Anisotropic horizontal viscosity for ocean models

A consistent mathematical formulation of a horizontally anisotropic friction operator is developed for use as a subgrid-scale turbulence parameterization in ocean general circulation models. It is derived as the divergence of a deviatoric stress tensor that represents the traceless part of the Reynolds stress tensor. The stress tensor is assumed to depend linearly on the velocity gradients, has the property that no stresses are generated by uniform rotation, and the viscous coefficients can be chosen to ensure the friction terms are purely dissipative of kinetic energy. The anisotropy is determined by an arbitrary field of horizontal unit vectors n̂ that specify a preferred direction at each point in space which breaks the transverse isotropy. This direction may be aligned with east–west (e.g., to describe anisotropy associated with the beta effect), with gradients of topography, with the mean flow, or with any other preferred direction which is expected to break the horizontal isotropy of turbulent momentum dissipation. The general anisotropic formulation involves four viscous coefficients, compared to only two in the isotropic case. A reduced two-coefficient anisotropic form is derived by neglecting terms in the energy dissipation rate proportional to the horizontal velocity divergence, which in geophysical flows is small compared to typical horizontal velocity gradients. This form involves different viscosities in the directions parallel and perpendicular to n̂. Experimental measurements of Reynolds stresses in self-preserving shear flows are shown to exhibit this type of anisotropy when n̂ is aligned with the mean flow. The two-coefficient form is evaluated in numerical simulations in a periodic zonal channel. A functional discretization of the operator which ensures positive-definite dissipation of kinetic energy is also presented in an appendix.