Validity of the Quasigeostrophic Model for Large-Scale Flow in the Atmosphere and Ocean

Abstract

The well-known quasigeostrophic system (QGS) for zero Rossby number flow has been used extensively in oceanography and meteorology for modeling and forecasting mid-latitude oceanic and atmospheric circulation. Formulation of QGS requires a (singular) perturbation expansion of a set of primitive equations at small Rossby number, and the quasigeostrophic equation expresses conservation of the zero-order potential vorticity of the flow. The formal expansion is justified by investigating the behavior of solutions of a set of primitive equations (PE) with a particular scaling, in the limit of zero Rossby number. This primitive model represents adiabatic, inviscid, incompressible flow with variable density and Coriolis force. Difficulties arise because PE, scaled for small Rossby number, contains unwanted solutions varying on a fast time scale with frequencies inversely proportional to the Rossby number. Without restrictions on the initial conditions, solutions of the scaled PE model do not necessarily converge to solutions of QGS in the singular limit. It is proven that, provided certain simple restrictions on the initial data are satisfied, solutions of QGS are valid approximations of solutions of the scaled PE model, with error on the order of the Rossby number. Going further in the PE expansion, the first correction to the QGS solution is obtained and it is shown that the improved approximation is second order accurate. The essential part of the analysis is to obtain energy estimates for the ageostrophic part of the solutions which allow suppression of the rapid growth. A new proof of the existence of solutions of QGS is also given.