Applied Mathematics Seminar

A finite-amplitude long wave evolution equation is derived to describe the effect of weak current shear on internal waves in a uniformly stratified fluid. For steadily propagating waves the evolution equation reduces to the steady version of the generalised Korteweg-de Vries equation. An analysis of this equation is presented for a wide range of possible shear profiles. The type of waves that occur are found to depend on the number and position of the inflexion points of the representation of the shear profile in amplitude space. The stability of these waves is generally found to decrease as the complexity of the waves increases. These solutions suggest that kinks and solitary waves with multiple lengthscales are only possible for shear profiles with a turning point, while instability is only possible if the shear profile has an inflexion point. The unsteady evolution of a periodic initial condition is considered and again the solution is found to depend on the inflexion points of the amplitude representation of the shear profile. The unsteady solutions demonstrate that finite-amplitude effects can act to halt the critical collapse of solitary waves which occurs in the context of the generalised Korteweg-de Vries equation. These solutions are then used to qualititatively relate previously reported observations of shock formation on the internal tide propagating onto the Australian North West Shelf to the observed background current shear.