On mathematical problems for the primitive equations of the ocean: The mesoscale midlatitude case
Continuing earlier work on the geostrophic asymptotics for the atmosphere [see, e.g., the authors, J. Math. Pures Appl., IX. Sér. 74, No. 2, 105-163 (1995; Zbl 0866.76025)], the authors study in this paper the geostrophic asymptotics of sea water equations. The article is oriented towards mathematical and modeling aspects. The mesoscale (or synoptic) midlatitude case is considered. The objectives are as follows: (a) to introduce systematically the geostrophic asymptotics for the mesoscale ocean, and (b) to perform the mathematical analysis of the resulting quasi-geostrophic equations and asymptotics. For objective (a), two types of models are considered. The first is the density-stratified primitive equations, the second is the primitive equations with double diffusions (for temperature and salinity functions). By analogy with Leray's formulation of weak solutions of Navier-Stokes equations, the mathematical analysis is proceeded for objective (b) as follows: 1. to start with the existence of solutions of quasi-geostrophic equations; 2. to obtain the weak formulations of quasi-geostrophic equations by projecting the equations onto the function spaces of geostrophic motions; 3. to prove the existence and properties of global (in time) weak solutions of quasi-geostrophic equations in both the density-stratified and double diffusive models via Galerkin methods; 4. in a particular case to derive a partial mathematical justification of the geostrophic asymptotics.
scientific article; zbMATH DE number 1472229
scientific article