Some Results On The Quasigeostrophic Equation

The well-known quasigeostrophic equation for small Rossby number flow has been used extensively in oceanography and meteorology for modeling and forecasting mid-latitude oceanic and atmospheric circulation, and for studies of stability, frontogenesis and turbulence.In this dissertation we discuss two types quasigeostrophic equations. We first consider the quasigeostrophic equation with no dissipation term which was obtained as an asymptotic model from the Euler equations with free surface under a quasigeostrophic velocity field assumption, it is called Hasegawa-Mima-Charney-Obukhov equation which also arise from plasmas theory. We use a priori estimates to get the global existence of strong solution for an Hasegawa-Mima-Charney-Obukhov equation.We next consider a model of fluid flow in a two-layer domain. The global solution, global attractor and exponential attractor are established for dissipa-tive quasigeostrophic equations of geophysical fluid dynamics in the space X~α ((1/2) < α≤ 1). Periodic boundary condition is imposed in x direction and y direction with ψi = △ψi = 0 (i = 1,2) conditions respectively. Furthermore, the synchronization of this model is obtained in rigorously mathematical analysis.