Waves and Mean Flows

Half-title......Page 3
Series-title......Page 4
Title......Page 5
Copyright......Page 6
Contents......Page 9
The aim of this book......Page 17
Contents outline......Page 19
Acknowledgements......Page 21
Part I Fluid Dynamics and Waves......Page 23
1.1.1 Mass, momentum and velocity......Page 25
1.1.2 Material trajectories and derivatives......Page 27
1.1.4 Evolution of material elements......Page 28
1.2.1 Euler's equation......Page 30
1.2.2 Constitutive relations......Page 31
1.2.3 The polytropic fluid model......Page 32
1.3 Conservation laws and energy......Page 33
1.4.1 Circulation theorem......Page 34
1.4.2 Vorticity and potential vorticity......Page 36
1.5 Rotating frames of reference......Page 38
1.6 Shallow-water system......Page 40
1.6.1 Available potential energy......Page 41
1.7 Notes on the literature......Page 42
2 Linear waves......Page 43
2.1 Linear dynamics......Page 44
2.1.1 Particle displacements and the virial theorem......Page 45
2.1.2 Vortical and wave modes......Page 46
2.1.3 Kinematics of plane waves......Page 47
2.1.4 Shallow-water plane waves......Page 49
2.1.5 Refraction......Page 52
2.1.6 WKB theory for slowly varying wavetrains......Page 54
2.1.7 Related wave equations and adiabatic invariance......Page 58
2.2 Notes on the literature......Page 59
3 Geometric wave theory......Page 60
3.1.1 Characteristics and Fermat's theorem......Page 61
3.1.2 Ocean acoustic tomography......Page 63
3.1.3 Ray tubes......Page 64
3.2.1 Green's function representation......Page 66
3.2.2 High-wavenumber boundary-value problem......Page 67
3.2.3 Stationary phase approximation......Page 68
3.2.4 Curved wave fronts and focusing......Page 70
3.2.5 The phase shift across caustics......Page 71
3.2.6 Solution directly on the caustic......Page 72
3.2.7 Non-smooth wavemakers and diffraction......Page 73
3.3 Notes on the literature......Page 74
4.1 Facets of group velocity......Page 75
4.1.2 Boundary forcing and radiation condition......Page 76
4.1.3 Asympotic solution to initial-value problem......Page 78
4.1.4 Asymptotic wave energy dynamics......Page 82
4.1.5 The case of equal-and-opposite frequencies......Page 84
4.2.1 Rotating shallow water......Page 86
4.2.2 Two-dimensional Rossby waves......Page 89
4.3 Ray tracing for dispersive wavetrains......Page 92
4.3.1 Model example......Page 94
4.3.2 Generic ray-tracing equations......Page 96
4.3.3 Symmetries and ray invariants......Page 98
4.4 Ray tracing in moving media......Page 99
4.4.1 Doppler shifting and the intrinsic frequency......Page 100
4.4.2 Refraction by the basic flow......Page 101
4.4.3 Fermat's principle for dispersive wavetrains......Page 102
4.4.4 Wave action conservation and amplitude prediction......Page 105
4.5 Wave activity conservation laws......Page 107
4.5.1 Ensemble conservation law in discrete mechanics......Page 108
4.5.2 Ensemble conservation law for linear waves......Page 109
4.5.3 Pseudomomentum and pseudoenergy......Page 110
4.5.4 Wave action for slowly varying wavetrains......Page 112
4.5.5 Moving media and several dimensions......Page 113
4.6 Notes on the literature......Page 115
Part II Wave-Mean Interaction Theory......Page 117
5 Zonally symmetric wave-mean interaction theory......Page 119
5.1.1 Small-amplitude wave-mean interactions......Page 120
5.1.2 Simple geometry......Page 122
5.1.3 Zonal averaging......Page 123
6.1 Boussinesq system and stable stratification......Page 125
6.1.1 Momentum, energy and circulation......Page 127
6.2 Linear Boussinesq dynamics......Page 128
6.2.2 Plane internal gravity waves......Page 129
6.2.3 Spatial structure of time-periodic waves......Page 132
6.3 Zonal pseudomomentum of internal waves......Page 133
6.3.1 Lagrangian and Eulerian pseudomomentum......Page 134
6.3.2 Forcing and dissipation of pseudomomentum......Page 137
6.4 Mountain lee waves and drag force......Page 138
6.4.1 Linear lee waves in two dimensions......Page 139
6.4.2 Hydrostatic solution using Hilbert transforms......Page 143
6.4.3 Drag force and momentum flux......Page 144
6.5.1 Eulerian-mean equations......Page 147
6.5.2 Mean buoyancy and pressure response......Page 148
6.5.3 Zonal mean-flow response......Page 151
6.5.4 Mass, momentum and energy budgets......Page 152
6.6 Wave dissipation......Page 155
6.6.1 Radiative damping and secular mean-flow growth......Page 156
6.6.2 Non-acceleration and the pseudomomentum rule......Page 157
6.7 Extension to variable stratification and density......Page 159
6.7.1 Variable stratification and wave reflection......Page 160
6.7.2 Density decay and amplitude growth......Page 161
6.8 Notes on the literature......Page 163
7 Shear flows......Page 164
7.1 Linear Boussinesq dynamics with shear......Page 165
7.1.1 Wave activity measures with shear......Page 166
7.1.2 Energy changes for a sheared wavetrain......Page 167
7.1.3 Rayleigh's theorem for shear instability......Page 168
7.1.4 Ray tracing in a shear flow......Page 169
7.2 Critical layers......Page 170
7.2.1 Validity of ray tracing in critical layers......Page 171
7.2.2 Failure of steady linear theory for critical layers......Page 172
7.2.3 Causal linear theory for critical layers......Page 174
7.2.4 Singular wave absorption by dissipation......Page 176
7.2.5 Strongly nonlinear critical layers......Page 178
7.2.7 Saturation parametrization of critical layers......Page 180
7.3 Joint evolution of waves and the mean shear flow......Page 182
7.3.1 Multi-scale expansion in wave amplitude......Page 183
7.3.2 Examples of joint wave-mean dynamics......Page 186
7.3.3 The quasi-biennial oscillation......Page 190
7.4 Notes on the literature......Page 192
8.1 Rotating Boussinesq equations on an f-plane......Page 193
8.2.1 Balanced vortical mode and Rossby adjustment......Page 194
8.2.2 Internal inertia-gravity waves......Page 197
8.2.3 Rotating lee waves and mountain drag......Page 200
8.3 Mean-flow response and the vortical mode......Page 202
8.3.1 Leading-order response and the TEM equations......Page 203
8.3.2 Forcing of mean vortical mode......Page 204
8.4.1 Stratification and rotation symmetry......Page 205
8.4.2 Wave-mean interactions in the slice model......Page 206
8.5 Notes on the literature......Page 207
9.1 Quasi-geostrophic dynamics......Page 208
9.1.1 Governing equations......Page 209
9.1.2 Conservation properties......Page 211
9.1.3 Quasi-geostrophic β-plane......Page 213
9.1.4 Response to effective zonal mean force......Page 214
9.2 Small amplitude wave-mean interactions......Page 215
9.2.2 Localized forcing and dissipation......Page 216
9.3.1 The Taylor identity for quasi-geostrophic dynamics......Page 218
9.3.2 Turbulent mixing of PV......Page 220
9.3.3 PV staircases and self-sharpening jets......Page 222
9.4 Notes on the literature......Page 225
10 Lagrangian-mean theory......Page 226
10.1 Lagrangian and Eulerian averaging......Page 227
10.1.1 Stokes corrections......Page 229
10.1.2 Stokes drift, pseudomomentum and bolus velocity......Page 231
10.2.1 Lifting map and Lagrangian averaging......Page 233
10.2.2 The mean material derivative and trajectories......Page 235
10.2.3 Mean mass conservation......Page 236
10.2.5 The divergence effect......Page 239
10.2.6 Mean surface elements and conservation laws......Page 242
10.2.7 Circulation and pseudomomentum......Page 245
10.2.8 Why pseudomomentum is conserved......Page 247
10.2.9 Vorticity and potential vorticity......Page 248
10.3 Wave activity conservation in GLM theory......Page 251
10.3.1 General wave activity equation......Page 252
10.3.2 Pseudomomentum and pseudoenergy......Page 254
10.3.3 Non-barotropic flows......Page 256
10.3.4 Angular momentum and pseudomomentum......Page 257
10.4 Coriolis forces in GLM theory......Page 259
10.4.1 Rotating circulation and pseudomomentum......Page 260
10.4.2 Wave activity relations......Page 261
10.4.3 Angular momentum and pseudomomentum......Page 262
10.4.4 Gauged pseudomomentum and the β-plane......Page 263
10.5 Lagrangian-mean gas dynamics and radiation stress......Page 267
10.5.1 Radiation stress and pseudomomentum flux......Page 268
10.6 Notes on the literature......Page 270
11.1 GLM theory for the Boussinesq equations......Page 271
11.1.1 Dissipative pseudomomentum rule......Page 274
11.1.2 Pseudomomentum with vertical shear......Page 275
11.2 Rotating Boussinesq equations on an f-plane......Page 276
11.2.2 EP flux in GLM theory......Page 277
11.2.3 Rotating vertical slice model in GLM theory......Page 278
11.3 Notes on the literature......Page 279
Part III Waves and Vortices......Page 281
12 A framework for local interactions......Page 283
12.1 A geometric singular perturbation......Page 284
12.2 Examples of mean pressure effects......Page 285
12.2.1 Mean-flow response to acoustic wavetrain......Page 286
12.2.2 Mean force on a wavemaker......Page 289
12.2.3 Large-scale return flow beneath surface waves......Page 292
12.3 Vortical mean-flow response......Page 296
12.3.1 Local interactions in shallow water......Page 297
12.3.2 Bretherton flow......Page 298
12.3.3 A wavepacket life cycle......Page 300
12.3.4 Strong interactions and potential vorticity......Page 302
12.4.1 Classical impulse theory......Page 303
12.4.2 Impulse and pseudomomentum in GLM theory......Page 307
12.5 Notes on the literature......Page 310
13.1 Wave-driven longshore currents......Page 311
13.2 Classic theory based on simple geometry......Page 313
13.2.1 Wave structure......Page 314
13.2.2 Mean-flow response......Page 316
13.3 Theory for inhomogeneous wavetrains......Page 320
13.4 Vorticity generation by wave breaking and shock formation......Page 321
13.5 Vortex dynamics on sloping beaches......Page 325
13.5.1 Impulse for one-dimensional topography......Page 326
13.5.2 Self-advection of vortices......Page 327
13.5.3 Mutual interaction of vortices and rip currents......Page 330
13.5.4 A statistical argument for vortex locations......Page 331
13.6 Barred beaches and current dislocation......Page 333
13.6.1 Current dislocation by vortex dynamics......Page 335
13.6.2 Bottom fricton and turbulence......Page 336
13.7 Notes on the literature......Page 337
14 Wave refraction by vortices......Page 339
14.1 Anatomy of wave refraction......Page 340
14.1.1 Refraction by a bath-tub vortex......Page 342
14.2 Remote recoil......Page 343
14.3 Wave capture of internal gravity waves......Page 346
14.3.1 Impulse and pseudomomentum for stratified flow......Page 348
14.3.2 Wavepacket and vortex dipole example......Page 351
14.3.3 Mean-flow response at the wavepacket......Page 352
14.4 Wave-vortex duality and dissipation......Page 355
14.5 Notes on the literature......Page 356
References......Page 357
Index......Page 361