  Oceanic internal tides and other inertia-gravity waves propagate in an
energetic turbulent flow whose lengthscales are similar to the wavelengths.
Advection and refraction by this flow cause the scattering of the waves,
redistributing their energy in wavevector space. As a result, initially plane
waves radiated from a source such as a topographic ridge become spatially
incoherent away from the source. To examine this process, we derive a kinetic
equation which describes the statistics of the scattering under the assumptions
that the flow is quasigeostrophic, barotropic, and well represented by a
stationary homogeneous random field. Energy transfers are quantified by
computing a scattering cross section and shown to be restricted to waves with
the same frequency and identical vertical structure, hence the same horizontal
wavelength. For isotropic flows, scattering leads to an isotropic wavefield. We
estimate the characteristic time and length scales of this isotropisation, and
study their dependence on parameters including the energy spectrum of the flow.
Simulations of internal tides generated by a planar wavemaker carried out for
the linearised shallow-water model confirm the pertinence of these scales. A
comparison with the numerical solution of the kinetic equation demonstrates the
validity of the latter and illustrates how the interplay between wave
scattering and transport shapes the wave statistics.
