Study On The Existence Of Solutions Of The Ocean Circulation Equation In Rotating Spherical Coordinate

The mathematical model of ocean circulation plays an important role in the study of ocean science and atmospheric science.It is of great scientific significance to further recognize ocean circulation and atmospheric dynamics.In this thesis,based on the inviscid Euler equation and the mass conservation equation,the governing equation is transformed into the nonlinear ocean vorticity equation by combining the hydrostatic pressure distribution,selecting the appropriate scale factor,introducing the flow function and applying the rank theorem.On this basis,the nonlinear ocean vorticity equation is transformed into the corresponding second order ordinary differential equation by using the stereographic projection transformation,and then the periodic,integral,non-local boundary value problem of the Antarctic Circumpolar Current and the asymptotic boundary value problem of the Arctic gyres are studied.The nonlinear ocean vorticity equation is transformed into the corresponding second-order elliptic equation by Mercator projection transformation,and the Dirichlet and Neumann boundary value problems for the Antarctic Circumpolar Current are studied.Firstly,the positive solution of the periodic boundary value problem of the Antarctic Circumpolar Current in the general nonlinear case is given,a method of calculation and the approximate solution format are presented,and the existence results of the solution in the weakly nonlinear and semilinear cases are provided,respectively.The explicit expression of Green’s function for the integral boundary value problem of the Antarctic Circumpolar Current is given.The existence and uniqueness of the solution are proved by using the mixed monotone operator theory and the cone fixed point theorem,and the sufficient conditions for the existence of the system solution are given.The existence of solution for nonlinear vorticity nonlocal boundary value problem is proved by using topological degree theory,zero exponent theory and fixed point technique.Secondly,the research method of boundary value problem for the Antarctic Circumpolar Current is extended to asymptotic boundary value problem of the Arctic gyres,and the explicit expression of solution of the Arctic gyres equation under linear vorticity is presented.Schauder’s fixed point theorem is used to study the existence and uniqueness of solution under nonlinear vorticity in local scope,and the growth condition of solution is given.Finally,the energy functional corresponding to Dirichlet boundary value problem of the Antarctic Circumpolar Current is constructed.By using Young inequality,Poincar′e inequality and Sobolev inequality,sufficient conditions for the existence of solutions to elliptic equations with two different nonlinear ocean vorticity are obtained according to Ekeland’s variational principle.Furthermore,the existence of infinite solutions to Neumann boundary value problem for the Antarctic Circumpolar Current is proved by the perturbation method,the truncation function,the deformation theorem and critical point theory.The results are helpful to explain and judge the rationality of ocean simulation,and provide theoretical guidance for recognizing the dynamic behavior of complex ocean circulation.