Study On Two Kinds Of Nonlinear Solitary Wave Models In Atmosphere And Ocean

For atmosphere and ocean motion,there are two important types of nonlinear waves due to the earth's the rotation and the gravity,they are large-scale nonlinear Rossby solitary waves and mesoscale nonlinear gravity solitary waves.Many dynamical problems of atmosphere and ocean can be attributed to the evolution of these two kinds of solitary waves.Additionally,the solitary wave in the actual atmospheric and ocean motion is affected by many physical factors,such as basic flow,topography,dissipation and external sources.Therefore,it is of great theoretical significance to establish a mathematical model to study the evolution mechanism of solitary wave under the action of many physical factors.In this thesis,on the one hand,based on the quasi-geostrophic theoretical model of large-scale atmosphere and ocean,including the barotropic,baroclinic and two-layer model,the multi-scale method and small parameter perturbation expansion method are adopted to characterize nonlinear Rossby solitary waves in barotropic fluids and stratified fluids under the action of multiple physical factors,where the(1+1)-dimensional and(2+1)-dimensional models are established,as well as coupling models in two-layer fluids.As results,analytical solutions or approximate calculations of the models are obtained by various methods,and the evolutionary mechanism of nonlinear Rossby solitary waves is studied in depth.On the other hand,based on the basic dynamics equations of atmospheric motion,and using weakly nonlinear theory,a(2+1)-dimensional model of nonlinear algebraic gravity solitary waves under the action of basic flow is obtained,as well as the mechanism of squall line weather phenomenon.The research can explain the propagation and evolution of nonlinear Rossby solitary waves and gravity solitary waves in the atmosphere and ocean in a straight line or a plane,providing a potential theoretical basis for weather phenomena,weather forecasting and meteorological dynamics.Firstly,starting from the barotropic quasi-geostrophic potential vorticity equation under the generalized beta plane approximation,the basic shear flow,topography,dissipation and external factors are considered,and the reduced perturbation method is adopted to obtain the nonlinear Boussinesq model with dissipation and external forcing,the forced modified Korteweg de Vries(fmKdV)model with dissipation and slowly varying topography,the new generalized(2+1)-dimensional mKdV-Burgers model and the new(2+1)-dimensional dissipative Boussinesq model with the beta effect that satisfy Rossby solitary wave amplitude.Solitary wave solutions were obtained using the modified Jacobi elliptic function expansion method,modified hyperbolic function expansion method,general mapping deformation method,and auxiliary equation method for different models.The formation and evolution mechanism of Rossby solitary waves under the action of different physical factors was studied according to the obtained nonlinear evolution model and solitary wave solution.Secondly,under the generalized beta plane approximation,a multiscale method and the perturbation expansion method are applied to establish a forced nonlinear Boussinesq model under the combined action of the topography and dissipation and a forced(2+1)-dimensional ZakharovKuznetsov(ZK)-Burgers models under the combined action of the slowly varying topography and dissipation,which characterize the evolution of nonlinear Rossby solitary waves in stratified fluids on straight lines and planes respectively.The model is used to analyze the solitary formation factors and conservation laws of waves.Similarly,modified Jacobi elliptic function expansion method,homotopy perturbation method,simplest equation method and the modified ansatz method are applied to obtain solitary wave solutions under different factors.The effects of topography,basic topography,slowly varying topography and dissipation on the evolution of solitary waves are studied further.Thirdly,the coupling model of nonlinear Rossby solitary wave amplitude evolution in the two-layer fluid is studied.Adopting two-layer baroclinic mode,Gardner-M¨orikawa transformation and small parameter perturbation expansion method are applied to derive the coupled nonlinear mKdV model under the effects of topography and dissipation.Besides,the necessary conditions and influencing factors of baroclinic instability are analyzed as well.The beta effect and Froude number,topography and dissipation on the evolution of solitary waves are discussed by solving the model.The coupled nonlinear KdV-mKdV model is also derived,which is analyzed that beta effect and basic shear flow are two significant factors for the generation of Rossby solitary waves,and Froude number is the necessary factor for the nonlinear coupling solitary waves,which has the coupling effect.The approximate solution of the coupled nonlinear KdV-mKdV model is solved by the variational iteration method.Alongside with graphic simulation,the wave-wave interaction in the formation and evolution of solitary waves in upper and lower fluid layers is discussed.Finally,this thesis studies the nonlinear algebraic gravity solitary wave model in a baroclinic atmosphere to explain the formation process of the squall line weather phenomenon.In the opinion of the baroclinic atmospheric non-static equilibrium equations,employing scale analysis,temporal and spatial multi-scale transformation,weak nonlinear method and with the aid of symbolic computation method,the(2+1)-dimensional integer order generalized Boussinesq-Benjamin-Ono(B-BO)model equation is established.Then the(2+1)-dimensional time-fractional order generalized B-BO equation is obtained through Agrawal method with the assist of semi inverse method and a fractional variational principle.Through an analytical solution and the conservation law,the relationship between the fission of algebraic gravity solitary wave and the formation of the squall line is generated.The formation mechanism of the squall line is explained theoretically,which provides a theoretical basis to the forecast of the squall line and other disaster weather phenomena.