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Numerical modeling of tsunami wave destruction and turbulent mixing at tsunami wave clash on the shore

Kshevetskii S., Vereshchagina I.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2016

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Vol. 20, No 2

157--169

EN

A numerical model of propagation of internal gravity waves in a stratified medium is applied to the problem of tsunami wave run-up onto a shore. In the model, the ocean and the atmosphere are considered as a united continuum in which the density varies with height with a saltus at the water-air interface. The problem solution is sought as a generalized (weak) solution; such a mathematical approach automatically ensures correct conditions of matching of the solutions used on a water-air interlayer. The density stratification in the ocean and in the atmosphere is supposed to be described with an exponential function, but in the ocean a scale of the density stratification takes a large value and the density changes slightly. The initial wave running to a shore is taken in the form of a long solitary wave. The wave evolution is simulated with consideration of the time-varying vertical wave structure. Near the shore, the wave breaks down, and intensive turbulent mixing develops in the water thickness. The wave breakdown effect depends on the bottom shape. In the case when the bottom slope is small and the inshore depth grows slowly with the distance from the shore, mixing happens only in the upper stratum of the fluid due to the formation of a quiet region near the bottom. When the bottom slope takes a sufficiently large value, the depth where fluid mixing takes place goes down up to 50 meters. The developed model shows that the depth of the mixing effects strongly depends on the bottom shape, and the model may be useful for investigation of the impact strong gales and hurricanes on the coastline and beaches.

2

Scattering of internal waves by vertical barrier in a channel of stratified fluid

Dolai P.

International Journal of Applied Mechanics and Engineering

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2015

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Vol. 20, no. 3

471--485

EN

The problem of two dimensional internal wave scattering by a vertical barrier in the form of a submerged plate, or a thin wall with a gap in an exponentially stratified fluid of uniform finite depth bounded by a rigid plane at the top, is considered in this paper. Assuming linear theory and the Boussinesq approximation, the problem is formulated in terms of the stream function. In the regions of the two sides of the vertical barrier, the scattered stream function is represented by appropriate eigen function expansions. By the use of appropriate conditions on the barrier and the gap, a dual series relation involving the unknown elements of the scattering matrix is produced. By defining a function with these unknown elements as its Fourier sine expansion series, it is found that this function satisfies a Carleman type integral equation on the barrier whose solution is immediate. The elements of the scattering matrix are then obtained analytically as well as numerically corresponding to any mode of the incident internal wave train for each barrier configuration. A comparison with earlier results available in the literature shows good agreement. To visualize the effect of the barrier on the fluid motion, the stream lines for an incident internal wave train at the lowest mode are plotted.

3

Internal wave diffraction by a strip of an elastic plate on the surface of a stratified fluid

Dolai P., Dolai D. P.

International Journal of Applied Mechanics and Engineering

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2013

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Vol. 18, no. 1

5--26

EN

The problem of internal wave diffraction by a strip of an elastic plate of finite width present on the surface of an exponentially stratified liquid is investigated in this paper. Assuming linear theory, the problem is formulated in terms of a function related to the stream function describing the motion in the liquid. The related boundary value problem involves a hyperbolic type partial differential equation (PDE), known as the Klein Gordon equation. The method of Wiener-Hopf is utilized in the mathematical analysis to a slightly generalized boundary value problem (BVP) by introducing a small parameter, and the problem is solved approximately for large width of the plate. In the final results, this small parameter is made to tend to zero. The diffracted field is obtained in terms of integrals, which are then evaluated asymptotically in different regions for a large distance from the edges of the plate and the results are interpreted physically.