Recent Advances Concerning Certain Class of Geophysical Flows

Jinkai Li () and Edriss S. Titi ()
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Jinkai Li: Weizmann Institute of Science, Department of Computer Science and Applied Mathematics
Edriss S. Titi: Weizmann Institute of Science, Department of Computer Science and Applied Mathematics

Chapter 19 in Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, 2018, pp 933-971 from Springer

Abstract: Abstract This chapter 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. It is 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.

Keywords: Anisotropic incompressible Navier-Stokes equations; Primitive equations for oceanic and atmospheric flow; Geophysical flow; Boussinesq equations; Tropical atmosphere; Hydrostatic approximation; Singular perturbation limit; 35A01; 35B45; 35Q30; 35Q86; 76D03; 76D09. (search for similar items in EconPapers)
Date: 2018
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DOI: 10.1007/978-3-319-13344-7_22

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