Abstract

Oceanic surface and internal waves often have curvilinear fronts and propagate over various depth-dependent currents. We study long weakly-nonlinear waves using a linear modal decomposition in the far-field set of Euler equations with the boundary conditions appropriate for oceanographic applications. The theoretical framework consists of a spectral problem defining three-dimensional modal functions and dispersion relation (angular adjustment equation in the form of a nonlinear first-order ODE ), as well as an amplitude equation (generally, 2+1-dimensional) with coefficients dependent on solutions of the modal equations.  I will discuss recent  analytical results for two- and three-layered fluids with basic shear flows suitable for the modelling of wind-generated currents, river inflows and exchange flows in straits.  I will also discuss numerical results  which reveal interesting features including the formation of transient dispersive shock waves only in the upstream part of the ring wave. The structure of the wave field in strongly three-dimensional. Singularities of the swallowtail-type may arise near the onset of the long-wave instability of the basic flow. Based on joint works with Curtis Hooper, Roger Grimshaw, Dmitri Tseluiko, Noura Alharthi and Ricardo Barros.