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Abstract We find a Green’s function solution for the propagation of two-dimensional oceanic lee waves in a linearly sheared background flow and use it to study the remote dissipation of waves. This Green’s function approach combines normal modes in the vertical direction with solving a coupled system of ODEs in the horizontal direction in an up-winding fashion. This leads to a solution for isolated topography that is valid on an infinite horizontal domain, in sharp contrast with the standard lee wave solution method based on horizontal Fourier series, which suffers from spurious resonances in a vertically bounded domain that must be controlled by artificial damping. Moreover, in the case of a negatively sheared background flow with an inertial critical layer present, the singularity normally associated with these layers is significantly modified when following the Green’s function approach. In this setting, we confirm ray-tracing estimates for lee wave absorption and highlight the ability for waves to propagate far from their generation site, even with critical layers present, which could help explain observed discrepancies between lee wave generation and dissipation. Elsewhere in the oceanographic literature, viscosity is used to deal with critical layers, and also more broadly to avoid resonant solutions in their absence, which can significantly affect the conclusions drawn if not handled correctly. This emphasizes the need for nonperiodic solvers in studies of oceanic lee waves and for care when incorporating any damping mechanism. 

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.