Statistical Simulation of the Magnetorotational Dynamo

Turbulence and dynamo induced by the magnetorotational instability (MRI) are analyzed using quasilinear statistical simulation methods. It is found that homogenous turbulence is unstable to a large-scale dynamo instability, which saturates to an inhomogenous equilibrium with a strong dependence on the magnetic Prandtl number (Pm). Despite its enormously reduced nonlinearity, the dependence of the angular momentum transport on Pm in the quasilinear model is qualitatively similar to that of nonlinear MRI turbulence. This demonstrates the importance of the large-scale dynamo and suggests how dramatically simplified models may be used to gain insight into the astrophysically relevant regimes of very low or high Pm.

Figure 1

Growth rate ${\gamma }_{\text{Dyn}}$ of the mean field, ${\overline{B}}_y={\overline{B}}_{y0}(z)e^{{\gamma }_{\text{Dyn}}t}$, as a function of the magnetic Reynolds number at $\mathrm{Pm}=1$, 2, 4, and 8.

Figure 2

Evolution of ${\overline{B}}_y$ as a function of $(z,t)$ at $\mathrm{Pm}=4$ for (a) $\mathrm{Rm}=4500$, time-independent saturated state, and (b) $\mathrm{Rm}=12000$, time-dependent saturated state. (c) Magnitude of ${\overline{B}}_y$ as measured by $({\overline{B}}_y{)}_{\mathrm{rms}}=(1/L_z\int dz{|{\overline{B}}_y|}^2{)}^{1/2}$ at saturation, as a function of Rm and Pm. The shaded regions illustrate the approximate maxima and minima of the time-dependent ${\overline{B}}_y$ when the system did not reach a time-independent statistical equilibrium. Gray points (point styles as for CE2 results) illustrate the mean values of equivalent driven nonlinear simulations, with error bars illustrating the approximate maxima and minima (the slight horizontal offset of $\mathrm{Pm}=1,4$ points is for clarity, and the same Rm is used for all Pm).

Figure 3

Angular momentum transport $⟨u_x{u}_y⟩-⟨b_x{b}_y⟩$ (including mean and fluctuating variables) as a function of time for $\mathrm{Rm}=12000$ and $\mathrm{Pm}=1\to 16$.