q 1997 American Meteorological Society A Simple Friction and Diffusion Scheme for Planetary Geostrophic Basin Models

A simple friction and diffusion scheme is proposed for use with the time-dependent planetary geostrophic equations, which in their proper asymptotic form cannot be solved in a closed basin. The resulting set of equations admits boundary conditions of no-normal flow and no-normal heat flux at all rigid boundaries, is amenable to efficient numerical solution, and may be solved with small heat (and salt) diffusivities. The scheme is formally a minor modification of several others that have recently been proposed but differs in significant details. Friction is represented by linear drag in the horizontal momentum equations, while the hydrostatic balance is retained exactly. The Laplacian vertical and horizontal heat (and salt) diffusion are supplemented by biharmonic horizontal diffusion. The latter is necessary in order that the smoothness of the solution can be maintained up to and along the boundary, which is particularly important because the no-normal-flow condition is enforced as a differential equation that must be solved along the boundary. The equations support a frictional western boundary current that is nearly adiabatic. 1