JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 98, NO. C2, PAGES 2543-2554, FEBRUARY 15, 1993

Trajectories in Gulf Stream Meanders

BENOIT CUSHMAN-ROISIN

Thayer School of Engineering, Dartmouth College, Hanover, New Hampshire

A simple model based on conservation of stream function and potential vorticity is used to derive theoretical estimates for the horizontal and vertical displacements of fluid particles in Gulf Stream meanders. These displacements are given as functions of the meander's curvature and meridional displacement. The theory also provides an estimate of the translation speed of the meander. Finally, comparisons with RAFOS float trajectories and observational estimates of the components of the potential-vorticity budget validate the theory.

1. INTRODUCTION

The discovery of the Trade Winds and the Gulf Stream quickly followed the discovery of the American continent about 500 years ago. These intense and persistent fluid motions, two of the most notable manifestations of geophys- ical flows, contributed immensely to the European explora- tion and colonization of the New World, the former by providing a means to reach the new destination, the latter a way back. Since Benjamin Franklin's first chart of the Gulf Stream in 1876, numerous conjectures and theories have been proposed as explanations for this major current; yet, it was not until the middle of the century that a first correct theory was formulated [Stommel, 1948]. Almost as old as the observation of the Stream itself are reports of its incessant meandering behavior, especially downstream from Cape Hatteras. Jonathan Williams, grandnephew of Benjamin Franklin, is credited with the first mention of what appears to have been a warm-core meander or ring [Williams, 1793; Richardson, 1983]. And, just as for the existence of the Stream, theories for its meandering behavior have been advanced only relatively recently. Thanks to the advent of abundant observations and of numerical models, meanders have now been well documented [Halliwell and Mooers, 1983; Tracey and Watts, 1986] and explained as a combina- tion of barotropic and baroclinic instabilities [Schmitz and Holland, 1982; Robinson et al., 1988].

Nevertheless, our understanding of the meandering pro- cess is still fragmentary. For example, very little attention has been paid to the motions of individual fluid parcels in the time-dependent jet. This lack of attention perhaps finds its root in the synoptic nature of cruise surveys and satellite images, and in the Eulerian character of most numerical models. It lies also perhaps in a tacit assumption that, because flowing particles define the jet, a meandering of the jet as a whole must necessarily translate into a similar meandering of particle trajectories. In other words, the path of the stream is easily taken as representative of particle trajectories. Recent observations, however, show that RAFOS floats, man-made surrogate water parcels, undergo large horizontal and vertical excursions as they enter and leave meanders [Bower and Rossby, 1989]. Such observa-

Copyright 1993 by the American Geophysical Union.

Paper number 92JC02059. 0148-0227/93/92JC-02059505.00

tions not only emphasize the vertical dimension of fluid trajectories, a facet too easily neglected in the analysis of traditional data, but also strongly challenge the assumption that jet-path and particle-trajectory meandering are essen- tially the same. The presence of significant lateral displace- ments, on the order of the width of the jet, even raises a semantic question: if, by definition, the jet consists of flowing particles, how can particles move across the jet? Obviously, the answer to this question lies in the fact that the jet is not always composed of the same particles; in other words, the particle with the maximum velocity at one location may slow down to become further downstream one particle on the flank, leaving to another particle the property of maximum speed. This being the case, it becomes impor- tant to distinguish clearly between the jet axis (e.g., the line joining at some instant the particles of maximum speed) and particle trajectories, and, if particles are observed to deviate substantially from the jet axis, as it is the case, then trajectories and jet paths are to be determined separately.

The purpose of the present study is to develop a simple conceptual model that permits us to discriminate clearly between jet axis and particle trajectories and to evaluate from simple formulas the lateral displacements of fluid particles from the jet axis in terms of the meander curvature and meridional amplitude. The theoretical developments also permit the estimation of the accompanying vertical excursions (upwelling in rightward curvature and down- welling in leftward curvature) and provide the basis for a close examination of the potential-vorticity budget. The theoretical predictions can then be confronted to estimates of the potential-vorticity budget components derived from observations [Bower, 1989].

The key theoretical premise is conservation of potential vorticity by individual fluid particles as they flow from a straight portion of the stream into a meander of given curvature and meridional amplitude. As it turns out, the statement of conservation of potential vorticity and stream function leads to a complete solution of the problem.

Two versions of the model are developed and solved. First, a one-layer version is derived to exemplify the ap- proach and to establish a procedure for solution. With the experience thus gained, an improved two-layer version of the model is then solved, the results of which are compared to observations. Both models rely on the quasi-geostrophic approximation, leaving extensions to primitive-equations physics for future investigation.

2543

2544 CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS

2. ONE-LAYER QUASI-GEOSTROPHIC MODEL

The Model

Let us consider a one-layer, reduced-gravity model as an idealization of the surface-intensified Gulf Stream and let us, moreover, restrict our attention to quasi-geostrophic dynam- ics. The governing equation is

Oq --+ J(½, q) - O, (1) Ot

where • is the stream function, proportional to the pressure in the upper, moving layer, and q is the potential vorticity of a water column in this layer, defined as

1

q = V 2 q• _ R-•d q• + fly. (2) In this expression, Rd = (#'H)i/2/fo, is the deformation radius constructed from the reduced gravity #', the average layer depth H, and the mean Coriolis parameter fo, while/3 = df/dy is the meridional gradient of the Coriolis parameter. In such model, a zonal current like a mean Guff Stream is represented by a stream function that decreases northward between two far-field, uniform values; the meridional gradi- ent -aq,/ay then provides the associated zonal current profile. Because the jet width is typically of the order of the deformation radius, the variation in stream function occurs on a scale on the order of R d. Meanders can be incorporated by adding a zonal variation to the stream function ½, provided that the generalized field meets (1)-(2).

Of particular interest to us in this simple theory are steadily translating meanders. Although such meanders may not strictly exist because of dispersion effects, instabilities and Rossby wave radiation, we nonetheless assume that a steadily translating solution is a valid meander representa- tion during the interval of time taken by a particle to negotiate one crest or one trough (a few days for a 400-kin- long meander and a particle speed exceeding 1 m s-I). A solution of type ½(x - ct, y), q(x - ct, y), where c is the eastward speed of meander propagation, transforms (1) into

J(q• + cy, q) = O. (3)

The modified stream function q• 4- cy, sometimes called the translating stream function [Flierl, 1981], is akin to the Bernoulli function. Indeed, under steady conditions, a Ber- noulli function is conserved by individual fluid particles, and, in the context of the quasi-geostrophic approximations, it can be shown that the stream function is the Bernoulli

function. For steadily translating flow patterns, (3) suggests that ½ 4- cy is the corresponding Bernoulli function. For simplicity, however, we shall call it the translating stream function, noted •:

•= qt + cy. (4)

It is well known that, in inviscid models like the present one, individual fluid particles conserve their potential vor- ticity q. Equation (3) can be regarded as a requirement that potential-vorticity lines must coincide with streamlines, which are particle trajectories. Therefore, in our model, fluid parcels conserve both their stream function defined in (4) and their potential vorticity, defined in (2). As we will see

trough

Fig. 1. Sketch of a meandering jet and some of the attendant notation. Two sections are considered, one in a straight portion of the jet, where the curvature is nil, and the other somewhere along a meander, where the curvature is 1/R and the northward displace- ment is Y. Both R and Y are much larger than the width L of the jet.

this tandem-conservation principle will allow us to deter- mine particle properties inside a meander of given curvature and meridional amplitude as functions of their properties in a straight portion of the jet.

Implementation of the Conservation Principles

Let us consider two separate sections of the jet, one between a meander crest and trough where the curvature is nil, heretofore referred to as the straight portion, and the other arbitrary, for example, somewhere in a crest where the curvature is clockwise. At each section, the local structure of the jet can be represented by a lateral profile of the stream function in terms of a cross-jet variable, ½(a) and ½(n), respectively (Figure 1). Because the jet is likely to undergo structural distortion in a meander, the function ½ in the meander is expected to differ from its analog ½, in the straight portion of the jet. We also distinguish between the cross-stream variables, a and n, so that these can be tagged with the particles' the particle at cross-section distance r7 in the straight portion of the jet is at cross-section distance n in the meander. At this point, there is some ambiguity regard- ing the origin of the • and n axes, since we have not yet defined precisely the jet center. A choice dictated by math- ematical convenience will be made later. We choose to

measure • and n positively to the left of stream, that is, in the general northward direction.

Since • and n are the cross-stream coordinates of the same

particle in both sections, the conservation principles simply require

ß (n) = X•(r7) (5)

q(n) = q(n), (6)

for the translating stream function and potential vorticity, respectively. The overbars denote variables in the straight portion of the jet, where the curvature is nil.

As an example of jet structure, we can take a hyperbolic stream function profile.

qt (n) = - UL tanh - (7) L'

representing a jet of width L (of the order of the deformation radius Re) and maximum speed U (at K = 0).

CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS 2545

In the meander, the stream is both curved and displaced meridionally (Figure 1). Typically, the radius of curvature R is much larger than the jet width, and a small number characteristic of the curvature,

L

= - << 1, (8) R

can be defined. Contrary to the tradition but for the sake of mathematical clarity, we will attribute a positive curvature to anticyclonic meanders, called crests, and vice versa, negative curvature to cyclonic meanders, or troughs. Thus R and e will be taken as positive for crests and negative for troughs.

The typical meridional displacement of a meander is, too, typically much larger than the jet width. Therefore all particles in the stream can be thought, in first approximation, as being moved meridionally between the straight-portion and meander sections by the same meridional distance Y. In a crest, Y is positive, while in a trough, Y is negative, but in any case I YI is much larger than L. Considering only the leading-order corrections due to curvature and meridional displacement, the expressions for the translating stream function and potential vorticity profile are in the straight section:

ß (ff) = ½(ff), (9)

d2• 1

•(•) • d• 2 R• •(•)' (10)

where a - L/Rd is a ratio measuring the jet width. This number and the scaled quantities c and Y are three dimen- sionless constants of the order of unity. By contrast, e is a small number. Examination of (13)-(14) makes clear that the meander induces perturbations of the order of e and that the approximations made in formulating (11)-(12), such as the replacement of R + n by R and the neglect of along-stream derivatives, all lead to errors of only the order of e 2. In dimensionless form, the prototypical stream profile (7) re- duces to

•(t7) = -tanh t7. (15)

Finally, the conservation principles (5)-(6), with the use of (13)-(14) lead to

E

½(n) + • c Y = •(t7) (16)

d2• d½ d2F) dn 2 + e dn a2tp(n) + e Y= d-•- 7 - a2•(t•). (17)

These two equations are the essence of the present model. They form a nonlinear problem for the two unknowns $(n) and ti(n), given the function $(17). Physically, the two equations provide the modified stream structure in the meander and a mapping function that relates particle posi- tions between straight-portion and meander sections. The system is highly nonlinear because the function $(t•) (see (15)) is not a linear function of the unknown t•.

and the meander section,

ß (n) = ½(n)+ cY (11)

d 2½ 1 de 1 q(n) = d-•-- 7 + ½(n) + t3 Y. (12) R dn R5

The most interesting case arises when the curvature and planetary effects are comparable, i.e., when (1/R)d½/dn BY, which occurs for meanders with meridional displace- ments Y of the order of U/fiR. The translation speed c is then expected to be of the order of the long Rossby wave speed/3R•, which is of the order of/3L 2. This makes the three corrections terms cY, (1/R)d½/dn, and BY all of the order of e compared to the remaining terms in their respec- tive expressions. Thus the smallness of e measures simulta- neously the curvature effect, the planetary (beta) effect, and the translation effect.

To show this even more clearly, let us nondimensionalize the variables as follows: the cross-jet distance n and t7 by the jet width L, the translating stream function •, ß and stream function ½, ½ by UL, the potential vorticity q, c7 by U/L, the mean meridional displacement Y by U/fiR, and the transla- tion speed c by /3R•. The resulting expressions for the straight-portion and meander sections are

E

ß (ff) = ½(ff) •(n) = ½(n) + --• cY (13)

d23 d2• d• q(t7) = d172 a 23(17) q(n) = +, dn

- a2tp(n) + e Y,

Expansion

When e = 0 (i.e., no curvature and no meridional displace- ment), the solution is obvious' t7 = n and ½(n) = ½(n); the stream structure is not perturbed and the particles do not move laterally with respect to one another. When e is not zero, our assumptions dictate that it nonetheless remain small, suggesting that we can seek a solution using a pertur- bation technique. Using for convenience the cross-section coordinate n of the meander section as the independent variable, we write

if(n) = n + en'(n) + O(e 2)

½(n) = •(n) + e ½'(n) + O(e 2),

(18)

from which follow the relations

(19)

•(17) = •[n + en' + O(e2)] = •(n) +

d•(ti)_(dn) -• d•(t7) d ff - •nn d n

de 2) •+ O(e dn

( )- dn ' 2) = l+e-•nn +O(e

( d•n.n• d n ' d • ß + e dn dn

d•7(n) d2•(n) (14) dn dn 2

d23 2)) dn 2 + O( e

• + O(• 2)

2546 CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS

and, similarly,

d2•(n) dff 2

d2F•(n) d37•(n) dn 2 + en dn 3 +O(e2)

Substitution in (16)-(17) and gathering the e-order terms yield two linear equations for the particle displacement n' and the stream function perturbation $', both in terms of the cross-stream variable n'

d$ cY • n'- ½' = • (20) dn a

( d3• 2 d•_••• d2, ' aS dn 3 - a n' •'• q_ 2½,__ . + r. Elimination of n' between (20) in (21) yields a single equation for ½' that can be cast as

d (d2• , dl• d,') cY d3• dn [ dn 2 ½ dn dn a 2 dn 3

+ + c)r + . With the choice (15) for the jet profile in the straight portion, this last equation can be simplified using $ as the indepen- dent variable and properties such as d$/dn = $2 _ 1 and d23/dn 2 = 2•(• 2 - 1). The result is

d (2•(•2 - 1)*'- (•2_ 1)2 85' I 2cY _ ag a, / a 2 (3•t2 1) + (1 + c)Y + (g2_ 1), (22)

leading to a first integration,

2½(•t 2- 1)½'-(•2_ 1)2 Me'_ 2cY _ _ de a 2 (iP3 •t) + (1 + c)Y(•- 1) + 1_•3_ • q_ 3 2_ 3 ' (23)

where the constant of integration was determined by requir- ing that far to the right of the stream (n --) -o•, $ --) + 1), the stream function perturbation $' be bounded. Imposing the same requirement on the other side of the stream (n --) + o•, O --> - 1) yields

(1 + c) Y = 32- . (24)

Translation Speed

The integrability condition (24) on (22) is interpreted as the formula providing the translation speed c in terms of the meander properties. Solving for c and returning to dimen- sional variables, we obtain

2Ra2U c = -/3Ra 2 + (25) 3RY '

displacement of the meander (positive northward). This expression for the meander's translation speed invites some discussion.

Because meander crests have positive curvature and pos- itive northward displacements, while troughs have negative curvature and negative displacements, the second term in (25) always takes positive values and suggests a tendency toward eastward, or downstream, propagation. This is the vortex-induction effect noted by Pratt [1988]. The first term, always negative, represents the westward propagation of Rossby waves under the planetary effect. (The fact that the expression is the asymptotic, long-wave limit is rooted in our choice of long radii of curvature.) Thus planetary and curvature effects induce opposite tendencies. If the meander curvature and amplitude are such that the two tendencies cancel, i.e., if

2U RY= •, (26)

the meander is stationary. Such stationary solution of a thin-jet model was used by Masuda [1982] in a theory for the semipermanent, large meander of the Kuroshio. More re- cently, Cushman-Roisin et al. [1992] derived a similar expression for thin jets of arbitrary profiles. When their expression is particularized to the quasi-geostrophic hyper- bolic-tangent profile, it reduces exactly to (26).

At large curvature radius and meridional displacement (R Y > 2 U/3/3), the curvature effect is weaker than the planetary effect, and the meander, taking the aspect of a Rossby wave, moves westward, or upstream. On the other hand, meanders of high curvature and weak meridional displacements (R Y < 2U/3/3) propagate under the vortex- induction mechanism [Pratt, 1988] downstream, or east- ward. To illustrate the situation, let us consider a sinusoidal meander of wavenumber k: Y = A sin kx; at the crests and troughs (sin kx = --1, cos kx = 0), the curvature 1/R = -(d 2 Y/dx2)/[(dY/dx) 2 + 1] 3/2 is Yk 2, and expression (26) yields the critical wavelength for a stationary meander:

A- k -2rr . (27) This wavelength is proportional to the so-called Rhines' [1975] scale (U/B)•/2. This is not surprising, since that scale is precisely that at which planetary and nonlinear-advection (e.g., curvature) effects play equally important roles [Cush- man-Roisin and Tang, 1990]. For typical values of the Gulf Stream downstream of Cape Hatteras (U = 1.2 m s -• and /3 = 2.0 x 10 -• m -• s-•), the critical wavelength is A = 1260 km. Typically, Gulf Stream meanders have shorter wavelengths and travel downstream [Halliwell and Mooers, 1983; Tracey and Watts, 1986], in agreement with the theory. The numerical factor 2•r • in (27) results from our one-layer quasi-geostrophic assumption. An improved two- layer model (see below) leads to a different numerical coefficient and thus to a different value of the critical

wavelength.

where/3 is the gradient of the Coriolis parameter, Ra is the radius of the deformation, U is the maximum speed in the straight portion of the jet, R is the meander's radius of curvature (positive for a crest), and Y is the meridional

Solution

Elimination of c according to (24) in equation (23) govern- ing the stream function perturbation •' yields

CUSHMAN-ROISIN'. TRAJECTORIES IN GULF STREAM MEANDERS 2547

n/L

left side side

Fig 3 Schematic plane view of particle trajectories in the meandering jet showing lateral divergence and convergence pat-

Fig. 2. Plot of the particle lateral displacement, n - if, between straight and meander sections versus n, the cross-jet coordinate in the meander section. The particle at distance n in the meander section was at distance ff in the straight portion of the jet. The curve is drawn for positive curvature (meander crest). Values change sign for negative curvature '(meander trough).

2½½'-(t• 2-- 1)-•-= for which the general solution is

1 2

•'= •0(t• 2-- 1) + 6 3a

where % is a constant of integration.

3•

Y

2 t 2, (28)

The determination of this constant involves the definition

of the origin of the n axis, which is related to the definition of the jet axis. There is come choice. For simplicity, we will choose the origin of the n axis in the meander to correspond to the particle that was at the jet center in the straight portion (g = 0, $ = 0). Note that this particle does not necessarily have the maximum speed in the meander, and thus n - 0 does not mark the center of the jet in the meander. With n = 0 and t7 = 0, (18) yields n' = 0, and (20) •' = -c Y/a 2 . This determines the constant of integration ($0 - 1/6), and solution (28) becomes

1 2 Y 1 2 Y

•, = • • 2 _ =- tanh2 n -•5 + •. (29) 3--• + • Z 6 3a a Equation (20) then provides the particle displacement n"

•2 1 n' = _ = -- sinh 2 n. (30)

6(• 2- 1) 6 Combining perturbations and basic-state profiles and re-

turning to dimensional variables, we obtain the jet profile in the meander section and the particle mapping function:

n UL 2 n 2R •U ½ (n) = - UL tanh - + tanh 2

L 6R L 3R

UL 3• + 13R,•Y + 0 •'-•-2-] (31)

L 2 - L(•_5) n(ff) ti + sinh 2 n = - + O (32) 6R L '

Figure 2 shows the particle displacement n - g as a function of the particle position a, for a positive meander curvature (crest). On the left side of the jet (g > 0, also left side of the figure, as if looking downstream), the function is positive, indicating that particles on the left flank of the stream are

terns. Divergence and convergence regions alternate according to the side of the jet (left or right) and the rate of change of curvature (from trough to crest or vice versa).

further to the left in the meander than they were in the straight portion. Similarly, positive values on the right imply that particles on the right flank of the stream are less to the right in the meander than they were in the straight portion. We thus conclude that as particles flow into a crest, they diverge on the left and converge on the right (Figure 3). Conversely, as particles leave the crest and flow toward a trough (negative R and e), they converge on the left and diverge on the right. This is in qualitative agreement with the findings of Bower and Rossby [ 1989], who report a systematic tendency among floats to leave the jet on the left side from trough to crest or on the right side from crest to trough, and to enter the jet elsewhere (see their Figure 13). This is also consistent with the inference of upwelling from observed enhancement of phytoplankton biomass and productivity in meander crests (D. B. Olson, The Synoptician, vol. 1, no. 3, unpublished newsletter, 1990). Finally, the convergence-divergence pattern conforms with the standard model for developing baroclinic waves in the atmo- spheric Jet Stream [Palm•n and Newton, 1969].

Figure 4 compares the stream function profiles of the straight-portion and meander sections of the jet, again for a positive curvature (meander crest). The vertical axis is pointing downward, so that curves can be viewed as pycnocline surfaces. (In our quasi-geostrophic model, the stream function is proportional to the layer depth anomaly, measured positively downward). Three features are worth noting: the net vertical displacements at large distances on

n/L xx•[, ' ' •'•,. 21 _•

left side 0.5. '"......••t..•_•ide 1.

1.5

Fig. 4. Comparison of the stream function profiles in the straight and meander sections of the jet. The vertical axis points downward, so that the curves can be viewed as pycnocline surfaces. The figure is drawn for a meander crest that propagates down- stream. Parameter values are a = 1.4, e - 0.4, and Y = 0.11.

2548 CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS

either side of the jet, the depth change at the center, and the asymmetry of the profile.

At large distances from the jet (n --> ___o•), the stream function perturbation is

UL 2 2R ,• U UL 2 lim [½(n)-t•(n)]=• •+13R•Y=•-cY,

n__>_+o• 6R 3R 6R (33)

which can be either negative or positive. For an eastward propagating crest (c > 0, R > 0, Y > 0), there is a deepening of the interface (½ larger than •p) if UL 2/6R > c Y. However, our asymptotic theory fails at large distances from the jet center (en' no longer much smaller than n), and no physical conclusion should be drawn from this result.

Of interest is the situation near the center of the jet. At n = 0, the stream function perturbation is

2R ,•U ½(0) - •(0) = -• + 13R,•Y = -c Y. (34)

3R

Eastward traveling crests (c > 0, Y > 0) cause upwelling of the interface (½ < ½), while eastward traveling troughs (c > 0, Y < 0) cause downwelling. The situation is reverse for the less common, westward traveling meanders. Bower and Rossby [1989] noted that floats near the jet center upwell as they go from trough to crest, and downwell from crest to trough. Because the meander wavelength at the time (about 500-600 km) indicates an eastward propagating meander, there is qualitative agreement between theory and observa- tions.

A quantitative comparison between theory and observa- tions can be attempted. Let us take numbers corresponding to the trajectory of float RAFOS 022 [Bower and Rossby, 1989; Bower, 1989]' f0 = 9 x 10 -5 s -• /3- 1.8 x 10 -• m -• s -• U = 1 2 ms -• Y= 60km, R = 120km #' = 0.02 m s-2 and H = 500 m. For these values we determine

Ra = 35 km,/•R• = 0.022 m s -1, 2R•U/3RY = 0.137 m s -• c = 0 137 - 0 022 = 0 115 m s -• (positive, thus indeed eastward), and ½(0) - •(0) = -6900 m 2 s -• . The corresponding upward interfacial displacement is Ah = -f0[ ½(0) - ½(0)]/#' = 31 m. According to the theory, this is the upwelling of the interface near the jet center between the straight portion and the meander crest. From trough to crest, the net upwelling is twice that and predicted to be about 62 m. This value is a gross underestimate of the 300- to 400-m upwelling observed by Bower and Rossby [1989]. The discrep- ancy is doubtlessly to be found in the model's restfiction to a single moving layer and/or to quasi-geostrophic dynamics.

Also, noteworthy in Figure 4 is the asymmetry of the stream function profile in the meander. This asymmetry is made clearer when we consider the derivative of the stream function.

From the cross-stream gradient of the stream function, geostrophic jet velocities can be derived:

a½(a) u

a(ti) = dn - cosh 2 ti/L (35a)

Straight portion

Meander

u(n) - • - cosl• 7 n/L 1 3R tanh . (35b)

.'/ I ',x,.

• I I / I i

2 I I -1 -2 left -.2. I. right

Fig. 5. Comparison of the velocity profiles in the straight and meander sections of the jet. The figure is drawn for a meander crest (R > 0). For a trough, the deviation is opposite. Note the acceleration of the jet and the fact that the fastest particle in the straight section is no longer the fastest in the meander, giving the appearance of a right shift of the jet. Parameter values are identical to those of Figure 4.

A graphical comparison is offered in Figure 5. Here we note an increase of the maximum speed in the meander. Relying on the smallness of e = L/R, we find Uma x -- /7ma x = UL2/36R 2. That is, the relative acceleration is about e2/36. This is small, perhaps too small to be noticed. (In her estimation of the various components of the vorticity budget, Bower [1989] assumed that the velocity profile in the meander is identical to that in the straight portion of the jet, thus not providing any estimate of a possible jet acceleration.) Of interest is the fact that the maximum velocity is displaced laterally. Figure 5 indicates that in a meander crest (R > 0), the particle with maximum velocity in the straight portion (n = 0) no longer has the maximum velocity in the meander but that this property now belongs to another particle L2/6R to its fight. This gives the appearance of a leftward shift of particles with respect to the jet axis. In a meander trough (R < 0), the situation is reverse, and near-center particles appear to shift to the fight of the stream. This is in qualitative agreement with the float observations [Bower and Rossby, 1989]. For the numbers quoted above and a jet width L = 50 km, the theory predicts shifts of the order of few kilometers. This may be small, but it must be kept in mind that the sharp variation of the lateral- displacement function on each side of the stream (Figure 2) contributes to make the apparent lateral drift of a fluid parcel with respect to the jet axis a very sensitive function of the distance between particle position and jet axis in the straight position of the stream. Not knowing precisely how close the floats [Bower and Rossby, 1989] were to the jet axis between meanders, we are unable to make a quantitative comparison.

3. Two-LAYER QUASI-GEOSTROPHIC MODEL

The Model

We now extend our previous model and method of solu- tion to include motions in two layers. The model then consists of two moving layers atop a third, infinitely deep layer at rest. Unlike the one-layer model, in which upwelled parcels must necessarily undergo squeezing against the surface, the two-layer model allows a decoupling between upwelling and squeezing for particles in the lower moving layer. With this model therefore we hope to shed light on the conclusions reached by Bower [1989], who suggested, after having made some assumptions, that Gulf Stream particles

CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS 2549

at a depth of about 600 m experience simultaneous upwelling and vertical stretching on the way from trough to crest.

With the same density difference across each of the two interfaces, the quasi-geostrophic equations are

Oql 1 +J($1, ql) =0 ql = V251-• 1 at R• (½ - ½2) +/3y

(36)

de 2 cY

dn a

( d3•l _ 2 dq71 dn 3 ot dn

aq7: + a: di' )n'l an 2

+. 2(q,' 1 - q,h)= d n

where the subscripts 1 and 2 refer to the upper and lower moving layers, respectively. The coefficient • = Hi/H2 is the ratio of mean layer depths, and the radius of deformation Rd = 07'H•) m/fo is defined with the reduced gravity (common to both interfaces) and the upper layer depth. In meanders steadily translating at eastward speed c, fluid particles in each layer conserve both their potential vorticity qi and their trans- lating stream function XIt i = Ipi + Cy (i-' 1, 2). Considering the water column at cross-jet distance n in the meander section, we trace the upper and lower fluid parcels to the respective cross-jet distances K• and K2 in the straight portion of the stream, and write the conservation principles:

O •(n) + • c Y = •(fi•) (38)

d2½• dn 2 + e dn

E

•t2(n) + • c Y = q•2(ff2) (39)

c•2[Ol(n ) - •t2(n)] + e Y

(40) d2•pl

-

•+ c•28[q•l(n) - 2½2(n) ] + e Y d21p2 diP2 dn 2 + e dn

(43)

(41) d2•p2 dff• + O• 2•$[•1(/•2)- 2•P2(I72)]'

using assumptions similar to those made previously, namely, that the meander curvature radius R and meridional dis-

placement Y are much larger than the jet width L. In writing (38)-(41), the two-layer extensions of (16)-(17), variables have also been nondimensionalized, according to the same scales as those used previously (n, •, and •2 by jet width L, all stream functions by UL, Y by U/DR, and c by The dimensionless coefficients are, as defined above, a = L/Ra = O(1) and e = L/R << 1.

Solution

Because the meander curvature parameter e is small, the solution is sought by the method of perturbations, which linearizes the system of equations:

d½• cY • n'• - ½'• = -• (42) dn a

+Y (44)

d•t2

- a28(½'•-2½•)= dn + Y, (45) where all variables are now expressed in terms of n, the cross-jet distance in the meander section. These equations are the two-layer extensions of (20)-(21). Elimination of the cross-jet displacements n'• and n• by use of the first two equations reduces the remaining two equations to a coupled set for the stream function perturbations ½• and

d2•t'• dn 2 -•+[dn] •d•5 +a dn],'•-ot •=

-- (d•11-' (d3•, dl•2 ) c Y dOl+(1 +cY)- 2 dn [ dn J • dn 3 + a dn • 2 d2 q. t• dn 2

(d•21-' ;d3•2 d•l 1 øt2t•*• + [ dn / • dn 3 + øt25 d;' J*• = -- (d•21-1 td332 dq•, 1 ½Y d•2 + (1 + 2/•c) Y - dn [ dn 2] [ dn '3 + øt215 dn J • '

To proceed with the solution of these equations, it is necessary to specify the functions ½• (n) and ½2 (n), i.e., the structure of the stream in its straight portion. For this, we again take a hyperbolic-tangent profile:

½'•(n) = -tanh n 02(n) = -K tanh n (46)

suitably scaled so that the maximum velocity in the upper layer is U• = U. The coefficient •( of proportionality between the two stream function profiles (• -- U2/U•) can be considered as a parameter representing the degree of baroclinicity in the stream. For • = 1, the flow is vertically uniform over the two layers (• = ½2), for • = 0 there is no flow in the lower layer, and for • < 0 the lower layer flow is counter to that in the upper layer. With ½• as the indepen- dent variable, the equations become

-(•- 1) 2 2•l(•p• 2 -1) _

+(6,• 2-2+ a2tr),'•- a24t•= 1--•Tj3• 2

(Y) + 1+ 1-tr+ c Y-1

d2q• dq• --(•12 -- l) 2 -- _ dq7? 2•l(• 2 - 1) dq7,

2550 CUSHMAN-ROISIN.' TRAJECTORIES IN GULF STREAM MEANDERS

to- cr---•/•l 2+ 1 + 2•--+ c Y- Inspection of these equations reveals that a solution qua- dratic in •l, as in (29), can be sought. We thus write

•'• = Aq7• 2 +B

Identifying the coefficients of similar powers of •1 leads to the following four equations for the four coefficients A-D'

(6+aeK)A+6B-aeC= 1 6cY

2 (47)

-42/5A + 6+ C+6D= K 6cY

2 (48)

2A+ (2- a2tc)B+ a2D= 1- [1 ( 1-•c+ c Y

(49)

42/5 ) 42/5B+2C+ 2 -- D

=•c- 1+ 2/5---+ Yo (50)

Elimination of B with (47) and D with (48) yields

a 2 1+ + (KA- C) = 1 4 4•c / 2

Y (51)

42/5 ( 42• 42/5) 3 1 +•+ (•A-C)=-m +-(1+ •c 4 4•c 2

(52)

This set of two equations for the single unknown •A - C admits a solution only if the right-hand sides meet the compatibility condition

1+-+/5c Y= • + (53)

This last relation provides the translation speed of the meander, extending relation (24) to the two-layer system. Solving for c and returning to dimensional variables, we obtain

c = • + U2)•,•,• + + . (54) 3R¾

This expression reduces to the one-layer equivalent when He = 0 and U• = Ue = U (that is, when the intermediate layer plays no role). Again, it can be seen that meanders propagate under two opposite tendencies, the eastward vortex-induction entrainment and the westward Rossby

wave drift. The critical wavelength at which these tenden- cies are equal, thus bringing the meander to rest, can be determined by setting c to zero. With 1/R = Yk 2 the result is

2•r ( 2(HiU12 + H2U22) ] 1/2

With H1 = H2 = 500 m, U1 = 1.2 m s -1 U2 = 0 6 m s -• and • = 2.0 x 10 -• m -• s -• the value is A = 1150 km This is the critical wavelength separating the meanders propagat- ing eastward (A < 1150 km) kom those propagating west- ward (A > 1150 km).

Because (51) and (52) are now identical, one additional equation is required. For this, we invoke the same boundary condition as for the one-layer case: we define the origin of the cross-stream axis in the meander (n = 0) as the location of the upper layer particle that was at the center of the jet in the straight section (g•: 0, • = 0). Equation (42) with n] = 0 and •l = 0 yields •](•l = O) = -cY/a 2 and, by virtue of the polynomial expression of •i,

cY

= 2. (56)

Equations (47), (48), (51), and (56) provide the solutions for the four constants A-D:

tr + (3 Y/2) c Y (3 Y/2) - 1

A= 6(1+ o') B= 2 C= •cA+ a a 2(1 + o')

D=B+-- •[o- + (3 Y/2)]

6(1 + o-)

(57)

[(3Y/2)- 1] [1 + (a 2/5/6t()] a 2( 1 -4-

where, for convenience, tr has been defined as

cr =• K+ . 4

(58)

Application

The number of parameters in the model (namely, a for the jet width, /5 for the depth ratio, • for the vertical velocity shear, Y for the scaled meridional displacement, and e for the relative curvature) does not permit a general discussion of the analytical solution. Therefore we set the parameters to specific values, which we choose to be representative of the trajectory of RAFOS float 022 documented by Bower [1989]' geography: latitude 38øN, f0 = 9.0 x 10 - 5 s - l, and/3 = 1.8 x 10 -ll m -1 s -l' Gulf Stream: 0' = 0.02 m s -2 H1 = 500 m, H1 +H2= 1000m, U1 = 1.2ms -1 U2 =06ms -1 and L = 50 km; and meander crest' wavelength 530 km (k = 1.2 x 10 -2 km-1), Y = 60 km, and R = 120 km (YR = l/k2). These numbers lead to Ra = 35 km, c = 0.28 m/s = 24 km/d (eastward), and interfacial depth variations across the jet given by

CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS 2551

3

n -n 2 2 EL

n/L

2 I -1 -2

left right

Fig. 6. Plot of particle lateral displacements, n - ffl and n - 52, between straight and meander sections of the jet, versus n, the cross-jet coordinate in the meander section. The curves are drawn for the parameter values quoted in section 3 (subsection on appli- cation). Positive values indicate that particles in both layers move leftward when they enter a crest. Values change sign for negative curvature (meander trough).

2 1

left

n/L •,h2

2 •-12=0 I

-.5

• I I I -1 -2

A(h 1)across jet = -- A(ip 1 -- IP2)across jet #'

2f0( U 1 - U2)L

! = 270 m,

f0

A (h I + h 2) across jet = •'• A ( Ip 2) across jet = • 2foU2L

= 270 m. #'

We also calculate the dimensionless ratios'

= =2.0 8 = H2

U2 K =--=0.5

U1

L =--=0.42,

R

and coefficients for the solution

(59)

A = +0.1037 B = -0.6833

C = -0.1370 D = -0.3370.

(60)

With these parameters, graphs can be constructed. Figure 6 displays the mapping functions or cross-jet displacements in each layer. Because of the similarity with Figure 2, we conclude again in a convergence-divergence pattern as de- picted in Figure 3 in each of the two layers. As with the one-layer model, lateral displacements are critically depend- ing on the distance between particle and jet axis in the straight-jet portion, and in the ignorance about the exact position of RAFOS float 022 with respect to the Stream axis, no quantitative inference can be made. The main conclusion, however, is that both layers undergo similar convergence and divergence from trough to crest and crest to trough. Another conclusion can be drawn from the fact that the

lower layer displacement n - /if2 is not zero at the origin, i.e., (n - •2)/eL = 0.69 or n - if2 = 14 km. This indicates that, near the jet center, the second layer moves to the left with respect to the top layer by about 28 km from tip of trough to tip of crest. Finally, because the n - /if2 curve lies everywhere above the n - g l curve, we conclude that the leftward shift of the second layer with respect to the top layer occurs throughout the jet width (albeit with varying

-.5

-1 -2

-.5 •-- hl+h 2

Fig. 7. Comparison of layer thickness and interface depths between straight and meander sections, versus n, the cross-jet coordinate in the meander section: (a) h l and K 1 , the top-layer thickness variation, also the depth variation of the first interface' (b) h 2 and K2, the second-layer thickness variation' and (c) h l + h2 and fi-1 + h-2, the depth variation of the second interface. Values are nondimensionalized by fo U1L/#'. Parameter values are identical to those chosen for Figure 6.

amplitude). This is in agreement with observations [Bower and Rossby, 1989].

Figure 7 displays the layer thickness and interface-depth variations from the state of rest for both the straight and meander sections of the jet. The layer thickness variations are determined using the quasi-geostrophic relations

f0 f0 hl = -- (•/1 -- •/2) h2 = -- (252 - •/l)

f0 (61) ha + h2 =•7 •/2.

Because the curves h 1 and h 1 + h 2 lie above those of •'1 and •'1 + •'2, respectively, the model predicts for both layers an upwelling of the isopycnals from trough to crest. This rise is modest, about 38 m near the jet center. Particles are uplifted to a greater extent, however, because of their concomitant leftward displacements, which cause them to "slide up" the isopycnical surfaces. For example, the second-layer parcel along the jet axis in the straight portion (•2 = 0, œ1 + h-2 = 0) finds itself at cross-stream coordinate n = 15 km at the peak of the crest, where the second-interface depth h I + h2 is 82.5 m less, or 31% of the total interfacial-depth variation across the jet.

2552 CUSHMAN-ROISIN' TRAJECTORIES IN GULF STREAM MEANDERS

I -1 -2

Fig. 8. Comparative plots of the velocity profiles in first and second layers (subscripts 1 and 2) and for both straight and meander sections of the jet (with and without overbar, respectively). Note the apparent rightward and leftward displacements of the velocity profiles in the first and second layer, respectively. Values are nondimensionalized by the upper layer maximum velocity U i. Parameter values are identical to those chosen for Figures 6 and 7.

From the depth profiles, we can also estimate the amount of stretching/squeezing undergone by second-layer parcels as they move laterally. The particle just mentioned sees its thickness vary from H 2 q- h2(h = 0) = 500 q- 0 = 500 m in the straight portion of the jet to H 2 + h2(n = 15 km) = 500 - 13.5 = 486 m. The situation is somewhat different for

second-layer parcels that are to the side of the jet in the straight portion. Take, for example, the particle some 25 km to the left of the jet axis in the straight portion (h2 = 25 km, H2 + h-2 = 500 m, Hi + K1 + H2 + h-2 = 937.6 m), in the meander crest, it is at n = 54.9 km where its thickness and depth are H 2 + h 2 = 395.2 m and H1 + hi + H2 + h2 = 844.2 m, respectively. Thus, this parcel is squeezed by 104.8 m while it upwells only 93.4 m. This model prediction of simultaneous upwelling and squeezing contradicts Bower's [1989] conclusion that Gulf Stream parcels at about 600 m undergo upwelling and vertical stretching on the way from trough to crest. Either the present model is faulty or some of Bower's assumptions are incorrect. We will return to this matter shortly in our discussion of the potential-vorticity budget.

Figure 8 displays the current profiles in the upper and lower layers, in the straight and meander portions of the jet. The maximum velocity in each layer is greater in the crest than in the straight portion, but only marginally so. We also note apparent displacements of the jet, which in terms of particle positions with respect to the jet axis (maximum velocity), translate into leftward displacements in the upper layer and rightward displacements in the lower layer. In a trough, this last conclusion is reversed.

A feature not readily apparent on these jet profiles but nonetheless present is the asymmetry brought on by the meander distortion, which results in an intensification of the velocity shear on one side of the jet and in a decrease on the other side. This point will be made clearer in the following analysis of the potential-vorticity budget where velocity shear plays the role of relative vorticity.

A tenet of the present model is conservation of potential vorticity, which consists of four separate contributions' relative vorticity, curvature vorticity, planetary vorticity, and vertical stretching. Of course, as a fluid parcel flows from a straight portion of the jet into a meander crest, those individual contributions to its potential vorticity vary, al- though they sum to the same value. For example, there is no

curvature vorticity in the straight portion, while there is one in the crest. To analyze the potential-vorticity budget, we dissect the potential-vorticity difference between crest and straight configuration, which is obviously nil, as follows'

O= ql(n) - ql(t71 )= • yr•5' y•2'/ + e dn shear curvature

+ a2[•l(hl) - •l(n) - q72(h•) + •t2(n)] + e Y (62)

stretching beta

for the upper layer, according to (40). The first group of terms represents the difference in relative vorticity (velocity shear) of the parcel between its two stages, which implicitly includes two separate contributions, the part due to the displacement of the parcel across the jet (difference between n and hi) and that due to the jet distortion itself (difference between ½• and ½1)- Because of insufficient data, Bower [1989] assumed that the jet profile was identical in both straight and meander sections, thus neglecting this latter possibility for relative vorticity modification. The second term represents the vorticity of the parcel as it turns follow- ing the meander curvature; its form, velocity d½l/dn divided by radius of curvature l/e, is reminiscent of the expression of vorticity in solid-body rotation. The third group of terms is proportional to fi-l(hl) - hi(n), thus representing, in a quasi-geostrophic sense, the contribution of fluid-parcel thickness. Finally, the last term, proportional to the merid- ional displacement between the two sections, is evidently the contribution of planetary vorticity to total vorticity.

Substituting the model solution for the stream functions qq(n) and q•2(n) and for the displacement function hi(n), the above contribution can be evaluated separately and as functions of the cross-stream coordinate n. The results are

presented graphically in Figure 9, where each curve corre- sponds to the grouping of terms bearing the same label in expression (62). The four curves add up to zero everywhere. The curvature contribution is negative throughout the jet because a crest means anticyclonic rotation; the values are larger near the center of the jet, where the speed is greater,

n/L beta

-,2'

Fig. 9. Potential-velocity budget for top-layer particles in a meander crest as a function of the cross-section coordinate. Values

are nondimensionalized by U1/R.

CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS 2553

and less on the flanks, where the speeds are lower. The stretching contribution is positive, indicating a squeezing of all water parcels from straight-jet to crest positions. The beta contribution is uniform across the jet because we have assumed a thin jet, i.e., a jet wherein all parcels essentially undergo the same meridional displacement in a meander; the contribution is positive because a crest implies a northward excursion (greater BY).

The last contribution, that of shear, could have hardly been anticipated. Near the center of the jet, the values are positive, signifying an increase in cyclonic velocity shear and thus an enhancement of the cyclonic vorticity on the jet's left side and a decrease of anticyclonic vorticity on the right side, from straight configuration to meander crest. On the flanks, the situation is reversed: on the left, the cyclonic vorticity is reduced, while on the right, the anticyclonic vorticity is increased.

In the shear variation, it is interesting to separate the contribution due to the particle shift across the jet from that attributable to the jet distortion. Let us write

shear modification = d2•l d2•l dn 2 dff l 2

/d2½1 = !dn2

d2•l dtl 2 a,) {d2tkl + !dn2

d2•l dill 2

particle shift jet distortion

(63)

= 2Aj•2(1 - 3• 2) + 2A(1 - j•2)(1 - 3312) + O(e),

where each term has been scaled by U i/R (as in Figure 9). At the center of the jet (n = 0, fil = 0, 61 = 0), the contributions are zero and 2A = 0.2074 (or 2.1 x 10 -6 s-l), for the particle shift and jet distortion, respectively. As we can therefore conclude, the assumption made by Bower [ 1989] that only particle shifts with respect to the jet account for relative vorticity changes is invalidated. Furthermore, in her calculations, the changes in vertical stretching were estimated by subtraction from balanced potential-vorticity budgets and are, too, questionable. This may explain the disagreement mentioned above regarding vertical stretching in the second moving layer. In turn, patterns of horizontal divergence/convergence as inferred by Bower from vertical squeezing/stretching are to be challenged.

Figure 10 displays the potential-vorticity contributions for particles in the second moving layer. Except for the stretch- ing term, all others exhibit behavior similar to those for the first layer. The beta contribution is obviously identical (the meander's meridional displacement is the same in both layers), the curvature variation is less (because the layer velocities are less), and the shear variation is intensified (in agreement with greater lateral displacements in the second layer).

Because RAFOS float 022 [Bower and Rossby, 1989; Bower, 1989] was submerged at an average depth of 600 m, we shall identify it with a particle in the second layer of the present model. Not knowing the exact position of the parcel with respect to the jet axis at any given time, but judging from Bower's Figure 8, we shall assume that it was situated, at the peak of the meander, approximately 50 km to the left of the stream axis and take our readings from the present

q2 q2

beta 2 I -1 -2

Fig. 10. Same as Figure 9 but for the second layer.

figures at cross-jet ordinate n/L = 1.0. For easier identifi- cation with Bower's time series (her Figure 7), we will focus on the float's passage from the tip of the first trough (day 292) to the peak of the next crest (day 297). In terms of model results, this requires doubling the variations from straight section (halfway between trough and crest) to crest.

From Figure 6 we estimate the particle's lateral displace- ment at 53 km to the left from trough to crest, while Bower quotes 70 km. As noted above, however, the sharp bends in the curves of Figure 6 make our estimate quite sensitive to the choice of coordinate n/L. From Figure 7 we infer an upwelling of 300 m, while Bower notes a rise in depth from 800 to 400 m, i.e., a 400-m upwelling. From Figure 10 we derive the following changes in the contributions to potential vorticity (all in units of 10 -5 s-l): beta +0.22, curvature -0.84, shear -0.15, and stretching +0.95, while Bower's estimates are +0.40, -1.24, +3.1, and -2.9, respectively. The agreement is thus mixed. Lateral and vertical displace- ments agree in directions and orders of magnitude, as do estimates of curvature-vorticity. The present value of the planetary-vorticity changes is underestimated because of the thin-jet approximation. (The lateral displacement of the parcel to the left of the jet causes an additional meridional displacement not counted in the model because its relative importance is of the order of e.) As argued above, the disagreement between the two shear-vorticity estimates is most likely a result of Bower' s neglect of the velocity-profile distortion in the meander. Finally, we also question Bower' s estimate of the vertical stretching to potential vorticity, since it is derived by subtraction, using the shear-vorticity esti- mate.

Obviously, it is difficult to reconcile the results from a thin-jet, quasi-geostrophic model of steadily translating me- anders with estimates derived indirectly from observations. In particular, the large value of • (=0.42, not much less than 1) makes the model only very approximate. The facts, however, that all predicted patterns are qualitatively correct and that the primary variables such as lateral displacements are in reasonable agreement with the observations, are very encouraging. The discrepancies, on the other hand, should serve as motivation for further studies, both theoretical and observational.

2554 CUSHMAN-ROISIN: TRAJECTORIES IN GULF STREAM MEANDERS

4. CONCLUSIONS

A theoretical investigation of particle displacements in Gulf Stream meanders was made possible by the combined assumptions of a thin jet, a steadily translating meander, and conservation of both translating stream function and poten- tial vorticity. For mathematical simplicity, two-layer quasi- geostrophic dynamics were selected, but in principle, the approach could be extended to the use of primitive equations and to a greater number of active layers.

Given the stream structure, in both vertical and cross-jet directions, in a nonmeander configuration, the model pro- vides the distorted stream structure and the particle dis- placements in both layers as a function of meander curvature and meander meridional excursion. Because curvature and

meridional excursion vary gradually from trough to crest and back to trough, the model results can be applied to any section along the stream's curved path.

As a bonus, the model also provides the zonal propagation speed of the meander. The drift can be eastward or west- ward, depending on the meander's curvature and meridional amplitude, although typical values suggest that most Gulf Stream meanders should propagate eastward.

A comparison with RAFOS float trajectories and infer- ences therefrom [Bower and Rossby, 1989; Bower, 1989] indicates that the model reproduces correctly the directions of the particles' lateral and vertical displacements and pro- vides reasonable orders of magnitude. This suggests that the model, although conceptually simple, retains the correct physical processes. Disagreement exists, however, between less direct estimates such as the contributions of lateral

shear and vertical stretching to potential vorticity. Although the present model is evidently an oversimplification of the Gulf Stream, it is also argued that Bower [1989] has over- looked an important aspect, namely, the distortion of the velocity profile across the Stream from trough to crest. As the model suggests, very small changes in maximum velocity may be accompanied by substantial changes in shear on the Stream's flanks, and Bower's estimate of shear-vorticity modification is most likely much too large.

The present model is but a first step in the investigation of fluid particles in a meandering jet based on dynamical principles, in addition to the pure kinematic approach [Bow- er, 1991]. It remains, however, to be improved by the use of less restricted dynamics and to be complemented by the results of numerical simulations.

University. The ensuing research was carried out at Dartmouth College under grant N-00014-91-J-1789 from the Office of Naval Research, which is gratefully acknowledged.

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B. Cushman-Roisin, Thayer School of Engineering, Dartmouth College, Hanover, NH 03755.

Acknowledgments. The ideas expressed in this article originated after a stimulating lecture given by T. Rossby at Florida State

(Received January 24, 1992; revised August 3, 1992;

accepted August 4, 1992.)