A direct wave model based on the one-dimensional nonlinear equations for potential waves is used for simulation of wave field development under the action of energy input, dissipation, and nonlinear wave-wave interaction. The equations are written in conformal surface-fitted nonstationary coordinate system. New schemes for calculating the input and dissipation of wave energy are implemented. The wind input is calculated on the basis of the parameterization developed through the coupled modeling of waves and turbulent boundary layer. The wave dissipation algorithm, introduced to prevent wave breaking instability, is based on highly selective smoothing of the wave surface and surface potential. The integration is performed in Fourier domain with the number of modes M = 2048, broad enough to reproduce the energy downshifting. As the initial conditions, the wave field is assigned as train of Stokes waves with steepness ak = 0.15 at nondimensional wavenumber k = 512. Under the action of nonlinearity and energy input the spectrum starts to grow. This growth is followed by the downshifting. The total time of integration is equal to 7203 initial wave periods. During this time the energy increased by 1111 times. Peak of the spectrum gradually shifts from wavenumber nondimensional k = 512 down to k = 10. Significant wave height increases 33 times, while the peak period increases 51 times. Rates of the peak downshift and wave energy evolution are in good agreement with the JONSWAP formulation.
REFERENCES
1.
M.
Onorato
, T.
Waseda
, A.
Toffoli
, L.
Cavaleri
, O.
Gramstad
, P. A. E. M.
Janssen
, T.
Kinoshita
, J.
Monbaliu
, N.
Mori
, A. R.
Osborne
, M.
Serio
, C. T.
Stansberg
, H.
Tamura
, and K.
Trulsen
, “Statistical properties of directional ocean waves: The role of the modulational instability in the formation of extreme events
,” Phys. Rev. Lett.
102
, 114502
(2009
).2.
A. V.
Babanin
, Breaking and Dissipation of Ocean Surface Waves
(Cambridge University Press
, 2011
), p. 480
.3.
N.
Mori
, M.
Onorato
, P. A. E. M.
Janssen
, A. R.
Osborne
, and M.
Serio
, “On the extreme statistics of long-crested deep water waves: Theory and experiments
,” J. Geophys. Res., C: Oceans
112
, C09011
, doi: (2007
).4.
A.
Ribal
, A. V.
Babanin
, I. R.
Young
, A.
Toffoli
, and M.
Stiassnie
, “Recurrent solutions of Alber equation initialized by JONSWAP spectra
,” J. Fluid Mech.
719
, 314
–344
(2013
).5.
M. A.
Donelan
, W. M.
Drennan
, and A. K.
Magnusson
, “Nonstationary analysis of the directional properties of propagating waves
,” J. Phys. Oceanogr.
26
, 1901
–1914
(1996
).6.
J. C.
Nieto Borge
, K.
Reichert
, and K.
Hessner
, “Detection of spatio-temporal wave grouping properties by using temporal sequences of X-band radar images of the sea surface
,” Ocean Modell.
61
, 21
–37
(2013
).7.
K.
Hasselmann
, R. P.
Barnett
, E.
Bouws
et al., “Measurements of wind-wave growth and swell decay during the Joint Sea Wave Project (JONSWAP)
,” Tsch. Hydrogh. Z. Suppl.
A8
(12
), 1
–95
(1973
).8.
J. A.
Battjes
, T. J.
Zitman
, and L. H.
Holthuijsen
, A reanalysis of the spectra observed in JONSWAP
,” J. Phys. Oceanogr.
17
, 1288
–1295
(1987
).9.
J. F.
Willemsen
, “Deterministic modeling of driving and dissipation for ocean surface gravity waves
,” J. Geophys. Res.
106
(C11
), 27187
–27204
, doi: (2001
).10.
J. F.
Willemsen
, “Deterministic modeling of driving and dissipation for ocean surface gravity waves in two horizontal dimensions
,” J. Geophys. Res.
107
(C8
), 30
–1
30
–17
, doi: (2002
).11.
V. P.
Krasitskii
, “On reduced equations in the Hamiltonian theory of weakly nonlinear surface waves
,” J. Fluid Mech.
272
, 1
–20
(1994
).12.
D.
Chalikov
, On the nonlinear energy transfer in the unidirected adiabatic surface waves
,” Phys. Lett. A
376
(44
), 2795
–2798
(2012
).13.
L.
Shemer
, H.
Jiao
, E.
Kit
, and Y.
Agnon
, “Evolution of a nonlinear wave field along a tank: Experiments and numerical simulations based on the spatial Zakharov equation
,” J. Fluid Mech.
427
, 107
–129
(2001
).14.
V. E.
Zakharov
, Stability of periodic waves of finite amplitude on the surface of a deep fluid
,” J. Appl. Mech. Tech. Phys.
9
, 190
–194
(1968
).15.
P. A. E. M.
Janssen
, “Nonlinear four-wave interactions and freak waves
,” J. Phys. Oceanogr.
33
, 863
–884
(2003
).16.
S. Yu.
Annenkov
and V. I.
Shrira
, Role of non-resonant interactions in the evolution of nonlinear random water wave fields
, J. Fluid Mech.
561
, 181
–207
(2006
).17.
D.
Chalikov
and D.
Sheinin
, “Numerical modeling of surface waves based on principal equations of potential wave dynamics
,” Tech. Note NOAA/NCEP/OMB
139
, 54
(1996
).18.
D.
Chalikov
and D.
Sheinin
, “Direct modeling of one-dimensional nonlinear potential waves
,” Adv. Fluid Mech.
17
, 207
–258
, (1998
).19.
D.
Chalikov
and D.
Sheinin
, “Modeling of extreme waves based on equations of potential flow with a free surface
,” J. Comput. Phys.
210
, 247
–273
(2005
).20.
J. W.
Miles
, “On the generation of surface waves by shear flows
,” J. Fluid Mech.
3
, 185
–204
(1957
).21.
R. L.
Snyder
, F. W.
Dobson
, J. A.
Elliott
, and R. B.
Long
, “Array measurements of atmospheric pressure fluctuations above surface gravity waves
,” J. Fluid Mech.
102
, 1
–59
(1981
).22.
S. V.
Hsiao
and O. H.
Shemdin
, “Measurements of wind velocity and pressure with a wave follower during MARSEN
,” J. Geophys. Res.
88
, 9841
–9849
, doi: (1983
).23.
D.
Hasselmann
and J.
Bösenberg
, “Field measurements of wave-induced pressure over wind-sea and swell
,” J. Fluid Mech.
230
, 391
–428
(1991
).24.
M. A.
Donelan
, A. V.
Babanin
, I. R.
Young
, and M. L.
Banner
, “Wave follower field measurements of the wind input spectral function. Part II. Parameterization of the wind input
,” J. Phys. Oceanogr.
36
, 1672
–1688
(2006
).25.
M. A.
Donelan
, A. V.
Babanin
, I. R.
Young
, M. L.
Banner
, and C.
McCormick
, “Wave follower field measurements of the wind input spectral function. Part I. Measurements and calibrations
, J. Atmos. Oceanic Technol.
22
, 799
–813
(2005
).26.
W. E.
Rogers
, A. V.
Babanin
, and D. W.
Wang
, “Observation-consistent input and whitecapping-dissipation in a model for wind-generated surface waves: Description and simple calculations
,” J. Atmos. Oceanic Technol.
29
(9
), 1329
–1346
(2012
).27.
D. V.
Chalikov
, “Numerical simulation of wind-wave interaction
,” J. Fluid Mech.
87
, 561
–582
(1978
).28.
D. V.
Chalikov
, “Numerical simulation of the boundary layer above waves
,” Boundary Layer Meterol.
34
, 63
–98
(1986
).29.
D.
Chalikov
, “The parameterization of the wave boundary layer
,” J. Phys. Oceanogr.
25
, 1333
–1349
(1995
).30.
D.
Chalikov
and V.
Makin
, “Models of the wave boundary layer
,” Boundary Layer Meterol.
56
, 83
–99
(1991
).31.
D.
Chalikov
and M.
Belevich
, “One‑dimensional theory of the wave boundary layer
,” Boundary Layer Meterol.
63
, 65
–96
(1993
).32.
H.
Tolman
and D.
Chalikov
, “On the source terms in a third‑generation wind wave model
,” J. Phys. Oceanogr.
26
, 2497
–2518
(1996
).33.
H. L.
Tolman
, “A mosaic approach to wind wave modeling
,” Ocean Modell.
25
(1–2
), 35
–47
(2008
).34.
D.
Chalikov
and S.
Rainchik
, “Coupled numerical modelling of wind and waves and the theory of the wave boundary layer
,” Boundary Layer Meteorol.
138
, 1
–41
(2010
).35.
C.-H.
Ting
, A. V.
Babanin
, D.
Chalikov
, and T.-W.
Hsu
, “Dependence of drag coefficient on the directional spreading of ocean waves
,” J. Geophys. Res.
117
, C00J14
, doi: (2012
).36.
A. V.
Babanin
, T.
Waseda
, T.
Kinoshita
, and A.
Toffoli
, “Wave breaking in directional fields
,” J. Phys. Oceanogr.
41
, 145
–156
(2011
).37.
A. Y.
Benilov
and Y.
Lozovatskiy
, “Semi-empirical methods of the turbulence description in the ocean
,” In: The Turbulent Diffusion of the Ingredients in the Sea, The coordination Center of COMECON Inf. Bull.
5
, 89
–97
(1977
).38.
A. Y.
Benilov
, T. G.
McKee
, and A. S.
Safray
, “On the vortex instability of the linear surface wave
,” Numerical Methods in Laminar and Turbulent Flow
, edited by C.
Taylor
(Pineridge Press
, UK
, 1993
), Vol. VIII
, Part 2, pp. 1323
–1334
.39.
F.
Qiao
, Y.
Yeli
, Y.
Yang
, Q.
Zheng
, C.
Xia
, and J.
Ma
, “Wave-induce mixing in the upper ocean: Distribution and application to a global ocean circulation model
,” Geophys. Res. Lett.
31
, L11303
, doi: (2004
).40.
A. V.
Babanin
and B. K.
Haus
, “On the existence of water turbulence induced by non-breaking surface waves
,” J. Phys. Oceanogr.
39
, 2675
–2679
(2009
).41.
I. B.
Savelyev
, E.
Maxeiner
, and D.
Chalikov
, “Turbulence production by non-breaking waves: laboratory and numerical simulations
,” J. Geophys. Res., [Oceans]
117
, C00J13
, doi: (2012
).42.
A.
Toffoli
, A. V.
Babanin
, and J.
McConochie
, “The effect of turbulence induced by non-breaking waves on the ocean mixed layer: Field observations on the Australian North-West Shelf
,” J. Geophys. Res.
117
, C00J24
, doi: 8p (2012
).43.
A.
Babanin
and D.
Chalikov
, “Numerical investigation of turbulence generation in non-breaking potential waves
,” J. Geophys. Res.
117
, C00J17
, doi: (2012
).44.
D.
Chalikov
and A. V.
Babanin
, “Simulation of wave breaking in one-dimensional spectral environment
,” J. Phys. Ocean
42
(11
), 1745
–1761
(2012
).45.
D.
Chalikov
, “Freak waves: Their occurrence and probability
,” Phys. Fluid
21
, 076602
(2009
).46.
M. P.
Tulin
and T.
Waseda
, “Laboratory observations of wave group evolution, including breaking effects
,” J. Fluid Mech.
378
, 197
–232
(1999
).47.
O.
Gramstad
and M.
Stiassnie
, “Phase-averaged equation for water waves
,” J. Fluid Mech.
718
, 280
–303
(2013
).48.
D.
Chalikov
and A. V.
Babanin
, “Three-dimensional periodic fully-nonlinear potential waves
,” in Proceedings of the ASME 2013, 32nd International Conference on Ocean, Offshore and Arctic Engineering, Structure, Safety and Reliability, ASME Digital Collection 2013 Vol. 2B, 9–14 July 2013, Nantes, France
(ASME
, 2013
), p. 8
.49.
M. A.
Donelan
, M.
Curcic
, 1S. S.
Chen
, and A. K.
Magnusson
, “Modeling waves and wind stress
,” J. Geophys. Res.
117
, C00J23
, doi: (2012
).50.
K.
Hasselmann
, “On the non-linear energy transfer in a gravity wave spectrum, Part 1
,” J. Fluid Mech.
12
, 481
–500
(1962
).51.
D.
Chalikov
, “Simulation of Benjamin-Feir instability and its consequences
,” Phys. Fluid
19
, 016602
–1
016602
–15
(2007
).52.
C.
Kharif
, E.
Pelinovsky
, and A.
Slunyaev
, A. Rogue Waves in the Ocean
(Springer
, 2009
).© 2014 AIP Publishing LLC.
2014
AIP Publishing LLC
You do not currently have access to this content.
