Fields Institute Workshop on Nonlinear Wave Dynamics

Carleton University, August 20-22 2008


Simon Clarke

School of Mathematical Sciences, Monash University

Finite amplitude two-layer flow past topography

We consider the problem of large-amplitude waves forced by the flow of a two layer fluid of finite depth past topography, such as would arise in exchange flows in geophysical fluid dynamics. Of particular interest is the long wave dynamics for this problem. We first consider the derivation of the governing equations; namely the forced Camassa--Choi (CC) equations. The various limits of this equation are then discussed. In the hydraulic limit these reduce to the two-layer shallow water (SW) equations, while for weakly nonlinear, near-resonant flows the limiting dynamics are governed by the forced extended Korteweg--de Vries (feKdV) equation. The phenomenology of the steady SW equations is then considered, which raises the fundamental question of what asymptotic solution results from a given initial problem, for example, the trivial initial condition. Numerical solutions of the finite-amplitude SW and CC equations are presented which demonstrate the evolution to hydraulic states. However, these cannot be efficiently used to determine the asymptotics for the full parameter space. Rather an alternative approach is outlined. In the weakly nonlinear limit the hydraulic regimes must match with the regimes which result from the feKdV equation. Thus we consider in detail the dynamics of the feKdV equation, in particular the weakly dispersive limit, and from these outline the hydraulic regimes which must result. \end{document}