Mathematical modelling of long ocean waves with a discontinuous density gradient

A mathematical model for studying the propagation of long internal ocean wa ves of finite amplitude is proposed. The vertical structure of the pressure perturbation is investigated and reduced to a Sturm-Liouville eigenvalue p roblem. In the continuous stratification case, the pattern of this vertical structure depends on the choice of the characteristic scale of a varying s tratification parameter, denoted by delta. As this parameter asymptotically approaches a critical value (i.e. delta --> delta (cri)), the amplitudes o f the solution's normal modes increase considerably. The internal waves bre ak and produce an unstable interface, which degenerates into a turbulent mi xed layer. These conditions correspond to the critical state of wave existe nce. When delta < delta (cri) a three-layer discontinuous gradient model is proposed to resolve the problem. It consists in specifying one solution wi thin a thin intermediary layer and two solutions on either side of this lay er. The results show that the use of appropriately matching interfacial con ditions allows to obtain generally matching solutions, even for small value s of the nonconstant stratification parameter delta.