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Abstracts

At the recent decades much attention of researchers has been attracted to essentially three-dimensional finite-amplitude disturbances (for instance, books [1–3]). However, most models (e.g., papers [4–6]) are applicable only to nonlinear waves propagating chiefly in one direction. Only in these cases the problem is reduced to one equation for the perturbation of the free surface. For this reason, finite-amplitude waves travelling simultaneously in different directions can be described only by systems of equations incorporating both the disturbance of the free boundary and the fluid velocity. In the systems proposed earlier (for example, [7–9]), even the linear terms of all equations involve terms depending on the fluid velocity. The new combined system of equations, which is more well-behaved was proposed in the paper [10].

There are assumed that liquid is incompressible, its stationary flow is absent, the disturbance amplitudes are small but finite, characteristic horizontal lengths of waves and of the bottom topography are larger and the thickness of unsteady viscous boundary layer is smaller than the fluid depth, and finally, capillary effects are moderate. The initial system of the Stokes equations and of the continuity equation for the shallow water above a gently sloping bottom was reduced to one basic nonlinear evolution equation for spatial perturbations of the free surface and two linear auxiliary differential equations for a determination of the horizontal velocity vector averaged over the layer depth which is contained in the main equation only in one term of the second order of smallness. The suggested model is suitable for finite-amplitude waves running on any angles. Even in the case of inviscid liquids this approach is in essence easier than known systems of equations, where all equations contain both linear and nonlinear items (e.g., [7–9]).

Some solutions of our model equations were found numerically. The calculations according to the model [6] were performed with the help of the implicit three-layer difference scheme, which is described in detail in the paper [11]. This scheme has the second order of approximation in all variables. The results of several numerical experiments for a transformation of initially plane moderately long nonlinear waves were adduced in the paper [11] too. A dynamics of the three-dimensional disturbance which is solitary in the space were demonstrated in the paper [12].

The calculations according to the model [10] were carried out in the following way. At the step “predicator” the calculations were made with the help of the simplest replacement for the velocity vector. At the step “corrector” the velocity vector was determined using the simple linear auxiliary equations. Poisson’s equation for a determination of the velocity vector was resolved by the method of the fast Fourier transformation by both horizontal coordinates on the each step of time.

Formally the evolution equation of the model [6] allows to study a collision of two plane waves running towards each other. But it is shown that at the point of time of their maximal interaction the calculation error may be equals 10 % approximately. A comparison of the numerical results for three-dimensional solitary in the space perturbations of small but finite amplitude was carried out too.

Some test solutions were found in the pools with different topographies. As it should be not only the changing of the wave velocities but also the intensification of disturbances moving towards the lower liquid depth and otherwise their weakening when the waves are moving to the deeper area were observed. It is seen, that the additional peaks and troughs took place over the bottom irregularities.

This work was supported partially by INTAS – SB RAS (Grant 06-9236) and RFBR (Project 07-01-00574).

References

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10. Arkhipov D. G., Khabakhpashev G. A. New approach to the description of spatial nonlinear waves in dispersive media. Doklady Physics. 2006. V. 51, no. 8. P. 418–422.

11. Litvinenko A. A., Khabakhpashev G. A. Numerical simulation of sufficiently long nonlinear two-dimensional waves on water in pools with a gently sloping bottom. Comput. Technol. 1999. V. 4, no. 3. P. 95–105.

12. Litvinenko A. A., Safarova N. S., Khabakhpashev G. A. Numerical simulation of essentially three-dimensional nonlinear disturbances of a free surface of shallow ponds with gently sloping bottom. In Abstracts of the IХ All-Russian Conf. «State-of-the-art Methods of Mathematical Simulation of Natural and Man-made Disasters». Barnaul: Altay State University, 2007. P. 65.

Note. Abstracts are published in author's edition

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