As stressed by Pedlosky pedlo96, the single gyre (as opposed
to the double gyre) Munk circulation faces the unique challenge that, in
terms of vorticity budget, all the wind input has to be fluxed out of the
domain by means of the viscous flux in order to yield a steady or
statistical mean solution. At equilibrium, the vorticity budget becomes:

To be precise, we refer to the single subtropical (anti-cyclonic) gyre problem in the northern hemisphere for the single gyre problem. In that particular case, the wind input to the vorticity budget, the first term in (5.1), is negative. Although the single gyre problem is extreme in that the vorticity input is one-signed, most people consider the double gyre problem (i.e., when the forcing integrates to zero) to be a special case. There is typically a net vorticity input of one sign or another into the ocean, and therefore, in the generic case, the system needs to dissipate some vorticity. The single gyre problem is certainly extreme, but it is argued after that some of its characteristics make this problem even more interesting and challenging. Moreover, for the single gyre forcing, there is a strong correspondence between the difficulty of balancing the vorticity budget and the strength of the overall circulation, since the second term in (5.1) links the magnitude of the eddy-viscosity, , to the importance of the normal derivative of the vorticity. This derivative is related to the strength of the circulation. As is reduced, the integral of the derivative must be augmented in proportion to yield an equivalent balance. The difficulty of balancing the vorticity budget is also dependent on the dynamical boundary condition. The vorticity balance is more difficult to achieve when free-slip boundary conditions are employed, as opposed to no-slip. Using free-slip conditions (see Chapter 1), the vorticity at the boundary is zero along straight walls and, under no-slip, it can reach large positive values, whereas the vorticity is mainly negative in the interior due to the negative wind input. Therefore the normal derivative of the vorticity, , is much higher in the no-slip case than in the free-slip case, fluxing more easily the vorticity through the boundaries. This enables gyres under no-slip to achieve weaker circulations compared to gyres under free-slip.

The importance of the nonlinear terms with respect to the viscous forces
is commonly scaled by the Reynolds number, ** R_{e}**. For Munk circulations,
it is more convenient to relate the Reynolds number to the dynamics of the
boundary layer (see Pedlosky).

where and are respectively the Munk and inertial numbers. They are defined as

where

For the wind forcing under consideration, m

comes from the evaluation of the balance between the advection and the viscous terms. These nonlinearities, and the presence of the inertial layer, introduce more difficulties in achieving a vorticity balance. For instance, in the absence of eddies, the inertial layer inhibits the transport of vorticity from the interior to the walls because, there, the streamlines and absolute vorticity contours are nearly parallel. Therefore, the negative input of vorticity in the interior of the ocean cannot be easily fluxed out. This favors an even more inertial and energetic interior flow and, when the Reynolds number is beyond a critical value, a Fofonoff-type gyre develops (as opposed to a Sverdrup interior) with unrealisticly large speeds of the order of 50 m/s. This is the so-called inertial runaway problem. According to Pedlosky, this scenario also occurs in the presence of no-slip boundary conditions, the no-slip only retarding the occurrence of the jump to the highly energetic branch (where the Fofonoff-type gyre lies). Moreover, he states that the inertial runaway is not just a feature of steady solutions but is prone to appear in unsteady solutions, as well.

Indeed, Ierley and Sheremet ierley95 observe this runaway
scenario for the free-slip condition in steady and unsteady circulations
in rectangular domains for single gyre forcing. Under free-slip, there is
no difference between unsteady and steady solutions because the eddy
activity is very weak in unsteady solutions. However, no-slip steady and
unsteady solutions are usually different. Nonetheless, Sheremet *et
al.* sheremet97 demonstrate that the same runaway problem
occurs in rectangular domains when the no-slip condition is applied to the
western and eastern walls (repeating the experimental setup of Bryan,
1963). They note that, after the unsteady and steady solutions
first depart, the strength of the circulation does not increase with
increasing Reynolds number because the eddies efficiently remove the
excess of vorticity produced in the boundary layer. However, past a
critical Reynolds number, they note that the mean circulation strengthens
again, the eddies being no longer efficient in removing the excess of
vorticity. Veronis veronis66 for the single gyre and Primeau
primeau98 for the double gyre demonstrate that the runaway
scenario is also observed for bottom friction only models.
Pedlosky (1996, p87) and Ierley and Sheremet are convinced that their
runaway scenario is universal, based on their experience with stratified
quasi-geostrophic (QG) unsteady simulations in idealized geometries.
According to them, no convergence of the statistical steady state can be
achieved with increasing Reynolds number, whatever the type of boundary
conditions. Of course, the latter argument conflicts with our day-to-day
experience. As far as we know, the Gulf Stream circulation has not blown
up! Nonetheless, these authors bring strong numerical evidences in favor
of their arguments. Therefore, where is the flaw ?

From the perspective of time-dependent simulations, one aspect of the
results of Sheremet *et al.* sheremet97
remains questionable. This is related to
the use of no-slip boundary conditions in unsteady solutions. The fact
that no-slip circulations are prone to barotropic instabilities cannot be
underestimated from the point of view of the inertial runaway. These
instabilities may be sufficient to produce eddies which would transport
the vorticity through the inertial layer to the viscous
sub-layer, where it can be fluxed across the wall. However, no-slip
circulations are very demanding in terms of
computer resources and, therefore, the issue is still unresolved.
One possibility is that we still need more resolution (to
achieve larger ** R_{e}**) in unsteady no-slip circulations. A second
possibility is related to the use of overly idealized geometries in the
aforementioned results. Finally, a third possibility is that the models
used in those results are too simple. From this last point of view, we
may lack certain physical processes which are important for the downward
cascade of energy. In favor of this argument, Scott and Straub
scott98 noted that, under no-slip, the Rossby number (which
scales the nonlinear terms to the Coriolis forces) increases quickly with
increasing Reynolds number. Since the QG approximation applies only for small
Rossby number,

In favor of these three arguments, recent high resolution (1/4 to 1/64
degree) simulations of the Atlantic were conducted using the MICOM model
(i.e., an isopycnal primitive equation model) and showed that the mean
circulation converges to a more and more realistic state with increasing
Reynolds number [Hulburt and Hogan, 2000]. The eddy-viscosity was lowered from 100
to 3 m** ^{2}**/s. The problem with this kind of experiment is that it is
difficult to distinguish which physical processes or technical details are
necessary to obtain the convergence with increasing Reynolds number. We
believe that one important distinction comes from the geometry.
Theoreticians typically focus on rectangular domains whereas primitive
equations models are generally run in more realistic geometries.
Irregular geometries may be sufficient by themselves to provide the
necessary source of eddies in order to get weaker and more realistic
circulations at high Reynolds number. An irregular geometry--especially
irregular along the western coastline where the currents are the
strongest--may also provide stronger interactions between geostrophic and
ageostrophic modes, and hence may facilitate a forward energy cascade.
The latter process is absent from idealized early experiments which are
based on the QG equation. Thus, the shallow water equations are a good
starting point for our investigation. Furthermore, we believe that having
irregular boundaries is more important than the choice on the type of
dynamical boundary conditions. In the context of the double gyre forcing
of the Munk problem, Scott and Straub scott98 show that the
increase in kinetic energy of non-symmetrical steady solutions and
time-dependent mean
solutions tends to level off as the Reynolds number increases for the two
boundary conditions. Therefore, the idealized double gyre experiment where
the wind input to the vorticity budget cancels may exhibit less
severe inertial
runaway. Contradictory evidence against inertial runaway has yet to be
found in single gyre circulations where the wind input of vorticity is
single signed. Therefore, we will conduct experiments using the free-slip
boundary condition since many evidences exist for a robust inertial
runaway under free-slip in rectangular basins. In fact, under the
free-slip boundary condition, irregular boundaries are the only way to
produce positive vorticity which is essential to the production of eddies.
The vorticity can be expressed using curvilinear coordinates following the
wall as

where

If the velocity at the wall is close to 1 m/s and the radius of curvature along the wall is of the order of the 10 km, is of the order of

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