vortex generation by topography in locally unstable baroclinic … · 2007. 1. 30. · with the...

JANUARY 2003 207B R A C C O A N D P E D L O S K Y

q 2003 American Meteorological Society

Vortex Generation by Topography in Locally Unstable Baroclinic Flows*

ANNALISA BRACCO AND JOSEPH PEDLOSKY

Department of Physical Oceanography, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

(Manuscript received 21 March 2002, in final form 9 July 2002)

ABSTRACT

The dynamics of a quasigeostrophic flow confined in a two-layer channel over variable topography on thebeta plane is numerically investigated. The topography slopes uniformly upward in the north–south direction(in the beta sense) and is a smooth function of the zonal coordinate. The bottom slope controls the localsupercriticality and is configured to destabilize the flow only in a central interval of limited zonal extent.Linearized solutions indicate that, for a wide enough channel, unstable modes exist for an arbitrary short intervalof instability, confirming previous analysis on disturbances with no meridional variation. For small local maximumsupercriticality, the instability is maintained by a short bottom-trapped wave localized at the downstream edgeof the unstable region and oscillating in phase with the upper-layer disturbance. When nonlinearity is retainedin the problem, the equilibration of the bottom-trapped wave is associated with the formation of coherent vortices.Both cyclones and anticyclones are formed continuously at the northeastern edge of the unstable interval. Throughvortex stretching mechanisms, dipoles inside the interval of instability can split upon reaching the northern wall:Anticyclones move downstream along the north wall and propagate into the downstream stable region, whilecyclonic structures tend to remain trapped inside the interval of instability. The authors suggest the relevanceof their results to the observed eddy field of the Labrador Sea.

1. Introduction

Coherent vortices are fundamental dynamic featuresof geophysical flows. The delineation of the factors gov-erning their formation, structure, and geographical dis-tribution is essential to our understanding of the oceanicand atmospheric circulation. Relevant literature sug-gests that strong eddy activity and enhanced cyclogen-esis are often associated with regions of high baroclin-icity. Hence, the general nature of localized baroclinicinstability has received much attention in the last twodecades. Merkine and Shafranek (1980) and Pierrehum-bert (1984) used the Wentzel–Kramers–Brillouin(WKB) technique to study flows in which the propertiesof the basic state change zonally on scales greater thanthose typical of the unstable disturbances. Frederiksen(1983) approached the problem of regional cyclogenesisnumerically for stationary zonally varying flows. Ped-losky (1989) derived a set of heuristic model equationsdescribing locally unstable flows and solved the eigen-value problem matching the solutions between the stable

* Woods Hole Oceanographic Institution Contribution Number10651.

Corresponding author address: Annalisa Bracco, Department ofPhysics of Weather and Climate, The Abdus Salam International Cen-tre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste 34014,Italy.E-mail: [email protected]

and unstable regions. Pedlosky (1992) and Oh et al.(1993) analyzed both linear and nonlinear solutions fora two-layer baroclinic flow for which stability propertiesare controlled by the friction coefficient of the bottomEkman layer. Samelson and Pedlosky (1990, hereafterSP) focused on local baroclinic instability of a quasi-geostrophic, zonal flow over a meridional bottom slope.

Topographic variations are likely to be responsiblefor the confinement of instabilities in most situations ofinterest to meteorology and oceanography. Indeed, thestability of atmospheric mean winds is affected by thepresence of coastal mountains that act as barriers. In theocean there are regions where, at least locally, the back-ground gradient of potential vorticity of the deep flowis controlled by varying topography, rather than by plan-etary curvature. Indeed the analysis in the present paperwas motivated in part by observations of strongly lo-calized eddy variability near the eastern boundary ofthe Labrador Sea, in a region where the continental slopevaries substantially in the alongshore direction (Lav-ender 2001; Prater 2002; Lilly et al. 2002, manuscriptsubmitted to J. Phys. Oceanogr.).

Samelson and Pedlosky chose a bottom slope such asto render the flow locally stable except in a central re-gion of limited zonal extent, in which the necessaryconditions for instability were satisfied and the localsupercriticality was positive. Their analysis was restrict-ed to perturbations with no meridional variation and thusto the linearized problem (in quasigeostrophic theory

208 VOLUME 33J O U R N A L O F P H Y S I C A L O C E A N O G R A P H Y

cross-stream variations are required for nonlinearity toappear). Surprisingly, SP found instabilities no matterhow short the length of the unstable interval. The naturalquestion that arises is the validation of such a resultwhen disturbances vary meridionally and the linearmodes are confined within a domain of finite width.

The purpose of the present work is to extend, throughnumerical analysis, the linear model of SP by confiningthe zonal flow in a channel on the beta plane. One reasonfor doing so is to mimic the limited lateral extent ofoceanic boundary currents, as in the Labrador Sea. Ad-ditionally this allows us to investigate the equilibrationof the nonlinear modes. Our second motivation is thatthe confinement of the flow leads to a new source ofvorticity in the disturbance field. Indeed, disturbancesthat depend on the cross-channel coordinate geostroph-ically produce zonal perturbation velocities in the lowerlayer that interact with the longitudinal gradient of thetopography. By contrast, disturbances with no meridi-onal variation sense only the latitudinal variation of thetopographic slope.

Solutions presented here show that for wide enoughchannels [i.e., for channels that satisfy the Phillips(1954) criterion for instability] unstable modes exist forarbitrarily small intervals. The equilibration of the lo-calized unstable modes generates coherent vortex struc-tures at the downstream edge of the interval of insta-bility. The formation, evolution, and stability of the eddyfield are consequently investigated.

In section 2 we introduce the mathematical model.Section 3 describes the growth of localized linear un-stable modes for several values of the local supercriti-cality, different channel widths, and various topographicprofiles. In section 4 we analyze the nonlinear equili-bration of the local instabilities, the formation of co-herent structures in the unstable interval, and theirdownstream propagation. The link between our simpli-fied model and more realistic configurations, possibleextensions of this work, and further applications arehighlighted in section 5. A review of the results followsin section 6.

2. Mathematical formulation of the model

Although the Labrador Sea example described aboveinvolves nonzonal currents, as a first step toward a morerealistic configuration, this paper focuses on a barocliniczonal flow confined in an infinitely long channel ofwidth Y*.

We consider a quasigeostrophic flow in a two-layermodel on the beta plane [see Pedlosky (1987) for acomplete derivation of the quasigeostrophic approxi-mation]. Within each layer the density is uniform andthe horizontal velocities are independent of depth. Forsimplicity we consider two layers of equal depth H1 5H2 5 H. Let U be the characteristic velocity associatedwith the vertical shear of the zonal current and LR theinternal Rossby deformation radius defined as

1/2L 5 (g9H) / f ,R o (1)

where g9 5 gDr/r is the reduced gravity and f o is thevalue of Coriolis parameter at the central latitude on thebeta plane. If U, LR, and LR/U are used to scale hori-zontal velocity, length, and time, the equations of mo-tion in nondimensional form are

]Qi 41 J(c , Q ) 5 n¹ c . i 5 1, 2, (2)i i i]t

where ci is the streamfunction in layer i, i 5 1, 2, with1 referring to the upper layer; Qi is the correspondingpotential vorticity given by

2Q 5 ¹ c 2 F (c 2 c ) 1 by (3a)1 1 1 1 22Q 5 ¹ c 2 F (c 2 c ) 1 by 1 h(x, y), (3b)2 2 2 2 1

where ¹2ci are relative vorticity contributions;n is thenondimensional coefficient of the horizontal turbulentmixing; and J is the Jacobian operator defined as J(a,b) 5 ]xa]yb 2 ]ya]xb. For two layers of equal depthsthe interface displacements coefficients Fi are

2 2f Lo RF 5 F 5 5 1. (4)1 2 g9H

The relative size between the potential vorticity gradientdue to planetary differential rotation and that associatedwith the vertical shear is measured by b 5 b* /U, with2LRb* being the dimensional northward gradient of the Cor-iolis parameter. In the absence of topographic relief andfor uniform vertical shear, the dispersion relation forplane waves requires b less than 1 for instability. Thelast term, h(x, y), is the nondimensional bottom slope,which contributes to the potential vorticity of the lowerlayer and is related to the dimensional amplitude of thetopography h* by

h* f Lo Rh(x, y) 5 ; (5)UH

in our model the bottom relief has the form

h(x, y) 5 g(x)y, (6)

where g is a function of the zonal coordinate such asto destabilize the flow over an interval of limited extent.

The streamfunctions of the flow can be written as

c (x, y, t) 5 2Uy 1 f (x, y, t) (7a)1 1

c (x, y, t) 5 f (x, y, t), (7b)2 2

where f1 and f2 are the perturbation streamfunctionsin each layer. In our nondimensionalization U has beenscaled out of the problem, and in the following we con-sider only U 5 1. We can now rewrite the equationsfor the evolution of the disturbances as


FIG. 1. Model geometry for the two-layer channel. The topography stabilizes the flow outsidethe region | x | # a. The mean flow U in the upper layer is uniform.

]q ]f ]q1 1 11 J(f , q ) 1 (b 1 U ) 1 U1 1]t ]x ]x45 n¹ f , (8a)1

]q ]f ]f dg2 2 21 J(f , q ) 1 (b 1 g 2 U ) 2 y2 2]t ]x ]y dx45 n¹ f , (8b)2

where2q (x, y, t) 5 ¹ f (x, y, t)i i

i1 (21) (f (x, y, t) 2 f (x, y, t))1 2

i 5 1, 2. (9)

We consider a zonal channel of width Y 5 Y*/LR .p/ , where Ỹ 5 p/ is the minimum width belowÏ2 Ï2which the flow is stable (Phillips 1954). In the inviscidlimit n 5 0 and for constant bottom slope g, the dis-persion relation for plane waves ensures stability for g. U 2 b for all wavenumbers, while a band of unstablewaves is found for g , U 2 b. We set g(x) to be apositive-definite function that changes zonally as

x 1 a x 2 ab tanh 2 tanh1 2s s

g(x) 5 c 2 . (10)a

2 tanhs

In Eq. (10), a is the half-length of the interval of insta-bility and s determines how smoothly the topography

varies between the stable and the unstable regions. In thelimit of abruptly varying topography, that is, for s → 0,c represents the constant value attained by the slope inthe stable region | x | . a, while b controls the value ofthe maximum local supercriticality within the interval ofinstability | x | # a. In the following we will consider c5 b 5 2 unless otherwise specified. With this choice, inthe limit s → 0, the bottom relief destabilizes the flowfor | x | # a, where g 5 gunstable 5 0, and stabilizes itfor | x | . a where g 5 gstable 5 2. The model geometryis schematically illustrated in Fig. 1.

3. The linear unstable modes

Before examining the eigenmodes for the linear prob-lem, we briefly discuss the appropriate boundary con-ditions at the channel walls y 5 0, Y. Following Oh etal. (1993), for localized instabilities the entire distur-bance field must vanish for large | x | . This conditionapplies together with the obvious requirement of no geo-strophic velocity normal to the boundary; that is,

]f ]f1 25 5 0 at y 5 0, Y. (11)]x ]x

Therefore, integrating (11) we must impose

f 5 f 5 0 at y 5 0, Y. (12)1 2

We numerically integrate the linearized form of equa-tions (8a) and (8b) constrained by (12) at the meridionalwalls and with horizontal periodic boundary conditions.We use a third-order Adams–Bashford integration


FIG. 2. Linear growth rate vi for U 5 1, b 5 0.04, n 5 5 3 1024, s 5 2p/4 vs a, the half-length of theinterval of instability, for Y 5 4p (dashed line), Y 5 2p (dotted line), and Y 5 p (dashed–dotted line). Thesolid line shows the linear growth rate in the limit Y → `, s → 0, and n 5 0. The open square indicatesthe growth rate for a 5 1 and Y 5 2p; the linear and nonlinear equilibrated solutions for this case arediscussed in detail in the paper.

scheme and a nonaliasing spectral transform method toevaluate the perturbation velocity fields. The compu-tational domain is a channel of width Y, with Y varyingfrom p/ to 4p, and of constant length 32p. TheÏ2meridional resolution of the uniform grid varies from128 to 256 grid points. In order to simulate the effectsof a zonally infinite domain, a sponge region covers thefirst 1/8 of the channel (i.e., 216p # x # 212p). Withinthis region, a damping with friction coefficient increas-ing from zero at the border to one at the center is ar-tificially introduced to absorb most of the energy leavingthe right-hand boundary of the domain. This allows usto suppress global modes that may arise in a periodicdomain if a wave packet passes repetitively through thesame unstable zone.

Random initial perturbations are allowed to evolveuntil the most unstable eigenmode emerges. This tech-nique restricts us to the most unstable eigenmode thatdominates the evolution after sufficiently long time. Thefrequency and growth rate of this mode are the real andimaginary part, respectively, of the complex eigenvaluev 5 vR 1 iv i. The growth rate vi is determined bycomputing a long-term average of the growth of theperturbation kinetic energy. The frequency vR is deter-mined by performing an exact fit to the local time evo-lution of the energy growth once the form of an eigen-mode has been achieved.

Figure 2 shows the growth rate obtained by numericalintegration of Eqs. (8a) and (8b) for U 5 1, b 5 0.04,n 5 5 3 1024, s 5 2p/4, and for three different valuesof the channel width Y; vi is plotted as a function of a,the half-length of the interval of instability. For com-parison, Fig. 2 also shows the growth rate calculatedusing the matching technique illustrated in SP under theassumption of perturbations with no meridional varia-tion and in the limits Y → `, s → 0, and n 5 0. Forwide enough channels, we find instabilities regardlessof the length of the interval where the flow is locallyunstable. Thus, localized modes exist when the intervalof instability is much smaller than the wavelength ofthe shortest possible plane wave of the classical problemat the same parametric supercriticality (l ; 2p). Thisholds even for small values of the local supercriticalityof the flow. When the interval of instability is compa-rable with the Rossby deformation radius, that is, a ;0.5, the growth rate remains a substantial fraction of itsmaximum value. The surprising result of SP is thereforeconfirmed when disturbances are allowed to vary in themeridional direction and the linear modes are meridi-onally confined.

In Fig. 2 the growth rates for the most unstable ei-genmode are shown for Y 5 4p, 2p, and p, respectively.Clearly, the channel width plays a crucial role in de-termining the eigenvalues of the problem. As Y ap-


FIG. 3. (a) Zonal profile of the topography field at y 5 2p, (b) zonal velocity disturbance, (c) potential vorticity disturbance in the upperlayer, (d) zonal velocity disturbance and (e) potential vorticity disturbance in the bottom layer, at t 5 300. Here a 5 1, b 5 0.04, s 5 2p/4, n 5 5 3 1024, and Y 5 2p. Bright and dark tones indicate positive and negative velocity (potential vorticity), respectively.

proaches the minimum width required for the flow tobe baroclinically unstable; that is, for Y → Ỹ 5 p/

in our nondimensional units (Phillips 1954), theÏ2growth rate vi rapidly decreases to zero. For Y 5 p anda , 0.5 the flow is stable only because the growth ratesof the most unstable modes are smaller than the decayrate due to the viscous term. On the other hand, for Y5 4p the eigenvalues are close to the ones attained withno meridional confinement on the modes.

a. The spatial structure of the modes

Samelson and Pedlosky noticed that the spatial struc-ture of the modes is depth dependent for short intervalsof instability and hence a function of x. When the flowis confined in a channel, we expect the term

]f dg22 y]y dx

to generate a latitudinal dependence as well. Indeed, themeridional variations of the disturbances geostrophi-cally induce zonal velocities in the bottom layer thatinteract with the longitudinal gradient of the topography.In Fig. 3 snapshots of the perturbation zonal velocityfield and of perturbation potential vorticity are shownfor the upper and lower layers for a 5 1, b 5 0.04, s5 2p/4, and Y 5 2p. In the upper layer, the velocity(Fig. 3b) and the potential vorticity (Fig. 3c) distur-bances are dominated by a quasi-stationary Rossbywave (RW in Fig. 3c), with wavenumber k ø (b/U)1/2,that decays downstream of the escarpment at the easternedge of the interval of instability. In the lower layer(Figs. 3d,e) the amplitude of the Rossby wave is neg-ligible in the stable region because the disturbance is indirect contact with the stabilizing bottom slope and de-cays more rapidly. In the unstable interval, zonal ve-locities in the two layers are nearly equal in magnitude,with a maximum attained near the topographic step in


the downstream direction, but different in structure. Anarrow oscillating disturbance that attains maximumamplitude near the northeastern corner of the unstableinterval distinguishes the evolution of the lower layerfrom that of the upper one. The signature of this wave-like structure is visible in the perturbation zonal velocityfield (Fig. 3d) and more clearly in the potential vorticityfield q2, where two patches of vorticity of opposite signare recognizable (Fig. 3e; label BW). The amplitude ofthis disturbance reaches its maximum at the northernedge of the channel, due to the interaction of the zonalvelocities with the longitudinal gradient of the topog-raphy that increases with latitude, and its wavenumberis not associated with the Rossby wave that dominatesthe upper-layer fields.

The analysis of SP demonstrated the bottom structureto be a short bottom-trapped wave that, in the limit of| v | → 0 and s → 0, has wavenumber

b 1 g 2 Uk ø 2 (13)

|v |

and amplitudes

2(b 1 g 2 U )A ø A . (14)2 12|v |

In the limit of abruptly varying topography, the bottom-trapped wave remains strongly localized at the topo-graphic escarpment that separates the unstable from thestable interval and oscillates in phase with the upper-layer flow. In the upper layer, however, the amplitudeof the disturbance is small when compared to the Rossbywave that dominates the dynamics. The short bottom-trapped wave balances the time rate of change of relativevorticity with motion in the ambient potential vorticitygradient. As already noted by SP, this short wave isfundamental in maintaining the instability for smallgrowth rates (i.e., for short intervals of instability orsmall local supercriticality). To better illustrate the evo-lution of the trapped-wave disturbance in the presenceof smoothly varying topography, Fig. 4 shows the lower-layer potential vorticity fields for b 5 0.95, Y 5 2p,and a 5 4, with s 5 2p/2 in (b) and s 5 2p/8 in (d).The high value of b ensures a small local supercriticalityin the unstable region and therefore a small growth rate(v i 5 0.008 for s 5 2p/2 and v i 5 0.011 for s 5 2p/8). At the same time, a relatively large value of a allowsa detailed resolution of the structure of the disturbanceswithin the unstable interval. In both simulations, thedynamics in the northern part of the channel are dom-inated by the bottom-trapped wave. The disturbance ap-pears in the northeastern corner of the unstable region,it moves downstream over the topographic step (thatseparates the unstable interval from the stable one), andit is pushed southward by the new wave that forms everyhalf period. This mechanism is illustrated in detail inFig. 4e, where an enlargement of the bottom-trappedwave is shown for s 5 2p/8. The bottom-trapped dis-

turbances is given by the two recirculation regions ofpotential vorticity of opposite sign, labeled as BW inthe figure. At the escarpment, upstream of the anticy-clonic patch, a new peak of the wave is forming (NBW).As the bottom-trapped wave BW moves downstream,the preexisting disturbance (OBW) is pushed southward,resulting in a meridional distribution of opposite signedstructures in the northern half of the channel. In the caseof smoother topography (Fig. 4b), one single peak ofthe bottom wave is identifiable. The peak extends overthe topographic step and penetrates into the stable in-terval. For an abrupt topography (Fig. 4d) the distur-bance retains a clear wavelike structure with two rec-ognizable peaks and its downstream motion stops overthe steep escarpment.

Upstream, at the entrance of the unstable interval, adifferent disturbance occupies the southern half of thechannel. Its signature is evident for s 5 2p/8 (Fig. 4d,label LW; see also Fig. 3e). When the topography issmoother the wave partially connects with the short bot-tom-trapped wave. Samelson and Pedlosky identified itas a long baroclinic wave that in the limit Y → `, s →0, and | v | → 0 has wavenumber

b 1 b 1 gk ø 2 |v | (15)

b(b 1 g 2 U )

and amplitude

bA ø 2 A . (16)2 1b 1 g

The long baroclinic wave propagates upstream and in-duces a disturbance in the potential vorticity field ofnegligible amplitude, compared to the one generated bythe bottom-trapped wave. Additionally, the disturbanceinduced by the wave in the upper-layer potential vor-ticity rotates in a sense counter to that in the deep flow.

It is worth mentioning that in case of westward flow(U 5 21), SP found the dynamics to be characterizedby the presence of long baroclinic waves propagatingdownstream and by a faster decay of the Rossby wavein the upper layer.

4. Nonlinear dynamics

In order to investigate how linear disturbances equil-ibrate, we numerically integrate Eqs. (8a) and (8b) re-taining full nonlinearity. We previously discussed theformation of bottom-trapped waves localized both in thezonal and meridional direction. The resulting distur-bance in the potential vorticity field resembled a vortexdipole (see Fig. 3e), and we might therefore expect theequilibration of the linear modes to generate coherenteddies in the interval of instability. It is worth recallingthat the interaction between meridional variations of theperturbation field and the longitudinal gradient of thetopography is responsible for the latitudinal localizationof the bottom-trapped wave.


FIG. 4. (a) (c) Zonal profile of the topography field at y 5 2p and (b) (d) snapshot of lower layer potential vorticity disturbance at timet 5 500 for a 5 4, b 5 0.04, n 5 5 3 1024, and Y 5 2p. In (a) and (b) s 5 2p/2 and in (c) and (d) s 5 2p/8. (e) An enlargement ofthe region around the downstream escarpment [inset in panel (d)]. Bright and dark tones indicate positive and negative potential vorticity,respectively.

Figure 5 illustrates the evolution of potential vorticityin the lower layer for a 5 1, b 5 0.04, n 5 5 3 1024,s 5 2p/4, and Y 5 2p. The initial conditions (Fig. 5b)are given by the linear solution at t 5 300 for the sameparameter values. The linear growth rate for this solutionis vi ; 0.13 (highlighted in Fig. 2). Amplitudes havebeen normalized to ensure that the initial conditions arewithin the linear regime.

Figure 5c shows the recirculation patches of oppositesigns generated by the bottom-trapped wave at thenortheastern corner of the interval of instability, CBWand ABW. These two patches maintain coherence, sep-arate, and move along the northern wall in oppositedirections. The cyclone–anticyclone separation is dueto the interaction of the disturbances with the free-slipwall. The boundary provides an ‘‘image effect’’ such


FIG. 5. Evolution of the bottom layer potential vorticity q2 for a 5 1, b 5 0.04, s 5 2p/4, n 5 5 3 1024, and Y 5 2p. (a) Zonal profile of thetopography field at y 5 2p. (b) Initial conditions. (c)–(i) Potential vorticity disturbance at time t 5 2, 4, 6, 8, 10, 16, and 22, respectively.

that each patch moves under the influence of its image,which has opposite sign vorticity. The anticyclone ABWforms a stable dipole with its cyclonic image at theboundary and in this configuration is able to climb theslope that separates the unstable from the stable region.The cyclonic bottom-wave CBW, on the other hand, isforced westward, toward the anticyclonic component ofthe Rossby wave (ARW) with which it creates a dipole.This new dipole moves southward (Fig. 5c), as expectedowing to its orientation, and bends slightly to the west(Fig. 5d). The physical mechanism that governs the evo-lution of a dipole on the beta plane over a flat bottom(inside the unstable interval) is the conservation of po-

tential vorticity along fluid elements, as stated by Eq.(1), neglecting dissipation. As a vortex tube is com-pressed or stretched, its absolute vorticity must decreaseor increase in proportion. Thus if a dipole moves south-ward, the conservation of potential vorticity forces itsnegative part to increase and its positive part to decreasein strength. The stronger anticyclone can then pull theweaker cyclone and bend it to the west, resulting in awestward displacement. The opposite behavior, an east-ward shift, characterizes dipoles moving northward(e.g., Nof 1985; Kloosterziel et al. 1993; Dewar andGailliard 1994; Reznik and Sutyrin 2001).

In Fig. 5d, the anticyclone (ARW), which has moved


FIG. 5. (Continued)

toward the cyclonic component of the Rossby wave(CRW), couples with it and the new dipole moves north-ward and approaches the wall. Figures 5e,f show a ‘‘vor-tex rebound’’ from the slip wall, as described by Car-nevale et al. (1997). The stretching of vortex tubes gen-erates the vorticity that allows the split of the dipoleinto two separate vortices. The anticyclone (cyclone)reaches the wall, is affected by the image vorticity ofopposite sign in the wall, and moves eastward (west-ward). The ‘‘liberated’’ anticyclone (ARW) under theinfluence of its image can now climb the downstreamtopographic step and penetrate into the stable interval.Once the anticyclone has reached the stable region, theslope slowly affects the location of its core and thevortex core shifts southward, toward deeper water. Aftersufficiently long time, the anticyclone ARW detaches

completely from the boundary and moves into the in-terior of the channel (Fig. 5i). Conservation of potentialvorticity is once again the physical mechanism that de-termines the propagation of an isolated eddy over aslope. As a general result, anticyclones move towarddeeper water and cyclones toward shallower water (e.g.,McWilliams and Flierl 1979; Smith and O’Brien 1983;Killworth 1986; Carnevale et al. 1988, 1991; Whiteheadet al. 1990; Dewar and Gaillard 1994; Jacob et al. 2002).The cyclone CRW moves upstream and interacts withthe Rossby waves and the long baroclinic waves. Thisinteraction leads to the formation of a new dipole thatmigrates southward. In the absence of a meridional wallthese dipoles will continue to travel southward whilemaintaining their coherence. Meanwhile dipoles arecontinuously created at the northeastern edge of the un-


FIG. 6. (a) Zonal profile of the topography field at y 5 2p and snapshot of the perturbation potential vorticity field at time t 5 200 forlayers (b) 1 and (c) 2. Here a 5 4, b 5 0.9, s 5 2p/8, and Y 5 2p.

stable interval, with a periodicity dictated by the bottom-trapped disturbance (Figs. 5g,h). Through a ‘‘rebound’’mechanism the two new vortices ABW2 and CBW2 splitand the anticyclone moves downstream into the stableregion (Fig. 5i).

We noticed before that the vorticity anomalies gen-erated in the lower layer by the bottom-trapped distur-bance are in phase with the upper-layer flow. It is worthmentioning that the stability and lifetime of a coherentstructure in the presence of topography are strongly af-fected by the baroclinicity of the vortex itself. Eddieswith a substantial barotropic structure, such as the onesgenerated here by equilibration of the bottom-trappedwave, are stable. On the contrary, vortices in which thedeep flow is counter to the sense of the shallow floware very unstable (Chassignet and Cushman-Roisin1991; Dewar and Killworth 1995; Kamenkovich et al.1996).

The scenario illustrated above is common to all thenonlinear solutions that we analyzed. The equilibrationof the bottom-trapped wave generates coherent dipolesat the northern edge of the unstable interval. These di-poles immediately split into two separate vortices: Theanticyclonic eddy propagates into the stable regionwhile the cyclonic one interacts with the waves insidethe region of instability and eventually forms a newdipole that propagates southward.

Snapshots of potential vorticity disturbances in layers1 and 2 are shown in Figs. 6 and 7 for Y 5 2p, b 50.9, s 5 2p/8, and a 5 4 and for Y 5 2p, b 5 0.02,

s 5 2p/2, a 5 6, and c 5 b 5 1, respectively. Thesesimulations are initialized with random initial condi-tions. Despite the differences in the parameter values,which results in a difference of an order magnitude inthe linear growth rates (0.012 and 0.19, respectively),the evolution of the flow is qualitatively similar. In bothsolutions, concentrations of vorticity of both signs oc-cupy the interval of instability. Only anticyclones prop-agate, as well defined coherent vortices, from the north-eastern corner of the unstable interval downstream intothe stable region, following a path along the wall. Ona timescale depending on the steepness of the topog-raphy, the anticyclones separate from the wall and movesouthward. The anticyclonic vortices are essentially bar-otropic, with a core in the upper layer slightly largerthan the bottom one. This is probably due to the sta-bilizing effect of the bottom slope over the edges of thevortices in the lower layer.

The short bottom-trapped disturbance, responsible forthe persistence of the instability for arbitrary small localsupercriticality, plays now the fundamental role of gen-erating continuously cyclonic and anticyclonic vorticesat the downstream edge of the unstable interval. Thosestructures are stable enough to maintain coherence overseveral rotation periods. The anticyclones propagate intothe stable region as isolated eddies, while the cyclonicvortices trigger the formation of dipoles inside the in-terval of instability.1

1 In some of the nonlinear solutions for smoothly varying topog-


FIG. 7. (a) Zonal profile of the topography field at y 5 2p and snapshot of the perturbation potential vorticity field at time t 5 80 forlayers (b) 1 and (c) 2. Here a 5 6, b 5 0.02, s 5 2p/2, Y 5 2p, and c 5 b 5 1.

5. Discussion

The mechanism of generation of stable coherent vor-tices explored above appears to be quite robust. It doesnot depend on the length of the unstable interval, noris sensitive to how abruptly the topography varies in thezonal direction. Additionally, only local maxima in su-percriticality are required for the existence of local un-stable modes and the growth rates are independent ofthe degree of stability of the flow outside the intervalof maximum instability (Pierrehumbert 1984; Pedlosky1989; Samelson and Pedlosky 1990). The bottom-trapped disturbance, responsible for the vortex forma-tion, grows to balance the variations in time of relativevorticity with the ambient gradient of potential vorticityand its basic physical behavior can be expected to occurin flow configurations more general than the one ana-lyzed in this work. The vortex propagation mechanism,however, seems to rely on the rebound of the dipolefrom the slip wall, with the penetration of anticyclonesinto the downstream stable interval due to the influenceof the cyclonic image in the wall. In oceanography, thechoice of slip boundaries has been motivated by theinability of resolving small-scale processes near the

raphy, small vortexlike structures were formed in lieu of the longbaroclinic disturbance in the upstream stable interval. Those struc-tures, in which the deep flow was always counter to the sense of theshallow circulation, were very unstable and quickly destroyed.

coast while modeling large-scale flows. This problemhas been addressed by assuming that the solution iscomposed of two parts, an outer solution with slipboundaries and a complicated boundary layer with no-slip conditions at solid surfaces (e.g., Pedlosky 1996).Nevertheless, we are reluctant to generalize this aspectof our results, and further studies, with nonuniformshear profiles, may help to elucidate the vortex propa-gation process.

Recently, Walker and Pedlosky (2002) have shownthat for meridional currents no minimum shear and nominimum channel width is required for instability. Inthe light of those results and given the original moti-vation of our analysis (a poleward boundary current inthe Labrador Sea), a natural question, which will beaddressed in a future work, concerns the generalizationof our results to a meridional current in a north–southchannel. Atmospheric flows along continental coasts areaffected by topographic gradients such as the ones gen-erated by coastal mountain barriers. In the context ofoceanic flows, obvious candidates for local instabilitycontrolled by variations in the bottom slope similar tothe model study presented, are coastal currents. Despitethe obvious model simplifications, the scenario we ad-dress, opportunely rotated, resembles qualitatively thetopography of the West Greenland continental slope.Indeed, a region of enhanced eddy kinetic energy (EKE)has been observed at the northern edge of the LabradorSea in the West Greenland Current near an abrupt


change in topography (Lavender et al. 2002; Pickart etal. 2002; Prater 2002). Interestingly, the high EKE sig-nal is not confined to the surface (Lavender 2001), andthe analysis of the eddy field performed by Lilly et al.(2002, manuscript submitted to J. Phys. Oceanogr.) in-dicates a predominance of anticyclones with a compos-ite double core. These vortices form in the West Green-land Current and propagate into the interior of the Lab-rador Sea. In our model, the southern wall limits thepropagation of the dipoles that form within the unstableinterval, and it bounds the deeper flow. Such a wallwould not exist in the oceanic context. Therefore, weexpect that in the absence of this artificial boundary,dipoles and anticyclones will move farther away fromthe northern boundary, filling the basin with vortices.

The predominance of anticyclones in our model de-serves an additional comment. In our calculations singleeddies are generated at the downstream escarpmentalong the northern wall (i.e., where the zonal topograph-ic variation dg(x)/dx attains its downstream maximum)through equilibration of the bottom-trapped disturbanceand vortex rebound from the slip boundary. As a generalresult, those vortices form over the topographic step thatseparates the unstable from the stable region. The di-rection of the basic flow determines the orientation ofthe dipole for the rebound to take place. Carnevale etal. (1999) have shown that the evolution of a coastalcurrent as it encounters a single escarpment dependsstrongly on the geometry of the topographic step. How-ever, regardless of the direction of the basic flow, onlyanticyclones can migrate as single coherent structurestoward deeper waters, that is, into the interior of thebasin in the case of a boundary current, because ofconservation of potential vorticity along fluid elements.

We anticipate that in the absence of the beta effectthe equilibration of the unstable modes is qualitativelysimilar to the cases previously illustrated, but the an-ticyclones in the upper layer are less strong and thereforeless stable. The influence of b oriented perpendicularlyto the topography, together with a more realistic con-figuration to model the dynamics in the West GreenlandCurrent, will be the subject of a future study.

6. Conclusions

Within the context of a simplified two-layer, quasi-geostrophic model, we have numerically investigatedthe dynamics of a uniform flow confined in a channelover meridionally variable topography on the beta plane.The bottom relief controls the local supercriticality ofthe flow. Stability is ensured everywhere except in aninterval of finite length at the center of the channelwhere localized baroclinic instability takes place. Wehave examined both the linear and the equilibrated be-havior of the most unstable normal modes. In a previousstudy, Samelson and Pedlosky (1990) analyzed linearsolutions for the same topographic configuration, in theabsence of meridional variations of the perturbation field

and without meridional confinement of the modes. Inaccordance with their results, in this work we show thatfor channels wide enough to sustain baroclinic insta-bility, unstable modes exist for an arbitrary short inter-val of instability. The interval can be, therefore, muchshorter than the shortest unstable wavelength for an in-finitely long domain at the same supercriticality.

For interval of instability comparable with the Rossbydeformation radius or greater, the most unstable modesare weakly trapped in the upper layer, and Rossby wavesdominate the eastward flow evolution. In the lower layerthe amplitudes of the disturbance decay faster beyondthe interval due to the stabilizing presence of the bottomslope. In the case of fairly small supercriticality, theinstability is maintained by a short bottom-trapped wavethat generates two recirculation zones of potential vor-ticity of opposite sign, trapped at the downstream edgeof the region of instability. The narrow wave in thelower layer grows in time and oscillates in phase withthe upper-layer flow. Upstream long baroclinic wavesdecay rapidly in both layers and oscillate with oppositesign.

When nonlinearity is retained in the problem, theequilibrated solutions predict the formation of coherentvortices within the interval of instability, for arbitrarilysmall local supercriticality. Both cyclones and anticy-clones are formed continuously at the northeastern edgeof the unstable interval, resulting from the equilibrationof the bottom-trapped wave. In the vortex generationprocess it is particularly important that the zonal com-ponent of the perturbation velocity can interact with thezonal gradient of the bathymetry. Indeed, this interac-tion ensures the meridional localization of the bottom-trapped disturbance. Via vortex stretching, dipolesformed at the northeastern edge of the interval of in-stability can rebound from the northern wall. Anticy-clones move downstream along the north wall under theinfluence of their image vortices and propagate into thestable region. Once over the meridional slope in thestable region, they propagate away from the wall, intodeeper water. Cyclones remain trapped inside the in-terval of instability, where they interact with the Rossbywave and the long baroclinic wave disturbances, andform new dipoles. Those coherent vortices are stableand are characterized by an upper core with circulationin the same sense as the deep one.

The bottom-trapped wave, responsible for the persis-tence of the instability for arbitrarily small supercriti-cality, is also responsible for the formation of stablecoherent vortices in the equilibrated solutions. Physi-cally, this wave balances the time rate of change ofrelative vorticity with the motion in the ambient gradientof potential vorticity. We may speculate that this simplemechanism of formation of coherent vortices is not lim-ited to our model study and applies to more generalbaroclinic flows of geophysical interest. The propaga-tion of the vortices formed in the unstable interval isstrongly affected by the meridional walls. It is natural


to suppose that in the absence of the free-slip southernboundary the coherent dipoles would move farther fromthe coast instead of rebounding from the meridionalwall, filling the basin with vortices as observed in theLabrador Sea. Further extensions of the model will beable to address this problem. Natural extensions of thiswork would thus include consideration of meridionalcurrents and laterally nonuniform vertical shear (i.e.,coastally confined currents).

Acknowledgments. Sincerest thanks to Robert Pickart,for showing us the relevance of this model to the eddyfield in the Labrador Sea. We are grateful to Joe LaCascefor stimulating discussions and to Nicolas Cauchy forhis careful editing of the manuscript.

A. B. is supported by the Postdoctoral Scholar Pro-gram at the Woods Hole Oceanographic Institution, withfunding provided by the USGS. J. P. is supported bythe National Science Foundation Grant OCE 9901654.

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