Two-way wave-vortex interactions in a Lagrangian-mean shallow water model

We derive and investigate numerically a reduced model for wave-vortex interactions involving non-dispersive waves, which we study in a two-dimensional shallow water system with an eye towards applications in atmosphere-ocean fluid dynamics. The model consists of a coupled set of nonlinear partial differential equations for the Lagrangian-mean velocity and the wave-related pseudomomentum vector field defined in generalized Lagrangian-mean theory. It allows for two-way interactions between the waves and the balanced flow that is controlled by the distribution of Lagrangian-mean potential vorticity, and for strong solutions it features a desirable exact energy conservation law for the sum of wave energy and mean flow energy. Our model goes beyond standard ray tracing as we can derive weak solutions that contain discontinuities in the pseudomomentum field, using the theory of weakly hyperbolic systems. This allows caustics to form without predicting infinite wave amplitudes, as would be the case in the standard ray-tracing theory. Suitable wave forcing and dissipation terms are added to the model and a numerical scheme for the model is implemented as a coupled set of pseudo-spectral and finite-volume integrators. Idealized examples of interactions between wavepackets and simple vortex structures are presented to illustrate the model dynamics. The unforced and non-dissipative simulations suggest a heuristic rule of "greedy' waves, i.e. in the long run the wave field always extracts energy from the mean flow.