Extreme dynamics of wave groups on jet currents

Rogue waves are known to be much more common on jet currents. A possible explanation was put forward in Shrira and Slunyaev [“Nonlinear dynamics of trapped waves on jet currents and rogue waves,” Phys. Rev. E 89, 041002(R) (2014)] for the waves trapped on current robust long-lived envelope solitary waves localized in both horizontal directions become possible, such wave patterns cannot exist in the absence of the current. In this work, we investigate interactions between envelope solitons of essentially nonlinear trapped waves by means of the direct numerical simulation of the Euler equations. The solitary waves remain localized in both horizontal directions for hundreds of wave periods. We also demonstrate a high efficiency of the developed analytic nonlinear mode theory for description of the long-lived solitary patterns up to remarkably steep waves. We show robustness of the solitons in the course of interactions and the possibility of extreme wave generation as a result of solitons' collisions. Their collisions are shown to be nearly elastic. These robust solitary waves obtained from the Euler equations without weak nonlinearity assumptions are viewed as a plausible model of rogue waves on jet currents

Sporadic wind wave horse-shoe patterns

International audienceThe work considers three-dimensional crescent-shaped patterns often seen on water surface in natural basins and observed in wave tank experiments. The most common of these 'horse-shoe-like' patterns appear to be sporadic, i.e., emerging and disappearing spontaneously even under steady wind conditions. The paper suggests a qualitative model of these structures aimed at explaining their sporadic nature, physical mechanisms of their selection and their specific asymmetric form. First, the phenomenon of sporadic horse-shoe patterns is studied numerically using the novel algorithm of water waves simulation recently developed by the authors (Annenkov and Shrira, 1999). The simulations show that a steep gravity wave embedded into widespectrum primordial noise and subjected to small nonconservative effects typically follows the simple evolution scenario: most of the time the system can be considered as consisting of a basic wave and a single pair of oblique satellites, although the choice of this pair tends to be different at different instants. Despite the effective low-dimensionality of the multimodal system dynamics at relatively sho ' rt time spans, the role of small satellites is important: in particular, they enlarge the maxima of the developed satellites. The presence of Benjamin-Feir satellites appears to be of no qualitative importance at the timescales under consideration. The selection mechanism has been linked to the quartic resonant interactions among the oblique satellites lying in the domain of five-wave (McLean's class II) instability of the basic wave: the satellites tend to push each other out of the resonance zone due to the frequency shifts caused by the quartic interactions. Since the instability domain is narrow (of order of cube of the basic wave steepness), eventually in a generic situation only a single pair survives and attains considerable amplitude. The specific front asymmetry is found to result from the interplay of quartic and quintet interactions and non-conservative effects: the growing and grown satellites have a specific value of phase with respect to the basic wave that corresponds to downwind orientation of the convex sides of wave fronts. As soon as the phase relation is violated, the satellite's amplitude quickly decreases down to the noise level

Surface gravity waves in deep fluid at vertical shear flows

Special features of surface gravity waves in deep fluid flow with constant vertical shear of velocity is studied. It is found that the mean flow velocity shear leads to non-trivial modification of surface gravity wave modes dispersive characteristics. Moreover, the shear induces generation of surface gravity waves by internal vortex mode perturbations. The performed analytical and numerical study provides, that surface gravity waves are effectively generated by the internal perturbations at high shear rates. The generation is different for the waves propagating in the different directions. Generation of surface gravity waves propagating along the main flow considerably exceeds the generation of surface gravity waves in the opposite direction for relatively small shear rates, whereas the later wave is generated more effectively for the high shear rates. From the mathematical point of view the wave generation is caused by non self-adjointness of the linear operators that describe the shear flow.Comment: JETP, accepte

Trapping and instability of directional gravity waves in localized water currents

The influence of localized water currents on the nonlinear dynamics and stability of large amplitude, statistically distributed gravity waves is investigated theoretically and numerically by means of an evolution equation for a Wigner function governing the spectrum of waves. It is shown that water waves propagating in the opposite direction of a localized current channel can be trapped in the channel, which can lead to the amplification of the wave intensity. Under certain conditions the wave intensity can be further localized due to a self-focusing (Benjamin-Feir) instability. The localized amplification of the wave intensity may increase the probability of extreme events in the form of freak waves, which have been observed in connection with ocean currents

Trapped waves on jet currents: asymptotic modal approach

AbstractAn asymptotic theory of surface waves trapped on vertically uniform jet currents is developed as a first step towards a systematic description of wave dynamics on oceanic jet currents. It has been shown that in a linear setting an asymptotic separation of vertical and horizontal variables, which underpins the modal description of the wave field on currents, is possible if either the current velocity is small compared to the wave celerity or the current width is large compared to the wavelength along the current. The scheme developed enables us to obtain solutions as an asymptotic series with any desired accuracy. The initially three-dimensional problem is reduced to solving one-dimensional equations with the lateral and vertical dependence being prescribed by the corresponding modal structure. To leading order in current magnitude to wave celerity, the boundary value problem specifying the modes and eigenvalues reduces to classical Sturm–Liouville type based upon the one-dimensional stationary Schrödinger equation. The modes, both trapped and ‘passing-through’, form a complete orthogonal set. This makes the modal description of waves on currents a mathematically attractive alternative to the approaches currently adopted. Properties of trapped eigenmodes and their dispersion relations are examined both for broad currents of arbitrary magnitude, where the modes are not orthogonal, and for weak currents, where the modes are orthogonal. Several model profiles for which nice analytical solutions of the leading-order boundary value problem are known were used to get an insight. The asymptotic solutions proved not only to capture qualitative behaviour well but also to provide a good quantitative description even for unrealistically strong and narrow currents. The results are discussed for various oceanic currents, with particular attention paid to the Agulhas Current, for which specific estimates were derived. For typical dominant wind waves and swell, all oceanic-jet-type currents are weak and, correspondingly, the developed asymptotic scheme based upon one-dimensional stationary Schrödinger equation for modes applies.</jats:p

Evaluation of Skewness and Kurtosis of Wind Waves Parameterized by JONSWAP Spectra

Abstract The authors consider the deviations of wave height statistics from Gaussianity, manifested in higher statistical moments of random wind-wave fields, namely, in the nonzero values of the skewness and the kurtosis. These deviations are examined theoretically under the standard set of assumptions used in the established statistical theory of water waves, in particular in the derivation of the Hasselmann kinetic equation. P. Janssen proposed integral representations of the skewness and the kurtosis in terms of multidimensional integrals of wave spectra. However, the use of these representations for broadband wind-wave fields proved to be challenging; it requires substantial computational resources, which is unsuitable for applications. Using specially designed parallel algorithms to evaluate the integrals, the authors provide a comprehensive picture of the behavior of the kurtosis and the skewness of wind waves in the multidimensional parameter space of the most commonly used Joint North Sea Wave Project (JONSWAP) parameterizations of wind-wave spectra. Except for very narrow angular distributions where the overall picture is qualitatively different, the behavior of the higher moments proved to be not sensitive to the particular form of the directional spectrum. On this basis for the broad angular spectra typical of the ocean, the study puts forward simple parameterizations of the skewness and the kurtosis in terms of the JONSWAP peakedness parameter γ and in terms of the inverse wave age. These parameterizations can be used in operational wave forecasting and other applications.</jats:p