Analytic Solution Computation For Boussinesq Approximate And Equatorial Beta-plane Approximate Type Equations

This paper, applying the theory of stratification, devotes to two kinds of approximate equations: the Boussinesq approximate type equations and the equatorialβ?plane approximate type equations, which are very important in the atmospheric science. We study the stability and the well-posedness of the problem for determining solution by analyzing the topological property of these equations. We give the solution approach of exact solution in detail in the class of the analytic functions for the well-posed problem and the methods of construction of the formal solutions for the ill-posed problem.The main results can be stated as follows:1. It is proved that the two-dimensional non-hydrostatic Boussinesq equations on the x ? z plane with the kinetic viscous term and the thermal dissipative term are unstable equations in the C 2 function class.2. If we replace the influence of the kinetic viscous term by Rayleigh friction and the thermal dissipative term by Newton cooling, the new generalized equations are stable equations in the C 1 function class. The construction of the solution space and the discriminating method for well-posedness of the problem of determining solution are given. The reasonableness of simplification and the fundamental reason of the changes of stability are analyzed. The cases are given for the generalized initial value problems in detail.3. The three-dimensional Boussinesq equations with no-viscosity and adiabatic under the hydrostatic equilibrium are studied. Especially, the generalized initial value problems are investigated and the cases are given in order to comparison with those under the non-hydrostatic equilibrium.4. The construction of the solution space for shallow-water equations on the equatorialβ?plane and the equations about the tropical circulation to heating are mainly studied. The solution method of exact solution in the analytic function class and the computational procedure to the series solution are expressed for the well-posed problem.