Oceanic Lee Waves in a Linearly Sheared Flow

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.