One-dimensional baroclinically unstable waves on the Gulf Stream potential vorticity gradient near Cape Hatteras

Dynamics of Atmospheres and Oceans, 11 (1988) 323-350 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands

323

ONE-DIMENSIONAL BAROCLINICALLY UNSTABLE WAVES ON THE GULF STREAM POTENTIAL VORTICITY GRADIENT NEAR CAPE HATrERAS

WILLIAM E. JOHNS *

Graduate School of Oceanography, Universityof Rhode Islana~ Narragansett, RI 02882 (U.S.A.) (Received December 22, 1986; revised July 23, 1987; accepted August 5, 1987)

ABSTRACT

Johns, W.E., 1988. One-dimensional baroclinically unstable waves on the Gulf Stream potential vorticity gradient near Cape Hatteras. Dyn. Atmos. Oceans, 11: 323-350. Application of linear baroclinic instability theory to the observed distributions of velocity, stratification, and potential vorticity in the Gulf Stream near 74 °W is successful in predicting the time and length scales of the most rapidly growing disturbances. A continuouslystratified, one-dimensional model with realistic bottom slope predicts propagation speeds of 10-50 cm s -1 associated with two regimes of rapid temporal growth centered at periods of 28 days and 5-7 days. This prediction is consistent with observations of the propagation and growth of Gulf Stream meanders derived from inverted echo sounder measurements in this region. The instability model also predicts that for realistic bottom slopes the baroclinic energy transfer should be weakly negative (eddy-to-mean) in deep water, but for low-frequency waves should change to significant positive (mean-to-eddy) transfer above depths of -1500 m, consistent with observations.

1. INTRODUCTION T h e i n s t a n t a n e o u s p a t h o f the G u l f S t r e a m a l o n g its e n t i r e length, f r o m the F l o r i d a Straits to the e v e n t u a l d e c a y region east o f the N e w E n g l a n d S e a m o u n t s , is c h a r a c t e r i z e d b y wavelike p e r t u r b a t i o n s k n o w n as G u l f S t r e a m m e a n d e r s . S o u t h o f C a p e H a t t e r a s , the r o o t - m e a n - s q u a r e lateral displacem e n t s o f the s t r e a m p a t h increase slowly f r o m 5 to 10 k m o f f F l o r i d a , to a local m a x i m u m o f 25 k m d o w n s t r e a m o f a t o p o g r a p h i c f e a t u r e o f f Charlestown, SC (the ' C h a r l e s t o w n B u m p ' , 31-32°'1q), in the lee o f w h i c h t h e y

* Now at: Rosenstiel School of Marine and Atmospheric Science, University of Miami, 4600 Rickenbacker Causeway, Miami, FL 33149-1098, U.S.A. 0377-0265/88/$03.50

© 1988 Elsevier Science Publishers B.V.

324

decay back to - 1 0 km r.m.s. (Bane and Brooks, 1979; Watts, 1983). Downstream of Cape Hatteras the envelope of meandering broadens rapidly from tens to hundreds of kilometers and the Stream can shift through more than its own width. Meanders which grow to large amplitude frequently separate from the Stream as isolated eddies, enclosing and transporting with them large masses of adjacent slope water or Sargasso seawater. Over the past several decades, a number of theories have been advanced to explain the origin and growth of meanders. The first class of theories, the so-called inertial jet theories, was introduced by Warren (1963), who developed a steady-state theory in which the Stream's path was determined by spatial variations in depth and Coriolis parameter downstream of an 'inlet' (e.g. Cape Hatteras) where the configuration of the path was assumed known. This theory was subsequently expanded by Robinson et al. (1975) who showed that the steady-state theory could adequately describe the mean path and envelope of the Gulf Stream, but could not account for the high curvatures and time-dependent aspects of the path variability observed by Hansen (1970). The generalization of Warren's problem to include time-dependence (Robinson et al., 1975) yielded a non-linear differential equation for the path which had unstable wave solutions, and whose linearized form was identical to Tareev's (1965) perturbation model of baroclinic instability. Hence the perception of Gulf Stream meanders as amplifying disturbances related to the presence of some baroclinic, barotropic, or mixed instability process is assumed to be the most valid approach. In this paper we examine the stability of the Gulf Stream in the vicinity of Cape Hatteras, based on the linearized equations expressing conservation of potential vorticity in a channel of variable depth. The model used is a continuously-stratified, one-dimensional system in which the fluctuations are assumed to be in approximate geostrophic and hydrostatic balance. This one-dimensional problem effectively isolates the baroclinic instability mechanism, in which the fluctuations draw their energy from the mean potential energy of the flow through down-gradient buoyancy fluxes. A major emphasis of this work concerns the sensitivity of the baroclinic instability problem to the assumed mean potential vorticity distribution of the flow. The ability of fluctuations to extract energy from the mean flow depends crucially on the actual mean potential vorticity gradient, and therefore effort has been directed here toward calculating the local mean potential vorticity gradient of the Gulf Stream and determining its uncertainty. The quasi-geostrophic theory for the above described one-dimensional system is by now well established (cf. Pedlosky, 1979), and will be reviewed only briefly below. It is also well known that the application of quasi-geostrophic theory to intense western boundary currents such as the Gulf

325

Stream is questionable, due to the high Rossby numbers associated with the near-surface flow. Formally, the quasi-geostrophic theory also requires that, in a two-layer sense, the change in depth of the upper layer 8h is small compared with its average depth H. This assumption is dearly violated in the Gulf Stream. There will be no attempt here to rationalize these shortcomings, except to point out the relative past success of linear quasi-geostrophic theory in predicting the dominant time and length scales of instabilities in both the atmosphere and the oceans (Pedlosky, 1979). A notable recent addition to this list of successes is Wright's (1981) application of the linearized theory in describing the characteristics of unstable disturbances in the Drake Passage, where the assumption 8 h / H << 1 is similarly invalid. The results of the present analysis suggest that the linear theory is also capable of predicting the dispersion and vertical structure of Gulf Stream meanders in their initial stages of evolution downstream of Cape Hatteras, NC. 2. THEORETICAL CONSIDERATIONS

We begin with a consideration of the first order vorticity and density, or adiabatic, equations D(0v Dt

Ox

~u) ay

0w + flv = f

Oz

(1)

Dp Dt

+

= 0

(2)

where/3 = d f / d y is the northward variation of the Coriolis parameter, and where D / D t = D/at + u a / a x + va/Oy. Under quasigeostrophic scaling, c = U / f L << 1, where U, L are velocity and length scales and f is the Coriolis parameter, a streamfunction p may be introduced such that (Pedlosky, 1979)

0=

1

-1

Po

Po

)px u=

/p,

in which case (1) may be written 1 {D

pof

b-Tt(v p) +

}

=f

Ow Oz

The proper form of (2) depends on the relative importance of vertical advection, measured by the Burgher number, Bu = h / L , where ~ = N H / f is the internal Rossby radius of deformation, N = ( - ( g / p o ) ~ } 1/z is the Brunt-Vaisala frequency, and H is the vertical scale. 'Long-wave' instability

326

theories and 'classical' instability theories have the two limits Bu = 0(e) and Bu = 0(1), corresponding to L >> ~ and L - ~, respectively. Recent observations east of Cape Hatteras indicate that there can be significant growth for fluctuations with length scales (wavelength/27r) of order ~ (Watts and Johns, 1982). Therefore, to properly investigate the instability of the flow in this region, we need to consider the 'classical' case of Bu = 0(1). Much of the previous work (Tareev, 1965; Robinson and Gadgil, 1970; Robinson et al., 1975) on Gulf Stream instability has focused on the long-wave limit of (2), which is strictly appropriate only for very long waves (wavelengths > 1000 km). Substitution of the vertical derivative of (2) into (1) and linearizing about the mean pressure field results in the well-known linearized quasi-geostrophic potential vorticity equation --

0,

+ U--

~7 2 p , + _ _

0z

pzj

+

PxQv = 0

(3)

where a

f2

)

is the cross-stream gradient of the basic state potential vorticity.* Boundary conditions appropriate for the ocean require that there be no flow normal to upper and lower rigid boundaries at z = H and z = 0, respectively. Variations in topography along the lower boundary are approximated here by a uniformly sloping plane boundary h (y), with depth increasing to the right of an observer facing downstream. At the upper (horizontal) boundary, z = H, the linearized boundary condition is

+ U-- p'=Uzp ~ z=H Ox

(4)

while at the lower (sloping) boundary, z = 0, it is

Ot +

Ox p ' = U z ( 1 - h y ) p "

z=0

(5)

where h * - N Z h v / f U , is the ratio of the bottom slope to the slope of the deep isopycnals. Y

.

* Strictly eq. (3) is correct only for zonal flows due to the presence of the fl term. However, it will be shown later that fl is negligible, being roughly two orders of magnitude smaller than the basic Gulf Stream potential vorticity gradient.

327 The remaining step needed to close the instability problem is the specification of lateral boundary conditions. In the general case, where the mean fields U, N and Qy may be functions of both y and z, (3) permits mixed barotropic and baroclinic instabilities. We will consider here only the formally y-independent (baroclinic) problem, in which fluctuations draw their energy exclusively from the mean potential energy of the flow. This eliminates a possible mechanism for determining internally the cross-stream scale of the fluctuations, and therefore the cross-stream scale must be externally imposed. The simplest choice is to confine the perturbation between channel walls located at y - - + L, upon which the normal velocity must vanish t

px--0

y=_L

(6)

Appropriate choices for the cross-stream scale L and the simplified, y-independent mean fields are discussed subsequently. Wavelike solutions, periodic in x, which satisfy (6) are of the form

p'(x, y, z, t) = i f ( z ) cos(\2LrrY) e ik(x-ct)

(7)

where +(z) and c are the (possibly complex) amplitude and phase speed c, + ici of the perturbation and k is the (real) downstream wavenumber. If ci is positive for a given wavenumber k, then the perturbation will grow exponentially in time as exp(kcit ). Spatially growing modes with k complex and c real are also possible (Hogg, 1976), but will not be treated here. Substituting (7) into (3), (4), and (5), with no y-dependence in U o r Qy, yields the system of equations o-S

-

4L 2

-( U- - c)

( U - c)~k, = U,~k z = H

(U-c)q~z=U,(1-h;)~k

¢=o

(8)

(9)

z=O

(10)

These equations form an eigenvalue/eigenvector problem which may be solved numerically for arbitrary vertical profiles of U, N, and Qy t o yield the structure qJ(z) and phase speed c of evolving perturbations. To the extent that the model is physically accurate, perturbations with c i > 0 should emerge rapidly from the background noise field as readily observable phenomena. Necessary conditions for instability in the simplified one-dimensional baroclinic instability model defined by (8)-(10) may be derived by multiplying (8) by the complex conjugate if*, integrating (by parts) over depth, and

328 substituting in the boundary conditions (9) and (10). The imaginary part of this expression is

N21U_cl2

, -

NZ I U - c l

z

o +

IU-cl

z dz

=0

(11)

where ~p2 has been used to designate the real, positive-definite quantity ~b*~k. Unstable waves, for which ci > 0, must satisfy this condition. For non-zero Qy, there are three basic possibilities:

(1) Qy changes sign between the top and bottom boundaries. (2) At some depth, Qy is of opposite sign to ~ at the surface. (12) (3) At some depth, Q v is of the same sign as Uz(1 - h*) at the bottom. A discussion of which of these conditions is relevant to a given problem necessarily depends on the specific U, N, and Qy profiles. This topic will be explored in the next section. The one-dimensional system of differential equations (8), (9) and (10) was solved numerically by decomposing (8) into a system of linear algebraic equations at evenly spaced levels chosen to adequately resolve the important features of the basic state U, N and Qy profiles. As is discussed in the next section, the smallest-vertical-scale variations in ay which are relevant to the instability problem are located in the main thermocline region and have vertical scales of 0(100 m). Thus for an ocean depth of 3000 m, a minimum of 30 levels would be required *. In the actual computations, shown later, 40 levels were used, which was found to be sufficient as this resulted in eigenvalue/eigenvector estimates consistently within 2% of values obtained using much finer vertical resolution. Before the actual calculations the model was verified against the basic Eady (1949) problem, with Qy = 0, and the one-dimensional numerical problem considered by Gill et al. (1974) for the instability of mid-ocean regions, which includes several cases of non-zero Qy and h y . 3. BASIC STATE PROFILES In simplifying the general instability problem to a one-dimensional problem the usual procedure is to specify the basic state profiles U, N, and Qy so that they are representative of the average properties of the flow over its cross-section (Bryden, 1979). Killworth (1980) has discussed the limited relevance of such local (i.e. y-independent) calculations in determining the * Of course, to minimizecomputation costs a model with variable depth incrementsmay be constructed, with larger depth incrementsused whereproperty gradients are weak.

329 stability properties of two-dimensional flows. In particular, he notes that for certain flows the solutions to the local eigenvalue problem at various y can be quite different, and that the actual two-dimensional solutions do not necessarily tend toward the local solutions, even for flows with cross-stream length scales large compared with the deformation radius. It is therefore not entirely clear what is gained by averaging the properties of the flow over the cross-section, except that this seems more relevant than doing strictly local calculations based on profiles measured at some arbitrary point in the cross-section. We will use the same approach here. The basic assumption, then, is that the gravest cross-stream mode instability should be relatively insensitive to small-scale variations in the cross-stream distribution of Qy, but must interact with and draw its energy from the integrated Qy profile over the Stream's width. To determine the extent to which this is actually true for a given flow would require two-dimensional calculations; however, for the purposes of this paper, we focus our attention on the simpler one-dimensional problem, so that the basic baroclinic instability mechanism and the means by which the necessary conditions for instability are satisfied in this case can be clearly illustrated. Two sources of data have been used to estimate the basic state profiles needed for the stability analysis. Firstly, two CTD sections to 2000 m depth across the Gulf Stream near 73°30'W taken during July 1982 were used (Johns, 1984). These sections were done in conjunction with 'Pegasus' velocity profiler surveys, so that simultaneous reference velocity observations were available. Additionally, four 'Pegasus' velocity sections (H.T. Rossby, personal communication, 1983) taken in May-November 1981 and extending to depths > 3000 m were used to define the deep (2000-3000 m) structure, and to provide needed redundancy over the upper water column. Each section was projected normal to the (known) Stream path to eliminate any broadening of property gradients due to oblique intersection. For the CTD data, velocity profiles relative to zero velocity at 2000 m were computed by the dynamic method, using station pairs which spanned the central baroclinic portion of the Stream. For the 'Pegasus' data, the average velocity profile was obtained by integrating the downstream velocity component across the Stream's width 1

t

U(z) = 7 fo V(y, z) dy Reference velocities, U(2000 m), obtained in this manner were added to the CTD shear profiles to obtain geostrophic velocity profiles. Bracketing stations were selected such that the onshore station had a 15°C depth of - 1 5 0 m and the offshore station had a 15°C depth of - 7 0 0 m. The cross-stream separation between these bracketing stations

330

Co) -50 0

(b)

U [cm/sl 0

50

100

N 2 ilOSs-21

(C) Oy icm-1 s 11 xlO-~1

150

500

/y

1000 E ..t...

I-'eL 1500 laJ t', 2000

i

'

I

2500 t .]

30001

Fig. 1. Ensemble-averaged basic state profiles: (a) velocity profile, U; (b) Brunt-Vaisala profile, N2; and (c) potential vorticity gradient profile, Qy. The dashed lines represent the + lo envelope about the ensemble-averaged profile.

ranged from 92 to 128 km, with a mean separation of 112 km. Choosing stations significantly onshore or offshore of these locations tends to artificially broaden the current and reduce its vertical shear. However, no significant differences in the vertical profiles resulted from choosing stations nearer to the center of the Stream, down to a separation of 60 kin. The mean and standard deviation of the individual stream-average velocity profiles are shown in Fig. l a. Stream-averaged sigma-theta (o0) profiles were computed for each section by averaging together the profiles at each station between and including the bracketing stations. (Intermediate stations were included to take into account the curvature of density surfaces across the Gulf Stream, and to provide added smoothing.) These were then differentiated to form N 2 profiles. The average N 2 profile and its standard deviation are shown in Fig. lb. An assumed TS relationship (Watts, 1983) was used to calculate density from the 'Pegasus' temperature data up to depths of 200 m, above which the computation of density from temperature becomes unreliable. As a result, the mean N 2 profile in the upper 200 m is based only on the two summer CTD sections; however, a consideration of the necessary conditions for instability (discussed later) indicates that seasonal variations in the upper 200 m are not important. (This has also been verified directly.) Neglecting horizontal shear, the basic cross-stream potential vorticity gradient is defined as Q Y = f l - ~z

Uz

331

Using the thermal wind relation, Uz = g-Py/Pof, and the definition of N 2, the second term may be rewritten

f which has the simple conceptual interpretation as the change in slope of the density surfaces with depth. For bracketing stations separated by a fixed distance 1, this can be expressed as

fo

Qy=fl+---[dzp(z)] l ~z where d z o is the difference in depth of a given density surface across the section, and for consistency the depth at which a certain dz 0 occurs is specified by the depth at which that particular p occurs in the section-average density profile just computed. This method of computing Qy eliminates variations which may be introduced by the different averaging used to obtain the U and N 2 basic state profiles, a difficulty which has been discussed by Bryden (1979). Before computing the Qy profiles, bracketing density stations were smoothed in the vertical by a least-squares cubic spline function which was allowed to deviate from the raw density profiles by an r.m.s, amount of 0.0005 g c m -3 for the C T D profiles and 0.0015 g c m -3 for the 'Pegasus' profiles. As with the velocity profiles, no significant differences resulted in computing the Qy profiles over different scale widths ranging from - 60-120 kin. The mean Qy profile and its standard deviation are plotted in Fig. lc. Two features shown there were consistently present in all profiles: (1) an upper region of pronounced positive Qy with peak values approaching 20 × 10 -az era-is -1 near 200 m, and (2) a region of negative Qy near 1300-1500 m with peak values of - - 5 × 10 -12 c m - 1 S-1. The Oy profile is also weakly negative in the 600-800 m range, below which it becomes slightly positive again before returning to negative values toward 1400 m. This triple-zero-crossing behavior occurred in five of the six individual profiles. In deep water below 1600-1700 m the Qy profiles were generally negative, with values near - 1 × 10-12 c m - 1 s- 1. It seems logical to ask whether the Uyy term, formally neglected in the y-independent problem, can be large enough to substantially alter the potential vorticity gradient arising from variations in layer thickness. Although Uyy may be large locally, for a stream-averaged profile the contribution to Qy by this term can be approximated as Uyy = - 3__U/2A2, where A is the half-width of a parabolic jet with speed Umax = 3U/2 at y = 0 and Umi~= 0 at y = +A. The size of Uyy(Z) is therefore proportional to U(z),

332 and for a consistent choice of A = 50-60 km ranges from - - 4 . 0 × 10 -12 cm -1 s -a near the surface to < - 1 . 0 x 10 -a2 cm -1 s -1 below a depth of - 800 m. This term is not large enough to change the sign of the basic Q~. profile of Fig. lc (except perhaps very near the surface) and otherwise serves to reinforce the negative Qy regions centered near 750 and 1400 m, which, as will be shown later, are essential for instability. It is also quite evident from Fig. lc that fl - 0.2 × 10 -12 is small compared with the basic potential vorticity gradient, even in deep water. 4. RESULTS 4.1. D i s p e r s i o n

Dispersion curves were computed for several cases of basic state profiles and bottom slope parameters hy by varying the wavenumber k incrementally through 0.001 < k < 0.06 km -1, thereby spanning wavelengths 2 ~ r / k of - 1 0 0 - 6 0 0 0 km. Figure 2 shows the dispersion results obtained for the average basic state profiles, with h* = 2 and L = 75 km (channel width = 150 km). This is considered to be the most realistic case. (In this figure, as in all subsequent dispersion diagrams, the parameters c r (propagation speed) and o = k c i (temporal growth rate) are plotted versus frequency f = k c r as the independent variable rather than wavenumber, in order to facilitate later comparisons with observational data for which frequency is the measured independent variable.) The choice of L = 75 km is somewhat arbitrary, although Johns and Watts' (1985, 1986) observations suggest that a crossstream scale of 50-100 km is probably most reasonable. Physically, variations in L are uninteresting here in the sense that changing L amounts to a simple shift of the results in wavenumber space, without changing the underlying dynamics or vertical mode structures. In theory the cross-stream scale could be infinite, though in practice choosing L to be much greater than 100 km (channel width of 200 kin) has a significant effect on the results only at very small k. The dispersion relation of Fig. 2 is characterized by three distinct regions of instability, centered near periodicities of 5, 7 and 28 days, which are separated by regions where no growth occurs. The center wavelengths associated with these modes are 180 km for the 5 day period, 220 km for the 7 day period, and 390 km for the 28 day period. The peak exponential growth rates for the 5 day and 28 day period instabilities are similar, approximately o = 0.075 day -a, while the peak growth rate for the 7 day instability is less, o = 0.05 day -1. These growth rates correspond to perturbation amplitude e-folding time scales of - 13 and 20 days, respectively.

333

I>., o'


o'

28

Sday 180 km i

390 .a~ km

i

//.""'"'",..,.

7 day 220/~""1km /

~ :

! o

O. GO

0.05

o',o

o'~s

o'~o

o'. 2S

o.3o

o'. ts

o ~o

FREQUENCY (CPD)

o

,o .........--*'°'""" ~.t ,S

E bJ (3.

°o. oo

o" os

o: ~o

o" ~s

FREQUENCY

o" 20 (CPD)

Fig. 2. Dispersion diagram for the ensemble-averaged basic state profiles, with L = 75 km and h* = 2. The lower and upper panels plot the real phase speed Cr, and the temporal growth rate a = kc i, as a function of frequency f = kc r. The solid, dashed, and dotted lines correspond to three distinct growth regions in different frequency bands.

The propagation speeds generally increase toward shorter periods, except in a limited region spanning frequencies of - 0.05 to 0.08 day-1 (periods of 1 3 - 2 0 days) where the propagation speed decreases slightly. In the lower frequency regime the propagation speeds range from - 9 to 18 cm s -1, beyond which there is a jump to propagation speeds within a range of - 32-43 cm s-1 for the higher-frequency modes. The sensitivity of these dispersion results to the observational uncertainty in the basic state Qr profile is illustrated in Fig. 3, where the two standard deviation envelopes of the 'observed' Qy profile have been used as input Q y profiles. The basic result shown there is that unstable modes grow faster and

334

O0

"[3

%~ oo

o~ os

o~ ,o

o' ~s

FREQUENCY

o'. =o

o~ 25

030

(CPD) °°..... ......

,n

,o°

..--°"

°°.o

hi 13-

°0. O0

O~ 05

FREQUENCY

(CPD)

Fig. 3. Comparison of dispersion results obtained using the ensemble averaged Qy profile (solid), and the left-hand (dotted) and right-hand (dashed) standard deviation envelopes of the average Qy profile, for L = 75 km and hy - 2.

occupy a broader frequency band when negative Qy values occur at shallower depths and span a greater depth range (e.g. when using the left-hand Qy envelope of Fig. l c as the input Qy profile). Conversely, unstable modes grow more slowly and are confined to a low frequency band when negative Qy values occur deeper in the water column and span a smaller depth range (e.g. when using the right-hand Qy envelope as input). The reasons for these variations are elaborated upon below. The above two cases should probably be considered as extremes, since the profiles represent a bias of the average Qy profile by one standard deviation to the left or right rather than an uncertainty distributed randomly over the water column.

335

4.2. Evaluation of the necessary conditions Considerable insight into how the unstable modes of Figs. 2 and 3 arise can be gained by considering three 'idealized' basic state Qy profiles, shown in Fig. 4. These Qy profiles are identical above 500 m and below 1300 m, but differ in shape within the 500-1300 m range, where each profile has a single zero-crossing located near depths of 650, 850 and 1100 m, respectively. For each profile the computed dispersion curves (shown in Fig. 5) are very similar for frequencies lower than - 0.05 day-1 but become different at higher frequencies. In each case the m a x i m u m propagation speed (hence highest frequency) associated with an unstable m o d e coincides exactly with the value of the m e a n downstream velocity at the level where the Qy profile crosses from positive into negative values. It m a y therefore be stated as a general result for these 'ideMized' profiles that only modes which have their 'steering' level (defined as the level at which c r = U; Pedlosky, 1979) within the region of negative Qy can be unstable. This is obviously not the only requirement, since there are no unstable modes found which have steering levels deeper than - 1800 m, despite the fact that Qy is negative throughout the deep water column. It is clear that there are distinct upper and lower limits imposed on the location of the steering level by the Qy distribution. This effect m a y be understood qualitatively by considering the necessary condition for instability (11). Since the Qy profile has both positive and

Oy( c m -I s - ; i 0

-2

-1

x l O -11 1

500.

1000

E "tI'- 1 5 0 0 0.. UJ a :2000

2500

3000

Fig. 4. 'Idealized' Qy profiles. The three profiles have different shapes within the main thermocline region (500-1300 m), but are identical above 500 m and below 1300 m. The dotted curve (denoted as the '1st' profile) changes sign near 650 m, the solid curve ('2nd' profile) near 850 m, and the dashed curve ('3rd' profile) near 1100 m.

336 o

"!i o

,,

.-'"-.

o

i

O. O0

O. 0 5

O. 10

O. 15

FREQUENCY

O~ 2 0

o: 25

0.30

0:25

o 30

(cP0)

...,,.'-""

o.

.,'" ....""

E ra ILl hi

~o.

/

,•.. ,,'"'•

,, ¢ ,,' /

n

o.

°o.oo

0:05

o: 1o

o'. ,5

o:2o

FREQUENCY (CPD) Fig. 5. Dispersion diagrams for the three 'idealized' Qy profiles, with L = 75 k m and hy - 2• The notation is as described in Fig. 4: 'lst' Qy profile (dotted), '2nd' Qv profile (solid), and '3rd' Qv profile (dashed)•

negative signs, all three of the general conditions (12) are satisfied. However, due to the presence of the I U - c l 2 factor in the denominator of each term in (11) it is also evident that the boundary terms can play an important role only if the steering level is located sufficiently close to a boundary. The absence of steering levels near the boundaries in the present problem implies that these instabilities are basically of 'interior' type. This may be estabfished more quantitatively by considering the relative size of the terms in

(11)• For the basic state under consideration, and for k, c, ~ spanning the entire range of values observed in the modes of Figs. 2 and 5, the size of the upper boundary term ranges from - 0 . 1 - 0 . 5 × 10 -1° s -1 and the lower boundary term ranges from - 1-10 × 10 -1° s -1. This lower boundary term

337

only becomes large for low-frequency waves, which have relatively deep steering levels. By contrast, a typical 100 m incremental contribution to the integral term at a depth near the steering level is of size 1-10 × 10 -1° s -1, assuming values of Qy = 1-10 × 10 -12 cm -1 s -1 and I U - cJ = 5-10 cm s -1. Owing to the [U-cJ 2 dependence, these contributions to the integral are necessarily largest near the steering level, even for quite large variations in Qy and ~k. * The constraints on the location of the steering level may now be stated as follows. Since the boundary terms are of the same order as (or smaller than) the incremental contributions to the integral term, it follows that the value of the entire integral must also be of the same size as its incremental contributions. This can only be accomplished by an adjustment of the steering level to a depth where Qy has both positive and negative values within a limited depth region; i.e. in order to satisfy the necessary condition for stability the steering level must be located 'near' sign changes in Qy. This is why there is a lower limit to the depth of the steering level. The reason why steering levels are always found within negative Qy regions is due to the extremely large positive Qy values found in the upper 500 m. To offset this positive contribution to the integral, the steering level must be located within a negative Qy region so that I U - c J is small there. Returning once again to the dispersion curve for the 'observed' basic state (Fig. 2), it is now possible to associate each instability which is found with a particular feature of the Qy distribution. As shown in Fig. 6, instabilities in the low-frequency regime, spanning frequencies of - 0 - 0 . 0 8 day -1, have their steering levels within the depth range of 1000-1500 m and are associated with the negative Qy region there. The higher-frequency modes, spanning frequencies of - 0.11-0.23 day-1, have their steering levels within the upper region of negative Qy between depths of 600 and 800 m. The absence of instabilities with steering levels in the 800-1000 m range is evidently due, for reasons discussed above, to the region of positive Qy values located within that depth range. A final point to be made here is that the low-frequency regime of instabilities appears to be associated with a significant feature of the Gulf Stream Oy distribution (the negative region near 1400 m depth), whereas the higher-frequency regime is associated with more uncertain qualities of the

* It will be shown later that the vertical structure qJ(z) is similar to a 1st baroclinic mode, and is observed to change remarkably little as the wavenumber is varied, except near growth discontinuities in the dispersion relationship. This fact allows one to gather considerably more intuition about the effects that variations in wavenumber have on the eigenvalue c than would be possible if both c and qJ varied significantly, and is rather crucial to heuristic arguments presented later concerning the effects of variation in bottom slope.

338 Oy (crn-1 s-~} ~10-"

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Fig. 6. Ensemble-averaged U and Qy profiles, illustrating the range of steering level depths (where c = U) for low-frequency modes (lower shaded region) and for the higher frequency modes (upper shaded region).

upper (600-1300 m) Qy distribution. The dispersion results are found to be quite sensitive, in fact, to small variations in the Qy profile within this upper (thermocline) region.

4. 3. Variations in bottom slope The effect of bottom slope on the stability of the flow was investigated by varying h~' over a range of roughly one-half to twice the actual bottom slope. For illustrative purposes we show in Fig. 7 the dispersion results for the 1st 'idealized' Qy profile (see Fig. 3), for h~ = 1, 2 and 4. The major effect is an increase in growth rate at low frequencies as the bottom slope is reduced. A secondary but noticeable effect is a slight shifting of the growth rate curves toward higher frequencies and wavenumbers. Notably, changes in bottom slope have almost no effect on the growth rates associated with the highest-frequency unstable modes near periods of 5 days. Another important qualitative effect is that there is actually reduced growth at the low-frequency end of the dispersion diagram ( f < 0.03 day-1) as the bottom slope is decreased from h~ = 2 to h~ = 1. Thus, although decreased bottom slope is usually destabilizing, it is not uniformly destabilizing. Furthermore, from the propagation speed curve it is apparent that the

339 o 6

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%.00

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Fig. 7. Dispersion diagrams for the 1st 'idealized' Qy profile, for three different values of hyW ,.hyW =1 (dotted), hyW = 2 (solid), and hyW = 4 (dashed).

case hy = 1 is markedly less dispersive at lower frequencies ( f < 0.01 day-1) than the other two cases. This peculiar effect at low-frequencies can be better understood by considering again the necessary condition for instability (11). Firstly, since the bottom slope h~ appears only in the second term (h* enters the problem through the lower boundary condition), its effect is large only for instabilities which have relatively deep steering levels, i.e. for which IU-c1-2 evaluated at the lower boundary is significant. This is why the higherfrequency, rapidly-propagating modes are only weakly affected by changes in the bottom slope. The second and related consideration is that for deep steering levels the principal balance in (11) is between the last two terms,

340 since the first term changes little as c r varies within its range for lowfrequency waves. Consider now the case for a bottom slope which is steeper than the slope of the isopycnals at the bottom (e.g. hy* = 2), in which case the second term in (15) is positive definite, i.e.

Note also that the steering level ranges from depths of - 700-1500 m (10 cm s - l < cr < 40 cm s-l), where the Qy profile is becoming increasingly negative with depth. As the wavenumber is decreased, the balance in (11) is maintained by a continuous adjustment of the steering level deeper, which has two effects: (1) an increased negative contribution to the integral (third) term due to the fact that smaller [U-c] values occur with more negative Q~ values; and (2) an increased positive contribution from the second term due to a decrease in [ U - c I at the lower boundary. Thus the balance can be maintained without requiring large changes in the vertical structure tp (z). As the bottom slope is further increased (e.g. hy* = 4), the second term becomes more positive for any given c, but the balance between the second and third terms is achieved by a slightly different 'initial' positioning of the steering level relative to the Qy profile. For the case h~ = 1, where the bottom slope is equal to the slope of the isopycnals at the bottom, the second term in (11) is then trivially zero. In this case, any deepening of the steering level beyond a critical point would increase the negative contribution from the integral term, without an offsetting contribution from the bottom boundary term. Clearly, unless significant changes in ~2 occur the steering level must be maintained within a limited depth range in order to retain the positive contribution to the integral term from the upper Qy region. Thus, one might expect to see a tendency toward less rapid dispersion at low frequencies, which actually occurs. In fact, the only way that the balance in (11) can be maintained for deeper steering levels is by a change in the vertical structure of the perturbation, which tends to result in structures less favorable to growth. A third case which is not observationally relevant but is of academic interest is the case where the bottom slope is less steep than the slope of isopycnals at the bottom. Figure 8 shows the results for h~ = 0, using each of the three 'idealized' Qy profiles. The striking result is that the flow is virtually stable to fluctuations with frequencies lower than - 0 . 0 1 day -a (periods longer than about 10 days), independent of which Qy profile is used. Considering again the necessary condition (11), where now the bottom boundary term is negative definite, it is clear that any deepening of the

341

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Fig. 8. Dispersion diagrams for the three 'idealized' Qy profiles, with hyl i t - 0. The notation is as described i n Fig. 4: 'lst' Qy profile (dotted), ' 2 n d ' Qy profile (solid), and ' 3 r d ' Qy profile (dashed).

steering level beyond a critical point would increase the negative contributions from both the second and third terms. There is therefore a strong 'resistance' to the dispersive character seen in the above cases, and much greater changes in vertical structure are required to satisfy condition (11) as the wavenumber is decreased. Eventually a point is reached where there are no unstable modes which can satisfy (11), beyond which only small, isolated regions of weak growth are found. These results emphasize the important fact that changes in bottom slope may be either stabilizing or destabilizing, depending upon the shape of the actual Qy profile. They also raise the interesting possibility that the initial evolution of long waves in the Gulf Stream depends, paradoxically, on the

342

presence of a rather strong bottom slope, which is normally perceived as a stabilizing influence. 4.4. Vertical structure

The amplitude ( 6 ) and phase (~) profiles for the unstable modes of Fig. 2 are shown in Fig. 9, where the solid, dashed, and dotted curves indicate the structure of the most unstable modes within the low-frequency regime [k -- 0.016 km -1, f = 0.035 day -1, Cr = 15.8 cm s -1, o ----0.078 day-l], and in the two higher-frequency regimes, [ k = 0 . 0 2 8 km -1, f = 0 . 1 3 4 day -1, cr = 34.7 cm s -I, o = 0.051 day -1] and [k = 0.035 km -1, f = 0.198 day -1, Cr = 41.1 cm s -1, o = 0.073 day-~], respectively. The amplitude structure for all three modes is similar, with a maximum occurring near 200 m depth. The only real difference between these modes is that the deep-water amplitude for the low-frequency mode ( 6 - 0.2) is about twice as large, in relative terms, as the deep-water amplitude for the higher-frequency modes ( ~ - 0.1). The primary difference between the higher and lower frequency modes emerges when considering the structure of phase with depth. The sense of the phase variation is such that a phase which decreases with depth implies wavefronts which tilt westward with height. This phase tilt, opposite to the sense of the mean vertical shear, is required to release the available potential energy from the mean flow (Pedlosky, 1979). The higher-frequency modes are characterized by a total phase shift of only - 10 o, with most of this

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343 AMPLITUDE

0.25 i

0,50 I

PHASE

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occurring at or just above the steering level, between depths of 500 and 700 m. The lower-frequency mode has a larger total phase shift, - 35 o, which occurs over a larger depth range but which is again concentrated near and above the steering level, in this case near 1100 m. An interesting feature that all three modes have in c o m m o n is that near 1500 m depth the phase reaches a minimum, beyond which it increases monotonically with depth. This effect is quite visible in the low-frequency mode, where the phase increases by 3 ° between 1500 m and the bottom, but is difficult to see on the scale shown for the higher-frequency modes, as the increase between 1500 m and the bottom for these modes is only 0.2-0.3 °. This implies a weak energy conversion which is of the opposite sign, from the fluctuations to the mean, in deep water. The effect of variable bottom slope on the vertical structure of perturbations is examined in Fig. 10, where the vertical structure of modes with the same frequency, f = 0.06 day-1 (also approximately the same wavenumber), for the 1st 'idealized' Qy profile is plotted for each of the b o t t o m slope cases hy = 1, 2 and 4. Increasing the bottom slope has two major effects, causing: (1) a decrease in the deep-water amplitude of the perturbations, and (2) a smaller total phase shift through the thermocline. Thus the perturbations typically become more baroclinic and grow more slowly as the b o t t o m slope is increased. A secondary but important effect is the change in the shape of the phase profile in deeper waters. For the case h* - 1 the phase decreases monotoni-

344 cally with depth, in contrast with the change in sign of dg,/dz at intermediate depth seen in the other two cases, h~ = 2 and 4, where the bottom slope is steeper than the slope of isopycnal surfaces at the bottom. For h*---2 the phase reaches a minimum at - 1 6 5 0 m, whereas for the case hi* = 4 this occurs at - 1 3 5 0 m. The effect of increasing bottom slope is therefore twofold. In addition to causing a decrease in the total phase shift through the thermocline, bottom slopes which are steeper than the deep isopycnal slopes effectively 'stabilize' the entire deep water column. Further increasing the bottom slope causes this stabilized region of weak eddy-mean energy transfer to penetrate to shallower depths. While this secondary effect has little impact on the overall energy transfer, it does have important implications concerning the inference of the overall stability of a flow from measurements taken even quite far up in the water column. This point is discussed further in the next section. 5. COMPARISON WITH OBSERVATIONS In this section, the model results are compared with direct observations of Gulf Stream meander variability immediately northeast of Cape Hatteras, NC. As with all linearized models, these comparisons are subject to the usual criticism that the fluctuations are in reality of finite amplitude, in which case non-linear effects may become important and could alter the 'basic state'. However, it should also be noted that meander amplitudes are observed to reach a local minimum of 5-10 km r.m.s, near Cape Hatteras (Watts, 1983), downstream of which they grow rapidly. The linearizing approximation of infinitesimal amplitude is therefore considerably less restrictive in this region than it would be farther downstream. In recent years, the observational data base for comparison has increased dramatically. Much knowledge has been gained from inverted echo sounder (IES) data about the propagation and growth characteristics of Gulf Stream meanders here. These observations include those reported by Watts and Johns (1982), as well as an additional year of measurements reported by Tracey and Watts (1986). Also, recent deep current-measurements (Johns and Watts, 1985, 1986) spanning a total duration of over 18 months have been collected in this region and are beginning to provide useful statistics on the energy transfer rates associated with growing meanders. Figure 11 shows two realizations of the 'observational' dispersion relationship for Gulf Stream meanders near 74 o W, each derived from a year-long time series of IES array measurements. These were obtained by tracking the position of the Gulf Stream's north wall at a series of IES sections spaced along the path, and using the observed path displacement spectra and phase lags to infer the propagation and growth rates occurring within various

345

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Fig. 11. Comparison of observational dispersion parameters and model dispersion results computed for the ensemble-averaged basic state profiles. The dotted and dashed lines represent IES-derived dispersion parameters for Gulf Stream meanders from 2 year-long deployments during 1979-80 and 1981-82, respectively.

frequency bands. The methodology is discussed thoroughly in Watts and Johns (1982). Estimated observational uncertainties in determining c r and o are roughly + 25 and + 50%, respectively. The observational growth rate curves suggest two regions of significant growth, a low-frequency region spanning frequencies less than f = 0.10 day -1 (periods > 10 days), and a higher-frequency region spanning frequencies of f = 0.15-0.25 day -1 (periods of 4 - 7 days). The two observational curves are quantitatively similar in the higher-frequency growth region, with peak growth rates of o = 0.13-0.16 day -1, but are quite different in the low-frequency region, where peak growth rates range from o = 0.04-0.17

346 day-~. Both of these growth rate curves indicate a relative growth minimum occurring near frequencies of 0.10 day -1, consistent with the results of the instability model. In interpreting the first year data, Watts and Johns (1982) pointed out that the growth minimum near f = 0.10 day-1 corresponded closely with the location of the high-wavenumber temporal growth cutoff found in Phillip's (1954) two-layer baroclinic instability model and also that found in Hogg's (1976) continuously-stratified model. (Pedlosky (1979) treats several baroclinic instability models, in all of which the perturbations propagate as exp [ik(x- ct)] with k real and c complex, and in which the temporal growth rate o = kc, exhibits a high-wavenumber cutoff.) Watts and Johns (1982) further suggested that the higher-frequency instabilities might be spatially-growing modes, consistent with Hogg's (1976) finding that instabilities which include complex k can exhibit spatial growth at wavenumbers greater than the temporal growth cutoff. An important result of the present model is that the observed separation of growth regions may plausibly be due to the particular structure of the Gulf Stream Qy profile, rather than the existence of a new class of instabilities characterized by spatial instead of temporal growth. * In addition to similarities in the growth rate curve, the propagation speed curve also compares favorably with the observations. At low frequencies the observed propagation curves differ somewhat, but there is good agreement between the model and the most recent set of observations. The observed propagation speeds in the high-frequency regime are consistent between the measurement periods, and are - 15-20% higher than predicted. It is also notable that the propagation speeds derived from the most recent set of observations show a rapid increase within the transition region between growth regimes, consistent with the 'jump' in propagation speeds predicted by the model near f = 0.10 day-1 The model therefore predicts not only the general dispersive character of the fluctuations but also quite accurately predicts the magnitude of the propagation speeds. This contrasts with Orlanski's (1969) model, for which the predicted propagation speeds are smaller than those observed by a factor of two to four. A second comparison between the model and the observations concerns the structure of the baroclinic energy transfer with depth. As shown in the previous section, low-frequency instabilities are predicted to have significant mean-to-eddy energy transfer extending to depths of 1500 m (below which

* This is not to say that spatially growing disturbances are absent. In fact, the observational data are more suggestiveof spatial growth, especiallyat the higher frequencies(see Watts and Johns, 1982).

347 v,T Correlation Coefficient 12- 48 DAY BAND -1.0 0

-0.5 r

0.5 r

4 - 10 DAY BAND 1.0

-1.0

-0.5

0,5

~

5

1.0

day 7

day

1000

0S 2000

3000

--.IL

- -

A

-.--.s__~

L____a~

Fig. 12. Comparison of vertical profiles of the velocity-temperature correlation coefficient (proportional to the down-gradient eddy heat flux) for the 28, 7 and 5 day modes, against observed (band-averaged) estimates from current meters (shown by triangles). The observed correlation coefficients are separated into two bands, spanning periods of 12-48 days and 4-10 days, chosen to roughly coincide with the two observed growth regions of Fig. 11. The error bars indicate the 95% confidence limits for the correlation estimates.

there is a reversal in the sign of the energy transfer), whereas the energy transfer for the higher-frequency instabilities is essentially confined to the upper 700-800 m. Figure 12 shows profiles of the observed versus predicted correlation coefficient between cross-stream velocity and temperature, for both the high and low-frequency growth regimes. A positive correlation indicates a down-gradient, or energy-releasing, eddy buoyancy flux. The observed correlation coefficients are band-passed estimates spanning periods of 12-48 and 4-10 days, respectively, chosen to roughly coincide with the two observed growth regions in Fig. 11. In the low-frequency (12-48 day) band there is a distinct trend toward positive correlation coefficients above 1500 m and negative correlation coefficients below - 2 0 0 0 m. (The exception to this is a single outlying estimate showing an apparently significant positive correlation at 2000 m depth.) In the high-frequency (4-10 day) band the observations indicate no such trend, with generally negative correlation coefficients occurring throughout the deep water column. Thus, although few of the correlation coefficients computed from individual current meter records are actually significant, the overall trends in the data are qualitatively consistent with the predictions of the model. Concerning future direct measurements, the model results suggest that it should be possible (if, in fact, these simple baroclinic

348 waves are at all representative of real Gulf Stream instabilities) to measure significant mean-to-eddy energy transfer for the low-frequency waves, but probably quite difficult (with typical moored record lengths) for the high frequency waves. Yet, as these calculations illustrate, even small correlation coefficients occurring over a limited depth range can result in quite rapidly growing disturbances. 6. SUMMARY A linear, one-dimensional, baroclinic instability model applied to crossstream averages of the observed velocity, stratification, and potential vorticity gradient fields in the Gulf Stream near Cape Hatteras is found to be surprisingly successful in predicting the time and length scales of growing meanders. The model predicts two regimes of instability: a low-frequency, long-wave regime with center period and wavelength (T, X) = (28 days, 390 km), and a higher frequency, short-wave regime with (T, X ) = (5-7 days, 180-220 kin). Consideration of the necessary conditions for instability shows that these unstable regimes are associated with two distinct minima in the mean potential vorticity gradient, occurring in and below the main thermocline at depths of - 6 0 0 - 8 0 0 and 1300-1500 m, respectively. The model also predicts that for realistic bottom slopes the baroclinic energy transfer should be weakly negative (eddy-to-mean) over approximately the lower half of the water column. This effect is most pronounced for lowfrequency disturbances and results from the bottom slope being steeper than the mean slope of the deep isopycnals, thereby forcing deep parcel motions along stabilizing trajectories. All of these predictions are qualitatively consistent with recent deep current-meter-measurements and empirically-derived dispersion curves for Gulf Stream meanders based on inverted echo sounder measurements downstream of Cape Hatteras. Several model runs were also undertaken to examine the sensitivity of the results to variations in the mean potential vorticity gradient profile and bottom slope. Larger bottom slopes were found to be typically stabilizing, although this was not true in general. Variations in bottom slope were also found to have little effect on the higher frequency disturbances. One unexpected result was that long waves were found to be" unstable only for fairly strong bottom slopes (i.e. such that the bottom slope is comparable to or larger than the slope of deep isopycnals), with a low-wavenumber cutoff emerging in the case of zero bottom slope. This is due to the particular structure of the mean Gulf Stream Qy profile, and is completely different from the behavior of uniform potential vorticity flows (cf. de Szoeke, 1975), where disturbances are found to become increasingly unstable as the bottom slope is decreased.

349 For realistic b o t t o m slopes, a class of long-wave instabilities was f o u n d to exist in all cases where the basic state was varied to account for measurement uncertainties. However, for this one-dimensional problem the dispersion results were found to be highly sensitive to uncertainties in the Qy profile in the thermocline region, where comparatively small variations can cause large changes in the properties of higher-frequency disturbances. The propagation and growth curves computed for the ensemble-averaged mean field distributions were nevertheless found to be in generally good agreement with those derived from observations. In summary, the linear, quasi-geostrophic theory appears capable of predicting major qualitative features of both the dispersion and vertical structure of Gulf Stream meanders downstream of Cape Hatteras. ACKNOWLEDGMENTS This work was supported by the Office of Naval Research under contract N00014-81C-0062 and by the National Science F o u n d a t i o n under G r a n t OCE82-01222 to the University of Rhode Island. The computer modeling was generously supported by the U.R.I. G r a d u a t e School of Oceanography. I also thank R a n d y Watts and T o m Rossby of U.R.I. for their helpful comments on the manuscript. REFERENCES Bane, Jr., J.M. and Brooks, D.A. Gulf Stream meanders along the continental margin from the Florida Straits to Cape Hatteras. Geophys. Res. Lett., 6: 280-282. Bryden, H.L., 1979. Poleward heat flux and conversion of available potential energy in Drake Passage. J. Mar. Res., 37: 1-22. De Szoeke, R.A., 1975. Some effects of bottom topography on baroclinic stability. J. Mar. Res., 33: 93-122. Eady, E.T., 1949. Long waves and cyclone waves, Tellus, 1: 33-52. Gill, A.E., Green, J.S.A. and Simmons, A.J., 1974. Energy partition in the large-scale ocean circulation and the production of mid-ocean eddies. Deep-Sea Res., 21: 499-528. Hansen, D.V., 1970. Gulf Stream meanders between Cape Hatteras and the Grand Banks. Deep-Sea Res., 17: 495-511. Hogg, N.G., 1976. On spatially growing baroclinic waves in the ocean. J. Fluid Mech., 78: 217-235. Johns, E., 1984. Velocity and potential vorticity in the Gulf Stream northeast of Cape Hatteras, NC. Ph.D Thesis, University of Rhode Island, RI. Johns, W.E. and Watts, D.R., 1985. Gulf Stream meanders: observations on the deep currents. J. Geophys. Res., 90: 4810-4832. Johns, W.E. and Watts, D.R., 1986. Time scales and structure of topographic Rossby waves and meanders in the deep Gulf Stream. J. Mar. Res., 44: 267-290. KiUworth, P.D., 1980. Barotropic and baroclinic instability in rotating stratified fluids. Dyn. Atmos. Oceans, 4: 143-184.

350 Orlanski, I., 1969. The influence of bottom topography on the stability of jets in a baroclinic fluid. J. Atmos. Sci., 26: 1216-1232. Pedlosky, J., 1979. Geophysical Fluid Dynamics. Springer, New York, 624 pp. Phillips, N.A., 1954. Energy transformations and meridional circulations associated with simple baroclinic waves in a two-level quasigeostrophic model. Tellus, 6: 273-286. Robinson, A.R. and Gadgil, S., 1970. Time-dependent topographic meandering. Geophys. Fluid Dyn., 1: 411-438. Robinson, A.R., Luyten, J.R. and Flierl, G., 1975. On the theory of thin rotating jets: a quasi-geostrophic time-dependent model. Geophys. Fluid Dyn., 6: 211-244. Tareev, B.A., 1965. Unstable Rossby waves and the instability of oceanic currents. Izv. Acad. Sci. USSR Atomos. Oceanic Phys., 1(4): 426-438 (English translation). Tracey, K.L. and Watts, D.R., 1986. On Gulf Stream meander characteristics near Cape Hatteras. J. Geophys. Res., 91: 7587-7602. Warren, B.A., 1963. Topographic influences on the path of the Gulf Stream. Tellus, 15: 167-183. Watts, D.R., 1983. Gulf Stream Variability. In: A.R. Robinson (Editor), Eddies in Marine Science. Springer, New York, pp. 114-144. Watts, D.R. and Johns, W.E., 1982. Gulf Stream meanders: observations on propagation and growth. J. Geophys. Res., 87: 9467-9476. Wright, D.C., 1981. Baroclinic instability in Drake Passage. J. Phys. Oceanogr., 11: 231-246.