Periodic magnetorotational dynamo action as a prototype of nonlinear magnetic-field generation in shear flows

The nature of dynamo action in shear flows prone to magnetohydrodynamc instabilities is investigated using the magnetorotational dynamo in Keplerian shear flow as a prototype problem. Using direct numerical simulations and Newton’s method, we compute an exact time-periodic magnetorotational dynamo solution to three-dimensional dissipative incompressible magnetohydrodynamic equations with rotation and shear. We discuss the physical mechanism behind the cycle and show that it results from a combination of linear and nonlinear interactions between a large-scale axisymmetric toroidal magnetic field and nonaxisymmetric perturbations amplified by the magnetorotational instability. We demonstrate that this large-scale dynamo mechanism is overall intrinsically nonlinear and not reducible to the standard mean-field dynamo formalism. Our results therefore provide clear evidence for a generic nonlinear generation mechanism of time-dependent coherent large-scale magnetic fields in shear flows and call for new theoretical dynamo models. These findings may offer important clues to understanding the transitional and statistical properties of subcritical magnetorotational turbulence.

Figure 1

Geometry of the shearing sheet. For a Keplerian flow the vorticity of ${\mathbf{U}}_s$ is anti-aligned with the rotation vector.

Figure 2

Suggested physical mechanism of the MRI dynamo. Full arrows: main dynamo loop. Dashed arrows: nonlinear regeneration of MRI-unstable fluctuations. The various colors are used to identify the active modes taking part in the cyclic MRI dynamo described in Fig. 5: red (top left box) and blue (top right box) denote axisymmetric field components, different colors in the bottom boxes denote successive nonaxisymmetric MRI waves.

Figure 3

Projection in the ${\overline{B}}_{0x}-{\overline{B}}_{0y}$ plane of a DNS trajectory approaching a nonlinear cycle for $(L_x,L_y,L_z)=(0.7,20,2)L$ and $\mathrm{Re}=70$, $\mathrm{Rm}=360$. The full blue circle whose coordinates are $(0.046,0)$ marks the position of the system at $t=0$ and the red one located at $(-0.046,0.997)$ marks the position at $t=457.1S^{-1}$. Typical amplitudes of the various components of the initial conditions used to obtain this kind of trajectory are given in Sec. 3c.

Figure 4

Volume renderings of $B_y$ and isosurfaces of $B=0.9$ colored by $B_y$ every $T/8$ (positive $B_y$ in red/light gray, negative $B_y$ in blue-violet/dark gray). The coordinates of the bottom left corner of each box are $x=0$, $y^{\prime }=0$, $z=0$, where the $y^{\prime }$ coordinate is defined as $y^{\prime }=y-(L_x/2)St$ (see Appendix). The arrows field in the $y^{\prime }=0$ plane traces nonaxisymmetric MRI velocity perturbations (velocity perturbations in the $y^{\prime }=L_y/2$ plane have opposite sign), whose effect is to distort and advect magnetic field lines of opposite polarities in opposite directions.

Figure 5

From top to bottom: evolution over two cycles of (a) ${\overline{B}}_{0x}$ (blue/dark gray) and ${\overline{B}}_{0y}$ (red/light gray); (b) $-{\overline{E}}_{0y}$ (blue/dark gray, source term for ${\overline{B}}_{0x}$) and ${\overline{E}}_{0x}$ (red / light gray, source term for ${\overline{B}}_{0y}$); (c) amplitudes of the three terms on the r.h.s. of the projection of the axisymmetric induction Eq. (8) on ${\mathbf{e}}_y$ (dotted blue: $\Omega$ effect, full line red: nonlinear EMF, dashed magenta: magnetic diffusion); (d) total energy of successive MRI shearing waves with $k_y=\pm 2\pi /L_y$ (rainbow colors, dashed/full line: leading/trailing phase. Each color represents a different shearing wave); (e), amplitude of the axisymmetric magnetic field modulation ${\overline{B}}_{\mathrm{mod}y}$.

Figure 6

Projection of the periodic MRI dynamo orbit in the ${\overline{E}}_x$ (blue/dark gray)|${\overline{E}}_y$ (red/light gray) vs. ${\overline{B}}_x$|${\overline{B}}_y$ planes.