Atmospheric Dynamics

Atmospheric Dynamics......Page 1
Atmospheric Dynamics......Page 3
Contents......Page 7
Preface......Page 11
1.1.2 Lagrangian vs. Eulerian descriptions of a fluid......Page 15
1.2 Laws of mechanics......Page 17
1.2.1 Inertial vs. non-inertial reference frames......Page 18
1.3 Equations of motion in a rotating reference frame......Page 19
1.3.1 General derivation......Page 20
1.3.2 Physical nature of the Coriolis force......Page 22
1.3.3 Equation of motion in spherical coordinates......Page 26
1.4.1 Gravitational force and gravity......Page 28
1.4.2 Pressure gradient force......Page 29
1.4.3 Molecular viscous force......Page 30
1.5 Conservation of mass......Page 31
Outline placeholder......Page 0
1.6.1 Equation of state......Page 33
1.6.2 First Law of Thermodynamics......Page 34
1.6.3 Second Law of Thermodynamics......Page 36
1.7 Stratification and baroclinicity......Page 37
1.8 Summary of the equations for a dry atmospheric model......Page 39
2.1 Sphericity of the Earth and thin-atmosphere approximation......Page 41
2.2 Hydrostatic balance, implications and applications......Page 42
(iii) Surface pressure tendency equation......Page 43
(iv) Thickness of an atmospheric layer in hydrostatic balance is proportional to its average temperature.......Page 44
(v) Sea-level pressure differential over continent versus over oceans......Page 45
2.2.2 Isobaric coordinates and governing equations......Page 47
2.2.4 Isentropic coordinates and governing equations......Page 50
2.3 Geostrophic balance......Page 52
2.4 Thermal wind relation......Page 53
2.5.1 Natural coordinates......Page 56
2.5.2 Gradient wind balance and special flows......Page 57
(ii) Cyclostrophic wind......Page 58
2.5.3 Dynamical nature of four conceivable configurations of circular flows......Page 59
2.6 Kinematic properties of wind......Page 62
2.6.1 Structural properties of a relevant idealized flow......Page 65
2.7 Divergent wind and vertical motion......Page 66
2.8 Summary: z-, p- and theta-coordinates and equations of balance......Page 68
Vorticity......Page 69
Circulation......Page 70
3.2 Relationship between vorticity and circulation......Page 71
3.3 Kelvin circulation theorem......Page 74
3.4 Dynamics of sea-breeze from the circulation perspective......Page 75
3.5 Tendency of relative circulation......Page 77
3.6 General vorticity equation......Page 78
3.7 Vorticity dynamics of a large-scale flow......Page 79
(iii) Tilting of the horizontal component of vorticity vector, $\\left( {\\bizeta _1 w_x + \\left( {\\bizeta _2 + h} \\right)w_y } \\right)$......Page 81
(iv) Baroclinic effect......Page 82
(v) Frictional effect......Page 83
3.8.1 Preliminary remarks......Page 84
3.8.2 Physical nature of PV of a compressible fluid......Page 85
3.8.3 Observed characteristics of potential vorticity in the atmosphere......Page 86
3.8.4 Physical nature of PV in a shallow-water model......Page 88
3.8.5 An intriguing deduction: eddy-driven jet......Page 89
3.8.6 General potential vorticity equation......Page 91
3.8.7 Dynamics of hurricane from the PV perspective......Page 92
3.9 Impermeability theorem and generalized potential vorticity......Page 93
3.9.1 PV substance and impermeability theorem......Page 94
3.9.2 Influence of boundaries and generalized potential vorticity......Page 96
4.1 Scale and estimate of the frictional force......Page 102
4.2 Concept of boundary layer......Page 103
4.3 Reynolds averaging......Page 104
4.4 Boussinesq approximation......Page 105
4.5 Flux-gradient theory of turbulence......Page 106
4.6 Types of boundary layer......Page 107
4.7 Atmospheric Ekman layer......Page 108
4.7.1 Hodograph of an Ekman layer......Page 110
4.7.2 Energetics of an Ekman layer......Page 111
4.7.3 Ekman suction/pumping......Page 112
How realistic is the Ekman solution?......Page 114
Influence of stable stratification on the spin-down process......Page 116
OEL Model formulation......Page 117
4.8.1 Mass transport in oceans......Page 118
4.8.2 Oceanic Ekman pumping/suction......Page 120
4.9 Surface layer......Page 121
4.10 Mixed layer......Page 122
5.1 Preliminary remarks......Page 125
5.3 Generic model of IGW......Page 127
5.3.1 Method of linearization......Page 129
Dispersion relation of IGW......Page 130
Phase velocity of IGW......Page 131
Group velocity of IGW......Page 132
5.4.1 Structure of prototype IGW......Page 133
5.4.2 Energy flux and group velocity of prototype IGW......Page 134
5.4.3 Graphical depiction of all properties of prototype IGW......Page 135
5.4.4.1 Sinusoidal orography......Page 137
Case (1)......Page 138
Case (2)......Page 139
5.4.4.2 Localized orography......Page 141
5.4.5 Inertio-Internal gravity wave......Page 142
5.5 Rudimentary characteristics of wave motions in large-scale flows......Page 143
5.6 Physical nature of Rossby waves in a simplest possible model......Page 145
5.7 Properties of prototype Rossby wave......Page 146
5.7.1 Phase velocity of prototype Rossby waves......Page 147
5.7.2 Momentum flux, energy flux and group velocity of prototype Rossby waves......Page 149
5.7.3 Graphical depiction of all properties of a prototype Rossby wave......Page 151
5.7.4 Dispersion of Rossby wave-packet......Page 152
5.7.5 Rosssby wave propagation through a background uniform zonal flow......Page 154
5.8 Forced orographic Rossby waves in a shallow-water model......Page 155
5.8.1 Forced disturbance excited by a westerly flow......Page 158
5.8.2 Forced disturbance excited by an easterly flow......Page 159
5.9 Some observed statistical properties of Rossby waves......Page 161
5.10 Edge waves......Page 163
6.1 Observed features of a synoptic disturbance......Page 167
6.2 Scale analysis......Page 169
6.3.1 Scale analysis of the vorticity equation......Page 171
6.3.2 Scale analysis of the thermodynamic equation......Page 172
6.3.3 QG systems of equations......Page 173
6.4.1 Omega-equation (secondary circulation)......Page 174
6.4.2 Qualitative deductions on the basis of the omega equation......Page 175
6.4.3 Q-vector......Page 178
6.5.1 QG-Streamfunction tendency equation for psi/t......Page 179
6.5.3 Potential vorticity perspective of the streamfunction tendency equation......Page 180
6.6.1 Wave modes in the absence of a basic flow......Page 182
6.6.2 Wave modes in the presence of a basic zonal flow......Page 183
6.7 Evolution of a baroclinic jet streak in a quasi-geostrophic two-layer model......Page 186
Structural properties of the initial flow......Page 189
Evolution of the jet streak......Page 190
Initial advective properties and their implications......Page 193
6.8 Influences of the Earths sphericity in the QG Theory......Page 196
6.8.1 Fundamental modes of quasi-geostrophic motion on a sphere......Page 198
7.1 Problem of rotational adjustment......Page 201
7.2 Rossby problem of geostrophic adjustment......Page 202
7.2.1 An exact analysis......Page 203
7.2.2 Illustrative sample calculations......Page 206
7.2.3 Essence of geostrophic adjustment......Page 207
7.2.4 Energetics of geostrophic adjustment......Page 209
7.3.1 An exact analysis......Page 210
7.3.2 Illustrative sample calculations......Page 212
7.4.1 Model formulation......Page 215
7.4.2 Analysis......Page 216
Case (1)......Page 217
Case (2)......Page 219
Concluding remarks......Page 221
Part 8A Small-scale and meso-scale instability......Page 223
8A.1 Static instability and the impact of damping......Page 224
8A.2 Inertial instability and an application......Page 227
8A.2.1 Lagrangian analysis......Page 228
8A.2.2 Eulerian analysis......Page 230
Illustrative analysis......Page 231
8A.3.1 Lagrangian analysis......Page 234
8A.3.1 Eulerian analysis......Page 235
8A.3.2 Symmetric instability of a meso-scale jet......Page 236
8B.1 Historical highlights of past studies......Page 239
8B.2 Aspects of barotropic instability......Page 241
8B.2.2 Modal instability properties and structure......Page 242
8B.2.3 Instability from the perspective of energetics......Page 246
8B.2.4 Instability from the perspective of positive feedback from constituent components (wave resonance mechanism)......Page 247
8B.3 Optimal growth of barotropic disturbance......Page 249
Illustrative calculations......Page 251
8B.4 Baroclinic instability in a two-layer QG model......Page 254
8B.4.1 Necessary condition for baroclinic instability......Page 255
8B.4.2 Physical nature of baroclinic instability mechanism......Page 256
8B.5.1 Modal baroclinic instability analysis......Page 259
8B.5.2 Instability properties......Page 260
How relevant are the instability results to cyclogenesis in the atmosphere?......Page 262
8B.5.3 Structure of the unstable baroclinic wave......Page 263
8B.5.4 Baroclinic instability from the perspective of energetics......Page 265
8B.5.5 Potential vorticity (PV) transport by unstable baroclinic wave......Page 267
8B.5.6 Baroclinic instability from the perspective of wave resonance mechanism......Page 268
8B.6 Transient growth......Page 270
8B.7 Optimal growth......Page 273
Illustrative calculation......Page 275
8B.8 Wave-activity density and general necessary condition for instability......Page 276
Concluding remarks......Page 278
8C.1 Nature and scope of the problem......Page 279
8C.2 Barotropic-governor effect......Page 281
8C.3 Instability of baroclinic jets......Page 284
Structure of the most unstable mode......Page 285
8C.3.3 Instability properties of a narrow baroclinic jet......Page 287
Structure of an unstable normal mode......Page 288
8C.4 Instability of a localized barotropic jet......Page 290
8C.4.2 Local energetics analysis......Page 292
8C.5 Instability of a localized baroclinic jet......Page 296
8C.5.1 Construction of the basic state......Page 297
8C.5.2 Instability properties......Page 299
Reduction of growth rate......Page 300
Structure of unstable disturbance......Page 301
8C.5.3 Local energetics analysis......Page 302
8D.1 Introductory remarks......Page 304
8D.2.1 Model formulation......Page 306
8D.2.3 General solution......Page 307
Analytic solution......Page 309
CISK-threshold and implication......Page 311
Solution for the usual case with a basic baroclinic shear......Page 312
Instability properties......Page 313
Structure of unstable modes......Page 317
Energetics......Page 320
Concluding remarks......Page 322
9.1 Observed characteristics of stationary planetary waves......Page 323
9.2 Introductory remarks about the dynamics of stationary waves......Page 325
9.3.1 Horizontal propagation of stationary waves......Page 327
Illustrative computation......Page 329
9.3.2 Vertical propagation of stationary wave......Page 331
9.4.1 Model formulation......Page 334
Method of analysis......Page 335
9.4.2 Simplest case......Page 336
9.4.3 Influence of the beta-effect alone......Page 338
9.4.4 Effect of a constant basic westerly flow alone......Page 339
9.4.5 Influence of baroclinic shear in a basic flow with beta-effect......Page 341
9.4.6 Influence of barotropic shear in a basic flow with beta-effect......Page 343
9.4.7 Dynamical nature of a general thermally forced steady disturbance......Page 344
9.5.1 Model formulation......Page 347
9.5.2 Effect of a uniform westerly basic flow alone......Page 348
9.5.3 Effect of a uniform basic flow with beta-effect......Page 351
9.5.4 Influence of baroclinic shear in a basic westerly flow with beta-effect......Page 352
9.5.5 Influence of barotropic shear in a basic westerly flow with beta-effect......Page 353
9.6 Illustrative application: mean Asian monsoonal circulation......Page 355
9.6.1 Model formulation......Page 356
9.6.3 Structure of the prototype mean ASM......Page 358
Concluding remarks......Page 362
10.1 Eulerian mean meridional overturning circulation......Page 364
10.2 Lagrangian mean meridional overturning circulation......Page 369
10.3.1 Residual circulation, Eliassen-Palm vector and transformed Eulerian mean equations......Page 370
10.3.2 Climatological distributions of eddy, diabatic and frictional forcing......Page 375
10.3.3 Structure of the annual mean residual circulation......Page 379
Winter characteristics......Page 383
Summer characteristics......Page 385
10.4 Non-Acceleration Theorem......Page 387
10.5 Stratospheric sudden warming......Page 389
10.5.1 Model formulation......Page 390
10.5.2 Matsunos model results......Page 392
11.1 Introductory remarks......Page 396
11.2 Rudiments of geostrophic turbulence in a two-layer model......Page 399
11.2.1 Conservation constraints......Page 400
11.2.2 Characteristics of barotropic triad interaction......Page 402
11.2.3 Characteristics of baroclinic triad interaction......Page 403
11.2.4 Influence of forcing and damping on turbulent cascades......Page 404
11.2.6 Spectra of energy and potential enstrophy......Page 406
11.3 Life cycle of baroclinic waves......Page 407
11.3.1 Influence of zonal barotropic cyclonic shear: LC2......Page 412
11.3.2 Influence of zonal barotropic anticyclonic shear: LC1......Page 413
11.4.1 Model formulation......Page 417
11.4.2 Model results......Page 418
11.5 Relative intensity of the winter storm tracks......Page 424
11.5.1 Model formulation......Page 426
(i) Model storm tracks......Page 428
(ii) Synoptic eddy fluxes of heat and potential vorticity......Page 429
(iii) Time mean PV and velocity fields of the equilibrated state......Page 431
12.1 Surface frontogenesis......Page 433
12.1.1 Two-dimensional semi-geostrophic model analysis of frontogenesis......Page 435
12.1.2 Illustrative calculation......Page 439
12.2 Hadley circulation......Page 442
12.2.1 Dynamics of the annual mean Hadley circulation......Page 443
Distribution of the model zonal velocity......Page 448
Distribution of the model vertical velocity......Page 449
Illustrative calculation......Page 450
12.2.2 Dynamics of the seasonal mean Hadley circulation......Page 452
12.3.1 Hamiltonian formulation of a model for nonsupercell tornadogenesis......Page 457
12.3.2.1 Intrinsic properties......Page 461
12.3.2.2 Nonhydrostatic barotropic instability......Page 463
12.3.2.3 Impact of nonhydrostatic process on the instability......Page 465
12.3.2.4 Energetics and vorticity diagnoses......Page 469
Preamble......Page 472
A.1 Vector analysis......Page 473
Cross product of two vectors......Page 475
A.3 Derivative and finite-difference......Page 476
A.5 Del operator: Gradient, Laplacian, divergence, curl......Page 477
A.7 Stokes theorem and Gauss theorem......Page 478
Case (A.1)......Page 479
Case (A.2)......Page 480
A.9 Matrix, eigenvalue, eigenvector, normal modes and non-normal modes......Page 481
Representation of a vector in terms of eigenvectors......Page 482
A.10 Fourier transform and spectral representation of a field......Page 483
A.11 Poisson problem......Page 484
A.12 Method of Greens function......Page 485
A.14 Calculus of variations......Page 486
A.15 Triad interactions......Page 488
References......Page 490
Index......Page 495