Self-Sustaining Nonlinear Dynamo Process in Keplerian Shear Flows

A three-dimensional nonlinear dynamo process is identified in rotating plane Couette flow in the Keplerian regime. It is analogous to the hydrodynamic self-sustaining process in nonrotating shear flows and relies on the magnetorotational instability of a toroidal magnetic field. Steady nonlinear solutions are computed numerically for a wide range of magnetic Reynolds numbers but are restricted to low Reynolds numbers. This process may be important to explain the sustenance of coherent fields and turbulent motions in Keplerian accretion disks, where all its basic ingredients are present.

Figure 1

2D-3C magnetic configuration induced by the artificial toroidal EMF (3), $\beta =1$ and perfectly conducting boundary conditions. $b_x$ is represented on a color scale (black to white from $-\max $ to max) and ($b_y$, $b_z$) by arrows. $\mathrm{Rm}=750$, $\mathrm{Re}=10$, $A=1.5$, $\max (b_x)=1.006$, and $\max (b_y)=0.00195$. $(N_y,N_z)=(32,32)$.

Figure 2

Growth rates of the most unstable three-dimensional MRI eigenmodes for the 2D-3C configuration with $O(1)$ toroidal field shown in Fig. 1, as a function of the streamwise wave number $\alpha$. The legend indicates the symmetries of each mode.

Figure 3

Magnetic field in the $x=0$ plane of the near marginal (R,S) MRI eigenmode with $\alpha =0.375$, depicted by a black dot in Fig. 2. Same representation and parameters as in Fig. 1.

Figure 4

Full line: normalized shearwise profile of the $k_z=\beta$ component of the toroidal EMF created by the self-interactions of the three-dimensional $\alpha =0.375$ MRI mode of Fig. 3. Dashed line: normalized shearwise profile of the artificial EMF (3).

Figure 5

$y$-integrated amplitude of the $k_x=0.375$, $k_z=0$ component of $v_y$ for three-dimensional steady forced solutions, as a function of the forcing amplitude $A$. $(N_x,N_y,N_z)=(12,32,32)$.

Figure 6

Cuts through the unforced ($A=0$) nonlinear steady solution of Fig. 5 ($\alpha =0.375$, black dot). Top: $b_z$ at $z=L_z/2$. Bottom: $b_x$ (color scale) and ($b_y$, $b_z$) (arrows) at $x=L_x/4$.

Figure 7

Continuation with respect to Rm for $\mathrm{Re}=10$, $(\alpha ,\beta )=(0.375,1)$. Inset: continuation with respect to Re for $\mathrm{Rm}=1500$ (full line), $\mathrm{Rm}=3000$ (dashed line). $(N_x,N_y,N_z)=(8,24,32)$.