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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

A high-accuracy method of computation of X-ray waves propagation through an optical system consisting many lenses

Kshevetskii S., Wojda P.

TASK Quarterly : scientific bulletin of Academic Computer Centre in Gdansk

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2016

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Vol. 20, No 2

171--186

EN

The propagation of X-ray waves through an optical system consisting of many X-ray refractive lenses is considered. Two differential equations are contemplated for solving the problem for electromagnetic wave propagation: first – an equation for the electric field, second – an equation derived for a complex phase of an electric field. Both equations are solved by the use of a finite-difference method. The simulation error is estimated mathematically and investigated. The presented results for equations show that in order to establish a high accuracy computation a much smaller number of points is needed to solve the problem of X-ray waves propagation through a multi-lens system when the method for the second equation is used. The reason for such a result is that the electric field of a wave after passing through many lenses is a quickly oscillating function of coordinates, while the electric field phase is a quickly increasing, but not oscillating function. Therefore, a very detailed difference grid, which is necessary to approximate the considered electric field can be replaced by not such a detailed grid, when computations are made for the complex wave of the electric field. The simulation error of both suggested methods is estimated. It is shown that the derived equation for a phase function allows efficient simulation of propagation of X-rays for the multi-lens optical system.

3

Efficient quadrature for fast oscillating integral of paraxial optics

Kshevetskii S., Wojda P.

Mathematica Applicanda

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2015

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Vol. 43, No. 2

253--267

EN

The study concerns the determination of quadrature for the integral solution of the paraxial wave equation. The difficulty in computation of the integral is associated with the rapid change of the integrand phase. The developed quadrature takes into account the fast oscillating character of the integrand. The presented method is an alternative to the commonly used methods based on the use of the Fourier transform. The determination of the quadrature is discussed on the example of the integral arisen in the theory of propagation and focusing on hard X-rays waves. Due to the generality of the presented quadrature, it may also be applied to issues related to standard optics and acoustics.

PL

Praca jest poświęcona wyznaczaniu kwadratury dla rozwiązań całkowych równania przewodnictwa cieplnego z zespolonym potencjałem. Trudność w wyznaczaniu tego typu całek jest związana z szybkimi oscylacjami funkcji całkowanej. Prezentowana metoda jest alternatywa dla powszechnie stosowanej metody opartej o zastosowanie transformacji Fouriera. Sprecyzowanie kwadratury jest przedyskutowane na przykładzie całek występujących przy badaniu teorii propagacji i skupiania promieniowania rentgenowskiego. Dzięki ogólności prezentowanej kwadratury, może być ona także zastosowana do zagadnień związanych z optyką i akustyką.