PREFACE ........................................................ XV
1 Fundamentals
1.1 Introduction ............................................ 1
1.2 Some Wave Mechanics ..................................... 9
1.2.1 Waves, Harmonics, and Modes ..................... 11
1.2.2 Frames of Reference ............................. 12
1.2.3 Wave Scales ..................................... 13
1.2.4 Wave Phase and Wave Speed ....................... 15
1.2.5 Group Velocity .................................. 18
1.2.6 Wave Dispersion ................................. 22
1.3 The Buoyant Force ...................................... 22
1.4 The Boussinesq Approximation ........................... 26
2 The Linear Theory
2.1 Introduction ........................................... 29
2.2 The Taylor-Goldstein Equation .......................... 31
2.3 A Simple Solution ...................................... 37
2.3.1 No Background Wind Speed ........................ 37
2.3.2 Constant Background Wind Speed .................. 42
2.4 The WKB or "Slowly Varying" Method ..................... 46
2.5 Energetics ............................................. 47
2.5.1 Wave Energy ..................................... 47
2.5.2 Wave-Activity Conservation Laws ................. 52
2.5.2.1 Wave Action ............................ 52
2.5.2.2 Pseudo-Energy and Pseudo-Momentum ...... 53
3 Mountain Waves
3.1 Introduction ........................................... 57
3.2 Uniform Flow Over a Surface Corrugation ................ 61
3.3 The Two-Dimensional Mountain ........................... 67
3.4 The Three-Dimensional Mountain ......................... 76
3.5 Nonorographic Gravity Waves ............................ 81
4 Ducted gravity Waves
4.1 Introduction ........................................... 87
4.2 Wave Reflection and Refraction at an Elevated Layer .... 89
4.3 Wave Trapping, Energy Flux, and Wave Resonance ......... 96
4.4 Reflection at the Ground Surface ....................... 98
4.5 Wave Ducts ............................................ 102
4.5.1 The Pure Temperature Duct ...................... 102
4.5.2 The Pure Wind Duct ............................. 105
4.5.3 Wind Spirals and Ducts ......................... 109
4.5.4 Mountain Lee Waves ............................. 112
5 Gravity wave instability and Turbulence
5.1 Introduction .......................................... 117
5.2 Parcel Exchange Analysis of Flow Stability ............ 118
Wave Instability ........................................... 121
5.3.1 Introduction ................................... 121
5.3.2 Kelvin-Helmholtz Instability ................... 122
5.3.3 The Stability of Shear Flows ................... 127
5.3.3.1 Inflection Point Instability .......... 128
5.3.3.2 Instability of Stratified Shear
Flows ................................. 130
The Critical Level .................... 133
Neutral, Stable, and Unstable Modes ... 140
Wave-Modulated Richardson Number ...... 143
Wave-Turbulence Coupling .............. 147
Jefferys' Roll-Wave Instability
Mechanism ............................. 151
6 Wave Stress
6.1 Introduction .......................................... 159
6.2 Mathematical Derivation ............................... 160
6.3 Variation of Wave Stress with Height .................. 162
6.4 Mountain Wave Stress .................................. 164
6.4.1 Wave Stress Over a Surface Corrugation ......... 164
6.4.1.1 Wave Stress Over an Isolated Two-
Dimensional Mountain .................. 166
6.4.2 Wave Stress Over Three-Dimensional Objects ..... 168
6.5 Secondary Effects of Wave Drag ........................ 174
6.5.1 Direction Forcing .............................. 174
6.5.2 Lee Wave Drag Over a Two-Dimensional Mountain .. 175
6.5.3 Momentum Flux Due to Mountain Lee Waves ........ 176
7 Gravity Waves in the Middle and Upper Atmosphere
7.1 Introduction .......................................... 181
7.2 Background ............................................ 182
7.3 Interia-Gravity Waves in the Middle Atmosphere ........ 187
7.4 Planetary Waves in the Middle Atmosphere .............. 190
7.4.1 Rossby Wave .................................... 190
7.4.2 Tropical Atmosphere ............................ 191
7.4.2.1 Vertically Propagating Rossby-
Gravity Waves ......................... 192
7.4.2.2 The Kelvin Wave ....................... 193
7.5 Midlatitude Wave Spectra .............................. 194
7.5.7 High-Frequency Waves: Ω  ƒ ................... 194
7.5.2 Mid-Frequency Waves: ƒ Ω TV .............. 195
7.5.3 Low-Frequency Range: Ω ~ ƒ ..................... 196
7.6 Modeling the Gravity Wave Fluxes in the MUA ........... 197
7.6.1 Hydrodynamic Models ............................ 197
7.6.2 Ray Tracing .................................... 197
8 Wave Stress Parameterization
8.1 Introduction .......................................... 201
8.1.1 Wave Breaking, Wave Saturation, and Eddy
Diffusivity .................................... 202
8.1.2 Wave Breaking Heights .......................... 205
8.2 Wave-Saturation Parameterization ...................... 208
8.3 Parameterization Methods .............................. 209
8.3.1 Sawyer Method .................................. 209
8.3.2 Lindzen-Holton Method .......................... 209
8.3.3 The Palmer Method .............................. 215
8.3.4 The McFarlane Method ........................... 217
8.3.5 The Schoeberl Method ........................... 221
8.3.6 The Terrain-Height Adjustment Scheme ........... 222
8.4 Saturation Limits and Other Problems .................. 226
9 Observations and Measurements of Gravity Waves
9.1 Introduction .......................................... 231
9.2 Ground-Based Measurements ............................. 233
9.2.1 Pressure ....................................... 234
9.2.1.1 Static Pressure Ports ................. 235
9.2.1.2 Noise Filtering ....................... 237
9.2.2 Sampling Arrays ................................ 245
9.3 Free-Balloon Soundings ................................ 249
9.3.1 Tethered Lifting Systems ....................... 252
9.3.2 Superpressure Balloons ......................... 253
9.3.3 Aircraft ....................................... 263
9.4 Remote Measurements ................................... 264
9.4.1 Radar .......................................... 265
9.4.2 Doppler Radar .................................. 266
9.4.3 FM-CW Radar .................................... 267
9.4.4 Sodar .......................................... 268
9.4.5 Lidar .......................................... 270
9.4.6 Airglow ........................................ 272
9.4.7 Satellites and Global Positioning Systems ...... 274
9.4.7.1 Photographic Analyses ................. 274
9.4.7.2 Sounding Rocket ....................... 274
9.4.7.3 GPS Radio-Occultation Sounding ........ 275
9.4.7.4 Microwave Limb Sounding ............... 276
10 Gravity Wave Analyses
10.1 Introduction .......................................... 279
10.2 Analyses of Tropospheric Gravity Waves ................ 280
10.2.1 Initial Steeps ................................ 281
10.2.2 Spectral Analysis ............................. 282
10.2.3 Wavelet Analysis .............................. 283
10.2.4 Band-Pass Filtering ........................... 288
10.2.5 Lag Analysis .................................. 289
10.2.6 Cross-Correlation Lag Analysis ................ 291
10.2.7 Beamsteering .................................. 293
10.2.7.1 Beamsteering in the Slowness Domain .. 293
10.2.7.2 Beamsteering in the Frequency
Domain ............................... 295
10.2.7.3 Array Response and Examples .......... 296
10.2.8 Impedance ..................................... 297
10.2.9 Analysis Results .............................. 299
10.3 Gravity Wave Analyses in the MUA ...................... 303
10.3.1 Wavelet Analysis ............................. 304
10.3.2 Hodograph Analysis ........................... 305
Appendix A
The Hydrostatic atmosphere
А.1 The Hydrostatic Approximation ......................... 309
А.2 The Scale Height of the Isothermal Atmosphere ......... 310
А.З Adiabatic Lapse Rate .................................. 310
А.4 Potential Temperature ................................. 310
А.5 Boussinesq Relations .................................. 311
A.6 The Geostrophic Wind .................................. 312
А.7 The Critical Level .................................... 313
A.8 Convolution ........................................... 315
A.9 The Eckman Wind Spiral ................................ 316
A.10 Numerical Methods ..................................... 317
A.10.1 Mountain Waves ................................. 317
A.10.2 Ducted Gravity Waves ........................... 320
Bibliography .................................................. 323
Index ......................................................... 343
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