Well-Posedness and Long-Time Dynamics of the Rotating Boussinesq and Quasigeostrophic Equations

Abstract

Most differential equations occurring in multiscale modelling of physical and biological systems cannot be solved analytically. Numerical integrations do not lead to a desired result without qualitative analysis of the behavior of the equation’s solutions. The authors study the quasigeostrophic and rotating Boussinesq equations describing the motion of a viscous incompressible rotating stratified fluid flow, which refers to PDE that are singular problems for which the equation has a parabolic structure (rotating Boussinesq equations) and the singular limit is hyperbolic (quasigeostrophic equations) 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 quasigeostrophic equations governing modons or Rossby solitons. The rotating Boussinesq 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. Thus the foundation for the study of the quasigeostrophic and rotating Boussinesq equations is the Navier-Stokes equations modified to accommodate the effects of rotation and stratification. 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 authors consider this model with Reynolds stress, which adds hyper-diffusivity terms of order 6 to the equations. This course focuses primarily on deriving the quasigeostrophic and rotating Boussinesq equations for geophysical fluid dynamics, showing existence and uniqueness of solutions, and outlining how Lyapunov functions can be used to assess energy stability. The main emphasis of the course is on Faedo-Galerkin approximations, the LaSalle invariance principle, the Wazewski principle and the contraction mapping principle of Banach-Cacciopoli. New understanding of quasigeostrophic turbulence called mesoscale eddies and vortex rings of the Gulf Stream and the Agulhas Current Retroflection could be helpful in creating better ocean and climate models.