Stabilization, limiting amplitude, and Galerkin method for solving initial boundary value problems in atmosphere and ocean dynamics

Certain problems in atmosphere and ocean dynamics, where horizontal dimensions are much greater than vertical dimensions, employ hydrodynamic systems that do not contain a full time derivative of the fluid velocity vector and, in the case of a viscous fluid, the third equation of motion does not include a Laplacian. Here, solutions to hydrodynamic systems of this kind are investigated, with particular attention given to the asymptotic properties of the solutions for large times.