Preface ........................................................ ix
Part I A grammar of turbulence
1 Introduction ................................................. 3
1.1 Turbulence, its community, and our approach ............. 3
1.2 The origins and nature of turbulence .................... 4
1.3 Turbulence and surface fluxes ........................... 5
1.4 How do we study turbulence? ............................ 10
1.5 The equations of turbulence ............................ 11
1.6 Key properties of turbulence ........................... 14
1.7 Numerical modeling of turbulent flows .................. 18
1.8 Physical modeling of turbulent flows ................... 20
1.9 The impact of Kolmogorov ............................... 20
2 Getting to know turbulence .................................. 27
2.1 Average and instantaneous properties contrasted ........ 27
2.2 Averaging .............................................. 28
2.3 Ergodicity ............................................. 34
2.4 The convergence of averages ............................ 35
2.5 The turbulence spectrum and the eddy velocity scale .... 38
2.6 Turbulent vorticity .................................... 43
2.7 Turbulent pressure ..................................... 44
2.8 Eddy diffusivity ....................................... 45
2.9 Reynolds-number similarity ............................. 49
2.10 Coherent structures .................................... 49
3 Equations for averaged variables ............................ 55
3.1 Introduction ........................................... 55
3.2 Ensemble-averaged equations ............................ 56
3.3 Interpreting the ensemble-averaged equations ........... 59
3.4 Space-averaged equations ............................... 65
3.5 Summary ................................................ 70
4 Turbulent fluxes ............................................ 75
4.1 Introduction ........................................... 75
4.2 Temperature flux in a boundary layer ................... 76
4.3 Mass flux in scalar diffusion .......................... 78
4.4 Momentum flux in channel flow .......................... 83
4.5 The "mixture length" ................................... 84
4.6 Summary ................................................ 85
5 Conservation equations for covariances ...................... 89
5.1 Introduction and background ............................ 89
5.2 The fluctuation equations .............................. 90
5.3 Example: The scalar variance equation .................. 91
5.4 The scalar flux and Reynolds stress budgets ............ 99
5.5 Applications .......................................... 101
5.6 From the covariance equations to turbulence models .... 106
6 Large-eddy dynamics, the energy cascade, and large-eddy
simulation ................................................. 115
6.1 Introduction .......................................... 115
6.2 More on space averaging ............................... 116
6.3 A "thought problem": equilibrium homogeneous
turbulence ............................................ 120
6.4 Application to flows homogeneous in two dimensions .... 127
6.5 The physical mechanisms of interscale transfer ........ 129
6.6 Large-eddy simulation ................................. 130
7 Kolmogorov scaling, its extensions, and two-dimensional
turbulence ................................................. 145
7.1 The inertial subrange ................................. 145
7.2 Applications of inertial-range scaling ................ 151
7.3 The dissipative range ................................. 153
7.4 Revised Kolmogorov scaling ............................ 159
7.5 Two-dimensional turbulence ............................ 163
Part II Turbulence in the atmospheric boundary layer ......... 173
8 The equations of atmospheric turbulence .................... 175
8.1 Introduction .......................................... 175
8.2 The governing equations for a dry atmosphere .......... 175
8.3 Accounting for water vapor, liquid water, and phase
change ................................................ 182
8.4 The averaged equations for moist air .................. 185
9 The atmospheric boundary layer ............................. 193
9.1 Overview .............................................. 193
9.2 The surface energy balance ............................ 197
9.3 Buoyancy effects ...................................... 198
9.4 Average vs. instantaneous structure ................... 204
9.5 Quasi-steadiness and local homogeneity ................ 204
9.6 The mean-momentum equations ........................... 205
10 The atmospheric surface layer .............................. 215
10.1 The "constant-flux" layer ............................. 215
10.2 Monin-Obukhov similarity .............................. 217
10.3 Asymptotic behavior of M-O similarity ................. 225
10.4 Deviations from M-O similarity ........................ 228
11 The convective boundary layer .............................. 241
11.1 Introduction .......................................... 241
11.2 The mixed layer: velocity fields ...................... 241
11.3 The mixed layer: conserved-scalar fields .............. 252
11.4 The interfacial layer ................................. 259
12 The stable boundary layer .................................. 267
12.1 Introduction .......................................... 267
12.2 The late-afternoon ABL transition over land ........... 273
12.3 The quasi-steady SBL .................................. 281
12.4 The evolving SBL ...................................... 286
12.5 Modeling the equilibrium height of neutral and
stable ABLs ........................................... 288
Part III Statistical representation of turbulence ............. 295
13 Probability densities and distributions .................... 297
13.1 Introduction .......................................... 297
13.2 Probability statistics of scalar functions of
a single variable ..................................... 298
13.3 Examples of probability densities ..................... 302
13.4 The evolution equation for the probability density .... 308
14 Isotropic tensors .......................................... 313
14.1 Introduction .......................................... 313
14.2 Cartesian tensors ..................................... 313
14.3 Determining the form of isotropic tensors ............. 314
14.4 Implications of isotropy .............................. 316
14.5 Local isotropy ........................................ 319
15 Co variances, autocorrelations, and spectra ................ 331
15.1 Introduction .......................................... 331
15.2 Scalar functions of a single variable ................. 331
15.3 Scalar functions of space and time .................... 337
15.4 Vector functions of space and time .................... 341
15.5 Joint vector and scalar functions of space and time ... 350
15.6 Spectra in the plane .................................. 351
16 Statistics in turbulence analysis .......................... 361
16.1 Evolution equations for spectra ....................... 361
16.2 The analysis and interpretation of turbulence
signals ............................................... 369
16.3 Probe-induced flow distortion ......................... 378
Index ......................................................... 387
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