The Magneto-rotational Instability near Threshold: Spatio-temporal Amplitude Equation and Saturation

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 (Pm), to a real Ginzburg-Landau equation for the disturbance amplitude. The saturation amplitude As is found to scale in this regime as P m, with 1/2 < δ < 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 Pm ≪ 1.