NH5.2

We study the self-modulation of flexural-gravity waves on a water surface covered by a compressed ice sheet. For weakly nonlinear long perturbations of the potential flow, we derive the nonlinear Schrödinger equation and investigate the conditions when a quasi-sinusoidal wave becomes unstable with respect to the amplitude modulation. The domains of instability are presented in the planes of parameters depending on the values of the local water depth, and the coefficients of ice rigidity/elasticity and of longitudinal stress. We show that under some conditions the occurrence of the modulational instability of oceanic waves under ice looks feasible and present estimates for the real oceanic conditions. Depending on the conditions, bright envelope solitons or dark solitons can emerge on the surface.

A.S. acknowledges the support from Laboratory of Dynamical Systems and Applications NRU HSE (the Ministry of Science and Higher Education of the Russian Federation Grant No. 075-15-2019-1931) and by the Russian Foundation for Basic Research (Grant No. 21-55-15008). Y.S. acknowledges the funding provided by the grant No. FSWE-2020-0007 through the State task program in the sphere of scientific activity of the Ministry of Science and Higher Education of the Russian Federation.

How to cite: Slunyaev, A. and Stepanyants, Y.: Self-modulation of oceanic waves on an ice-covered surface, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-1533, https://doi.org/10.5194/egusphere-egu22-1533, 2022.

Extreme waves can randomly arise in crossing seas with waves coming from two or more main directions. Linear superposition with weak nonlinearity has been proposed to explain "every-day'' extreme waves, i.e. waves with an amplitude of at least twice the amplitude of those in the ambient sea. Alternatively, higher-order nonlinear effects in statistical distributions have been simulated and observed to lead to extreme waves. In the former case, it has been proposed to reserve the term "rogue waves'' for very high and steep waves. Hence, we have investigated exact and numerical "rogue-wave'' solutions of water-wave equations for crossing seas but in a deterministic manner. Two exact web-solitons have been analysed for the unidirectional Kadomtsev-Petviashvili equation (KPE) and numerical solutions have been simulated for the bi-directional, higher-order Benney-Luke equations (BLE) in two horizontal dimensions, the latter seeded at an initial time by either one of these two exact solutions of the KPE. The first exact solution of the KPE is well-known and consists of two main soliton branches of amplitude A, interacting under an angle, to lead to a branch with amplitude 4A. The second exact solution of the KPE is less well-known and consists of three main soliton branches, each with far-field amplitude A, involving waves coming from three directions, and it leads at one point in space and time to an extreme-wave splash. We analyse this exact three-line soliton solution under a symmetric set-up and show in a novel analysis that its maximum, limiting amplification is 9A for a certain angle between the main solitary wave travelling in the positive x-direction and the two other symmetrically-aligned solitary waves. Due to a phase shift, it is only possible to reach the ninefold amplification asymptotically, given a suitable small parameter.

Simulation of these solutions is cumbersome given that they travel fast and exist on an infinite horizontal plane. Given the symmetry in both solutions, we artificially place them as initial condition on a sufficiently-large domain periodic in the x,y-directions, and show, by using further (approximate) symmetry around a y-level, that half a domain suffices with two solid walls and periodicity in the x-direction. Effectively, we have thus created cnoidal-wave solutions of crossing seas. Hence, we seed simulations of the BLE with the exact solutions at some initial time and use geometric or variational (finite-element) integrators to discretise the BLE in space and time, thus preserving phase-space volume, mass and keeping energy oscillations small and bounded, a methodology geared to avoid artificial numerical damping of wave amplitudes. Simulations at different resolutions (using polynomial or p-refinement) reveal that these two types of extreme waves or web-solitons reach maximum amplifications of circa 3.65 to 4.0 as well as circa 7.8, respectively, for maximum amplifications of 4.0 and circa 8.4 in the exact KPE solutions. Deviations of the exact solutions emerge also because of minor secondary waves created after the initial time, which cause the far-field soliton(s) of original amplitude A to oscillate in amplitude a bit, diffusing the definition of what the maximum amplification is.

How to cite: Bokhove, O., Kalogirou, A., and Choi, J.: Analysis and numerical experiments on extreme waves through oblique interaction of solitary waves, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-4143, https://doi.org/10.5194/egusphere-egu22-4143, 2022.

Reducing nonlinear evolution equations into integrable systems has emerged as a practical approach to probing nonlinear water waves. As a widely-accepted result, rogue waves can generate from the state changes of other types of wave solutions, for instance, a breather whose period approaches infinity. Crucially, the physical parameters varying across time and space can induce state transfers between different nonlinear wave solutions. More precisely, the coefficients of integrable systems are closely related to the real-world physical context. Especially in oceanography, the physical parameters concerned have large scale distribution in space and time, which implies that the simple iso-spectral models are out of reach.

The non-isospectral generalization of the (2+1)-dimensional Gardner equation adequately describes the nonlinear oceanic waves' abundant phenomena and characteristics. For this sake, we propose an extension of the Bell polynomial approach to derive its integrability features. Moreover, we obtain the explicit solution of corresponding non-isospectral solitary interactions, which indicates the drifting and alternating mechanisms of the wave envelopes in non-isospectral kink collisions.

How to cite: Zhang, L., Duan, W., and Li, J.: Non-isospectral modelling in probing nonlinear water waves, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-1680, https://doi.org/10.5194/egusphere-egu22-1680, 2022.

A depth-dependent shear current can play a significant role in modifying the statistics of wave surface elevation, including the probability of rogue waves. This work develops a second-order theory in wave steepness for the prediction of a train of weakly nonlinear irregular waves, which extends Dalzell (1999) to allow for the influence of a depth-dependent background flow and does not rely on the assumption of irrotational flows.

The theory was implemented numerically and we focus on the analysis of statistical proprieties of random waves. The linear dispersion relation of the coupled wave-current system is solved implicitly by a direct integration method (Li & Ellingsen 2019).

Both a JONSWAP spectrum and a directionally spread distribution are used to generate random linear waves, to which second-order modifications are obtained in the presence and absence of a depth-dependent shear current. The resulting probability distributions of wave crest are found to be greatly different from the Tayfun distribution (Tayfun 1980). The shear current with positive or negative vorticity leads to an increase or decrease in the probability of rogue waves, respectively. We also analyse the effects of different shear current profiles on the wave group averaged from a number of largest waves, the spectral change, and skewness and kurtosis of wave elevation.

Key words: waves/free-surface flow, ocean surface waves, wave-current interaction

References

Dalzell, J. F. (1999) "A note on finite depth second-order wave–wave interactions." Applied Ocean Research 21 105-111.

Li, Y. and Ellingsen, S. Å. (2019) "A framework for modeling linear surface waves on shear currents in slowly varying waters." Journal of Geophysical Research: Oceans 124 2527-2545.  

Tayfun, M. (1980) "Narrow-band nonlinear sea waves. " Journal of Geophysical Research 85 1548-1552..

How to cite: Zheng, Z., Li, Y., and Ellingsen, S.: Extreme waves on vertically sheared flows: Statistical analysis of weakly nonlinear waves, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-8285, https://doi.org/10.5194/egusphere-egu22-8285, 2022.

It is known that the kurtosis of wave elevations can reach a maximum near the top of abrupt depth transitions (Zeng, et al. 2012), which can be explained due to a mechanism of the interaction between free and second-order bound waves due to a depth transition (Li, et al., 2021). The horizontal velocity at a still surface can show significantly different statistics from that of surface displacement for an irregular train of random long-crested waves atop a submerged bar (Trulsen, et al. 2020). Motivated by the latter, this work focuses on effects of a submerged bar and trench on the main properties of weakly nonlinear surface gravity waves in two dimensions. The analysis is based on a novel theoretical framework that allows for narrow-banded surface waves experiencing a step-type seabed with two sudden depth transitions, correct to the second order in wave steepness. Such a seabed is modeled both as a submerged trench and bar. To reveal the fundamental physics, the evolution of a wavepacket that experiences abrupt depth transitions are examined in detail; (a) we show the differences of the release of free waves at second order in wave steepness both for the super-harmonic and sub-harmonic or ‘mean’ contents between a submerged bar and trench; (b) we also show the differences between the spatial distributions of horizontal velocity field induced by a narrowband wavepacket over a bar and a trench; (c) furthermore, we examine which parameters affect the release of free waves and the distributions of the horizontal velocity. The novel physics has implications for wave statistics for long-crested irregular waves experiencing a submerged bar as investigated experimentally by Trulsen et al. (2020) and numerically by Laurence et al. (2021) and Zhang et al. (2021).

References

Lawrence, C., Trulsen, K. & Gramstad, O. 2021 Statistical properties of wave kinematics in long-crested irregular waves propagating over non-uniform bathymetry. Phys. Fluids 33 (4), 046601.

Li, Y., Draycott, S., Zheng, Y., Lin, Z., Adcock, T.A.A. & Van Den Bremer, T.S. 2021b Why rogue waves occur atop abrupt depth transitions. J. Fluid Mech. 919, R5.

Trulsen, K., Raustøl, A., Jorde, S. & Bæverfjord Rye, L. 2020 Extreme wave statistics of long-crested irregular waves over a shoal. J. Fluid Mech. 882, R2.

Zeng, H. & Trulsen, K. 2012 Evolution of skewness and kurtosis of weakly nonlinear unidirectional waves over a sloping bottom. Nat. Hazards Earth Syst. Sci. 12 (3), 631–638.

Zhang, J. & Benoit, M. 2021 Wave–bottom interaction and extreme wave statistics due to shoaling and de-shoaling of irregular long-crested wave trains over steep seabed changes. J. Fluid Mech. 912, A28.

How to cite: Li, Y.: Effects of a submerged bar and trench on weakly nonlinear surface gravity waves, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-1772, https://doi.org/10.5194/egusphere-egu22-1772, 2022.

The Galilei transformation (GT) is a universal operation connecting the fixed and translated co-ordinates of a dynamical system. When considering exact wave envelope solutions of the nonlinear Schrödinger equation (NLSE) propagating with a relative Galilei speed (GS), the GT imposes a frequency shift to satisfy the symmetry. This limits the applicability of the GT to nonlinear dispersive waves. We show that the Galilei-transformed envelope solitons and Peregrine breathers generated in a wave tank clearly deviate from their respective pure NLSE hydrodynamics. The type of discrepancies depends on the sign of the GS while these can be still quantified by the modified NLSE or solving the Euler equations. Furthermore, Galilei-transformed envelope solitons with positive GSs exhibit self-modulation. With designated GS and steepness values, such solitons can be transformed to follow the exact dynamics of higher-order solitons, which under specific circumstances are responsible for the generation of supercontinua.

How to cite: He, Y., Ducrozet, G., Hoffmann, N., Dudley, J. M., and Chabchoub, A.: A numerical and experimental study of Galilei-transformed nonlinear wave groups, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-8981, https://doi.org/10.5194/egusphere-egu22-8981, 2022.

The present study deals with the generation and the propagation of unidirectional irregular waves in experimental or numerical wave tanks. It introduces, tests and validates a new wave generation procedure, allowing for the accurate control of a generated sea state at any position of the domain.

A sea state represents the wave conditions observed at sea. It is defined with a wave spectrum and a certain duration, associated to the probability of occurrence of the extreme events. A sea state can be reproduced at model scale in wave tanks, usually for wave structure interaction tests with a typical duration of 3hours at full scale.

To generate a sea state, the usual procedures rely on a stochastic approach. Long-duration free surface elevation time-series (realizations) are generated by the wave maker. They are built in the Fourier space using the linear dispersive waves theory. The amplitudes are set using the wave spectrum of interest, and the phases are random. The quality of the generated wave field is then assessed at a target position Xt (usually the position of a tested marine structure). The qualification of the sea state at Xt mainly rely on the mean wave spectrum (over all the realizations) and the ensemble crest heigth distribution (considering all the realizations).

However, nonlinear phenomena occur from the wave maker to the target position. They affect the shape of the spectrum and deviate the crest distributions from its references. The most advanced state-of-the-art wave generation procedures iterate on the wave maker motion to correct the spectrum at Xt. However, recent studies [1] show that using such methods, the crest statistics can strongly vary with Xt.

On those grounds, our new procedure complete the state-of-the-art methodologies. It allows for the control of the spectrum together with the crest distributon at any target position. The method is numerically tested and validated with the nonlinear wave propagation solver HOS-NWT [2]. And some preliminary experimental results are presented.

[1] Canard, M., Ducrozet, G., and Bouscasse, B., 2022 (accept-edfor publication). Varying ocean wave statistics emerg-ing from a single energy spectrum in an experimental wavetank. Ocean Engineering.

[2] Ducrozet, G., Bonnefoy, F., Le Touzé, D., & Ferrant, P. (2012). A modified high-order spectral method for wavemaker modeling in a numerical wave tank. European Journal of Mechanics-B/Fluids, 34, 19-34.

How to cite: Canard, M., Ducrozet, G., and Bouscasse, B.: Control of crest heigth statistics at a target position in a wave tank environement, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-11154, https://doi.org/10.5194/egusphere-egu22-11154, 2022.

Freak waves are extreme events in the ocean, usually defined as waves with crest-to-trough excursion higher than twice the significant wave height of the ambient sea-state. In recent decades, freak waves have attracted considerable attention of oceanographers, as they seem to manifest with unexpected high frequencies and resulted in numerous tragedies.  

The dominant formation mechanisms of freak waves are still an open question, and various hypotheses have been put forward. The modulation instability (MI) is the most well-known prototype of freak wave in deep water. However, some researchers argue that the role played by MI is insignificant in real ocean, where the sea-state could be stabilized by the wave directionality, finite spectral width and shallow water depth. The non-equilibrium dynamics (NED) induced by significant depth variations could explain freak waves occurring in coastal areas where the MI is absent.

The NED manifests when an incident quasi-equilibrium sea-state undergoes a rapid depth decrease, and propagates in the new shallower water depth. Before reaching the new equilibrium state after depth transition, the wave evolution is characterized by strong non-Gaussian behavior. Previous studies mainly focused on the NED effects over an extent of a couple of wavelengths after the depth variation, showing the local enhancement of skewness and kurtosis (third- and fourth-order moments of the free surface elevation), the excitation of bound super-harmonics, and the intensified occurrence probability of freak waves. However, very few studies discuss how NED fades away at larger scale, and what is the equilibrium state established in the shallower region.

The present work numerically investigates the experiments reported by Trulsen et al. (J. Fluid Mech., vol. 882, R2, 2020) using a fully nonlinear potential wave model. We extend the analysis of NED to a longer spatial extent in the shallow water area, from O(L_p) to O(10² L_p), with L_p being the spectral peak wavelength. It is found that the NED affects the shallow water wave evolution in two spatial scales: (i) in the shorter scale O(L_p), as reported in the literature, the sea-state undergoes fast changes of wave statistics (skewness, kurtosis and probability of freak waves); (ii) in the longer scale O(10² L_p), the NED results in strong modulation of the spectral shape. The analysis of the wave height distribution shows that, in the long scale, the sea-state is still non-Gaussian, and the freak wave occurrence probability is lower than the linear expectation. So the NED effects could “protect” the structures and ships from freak waves, at long distances from abrupt depth transitions. The output of the current work allows for a better assessment of the risk of freak waves in coastal areas, with practical benefits for coastal engineers and oceanographers.

How to cite: Zhang, J., Benoit, M., and Ma, Y.: Wave equilibration process of a non-equilibrium sea-state in shallow water after strong depth variation, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-1977, https://doi.org/10.5194/egusphere-egu22-1977, 2022.

Non-equilibrium evolution of wave fields due to shoaling can amplify rogue wave statistics. This process is studied by the analysis of spatial variations in the energy density and time averages, affecting the Khintchine theorem. The resulting probability model reproduces the heavy tail of the probability distribution of unidirectional wave tank experiments, describes why the peak of rogue wave probability appears atop the shoal, and explains the influence of depth on variations in peak intensity.

How to cite: Mendes, S., Scotti, A., Brunetti, M., and Kasparian, J.: Non-homogenous analysis of shoaling rogue wave statistics, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-3107, https://doi.org/10.5194/egusphere-egu22-3107, 2022.

The hydrodynamics of potential flows of a 3D ideal incompressible fluid with a free surface in a gravitational field is considered in the approximation of the Zakharov equation. In the case of one-dimensional waves a special property of the four-wave interaction coefficient in the Zakharov equation allows it to be written in a simple form of the so-called supercompact equations for one-dimensional counterpropagating waves. We generalize the system of equations to the case of two-dimensional surface waves and study the problem of modulation instability for a monochromatic and standing wave, as well as resonant interactions of such waves within the framework of this model. The work was supported by Grant No. 19-72-30028 of the Russian Science Foundation.

How to cite: Dremov, S., Kachulin, D., and Dyachenko, A.: The System of Supercompact Equations for Two-Dimensional Waves Propagating on the Surface of a Three-Dimensional Deep Fluid, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-9535, https://doi.org/10.5194/egusphere-egu22-9535, 2022.

We perform the direct numerical simulation of surface gravity waves in the deep sea with the initial conditions specified by waves with a given JONSWAP spectrum and the directional spreading according to the cos² distribution. The High Order Spectral Method employed for the simulation, allows to control the order of nonlinearity through the parameter of the scheme, M. In particular, the value M = 1 corresponds to the linear solution, M = 3 – to the account of the cubic nonlinearity due to the four-wave nonlinear interactions. Most of the direct numerical simulations of the HOSM available in the literature, are performed with the parameter M = 3, which is sufficient to take into account the modulational instability. In this work we examine the role of even higher order nonlinear effects due to 5-wave interactions. To this end, a series of comparative numerical simulations have been performed with M = 3 and M = 4. The obtained wave data are examined with respect to the probability distribution functions for the wave heights, and the typical rogue wave shapes. So far, no new dynamical effects between waves associated with the high-order nonlinearity is found. The high-order nonlinearity seems to affect the dynamics of very steep waves leading to the generation of even slightly higher waves. The main part of the wave height probability distribution function remains unchanged.

The research is supported by the RFBR grants Nos. 20-05-00162 and 21-55-15008.

[1] A. Sergeeva (Kokorina), A. Slunyaev, Rogue waves, rogue events and extreme wave kinematics in spatio-temporal fields of simulated sea states. Nat. Hazards Earth Syst. Sci. 13, 1759-1771 (2013).

[2] A. Slunyaev, A. Sergeeva (Kokorina), I. Didenkulova, Rogue events in spatiotemporal numerical simulations of unidirectional waves in basins of different depth. Natural Hazards 84(2), 549-565 (2016).

[3] A. Slunyaev, A. Kokorina, Account of occasional wave breaking in numerical simulations of irregular water waves in the focus of the rogue wave problem. Water Waves 2(2), 243-262 (2020).

[4] A. Slunyaev, A. Kokorina, Numerical Simulation of the Sea Surface Rogue Waves within the Framework of the Potential Euler Equations. Izvestiya, Atmospheric and Oceanic Physics 56, No. 2, 179–190 (2020).

How to cite: Kokorina, A. and Slunyaev, A.: Numerical simulation of directional JONSWAP sea waves taking into account four- and five-wave nonlinear resonances, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-3393, https://doi.org/10.5194/egusphere-egu22-3393, 2022.

The numerical direct and inverse scattering transform applications represent a broad topic of nonlinear wave field studies [1]. Here we investigate various stochastic nonlinear wavefields with the dominant role of a large number of solitons within the one-dimensional nonlinear Schrodinger equation model. First, applying the recently developed direct scattering transform numerical scheme allowing accurate identification of the complete wavefield scattering data [2,3], we find distributions of all soliton parameters (amplitudes, velocities, positions, and phases). Then, using the previously developed numerical tools of solving the inverse scattering problem for a large number of solitons [4], we reconstruct the solitonic content of the initial wave field, which allows us to estimate the role of solitons in the initial wave field composition. Finally, we discuss the obtained scattering data distributions, paying particular attention to the correlations in parameters of different solitons. The presented accurate characterization of soliton positions and phases in stochastic nonlinear wavefields can be used in further studies of such important realms of nonlinear physics as spontaneous modulation instability development [5] and integrable turbulence growing [6].

The work was supported by Russian Science Foundation grant No. 20-71-00022.

[1] A. Osborne, Nonlinear Ocean Waves (Academic Press, New York, 2010).

[2] A. Gelash, and R. Mullyadzhanov, Physical Review E, 101(5), 052206, 2020.

[3] R. Mullyadzhanov, and A. Gelash, Optics Letters, 44(21), 5298-5301, 2019.

[4] A. A. Gelash and D. S. Agafontsev, Physical Review E 98, 042210, 2018.

[5] D. S. Agafontsev and V. E. Zakharov, Nonlinearity 28, 2791, 2015.

[6] D. S. Agafontsev and V. E. Zakharov, Low Temperature Physics, 46(8), 786-791, 2020.

How to cite: Gelash, A.: Direct and inverse scattering transform analysis of stochastic nonlinear wavefields, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-4744, https://doi.org/10.5194/egusphere-egu22-4744, 2022.