We show applications of a discontinuous spectral element model to the problem of the inertial runaway under the free-slip condition in irregular geometries. We first show that more traditional numerical methods, such as the finite difference methods, fail to converge in irregular domains for the boundary condition under interest. Second, the main results of this application of a spectral element model show that, in the presence of irregular boundaries, the jump to the high energetic branch is considerably retarded, occurring at a higher boundary layer Reynolds number. The presence of smooth bumps along the coastline introduces a source of positive vorticity and thus a source of production of eddies through barotropic instabilities. From the point of view of the vorticity budget, positive vorticities along the walls ease the process of balancing the wind input with stronger viscous fluxes of vorticity at the walls. Eddies are also important to the vorticity budget because they transport the vorticity through the inertial layer to the viscous sublayer where it can be dissipated. However, we noted that the eddies do not play a large role in the vorticity budget. We also noted the presence of strong Kelvin waves that may provide a mechanism for transferring energy to smaller scales and dissipate it. Of course, as the Reynolds number is increased (and decreased), these Kelvin waves are no more sufficient to dissipate the energy. Then, other nonlinear processes must come into place, such as triad interactions developed by Bartello bartello95 between low and fast modes. Of interest is to note that the main Rossby mode of oscillation of the basin contrasts with that in rectangular geometries where it is usually observed that the main mode of oscillation is a basin scale Rossby wave of large amplitude (observable in both QG and SW models). The weak presence of such a mode in our simulation may mean that this mode is damped by the complex geometry of the basin.
The assessment of our scaling arguments brings up some interesting issues. First, our scaling arguments are surprisingly close to the numerical results despite obvious theoretical weaknesses. Because those arguments assumed laminar boundary layers, this implies that production of eddies was insufficient, not only to invalidate our scaling, but also to prevent inertial runaway. It is worth commenting that the double gyre circulation usually induces many more eddies. In order to get more eddies, the single gyre circulation would require more curved boundaries. For instance, it would be interesting to investigate what sort of equilibrium can be reached in basins were the boundaries are so irregular that free-slip flows have no choice but to separate from the boundaries at each bump. In such a case, it is however likely that the assumptions on which the SW equations are based would be no longer valid. For instance, the fact that the region around Bump 2 requires a resolution below 1 km implies that the SW assumption is breaking down. Moreover, the small eddy production points out to lack of physical processes represented by the SW models. Baroclinic instability, for instance, which is a main contributor to turbulence in both the atmosphere and the ocean seems to be needed in order to definitively close the inertial runaway issue in single gyre experiment. In such a case, we would need to run expensive three-dimensional baroclinic models. Finally we demonstrate the use of an adaptive strategy in ocean modelling. We note however that the cost of such a method is higher than using fixed meshes in time (see Section 3.4). Nonetheless, it provides an automated procedure for resolving and localizing fronts and strong nonlinear currents which would otherwise require tedious manual remeshing. For instance, we noted that the results from the adapted mesh yield stronger Kelvin waves, apparently related to the increased resolution in the regions of Kelvin wave production.
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