Fundamentals of numerical weather prediction

Cover......Page 1
Fundamentals of NumericalWeather Prediction......Page 4
©......Page 5
Contents......Page 6
Foreword to the French Edition......Page 9
Foreword to the English Edition......Page 11
 Preface......Page 12
 Acknowledgments......Page 14
Latin letters......Page 16
Gothic letters......Page 19
Greek letters......Page 20
Generalized vectors, matrices, and operators......Page 21
Various mathematical notations......Page 22
1.1 Introduction......Page 24
1.2 The early days......Page 25
1.3 Half a century of continual progress......Page 26
1.3.1 The need to be fast and accurate......Page 27
1.3.3 Back to the primitive equations and initialization......Page 29
1.3.4 Global processing and the spectral method......Page 30
1.3.6 Algorithms for an increased time step......Page 31
1.3.8 Physical processes......Page 32
1.3.9 Objective analysis and data assimilation......Page 33
1.4.1 Computing power accompanies progress......Page 34
1.4.2 From the ENIAC to scientific mainframes......Page 35
1.4.4 Massively parallel computers......Page 36
1.4.5 Software advances......Page 37
2.1 Introduction......Page 38
2.2.1 The general form of the equations......Page 39
2.2.2 The traditional approximation and the nonhydrostatic equations......Page 40
2.2.3 The hydrostatic assumption and the primitive equations......Page 41
2.2.4 The primitive equations in the pressure coordinates......Page 42
2.2.5 The shallow water model equations......Page 44
2.2.6 The zero divergence model equation......Page 46
2.3.1 Vector operators in curvilinear coordinates......Page 47
2.3.2 The equations in geographical coordinates......Page 48
2.3.3 Formulation of the equations for a conformal projection......Page 50
2.4.1 Polar stereographic projection......Page 52
2.4.2 The Mercator projection......Page 54
2.4.3 The Lambert conical projection......Page 55
2.4.4 The conformal transformation of the sphere onto itself......Page 57
3.2.1 Computational principle, order of accuracy......Page 62
3.2.2 Common notations for finite differences......Page 64
3.2.3 The accuracy of finite difference schemes......Page 66
3.3 The grids used and their properties......Page 69
3.3.2 The A-type grid......Page 70
3.3.3 The B-type grid......Page 71
3.3.4 The C-type grid......Page 72
3.3.5 The D’-type staggered grid (Eliassen grid)......Page 73
3.3.6 The properties of the various grids......Page 74
3.3.7 Spatial filtering......Page 76
3.4 Conclusion......Page 78
4.2.1 General remarks on Galerkin methods......Page 80
4.2.2 Using finite elements for the advection equation......Page 81
4.3.2 The basis of surface spherical harmonics......Page 84
4.3.3 The properties of spherical harmonics......Page 86
4.3.4 Expanding a spherical field......Page 88
4.3.5 Truncating the expansion......Page 89
4.3.6 Calculating linear terms and application to wind calculation......Page 90
4.3.7 Calculating nonlinear terms......Page 92
4.3.8 Practical implementation of the spectral method......Page 95
4.4.1 Constructing a doubly periodic domain......Page 96
4.4.2 Basis functions......Page 97
4.4.3 Elliptical truncation......Page 98
4.4.5 Calculating nonlinear terms......Page 100
4.4.6 The advantage of the method......Page 101
5.2.1 The equations for the perturbations......Page 102
5.2.2 The analytical solutions of the linearized model......Page 104
5.3.1 General principle......Page 105
5.3.2 Application to the various grids......Page 106
5.4.1 The Euler explicit scheme......Page 110
5.4.2 The centred explicit scheme......Page 113
5.4.3 The centred semi-implicit scheme......Page 115
5.4.4.1 Implementation with perfect interpolation......Page 118
5.4.4.2 Implicit treatment of the Coriolis parameter......Page 121
5.4.4.3 The effects of interpolation in the semi-Lagrangian scheme......Page 122
5.5 Time filtering......Page 125
5.6.1 The case of finite difference models......Page 126
5.6.2 The case of spectral models......Page 128
6.1.1 The zero divergence model......Page 129
6.1.2 Introducing a divergence term......Page 130
6.1.3 Nonlinear instability and how to prevent it......Page 132
6.2.1 The properties of the shallow water model......Page 137
6.2.2 Discretization of the equations on a C grid......Page 138
6.2.3 The centred explicit scheme......Page 141
6.2.4 The centred semi-implicit scheme......Page 142
6.2.5.1 The centred scheme and determination of the particle origin point......Page 144
6.2.5.2 Variants of semi-Lagrangian processing......Page 148
6.3.1 Formulation of the equations......Page 150
6.3.2 Semi-implicit processing......Page 153
6.3.3 Semi-Lagrangian processing......Page 154
6.4 Practical use of the shallow water model......Page 158
7.2.1 The transformation formulas......Page 159
7.3.1 The hydrostatic equation......Page 160
7.3.3 The continuity equation......Page 161
7.3.4 The surface pressure tendency equation......Page 162
7.4.2 The sigma coordinate......Page 163
7.4.3 The progressive hybrid coordinate......Page 164
7.5.1 The role of ‘hydrostatic pressure’......Page 169
7.5.2 The normalized ‘hydrostatic pressure’ hybrid coordinate......Page 170
7.5.3 A comprehensive synthetic formulation of the equations......Page 172
7.6.1 The expression of global parameters......Page 174
7.6.3 Conservation of angular momentum......Page 175
7.6.4 The conservation of energy......Page 176
7.7 Conclusion......Page 178
8.2.1 The equations......Page 180
8.2.2 The layers, levels, and positions of variables......Page 181
8.3.1 Vertical advection......Page 183
8.3.4 Diagnostic equation for geopotential......Page 184
8.3.5 Pressure force term......Page 185
8.3.6 Energy conversion term......Page 186
8.3.7 Location of the pressure levels......Page 188
8.3.8 Alternative solutions for vertical discretization......Page 189
8.4.1 Simplifications with the pure sigma coordinate......Page 190
8.4.2 The location of variables on the C grid......Page 191
8.4.3 The discretized equations......Page 192
8.4.4 Explicit time integration of the model......Page 195
8.4.5 Implementation of semi-implicit time integration......Page 198
8.5.1 General formulation of the algorithm......Page 202
8.5.2 Interpretation of the semi-implicit method......Page 203
8.6.1 The equations......Page 205
8.6.2 Explicit time integration of the model......Page 206
8.6.3 Implementation of semi-implicit time integration......Page 210
8.7 Lagrangian advection in baroclinic models......Page 213
9.1 Introduction......Page 215
9.2.1 Schematic framework of interaction among constituents......Page 218
9.2.2 The equations in conservative form......Page 220
9.3.1 General points......Page 222
9.3.2 Allowance for the effects of radiation in the atmosphere......Page 223
9.3.3 Two-flux approximation and integration over a layer......Page 224
9.3.4.1 The case of gases......Page 227
9.3.5.1 The case of solar fluxes......Page 229
9.3.5.2 The case of thermal fluxes......Page 230
9.3.6 Processing of clouds......Page 231
9.3.6.2 The maximum-random overlap hypothesis......Page 232
9.3.6.3 Calculation of optical depths allowing for cloudiness......Page 233
9.3.6.4 Calculating cloudiness......Page 234
9.4.2 Parameterization of turbulent surface fluxes......Page 236
9.4.3 Planetary boundary layer fluxes......Page 240
9.4.4 Allowance for shallow convection......Page 242
9.4.5.1 Evolution of surface temperature......Page 243
9.4.5.2 Evolution of soil moisture......Page 245
9.4.6 Allowance for fluxes, vertical diffusion......Page 246
9.5.2 Calculating precipitation in an atmospheric layer......Page 248
9.5.3 Evaporation of precipitation during fall......Page 249
9.5.4 The melting of snow as it falls......Page 251
9.6.1 General points......Page 254
9.6.2 The problem of causality for convection......Page 255
9.6.3 The basic equations......Page 256
9.6.4 The characteristic profiles of the cloud and the influence of microphysics......Page 258
9.6.6 The closure relation......Page 259
9.7 Effect of sub-grid orography......Page 260
9.7.1 The wave momentum flux induced by orography......Page 261
9.7.2 Effects of resonance and trapping of the wave......Page 262
9.7.3 Consequences of partial blocking of flow......Page 263
9.8 Horizontal diffusion......Page 264
9.9 Validation of physical parameterizations......Page 266
10.2.1 The Global Earth Observation System......Page 268
10.2.2 In situ observations......Page 269
10.2.3 Remote sensing observations......Page 270
10.3.1 Introduction......Page 271
10.3.2 The successive correction method......Page 272
10.3.3 The statistical approach and error modelling......Page 273
10.3.4 Statistical interpolation by the least squares approach......Page 274
10.3.5 Optimal interpolation......Page 275
10.3.6 The 3D variational method......Page 276
10.3.7 The incremental variant......Page 277
10.3.8 4D variational assimilation......Page 278
10.3.9 Kalman filtering......Page 280
10.4.1 Early attempts at initialization......Page 282
10.4.2 The principle of using normal modes......Page 283
10.5.1 Limited area models......Page 286
10.5.2 The classical treatment of lateral boundary conditions......Page 287
10.5.3 The principle of nested models......Page 288
10.6 Post-processing of model output......Page 289
10.6.2 The various interpolations......Page 290
10.7.1 The principle of statistical adaptation......Page 291
10.7.3 The model output statistics method......Page 292
10.8.1 The various tasks of the forecast suite......Page 293
10.9.1 Basic definitions......Page 294
10.9.2 Some standard verification scores......Page 295
10.9.3 The verification of categorized events......Page 298
10.10.1 The limit of predictability......Page 299
10.10.2 Using an ensemble......Page 300
10.10.3 Presentation of forecasts......Page 302
10.11 International cooperation......Page 303
10.12.2 Data assimilation......Page 304
10.12.4 Specialized application models......Page 305
10.12.6 Ensemble forecasting......Page 306
10.12.7 Conclusion......Page 307
A.2.1 Introduction......Page 308
A.2.2 The equations used......Page 309
A.2.3 Vertical discretization......Page 311
A.2.4 The choice of the linear part of the model......Page 312
A.2.6 Semi-Lagrangian semi-implicit time integration......Page 314
A.2.7 Processing horizontal diffusion......Page 315
A.2.9.3 The boundary layer and turbulent diffusion......Page 316
A.2.9.5 Microphysics......Page 317
A.2.10 Data assimilation and initialization......Page 318
A.3.1 Introduction......Page 319
A.3.2 The equations used in the WRF/ARW model......Page 320
A.3.3 The equations for the perturbations......Page 322
A.3.4.1 The third-order Runge-Kutta scheme......Page 323
A.3.4.2 The equations used for small time step integration......Page 324
A.3.4.3 The successive steps of time integration......Page 326
A.3.5.1 The grids used and the arrangement of variables......Page 327
A.3.5.2 Discretization of non-advective terms......Page 328
A.3.5.3 Discretization of advection terms......Page 329
A.3.5.5 Processing horizontal and vertical diffusion......Page 330
A.3.6.3 Off-centering of the semi-implicit scheme......Page 331
A.3.7.2 Implicit Rayleigh damping......Page 332
A.3.8 Physics......Page 333
A.3.8.5 Microphysics......Page 334
A.3.10 Data assimilation......Page 335
A.3.12 Conclusions......Page 336
Dynamic meteorology......Page 338
Physical processes and parameterizations......Page 339
References......Page 341

Index......Page 360