Instability of unbounded transverse mixing layer in shallow waters

Abstract

This paper examines the role of waves in the development of the instability of an unbounded transverse mixing layer in shallow waters. In supercritical state waves become significant. The matching of the wave-like asymptotic solution in the far field to the solutions near the inflection and the returning surfaces poses a difficult mathematical problem. We therefore conducted numerical simulations to find solutions directly from the shallow water equations. The simulations found that the growing waves in the unstable current had a consistent structure throughout the linear stage of the instability’s development. The excitation, reflection, and transmission of waves produced a modulating instability that maintained a consistent structure as the amplitude of the instability increased over many orders of magnitude. This role of the waves on the supercritical instability, as delineated by our numerical simulations, is not solely describable by the classical analytical method.