In the weakly nonlinear limit, oceanic internal solitary waves for a single linear long wave mode are described by the KdV equation, extended to the Ostrovsky equation in the presence of background rotation. In this paper we consider the scenario when two different linear long wave modes have nearly coincident phase speeds and show that the appropriate model is a system of two coupled Ostrovsky equations. These are systematically derived for a density-stratified ocean. Some preliminary numerical simulations are reported which show that, in the generic case, initial solitary-like waves are destroyed and replaced by two coupled nonlinear wave packets, being the counterpart of the same phenomenon in the single Ostrovsky equation.
Received 05 March 2013Accepted 15 May 2013Published online 05 June 2013
Lead Paragraph: Large amplitude internal solitary waves in the coastal ocean are commonly modelled with the Korteweg-de Vries equation or a closely related evolution equation. The characteristic feature of these models is the solitary wave, and it is well documented that these provide the basic paradigm for the interpretation of oceanic observations. However, often the internal waves in the ocean survive for several inertial periods, and in that case, the Korteweg-de Vries equation is supplemented with a linear non-local term representing the effects of background rotation, commonly called the Ostrovsky equation. This equation does not support solitary wave solutions, and instead a solitary-like initial condition collapses due to radiation of inertia-gravity waves, with instead the long-time outcome being a nonlinear wave packet. The Korteweg-de Vries equation and the Ostrovsky equation are formulated on the assumption that only a single vertical mode is used. In this paper we consider the situation when two vertical modes are used, due to a near-resonance between their respective linear long wave phase speeds. In this case, we derive two coupled Ostrovsky equations and demonstrate through numerical simulations that typical solitary-like initial conditions collapse into two nonlinear wave packets, propagating usually with distinct speeds.
A. Alias is supported by Universiti Malaysia Terengganu and the Ministry of Higher Education of Malaysia. We are grateful to the referees for some helpful and insightful comments.
Article outline: I. INTRODUCTION II. FORMULATION A. Asymptotic derivation B. Coupled Ostrovsky equations C. Three-layer example III. LINEAR DISPERSION RELATION IV. NUMERICAL RESULTS A. Symmetric case B. Non-symmetric cases V. DISCUSSION