Friction-induced scale-selection in the extended Cahn-Hilliard model for zonal staircase

In this work, we describe a possible mechanism to set the radial scale of zonal flows, which may be applicable to the $E\times B$ staircase found in the global full-f simulations such as [G. Dif-Pradalier et al. Phys. Rev. Lett. 114, 085004 (2015)]. 1D numerical simulation results of the Cahn-Hilliard model - extended to include zonal flow friction - can be understood from a heuristic nonlinear analysis. The staircase step-size $\mathrm{\Delta }$ is found to decrease as the dimensionless zonal flow friction $\mu$ increases. It scales like $\mathrm{\Delta }\sim \log {\mu }^{\alpha }$, with $\alpha \simeq -0.41$, up to a constant offset.