1 Hydrodynamics of tsunami waves by Efim Pelinovsky ............ 1
1 Introduction ................................................. 1
2 Parameters of tsunami waves in the source .................... 2
2.1 Tsunamis of seismic origin .............................. 3
2.2 Tsunamis from underwater explosions ..................... 4
2.3 Tsunamis generated by landslides ........................ 6
3 Shallow water equations ...................................... 7
4 Tsunami generation and propagation in the shallow sea
of constant depth (linear approximation) ..................... 9
Vertical bottom displacement ................................. 9
Landslide motion ............................................ 12
5 Effects of finite depth for tsunami waves of seismic
origin ...................................................... 13
6 Explosive generated tsunamis (deep-water approximation) ..... 18
7 Nonlinear-dispersive theory of tsunami waves ................ 21
8 Tsunami waves in the ocean of variable depth ................ 25
9 Tsunami wave runup on the coast ............................. 34
10 Practice of tsunami computing ............................... 40
11 Conclusion .................................................. 45
Bibliography ................................................ 46
2 Weakly nonlinear and stochastic properties of ocean wave
fields. Application to an extreme wave event by Karsten
Trulsen ..................................................... 49
1 Introduction ................................................ 49
2 Empirical description of the Draupner "New Year Wave" ....... 52
3 The governing equations ..................................... 55
4 Weakly nonlinear narrow banded equations .................... 57
4.1 The bandwidth .......................................... 58
4.2 Derivation of higher-order nonlinear Schrödinger
equations .............................................. 58
4.3 Deep water time evolution in terms of velocity
potential .............................................. 60
4.4 Deep water space evolution in terms of the velocity
potential .............................................. 61
4.5 Deep water time evolution in terms of the surface
elevation .............................................. 61
4.6 Deep water space evolution in terms of the surface
elevation .............................................. 62
4.7 Finite depth ........................................... 63
5 Exact linear dispersion ..................................... 63
6 Properties of the higher order nonlinear Schrödinger
equations ................................................... 65
6.1 Conservation laws ...................................... 66
6.2 Modulational instability of Stokes waves ............... 67
7 An application of the higher-order nonlinear Schrödinger
equations: Deterministic wave forecasting ................... 71
8 Stochastic description of surface waves ..................... 73
9 Theory of stochastic variables .............................. 75
9.1 Theory of a single stochastic variable ................. 75
Example: Gaussian or normal distribution ............... 78
Example: Uniform distribution .......................... 78
Example: Rayleigh distribution ......................... 79
Example: Exponential distribution ...................... 79
9.2 Theory for several stochastic variables ................ 79
Example: Multi normal distribution ..................... 81
9.3 The Central Limit Theorem .............................. 81
10 Theory for stochastic processes ............................. 83
Example: Simple harmonic wave with random phase ........ 86
Example: Third order Stokes wave with random phase ..... 87
Example: Simple harmonic wave with random amplitude
and phase ..................................... 88
11 The spectrum ................................................ 89
11.1 Definition of frequency spectrum ....................... 89
Example: Periodic oscillation with random amplitude
and phase ..................................... 91
11.2 Definition of wave spectrum ............................ 91
Example: Linear waves with random phase ................ 92
Example: Linear waves with random amplitude and
phase ......................................... 93
11.3 An estimator for the spectrum .......................... 93
11.4 The equilibrium spectrum ............................... 94
12 Probability distributions of surface waves .................. 95
12.1 Linear waves ........................................... 95
12.2 Linear narrow banded waves ............................. 97
12.3 Second order nonlinear narrow banded waves with
Gaussian first harmonic ................................ 98
12.4 Broader bandwidth and non-Gaussian first harmonic ...... 99
13 Return periods and return values ........................... 101
13.1 How unusual is the Draupner "New Year Wave"? .......... 101
14 Conclusion ................................................. 102
A Continuous and discrete Fourier transforms .............. 103
A.l Continuous Fourier transform of a function on an
infinite interval ................................... 103
A.2 Fourier series of a function on a finite length
interval ............................................ 103
A.3 Discrete Fourier Transform (DFT) of a finite
series .............................................. 104
Bibliography ............................................... 105
3 Freak waves phenomenon: Physical mechanisms and
modelling by Christian Kharif and Efim Pelinovsky .......... 107
1 Introduction ............................................... 107
2 Freak wave observations .................................... 108
3 A brief description of the main physical mechanisms of
freak wave generation ...................................... 110
3.1 Wave-current interaction .............................. 1ll
3.2 Geometrical focusing .................................. 112
3.3 Spatio-temporal focusing .............................. 112
3.4 Modulational instability .............................. 112
3.5 Soliton interaction ................................... 112
3.6 Wind effect ........................................... 113
4 Freak wave definition ...................................... 113
5 Governing equations ........................................ 114
6 Linear approaches to the problem ........................... 115
6.1 Wave trains in inhomogeneous moving media ............. 115
Wave kinematics ....................................... 115
Wave dynamics ......................................... 116
6.2 Wave-current interaction .............................. 119
6.3 Dispersion enhancement of transient wave packets ...... 123
7 Nonlinear approaches of the problem ........................ 127
7.1 Weakly nonlinear freak wave packets in deep and
intermediate depths ................................... 127
The one-dimensional nonlinear Schrödinger equation .... 127
The two-dimensional nonlinear Schrödinger equation .... 139
The Davey-Stewartson system ........................... 145
7.2 Extended nonlinear models for freak waves ............. 147
7.3 Weakly nonlinear freak waves in shallow water ......... 151
7.4 The fully nonlinear equations ......................... 160
8 Experiments ................................................ 166
9 Conclusion ................................................. 166
Bibliography ............................................... 167
4 Rapid computations of steep surface waves in three
dimensions, and comparison with experiments by John Grue ... 173
1 Introduction ............................................... 173
2 Efficient solution of the Laplace equation ................. 175
3 Successive approximations .................................. 177
4 Effect of a finite depth ................................... 178
5 Time integration ........................................... 179
6 Nonlinear wave generation and absorption ................... 180
6.1 Generation ............................................ 180
6.2 Absorbing conditions .................................. 181
7 Convergence ................................................ 182
7.1 Integration constants ................................. 182
7.2 Convergence test ...................................... 182
8 Numerical examples of rogue waves. Comparison with
experimens ................................................. 186
8.1 Very steep wave events. Comparison with PIV-
experiments ........................................... 186
Particle Image Velocimetry (PIV) ...................... 186
Wave induced velocity vectors ......................... 187
The wave propagation speed ............................ 187
Acceleration vectors .................................. 187
8.2 Kinematics of the Camille and Draupner waves .......... 190
9 Computations of tsunami waves in three dimensions .......... 190
10 Computations of three-dimensional wave patterns ............ 191
10.1 The stability analysis by McLean et al. (1981) ........ 191
10.2 Computations of the classical horseshoe pattern ....... 194
10.3 Oscillating horseshoe pattern. Computations of
the experiments by Collard and Caulliez ............... 197
10.4 Other features of class II instability ................ 200
Class I instability may restabilize class II
instability .................................. 200
Class II instability may trigger class I
instability, leading to breaking ............. 200
Class I instability may trigger class II
instability, leading to breaking ............. 201
Class II leading to breaking .......................... 201
Predominance of class I and class II instabilities.
Recurrence vs. breaking. Wave slope thresholds ........ 202
Bibliography .......................................... 203
5 Very large internal waves in the ocean - observations
and nonlinear models by John Grue .......................... 205
1 Introduction ............................................... 205
1.1 The dead-water phenomenon ............................. 206
1.2 The discovery of internal tides ....................... 207
1.3 Internal waves in the ocean. Research up to 1960 ...... 208
1.4 Loss of submarines .................................... 208
1.5 Very large internal waves ............................. 208
1.6 Mechanisms for internal wave - surface wave
interaction ........................................... 211
Reduction of the surface wave amplitude caused
by internal wave induced surface current .............. 212
The effect of surface active films .................... 214
1.7 Transportation of biological and geological
material .............................................. 214
1.8 Breaking internal waves and energy dissipation in
the World Ocean ....................................... 215
1.9 Strong bottom currents due to internal waves .......... 215
The gas-field Ormen Lange ............................. 218
2 Long wave models ........................................... 218
2.1 The Korteweg-de Vries equation ........................ 219
Continuous stratification ............................. 219
Two-layer (interfacial) case .......................... 221
2.2 The Benjamin-Ono equation ............................. 222
2.3 The intermediate-depth equation ....................... 223
2.4 Weakly nonlinear solitary waves ....................... 223
KdV soliton. Stratified case .......................... 223
Interfacial KdV soliton ............................... 223
Algebraic soliton ..................................... 224
Intermediate depth soliton ............................ 224
3 Fully nonlinear interfacial solitary waves ................. 225
3.1 Solution of the Laplace equation ...................... 226
Numerical procedure for the fully nonlinear two-
layer model ........................................... 227
3.2 Fully nonlinear computations in the small amplitude
limit ................................................. 228
3.3 Solitary waves of large amplitude ..................... 228
3.4 Solitary waves of maximum amplitude ................... 231
3.5 Overhanging waves ..................................... 231
4 Transient computations of interfacial motion ............... 237
4.1 Two-dimensional transient model ....................... 237
4.2 Solution of the Laplace equation ...................... 238
4.3 Solitary wave generation .............................. 239
Simulations of the waves observed upstream at Knight
Inlet ................................................. 241
Simulations of the waves in the Sulu Sea .............. 242
4.4 Upstream waves: geometry in the thin layer ............ 242
4.5 Fully nonlinear interfacial motion in three
dimensions ............................................ 249
Final set of equations ................................ 251
Global evaluation using FFT ........................... 251
Local, truncated integration .......................... 251
5 Fully nonlinear wave motion in a continuously stratified
fluid ...................................................... 252
5.1 Basic equations ....................................... 252
5.2 The vorticity ......................................... 254
5.3 The local Richardson number ........................... 255
5.4 The field equation .................................... 256
5.5 The linear long wave speed ............................ 256
Three-layer case ...................................... 256
Two-layer case ........................................ 271
5.6 Nonlinear three-layer wave motion. Solution by
integral equations .................................... 257
5.7 Wave motion along a thick pycnocline .................. 259
6 Concluding remarks ......................................... 261
A Inverse scattering theory. Lax pairs .................... 264
A.l Laboratory waves ................................... 264
A.2 Brief history of solitons and inverse scattering
theory ............................................. 264
Bibliography ............................................... 265
6 Internal tides. Global field of internal tides and
mixing caused by internal tides by Eugene Morozov .......... 271
1 Global field of internal tides ............................. 271
1.1 The model ............................................. 272
1.2 Measurements .......................................... 275
Henderson seamount in the Eastern Pacific (25°N,
119°W) ................................................ 276
Mascarene Ridge in the western Indian Ocean ........... 276
Region, 600 km south of the Mendocino Ridge, 700 km
west of San Fransisco ................................. 277
East of Macquarie Island and south of New Zealand ..... 279
The North Atlantic (2925°N), east of the Mid-
Atlantic Ridge (MAR) .................................. 279
Four sites near the equator of the Indian Ocean:
85°E, 75°E, 65°E, and 55°E ............................ 279
The South Atlantic (21°S) near Brazil, Trinidad and
Martin Vaz Islands .................................... 279
Kusu-Palau Ridge south of Japan (26°N) ................ 279
South of Iceland (54°N, 27°W) ......................... 279
Region east of the Great Meteor banks in the North
Atlantic (31°N, 26°W) ................................. 279
Northwestern Pacific region ........................... 280
Atlantic Polygon-70 with 17 buoys deployed in 1970
and Mesopolygon with 70 buoys deployed in 1985
almost in the same region 16-20°N, 33-37°W ............ 280
Madagascar Basin ...................................... 280
Sargasso Sea, POLYMODE, Array-1, and Array-2 .......... 280
Crozet Bazin north of Kerguelen Island ................ 280
1.3 Discussion about the global field of internal tides .... 280
2 Internal tide at high latitides ............................ 283
2.1 Numerical model ....................................... 284
2.2 Numerical experiments to study internal tides ......... 285
3 Internal tides in the Kara Strait .......................... 291
4 Internal tides in the Strait of Gibraltar .................. 298
5 Application of WOCE sections to a global view of mixing
in the Atlantic Ocean ...................................... 305
5.1 Dropped spectra of CTD profiles ....................... 305
5.2 Analysis of data ...................................... 306
5.3 Topographic influence on vertical wavenumber
spectra ............................................... 307
5.4 Topographic influence of submarine ridges in
the water column 600 dbar above the bottom ............ 308
5.5 Topographic influence of submarine ridges in the
water column between 2000 and 3000 dbar ............... 309
5.6 Spreading of Antarctic Bottom Water in the Vema and
Equatorial channels ................................... 311
5.7 Frontal zone of the North Atlantic Current ............ 313
5.8 Influence of the Mediterranean outflow in
the Atlantic Ocean .................................... 313
5.9 Spreading of the North Atlantic deep water ............ 317
6 Several approaches to the investigation of tidal
internal waves in the northern part of the Pacific Ocean ... 320
6.1 Moored data analysis .................................. 321
6.2 Numerical modeling .................................... 323
6.3 Analysis of data from sections made with expandable
bathythermographs (XBT) ............................... 325
6.4 Analysis of CTD sections data ......................... 326
6.5 Data of drifters ...................................... 328
Bibliography .................................................. 330
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