S H O R T E R C O M M U N I C A T I O N

Amplification of transient response of the ocean to storms by the effect of bottom topography

( R e c e i v e d 13 N o v e m b e r 1958)

VFaONtS and STOMMEL (1956) showed that the ocean responds barotrophically to short-period large- scale f luctuations in the winds. Since mot ions in such a case extend all the way to the bo t tom it is natural to enquire whether irregularities in the topography of the bo t t om migh t modify the na ture o f the transient response; but VERONm and STOMMEL did not consider this question.

The actual ocean bo t tom is very complicated in form. We do not propose to try to approximate this complexity in the following analysis. Also, it seems certain that irregularity o f the bo t t om enhances the baroclinic mode ' s response (CHARNE¥, 1957), but we consider here merely a barotropic

model. It is well known that in a homogeneous model the /J-effect can be approximated by in t roducing

a slope o f the bo t tom upward in the y-direction. The equat ions which we will use are

- - - - J i . . . . g - - ( I ) ,~t 3x

--~fu (2) s t - - - g T y

~-7 + ~ (hu) + ~ (by) = s (x, >,, t) (3)

where u and v are velocity components , ~ is the displacement o f the free surface, f the Coriolis para- meter , and S (x, y, t) a driving agency - in this case explicitly a variable mass source : for e x a m p l e

precipitation. There is no problem in mak ing this mass source absorb the effects of o ther current producing mechanisms such as normal pressure, wind-stress, etc. (see STOMMEL, 1957). The quant i ty h is the variable depth of the ocean.

Assuming that the funct ion S is of the form S (x, y ) e ic~t we can eliminate u and v f rom the above

equat ions and obtain

[ - - hV 2 t i ~- i g f J ( h , ) t c, ( f 2 _ a2) ~ = ( f 2 _ e2) S (x, y) (4) °g ,,, ~>, , , x ~

where the factor p let has been removed f rom ~, The depth h may be written as

h = H ( I -- 3),-! , b ( x , y ) ) (5)

This form is convenient because it separates the scale depth H f rom the small gentle slope 3H (which we introduce to s imulate the fl-effect) and the bo t t om irregularity H ~ b (x, y) where ~ is a per turbat ion parameter which we define as the fraction o f the total depth occupied by the submar ine topography, and b (x ,y) is the funct ional fo rm of the bo t tom topography with an ampl i tude o f unity. This subst i tut ion makes equat ion (4) a linear partial differential equat ion in ~ with n o n - c o n s t a n t coefficients. If wc regard ¢ as a small per turbat ion number (¢ ~. l) and assert that the solut ion is o f the form

+~. ! ~ t i . . . we obtain the following equat ion

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Shorter Communication 313

(L + • B) (go + ~ ¢1 + • • .) = ( f~ -- ~2) S (x, y) (6)

where L is the linear operator

L = { - - a g H [(l --~v) V ~ 8~-y] + i g f H 3 3 } _ __ + . ( f 2 _ o~) (7)

and the non-constant coefficients are contained in the operator B

B = - - o g H bV 2+~y ,~ - - - fy+~x~x j + ig fHJ(b , ) (8)

To zero order, we therefore have the equation

L g0 = ( f 2 _ ~2) S (x, y) (9)

If S (x, y) is taken to be of the form S o ei(U x+ vy the solution of (9) is

go = al et(ux+ vy)

S o ( 1 - s 2) a°=f{sA~t~2- t v 2-~ i ~ v ) - A 2 ~ ~ s ( I - s ~ ) } (10)

A~ gH a f2 f

This expression contains the effect of the large scale gentle slope -- 3H; and if s < I we are dealing with quasi-geostrophic motion of essentially the same kind as the planetary (Rossby, second kind) waves deduced by VERONIS and STO~MEL (1956, equation 4.3).

The first order perturbation effect of the bottom topography on ~ is given by the following equation equation with constant coefficients

L g l = - - B g 0 (11)

Consider a cellular form for the bottom topography : b (x, y) : d(mx+ny ). In this cas~ the solution of equation (11) is

~1 = al el[(~+m)x+(v+n)Y]

- - s [t~ 2 + v 2 + nv + mt~] - - imu + int~ a t = - - × a 0 (12)

S s [ ( t ~ + m ) 2 + ( v + u) 2 - i 3 ( v + n ) ] -- 8 ( t L + m ) + ~ ( I - - s z)

Since we imagine that the scale of the bottom topography is smaller than the scale of the moving wind systems we must regard at least one of the quantities m and n as being larger than either t~ or v. The perturbation method works only for ranges of parameters where ~ ~1/.~0 "< 1, or ~ alia o < I. On the other hand, the ratio A of the velocities produced by the bottom topographic perturbation to those produced on the flat-bottom model is of the form

A =-- "a---~l (m + n~ ao \ ~, + ~ 2

and can be considerably greater than unity without violating the conditions of the perturbation method. The quantity A may be called an amplification factor because it expresses the degree to which bottom topography can produce small scale geostrophic motions with greater amplitude than the planetary geostrophic motions of the scale of the wind-system.

By addition of elementary solutions (12) with various combinations of signs for t~, v, m, n, a, it is possible to construct solutions for all kinds of progressive and standing storm shapes and for sub- marine mountains with various configurations of crests. By Fourier superposition more elaborate forms can be obtained, but if the motive is comparison with transient current observed in nature, this model does not merit such elaboration. We are content merely to obtain an order of magnitude

314 Shorter Communicat ion

indication of whether the perturbation method works for the range of scales, etc., o f interest, and roughly what the amplification of velocities is likely to be. Therefore we will restrict our attention to a definite range of scales and frequencies.

Consider a mid-latitude region where f = 10 -4 sec --a and the mean depth H is 4 >: 105 cm. The slope 3 corresponding to the equivalent fl-effect is 10 -gcm -x. Gravity is 10acmsec -2 thus ,~2 4 × l016 cm 2.

The dimensions of storms are in the neighbourhood of 3000 km wavelength both east-west and north-south. Therefore we take tz ~, : 2 x 10 - s c m - L We restrict the range of frequencies to 3 to 30 days (2 × 10 -2 < s < 2 ~.~ 10-1). Furthermore we consider mountains on the bot tom with a range of wavelengths from 6 to 600 km (10-Scm -1 ~ m o r n ~ 1017 cm-1). Within this range the value of the ratio of amplitude of the vertical displacement is

~a I 2i~

] a01 sn

where we can interchange p. and v or m and n. If the quantity should exceed, say, 0.2, the pertur- bation analysis would not be applicable, but this does not happen in the range under consideration. The amplification factor A for the velocities is

,a 1 (m + n I 2i A ~ - - ~ - - ~

- - l a o l \ ~ ' + ~ / s

in the range under consideration. It is noteworthy that there is no strong effect o f the wave-length of the mountains on the amplification factor. The only resonance frequency introduced by the bot tom is at approximately s = 8/m from equation (12) and this is well outside the range of parameters under consideration. Low frequencies are favoured.

Dr. JOHN C. SWALLOW'S measurements of transient currents off Portugal in May-June 1958 are compatible with this result. Take c = 0.20, and the wavelength o f the low topographic elements as 60 km (m and n = l0 -6 cm-1). If the moving wind system occurs at bi-weekly intervals (s = 0.05), the amplification factor is 8, hence the velocities o f the small scale geostrophic motions associated with the bot tom topography are of an order of magnitude greater than those with the scale of the moving wind-system; an estimate o f the latter f rom Figure 2 of V~RONIS and STOMMEL gives deep layer velocities of about 0-1 cm sec -1 so that we could expect geostrophic eddies o f 60 km scale with velocities o f around 1.6 cm sec -1, and a frequency of once every two weeks. Shorter periods are not proportionately amplified. One concludes therefore that the irregularity o f the bot tom topography which actually occurs in the ocean is o f dominant important in fixing the scale and amplitude of the transient barotrophic motions induced by storms.

Woods Hole Oceanographic Institution ALLAN ROmNSON Woods Hole Mass., U.S.A. HENRY STOMMEL

Contribution flora Woods Hole Oceanographic Institution No. 1004.

R E F E R E N C E S

CHARr~E¥ J. G. (1955) T h e gene ra t ion o f ocean cu r r en t s by wind . J . Mar. Res. 14 (4) : 477-498.

STOr~IEL H. (1957) A survey o f o c e a n cu r r en t theory . Deep-Sea Res. 4, 149-184. VERONtS G. a n d STOMMEL H. (1956) T h e ac t ion o f var iab le w i n d stresses on a s t ra t i f ied

ocean . J. Mar. Res. 15 ( l ) : 43-75. SWALLOW J. C. (1958) Pr ivate c o m m u n i c a t i o n .