  In rotating stratified flows including in the atmosphere and ocean,
inertia-gravity waves (IGWs) often coexist with a geostrophically balanced
turbulent flow. Advection and refraction by this flow lead to wave scattering,
redistributing IGW energy in the position--wavenumber phase space. We give a
detailed description of this process by deriving a kinetic equation governing
the evolution of the IGW phase-space energy density. The derivation relies on
the smallness of the Rossby number characterising the geostrophic flow, which
is treated as a random field with known statistics, and makes no assumption of
spatial scale separation.
  The kinetic equation describes energy transfers that are restricted to IGWs
with the same frequency, as a result of the timescale separation between waves
and flow. We formulate the kinetic equation on the constant-frequency surface
-- a double cone in wavenumber space -- using polar spherical coordinates, and
we examine the form of the two scattering cross sections involved, which
quantify energy transfers between IGWs with, respectively, the same and
opposite directions of vertical propagation. The kinetic equation captures both
the horizontal isotropisation and the cascade of energy across scales that
result from scattering. We focus our attention on the latter to assess the
predictions of the kinetic equation against direct simulations of the
three-dimensional Boussinesq equations, finding good agreement.
