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The stability of an exponential current in water to infinitesimal perturbations in the presence of gravity and capillarity is revisited and reformulated using the Weber and Froude numbers. Some new results on the generation of gravity-capillary waves are presented, which supplement the previous works of Morland et al. ["Waves generated by shear layer instabilities," Proc. Math. Phys. Sci. 433, 441–450 (1991)] and Young and Wolfe ["Generation of surface waves by shear-flow instability," J. Fluid Mech. 739, 276–307 (2014)] on finite depth. To consider perturbations at much larger scales, special attention is given to the stability of exponential currents only in the presence of gravity. More precisely, the present investigation reveals significant insights into the stability of exponential shear currents under different environmental conditions. Notably, we have identified that the dimensionless growth rate increases with the Froude number, providing a deeper understanding of the interplay between shear layer thickness and surface velocity. Furthermore, our analysis elucidates the dimensional wavelength of the most unstable mode, emphasizing its relevance to the characteristic shear layer thickness. Additionally, within the realm of gravity-capillary instabilities, we have established a sufficient condition for the stability of exponential currents based on the Weber number. Our findings are supported by stability diagrams at finite depth, showing how the size of stable domains correlates with the characteristic thickness of the shear layer. Moreover, we have explored the stability of a thin film of liquid in an exponential shearing flow, further enriching our understanding of the complex dynamics involved in such systems. [ABSTRACT FROM AUTHOR]

The bifurcation of two-dimensional gravity–capillary waves into solitary waves when the phase velocity and group velocity are nearly equal is investigated in the presence of constant vorticity. We found that gravity–capillary solitary waves with decaying oscillatory tails exist in deep water in the presence of vorticity. Furthermore we found that the presence of vorticity influences strongly (i) the solitary wave properties and (ii) the growth rate of unstable transverse perturbations. The growth rate and bandwidth instability are given numerically and analytically as a function of the vorticity. [ABSTRACT FROM AUTHOR]

A nonlinear Schrödinger equation for the envelope of two-dimensional gravity- capillary waves propagating at the free surface of a vertically sheared current of constant vorticity is derived. In this paper we extend to gravity-capillary wave trains the results of Thomas et al. (Phys. Fluids, 2012, 127102) and complete the stability analysis and stability diagram of Djordjevic & Redekopp (J. Fluid Mech., vol. 79, 1977, pp. 703-714) in the presence of vorticity. The vorticity effect on the modulational instability of weakly nonlinear gravity-capillary wave packets is investigated. It is shown that the vorticity modifies significantly the modulational instability of gravity-capillary wave trains, namely the growth rate and instability bandwidth. It is found that the rate of growth of modulational instability of short gravity waves influenced by surface tension behaves like pure gravity waves: (i) in infinite depth, the growth rate is reduced in the presence of positive vorticity and amplified in the presence of negative vorticity; (ii) in finite depth, it is reduced when the vorticity is positive and amplified and finally reduced when the vorticity is negative. The combined effect of vorticity and surface tension is to increase the rate of growth of modulational instability of short gravity waves influenced by surface tension, namely when the vorticity is negative. The rate of growth of modulational instability of capillary waves is amplified by negative vorticity and attenuated by positive vorticity. Stability diagrams are plotted and it is shown that they are significantly modified by the introduction of the vorticity. [ABSTRACT FROM AUTHOR]

Interpretation of random wave field on a shallow water in terms of Fourier spectra is not adequate, when wave amplitudes are not infinitesimally small. A nonlinearity of wave fields leads to the harmonic interactions and random variation of Fourier spectra. As has been shown by Osborne and his co-authors, a more adequate analysis can be performed in terms of nonlinear modes representing cnoidal waves; a spectrum of such modes remains unchanged even in the process of nonlinear mode interactions. Here we show that there is an alternative and more simple analysis of random wave fields on shallow water, which can be presented in terms of interacting Korteweg–de Vries solitons. The data processing of random wave field is developed on the basis of inverse scattering method. The soliton component obscured in a random wave field is determined and a corresponding distribution function of number of solitons on their amplitudes is constructed. The approach developed is illustrated by means of artificially generated quasi-random wave field and applied to the real data interpretation of wind waves generated in the laboratory wind tank. [ABSTRACT FROM AUTHOR]

Subaerial landslides are common events, which may generate very large water waves. The numerical modelling and simulation of these events are thus of primary interest for forecasting and mitigation of tsunami disasters. In this paper, we aim at describing these extreme events using a simplified shallow water model to derive relevant scaling laws. To cope with the problem, two different numerical codes are employed: one, SPHysics, is based on a Lagrangian meshless approach to accurately describe the impact stage whereas the other, Gerris, based on a two-phase finite-volume method is used to study the propagation of the wave. To validate Gerris for this very particular problem, two numerical cases of the literature are reproduced: a vertical sinking box and a 2-D wedge sliding down a slope. Then, to get insights into the problem of subaerial landslide-generated tsunamis and to further validate the codes for this case of landslides, a series of experiments is conducted in a water wave tank and successfully compared with the results of both codes. Based on a simplified approach, we derive different scaling laws in excellent agreement with the experiments and numerical simulations. [ABSTRACT FROM AUTHOR]

A nonlinear Schrödinger equation for the envelope of two dimensional surface water waves on finite depth with non-zero constant vorticity is derived, and the influence of this constant vorticity on the well-known stability properties of weakly nonlinear wave packets is studied. It is demonstrated that vorticity modifies significantly the modulational instability properties of weakly nonlinear plane waves, namely the growth rate and bandwidth. At third order, we have shown the importance of the nonlinear coupling between the mean flow induced by the modulation and the vorticity. Furthermore, it is shown that these plane wave solutions may be linearly stable to modulational instability for an opposite shear current independently of the dimensionless parameter kh, where k and h are the carrier wavenumber and depth, respectively. [ABSTRACT FROM AUTHOR]

The modulational instability of gravity wave trains on the surface of water acted upon by wind and under influence of viscosity is considered. The wind regime is that of validity of Miles' theory and the viscosity is small. By using a perturbed nonlinear Schrödinger equation describing the evolution of a narrow-banded wavepacket under the action of wind and dissipation, the modulational instability of the wave group is shown to depend on both the frequency (or wavenumber) of the carrier wave and the strength of the friction velocity (or the wind speed). For fixed values of the water-surface roughness, the marginal curves separating stable states from unstable states are given. It is found in the low-frequency regime that stronger wind velocities are needed to sustain the modulational instability than for high-frequency water waves. In other words, the critical frequency decreases as the carrier wave age increases. Furthermore, it is shown for a given carrier frequency that a larger friction velocity is needed to sustain modulational instability when the roughness length is increased. [ABSTRACT FROM AUTHOR]

Abstract: This paper reports on a series of numerical simulations designed to investigate the action of wind on steep waves and breaking waves generated through the mechanism of dispersive focusing on finite depth. The dynamics of the wave packet propagating without wind at the free surface are compared to the dynamics of the packet propagating in the presence of wind. Wind is introduced in the numerical wave tank by means of a pressure term, corresponding to the modified Jeffreys'' sheltering mechanism. The wind blowing over a strongly modulated wave group due to the dispersive focusing of an initial long wave packet increases the duration and maximal amplitude of the steep wave event. These results are coherent with those obtained within the framework of deep water. However, steep wave events are less unstable to wind perturbation in shallow water than in deep water. Furthermore, a comparison between experimental and numerical wave breaking is presented in the absence of wind. The numerical simulations show that the wind speeds up the wave breaking and amplifies slightly the wave height. The wall pressure during the runup of the steep wave event on a vertical wall is also investigated and a comparison between experimental and numerical results is provided. [Copyright &y& Elsevier]

The interaction of wind and water wave groups is investigated theoretically and numerically. A steep wave train is generated by means of dispersive focusing, using both the linear theory and fully nonlinear equations. The linear theory is based on the Schrödinger equation while the nonlinear approach is developed numerically within the framework of the potential theory. The interaction between the chirped wave packet and wind is described by the Miles' mechanism. The differences between both approaches are discussed, and the influence of nonlinearity is emphasized. Furthermore, a different mechanism is considered, described by the modified Jeffreys' sheltering theory. From comparison between the two mechanisms, it is found that the persistence of the steep wave group depends on the physical model used, and is significantly increased when we use the latter mechanism. [ABSTRACT FROM AUTHOR]

Freak waves are generated numerically by means of modulational instability. Their interaction with wind is investigated. Wind is modeled as Jeffreys’ sheltering mechanism. Contrary to the case without wind, it is found that wind sustains the maximum of modulation due to the Benjamin-Feir instability. The general kinematic behavior observed for freak waves due to dispersive focusing is recovered here, even if the underlying physics are different in both cases. [ABSTRACT FROM AUTHOR]

Nearly 40 years have passed since V. I. Talanov discovered the nonlinear parabolic equation which played an important role in the nonlinear optics. It was very quickly understood that this equation could also be adapted for nonstationary wave packets of different physical nature and of any dimension. Under the later name of the nonlinear (cubic) Schrödinger equation, it became a fundamental equation in the theory of weakly nonlinear wave packets in media with strong dispersion. The article is devoted to only one application of the nonlinear Schrödinger equation in the theory of the so-called freak waves on the sea surface. In the last five years a great boom has occurred in the research of extreme waves on the water, for which the nonlinear parabolic equation played an important role in the understanding of physical mechanisms of the freak-wave phenomenon. More accurate, preferably numerical, models of waves on a water with more comprehensive account of the nonlinearity and dispersion come on the spot today, and many results of weakly nonlinear models are already corrected quantitatively. Nevertheless, sophisticated models do not bring new physical concepts. Hence, their description on the basis of the nonlinear parabolic equation (nonlinear Schrödinger equation), performed in this paper, seems very attractive in view of their possible applications in the wave-motion physics. [ABSTRACT FROM AUTHOR]

Being considered as a prototype for description of oceanic rogue waves, the Peregrine breather solution of the nonlinear Schrödinger equation has been recently observed and intensely investigated experimentally in particular within the context of water waves. Here, we report the experimental results showing the evolution of the Peregrine solution in the presence of wind forcing in the direction of wave propagation. The results show the persistence of the breather evolution dynamics even in the presence of strong wind and chaotic wave field generated by it. Furthermore, we have shown that characteristic spectrum of the Peregrine breather persists even at the highest values of the generated wind velocities thus making it a viable characteristic for prediction of rogue waves. [ABSTRACT FROM AUTHOR]

The dynamics of surface gravity water waves can be described by the self-defocusing nonlinear Schrödinger equation. Recent observations of black solitons on the surface of water confirmed its validity for finite, below critical depth. The black soliton is a limiting case of a family of gray soliton solutions with finite amplitude depressions. Here, we report observations of gray solitons in water waves, thus, complementing our previous observations of black solitons. [ABSTRACT FROM AUTHOR]