Simulation of internal undular bores of depression propagating over a slowly varying region

Abstract

Internal undular bores have been observed in many parts of the world. This paper intends to look at the variable topographic effects on the evolution of the internal undular bores of depression. Here, the internal undular bores is considered to be propagating in a two-layer fluid system. The topography is assumed to be slowly varying. Therefore, the appropriate mathematical model is the variable-coefficient extended Korteweg-de Vries equation. The governing equation is solved numerically using the method of lines. We are especially interested in looking at the transformation of two types of internal undular bore of depression, which are Korteweg de Vries-type and table-top internal undular bore. Our numerical results show that the internal undular bore of depression transformed into a positive undular bore as it propagates over a slowly increasing slope and when it involves polarity change. In front of the transformed internal bore, a series of isolated solitary waves or a solitary wavetrain is generated as a non-adiabatic response to the interaction between the internal undular bore with the changing bottom surface. The solitary wavetrain is observed to be climbing the negative pedestal. As time increases, the amplitude of the individual solitary wavetrain is decreasing and finally the solitary wavetrain will die out due to the pedestal. However, if there is no polarity change, the internal undular bores deforms adiabatically and its amplitude decreases slowly until it reaches new limiting amplitude value due the increasing bottom surface. On the other hand, when the slope is decreasing slowly, the internal undular bore deforms adiabatically where its amplitude increases slowly. There is a multi-phase behaviour is observed during the evolution as the results of the interaction between the internal bore and the varying slope. In this case, there is no polarity change. The transformation of internal undular bores of depression depends on the nature of the topographic change.