation, Hence, we

3 *
formally match e273*y with V Then k = 1/3. and the

UDESE Ifo)

equation becomes

Vipces ~ Yip = 2008 7 Sin patie :
Ay
are sin) ) &° ote =). (33)
V3

The term Vip can now be exnvanded in an asymptotic series
in € and only the first terms will be kepte Since the inhomo-
geneous term of (33) contains only exponential and trigonometric

funeccrons, let us Ury a solution of the form

Vay = 6 COs we Silin Inshc weve cos( V3) 7° sin( £15) ge
(34)

where vy and vo are the first terms of asymptotic expansions

and are to be determined.
If Vi, 2s ae by (34) be substituted into (33) and
if coeificients of Sea (S BV3) be equated, two simultaneous differ-

ential equations with constant coefficients result.

es O O a VA =e BV Sey 4 2}
2 Vip 3 Vier 4 Viger 7 : Le +35 Vice = i 2 DE

2
3V3 y 3 ey y° -3 v9 235
Vee asa Lene, 2

* The fact that k = 1/3 indicates that the thickness of the
boundary layer is of the same order of magnitude in the zero
and first order solution, as was anticipated.

‘poses * die ent Aa

owe Ce ae
ire

rata

Peep ve hemi aidl ck
Ok EY SO CEIMAESR teat

By
ae

A= TOM 7

Particular solutions of (35), (36) are

v4 = ea cae 3 , Vo = ee Se Ee
3 i 3 V3

The homogeneous solutions may be derived by letting

Vi = eS Vv

0
ali
Then (35), (36) become

eee ee ie QO (7

ee a Ne) ee

]
Gx3 eels x + Bi-=n - 3X Pel = @, (33)

Hence, since the determinant of these two simultaneous
- equations must vanish, we have

3. 2

Bz
OF - 30° -30)° + 22 (y- 07)? = 0, (39)

The roots are

= 0,0. Seem A —_ ; VAS & V2 a (0)
cnerar
1/3 3+ V3i Vong euslsty 7 3}3b 15
Ou a 2 7+ Ane +A e Aye ee
—6) 4 ees Say V3it sau ne - Y3ik
Vo = ==-—- 6 + Bee D + Boe +B, € 2 By, By
moe
Hence, from (3+)
Vip = weost sin nsy e234 5 (35 fe 277 eV 31E+ Aye V32E 4 1

1/3 Voit) = Vai 1
+ sin Sy aie E+ Boe +B),€ : +B, | Me
3 V3 ;

‘

Hy iy CRAM

one
baa

Crt a ie

1

MLASSOR. 48

where we have set Ar = Ay = ie B, = 0 since the contributions
of the terms with those coefficients do not tend to zero as
E—-> ©,

When (37) and (38) are used to get a relationship be-
tween the A; and the Bay then the final form for Vj, near x = 0

LG} atoll WO foe)

he W/3
oe =a COS Sin ensy ve 23d M3 5 ree sdeos(lab.)

1/3
+ a Sa © 3) sing a (42)

where Cc, and Oe are arbitrary functions of y and t and must be
| found by applying the boundary conditions to the complete solu-
tione

In a similar manner, if we make the following two sub-=

stitutions for the right (eastern) boundary

h
(x-r) =e!
h=1/3

-1/3 ne

Vay =) GOS % Salia migyy & e [v$ Tiialiove Ms
*
Wemmind that lk = 1/3) and
-1/

Woe = 6 COS a Silla inca eS 3f 3 + A, (y,t)le" g fis)

We have used the fact that V.,.—>0 as n~- ©. (As stated in

1b
the appendix, 1 ->~q@ when the boundary on the right is under
consideration, since the boundary layer solutions must become

* The same remark applies to the value of h as previously made
Hors che awwe VOm | Ke.

‘don ddtaete? ‘aah « ube th a
ag ih vg Rll hint: uN

Nd th

ay mero

AGT AUS 9

negligibly small as the distance from the boundary increases,
ican TS) Ol Xe Ceereascss
If the three contributions (29), (42), (43) to the com-

plete solution for V, be added, the final form for Ve is

dL
Vi = SEC OS8 (See eee 13) ty?ns + @n“s-) sin nsy + y cos nsy]
i.)
=/3 -1/3
-2
+ @ COS tSin nsy « 13 jee 2 4 Co(y))eos(= eas
3
-1/3 x Vie ae a = zene
+ (Sie ee las x +C 3(y))sin(iets-
3 V3
-1/3
(a
W/o l/s yi (x-r)e
+ @cOS tT Sin nsy ¢€ ae E + A, (y) io .
: (44)
By means of the continuity equation we then find
UL =e ao [2nsy sin nsy + Gan 25 oa Der 6n3s3)cos nsy |

2 ’ 1/3
[- oe ar (( we) ] + Ci(yb)- = sy) sin nsy e-/ 36 ==2)E

173} (xer)e7l/3

- acos Tt ns cos nsy[A- : a. e
-1/3
A FS
= cos GF Sin nsy = A mE - oo ialenye habia sash S
-1
-1/3 1/3 -1/3 eee
T(rel/3 - pee) ay ie - #&— sinha ) Ie -
73 2/3 -1/3
- a@cos tT ns cos nsyé 20 EXC Sg Sed fe 1S) ogee oe)
3 3 3 2
1/3
2/3 ELA) iieny Se a
+ (ea gee? Soe gems Vet dle 2
Va woos 3 V3 2

-1/3 173
“a tc, V2 ¢ 3) sin ney Cog WBE Es) +

=273))) eycecmas
- V3 C,,)sin nsy sin (Sie ——) e 2 5 CD

+ aecost

+

1 0kt
PAL Wi

} sii an os
Mm a

mM ae
SEN ha

”

{yet Oy “Wy, ean ae ie 4.

ALT oWOal 50

The arbitrary functions of y can be evaluated by means

of the boundary conditions U, = V, = 0 on x = O,r. We have
2/3 2 2?
sin nsy Cy = SS [Cay + Q@n°s“)sin nsy + y cos nsy] (46 )
4 e2/3 2 2 2
sin nsy A, = =a (Las + Qn°s-)sin nsy + y cos nsy] (4-7)

C, = fe! a.cget [onsy sin nsy + (y°n 25> 4 2 + On3s3) »

—

2 iL 2 2
’ “GOS nsy]{5- = Pe /3 + € (3y -€ 13 (828 + 1l)cos nsy f (48)
sin nsy C, = [2e/3(y*ns +2 4 en2s®)(rel/3 = 22 - 62/3) +
F | ns
2€ r9et/3 bh 2/3
+ S& . £2£_-Jsin nsy + (Sy cos nsy - — sin nsy)re ~
3 3 ns
2 2
- (9y cos nsy - ee sin nsy)e eel: [Sy (ee oneacyneae ih nsy -
(49)
The first-order contribution to H can be found from
equations (12), (13). The first order equations are
ou 0H,
ns ee nsy va + Q erin aseAU,
aV 6H
ale pune
ns rem ash We, ae © arte nseAV,
from which Hy is found to be
Hy = peel. + y “Joos nsy + (y3 ns + yon® s =) sin nsy

- r)(x + swe) J + 4. cos nsy +

2/3 a
(3 (G08 4 » De ae nsy , ees yb,
nas

tt Rabat Seo Sita

Cady

(fat)

ab cw tn

en rn 4 ite

NTT Wea Byal

4+ 2 o98 ~ nsy sin nsy e 195 [8 yeremee _ rx ae 3
i op.
1/3} 2/3 SS Sil 3} 1/ as
= eee 2 e x Vale E 3
ieee

Wy 3}

aed a) een
+ (C. = |B C.) sin( 11 be 2 + aaa nsy sin nsy *°

~ = 3}

1 I/ Be) = 7/3 | (eerde
N= (x = r-=-c€ ) + A, pe :

: J
The terms Uy and Vi do not satisfy the boundary con-
6U

ditions Wa = = =Oony =0,le We must recall that these
boundary conditions were chosen rather arbitrarily as being

(50)

plausible ones for the type of wind distribution specified, and
the y dependence of the zero-order solution was accordingly
chosen as sin nsy.e We cannot expect such a y dependence to
satisfy all the conditions for each set of equations. The fact
that U, and Vi do not satisfy the boundary conditions does not
seem to be very serious since we do not really know what con-=
ditions are appropriate.

If we next consider the equations resulting from equat-

ing the coefficients of 8° to zero, we obtain from (8) and (11),

e[ V +V Sr) 20

2XXX Qxyy 2xxy 2yvyy lec a = Wage) = Ujyy-vHy)

A
Ope Ve = 2 Bac

In the boundary layer, near x = 0, V,, is of order au

L

Thus we can expect V5 to be of order ST in way 2wegiom, By 2

similar argument, we can expect V3 to be of order omy Vj, to

-5/3

19S) Oi Oiler Ee AGwOs dh? WE) WhaSwEKOME WeIwe OCG wine Seieles

ny
ae Mt
yy,

AVS uOul 52

WS Ve Oa ae 4 eV, soy, fe) aie

we have in terms of orders of magnitude near x = O,
Vie O(en’ 2) + be7 V3 o(e- V3) 4

+ soerls o(e7 1/3) + ose7t o¢e7 3) hen

-1
or factoring out the O(e ey we have

=I
VS o(e7 2/3) [4 te gen l/3 + (d€ £32 saa, JER

The perturbation scheme may be expected to be valid

1/3

provided d¢ < 1. We can expect a fairly good approximation

from only the first two terms provided the more stringent con-

~1/
1/3 << il ale anoseels ihe oe : = 1/95 wae) Sieicor9

Gustaom “S/e )
involved in neglecting the third term is no larger than 5% of
the first term

“V3 2 1/6. Hence

For yearly variation of the wind, de
we shall keep only the first two terms of the series, It should
be noted that a determines the magnitude of the effect of the
perturbation but it has no bearing on the validity of the ex-

pansion.

Numerical Example
In order to discuss the above solution, we shall pre-=

scribe numerical values for the constants of the problem Let

Ste ys
ry = 6.5 x 119° aya ) = 2 sc lO ent seers
8

S |= 5 xc lO om D

i

5) 3s 10 tem(C = QO, G = GOOm ))

" mn
ee a ap ih eh

cei bipe vi
ty us “a

ne ee
sited

rt anna

Pa Os

ia ¢ ds we

cine gy

Pee

ele Wen 4 vf aesah aad ri

LAO 53

25 _1*
2 sect Qa 3 LO Uae 1

7
WM = 5 se MO) olin

ae 2

1 = 2n/s Wt = 0.65 gm emy- sec - «

The magnitudes of r,, s, A, D correspond roughly to the
Atlantic Ocean parameters. The value of B is chosen so as to
give the best approximation to the Coriolis parameter in the
laticude of Cape Hatteras. The equality mn = 2/5 corresponds
roughly to the east-west components of the trades and the west-
erlies. The value of w corresponds to yearly frequency of the

“2 is the value used by

wind variation and W! = 0.65 gm em tsec
Munk [5] for the wind stress.

Then the dimensionless constants have the values

We=odx 1072 ing = Zac

oO
UH

fe
% D
€ = A. = 2 x 10 6 e= = = Op 123)
Bs? Bs
r= NG 3}

Also I! has been chosen so that
a= Oo 2

The results for this numerical example are shown in
Ries. 2 =) Os

In Fig. 2 the non-dimensional, north-south component, V,
of the mass transport is plotted against x' near x! = O for the

value y' = 0.25. The region of large V corresponds to the Gulf

* Corresponding to an annual period for the wind fluctuation,

ik
Me 5)

hon

Yih

ih

CMW

d if

a,
ALMA
Pai

a faye | Bh wih eed j aa
Pe Mae eae |

{
ny

A11-101 54

Stream and the section adjacent to the Gulf Stream, with nega-
tive V, corresponds to the off-shore counter-current,

For the Gulf Stream, the extreme values of V are in
phase with the extreme values of the wind. However, for the
points between the maximum and minimum values of wind strength,
the transport lags behind the wind.

During one cycle of wind variation the following result
is found. The transport and wind both have maximum values at
t =7/2. Immediately after t=m2/2, the wind begins to decrease.
The transport also decreases but it lags behind the wind. At
tT =m the wind has reached its mean amplitude and the lag of the
transport is greatest, vize, an interval of 9 days” elapses
between the time the wind reaches its mean amplitude and the
time at which the transport reaches its mean amplitude, After
t =n, the transport begins to gain on the wind until at
t= 3n/2, the two are again in phasee The wind and the trans~
port now begin to increase and the transport again lags behind
the wind, The maximum lag is reached at t = 2x at which point
the transport begins to catch up to the wind. They are in phase
again at t = 5n/2, This cycle is repeated indefinitely.

The discussion presented here is based on the assumption
that the first two terms of the series represent, in a sufticient—
ly accurate manner, the complete solution. One result of this
assumption is that transport reaches its maximum value at t =7/2.

* It is shown later that the value 9 days is independent of the
specific value of the fredueney for slowly varying winds.

meiyin® aig Pk a nae nt oy 4 Homsatn

iy

ies dee ii ted pace Phy oH & i: a he a

“tio rae, ae, nie anh

}

Wye 4
“adn! " ate

h

? et i

* 5 ’ j

or “ge ati 4 mah , 1,

jte dents hoa ak wa

SA nia
i Be a8) ba ; sie eae OP

Wire

cae |" i. | ant ; bts ae iit ene

i
We

pried: ons
, © Dele an pie A ae me
Pa tes ely
OROMy ht
ct Ae
rr
f
NOE Guta pa

Po mprroeg ls msi is

ta ce, Me ree, i
bg ’ : S 4 N, % 5 N if Wi Went i i "
~ y (oa

Me aad eta PARES, 4 rae he a aap
f +I
ue i

Auta OH BEI

The perturbation contribution vanishes at that instant since

its coefficient is cos gy Thus, no matter what the value of 6
(essentially, the frequency), as long as it lies within the
limits necessary for the validity of the above method of solu-
tion, the maximum value of the transport will occur at t = mn/2,
m= 1, 5, 9 ««. 5 and its value is given by 1 +a times the
steady transport value.

The interval of 9 days between the time at which the
wind reaches its mean amplitude and the time at which the trans-
port reaches its mean amplitude is also independent of the fre-
quency. To show this let Vp = (1+ a sin t)Q and V, = al cost.
Then V = (1 +a sint)Q + 8a Lcost. Since the mean value of

the transport is V = Q, we can find the time at which this occurs

by setting
(1 + a sin t)Q + aL cos t = Q
or Ls
(Geia ~ S53 5 4
Q
since 7 is small, we can write tant 7 7 and therefore
L
Te - a.

Substituting t =wt and 8 = w/Bs, we have finally

Rare eic eT]
oa0 ace

which is independent of frequency anda.
It is apparent from Fig. 2 that the out-of-phase effect
is of relatively greatest importance in the counter-current

rather than in the main stream The graph shows the various

en

4

ie WS Vinton
1g, he on set, odd ‘pat

pid siete Poe aie a
athe ce ‘bonita vols ’ te i
Ry Sane ee + mo 3h bN Preoged 15
hi. ol eee Bout. 2s

0 ine)

‘ ae “enlt plotras 2
aus alt fia « 2

ao ase, ao c=) fm ta |

sf es

us oie enn aR Wie

} pe 1 an oer
Sen ive’ yh Sia! AT

ye

, . ie Vea
ee £AW A2eiiG~ atau ras
AY

aie
eM
on
|

ais |

Sree y thi wy airs ye"

satis yew ‘dptd WA

Aaah 56

effects only up to the eastern edge of the counter-current at

x' =O.le For x! > 0.1 only the mean position of the transport

is plotted since the deviations from this mean position are very

small.

Near the eastern boundary of the ocean (Fiz, 3) and in
the counter-current region (Fig. 2), the absolute magnitude of
the extreme values of the transport (which is now negative) are
also in phase with the extreme values of the wind and the trans-
port lags behind the wind at all other times.

Figures 4, 5, and 6 show surface contours for the
southern half of the rectangular ocean for t= 0, ®/2, T, 37/2.
The contribution of 5H; is very small throughout the ocean** and
has therefore been neglected. Thus the graphs for t= 0 and
T=" coincide. This result is based on the assumption that D
is 500 meters in thickness.e If D were increased the above re=
marks would be even more appropriate. If D were decreased, the
contribution of the perturbation term would be larger and we
would therefore have to account for ite The value of the first-
* If we define the thermocline as the surface at z = T - d/2,

then the contour lines of Figs. 4+, 5, and 6, multiplied by?
-200 represent the deviation of the /Hnermoel ine from its
equilibrium position at z=-C-d/2=.D,

** Tf for any of the variables the magnitude of the coefficient
of 8 in the perturbation solution is of the same order as that
of the zero-order term, the coefficient 8 = 0,002 renders such
a correction negligibles Throughout the present example, the
only sizable contribution of the out-of-phase term is found
in the north-south transport V in the boundary layer where
the function V increases by order e7!/3, However, Ho and Hy

have the same order of magnitude throughout the ocean so that
the first-order correction H, can be neglected throughoute

ee “te deivacesinive Ar ‘ep eo ir 0 ad
“Vatineot bial ha bal Aor ,

ripe ny wai Hy

We her iate 03
4B. tantseden: sf
a « seaeuired ual pes ik

G ait tb, ie! r $ ions Reta

Aire ec! ad 0 | bees

; aban neste J
a it), aa i; at
j Ye

Ir OL PSS ;
Mees ne shia Be
ae oe tay Sow s0
WG yp BE UitEe E EA :

Pere Ql wits a: if

eratw mes ryt myearbriae
ie, BTA th! OV RANE a eS
tm tiny nite Sienna,
reich tans Any bet atl Mi afi eat wf

cat Co
ref mis a

aa a aca (70)

Aes ONL 57

order velocities would also be altered when © is changed. We
shall consider several values of 6 when we discuss the deflection
of the thermocline in the steady two-layer ocean.

The meanmass transport of the Gulf Stream (corresponding to
the steady problem) is 26.6 x 10° metric tons per second as com-

6

pared to Munk's value [5] of 36 x 10° and the observed value of
Y2=eo x 10° metric tons per second. Munk [5] used the east-west
component of an empirical wind system and the discrepancy is
therefore due to the difference between the two wind systems,

At the time of maximum (minimum) wind the transport is 20% higher
(lower) in accord with the remarks made previously in this sec-
‘tion. In the counter-current the steady mass tranport is 4.61 x
10° metric tons per second.

The difference between the computed and the observed
values is not surprising when one considers the many idealizing
assumptions made. Such features as the straight coast lines,
the simplified theory of turbulence used, the neglect of the non-
Winear terms, and a more realistic stress-effect of the wind on
the water could well change the quantitative results by a factor
of two or three.

The problem as stated and solved by the above method
gives no sensible east-west variation in the position of the
Gulf Stream, but a careful investigation of the eastern boundary
of the Gulf Stream shows a very small narrowing of the stream.

How well such a result agrees with field evidence is uncertain

since our solution yields no inshore counter-current,.

Le el em a Set com i ET, fF Pie ke SAE YOY ENA TRIS ACCT es i
eee ee
ae iapralie ae © pth hove! fi ‘eat it: a
fon ‘sit bidsamtes ow ma wide ¥

‘i fans 5 Bibl ot) 0 + foe
| Maen toy ener artiste a dk shy AS, a
v abe Govorgs | itt iat My oe * a4 Ay

it 7 | Pease: ott Roan LPT asacse, whicoaien 0g eowy
' 7 a ME regen. AY Bio ie ad
a CBee an wit c ee

ee aay a fi

ay

Pic

. ag ‘wine iH ihe, to = “ne

tet wes: pe tc Bit baa ie bisa
‘eon aie ae ¥ Aeaoton'sy cn

oe tet ae ¥ gion rs tyes ions ;

| o aa baled. aint ‘Meats Delite Te wii. ened oat fii eg

thts bBebL ‘yea f ot * wah lanée | bens “teat oo dap at
y peat doaba wi ly Mk i oe ana ad ‘einen te gest mae

| Shari ttt. “tna: Ayer is te fad if “lone dieci ey say

MG! Det weit iy d- i Pa bonoe sbiute) ne sini 1

a | mena * ‘yal pitt asin elite treney pi annie ta: 1 hte ’

horas own: fé. ‘pale Ge Site ae

wea,
_ : aly Fo Oke foe ne 8) i Nebo | teasintang Hiaesaey it

Beabhned wmxady iy wht Sa teh i Magid Suairad hudwuee Le , and fe
aa) ost 3) whith FL Pie hi a toownd stot
bse onN at 6 oie se whi ba’ edi van ahi

iter a 1) ahitnai: R

Al1-101 58

It would be interesting to ascertain how well our pre-~
dicted results agree with observation; specifically, if the mass
transport of the Gulf Stream responds as indicated to variations
in the wind and if the lag of the transport is independent of

the frequency.

5. Methods of Solution for Problem 2. The equations
(3.25) - (3-30) are non-dimensionalized below in order that
boundary layer theory may be employed. Using the arguments of

Section 4 for the method of non-dimensionalizing, we have

Sey ESaeple Ki = gw
= ‘ = Bs
aor
ae = ISIN
ng(D, - Do)
tT =wt, Os 5 9
2 Bs?
Vv 2 ap _ B
aie Ac ome
ue Bas
me A
W Bs
a Uap ee
NE sre Bs ?
Uop rota
aa: at
: B°sp 174 a0
i 5 ew ’ a ’
a) ip
H a eOSEe 2s bie eee
2 W Py

Then equations (3.25) = (3.30) become

ew vai

he

Nl dal Tay
benpats hey, re tb 4 atelier

ly %: oa re sre 4X) te

if , i

LON EN

NAL ALO)al 59

aU aH
1 a ee US ab
nsdé -—=- - nsyVy = - 9 = = - Steet
Ot ae ax a Behn Sects :
+ nseAU, - (1 + a sin t)cos nsy (aL)
av 0H a 6H
ns ——+ + nsyJ, = - 09 —1 - S84; H, - a,j} +
aa yUy ay [ Bel a + nseAV, (2)
0U aV
pal per apeles acl 2 0) =
ae ae See [Hy aH, J (3)
Oa = Vg = — ta (Hy + DHS) = AH (Hy + BHD) FeAUR()
ov.
one} OV 0H
See a ee eRe I, geen 6
Ox ‘i Oy e @) 45; “ey

Let us first treat the case of a steady wind, Gan
a= 0 and Ydt =43/dt = O, and let us assume that, in the case
of steady motion, there are no velocities, and hence no horizon-=
tal pressure gradient, in the bottom layere Equations (4) =

(6) are then satisfied immediately by

ae Del
Uy Sg = On By So ae (7)

i fe) a AGN ee ;
- nsyV] = - se [ OH, — ad + nseAU, ~ cos nsy (3)
jee SO ca, 2 BES ge J esi (9)
ego PON wagag ert ee
aU, Vy

——= = 0, (10)

Al1-101 60

Differentiating (8) with respect to x, (9) with respect

to y, and substracting, we have

e[V We Se Galigy ney (11)

Lexx 7” "ery ~ Uy 7 U1yyy il

which is equation (4,16) with a = 0

Thus the transport distribution for the steady case is
preewcely, the same) aseit is) an Problem i, “Lherdin semencio sam
behavior enters into the non-steady case when the motion of the
interface affects the motion of the water in the top layers

If we set a= 0, then equations (4,22) and (4.23) are
the solutions for the present Ce ale Similarly with a = 0,

from equations (8) and (9) above

MSs 62
Bisa Hy Heh = OH,

where H, is given by (4,26). Then H, may be written

(je ae ze ask it

Hy = +--+ (2)
ost
b

However, if 2nX\/@b Hp < 1, then H, may be written

approximately
ee fee ss Ho]
Hy itu teats ee ote RENE se ES Ho e Gis a)
b
Hy can then be evaluated by
Hee yl
i =e (7)

If the dimensional constants* which were used in Problem

* The depth (D,-D5) is given the same value as D in Problem l»

‘Papimoe MR LED ahh a donee ae
iM ‘ ; i le x nt i i ; i eve

Wy

ens a

pA

uid! is

ys: wes!

MEE,
ane

oe

chad bin

Ta Or tack CET
i
i

ilies bi | Atul
i th Shs 6

5

Na cdt)
eit

@otdoy

Al1l-101 61

fare used here, and Wf we put b = .005, then (12.4) ds conrect
to 0(10°°), The streamlines and the thermocline, Ho are shown
alin Males, V7 ial Gy

In Fig. 8 it can be seen from the contour lines of the
thermocline that there is not much deviation of the thermocline
from its equilibrium position. In particular, if the initial
depth be 500 meters, the thermocline does not fall more than 35
meters below its average depth in the southern half of the ocean.

In checking our results with observation, we find that
quantitatively this result is in poor agreement with field evi-
dence. The definition of the thermocline in the real ocean is
vague, however, and hence the two parameters @ (corresponding
to the average thickness of the top layer) and b (the density
difference) are not clearly determined, In fact, they may vary
over a wide range giving rise to a very considerable variation
in the deflection of the thermocline.

In Fig. 9, the vertical cross section of the ocean at
y' = 0,25 is shown for four combinations of © and be If we
consider the curve with 9 = 0.0492 (Dy - Do = 200 m) and
b = 0.0025, our result is in good qualitative agreement with
measurements of the thermocline off Chesapeake Bay [10], Quan-
titatively, the values are out by a factor of (approximately )
thr eee

Our solution shows a tendency for the thermocline to
approach the surface in the northern part of the ocean (Fig, 8).

As a matter of fact, if @ and b be chosen small enough, the

hag ya hh
‘ Ws i ive
! y i ar ( a) ,

i } ;

; " i 7 ; } y ‘ 1

: : iy ia

4 ' ys } 1 : if
J ’ y eS rae.

— Yeerrinn, ag tind iy aad yer ey "

) ea

oh mes ote oo” ‘at ihr

in et fenen ,

Boni: dealt ‘a: ™ ond ae et site a7 7H
Di 5 pedals le gis tg sl
ot i Patil per age get ri rit. id ie ae

ann, ay a ie st

9p a apoyo

» Dail Ce iy Sa , “a ator ‘ait o. mee

ee ppukpire PRMD of Bi stpesiaueniy ian itt jones, Deon
| ins |

ac Nahe inti: at 6 wet ioyot ie i HN Lieeiiend one

Leet ¥ i divas

maw es ae

hit

aoa ‘ i HS ay LCOS,

a
>

Beit) aig tay. cha e80 ahs = ian
wy ES, i? hive a ta avo} Yam mann ae ne oo
fe revi iis INE by oft ie yh s ato 1
eA HS crea ott nia . k fy BY ‘ beng, at nee ‘th ; ne

7 a bie ON MSs ete pabeonae ait as (we ve iting nl 0 uy

atorviindarren il) ha apiee? 6 ON ihPe: (2, alta ont

i

i
a 6,

net t { $) myer) ir sd hd Kote: a sey i i b “ead tte Ute i |

a

one i aA sa
19) ee ee Gener Fie N ny Ai er Cuda, ne
J) an yh 2 ae gree ' ie ie
Bah ‘

ph TAN | LAN pe + 1a

+

Al1-101 62

interface lies above the free surface! Such a result is absurd,
of course, but the tendency of the thermocline to approach the
surface in the northern part of the ocean is clearly indicated.
This fact agrees with observation since the thermocline actually

reaches the surface in the north.

Non=Steady Wind

In the treatment of the non-steady, two-layer problem,
we shall neglect the terms with coefficient \ in equations (1),
(2), (4+), (5). For the steady problem, if @ and b are chosen
appropriately, it has been shown (eduation (12.a) that the error
involved herein is small.

Two methods of attack have been applied to the lineariz-
ed equations of (1) - (6). Our first procedure is that used
in Problem 1, viz, a perturbation in 6 followed by a boundary
layer analysis.

The ditticuilty anvthe fanrst method vom solutvonvarisers
from the fact that the quantities with coefficient 6 are no
longer small, i.e., the magnitude of the terms is no longer
governed by 6 In particular, in the continuity equation (3),
the term on the right hand side has magnitude 5/b H, (based on
the steady solution). In the interior of the ocean where Uy and
Wa) are (OC) and Hy = oer) in order for the perturbation in 6
to be valid, we must have 8 < <1/0b. ‘ith the dimensional con-
Stans on eroblem al thas means oO (<< 1onts Such a value corres-=
ponds to a wind period of one hundred years or morcee

If the above results were the only objection to the

bi { |

i nee ws. 1h eed m teat, Hiaiten' ata eee ‘hae? be ow |
petit chomema gay ah on ehenhient v. sated, | 5, Apes onan ae
do ae nee: bia ity wi “gat he Stat a oo

ih
i
ae

. Sen ssa na

ip Wpersitine ai 4 ton fol iets pin Antes vt ota
prado: win a ha ite TE gmely when meres wt 0%

=
x

me delt.2 Bie betiggs nok lah) ieee SOR
age Doe Seis Wk wate pou sak i a ;
, Wis. # ith 7 a ae we ‘We: epee aul

Ree AS. Hospi Set treats cin) herd. eps

gis

i Ba oUF fe ‘taker +s 7 ‘ AAG Chi me The ay iis Hi

BOARS: Hi-Res orl) TC obeAncA 8

y ¥

ight) Rola Vain oat ee tition
Ay 7 a : / 5 ' 5 by

ea ined) A d\ot Shahin aed fyge diet Sih

Bee PY OUR ads ONS Towietg, asht wl

OB? Hilews Tay Wa eT eh NE gh Ge

Bl eatit hy) Ga MAY AG > 0. ean

ie pen a ee
fy GH 1 Av Th ey ne / en Lf it bee ee) ey ij Tove, ai
ry en

abner meus Vanes Be TRH Bien a Na Let eng:

an) a! iW reer re ) it ad A +2 Masia wy nd :

A11=101 63

analysis, the problem as defined thus far might still have some
Qualitative value. Unfortunately, for such a small value of 6,
the terms in the equations of motion which involve a time-~
derivative become very small, and we are wholly unjustified in
neglecting the non=-linear terms while still retaining these
time dependent terms.

In spite of these objections, the analysis fcr Problem
2 by the first method was carried through but the results were
not computed numerically. The analytical results are listed

in the next few pages,

'!
<j

Uy = + 6U V

iLiby a

leh VE el + 6H

U 21) a = 56) 21? ao 21

U + 5U

2 20

7
where Ung=Vog = 0 by equation (42), Uios Vio» Hy are given by
equations (4.22), (4. 23) and (4.26) and the remaining values are

given below.

2
Vi, = SOS 4x tr - e/3) (ens? + ==)sin nsy + ~ cos nsy |

7/3

(x-r)e~

2/3) nsy*, FLOMe hey |) oS

+ sin ns €
a cos T y L¢ =a 3

2/3 73 -1/3

Bad )x cos X¥3E )

le Taree: D

UICC OS) usw MS Te

\

= 15 q eee
4 Game asin AE ae 2 KG GOS sin nsy °

Meibiee /Ae) 21/3 os ec) ig ss

r a) ll) 3 a
ae gies mice et
+ @cos Tt sin nsy C2 cos( 5 )+C.sin( 5 re

}

—

a ¥ SUE Eee
3 5

y is yen ee a ibohan’ men ‘ytd nee “00

ey eee oy iebonatt’ Ye ed ‘wii io
Lomein Sa sin, wi’ iow Haan sey a
voge seatingentinanes Pe hoe wh arin Met

tate mt ators ond eek oe bo |

“ets ie ep vit toneate hs ry tek a

es ea des Gt yl ne ves’. au jolts
1 eta 7 nah tan fn Lon seine [ 3p

S

+ - ba fone el ae

: win Fj (oa 5 sh > ty 4 pie ‘

er: te wee is a ee
Ne ACmn) nh a * Ng 4

ae)

\,
' me vy

BBE wide (2% +
; Maes ¢ ff

oy y! +

Le
iy ght )s Bt . a es 73 + A ae
- a yh a NT oleatiaas . - Ty

* Ves) ft ge. i : Seis Mo iy _ a { anal He o

ae ell iNet a Be aie
OY ve ORT ie. os (nM y Tee sine > : home fe es
see Nig . 8

my Hd vm,
orm | tN | ae

vz wy fu Ba rate eh Sete

All-

Iki

pee Ree NC.
Ay acos t ns

LO1 64.

2
=e ACOS TE (_ a bape xo/3){0n353 & ni s"y> 42)c0s nsy

2/3, (x=r) &

eyns
uae

Sulial nsy} + acestC(y) - ore sin nsy |e

+ [ Pe sine Ve 7) s(rel/3.262/3)cos(HVe >) Jo
3 2 D

2 73
SG GOene | sin sire DSU, Eo) Ge eee V3)
Oy be anid
71/3 -1/3

1 r
+¢ Al] Fe = a cos a2 habia inhye |= see
uy

SS Sees Sarees

1/3 fa 1/3
- @ cos T ae = {stn nsy[( V3 C, -C oe a -)
75s ee £3

= (va C, +C€ cos Gof et Ife ve ‘

—

2 = cE UaxPar®)+(nee af eos

nsy?

_

cos nsy

3ns

Ila sin aSVae Jb GOS INsiz
ns Tas

“1/3 neyoel/e 1/3

=- E
aa aie

+ @n“sy) sin nsy + (@ns + y—)cos nsy t+

y eee Bey pane fcos nsy -
joel, a u

+ nsy-sin nsy] + y sin nsy Ce

—

ane b, ay or Vt hen x
be Satin) (: P | dn WX, %
; : oh i hd t i i]
we

. f : v
Lewvepmtiodhy gh i tdemmts an
( ie

ov)
Al1-101
1/3)

=r len | v3 nsy
eek be i) 8 + y sin nsy [@ pa ) =Ss"
_ 1/3 -1/3
ae -2/3 :
= ney i sin eee + aaa oe Ss
3 V3 ie ;
1/3 zu : ace ove i/3
+ 2g - =) i) x cos ( ———=)) + |i ae:
a 2
ve be) Sin ae ) tae : - Be) - 4+] cos»
~ “be 3 3 2 be 3
-1/3
- -1/3 - Xf 0 173 k
pn ESE Ay gee - =— y—sin-nsy [(C, + 73 C3 ) cos
-1/3 Xe wane)
: ee ae + (C, -\/3 Co jsinew ees ya}er ‘
1/3
| We. = Gh@OSE4 (Gp GOS acy + mene ea ashe oie sb 8 ~ )
a i wW/S}
Ree ea % | (x-r)e
+ @ cos T a sin noy | 3 + Or e
F -1/3
e 23
+ a@cos toaaesin v4 3 cos(= oi As) ue

Be olly/Siesen ol
-1/3 nV S\e { D
ADEE: oe fa (oS ae
+ ¢ 3 ) a s 5 Le

[ Be) i Be) -
e j i 6
eee 2 ee,
U.. = 2<os 4-2 x - eee!) [enya lene nsy+2yns sin nsy]
hs mop 2]
; 2/3
2
Co (y) ) | nsy@ mt (22
+ a cos 7 Se - acos | es sin nsy [ 3
5 BEL eoae VEL ) pe — si ( 5
3} ae
/3
Can eee sin(&Vie ee - £) x cos *

at
3 V3

ey A.
VO ie Aeaat

‘ot awe ait oF a
: Jip t i er \ pe ;

5 typ ete!

(ek ae 4 Veins te

ie ce Pon a m .
iat a3 t hsbetie npn a 3
BH ai ~{ SY

me a Nee
- “i ace Gee

All-101 66

x

\
—_ SiG COs 4 — > Je

SW NY
(36 lie : 2 fe’? sin ney .
: | 2 os

= =1/3
1CV3 Coo = Cy5)sin (eves mad) Gr (Cinya. ar V3 Cy5)cos *

Bren 3 x Roce 1/3
‘ BAU Sena), = =O COS) o = Es Salim ialsyp(@sak a=

a are ee nec eae

ny = ao scos «+ y sin nsy[e7

=1/3 -1/3

He cil eee (> = Ave

V8) B73

3} (xer)e

+ ¢ Ap Je

iy
cos(Z¥ie. +) | e 2 y sin nsy [(Cn. 1 W/A C.)

32

A 1/3 = -1/3 L_ a
Cos (eae Se. BaeOY) eee oa V3Cy9) sin )Je 2

Se 73),
* Gcos t nsy sin nsy OS fae ee € \ ral

be \ 3 eee:

\ /
ns) g COS 1 nsysin nsy

+ =
9 a)

BS] ¢ + (

a
L

Ee San CAS cite | top = V2 € 32) cost

Fie’ ve een
ae (C5 = V3 Cop) sin(= =.) Fe

oie thy
ow

el

+ nla w: ide? oF al ( Pane

ae

i ne seat 8

aaah wm atta

Coie, fn Es ‘ me 7

he.
fi

Se € ‘ ‘te eae i ty ; i oo ne
°c a Hie “Jive eon car a

»
b
a
\,

A11-101 67

j

aaa j [2G ar r-) = ( 73 - r) (x on el/3) ] 9
Ae

sin nsy + ns cos nsy ] - (cos = + y sin nsy)}

3

= co ¢ cas [5 (x 4 nr?) 4 ce3 - r)(x + e1/3)] :

3

Sensyasan! nsy. yo COs! nisys iy cee oieG Ga eS) °

Tearomne

C [ uy@ cos nsy + nsy3 sin nsy ~ ae OY vey EE Coie Iss es | ‘

The functions AyAs, Cy Cys Cos Cas Coos C35 are determined
by applying the boundar: conditions Uy = Vy = U5 = V, = © Om

xX = O,Ye

a
ae fod) mye

"9 1d!

Al1l-101 68
AS ar 2 5
A sin ney = £1 (on2s° + 218) cin ney + LOO8 OY F
Cry l b J
1/3-r f @ e /
i any ah (ae oe uae El AMINES oe Wp COS ashy
C, sin nsy eee eae s cere ane nsy + =
ee 3 Bs Ba) nese 2
CQ) = al - Te ahem) {en oak ae —=)OOS iashy
ae ANS 2 3 2/3
eyns SAMS BT ae n@ ey E
ae sin nsy - ae nsy + ag se
ns
- ==)cos ns
BU a
1/3 L/ ji 2
€ — ms Sep al ae = 8 he 2/3 2] 2, Al. ns
V3 sin nsy C. AGS re ) {on + =\ y
4 2/3 2 22 :
ZS os € 3 nsy Qn S_)as
A a + Soy ee en Cs ee i
ae | sin nsy | a 5 )sin nsy + Bpeos nsy:
1/3 2 :
re iL pnsy INOS 3 Se:
+e ice St ee )sin nsy sR COS nee

nsy“sin nsy A>

sin nsy Coo

or

wat 3
—-Sin nsy eae

Vc

- (r - o/3)(y eos nsy + nsy

2

2 ee

el/3(y cos nsy + nsy* sin nsy)

sin nsy)

rates ay 3
(75 ee
03 (at Sin nsy + cos sy oc ar cos nsy ]

i]

2
ae
C5

ao / 3 (eee a
5
/3

re

ey

i

3

=

2
nsy

sin nsy)

Nee:
(nsy sin nsy + 3y cos nsy)e

- r el/3) [nsy“sin nsy + £52 O87]
n

+ Sie + yon*s“)cos nsy + 2yns sin nsy ]

i in) she set Fe ee! ; o. & Bae ix
a | %

2 Laois a th icant Ay Rea pe Bie, S
ee too: > tae e, pay ek oy

sib. ea. he. sie wong ae ae if

: ie 4 < P We : is 3 My EPO es Sse mn : Geri
5 yen Boe Rae x ey - PON aaa PER Meat ii

Lmao oie 7 ed 0 ana yh, -

r i ple
Pagwe Maes WIGS YRC Keot oe yy CEs
bY “\ fy, he io w ahs i Bt oe w wba ads: Be a

sire A €
AO Bs bess 4 nv ees

ie ee

Al1<101 69

The second method of attack on the non-steady two-layer
problem consists of separating the expression for the wind-~
stress into its steady and periodic parts, ieee, (1 + a sin 7)
cos nsy = cos nsy + a sin acos nsy, and treating each problem
separately, This method of solution was also attempted in the
One-layer problem The resulting equations could not be solved,
however, without recourse to numerical methods. In the present
case, we hope to make use of the smallness of the parameter b
in seeking a solution.

In equation (3) the right hand side may be approximated
by d/ dt (= Hy + aHy) & 3/dt (- Hy + H5)s% OHo/ 0%.

The steady problem with cos nsy as the wind-stress term
has been solved previously. For the time-dependent problem, we
write

H, = 0H,, Hy = ObH,, t=

Then, with the time-dependent part of the wind=-stress only, (1)
(6) become

Uy 3H, i
Mo) =i ASVVE) = = + MSicAUS T=) 1G) Saint a icoysisnis

6H
mem 2 iE misy Ua p=) ese nse Ay
Ot 1 ay 1
av aH
Ue mee
Ox 6) OT

silt saan sara ‘e

ere ‘a ‘0 ine Aides
ie hadcke ¥ oe hitan ee ash ct

oe Bie at

‘daichii eset to ayer 0 ‘aay “etiam: of asd We

= ‘
| is i : i ‘ %/ : i eh)

smoniiaten, i.

thei: ime, ae ton 9 | ore

ees nner aaact ote i i 5 eae ene rate wiotong shoei a
ew aianiita trahngo en we mot — bevton

den goa 7 ahem (VA oy Be
. ee ae

fj ma a Peasy 6 ze i ge ou Puna MP
eet en

Al1l-1LOl 70

av oH, oH,
d—£ + yU, =-2 (—t + 2) + cAV
Barer ay ae ee
CUB Ue aoe
ox cy Y ar

Next let us write the wind-stress as the imaginary part

LG Hi é :
of ae?” cos nsye Then if we take only the imaginary terms in

the remaining parts of the equation, the results will be the
same as those above.

Define

seas et Mee! Gian Hee Nace | ee kl
Uae = ae YU, o(%sy) Nae = ge Vy ol%sy)y Hy p=ae Hyp le,y)s
The equations become
mee sh dhy ik
SIS OMG Re SYN T= ee + nse Uy - cos nsy

a ah
insdv, eS Va aval + nseAVyz

Unait Vay = ly hy

- Fat , dha,

16U5 -_ YV5 = An Pp ae) op EAUy

‘ay ps én; 6) ) A

al Vo qP YU5 3S 2 7 aa ap Vo
v So Fl yh a

Case + Voy al vf 2)

The above equations must be solved for the six unknowns,
The difficulty arises in trying to match the boundary layer con-
tirbution with the interior solution. To conserve space, we

shall not give the entire analysis here, but shall confine

AN : : ay,
; an wi hk ik rs

HAN, ’ :
7 i cine

ig oo... nN cine vite ide
“watt oy bau haa or. at mires

a Nee e

‘vee ae: oe i, ei a Deal wy
“eri waved iobutin a
(RY opm ‘wien a
» eka ae tod: 1h wir ati

Ser and tiny a alr
ah

j

fis Y Fl : .. oh ery OTs oma tLe } i ie ae sit )
ae ee: ee mee Uh in

Al1l-101 TAs

ourselves to the determination of the boundary layer contribu-
tion and to an indication of the ensuing difficulties.
Carry out the following three steps:
(a) Let x = oe ise, stretch x coordinate near x = 0.
memes) el o=
(bo) Substitute Mame = 6 ee yee lye

(c) Keep the leading terms of the equations.

The equations then reduce to

nsyV, = Hie
insbe /3y, cp aR Bly SS Ay + NSVI xe
Wie + Vay = iyhy
YV5 = y (hy ¢ + Nyy)
Uy te + YU5 = - R (hy, srl) Vorr
Ug + Voy = - iyh, e

Eliminating all the unknowns except h5, we find

-— - = O
Psa ee esg AG 12
where
Sy ee yg hace Ol)
1 iy
Solutions are
y D,&
No = No Cree
wae i
eile

where the Di are the roots of

Bre b4D° Sp 4 GS On

ee oak

se det ‘woniene ont e ppamntes we: « bin 9
ne ent? yeot wanton bey he, oie |

we elon qeeah oe A 1 ow fer te

noe (EO, a Sn

2 cs eetonis eu to ii pane ute fhe 7

oe ae “Day 2 ie ae sta ten 4 ana eet

: } :

ae im ee mye a Fi

a. | eh gt La

: ie [

- i ee | beh ‘be ee heap pypnwact vires avtd. iw pendombestit
a, : i ‘ he i es : a

a) 1 6 | i pif 2 Hi ts : ee Fl

iW? 7 : ; ; TiS : F ‘ y G > tt e <i]
| | ya) arte ah NE ey

gO ee a

: : ; i it + i
Cid yn | rae
Ty Amare Tas Vaan

All-1LOL 72

They are ia oa ER
= (2 6, +2A+2B) aye b, +2A42B)* - 2(A+B = oe)
Dea oars a 3 3
U2 ree eae at a EAT EaTE ay
2]
‘ : 4 64 +2A+2B = & b1+2A+2B)* - 2(A+B- OMe 2)
35 2
where mined eee ——, 4)
ea a Rey Ar 5
3 2 3
a ee / 1 SK a, a al
A = a oF 3 39 oF - [SCL oa) | Fs ara - 37)

3 Mii a ha iy moe 3
USeuen mes b EBulavn Sages
Wea) Te ge elk. ewe a rel ai eh ell ileal RT|
i) = {4 ae cae a - \ fn oF 3 )] a + + | °

The above solution for ho must now be substituted into
the previous six equations and the boundary layer contributions

eon wh can be derived by keeping the parts which

Te ihe ual
—+> O0as —€—5o. If the interior and boundary layer solutions
are added, the Cy can be evaluated by means of the boundary
Comditions Uy = Vi = Uo = Vo = 0 en x = Or,

Practically, this is an almost impossible task, and
numerical methods must be employed for the whole procedures
In view of this fact, nothing is gained by the analysis and the

entire solution might as well be carried out numerically from

the very beginning.

tat: eter ous, ‘ed “what Pen. cil, star ies a ldapde. ts: is ae
“ ahabaitbta0 shane Peto: wit es ue henge, abe: suob tat ; |
wokity agony ait ‘grbgenl Wi bor beat get nih, a th te hee pe
siakmitin ens whocrwed Bea: dakuited oo Nie ee.
arn et on te anes “ bot care, ed Has e ‘
ms): ee a ah G4 oY 2 oe we ar, |
brie, sens ota west a fon th 23 ott ees
eR ony stonive evi ‘ttt Wee ie ;
one fury: CLev tunes yt Ae Peony, wa wiih iid-0h eata i a
SortguLantreme ee re wean, ai stow ate ad vito

PY . f
if

ny °

os y :

: S
ey. see tio ce’ (oe ee i | SAORI Ly ie: |

Al1-101 73

Since we have been unable to arrive at a useful solu-
tion for the non-steady ocean circulation without assuming
negligible velocities in the bottom layer, we have no assurance
that our analysis is valid. Reliable observational data which
might guide us in this matter are not available, We may per~
haps gain a little more confidence in the results of this inves-
tigation by the following considerations.

For the formulation of Problem 1 it was assumed that
the velocities, and hence the horizontal pressure gradient,
vanish in the bottom layer. This, together with the hydrostatic
pressure law, immediately led to the conclusion that the thermo-
celine responds instantaneously to any motion of the free surface.
Natumaity, this can hold, if at all, only for surticienbly,
slowly varying circulation.

Some investigators are of the opinion that the very
opposite situation actually exists, i.¢e, the thermocline re-
mains essentially fixed and does not respond to wind variations
of, say, seasonal or annual periods, This is perhaps a more
reasonable assumption because it is based on the idea that the
frequency of wind variation is much greater than the important
frequencies of free oscillations of the bottom layer.

Let us assume, therefore, that the shape of the thermo-
cline remains roughly fixed in such a manner as to result ina
vanishing time-average horizontal pressure gradient in the
bottom layer. That is to say, the thermocline adjusts itself

to the mean wind distribution so as to give zero pressure

4

| | ait oa Auhaiegt.. i on wiebrse wy pid
ae | salou etna pe i Teheente
‘ ror tonae att toy ang mA
“eb atl inh Legos brand, wiatnitaa:

atten ent al geese ys Fo on.

“suvnit iad ite, ae tiaate ere iii,

Bah: ey

}

aes ov ey iat

5 ing x

rh oi 08 Th) wrath oy

Aaee, ¥ 6 ‘Sct,

aes rEagk ey.

I *

pone ‘i
a i 13s

>
bene
ere

fold ate

SRN tasters |B ca aE sgt &

4 meee fy Prog ioh ake
da pein 2A oh i eae eng Pe Sods ies abel ee tail,
vi teers: awine4 prtaishapd ady so
| _wbtateor iy wat De ae ike epee tbe: os at
| F Sec ede ald aR WORE. di oS. Dus ete! tii . cf
Pes yee piht ue i) he man Gh ie ree ae a f i
eM Soo 9 aot whe: i ER: t 4 pide Cis ae
alae 4 eae i vot tinea ¥ Ma 1 ay
er ae? vais eat We: 0 at HO ke tpi 2 tenga she |
wer sib vonaaity wile GO piaie nei a bonis
Sahih ma aac bg yd i Tans tt heap Precie, fae ihe ne anek aan

ae gion 5 URRY, ts, prey | ged Acs E Gh

‘ 3 ath thet site ay $e Bonin Pek 3 Bouin oot

f Jit yea’ t ici pine ot Feo, an bin wil the so

7, | “<cateal a . od Yn Katies banat

: oF Spee as neha ey " a seen T TT

) ae ee EO ae ae i: tip |B Iowa! ah ‘Give

A Rie ieg soloing: Wsihenindy Hedin aE a

’ roett in deen fhe Wigs agaist a fai en, Qi’

| eyeing ot Na ¢ meni ei ny albe | act oe ere

MTL SUCH, 74

gradient for the case of a steady wind having this mean distri-
bution, If we now have a time-dependent wind, we will have
non-vanishing pressure gradients in the bottom layer as a result
of changes in the free surface shape. The resultant velocities
in the bottom layer will tend to be uniform vertically (except
as influenced by friction) provided the bottom layer has fairly
uniform density so that the pressure gradient is independent of
depthe

Suppose we have a two-layer ocean and integrate over the
top layer only. If we make use of the assumption of a station-
ary thermocline, and if the effect of friction at the thermo-
cline on the transport in the top layer is negligible, then the
resulting transport equations are essentially the same as those
attained in Problem 1. Henee, the distribution of mass trans-
port obtained in Problem 1 may be expected to be valid now,
provided it is interpreted as the distributions of transport
above the thermocline. Since this is the transport usually

measured, we may still hope that the results are useful.

66 Conclusionss If the velocities in the depths of
the ocean are negligible, then the horizontal pressure gradients
are also negligible and the thermocline responds immediately to
a change in the free surface height provided the hydrostatic
pressure equation is valid. For such a case, the following
results appear to be valid (within the framework of subsequent
approximations made in this report):

(i) For a varying wind with a period of three months or

D,: “aie Asha sly anevart balw ghee)> 4s * Weiee ‘eat Ara
‘ we hah Litw ‘ow. “hon nn p Sound whey ow Ss ‘a
Feinen x i ‘seis: ota eri wt a¢ nen pen wt
wetrisonay srusivens Bey ete WOR CIS tent! ‘one mM :
Hqoony) rettantioge’ mat bit oth ag paca ptew. eogat : ;
cat eae ‘sie Imeedprt aid oh ven + Get gahgt agi’
ote Duoberacrs bak at tie! bess wieiey od ) its 0 ethes
a “ egttcs biog ys AESO Tees my GY met ay HeN aa
ah ihe) one tie dp A be) wolitqauniays pat aa eye © Pn: ‘
‘de Y) orcse “ei fr wokbaln® ae ), an ae sas Pith
, Nail a pattigtigen: ee oe es arte i
eens be sepia’ at mt phen Bate ee ) Bnet
i ae eagle te ‘ons pdbmdnkh ety cs
un Mi aot Sh.bay wh pe Prd oeane- ‘a
“Progenot | Ae bite c sido ek ord sto i

= tina whe pat Ane) Da = th ic Unt «ae pout J ft it

‘Ms ~ widen brailius beats pate ue se
| nother wien Ort. Lats pli e a 7 chines tout yaaa ia
: 9 osntine ip by one acy scare eee th. Pipa pee |
pbaeincocbe ung Rava Ve ma aly Ra Shenk" SEY _ we
a ae jane sy) fours eit Lah Cat, YB TU ane

i . hee nye Oo me i ~ LY sili nan

Nabaleesll Os 5

more, the mass transport through the Gulf Stream responds to
the wind but lags behind it at all times except at the instants
of extreme wind variation when the two are in phase.

(ii) The maximum lag appears when the wind is in its
mean position and an interval of about nine days elapses between
the time at which the wind reaches its mean value and the time
at which the transport reaches its mean value, The actual
length of the interval, oe nine days, is independent of the
frequency of the wind variations

(iii) The value of the maximum mass transport through the
Gulf Stream does not depend on the frequency but only on the
maximum strength of the wind.

(iv) The Gul? Stream does not undergo any noticeable east-
west shift nor is its width altered because of the wind variation,

For the steady two-layer problem, the streamline pattern
coincides with that of the one-layer casee The computed steady
position of the thermocline can be made to agree qualitatively
with the position of the observed thermocline provided the two
parameters (a) the thickness of the top layer and (b) the density
difference, are chosen appropriately.

At the outset of our investigation we had hoped to solve
the linearized, non-steady, two-layer problem with no a priori
assumption concerning the vertical distribution of velocity.
However, we were unsuccessful in doing so except for the case
of a wind with a period of oscillation of 100 years or mores

For such a low frequency, the retention of the time derivative

‘

“eubtngiey Abie: ond eee ee prods tobi eat ak oa bie

a

iy

ex
628

2 Tithe:

fy <w j
nro emt oe ion gel 16. sey eo

NG

‘atts aaa 3

it 19, Aaa 4

ee aS (a Beirne) a Me | ee
sonst aidedots ran ‘tan esata ball agicihy sagt sha oat (we

ai

¢

PRE AS: ont tie one
‘aba bovine “antl waa oust Semi a ae ame — ne
wieviied Leino | conga ) na

: ane. add sbi

ye Hawoh en zie \ £3 bite

bene!
px

pte eosade, saaiek na

ae ty Nya bakes aa sors a a) em. Key sonst

abel jee jb eNOS sheet iy

Pistia). a eb ie es Hy ort

Aboot aw Nt peu ni —

Ae ee aes

Bee enle wom t40creii 3

i)

satrom Hi acme ome: | acim Me, by Hon ve

Al1-101 7%

terms in favor of the non-linear terms seems wholly unjustified.
The only conclusion (which may not be justified because of the
previous statement) resulting from this last investigation is
that the transports in the lower layer are ef the same order of
magnitude as the out-of-phase transports of the upper layer.

In view of the statements made at the end of Section 5,
the results listed for the one=layer problem are approximately
valid for the non-steady two-layer problem provideds

(a) The thermocline adjusts itself to the mean wind dis-
tribution and remains fixed.

(b) The mass transports of Problem 1 are interpreted as
the transports in the upper layers

The assumption of hydrostatic pressure is not necessary
for the solution of the mass transports in the steady problem.

Wherever the results of this analysis permit a compari-
son with observation, good qualitative agreement is achieved,
but the quantitative results are off by a factor of about three.
In view of the many idealizing assumptions made, however, no more
than qualitative agreement could be hoped for.

A number of features have been left out of the present
model. Changing topography, non-linear terms, variable eddy
viscosity and many other features could combine to change the
results noticeably. However, the analysis of the problem in=
eluding most of the features which were omitted in our model

would probably require a numerical treatment.

i oo aired bere ioctl nine
ante h sith aoe

: Ms ; | oy ; |
Ma = BAAN ' i

oe ) oath

{ Cie uit ine ae
nant DoE ype tf hes
Pi aaah <0. ts :

he A nate’

p)
Cy

a

“i “oh

‘ny x

} sa het iced

aoe.

fn sae. noe. , ;

t

at oie

4 Heb. tate sey

ete writ Min tte:

* en Bis 3 Teton ee uh

i w ne 1 *!
ce Per i NW
TY 0G
Ln | i:
anit ft Hi a
ke *)
wih i we 6 iT Wy 7

Al1-101 aq

CU

[2]

[3]

a
ul
[6]

[7]

[8]

[9 ]

Ld

1)

(2)

Bibliography

Ekman, V.W.. On the influence of the earth's rotation on
ocean currents. Arkiv for Mathematik, Astronomi, and
ysaiken 2) lO Odys

Sverdrup, H.U. Wind-driven currents in a baroclinic
ocean; with application to the equatorial currents of
vee Rastern Pacific, Proc, Nat .Acad.sei., 33, Nol,
Os

Reid. RO. The equatorial \cumments yor taegkasrern
Pacific as maintained by the stress of the wind, J.
Marine Res., 7, No. 2, 1948.

Stommel, H. The westward intensification of wind-driven
ocean currents. Trans.Amer.Geo.Union, 29, No. 2, 1948.

Munk, W.H. On the wind-driven ocean circulation. J.
Meteorology, 7, No. 2, 1950,

Munk, W.H., and Carrier, G.F. The wind-driven circulas
tion in ocean basins of various shapes. Tellus, 2, 1950.

Munk, W.H., Groves, G.W., and Carrier, G.F. Note on
the dynamics of the gulf stream. J. Marine Res., 9,
WO5 Boy LODO

Carrier, G.F. Boundary layer problems in applied
mechanics, Adv. Appl. Mech., Vol. 3, Academic Press
TING aq N. Moa UO53. pp. 1-19.

Lamb, H. Hydrodynamics. Dover, ia welled, IUS3\2.
pp. oa7)

Iselin, C. O'D. A study of the circulation of the
Western North Atlantic. Papers in Physical Oceano-
graphy and Meteorology, 4, No. 4+, 1926.

Iselin, C. O'D. Preliminary report on long-period
variations in the transport of the gulf stream systems.
Papers in Physical Oceanography and Meteorology, 8,

Wo. 1, 1940.

Hidaka, K. Drift currents in an enclosed ocean. Part
I, Geophysical Notes(Geophysical Inst., Tokyo Univ.),
Ao WO, 235 LEO.

_
rh

a G/M vel vient

A per vigil at? sibs mn vogina aa

ere aided diay

ny
ie

"a itt per date
-diaele at deta erie be! REDON s treo

Ok ceca yee pot is ee aa Mati a
a 8 me 8 PACE ola dents edertsnt os
1 if eS aot %, Ey ee

baht 2 ‘isa olay cn
ie Ris si mA ee — ae 34 a i ” t Babi
alt Riso babe re Thy

ats) Jit i i psn phi 1 dies ee i! 20 3 ¥
ee jes iat “ee ee eee Kabeed akeoe it

y paerore:
4 ap he Nihs gaae

f Fo y

- ae ie ssathderna hs i Penner ee :

by, Seiaine toe ae Latha ow

\ E05 ns on 41
ptoe bl

echt Sis Hots Peta sbi ee a. hs
=m Th Bray ai a anhawe £ nt: re he ge
| eaten io em, il

FOLMWE DDT | po jeouel eee oe et mig
etiit WO te it °F 33) Ls ta i iin 4 , sith: hits) te mo hit Ming ae | ; ot 1 re aii ;
A ; PHU BARN sehapneT 9) tun Laue

3

eet siete india Lage tL ah Som ne art.
j : A’ " y - bya! all ,
ge VS. OYaaT i tating sian | ia

A11-101 78

{13} Hidaka, K. Drift currents in an enclosed ocean, Part
II, Geophysical Notes(Geophysical Inst., Tokyo Univ.),
As INOo Bo8q USOC

[4] Hidaka, K. Drift currents in an enclosed ocean, Part
III. Geophysical Notes(Geophysical Inst., Tokyo Univ.),
Pe NOS) Sha) OSI

il Hadaka, Ky, and Miyosha, Hs On the neglect of the
inertia terms in dynamic oceanograpny. Geophysical
Notes(Geophysical Inst., Tokyo Univ.), 2, No. 22, 1949.

[16] fichaye, 2. On the variavionyor (oceanic eirculation.
The Oceanographic Magazine, 3, No. 2, 1951.

i

oe : iV a wir

Sain a sivée, bhai iss ve Ren
pu al atch

mar) 1 a ia

=

is i Wor Seid: wale eileen ‘a “i
ee il | patent babi: ie eal Lave

sii Aha Dee i i ihe ee als is may ie
Lao. tie? et a pleco 4 Bet
Wyo fi. ar + net 4 rus « he ving

nest smgyy shy tw abate 8 be

ee eve Hi
pene Noe ae ee yan ‘G aie a hat

THANE. oy ie

aa

eu

i ny
= {
i!
-
\
.
n
x
i p 1
Ay
~ es
4
Ke 4
i
{
ph
habated
ie
f La
7 oe
» 4 Fi
Tt
, at TE
]
cd
Bee:
u
iy
Viale
OW rae | ee Te |

Al1-101 79

Appendix 1. Transformation of the Differential Equations from

Spherical to Rectangular Coordinates.

Consider a rotating spherical coordinate system; let
r be the radial distance from the center of the sphere, 9 the

colatitude, » the meridianal angle. The equations of motion are*

meee sine - 2v@sin @ = - 1B - gi +i (vt apy) u
= + u sett 2 — on + x & weot @ +2°r sin © cos 0
- WwQceos 90=-41 804 liye Ay VY) Vv
i 6) p
ee eo + etre ce ee ae ee

tA UO Saer ale es 5 feta dil

Opylcy: a,
sin 9 d9 i vom

where 24 is the material derivative of the radial velocity in
terms of spherical coordinates

g! denotes the gravitational force

2 * and Y> denotes the

BS Ny nee ee Ao)

a! A; V = AV is a hey
Laplacian operator for the two dimensions @ and 9.

We shall neslect the radial acceleration and shear terms

arising as a result of the velocities relative to the rotating

ed

eee re ee ee ee ee

emery re ee

* : , i
We shall not consider the non-linear terms or the viscous
terms in the radial equation of motion; hence, this equation
is written in operator form only.

Bekah 2A ‘e paltwegam
aii pa ns |

a ‘f ae ide

poe)
: ia | Dt
janes Supe aR i% ce yas ue #2

Ml

;

te

idan petae)

Op Reng

" \ AS

vA 0g ae
4 si Ms vont Rome Age py vaio. EA i

i
AY

te tienes sno tpn wine mths, seit tn eas i

ties

alia bes ‘noite ak ae oe cnt
b ; ; . ay ver ; if
nt ca ‘aaaper ye bP Le: a eT eee

yi

An
BS
' ayy %

4 ial stove on std”,
} en

Teeter Bilton ee nen
ay +, ae NRE: alk wh bi Sue Ne Rae
itn Aredia 4

ry ; » i nt i a

A11-101 80
sphere. We then have
L Op - -
ar g (1)
IVA OM, en, VON gw Ov 4 UV weeot © Zaye
at ie ee) isin ea Tr pegt oan Warerereree al Orig “Gialial ©) Cols |S
BO ocos © S 5 Lt wo a Cy My Bin
: p r a9 et p a8 ar )
COW OM) a VOW wow! Gu wu nvwalcomme
ot dr r a0 r sin 0 0 a anes Ae
+ 2 ©) shane Se a ay Ay OL ow
i eS oe 7 p or ae (3)

where g =
foree. The viscous terms for equations
2M av, 8° Ge
KK WYaSy = 55 cot © QV aa) + SOT a,
iy de ae sin-Q 3dg°
Peer ts ce Bw) o Olw a i ew
re a 88 ga sin°o ep*
Since the

thin layer on the

D)
g' - 2.6 r° sin? 0) is the apparent gravitational

(2)) eiiael (3) anee

a (2ECOsie aw
sin-@ sin9 o9
Wi) /2acoske au:
sin? sin°o Oy

region Of interest fo us consists of a) very

surface of the globe, we shall approximate

r by R, the mean radius of the earth, whenever r appears in

undifferentiated form. At the

east-west coordinate by x =9R sin Q,
by y = RCS - @) and a vertical
equations (1)-(3) become

9p
Oz

Dlr

Cooma loi 4 = Ww,

same time let us define a new

a north-south coordinate

Then

(+)

Oo pha lngings Hoenig be aP LA Rey @ 9

w | " 4 | a

Baars on sts |

x v es mr ae 7 i
a ae “68 a Paty . er |

= he nt pen nie oe sips rin
WER, 0 sanyo. at tad art am pan m da.

gprs ah rth Wy Praadkioos + oe fe

Al1-101 81

ov Ov av OV , uv wee ie 2 ‘
at 2 Us = ay CW aS + = Tae Oe Q“r sin 0 cos 90
{Pi Coe 6) = 24 eld aN ‘e cot 9 av 4 av i ey ee
p oy Ri Ova ton, ax° Resin?
~2.cot 8 dw i Av
Ow Ow WwW
3 + uss - ee + vel + = uw cot © + 2v 2 cos © - 2uQsin ©
Eiietsap) s ) 2 2 eot.6 Bw 4 foow lonn) enue OMCoLmelan }
p Ox R Oy ay< axe R-sin°-9 R Ox mi
os do (A, ow) (6)
Pp OZ 3 8z

Since R is very large, we shall neglect terms divided
by R. We can do this provided the region is sufficiently far
removed from the poles (9 = 0,x) where cot @ becomes infinite.
The velocity component u is assumed to be much smaller than the
components v and w so that we can neglect u throughout the
equations of motion,

Ordinarily, one uses the velocity components u,v,w to
correspond to the directions x,y,z respectively. In equations
(4)-(6), u,-v,w correspond to z,y,x respectively. The negative
sign was carried over from the definition of v which was defined
positive southward. If we revert to the more familiar nota-
tions and write u' = w, v' = -v, w' = u, we have for equations
(4)-(6)(with the terms with coefficient = and all terms con-

taining w! neglected)

-

LE} bie inna: Hotsen St Wels Ce Bin eet Hn am soatt

me i ae a ig? rie i A peadaget eit? Duin OR» & ty oh nei aie.

px

eal pha hae oats a sity ite scale Cit Oe @) palog wit BONY) :

OG Snotgnne ‘a sootey
ei “aioli * iv. Ponlgn Car ee: 2)

atte Poritd in Laine. bien at iad fromive a

OPW aM asi stan 0 ope tants We Eid. var

§

nAOEAeH pS oct: anit whe oo ' oe Arie Dye

Visa a" ie ane Reyes G2
Penkieh oe Rodi, ane at bas, co wong

white <eREL wi sine pate og See Tey

ARS ars Tor Sw a ow i Sa q ive 2 . Ww

‘

foo ater Ate bine

& dane eh ical. ie Rural oe it cs

A11-101 30

Ou! Ou! 1 Ou
eee a 2Qv' gin (e)
2
~- 19044 at, evil ia (13.224) (7)
p Ox ay* | p 02 02
av! ! =
- + wt Su + aw + 2Qu' sin (2) = Oon sin 9 cos ©
a5 £ OD ah avi, av! ¢ tb & (4 Qv') (8)
p ay ax ay? i ks 2) ae
; ge = =f (9)

If the above procedure be carried out for the con-

tinuity equation, the latter becomes

Ons Gy a (10)
x y Zz

In making the transformation from spherical to
rectangular coordinates, we must consider the distortion of
the spherical surface as a result of the mapping process.
opecifdeally, a rectangle in the rectangular system maps! anito
a region on the sphere in such a manner that the east-west
distance remains constant and the right angles between the
lsimes) x)= Const, and y = const. map anto jobtuse angiles pe=
tween the lines on the sphere corresponding to x = const. and
Ve CoOnsic.e) Laus, tie Mapping is Mot Comlormerk

With the above transformation we have mapped a
spherical surface onto the plane. Our real aim, however, is

to map the equilibrium surface which establishes itself as a

f iy

Ss Wear

a e 209 @ she SG) “4. dp ste he 2S oh

ae | TY oF

a
7A

Sota Kady a .

cote: TG a seen et a ih ay
Bay mee Tt he Me tates ap | Ry

y

Brey a hy: or aptienins cage th
i) ae wotitrmpandly. ‘pid obs Stawe SD Hh at 489 a

Nene gh incon. Yt, 1 eee BRR nevis ares

nat ati marge Fog ba cits +a styrene
“Baton pay ‘egy pat eaves a Aue ot ‘earayiye tate i;
a) asin. nature Piet ad9 ins duedsqem
“08 oie oairago ‘oan co ier ae y ane gs
Tt aes Pei, ae % obs sant bade patoost q si.
* | + Late wa he aha rae: she on
sh ae op onan bw olin wna SLAMS pAb) 1a,
at (Moen okey te .’. sacr wane he oP OE.
; ee eid Hei: pon ns wintad hae

. we

A11-101 83

result of the interaction of centripetal acceleration and
gravity, onto the plane, We shall, therefore, neglect the
small difference between the true equilibrium surface and
the sphere.

The apparent gravity, g, in (1) acts perpendicular
to the spherical surface. We shall now consider g to act
perpendicular to the equilibrium surface. We must then
drop the term Q°r sin @ cos © from the 9 equation since, in
reality, this force combines with g acting normal to the
spherical surface, to give rise to a resultant normal to the
equilibrium surface. Finally, g = g!' = B-(4 2° r° sin®e) is
assumed constant. The final result of the approximate trans-
formation is to map the equilibrium free surface of the

ocean onto the x-y plane, with the apparent force of gravity

acting normal to this plane,

ip
i Pri ' rt
: / i
. 1 ee ; 4 Fa) >.
; Hite OAs wae a
A. " * vat: nec Tn ee aed et a i a). aay Bis
wey seo eae Ps 1 | mend, Lihat gihhs
p ou es dk De r afk Ls ‘ i <i
Win CSireye Loe ET Hi
, Woe
Pi Aeris eee te Ee 4
j y ‘ " = sh Kes
_ weiter: Eiah + st A Phcoes.) Rkret ‘ caine ¢ oO
‘VG Fae epteaty [ERY
: . ;
iti Ge sit Wey Ar cho
Wa CME eT REED ey AF Soy OMay ap
rf " ey aS, Pe ra, ¢
Pie * svi les
f hay i « ,
i Lee Te
. Vege e Nr ae ad Ak
z

Vir ee’ ie HT io Weta ye a ee g!

NSTC Su
Appensix 2, Neglect of the Non-Linear Terms.

Rae ate = mae

vonsider the integrated equetions of motion of section

i) eat (4
on Ga & | mo ay Geir i yo ela = py | vicz

ot Sie 0
=a -h @ ja y <1
== 2D ga + A ) Au' az + ty (1)
Riou! qn 6 CY ae x | 1 Ov! 7 "4
se Oe | vt SY" az | ye dz + oh v'dz
-h =< U = =a)
So & 1) see Wis dz +ty (2)
-h

where we heve linearized the pressure term in accordance with

remarks to be made later in sections 3 and 5. are now

Tien 8
2 Cea
the wind-stress components of section 3 divided by p. Assume

okZ

i.e. the velocities decay cxnohnentially with denth,

Then,
a 2g 1- ou 2kz Wo Su Bim elke 1
ae” Uys eee are ae
=a -h <1
a 1) Ul

So 0) Sil es Aa a +t, G)
. -h

i :

' a

ae ite ey ee lv
ERTS Peis att ae

gt a Bc nr

AL1L-=1LOL

av ok i

Sonne

3
ss
Ne)

a passes Phi
7 oy 2)\kz,
Oy

ct

-h

=
}
Q

129) sin(t) Uo

-h

Avproxinate the exnonentials at their limits by

coe
CS) A nS h O. Then (3) and (4) become
EE Ol Bl ol 2 ae) a, Selo Suetianl =
ae” Bp Dae Py af 2Q sin(¥) Ww so @ 1D) suk + AAU + 7T
oe be OY A 7 oe RCo ys oe mun 3
a Ss ae tS - + 2Q sin(s) His o 2D oe + AAV +7

Linearize the Coriolis nareneter by 22 sin(<) ~
L
2

where B = =e Teking the derivative of (6) with respec

x and the derivative of (5) with respect to y and subtracting,
we have
a on Hy es _ aa STi _.2- ne _ O=
Hoa. aly 4 1 | ou oN 4 fy Oa 2 ow Oi cave 2 one cae
at Xe ay Z| Oi Or axe Ox OY Oxdy ax OY axay
av ou _ sou a 4. Oly 2 Ge av _ au)
7 oe sel Gere om gpa GR YN ee oe
9 OT OT
cee Ve ES
PU are Sy ) (7)
Choose tis Olt e te Ne (wi +P" sin wit) cos ny.

85

Gin, a == eon}
gD ae k + MAve | +t k (4)

Hic GD
ye (6)
By

18 WO)

We shall non-dimensionalize the velocities so that they

are of order unity in the interior of the ocean. It is

venient* to choose

aT SEE AF

*The choice of the non-dimensional quantities is motivat
section 44.

con-=-

ed in

» Aey ih a “a Mees r y rt te be Re ee mee AY Te th:
hah PK ok:

ae A
y ae sie! { hie Ave ptabisuonens % walt

hae . mse? 7) tee Fierce Te Ms
Hey he ‘

ew ‘ye ri cn , “ah Ge Eup Be 8 Ee

Bayi )

i q aq + i ; igs | oo a ; ze Merl ae -
y Coy x > Gin os “ a eB qe). ue « “ va € e e.

J se fis ine we to ahd brie ; eMe by ik cule sake i

©.

7 pace A ar es) a) hic awe pet? 9

,

“gaieser fut » oy ia Kibw fe) aaa
ae a ae 1 "
a hig nas >
ea ; aad it a:

Fy my ror Pi Wm: is «et nee
waitd Ren! oe’ nied ioe wit’ Gabba! beyelted Bette oD
che Bd HY ready eon Ne theta at Witch ee

Al1-101 86

2 2
" S aes i AL. lal Bos 9 W = nw! PS iain
9 By) 3 ’ W ») b)
2B s Bs :

In this notation and with the prescribed form for t.. and t

x Yi
equation (7) becomes

Ta "iF 2
aia a ai ou av Ay Gl Al a“
5 =) eto a ew oo ee SIQIEE) 3 ONGOM —=
at ox! a8, 1 Oe ax! ax! av Ane - x' ay! af Vaxtoyl
Soy | ew A Ia ge eo
x'dy'! ax'ay' dy'dy'! ay"?

Bly, | 2u,+ a | AW sen! | ee | = (1 +q sin t)isim nsy!

x vi x! dy!
(8)
The integrated, non-dimensionalized continuity equation be-
comes
oie OMe Sy oe (9)

ga Oy

iijwe expand thellvelocitimestand Gacy edie lita a Hemanmc
Series in ©, then the solution can be looked upon as the sum
of a quasi-steady part plus a number of out-of-phase contribu-
tions. If 6 is small enough we may be justified in keeping
only the first two terms of such a series as a fairly accurate
representation of the complete series.

Hence, let

Tews 15 Uy + OSU kls 4 V = Vo ON \y @ Baal Man

Fs ad EG ae a

Ny. Hee uu’ we oh ouvir ons Jae 2

inher. ny 4 ney

mie cola a .

gH des ish) | vi howe 0 " is

PMs eS Ke
9 ; ‘3
i
oy
'
eve ¥ el 5

Aue @a 87

Then for the equations of zero-order in 8, we have*

2
GU SoM 6~U av au
OY eo) ev tere) a One fe)
= (i acy seine isa anisiy.! (10)
OU OV
OQ QO _
ax! + ay! = © (CaLaL
The first order equations in 6 are:
av, au OU Oy, BU, ay acy acy
10s (228 ) al iL se} ons + *U O

we se) +¥ (eS ee ee Ss
OR) Ope! dy! Geax! ex! Osa Ox! ax! 1 gx!

fe) av, aU
ch) Wovens ~ SEO Pea = ' set Vai brated 2
] his 1 =e A er aD (GZ)
Se Ess (13)
Oza! dy! OT

Munk, Groves, and Carrier [7] have shown that the effect
of the non-linear terms in [10] is quantitative and that these
non-linear terms can be neglected as compared to the Coriolis
term, Vo. The relationship of the non-linear terms to the
Coriolis term in equation (12) is essentially the same as that
in equation (10). This fact can be shown by considerations

based on orders of magnitude. We choose a typical non-linear

au er, BU, ov;
term in each equation, y yr ggr in (10) and y aon age ee

(2), and compare it to the Comiolis terms in that equation,
Neuen GLO) ernie: Vey an \Cr2)).
In the solution it is shown that Up, Vo, Uj, Vy and all

their derivatives are of order unity in the interior of the

ee rs ener ee eS

*Rquation acy with @ = 0 is the same as that of Munk, Groves,
Garner 7)

| : : vi | i ) | | | | | nt
aint ii: wl al meth nrtan e ait il

ae, nn it } no ea el if
a it ae ae

re ae
dine i ) . ; lars Ie
He Sak enokdah wn me Pith oo) att

we ae bck ts ita :

AA Basra * NE irr i
= } I ty a
i wg
baton 7a

Pg ae eR

: doonty, le nae sii i eavrset ae webis tit
7 a Be, abbeer ivan ei oe ee |
Be ipl wit dict ae Bareutyeay id ‘nye “tet ovata on
- i ou ery i aes Maan hones’ etl bag! gs Avie tz oat on eat | he

va AT

este Pant eu dae ee ns bi daradi wRE, " om fit 1
a Beets ay thk ames oe vat tn etait sail ai

iaerettaiae eee a ee nil ee aunt.
| “he Be A ae
ae ele : ee * , ice Th pi | (a 4 Sli 5 he rf
‘» u 1. Wy 7 of e

ered TE jpe. a: et fue ry bthad

' i ty : as ‘ a a ‘ b* 4 , 7 y
- = 4¢ i: _ ef fh, “alt oY TAA i feist 7 er
; pa he bot va gine sd ni, rr WN
geavnn” hey HR fe vibe va "a ha
i ob

Al1-101 88

ocean, Near the boundary x' = 0, it is shown that Up = 0(1),

lig SOE )y ly = Olee =), th = Ole=/2)) and £, has the

effect of multiplying the magnitude of a term by o(e-1/3),
Based on these results the terms to be compared are

given in the table below.

Interior Nears ©
Vo = 0(1) lip = Oe 2)
CW OW CW BW
a Oy SO. bd OMRON NE -1
Y Ona aac! yo(1) el @)sicl yote )
Va = OC) Vy = 0(e 72/3)
OURO
One ieee OU, OVy hy
axt age (Y (OM) at a ae

Thus, in the interior in each case we haveO(1) vs.
yO(1) . Near the boundary x! = QO, in cach case we must com-
pare O(1) vs. yO(e7-/3), Henee, the relationship of the non-
linear terms to the Coriolis terms is essentially the same
in the two sets of equations. It would seem therefore that,
if the non-linear terms can be neglected in the steady equation
(10), they can also be neglected in the first-order, non-steady

equation (12).

fi (oie vi
ee

| ef . Sa
¢' J; |
Wee ae owas, anapisn ah ” aliba ;
beak ee Oe
' fala ‘ed eine a ‘to oh Pe a
i “« fiat. | ga aan
7 mm sienna ot We were iis Pisin
ra Dy a
sg , -
ze)
ms we cz Me ayo i” ie
| : le’ ato a Hie a
a | i
it)
sa come hi) we y :
Ph i at F

r vs i

ro De sb av ‘avat Haec Wa Aig ae,

_—- u _ é,
pe ae
fe iy

H PEC a0 } of)
pl On aa
; Use
Pees:

aT rs

C206 ge a

nH ff
sh “a eli ay wean hipae ; te ng ete 28) a
‘wit aad * ica) BG ee ad» gon CE atch Phew tio
Cd salt “eh hetanaees Py Hig bi Rthe cate i 1 a
gia hs ‘arpa A bow Fh sk HBP ae Rin
meebo wi. dot. ‘an a | tae aah ick teh
ys eer} eR. te ry pein: Chane tie ont aah, ,
- ~ a) R a :
: ati, LCRA ih hie aaa

Al1-101 89

Pressure Assumption.

The results in the main body of the report are based on
the assumption that the vertical cquation of motion can be
approximated by the hydrostatic pressure equation. Although
this approximation is probably sufficiently accurate for the
problem under consideration, it may warrant a few further
remarks.

Consider the steady, lincarized problem. The equations

Of motion with a linearized Coriolyvs, term are

Ee i= 2 26D (eae Or gu'
B yv j ey 576 A3 ae (1)
b= 4b fo Gn ele ov!
Byu x oe AAv' + 5 EES ae ) (2)
and the continuity equation is
ae mn @)
Equations (1) and (2) ean be multiplied by the density
to yield
a ty eo Le ' moe gu!
BT Do ax oe von yt ae OZ )
acest eeacly A iol ov!
Byu'p ay + AA (pv') + =o 6A3 ae )

where we have written AA(pu') for ApM' and AA (pv') for
ApAv'. This approximation is certainly permissible since
these terms represent, in the first instance, only very rough

approximations to the true state of affairs in turbulent motion.

“ae
-

aes
—

Ite host afte peat rere Raw tees ni. aati ode
“a om Lars dipeialliel Kapbiien ss podd oma

iu Bae Bide ecrriv sath Wh Pe halon
. “pbeakal te oie ide) ont es, soluneathida a
Tee fet a tg Abie a pnettn toned Be) ope on Raertel
| Ne «oil Yl

ait

ans re oe ation Moe bs@e ota wig whey! adds ag b Ree ey

Sic anal pte gion toa ee i be | notte ‘%
» $5 ; : me es |
a * ‘ote ais oe 1
Mi L an t babes is i. yf t i i 16 bd Oa
Lgphy:
4 oh Pima a 2
at ea | C hes ve: hits BO ta
| = {

mt Mey) mA ae ‘ial rn Cy hast a A ors tM ead ot |

" nae RG bea Hidde, 4 vba oo t 4 apa ra ele

doo xeon fhe wena baie ky Pats

PS es a ies cen anny ett)

Al1-101 90

If we integrate (1)-(3) from a depth z = - h(x,y,t)
where the motion is assumed negligible to the free surface

Z =) (x,y, ¢), then

Ean irae eee lemme 7 ou! n
By F Op az + AAD + a, 22 | (4)
= -h
= Oee se) =a 4) Goya eevaaT ol gy! 7 (5)
y ay 5) Oz
= Ge =e (6)

where the non-lincar torms resulting from the interchange of
derivatives and integrals in the viscous terms have been
neglected.

q

The terms Aa ou!

SS = 5 nda iv! =
az ops #3 az ‘

-h -h :
provide the wind-stress components at the free surface (see
Seey So Of report). The depthy z=) —myhacn peenuchosenias atlas
depth where the velccitics are negligible so that the econtribu-
tions of the above terms at the lowor limit are nogligible.
When the oul term in the continuity equation is integratcd,
it provides a contribution involving a time-derivative, viz.,

“Mee , so that it vanishes in the present problem.

The pressure terms are

oo)
1
=)
Ig
Q
N
!
lo
eS)
o>)
S
oY)
N
f
Q
3
ro}
>)
1
Q
(Sy
‘a

(ae
=
Ig
fen
N
!
lo
—>
|
Oo
ion
N
I
ic
rs
1
i@
iS
oS

Og

Ci eed wile * NBR ‘ek had ty vse sail boul a

nated ‘nit ne pd bid gky 0 hierar ea

tw)

‘int | stant » Bectt Rear

i eeanlioty )
| ‘tne Witwer ont

Bc sine
ot pep Ce ernyt racy

re : t
A, a ae
poi é tty } o a ‘ his Re i
‘ ee ‘ ei:
saris ai eek is n oe tid, ite . ’ Se a

apd bats | i eat ita ae

he ‘one
” me oie

wf sent tn us opi ; tee

aN a aaa

geidight gue ate
gderasy ivy

. ae! oe ad
getty 2 UES on Ltn OF fi * ae fet

a VERN al fe

a oie

Le ea

oft!

‘ AO. ” f a, w wh
rt ae ww r { vs

if Ee

a velba: ‘nats B
i geek a
‘ me i . me: mits oy

int fotton 4

wart Foie site 4 ay)

tat ‘ihe.

Al11-101 91

where Py is p evaluated at z= , and io is p evaluated at
Zi a—alalt

If the free surface be considered a surface of zero
pHessumme,, then Py = 0%

Defining

P =

( 10), Gle

we have for equation (4) and (5)

-B. V=- OF » Sh u
By V Ae ee Dain SMU ty (7)
U Ss 3 SP + oh V
By ; 7 Poy AAV + Ty (8)
A stream function can be defined by U = - z > V=+4 a

so that (6) is satisfied identically. Taking the derivative of

(7) with respect to y and (8) with respect to x and subtracting,

we obtain
dh 9P_-h dh OP_h OT > Oty
= ey el, =
A DAW Bip, ae 5 5 5 (9)
Since z = -h is the depth where the velocities are

negligible, the third equation of motion below this depth re-

duces to the hydrostatic pressure equetion, - op EO Lk (p

iSMCOnsizant alongs) — =e. you ie aera a = ole 55

= gps . With these results substituted into (9), we have

OT AT
AAA) = By, = — = ee" (10)

Da Tnbettaryy 4

ae

Pe yay

beets: Pigg eens eet | hie

A! ; i an ve am ‘ i! "ee Hs ef 3 ~ nt
|) ree

vais Ga Hy ‘oktonne oe ;

Re Mi tduviinl iy) sate teh sscont cath

ameros tion An Md) of i ey (2) ee tenis nit

oy |

# tH 3

7 a: a. *. ane - ig on f: i ack : made 4 4 ait cil ;

Pe ie et af ) am 4
*: ete op be as Bei, : ie aes 5 ee fe
Ka ¥/ on ; wT) : . fe na

ovo" OW iy CF) arg iat Lan i$ tansy wi Time wis eee

on) | Sy Re i. ai

Al11-101 92

If boundary conditions are imposed and if 7 and Ui
are specified, the problem defined by (10) can be solved (see
Appendix 5). Thus for the analysis of the steady state prob-
lem, the only necessary assumption concerning the pressure and
the density is that the density be constant along the surface
below which the velocities are negligible.

If the height z = -h is approximated by a constant,
then the derivatives of the pressure terms in (9) vanish and

no assumption need be made concerning the density along the

sumtace 7 = <—h.

ele hh
anime eal
ae
‘ ne A)
f 1
|
I *
J ‘ nd
uu
5 ] - Va
cw a De '
A
Ve

: bus ‘Some er ee rei i bomen Peer pap
pi ni ae i «hein add to ae
) aeirsoail Bild Bad |
‘ebative ata | a dun?
_auatenes A we eve aad aa ft dee:

an Hasina ey We eines! 4 poly % corte “ob _

A11-101 23

Appendix 4(a). Derivation of Relationship Between T andn.

With the density distribution given by
PSP 6 i) 2 pF ae
Oa ha ot) | Po BS Ww = él
P=P _» =P, [1 + cal] He eG Ss

we can find a relationship between T and n by considering the

conditions

i OP =
‘Soe os

p=e i pat
Z
; 16 ZO ep 23
“BRANT Bae 8B

But

Hence,

Lael

} KM

Ne

i Ay)

A11L-10O1
1 Sag a Bp
= QP = pc) OE ae ae ea
Poe Vash | PSR) eq Pon ox
or
eae oe Sto
Ox Ox
OE 5 2h en eoren
Ox ed ox 9 V Ap Gx
where Ap = Pie Piece
Similarly,
an. Eo on

(Oy us Ap oy

imbvecnracane () and

rivation of Integra

In order to compute the terms a ,
=

oh

(1)

(2)

we must divide the region of integration into three scparate

uss Walon

(1)

ChE nape (at)
i

Ua)

hae
. K

Al1-101

Using the values of op
ex

4(a), we have

for the three layers listed in Appendix

1 20 av = ar é
(ee aha cla z < T-d
eb OT
= Dee ae (T-z) Ted 6 Zi <2
= 0 NP Leh 2
Then
(T-d aq]
| 2p \ 22 at jaz = 2 Co 82 ited 2
alee pl ile Clee l+ed 0x [ ]
g ; DG Saar aas eo) ae eae
mr Pi Ox " Ox 7 2 escue mc mncaenl
T-d Z
nq} n
ei S(t S26elaz = ¢
TP Jy OX
1 1
3 al Bye mln Pele 5 ee Cin ingles
BP ae ie p ae a x l+cd [rca 2 CX C 8 (cae
+ g 20 (n-T).
Ox
Let us put these values into (1) and at the same time
use
fe eye On Oy eels ell] Eo bees
Onan PeANP xx @Gl Ox rete Ap Wage
ae ENE dn 1 Ones
+ OP Se) oi) imecienl) 2 e@ au T-d+h
; 1G, p Ox of © Ox ite ! ° Sx Ited [ J
5 8 one 2 Lot ei nee( 5) 22 2 Sn tes(—)
= ai - Ap ¢ Ox 08 (aT > 6 Ox soar

|
‘>
;
it
{
‘
1

Al1-101

Ap Ox Oo 7 P Lh
But,
Ap+
Log PO] So tee Bake se ge eho == ioe (No 2 i)
p p
-h fo) Po Po

Since the term Be is small we can write
fo)

log(1 + Ag) = AO = J (Ag)?
Po M5 2 Be

Henee (2) becomes

Similarly,

where D=C + $ 5

96

(2)

fa EAB AOH
97

Appendix 5. An Uxample of Boundary Layer Tochniquc. *

In this section we shall discuss the application of
the boundary layer technique to the solution of the problem

defined by the equation
eAA = 4h. = (l+a sint) sin nsy (aL)
and the boundary conditions
VS Wi = Oa x = On: (2)
Wi =o = © ony, Onl

The nature of the boundary layer problem is characterized
by three features: (1) the problem is non-dimensionalized so
that the size of the domain has lengths of order unity; (2)
ihiescoeiiicient of the most highly ditrerenttateds term aks
small compared to unity; (3) the remaining terms have coeffi-
cients of order unity. The problem to be considered here has
already been put into a suitable non-dimensional form.

If ~ were everywhere a smooth** function of its arguments
and of order unity, then it should be possible to determine a
good approximation to the solution by neglecting the term with

coefficient e(e <<1) and by considering the remaining equation

* For an interesting account of boundary layer technique, in-
cluding the treatment of non-linear problems, the reader is
referred to [8] .

aN By"smooth"' we mean that )p has no larze derivatives, i.e.,

vp, Vx5 iW) eeeecey ete. are all of the same order of magnitude.

| et
wouue Res

ACs evn

ie Revel

Yay

Ohh batch
Ns

ii ows,
ee

ee cary
LAN ty

Al1-101 98
b, = - (lta sin t) sin nsy (3)
Thus, a possible solution is
Wa SS (Cite Salta a5) Gaia tas lo be 2 Craigs) Je (4)

We are now faced with a dilemna, however, w as given
in (4) provides one arbitrary function of y and t to satisfy
the four conditions on the boundaries x = 0, x=r, If our
assumption that ~» is everywhere a smooth function is correct,
then we are at a loss to find a complete answer to the problem,
For if ) and its derivatives have the same order of magnitude
everywhere, the only possible solution is of the form
vs PIOCE) and at ds not possible Go savicimy elalibounderivacon.
ditions.

It is obvious, therefore, that ) cannot be smooth
everywhere. la particular, in order fom che full solutuon
to be different from; + O(e€), at least one of the terms,
or ¥ must be of order e7! in some part of

VS
the domain under consideration so that the approximation of

ieeee Weer

neglecting torms of order € will not reduce the order of the
differential equation. If ) is smooth away from the boundaries
and if derivatives with respect to x are large, so that

Vesa
of the boundary layer type. We shall proceed formally on the

LS Ort Orgelicns eae meee 26 = Op, waleial wae joieolollem Is Cia

assumption that this is true, realizing that if it is not the

case, we shall be led to a contradiction.

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Al1-101 22)

The solution may now be written as the sum of two parts-
ve given by (4)(the "interior solution"), by being sensibly
large only near the boundary and negligibly small in the in-
terior, (the "boundary layer contribution"). We must now try
to determine the boundary layer contribution.

Rie Watune of the tobe) solurdion ittcelin acm mle me Mell rere
factor in the investigation. We have supposed that near the
boundaries x = O,r, has large derivatives with respect to x
while ); is everywhere smooth and of order unity. Thus, if we

wire your solution in two parts, dies, we

ie: Vos the differential

equation can be written in the form

eAA; + cAA hb, - Wy - bp, = (Ita sin t)sin nsy.

Now the term eAA) ; iS Or Order ie, whe GeRmsmundermlsined sami ace

are of order unity and the order of magnitude of the terms
underlined once is as yet undetermined. Since the terms in

vy are to have derivatives with respect to x which are (assumed)
large, we have Vox >> il. Hence, ab leas th Onew oO ice mnermsiaon,
cAAy, must be as large as Vox in order to balance this term,

The equation will then be satisfied approximately if we write

2 Wa (1l+a sin t) sin nsy

and

= 0 (5)

eAAb, - Vox

We must now integrate these equations and then add the two

solutions p~; and p to form the complete solution Wo

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mine) atid matt of rodioy ou. ol Game tia ‘oe

>») *w Nv b} eon : et th #44 { an ik te i)
< i a "
Wii nie fe aie eee ee ie
=. 4 . TES Re

A11-101 100

The solution to the first of the two equations is given
by (4). Since the complete solution will only be approximate,
in that terms of order ¢ have already been neglected, Vy need
only be determined approximately.

It is suggested by the above considerations that we
find a formal method for writing our equation so that the
magnitudes of the terms are expressed by the coefficients and
that the derivatives, etc., be of order unity. We ean do this
by stretching the x coordinate near the boundary i.e., by de-
fining a new x coordinate so that a particuler distance in x
becomes a much larger distance in the new coordinate.

Formally, wc operate as follows. Let x be replaced by

the coordinate & such that
% = 6
where n is to be determined. Then the equation (5) becomes

o,rentl

-~ ntl vy i

=n
the yy + "Yyyyy “© Poe = 2

In choosing n we note that it must be positive if the
x coordinate is to be stretched. Thus of the terms which
originally had coefficient e€, e7 tnt Vorcer is the largest
since it has the largest coefficicnt (n.b. Ye» saad
VeEEyy? Wi cemey ere the same order of magnitude). This term
is matched with e~” YoEs the remaining large term in the

differential equation, and by equating the coefficients of the

above two terms, we have n = 1/3.

OOF

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A11-101 1O1
Thus we get

=i1/3) 1/3 =il/
i Vinggee ee = Unaaan & ee are bye = 0

E ~
yyy

or

Qi
Tpece = Wye 7 Oe =)!

Now if vy be expanded into an asymptotic series and if
we keep only the first term in the series (for all practieal
purposes, this amounts to neglecting the o(e2/3) terms), we

have

Veeco 7 ie = © ce)

ae Golliteaoia wo) (Uo) abs
E ge
Wy = Cyoly,t) + Conly,t)e” + Cao(yyt Je 3

Ur i
te he
Flea Cate eae
We have specified that this solution is to become
negligibly small as the distance from the boundary increases.
Thus letting Eo, we note that it is necessary that Cj5 =
Con = O since neither C,5 nor eb GeUCls 10) Wei, Inlswle, ioe

the region near x = 0, we have

fo

b= Coen 8 | eo yaiepE

or, changing our coordinates back to x by means of x =€ oe

}
iy
¢
ew
1 as ‘
i tai
i 4
1 ie
; | MY, rh iy ie
a7 SN Eb, MARE

ion EH!
ey

at y+

Ania

4

inet

miki i Gata
Be BB, wieae

i

Al1-101 102

ua em 2 Ugi
by = C,o(y;t)e™ Bo: 3 + Go(y,7)e wy © 3

For the boundary near x =r, we now define & by
(x-r) = 6%

and specify that the solution vanish as &-@, i.e., as the
distance into the interior part) of the ocean anereases By
acimilarvanailliy sis, we find that near a =n.
r ae ani
bp = C13 (y,7) 4 Co, (y, te + C67) Te 3
Lad
e —<——<—
FO(¥, ne" 3
Iinvonder tor by, to tend) to zero ase — >i) Go aGes
= = = il
necessary that C13 ©33 C13 O. Hence

any f=173
b, = On,(y, te" 2 C,,(y, en se)

The total boundary layer solution can be written
eee L/S , 208
by = Cy (y, ve r) + Cay, te 3

=1/3 2 tad
+ CCy, we 3 (7)
The solution throughout the domain consists of (4) and

(7), oe
v=, +p = (2%: Silin ae )) Satia nsyfl - x + Ci(y, 2 J

Cane ta

a i
+ Coly,t)e + Ca(y;t)e c

(8)

1/35 bad
HCAG ee i ic ten j

=

.

'

"

4

j Pee

o

y

) es eye

Al1-101 103

An application of the boundary conditions, = eS

onl) x = O57), yields

3 4 al/3 Gereq2

~= (1l+asin ct) sin nsy eee

4 [6et-r)e0s cae: ea) + (3 1/3 - ao cxVael43,

Tt ee i

en

eran eee Sl

The term 1 is valid throughout the ocean. Near x = On 3) be=
comes as important as 1 and gets negligibly small as x in-
Creases. Near x =r, 2 and 1 together form the solution but
« tends to zero as x decreases,

Perhaps a few remarks should be made as to the specific
choice of sin nsy for the total y dependence of the solution.
“The particular choice of sin nsy satisfies the boundary con-
ditions v = Uae = © ony = 0.) y=) ond) us Ssmppomce dem yaniume
specified wind distribution. Thus we were not forced to resort
0 a boundary layer analysis to satisfy the four boundary
conditions. Of course, such a simple choice is not always
possible, and one might have to resort to methods for refining
the interior solution in other problems in order to satisfy

the necessary boundary conditions.

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620, 798, tm rp. The correction cf the perturbation terms is negiiaible.

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