  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.
