Review of the Single Gyre Problem with Free-Slip Boundary Conditions

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:

$$\oint \frac{\mbox{\boldmath$\tau$ }}{h} \cdot \mbox{\bf dl}+ \nu \oint \frac{\partial \zeta}{\partial n} dl =0 \mbox{~.}$$ (5.1)

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, $\nu$, to the importance of the normal derivative of the vorticity. This derivative is related to the strength of the circulation. As $\nu$ 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, $\partial \zeta /\partial n$, 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). R_e is therefore defined as

$$R_e = \left( \frac{\delta_I}{\delta_M} \right)^3$$ (5.2)

where $\delta_M$ and $\delta_I$ are respectively the Munk and inertial numbers. They are defined as

$$\delta_M = \left( \frac{\nu}{\beta L^3} \right)^{1/3} \mbox... ...ta_I = \left( \frac{V_{Sv}}{\beta L^2} \right)^{1/2} \mbox{~,}$$ (5.3)

where V_Sv is the Sverdrup velocity and L is the width of the basin. In scaling arguments, V_Sv is usually taken as the maximum value observed in the interior away from the boundary layers. We prefer to use the mean value of the V_Sv which can be obtained by integrating the Sverdrup relation over the whole domain except for the boundary layers/footnoteThis choice is motivated by the quantitative estimations of coming Section 5.4 which are based on vorticity budgets arguments and are better approximated by using the mean rather than the maximal Sverdrup velocity. There is also a a posteriori and cosmetic argument related to the fact that the transition between the Sverdrup interior solution to a Fofonoff-type interior solution (explained at the end of this paragraph) occurs at $R_e \sim 1$ if a mean Sverdrup velocity is chosen but will occur at $Re \sim 4$ if the maximum Sverdrup velocity is chosen.:

$$V_{Sv} = \frac{1}{\beta L^2} \oint \frac{\mbox{\boldmath$\tau$ }}{h} \cdot \mbox{\bf dl}\mbox{~.}$$ (5.4)

For the wind forcing under consideration, $\beta=1.6 \times 10^{-11}$ m^-1s^-1and assuming that $h \sim H$ along the walls, the mean V_Sv is approximately $1.25\times 10^{-2}$ m/s. This choice will lead to smaller values of R_ewhen compared to other authors' results. Another important remark concerns the physical meaning of these two numbers. $\delta_M$ and $\delta_I$, multiplied by the width of the basin, L, yield respectively the thickness of the Munk and inertial layers. These are the lengths at which the vorticity varies in order to yield a balance between the viscous terms and the beta term, and a balance between the advection and the beta term, respectively. The Munk layer exists only for weak nonlinear terms. When these nonlinear terms are large enough ( $R_e \sim 1$ and beyond), the inertial boundary layer prevails along the western boundary. In such a case, the Munk layer is replaced by a viscous sublayer whose thickness is given by $L\delta_M'$ where

$$\delta_M'= \frac{\delta_I}{\sqrt{R_e}} \mbox{~.}$$ (5.5)

$\delta_M'$ 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, R_o, large R_o 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.

  

Figure 5.1: Notation corresponding to the curvilinear coordinates.

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 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

$$\zeta = \frac{\partial v_s}{\partial n} -\frac{\partial v_n}{\partial s} + \frac{v_s}{R_s} - \frac{v_n}{R_n} \mbox{~,}$$ (5.6)

where (s,n) are respectively the coordinate along and normal to the wall, v_s and v_n are the velocity components respectively along sand n and R_s and R_n are the respective radii of curvature of the axes along the wall and normal to the wall. (notation is shown in Figure 5.1). Right at the wall and under the free-slip boundary condition (as defined in Chapter 1), the equation reduces to

$$\zeta = \frac{v_s}{R_s} \mbox{~.}$$ (5.7)

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, $\zeta$ is of the order of 10^-4 s^-1, that is, of the order of f₀ ^5.1. One way to evaluate the Rossby number is to measure the ratio of $\zeta /f_0$. Therefore, if the radius of curvature is of the order of 10 km, we can obtain Rossby numbers of the order of unity; that is, well beyond the range for which the QG approximation applies. This stresses again the need to use the primitive equations. 10 km is also somewhat below the radius of deformation for the first baroclinic mode given the value of the parameters we use ( $L_{R} \sim 31$ km). This means for instance that Kelvin waves may encounter difficulties in going around such geometrical features.

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