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 Schrödinger (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.
keywords:Rogue waves, Water waves, Wind forcing
The investigation of the physical mechanisms for the generation of ocean waves by wind has a long history which starts at the beginning of the 20th century 1925RSPSA.107..189J () and is still ongoing. The problem is highly nonlinear JanssenBook () and the feedback at the air-water interface between wind and water waves is difficult to study experimentally and theoretically because of turbulence in both fluids.
The problem can be simplified at first by neglecting currents in the water and by considering the so-called wind-driven wave regime which is characterised by growing seas with wave ages , where is the phase velocity of the water waves and is the friction velocity of wind over water waves Sullivan2010 (). Direct field measurements of the pressure induced by airflow on waves are rare, thus there is no agreement in the scientific community on the underlying mechanisms leading to wave amplification (for a review see (JanssenBook, , Chap. 3) and Sullivan2010 ()).
In the shear flow model introduced by Miles Miles1957 (); Janssen1991 () the rate of energy transfer from the wind to a wave propagating at phase velocity is proportional to the wind profile curvature at the critical height where the wind speed equals the phase velocity of the wave, . The Miles mechanism has been recently confirmed in field experiments, in particular for long waves Hristov2003 (). For a logarithmic velocity profile in the boundary layer, the Miles growth rate results in Miles1957 (); Banner2002 (); Kharif2010 ()
where is the wave energy, is the frequency of the carrier wave, is the density ratio ( between air and water), and is an empirical constant of the order of 32.5 in the wind-driven wave regime Banner2002 (). The pressure induced at the water surface then depends on the surface elevation as follows Miles1957 (); Kharif2010 ()
Typical values of are shown in Fig. 1 of Banner2002 () (in that figure ) or in Fig. 1 of Farrell2008 () (where ) as a function of the wave age . They range from - for fast-moving waves () to -1 for slow-moving waves and laboratory tank experiments (). Thus, the growth rate can be regarded as a small parameter in the wind-driven wave regime and generally it is assumed that , where is the wave steepness, being the amplitude of the vertical water displacement and the wavenumber of the water wave. For weak-nonlinear waves the steepness is indeed small and in ocean waves it is smaller than 0.55, the value for which wave-breaking occurs Toffoli2010 (). The case gives rise to the following damped/forced nonlinear Schrödinger equation Leblanc2007 (); Kharif2010 (); OnoratoProment2012 ()
where is the kinematic viscosity. Thus, the case describes the quasi-equilibrium between wind and damping effects due to viscosity.
In this Letter, we consider the case , corresponding to stronger winds, the effect of which overcomes the dissipation due to viscosity. This case turns out to be relevant for explaining experimental results obtained in the context of dispersive focusing of waves under the action of wind 2006EJMF…25..662T (); 2008NPGeo..15.1023T (). We will insert the aerodynamic pressure term, given in eq. (2), into the Bernoulli equation evaluated at the ocean surface and we will use the method of multiple scales to obtain the corresponding nonlinear Schrödinger equation in the case of fast-growing waves. Due to the universality of the NLS equation in many other fields of physics, the considered case can in principle be of interest in other physical situations where the multiple-scale method can be applied and the forcing term is introduced at first order in the development parameter.
2 Governing equations and the method of multiple scales (MMS)
We recall here the equations governing the propagation of surface gravity waves in the presence of wind and the main assumptions used in the method of multiple scales for deriving the NLS equation.
At low viscosity the water-wave problem can be set within the framework of potential flow theory 2008Dias () and the two-dimensional flow of a viscous, incompressible fluid is governed by the Laplace equation
where is the velocity potential. This equation is solved together with the kinematic boundary condition at the free surface
and at the bottom
The other boundary condition is given by the Bernoulli equation which at the free surface takes the form
where is the gravity acceleration and is the excess pressure at the ocean surface in the presence of wind, given by eqs. (2) in the context of the Miles mechanism.
We use the method of multiple scales (MMS) to find the terms in the NLS equations which are related to the wind forcing with a growth rate of first order in the wave steepness, . This method is based on the fact that temporal and spatial scales of the carrier wave are much smaller than those of the envelope. MMS has been used for deriving the NLS equation under the assumption of small nonlinearity, , and narrow spectral width 1972Hasimoto () and successfully applied for including high-order nonlinear terms Slunyaev2005 () and constant vorticity in water waves 2012PhFl…24l7102T (), or in other physical contexts. For example, in the context of the propagation of optical waves in nonlinear materials 1994PhRvA..49..574K (), this method is also known as the slowly varying envelope approximation (SVEA) 2007PhR…441…47C (); 2007RPPh…70.1633B ().
where the second index in the amplitudes refers to the harmonics
The velocity potential at the free surface, , is written as a Taylor expansion around :
The operators for the derivatives are replaced by sums of operators
corresponding to fast and slow temporal derivatives, and analogously for . We use the same notation as in Ref. 2012PhFl…24l7102T () (note however that the order of indices in the amplitudes is inverted).
3 Wind-forced NLS equation
In this section we apply the method of multiple scales for developing the governing equations in terms of the expansion parameter . Terms of linear order in give the dispersion relation , where , and they are not affected by wind forcing. The wind forcing terms appear in the expansion at second order in the following relations:
where and .
At third order, the new terms are in the kinetic boundary condition (5), which must be evaluated using eq. (14), and in the Bernoulli equation at , eq. (7). Including these terms finally gives the wind-forced NLS equation in the limit of deep-water waves, :
where , , , and .
Note that the equation that we obtain for differs, as it should, from the usual equation obtained assuming . When , eq. (15) reduces to eq. (3). Indeed, the terms proportional to in eqs. (13)-(14) become of higher order. Moreover, in the Bernoulli equation at , the term becomes . This term in the forced NLS equation reduces for to the term , which corresponds to the one on the right-hand side of eq. (3). As we will see in the next section, the two terms in the NLS equation (15) due to wind forcing correspond to a variation of the dispersion term and of the phase of the wave field.
It is interesting to calculate the energy evolution. The Miles growth rate is recovered from the relations at second-order expansion. Indeed, multiplying eq. (13) by the complex conjugate , adding the obtained equation to its complex conjugate and integrating by parts yields
where the wave energy is defined as and is the energy density. From the same procedure, but starting from the wind-forced NLS equation (15) at third-order expansion we obtain
where is the energy flux
The second term in eq. (17) disappears if we assume that there are no incoming or outgoing waves at infinity and we get
At this third order, the wave energy is dissipated due to viscosity effects (second term on the right-hand side) and it is amplified (for ) under the action of wind (first term of the right-hand side). For comparison, in the case one obtains
4 Reduction of the wind-forced NLS to the standard NLS
In the previous section we have derived the NLS equation in the case where the growth rate of the wave energy due to the wind effect is of first order in the wave steepness, while viscosity is of second order, eq. (15). In this section we define a coordinate transformation to obtain the standard NLS equation with constant coefficients in order to use its well-known solutions.
We neglect the viscosity term (since it was already discussed in OnoratoProment2012 ()) and obtain the following equation
where , , and . We scale the envelope amplitude as
by changing its phase. Thus eq. (21) becomes
The coordinate transformation which directly reduces eq. (23) to the standard NLS equation with constant coefficients is given by
Indeed, after this transformation eq. (23) results in
Thus we have formally mapped the wind-forced NLS equation, eq. (21), into the standard NLS equation, which has a number of known analytical soliton solutions (Peregrine, Akhmediev and Kuznetsov-Ma solutions), which we will discuss in the next section.
Alternatively, another coordinate transformation is useful for understanding the physical content of eq. (21). We consider the coordinate transformation , where . The derivatives become
and eq. (23) reduces to
The factor modulates the dispersion term and consequently affects the focusing properties of the system. We check that for , the term and we recover the standard NLS equation.
5 Soliton solutions
Eq. (25) being the standard NLS equation, its solutions include the Peregrine, the Akhmediev, and the Kuzbetsov-Ma solutions. Here we discuss how the coordinate transformation (24) modifies these solutions.
In Fig. 1 we show the Peregrine solution for and . In panels - we show the unperturbed Peregrine solution, eq. (29) with and , while in panels - and - we show the wind-forced solution with and , respectively. The effect of the wind is to break the symmetry along the direction of wave propagation and to increase the temporal duration of the rogue wave event. The lifetime and the maximum amplitude of the Peregrine soliton under the effect of wind are shown in Fig. 2 (blue solid and dotted lines). The lifetime of the soliton, defined as the period of time where the rogue wave criterium is satisfied, increases as the growth rate increases, while its maximum amplitude remains constant for and slightly increases for . At the maximum splits into two peaks symmetrical with respect to the line111It is interesting to note that the values of both amplitudes and lifetimes scale with for the Peregrine soliton, so that the results shown in Figs. 1 and 2 are valid for all . This is not true for the Akhmediev soliton.. Enhancement of both amplitudes and lifetimes of rogue waves under the effect of wind has indeed been observed in tank experiments and in numerical simulations of dispersive focusing 2006EJMF…25..662T (); 2008NPGeo..15.1023T () and nonlinear focusing TouboulKharif2006 (). In these papers, the parameters are chosen so that the condition is satisfied, ensuring that their results can be compared to the ones presented here. Note that in the case only the soliton maximum amplitude increases, while its lifetime is not affected by the wind OnoratoProment2012 (). Thus, growth rates of the same order as the steepness are required to reproduce the experimental and numerical results shown in 2006EJMF…25..662T (); 2008NPGeo..15.1023T ().
The Akhmediev solution 1987TMP….72..809A (), which for large negative times corresponds to a perturbed Stokes wave, represents the nonlinear evolution of the modulational instability. It is periodic in space and its analytical form 2013PhR…528…47O () becomes for fast-growing waves
where ( is the wavenumber of the perturbation), and . In Fig. 3 we show the unperturbed Akhmediev solution for , and in panels -, and the wind-perturbed solution in panels - and - for and , respectively. The effect of the wind is similar to the case described for the Peregrine soliton, as can be expected from the interrelation between the first-order solutions of the NLS equation described in Ref. Akhmediev2009 (), although it becomes significant at a lower wind forcing for the same steepness. The shape of the wave along the direction of wave propagation is distorted, the maximum amplitude splits into two peaks symmetrical with respect to the line at (for ) and then it slightly increases for larger values, and the lifetime increases as the growth rate increases, as shown in Fig. 2 (red thick-solid and dashed lines) and Fig. 3. The spatial periodicity is not affected.
The Kusnetsov-Ma solution 1979StAM…60…43M () is periodic in time. While the large-time limit for the Akhmediev solution is a small perturbation of a plane wave, the Kusnetsov-Ma solution is never small and cannot grow from the modulational instability. Its analytical form 2013PhR…528…47O () becomes for fast-growing waves
where , and . In Fig. 4 we show the unperturbed Kusnetsov-Ma soliton for , and in panels -, and the wind-perturbed solution in panels - for . Even in this case, the effect of the wind is very strong: the amplitude of the wave is modulated in time with an envelope that diverges as . Therefore this solution is eliminated by imposing the boundary condition that the ocean surface is unperturbed at .
The forcing term in the physical context of water surface waves is the wind. Different mechanisms have been proposed in the literature for leading to wave amplification under the action of wind. In this Letter we have considered a weakly nonlinear model (the NLS equation) and the Miles mechanism for the wind-wave coupling. The growth rate of the wave energy has been compiled by different authors Banner2002 (); Farrell2008 () and its values range from - for fast-moving waves () to -1 for slow-moving waves and laboratory tank experiments (). This prompted us to investigate the case , where is the wave steepness, which both in ocean and in tank experiments is of the order of 0.1 or less.
We have used the method of multiple scales for deriving the wind-forced NLS equation with wave growth rate of first order in the wave steepness. Beside wave amplification, the effect of the wind is to modify the dispersion term. A simple coordinate transformation reduces the wind-forced NLS equation into the standard NLS equation with constant coefficients. We have thus shown that the soliton solutions (Peregrine, Akhmediev and Kuznetsov-Ma solutions) are modified in the presence of wind. In particular, the lifetime of both the Peregrine and the Akhmediev solitons increases for large growth rates. We find that the maximum amplitude of these solitons slightly increases for growth rates larger than a certain value, characterised by the fact that two maxima appear at opposite positions with respect to the line. The enhancement of both lifetime and maximum amplitude of rogue waves under the action of wind has been observed in tank experiments and numerical simulations of dispersive focusing 2006EJMF…25..662T (); 2008NPGeo..15.1023T (), thus confirming the relevance in this context of the case with respect to for which the soliton lifetime does not change under the action of wind OnoratoProment2012 ().
The results presented here should be tested in wind-wave tank experiments with different ranges of growth rate and steepness to characterise the transition between the two different regimes and .
We would like to thank Jean-Pierre Wolf and Martin Beniston for interesting discussions and the two anonymous referees for useful comments. We acknowledge financial support from the ERC advanced grant ”Filatmo” and the CADMOS project.
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