JUNE 2000
LACASCE AND BRINK
1305
Geostrophic Turbulence over a Slope
J. H. LACASCE
AND
K. H. BRINK
Woods Hole Oceanographic Institution, Woods Hole, Massachusetts
(Manuscript received 19 March 1998, in final form 20 July 1999)
ABSTRACT
The authors examine freely evolving geostrophic turbulence, in two layers over a linearly sloping bottom.
The initial flow is surface trapped and subdeformation scale. In all cases with a slope, two components are
found: a collection of surface vortices, and a bottom-intensified flow that has zero surface potential vorticity.
The rate of spinup and the scale of the bottom flow depend on L [ F 2 U1 /b 2 , which measures the importance
of interfacial stretching to the bottom slope, with small values of L corresponding to a slow spinup and stronger
along-isobath anisotropy. The slope also affects the mean size of the surface vortices, through the dispersal of
flow at depth and by altering vortex stability. This too can be characterized in terms of the parameter, L.
1. Introduction
In a previous work, one of the authors examined how
a topographic slope affects an isolated vortex (LaCasce
1998, hereafter L98). A slope favors dispersal of the
deep vortex flow by topographic waves when U 2 ,
b 2 r 2 , where U 2 is the magnitude of the deep velocity,
r the vortex radius, and b 2 the slope (see below). Unlike
on the barotropic b plane, an initially barotropic vortex
leaves behind a surface vortex because topographic
waves have no potential vorticity (PV) at the surface.
This residual vortex always has zero flow in the lower
layer, or is compensated (e.g., McWilliams and Flierl
1979); it owes its existence to the vertically asymmetric
influence of topography in the presence of stratification.
The slope also affects vortex stability. Over a flat
bottom, a compensated vortex larger than deformation
scale is baroclinically unstable, and tends to break into
smaller vortices (e.g., Flierl 1988; Helfrich and Send
1988). But, as suggested originally by Hart (1975), a
bottom slope can stabilize the vortex. In particular, when
the parameter L [ F 2 U1 /b 2 is less than unity (F 2 being
the inverse squared deformation radius at depth), unstable growth is inhibited (L98).
While of interest, L98 had several conceptual shortcomings. For one, the initial vortex was (necessarily)
specified and perturbations were always assumed weak.
One could well ask what happens when vortices form
over topography? And further, what happens when the
perturbations are not so small?
Corresponding author address: Joe LaCasce, Woods Hole Oceanographic Institution, M.S. 29, Woods Hole, MA 02543.
E-mail: jlacasce@whoi.edu
q 2000 American Meteorological Society
We consider one such case where vortices form and
are perturbed severely: freely evolving geostrophic turbulence over a slope. It is known that vortices emerge
from random initial flows in many instances (McWilliams 1984) and that they continually merge, producing still larger vortices. We seek to find how a slope
alters this process.
The turbulent 2D evolution from random initial phases can be regarded from two complimentary viewpoints.
The first focuses on the energy, which shifts to larger
scales [Fjortoft (1953), Onsager (1949); for reviews, see
Rhines (1979) or Kraichnan and Montgomery (1980)].
In numerical models, this ‘‘inverse cascade’’ continues
until the scales of motion are comparable to that of the
domain (e.g., Smith and Yakhot 1994). However, as
discussed by Rhines (1975), Holloway and Hendershott
(1977), Vallis and Maltrud (1994), and others, the cascade can stall at a smaller scale in the presence of the
b effect due to Rossby waves, which are inefficient in
terms of energy transfer (Rhines 1975). With two layers
and a flat bottom, baroclinic energy cascades to the deformation radius, whether from small or large initial
scales (e.g., Rhines 1977; Salmon 1978) and thereafter
the cascade is primarily barotropic. This is a special
case of the more general shift to graver horizontal and
vertical wavenumbers with continuous stratification,
postulated by Charney (1971).
Alternately, one may focus on the vorticity fields. In
f plane, barotropic turbulence, vortices often emerge
from a random initial field to dominate the flow at later
times (e.g., McWilliams 1984, 1990a; Benzi et al. 1986).
The inverse cascade occurs as a result of vortex mergers,
and the b effect can defeat the cascade by favoring
Rossby wave dispersion of vortices larger than a certain
scale (e.g., McWilliams 1984). The shift to more baro-
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JOURNAL OF PHYSICAL OCEANOGRAPHY
tropic flow moreover may be accomplished by the vertical alignment of vortices at different levels (Polvani
1991; McWilliams 1990b; see also below).
We examine the two-layer, f -plane case, but with a
linear slope in the lower layer. Conceptually, one expects a response like that in two layers with planetary
b, except that the inverse cascade will be inhibited by
topographic waves rather than planetary waves. However, there is a catch: topographic waves require motion
at depth, so if the flow is initially surface trapped, there
must be a vertical transfer of energy; the slope itself
might affect this transfer. Furthermore, there is the question of the vertical distribution of vortices because topographic waves selectively disperse those at depth.
We present a set of ‘‘observations’’ from numerical
experiments of freely evolving turbulence and attempt
to rationalize the findings with heuristic arguments,
which derive from previous studies of two-dimensional
turbulence. Though we limit ourselves to freely cascading (unforced) turbulence from a surface-trapped
flow of small scales, we suspect the general characteristics will apply to other systems.
Stratified geostrophic turbulence over topography has
been considered before. Rhines (1977) examined freely
evolving turbulence over a bumpy bottom, and Treguier
and Hua (1988) examined a similar system with forcing.
Vallis and Maltrud (1994) considered forced turbulence
over sinusoidal topography. Their results are discussed
in more detail later on.
The paper is organized as follows: the equations and
numerics are presented first, with the initial spectrum.
We then examine two examples: a run with a weak slope
and second with a strong slope. These runs suggest that
there are two dominant physical elements, a collection
of surface vortices and second component (hereafter
called the ‘‘deep mode’’). We consider each in turn before commenting on the results.
VOLUME 30
tential vorticities2 and c i the velocity streamfunctions
for layers i 5 1, 2, with u i 5 k 3 =c i . The Jacobian
is defined as
J(a, b) [
1]x ]y 2 ]x ]y2
]a ]b
]b ]a
and ¹ 2 the horizontal (two-dimensional) Laplacian operator. The (squared) inverse of the deformation radius
in each layer is F i 5 f 2 /g9H i , with g9 the reduced gravity, f the Coriolis parameter, and H i the depth of the
layer. The layer depths are taken to be equal unless
indicated otherwise, with F1 5 F 2 5 100.
The linear slope yields a constant PV gradient of b 2
[ 2 f (] y H 2 )/H 2 in the lower layer. Quasigeostrophy requires that |(] y H 2 )L| K H 2 (if L is the scale of motion),
so the slope is fundamentally weak. However, this restriction does not preclude a strong relative topographic
influence (L98).
Equations (1)–(2) were solved with the Fortran code
used in Flierl et al. (1987) and in L98. The code employs
a doubly periodic domain (e.g., Canuto et al. 1988), so
all variables are Fourier decomposed in both directions:
(q̂i , û i ) 5
1
Nx Ny
O O (q , u ) exp(ikx 1 ily),
k max
l max
i
i
(3)
k5k min l5l min
with q i [ ¹ 2 c i 1 F i (c 32i 2 c i ) the perturbation po-
where N x 5 N y are the number of Fourier modes and
kmin 5 2kmax 1 1 and lmin 5 2lmax 1 1. We nondimensionalize length scales so that the domain has dimensions 2p 3 2p, a common choice that yields integral
wavenumbers and (kmax , lmax ) 5 (N x /2, N y /2). Variables
are alternately referred to by their Fourier and real representations, and the caret distinguishes the Fourier
transform (as in 3). We employed 256 2 modes (kmax 5
lmax 5 128). The deformation wavenumber of 10 was
thus reasonably well resolved.
Time stepping was via a standard leapfrog, with an
Euler step every 50 steps to suppress computational
errors. Spectral dealiasing (Patterson and Orszag 1971)
had little impact on the results but greatly increased the
computational cost and therefore was not used. Numerical stability was achieved via an exponential cutoff
filter with exponent 4 (Canuto et al. 1988) and cutoff
wavenumbers of 84 (and 42). Potential vorticity was
better conserved and horizontal gradients better resolved
under the action of such a filter than with the use of the
more traditional Laplacian (n 2¹ 2 q i ) or ‘‘biharmonic’’
damping (n 4¹ 4 q i ) schemes. Details are given in LaCasce
(1996).
The initial flow was isotropic and surface trapped,
with a relatively narrow spectral peak at a scale smaller
than the deformation radius. Such a spectra facilitates
observing the ‘‘arrival’’ of energy at the deformation
radius and the spinup of the lower layer, but there is no
1
The orientation of the slope is irrelevant on the f plane, which
is rotationally invariant.
2
Hereafter PV refers to the total potential vorticity, i.e., q1 or q 2
1 b 2 y.
2. Equations, numerics, and initial conditions
The model is the same as that in (L98): it employs
two layers, quasigeostrophic (QG) dynamics, and a linear slope. The planetary gradient is assumed to be zero;
that is, b 5 0, and the implications of this omission are
discussed later on. Assuming the bottom shoals to the
north,1 the inviscid equations for the layerwise PV are
(e.g., Pedlosky 1987)
]
q 1 J(c1, q1 ) 5 0
]t 1
(1)
]
]
q 1 J(c2 , q2) 1 b 2 c2 5 0,
]t 2
]x
(2)
JUNE 2000
LACASCE AND BRINK
compelling geophysical reason for the choice. We also
examined initial flows of different scale and vertical
structure, but the essential aspects are captured by this
particular combination. Rhines (1977) used a similar
initial field in his study of two-layer turbulence on the
planetary b plane.
The initial surface kinetic energy spectrum is defined:
1307
where the domain widths are L x 5 L y 5 2p, as stated.
Condition (4) yields an rms surface velocity of one. The
spectrum is the same as that of McWilliams (1990a);
the steep tail was chosen to reduce energy dissipation
at early times.
The deformation scale was 1/10 the domain length,
roughly one unit. Assuming the deformation radius is
O|10| km and the surface velocities O|10| cm s21 , the
time unit }L/U is about 1 day.
turbation PV (e.g., McWilliams 1984) exceeds 50 in the
upper layer at t 5 40, but is approximately 3 in the
lower layer (see below). The major differences between
the strong and weak slope cases are that the vortices
(which we tentatively identify as the residual vortices
described above and in L98) are somewhat smaller and
the lower-layer PV more severely perturbed with the
weak slope.
Because the initial flow is surface trapped, the deep
perturbation PV is initially nonzero due to interfacial
stretching. With the weak slope the stretching is clearly
visible, but is overwhelmed by the strong slope. By
scaling, the relative importance of the stretching to the
slope PV at t 5 0 can be gauged by the parameter L
5 F 2 U1 /b 2 (L98); in Figs. 1 and 3, L 5 20, whereas
in Figs. 2 and 4, L 5 1/8.
How can the PV and streamfunction fields differ so
profoundly? One possibility is if the streamfunctions
are dominated by a component that has zero PV in one
layer. Topographic waves, with zero surface PV, are an
example (L98); the surface PV ‘‘filters’’ out the waves,
and what we see is what is left behind: here, vortices.
Thus we tentatively identify two flow components: surface vortices and a deep mode (perhaps topographic
waves or jets), which may have flow in both layers (e.g.,
Fig. 1).
Hereafter we consider these two modes in more detail.
In particular, we ask: 1) What sets the scales of the deep
mode? 2) How quickly does it spin up? 3) What role
do the vortices play? and 4) How do these aspects depend on stratification, bottom slope, and flow intensity?
We focus first on the deep mode and then the vortices.
3. Sample results
4. Deep mode
To illustrate the possible responses, we present two
cases: one with a weak bottom slope (by a measure
defined later) and one with a strong slope. The streamfunctions at various times are shown in Figs. 1 and 2.
Both cases exhibit a shift to larger scales, consistent
with an inverse cascade, but otherwise are quite different. With a weak slope, the fields are largely barotropic
by t 5 5 (i.e., roughly 5 days), and by t 5 40 the
dominant structures are greatly zonally elongated (recall
that zonal here means along-isobath). Moreover, westward phase propagation is evident in successive plots,
suggestive of (topographic) Rossby waves. In contrast,
the strong slope fields at t 5 5 are markedly baroclinic,
with isotropic structures at the surface and bands of
zonal flow at depth. By t 5 40, these bands have intensified and also are evident at the surface. However,
the isotropic surface features persist, presenting a Jupiter-like mixture of circular (vortices) and bands.
The PV fields are rather different (Figs. 3 and 4). In
neither case does the PV become barotropic. Rather, the
surface PV is dominated by vortices, whereas the deep
PV is devoid of vortices and dominated by the slope
contribution, b 2 y. In both cases the kurtosis of the per-
a. Scale selection
0
KE 1 (k) 5
if k , k 0

KE 01k

if k $ k 0 ,
18
(k 1 k 0 )

6
where k 2 5 k 2 1 l 2 and k 0 was taken to be 14, that is,
somewhat larger than the deformation wavenumber of
10. The coefficient KE 01 was determined by demanding
1
Lx Ly
EE
O O KE (k)
1
5
O O k |ĉ |
N N
(u12 1 y 12 ) dx dy 5
1
Nx2 Ny2
1
k
l
k
l
2
2
x
5 1,
2
y
2
1
(4)
Certain spectral characteristics of the deep flow are
predictable. The following derivation follows that of
Vallis and Maltrud (1994) for the anisotropic spectral
arrest in barotropic b turbulence. As shown by those
authors, one can identify a spectral boundary in wavenumber space that separates harmonics which are
‘‘wavelike’’ from those which are more nonlinear. To
this end, one compares a turbulence ‘‘frequency’’ (Uk)
to the Rossby wave frequency (bk/k 2 ) at each harmonic,3 and modes with higher wave frequencies are
presumedly more linear. The boundary is found by
equating the two frequencies. If energy is initially predominantly in the nonlinear modes, it will cascade to
smaller wavenumbers to accumulate (roughly) at the
boundary. The boundary is found to be
3
There are several alternate means of estimating the turbulence
frequency, such as the inverse rms vorticity. All yield similar results
for freely evolving turbulence, which is generally dominated by a
single scale of motion (Vallis and Maltrud 1994).
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VOLUME 30
FIG. 1. Upper (left) and lower (right) layer streamfunction fields for the case L 5 20, at various times.
The contour intervals are 6[0.05, 0.1, 0.2, 0.4, 0.8, and 1.6].
k b2 5 C1
12
b
cos(u ),
U
(5)
where u 5 tan21 (l/k) is the angle the wavevector makes
with the l 5 0 axis and C1 is an unknown constant of
order unity. The original arrest parameter of Rhines
(1975) resembles this, except that Rhines assumed isotropy at arrest [by setting cos(u) 5 ½]. The curve is
theoretically fixed without forcing or dissipation because U is constant.
To extrapolate to two layers with a linear slope, one
would probably use the velocity at the bottom and the
topographic wave frequency (e.g., Rhines 1970):
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LACASCE AND BRINK
FIG. 2. The streamfunction fields with L 5 0.125. The contour values are as in Fig. 1.
v52
1
2
b2k
k 2 1 F1
.
k 2 k 2 1 F1 1 F2
(6)
This resembles the barotropic Rossby frequency, but for
the scale- and stratification-dependent correction (in parentheses). The latter varies from ½ to 1 when the layer
depths are equal, and so can be assumed to be unity for
the purposes of this scaling argument. The topographic
wave boundary then follows directly from (5):
k b2 2 5 C2
1U 2 cos(u ),
b2
(7)
2
where the constant C 2 is unknown. The key difference
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VOLUME 30
FIG. 3. The potential vorticity fields for the L 5 20 case. The contour values are 6[20, 40, 60, 80, 100,
120, 140, 160, 180, 200].
between (5) and (7) is that U 2 in the latter is generally
not constant. For example, if the lower layer is initially
at rest, kb2 decreases from infinity as the lower layer spins
up. This frustrates an a priori prediction of the arrest scale.
But such a prediction can be obtained from the following heuristic argument. Suppose the lower layer is
initially at rest and the upper layer active. Then we could
write c1 5 C1 1 c19 and c 2 5 c92 , where the primed
quantities are initially small and related to the deep flow.
Then the lower-layer PV equation, (2), is a forced nonlinear wave equation:
D
]
D
q92 1 b 2 c29 5 2F2
C,
Dt 2
]x
Dt 2 1
(8)
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LACASCE AND BRINK
FIG. 4. The potential vorticity fields for the L 5 0.125 case. The contour values are as in Fig. 3.
where D/Dt 2 [ ] t 1 u92 · = . If the surface flow is unsteady, the undulating interface will drive flow at
depth. Weak forcing presumedly produces topographic
waves, but stronger forcing affords a more nonlinear
response; which one occurs will likely depend on the
ratio of the forcing frequency to the topographic wave
frequency at a given harmonic. From (8), the former
scales as F 2 U1 , leading to an alternate wave boundary
prediction:
gb 2 (u ) [ C3
1F U 2 cos(u ) 5 1 L 2 cos(u ),
b2
2
C3
(9)
1
with C 3 an unknown constant. In contrast to (7), ex-
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VOLUME 30
FIG. 5. The lower-layer kinetic energy spectra at two times, for various L. The spectra at t 5 1 represent
the means of three runs; the spectra at t 5 40 are from 1 of the 3. Also shown are two arrest ‘‘boundaries,’’
g b 2 (solid) and k b 2 (dashed), both defined in the text. The spectra contour values in each case are [0.1, 0.3,
0.5, 0.7, and 0.9] times the maximum value of KE 2 (k, l ) in each case.
pression (9) can be evaluated a priori. Except for different dependences on b 2 , both expressions produce a
‘‘dumbbell’’ shaped barrier. Moreover, both predict a
privileged role for modes with k 5 0, where the Rossby
wave frequency is zero, because energy may be transferred there without impedance from waves (Rhines
1975; Vallis and Maltrud 1994). The dependence on L
is perhaps unsurprising, given it measures the relative
importance of the interfacial stretching and slope contributions to the deep PV. In this context, small values
of L imply modes are more wavelike.
Shown in Fig. 5 are the two-dimensional spectra of
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LACASCE AND BRINK
deep kinetic energy from numerical runs with different
values of L. The KE 2 spectra are defined as
KE 2 (k, l, t) [ k 2 |ĉ 2 (k, l, t)| 2 ,
(10)
where again k 2 5 k 2 1 l 2 , and the sum over all wavenumbers is the mean kinetic energy, that is, KE 2 (t) 5
1/(N 2xN 2y) S S KE 2 (k, l, t). The (t 5 1) spectra were averaged from a small ensemble (3) of runs with identical
initial spectra but different random phases, and the (t
5 40) spectra are from one run at a later time. Also
shown are two ‘‘wave boundaries’’: k b 2 (u) from (7) (the
dashed line) and g b 2 (u) from (9) (solid line), with the
multiplicative constants, C 2 and C 3 , each set to one.
Expression (9) was evaluated using the initial surface
velocity (51), and k b 2 (u) derives from the instantaneous
mean deep velocity.
The active modes at t 5 1 generally lie outside the
solid curve, g b 2 (u). With L 5 4, there are few wavelike
modes, and many wavenumbers smaller than the deformation wavenumber, 10, are active. With smaller values of L, more modes are apparently excluded. This is
not to say the wavelike modes are inactive, but the energetic peak lies outside the curve in all cases.
Unlike the solid curve, the Rhines parametric curve,
k b 2 (u), shifts as the lower layer spins up. In all cases,
it comes to rest near the solid curve, g b 2 (u) . With L 5
4, k b 2 (u) . g b 2 (u) for all u at t 5 40, and the peak of
KE 2 arguably lies closer to k b 2 (u) than g b 2 (u). This is
to be expected, had the flow become barotropic prior
to arrest, and indeed the streamfunction fields here are
nearly barotropic (Fig. 1). With L 5 ¼, k b 2 (u) instead
sweeps in to lie just inside of the peak, which was already next to g b 2 (u). This alignment occurs because the
peak intensifies during the run, but modes with k ,
g b 2 (u) do not.
As noted, the k 5 0 modes are always accessible
because the wave frequency is zero there. Because fewer
harmonics are accessible with larger slopes, the k 5 0
modes are more likely populated, and the flow is more
jetlike. The meridional scale selected is probably related
to the choice of initial spectrum, peaked at k 5 14.
The surface kinetic energy spectra, defined as
KE1 (k, l) [ k 2 |ĉ1 (k, l)| 2 ,
(11)
are shown in Fig. 6. Early on, KE1 (k, l) is approximately
isotropic and greatest near the deformation wavenumber,
k 5 10, consistent with an initial cascade to the deformation radius (e.g., Rhines 1977; Salmon 1978). The (t
5 40) peaks resemble those in KE 2 at the same time
(Fig. 5), suggesting they are related. They are also somewhat weaker, which implies bottom intensification.
4
Assuming for the moment that the flow at depth was
due entirely to topographic waves, its surface expression
could be found. Exploiting the fact that topographic
waves have zero PV (L98), one can write:
ĉ1d (k, l) 5
F1
ĉ (k, l).
F1 1 k 2 2
(13)
This in turn can be used to calculate the surface kinetic
energy related to the deep mode, KE1d (k, l) (right column).
The positions and amplitudes of the peaks in KE1 (k, l)
(second column) are reproduced in KE1d (k, l), supporting
the notion that they are related to the deep mode and that
the latter has zero surface PV [the necessary condition for
(13)]. Note the peaks coincide even for ‘‘zonal’’ modes
(k 5 0), that is, those that definitely are not wavelike.
Thus the surface flow is eventually dominated by the
surface portion of the deep mode in all cases. The next
question then is: How quickly does the deep mode intensify?
b. Spinup
Varying L also alters the rate of energy transfer to
the lower layer, as can be seen in plots of the upperand lower-layer kinetic energies versus time for various
L (Fig. 7). If the bottom is flat (L 5 `), KE1 (and the
potential energy, not shown) decrease as KE 2 increases,
though KE1 . KE 2 throughout. If L . 1 (weak slope),
the rate of increase of KE 2 is the same, but KE 2 asymptotes to a larger value. When L , 5 or so, KE 2
exceeds KE1 at late times. So a weak slope enhances
the total transfer of energy, without greatly altering the
rate of transfer.
Over larger slopes (L , 1), KE 2 may become as large
as in the weak slope cases at late times, but the deep
spinup is retarded. This is qualitatively consistent with
the observation that surface vortices are stabilized by a
slope when L , 1 (L98). To quantify how this retardation
varies with L, several short duration runs were performed
in which the slope, stratification, and initial surface kinetic energy were varied independently (appendix A).
The results suggest ] t KE 2 (t 5 0) } L when L , 1, so
L also determines the spinup rate over strong slopes.
Thus decreasing L has a nonmonotonic effect on the
vertical transfer of energy. Decreasing L to one increases
the total transfer from that over a flat bottom, but decreasing it further slows the rate of transfer. It is important
to note that the lower layer always spins up; what varies
is the rate. A similar nonmonotonic dependence on L is
observed with the vortex statistics, as discussed next.
5. Surface vortices
4
With potential energy defined:
PE(k, l) [ F1 |ĉ1 2 ĉ 2 | 2 ,
(12)
the total energy, S k S l H1 KE1 1 H 2 KE 2 1 H1 PE (e.g., Rhines 1977),
is conserved in an inviscid flow.
As noted in section 3, we find vortices, or long-lived
intense structures, in the surface PV in all cases. First,
we attempt to assess their vertical and horizontal extent
and then comment on their importance relative to the
deep mode.
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a. Vertical structure
The vortices appear to be surface trapped, but determining their vertical structure is somewhat difficult because the deep mode so dominates the deep streamfunction. So we tested a couple of possible vertical
structures to see which gave the most reasonable separation of fields.
To this end, we first assume that the deep mode has
zero surface PV, or equivalently that the vortices account
for all the surface PV (q1 5 q1y ). The results of section
4a support this. To infer the vortex flow requires additional information, either the deep vortex PV or their
streamfunction in either layer. Since we do not know
either of these, we make two educated guesses.
For the first, we assume the vortex has zero deep PV.
This follows from the knowledge that deep potential
enstrophy (squared perturbation PV) is conserved with
a flat bottom without dissipation and decreases monotonically with dissipation. With this initial flow, the initially weak deep PV only gets weaker, and the total
surface enstrophy far exceeds that at depth at late times.
In fact, we observe two to three isolated vorticity maxima at depth, but these are weak and exceptional.
For the second, we assume the vortex has no deep
flow (are compensated). This follows from the observation that the deep flow of isolated vortices over intermediate and strong slopes is quickly dispersed (L98).
Note that this assumption is equivalent to saying that
the deep mode constitutes all the lower-layer flow, that
is, the assumption used in predicting the surface kinetic
energy of the deep mode (Fig. 6).
With these assumptions, and that q1 5 q1y , we can
find all the vortex fields. To obtain the surface streamfunction, we invert the expressions for the Fourier-transformed perturbation PV (section 2); this yields either
ĉ1y (k, l) 5
2(k 2 1 F2 )q̂1 (k, l)
,
k 2 (k 2 1 F1 1 F2 )
(14)
in the case of zero deep vortex PV (q 2y 5 0), or
ĉ1y (k, l) 5
2q̂1 (k, l)
k 2 1 F1
(15)
for zero deep flow (c 2y 5 0). The resulting vortex surface streamfunctions and the residuals, c1 2 c1y are
shown in Figs. 8–10.
With a flat bottom (Fig. 8), the assumption that q 2y
5 0 yields strong surface vortices and almost no residual
surface flow; that is, the vortices are the only flow constituent. In contrast, assuming c 2y 5 0 yields weak vortices and a second, larger-scale component that is bottom
intensified. It is difficult to explain such a flow, given
the layer symmetry, so we conclude that the vortices
are more likely to have zero deep PV than zero deep
flow.
With a weak slope L 5 4, assuming c 2y 5 0 yields
weak surface vortices and a strong bottom-intensified
component (Fig. 9). But in this case we have the slope
VOLUME 30
to break the layer symmetry, permitting a second independent component (the deep mode). The residual
streamfunction at the surface has westward phase propagation, as does c 2 , suggestive of topographic waves.
The q 2y 5 0 decomposition mixes the two fields so that
the vortices appear to have a large-scale component with
westward phase propagation. Thus these vortices are
more likely to be compensated.
Over a strong slope (L 5 ¼), the c 2y 5 0 decomposition yields isotropic surface vortices, and jets that
resemble similar features in the lower layer (Fig. 10).
The q 2y 5 0 decomposition in contrast yields vortices
embedded in an unusual larger-scale flow with a marked
meridional anisotropy. The latter appears also in the
residual. So again the c 2y 5 0 assumption produces a
more believable field separation.
Therefore, vortices over both strong and weak slopes
appear to be compensated while those over the flat bottom
have no deep PV and strong deep flow. Why is the flat
bottom different? In fact, in both topographic cases, the
slope is strong enough so that the cascade in the lower
layer arrests (Fig. 5). From scaling, an arrest implies the
dispersal of vortices; from (7), the scale of motion is such
that L ø (U 2 /b 2 )1/2 , or equivalently U 2 ø b 2 L 2 , which
is the same condition for dispersal of vortices of scale L
(e.g., L98). So a deep arrest goes hand and hand with
surface vortex compensation. Of course, an arrest (and
vortex compensation) could have been avoided had the
slope been much weaker (see also below).
b. Vortex area and number
We observed that the mean vortex area (its surface
PV) also varied with bottom slope. To quantify this, we
employed an automated vortex census similar to that of
McWilliams (1990a). The routine locates simply connected regions in which the surface PV exceeds a fixed
threshold value, here taken (arbitrarily) to be q1 5 20.
It then rejects regions that are too small (less than five
grid points) and/or too elongated (a horizontal aspect
ratio greater than 2), the latter to exclude filaments.
Finally it counts the vortices and determines their mean
area, circulation, etc.
In spectral numerical simulations, vortex mergers decrease the total number of vortices because stronger
vortices shear out weaker ones (McWilliams 1990a).
The McWilliams results suggest a power-law decay in
the number of vortices, an observation which led to the
scaling theory of Carnevale et al. (1991).5
Using the vortex counting routine, we can examine
how the slope impacts the vortex statistics. The number
and mean areas of vortices in experiments with various
L are shown in Figs. (11) and (12). For comparison, we
5
The decrease in the number of vortices may depend on the lateral
viscosity, because vortex mergers may yield more than one or two
vortices in the inviscid limit (Dritschel 1992).
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LACASCE AND BRINK
1315
FIG. 6. The surface kinetic energy as a function of L. Again, the spectra on the left are ensemble averages from
three runs; note these fields were from t 5 2. The spectra at t 5 40 are from one of the three. The dashed quarter
circle indicates the deformation wavenumber, and the contour values are [0.1, 0.3, 0.5, 0.7, and 0.9] times the maximum
value of KE1 (k, l) in each case.
include the results from a flat-bottom experiment, from
one with 1.5 layers (a stationary lower layer, e.g., Pedlosky 1987), and one from a run in which the upper layer
is 1/5 as thick as the lower layer (labeled L* 5 0.1).
The number of vortices decreases in time (Fig. 11),
with a power-law decay similar to that of McWilliams
(1990a), N } t 0.7 . This is approximately true for most
values of L, so varying the slope apparently induces
only minor changes in the vortex merger rate; this is
probably due to the high density of vortices in these
simulations. The exception is the run, L 5 0.125, which
exhibits a higher vortex mortality rate (see below).
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VOLUME 30
(perhaps coincidentally) at a rate comparable to that of
McWilliams (1990a).
The differences in mean area are reflected in the PV
kurtosis because with more compact vortices the kurtosis is higher. The upper- and lower-layer perturbation
PV kurtoses are shown in Fig. 13. The upper-layer kurtosis exceeds the Gaussian value of three and grows in
time as dissipation selectively removes the background
PV around the vortices. The PV kurtosis exhibits the
same nonmonotonic dependence on L as does the mean
area, with the largest values occuring when L is near
unity, that is, when the vortices are the smallest.
The lower-layer PV kurtoses (lower panel) in contrast
are generally near three, consistent with a lack of coherent structures. The kurtosis is somewhat elevated
when L 5 20, which is perhaps evidence of deep vortex
flow; however, it is not greatly so. The flat bottom case
is obviously different, with the kurtosis moving offscale;
this is related to two or three weak vortices at depth,
alluded to earlier.
Thus vortex mergers over a slope do not necessarily
produce larger and larger vortices as they do in a barotropic fluid. The growth depends on the slope, and like
the mean deep kinetic energy (section 4b), the variation
is nonmonotonic with increasing slope severity. The total vortex energy (assuming vortex compensation) can
be shown to display a similar dependence, with the
weakest vortex energies occuring when L ø 1. To rationalize this nonmonotonic dependence, one must recognize the dual role played by the slope: in permitting
deep dispersion but also in altering vortex stability
(L98).
c. The variation with L
FIG. 7. The total upper-and lower-layer kinetic energies as a function of time for various values of L.
The mean vortex area is shown in Fig. (12), and here
we find noticeable variations. Over a flat bottom, the
mean area increases in time such that a } t 0.12 . This is
somewhat slower than McWilliams’s rate of a } t 0.37 .
The reason for the difference is not known, but may be
related to the different types of numerical damping used
in our and his studies (see also LaCasce 1996). With
finite values of L, we find smaller vortices, with a mean
area that is either approximately constant or decreasing
in time. However, for values of L , 1, the mean area
is larger, though also usually decreasing in time. The
run with a shallower upper layer (L* 5 0.1) has still
larger vortices, and the largest are found in the 1.5-layer
case. The area increases fastest in this last case, and
Consider the progression from a flat bottom to a steep
slope. Mergers in the flat bottom case produced vortices
with zero deep PV. Such vortices become more barotropic as their scale increases; inverting the perturbation
PV, with q 2 5 0 yields
ĉ1 (k, l) 5
2(k 2 1 F2 )q̂1 (k, l)
and
k 2 (k 2 1 F1 1 F2 )
(16)
ĉ2 (k, l) 5
2F2 q̂1 (k, l)
.
k 2 (k 2 1 F1 1 F2 )
(17)
Plainly, c 2 ø c1 if k 2 , F 2 , that is, if the vortex scale
exceeds the deep deformation radius.6 Vortices with
strong barotropic circulation are more likely to be baroclinically stable (e.g., Flierl 1988), so the vertical transfer of energy is curtailed later on.
Incidentally, the inverse cascade with zero or nearzero deep PV is purely a two-dimensional merger pro-
6
Although it is interesting to note |c1| . |c 2| at all scales, which
is why KE1 . KE 2 at all times in Fig. (7).
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LACASCE AND BRINK
1317
FIG. 8. Decomposing the surface streamfunction under the assumption of (a) zero deep PV and (b) zero
deep flow for the flat bottom case at t 5 20. Shown are the ‘‘vortex’’ c1 , or the streamfunction associated
with the surface PV, the remainder (which corresponds to a second component with zero surface PV), and
the deep streamfunction. The contour values are 6[0.11, 0.32, 0.53, 0.74 and 0.95].
cess. The flow becomes more barotropic simply because
larger vortices are more barotropic. No ‘‘alignment’’ of
vortices (e.g., McWilliams 1990b) is required for the
increase of barotropy.
With a weak slope, deep dispersal prevents vortices
from becoming barotropic, so mergers produce more
and more unstable surface vortices. The latter break into
smaller vortices or possibly only a smaller single vortex
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 30
FIG. 9. As in Fig. 8 but with L 5 4. The contour values are 6[0.05, 0.15, 0.23, 0.33, 0.43].
(L98), which are then free to merge again. So there is
repeating cycle of merger, instability, and radiation,
which in turn prolongs the transfer of energy to the
lower layer (e.g., Fig. 7). The prolonged transfer is also
evident in spectral energy fluxes as a ‘‘topographic flux’’
to the baroclinic mode, like that discussed by Treguier
and Hua (1988); see LaCasce (1996) for details.
Over strong slopes, deep dispersion occurs so rapidly
that unstable growth is suppressed. To understand why
this then affords larger vortices, it helps to consider the
extreme case of 1.5 layers, where there is no deep motion at all. In 1.5 layers, vortices are by definition compensated. Such a vortex has an azimuthal velocity proportional to K1 (r/l ) in the far field, where K1 is a mod-
JUNE 2000
LACASCE AND BRINK
1319
FIG. 10. As in Fig. 9 but with L 5 0.125. The contour values are 6[0.04, 0.12, 0.20, 0.28, 0.36], for the
right panels and the lower-left panel, and three times those values for the upper left and middle left panels.
ified Bessel function and l is the deformation radius
(e.g., Flierl et al. 1980). So the compensated vortex
velocity decays exponentially beyond a deformation radius from its center. In contrast, the barotropic vortex
velocity decays as 1/r in the far field, and thus has a
much greater range of influence when the deformation
radius is smaller than the domain. The more compact
velocity field results in less filamentation in the 1.5layer case (Polvani et al. 1989), and thus larger vortices.
Moreover, it means that, when these vortices reach the
deformation scale, they cease to interact with one another, greatly slowing the inverse cascade (see Larichev
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JOURNAL OF PHYSICAL OCEANOGRAPHY
VOLUME 30
FIG. 11. The number of surface vortices as a function of time, for
various L. The vortices are identified as defined in the text.
and McWilliams 1991; Polvani et al. 1994). A decrease
in merger rate is difficult to see in Fig. 11 only because
the density of vortices is high, as noted before.
The same presumedly applies with an infinite slope,
but with finite slopes, energy is always lost to the deep
mode. Because the latter has an (intensifying) surface
expression, it can advect and even shear out surface
vortices. This is, in fact, why the mean area often appears to decrease more rapidly later on in these experiments (Fig. 12). As a check, we made an additional
FIG. 13. The kurtoses of the upper- and lower-layer perturbation
PV vs time for various values of L.
FIG. 12. The mean area of surface vortices as a function of time
for various L.
run with a weaker deep mode; with a shallower surface
layer, the available potential energy is less, as is the
eventual strength of the deep mode. In the L* 5 0.1
run (H1 /H 2 5 1/5), we indeed observe larger vortices
(Fig. 12), although here too there was evidence for vortex shearing.
Ideally, one could construct a set of experiments in
which vortices were not sheared like this, to isolate the
effect of variable stability. In fact, this is possible if one
replaces the bottom slope with a bottom Ekman layer
(appendix B). It has long been recognized that weak
damping that acts only at depth can be destabilizing,
whereas strong damping is stabilizing (Holopainen
1961; Pedlosky 1983). In other words, bottom drag
ought to have a similar nonmonotonic effect on surface
vortices. Substituting a bottom Ekman layer for the
slope does indeed yield similar vortex statistics, with
the exception that the mean area converges to the 1.5-
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LACASCE AND BRINK
layer case in the limit of strong damping (appendix B).
That this is true suggests that our present conclusions
about the vortex statistics over a slope are correct.
What then can we conclude about the role of the
vortices? Over a flat bottom, the vortices are the only
dynamical element and determine the flow in both layers. With a slope, energy is transferred to the deep mode.
The latter always dominates the lower layer, so long as
the slope is strong enough to disperse deep vortex flow,
and also wins out at the surface at later times, advecting
and shearing the vortices. The vortices then are most
important in the upper layer at intermediate times. At
late times, their primary role perhaps is to induce anomalously strong velocities intermittently at the surface.
6. Discussion
The present system in some ways resembles an f
plane stacked on top of a b plane. As in f plane turbulence, vortices emerge in the upper layer and merge
at a comparable rate. As with b turbulence, the lower
layer experiences a spectral arrest, and most vortices are
dispersed by Rossby wave radiation.
Were the interface rigid, the analogy would be perfect,
but the present system differs because the two layers
interact. Energy is transferred from the surface to the
bottom, or alternately from the vortices to the deep
mode. This transfer, affected primarily through the radiation of the deep portion of the vortices, has a rate
that depends on the parameter, L 5 F 2 U1 /b 2 ; if L ,
1, the transfer is inhibited. However, energy is always
transferred eventually so that the deep mode dominates
in the end.
There are other differences. Unlike with barotropic b
turbulence, the energy level and hence the effective arrest scale change in the lower layer as it spins up. As
such, the traditional barotropic arrest scale (Rhines
1975; Vallis and Maltrud 1994) cannot be evaluated a
priori, and a modified one is required (section 4a).
Vortices over a slope differ from their barotropic
counterparts in three ways. Deep dispersal, which can
either destabilize or stabilize (depending on the severity
of slope) impacts the size of vortices. It also affects the
way they interact because compensated vortices have a
far smaller radius of influence than do barotropic vortices. Because the deep mode penetrates the upper layer,
it can shear out vortices, thereby affecting their size and
number.
Clearly L is the most important parameter in these
experiments. What are typical oceanic values of L? Rewriting L as
L5
F2 U1
f 2 U1 H2
fU1
5
5
b2
g9H2 f ]y H
g9]y H
(18)
and substituting typical values for f 5 1 3 1024 s21
and g9 5 2.5 3 1022 m s22 yields L ø 0.004U1 /] y H.
For 1 m s21 surface velocities and a 1% slope, L 5 0.4;
if the velocity scale is more like 10 cm s21 , a 0.1% slope
1321
yields the same value of L. This is a fairly modest slope,
suggesting that the strong slope scenario may be fairly
common. If so, a sloping bottom is a plausible explanation for why oceanic vortices tend to be intensified
at the surface (or middepth).
If deep dispersal is common, vortex velocities will
typically span an area smaller than the deformation radius. McWilliams (1985) suggests that submesoscale,
coherent vortices are common in the ocean. The sloping
bottom may help explain their size.
How do the present results relate to previous studies
of turbulence over bumpy or irregular terrain (e.g.,
Rhines 1977; Treguier and Hua 1988; Vallis and Maltrud
1994)? With bumps, there is a spectrum of topographic
slopes, but topographic waves nevertheless. So the ratio
of interfacial stretching and topographic wave frequencies is still relevant, although one would likely consider
an rms wave frequency. Vortices still would be favored
at the surface, and their deep flow probably dispersed
over the bumps.
With bumps one often obtains a large-scale flow
whose streamfunction is anticorrelated with the topography (e.g., Bretherton and Haidvogel 1976; Herring
1977; Holloway 1978). This so-called ‘‘Neptune effect’’
(Holloway 1992) is not possible in a doubly periodic
domain with a linear slope because a geostrophic mean
flow cannot be supported. But given the flow is along
isobaths and suffers no competition from topographic
waves, it is in principle like the k 5 0 modes in the
present context.
A critical difference between the Treguier and Hua
(1988) and Vallis and Maltrud (1994) studies and our
own is that their systems were forced. With forcing, U1
(and L) can change, which obviously will impact the
evolution. One expects that L in a statistically stationary
state will depend on the forcing and damping. One might
speculate that L would adjust until the rate of energy
transfer was sufficient to close the energy budget; this
could easily be checked with forced experiments. Surface vortices might still be found, provided they are not
disrupted by the forcing (e.g., Herring and McWilliams
1985).
The planetary b effect likewise could alter the evolution; to what extent will likely depend on the bottom
slope. With a weak slope (e.g., b 2 comparable to b),
there will be a fast, pseudobarotropic wave (Rhines
1970), which could feasibly disperse surface vortices.
If the slope is strong, planetary b adds a surface-trapped
Rossby wave that also could disperse vortices; however,
the latter has very slow phase speeds and probably
would only affect weak vortices. Here too further experiments are warranted.
Lastly, the oceanic kinetic energy field is strongly
inhomogeneous, unlike in these experiments. Inhomogeneity itself can adversely affect the inverse cascade
(Rhines 1977), but could alter the present picture as
well by permitting the radiation of waves away from
the turbulent ‘‘site.’’ As such, the surface vortices might
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VOLUME 30
FIG. A1. The dependence of the deep spinup on L, when the latter is less than unity. At upper left is
KE 2 (t) for various L. In the other panels, we plot the nondimensionalized initial growth rate, ] t KE 2 (t 5 0),
vs slope, mean surface velocity, and the ratio of layer depths, respectively. Linear and inverse linear relations
are indicated by the lines.
be left behind, without suffering the same degree of
shearing as described here. Moreover, inhomogeneity
could substantially alter the character of the deep mode,
favoring long waves rather than jets. Such may be the
case near the Gulf Stream, where ring–slope interactions
(e.g., Louis and Smith 1982) and meanders (e.g., Pickart
1995) routinely radiate topographic waves to the far
field.
7. Conclusions
Topography plays an important role in turbulent flows
such as these. Even if there is no flow initially at depth,
the topography can alter the flow character so long as
the surface currents are unsteady. We have identified an
easily evaluated parameter, L, which can be used to
assess the extent of this topographic influence; the hope
is that it will be of use in interpreting more complex
stratified flows over topography.
Acknowledgments. This work was part of the thesis
of one of the authors (JHL) while in the MIT/WHOI
Joint Program in Physical Oceanography. We are grateful to Joe Pedlosky for comments, and to Glenn Flierl
for advice and the use of his numerical code. The numerical filter was implemented by Audrey Rogerson.
JHL is also grateful to committee members Dave Chapman and Steve Lentz for their many suggestions. The
manuscript was substantially revised after comments
from anonymous reviewers. Funding for this work was
provided by Grants N00014-92-J-1643 and N00014-92J-1528 from the U.S. Office of Naval Research (coastal
dynamics code).
APPENDIX A
Suppression of Deep Spinup by Bottom Slope
A number of low-resolution experiments (128 2
modes) were run to quantify the suppression of the deep
spinup by the bottom slope when L , 1. Several examples are shown in the upper-left panel of Fig. A1.
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LACASCE AND BRINK
We observe that when L is decreased, KE 2 increases
more slowly.
The dependence of the rate of spinup on L was quantified by comparing ] t KE 2 (t 5 0) from various runs, via
a least squares fit. A suitable rescaling of the initial
energy and the time step were required to permit the
comparison. The initial energy spectrum may be written
as
E(k, l, t 5 0) 5 E 0 (dk 2 |ĉ1 | 2 1 F2 |ĉ1 | 2 )
5 E 0d (k 2 1 F1 )|ĉ1 | 2 ,
(A1)
where d [ H1 /H 2 and E 0 } U 21, which implies that the
energy varies with d and with the square of U1 . The
dependence on d is slightly more complicated because
F1 is also a function of d, but with this initial spectrum
(k 0 . ÏF1), this is probably unimportant. Were the
initial potential energy larger, a d/(1 1 d ) scaling might
be more appropriate.
The timescale was nondimensionalized by (k 0 U1 )21 ,
where k 0 is the initial peak wavenumber. Combining
this with the energy scaling yields a nondimensional
scaling for the derivative of the deep kinetic energy of
dk 30 U 31.
Increasing the bottom slope alone (upper-right panel)
causes a decrease in ] t KE 2 (t 5 0), and moreover ] t KE 2 (t
5 0) } b21
2 , approximately. Similarly, ] t KE 2 (t 5 0) is
roughly linearly proportional to both the layer thickness
ratio (and thus to F 2 5 H1 F1 /H 2 ) and to U1 . The deviation from linearity in the former is due to the dependence of the initial PE on the ratio of layer depths,
as noted above. There are also slight departures at smaller b 2 and larger U1 in the region where L → 1. But on
the whole, these results suggest a linear dependence of
] t KE 2 (t 5 0) on L.
APPENDIX B
Vortex Statistics with a Bottom Ekman Layer
Similar vortex statistics are found when the bottom
slope is replaced with a bottom Ekman layer with various drag coefficients. The difference is that the Ekman
layer causes dissipation at depth rather than dispersion,
but both act asymmetrically.
We conducted a number of lower resolution (128 2
modes) runs to obtain solutions to Eqs. (1) and (2) where
the latter was altered thus:
]
q 1 J(c2 , q2 ) 5 2r¹ 2c2 ,
]t 2
(B1)
and various values of r were used. Identical initial conditions were used, that is, a surface-trapped initial flow
with random phases and a peak wavenumber larger than
the deformation wavenumber.
As with a slope, vortices emerge only in the upper
layer, and their mean area depends nonmonotonically
on r (Fig. B1). Specifically, increasing r from 0 to ;1
FIG. B1. The mean area of vortices as a function of time, for various values of the bottom drag coefficient.
yields progressively smaller vortices, which moreover
shrink in time, while larger vortices are found for larger
values of r. When r 5 100, the result resembles that
found in 1.5 layers (Fig. 12), that is, the vortices are
large and increasing such that N } t.37 .
As with a slope, the nonmonotonic variation is due
to the dual role played by the Ekman layer: for dissipation of the deep vortex flow and for the stabilization
of surface flows. Bottom damping favors compensated
vortices, so mergers in the presence of an Ekman layer
produce a prolonged vertical transfer of energy and vortex break up. However, severe bottom damping quenches the growth of barotropic disturbances, and so stabilizes large surface vortices.
The difference with the slope runs is that here there
is no secondary flow component that can adversely affect the vortices. So we obtain convergence to the 1.5layer results in the limit of strong damping, because
vortices are not being sheared out.
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