Multi-Scale Singular Limits Of Hydrodynamics In Atmosphere And Ocean

The thesis mainly investigates the three-scale singular limits of the rotating stratified Boussinesq equations in the atmosphere and ocean.It is composed by the following four chapters:In the first chapter,we introduce the research background and status of Boussinesq equations and the related rotating fluid equations.We then give an overview of the mathematical theory of singular limits for quasi-linear symmetric hyperbolic systems.In the second chapter,we consider the rotation-dominant limit(Rossby num-ber is the high order infinitesimal of Froude number)and stratification-dominant limit(Froude number is the high order infinitesimal of Rossby number)of the inviscid Boussinesq equations in a periodic domain.In these two singular limits,the system has three different time scales.When the data are well-prepared,we prove rigorously the convergence of the strong solutions for Boussinesq equations in the two limits by the energy method,and obtain the rotation-dominant lim,it equations and the stratification-dominant limit equations,respectively.When the data are ill-prepared,we develop a three-scale fast averaging method through which we prove that the fast part of the strong solutions to Boussinesq equations converges weakly to zero,and the slow part converges strongly to the two limiting systems derived above.In the third chapter,we consider three different dynamic regimes(the quasi-geostrophic,the rotation-dominant and the stratification-dominant)for the weak solutions to the viscous Boussinesq equations with stress-free boundary and ill-prepared data in T2 ×(0,?).First we construct an asymptotic profile to the weak solutions in a specific space and prove the well-posedness of the profile.Then we prove that the profile has the same asymptotic behavior to the solutions of the original system by energy estimates.Finally we validate the singular limits of the profile for the three physical regimes respectively by the three-scale fast averaging method.The methods utilized in this chapter are different from that in the two-scale singular limits of weak solutions used before.In the fourth chapter,we consider the Ekman boundary layer of the anisotrop-ic Boussinesq equations with no-slip boundary and almost ill-prepared data.The dynamic regime considered is rotation-dominant limit.The spatial domain con-sidered is T2 ×(0,?).The main difficulty in this problem comes from the coupling of the initial layer and the boundary layer.Due to this reason,we first introduce a new asymptotic profile so as to decouple the initial layer and the boundary layer and as a result we could deal with the boundary layer and the singular limits respectively.Then we construct the Ekman boundary layer and prove the con-vergence from the global weak solutions of the original system to the asymptotic profile.Finally we validate that the profile converges to a damped 2-D incom-pressible Navier-Stokes equations through the spectrum method.