Magnetic eddy viscosity of mean shear flows in two-dimensional magnetohydrodynamics

Magnetic induction in magnetohydrodynamic fluids at magnetic Reynolds number (Rm) less than 1 has long been known to cause magnetic drag. Here we show that when $\mathrm{Rm}\gg 1$ and the fluid is in a hydrodynamic-dominated regime in which the magnetic energy is much smaller than the kinetic energy, induction due to a mean shear flow leads to a magnetic eddy viscosity. The magnetic viscosity is derived from simple physical arguments, where a coherent response due to shear flow builds up in the magnetic field until decorrelated by turbulent motion. The dynamic viscosity coefficient is approximately $(B_p^2/2{\mu }_0){\tau }_{\mathrm{corr}}$, the poloidal magnetic energy density multiplied by the correlation time. We confirm the magnetic eddy viscosity through numerical simulations of two-dimensional incompressible magnetohydrodynamics. We also consider the three-dimensional case, and in cylindrical or spherical geometry, theoretical considerations similarly point to a nonzero viscosity whenever there is differential rotation. Hence, these results serve as a dynamical generalization of Ferraro's law of isorotation. The magnetic eddy viscosity leads to transport of angular momentum and may be of importance to zonal flows in astrophysical domains such as the interior of some gas giants.

Figure 1

Schematic showing the magnetic response to a mean shear flow expected in a turbulent regime. As $\mathrm{Rm}$ increases, the magnetic response transitions from a magnetic drag (for flow perpendicular to the magnetic field) for $\mathrm{Rm}<1$ to a magnetic eddy viscosity for $\mathrm{Rm}>1$, assuming also $\mathcal{A}\ll 1$. As $\mathrm{Rm}$ increases further, the magnetic energy rises to a value comparable to the kinetic energy, $\mathcal{A}=1$. If the mean shear flow is turbulence driven, such as zonal flow, then the mean flow is suppressed.

Figure 2

Change in the magnetic field in a short time increment $\mathrm{\Delta }t$ due to a mean shear flow for two different initial configurations, neglecting magnetic diffusivity. The modification of the field has been exaggerated for visibility. The resulting $x$ component of the zonally averaged Lorentz force ${\overline{F}}_x$ acts in both cases as a viscosity to reduce the shear flow. The decomposition into magnetic tension and pressure forces ${\overline{F}}_{Tx}$ and ${\overline{F}}_{Px}$, respectively, is also shown. The top row shows that from an initially straight field line, ${\overline{F}}_x$ is due solely to tension (at first order in $\mathrm{\Delta }t$). The bottom row shows that when the magnetic field is not straight, ${\overline{F}}_x$ will in general have contributions from both tension and pressure.

Figure 3

Results from three representative simulations. The left column shows space-time plots of the zonal flow, and the right column shows time traces of averaged energy densities associated with the zonal flow (ZKE) and the hydrodynamic eddies (EKE), the background magnetic field (BME), the zonally averaged magnetic field (ZME), and the eddy magnetic field (EME). (a) and (b) Case at ${\mathrm{Rm}}_0=0.3$ in which magnetic fluctuations are too weak to have an effect on the hydrodynamic flow. (c) and (d) Effect of magnetic field on the zonal flow with toroidal background field and ${\mathrm{Rm}}_0=300$. (e) and (f) Case with poloidal background field and ${\mathrm{Rm}}_0=30$.

Figure 4

Results for the simulation with toroidal $B_0={10}^{-2}$, $\eta ={10}^{-3}$ (${\mathrm{Rm}}_0=30$), and $\mathcal{A}=0.005$. The left and right columns show short-time and long-time averages, respectively. (a) and (b) Zonally averaged magnetic eddy energy components $\overline{B_x^{\prime 2}}$ and $\overline{B_y^{\prime 2}}$. (c) and (d) Zonal flow shear ${\partial }_{y}U$. (e) and (f) Maxwell stress $\overline{B_x^{\prime }{B}_y^{\prime }}$ and the prediction of Eq. (11). (g) and (h) Structure of $-U$ and ${\partial }_y^{2}U$. (i) and (j) Divergence of the Maxwell stress ${\partial }_y\overline{B_x^{\prime }{B}_y^{\prime }}$ and the derivative of the prediction of Eq. (11). The proportionality constant $\alpha$ is 1.02 for the short-time average and 1.04 for the long-time average.

Figure 5

Same as Fig. 4, for the simulation with poloidal $B_0={10}^{-3}$, $\eta ={10}^{-3}$ (${\mathrm{Rm}}_0=30$), and $\mathcal{A}=0.006$. In (i) and (j) the term ${\partial }_y\left({\overline{B}}_x{B}_{0y}\right)$ is also shown. The proportionality constant $\alpha$ is 1.05 and 1.17 for the short- and long-time averages, respectively.

Figure 6

Results for several simulations with varying $\eta$ (${\mathrm{Rm}}_0=0.03/\eta$). The left column shows the case with background toroidal field and the right column with poloidal field. (a) and (b) Steady-state energies (see the text for definitions of the various terms). $\mathcal{A}$ is small for ${\mathrm{Rm}}_0\lesssim 300$. (c) and (d) Fraction of kinetic energy in a coherent zonal flow, where ${\langle ·\rangle }_t$ is a time average. (e) and (f) Correlations of the magnetic stresses observed in the simulations with the prediction of Eq. (11). For the case of a poloidal field, the correlation of ${\partial }_y\left({\overline{B}}_x{B}_{0y}\right)$ with $-U$ is 1 for ${\mathrm{Rm}}_0<1$; this is magnetic drag. (g) and (h) The rms (over $y$) of the relevant stresses, showing that the Maxwell stress rises to be sufficiently strong to counteract the Reynolds stress at large enough ${\mathrm{Rm}}_0$. (i) and (j) Best-fit proportionality constants $\alpha$ and $\gamma$, where $\alpha$ is shown only for the cases in the magnetic eddy viscosity regime: ${\mathrm{Rm}}_0>1$, but small enough such that a coherent mean zonal flow still exists. All quantities were averaged over $2000\le t\le 3000$.

Figure 7

Simulation without frictional drag, with toroidal $B_0={10}^{-2}$ and parameters $\nu ={10}^{-4}$ and $\eta ={10}^{-4}$. In this simulation, $L_{\mathrm{turb}}=0.56$, $v_{\mathrm{turb}}=0.46$, $\mathrm{Rm}\approx 2600$, and $\mathcal{A}\approx 6\times {10}^{-4}$. See the text for more details.

Figure 8

Comparison of the zonally averaged Reynolds stress in a simulation with the prediction of Eq. (13) and the refined prediction of Eq. (16). Shown are (a) the Reynolds stress and (b) its divergence. This simulation used a toroidal background magnetic field $B_0={10}^{-2}$ and $\eta ={10}^{-2}$ (${\mathrm{Rm}}_0=3$). The proportionality constants are $\gamma =0.08$ and ${\gamma }_2=0.13$. Quantities were averaged over $2000\le t\le 2500$.