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Table of Contents
Preface; Contents; 1 Introduction; 2 Spaces of Functions on a Sphere; 2.1 Spherical Harmonics; 2.2 Geographical Coordinates Maps; 2.3 Orthogonal Projections on Hn and Fractional Derivatives; 2.4 Hilbert Spaces Hs on a Sphere; 2.5 Space C(S) of Continuous Functions on a Sphere; 2.6 Some Estimates in the Norms of Lp(S) and Lp(0,T; X); 3 Solvability of Vorticity Equation on a Sphere; 3.1 Vortex Dynamics of Viscous Incompressible Fluid; 3.2 Properties of Jacobian Determinant; 3.3 Unique Solvability of a Non-Stationary Problem; 3.4 Solvability of a Stationary Vorticity Equation
4.7 Distance Between Solutions4.8 Euler Angles; 5 Stability of Rossby-Haurwitz (RH) Waves; 5.1 Conservation Laws for Arbitrary Perturbations to RH Wave; 5.2 Invariant Sets, Quotient Space and Norm of Perturbations; 5.3 A Hyperbolic Law for Perturbations from M-n and M+n; 5.4 Geometric Interpretation of Variations in the Perturbation Energy; 5.5 Liapunov Instability of Non-Zonal RH Wave; 5.6 Exponential Instability of RH Wave; 5.7 Normal Mode Instability of Zonal RH Waves and LP Flows; 6 Stability of Modons and Wu-Verkley Waves; 6.1 Steady Wu-Verkley Waves and Modons
6.2 Conservation Law for Disturbances of WV Wave and Modon6.3 Conditions for Exponential Instability of WV Waves and Modons; 6.4 Bounds of Growth Rate and Orthogonality of Unstable Modes; 6.5 Dipole Modons Moving Along the Same Latitudinal Circle; 6.6 Liapunov Instability of Dipole Modons; 7 Linear and Nonlinear Stability of Flows; 7.1 Shear Flow Stability; 7.2 Linear Stability of Zonal Flows; 7.3 Nonlinear Stability; 7.4 Instantaneous Evolution of Kinetic Energy of Perturbations; 7.5 The First Mechanism of Generation of the Energy of Perturbation Near a Zonal Jet
7.6 Generalized Eliassen-Palm Flux and the Eigenvalue Problem Method7.7 Numerical Example: Analysis of Climatic January Circulation; 8 Numerical Study of Linear Stability; 8.1 Method of Normal Modes; 8.2 Spectrum of Linearized Operator for Viscous Fluid; 8.3 One Estimate in Terms of the Graph Norm of Operator; 8.4 Spectral Approximation; 8.5 Rate of Convergence Estimates; 8.6 Spectrum of Linearized Operator for Ideal Fluid; 8.7 Stability Matrix in the Basis of Spherical Harmonics; 8.8 Stationary States Having Block Diagonal Structure of Stability Matrix
4.7 Distance Between Solutions4.8 Euler Angles; 5 Stability of Rossby-Haurwitz (RH) Waves; 5.1 Conservation Laws for Arbitrary Perturbations to RH Wave; 5.2 Invariant Sets, Quotient Space and Norm of Perturbations; 5.3 A Hyperbolic Law for Perturbations from M-n and M+n; 5.4 Geometric Interpretation of Variations in the Perturbation Energy; 5.5 Liapunov Instability of Non-Zonal RH Wave; 5.6 Exponential Instability of RH Wave; 5.7 Normal Mode Instability of Zonal RH Waves and LP Flows; 6 Stability of Modons and Wu-Verkley Waves; 6.1 Steady Wu-Verkley Waves and Modons
6.2 Conservation Law for Disturbances of WV Wave and Modon6.3 Conditions for Exponential Instability of WV Waves and Modons; 6.4 Bounds of Growth Rate and Orthogonality of Unstable Modes; 6.5 Dipole Modons Moving Along the Same Latitudinal Circle; 6.6 Liapunov Instability of Dipole Modons; 7 Linear and Nonlinear Stability of Flows; 7.1 Shear Flow Stability; 7.2 Linear Stability of Zonal Flows; 7.3 Nonlinear Stability; 7.4 Instantaneous Evolution of Kinetic Energy of Perturbations; 7.5 The First Mechanism of Generation of the Energy of Perturbation Near a Zonal Jet
7.6 Generalized Eliassen-Palm Flux and the Eigenvalue Problem Method7.7 Numerical Example: Analysis of Climatic January Circulation; 8 Numerical Study of Linear Stability; 8.1 Method of Normal Modes; 8.2 Spectrum of Linearized Operator for Viscous Fluid; 8.3 One Estimate in Terms of the Graph Norm of Operator; 8.4 Spectral Approximation; 8.5 Rate of Convergence Estimates; 8.6 Spectrum of Linearized Operator for Ideal Fluid; 8.7 Stability Matrix in the Basis of Spherical Harmonics; 8.8 Stationary States Having Block Diagonal Structure of Stability Matrix