Modeling mesoscale eddies

Mesoscale eddies are not resolved in coarse resolution ocean models and must be modeled. They affect both mean momentum and scalars. At present, no generally accepted model exists for the former; in the latter case, mesoscales are modeled with a bolus velocity u(*) to represent a sink of mean potential energy. However, comparison of u(*)(model) vs. u(*)(eddy resolving code, [J. Phys. Ocean. 29 (1999) 2442]) has shown that u(*)(model) is incomplete and that additional terms, "unrelated to thickness source or sinks", are required. Thus far, no form of the additional terms has been suggested. To describe mesoscale eddies, we employ the Navier-Stokes and scalar equations and a turbulence model to treat the non-linear interactions. We then show that the problem reduces to an eigenvalue problem for the mesoscale Bernoulli potential. The solution, which we derive in analytic form, is used to construct the momentum and thickness fluxes. In the latter case, the bolus velocity u(*) is found to contain two types of terms: the first type entails the gradient of the mean potential vorticity and represents a positive contribution to the production of mesoscale potential energy; the second type of terms, which is new, entails the velocity of the mean flow and represents a negative contribution to the production of mesoscale potential energy, or equivalently, a backscatter process whereby a fraction of the mesoscale potential energy is returned to the original reservoir of mean potential energy. This type of terms satisfies the physical description of the additional terms given by [J. Phys. Ocean. 29 (1999) 2442]. The mesoscale flux that enters the momentum equations is also contributed by two types of terms of the same physical nature as those entering the thickness flux. The potential vorticity flux is also shown to contain two types of terms: the first is of the gradient-type while the other terms entail the velocity of the mean flow. An expression is derived for the mesoscale diffusivity kappa(M) and for the mesoscale kinetic energy K in terms of the large-scale fields. The predicted kappa(M)(z) agrees with that of heuristic models. The complete mesoscale model in isopycnal coordinates is presented in Appendix D and can be used in coarse resolution ocean global circulation models. (C) 2003 Elsevier Ltd. All rights reserved.