A divergent quasi-geostrophic model for wind-driven oceanic fluctuations in a closed basin

Dytlaknics of Atmospheres and Oceans, 14 (1990) 259-277

259

Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

A DIVERGENT Q U A S I - G E O S T R O P H I C M O D E L FOR WIND-DRIVEN OCEANIC FLUCTUATIONS IN A CLOSED BASIN

STEFANO PIERINI *

Center for Meteorology and Physical Oceanography, Massachusetts Institute of Technology, Cambridge, MA 02139 (U.S.A.)

ABSTRACT Pierini, S., 1990. A divergent quasi-geostrophic model for wind-driven oceanic fluctuations in a closed basin. Dyn. Atmos. Oceans, 14:259-277. The oceanic current and sea-level fluctuations driven by the fluctuating component of a wind stress field are analyzed by considering a linear, deterministic, barotropic, quasi-geostrophic model on the fl plane in a circular domain. Divergent and non-divergent forced solutions are obtained analytically and their structure in different frequency ranges is discussed. For parameter values roughly representative of the Mediterranean Sea, divergent oscillations with a clear boundary layer character are found in the range T < O(1 month) (i.e. below the fundamental eigenperiod of the system), while Rossby wave-like behavior is observed for higher forcing periods. Moreover, the comparison between divergent and non-divergent solutions reveals the inadequacy of the rigid-lid approximation for frequencies larger than the highest eigenfrequency, even in cases in which the divergence parameter is much less than unity.

1. INTRODUCTION

The fluctuating component of a large-scale wind stress field, together with internal oceanic mechanisms, is a source of energy for mesoscale oceanic variability. The oceanic response to a time-dependent wind at subinertial frequencies in mid-latitudes typically takes the form of Rossby, coastally and topographically trapped ways (e.g., see Philander (1978) and Gill (1982) for a review) that can also feed a mesoscale eddy field. Both deterministic (e.g., Pedlosky, 1965, 1967; Phillips, 1966; Flied, 1977; Leetma, 1978; Harrison, 1979) and stochastic (WiUebrand et al., 1980; Muller and Fran* Present address: Istituto di Oceanologia, Istituto Universitario Navale. Via Acton, 38-80133 Napoli, Italy. 0377-0265/90/$03.50

© 1990 Elsevier Science Publishers B.V.

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s. PIERINI

kignoul, 1981) models have been proposed for wind-driven oceanic fluctuations in open oceans and closed basins. Miller et al. (1987) discuss the importance of internally generated fluctuations in comparison to wind-driven fluctuations. All the literature devoted to the subject deals either with open ocean basins or with closed basins on the scale of large oceans. In this study we are concerned with the western and eastern Mediterranean basins, which are 'oceans' of very limited extent. Obviously the underlying dynamics is unchanged, but particular attention should be given to the fact that, owing to the small size of the basins, the role the coasts play in shaping the forced response is expected to be particularly significant. The aim of this investigation is to present a process study of wind-induced barotropic, quasi-geostrophic fluctuations in a simple closed domain in which particular attention is devoted to the boundary conditions that, in a study of this nature, can prove to be critical. Pedlosky (1965, 1967) has studied the linear response to a fluctuating wind stress in a square domain analytically, along with the corresponding rectified circulation, in a non-divergent quasi-geostrophic model on the t-plane, while Flierl (1977) has obtained the linear forced response in a circle on the f-plane in a divergent model with appropriate boundary conditions. In this study we find an analytical solution to the linear forced problem in a circular basin on the t-plane with divergence (section 2). Thus the present solution yields a Rossby wave-like structure and resonances, as in Pedlosky's studies (it is to be observed, however, that here the resonances cannot predict the amplitude of the response because no limiting mechanisms have been taken into account) and also models correctly the strong oscillations near the coasts when the forcing frequency is sufficiently large, as in Flierl's study (section 3). Hence this solution makes it possible to pass in a continuous way from one extreme case to the other. Some qualitative information on the character of the forced response in the Mediterranean Sea may also be drawn. Moreover, the non-divergent problem has also been solved (section 4) and a comparison between divergent and non-divergent solutions is presented (section 5). The two solutions are found to be substantially different for frequencies larger than the highest eigenfrequency of the system, despite the small value of the divergence parameter ( O ( 1 0 - 1 ) ) in the example considered. A criterion making it possible to determine when the non-divergent approximation is acceptable is then proposed and checked in two significant examples. 2. FORCED DIVERGENT ROSSBY WAVES IN A CIRCULAR DOMAIN

The current fluctuations induced in the sea by a large-scale fluctuating wind stress with forcing periods fifo1 << T < O < 100 days) are depth-inde-

QUASI-GEOSTROPHIC M O D E L FOR OCEANIC F L U C T U A T I O N S

261

pendent and essentially linear (Philander, 1978). Their interaction with the steady circulation driven by the mean field is assumed negligible. Therefore we shall consider the linearized, forced quasi-geostrophic equations for a barotropic fluid with free surface for the fluctuating streamfunction ~b(x, t)(x = (x, y))

Ot

L~

+(fl+dY)~/x-dx~y =F

F(x, t) = curlz~/pD

(1)

f0 g where u is the velocity, f the free surface elevation, f = fo + flY the Coriolis parameter, d(x) the bottom topography, ~- the wind stress, O the water density, D the water depth, g the acceleration due to gravity and L = (gD)l/2/f o the external Rossby deformation radius. It is to be observed that eqn. (1) also describes the quasi-geostrophic baroclinic motion in a continuously stratified ocean, provided that L is the internal Rossby deformation radius of a given vertical mode. Hence, the results discussed in this study are also applicable to the baroclinic response, which is forced by lower frequency forcings. The free-slip condition requires that the solid boundary of a closed domain E coincides with a streamline

+Joe = C(t)

(2)

where c(t) is an unknown function of time. The system eqns. (1) and (2) requires one further condition in order to be well posed, namely the conservation of mass d ~--;t~ b d × dY = 0 which together with eqn. (1) gives

d ~ E v + . n ds= £ F d × dy

(3)

The free version of this condition ( F = 0) was first proposed by Larichev (1974) and (3) was discussed by McWilliams (1977) and Flierl (1977).

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Let us consider a circular basin E of radius R and constant d e p t h D ( d = 0) and search for the Fourier c o m p o n e n t of frequency f~ of the forced solution ~(X, t) = ~(X) e -mr ( F ( x , t) = if(x) e -i~t) With these positions eqns. (1-3) reduce to

i

i

V 2c~ - d#/L 2 + -~fl¢~ = - - f f

(4a)

~ l a e = c = const

(4b)

~ev~.nds=-~fEffdxdy

(4c)

In order to solve eqn. (4) we write ~ as the sum (I) ~-- (I)(1) -.[.- (I)(2)

where ~(1) is the solution of the forced equation with h o m o g e n e o u s b o u n d a r y conditions ~ V(I )(1). n ds = 0

(5)

E

and ¢(2) is the solution of the h o m o g e n e o u s equation

i

V2(~ (:) - ~(:)/L2 + ~ f l ~ 2 ) = 0

(6)

with n o n - h o m o g e n e o u s b o u n d a r y conditions. The solution ~(1) is easily found when wind stress curl is constant L2 (7)

(I)(1) = - - P

The second term ~(2) can be obtained as follows: Larichev (1974) has shown that the function oo F(x, fl) = ~'.

2

m=O am

K= B/2a X2= K 2 _ L - 2

a, = { 2 , 1,

m=O m>~l

im

Jm(KR)

Jm()~r) cos m 8 e -ix'c°s 0

Jm(XR)

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QUASI-GEOSTROPHIC MODEL FOR OCEANIC F L U C T U A T I O N S

(where r and O are polar coordinates and J,,, is the Bessel function of the first kind) is the solution of eqn. (6) and satisfies the boundary condition (eqn. (4b)) with c = 1. It should be noted that c is arbitrary in the homogeneous problem, while in the present forced case it can be determined by imposing condition (4c). Substituting ~(2)= cf(x, ~) in eqn. (4c), one obtains ~E ~71~(2)

oo im J~(KR)J,~(XR) • u ds = 2 c X R m=O y'~ a,,, J,,,(~.R)

•

fo 2~

cos

mOe -iK"c°s°

dO

i

=

--,trRZff f~

Now, taking the relation 1 2rr J m ( z ) = 2¢r(_i)mf0 cosm0e-i~c°S°d0 into account we finally obtain C=

iPR

4ftXH(~) = Z m=0

j2(KR) a,,,

J'(XR) J,,,(•R)

The complete solution will thus read as follows qJ(x, t ) =

iffL 2 f~

R

f(x, f~)

4XL 2 H(f~)

l 1j e -ia'

(8)

where the real part of the fight-hand side is implied. We note that in the problem without forcing, • is given by ~F=0 = f ( x , f~u)=fu(x)

(9a)

H(flu) = 0

(9b)

or by any linear combination of fn- Condition (9b) ensures mass conservation and also represents the dispersion relation of the system. The function f gives a vanishing integral over the basin only for those values of f~ that correspond to the eigenfrequencies. Knowledge of the normal modes allows one to solve the free initial boundary-value problem thanks to the orthogonality and completeness of the set (fn } and to the reality of the ~2n (see Pedlosky (1987, section 3.25) for details). In the forced problem, after

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s. PIERINI

an initial transient period following the onset fo the atmospheric forcing, the solution reads OD

xIt(Xlt ) = (I)f(X) e-iflt + E n=0

au(t)fu(x)

e-in"t

where ~f is the forced solution. In this case, however, the slowly varying functions of time an(t ) , dependent on the initial condition, decay because of bottom friction. Returning to the forced solution (eqn. (8)) we note that ~k yields resonances for the eigenfrequencies of the free case, as natural. Moreover the spatial structure of ~ will resemble that of the mode for which fin is closer to the forcing frequency fL Clearly, since no limiting mechanisms, such as bottom friction and non-linearities, have been taken into account, solution (8) is physically acceptable only for fl sufficiently far from an f~n. In the next section, numerical computations of eqn. (8) will be discussed and some conclusions on the structure of wind-induced current fluctuations of the sort to be expected in the Mediterranean Sea will be drawn. 3. DISCUSSION OF THE SOLUTION

In this section we will study solution (8) by choosing parameter values that are roughly representative of the two Mediterranean basins. Obviously, an idealized model, such as that described in the previous section, cannot give any quantitative information on wind-induced current fluctuations in a situation with real coasts, bottom topography and wind stress; nevertheless some general properties will be derived that are likely to hold even in more realistic situations, so that it may be possible to draw some qualitative conclusions for the Mediterranean Sea. Let us choose D = 2000 m and R = 500 km, f0 -- 10-4 rad s -1, fl = 1.6 X 10 -11 rad m -1 s -1 (L = 1400 km). Table I shows the first 10 eigenperiods given by eqn. (9b) ( H and f have been computed retaining the first 12 terms in the sum). Figures 1 and 2 give the contours of constant elevation and velocity at t = 0, 7"/8, T / 4 for two different forcing periods: the first, T = 10 days, smaller than the lowest eigenperiod TO and the second TO< T --50 days < Tr The wind stress was taken as "r/pD = A y i (i is the unit vector in the x direction, A is a constant), and A was chosen to as to give [~-(y = + R ) [ = 1 dyn cm -2. The behavior in the two cases is quite different. For T = 10 days (case (a)) the energy is mainly confined to the vicinity of the boundary and its spatial density decreases rapidly with decreasing r (see Fig. lc, where the m a x i m u m amplitude is reached). Wave-like behavior can only be observed in the central region of the basin, but the energy associated with it is virtually

QUASI-GEOSTROPHIC

MODEL

FOR OCEANIC

FLUCTUATIONS

265 e..

r.,

~E Ec~ O

,"

z~

U

~E ~E

A

/ E ~ c~

I

e...

~..=o

~

.=_

@

Jjf(((oJJ}ll0

Fig. 2. As in Fig. 1 but with T = 50 days. Elevations are in m m a n d velocities in 10 - 2 c m s - l .

i

i

I'O O~

QUASI-GEOSTROPHIC MODEL FOR OCEANIC FLUCTUATIONS

267

TABLE I Eigenpedods of the free surface problem com~oi~l~d with the values of section 3 (Kn - - f l / 2 (days)

n

Rk n

T.

0 1 2 3 4 5 6 7 8 9

2.42 3.84 5.i5 5.53 6.39 "/,0"2 7.59 8.42 8.66 8.78

44.0 69.8 93.6 100.5 116.1 127.6 138.0 153.1 157.4 159.6

negligible. Such an elevation and current patterns with a boundary layer structure can be explained as follows: T is so much less than TO that the wave-like character typical of the normal modes, and due to the carrier wave e x p - i(kx + i~t) modulating the field, is only partly present in the forced solution. One would rather expect a pattern more similar to the solution of the same problem on the f-plane, for which the/3 effect is negligible. Indeed, when k = 0, )~ = i / L and taking into account some properties of the Bessel functions J, and I., eqn. (8) reduces to

AL2[I(),0,rJ,, ]

~P= i---if- -2 L

II(R/L )

1 e -iu'

(10)

which is the solution found by Flierl (1977) by neglecting the t-effect altogether. The value ~2 = 0 corresponds to the threshold period T ' = 6.5 days, therefore for T < T ' the solution loses any wave structure and gives a standing oscillation close to eqn. (10) (the steadiness of the oscillation is a consequence of the particular type of forcing--for a travelling weather system the forced solution would 'follow' the f o r c i n g - - b u t the boundary layer character would still be present). However, for any T < O(1 month) basically the same behavior is found. For T = 50 days (case (b)), however, the elevation and velocity patterns travel westward in a normal mode-like propagation (Fig. 2). Solutions of this kind are observed for any T >~O(To). In this case the response in the interior of the basin is significantly larger than that for T = 10 days (by a factor O(10) in the elevation), while it is only moderately larger along the coast (by a factor of two), where, therefore, the sea-surface fluctuations are of comparable amplitude in the two cases.

268

s. PIERINI

Note that in the Atlantic Ocean typical values of D = 5000 m and R = 2000 km (L = 2200 km) give T ' - 4 days and TO= 11 days so that a forced response like that of case (a) is possible only in a narrow, very high-frequency range (of a few days). Most of the wind-induced barotropic energy would take the form of a Fourier combination of solutions like that of case (b). In fact this is what Willebrand et al. (1980) observed by studying the barotropic response to a broadband stochastic wind stress: an analysis of the numerical signal revealed a strong coherence between current and wind stress only for T < 10 days, while no significant coherence was found for greater periods, being destroyed (also) by the wave structure of the forced response. For the Mediterranean Sea, however, the discussion presented above suggests that in the range fifoI << T < 0(1 month) the wind-induced current and sea-level fluctuations will tend to be confined to a boundary layer, with very little Rossby propagation (but they can propagate with the forcing if the latter is travelling). It is to be noted that in this frequency range a peak in the wind stress spectrum is recorded at around T = 10 days in the central Mediterranean Sea (Grancini and Michelato, 1987). 4. N O N - D I V E R G E N T

SOLUTION

In most of the studies on free and forced quasi-geostrophic equations in a closed domain, the divergent term - 6 / L 2 is neglected. This omission brings about the so-called 'rigid-lid' approximation for which the streamfunction is no longer proportional to the surface elevation (being set equal to zero) and is therefore defined within a constant. Thus it is possible to impose a simpler boundary condition so that problem (4) reduces to i i V2# + _ 8 ~ x = _ f f o)

~)IoE = 0

(lla)

6o

(llb)

System (11) is much simpler than system (4), especially because of the absence of a condition such as eqn. (4c), which is usually difficult to impose, even in numerical models. However, Larichev (1974) pointed out that the rigid-lid approximation cannot model the oscillations close to shore connected with the presence of excited Rossby modes within the basin. In fact, using eqn. (4) instead of eqn. (11) would allow one to infer the presence of a Rossby wave by analyzing long time series taken from coastal wave recorders. Analogous conclusions were drawn by Flierl (1977) for multiply connected domains.

QUASI-(3EOSTROPHIC MODEL FOR OCEANIC FLUCTUATIONS

269

We have therefore solved eqn. (11) and compared the non-divergent solution with eqn. (8) in order to assess the importance of the divergent effects (i.e., of the free surface in the barotropic case) in the present forced problem. Even for the parameter values of section 3, for which the divergence parameter ~ = R 2 / L 2 (measuring the relative importance between - ~ k / L 2 and vz~k, is only ~m0.13, the two results are very different for T < T0, both quantitatively and qualitatively, so that any quasi-geostrophic model used to resolve wind-driven oscillations in the Mediterranean Sea at forcing periods T < O(1 month) should include the divergent term, along with condition (4c). This will be discussed in the following section. The solution to eqn. (11) can be determined as follows. Let G(x, x') be the Green's function of the system i 1 v 2 G + -/3Cx = - a ( x - x') ~o ¢o GI~E=0

(12)

then the solution will be given by , ( x ) = ifEG(x, x ' ) F ( x ' ) d x ' d y '

(13)

G can be projected on to the set of the non-divergent normal modes ~,, satisfying eqn. (11) with i f = 0 and w = 0~, - - t h e (real) eigenfrequencies of the system G(x, x') = • Au(x'),~u(x )

(14)

u=0

Now, substituting eqn. (14) into eqn. (12) and taking into account the orthogonality relation

one obtains G(x,x')= u=o

-

,o)

(15)

Substituting (15) into (13) one gets the final result

Du

~uoudiv(X,t ) = i u=0ECu(OJu_O)) ~u(X) e-iwt (16)

D u = rE,/,* (x')ff(x') d x ' d y '

270

S. PI E RI N I

TABLE II Eigenperiods of the non-divergent problem n

Rk n

T. (days)

(0, 1) (1, 1) (2, 1) (0, 2) (3, 1) (1, 2) (4, 1) (2, 2)

2.40 3.83 5.13 5.52 6.38 7.01 7.59 8.41

43.7 69.6 93.4 100.3 116.0 127.5 137.9 153.0

(0, 3) (5, 1)

8.65 8.77

157.3 159.4

The non-divergent normal modes in a circular basin have been calculated by Longuet-Higgins (1964) q~u(x) - q~(m~)(x) = Jm(K,,,,r) e i(+rnO-K..,x) where the eigenvalues K,,,t = fl/2o~,,,t are given by

J,, ( KmtR ) = 0 In eqn. (16) the sum is over any n = (m, l, _). The first 10 eigenperiods are given in Table II. As one can see they are very close to those of the divergent modes (Table I). However, this does not necessarily imply that the solutions are close, not even for small/~ values, as we will see in the next section. 5. COMPARISON BETWEEN DIVERGENT A N D N O N - D I V E R G E N T SOLUTIONS

As noted above, problem (4) can be used to explain tide gauge records taken close to shore, as far as Rossby waves in a closed basin are concerned, while problem (11) is useless in this respect. However, a question arises: do forcing frequencies exist for which the non-divergent solution approximates the divergent solution in a large internal portion of the domain? The answer to this question would be helpful in studying realistic cases by means of numerical models because one would know in advance when and if eqn. (11) can be used instead of the much more complex eqn. (4). The answer can be found simply by analyzing the divergent solution (eqn. (8)). Let us restrict ourselves to small basins like those of the Mediterranean Sea, for which/.t << 1. The fact that ~ << 1 can erroneously lead us to believe that eqns. (4) and (11) have close solutions because the term - q , / L 2 is negligible in the potential vorticity. However, the boundary conditions may be substantially

271

Q U A S I - G E O S T R O P H I C M O D E L FOR OCEANI C F L U C T U A T I O N S

~b

50

IT 0

100

(clayu) Fig. 3. Surface elevation amplitude at the boundary (r =1) versus forcing period, derived from formula (8) with a wind stress as in section 3.

different for eqns. (4) and (11), even if /~ << 1. Two cases can be distinguished. (1) For forcing periods T for which the amplitude of the free surface along the coast adjusts to a constant c (in eqn. (4b)) that is comparable to a typical amplitude in the interior of the domain, the two b o u n d a r y conditions are essentially different. Therefore, divergent and non-divergent solutions m a y be different, even far from the boundary. (2) For forcing periods T for which c is m u c h less than a typical amplitude scale in the interior of the domain, conditions (4b) and ( l l b ) are basically equivalent, therefore the two solutions are expected to be close to each other sufficiently far from the boundary. This can be expressed quantitatively by introducing a function b(T) defined as follows

b(T) = I ~ l a ~ r l / I max ~ ( x ) l E

(17)

where tb is given by eqn. (8). With this notation, case (1) corresponds to b(T) = O(1), while for case (2), b(T) << 1. Figure 3 shows the elevation amplitude along the b o u n d a r y as a function of the forcing period and Fig. 4 gives the function b(T). It appears clear that case (1) corresponds to periods smaller than the lowest eigenperiod To, while the response at higher frequencies falls into case (2). Let us now show that this criterion proves correct in the two cases considered in section 3, for which/~ --- 0.13.

272

s. PIERINX 1~7') 1

.5

i

J'/'

0

100

(days)

Fig. 4. Ratio between amplitude at the boundary and maximum response in the basin (as defined in eqn. (17)) versus forcing period, derived from formula (8).

For T = 10 days, b = 1 (i.e. the amplitude at the boundary has the largest value in the whole domain), hence the non-divergent solution is expected to be incorrect. This can be checked by comparing eqn. (8) for t = T/4 (Fig. lc) with eqn. (16) for t --- 7"/4 (Fig. 5a: the first 10 modes reported in Table II have been considered in the sum). Compare also Fig. 6a with Fig. 6b, in which the surface elevation along y = 0 (the circle is centered at the origin of the axis) is given for t - 0, 7"/8 and T/4 for the divergent and non-divergent cases, respectively. The difference is striking for any x value. In particular, for t --- 7"/8 and T/4, aJ/~x (and so the longshore velocity) has a different sign near the boundary in the two cases. In the absence of a length scale imposed by the wind forcing (being basinwide) or by the form of an eigenfunction (being T < To), the typical scale of the response would be L if it was L << R. However, since/x << 1, the length scale is forced to be R, the basin radius. This explains why a large difference between divergent and non-divergent responses near the coastline is felt basinwide. For T = 50 days, however, b --- 0.1, thus the two solutions are expected to be dose. This can be checked by comparing Fig. 2c with Fig. 5b, and Fig. 6c with Fig. 6d. The agreement is sufficiently good both qualitatively and quantitatively. A hybrid model has also been considered in previous studies (LonguetHiggins, 1965; Flierl, 1977) in which free surface effects are taken into account, but for which condition (11b) is imposed instead of (4b). This is not physically justified because mass is not conserved in this case, nevertheless such a model on the fl-plane is able to reproduce nearly correct

273

QUASI-GEOSTROPHIC MODEL FOR OCEANIC FLUCTUATIONS

b

Fig. 5. Non-divergent fields computed using formula (16) at t = T/4 (note that here f is still defined as in eqn. (1) but does not represent an elevation, the rigid-lid approximation being assumed) for T= 10 days (a) (compare with the corresponding divergent response, Fig. lc) and for T= 50 days (b) (compare with Fig. 2c). ~" is in mm and U is in 10 -2 cm s -1.

eigenfrequencies either for/.t << 1 and for ~ >> 1 (Flied, 1977). For/.t >> the correct divergent and 'hybrid' forced solutions are found to be very different in the f-plane. F o r # << 1, however, Flied (1977) suggests that the two solutions are nearly the same. Now, for small divergence the hybrid model reduces to eqn. (11), moreover for T < TO the fl effect is virtually negligible, so that solutions (8) and (16) will, with these restriction, be very close to those respectively considered b y Flied in the f plane. Yet, the difference between eqns. (8) and (16) is substantial in this frequency range (as dis-

274

S. PIERINI

cussed above for T= 10 days), although it may be smaller than that for p S- 1. This is to emphasize once more that a non-divergent treatment (or a hybrid one) never models the correct forced response accurately for high frequencies (0 > Qt,), not even when p -SE1. 6. CONCLUSIONS

The solution of problem (4) given by eqn. (8) describes the linear, inviscid response of a homogeneous layer of fluid with free surf~e in a circular domain to wind fluctuations in the range f;;’ * T -Z 6(100 days). In this

5

(k%

-500

500

I.

(k:)

Fig. 6. Sections of [ taken at y = 0 of Fig. l(a-c) corresponding non-divergent fields are given.

(a) and of Fig. 2(a-c) (c). In (b) and (d) the

QUASI-GEOSTROPH1C MODEL FOR OCEANIC FLUCTUATIONS

275

100

o

\ /

-100

(k~)

-500

100

~wmm

....

C

o

--100 -500

0 (kin)

500

Fig. 6 (continued).

study, unlike most of the previous quasi-geostrophic studies, full account has been taken of the oscillations near the boundary by virtue of condition (4c). This is found to be more important for forcing periods T smaller than the lowest eigenperiod To, for which the non-divergent solution fails to model the field accurately even far from the boundary, even for small values of the divergence parameter. For higher forcing periods, however, a non-divergent treatment can be acceptable. In the first (high) frequency range the solution has virtually no Rossby wave-like behavior but rather a boundary layer structure. In the second (low) frequency range the solution yields a Rossby mode-like propagation.

276

s. PIERINI

While the high-frequency range for a large basin like the Atlantic Ocean is limited to periods T < O(10 days), for smaller basins like the Mediterranean Sea it extends up to T = O(1 month), which includes a peak in the wind stress spectrum. This implies a qualitative difference in the barotropic response to wind stress fluctuations between the Atlantic Ocean and the Mediterranean Sea. The problem of wind-driven fluctuations in a realistic situation should include several effects that have been neglected here: real coasts and b o t t o m topography that affect the form of the normal modes and the value of the eigenfrequencies (e.g., Ripa, 1978; Miller, 1986), non-linearities that can produce rectified m e a n flows (e.g., Pedlosky, 1965; Veronis, 1966; Haidvogel and Rhines, 1983) and an energy cascade from the baroclinic modes if stratification is considered (e.g., Treguier and Hua, 1987), interaction between fluctuations and m e a n flow, frictional effects, etc. However, for sufficiently high frequencies the response is usually linear and barotropic so that the qualitative information concerning the spatial distribution of energy in the two frequency ranges obtained in the present simple-process study could also be c o m m o n to more general cases. It is in this spirit that we have used parameter values roughly representative of a real basin (the Mediterranean Sea) in the examples of sections 3-5. More complex process models and sophisticated eddy-resolving general circulation models will be required to describe wind-induced mesoscale variabilities in the Mediterranean Sea. ACKNOWLEDGMENTS I should like to thank G.R. Flierl for useful discussions on the subject. This work was supported by F O R M E Z (Centro di Formazione e Studi per il Mezzogiorno, Italy), and was carried out at the D e p a r t m e n t of Earth, Atmospheric and Planetary Sciences of the Massachusetts Institute of Technology, the use of whose facilities is gratefully acknowledged. REFERENCES Flierl, G.R., 1977. Simple applications of McWilliams, "A note on a consistent quasigeostrophic model in a multiply connected domain". Dyn. Atmos. Oceans, 1: 443-453. Gill, A.E., 1982. Atmosphere-Ocean Dynamics. Academic Press, New York. Grancini, G.F. and Michelato, A., 1987. Current structure and variability in the strait of Sicily and adjacent area. Ann. Geophys., 5B: 75-88. Haidvogel, D.B. and Rhines, P.B., 1983. Waves and circulation driven by oscillatory winds in an idealized ocean basin. Geophys. Astrophys. Fluid Dyn., 25: 1-63. Harrison, D.E., 1979. On the equilibrium linear basin response to fluctuating winds and mesoscale motions in the ocean. J. Geophys. Res., 84: 1221-1224.

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