Abstract
The numerical simulation of turbulent oceanic flows is susceptible to the appearance of instabilities associated with the misrepresentation of nonlinear interactions among small-scale motions. Specialized filters and differencing schemes have been successfully used in the past to suppress the growth of these instabilities in finite-difference ocean models. Here, we introduce a new filtering procedure designed to control the growth of nonlinear instabilities in the spectral element solution of nonlinear oceanic flows. The new procedure involves two separate steps. First, a spectral filter is applied to the vorticity and divergence fields to damp oscillations in high-gradient regions and to restore spectral accuracy away from them. Second, the associated velocity field is computed from a set of Poisson equations, and its boundary conditions and interelement continuity are restored. This two-step strategy avoids the loss ofC0continuity and the weakening of Dirichlet boundary conditions that can result when the filter is directly applied to the velocity field.
The behavior of the filter is investigated numerically on the canonical problem of the double-gyre wind-driven circulation in a rectangular basin using a spectral element shallow water model. The parameters of the simulation are chosen to produce mesoscale eddies. The filter is able to stabilize the simulation even at coarse resolution and to recover the “correct” statistical behavior with as few as two grid points per Rossby deformation radius. Finally, a simulation of the wind-driven circulation in the North Atlantic Ocean is performed to illustrate the effectiveness of the filter in realistic settings.
