On the qualitative theory of the rotating boussinesq and quasi-geostrophic equations

Abstract

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., a Lagrangian and Eulerian analysis of a geophysical fluid flow. This article focuses on the initial boundary value problem to the rotating Boussinesq equations. The equations are derived from conservation laws in continuum physics followed by the formulation of the problem as initial value problem on Hilbert spaces. By using Faedo-Galerkin approximation and semigroup technique, existence and uniqueness of solutions are proved. Additionally, the manuscript outlines how Lyapunov functions can be used to assess energy stability criteria. The author also addresses singular problems for which the equation has parabolic structure (rotating Boussinesq equations for the atmosphere and ocean) and the singular limit is hyperbolic (quasi-geostrophic equations for the atmosphere and ocean) in the asymptotic limit of small Rossby number. In particular, this approach gives as a corollary a constructive proof of the well-posedness of the problem of quasi-geostrophic potential vorticity equations governing modons or Rossby solitons.

Keywords: Rotating Boussinesq equations, quasi-geostrophic equations, existence and asymptotic stability