Nonlinear fast growth of water waves under wind forcing

In the wind-driven wave regime, the Miles mechanism gives an estimate of the growth rate of the waves under the effect of wind. We consider the case where this growth rate, normalised with respect to the frequency of the carrier wave, is of the order of the wave steepness. Using the method of multiple scales, we calculate the terms which appear in the nonlinear Schrodinger (NLS) equation in this regime of fast-growing waves. We define a coordinate transformation which maps the forced NLS equation into the standard NLS with constant coefficients, that has a number of known analytical soliton solutions. Among these solutions, the Peregrine and the Akhmediev solitons show an enhancement of both their lifetime and maximum amplitude which is in qualitative agreement with the results of tank experiments and numerical simulations of dispersive focusing under the action of wind.

[1]  Jinbao Song,et al.  On Determining the Onset and Strength of Breaking for Deep Water Waves. Part II: Influence of Wind Forcing and Surface Shear , 2002 .

[2]  Hiroaki Ono,et al.  Nonlinear Modulation of Gravity Waves , 1972 .

[3]  A. Couairon,et al.  Femtosecond filamentation in transparent media , 2007 .

[4]  B. Johansson,et al.  Dynamical coupling of wind and ocean waves through wave-induced air flow , 2003 .

[5]  H A Haus,et al.  Nonlinear Schrödinger equation for optical media with quadratic nonlinearity. , 1994, Physical review. A, Atomic, molecular, and optical physics.

[6]  Miguel Onorato,et al.  Approximate rogue wave solutions of the forced and damped nonlinear Schrödinger equation for water waves , 2012, 1203.4735.

[7]  A. I. Dyachenko,et al.  Theory of weakly damped free-surface flows: A new formulation based on potential flow solutions , 2007, 0704.3352.

[8]  N. Akhmediev,et al.  Exact first-order solutions of the nonlinear Schrödinger equation , 1987 .

[9]  Efim Pelinovsky,et al.  Freak waves under the action of wind: experiments and simulations , 2006 .

[10]  J. Touboul,et al.  On the interaction of wind and extreme gravity waves due to modulational instability , 2006 .

[11]  Christian Kharif,et al.  The modulational instability in deep water under the action of wind and dissipation , 2010, Journal of Fluid Mechanics.

[12]  J. Miles.  On the generation of surface waves by shear flows , 1957, Journal of Fluid Mechanics.

[13]  E. Pelinovsky,et al.  On the interaction of wind and steep gravity wave groups using Miles' and Jeffreys' mechanisms , 2008 .

[14]  J. McWilliams,et al.  Dynamics of Winds and Currents Coupled to Surface Waves , 2010 .

[15]  P. Janssen.  Quasi-linear Theory of Wind-Wave Generation Applied to Wave Forecasting , 1991 .

[16]  S. Skupin,et al.  Ultrashort filaments of light in weakly ionized, optically transparent media , 2007 .

[17]  J. Soto-Crespo,et al.  Extreme waves that appear from nowhere: On the nature of rogue waves , 2009 .

[18]  D. H. Peregrine,et al.  Water waves, nonlinear Schrödinger equations and their solutions , 1983, The Journal of the Australian Mathematical Society. Series B. Applied Mathematics.

[19]  Stéphane Leblanc,et al.  Amplification of nonlinear surface waves by wind , 2007 .

[20]  B. Farrell,et al.  The Stochastic Parametric Mechanism for Growth of Wind-Driven Surface Water Waves , 2008 .

[21]  Harold Jeffreys,et al.  On the Formation of Water Waves by Wind , 1925 .

[22]  Roger Grimshaw,et al.  Water Waves , 2021, Mathematics of Wave Propagation.

[23]  Takuji Waseda,et al.  Maximum steepness of oceanic waves: Field and laboratory experiments , 2010 .

[24]  A. Slunyaev.  A high-order nonlinear envelope equation for gravity waves in finite-depth water , 2005 .

[25]  Yan‐Chow Ma,et al.  The Perturbed Plane‐Wave Solutions of the Cubic Schrödinger Equation , 1979 .

[26]  P. Janssen.  The Interaction of Ocean Waves and Wind , 2004 .

[27]  C. Kharif,et al.  A nonlinear Schrödinger equation for water waves on finite depth with constant vorticity , 2012, 1207.2246.

[28]  Umberto Bortolozzo,et al.  Rogue waves and their generating mechanisms in different physical contexts , 2013 .