Atmospheric Dynamics

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
1.1.2 Lagrangian vs. Eulerian descriptions of a fluid......Page 17
1.2 Laws of mechanics......Page 19
1.2.1 Inertial vs. non-inertial reference frames......Page 20
1.3 Equations of motion in a rotating reference frame......Page 21
1.3.1 General derivation......Page 22
1.3.2 Physical nature of the Coriolis force......Page 24
1.3.3 Equation of motion in spherical coordinates......Page 28
1.4.1 Gravitational force and gravity......Page 30
1.4.2 Pressure gradient force......Page 31
1.4.3 Molecular viscous force......Page 32
1.5 Conservation of mass......Page 33
1.6.1 Equation of state......Page 35
1.6.2 First Law of Thermodynamics......Page 36
1.6.3 Second Law of Thermodynamics......Page 38
1.7 Stratification and baroclinicity......Page 39
1.8 Summary of the equations for a dry atmospheric model......Page 41
2.1 Sphericity of the Earth and thin-atmosphere approximation......Page 43
2.2 Hydrostatic balance, implications and applications......Page 44
(iii) Surface pressure tendency equation......Page 45
(iv) Thickness of an atmospheric layer in hydrostatic balance is proportional to its average temperature.......Page 46
(v) Sea level pressure differential over continent versus over oceans......Page 47
2.2.2 Isobaric coordinates and governing equations......Page 49
2.2.4 Isentropic coordinates and governing equations......Page 52
2.3 Geostrophic balance......Page 54
2.4 Thermal wind relation......Page 56
2.5 Balanced flows......Page 58
2.5.1 Natural coordinates......Page 59
(ii) Cyclostrophic wind......Page 60
(iii) Inertial oscillation......Page 61
(i) Cyclonic flow around a low-pressure center in the northern hemisphere (NH)......Page 62
(ii) Anticyclonic flow around a low-pressure center in NH......Page 63
(iv) An anticyclonic flow around a high-pressure center in NH......Page 64
2.6 Kinematic properties of wind......Page 65
2.6.1 Structural properties of a relevant idealized flow......Page 68
2.7 Divergent wind and vertical motion......Page 69
2.8 Summary: z-, p- and theta-coordinates and equations of balance......Page 70
Vorticity......Page 71
Circulation......Page 72
3.2 Relationship between vorticity and circulation......Page 73
3.3 Kelvin circulation theorem......Page 76
3.4 Dynamics of sea-breeze from the circulation perspective......Page 77
3.5 Tendency of relative circulation......Page 79
3.6 General vorticity equation......Page 80
3.7 Vorticity dynamics of a large-scale flow......Page 81
(iii) Tilting of the horizontal component of vorticity vector,.........Page 83
(iv) Baroclinic effect......Page 84
(v) Frictional effect......Page 85
3.8.1 Preliminary remarks......Page 86
3.8.2 Physical nature of PV of a compressible fluid......Page 87
3.8.3 Observed characteristics of potential vorticity in the atmosphere......Page 88
3.8.4 Physical nature of PV in a shallow-water model......Page 90
3.8.5 An intriguing deduction: eddy-driven jet......Page 91
3.8.6 General potential vorticity equation......Page 93
3.8.7 Dynamics of a hurricane from the PV perspective......Page 94
3.9 Impermeability theorem and generalized potential vorticity......Page 95
3.9.1 PV substance and impermeability theorem......Page 96
3.9.2 Influence of boundaries and generalized potential vorticity......Page 98
4.1 Scale and estimate of the frictional force......Page 104
4.2 Concept of boundary layer......Page 105
4.3 Reynolds averaging......Page 106
4.4 Boussinesq approximation......Page 107
4.5 Flux-gradient theory of turbulence......Page 108
4.6 Types of boundary layer......Page 109
4.7 Atmospheric Ekman layer......Page 110
4.7.1 Hodograph of an Ekman layer......Page 112
4.7.2 Energetics of an Ekman layer......Page 113
4.7.3 Ekman suction/pumping......Page 114
How realistic is the Ekman solution?......Page 116
Influence of stable stratification on the spin-down process......Page 118
OEL model formulation......Page 119
4.8.1 Mass transport in oceans......Page 120
4.8.2 Oceanic Ekman pumping/suction......Page 122
4.9 Surface layer......Page 123
4.10 Mixed layer......Page 124
5.1 Preliminary remarks......Page 127
5.3 Generic model of IGW......Page 129
5.3.1 Method of linearization......Page 131
Dispersion relation of IGW......Page 132
Phase velocity of IGW......Page 133
Group velocity of IGW......Page 134
5.4.1 Structure of prototype IGW......Page 135
5.4.2 Energy flux and group velocity of prototype IGW......Page 136
5.4.3 Graphical depiction of all properties of prototype IGW......Page 137
5.4.4.1 Sinusoidal orography......Page 139
Case (1)......Page 140
Case (2)......Page 141
5.4.4.2 Localized orography......Page 143
5.4.5 Inertio-internal gravity wave......Page 144
5.5 Rudimentary characteristics of wave motions in large-scale flows......Page 145
5.6 Physical nature of Rossby waves in a simplest possible model......Page 147
5.7 Properties of prototype Rossby waves......Page 148
5.7.1 Phase velocity of prototype Rossby waves......Page 149
5.7.2 Momentum flux, energy flux and group velocity of prototype Rossby waves......Page 151
5.7.3 Graphical depiction of all properties of a prototype Rossby wave......Page 153
5.7.4 Dispersion of Rossby wave-packet......Page 154
5.7.5 Rossby wave propagation through a background uniform zonal flow......Page 156
5.8 Forced orographic Rossby waves in a shallow-water model......Page 157
5.8.1 Forced disturbance excited by a westerly flow......Page 160
5.8.2 Forced disturbance excited by an easterly flow......Page 161
5.9 Some observed statistical properties of Rossby waves......Page 163
5.10 Edge waves......Page 165
6.1 Observed features of a synoptic disturbance......Page 169
6.2 Scale analysis......Page 171
6.3.1 Scale analysis of the vorticity equation......Page 173
6.3.2 Scale analysis of the thermodynamic equation......Page 174
6.3.3 QG system of equations......Page 175
6.4.1 Omega equation (secondary circulation)......Page 176
6.4.2 Qualitative deductions on the basis of the omega equation......Page 177
6.4.3 Q-vector......Page 180
6.5.1 QG-streamfunction tendency equation for.........Page 181
6.5.3 Potential vorticity perspective of the streamfunction tendency equation......Page 182
6.6.1 Wave modes in the absence of a basic flow......Page 184
6.6.2 Wave modes in the presence of a basic zonal flow......Page 185
6.7 Evolution of a baroclinic jet streak in a quasi-geostrophic two-layer model......Page 188
Structural properties of the initial flow......Page 191
Evolution of the jet streak......Page 192
Initial advective properties and their implications......Page 195
6.8 Influences of the Earth´s sphericity in the QG theory......Page 198
6.8.1 Fundamental modes of quasi-geostrophic motion on a sphere......Page 200
7.1 Problem of rotational adjustment......Page 203
7.2 Rossby problem of geostrophic adjustment......Page 204
7.2.1 An exact analysis......Page 205
7.2.2 Illustrative sample calculations......Page 208
7.2.3 Essence of geostrophic adjustment......Page 209
7.2.4 Energetics of geostrophic adjustment......Page 211
7.3.1 An exact analysis......Page 212
7.3.2 Illustrative sample calculations......Page 214
7.4.1 Model formulation......Page 217
7.4.2 Analysis......Page 218
Case (1)......Page 219
Case (2)......Page 221
Concluding remarks......Page 223
Part 8A: Small-scale and meso-scale instability......Page 225
8A.1 Static instability and the impact of damping......Page 226
8A.2 Inertial instability and an application......Page 229
8A.2.1 Lagrangian analysis......Page 230
8A.2.2 Eulerian analysis......Page 232
Illustrative analysis......Page 233
8A.3.1 Lagrangian analysis......Page 236
8A.3.2 Eulerian analysis......Page 237
8A.3.3 Symmetric instability of a meso-scale jet......Page 238
8B.1 Historical highlights of past studies......Page 241
8B.2 Aspects of barotropic instability......Page 243
8B.2.2 Modal instability properties and structure......Page 244
8B.2.3 Instability from the perspective of energetics......Page 248
8B.2.4 Instability from the perspective of positive feedback from constituent components (wave resonance mechanism)......Page 249
8B.3 Optimal growth of barotropic disturbance......Page 251
Illustrative calculations......Page 253
8B.4 Baroclinic instability in a two-layer QG model......Page 256
8B.4.1 Necessary condition for baroclinic instability......Page 257
8B.4.2 Physical nature of baroclinic instability mechanism......Page 258
8B.5.1 Modal baroclinic instability analysis......Page 261
8B.5.2 Instability properties......Page 262
How relevant are the instability results to cyclogenesis in the atmosphere?......Page 264
8B.5.3 Structure of the unstable baroclinic wave......Page 265
8B.5.4 Baroclinic instability from the perspective of energetics......Page 267
8B.5.5 Potential vorticity (PV) transport by an unstable baroclinic wave......Page 269
8B.5.6 Baroclinic instability from the perspective of wave resonance mechanism......Page 270
8B.6 Transient growth......Page 272
8B.7 Optimal growth......Page 275
Illustrative calculation......Page 277
8B.8 Wave-activity density and general necessary condition for instability......Page 278
Concluding remarks......Page 280
8C.1 Nature and scope of the problem......Page 281
8C.2 Barotropic-governor effect......Page 283
8C.3 Instability of baroclinic jets......Page 286
Structure of the most unstable mode......Page 287
8C.3.3 Instability properties of a narrow baroclinic jet......Page 289
Structure of an unstable normal mode......Page 290
8C.4 Instability of a localized barotropic jet......Page 292
8C.4.2 Local energetics analysis......Page 294
8C.5 Instability of a localized baroclinic jet......Page 298
8C.5.1 Construction of the basic state......Page 299
8C.5.2 Instability properties......Page 301
Reduction of growth rate......Page 302
Structure of unstable disturbance......Page 303
8C.5.3 Local energetics analysis......Page 304
8D.1 Introductory remarks......Page 306
8D.2.1 Model formulation......Page 308
8D.2.3 General solution......Page 309
Analytic solution......Page 311
CISK-threshold and implication......Page 313
Solution for the usual case with a basic baroclinic shear......Page 314
8D.2.5 Instability properties......Page 315
Structure of unstable modes......Page 319
8D.2.6 Energetics......Page 322
Concluding remarks......Page 324
9.1 Observed characteristics of stationary planetary waves......Page 325
9.2 Introductory remarks about the dynamics of stationary waves......Page 327
9.3.1 Horizontal propagation of stationary waves......Page 329
Illustrative computation......Page 331
9.3.2 Vertical propagation of stationary wave......Page 333
9.4.1 Model formulation......Page 336
Method of analysis......Page 337
9.4.2 Simplest case......Page 338
9.4.3 Influence of the beta-effect alone......Page 340
9.4.4 Effect of a constant basic westerly flow alone......Page 341
9.4.5 Influence of baroclinic shear in a basic flow with the beta-effect......Page 343
9.4.6 Influence of barotropic shear in a basic flow with the beta-effect......Page 345
9.4.7 Dynamical nature of a general thermally forced steady disturbance......Page 346
9.5.1 Model formulation......Page 349
9.5.2 Effect of a uniform westerly basic flow alone......Page 350
9.5.3 Effect of a uniform basic flow with the beta-effect......Page 353
9.5.4 Influence of baroclinic shear in a basic westerly flow with the beta-effect......Page 354
9.5.5 Influence of barotropic shear in a basic westerly flow with the beta-effect......Page 355
9.6 Illustrative application: mean Asian monsoonal circulation......Page 357
9.6.1 Model formulation......Page 358
9.6.3 Structure of the prototype mean ASM......Page 360
Concluding remarks......Page 364
10.1 Eulerian mean meridional overturning circulation......Page 366
10.2 Lagrangian mean meridional overturning circulation......Page 371
10.3.1 Residual circulation, Eliassen-Palm vector and transformed Eulerian mean equations......Page 372
10.3.2 Climatological distributions of eddy, diabatic and frictional forcing......Page 377
10.3.3 Structure of the annual mean residual circulation......Page 381
Winter characteristics......Page 385
Summer characteristics......Page 395
10.4 Non-Acceleration Theorem......Page 397
10.5 Stratospheric sudden warming......Page 399
10.5.1 Model formulation......Page 400
10.5.2 Matsuno´s model results......Page 402
11.1 Introductory remarks......Page 406
11.2 Rudiments of geostrophic turbulence in a two-layer model......Page 409
11.2.1 Conservation constraints......Page 410
11.2.2 Characteristics of barotropic triad interaction......Page 412
11.2.3 Characteristics of baroclinic triad interaction......Page 413
11.2.4 Influence of forcing and damping on turbulent cascades......Page 414
11.2.6 Spectra of energy and potential enstrophy......Page 416
11.3 Life cycle of baroclinic waves......Page 417
11.3.1 Influence of zonal barotropic cyclonic shear: LC2......Page 422
11.3.2 Influence of zonal barotropic anticyclonic shear: LC1......Page 423
11.4.1 Model formulation......Page 427
11.4.2 Model results......Page 428
11.5 Relative intensity of the winter storm tracks......Page 434
11.5.1 Model formulation......Page 436
(i) Model storm tracks......Page 438
(ii) Synoptic eddy fluxes of heat and potential vorticity......Page 439
(iii) Time mean PV and velocity fields of the equilibrated state......Page 441
12.1 Surface frontogenesis......Page 443
12.1.1 Two-dimensional semi-geostrophic model analysis of frontogenesis......Page 445
12.1.2 Illustrative calculation......Page 449
12.2 Hadley circulation......Page 452
12.2.1 Dynamics of the annual mean Hadley circulation......Page 453
Distribution of the model zonal velocity......Page 458
Distribution of the model vertical velocity......Page 459
Illustrative calculation......Page 460
12.2.2 Dynamics of the seasonal mean Hadley circulation......Page 462
12.3.1 Hamiltonian formulation of a model for non-supercell tornadogenesis......Page 467
12.3.2.1 Intrinsic properties......Page 471
12.3.2.2 Non-hydrostatic barotropic instability......Page 473
12.3.2.3 Impact of the non-hydrostatic process on the instability......Page 475
12.3.2.4 Energetics and vorticity diagnoses......Page 479
Preamble......Page 482
A.1 Vector analysis......Page 483
Cross product of two vectors......Page 485
A.3 Derivative and finite-difference......Page 486
A.5 Del operator: gradient, Laplacian, divergence, curl......Page 487
A.7 Stokes theorem and Gauss theorem......Page 488
Case (A.1)......Page 489
Case (A.2)......Page 490
A.9 Matrix, eigenvalue, eigenvector, normal modes and non-normal modes......Page 491
Representation of a vector in terms of eigenvectors......Page 492
A.10 Fourier transform and spectral representation of a field......Page 493
A.11 Poisson problem......Page 494
A.12 Method of Greens function......Page 495
A.14 Calculus of variations......Page 496
A.15 Triad interactions......Page 498
References......Page 500
Index......Page 505