This book offers an overview of advanced techniques to study atmospheric dynamics by numerical experimentation. It is primarily intended for scientists and graduate students working on interdisciplinary research problems at the intersection of the atmospheric sciences, applied mathematics, statistics and physics. Scientists interested in adopting techniques from the atmospheric sciences to study other complex systems may also find most of the topics covered in the book interesting. The specific techniques covered in the book have either proven or potential value in solving practical problems of atmospheric dynamics.
Preface vii
1 Governing Equations 1 (188)
1.1 Introduction 1 (1)
1.2 Primitive Equations 2 (44)
1.2.1 The Equations 3 (11)
1.2.2 Eulerian Form of the Equations 14 (4)
1.2.3 Scale Analysis of the Momentum 18 (10)
Equation
1.2.4 Diabatic Heating 28 (7)
1.2.5 Atmospheric Constituents 35 (4)
1.2.6 Boundary and Initial Conditions 39 (7)
1.3 Representation of the Location with 46 (27)
Coordinates
1.3.1 Spherical Coordinates 46 (14)
1.3.2 Map Projections 60 (10)
1.3.3 Cartesian Coordinates 70 (3)
1.4 Alternate Vertical Coordinates 73 (44)
1.4.1 General Formulation 74 (6)
1.4.2 Pressure Vertical Coordinate 80 (19)
1.4.3 Sigma Vertical Coordinate 99 (2)
1.4.4 Isentropic Vertical Coordinate 101(9)
1.4.5 Hybrid Vertical Coordinates 110(2)
1.4.6 Pseudo-Height and Log-Pressure 112(5)
Vertical Coordinates
1.5 Vorticity and Divergence Equations 117(34)
1.5.1 Vorticity, Absolute Vorticity and 118(1)
Divergence
1.5.2 Vorticity Equations 119(7)
1.5.3 The Vorticity and the Divergence as 126(1)
Prognostic State Variables
1.5.4 The Vorticity and the Divergence 127(8)
Equation in Pressure Coordinate System
1.5.5 Reduced Forms of the Vorticity and 135(16)
the Divergence Equations
1.6 Potential Vorticity (PV) 151(26)
1.6.1 General Case 151(4)
1.6.2 Hydrostatic Case 155(9)
1.6.3 Computation of the Potential Vorticity 164(3)
1.6.4 Vertical Structure of the Potential 167(7)
Vorticity Field
1.6.5 Potential Vorticity Inversion and 174(3)
"PV-thinking"
1.7 Integral Invariants 177(12)
1.7.1 Hamiltonian Form of the Governing 177(6)
Equations
1.7.2 Energy, Momentum, and Angular Momentum 183(1)
1.7.3 Integrals of the Potential Vorticity 183(1)
1.7.4 Integral Invariants of the Simplified 184(5)
Equations
2 Perturbation Dynamics 189(98)
2.1 Introduction 189(2)
2.2 Zonal-Mean Structure of the Atmosphere 191(8)
2.2.1 Zonal-Mean Temperature Field 192(1)
2.2.2 Zonal-Mean Potential Temperature Field 193(2)
2.2.3 Zonal-Mean Wind Field 195(3)
2.2.4 Available Potential Energy 198(1)
2.3 Quasi-Geostrophic Baroclinic Equations 199(14)
2.3.1 General Assumptions 200(4)
2.3.2 Quasi-Geostrophic Potential Vorticity 204(2)
2.3.3 Quasi-Geostrophic w-Equation 206(1)
2.3.4 Quasi-Geostrophic Baroclinic Model 206(7)
Equations
2.4 Atmospheric Waves 213(74)
2.4.1 General Formulation 214(18)
2.4.2 Large and Synoptic Scale Waves: 232(24)
Rossby Waves and Unstable Baroclinic Waves
2.4.3 Techniques to Detect Synoptic-Scale 256(7)
Wave Packets
2.4.4 Eddy Kinetic Energy Equation 263(13)
2.4.5 Shallow-Water Waves with Constant 276(3)
Amplitude
2.4.6 Convectively Coupled Equatorial 279(8)
Waves: Shallow-Water Waves with Latitude
Dependent Amplitude
3 Numerical Models 287(118)
3.1 Introduction 287(2)
3.2 Dynamical Cores 289(1)
3.3 Spatial Discretization 290(33)
3.3.1 Nonlinear Interactions in the 291(10)
Horizontally Discretized Equations
3.3.2 Three- and Two-Dimensional Turbulence 301(7)
3.3.3 Spectral Transform Method 308(10)
3.3.4 Finite-Difference Methods 318(4)
3.3.5 Finite-Volume and Spectral-Element 322(1)
Schemes
3.4 Temporal Discretization 323(8)
3.4.1 Explicit and Implicit Schemes 323(3)
3.4.2 Semi-Implicit Schemes 326(1)
3.4.3 Semi-Lagrangian Schemes 327(4)
3.5 Parameterization Schemes 331(4)
3.5.1 Radiative Processes 332(1)
3.5.2 Boundary Layer Turbulence and 333(1)
Ocean-Land-Atmosphere Interactions
3.5.3 Convective Processes 333(1)
3.5.4 Microphysics 334(1)
3.5.5 Orographic Drag 334(1)
3.6 State-of-the-Art Numerical Models 335(3)
3.6.1 Global Models 335(1)
3.6.2 Limited Area Models 336(2)
3.7 Simplified and Idealized Numerical Models 338(10)
3.7.1 Simplified Models 339(4)
3.7.2 Idealized Models 343(5)
3.8 Measures of Forecast Error 348(14)
3.8.1 Root-Mean-Square Error 350(7)
3.8.2 Anomaly Correlation 357(5)
3.9 Models as Dynamical Systems 362(43)
3.9.1 Finite-Dimensional State Vector 362(1)
3.9.2 Nonlinear Models 363(5)
3.9.3 Linearized Models 368(10)
3.9.4 Lyapunov Exponents and Vectors 378(12)
3.9.5 Transient Perturbation Growth 390(4)
3.9.6 Forecast Ensembles 394(11)
4 Data Assimilation 405(132)
4.1 Introduction 405(1)
4.2 General Formulation for Normally 405(40)
Distributed Observation Errors
4.2.1 The Cost Function 405(4)
4.2.2 Behavior of the Cost Function 409(5)
4.2.3 Sequential Formulation for the Linear 414(7)
Case: Kalman Filter
4.2.4 Computation of the Kalman Gain Matrix 421(3)
4.2.5 Sequential Formulation for the 424(3)
Nonlinear Case: Extended Kalman Filter
4.2.6 Serial Processing of the Observations 427(3)
4.2.7 Sensitivity to Nonlinearities: 430(10)
Simulated Observa- tions Experiments
4.2.8 Robust Statistics 440(2)
4.2.9 The Sequential Cost Function and 442(3)
Bayes' Rule
4.3 3-Dimensional Schemes 445(16)
4.3.1 General Formulation 445(2)
4.3.2 Optimal Interpolation 447(3)
4.3.3 3-Dimensional Variational Schemes 450(8)
4.3.4 Proxies for the Background Error 458(2)
4.3.5 Balance Constraints 460(1)
4.4 4-Dimensional Algorithms 461(23)
4.4.1 4-Dimensional Variational Schemes 462(2)
4.4.2 Ensemble-based Kalman Filtering (EnKF) 464(19)
4.4.3 Hybrid Schemes 483(1)
4.5 Accounting for Model Errors and 484(18)
Observation Bias
4.5.1 Model Errors 485(1)
4.5.2 Modifying the Observation Function 486(2)
4.5.3 Modifying the Model Dynamics 488(1)
4.5.4 Modifying the Observation Error 489(1)
Statistics
4.5.5 Sequential Schemes 490(11)
4.5.6 Weak Constraint 4D-Var 501(1)
4.6 Assimilating Satellite-based Observations 502(16)
4.6.1 Radiative Transfer in the Infrared 504(3)
and Microwave Ranges
4.6.2 Assimilating Radiance Observations 507(4)
4.6.3 Assimilating Retrievals 511(7)
4.7 Frequently Assimilated Observation Types 518(1)
4.7.1 In Situ Observations 518(13)
4.7.2 Satellite-based Observations 520(7)
4.7.3 Diagnosing and Predicting the 527(4)
Forecast Effect of Observations
4.8 Reanalysis Data Sets 531(6)
4.8.1 First Generation Data Sets 532(2)
4.8.2 Second Generation Data Sets 534(1)
4.8.3 Third Generation Data Sets 535(2)
Bibliography 537(20)
Index 557
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