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:
~~
Y.~ Gomputatten of Sea Surface Slope and Horizontal Variation of
ere
oes se h%
Po y
il
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: ¢ , ‘ a . 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.
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Het Ri ro
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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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Mb tidnosio wh moth oes, ‘sao dealt ny
a ‘od? deen ot aes Saat vet te decks any afi os ‘ovo bs i
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) ww
ey
a me
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oe Ay A Rad LNT ae nee tite ae patient wo ftayaie cis mdre ete ”" wotiotideare’ of
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4
"Pina lepton sts So. yore tee ita OS CD tothe: aun 20 [ybtucentn
‘vets att emt mr gone AF OD SRSA IRS weet eat aditenpoey add
fn " en eS me ¥ * c~ bch, a: ~? gg oth ee se viene Ny 1 , ae we 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
do gata ati eS
; Hin Van -odnt ddan ot io @ ete a f a 4 wi ‘e 92K 2 Sy HK Ane) ba 2
ae pied; wor eobiabnuod wd paota bof ven cd amats toaon 4 , 3)
(en
oe Fallin a Ot.
bin lal VERSED”
( set a) uae a, Ash
Y i : A \ ‘i ae 5 au vee nena bi? re * in : On Ms i ny) ate ri ake (aye sigh ay ae ah res
aitce
> +++ 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 %
ate erat tb een | his - Felaw in’ ; bases sana ai 0: ser
{
; | Set ott i Me tii net noteowae ye. ie astiwe=\ ag a eas (i eneer ir ae
oe ysl va GN hus ats #9 We
c
‘ i ‘ F “As Ls, A i Py eR
nN : WE is i eon 0 { pest alba NA “ apes eae
q we a.) or
4 oe orm IY cs
gi 8ie3 e Seen :
ian bDiatiete 2, wt”. aM Yo als ont
sf aN. En, an
(ee
vial A AF eu eh pal pa sree eRe Moy gts irm
tm 4 > j aw
J
‘esa
eSmatioses oe! emote hs 25K. Ply Set UE Peostieh ee sais Bn Suatose ole 9 Raleah RO Smcre are 5 7s oa
D ¢ ger hdc nig ends t se
* Pee oe en ‘. PS ow y a 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
~ ! o . a tw ; ios . 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
“10s
7 Zac yiwads et fo Liey eaten aif #yeroe abireges guibeneng'¢ ] | Mit aory ecso islsegs o ae <igneasn sodtus orf oar brave ih,
a = ai f
\) . = vet as : a ; mn oe < j hat - Se Soen- fat> ul eecel yee ni ¢sa odsis
A. -bewbtacce arr (01) So sede Geelditigh af al ores ivoos 6
« Gas “q soolsr ~ Mace ais galviavel colisteqao> sootiewe et gees i = : 1677 lie Beyeities od 10 ly -e7oté ag ad wet a 4&2 [ ) 2° C2 aan cuit at eriy Lia a tastuc af vue al w= = wk ESiis eerie ote cnt
: > © @eeitera «. Sees arc, Sst 22 Tia SO Te Pr . = , Roos x= 7 i ». =a25 uM i a - q . £ £028 Pe
- ’ - 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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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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