Understanding large-scale dynamos in unstratified rotating shear flows

Abstract

We combine simulations with new analyses that overcome previous pitfalls to explicate how nonhelical mean-field dynamos grow and saturate in unstratified, magnetorotationally driven turbulence. Shear of the mean radial magnetic field amplifies the azimuthal component. Radial fields are regenerated by velocity fluctuations that induce shear of radial magnetic fluctuations, followed by Lorentz and Coriolis forces that source a negative off-diagonal component in the turbulent diffusivity tensor. We present a simple schematic to illustrate this dynamo growth. A different part of the Lorentz force forms a third-order correlator in the mean electromotive force that saturates the dynamo.

Figures (6)

• Figure 1: (a) Time evolution of magnetic energy densities (scaled by two) for $xy$-averaged large-scale and fluctuating fields. (b), (c) Spatial variation of individual terms from the vertical gradient of $\mathcal{\bar{E}}_y$ (Eq. \ref{['eq:meanBx_xy_2']}) responsible for generating $\bar{B}_x(z)$, shown during (b) exponential growth ($t/T_{\text{orb}} = 3.34$) and (c) nonlinear stage ($t/T_{\text{orb}} = 60$). Line styles and colors are consistent in (b) and (c); legend is shown in panel (c). (d) Time evolution of the off-diagonal turbulent diffusivity $\eta_{yx}$ (Eq. \ref{['eq:eta_yx']}). In (a) and (d), additional averaging along $z$ is applied.
• Figure 2: Spatial variation of individual terms in the evolution equations for fluctuating velocity fields and mean Faraday stress during the MRI growth phase, evaluated at $t/T_{\text{orb}} =3.34$. Panels (a, b) show terms for $u_y$ and $u_x$ (Eq. \ref{['eq:fluctuatingU']}); panels (c, d) for $\bar{F}_{yz}$ and $\bar{F}_{xz}$ (Eq. \ref{['eq:Fij_exact']}). To ensure interpretability, we multiply $u_i$ and $\bar{F}_{ij}$ on both sides of their respective equations so that source terms remain positive (not shown in legends).
• Figure 3: Streamlines of in-plane fluctuating velocity fields during the exponential growth phase of MRI at $t/T_{\text{orb}} =3.34$. (a) Streamlines of ($u_y$, $u_z$) in the y-z plane at $x=0.25$, overlaid on a colormap of the magnetic tension term $b_z \partial_z B_y$, which drives $u_y$. (b) Streamlines of ($u_x$, $u_z$) in the x-z plane at $y=0.25$, with a colormap of the Coriolis term, which converts $u_y$ into $u_x$.
• Figure 4: Colormaps with contours of fluctuating fields in the x-z plane during the exponential growth phase of MRI, at $t/T_{\text{orb}} =3.34$. (a) Fluctuating velocity $u_x$, (b) fluctuating magnetic field $b_z$, and (c) their product $u_x b_z$, with solid (dashed) contours denoting positive (negative) values. While fluctuations $u_x$ and $b_z$ individually average out when integrated over $x$, their correlator $u_x b_z$ remains finite, indicating a nonzero mean stress.
• Figure 5: Schematic illustration of the rotation-shear-current effect. (a) Two initially vertical magnetic field sectors with opposite polarity. (b) An $x$-dependent perturbation with a phase shift is introduced. (c) Background shear $(\partial U^0_y / \partial x < 0)$ stretches the field lines and shifts the sectors relative to each other. The resulting field line bending produces radial currents $J_x \sim - \partial_z B_y$. (d) Magnetic tension from field curvature induces $u_y$ fluctuations, which are converted into $u_x$ via the Coriolis force. (e) The resulting vectors of $- u_x b_z$ are shown, with $z$-layers (1 to 6) indicating matched heights across stacks. The EMF component $\mathcal{E}_y \sim - u_x b_z$ aligns within each $x$-$y$ plane at fixed $z$, but reverses sign between adjacent $z$-layers. After $xy$-averaging, $\mathcal{\bar{E}}_y (z)$ exhibits an alternating pattern: $\mathcal{\bar{E}}_y < 0$ at layer 1, $\mathcal{\bar{E}}_y > 0$ at layer 2, $\mathcal{\bar{E}}_y < 0$ at layer 3, and so on. The induced mean-field satisfies $\partial_t \bar{B}_x = - \partial_z \mathcal{\bar{E}}_y$, generating a coherent $\bar{B}_x (z)$ that reverses in $z$, consistent with the fastest-growing MRI mode.