Atmospheric Modeling, Data Assimilation and Predictability

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Foreword......Page 13
Acknowledgements......Page 17
Abbreviations......Page 19
Variables......Page 23
1.1 Introduction......Page 25
1.2 Early developments......Page 28
1.3 Primitive equations, global and regional models, and nonhydrostatic models......Page 34
1.4 Data assimilation: determination of the initial conditions for the computer forecasts......Page 36
1.5 Operational NWP and the evolution of forecast skill......Page 41
1.6 Nonhydrostatic mesoscale models......Page 48
1.7 Weather predictability, ensemble forecasting, and seasonal to interannual prediction......Page 49
1.8 The future......Page 54
2.1 Governing equations......Page 56
2.2 Atmospheric equations of motion on spherical coordinates......Page 60
2.3 Basic wave oscillations in the atmosphere......Page 61
2.3.1.1 Pure sound waves......Page 63
2.3.1.2 Lamb waves (horizontally propagating sound waves)......Page 64
2.3.1.4 Inertia oscillations......Page 65
2.3.1.5 Lamb waves in the presence of rotation and geostrophic modes......Page 66
2.3.2 General wave solution of the perturbation equations in a resting, isothermal atmosphere......Page 67
2.3.2.1 External waves......Page 69
2.3.3 Analysis of the FDR of wave solutions in a resting, isothermal atmosphere......Page 70
2.4.1 Quasi-geostrophic approximation......Page 71
2.4.2 Quasi-Boussinesq or anelastic approximation (Ogura and Phillips, 1962)......Page 73
2.4.3 Hydrostatic approximation......Page 74
2.5 Shallow water equations, quasi-geostrophic filtering, and filtering of inertia-gravity waves......Page 77
2.5.1 Quasi-geostrophic scaling for the SWE......Page 80
2.5.2 Inertia-gravity waves in the presence of a basic flow......Page 82
2.6.1 General vertical coordinates......Page 84
2.6.2 Pressure coordinates......Page 87
2.6.3 Sigma and eta coordinates......Page 88
2.6.4 Isentropic coordinates......Page 90
3.1.1 Reminder about PDEs......Page 92
3.1.2 Well-posedness, initial and boundary conditions......Page 93
3.2.1 Finite difference method......Page 96
3.2.3 Convergence and stability criteria......Page 97
3.2.4 Implicit time schemes......Page 109
3.2.5 Semi-implicit schemes......Page 112
3.3.1 Space truncation errors. Computational phase speed. Second and fourth order schemes.......Page 115
3.3.2 Galerkin and spectral space representation......Page 118
3.3.3 Semi-Lagrangian schemes......Page 122
3.3.4 Nonlinear computational instability. Quadratically conservative schemes. The Arakawa Jacobian......Page 124
(a) Filtering out high wavenumbers.......Page 126
(b) Using quadratically conserving schemes......Page 127
3.3.5 Staggered grids......Page 132
3.3.6 Finite volume methods......Page 136
3.4.1 Introduction......Page 138
3.4.2 Direct methods for linear systems......Page 140
3.4.3 Iterative methods for solving elliptic equations......Page 141
Alternating Direction Implicit (ADI)......Page 142
Krylov subspace methods......Page 143
3.5.1 Introduction......Page 144
3.5.2 Lateral boundary conditions for one-way nested models......Page 145
(a) Pseudo-radiation boundary conditions......Page 146
(d) Flow relaxation scheme (Davies 1976, 1983)......Page 147
3.5.4 Two-way interactive boundary conditions......Page 149
4.1 Introduction......Page 151
4.2 Subgrid-scale processes and Reynolds averaging......Page 153
4.3 Overview of model parameterizations......Page 156
5.1 Introduction......Page 160
5.2.1 Successive corrections method (SCM)......Page 164
5.2.2 Nudging......Page 165
5.3.1 Least squares method......Page 166
5.3.2 Variational (cost function) approach......Page 168
5.3.3 Simplest sequential assimilation and Kalman filtering for a scalar......Page 169
5.4 Multivariate statistical data assimilation methods......Page 173
5.4.1 Optimal interpolation (OI)......Page 174
5.4.2 Approximations made in the practical implementation of OI......Page 181
5.4.3 Bratseth’s iterative method for OI......Page 189
5.4.4 One-dimensional example of OI and comparison with Bratseth’s scheme......Page 190
5.5 3D-Var, the physical space analysis scheme (PSAS), and their relationship to OI......Page 192
5.5.2 Physical space analysis system (PSAS)......Page 195
5.5.3 Final comments on the relative advantages of 3D-Var, PSAS and OI......Page 196
5.6 Advanced data assimilation methods with evolving forecast error covariance......Page 199
5.6.1 Extended Kalman filtering......Page 201
5.6.2 Ensemble Kalman filtering......Page 204
5.6.3 4D-Var......Page 205
5.6.4 Method of representers......Page 208
5.7 Dynamical and physical balance in the initial conditions......Page 209
5.7.1 Geostrophic adjustment and the relative importance of different observations......Page 210
(b) Data coverage, both in the horizontal and in the vertical......Page 212
(c) Physical model adjustments and model spin-up......Page 213
5.7.2 Normal modes initialization......Page 214
5.7.3 Dynamic initialization using digital filters......Page 218
5.8 Quality control of observations......Page 222
6.1 Introduction to atmospheric predictability......Page 229
6.2 Brief review of fundamental concepts about chaotic systems......Page 232
6.3 Tangent linear model, adjoint model, singular vectors, and Lyapunov vectors......Page 236
6.3.1 Tangent linear model and adjoint model......Page 237
6.3.2 Singular vectors......Page 239
6.3.3 Lyapunov vectors......Page 244
6.3.4 Simple examples of singular vectors and eigenvectors......Page 247
6.4 Ensemble forecasting: early studies......Page 251
6.4.1 Stochastic-dynamic forecasting......Page 252
6.4.2 Monte Carlo forecasting......Page 253
6.4.3 Lagged average forecasting......Page 255
6.5 Operational ensemble forecasting methods......Page 258
6.5.1 Breeding......Page 261
6.5.2 Singular vectors......Page 268
6.5.3 Ensembles based on multiple data assimilation......Page 271
6.5.4 Multisystem ensemble approach......Page 272
6.6 Growth rate of errors and the limit of predictability in mid-latitudes and in the tropics......Page 273
6.7 The role of the oceans and land in monthly, seasonal, and interannual predictability......Page 278
6.8 Decadal variability and climate change......Page 282
History of NWP......Page 285
Why Sweden?......Page 286
More provocative ideas......Page 287
Appendix B Coding and checking the tangent linear and the adjoint models......Page 288
B.2 Example of FORTRAN code......Page 292
C.1 Model Output Statistics (MOS)......Page 300
C.2 Perfect Prog......Page 303
C.3 Adaptive regression based on a simple Kalman filter approach......Page 304
References......Page 307
Index......Page 352