Parameterizing mesoscale eddies with residual and Eulerian schemes, and a comparison with eddy-permitting models

Abstract

In this paper we explore and test certain parameterization schemes that aim to represent the effects of unresolved mesoscale eddies on the larger-scale flow. In particular, we examine a scheme based on the residual or transformed Eulerian mean formulation of the equations, in which the eddies are parameterized by a large vertical viscosity in the momentum equations, with no skew flux parameterization appearing in the tracer (e.g., temperature or salinity) evolution equations, although terms that parameterize diffusion along isopycnal surfaces remain. The residual scheme is compared both to a conventional parameterization that uses a skew diffusion (or equivalently advection by a skew velocity), and to eddy-permitting calculations. Although in principle almost equivalent to certain forms of skew flux schemes, the residual formulation is found to have certain practical advantages over the conventional scheme in some circumstances, and in particular near the upper boundary where conventional schemes are sensitive to the choice of tapering but the residual scheme is less so. The residual scheme also enables the horizontal viscosity – which is mainly applied to maintain model stability – to be reduced. Finally, the residual scheme is somewhat easier to implement, and the tracer transport is easier to interpret. On the other hand, the residual scheme gives, at least formally, a transformed velocity, not the Eulerian velocity and will not be appropriate in all circumstances.