On Parameterizing Vertical Mixing of Momentum in Non-eddy Resolving Ocean Models

Abstract

We investigate the consequence, at small Ekman number, of adding vertical mixing of momentum terms to the incompressible thermocline equations. We find that choosing the vertical eddy viscosity, ? = Af2/N2, where f is the Coriolis parameter and N is the local value of the buoyancy frequency, leads to isopycnal mixing of fQ, where Q is the reciprocal of potential vorticity, provided A is independent of the vertical coordinate. If, additionally, A is also independent of the north?south coordinate, then on a beta-plane, this implies homogenization of potential vorticity, q, within closed q-contours on isopycnal surfaces. This conclusion extends to spherical geometry if ? is also inversely proportional to ?, the gradient of f with respect to latitude, i.e. ? = Af2/(N2?). The connection with the recent work of Gent and McWilliams and the consequences for coarse resolution numerical model studies are discussed.