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 in Dif-Pradalier et al., [Phys. Rev. Lett. 114, 085004 (2015)]. One-dimensional 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.

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• Drift waves
• Magnetic confinement fusion
• Nonlinear phenomena in plasmas
• Pattern formation
• Plasma transport
• Plasma turbulence
• Shear flows