The Stability Of Boussinesq Equations

Boussinesq equation is a simplified model of the atmospheric movement equation . It is applicable to mesoscale, non-static equilibrium, quasi-incompressible fluid movement. Its specific features are as follows:1. Ignoring the individual change of density in continuity equations, and thus making it approximately incompressible fluid;2. Partially considering the effect brought about by the change of density in the movement equation related to gravity (vertically upward);3. Taking into account the effect of density change in status equation and thermo-dynamics equation, while regarding the density change as the result of change in temperature only, without any thought being given to pressure;4. Considering the coefficients of viscosity and thermo-conductivity as constants.5. The functions of Coriolis force are being taken into consideration;6. Non-static equilibrium. Ordinarily, it takes the following form:If the influence of moisture is ruled out for consideration, the moisture equation can be taken away. In movement equations, the viscosity of turbulent flow is mainly considered. The coefficients of viscosity and thermo-dissipation in horizontal and vertical direction are regarded as different. The temperature change caused by the vertical motion is also taken into account. Under such circumstances, the equation goes like this:If the viscosity of turbulent flow is replaced by Reyleigh friction and the thermo-dissipation replaced by Newton cooling, we come to the model below:The main work in this paper is to study the stability of the above three models, the suitability of the pose of the initial (boundary) value problem of those models are also discussed.In Chapter two, the specific terms, basic concepts, definitions and theorems of stratification are introduced. Examples are given to explain, test and verify. Some common equations are taken as examples, their "pre-gradue" and "gradue" are found out. Their L - simplicity are discussed.In Chapter three, the stability of the above three models, through the application of stratification theory, are studied and analyzed. The results are:1. The system of equations (A) is 0 - simple; S'3,k-1(D) ≠ΦThe system of equations (A) is stable.There exists a unique stable solution to suitable initial (boundary) valueproblem.As to the initial value given on {t = t0}, the related problem is ill-posed. The structure of solution space is given.