Evolution of internal cnoidal waves with local defects in a two-layer fluid with rotation

Abstract

Internal waves in a two-layer fluid with rotation are considered within the framework of Helfrich's f-plane extension of the Miyata-Maltseva-Choi-Camassa (MMCC) model. Within the scope of this model, we develop an asymptotic procedure which allows us to obtain a description of a large class of uni-directional waves leading to the Ostrovsky equation and allowing for the presence of shear inertial oscillations and barotropic transport. Importantly, unlike the conventional derivations leading to the Ostrovsky equation, the constructed solutions do not impose the zero-mean constraint on the initial conditions for any variable in the problem formulation. Using the constructed solutions, we model the evolution of quasi-periodic initial conditions close to the cnoidal wave solutions of the Korteweg-de Vries (KdV) equation but having a local amplitude and/or periodicity defect, and show that such initial conditions can lead to the emergence of bursts of large internal waves and shear currents. As a by-product of our study, we show that cnoidal waves with expansion defects discussed in this work are generalised travelling waves of the KdV equation: they satisfy all conservation laws of the KdV equation (appropriately understood), as well as the Weirstrass-Erdmann conditions for broken extremals of the associated variational problem and a natural weak formulation. Being smoothed in numerical simulations, they behave, in the absence of rotation, as long-lived states with no visible evolution, while rotation changes this behaviour and leads to the emergence of strong bursts.