Analysis On Periodic Solutions And Stability Of Atmospheric And Oceanic Dynamical Systems

This thesis mainly focus on introducing a nonautonomous atmosphere-ocean model with initial value condition or boundary condition.This model is mainly used to describe the energy and materials exchange at the air-sea surface,as well as the dynamical behavior of the flow in different positions.As the basis of understanding this system,three types of fundamental equations,Euler equation,Navier-Stokes equation and Quasi-Geostrophic equation and their relative results are introduced in the first place,including their setup process,well-posedness,asymptotic behavior of the solutions,and the comparative analysis among them,etc.Without any supportive physical background and theory,no simulation of the evolution of natural phenomenon will proceed successfully.The atmosphere-ocean model,and even the three fundamental equations are all constructed upon energy conservation and mass conservation.We estimate their solutions asymptotically and,in the meanwhile,deduce many energy equalities and inequalities for explaining the actual situation theoretically.It is well known that the energy for ocean motion mainly have two sources,the wind-driven force and the solar radiation.The winddriven force act directly on the fluid surface.The kinetic energy of wind turns into the kinetic energy of fluid.However,the solar radiation works in an indirectly way.It heats up the shallow fluid and reduce its density,and the heat is transferred to lower fluid.The temperature and density difference lead to a large scale of current migration.The internal energy of atmosphere turns into the kinetic energy of fluid.All above are the physical background of atmosphere-ocean model.During the productive progress related to the ocean,one thing people mostly concerned is how to predict the marine climate in the future.One of the most important measure is to predict the long-time asymptotic ocean motion or find any periodic changing pattern.In the concept of dynamical system,that means to construct a proper nonautonomous dynamical model and find the variation tendency of solutions with respect to time.In order to do so,we emphasize the stability and periodicity of atmosphere-ocean model with external force.Because of the existence of nonlinear operator,fluid equations are more complicated.Any progress or any new result will make a great difference in this area.Due to our lack of method and immaturity of technique,we still have a long way to go for investigate such a sophisticated locomotory mechanism.The work has been done now could only be used in some certain conditions,and they need to be further improved.Besides,some characters of these motions,although already have been seen as common senses,still need to be proved theoretically.