1. Nonlinear Equations of the Atmospheric and the Oceanic Motions 1.1 Basic Equations of the Atmospheric and the Oceanic Motions 1.1.1 Basic Equations of the Atmosphere 1.1.2 Basic Equations of the Oceans 1.2 Equations of the Atmosphere and the Oceans in the Sphere Coordinate Frame 1.2.1 Equations of the Atmosphere in the Sphere Coor- dinate Frame 1.2.2 Equations of the Oceans in the Sphere Coordinate Frame 1.3 Equations of the Atmosphere in Atmospheric Pressure Coordinate Frame 1.4 Equations of the Atmosphere in the Topography Coordi- nate Frame 1.5 Equations of the Atmosphere and the Oceans in Local Rectangular Coordinate Frame under/%Plane Approxima- tion 1.6 Equations of the Atmosphere and the Oceans under Sati- fication Approximation 1.7 Boundary Conditions 1.7.1 The Lower Boundary Conditions of the Atmosphere 1.7.2 The Upper Boundary Conditions of the Atmosphere 1.7.3 The Boundary Conditions of the Oceans 2. Some Quasi-Geostrophic Models 2.1 The Barotropic Model and the Two-Dimensional Quasi- Geostrophic Equation 2.1.1 The Barotropic Model 2.1.2 The Two-Dimensional Quasi-Geostrophic Equation 2.2 Three-Dimensional Quasi-Geostrophic Equation 2.3 The Multi-Layer Quasi-Geostrophic Model 2.4 The Surface Quasi-Geostrophic Equation 2.4.1 Introduction of Surface Quasi-Geostrophic Equation 2.4.2 Some Results of Surface Quasi-Geostrophic Equation 3. Well-Posedness and Global Attractors of the Primitive Equations 3.1 Existence of Weak Solutions and Trajectory Attractors of the Moist Atmospheric Equations 3.1.1 Primitive Equations of the Large-Scale Moist At- mosphere 3.1.2 The Global Existence of Global Weak Solutions of Problem IBVP 3.1.3 Trajectory and Global Attractors for the Moist At- mospheric Equations 3.2 Long-Time Behavior of the Strong Solutions of the Primi- tire Equations of the Large-Scale Moist Atmosphere . . . 3.2.1 The Primitive Equations of the Large-Scale Moist Atmosphere 3.2.2 Main Results 3.2.3 Uniform Estimates of Local Strong Solutions in Time 3.2.4 Global Existence and Uniqueness of Strong Solutions 3.2.5 Some Preliminaries about the Infinite-Dimensional Dynamical System 3.2.6 The Existence of Global Attractors 3.3 The Global Well-Posedness of the Primitive Equations . . 3.3.1 Main Results i 3.3.2 The Global Well-Posedness of IBVP 3.3.3 The Global Existence of Smooth Solutions of IBVP 3.4 Global Well-Posedness of Primitive Equations of the Oceans 3.4.1 The Viscous Primitive Equations of the Large- Scale Oceans 3.4.2 Main Results 1 3.4.3 The Local Existence of Strong Solutions 1 3.4.4 The Global Existence and Uniqueness of Strong Solutions 4. Random Dynamical Systems of Atmosphere and Ocean 4.1 Random Attractors of Two-Dimensional Quasi-Geostrophic Dynamical System 4.1.1 Model 4.1.2 Global Existence and Uniqueness of Solutions 4.1.3 Preliminaries of Random Attractors 4.1.4 Existence of Random Attractors 4.1.5 Stochastic Quasi-Geostrophic Equations on Unbounded Domains 4.2 Global Well-Posedness and Attractors of 3D Stochastic Primitive Equations of the Large-Scale Oceans 4.2.1 Three-Dimensional Stochastic Equations of the Large-Scale Oceans 4.2.2 New Formulation of IBVP 4.2.3 Existence and A Priori Estimate of Local Solutions 4.2.4 The Global Well-Posedness of IBVP 4.2.5 Existence of Random Attractors 4.3 The Primitive Equations of Large-Scale Oceans with Ran- dom Boundary 4.3.1 Model 4.3.2 The New Formulation of the Initial-Boundary Value Problem (4.3.10)-(4.3.17) 4.3.3 The Global Well-Posedness of the Primitive Equa- tions with Random Boundary 4.3.4 Existence of Random Attractors 5. Stability and Instability Theory 5.1 Stability and Instability of Gravity Waves 5.1.1 Stability and Instability of Internal Gravity Waves in Stratified Flows 5.1.2 Stability of General Internal Gravity Wave 5.1.3 Stability of General Inertial Internal Gravity Wave 5.2 Instability of Rossby Waves 5.2.1 Necessary Conditions of Linear Instability 5.2.2 Linear Instability of Barotropic Rossby Waves . 5.2.3 Linear Instability of Baroclinic Rossby Waves . 5.3 Stability of Rossby Waves 5.3.1 Stability of Bidimensional Quasi-Geostrophic Flow 5.3.2 Stability of Saddle-Point-Type Two-Dimensional Quasi- Geostrophic Flows 5.4 Critical Rayleigh Number of Rayleigh-Benard Convection 5.4.1 Linear Stability 5.4.2 Nonlinear Stability for Ra 5.4.3 Nonlinear Instability for Ra> R*a References
