A variational formulation for barotropic quasi-geostrophic flows

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 quasigeostrophic equations can be derived systematically in the context of Lagrangian variational principles.