  The formation of zonal flows from inhomogeneous drift-wave (DW) turbulence is
often described using statistical theories derived within the quasilinear
approximation. However, this approximation neglects wave--wave collisions.
Hence, some important effects such as the Batchelor--Kraichnan inverse-energy
cascade are not captured within this approach. Here we derive a wave kinetic
equation that includes a DW collision operator in the presence of zonal flows.
Our derivation makes use of the Weyl calculus, the quasinormal statistical
closure, and the geometrical-optics approximation. The obtained model conserves
both the total enstrophy and energy of the system. The derived DW collision
operator breaks down at the Rayleigh--Kuo threshold. This threshold is missed
by homogeneous-turbulence theory but expected from a full-wave quasilinear
analysis. In the future, this theory might help better understand the
interactions between drift waves and zonal flows, including the validity domain
of the quasilinear approximation that is commonly used in literature.
