Simulation of Internal Undular Bores Propagating over a Slowly Varying Region

Abstract

Internal undular bores have been observed in many parts of the world. Studies have shown that many marine structures face danger and risk of destruction caused by internal undular bores due to the amount of energy it carries. This paper looks at the transformation of internal undular bore in two-layer fluid flow under the influence of variable topography. Thus, the surface of the bottom is considered to be slowly varying. The appropriate mathematical model is the variable-coefficient extended Korteweg-de Vries equation. We are particularly interested in looking at the transformation of KdV-type and table-top undular bore over the variable topography region. The governing equation is solved numerically using the method of lines, where the spatial derivatives are first discretised using finite difference approximation so that the partial differential equation becomes a system of ordinary differential equations which is then solved by 4th order Runge-Kutta method. Our numerical results show that the evolution of internal undular bore over different types of varying depths regions leads to a number of adiabatic and non-adiabatic effects. When the depth decreases slowly, a solitary wavetrain is observed at the front of the transformed internal undular bore. On the other hand, when the depth increases slowly, we observe the generation of step-like wave and weakly nonlinear trailing wavetrain, the occurrence of multi-phase behaviour, the generation of transformed undular bore of negative polarity and diminishing transformed undular bore depending on the nature of the topography after the variable topography.