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:
The importance of the nonlinear terms with respect to the viscous forces
is commonly scaled by the Reynolds number, Re. For Munk circulations,
it is more convenient to relate the Reynolds number to the dynamics of the
boundary layer (see Pedlosky). Re is therefore defined as
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 Re) 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, Ro, large Ro means that the rather inexpensive QG models cannot be used for even such idealized experiments, but have to be replaced by, at a minimum, more costly shallow water models.
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 m2/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
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
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
Get the PS or PDF version here