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

TL;DR

This paper investigates how friction influences the scale of zonal flows in a modified Cahn-Hilliard model, revealing a logarithmic relationship between flow friction and staircase step size through numerical and heuristic analysis.

Contribution

It introduces a friction-inclusive extension of the Cahn-Hilliard model to explain the scale selection mechanism of zonal flows, supported by numerical simulations and heuristic analysis.

Findings

Step size decreases with increasing zonal flow friction.

Step size scales logarithmically with friction, Δ ∼ log(μ)^α.

Numerical results align with heuristic nonlinear analysis.

Abstract

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 $\Delta$ is found to decrease as the dimensionless zonal flow friction $\mu$ increases. It scales like $\Delta \sim \log \mu^\alpha$, with $\alpha \simeq -0.41$, up to a constant offset.