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JANUARY 2003 BRACCO AND PEDLOSKY 207 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 the beta 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 local supercriticality 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 interval of instability, confirming previous analysis on disturbances with no meridional variation. For small local maximum supercriticality, the instability is maintained by a short bottom-trapped wave localized at the downstream edge of the unstable region and oscillating in phase with the upper-layer disturbance. When nonlinearity is retained in 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. Through vortex 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, while cyclonic structures tend to remain trapped inside the interval of instability. The authors suggest the relevance of their results to the observed eddy field of the Labrador Sea. 1. Introduction Coherent vortices are fundamental dynamic features of geophysical flows. The delineation of the factors governing their formation, structure, and geographical distribution is essential to our understanding of the oceanic and atmospheric circulation. Relevant literature suggests that strong eddy activity and enhanced cyclogenesis are often associated with regions of high baroclinicity. Hence, the general nature of localized baroclinic instability has received much attention in the last two decades. Merkine and Shafranek (1980) and Pierrehumbert (1984) used the Wentzel–Kramers–Brillouin (WKB) technique to study flows in which the properties of the basic state change zonally on scales greater than those typical of the unstable disturbances. Frederiksen (1983) approached the problem of regional cyclogenesis numerically for stationary zonally varying flows. Pedlosky (1989) derived a set of heuristic model equations describing locally unstable flows and solved the eigenvalue problem matching the solutions between the stable * Woods Hole Oceanographic Institution Contribution Number 10651. Corresponding author address: Annalisa Bracco, Department of Physics of Weather and Climate, The Abdus Salam International Centre for Theoretical Physics (ICTP), Strada Costiera 11, Trieste 34014, Italy. E-mail: [email protected] q 2003 American Meteorological Society and unstable regions. Pedlosky (1992) and Oh et al. (1993) analyzed both linear and nonlinear solutions for a two-layer baroclinic flow for which stability properties are controlled by the friction coefficient of the bottom Ekman layer. Samelson and Pedlosky (1990, hereafter SP) focused on local baroclinic instability of a quasigeostrophic, zonal flow over a meridional bottom slope. Topographic variations are likely to be responsible for the confinement of instabilities in most situations of interest to meteorology and oceanography. Indeed, the stability of atmospheric mean winds is affected by the presence of coastal mountains that act as barriers. In the ocean there are regions where, at least locally, the background gradient of potential vorticity of the deep flow is controlled by varying topography, rather than by planetary curvature. Indeed the analysis in the present paper was motivated in part by observations of strongly localized eddy variability near the eastern boundary of the Labrador Sea, in a region where the continental slope varies substantially in the alongshore direction (Lavender 2001; Prater 2002; Lilly et al. 2002, manuscript submitted to J. Phys. Oceanogr.). Samelson and Pedlosky chose a bottom slope such as to render the flow locally stable except in a central region of limited zonal extent, in which the necessary conditions for instability were satisfied and the local supercriticality was positive. Their analysis was restricted to perturbations with no meridional variation and thus to the linearized problem (in quasigeostrophic theory 208 JOURNAL OF PHYSICAL OCEANOGRAPHY cross-stream variations are required for nonlinearity to appear). Surprisingly, SP found instabilities no matter how short the length of the unstable interval. The natural question that arises is the validation of such a result when disturbances vary meridionally and the linear modes are confined within a domain of finite width. The purpose of the present work is to extend, through numerical analysis, the linear model of SP by confining the zonal flow in a channel on the beta plane. One reason for doing so is to mimic the limited lateral extent of oceanic boundary currents, as in the Labrador Sea. Additionally this allows us to investigate the equilibration of the nonlinear modes. Our second motivation is that the confinement of the flow leads to a new source of vorticity in the disturbance field. Indeed, disturbances that depend on the cross-channel coordinate geostrophically produce zonal perturbation velocities in the lower layer that interact with the longitudinal gradient of the topography. By contrast, disturbances with no meridional variation sense only the latitudinal variation of the topographic slope. Solutions presented here show that for wide enough channels [i.e., for channels that satisfy the Phillips (1954) criterion for instability] unstable modes exist for arbitrarily small intervals. The equilibration of the localized unstable modes generates coherent vortex structures at the downstream edge of the interval of instability. The formation, evolution, and stability of the eddy field are consequently investigated. In section 2 we introduce the mathematical model. Section 3 describes the growth of localized linear unstable modes for several values of the local supercriticality, different channel widths, and various topographic profiles. In section 4 we analyze the nonlinear equilibration of the local instabilities, the formation of coherent structures in the unstable interval, and their downstream propagation. The link between our simplified model and more realistic configurations, possible extensions of this work, and further applications are highlighted in section 5. A review of the results follows in section 6. 2. Mathematical formulation of the model Although the Labrador Sea example described above involves nonzonal currents, as a first step toward a more realistic configuration, this paper focuses on a baroclinic zonal flow confined in an infinitely long channel of width Y*. We consider a quasigeostrophic flow in a two-layer model on the beta plane [see Pedlosky (1987) for a complete derivation of the quasigeostrophic approximation]. Within each layer the density is uniform and the horizontal velocities are independent of depth. For simplicity we consider two layers of equal depth H1 5 H 2 5 H. Let U be the characteristic velocity associated with the vertical shear of the zonal current and L R the internal Rossby deformation radius defined as VOLUME 33 L R 5 (g9H)1/2 / f o , (1) where g9 5 gD r/r is the reduced gravity and f o is the value of Coriolis parameter at the central latitude on the beta plane. If U, L R , and L R /U are used to scale horizontal velocity, length, and time, the equations of motion in nondimensional form are ]Q i 1 J(c i , Q i ) 5 n¹ 4 c i . ]t i 5 1, 2, (2) where c i is the streamfunction in layer i, i 5 1, 2, with 1 referring to the upper layer; Q i is the corresponding potential vorticity given by Q 1 5 ¹ 2 c1 2 F1 (c1 2 c 2 ) 1 by (3a) Q 2 5 ¹ 2 c 2 2 F2 (c 2 2 c1 ) 1 by 1 h(x, y), (3b) where ¹ 2 c i are relative vorticity contributions;n is the nondimensional coefficient of the horizontal turbulent mixing; and J is the Jacobian operator defined as J(a, b) 5 ] x a] y b 2 ] y a] x b. For two layers of equal depths the interface displacements coefficients F i are F1 5 F2 5 f o2 L 2R 5 1. g9H (4) The relative size between the potential vorticity gradient due to planetary differential rotation and that associated with the vertical shear is measured by b 5 b*L R2 /U, with b* being the dimensional northward gradient of the Coriolis parameter. In the absence of topographic relief and for uniform vertical shear, the dispersion relation for plane waves requires b less than 1 for instability. The last term, h(x, y), is the nondimensional bottom slope, which contributes to the potential vorticity of the lower layer and is related to the dimensional amplitude of the topography h* by h(x, y) 5 h* f o L R ; UH (5) 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 as to destabilize the flow over an interval of limited extent. The streamfunctions of the flow can be written as c1 (x, y, t) 5 2Uy 1 f1 (x, y, t) (7a) c 2 (x, y, t) 5 f 2 (x, y, t), (7b) where f1 and f 2 are the perturbation streamfunctions in each layer. In our nondimensionalization U has been scaled out of the problem, and in the following we consider only U 5 1. We can now rewrite the equations for the evolution of the disturbances as JANUARY 2003 209 BRACCO AND PEDLOSKY FIG. 1. Model geometry for the two-layer channel. The topography stabilizes the flow outside the region | x | # a. The mean flow U in the upper layer is uniform. ]q1 ]f ]q 1 J(f1, q1 ) 1 1 (b 1 U ) 1 1 U ]t ]x ]x 5 n¹ 4 f1, (8a) ]q 2 ]f ]f dg 1 J(f 2 , q 2 ) 1 2 (b 1 g 2 U ) 2 2 y ]t ]x ]y dx 5 n¹ 4 f 2 , (8b) where q i (x, y, t) 5 ¹ 2 f i (x, y, t) 1 (21) i (f1 (x, y, t) 2 f 2 (x, y, t)) i 5 1, 2. 3. The linear unstable modes (9) We consider a zonal channel of width Y 5 Y*/L R . p/Ï2, where Ỹ 5 p/Ï2 is the minimum width below which the flow is stable (Phillips 1954). In the inviscid limit n 5 0 and for constant bottom slope g , the dispersion relation for plane waves ensures stability for g . U 2 b for all wavenumbers, while a band of unstable waves is found for g , U 2 b. We set g (x) to be a positive-definite function that changes zonally as 1 b tanh g (x) 5 c 2 x1a x2a 2 tanh s s a 2 tanh s 2 varies between the stable and the unstable regions. In the limit of abruptly varying topography, that is, for s → 0, c represents the constant value attained by the slope in the stable region | x | . a, while b controls the value of the maximum local supercriticality within the interval of instability | x | # a. In the following we will consider c 5 b 5 2 unless otherwise specified. With this choice, in the limit s → 0, the bottom relief destabilizes the flow for | x | # a, where g 5 g unstable 5 0, and stabilizes it for | x | . a where g 5 g stable 5 2. The model geometry is schematically illustrated in Fig. 1. Before examining the eigenmodes for the linear problem, we briefly discuss the appropriate boundary conditions at the channel walls y 5 0, Y. Following Oh et al. (1993), for localized instabilities the entire disturbance field must vanish for large | x | . This condition applies together with the obvious requirement of no geostrophic velocity normal to the boundary; that is, ]f 1 ]f 2 5 50 ]x ]x at y 5 0, Y. (11) Therefore, integrating (11) we must impose . (10) In Eq. (10), a is the half-length of the interval of instability and s determines how smoothly the topography f1 5 f 2 5 0 at y 5 0, Y. (12) We numerically integrate the linearized form of equations (8a) and (8b) constrained by (12) at the meridional walls and with horizontal periodic boundary conditions. We use a third-order Adams–Bashford integration 210 JOURNAL OF PHYSICAL OCEANOGRAPHY scheme and a nonaliasing spectral transform method to evaluate the perturbation velocity fields. The computational domain is a channel of width Y, with Y varying from p/Ï2 to 4p, and of constant length 32p. The meridional resolution of the uniform grid varies from 128 to 256 grid points. In order to simulate the effects of a zonally infinite domain, a sponge region covers the first 1/8 of the channel (i.e., 216p # x # 212p). Within this region, a damping with friction coefficient increasing from zero at the border to one at the center is artificially introduced to absorb most of the energy leaving the right-hand boundary of the domain. This allows us to suppress global modes that may arise in a periodic domain if a wave packet passes repetitively through the same unstable zone. Random initial perturbations are allowed to evolve until the most unstable eigenmode emerges. This technique restricts us to the most unstable eigenmode that dominates the evolution after sufficiently long time. The frequency and growth rate of this mode are the real and imaginary part, respectively, of the complex eigenvalue v 5 v R 1 iv i . The growth rate v i is determined by computing a long-term average of the growth of the perturbation kinetic energy. The frequency v R is determined by performing an exact fit to the local time evolution of the energy growth once the form of an eigenmode has been achieved. VOLUME 33 Figure 2 shows the growth rate obtained by numerical integration of Eqs. (8a) and (8b) for U 5 1, b 5 0.04, n 5 5 3 10 24 , s 5 2p/4, and for three different values of the channel width Y; v i is plotted as a function of a, the half-length of the interval of instability. For comparison, Fig. 2 also shows the growth rate calculated using the matching technique illustrated in SP under the assumption of perturbations with no meridional variation and in the limits Y → `, s → 0, and n 5 0. For wide enough channels, we find instabilities regardless of the length of the interval where the flow is locally unstable. Thus, localized modes exist when the interval of instability is much smaller than the wavelength of the shortest possible plane wave of the classical problem at the same parametric supercriticality (l ; 2p). This holds even for small values of the local supercriticality of the flow. When the interval of instability is comparable with the Rossby deformation radius, that is, a ; 0.5, the growth rate remains a substantial fraction of its maximum value. The surprising result of SP is therefore confirmed when disturbances are allowed to vary in the meridional direction and the linear modes are meridionally confined. In Fig. 2 the growth rates for the most unstable eigenmode are shown for Y 5 4p, 2p, and p, respectively. Clearly, the channel width plays a crucial role in determining the eigenvalues of the problem. As Y ap- FIG. 2. Linear growth rate v i for U 5 1, b 5 0.04, n 5 5 3 10 24 , s 5 2p/4 vs a, the half-length of the interval of instability, for Y 5 4p (dashed line), Y 5 2p (dotted line), and Y 5 p (dashed–dotted line). The solid line shows the linear growth rate in the limit Y → `, s → 0, and n 5 0. The open square indicates the growth rate for a 5 1 and Y 5 2p; the linear and nonlinear equilibrated solutions for this case are discussed in detail in the paper. JANUARY 2003 BRACCO AND PEDLOSKY 211 FIG. 3. (a) Zonal profile of the topography field at y 5 2p, (b) zonal velocity disturbance, (c) potential vorticity disturbance in the upper layer, (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 10 24 , and Y 5 2p. Bright and dark tones indicate positive and negative velocity (potential vorticity), respectively. proaches the minimum width required for the flow to be baroclinically unstable; that is, for Y → Ỹ 5 p/ Ï2 in our nondimensional units (Phillips 1954), the growth rate v i rapidly decreases to zero. For Y 5 p and a , 0.5 the flow is stable only because the growth rates of the most unstable modes are smaller than the decay rate due to the viscous term. On the other hand, for Y 5 4p the eigenvalues are close to the ones attained with no meridional confinement on the modes. a. The spatial structure of the modes Samelson and Pedlosky noticed that the spatial structure of the modes is depth dependent for short intervals of instability and hence a function of x. When the flow is confined in a channel, we expect the term 2 ]f 2 dg y ]y dx to generate a latitudinal dependence as well. Indeed, the meridional variations of the disturbances geostrophically induce zonal velocities in the bottom layer that interact with the longitudinal gradient of the topography. In Fig. 3 snapshots of the perturbation zonal velocity field and of perturbation potential vorticity are shown for the upper and lower layers for a 5 1, b 5 0.04, s 5 2p/4, and Y 5 2p. In the upper layer, the velocity (Fig. 3b) and the potential vorticity (Fig. 3c) disturbances are dominated by a quasi-stationary Rossby wave (RW in Fig. 3c), with wavenumber k ø (b/U)1/2 , that decays downstream of the escarpment at the eastern edge of the interval of instability. In the lower layer (Figs. 3d,e) the amplitude of the Rossby wave is negligible in the stable region because the disturbance is in direct contact with the stabilizing bottom slope and decays more rapidly. In the unstable interval, zonal velocities in the two layers are nearly equal in magnitude, with a maximum attained near the topographic step in 212 JOURNAL OF PHYSICAL OCEANOGRAPHY the downstream direction, but different in structure. A narrow oscillating disturbance that attains maximum amplitude near the northeastern corner of the unstable interval distinguishes the evolution of the lower layer from that of the upper one. The signature of this wavelike structure is visible in the perturbation zonal velocity field (Fig. 3d) and more clearly in the potential vorticity field q 2 , where two patches of vorticity of opposite sign are recognizable (Fig. 3e; label BW). The amplitude of this disturbance reaches its maximum at the northern edge of the channel, due to the interaction of the zonal velocities with the longitudinal gradient of the topography that increases with latitude, and its wavenumber is not associated with the Rossby wave that dominates the upper-layer fields. The analysis of SP demonstrated the bottom structure to be a short bottom-trapped wave that, in the limit of | v | → 0 and s → 0, has wavenumber b1g2U |v | (13) (b 1 g 2 U ) 2 A1. |v | 2 (14) kø2 and amplitudes A2 ø In the limit of abruptly varying topography, the bottomtrapped wave remains strongly localized at the topographic escarpment that separates the unstable from the stable interval and oscillates in phase with the upperlayer flow. In the upper layer, however, the amplitude of the disturbance is small when compared to the Rossby wave that dominates the dynamics. The short bottomtrapped wave balances the time rate of change of relative vorticity with motion in the ambient potential vorticity gradient. As already noted by SP, this short wave is fundamental in maintaining the instability for small growth rates (i.e., for short intervals of instability or small local supercriticality). To better illustrate the evolution of the trapped-wave disturbance in the presence of smoothly varying topography, Fig. 4 shows the lowerlayer 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 supercriticality in 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 allows a detailed resolution of the structure of the disturbances within the unstable interval. In both simulations, the dynamics in the northern part of the channel are dominated by the bottom-trapped wave. The disturbance appears in the northeastern corner of the unstable region, it moves downstream over the topographic step (that separates the unstable interval from the stable one), and it is pushed southward by the new wave that forms every half period. This mechanism is illustrated in detail in Fig. 4e, where an enlargement of the bottom-trapped wave is shown for s 5 2p/8. The bottom-trapped dis- VOLUME 33 turbances is given by the two recirculation regions of potential vorticity of opposite sign, labeled as BW in the figure. At the escarpment, upstream of the anticyclonic 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 signed structures in the northern half of the channel. In the case of smoother topography (Fig. 4b), one single peak of the bottom wave is identifiable. The peak extends over the topographic step and penetrates into the stable interval. For an abrupt topography (Fig. 4d) the disturbance retains a clear wavelike structure with two recognizable peaks and its downstream motion stops over the steep escarpment. Upstream, at the entrance of the unstable interval, a different disturbance occupies the southern half of the channel. Its signature is evident for s 5 2p/8 (Fig. 4d, label LW; see also Fig. 3e). When the topography is smoother the wave partially connects with the short bottom-trapped wave. Samelson and Pedlosky identified it as a long baroclinic wave that in the limit Y → `, s → 0, and | v | → 0 has wavenumber kø2 b1b1g |v | b (b 1 g 2 U ) (15) b A. b1g 1 (16) and amplitude A2 ø 2 The long baroclinic wave propagates upstream and induces a disturbance in the potential vorticity field of negligible amplitude, compared to the one generated by the bottom-trapped wave. Additionally, the disturbance induced by the wave in the upper-layer potential vorticity 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 characterized by the presence of long baroclinic waves propagating downstream and by a faster decay of the Rossby wave in the upper layer. 4. Nonlinear dynamics In order to investigate how linear disturbances equilibrate, we numerically integrate Eqs. (8a) and (8b) retaining full nonlinearity. We previously discussed the formation of bottom-trapped waves localized both in the zonal and meridional direction. The resulting disturbance in the potential vorticity field resembled a vortex dipole (see Fig. 3e), and we might therefore expect the equilibration of the linear modes to generate coherent eddies in the interval of instability. It is worth recalling that the interaction between meridional variations of the perturbation field and the longitudinal gradient of the topography is responsible for the latitudinal localization of the bottom-trapped wave. JANUARY 2003 BRACCO AND PEDLOSKY 213 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 time t 5 500 for a 5 4, b 5 0.04, n 5 5 3 10 24 , and Y 5 2p. In (a) and (b) s 5 2p/2 and in (c) and (d) s 5 2p/8. (e) An enlargement of the 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 vorticity in the lower layer for a 5 1, b 5 0.04, n 5 5 3 10 24 , 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 same parameter values. The linear growth rate for this solution is v i ; 0.13 (highlighted in Fig. 2). Amplitudes have been normalized to ensure that the initial conditions are within the linear regime. Figure 5c shows the recirculation patches of opposite signs generated by the bottom-trapped wave at the northeastern corner of the interval of instability, CBW and ABW. These two patches maintain coherence, separate, and move along the northern wall in opposite directions. The cyclone–anticyclone separation is due to the interaction of the disturbances with the free-slip wall. The boundary provides an ‘‘image effect’’ such 214 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 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 the topography 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 ABW forms a stable dipole with its cyclonic image at the boundary and in this configuration is able to climb the slope that separates the unstable from the stable region. The cyclonic bottom-wave CBW, on the other hand, is forced westward, toward the anticyclonic component of the Rossby wave (ARW) with which it creates a dipole. This new dipole moves southward (Fig. 5c), as expected owing to its orientation, and bends slightly to the west (Fig. 5d). The physical mechanism that governs the evolution 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 compressed or stretched, its absolute vorticity must decrease or increase in proportion. Thus if a dipole moves southward, the conservation of potential vorticity forces its negative part to increase and its positive part to decrease in strength. The stronger anticyclone can then pull the weaker cyclone and bend it to the west, resulting in a westward displacement. The opposite behavior, an eastward shift, characterizes dipoles moving northward (e.g., Nof 1985; Kloosterziel et al. 1993; Dewar and Gailliard 1994; Reznik and Sutyrin 2001). In Fig. 5d, the anticyclone (ARW), which has moved JANUARY 2003 BRACCO AND PEDLOSKY 215 FIG. 5. (Continued) toward the cyclonic component of the Rossby wave (CRW), couples with it and the new dipole moves northward and approaches the wall. Figures 5e,f show a ‘‘vortex rebound’’ from the slip wall, as described by Carnevale et al. (1997). The stretching of vortex tubes generates the vorticity that allows the split of the dipole into two separate vortices. The anticyclone (cyclone) reaches the wall, is affected by the image vorticity of opposite sign in the wall, and moves eastward (westward). The ‘‘liberated’’ anticyclone (ARW) under the influence of its image can now climb the downstream topographic step and penetrate into the stable interval. Once the anticyclone has reached the stable region, the slope slowly affects the location of its core and the vortex core shifts southward, toward deeper water. After sufficiently long time, the anticyclone ARW detaches completely from the boundary and moves into the interior of the channel (Fig. 5i). Conservation of potential vorticity is once again the physical mechanism that determines the propagation of an isolated eddy over a slope. As a general result, anticyclones move toward deeper water and cyclones toward shallower water (e.g., McWilliams and Flierl 1979; Smith and O’Brien 1983; Killworth 1986; Carnevale et al. 1988, 1991; Whitehead et al. 1990; Dewar and Gaillard 1994; Jacob et al. 2002). The cyclone CRW moves upstream and interacts with the Rossby waves and the long baroclinic waves. This interaction leads to the formation of a new dipole that migrates southward. In the absence of a meridional wall these dipoles will continue to travel southward while maintaining their coherence. Meanwhile dipoles are continuously created at the northeastern edge of the un- 216 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33 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 for layers (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 bottomtrapped disturbance (Figs. 5g,h). Through a ‘‘rebound’’ mechanism the two new vortices ABW 2 and CBW 2 split and the anticyclone moves downstream into the stable region (Fig. 5i). We noticed before that the vorticity anomalies generated in the lower layer by the bottom-trapped disturbance are in phase with the upper-layer flow. It is worth mentioning that the stability and lifetime of a coherent structure in the presence of topography are strongly affected by the baroclinicity of the vortex itself. Eddies with a substantial barotropic structure, such as the ones generated here by equilibration of the bottom-trapped wave, are stable. On the contrary, vortices in which the deep flow is counter to the sense of the shallow flow are very unstable (Chassignet and Cushman-Roisin 1991; Dewar and Killworth 1995; Kamenkovich et al. 1996). The scenario illustrated above is common to all the nonlinear solutions that we analyzed. The equilibration of the bottom-trapped wave generates coherent dipoles at the northern edge of the unstable interval. These dipoles immediately split into two separate vortices: The anticyclonic eddy propagates into the stable region while the cyclonic one interacts with the waves inside the region of instability and eventually forms a new dipole that propagates southward. Snapshots of potential vorticity disturbances in layers 1 and 2 are shown in Figs. 6 and 7 for Y 5 2p, b 5 0.9, 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. These simulations are initialized with random initial conditions. Despite the differences in the parameter values, which results in a difference of an order magnitude in the linear growth rates (0.012 and 0.19, respectively), the evolution of the flow is qualitatively similar. In both solutions, concentrations of vorticity of both signs occupy the interval of instability. Only anticyclones propagate, as well defined coherent vortices, from the northeastern corner of the unstable interval downstream into the stable region, following a path along the wall. On a timescale depending on the steepness of the topography, the anticyclones separate from the wall and move southward. The anticyclonic vortices are essentially barotropic, with a core in the upper layer slightly larger than the bottom one. This is probably due to the stabilizing effect of the bottom slope over the edges of the vortices in the lower layer. The short bottom-trapped disturbance, responsible for the persistence of the instability for arbitrary small local supercriticality, plays now the fundamental role of generating continuously cyclonic and anticyclonic vortices at the downstream edge of the unstable interval. Those structures are stable enough to maintain coherence over several rotation periods. The anticyclones propagate into the stable region as isolated eddies, while the cyclonic vortices trigger the formation of dipoles inside the interval of instability.1 1 In some of the nonlinear solutions for smoothly varying topog- JANUARY 2003 BRACCO AND PEDLOSKY 217 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 for layers (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 vortices explored above appears to be quite robust. It does not depend on the length of the unstable interval, nor is sensitive to how abruptly the topography varies in the zonal direction. Additionally, only local maxima in supercriticality are required for the existence of local unstable modes and the growth rates are independent of the degree of stability of the flow outside the interval of maximum instability (Pierrehumbert 1984; Pedlosky 1989; Samelson and Pedlosky 1990). The bottomtrapped disturbance, responsible for the vortex formation, grows to balance the variations in time of relative vorticity with the ambient gradient of potential vorticity and its basic physical behavior can be expected to occur in flow configurations more general than the one analyzed in this work. The vortex propagation mechanism, however, seems to rely on the rebound of the dipole from the slip wall, with the penetration of anticyclones into the downstream stable interval due to the influence of the cyclonic image in the wall. In oceanography, the choice of slip boundaries has been motivated by the inability of resolving small-scale processes near the raphy, small vortexlike structures were formed in lieu of the long baroclinic disturbance in the upstream stable interval. Those structures, in which the deep flow was always counter to the sense of the shallow circulation, were very unstable and quickly destroyed. coast while modeling large-scale flows. This problem has been addressed by assuming that the solution is composed of two parts, an outer solution with slip boundaries and a complicated boundary layer with noslip conditions at solid surfaces (e.g., Pedlosky 1996). Nevertheless, we are reluctant to generalize this aspect of our results, and further studies, with nonuniform shear profiles, may help to elucidate the vortex propagation process. Recently, Walker and Pedlosky (2002) have shown that for meridional currents no minimum shear and no minimum channel width is required for instability. In the light of those results and given the original motivation of our analysis (a poleward boundary current in the Labrador Sea), a natural question, which will be addressed in a future work, concerns the generalization of our results to a meridional current in a north–south channel. Atmospheric flows along continental coasts are affected by topographic gradients such as the ones generated by coastal mountain barriers. In the context of oceanic flows, obvious candidates for local instability controlled by variations in the bottom slope similar to the model study presented, are coastal currents. Despite the obvious model simplifications, the scenario we address, opportunely rotated, resembles qualitatively the topography of the West Greenland continental slope. Indeed, a region of enhanced eddy kinetic energy (EKE) has been observed at the northern edge of the Labrador Sea in the West Greenland Current near an abrupt 218 JOURNAL OF PHYSICAL OCEANOGRAPHY change in topography (Lavender et al. 2002; Pickart et al. 2002; Prater 2002). Interestingly, the high EKE signal is not confined to the surface (Lavender 2001), and the analysis of the eddy field performed by Lilly et al. (2002, manuscript submitted to J. Phys. Oceanogr.) indicates a predominance of anticyclones with a composite double core. These vortices form in the West Greenland Current and propagate into the interior of the Labrador Sea. In our model, the southern wall limits the propagation of the dipoles that form within the unstable interval, and it bounds the deeper flow. Such a wall would not exist in the oceanic context. Therefore, we expect that in the absence of this artificial boundary, dipoles and anticyclones will move farther away from the northern boundary, filling the basin with vortices. The predominance of anticyclones in our model deserves an additional comment. In our calculations single eddies are generated at the downstream escarpment along the northern wall (i.e., where the zonal topographic variation dg (x)/dx attains its downstream maximum) through equilibration of the bottom-trapped disturbance and vortex rebound from the slip boundary. As a general result, those vortices form over the topographic step that separates the unstable from the stable region. The direction of the basic flow determines the orientation of the dipole for the rebound to take place. Carnevale et al. (1999) have shown that the evolution of a coastal current as it encounters a single escarpment depends strongly on the geometry of the topographic step. However, regardless of the direction of the basic flow, only anticyclones can migrate as single coherent structures toward deeper waters, that is, into the interior of the basin in the case of a boundary current, because of conservation of potential vorticity along fluid elements. We anticipate that in the absence of the beta effect the equilibration of the unstable modes is qualitatively similar to the cases previously illustrated, but the anticyclones in the upper layer are less strong and therefore less stable. The influence of b oriented perpendicularly to the topography, together with a more realistic configuration to model the dynamics in the West Greenland Current, will be the subject of a future study. 6. Conclusions Within the context of a simplified two-layer, quasigeostrophic model, we have numerically investigated the dynamics of a uniform flow confined in a channel over meridionally variable topography on the beta plane. The bottom relief controls the local supercriticality of the flow. Stability is ensured everywhere except in an interval of finite length at the center of the channel where localized baroclinic instability takes place. We have examined both the linear and the equilibrated behavior of the most unstable normal modes. In a previous study, Samelson and Pedlosky (1990) analyzed linear solutions for the same topographic configuration, in the absence of meridional variations of the perturbation field VOLUME 33 and without meridional confinement of the modes. In accordance with their results, in this work we show that for channels wide enough to sustain baroclinic instability, unstable modes exist for an arbitrary short interval of instability. The interval can be, therefore, much shorter than the shortest unstable wavelength for an infinitely long domain at the same supercriticality. For interval of instability comparable with the Rossby deformation radius or greater, the most unstable modes are weakly trapped in the upper layer, and Rossby waves dominate the eastward flow evolution. In the lower layer the amplitudes of the disturbance decay faster beyond the interval due to the stabilizing presence of the bottom slope. In the case of fairly small supercriticality, the instability is maintained by a short bottom-trapped wave that generates two recirculation zones of potential vorticity of opposite sign, trapped at the downstream edge of the region of instability. The narrow wave in the lower layer grows in time and oscillates in phase with the upper-layer flow. Upstream long baroclinic waves decay rapidly in both layers and oscillate with opposite sign. When nonlinearity is retained in the problem, the equilibrated solutions predict the formation of coherent vortices within the interval of instability, for arbitrarily small local supercriticality. Both cyclones and anticyclones are formed continuously at the northeastern edge of the unstable interval, resulting from the equilibration of the bottom-trapped wave. In the vortex generation process it is particularly important that the zonal component of the perturbation velocity can interact with the zonal gradient of the bathymetry. Indeed, this interaction ensures the meridional localization of the bottomtrapped disturbance. Via vortex stretching, dipoles formed at the northeastern edge of the interval of instability can rebound from the northern wall. Anticyclones move downstream along the north wall under the influence of their image vortices and propagate into the stable region. Once over the meridional slope in the stable region, they propagate away from the wall, into deeper water. Cyclones remain trapped inside the interval of instability, where they interact with the Rossby wave and the long baroclinic wave disturbances, and form new dipoles. Those coherent vortices are stable and are characterized by an upper core with circulation in the same sense as the deep one. The bottom-trapped wave, responsible for the persistence of the instability for arbitrarily small supercriticality, is also responsible for the formation of stable coherent vortices in the equilibrated solutions. Physically, this wave balances the time rate of change of relative vorticity with the motion in the ambient gradient of potential vorticity. We may speculate that this simple mechanism of formation of coherent vortices is not limited to our model study and applies to more general baroclinic flows of geophysical interest. The propagation of the vortices formed in the unstable interval is strongly affected by the meridional walls. It is natural JANUARY 2003 BRACCO AND PEDLOSKY to suppose that in the absence of the free-slip southern boundary the coherent dipoles would move farther from the coast instead of rebounding from the meridional wall, filling the basin with vortices as observed in the Labrador Sea. Further extensions of the model will be able to address this problem. Natural extensions of this work would thus include consideration of meridional currents 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 eddy field in the Labrador Sea. We are grateful to Joe LaCasce for stimulating discussions and to Nicolas Cauchy for his careful editing of the manuscript. A. 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