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The scattering of three-dimensional inertia-gravity waves by a turbulentgeostrophic flow leads to the redistribution of their action through what isapproximately a diffusion process in wavevector space. The correspondingdiffusivity tensor was obtained by Kafiabad, Savva & Vanneste (2019, J. FluidMech., 869, R7) under the assumption of a time-independent geostrophic flow. Werelax this assumption to examine how the weak diffusion of wave action acrossconstant-frequency cones that results from the slow time dependence of thegeostrophic flow affects the distribution of wave energy. We find that thestationary wave-energy spectrum that arises from a single-frequency waveforcing is localised within a thin boundary layer around the constant-frequencycone, with a thickness controlled by the acceleration spectrum of thegeostrophic flow. We obtain an explicit analytic formula for the wave-energyspectrum which shows good agreement with the results of a high-resolutionsimulation of the Boussinesq equations.

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