Eddy Viscosity of the Anticipated Potential Vorticity Method

Abstract

The anticipated potential Vorticity method (APV) is a parameterization of the effects of subgrid or unresolved scales on those explicitly resolved for barotropic, quasi-geostrophic and certain types of primitive equation models. One novelty of the method lies in the fact that it exactly conserves energy, while still dissipating the enstrophy. (Diffusion of potential Vorticity, on the other hand, conserves neither.) We have numerically evaluated the effective eddy diffusivity associated with such parameterizations. Additionally, we have evaluated the effective eddy diffusivity explicitly, i.e., by direct numerical simulation. We find, in accord with closure calculations, that explicit simulations give cusplike behavior near the cut-off wavenumber kmax. This is induced by the continuous interaction of scales on both sides of kmax, transferring enstrophy to higher wavenumbers. The APV method reproduces this, provided that the lag or anticipation times of the vorticity are suitably (but perhaps arbitrarily) chosen. In particular, it is necessary for the anticipation time to increase rapidly with wavenumber. This in turn necessitates extra boundary conditions at walls. At low wavenumbers, the eddy viscosity produced by the APV method, predicted by closure theory and directly calculated are all negative. The test-field model prescribes a saturation toward a constant negative eddy viscosity for k ? kmax. This is qualitatively verified by explicit simulations. The APV method is consistent for wavenumbers in the inertial range. For the very lowest wavenumber, when the energy at the lowest wavenumbers is small, the method produces large negative values, producing another cusp at the largest scales resolved by the model. Explicit simulations show similar behavior. (Diffusion of potential vorticity, on the other hand, is similar to a constant, positive, eddy viscosity.) Some numerical simulations of baroclinic forced dissipative flows are presented. At medium and high resolutions the APV method is successfully able to produce a fairly ?flat? inertial range. At low resolutions, when the maximum wavenumber is in the energy containing range, the APV method either produces unrealistically high energy levels at low wavenumbers, or else the simulation is too energetic at all scales, depending on the strength of the parameterization. An energy conserving parameterization is not necessarily appropriate here. Overall, in terms of eddy viscosities, the APV method performs as well as or better than more conventional schemes using prescribed eddy diffusivities. If the resolution of the model extends into the inertial range, the APV method apparently performs very well, although the effects of a lack of Galilean invariance remain unresolved.