Zonal-flow dynamics from a phase-space perspective*

The wave kinetic equation (WKE) describing drift-wave (DW) turbulence is widely used in the 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 (GO) limit. Here we present a new theory that captures both of these effects, while still treating DW quanta (``driftons") as particles in phase space [Ruiz {\it et~al.}, Phys. Plasmas {\bf 23}, 122304 (2016)]. In this theory, the drifton dynamics is described by an equation of the Wigner--Moyal type, which is analogous to the phase-space formulation of quantum mechanics. The ``Hamiltonian" and the ``dissipative" parts of the DW--ZF interactions are clearly identified. Moreover, this theory can be interpreted as a phase-space representation of the second-order cumulant expansion (CE2). In the GO limit, this formulation features additional terms missing in the traditional WKE that ensure 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 traditional WKE. Numerical simulations are presented to illustrate the importance of these additional terms.