Andrey Kurkin

The 1867 Virgin Island tsunami: observations and modeling

The 1867 Virgin Island tsunami was of a great effect for the Caribbean Islands. A maximal tsunami height of 10 m was recorded for two coastal locations (Deshayes and St. Rose) in Guadeloupe. The historical data of this event for the Caribbean Sea are discussed. The modeling of the 1867 tsunami is performed in the framework of the nonlinear shallow-water theory. The four different orientations of the tsunami source in the Anegada Passage are examined. The directivity of the tsunami wave in the Caribbean is investigated. The time histories of water surface fluctuations are calculated for several coastal locations on the coasts of the Caribbean Sea. Results of the numerical simulations are in reasonable agreement with data of observations.

Analytical and numerical study of nonlinear effects at tsunami modeling

The giant tsunami occurred in the Indian Ocean on 26th December 2004 pays attention to this natural phenomenon and the possibility to use the analytical methods. The given paper demonstrates the role of nonlinear effects in the computed dynamics of the tsunami waves in shallow seas and the applicability of the rigorous and approximated solutions of the nonlinear theory of water waves to explain the results of the numerical simulation.

Run-up of Solitary Waves on Slopes with Different Profiles

The problem of sea-wave run-up on a beach is discussed within the framework of exact solutions of a nonlinear theory of shallow water. Previously, the run-up of solitary waves with different forms (Gaussian and Lorentzian pulses, a soliton, special-form pulses) has already been considered in the literature within the framework of the same theory. Depending on the form of the incident wave, different formulas were obtained for the height of wave run-up on a beach. A new point of this study is the proof of the universality of the formula for the maximum height of run-up of a solitary wave on a beach for the corresponding physical choice of the determining parameters of the incident wave, so that the effect of difference in form is eliminated. As a result, an analytical formula suitable for applications, in particular, in problems related to tsunamis, has been proposed for the height of run-up of a solitary wave on a beach.

Focusing of edge waves above a sloping beach

The mechanism of the spatial-temporal focusing (dispersion enhancement) of the edge waves in the shelf zone is studied in the framework of linear shallow water theory. The multi-modal Stokes edge waves are considered as an example of the shallow waves propagated in the coastal zone above uniformly sloping plane beach. The method to find the localized anomalous high wave generated in the process of the wave packet focusing is suggested. The characteristics of the wave trains evolving into the anomalous large wave are discussed.

Steepness and spectrum of a nonlinearly deformed wave on shallow waters

The process of nonlinear deformation of a surface wave on shallow waters is investigated. The main attention is given to the relationship between the wave Fourier spectrum and the steepness of wave front slope. It is shown that an unambiguous relationship couples these quantities in the case of an initially sinusoidal wave, which allows estimation of the spectral composition of the wave field from the observed wave steepness.

Solitary wave dynamics in shallow water over periodic topography

The problem of long-wave scattering by piecewise-constant periodic topography is studied both for a linear solitary-like wave pulse, and for a weakly nonlinear solitary wave [Korteweg-de Vries (KdV) soliton]. If the characteristic length of the topographic irregularities is larger than the pulse length, the solution of the scattering problem is obtained analytically for a leading wave in the framework of linear shallow-water theory. The wave decrement in the case of the small height of the topographic irregularities is proportional to delta2, where delta is the relative height of the topographic obstacles. An analytical approximate solution is also obtained for the weakly nonlinear problem when the length of the irregularities is larger than the characteristic nonlinear length scale. In this case, the Korteweg-de Vries equation is solved for each piece of constant depth by using the inverse scattering technique; the solutions are matched at each step by using linear shallow-water theory. The weakly nonlinear solitary wave decays more significantly than the linear solitary pulse. Solitary wave dynamics above a random seabed is also discussed, and the results obtained for random topography (including experimental data) are in reasonable agreement with the calculations for piecewise topography.

Resonance three-wave interactions of stokes edge waves

Nonlinear three-wave interactions of Stokes edge waves propagating both in one direction and in opposite directions along a uniformly sloping shelf are considered. In the cases when only the lowest four modes participate in interaction, the synchronism conditions are determined and interaction coefficients are calculated. It is shown that the interaction coefficients of unidirectional edge-wave modes can vanish for certain triads. The spatiotemporal dynamics of a triad of edge waves is investigated. In addition, expressions are given for the resonance interaction coefficients of edge waves over the bottom of an arbitrary profile.

Modeling the dynamics of intense internal waves on the shelf

The transformation of the internal wave packet during its propagation over the shelf of Portugal was studied in the international experiment EU MAST II MORENA in 1994. This paper presents the results of modeling of the dynamics of this packet under hydrological conditions along the pathway of its propagation. The modeling was performed on the basis of the generalized Gardner-Ostrovskii equation, including inhomogeneous hydrological conditions, rotation of the Earth, and dissipation in the bottom boundary layer. We also discuss the results of the comparison of the observed and simulated forms and phases of individual waves in a packet at reference points.

Dynamics of solitons in a nonintegrable version of the modified Korteweg-de Vries equation

Nonlinear wave dynamics is discussed using the extended modified Korteweg-de Vries equation that includes the combination of the third- and fifth-order terms and is valid for waves in a three-layer fluid with so-called symmetric stratification. The derived equation has solutions in the form of solitary waves of various polarities. At small amplitudes, they are close to solitons of the modified Korteweg-de Vries equation. However, the height of large-amplitude solutions has a limit approaching which solitary waves widen and acquire a table like shape similar to soluitons of the Gardner equation. Numerical calculations confirm that the collision of solitons of the derived equation is inelastic. Inelasticity is the most pronounced in the interaction of unipolar pulses. The direction of the shift of the phase of the higher-amplitude soliton owing to the interaction of solitons of different polarities depends on the amplitudes of the pulses.

Shallow-water edge waves above an inclined bottom slowly varied in along-shore direction

Abstract The dynamics of the shallow-water linear edge waves above an inclined bottom slowly varied in an along-shore direction is studied. By using the asymptotic method, the offshore structure of the edge waves and their dispersion relation are determined in the leading order, and the wave amplitude – in next order. The asymptotic theory confirms the “energetic” approach that the wave amplitude can be derived from the energy flux conservation in the leading order. Three different offshore bottom profiles are considered: the beach of constant slope, exponential shelf, and step-shelf. The variations of the wave amplitude and along-shore wave number are calculated in details for the cases when the parameters of the shelf zone are changed slowly in the along-shore direction.