Frictionless Zonal Flow Saturation by Vorticity Mixing

Consideration of wave--flow resonance addresses the long-standing problem of how zonal flows (ZF) saturate in the limit of weak or zero frictional drag, and also determines the ZF scale. For relevant magnetic geometries, the frequently quoted tertiary instability requires unphysical enhancement of ZF shear and thus is irrelevant to the near-marginal, frictionless regime. We show that resonant vorticity mixing, which conserves potential enstrophy, enables ZF saturation in the absence of drag, and so is effective in the Dimits up-shift regime. Vorticity mixing is incorporated as a nonlinear, self-regulation effect in an extended 0D predator--prey model of drift--ZF turbulence. This analysis determines the saturated ZF shear and shows that the mesoscopic ZF width scales as $L_{ZF}\sim f^{3/16} (1-f)^{1/8} \rho_s^{5/8} l_0^{3/8}$ in the relevant adiabatic limit (i.e., $\tau_{ck} k_\|^2 D_\| \gg 1$). $f$ is the fraction of turbulence energy coupled to ZF and $l_0$ is the mixing length absent ZF shears. We calculate and compare the stationary flow and turbulence level in frictionless, weakly frictional, and strongly frictional regimes. In the frictionless limit, the results differ significantly from conventionally quoted scalings derived for frictional regimes. The flow is independent of turbulence intensity. The turbulence level scales as $E \sim (\gamma_L/\varepsilon_c)^2$, which defines the extent of the "near-marginal" regime to be $\gamma_L < \varepsilon_c$, for the case of avalanche-induced profile variability. Here, $\varepsilon_c$ is the rate of dissipation of potential enstrophy and $\gamma_L$ is the characteristic linear growth rate of fluctuations. The implications for dynamics near marginality of the strong scaling of saturated $E$ with $\gamma_L$ are discussed.