Recent Advances Concerning Certain Class of Geophysical Flows

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.