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