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Amplification of transient response of the ocean to storms by the effect of bottom topography
SHORTER
COMMUNICATION
Amplification of transient response of the ocean to storms by the effect of bottom topography (Received
13 N o v e m b e r
1958)
VFaONtS a n d STOMMEL (1956) showed that the ocean responds barotrophically to short-period largescale fluctuations in the winds. Since m o t i o n s in such a case extend all the way to the b o t t o m it is natural to enquire whether irregularities in the t o p o g r a p h y o f the b o t t o m m i g h t modify the n a t u r e o f the transient response; but VERONm a n d STOMMEL did not consider this question. The actual ocean b o t t o m is very complicated in form. W e do not propose to try to a p p r o x i m a t e this complexity in the following analysis. Also, it seems certain that irregularity o f the b o t t o m enhances the baroclinic m o d e ' s response (CHARNE¥, 1957), but we consider here merely a barotropic model. It is well known that in a h o m o g e n e o u s model the /J-effect can be a p p r o x i m a t e d by introducing a slope o f the b o t t o m upward in the y-direction. T h e equations which we will use are --
,~t
--Ji ....
--~fu st
-- -
g--
(I)
3x
(2)
g Ty
(3)
~-7 + ~ (hu) + ~ (by) = s (x, >,, t)
where u and v are velocity c o m p o n e n t s , ~ is the displacement o f the free surface, f the Coriolis parameter, a n d S (x, y, t) a driving agency - in this case explicitly a variable m a s s source : for e x a m p l e precipitation. There is no problem in m a k i n g this m a s s source absorb the effects of other current producing m e c h a n i s m s such as n o r m a l pressure, wind-stress, etc. (see STOMMEL, 1957). T h e quantity h is the variable depth of the ocean. A s s u m i n g that the function S is of the form S (x, y ) e ic~t we can eliminate u a n d v f r o m the above equations and obtain
[--
°g
hV 2 t
,,, ~>,
i
,,x~
~- i g f J ( h , )
t c, ( f 2 _ a2)
~ = ( f 2 _ e2) S (x, y)
(4)
where the factor p let has been removed from ~, T h e depth h m a y be written as h =H(I
-- 3),-! , b ( x , y ) )
(5)
This form is convenient because it separates the scale depth H f r o m the small gentle slope 3 H (which we introduce to simulate the fl-effect) a n d the b o t t o m irregularity H ~ b (x, y) where ~ is a perturbation parameter which we define as the fraction o f the total depth occupied by the s u b m a r i n e t o p o g r a p h y , a n d b ( x , y ) is the functional f o r m o f the b o t t o m t o p o g r a p h y with an amplitude o f unity. This substitution m a k e s equation (4) a linear partial differential equation in ~ with n o n - c o n s t a n t coefficients. If wc regard ¢ as a small perturbation n u m b e r (¢ ~. l) a n d assert that the solution is o f the f o r m +~. ! ~ t i . .. we obtain the following equation
312
313
Shorter Communication (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=
--ogH
b V 2 + ~ y , ~ - - - f y + ~ x ~ x j + igfHJ(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(Ux+ vy the solution of (9) is
go = al et(ux+ vy) S o ( 1 - s 2) a ° = f { s A ~ t ~ 2 - t v2-~ i ~ v ) - A 2 ~
A~
gH f2
~ s(I-s~)}
(10)
a 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 Lgl= --Bg0 (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~] - - i m u + int~
at = - s [ ( t ~ + m ) 2 + ( v + u)2 - i 3 ( v
+ n ) ] -- 8 ( t L + m )
S
+ ~(I
× a0
(12)
- - 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 submarine 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 Communication
indication o f whether the perturbation method works for the range o f scales, etc., o f interest, and roughly what the amplification o f velocities is likely to be. Therefore we will restrict our attention to a definite range o f 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 1 0 - g c m -x. Gravity is 1 0 a c m s e c -2 thus ,~2 4 × l016 cm 2. The dimensions of storms are in the neighbourhood o f 3000 km wavelength both east-west and north-south. Therefore we take tz ~, : 2 x 10- s c m - L We restrict the range o f frequencies to 3 to 30 days (2 × 10 -2 < s < 2 ~.~ 10-1). Furthermore we consider mountains on the bottom with a range of wavelengths from 6 to 600 km ( 1 0 - S c m -1 ~ m o r n ~ 1017 cm-1). Within this range the value of the ratio o f amplitude o f the vertical displacement is ~aI
2i~
] a01
sn
where we can interchange p. and v or m and n. If the quantity should exceed, say, 0.2, the perturbation analysis would not be applicable, but this does not happen in the range under consideration. The amplification factor A for the velocities is
A~
,a 1 ( m + n I 2i -~--~
-- laol \~' + ~/
s
in the range under consideration. It is noteworthy that there is no strong effect o f the wave-length o f the mountains on the amplification factor. The only resonance frequency introduced by the bottom is at approximately s = 8/m from equation (12) and this is well outside the range o f parameters under consideration. Low frequencies are favoured. Dr. JOHN C. SWALLOW'S measurements o f 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 bottom topography are o f an order o f magnitude greater than those with the scale o f the moving wind-system; an estimate o f the latter from Figure 2 o f 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 k m scale with velocities o f around 1.6 cm sec -1, and a frequency o f once every two weeks. Shorter periods are not proportionately amplified. One concludes therefore that the irregularity o f the bottom topography which actually occurs in the ocean is o f dominant important in fixing the scale and amplitude o f the transient barotrophic motions induced by storms.
Woods Hole Oceanographic Institution Woods Hole Mass., U.S.A.
ALLAN ROmNSON HENRY STOMMEL
Contribution flora Woods Hole Oceanographic Institution No. 1004. REFERENCES CHARr~E¥ J. 477-498. STOr~IEL H. VERONtS G . o c e a n . J.
G. (1955) T h e g e n e r a t i o n o f o c e a n c u r r e n t s by w i n d .
J. Mar. Res. 14 (4) :
(1957) A survey o f o c e a n c u r r e n t t h e o r y . Deep-Sea Res. 4, 149-184. a n d STOMMEL H . (1956) T h e a c t i o n o f v a r i a b l e w i n d stresses o n a stratified Mar. Res. 15 ( l ) : 43-75. SWALLOW J. C. (1958) Private c o m m u n i c a t i o n .
COMMUNICATION
Amplification of transient response of the ocean to storms by the effect of bottom topography (Received
13 N o v e m b e r
1958)
VFaONtS a n d STOMMEL (1956) showed that the ocean responds barotrophically to short-period largescale fluctuations in the winds. Since m o t i o n s in such a case extend all the way to the b o t t o m it is natural to enquire whether irregularities in the t o p o g r a p h y o f the b o t t o m m i g h t modify the n a t u r e o f the transient response; but VERONm a n d STOMMEL did not consider this question. The actual ocean b o t t o m is very complicated in form. W e do not propose to try to a p p r o x i m a t e this complexity in the following analysis. Also, it seems certain that irregularity o f the b o t t o m enhances the baroclinic m o d e ' s response (CHARNE¥, 1957), but we consider here merely a barotropic model. It is well known that in a h o m o g e n e o u s model the /J-effect can be a p p r o x i m a t e d by introducing a slope o f the b o t t o m upward in the y-direction. T h e equations which we will use are --
,~t
--Ji ....
--~fu st
-- -
g--
(I)
3x
(2)
g Ty
(3)
~-7 + ~ (hu) + ~ (by) = s (x, >,, t)
where u and v are velocity c o m p o n e n t s , ~ is the displacement o f the free surface, f the Coriolis parameter, a n d S (x, y, t) a driving agency - in this case explicitly a variable m a s s source : for e x a m p l e precipitation. There is no problem in m a k i n g this m a s s source absorb the effects of other current producing m e c h a n i s m s such as n o r m a l pressure, wind-stress, etc. (see STOMMEL, 1957). T h e quantity h is the variable depth of the ocean. A s s u m i n g that the function S is of the form S (x, y ) e ic~t we can eliminate u a n d v f r o m the above equations and obtain
[--
°g
hV 2 t
,,, ~>,
i
,,x~
~- i g f J ( h , )
t c, ( f 2 _ a2)
~ = ( f 2 _ e2) S (x, y)
(4)
where the factor p let has been removed from ~, T h e depth h m a y be written as h =H(I
-- 3),-! , b ( x , y ) )
(5)
This form is convenient because it separates the scale depth H f r o m the small gentle slope 3 H (which we introduce to simulate the fl-effect) a n d the b o t t o m irregularity H ~ b (x, y) where ~ is a perturbation parameter which we define as the fraction o f the total depth occupied by the s u b m a r i n e t o p o g r a p h y , a n d b ( x , y ) is the functional f o r m o f the b o t t o m t o p o g r a p h y with an amplitude o f unity. This substitution m a k e s equation (4) a linear partial differential equation in ~ with n o n - c o n s t a n t coefficients. If wc regard ¢ as a small perturbation n u m b e r (¢ ~. l) a n d assert that the solution is o f the f o r m +~. ! ~ t i . .. we obtain the following equation
312
313
Shorter Communication (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=
--ogH
b V 2 + ~ y , ~ - - - f y + ~ x ~ x j + igfHJ(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(Ux+ vy the solution of (9) is
go = al et(ux+ vy) S o ( 1 - s 2) a ° = f { s A ~ t ~ 2 - t v2-~ i ~ v ) - A 2 ~
A~
gH f2
~ s(I-s~)}
(10)
a 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 Lgl= --Bg0 (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~] - - i m u + int~
at = - s [ ( t ~ + m ) 2 + ( v + u)2 - i 3 ( v
+ n ) ] -- 8 ( t L + m )
S
+ ~(I
× a0
(12)
- - 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 submarine 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 Communication
indication o f whether the perturbation method works for the range o f scales, etc., o f interest, and roughly what the amplification o f velocities is likely to be. Therefore we will restrict our attention to a definite range o f 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 1 0 - g c m -x. Gravity is 1 0 a c m s e c -2 thus ,~2 4 × l016 cm 2. The dimensions of storms are in the neighbourhood o f 3000 km wavelength both east-west and north-south. Therefore we take tz ~, : 2 x 10- s c m - L We restrict the range o f frequencies to 3 to 30 days (2 × 10 -2 < s < 2 ~.~ 10-1). Furthermore we consider mountains on the bottom with a range of wavelengths from 6 to 600 km ( 1 0 - S c m -1 ~ m o r n ~ 1017 cm-1). Within this range the value of the ratio o f amplitude o f the vertical displacement is ~aI
2i~
] a01
sn
where we can interchange p. and v or m and n. If the quantity should exceed, say, 0.2, the perturbation analysis would not be applicable, but this does not happen in the range under consideration. The amplification factor A for the velocities is
A~
,a 1 ( m + n I 2i -~--~
-- laol \~' + ~/
s
in the range under consideration. It is noteworthy that there is no strong effect o f the wave-length o f the mountains on the amplification factor. The only resonance frequency introduced by the bottom is at approximately s = 8/m from equation (12) and this is well outside the range o f parameters under consideration. Low frequencies are favoured. Dr. JOHN C. SWALLOW'S measurements o f 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 bottom topography are o f an order o f magnitude greater than those with the scale o f the moving wind-system; an estimate o f the latter from Figure 2 o f 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 k m scale with velocities o f around 1.6 cm sec -1, and a frequency o f once every two weeks. Shorter periods are not proportionately amplified. One concludes therefore that the irregularity o f the bottom topography which actually occurs in the ocean is o f dominant important in fixing the scale and amplitude o f the transient barotrophic motions induced by storms.
Woods Hole Oceanographic Institution Woods Hole Mass., U.S.A.
ALLAN ROmNSON HENRY STOMMEL
Contribution flora Woods Hole Oceanographic Institution No. 1004. REFERENCES CHARr~E¥ J. 477-498. STOr~IEL H. VERONtS G . o c e a n . J.
G. (1955) T h e g e n e r a t i o n o f o c e a n c u r r e n t s by w i n d .
J. Mar. Res. 14 (4) :
(1957) A survey o f o c e a n c u r r e n t t h e o r y . Deep-Sea Res. 4, 149-184. a n d STOMMEL H . (1956) T h e a c t i o n o f v a r i a b l e w i n d stresses o n a stratified Mar. Res. 15 ( l ) : 43-75. SWALLOW J. C. (1958) Private c o m m u n i c a t i o n .