Characteristics of tsunami motion and energy budget during runup and rundown processes over a plane beach

Abstract

Motions of tsunami waves during runup and rundown processes on the uniform sloping beach are studied numerically by the fully nonlinear and highly dispersive Boussinesq equations. We first study the leading-depression N-wave defined by solitary theory. The shoreline movement and its moving speed are analyzed during the wave approaching the coastal area. Details on the flow field and the energy transformation are obtained in terms of the reconstruction of the full velocity field by Boussinesq equations. In addition, solitary wave is studied as a comparison. The different energy budget explains the phenomenon that the N-wave leads much larger runup than the solitary wave in some specific situations. The investigation is then extended to the patterns of the energy transformation of N-shape waves, hump-like waves, and sinusoidal long waves. These waves are of the same order in scale with the recent giant tsunamis. The results show that the potential energy of the N-shape tsunami waves nearly reaches the maximum while that of the hump-like tsunami waves does not reach the maximum at the maximum runup, meanwhile the kinetic energy of both waves does not go to zero. For the sinusoidal wave train, however, its potential energy reaches the maximum and its kinetic energy goes to zero exactly at the maximum runup. To understand the responses caused by the variation of the waveforms, effects of the nonlinearity and the dispersion on the energy budget are studied in the geophysical tsunamis order. Moreover, the regularities of the energy budget of some general cases including the leading-depression N-waves (LDNs), the leading-elevation N-wave, and generalized LDNs are investigated extensively. The mechanism of the energy budget of these waves is quite different as just considering the potential energy at the maximum runup.