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9789056991647

Mathematical Problems and Methods of Hydrodynamic Weather Forecasting

by ;
  • ISBN13:

    9789056991647

  • ISBN10:

    9056991647

  • Format: Hardcover
  • Copyright: 2000-09-20
  • Publisher: CRC Press

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Summary

The material provides an historical background to forecasting developments as well as introducing recent advances. The book will be of interest to both mathematicians and physicians, the topics covered include equations of dynamical meteorology, first integrals, non-linear stability, well-posedness of boundary problems, non-smooth solutions, parameters and free oscillations, meteorological data processing, methods of approximation and interpolation and numerical methods for forecast modelling.

Author Biography

Vladimir Gordin is Professor in Mathematics at the Hydrometeorological Centre of Russia.

Table of Contents

Introduction xv
Equations of Dynamical Meteorology
1(102)
Equations of the Hydrothermodynamics
2(16)
Continuous Medium
2(1)
Mass Conservation Law
3(1)
Momentum Conservation Law
4(2)
Energy Conservation Law
6(2)
Thermodynamic Functions
8(1)
Ideal (Perfect) Gas
9(3)
Hydrothermodynamic Equations in Spherical Coordinates
12(2)
Gravity Force and Coriolis Force
14(1)
Physical Limitations of Model
15(1)
Free Atmosphere on l- and β-Planes
16(2)
First Integrals of Evolutionary Systems
18(21)
Total Derivative of Functional in virtue of a System
19(3)
First Integrals of Degree 0 for a First Order Quasi-linear System
22(1)
First Integrals of Degree 0 for System (1.32)
22(7)
First Integrals of Degree 0 for the Spherical Model
29(4)
First Integrals and Hydrodynamic Stability
33(2)
Stable Steady---State Solutions for the System of Hydrothermodynamic Equations
35(3)
Stable Zonal Flow
38(1)
Properties of Evolution Equations Systems
39(22)
The Cauchy-Kovalevskaya Theorem
41(2)
The Cauchy Problem for Systems of Linear Ordinary Equations with Constant Coefficients
43(2)
Applications of Formula (1.66): The Splitting Method and a Lax Pair
45(1)
Systems of Linear Partial Differential Equations with Constant Coefficients
46(3)
First Order Hyperbolic Systems
49(2)
Parabolic Systems
51(3)
Mixed Problem for Evolutionary Systems
54(6)
Systems with Discontinuous Coefficients
60(1)
Quasi-linear Hyperbolic Systems and Gas Dynamics
61(42)
Cauchy Problem for the Non-linear Transport (Advection) Equation (1.40)
62(4)
Discontinuous Solutions and the Hugoniot---Rankine Condition
66(2)
Choice of a Unique Weak Solution
68(6)
Discontinuous Solutions for More General Systems and Equations
74(3)
Jump Conditions for Hydrothermodynamic Equations
77(3)
Riemann Invariants and Characteristics
80(7)
Systems of Conservation Laws with Generalized Entropy and Symmetrization
87(7)
Propagation of Disturbances
94(2)
Examples of Tangential Discontinuities' Instability
96(7)
Small Parameters and Small Oscillations
103(76)
The Hydrostatic Equation
105(15)
The Small Parameter in System (1.32) Describing Dynamics of the Atmosphere
106(5)
p-Coordinate System
111(2)
Boundary Conditions for the Hydrostatic System of Equations with in the p-Coordinate
113(4)
Phillips' σ-Coordinate System
117(1)
&thetas;-Coordinate System
118(2)
Free Oscillations over the l-Plane
120(27)
Reduction of the System to an Ordinary Second Order Differential Equation Depending on a Spectral Parameter
121(4)
Separation of Variables
125(2)
Specific Values of the Spectral Parameter λ
127(4)
Boundary Conditions for the Function ψ
131(8)
Free Oscillations in the Quasi-static Approximation
139(8)
Small Oscillations of the Atmosphere over the Spherical Surface
147(21)
Linearization of the Hydrodynamic Equations and Separation of Variables
147(5)
Eigen-functions of the Laplace Tidal Equation (the Hough Functions)
152(10)
Eigen-functions of the Model over the β-Plane
162(2)
Initialization Using the Normal Modes Technique
164(4)
Atmospheric Front and Stability of Small Oscillations in its Vicinity
168(11)
Geostrophic Stationary Solutions
168(4)
Small Oscillations. A Basic State as a Perturbation of a State in Which Separation of Variables is Allowed
172(2)
The Spectrum Perturbation of Geostrophic Modes
174(3)
Perturbation of the Spectrum of Gravitational and Acoustic Modes
177(2)
Meteorological Data Processing
179(150)
Sources of Meteorological Data
181(10)
Types of Measurements
184(1)
The Density of Meteorological Data Covering Different Areas and Different Vertical Levels
185(2)
Bias and Precision of Measurements
187(4)
Optimum Interpolation Method
191(41)
Optimum Interpolation of Scalar Fields (e.g., of Geopotential Field)
192(2)
Choice of a Correlation Function. Phillips Theorem
194(2)
Optimum Integration over Sphere S2
196(7)
Optimum Differentiation on Sphere S2
203(6)
Optimum Interpolation of Wind Field and Multivariate Optimum Interpolation of Geopotential and Wind
209(6)
Selection of Relevant Stations (Example of Algorithm)
215(2)
Objective Analysis of Atmospheric Fronts
217(2)
Choice of a Configuration Space and Correlation Functions for Optimum Interpolation
219(2)
Non-Homogeneity of 3D Correlation Functions
221(2)
Optimal Approximation of a Correlation Function by Self-similar Ones
223(3)
Vertical Coordinate where a given Random Field is Closest-to-Homogeneous
226(2)
Objective Analysis of Tropopause
228(4)
Variational Adjustment of Meteorological Fields
232(19)
Adjustment of Geopotential and Temperature Profiles
234(2)
Variational Adjustment of Wind and Geopotential Fields with the Geostrophic Relationships as Constraints
236(1)
Variational Adjustment of Wind and Geopotential Fields with the Linear Balance Equation as a Constraint
237(5)
Variational Adjustment of Wind and Geopotential Fields with the Non-Linear Balance Equation as a Constraint
242(1)
Variational Adjustment of Wind's Speed
243(3)
Variational Adjustment of the Covariance Functions of Geopotential and Temperature
246(2)
Improvement in Small Non-positive Definiteness of a Covariance Matrix. Variational Approach
248(3)
Solvers for Elliptic Problems and Functional's Minimization
251(36)
Matrix Double-Sweep Method and Fast Fourier Transform for Numerical Solution of Elliptic Boundary Value Problem (e.g., for system (3.41)-(3.44))
252(8)
Simple Iteration Method
260(3)
Chebyshev Acceleration, Projections, and Energy Equivalent Operators
263(5)
Relaxation and Multi-grid Methods
268(13)
Minimizing of Non-linear Functionals
281(6)
Block Relaxation Method
287(1)
Quality Control and Reconstruction of Missing Meteorological Data
287(10)
Climatic Checking
288(2)
Horizontal and Temporal Verifications
290(1)
Vertical Checking
291(4)
Complex (Multi-component) Quality Control
295(2)
4D Data Assimilation
297(32)
Lyapunov Vectors
300(4)
The Continuous 4D Data Assimilation
304(3)
The Linear Transport Equation
307(2)
Finite-Difference Approximation with Respect to Time
309(3)
Ellipticity Generically is Violated
312(1)
Hypoellipticity for Generic Wind Fields
313(3)
The Problem for Three-Time-Level Schemes
316(1)
The Problem for Other Equations
317(1)
Characteristics and Optimum Interpolation
317(4)
Variational Adjustment of Potential Vorticity and Wind Fields
321(2)
Forecast Equations as Constraints
323(6)
Numerical Methods for Prognostic Systems
329(198)
Spectral, Spectral-Difference, Finite-Element and Finite-Difference Methods for Spatial Discretization
332(46)
Spectral Methods
332(6)
The Finite-Element Method
338(1)
Projection Methods for Solution of Dissipative Equations. Stability and Convergence
339(2)
Projection Methods Application in Non-linear Problems
341(2)
Finite-Difference Methods of Differential Operators Approximation
343(5)
Equations, that are Differential with Respect to Time and Finite-Difference with Respect to Space
348(7)
Finite-Difference Spatial Approximation of Conservation Laws. Divergence Schemes
355(2)
Approximation, Stability, Convergence
357(4)
Mixed Initial-Boundary Value Problem
361(8)
Finite-Difference Approximation of Non-linear Operators. Lattices. Non-linear Instability
369(9)
Vertical Discretization of the Hydrostatic System and the Barotropic Models of Atmosphere
378(33)
The Projection Method of the Vertical Discretization of the Quasi-Static System (2.8), (2.9), (2.13)
378(8)
Difference Approximation of the Hydrostatic System along the Vertical Coordinate
386(7)
The First Integrals and Exact, Stable According to Lyapunov, Solutions for Barotropic Models of Atmosphere
393(11)
Boundary Conditions along the Horizontal Variables
404(7)
Linear Schemes of Integration of Evolutional Equations
411(70)
Linear Schemes
412(1)
Explicit and Implicit Schemes; Examples
413(3)
Multi-Step Schemes
416(5)
The Order of Approximation with Respect to Time and the Classification of the Schemes of Maximal Order of Approximation with Respect to Time
421(1)
Computational Waves in Multi-Level Schemes and their Suppression
422(7)
Stability and Convergence
429(20)
The Schemes Using Staggered (Chess-board) Grids
449(3)
Schemes with Averaging and Phase Error Minimization
452(6)
Multi-Splitting Schemes
458(5)
Mixed Boundary-Value Problem for Finite-Difference Equations and Systems
463(7)
Boundary Conditions of the Total Absorption of the Waves Coming out of the Prognostic Area (the Sommerfeld Condition of Radiation, the Conditions Simulating Cauchy Problem, or the Open Boundary Conditions)
470(8)
Spectral Models
478(3)
Non-linear Schemes for Integration of Conservation Laws Systems
481(46)
Schemes for a Scalar Conservation Law
482(3)
Monotone Schemes
485(6)
Consistence with Entropy Condition
491(1)
Godunov Scheme
492(2)
Godunov-Type Schemes
494(3)
Total Variation Non Increasing Schemes (or Total Variation Diminishing Schemes) and Monotonicity Preserving Schemes
497(10)
Numerical Experiments with a Generalized Solution of Eq. (1.40)
507(6)
Generalization to the Multi-dimensional Case
513(2)
Systems of Conservation Laws
515(7)
Scheme Conserving the Energy and Diminishing the Entrophy
522(5)
Appendix 1. Analysis on Metric Spaces and Manifolds 527(20)
Appendix 2. Calculus of Variations 547(14)
Appendix 3. Distributions. Integral Transforms. Pseudo-differential Operators 561(28)
Appendix 4. Hamiltonian Structures. Finite and Infinite Dimensional Cases 589(28)
Appendix 5. Stability and Perturbation Theory for Differential Equations 617(30)
Appendix 6. Eigen-functions of Linear Differential Equations 647(18)
Appendix 7. Polynomial and Spline Interpolation, Rational Chebyshev Approximation, and Quadrature Formulae 665(22)
Appendix 8. Probability, Random Fields, and the Optimal Operator Synthesis in a Statistical Sense 687(22)
Appendix 9. Pade Approximation and its Generalizations 709(10)
Appendix 10. Mean Values of Meteorological Fields 719(8)
Appendix 11. Models and Results in Leading Meteorological Centers 727(8)
Notations 735(4)
Bibliography 739(72)
Subject Index 811

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