Three-Dimensional Long Interfacial Wave Propagation Near the Critical Depth Level

摘要

In this paper, the propagation of interfacial waves near the critical depth levelin a two-layered fluid system is investigated. We first derive an evolution equation for weakly nonlinear and dispersive interfacial waves propagating predominantly in the longitudinal direction of a slowly rotating channel with gradually varying topography and sidewalls. The new evolution equation includes both quadratic and cubic nonlinearities. For interfacial waves propagating in certain type of non-rotating channels with varying topography, we find two families of periodic solutions, expressed in terms of the snoidal function, to the variable coefficient equation. As the limiting cases of these periodic-wave solutions, a family of solitary-wave solutions and an isolated shock-like wave solution are also obtained. In a uniform rotating channel, our small-time asymptotic analysis and numerical study show that depending on the relative importance of the cubic nonlinearity to quadratic nonlinearity, the wavefront of a Kelvin solitary wave will curve either forwards or backwards, trailed by a small train of Poincare' waves. When these two nonlinearities almost balance each other, the wavefront becomes almost / straight-crested across the channel, and the trailing Poincare waves diminish.