The magneto-rotational instability near threshold: spatio-temporal amplitude equation and saturation

TL;DR

This paper derives a Ginzburg-Landau equation for the amplitude of axisymmetric MRI in thin-gap Taylor-Couette flow at low magnetic Prandtl numbers, revealing how saturation scales and its implications for transport are affected.

Contribution

It introduces a weakly nonlinear analysis leading to a Ginzburg-Landau equation for MRI amplitude near threshold, with scaling laws for saturation depending on boundary conditions.

Findings

Saturation amplitude scales as P^ with 1/2<<2/3.

Numerical simulations confirm asymptotic predictions.

Transport due to MRI may be negligible for very small P_m.

Abstract

We show, by means of a perturbative weakly nonlinear analysis, that the axisymmetric magneto-rotational instability (MRI) in a magnetic Taylor-Couette (mTC) flow in a thin-gap gives rise, for very small magnetic Prandtl numbers (P_m), to a real Ginzburg-Landau equation for the disturbance amplitude. The saturation amplitude A_s is found to scale in this regime as P^\delta, with 1/2<\delta<2/3 (depending on the boundary conditions adopted). The asymptotic results are shown to comply with numerical calculations performed by using a spectral code. They suggest that the transport due to the nonlinearly developed MRI may be vanishingly small for P_m << 1.