  Inhomogeneous drift-wave turbulence can be modeled as an effective plasma
where drift waves act as quantumlike particles and the zonal-flow velocity
serves as a collective field through which they interact. This effective plasma
can be described by a Wigner-Moyal equation (WME), which generalizes the
quasilinear wave-kinetic equation (WKE) to the full-wave regime, i.e., resolves
the wavelength scale. Unlike waves governed by manifestly quantumlike
equations, whose WMEs can be borrowed from quantum mechanics and are commonly
known, drift waves have Hamiltonians very different from those of conventional
quantum particles. This causes unusual phase-space dynamics that is typically
not captured by the WKE. We demonstrate how to correctly model this dynamics
with the WME instead. Specifically, we report full-wave phase-space simulations
of the zonal-flow formation (zonostrophic instability), deterioration (tertiary
instability), and the so-called predator-prey oscillations. We also show how
the WME facilitates analysis of these phenomena, namely, (i) we show that
full-wave effects critically affect the zonostrophic instability, particularly,
its nonlinear stage and saturation; (ii) we derive the tertiary-instability
growth rate; and (iii) we demonstrate that, with full-wave effects retained,
the predator-prey oscillations do not require zonal-flow collisional damping,
contrary to previous studies. We also show how the famous Rayleigh-Kuo
criterion, which has been missing in wave-kinetic theories of drift-wave
turbulence, emerges from the WME.
