A Consistent Formulation of the Anisotropic Stress Tensor for Use in Models of the Large-Scale Ocean Circulation

Abstract

Subgrid-scale dissipation of momentum in numerical models of the large-scale ocean circulation is commonly parameterized as a viscous diffusion resulting from the divergence of a stress tensor of the form σ = A:∇u. The form of the fourth-order coefficient tensor A is derived for anisotropic dissipation with an axis of rotational symmetry. Sufficient conditions for A to be positive definite for incompressible flows, so guaranteeing a net positive dissipation of kinetic energy, are found. The divergence of the stress tensor, in Cartesian and spherical polar coordinates, is given for A with constant and spatially varying elements. A consistent form of A and σ for use in models based on the Arakawa B- and C-grids is also derived.