Turbulence in the Atmosphere

Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Contents......Page 7
Preface......Page 11
Part I A grammar of turbulence......Page 15
1.1 Turbulence, its community, and our approach......Page 17
1.2 The origins and nature of turbulence......Page 18
1.3 Turbulence and surface fluxes......Page 19
1.4 How do we study turbulence?......Page 24
1.5 The equations of turbulence......Page 25
1.6.1 Vortex stretching and tilting: viscous dissipation......Page 28
1.6.3 The effective diffusivity......Page 29
1.6.4 Range of scales: the turbulence Reynolds number......Page 30
1.6.5 Mathematical intractability......Page 31
1.7 Numerical modeling of turbulent flows......Page 32
1.7.2 Space-averaged turbulence models......Page 33
1.9 The impact of Kolmogorov......Page 34
Questions on key concepts......Page 36
Problems......Page 37
References......Page 39
2.1 Average and instantaneous properties contrasted......Page 41
2.2 Averaging......Page 42
2.2.1 The ensemble average......Page 44
2.2.3 The ensemble-averaging rules......Page 45
2.2.4 A simple example of ensemble averaging......Page 46
2.3 Ergodicity......Page 48
2.4 The convergence of averages......Page 49
2.5.1 The spectrum of a one-dimensional, real, random, homogeneous scalar function......Page 52
2.5.2 Extension to three dimensions......Page 54
2.5.3 Application to a homogeneous velocity field......Page 55
2.5.4 The eddy velocity scale u(r)......Page 56
2.6 Turbulent vorticity......Page 57
2.7 Turbulent pressure......Page 58
2.8 Eddy diffusivity......Page 59
2.10 Coherent structures......Page 63
Questions on key concepts......Page 64
Problems......Page 65
References......Page 67
3.1 Introduction......Page 69
3.2 Ensemble-averaged equations......Page 70
3.2.1 Example: Steady turbulent flow in a channel......Page 71
3.3.1 Example: A conserved scalar diffusing from a point source......Page 73
3.3.2 Enhancement of molecular diffusion by turbulence......Page 76
3.4 Space-averaged equations......Page 79
3.4.1 The generalization to spatial filtering......Page 80
3.4.2 Spatially filtered governing equations......Page 81
3.4.3 Resolved and subfilter-scale turbulent fluxes......Page 83
3.5 Summary......Page 84
Problems......Page 85
References......Page 86
4.1 Introduction......Page 89
4.2 Temperature flux in a boundary layer......Page 90
4.3.1 Laminar flow......Page 92
4.3.2 Turbulent flow......Page 93
4.4 Momentum flux in channel flow......Page 97
4.5 The ``mixture length\'\'......Page 98
Questions on key concepts......Page 99
Problems......Page 100
References......Page 101
5.1 Introduction and background......Page 103
5.2 The fluctuation equations......Page 104
5.3.1 Derivation and interpretation......Page 105
5.3.2 Scaling guidelines......Page 108
5.3.3 Quasi-steadiness, local homogeneity......Page 109
5.3.4 Interpreting the molecular destruction term......Page 110
5.3.5 Simple limiting cases......Page 112
5.4.1 The cui budget......Page 113
5.4.2 The uiuk budget......Page 114
5.5.1 The TKE budget......Page 115
5.5.2 The mean-flow kinetic energy equation......Page 117
5.5.3.2 TKE budget......Page 118
5.5.3.3 The role of turbulent pressure......Page 119
5.6.1 Background......Page 120
5.6.2 History, status, and outlook......Page 121
Appendix Scaling the molecular-diffusivity terms in the budgets of scalar flux and Reynolds stress......Page 123
Questions on key concepts......Page 124
Problems......Page 125
References......Page 127
6.1 Introduction......Page 129
6.2.1 A one-dimensional, homogeneous example......Page 130
6.2.2 The generalization to spatial filtering......Page 131
6.2.3 The wave-cutoff filter......Page 132
6.2.4 The Gaussian filter......Page 133
6.3.1 The filtered equations......Page 134
6.3.2.2 The resolvable-scale TKE budget......Page 137
6.3.2.3 The subfilter-scale TKE budget......Page 138
6.3.3 Completing the resolvable and subfilter-scale TKE budgets......Page 139
6.3.5 The flow of scalar variance from larger scales to smaller......Page 140
6.4 Application to flows homogeneous in two dimensions......Page 141
6.5 The physical mechanisms of interscale transfer......Page 143
6.6.1 The concept......Page 144
6.6.2.1 Stress......Page 146
6.6.2.3 Scalar flux......Page 147
6.6.3 Modeling subfilter-scale fluxes......Page 148
6.6.4 Measuring subfilter-scale fluxes......Page 149
Appendix 6.1 TKE decomposition......Page 151
Appendix 6.2 Interscale transfer......Page 152
Questions on key concepts......Page 154
Problems......Page 155
References......Page 157
7.1.1 Energetics......Page 159
7.1.2 The velocity spectrum......Page 160
7.1.4 The scalar spectrum......Page 161
7.1.5 The scalar spectrum beyond the inertial subrange......Page 162
7.1.5.2 The viscous-convective subrange: v/y >>1......Page 163
7.2.1 Atmospheric diffusion......Page 165
7.2.2 Structure-function parameters......Page 166
7.3 The dissipative range......Page 167
7.3.1 Local isotropy......Page 168
7.3.2 Structure of the dissipative regions......Page 170
7.4 Revised Kolmogorov scaling......Page 173
7.5.1 Vorticity and conserved scalars......Page 177
7.5.2 Interscale transfer of enstrophy......Page 178
7.5.3 Inertial-range cascades and scaling......Page 179
7.5.4 Forward and inverse cascades......Page 180
7.5.5 Atmospheric observations......Page 181
Problems......Page 182
References......Page 184
Part II Turbulence in the atmospheric boundary layer......Page 187
8.2.1 An isentropic, hydrostatic base state......Page 189
8.2.2 Flow-induced deviations from the base state: mass conservation......Page 190
8.2.3 Dynamics and thermodynamics......Page 192
8.2.4 Scalar-constituent conservation......Page 195
8.3.1 The effects of water vapor......Page 196
8.3.3 A conserved temperature in cloud air......Page 198
8.4 The averaged equations for moist air......Page 199
8.4.1 The general case......Page 200
8.4.2 The ensemble-averaged equations......Page 201
Problems......Page 203
References......Page 205
9.1 Overview......Page 207
9.2 The surface energy balance......Page 211
9.3 Buoyancy effects......Page 212
9.3.2 Stable and unstable stratification......Page 213
9.3.3 The convective ABL......Page 214
9.3.4 The stable ABL......Page 215
9.5 Quasi-steadiness and local homogeneity......Page 218
9.6 The mean-momentum equations......Page 219
9.6.2 The baroclinic case......Page 222
9.6.3 Ekman pumping......Page 224
Questions on key concepts......Page 225
Problems......Page 226
References......Page 228
10.1 The ``constant-flux\'\' layer......Page 229
10.2.1 Dimensional analysis......Page 231
10.2.2 M-O governing parameters......Page 232
10.2.3 The application of M-O similarity......Page 234
10.2.4.2 Inferring a surface-layer property from measurements of others......Page 236
10.2.4.4 Physical interpretation of L......Page 237
10.2.4.6 The Richardson number as an alternative stability variable......Page 238
10.3.1 The asymptotic unstable state......Page 239
10.3.2 The asymptotic stable state......Page 241
10.4 Deviations from M-O similarity......Page 242
10.4.2 Conserved scalar variance budgets......Page 245
10.4.3 The shear-stress budget......Page 246
10.4.4 The wtheta budget......Page 247
Appendix Length scales near the surface......Page 248
Questions on key concepts......Page 250
Problems......Page 251
References......Page 252
11.2.1 Mixed-layer similarity......Page 255
11.2.3 The mean momentum balance......Page 257
11.2.3.1 The barotropic case......Page 258
11.2.3.2 The baroclinic case......Page 260
11.2.3.3 Direct testing of eddy-diffusivity models......Page 262
11.2.3.4 Inferences from the stress budgets......Page 263
11.3.1 ``Top-down\'\' and ``bottom-up\'\' diffusion......Page 266
11.3.1.1 Early LES results......Page 268
11.3.1.2 Scalar variances......Page 270
11.3.2 Generalizations of K-closure for scalars......Page 271
11.3.3 Generalized mixed-layer similarity......Page 272
11.4.1 The entrainment flux......Page 273
11.4.2 The entrainment velocity......Page 275
Questions on key concepts......Page 276
Problems......Page 277
References......Page 278
12.1 Introduction......Page 281
12.1.1 Energetics of stably stratified turbulence......Page 282
12.1.2 Second-order-closure modeling of the SBL......Page 285
12.1.3 Large-eddy simulation of the SBL......Page 286
12.2.1 The near-surface response......Page 287
12.2.2 The inertial oscillation aloft......Page 289
12.2.3 The effects of sloping terrain......Page 291
12.2.4 An approach to treating gravity waves......Page 292
12.3.1 The stable surface layer limit......Page 295
12.3.3 Equilibrium height of the nocturnal SBL......Page 296
12.3.4 A constraint on the maintainence of turbulence in the SBL......Page 298
12.3.5 Insights from LES......Page 299
12.4.1 Structure......Page 300
12.5 Modeling the equilibrium height of neutral and stable ABLs......Page 302
Questions on key concepts......Page 303
Problems......Page 304
References......Page 306
Part III Statistical representation of turbulence......Page 309
13.1 Introduction......Page 311
13.2.1 Probability densities and distributions......Page 312
13.2.2 Moments and characteristic functions......Page 313
13.2.3 Joint probability densities and distributions......Page 314
13.2.4 Stationarity......Page 315
13.3.1 The Gaussian......Page 316
13.3.2.1 Velocity derivatives......Page 317
13.3.2.2 Vertical velocity in the CBL......Page 319
13.3.3 Joint probability densities......Page 320
13.4 The evolution equation for the probability density......Page 322
Questions on key concepts......Page 324
Problems......Page 325
References......Page 326
14.2 Cartesian tensors......Page 327
14.3.1 Single-point tensors......Page 328
14.4.2 Rates of molecular destruction of covariances......Page 330
14.4.3 Higher order examples......Page 332
14.5.1 The concept......Page 333
14.5.2 Evidence......Page 334
14.5.3 The maintenance of local anisotropy......Page 336
14.5.4 Gauging local anisotropy......Page 339
14.5.5 Local isotropy in turbulence models......Page 340
Problems......Page 341
References......Page 343
15.2.1 The autocorrelation function......Page 345
15.2.2 Fourier representation of a real, stochastic scalar function......Page 346
15.2.4 Cross correlations and cross spectra......Page 348
15.3.1 Extending the formalism......Page 351
15.3.2 Application to typical measurements......Page 352
15.3.3 Isotropy......Page 354
15.4 Vector functions of space and time......Page 355
15.4.1 The covariance, spectral density pair......Page 356
15.4.2 Isotropy......Page 357
15.4.2.1 The spectral density tensor......Page 358
15.4.2.2 The correlation tensor......Page 359
15.4.3 The inertial subrange......Page 362
15.5.2 Isotropy......Page 364
15.6.1 The concept......Page 365
15.6.2 The inertial range......Page 366
15.6.2.1 Vertical velocity......Page 367
15.6.2.2 Horizontal velocity......Page 369
Questions on key concepts......Page 370
Problems......Page 371
References......Page 373
16.1.1.1 The spectral equation......Page 375
16.1.2 A scalar in a steady, horizontally homogeneous boundary layer......Page 380
16.2 The analysis and interpretation of turbulence signals......Page 383
16.2.1 Spatial averaging of conserved scalars......Page 384
16.2.2 Response of scalar structure-function sensors......Page 386
16.2.4 Application to velocity fields......Page 390
16.2.5 Measuring resolved and subfilter-scale variables......Page 391
16.3 Probe-induced flow distortion......Page 392
16.3.1 A simple approach for the velocity field......Page 393
16.3.2.1 Steady laminar flow......Page 395
16.3.2.2 Quasi-steady turbulent flow, low frequencies......Page 396
Questions on key concepts......Page 397
Problems......Page 398
References......Page 399
Index......Page 401