FEBRUARY 2008
LACASCE ET AL.
517
Asymmetry of Free Circulations in Closed Ocean Gyres
J. H. LACASCE
Institute of Geophysics, University of Oslo, Oslo, Norway
O. A. NØST
Norwegian Polar Institute, Tromsø, Norway
P. E. ISACHSEN
Norwegian Institute for Water Research, Oslo, Norway
(Manuscript received 1 March 2007, in final form 24 April 2007)
ABSTRACT
Steady flows in regions of closed geostrophic contours, such as in ocean basins or over seamounts, are
examined by calculating solutions to the nonlinear quasigeostrophic and shallow-water equations with
topography. For oceanically realistic choices of parameters, the solutions are asymmetric in that those with
cyclonic circulation resemble the topography while those with anticyclonic circulation exhibit small-scale
structure and cross-isobath flow. These small-scale structures reflect topographic wave modes, which are
stationary with the anticyclonic circulation. The implication of the asymmetry is that random wind forcing
is much more likely to excite persistent cyclonic than anticyclonic flow. This may explain why the circulation
in the Norwegian and Greenland Gyres is most often cyclonic.
1. Introduction
A distinguishing feature of the Nordic seas and Arctic Ocean is that the geostrophic or “f /H ” contours are
primarily closed due to the deep basins there (e.g., the
Norwegian and Greenland Basins). Closed geostrophic
contours permit inviscid, steady flows without forcing
(Welander 1968), and this has a major effect on the
forced response. With blocked contours, a divergence
in the surface Ekman transport can be balanced by
cross-contour flow, the Sverdrup transport. With closed
contours, the surface divergence can be balanced instead by a convergence in the bottom Ekman layer, and
this requires along-isobath flow.
A second feature of the Nordic seas and Arctic is that
the stratification is comparatively weak and the variability predominantly barotropic. Exploiting these features, Isachsen et al. (2003) derived a barotropic model
with closed f /H contours to predict the wind-driven cur-
Corresponding author address: J. H. LaCasce, Institute of Geophysics, University of Oslo, P.O. Box 1022, Blindern, 0315 Oslo,
Norway.
E-mail: j.h.lacasce@geo.uio.no
DOI: 10.1175/2007JPO3789.1
© 2008 American Meteorological Society
JPO3183
rents here.1 They found that the predicted currents
compared remarkably well with observations from altimetry and current meters. For example, correlations
between predicted transports and those derived from
the principal EOF of sea surface height yielded coefficients of r ⫽ 0.76 in the Norwegian Gyre and r ⫽ 0.88
in the Greenland Gyre. Likewise, the correlation between the Norwegian gyre transport from the analytical
model and that from a full primitive equation model
was r ⫽ 0.75. So the analytical model appears to capture
the dominant dynamics in these gyres.
However, there was one curious feature: the gyre
flows were more often cyclonic than anticyclonic (see,
e.g., Fig. 6 of Isachsen et al. 2003). As no such asymmetry exists in the linear solutions, the authors concluded it stemmed from an asymmetry in the wind curl.
The curl over the Nordic seas is mostly positive in the
winter, which would favor cyclonic flow.
However, a similar asymmetry was subsequently
1
Nøst and Isachsen (2003) developed a similar model to explain
the steady, baroclinic flow in the region, using hydrographic observations and winds.
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JOURNAL OF PHYSICAL OCEANOGRAPHY
found in numerical experiments of flow over seamounts
by Nycander and LaCasce (2004). In their spindown
experiments, a wide range of initial states produced
topography-following anticyclonic flows while none
yielded cyclonic circulations. The only exception was in
the special case of a perfectly circular seamount; then
either sign circulation could result. The authors rationalized the asymmetry using nonlinear stability arguments (in the sense of Arnol’d 1966). In particular, only
an anticyclonic flow over a seamount is demonstrably
stable.
Recently, Nøst et al. (2008) examined idealized basin
flows in the laboratory and numerically. Unlike the
spindown experiments of Nycander and LaCasce, these
employed both surface forcing and bottom friction. The
authors found that either anticyclonic or cyclonic flow
could be obtained. Nevertheless, there were differences. While the cyclonic flows were strictly isobathfollowing, the anticyclonic flows exhibited regions of
strong across-slope flow. Nøst et al. suggested the departures were a consequence of nonlinearity, as they
appeared only when the bottom friction was weak
(when the Rossby number was larger than the Ekman
number). We see that Nøst et al.’s results are in line
with the Isachsen et al. (2003) results, where there is no
asymmetry, when the damping is strong and consistent
with the Nycander and LaCasce (2004) experiments
when the damping is weak.
Nøst et al. (2008) also rationalized the asymmetry at
larger Rossby numbers using nonlinear stability arguments. In this case, only a cyclonic basin flow is demonstrably stable. But, they also suggested another possible
reason: that there is actually an asymmetry in the
steady-state solutions. They argued in particular that
the different sign flows would behave differently near
topographic irregularities due to differences in the advection of relative vorticity.
We demonstrate that there is, indeed, a difference in
the steady solutions. While cyclonic basin solutions
generally follow the topography, anticyclonic solutions
have significant small-scale structure and diverge from
the isobaths. So nonlinear stability need not come into
play at all because large-scale forcing is much more
likely to excite cyclonic steady flows than anticyclonic
ones.
2. Theory
Much of the background for this problem is laid out
in the seminal work of Carnevale and Frederiksen
(1987, hereafter CF). Flows with a small Rossby number and weak topographic variations conserve their
quasigeostrophic (QG) potential vorticity in the absence of forcing or dissipation (Pedlosky 1987):
冉
VOLUME 38
冊
⭸
⭸
⫹ u · ⵱ 共ⵜ2␺ ⫹ h兲 ⫽ q ⫹ J共␺, q兲 ⫽ 0,
⭸t
⭸t
共1兲
where ␺ is the velocity streamfunction, q is the QG
potential vorticity (PV), and J(·, ·) is the Jacobian function. Here the depth is written
H共x, y兲 ⫽ H0 ⫺ ␩共x, y兲,
under the rigid-lid assumption. QG requires that the
nondimensional topographic height in (1), h ⫽ ␩/H0 , is
on the order of the Rossby number. Hereafter we consider vorticity variations due solely to the topography;
that is, we neglect the variation of the Coriolis parameter.2
Equation (1) guarantees that all integrals of the form
冕冕
F 共q兲 dx dy
共2兲
are conserved, where F(q) is any function of the PV.
The most familiar such function is the square of the PV,
the potential enstrophy:
1
2
冕冕
q2 dx dy.
共3兲
From (1) it also follows that the energy is conserved:
1
2
冕冕
共⵱␺兲2 dx dy.
共4兲
Based on the existence of these conserved quantities,
the theory of Arnol’d (1966) can be used to prove the
nonlinear stability of the stationary state given by
q ⫽ ⵜ2␺ ⫹ h ⫽ ␮␺,
共5兲
where ␮ is a constant. This solution represents one family of all the possible solutions to the steady version of
(1):
J共␺, q兲 ⫽ 0,
共6兲
which is satisfied by any flow for which
q ⫽ F 共␺兲,
共7兲
where F can be any function, possibly a nonlinear one
(Fofonoff 1954).
Thus, relation (5) corresponds to having a linear
F (␺). As shown by Fofonoff (1954), one can thus find
solutions to the nonlinear problem (6) by solving a linear equation. The character of the solutions to (5) depends on the sign of ␮. If ␮ is positive, (5) is like a
diffusion equation, implying that the streamfunction
2
The present results pertaining to cyclonic/anticyclonic flows in
a basin apply equally to anticyclonic/cyclonic flows over a seamount. We refer only to the basin case, for brevity.
FEBRUARY 2008
LACASCE ET AL.
will resemble the forcing function (the topography). If
instead ␮ is negative, (6) is a Helmholtz equation and
possesses normal mode solutions (like a drum). These
modes can be excited “resonantly” by the topography
for specific values of ␮. Near such resonances, one expects very energetic flows.
Solutions to (5) can be obtained easily following a
Fourier transform. Consider a rectangular domain with
sides Lx and Ly, where the streamfunction vanishes at
the walls. Then we can write
n⫽N m⫽M
␺共x, y兲 ⫽
兺 兺
␺ˆ 共n, m兲 sin
n⫽1 m⫽1
冉 冊 冉 冊
m␲y
n␲x
sin
,
Lx
Ly
ĥ共n, m兲
共n␲ⲐLx兲 ⫹ 共m␲ⲐLy兲2 ⫹ ␮
2
.
In the limit of infinite resolution, the only stable solutions are those with ␮ as in (10). These have cyclonic
circulation in a basin because the denominator of (9) is
positive definite so that ␺ has the same sign as h.5 As
argued above, these solutions should resemble the topography.
Nonlinearity stability says nothing about the range
for which
⫺共N␲ⲐLx兲2 ⫺ 共M␲ⲐLy兲2 ⬍ ␮ ⬍ ⫺共␲ⲐLx兲2 ⫺ 共␲ⲐLy兲2.
共12兲
共8兲
where N and M are the maximum wavenumbers resolved by the model. Transforming (5), we obtain
␺ˆ 共n, m兲 ⫽
519
共9兲
CF used Arnol’d’s (1966) method to show that the
solutions to (9) with
␮ ⬎ ⫺共␲ⲐLx兲2 ⫺ 共␲ⲐLy兲2
共10兲
␮ ⬍ ⫺共N␲ⲐLx兲2 ⫺ 共M␲ⲐLy兲2
共11兲
or
are nonlinearly stable. The second condition (11) applies effectively to wavenumbers unresolved by the
Fourier decomposition and thus becomes irrelevant as
the grid size decreases to zero (i.e., as N, M → ⬁). Note
also that as the scale of the domain is increased (Lx,
Ly → ⬁), the first condition (10) implies simply that ␮
is positive.
Steady solutions that satisfy (6) are extrema in that
they correspond to the vanishing of the first variation of
the energy (CF). Their stability is evaluated by examining the second variation. Solutions with ␮ in the two
ranges shown above correspond to energy minima.3
Perturbing the state produces a positive definite change
in the energy, implying that an arbitrary disturbance
cannot grow in time without violating energy conservation.4
These solutions may be stable or not, depending on the
character of the disturbances. The anticyclonic basin
solutions occur in this range, as shown hereafter.
Significantly, this range exhibits singularities because
expression (9) is infinite when ␮ matches one of the
squared wavenumbers. This was noted by CF and is
expected for a Helmholtz-type equation, as noted
above. Physically, the singularities represent topographic wave modes that are stationary for the given
mean flow. Without a mean flow, there are an infinite
set of topographic modes (e.g., Bokhove and Johnson
1999) and these exhibit cyclonic propagation, with shallow water to their right. As such, they can be stationary
only in an anticyclonic mean flow. The gravest topographic mode is the fastest, and the higher modes have
phase speeds that decrease monotonically with wavenumber. Thus the weaker the mean flow, the smaller
the scale of the stationary mode. With infinite resolution there will always be a mode whose phase speed
approximately matches the mean flow speed. But with
finite resolution, one may obtain mean flows that are
slower than the smallest resolved topographic mode
[these correspond to the range of ␮ in (11)]. In the
absence of damping, the flow will be very energetic in
the vicinity of the topographic singularities.
Thus we expect that cyclonic basin solutions should
resemble the topography and are nonlinearly stable.
The anticyclonic solutions, on the other hand, may or
may not be stable and may not resemble the topography at all.
3. QG solutions
3
CF suggested incorrectly that the range in (11) corresponds to
an energy maximum. However, this does not affect their conclusion with regard to the stability.
4
An alternate variational approach, used by Bretherton and
Haidvogel (1976), is to minimize the enstrophy while keeping the
energy constant. Their idea was to mimic the effect of twodimensional turbulence, where enstrophy is dissipated at small
scales while energy is preserved at large scales. CF showed that
solutions with (10) have minimum enstrophy while those with (11)
have maximum enstrophy.
We now examine the solutions to (5). As stated,
these represent only a subset of the possibly infinite
number of solutions to (6). We assume that the linear
solutions, which are relatively easily obtained, are to
some degree representative of the fuller set. The linear
5
Note that h is negative for a basin, implying a negative streamfunction and hence a positive vorticity, using our sign conventions.
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FIG. 1. The total kinetic energy for an elliptical basin, plotted as a function of the parameter
␮ (solid line). The dashed line indicates the two-dimensional correlation between the streamfunction and the topography. The correlation is 1 for positive ␮, indicating a topographyfollowing cyclonic flow. It approaches ⫺1 in the negative ␮ range between the singularities.
solutions are, in addition, the relevant ones in terms of
minimum enstrophy and maximum enstrophy considerations (CF).
We obtained the solutions with various bottom topographies using the MATLAB and COMSOL Multiphysics software packages. We first consider an elliptical basin:
关⫺共xⲐDx兲2⫺共yⲐDy兲2兴
h共x, y兲 ⫽ ⫺Ae
,
共13兲
with A ⫽ 0.3, Dx ⫽ 0.5, and Dy ⫽ 0.25. We use an
elliptical basin because this is asymmetric in the sense
discussed by Nycander and LaCasce (2004) and Nøst et
al. (2008) (a circular basin is a special case because any
azimuthally symmetric streamfunction is a solution to
6). In the MATLAB solutions, the basin was contained
in a larger, rectangular basin with no normal flow
through the walls. The COMSOL software uses a variable mesh and was configured to encompass only the
elliptical basin. The results using the two approaches
were nevertheless consistent.
Solutions were obtained by varying ␮ over a range of
values. We calculated the total kinetic energy of the
solutions and plot the results in Fig. 1. The kinetic energy is greatest near the value of ␮ corresponding to the
gravest mode in the system, as discussed below. It decreases monotonically for larger values of ␮, while for
negative values of ␮ there are discrete peaks.
The latter are the topographic singularities discussed
in section (2) and by CF. The peaks occur at wavenumbers n, m such that
␮⫽⫺
冉 冊 冉 冊
n␲
Lx
2
⫺
m␲
Ly
2
.
共14兲
The energy also decreases as ␮ becomes more negative,
reflecting a weaker topographic projection, ĥ(n, m), at
larger wavenumbers.
Also plotted in Fig. 1 is the two-dimensional correlation between the topography and the streamfunction:
冕冕
冕 冕公
␺h dx dy
c⫽
.
共15兲
␺ h dx dy
2 2
For positive ␮, the correlation approaches 1, indicating
a cyclonic, isobath-following solution. For negative ␮,
the correlation varies substantially, approaching zero
near the singularities and falling toward ⫺1 between
the singularities. The latter is indicative of anticyclonic
flow.
Another way to determine the sign of the flow is
simply to average the streamfunction. This is shown, as
a function of ␮, in Fig. 2. The resulting curve exhibits a
single discontinuity. It is negative (cyclonic flow) for
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LACASCE ET AL.
tion is very like that in the upper-left panel, albeit
weaker.
The solution with ␮ ⫽ ⫺103 (lower-right panel) is
quite different. There is a weak along-isobath background flow, which is anticyclonic, but superimposed
on this are small-scale structures.6 These structures reflect the Fourier mode that is nearly singular. In this
case, we could write
␮ ⫽ 共n␲ⲐLx兲2 ⫹ 共m␲ⲐLy兲2 ⫾ ␦,
where n and m are much larger than one. Assuming ␦ is
small, the streamfunction is dominated by that mode:
␺ˆ 共n, m兲 ⫽ ⫾
FIG. 2. The mean streamfunction in an elliptical basin as a
function of the parameter ␮.
larger values of ␮ and positive (anticyclonic flow) for
lesser values. The singularity occurs at the wavenumber
corresponding to the gravest mode in the system, which
resembles the basin itself.
We contour solutions with various values of ␮ in Fig.
3. The upper panels show two solutions with values of
␮ slightly greater and less than the transition value. In
the cyclonic solution (upper-left panel of Fig. 3), the
flow is predominantly along-isobath and the streamfunction closely resembles the basin. The same is true
with the anticyclonic solution (upper-right panel). The
solutions with ␮ near the critical value are therefore
symmetric, with both anticyclonic and cyclonic solutions having predominantly along-isobath flow.
These solutions can be understood as follows. If ␮ is
on either side of the gravest mode, then we have
␮ ⫽ 共␲ⲐLx兲2 ⫹ 共␲ⲐLy兲2 ⫾ ␦.
This implies, from (9), that
␺ˆ 共1, 1兲 ⫽ ⫾
ĥ共1, 1兲
.
␦
共16兲
Assuming ␦ K 1, the streamfunction will be large and
will mirror the gravest mode. As the gravest mode resembles the basin itself, we obtain approximately isobath-following flow.
The lower panels show two solutions with ␮ far from
the streamfunction transition value. The solution with
␮ ⫽ 103 (lower-left panel) is cyclonic and the flow is
again predominantly along-isobath. In this case, the
gravest mode, ĥ(1, 1), dominates the transform, but the
large value of ␮ weakens the amplitude. So, the solu-
ĥ共n, m兲
.
␦
共17兲
As before, the sign of ␺ˆ (n, m) is the same as that of ␦.
The difference is that this is a much higher mode, so
changing the sign simply turns the small-scale cyclones
to anticyclones, and vice versa.
Unlike with the corresponding cyclonic solution, the
basin-average kinetic energy for the negative ␮ solution
is usually large, due to the singularities. However, because the singularities have a sinusoidal structure, with
both positive and negative signs, their basin average is
approximately zero. So the average streamfunction reflects instead the anticyclonic background. This explains the discrepancy between the kinetic energy diagram in Fig. 1 and the mean streamfunction curve in
Fig. 2.
There are regions of the kinetic energy curve between the singularities where the energy falls to lower
values. As can be deduced from the correlation curve in
Fig. 1, these flows are more dominated by the alongisobath component. However, the small-scale structures are generally visible and prevent the correlation
from relaxing completely to ⫺1. So the cross-isobath
flow is essentially always present in the anticyclonic
range. In addition, the spacing between the singularities
decreases to zero in the limit of a large domain (Lx,
Ly → ⬁), so the relevance of these gap regions in more
realistic settings is likely to be minor.
Thus for most values of ␮ less than the streamfunction transition value, the anticyclonic mean states are
contaminated by small-scale structures. The important
question then is what range of ␮ is realistic for the
ocean? If ␮ is typically small, the solutions will lie near
the gravest topographic mode and the solutions would
be approximately symmetric. To see, we turn to the
6
Similar structures were noted in the Fofonoff-like solutions in
a flat-bottom, ␤-plane basin by Marshall and Marshall (1992).
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FIG. 3. The solutions to (5) with various values of ␮: (top) solutions near the critical value of ␮, where
there is a transition from cyclonic to anticyclonic flow in Fig. 2, and (bottom) ␮ far from the critical value.
(left) Cyclonic and (upper right) anticyclonic flow. (lower right) Superposition of a weak anticyclonic
background flow with small-scale features.
shallow-water formulation of the problem, which permits realistic topographic variations.
Q⫽
Q⫽
The conservation of potential vorticity in the shallow-water (SW) system with a rigid lid can be written
冉
冊冉 冊 冉
冊
␨⫹f
⭸
⬅
⫹ u · ⵱ Q ⫽ 0,
H
⭸t
or
共18兲
where
␨⫽
⭸
⭸
␷⫺ u
⭸x
⭸y
is the relative vorticity, H ⫽ H(x, y) is the water depth,
and Q is the SW potential vorticity. The continuity
equation is
⵱ · 共Hu兲 ⫽ 0.
共19兲
So the steady version of (18) can be written
J共⌿, Q兲 ⫽ 0,
共21兲
Following section (3), we will focus on the linear version of (21):
4. Shallow-water solutions
⭸
⫹u·⵱
⭸t
␨⫹f
⫽ F 共⌿兲.
H
共20兲
where ⌿ is the transport streamfunction. Solutions to
this have
⵱·
⵱ · 共⵱⌿ⲐH兲 ⫹ f
⫽ ␭⌿
H
冉 冊
⵱⌿
⫺ ␭H⌿ ⫽ ⫺f.
H
共22兲
共23兲
It is straightforward to show that we recover the QG version of the problem, (5), if the (scaled) vorticity and topographic variations are O(Ro) and if H 2f (⫺1)UL␭ → ␮.
We obtained solutions to (23), again using the
COMSOL software package. In keeping with the preceding section, we first examine an elliptical basin. The
basin was 1000 m deep with dimensions of 200 km by
100 km. The mean streamfunction (not shown) exhibited the same parametric dependence as the QG calculation, with a single discontinuity separating cyclonic
solutions for positive ␭ and anticyclonic ones for negative ␭.
Examples of the solutions are shown in Fig. 4. The
upper panels correspond to solutions near the streamfunction singularity (with ␭ ⫽ ⫾10⫺14). The cyclonic
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523
FIG. 4. The solutions to (23) with a 50-km-wide elliptical basin: (left) cyclonic and (right) anticyclonic
circulation. The solutions have ␭ as indicated.
solution is nearly isobath-following, while the anticyclonic solution already deviates significantly from the
isobaths. The tendency as one moves away from the
singularity is as in the QG solutions. With ␭ ⫽ 10⫺13,
the cyclonic solution follows the topography while the
anticyclonic one exhibits small-scale structures. With
␭ ⫽ 10⫺12 (lowest panels), the anticyclonic solution is
riddled with small structures.
The central point of interest though is the magnitude
of the velocities. The best way to gauge these is to use
the cyclonic solutions because the velocities decrease
monotonically with increasing ␭. With ␭ ⫽ 10⫺14 (the
upper panels), the maximum velocities are roughly 1.5
m s⫺1. With ␭ ⫽ 10⫺13, they are 55 cm s⫺1 and with ␭ ⫽
10⫺12 they are 20 cm s⫺1. So the solutions that are most
“realistic” for oceanic flows are those with |␭| ⬇ 10⫺12.
We calculated additional solutions, with more realistic topography. For this we used the Nordic seas topography, derived from 5-minute gridded elevations/
bathymetry for the world (ETOPO5) and modified to
lie in a square domain (we set all depths less than 750 m
equal to 750 m and enclosed the resulting basin with
square “walls”). In reality the contours corresponding
to depths less than 750 m extend into the Arctic and out
into the North Atlantic, but as we are interested in the
flow in the three enclosed gyres (the Norwegian, the
Greenland, and the Lofoten), this representation suffices.
The flows with various ␭ are shown in Figs. 5 and 6.
With ␭ ⫽ 10⫺17 (upper left panel of Fig. 5), the cyclonic
solution has intensified flow in the regions of the three
gyres. The anticyclonic solution (right panel), on the
other hand, deviates significantly from the topography.
With ␭ ⫽ ⫾10⫺16 (lower panels), the cyclonic solution
has three recognizable gyres, while the anticyclonic solution has significant small-scale structure. Decreasing
␭ further (Fig. 6) yields even smaller scales in the anticyclonic solutions.
Again, we can gauge the strength of the flows by
examining the maximum velocities in the cyclonic solutions. With ␭ ⫽ 10⫺17, the maximum velocities are
roughly 15 m s⫺1 (a respectable value even by atmo-
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FIG. 5. The solutions to (23) with a representation of the topography of the Nordic seas:
(left) cyclonic and (right) anticyclonic circulation for (top) ␭ ⫽ ⫾10⫺17 and (bottom) ␭ ⫽
⫾10⫺16. Depth contours less than 750-m depth have been set to 750 m.
spheric standards). For ␭ ⫽ 10⫺16 the maximum is
about 4.5 m s⫺1, while at ␭ ⫽ 10⫺15 it is 1.2 m s⫺1. The
maximum with ␭ ⫽ 10⫺14 is 30 cm s⫺1.7
As noted earlier, these velocities relate to the phase
speeds of the corresponding topographic modes. The
small-␭ solutions correspond to the gravest modes, with
the fastest phase speeds. The speeds are evidently on
7
A prominent feature of the cyclonic solutions is the very energetic flow around the island of Jan Mayen, near the center of the
domain. We excluded this when estimating the maximum velocities but the maximum here with ␭ ⫽ 10⫺14 is over 1 m s⫺1. In the
present context Jan Mayen is a seamount, with anticyclonic flow,
and the velocities are large because the topography is steep. Such
an intense flow is probably unrealistic and is perhaps limited by
bottom friction.
the order of meters per second for these basins. The
higher modes have smaller speeds and, of course, more
small-scale structure.
So wind forcing that would produce 30 cm s⫺1 velocities or less would excite solutions with ␭ ⬇ 10⫺14. Of
course the velocities in the anticyclonic solutions do not
decrease monotonically with decreasing ␭ due to the
energetic contribution from the small scales. Nevertheless, wind forcing that would produce a 30 cm s⫺1 alongisobath anticyclonic flow would necessarily be compared to the solution with ␭ ⫽ 10⫺14. This solution is far
from isobath-following.
5. Discussion
Steady solutions in closed basins that are of realistic
strength are asymmetric in that cyclonic solutions are
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525
FIG. 6. The solutions to (23) with the Nordic seas topography: (top) ␭ ⫽ ⫾10⫺15 and
(bottom) ␭ ⫽ ⫾10⫺14.
nearly isobath-following, while anticyclonic solutions
exhibit small-scale structure and cross-isobath flow.
The small-scale structure reflects stationary topographic wave modes, which due to their cyclonic propagation necessarily occur with anticyclonic mean flows.
Similar comments apply with seamount-trapped flows
with the signs of the circulation reversed.
This sign asymmetry is not present in the linear equations of motion where any topography-following flow is
steady. The linear form of Eq. (5), which neglects the
relative vorticity contribution, has solutions that are
just functions of the topography and can have either
sign. The nonlinear case opens the possibility of the
coexistence of a mean flow and topographic modes.
The linear solutions, in fact, correspond to the nonlinear solutions with large positive values of ␮, or large
negative values of ␮ between the singularities. As we
have seen, the latter are tightly clustered, and we rarely
find instances of anticyclonic flow without small-scale
structure. In a linear stability analysis of a basintrapped flow, one would likely start with topographyfollowing flow. But such a flow in the anticyclonic case
is difficult to find.
Nevertheless, forcing can produce anticyclonic basin
flows (Isachsen et al. 2003; Nøst et al. 2008). Nøst et al.
(2008) found that the streamlines in such cases diverged
from the isobaths in localized regions (see, e.g., their
Fig. 8) but otherwise followed the topography. This implies that the balance of forcing and dissipation, integrated over the closed contours, is selecting a certain
circulation. The present results however tell us how the
systems will behave once the forcing is switched off. If
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the flow is similar to a free solution, it ought to persist,
spinning down due to bottom friction. The anticyclonic
forced solutions, on the other hand, would necessarily
evolve in time, not being near a steady solution. We
emphasize that the breakup of the anticyclonic flow
occurs because the flow is not steady, not due to instability.
Of course, the wind could excite one of the free anticyclonic modes if it had precisely the same small-scale
structure as the mode (e.g., as in Fig. 6). This is a fairly
unlikely event. But if this were to occur, the flow would
likely be unstable in any case.
Interestingly, there is a possibility of obtaining weak,
topography-following, anticyclonic circulations in numerical models with finite resolution. This will occur
for flow in which the mean speed is slower than that of
the smallest resolved topographic mode (corresponding
to the range of ␮ shown in 11, as noted by CF). So
coarse-resolution ocean models will permit steady anticyclonic circulations that would otherwise not exist.
As these unresolved solutions are also Arnol’d stable
(CF), they could persist for long periods of time. It is
thus worth checking such model results with higherresolution experiments.
This study was motivated by considerations of the
Nordic seas, where the stratification is relatively weak,
but the asymmetry could be expected to carry over to
the stratified case as well. In this case, the topographic
waves would be bottom-intensified. But they would still
propagate cyclonically in a basin, and thus could be
stationary only in an anticyclonic flow.
VOLUME 38
Acknowledgments. We are grateful to George
Carnevale, Johan Nilsson, and two anonymous reviewers for their comments.
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