Mathematics of Climate Modeling

1. Dynamical Systems. Attractors, Invariant Measures.- 1.1 Metric Spaces. Compactness.- 1.2 Dynamical Systems. Main Properties.- 1.3 Invariant Sets.- 1.4 Classification of Motions.- 1.5 Recurrence of Domains.- 1.6 Measure. Krylov-Bogolyubov Theorem.- 1.7 Dynamical Systems with Invariant Measure.- 1.8 Nonlinear Dissipative Systems.- 1.9 Inertial Manifolds of Dissipative Systems.- 2. Non-Autonomous Dissipative Systems, their Attractor and Averaging.- 2.1 Introduction.- 2.2 Processes and their Attractors. Kernel of Processes, Section of Kernel.- 2.3 Families of Processes and their Attractors.- 2.4 Family of Processes and Semigroups.- 2.5 Averaging of Nonlinear Dissipative Systems. Closeness between Attractors of Original and Averaged Systems.- 2.6 On Closeness of Solutions of Original and Averaged Nonlinear Dissipative Systems on Infinite Time Interval.- 3. Analysis of Barotropic Model.- 3.1. Existence of Global Attractor.- 3.2 Estimate of Dimension of Attractor.- 3.3 Statistical Solutions and Invariant Measures on Attractor.- 3.4 Estimate of Attractor Dimension with Respect to Orography.- 3.5 Galerkin Approximations.- 3.6 Existence of Inertial Manifold.- 4. Discretization of Systems Possessing Attractor.- 4.1 Discretization of Systems Possessing Inertial Manifolds.- 4.2 Time-Space Discretization of Systems Possessing Attractor.- 4.3 Globally Stable Difference Schemes for Barotropic Vorticity Equation.- 5. Numerical Study of Structure of Attractor Generated by Barotropic Equations on Sphere.- 5.1 Equations and Parameters of Model. Methods of Solving of Stationary and Nonstationary Problems.- 5.2 Statistical Stationary Solution and Stationary Points.- 5.3 Lyapunov Exponents and Attractor Dimension.- 5.4 Analysis of Analytical Estimates of Attractor Dimension of Barotropic Atmospheric Equations.- 6. Two-Layer Baroclinic Model.- 6.1 Two-Layer Baroclinic Model.- 6.2 Estimate of Attractor Dimension.- 6.3 Numerical Investigation of Attractor. Characteristics of Two-Layer Baroclinic Model.- 7. Investigation of Structure of Climate Attractors by Observed Data Series.- 7.1. Correlation Dimension of Attractor.- 7.2. Calculation of Lyapunov Exponents.- 7.3 Statistically Independent Degrees of Freedom and Attractor Dimension.- 8. Regimes of Atmosphere Circulation.- 8.1 Definition of Atmosphere Circulation Regimes.- 8.2 Dynamical Theory of Two-Regime Barotropic Circulation.- 8.3. Statistical Theory of Two-Regime Barotropic Circulation.- 8.4 S-Regimes of Atmosphere Circulation.- 9. Solvability of Ocean and Atmosphere Models.- 9.1 Introduction.- 9.2 Solvability of Ocean and Atmosphere Models in Bounded Domains.- 9.3 Solvability of Ocean and Atmosphere Models on Sphere in p-System of Coordinates.