Shallow water equations on a movable bottom

Abstract

Nonlinear dispersion Zheleznyak-Pelinovskii equations describing surface water waves are derived in the paper for the case of an unsteady bottom surface. It is shown that the obtained equations and the well-known Green-Naghdi equations are different forms of the same system of shallow water equations of the second approximation taking into account the bottom variations. The issue of wave generation by a movable bottom is rather topical nowadays. It is related to mathematical simulations of long surface waves (tsunami) caused by the sea bottom fluctuations due to landslides or the formation of extended clefts (3, 4, 7). This class also includes the problems of motion of bodies over a reservoir bottom, which may cause waves on the water surface (2). The simulation of surface waves using complete hydrodynamic models requires much computation time (9, 10). Thus, wave patterns appearing in the motion of an underwater landslide were studied numerically in (9). The specifics of simulation of such waves is determined by a large time period of the landslide movement from the shore to the deep part of the reservoir, by the sufficiently high speed of its movement, and, as a consequence, by the necessity to use extended calculation domains and grids with a large number of nodes. As a result, computation of a single variant by the complete model takes several hours. On the other hand, approximate models require a few minutes of computation and the general picture of the wave modes is satisfactory described even by simplest models. However, it has been noted that a detailed simulation of a long-time phe- nomenon requires models capable to reproduce the variance and the inhomogeneity of the process in the vertical direction. For example, this was shown in (7, 14) where various approximate and complete hydrodynamic equations were used for the study of the generation of waves by a solid rigid body moving along a slope. Moreover, experimental physical data were used in (7) for the study of the formation of waves. Comparative analysis shows that in the case of a long thin landslide at the ini- tial stage of the process all models, from the classic shallow water equations to the complete model of ideal liquid flows, sufficiently well describe the most noticeable wave-forming characteristics observed in experiments. Even the linear shallow wa- ter model gives a pattern sufficiently close to experimental data at the initial stage � Institute of Computational Technologies, Siberian Branch of the Russian Academy of Sciences,