Discontinuous Galerkin methods for multi-layer ocean modeling: Viscosity and thin layers

Abstract

The work described here is part of a continuing project to develop, analyze, and test some procedures for using discontinuous Galerkin (DG) numerical methods in multi-layer, isopycnic models of ocean circulation. The steps taken in the present paper include the following.(1) Develop an implementation of horizontal viscosity for usage in DG methods for multi-layer models. This step involves a formulation of the local DG method that can be used in the context of barotropic-baroclinic splitting, a widely-used approach to handling the multiple time scales in ocean circulation models.(2) Develop techniques that enable a layered model to exhibit thin layers without computational failures. Layers with negligible thickness can develop in situations that include coastal upwelling, outcropping of surfaces of constant density to the upper boundary of the fluid due to lateral variations in temperature, or intersections of density surfaces with bottom topography. For the sake of DG computations involving thin layers, this paper develops (i) implementations of wind stress, bottom stress, and interfacial shear stress that do not provoke spuriously large velocities, and (ii) a limiter that maintains nonnegative layer thicknesses in DG solutions.(3) Test the above techniques in numerical experiments involving model problems.