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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, and 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 coastal structures.

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Co-organized by NP7/OS2
Convener: Alexey Slunyaev | Co-conveners: Amin Chabchoub, Henrik Kalisch, Efim Pelinovsky
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| Mon, 04 May, 08:30–10:15 (CEST)

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Chat time: Monday, 4 May 2020, 08:30–10:15

Chairperson: Alexey Slunyaev
D2033 |
EGU2020-16801
| Highlight
Dion Häfner, Johannes Gemmrich, and Markus Jochum

Despite several strong hypotheses on how rogue waves can be generated in idealized conditions, the actual real-world causes of these waves are still largely unknown. We credit this to insufficient amounts of observational data and a missing robust probabilistic framework to analyze the available data.

We adress these issues by processing over 1 billion waves measured in the North Pacific and organizing them into a comprehensive catalogue. Through a robust, machine-learning driven analysis, we then identify several characteristic sea conditions that lead to significantly higher risks to encounter a rogue wave. This, in turn, yields quantitative evidence on the relative importance of the underlying physical mechanisms.

D2034 |
EGU2020-1131
James Herterich, Francesco Fedele, Aziz Tayfun, and Frederic Dias

We present a statistical analysis of nearshore waves observed during two major north-east Atlantic storms in 2015 and 2017. Surface elevations were measured with a 5-beam acoustic Doppler current profiler (ADCP) at relatively shallow waters off the west coast of Ireland. To compensate for the significant variability of both sea states in time, we consider a novel approach for analyzing the non-stationary surface-elevation series and compare the distributions of crest and wave heights observed with theoretical predictions based on the Forristall, Tayfun and Boccotti models. In particular, the latter two models have been largely applied to and validated for deep-water waves. We show here that they also describe well the characteristics of waves observed in relatively shallow waters. The largest nearshore waves observed during the two storms do not exceed the rogue thresholds as the Draupner, Andrea, Killard or El Faro rogue waves do in intermediate or deep-water depths. Wave breaking limits wave growth and impedes the occurrence of rogue waves. Nevertheless, our analysis reveals that modulational instabilities are ineffective, third-order resonances negligible and the largest waves observed here have characteristics quite similar to those displayed by rogue waves for which second order bound nonlinearities are the principal factor that enhances the linear dispersive focusing of extreme waves.

Fedele, F., Herterich, J., Tayfun, A., & Dias, F. (2019). Large nearshore storm waves off the Irish coast. Scientific reports, 9(1), 1-19.

D2035 |
EGU2020-21483
Mark McAllister and Ton van den Bremer

Wave-following buoys are used to provide measurements of free surface elevation across the oceans. The measurements they produce are widely used to derive wave-averaged parameters such as significant wave height and peak period, alongside wave-by-wave statistics such as crest height distributions. Particularly concerning the measurement of extreme wave crests, these measurements are often perceived to be less accurate. We directly assess this through a side-by-side laboratory comparison of measurements made using Eulerian wave gauges and model wave-following buoys for directionally spread waves representative of extreme conditions on deep water. Our experimental measurements are compared to exact (Herbers and Janssen 2016, J. Phys. Oceanogr, 46, 1009-1021) and new approximate expression for Lagrangian second-order theory derived herein. We derive simple closed-form expressions for the second-order contribution to crest height representative of extreme ocean waves. Our experiments confirm that the motion a wave-following buoy should not significantly affect the measurements of wave crests or spectral parameters, and that discrepancies observed for in-situ buoy data are most likely a result of filtering. This filtering occurs when accelerations that are measured by the sensors within a buoy are converted to displacements. We present an approximate means of correcting the resulting measured crest height distributions, which is shown to be effective using our experimental data.

D2036 |
EGU2020-18509
Dylan Barratt, Harry B. Bingham, Paul H. Taylor, Ton S. van den Bremer, and Thomas A. A. Adcock

We have performed numerical simulations of steep three-dimensional wave groups, formed by dispersive focusing, using the fully-nonlinear potential flow solver OceanWave3D. We find that third-order resonant interactions result in directional energy transfers to higher-wavenumber components, forming steep wave groups with augmented kinematics and a prolonged lifespan. If the wave group is initially narrow banded, quasi-degenerate interactions resembling the instability band of a regular wave train arise, characterised by unidirectional energy transfers and energy transfers along the resonance angle, ±35.26°, of the Phillips ‘figure-of-eight’ loop. Spectral broadening due to the quasi-degenerate interactions eventually facilitates non-degenerate interactions, which dominate the spectral evolution of the wave group after focus. The non-degenerate interactions manifest primarily as a high-wavenumber sidelobe, which forms at an angle of ±55° to the spectral peak. We consider finite-depth effects in the range of deep to intermediate waters (5.592 ≥ kpd ≥ 1.363), based on the characteristic wavenumber (kp) and the domain depth (d), and find that all forms of spectral evolution are suppressed by depth. However, the quasi-degenerate interactions exhibit a greater sensitivity to depth, suggesting suppression of the modulation instability by the return current, consistent with previous studies. We also observe sensitivity to depth for kpd values commonly considered "deep", indicating that the length scales of the wave group and return current may be better indicators of dimensionless depth than the length scale of any individual wave component. The non-degenerate interactions appear to be depth resilient with persistent evidence of a ±55° spectral sidelobe at a depth of kpd =1.363. Although the quasi-degenerate interactions are significantly suppressed by depth, the interactions do not entirely disappear for kpd =1.363 and show signs of biasing towards oblique, rather than unidirectional, wave components at intermediate depths. The contraction of the wavenumber spectrum in the ky-direction has also proved to be resilient to depth, suggesting that lateral expansion of the wave group and the "wall of water" effect of Gibbs & Taylor (2005) may persist at intermediate depths.

D2037 |
EGU2020-11732
Karsten Trulsen

In Trulsen et al. (2020) we reported that when irregular waves propagate over a shoal the extreme wave statistics of surface elevation and water velocity can be dramatically different:  The surface elevation can have a local maximum of kurtosis some distance into the shallower part of the shoal, while it relaxes to normality after the shoal.  The velocity field can have a local maximum of kurtosis after the shoal, while it is close to normality over the shallower part of the shoal.  These two fields clearly do not coincide regarding the location of increased probability of extreme waves.

Here we consider the evolution of the irregular waves over the shoal as a multivariate stochastic process, with a view to reveal the evolution of the joint statistical distribution of surface elevation and water velocity.  Higher order multivariate moments, coskewness and cokurtosis, more commonly seen in mathematical finance theory, are employed to describe the joint extreme wave statistical distribution of the elevation and the velocity.

Trulsen, K., Raustøl, A., Jorde, S. & Rye, L. B. (2020) Extreme wave statistics of longcrested irregular waves over a shoal.  J. Fluid Mech. 882, R2.

D2038 |
EGU2020-6562
Dmitry Kachulin, Alexander Dyachenko, and Vladimir Zakharov

Using the Hamiltonian formalism and the theory of canonical transformations, we have constructed a model of the dynamics of two-dimensional waves on the surface of a three-dimensional fluid. We find and apply a canonical transformation to a water wave equation to remove all nonresonant cubic and fourth-order nonlinear terms. The found canonical transformation also allows us to significantly simplify the fourth-order terms in the Hamiltonian by replacing the coefficient of four-wave Zakharov interactions with a new simpler one. As a result, unlike the Zakharov equation (written in k-space), this equation can be written in x-space, which greatly simplifies its numerical simulation. In addition, our chosen form of a new coefficient of four-wave interactions allows us to generalize this equation to describe two-dimensional waves on the surface of a three-dimensional fluid. An effective numerical algorithm based on the pseudospectral Fourier method for solving the new 2D equation is developed. In the limiting case of plane (one-dimensional) waves, we found solutions in the form of breathers propagating in one direction. The dynamics of such nonlinear traveling waves perturbed in the transverse direction is numerically investigated.

The work was supported by the Russian Science Foundation (Grant No. 19-72-30028).

D2039 |
EGU2020-13467
Constance Schober

 The spatially periodic breather solutions (SPBs) of the nonlinear Schrödinger (NLS) equation, i.e. the heteroclinic orbits of unstable Stokes waves, are typically unstable. In this talk  we examine  the effects of dissipation on the  one- mode SPBs  U(j)(x,t) as well as multi-mode SPBs U(j,k)(x,t) using a damped  NLS equation which incorporates both uniform linear damping and nonlinear damping  of the mean flow,
for a range of parameters typically encountered in experiments. The damped wave dynamics is viewed as near integrable, allowing one to use the spectral theory of the NLS equation to interpret the perturbed flow. A broad categorization of how the route to stability for the SPBs  depends on the mode structure of the SPB and whether the damping is linear or nonlinear is obtained 
as well as the distinguishing features of the stabilized state.  Time permitting, a reduced, finite dimensional dynamical system that goverms the linearly damped SPBs will be presented 

D2040 |
EGU2020-20079
Alfred Osborne

I give a new perspective for the description of nonlinear water wave trains using mathematical methods I refer to as nonlinear Fourier analysis (NLFA). I discuss how this approach holds for one-space and one time dimensions (1+1) and for two-space and one time dimensions (2+1) to all orders of approximation. I begin with the nonlinear Schroedinger (NLS) equation in 1+1 dimensions: Here the NLFA method is derived from the complete integrability of the equation by the periodic inverse scattering transform. I show how to compute the nonlinear Fourier series that exactly solve 1+1 NLS. I then show how to extend the order of 1+1 NLS to the Dysthe and the extended Dysthe equations. I also show how to include directional spreading in the formulation so that I can address the 2+1 NLS, the 2+1 Dysthe and the 2+1 Trulsen-Dysthe equations. This hierarchy of equations extends formally all the way to the Zakharov equations in the infinite order limit. Each order and extension from 1+1 to 2+1 dimensions is characterized by its own modulational dispersion relation that is required at each order of the NLFA formalism. NLFA is characterized by its own fundamental nonlinear Fourier series, which has particular nonlinear Fourier modes: sine waves, Stokes waves and breather trains. We are all familiar with sine waves (known for centuries) and Stokes waves (known since the Stokes paper in 1847). Breather trains have become known over the past three decades as a major source of rogue or freak waves in the ocean: Breather packets are known to pulse up and down during their evolution. At the moment of the maximum amplitude the largest wave in a breather packet is often referred to as a “rogue” or “freak” wave. Such extreme packets are known to be “coherent structures" so that pure linear dispersion does not occur as in a linear packet. Instead the breather packets have components that are phase locked with each other and hence remain coherent and are “long lived” just as vortices do in classical turbulence. Because the breathers live for a long time, the notion of risk based upon linear dispersion, as used in the oil and shipping industries, must be revised upwards. I discuss how to apply NLFA to (1) nonlinearly Fourier analyze time series, (2) to analyze wave fields from radar, lidar and synthetic aperture radar measurements, (3) how to treat NLFA to describe nonlinear, random wave trains using a kind of nonlinear random phase approximation and (4) how to compute the nonlinear power spectrum in terms of the parameters used to describe the rogue wave Fourier modes in a random wave train. Thus the emphasis here is to discuss a number of new tools for nonlinear Fourier analysis in a wide range of problems in the field of ocean surface waves.

D2041 |
EGU2020-8594
Andrey Gelash

The key result of this work is the first theoretical computation of exact expressions describing space and phase shifts acquiring by breathes after mutual collisions in the framework of the focusing one-dimensional nonlinear Schrödinger equation (NLSE) model [1]. Similar expressions are the backbone of soliton theory, where they allow to predicts soliton interaction dynamics and introduce statistical description of soliton gas in terms of kinetic equation. Theory of breathers – solitary type wave groups on an unstable background – has been developing almost as long as theory of solitons. However, up to now, this important part of theory has been missing.

In our work we present space and phase shift formulas for the NLSE breathers and demonstrate how they can be used to go deeply in understanding of an intriguing nonlinear phenomena – formation rogue waves from a calm background. With these formulas we show that synchronized collisions of breathers are the central mechanism of extreme amplitude wave formation as a result of modulation instability development. We illustrate this conclusion by particular examples of multi-breather dynamics as well as by statistical analysis of multi-breather interactions. In comparison to the work [1], here we also analyse the impact of the effects lying beyond the NLSE model on the multi-breather synchronization. Finally, we present new scenarios of the synchronised multi-breather interactions, that can be observed in laboratory experiments.

The work was supported by the RFBR grant No. 19-31-60028.

[1] A. A. Gelash, Formation of rogue waves from a locally perturbed condensate, Phys. Rev. E 97, 022208 (2018).

D2042 |
EGU2020-19355
Sergey Dremov, Dmitriy Kachulin, and Alexander Dyachenko

        The present work is devoted to the study of coherent structures collisions dynamics in the models of deep water waves equations: the model of a supercompact equation for deep water unidirectional waves (SCEq) and the model of Dyachenko equations for potential flows of incompressible fluid with free surface. In these models there are special solutions in the form of coherent wave structures called breathers. They can be found numerically by using the Petviashvili method. One can consider the combination of such breathers as a model of rarefied soliton gas, and their paired collisions in this case are a key feature in forming of dynamics and statistics in the model. To describe statistical characteristics of breathers collision Probability Density Function (PDF) is used. PDF of breathers wave amplitudes during their collision was calculated and compared with the known results in the model of Nonlinear Schrodinger equation (NLS). In contrast to the NLS model there is a number of interesting features in the model of SCEq. For instance, the amplitude maximum of wave arising during the collision can exceed the sum of interacting breathers amplitudes, what cannot happen in NLS model. Moreover, it depends on the initial breathers steepness. In addition, it is shown that the breathers acquire phase and space shifts after each collision, and thus their velocity also changes. Depending on the relative phase breathers can give their energy or take it, and as a result their amplitude can be decreased or increased respectively. The same situation can be seen in the model of equations for potential flows of incompressible fluid with free surface. In addition to the dependence on relative phase the duration of the collision also affects the energy exchange. Breathers collisions are accompanied by appearance of little radiation, and its value is relatively less than the value of energy exchange. The results of statistics calculating and dynamics studying in the rarefied gas of coherent structures will be shown in the present work.

           The work was supported by Russian Science Foundation grant № 18-71-00079.

D2043 |
EGU2020-641
Oleg Didenkulov

The numerical simulation of the so-called breather turbulence (dynamics of breather ensembles) is performed within the integrable model of modified Korteweg – de Vries equation. Two-breather interaction is considered as an elementary act of the breather turbulence. The possible modes of breather-breather or breather-soliton interactions have been investigated. The influence of such interactions on multi-breather dynamics and its statistical characteristics has been analyzed. The results of direct numerical simulation of breather focusing with “non-optimal” phases allow finding the probability of a possible collision of more than two breathers. The formation of abnormally large pulses (rogue waves) in random ensembles of breathers has been demonstrated.

 

D2044 |
EGU2020-4457
| Highlight
Ekaterina Didenkulova

A catalogue of anomalously large waves (rogue or freak waves) occurred in the World Ocean during 2011-2018 reported in mass media sources and scientific literature has been compiled and analyzed. It includes 210 hazardous events caused damages or human losses. The majority of events is based on eyewitness accounts, and as a rule is not confirmed by direct measurements. All collected events divided into deep water cases, shallow water cases and occurrences on the coast (gentle beach or rocks). The following parameters have been determined: date, location, damage, description, reference, and weather conditions. The most dangerous areas in the World Ocean in terms of freak waves are highlighted.

This work was supported by the Russian Science Foundation (project No. 18-77-00063).

D2045 |
EGU2020-4708
Anna Kokorina and Alexey Slunyaev

Direct numerical simulations of the directional sea surface gravity waves are carried out within the framework of the primitive potential equations of hydrodynamics using the High Order Spectral Method. The data obtained for conditions of deep water, the JONSWAP spectrum, and various wave intensities are processed and the results are discussed. The statistical and spectral characteristics of the waves evolve over a long period. The particular asymmetry of the profiles of rogue waves is highlighted. We show that besides the conventional crest-to-trough asymmetry of nonlinear Stokes waves, the extreme events are characterized by a specific combination of the troughs adjacent to the large crest, so that the trough behind the crest is typically deeper than the preceding trough. Surprisingly, the extreme wave crest-to-trough asymmetry and the discrimination between the extreme wave troughs exhibit the tendency to grow when the angle spectrum broadens. This effect contradicts the expectation based on the Benjamin – Feir Index that broad-banded waves should behave similar to linear waves, and hence the asymmetries should diminish.

                                                                 

The research is supported by the RSF grant No. 19-12-00253.

 

A. Kokorina, A. Slunyaev, The effect of wave nonlinearity on the rogue wave lifetimes and shapes. Proc. 14th Int. MEDCOAST Congress on Coastal and Marine Sciences, Engineering, Management and Conservation (Ed. E. Ozhan), Vol. 2, 711-721 (2019).

D2046 |
EGU2020-5348
Alexey Slunyaev

The dynamic kurtosis (i.e., produced by the free wave component) is shown to contribute essentially to the abnormally large values of the full kurtosis of the surface displacement, according to the direct numerical simulations of realistic directional sea waves within the HOSM framework. In this situation the free wave stochastic dynamics is strongly non-Gaussian, and the kinetic approach is inapplicable. Traces of coherent wave patterns are found in the Fourier transform of the directional irregular sea waves. They strongly violate the classic dispersion relation and hence lead to a greater spread of the actual wave frequencies for given wavenumbers.

The research by is supported by the RSF grant No. 16-17-00041.

Slunyaev, A. Kokorina, The method of spectral decomposition into free and bound wave components. Numerical simulations of the 3D sea wave states. Geophysical Research Abstracts, V. 21, EGU2019-546 (2019).

A.V. Slunyaev, A.V. Kokorina, Spectral decomposition of simulated sea waves into free and bound wave components. Proc. VII Int. Conf. “Frontiers of Nonlinear Physics”, 189-190 (2019).

Slunyaev, A. Kokorina, I. Didenkulova, Statistics of free and bound components of deep-water waves. Proc. 14th Int. MEDCOAST Congress on Coastal and Marine Sciences, Engineering, Management and Conservation (Ed. E. Ozhan), Vol. 2, 775-786 (2019).

Slunyaev, Strongly coherent dynamics of stochastic waves causes abnormal sea states. arXiv: 1911.11532 (2019).

D2047 |
EGU2020-5740
Alexey Slunyaev, Guillaume Ducrozet, and Yury Stepanyants

The problem of the weakly nonlinear wave transformation on a bottom step is studied analytically and numerically by means of the direct simulation of the Euler equation. It is assumed that the quasi-linear wave packets can be described by the nonlinear Schrödinger equation for surface waves in finite-depth water. The process of wave transformation in the vicinity of the bottom step can be described within the framework of the linear theory and the transformation coefficients (the transmission and reflection coefficients) can be determined by the approximate formula suggested in [1]. The fate of transmitted and reflected wave trains emerging from the incident envelope soliton can be determined with the help of the Inverse Scattering Technique [2, 3].

The parameters of secondary envelope solitons (their number, amplitudes, and speeds) asymptotically forming in the far-field zone are obtained analytically and compared against the numerically calculated ones, as the functions of the depth drop h2/h1, where h1 and h2 are the undisturbed water depths in front of and behind the bottom step, respectively. It is shown that the wave amplitudes can notably increase when the envelope soliton travels from the relatively shallow to much deeper water. The amplitudes of secondary solitons can exceed more than twice the amplitude of the incident wave.

The direct numerical simulation of envelope soliton transformation was undertaken by means of the High Order Spectral Method [4, 5]. The comparison of approximate analytical solutions with the results of numerical simulations reveals the domains of very good agreement between the data where the approximate theory is applicable. In the meantime, the noticeable disagreement between the approximate nonlinear theory and the direct simulations is found when the theory is inapplicable.

The research by A.S. is supported by the RFBR grant No. 18-02-00042; he also acknowledges the support from the International Visitor Program of the University of Sydney and is grateful for the hospitality of the University of Southern Queensland. The research of Y.S. was support by the grant of the President of the Russian Federation for State support of scientific research of leading scientific Schools of the Russian Federation NSh-2485.2020.5.

[1] Kurkin, A.A., Semin, S.V., and Stepanyants, Yu.A., Transformation of Surface Waves over a Bottom Step. Izvestiya, Atmospheric and Oceanic Physics, 2015, Vol. 51, 214–223.

[2] Zakharov, V.E., Shabat, A.B., Exact theory of two-dimensional self-focussing and one-dimensional self-modulation of waves in nonlinear media. Sov. Phys. JETP, 1972, Vol. 34, 62-69.

[3] Slunyaev, A., Klein, M., Clauss, G.F., Laboratory and numerical study of intense envelope solitons of water waves: generation, reflection from a wall and collisions. Physics of Fluids, 2017, Vol. 29, 047103.

[4] West, B.J., Brueckner, K.A., Janda, R.S., Milder, D.M., Milton, R.L., A new numerical method for surface hydrodynamics. J. Geophys. Res., 1987, Vol. 92, 11803-11824.

[5] Ducrozet, G., Gouin, M., Influence of varying bathymetry in rogue wave occurrence within unidirectional and directional sea-states. J. Ocean Eng. Mar. Energy, 2017, Vol. 3, 309-324.

D2048 |
EGU2020-5458
Maura Brunetti, Alexis Gomel, Andrea Armaroli, Amin Chabchoub, and Jérôme Kasparian

The nonlinear Schrödinger equation is a robust model for describing the evolution of surface gravity wave-packets over arbitrary bathymetry. Both the dispersive and the nonlinear coefficients turn out to depend on the fluid depth [1,2]. Its variation along the propagation direction provides a new degree of freedom to tailor the wave-packet evolution, in analogy to what has been obtained in optical fibers with varying dispersion [3].

We investigate how the nonlinear stage of modulation instability can be frozen by varying the water bottom from intermediate to large depth giving rise to an increase of the magnitude of the nonlinear coefficient within the focusing regime. We consider the case of an abrupt bathymetry change at the maximal focusing point. With the help of a three-wave truncation, we provide analytical conditions on the occurrence of freezing. We present numerical simulations of the full model and the experimental confirmation in a water wave flume experiment. We show that the effects of high-order nonlinear terms and dissipation do not dominate the evolution, making the freezing quite a robust phenomenon.   

Our results help clarify how the breathing evolution of water wave-packets can be dynamically controlled and to understand the impact of bathymetry on extreme-wave lifetimes.

References

[1] H. Hasimoto, and Hiroaki Ono. "Nonlinear modulation of gravity waves." Journal of the Physical Society of Japan 33, 805-811 (1972)

[2] V. D. Djordjevic, and L. G. Redekopp, “On the development of packets of surface gravity waves moving over an uneven bottom” Journal of Applied Mathematics and Physics (ZAMP) 29, 950–962 (1978)

[3] A. Bendahmane et al., “Experimental dynamics of Akhmediev breathers in a dispersion varying optical fiber” Optics Letters 39, 4490-4493 (2014)

 

D2049 |
EGU2020-7468
Yan Li, Samuel Draycott, Yaokun Zheng, Thomas A.A. Adcock, Zhiliang Lin, and Ton S. van den Bremer

This work focuses on two different aspects of the effect of an abrupt depth transition on weakly nonlinear surface gravity waves: deterministic and stochastic. It is known that the kurtosis of waves can reach a maximum near the top of such abrupt depth transitions. The analysis is based on three different approaches: (1) a novel theoretical framework that allows for narrow-banded surface waves experiencing a step-type seabed, correct to the second order in wave steepness; (2) experimental observations; and (3) a numerical model based on a fully nonlinear potential flow solver. To reveal the fundamental physics, the evolution of a wave envelope that experiences an abrupt depth transition is examined in detail; (a) we show the release of free waves at second order in wave steepness both for the super-harmonic and sub-harmonic or ‘mean’ terms; (b) a local wave height peak that occurs near the top of a depth transition – whose exact position depends on several nondimensional parameters – is revealed; (c) furthermore, we examine which parameters affect this peak. The novel physics has implications for wave statistics for long-crested irregular waves experiencing an abrupt depth transition. We show the connection of the second-order physics at work in the deterministic and stochastic cases: the peak of wave kurtosis and skewness occurs in the neighborhood of the deterministic wave peak in (b) and for the same parameters set composed of a seabed topography, water depths, primary wave frequency and steepness, and bandwidth.

D2050 |
EGU2020-6007
Christopher Lawrence, Karsten Trulsen, and Odin Gramstad

It was shown experimentally in Trulsen et al. (2012) that irregular water waves propagating over a slope may have a local maximum of kurtosis and skewness in surface elevation near the shallower side of the slope. Later on, Raustøl (2014) did laboratory experiments for irregular water waves propagating over a shoal and found the surface elevation could have a local maximum of kurtosis and skewness on top of the shoal, and a local minimum of skewness after the shoal for sufficiently shallow water. Numerical results by Sergeeva et al. (2011), Zeng & Trulsen (2012), Gramstad et al. (2013) and Viotti & Dias (2014) support the experimental results mentioned above. Just recently, Jorde (2018) did new experiment with the same shoal as in Raustøl (2014) but with additional measurement of the interior horizontal velocity. The experimental results from Raustøl (2014) and Jorde (2018) were reported in Trulsen et al. (2020) and it was found the evolution of skewness for surface elevation and horizontal velocity have the same behaviour but the kurtosis of horizontal velocity has local maximum in downslope area which is different with the kurtosis of surface elevation. In present work, we utilize numerical simulation to study the effects of incoming significant wave height, peak wave frequency on evolution of wave statistics for both surface elevation and velocity field with more general bathymetry. Numerical simulations are based on High Order Spectral Method (HOSM) for variable depth Gouin et al. (2016) for wave evolution and Variational Boussinesq model (VBM) Lawrence et al. (2018) for velocity field calculation.

References
GOUIN, M., DUCROZET, G. & FERRANT, P. 2016 Development and validation of a non-linear spectral model for water waves over variable depth. Eur. J. Mech. B Fluids 57, 115–128.
GRAMSTAD, O., ZENG, H., TRULSEN, K. & PEDERSEN, G. K. 2013 Freak waves in weakly nonlinear unidirectional wave trains over a sloping bottom in shallow water. Phys. Fluids 25, 122103.
JORDE, S. 2018 Kinematikken i bølger over en grunne. Master’s thesis, University of Oslo.
LAWRENCE, C., ADYTIA, D. & VAN GROESEN, E. 2018 Variational Boussinesq model for strongly nonlinear dispersive waves. Wave Motion 76, 78–102.
RAUSTØL, A. 2014 Freake bølger over variabelt dyp. Master’s thesis, University of Oslo.
SERGEEVA, A., PELINOVSKY, E. & TALIPOVA, T. 2011 Nonlinear random wave field in shallow water: variable Korteweg–de Vries framework. Nat. Hazards Earth Syst. Sci. 11, 323–330.
TRULSEN, K., RAUSTØL, A., JORDE, S. & RYE, L. 2020 Extreme wave statistics of long-crested irregular waves over a shoal. J. Fluid Mech. 882, R2.
TRULSEN, K., ZENG, H. & GRAMSTAD, O. 2012 Laboratory evidence of freak waves provoked by non-uniform bathymetry. Phys. Fluids 24, 097101.
VIOTTI, C. & DIAS, F. 2014 Extreme waves induced by strong depth transitions: Fully nonlinear results. Phys. Fluids 26, 051705.
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, 631–638.

D2051 |
EGU2020-11760
Andrey Serebryany, Valeriy Bondur, and Viktor Zamshin

When conducting work in the fall of 2015 on the Black Sea northeast shelf, we recorded internal waves, the unusualness of which attracts special attention for the following reasons. For the first time in 40 years of internal waves observations in the Black Sea, such high waves with amplitudes of 14–16 m were measured. The generation of these anomalous waves was connected with a cold atmospheric front passing over the sea. It was the first experimental evidence in the sea of such mechanism for internal waves generation. The observed internal waves had a clear seen character of nonlinear soliton-like waves.

We met the train of internal solitons during a sub-satellite survey conducted in the sea from a motor yacht equipped with ADCP “Rio Grande 600 kHz” in the waters near Cape Tolsty. The train was found at a point of the sea with a depth of 33 m and then was recorded on seven multidirectional tacks oriented normal to the coast. It moved across the shelf to the coast along the bottom thermocline, while the bottom currents accompanying it had a northwestern coastal orientation. The train included four waves of a soliton-like shape with sharpened crests and flattened troughs. Their lengths were 100-110 m, heights up to 14-16 m, vertical velocities in orbital currents reached 0.15-0.20 m/s. Another property of nonlinear waves was also expressed - the amplitude ranking of waves in the train. Traced on successive tacks for 2.5 hours, internal waves had preserved the soliton-like shape and as well the strong vertical component in their orbital currents. Despite the fact that the train was moving along the bottom thermocline, the effect of internal waves was sufficient to appear on satellite radar images of the sea surface of the study area. The performed processing of satellite images confirmed the wave parameters measured by contact methods.  An interesting fact of a long accompaniment of internal solitons by a school of fish was discovered. Fishes were concentrated in areas where internal waves carried the components of fish food supply to the surface from the bottom layers. The work was partially supported by RFBR grant 19- 05-00715.

D2052 |
EGU2020-11200
Gozde Guney Dogan, Pamela Probst, Bora Yalciner, Alessandro Annunziato, Narcisse Zahibo, and Ahmet Cevdet Yalciner

Tropical cyclones can be considered one type of extreme event, with their destructive winds, torrential rainfall and storm surge. Every year these natural phenomena affect millions of people around the world, leaving a trail of destruction in several countries, especially along the coastal areas. Only in 2017, two devastating major hurricanes (Irma and Maria) moved across the Caribbean and south-eastern USA, causing extensive damage and deaths. Irma formed in the far eastern Atlantic Ocean on 30 August 2017 and moved towards the Caribbean islands during the following week, significantly strengthening, becoming a Category 5 Hurricane. It caused wide-ranging impacts such as significant storm surge (up to 3m according to US National Oceanic and Atmospheric Administration, NOAA report) to several islands in the Caribbean and Florida. On the second half of September, 2017, another strong Category 5 Hurricane named Maria formed over the Atlantic and moved west towards the Caribbean Sea. Maria also caused several impacts and severe damage in Caribbean Islands, Puerto Rico and the U.S. Virgin Islands due to high speed winds, rainfall, flooding and storm surge with a maximum runup of 3.7 m (US NOAA) on the southern tip of Dominica Island. The most recent devastating event for the Atlantic is Hurricane Dorian. It formed on August 24, 2019 over the Atlantic Ocean and it moved towards the Caribbean islands, as getting stronger as moving, becoming a Category 5 before reaching the Bahamas, where it left a trail of destruction after its passage. The major effect of Dorian was on north-western Bahamas with very strong winds, heavy rainfall and a large storm surge.

In this context, a rapid and reliable modeling of storm surge generated by such kind of events is essential for many purposes such as early accurate assessment of the situation, forecasting, estimation of potential impact in coastal areas, and operational issues like emergency management.

A numerical model, NAMI DANCE GPU T-SS (Tsunami-Storm Surge) is developed building up on tsunami numerical model NAMI DANCE GPU version to solve nonlinear shallow water equations, using the pressure and wind fields as inputs to compute spatial and temporal distribution of water level throughout the study domain and respective inundation related to tropical cyclones, based on the equations used in the HyFlux2 Code developed by the Joint Research Centre of the European Commission. The code provides a rapid calculation since it is structured for Graphical Processing Unit (GPU) using CUDA API.

NAMI DANCE GPU T-SS has been applied to many cases as regular shaped basins under circular static and dynamic pressure fields separately and also different wind fields for validation together with combinations of pressure and wind fields. This study has been conducted to investigate the potential of numerical modeling of tropical cyclone generated storm surge based on recent events Irma, Maria and Dorian. The results are presented and discussed based on comparison with the measurements and observations. The study shows promise for developing a cyclone modeling capability based on available measurement and observational data.

D2053 |
EGU2020-6092
Ahmed Abdalazeez, Denys Dutykh, Ira Didenkulova, and Céline Labart

The runup of initial Gaussian narrow-banded and wide-banded wave fields and its statistical characteristics are investigated using direct numerical simulations, based on the nonlinear shallow water equations. The bathymetry consists of the section of a constant depth, which is matched with the beach of constant slope. To address different levels of nonlinearity, the time series with five different significant wave heights are considered. The total time of each such calculated time-series is 1000 hours.

It is shown for narrow-banded wave signal that runup oscillations are no more distributed by the Gaussian distribution. The distribution is shifted to the right towards larger positive values of wave runup. Its mean value increases with an increase in nonlinearity, which reflects the known phenomenon of wave set-up. The higher moments of runup oscillations, skewness and kurtosis are negative. The skewness is decreasing with an increase in wave nonlinearity, while kurtosis is negative and varies non-monotonically with an increase in wave nonlinearity. For Gaussian wide-banded signal, the runup oscillations also deviate from Gaussian distribution. The distribution is also shifted to the right towards larger positive values of wave runup. Its mean values increase with an increase in nonlinearity, while all other higher moments change non-monotonically.     

For the extreme wave runup heights, we conclude that the tail of the probability density function behaves like a conditional Weibull distribution if the incident random waves are represented by Gaussian narrow-banded or wide-banded spectrum. This distribution can be used for evaluation of wave inundation during extreme floods (rogue runups). 

D2054 |
EGU2020-5096
Nazmi Postacioglu, M. Sinan Özeren, and Ebubekir Çelik

Investigation of the behavior of various types of Tsunami wave trains entering bays is of practical importance for coastal hazard assessments. The linear shallow water equations admit two types of solutions inside an inclined bay with parabolic cross section: Energy transmitting modes and decaying modes. In low frequency limit there is only one mode susceptible of transmitting energy to the inland tip of the bay. The decay rates of decaying modes are controlled by the boundary conditions at the sides of the bay. Therefore a complicated eigenvalue problem needs to be solved in order to compute these decay rates. To determine the amplitude of the energy transmitting mode one should solve an integral equation, involving not just the energy transmitting mode but also decaying modes, the scattered field into the open sea, the incident wave and the reflected wave in the open sea. However, in the long wave limit, all these complications can be avoided if one applies the Dirichlet boundary conditions at the open boundary. That is to take the displacement of the free surface at the open boundary being equal to the twice of the disturbance associated with the incident wave in the open sea, just like a wall boundary condition. The runup produced by the solution obtained from this Dirichlet boundary condition, can be easily calculated using a series of images. In this model no energy is allowed to escape from the bay therefore the error arising from the simplification of the boundary conditions at the open boundary grows with time. Nevertheless the maximum runup occurs before this error becomes significant. If the characteristic wavelength of the incident wave train is equal to 5 times the width of the bay then this simple solution overestimates the first maximum of the runup only by %15 compared to the “exact” solution derived from the integral equation. This overestimation is partly due to the fact that Dirichlet boundary conditions violates the continuity of depth integrated velocities. The solution associated with Dirichlet boundary condition is perturbed in order to match fluxes inside and outside of the bay. This perturbation does not use the decaying modes inside the bay. The height of the first maximum of the runup coming from the perturbation theory is in excellent agreement with that obtained using the integral equation. This perturbation theory can also be applied to narrow bays with arbitrary cross section as long as their depth does not not change in the longitudinal direction.