Inertia-gravity-wave diffusion by geostrophic turbulence: the impact of flow time dependence

The scattering of three-dimensional inertia-gravity waves by a turbulent geostrophic flow leads to the redistribution of their action through what is approximately a diffusion process in wavevector space. The corresponding diffusivity tensor was obtained by Kafiabad, Savva & Vanneste (2019, J. Fluid Mech., 869, R7) under the assumption of a time-independent geostrophic flow. We relax this assumption to examine how the weak diffusion of wave action across constant-frequency cones that results from the slow time dependence of the geostrophic flow affects the distribution of wave energy. We find that the stationary wave-energy spectrum that arises from a single-frequency wave forcing is localised within a thin boundary layer around the constant-frequency cone, with a thickness controlled by the acceleration spectrum of the geostrophic flow. We obtain an explicit analytic formula for the wave-energy spectrum which shows good agreement with the results of a high-resolution simulation of the Boussinesq equations.