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

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

Figures (4)

• Figure 1: The linear dispersion relation, Eq.(\ref{['lindisp']}) is shown. The black horizontal line shows that for a given value of the zonal flow friction $\mu$, there are two possible radial wavenumbers: ${q_r}_+$ and ${q_r}_-$. The dashed line shows the maximal value of friction $\mu_c = \frac{1}{4}$, above which zonal flows are totally suppressed.
• Figure 2: The scale-selection is shown graphically: Semi-log plot of staircase step-size $\Delta_{\rm stair}$ v.s. ZF friction $\mu$v.s. ZF friction $\mu$ (solid-blue) evaluated using the formula of Ref.formula-sheet at 2nd order in $1-\kappa(\mu)$. The dashed-line is a semi-log fit.
• Figure 3: The time-averaged zonal flow shear profile $Z=\partial_x V_{\rm ZF}$ - normalized to its amplitude - is shown for different values of ZF friction $\mu=10^{-3}$ (blue), $\mu=0.1$ (red) and $\mu=0.2$ (yellow).
• Figure 4: Staircase nonlinear dispersion relation: Semi-log plot of $\Delta_{\rm stair}$ v.s. ZF friction $\mu$ from the 1D numerical simulation for $N_x=512$ radial points. The dashed-line is a semi-log fit of $\Delta_{\rm stair}$. The black solid-line shows the neutral-stability curve: $\frac{1}{{q_r}_\pm}$ v.s. $\mu$.