  The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is
widely used in studies of zonal flows (ZFs) emerging from DW turbulence.
However, this formulation neglects the exchange of enstrophy between DWs and
ZFs and also ignores effects beyond the geometrical-optics limit. We derive a
modified theory that takes both of these effects into account, while still
treating DW quanta ("driftons") as particles in phase space. The drifton
dynamics is described by an equation of the Wigner-Moyal type, which is
commonly known in the phase-space formulation of quantum mechanics. In the
geometrical-optics limit, this formulation features additional terms missing in
the traditional WKE that ensure exact conservation of the total enstrophy of
the system, in addition to the total energy, which is the only conserved
invariant in previous theories based on the WKE. Numerical simulations are
presented to illustrate the importance of these additional terms. The proposed
formulation can be considered as a phase-space representation of the
second-order cumulant expansion, or CE2.
