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Extreme events in sea waves: physical mechanisms and mathematical models

The scope of this session includes different aspects of large-amplitude wave phenomena in the ocean such as freak or rogue waves, surface and internal waves, as well as waves trapped by currents and bathymetry. The session is focused on the understanding of the physical mechanisms which cause extreme events, the derivation of appropriate mathematical models for their description and advanced methods for their analysis. An essential part of such studies is the validation of new models and techniques versus laboratory and in-situ data. Special attention is paid to the description of wave breaking processes, and the interaction of large-amplitude waves with marine structures in offshore and coastal areas

Convener: Alexey Slunyaev | Co-conveners: Amin Chabchoub, Henrik Kalisch, Yan LiECSECS, Efim Pelinovsky
| Mon, 23 May, 15:10–18:30 (CEST)
Room 1.61/62

Mon, 23 May, 15:10–16:40

Chairpersons: Amin Chabchoub, Yan Li


Johannes Gemmrich et al.

Direct observations of rogue waves in high sea states are rare. However, rogue waves can pose a danger to marine operations, onshore and offshore structures, and beachgoers. Here we report on a 17.6m rogue wave in coastal waters with crest height about twice, and wave height almost three times the significant wave height. These are amongst the largest normalized heights ever recorded.

Simulations of random superposition of Stokes waves in intermediate water depth show good agreement with the observation, whereas non-linear wave modulational instability did not contribute significantly to the rogue wave generation. We present a spectral parameter that can easily be derived from a routine wave forecast as an indicator of rogue wave risk. These results confirm that probabilistic prediction of oceanic rogue waves based on random superposition of steep waves are possible and should replace predictions based on modulational instability. Furthermore, large individual waves offshore do not necessarily result in extreme runup on the beach.

How to cite: Gemmrich, J., Cicon, L., and Holmes-Smith, C.: Observation and prediction of a coastal rogue wave, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-3231, https://doi.org/10.5194/egusphere-egu22-3231, 2022.


Ekaterina Didenkulova et al.

Rogue waves are abnormally large waves in the ocean that are at least twice as large as their surrounding waves. The present work combines existing data of rogue wave events, which have been reported in mass media sources. These rogue events caused damages of ships, oil platforms, coastal structures, and human losses [1-6]. Evidences of this phenomenon are widely spread around the globe.

The database includes 431 rogue events registered during the period 2005-2021. The following information about each event is available: data and location, description, reported wave height of the event (not always), damages, link to the source.

Locations of the events have been determined approximately based on the eyewitnesses’ reports. The water depth for each event has been taken from the GEBCO database. Based on this water depth, all events have been separated into groups based on the depth of their occurrence: deep water (depth is more than 50m), shallow water (depth is less than 50 m), and coast. The latter represented either gentle beaches or high rocky coasts.

Using the data from global atmospheric and ocean reanalysis ERA5, the characteristics of background waves and maximal individual waves in the area as well as meteorological conditions have also been determined and analyzed. This includes wind speed, gust, significant wave height, maximum individual wave height, peak wave period, and spectra. According to these data, the freak events that satisfy the criterion of modulation instability kh>1.363 (where h is the water depth and k is the wave number) have been distinguished.

According to the events’ descriptions and ERA5 information, all rogue wave events have been divided into two groups: “true” and “possible”. For true events the wave description satisfies the freak wave conditions: to be unexpected and abnormally high – twice larger than the background waves. The events, which could not be classified with certainty as “true” due to the lack of data, but which could still be related to rogue wave events, have been considered as “possible”.

Based on the available data the conclusions about characteristics of a rogue wave, associated to accidents, their occurrence, and their statistics are drawn.

This work was supported by the Russian Science Foundation (project No. 21-77-00003).


1) Didenkulova I, Slunyaev A, Pelinovsky E, Kharif Ch (2006) Freak waves in 2005. Nat Hazard Earth Syst Sci 6:1007-1015

2) Nikolkina I, Didenkulova I (2011) Rogue waves in 2006 – 2010. Nat Hazards Earth Syst Sci 11: 2913–2924

3) O'Brien L, Renzi E, Dudley J M, Clancy C, Dias F (2018) Catalogue of extreme wave events in Ireland: revised and updated for 14680 BP to 2017. Nat Hazards Earth Syst Sci 18:729-758

4) García-Medina G, Özkan-Haller H T, Ruggiero P et al. (2018) Analysis and catalogue of sneaker waves in the US Pacific Northwest between 2005 and 2017. Nat Hazards 94: 583–603

5) Didenkulova E (2020) Catalogue of rogue waves occurred in the World Ocean from 2011 to 2018 reported by mass media sources. Ocean and Coastal Management 188: 105076

6) Didenkulova I, Didenkulova E, Didenkulov O (2022) Freak wave accidents in 2019-2021. Proceedings of OCEANS 

How to cite: Didenkulova, E., Didenkulova, I., Didenkulov, O., and Medvedev, I.: Analysis of rogue wave events in 2005-2021, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-5640, https://doi.org/10.5194/egusphere-egu22-5640, 2022.


Efim Pelinovsky et al.

As it is known, the problem of finding traveling waves in 1D nonlinear and dispersive media may be reduced to solving a system of ordinary differential equations. If the order of the system is large, then the internal wave structure can be very complicated and even random. If the medium is inhomogeneous, it is natural to expect the absence of solutions in the form of traveling waves due to the reflection and multiple reflection effects. If, however, the medium parameters change slowly (in comparison to the wavelength), and the reflection is weak, it becomes possible to construct an approximate solution in the form of a traveling wave with a variable amplitude and phase by using asymptotic methods (WKB, geometric optics or acoustics). For the media with a monotonic change in parameters, such solutions demonstrate the highest gain and the ability to transmit a signal over long distances without distortion.

It turns out to be possible to find exact solutions in the form of traveling waves with variable amplitude and phase in highly inhomogeneous media under certain assumptions on the medium parameters. Our paper reviews possible approaches to finding travelling reflectionless waves in the shallow water channels with variable cross-sections and currents. The basic equations are the classical 1D nonlinear shallow-water equations for water displacement and velocity averaged through the cross-channel. Mathematical procedure to get the solutions in the form of travelling reflectionless waves is based on the transformation of the original equations with variable coefficients to the constant-coefficient PDE. We first demonstrate this procedure using the example of the classical linear wave equation with variable coefficient when it can be reduced to the Klein-Gordon equation with constant coefficients. This gives rise to an ordinary second-order differential equation for finding a variable coefficient (the wave speed), so that traveling waves exist in a wide class of inhomogeneous water channels. The second procedure is the reducing of variable-coefficient 1D wave equation to the spherical symmetric wave equation in the odd-dimensional space, where waves traveling to and from the center are separated. More complicated procedure is developed for the channels with non-uniform current. In conclusion, we discuss the effectiveness of this procedure in the framework of Boussinesq systems.

The study is supported by grants RFBR (20-05-00162, 21-55-15008, 19-35-60022), President of the RF for the state support of Leading Scientific Schools of the RF (Grant No. NSH-70.2022.1.5).

Recent publications:

  • Didenkulova I. and Pelinovsky E. On shallow water rogue wave formation in strongly inhomogeneous channels. Journal of Physics A: Mathematical and Theoretical, 2016, vol. 49, 194001.
  • Pelinovsky E., Didenkulova I., Shurgalina E., and Aseeva N. Nonlinear wave dynamics in self-consistent water channels. J Phys. A, 2017, vol. 50, 505501.
  • Pelinovsky E., Talipova T., Didenkulova I., Didenkulova E. Interfacial long traveling waves in a two-layer fluid with variable depth. Studies in Applied Mathematics, 2019, vol. 142, No. 4, 513–527.
  • Churilov S.M., Stepanyants Yu.A. Reflectionless wave propagation on shallow water with variable bathymetry and current. J. Fluid Mech., 2022, vol. 931, A15.

How to cite: Pelinovsky, E., Talipova, T., Didenkulova, E., Kaptsov, O., Stepanyants, Y., Churilov, S., and Didenkulova, I.: Travelling reflectionless waves in shallow water channels with variable cross-section and current, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-1541, https://doi.org/10.5194/egusphere-egu22-1541, 2022.


Alexey Slunyaev and Yury Stepanyants

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.


Onno Bokhove et al.

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.


Longyuan Zhang et al.

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.


Zibo Zheng et al.

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



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.


Yan Li

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).


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.


Alexis Gomel et al.

We consider the stabilization of modulationally unstable wave packets by designing and abruptly changing the floor topography within the framework of the nonlinear Schrödinger equation in variable water depth [1]. This is achieved as a result of abrupt expansion of a homoclinic Akhmediev breather orbit and its into an elliptic fixed point [2,3].

We experimentally demonstrate this phenomenon process in a water wave tank and provide a rigorous theoretical description of this process. The low-dimensional theoretical predictions and measurements show that the relative phase among the side-bands locks to π and their relative amplitudes oscillates around a finite value. Apart from a 10% conversion to higher-order side-bands, this implies that the breathing stage of modulation instability (MI) is indeed frozen. This phenomenon has been also verified in an optical fiber experiment [4].

We confirm that this complex wave dynamics is robust and such control of MI processes is feasible in a realistic experimental system. Our results highlight the influence of topography and how waveguide properties can influence and manipulate the lifetime of nonlinear and extreme waves.



  • [1] Djordjevic, V. and Redekopp, L., On the development of packets of surface gravity waves moving over an uneven bottom, ZAMP 29, 950-962 (1978).
  • [2] Armaroli, A. et al., Stabilization of uni-directional water wave trains over an uneven bottom. 2021, Nonlinear Dynamics 101 , 1131-1145 (2021).
  • [3] Gomel, A. et al., Stabilization of Unsteady Nonlinear Waves by Phase-Space Manipulation, Physical Review Letters 126, 174501 (2021).
  • [4] Bendahmane, A. et al., Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber, Optics Letters 39, 4490-4493 (2014).


How to cite: Gomel, A., Chabchoub, A., Brunetti, M., Trillo, S., Kasparian, J., and Armaroli, A.: Stabilization of Unsteady Nonlinear Waves by Phase-Space Manipulation, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-6711, https://doi.org/10.5194/egusphere-egu22-6711, 2022.


Yuchen He et al.

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.


Maxime Canard et al.

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.


Mark McAllister et al.

Breaking poses an upper limit to how large an individual wave can become and is the main mechanism of dissipation of wave energy in the ocean. Understanding how and when waves break is essential for forecasting extreme waves and predicting the resulting loads they exert on fixed structures and floating bodies. When modelling and forecasting extreme wave heights, to predict when a wave may break, parametric wave breaking criteria are currently used. These criteria use properties such as the steepness or the ratio of the fluid speed to crest speed to determine whether a wave will break. Current state-of-the-art wave breaking criteria are capable of predicting when waves will break, when the waves travel in a single mean direction (`following-seas'). In the oceans, it is common to have remotely generated `swell waves', in combination with locally- generated wind-waves that travel in the direction of the wind, resulting in crossing when the two are not aligned. In such `crossing-seas', state-of-art breaking criteria become challenging to apply and may lose their predictive power. We present a series of experiments in which we create crossing and highly directionally spread breaking waves. We examine the effects of crossing and high degrees of spreading on the onset and intensity of wave breaking. Our results show that the onset of breaking, is strongly influenced by directional spreading (both spreading width and crossing angle). As the degree of directional spreading, or crossing angle increases there is a sharp rise in the amplitude at which the onset of wave breaking occurs.

How to cite: McAllister, M., Draycott, S., Calvert, R., Davey, T., Dias, F., and van den Bremer, T.: An Experimental Study of Wave Breaking in Crossing Seas, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-9969, https://doi.org/10.5194/egusphere-egu22-9969, 2022.


Mon, 23 May, 17:00–18:30

Chairpersons: Amin Chabchoub, Yan Li

Henrik Kalisch et al.

Some rock coasts around the world feature very steep coastal profiles which often continue past the waterline, and may quicly drop to one or two-hundred meters depth. We consider the interaction of ocean waves with such steep coastal topography. It has been shown that shoaling ocean waves may experience significant amplification in the last 50 to 100 meters before they run up on the shore, leading to potentially hazardous run-up events even under relatively calm conditions. We study the distribution of simulated run-up heights on steep and gentle slopes for a range of sea states, and compare the results to the distribution of offshore wave elevation.

How to cite: Kalisch, H., Lagona, F., Roeber, V., and Bjørnestad, M.: Statistical Analysis of Extreme Wave Run-up on Steep Slopes, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-13238, https://doi.org/10.5194/egusphere-egu22-13238, 2022.


Jie Zhang et al.

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(Lp) to O(102 Lp), with Lp 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(Lp), 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 Lp), 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.


Saulo Mendes et al.

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.


Samuel Draycott et al.

Abrupt depth transitions (ADTs) exist in the form of natural and man-made bathymetric features, such as seamounts, continental shelves, steep beaches, reefs, and breakwaters. ADTs have been shown to induce the release of bound waves into free waves, which results in spatially inhomogeneous wave fields atop ADTs and regions where rogue waves are significantly more likely than predicted by linear theory.  Herein, we examine the role of free-wave release in the generation and spatial distribution of higher-harmonic wave components and in the onset of wave breaking for very steep periodic waves upon interaction with an ADT. We utilise a Smoothed Particle Hydrodynamics (SPH) model, making use of its ability to automatically capture breaking and overturning surfaces. We validate the model against experiments. The SPH model is found to accurately reproduce the phase-resolved harmonic components up to the sixth harmonic, particularly in the vicinity of the ADT. For the cases studied, we conclude that second-order free waves released at the ADT, and their interaction with the linear and second-order bound waves (beating), drive higher-order bound-wave components, which show spatial variation in amplitude as a result.  For wave amplitudes smaller than the breaking threshold, this second-order beating phenomenon can be used to predict the locations where peak values of surface elevation are located, whilst also predicting the breaking location for wave amplitudes at the breaking threshold.  Beyond this threshold, the contributions of the second-order and higher harmonics (second–harmonic amplitudes are up to 60% and sixth-harmonic up to 10% of the incident amplitude) cause breaking to occur nearer to the ADT, and hence the wave breaking onset location is confined to the region between the ADT and the first anti-node location of the second-order components.  Counter-intuitively, we find that, at the point of breaking, steeper incident waves are found to display reduced non-linearity as a result of breaking nearer to the ADT.

How to cite: Draycott, S., Li, Y., Stansby, P., Adcock, T., and van den Bremer, T.: Harmonic-induced wave breaking due to abrupt depth transitions, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-9460, https://doi.org/10.5194/egusphere-egu22-9460, 2022.


Sergey Dyachenko et al.

We consider the classical problem of 2D fluid flow with a free boundary. Recent
works strongly suggest that square--root type branch points appear naturally in
2D hydrodynamics. We illustrate how the fluid domain can be complemented by a
"virtual'' fluid, and the equations of motion are transplanted to a branch cut (a vortex sheet)
in the conformal domain. A numerical and theoretical study of the motion of
complex singularities in multiple Riemann sheets is suggested.

Unlike preceding works for dynamics of singularities: the short branch cut approximation,
and the study of viability of meromorphic solutions in fluid dynamics, the present approach
neither simplifies the equations of fluid flow, nor uses local Laurent expansions. Instead
the new approach is based on analytic functions and allows construction of global solutions
in 2D hydrodynamics.

A natural extension of the approach considers fluid flows described by many pairs
of square--root branch points.

Supported by RSF Grant 19-72-30028

How to cite: Dyachenko, S., Dyachenko, A., and Zakharov, V.: Singularities in 2D flows: The Tale of Two Branch Points, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-10424, https://doi.org/10.5194/egusphere-egu22-10424, 2022.


Sergey Dremov et al.

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.


Thomas Adock et al.

When investigating large ocean waves we frequently run simulations or experiments with random waves. For practical reasons such seastates are often initalised without the full spectral tail which would normally be present in the ocean. The change in the distribution of energy associated with this is generally very small and it impact on subsequent wave statistics presumed negligible. Here we investigate this by running firstly fully non-linear simulations of the sea-state considered in the experiments of Latheef & Swan (2013) and secondly Modified non-linear Schrödinger equation simulations of the experiments of Onorato et al. (2009). We find a consistent pattern across our simulations (which are also consistent with other results in the literature). We find that the curtailing of the spectrum in the initial conditions has a significant impact on the subsequent wave statistics—cutting off the spectrum leads to significantly more rogue waves than would otherwise be expected. We attribute this to the curtailed spectrum being far from equilibrium and this driving a strongly non-linear response. The results presented here are only for directionally spread waves—unidirectional waves are not expected to show the same physics.

How to cite: Adock, T., Barratt, D., Tang, T., Bingham, H., and van den Bremer, T.: The influence of spectral cutoff in random wave simulations, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-5837, https://doi.org/10.5194/egusphere-egu22-5837, 2022.


Anna Kokorina and Alexey Slunyaev

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.


Nikolay Makarenko et al.

We consider an analytical model of internal waves propagating in a weakly stratified two-layer fluid. A new model equation extents the long-wave approximation suggested in [1,2] for the non-linear Dubreil-Jacotin - Long equation. This model takes into account a slight density gradient in stratified layers, which can be comparable with the density jump at the interface between layers. Parametric range of solitary waves is determined in the framework of considered mathematical model. We demonstrate that solitary wave modes can be subject to the Kelvin-Helmholtz instability arising due to wave-induced velocity shear in layered flow. Such a marginal stability of internal waves could explain the formation mechanism of billow trains leading to the mixing in abyssal near-bottom flows.

This work was supported by the grant of the Russian Science Foundation (Project No 21-71-20039).


[1] Makarenko N.I., Maltseva J.L., Morozov E.G., Tarakanov R.Yu., Ivanova K.A. Internal waves in marginally stable abyssal stratified flow, Nonlin. Proc. Geophys. 2018, 25, 659-669

[2] Makarenko N.I., Maltseva J.L., Morozov E.G., Tarakanov R.Yu., Ivanova K.A.  Steady internal waves in deep stratified flow, J. Appl. Mech. Tech. Phys. 2019, 60(2), 248-256

How to cite: Makarenko, N., Maltseva, J., and Cherevko, A.: Non-linear internal waves in two-layer stratified flows, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-3452, https://doi.org/10.5194/egusphere-egu22-3452, 2022.


Andrey Serebryanyy

Unlike surface waves, whose generation is mainly associated with wind action, internal waves, which are widespread in the oceans and seas, have many sources of generation. The knowledge about the parameters of internal waves in the ocean and their structure is quite complete by now, however, the area concerning the causes of the appearance of internal waves is still poorly understood. Only one mechanism is well known that leads to the formation of intense internal waves. It is associated with the collision of tidal currents with a continental slope or underwater ridges. The talk will present the processes responsible for the generation of intense soliton-like internal waves on the shelf. This information has been collected over 40 years of experimental research by the author. Eleven main processes were identified that are responsible for the generation of intense short-period internal waves on the shelf. Most of them were observed in the non-tidal Black Sea. These processes are as follows: 1. Generation by a local hydrological front moving towards the coast in the post-storm period; 2. Ggeneration by subsurface intrusion of waters returning to the coastal zone in the post-storm period; 3. Generation of a submesoscale eddy during the passage of a submesoscale eddy on the shelf; 4. Generation by internal inertial waves approaching the coastal zone; 5. Generation by a cold atmospheric front passing over the sea; 6. Generation by river runoff at the places where large rivers flow into the sea; 7. Generation associated with the upwelling process in the coastal zone; 8. Generation of internal soliton-like waves by moving intrusion of surface freshened waters; 9. Generation of internal soliton-like waves in the collision of currents; 10. Generation of packets of internal waves when tidal internal waves enter the shelf; 11. Generation of packets of internal waves when internal seiches enter the shelf. This work was supported by RFBR grant No. 19-05-00715.

How to cite: Serebryanyy, A.: Processes responsible for the generation of internal solitons on a shelf: experimental evidences, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-6963, https://doi.org/10.5194/egusphere-egu22-6963, 2022.


Constance Schober

 The spatially periodic breather solutions (SPBs)  of the nonlinear Schr\"odinger equation, prominent in modeling rogue waves, are
 In this paper we numerically investigate the effects of nonlinear dissipation and higher order nonlinearities  on the routes to stability of the SPBs in the 
framework of the nonlinear damped higher order nonlinear Schr\"odinger (NLD-HONLS) equation. 
We appeal to the  Floquet spectral theory of the NLS equation to interpret and provide a characterization of the perturbed dynamics in terms of nearby solutions of the NLS equation. The number of instabilities of the background Stokes wave, the damping strength, and the time of onset of  nonlinear damping  are varied. 
A broad categorization of the routes to stability of the SPBs and the novel features related to the effects of nonlinear damping will be discussed.

How to cite: Schober, C.: Nonlinear damped higher order nonlinear Schrodinger dynamics, EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-6806, https://doi.org/10.5194/egusphere-egu22-6806, 2022.


Andrey Gelash

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.


Alfred Osborne

The goal of this talk is the study of the non-integrable, physical hierarchy of equations in 2+1 dimensions: Nonlinear Schroedinger, Dysthe, Trulsen-Dysthe and Zakharov. These equations describe, with increasing order, the dynamics of water waves in two dimensions. I demonstrate a procedure to determine a nearby hierarchy of these equations, which is Lax Integrable and the Lax pairs and the Its-Matveev formulae are given. I am then able to show that the solutions of each of these equations can be reduced to a quasiperiodic Fourier series with coherent structure basis functions. These include Stokes waves, envelope solitons and breather packets. In order to return to the original hierarchy of the Nonlinear Schroedinger, Dysthe, Trulsen-Dysthe and Zakharov equations I make a Hamiltonian perturbation of the Lax integrable hierarchy. I then apply a theorem of Kuksin, and a further theorem of Baker and Mumford, to write the algebraic geometric solutions. These steps provide me with an approach for the hyperfast numerical integration of these equations for physics and engineering purposes, and for the analysis of recorded data in one and two dimensions through a procedure I refer to as nonlinear Fourier analysis.

How to cite: Osborne, A.: Solving the Physical Hierarchy of Nonlinear Water Wave Equations via a Nearby Lax Integrable Hierarchy with Hamiltonian Perturbations , EGU General Assembly 2022, Vienna, Austria, 23–27 May 2022, EGU22-6076, https://doi.org/10.5194/egusphere-egu22-6076, 2022.