A weakly nonlinear model of long internal waves in closed basins

Abstract

A simple model is developed, based on an approximation of the Boussinesq equation, that considers the weakly nonlinear evolution of an initial interface disturbance in a closed basin. The solution consists of the sum of the solutions of two independent Korteweg–de Vries (KdV) equations (one along each characteristic) and a second-order wave–wave interaction term. It is demonstrated that the solutions of the two independent KdV equations over the basin length [0, L] can be obtained by the integration of a single KdV equation over the extended reflected domain [0, 2L]. The main effect of the second-order correction is to introduce a phase shift to the sum of the KdV solutions where they overlap. The results of model simulations are shown to compare qualitatively well with laboratory experiments. It is shown that, provided the damping timescale is slower than the steepening timescale, any initial displacement of the interface in a closed basin will generate three types of internal waves: a packet of solitary waves, a dispersive long wave and a train of dispersive oscillatory waves.