On the Qualitative Theory of the Rotating Boussinesq Equations

An alternative title for this paper would perhaps be ?Well-posedness of the problem of beta-plane ageostrophic flows for meteorology and oceanography.? The author studies the rotating Boussinesq equations describing the motion of a viscous incompressible stratified fluid in a rotating system which is relevant, e.g., for Lagrangian coherent structures. These equations consist of the Navier-Stokes equations with buoyancy-term and Coriolis-term in beta-plane approximation, the divergence-constraint, and a diffusion-type equation for the density variation. They are considered in a plane layer with periodic boundary conditions in the horizontal directions and stress-free conditions at the bottom and the top of the layer. Additionally, the author considers this model with Reynolds stress, which adds hyper-diffusivity terms of order 6 to the equations. This manuscript focuses primarily on deriving the rotating Boussinesq equations for geophysical fluids, showing existence and uniqueness of solutions, and outlining how Lyapunov functions can be used to assess stability. The main emphasis of the paper is on Faedo-Galerkin approximations as well as LaSalle invariance principle.