A variational formulation for barotropic quasi-geostrophic flows

Abstract We derive simplified balanced equations by introducing perturbation expansions in the variational principle of a low-order fluid model and in that of the rapidly rotating shallow-water equations. In the case of the shallow-water equations, this provides a constrained variational principle for the barotropic quasi-geostrophic equations which is based on the Lagrangian description of the fluid. Our results thus show that the quasi-geostrophic equations can be derived systematically in the context of Lagrangian variational principles.