Wind-driven sea level change of a shallow sea over a continental shelf

OCEANOGRAPHIC SURVEY OF THE
GULF OF MEXICO

Office of Naval Research Navy Department
Contract N7 onr-48702 Project NR 083 036
Bureau of Ships Technical Report No. 9
NE 120219-5 October 1953

WIND-DRIVEN SEA LEVEL CHANGE OF A SHALLOW SEA
OVER A CONTINENTAL SHELF

Koji Hidaka

Research Conducted through the

Cevas A.& M. Research Foundation

COLLEGE STATION, TEXAS

WEA WA

O 0301 OO44?74S 6

Ty AGRICULTURAL AND MECIIANICAL COLLEGE OF TEXAS
Department of Oceanography
College Station, Texas

‘Research conducted through the
Texas A. & M, Researeh Foundation

Projeet 24

WIND-DRIVEN SEA LEVEL CHANGE OF A SHALLOW SKA.
OVER A GONTINENTAL SHELF

Project 24 is an Oceenographie Survey of the Guif

of Mexico sponsored by the Office of Naval Research
(Projeet NR 083 036, Contract N7 onr-487 T.0. 2)

and the Bureau of Ships (HE 120219-5). Presentation
of material in this report is not considered to con=
stitute final publication.

Report prepared 18 September 1953
Koji Hidaka

Dale F. Leipper
Project Supervisor

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Relation Between the Wind Stress and the Slope of the Water

SUPPACO. 6. ssc orencnsaceecerseeerorenersuasorocounn scar oes:

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Y.~ Gomputatten of Sea Surface Slope and Horizontal Variation of

ere

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rstract: ePenaranoeruseces sineiaieinve DeOPVECH HEH OL ZUCOTOSHebRDetUECRDEASOn

x Sea Lev@locecossenaesssousssenscrcoreres OPCS SSS SS Pe a oe

-_ 4 -

iiseaestan. of the Earth's Rotation as a Factor Influencing ~

the SLOPE. cocccenevecaseceseoosvaccesouccsvoceencvecceeucos

Computation of the Sea Surface Slope. eceoasscoseoocaneacnalent

Change of Sea ‘Level in an oft ‘shore Directioncs su. 0 does<Oe. seed

Relation Between the Wind Direction and the Sea Level Change...

d eknowledgements. PBHOFOOKHFEOE HOR uA EHOSRORr2OGDOCOHeHGoBOOAGHoDSOEEGASES

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TABLES

Offshore Slope of the Sea Surface Af. indueed by a Wind Perpendicular

dx
to the Goast ( “% ), computed at Different Distances ean from the

Coast.

Offshore Slope of the Sea Surface, £2 induced by a Wind Parallel to

vp ¥.

the Coast ( 1 ), Computed at Different Distances */~ from the coast.

Sea Level Difference between the Coast and a Point Distant x from the
Coast, induced by am Offshore Wind Stress 7; o |

Sea Level Difference between the Coast and a Point Distant x from the
Coast, induced by a Longshore Wind Stress (en °
Function ¥ (= 2 ) for Computing the Sea Surface Slope for Uniform
Wind Stress. os

Fimetion — re eS ~ * } for Computing the Sea Surface Slope for Uniform
Wind Stress. |

FIGURES
Coast, Wind Zone and Calm Area.
h h
Depend=nce of the Function k Gz ya ) Upon the Ratio D °

. ahe gis
Dependence of the Function Te ie) A) Upon the Ratio re

Sea Level Difference Between the Coast and a Point A Distance *

from the Coast Produced by the Offshore Component T, of the Wind
Stress Conpiited for Several Different V-l ues a e

Sea Level Difference Between the Coast and a Point a Distance ¥%
from the Coss% Produced by the Longshore Component Tq of the Wind
Stress Computed for Several Different Values of * °

Schematic Diagrams Showing the Relation Bstween the Direction of Wind

and Sea Level Change.

iii

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cea as ND

Wind-Driven Sea Level Change of a Shallow Sea
Over a Continental Shelf
by
Koji Hidaka

ABSTRACT

A theory of wind-driven surface slope and level change in a shallow
sea close to the coast is given taking into account the earth's rotation
and both vertical and horizontel mixing, A wind zone of finite width
extending from the coast is assumed and the surface slopes in a steady
state are computed at several distances from the coast. If these are
pieced together, we can give the surface water level change as a function
of the distance from the coast. This research represents a portion of a
voluminous work which the author is carrying out concerning the three=
dimensional steady motion of water and the surface-contours as generated ~-
by a steady wind.

I, Introduction.
The concept of horizontal mixing introduced by C.=<G. Rossby (1936)

and subsequently developed by R. B. Montgomery and H. U. Sverdrup hag
presented several important changes and advantages in the dsteal explan=
Peicn of various meteorological and oceanographical phenomena which had
hitherto been very hard to explain. Montgomery mentioned various evie
dences which showed that some oceanographical phenomena cannot be explained

without taking this concept into account. We can mention the successful

1 Contribution from the Department of Oceanography of the Agricultural
and Mechanical College of Texas, Oceanographic Serics No. 000 3 based in
part on investigations conducted through the Tems A. & M. Research Founda
‘tion, under the sponsorship of the U. S. Navy Office of Naval Research, Con=
tract, Nfonr-48702.

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explanation of the westward intensification of the Gulf Stream and the
Kuroshio by this idea demonstrated several years ago by Henry Stonnel
(1948), Walter H. Munk (1950) and Koji Hidaka (1951). A theory of upwell~
ing recently worked out by Hidaka (1953) 48 also based on this consider~
ation. The present discussion also consists of an application of this
concept and treats the surface form of the sea off a straight coast
developed by the effect of steady winds blowing in a certain direction

in a finite band within a certain distance from the coast.

The theory of aarinees of water on the coast by the action of the
wind was first treated by V. W. Ekman. Ilis explanation consisted of the
fact that very close to the coast the steady flow of water driven by wind
toward or away from the coast just balances the flow due to the slope
current produced by the piling-up on or taking-away of water from the
coast. This seems to have been successful in predicting the slope and
of the water surface approximately. But since his theory assumes that the
velocity and surface slope are uniform in horizontal directions the diffi-e
culty is that of how far from the coast the predicted slope is. Present
research shows that the slope and level change of the water surface occurs
mostly belcw the wind zone only. Further, Ekman's theory is Gani to sey
how the height of the sea surface varies as we are removed away from the
coast. This is mathematically impossible because only the vertical
momentum transfer is taken into account and the velocity components and
slcpe of water surface are functions of vertical coordinates x alons.

In order to discuss the horizontal variation of these quantities, however,
4% is necessary to consider horizental mixing.
The following theory is nothing but a modification of Honan's theory

of wind-driven currents made by introducing the effect of horizontal eddy

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viscosity. Still the result has some advantages over the classical theory
in explaining various fentures encountered in the actual sea, especially
in enabling us to Imow the horizontal variation of the velocity components
and surface slope. If the complete mmerical computation could be worked
out, this problem would give a complete structure of water motion produced
by the stress of the winds in both decp and shallow seas. However, this
would require a great amount of tedious calewlation so the complete dis=
eussion is left for the future and only the digeriuation of the surface
slope and the change of level in an offshore direction will be treated in
this paper. It gives the steady surface seve developed by wind in a sea
of finite depth and will be especially applicable to the problem of wind~
produced piling=-up or lowering of water in continental shelves such as
found in the Gulf of Mexico or the North Siberian Shelf.

II. Theory

Consider a straight coast coincident with the axis of y, with the .
x-axis perpendicular to it in the offshore direction. (Figure 1) Suppose
a wind of constant. force and direction is bl. wing steadily in ae belt of
limited width L at a certain angie with the coast. Take the | ieee |
vertically downward.

If a constant wind blews for a sufficiently long time, a steady state
will be attained in which the motion of water is independent of time. We
assune that the wind stress cannot vary in the y-direction, but may be a
function of Z . This means that the wind can vary in an offshore direction
only. In such a steady state all the vertical and horizontal components of
the currents can be detcrmined as functions of x and g only. Of course, the

ate

i 1 tid Sy he
i Ha

; wit By) ;

WIND ZONE

WIND

SLAIN wittl
mat es Pak yite

(Stace Ore
- ke Beg

Nes

wR

aMiw’;Teagd.

ona's

surface of the sea will not be a plane, but have a slope in the offshore
direction, the amount of slope warying as a function of the distance zx from
the coast. In such a case ‘the hydrodynamical equations of motion are, after

several reasonable simplifications,

An ou Ar dU Aug Lt =o0
pie Pe cea a)

where y and y are the horizontal components of the current velocity in the
xe and y= ‘irections, C the surface elevation depending on x only, |
the density, Ay and | An the coefficients of wertical and horizontal mixing
(eddy viscosity) of sea water, C.) the angular velccity of the earth and
ih the geographic Intitude. In addition to these, we have the equation

of continuity in the form

Be OW. =

Gh 22
(2)

av |
since a7 =O . Here Ur is the vertical component of currents.

Let the components of wind stress be given by Cn and c « Because
the sone of wind is of finite width within a distance J) from the coast,
the conditions to be satisfied on the surface of the sea are therefore

au Tv Zin <
a ga =,
v2 /, awe foy
= =6 eo
5 =O), # ry We a OF iS
ov" we ay & VA
-AS/ =% fer O87
OA 7a
oD
Ce ee (3)
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Ny 1 , ae
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M/ a
wee My

ae

pedicure: to Sicmengenten, Lemstiiow ORF we POET a sn
pe Vacaasels a eo ie) yi ¢ o me Kees, diet yen cl Phys shes ro tise tes ager itd wee

gdteanco Cnt mary ha dedatetn oe eke e ) MBE of ett te Bebe To al n
exetorioiis oe Bee! pith Le wenarithsa aul? Ho So Mtnrton ot Pe anos Sey, if

where (eB and Cs may be either ‘constants or functions of X only. On
the bottom Z= A we mst have

ie = =O
“4 u=Y=a hs

beeause of the vertical friction. Along the coast which for the sake of
simplicity may be supposed to be vertical

} ) (5)
because of horizontal friction. In the region very far from either coast

or the wind region, we have
A=oe ® Uh = Va A)

let us define D and 1X ty

, Ly = m/ Ar/Peose 3 ae me] frag :
(7)

D: 4s the "depth of frictional influence" defined by Ekman (1905) in his

theory of wind-driven ocean currents, and D is a quantity having a die
mension of length and may be called "frictional distance". This is a
measure of the horizontal turbulence.

If we put

(8)

Bey wliteS sai
_ . i . }
; 7 » raking 4
won . at

*

(8)

1 vi Ae ws yi rs ‘di vA

+ oe

ve ASN: “8s Ass aa

Me psiae ets

west fs Sibes sprees: ott arth este te fogihircov ait Ver a:

Sop stra oo beware el Wont chokes

| ES ok Seer se 8
Peas aeitie avr 162 yiov notger et of .aobtofrt Lntonateed : shee

erat oy. .totgey . baty

| a
ta hy mi)

2@

and if (20¢2) mores, ne: een Precinct fecebie rs} ‘to ; itqen” eds a Wo,
wit F.) gaivad whideaup Bak me bas shits une Passed meviabeticke to it
; a ic we i
oe Te elit sSeaainta ts facehayth® bal ien od vet bas soguad 26 a

cepintarent tation Brod ort to-<

an on

the equations in (1) now become

aay: pel ema ee sr ue AG Bs

7 Ez aie GS Stn EOIN she

2 ae ih

wY , pee _ wy =o }

ore d . (9)

In order to solve these equations (9), suppose with Takegani (1934)

1 = 3 [Ug Bed E4A ,
| (20)
U, ae Uld, r, Z)Suw AKA i

a 5 [ab acre ap, ‘
tie [via ry Re AAA | (a)
a i 2 (rr) se ngads
va) Le Nha a2)
Next suppose for the wind prone
Aas ie An aph- 9/2 A)surg a (13)
od Ape
Tiv= fr Ay il. Ars hd g ee
ee ee
a igi ea (a)

» (Werte oo

MARGL): Erente’t ‘ake

ta

Ashi RYE \

La.

re

ni hektw Sib Ot deotent

if Cy and Ty are independent of Z . Substituting (10), (12), and (12)
into (9) and writing

U4 +vv,= WW,

(15)
the two equations (9) can be combined into
pee ee ee we yl =v
(16)
and the conditions to te satisfied along the boundaries now become
Z
We a /- coy (A /0,)
“A Gh SG 46%)
(17)
and
(2)
W ae
(18)

The solution of equation (16) subject to conditions (17) and (18) is ,

Pe ora) 6 read ata
or N27 \eosu.7 cork (Meme -*/D. )

TetiTG De buk(virane BZ) Lem(4p)
Neer Av cork [Rare */0,) = X09)

ays

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92K 2 Sy HK Ane) ba 2

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3)

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i ny) ate ri ake (aye sigh ay ae ah res

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> +++ ane 94 us a
If we separate the real part | of VA N62 ae from (he imaginary

part Al » we have

eae —
ee
NO OR ee ee my |

iP. [Axe g = |vAtert *

0 en mee

ree

A“ 2 2) & Sent danke rn aed sae
(20)
wes the real part of vier ee eM Ae is always pres iter ‘jnan 7T .

Tit. Reintion between the Wind Stress and the Slope | | of | bs Weter Surface.
Now Jt may be shown that we can establish a definite relation between
the wind siress and the slope of the water surface. In 4 steady state we

have no vertical motion of water on the surface of the sta. So we have

vin ates Up = a

since the vert cal velocity always vanishes on the bottem. Integrating
aw equation of contimity (2) with respeet to ie fror the surface down

to the bottom, we heve

Thig means that the ‘ntegral ud y is independent »f 7% , or there-
fs) he
fore a constant. Buy as this integral must vanich along the coast or for

Y¥= ODO 9» we must have
Fa

ai
aes

(21)

Cmte Tan) Sy

always, Integrating (19) with respect to i fiom 2 to £ p and

following expression for ee A )
yo)= =; & je = */04.) £ x

ae cosh (PY Jeas(q: Ve) 4 2PR cal Yes Jn oe
SP Y/a,) po07(@ Sa TRE uh

equating the real part of the resulting equation to zero, we have the |
:

et PS SPA DSL (php, i (7p) +3 Pat
T ee Ms ca Pee i,

+ 85h d Dek |
pa Auhl PI Min oc) 3a . cork (P (P. 2h) oy ‘

am pit Psa Ph (P 79, \0ok ( (Bip, area la ea ne |
oT Dy | (FART ae (Te) Fen a Yo yap. Ad |
(22) ie
If we substitute this expression in (12) or

We

: fad ar Tian \E AA,

(23)
we have the surface slope as induced by the wind stress whose components
are Tr and 1G respectively.

Ones the expression for 4 A ) is determined, we can obtain 7/ :

and VU ‘ty substituting y ( ‘\) in (19). Further substitutions of (4, and 4
in expressions (10) and (12) will give the horizontal components of velocity.
The vertical velocity can be er fron

et x oe (24)

r i r %

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{

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a eas (i eneer ir ae

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‘ i ‘ F “As
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WE is i eon 0 { pest alba NA
“ apes eae

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an

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9 Raleah RO Smcre are 5 7s oa

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Shay HAGSwO Sen ch). hue ce ep ase, CE “4 ees rs poy
a " Be g ‘ ee a

Bhs a5 a eeary : nr ayy tar as S Tr ee ee =
Para a ae ay S- my ant Jig Oe Se kes M4 ve wag Mo sien Wa pe i 1 EN?h NOB: sili linea if

* = MORES Riberd acho eT see int ey Tinto 4

~
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a tw
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. is i 7
‘ ry ‘ god enine Pa
a i \ alte wh
Fh ty \ r a
\ Ae \\ ? %
* Fi
.
Ss
4
sho
h ’

an equation derived by integrating the equation of continuity (2) from
7 the surface down to a depth a6
The preceding vnalysis covers ‘tthe principal part of the theory of
upwelling discussed by the author recently as a special case when the
depth of the ses is very large, In that case we had Y¥ (\) > Z 80
only the secozd term in the right-hand member of (19) wes considered. A
complete nursrical computation involving three components of velocity and
the surfaces slope wild be achieved oniy acter a tedious work of very long
period. We shall give in this report only the computations as to how the
slope «f the sea surface varies as we go away from the coast. The author
pents to express his desire tio extend the computations to all three com=
ponents of the motion of water in the future because this promises a great
number cf practical applications, The comparison of computed motion to
that, actually observed wiil enable us to estimate the approximate intensity
of botn vertical and horizontal turbulence in the sea, thus making it
possible to predict the wind currents in the sea more eccurately.
IV. Gomputation of Sey Surface Slope and Horizontal, Veriation of Sea Level.
It is a question of practical calculations to cerry the analysis to

numerical resvits, A rather elaborate computation has been carried out by
the author durixg the summer of 1953 when he stayed in the Department of
Ocean-craphy, Agricultural and Mechanical College of Texas.

The greater part of the work consisted of mmerical evaluation of the
function OX as given by (22). Because the components of wind stress are
given in advances we have only to evaluate the two functions EG. N) and

LG, A) given by (26) and (27). They have been computed for

& = 1/16, 1/8, 4, 1/2, 1, 2 and 4
yee

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et gees i = : 1677 lie Beyeities od 10 ly -e7oté ag
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tastuc af vue al w= = wk ESiis eerie ote cnt

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. =
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- ’ - ieee i Sets a]
—, a oo) é %.
ae = J 1Oe Soa 2 ¥ ie ok ;
} - : z Lie ge tPogiq: 2
B~» - —Srorse5 ~" + ee oe
< - - =, a °.
, io) = we * - ‘
4,7 oy > Pes = »
354 * Z 2 ?
- . 3 ¥ 4 o 7
* + =o
es 3 ‘ “ Pec c at -
2a Ss

4 - ‘ ee ‘ its
: ;
e J \
.
: 2

Teble I

Offshore Slope of the Sea Surface > te ST isiicod
By a Wind Perpendicular to the Coast (7x ), computed
at heehee es eB pA >, from 7 Coast

(units 7 /ps+ ri =
Yp,= Wie 1/8 7) 1/2 1

M

O 0 0 O ry) 0
Ol 1.5072 3.4883 1.4012 41.3274 41.6308
0.2 1.5226 1.2915 1.4720 +1.3203 40.9259
03 ile 5239 1.4984 1.4791 +1.3076 40.6871
0.4 1.4999 24970 1.4467 +1.3270 +1.,2606
005 0.7503 40.7413 40.7425 40.6651 40.3380
0.6 00045 +0..0091 40,0362 <0,0511 0.8992
0.7 =0,0069 40,0088 40.0064 0.0324 =0 04360
0.8 0.0096 0065 -0,0002 0027 40,0025
0.9 +0..0036 40,0002 -0.0001 20163 20.3723
1.0 -0,0040 40,0054 40.0005 0.0318 -0.2729
ibsal 0.0007 = ,00L4 40,0007 +70.0354 402424
1.2 =0,0026 ~0 0049 -0,0022 40.0088 40,0812
1.3 40.0007 0026 0.0004 20217 -0.1910
1.4 __ +0.0000 0042 40.0000 0.9077 40,0483
755 40.0076 40,0003 40.0022 40,0157 40,1036

°o oo00g00 0000

Sc atoeorl are
OU RWNEFO OBYHRH KRY

9

o ¢ 8 &¢ 6

Mr Heke BDGOOGS

°

Table II
NE *
Offshore Slope of the Sea Surface, A induesd
by a Wind Parallel to the Coast ( Ti), Computed
at Different Distances X/Dp, from the coast.

(units T/foA, L= £0,

as) SRVALS 1/8 V4 1/2 oe a Longshore
Wind Stress
0 a) 0 0 0 Ty
OsO0442 —«-—«s@ 13947 1.2577 ~0.2173 =1 8402 Ty
0.0477 0.1745 1.5085 +1.1535 +1.9789 Ty
0.0482 0.1813 1.5815 +1.,6083 +3.2510 Ty
0.0460 0.1622 1.4528 40.6742 +0. 3257 TC
0.0241 0.0958 0.8432 +0.9606 1.8433 1/2 %
O o C021 0 D>) 0292 2 0 Rc} 2663 at 0698 ho 7393 0
0.0004 - 9.0089 0.1163 0.7947 2.2709 0
00002 0.0012 0.0179 0.0527 0.0622 O°
0.0001 0.0013 0.0512 0.5501 1.7700 0
=0,.0001 0.0004 . 0.0335 0.3901 1.2799 9)
0.0000 = ,0007 =O ,0326 0.3542 1.1416 (0)
=) 0002 =) 0012 of) .0104 0.1182 0.3825 9)
0.0000 40.0005 40.0248 40.2778 40.9025 0
6.0000 =0,,0002 =0.0067 =Os071, 0.2291 9)
0.0002 af) 0008 =<),0130 20,1493 A) 4765 0

and for a mumber of necessary values of ¢

V. Discussion of the Earth's Rotation ag a Factor Influencing the Slope.
Expression (23) can Ne written

s 6)
ae % fe Sn acme ne
an” Pgh or fe ND, a)

Sei ig d) seen, Mu h% Ad

(25)
where
By Dee ae
EE) Ww he Mom ae Dy?
(26)
£ Ge raed a oy S48
_( 1) n & f_Mom- Neon Pv os
and
(28)
ob Me aud (Po, Jace (0-00)
(asian prt (2.4/9,)+c00*(Q: £/p,,.) iz

apc

‘ aif o Be

Mz.

NE)

(\) =
=
Ym () =
m ON) =

(ae
-

yx AAP, cote P%,)
dui h=( ‘ */0, ) reed? (Q: ip.) ?

awh (2%, 00 (a-%p,)

—
—

P-- Q?

(P% gr)e’
ZPR
(Pao )=

po SPE
(Ps a)?

SPa-@
(Psa)

JX +X?
Ls

Nevrrt a >
Be

oleae

Prec SG Sh eat
rile (P.*FID.) +003 (Q-*/p,)

(32)

(32)

(33)

(34)

(35)

pts,’
iH viata teary rie

aN

t aay pte
HPs uy uy Ty | Ui,
I y i i 1
‘ : i '
i
} 1 i i PVE
Hvih i i
’ Y 0 i ,
eth a Roem I
i mal

Scooby!

7a ee
being real ard imginary parts of V/A + Dry 4 vespectively.

Thus Rie. » ) and LG. r) depend upon ah and >) only, while

S Dy
ey Ty, As 4 fund he are of course given quantities.
Next we have
\ [ A
I end (A/D )
ae A ig ip

7A

Racha 7D, iia) teeter ee. =e a

SEMEMER LE URURCRST ORE MATICCUREC Aas eT eNO A ESI ETN Se
ev mene

‘eNO Vo

rey <

t=/23 A a)

j= 71D ,p

Oa, (eg ae (37)

Figures 2 and 3 give the grapls of the functions ie (es {Pan oA) and

where

{|
we)

(x- LY, , pr= Ok TAD

is (s., nN ) respectively. These two functions have been estimated for
values of

=D) = V6, 1/8, 4, 1/2, 1, 25 and 4

and some values of A necessary for furthering the computations. Only the

curves for K/, p= = 1/16, 1/8, 1/4, 1/2, and.1 are given in these figures.
From these diagrams we recognize that the value of the function

o13=

Table TIT

Sea Level Difference between the Coast and a

Point Distant °~ from the Coast, induced by an

Offshore Wind Stress {_. (Unite To Vs /ag/,
j ‘~ yy J w
i

ae

Ys Ve V2
0 0 0
0.074 0.071 0.066 0,082
90223 Q.215 0,199 0.209
0.373 0.362 0,338 0,290
90523 0.509 0,462 0.387
0,635 0.618 0.512 0.467
0.672 0,657 0.592 0.439
0.673 0,659 0.588 0.373
0.673 0.659 0, 587 0.351
0.673 0.659 0.586 0.332
0.673 0.659 0.583 0.300
0.673 0.659 0+ 584 0.299
0.673 0.659 0. 0.315
0.673 0.659 0,562 0.309

ww

© C0000 o000"

em

ran a

seal |
me ae i} 3
aly, *

Table IV

Sea Level Difference between tre Coast and a
Point Distant X% from the Coast, induced by a
Longshore Wind Stress ck (Unite GY els he

1/8 Vike % 0, wae i
oO 0 fy) 0
0,007 0.063 0.012 ~0,092
0.022 0.201 “40.036 085
O,020° °° -OedSer i OE a7y 40.176
@5057, 06507 +0..288 40,358
Dolonaw |) 0se22 40.370 40.470
< 0.076 ue 0.678 +0501 40.799
0.078 a 0.697 40. 625 +1.150
0.079 0,703 40.667 — 1.266
O.079%, + 10-767 40. 697 < 1.358
0-079 7) OnTLE O.%hh 1.510
0,079 0.711 0 0.746 Lol?
0:079 10.709 0.722 1.441
6,079 0.710 * 0.730 1.462
0.079 0.711 0.741 1.496
1.461

0.079 - 0.710 0.730

ans
bit

Pa
Pavan lke
CK 4 A Cake

bids

Ps es
ie)
pret

7]

pet a

A A
qT +

A A
(x NOILONNS 3HL JO 3ONVGN3daq © 914 (xh) NOILONM4 BHL JO 3ONVGNadaa 2°9l4

Oo! OS O OS] OO! OS O

g

Lie DEwEMOVACE Ob ite, Leweow 1

INE wyss0

: Bees elstcaerectinmaerstigeeee
en i :
iar heuer cpeimmemnbenaiied [ea cient ees eal decal fect nla Ualaddah

ge
a ’

Poe. Awe dv2ie =

related to is or the offshore component of wind stress does not show any
marked variation for either7y, or )\ except that its value suddenly falle
to zero at d =O for larger values of /p) )-» while the function lL E a } d)
related to Cy or the longshore component of wind stress has a very large
variation, For a smaller value of ”/, Di, this function varies slightly
and smoothly. When n/ D increases, however, its value at y, =O increases
very rapidly. This function thus has always a peak at =O ait
height of this peak increases proportionally to the square of A/p, when

£. is small and is directly proportional to 2, wren it is large. At
any rate this shows a rapid increase of the function Ts ie Dy! d) around

A =() when hy Dy increases from a small value to a larger value.
Now since the function of the type

Binhe rf

always has a largest value (=p) at y =O and falls rapidiy as
increases, it can be anticipated that the contributions of the functions

K(M%,, r) and / (Yo. P 2 the integrals

and

bo
fz (p,, \) i aA

will be largest at r =6 . This fact clearly shows that the value of

~l4=

ya T ay ; : tay

ae cy

inten Ais

fay wit ine Rey

the integral
bo ye
/— coo (NY;
[BE Oa a
does not ea with h/ D, while the integral

fh, yee eee Py Na d

increased greatly as Af; Pe inereases, In other words, the influence

of the earth's rotation ls more conspicuous in produeing the slope of water
surface wivn it is induced by 1 wind parallel to the coast then by a wind
perpendi:ilar to the coast.

VI. Grnpute Bion of the Sea : Svrface Slope.

Now Bet satin (25) becones

Dy |
hd Ay (38)

jnere

Smor-fele ny LeathZe (bun, A

(39)
; Aa ~ y= ceol(h De) xX d
Gh fi S20 aa Ma
(40)

Ae

el)
sili

AS

ME roof do

+h aN

at

‘tees Sowa
wh \tutyere eRe ay

ia
Apa Mann ad CAITR

i i
Maen
Ne a

It is quite easy to compute these integrals if the functions

Sf oy /

(41)

V(E x)= -/uté hed FA]

(42)
are computed and compiled. Tables ¥ and VI give parts of such compilations.
For example, when we want to compute the integral (39) for oF, D, = 0:3
eesuning T= Dike > we have simply by (36) and (37) to make a

X (510-3) as XG, 103-0.) -2x(4 oaros)
2% (03) 4+K(O:2)— K(0.8)
a

for a given value of R/ Dp » because X%, io) is an even function
of XY . The same applies for tts function ¥ ce JD, ee) represented by
the sums of integrals of the form (42). Tables Vand “on will enable us to
ae conputations for other values of the ratio b/ dD, °
By this process, we can compute very easily the slope of water surface

induced by both offshore and longshore vind stress components.

‘The following computation was made vhen the width J of the wind zone
is half as large as the frictional distance [ . So we have L/p, =0.0

~L6=

y

Se oS a

|
Ms at a t Say dl
iy 4 AY’ é

1 be

phe a NEe Tes

1/8

0
1.4856
1.4948
1.4972

1.5043

1.4937
104989

1.5039

1.4926
1.5001

1. 5025

1.4940

i 1.4955 +

1, 5009

1.4955

1.5006
1.4980
1.4969
1.5000

1.4993

1.4980

‘S, 1X ve for. cence ia Sea Surface
tees for Uniform Wind Stress

0.2838
0.0953
0.1933

| 0.0228

: -—)
0.1104 i oe
0.0775 *
0.0328

. bles

40.0234 +)

2 40.0422 s¥
- =0 0460 7
=0..0034

40.0372 :
0249

i o.095 45

eee

¥
oy ips Ps

o c ©

oe

CL iGO Nay ta

i}

Oh 201 56.

oa

aq
OMIT FWHHO OMYRUN RUNEO

NY HEME PREP OOO0O0 02000

=)

9

Table VI

Function Y (8) eor Computing the Sea Surface
Slope for Uniform Wind Stress

us = 116 1/8 Ws V2

0 0 0 0
0.0442 0.1363 1.3230 0.4856
0.0475 0.1762 1.5620 1.6098
0.0484 0.1893 1.6717 2.1035
0.0481 0.1904 1.6760 1.8909
0.0482 0.1923 1.7339 2.4812
0.0482 0.1936 1.8064 3.2966
0.0481 0.1928 1.7787 3.0161
0.0482 0.1922 1.7425 2,6002
0.0482 0.1928 ay ave 2.9189
0.0482 0.1930 1.7817 3.0411
0.0482 0.1925 1.7573 2.7681
0.0482 0.1916 1.7628 2.8330
0.0482 0.1929 1.7774 2.9915
0.0482 0.1926 1.7644 2.8469
0.0482 0.2928 1.7624 2.8209
0.0482 0.1928 1.7735 2.9479
0.0482 0.1927 4.7677 2.8861
0.0482 0.1926 1.7628 2.8273
0.0482 0.1928 1.7709 2.9177

0.0482 0.1942 1.7691 2.8993

GOR

and B slope of the water surface was Greet ae devant] distances from ri
- the coast. Both the surface slopes induced by the offshore and longshore -
"vind stress components are given in Tables I and II.

From these results it can be concluded that the slope of the water
face is chiefly found in the wind zone and it is mostly very small
side the latter. However, the Tannen of increase of the slope of the
eer surface with tee ratio ke / Pp is much different between the
offshore and longshore winise In case of the offshore wind (x the i
8] pe indueed by it dees not wary much “with the ratio A ie ope > Their

es within the wind zone lie mostly between | |
-

to Ga At iad = 1, the slope varies rather wee g This

be dangerous to believe this result to be very accurate. At any rate

=o ee

4 From this result it can be concluded that the slope of the sea surface

3 ead by wind stresses is proportional to the wind stress Tz _ and G
recly to the depth b of the sea provided the ratio By Die asia

are nearly independent of the smal tele ie or the a turbulence.

a]J=

i

Ty hee

mot

If we take Vt =1, t = 50 meters, then we shall have

a QbK/d A 3.0K/0 a7

This is a slope about 3 em per 100 km, or about 2 inches per 100 nautical
miles. The stress Tie = 1 corresponds to a wind of speed about 6 or 7 m/sec.
’ When h = 100 m the slopes is half as large.

The fact that the slope is very smal] when b/Dy is small, means that
the influence of the sarth’s rotation is lergely pressed down by the bottom
friction. As the depth of the water approaches Dy gradually, the earth's
rotation becomes a more and more important factor.

Although these results are all purely theoretical ones, there ia no
réason a they are of no practical application. Comparison with great
mmbers of observations will give som idea about the magnitudes of both
horizontal and vertical mixing eefficients.

VII. Changs of Sea Level in an Offshore Direction.

Determination of the slope of the sea surface enables us to know how
the surface of the sea rises or falls as we are removed from the coast.
Because the water surfaces is assumed to neither rise nor fall in e direction
parallel to the coast, we have only to check the chatige of sea level in an
offshore direction.

An approximate formula to compute a curve y = F{x) from the values of

uy pesos at two points separated by AX is
qa +2 f(A) 1B), a2

4i-]

where (24) ws (Ft £) are the values of y at 7 and X)

Kae -/ Sh
=lB=

i

DD

an

ead ssh Phlget

in ¥ ae wht | ‘im

im itor: ae

| iy ve a
ho Tener “Tatas «3
f % cay poe

Chee Oa

ur Te tet

SAN HS RT Rt

pi f
i i
i % i
j i
Ata ;
il A i
5 t
oie ‘
Pe

nd

Rae aah

separated by 4X eo Assuming we have a water height ig on the coast,

we have for the change of level produced by an offshore stress

C= 6, acer
Z - f+ BE IS, (ea) Hie Gomf,

}

_
—

and so on. The same epplies to the slope induced by longshore stress Cy 6
. ‘By this process it will be possible for us to derive the sea surface profiles
produced by both offshore and longshore wind stresses. Actual sea level
consists of the sum of these two. Tables v and VI give the results for both
of these stress components ee tively. These are also illustrated by
Figures 4 and 5. |
Looking at Tables III and IV and the two diagrams (Figures 4 and 5) we

at once notice that there is practically no change in sea level outside the .
wind zone within a width je from the coast. .

; For a longshore wind blowing in such a manner that for an observer _
ioekiue in the direction of the wind with the sea on his right hand side,
the sea level rises nearly linearly age wé are removed away from the coast
until we arrive at the end of the wind zone. This tendency is common to the
cases h/D, = 1/16, 1/8, 1/4 but some irregularities occur when rd, =1/2.
Tt win be hard to know if those irregularities really exist or if they are
actually due to some incompleteness of the procedure of numerical integration.
Perhaps the latter explanation holds batters In any event, the general
tendency is that the sea surface outside the wind zone suffers no appreciable
-lewel change. Now since the sea is supposed to extend infinitely, the change
of the sea level in a finite area will not affect the level in an infinitely

=19=

A

ME Ati

i

Asoo (as

Bari thy

yah

Gn antic hy
ba aly Hh

ont
UME

FIG. 4

0.5D, SD. I.5D,

SEA LEVEL DIFFERENCE BETWEEN THE COAST AND A POINT A
DISTANCE x FROM THE COAST PRODUCED BY THE OFFSHORE

COMPONENT T, OF THE WIND STRESS COMPUTED FOR SEVERAL

DIFFERENT VALUES OF a
Vv

ee cc a ery

t

- snpaivae 07 asTuaMos ; azaare. “ad

/ ni 1

One ie. ae
pgh

GIL 1.2-—

hse
WIND ZONE

fe) 0.5D, D, 15D,

FIG. 5 SEA LEVEL DIFFERENCE BETWEEN THE COAST AND A POINT A
DISTANCE x FROM THE COAST PRODUCED BY THE LONGSHORE
COMPONENT T, OF THE WIND STRESS COMPUTED FOR SEVERAL
DIFFERENT VALUES OF =

i” o

i
f
1
ba sshincnbetemn bey aly Setee of
i
{
I

as
'
|

f= le

wide area outside the wind zone, This means that when the wind blows in the
above manner, we can expect a depression of sea level beneath the wind-swept
. area deepening linearly toward the coast. The maximum level fall occurs

of course along the beach. If the wind blows in the opposite direction, there
will occur an elevation of the sen surface toward the coast. The magnitude

of these depressions and elevations of course depends upon the ratio h/D, and

‘LO __, the wiath of the wind zone.
For an offshore wind blowing in such a manner that the observer looking
towards the sea has the wind on his back, the same sort of depression takes
place, of course, the manner of its dependence upon b/D, differing from the
ease of longshore oak If the wind blows from the sea to land there will
occur an elevation beneath the area swept by the wind.

These details are illustrated by the diagrams in Figure 6.

.

‘VIII. Relation Between the Wind Direction and the Sea Level Change.

The diagrams in Figure 6 give us an approximate idea of the relation=
ship between the direction of the wind stress and the sea level change in
a steady state. The sea level rises approximately linearly as we ace removed
ay from the coast. No slope of the sea surface is seen outside the wind
zone, The sea level responds to the offshore and longshore wind in different
ways. For example, in the area of° California, a north wind lowers the sea
Level below the wind zone and a south wind raises it. On the other hand an
east wind raises the level and a west wind lowers it. Thus is can be con-
eluted that for some direction of wind and for some ratio bh/Dy there will
occur neither rise nor fall of the sea level however strong the wind my Bie.
Such directions will be found in the sectors between north and west and Bonkh
and east.

On the contrary there will be a wind direction which fives a maximum

rise or fall of the sea level. This direction mst of course depend upon

=20=

il

@
COAST
co

<<_——

es fe
—>$ ss <—

| <—_ <—
e—

SE rm a

2 :

(op) 1

I
Dez
oO ee
— J

—)

\

<€h | LEVEL
(A)
BOTTOM
Lacan
SEA LEVEL
(B)
BOTTOM
ag
ee = 03b
(C)
BOTTOM
| —
SE aT LEVEL
(D)
BOTTOM

7

NN

ZZ

FIG. 6 SCHEMATIC DIAGRAMS SHOWING THE RELATION BETWEEN

THE DIRECTION OF WIND AND SEA LEVEL CHANGE.

nn lem ienrigpieritod a!

aie Jay:

egg sth

bay ner enigma eer
ee 6 we nats ih

te My tik

=
ae
2 err t 4
: fh ‘Leah
i ~ ;
, \ ‘Aro Ha!
i : ;
Wh anys hina py i
af Dd %
MiTAC ALM, Hkh, ray
re ‘ Vin, A ~~,"
% Ma tsi tn
eh NY

ee tt a Tip ai os) Bw) m
ey Sy hale

rel ie3,

eA Me ee ae Bed CHAA
» warins fF we aw wh ag

the ratio h/D,, that is to say, on the square root of the mixing coefficient,

providing the depth to the bottom is given, Off Texas and Louisiana eoasta,
the wind from east to southeast and from opposite directions will not be
effective in raising or lowering the sea level on the continental shelf, On
the contrary north or south winds are expected to produce strong falls or
rises in the sea level on the shelf. |

Summary. The theory of the wind-driven currents in a shallow sea is considered
taking into account the effect of horizontal momentum transfer. Other assump-
tions and conditions are nearly similar to Ekman’s work except that we assume
an infinite straight vertical barricr for the coast. The complete solution
involving the expressions for the three components of velocity and the varia-
tion of the surface slope at different distances front the coast appears to
take a very long time and require tedious computations. For this resson only
the result for the slopes of the sea surface is given in this paper. The
following conclusions have been drawn.

(1) Due to the stress of wind there occurs a rise or fall of the sea
surface, When the wind blows within a finite sone from the coast, this sur=
face slope occurs only in this zone and no slope is seen outside it.

(2) When the wind is wmiforn, bin Weir eee oa tidacehe int ene ieee
or falls linearly toward the coast.

(3) For a certain wind direction and for certain eee of the ratio
k / D rn? no rise or fall of the sea level will occur. On the contrary, there
will be directions of winds for which we have a maximum rise or fall of the
sea level, These features will depend upon the direction of wind, depth to the
bottom latitude and the vertical mixing coefficient.

(4) Complete numerical solution of this problem for the three dimensional
water movement is intended by the author for a future opportunity.

=F

at Lest

‘ nice ets eatta

ip

ial ’ tat, wa aah

ay ile eth say Air

1a oe bec
‘ oh
omg] SEM eves hh or

a Aina re old

ance sn on naa

lay ite
ere! |
Die

Acknowledgements, Generous help by the staff of the Department of Oceanography,
Agricultural and Mechanical College of Texas, College Station, Texas, hae

been given to the author which enabled him to carry out the numerical compu-
tations. The author is especially grateful to Dr. Dale F. Leipper, Head of

the Department, and to Mrs, Robert Shrode of this department.

~i2e

e

i
pers
Baily
me easy
Wyte pe
1

References

Elman, V. W., 1905

i On the influence of the earth's rotation on ocean currenta.
Arkiv f6r Matematik, Astronomi och Fysik, Stockholm 1905-06, Vol. 2,
No. 11.

| Hidaka, Koji, 1951

| Drift ae in an enclosed ocean part /ixT, ESE Notes,
Tokyo Uhiversity. Vol. 45 No. 3.

5993

A Goneriiation to the theory of upwelling and coastal currents.
American Geophysical Union Transactions. (an press).
Montgonery, RK, B., 1938
Circulation in upper layers of southern North atlantic deduced
a: with use of isentropic analysis. Papers in Physical Oceanography and
Meteorology, Published by Massachusetts Institute of Technology and
Woods Hole Oceanographic Institution. Vol. 6, No. 2.
= i » 1939
Ein Versuch, den vertikalen und seitlichen Austausch in der Tefe
der Sprungschicht im aquatorialen Atlantischen Ozean zu beatimmen, Ann.

der Hydrog. usw. Vol, 67, pp 242=246.

ripe he PA9 aaa ;

The present evidence on the importance of lateral mixing processes
in the ocean, American Meteorological Society Bulletin, Vol. 21, pp 87=
9h. |

Munk, Walter H., 1950

Wind=driven ocean circulation. Journal of Meteorology, Vol. 7,

111-133.
=a

oy
a)

i ; f if } ; ; a in 1) "1
rein hia CURVE r 0 ete fas Fad fe ica OLS Re be f
r 7 au ra | Me ig ea a
| nik cay! as ne Pe es at
i ve ;

AE) ke th aR

Pt de

Rossby, ©.-G., 1936
Dynamics of steady ocean eurrents in the light of expsrimentel
fluid Mechanics. Papers in Physical Oceanography and Meteorology,
‘Massachusetts Institute of Technology and Woods Hole Oceanograpl te
Institution, Vol. 5, no. 1.
Stommel, Henry, 1948
The westward intensification of the wind-driven ocean currents.
American Geophysical Union Transactions Vol. 29, pp 202-206.
Sverdrup, H. o% 1939
Lateral Mixing in deep water of the South Atlantic Ocean. Journal
of Marine Research, Vol. 2, pp 195=207.
Takegami, T.y 1934
The boundary value problem of the wind current in a lake or a
sea. Memoirs of the Kyoto Imperial University, Series A, Vol. 27,

No. 59 PP 305=316.

ee et ae

- 1 i PN til © * af q
* ‘ a ar
+ ier m bl > hon dart, Se
‘ F BiG i Na file et Y
i 1 i,
i nN i jay 5, t i
’ ws i 4
4 i nt ore Aa f
{ P ’ oy
Eas hid :
‘ 1 bo |
T J, ‘t
4 4 D {
Wy EA Ai) i
; Bay i oa ‘
i f 1 i A
cy ie te ‘
ray, 1 y Y \ r i
| Ae D e Ah
i int i if : ‘

ae

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Texas Game and Fish Commission
Rockport, Texas

California Academy of Sciences
Golden Gate Park

San Francisco, California
Attn: Dr. R. C, Miller

Director
Institute of Marine Science
Port Aransas, Texas

Director
Gulf Coast Research Laboratory
Ocean Springs, Mississippi

Director
Narragansett Marine Laboratory
Kingston, Rhode Island

1
Director, Hawaii Marine Laboratory

University of Hawaii
Honolulu, T. H,

Director, Marine Laboratory
University of Miami
Coral Gables, Florida

American Museum of Natural History
Central Park West at 74th Street
New York 24, New York

Director

Lamont Geological Observatory
Torrey Cliff

Palisades, New York

Department of Conservation
Cornell University

Ithaca, New York

Atf: Dr. J. Ayers

H

Addressee

Department of Zoology
Rutgers University

New Brunswick, New Jersey
Attn: Dr. H, H. Haskins

Scripps Institution of Oceanography
Ia Jolla, California
Attn: Director (1)

x Library (1)

The Oceanographic Institute
Florida State University
Tallahassee, Florida

Department of Engineering, Dean
University of California
Berkeley, California

Director
Woods Hole Oceanographic Institute
Woods Hole, Massachusetts

Director

Chesapeake Bay Institute
Box 426A RFD 2
Annapolis, Maryland

Head, Department of Oceanography
University of Washington
Seattle, Washington

Bingham Oceanographic Foundation
Yale University
New Haven, Connecticut

Allen Hancock Foundation
University of Southern California
los Angeles 7, California

Head, Department of Biology
Texas Christian University
Ft. Worth 9, Texas

Head, Dept. Meteorology 4 Ocean,
New York University
New York, New York

Bear's Bluff Laboratory
Wadmalaw Island, South Carolina
Attn: Dr. Robert Lunz

Department of Biology
University of Florida
Gainsville, Florida
Attn: Mr. E, Lowe Pierce

Director, Virginia Fisheries Labor.
College of William and Mery
Gloucester Point, Virginia

Director, University of Florida
Marine Biological Station
Gainsville, Florida

Director, Alabama Marine Laboratory
Bayou La Batre, Alabama

Sout .ern Regional Education Board
Marine Sciences
&30 West Peachtree Street, Ne We
Atlanta, Georgia

Director, Louisiana State University
Marine Laboratory
Baton Rouge, Louisiana

Director, Institute of Fisheries
Research

University of North Carolina

Morehead City, North Carolina

Director
Duke University Marine Laboratory
Beaufort, North Carolina

Institute of Engineering Research
244 Hesse Hall

Berkeley 4, California

Attn: Prof. J. W. Johnson

Chief, Air Neather Service
Department of the Air Force
Washington, D. C,

Head, Department of Oceanography
Brown University
Providence, Rhode Island

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