Download Vortex Generation by Topography in Locally Unstable Baroclinic

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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
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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
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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
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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
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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.
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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
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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.
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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
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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
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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-
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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. B. is supported by the Postdoctoral Scholar Program at the Woods Hole Oceanographic Institution, with
funding provided by the USGS. J. P. is supported by
the National Science Foundation Grant OCE 9901654.
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