Nonlinear Stability Of Quasi-Geostrophic Flow

This thesis aims to study the nonlinear stability of quasi-geostrophic flow. This paper is divided into three parts.In the first part, we present the research background of nonlinear stability of quasi-geostrophic flow and some preliminary knowledge.In the second part, we discussed the nonlinear stability of time-independent basic state. For the two-dimension quasi-geostrophic flow, we emphasis on the two-layer model on β-plane. The minimum enstrophy, subject to energy, momentum and circulation as constraints, is found by minimizing the functional. This minimum enstrophy flow is either a parallel flow or a finite-amplitude Rossby wave depending on the aspect ratio of the channel. If the zonal length of the channel is greater than 4.6188 times its meridional width, then the solution which minimizes the enstrophy is always a finite-amplitude Rossby wave except a special case: If the momentum is zero and the circulation is nonzero, especially, the aspect ratio of lower velocity to upper velocity is not equals to zero, the solution is a parallel flow. When this inequality is reversed, the form of the solution is decided by the ratio of the energy to the square of the momentum: E/M~2. When this parameter is below a critical value the minimum enstrophy solution is again a parallel flow. While if this critical value is exceeded, the minimum enstrophy solution is a finite-amplitude Rossby wave.Nonlinear stability theorems for the three-dimension continuously stratified quasi-geostrophic flow, especially for quasi-geostrophic zonally symmetric flow, are improved by establishing an optimal Poincare inequality. The inequality is derived by a variational calculation considering a minimizational problem in three-dimensional space with the additional invariant of zonal momentum. We focus on the application of the theorem to horizontally uniform potential vorticity. For the classical Eady model in a periodic channel with finite zonal length, the improved nonlinear stability criterion is identical to the linear normal-mode one provided the channel meridional width and its length satisfied somesubstantial relation. For the generalized Eady model, two cases were discussed according to the prescribed reference-state density function are constant function and exponential function, respectively. Both nonlinear stability theorem and the condition of the linear criterion coincide with the nonlinear one are obtained. A nonlinear stability theorem was also established for three-dimensional quasi-geostrophic motions in spherical geometry by establishing an optimal Poincaré inequality. Moreover, explicit upper bounds for the disturbance energy, the disturbance potential enstrophy, and the disturbance boundary energy on the rigid lids were also established.In the third part, the baroclinic instability of the generalized Phillips model on the beta-plane, where the top and bottom surfaces are either rigid or free, has been studied include the effects of time-varying baroclinic shear in the neighborhood of the classical threshold of instability. Parametric instability changes a lot because of the additional free surface parameter α. For the linear problem, due to the parametric instability the disturbance will exponentially grow for any α. So the influences of the free surface parameter α may be ignored in this case. For the nonlinear problem, the nature of finite-amplitude behaviour of the baroclinc waves when the vertical shear of the flow is a periodic function of time, was discussed by choosing a simple dissipation mechanism. The focus of the paper is on the influences of the free surface parameter. Three cases are discussed according to supercritical states, precisely marginal states and subcritical states, respectively. And we found the influences of the free surface parameter α can not be ignored in nonlinear problem.