Evolution of internal cnoidal waves with local defects in a two-layer fluid with and without rotation

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 model. We develop simultaneous asymptotic expansions for the evolving mean fields and deviations from them to describe a large class of uni-directional waves via the Ostrovsky equation, which fully decouples from mean-field variations. The latter generate additive inertial oscillations in the shear and in the phase of both the interfacial displacement and shear. Unlike conventional derivations leading to the Ostrovsky equation, our formulation does not impose the zero-mean constraints on the initial conditions of any variable. 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 with local defects, both with and without rotation. We show that rotation leads to the emergence of bursts of internal waves and shear currents, qualitatively similar to the wavepackets generated from solitons and modulated cnoidal waves in earlier studies, but emerging much faster. We also 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 corner condition 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 leads to the emergence of strong bursts.