Mathematical Modelling of Nonlinear Internal Ring Waves in Three-Layered Fluids

Abstract

Oceanic internal waves generated in straits, river-sea interaction zones, and by interaction with localised topographic features often look on satellite images as a part of a ring. We study long weakly-nonlinear mode-I and mode-II internal ring waves in a three-layered fluid in the presence of a parallel linear current (i.e. a shear flow with constant vorticity), distorting the wavefronts. Previous research has shown that there exists a linear modal decomposition in the far-field set of Euler equations describing the waves, which is used to describe the wavefronts and modal structure of these internal ring waves, and to obtain an appropriate weakly-nonlinear model for the amplitudes of the waves. The curvilinear wavefronts of waves in cylindrical geometry are described by the singular solution of a nonlinear first-order differential equation. We show that there are significant differences in the shapes of the wavefronts of mode-I and mode-II internal waves propagating over the same linear shear current: the wavefront of the mode-I wave is elongated in the direction of the current, while the wavefront of the mode-II wave is squeezed in this direction. In addition, we use a finite-difference scheme to solve the derived cKdV-type amplitude equation numerically. Numerical results are obtained for mode-I and mode-II internal ring waves generated by a localised source and initially propagating in the absence of a current. We assume that a part of the ring wave then starts propagating over the linear shear current, and our modelling within the scope of the cKdV-type equation can be applied to describe the behaviour of the relevant part of the ring wave. Numerical modelling confirms the analytical predictions concerning the shape of the wavefronts, and reveals very strong dispersive effects in the upstream direction.