Nonaxisymmetric magnetorotational instability in spherical Couette flow

We investigate numerically the flow of an electrically conducting fluid in a rapidly rotating spherical shell where the inner boundary spins slightly faster than the outer one. The magnetic field evolves self-consistently from an initial dipolar configuration of weak amplitude, and a toroidal field is produced by winding this poloidal field through the internal differential rotation. First, we characterize the axisymmetric field solutions obtained at long times when the Lorentz force is negligible and the flow follows the steady, purely hydrodynamical solution. We then examine the stability of these solutions, focusing on the regime of large magnetic Reynolds numbers where the field is dominantly toroidal. When the ratio of the azimuthal Alfvén frequency to the rotation frequency exceeds a certain value, a nonaxisymmetric instability develops. We show that the instability properties are compatible with those expected for the magnetorotational instability. Finally, we compare the instability properties with predictions obtained from a local linear stability analysis. The linear analysis agrees well with the numerical simulation results, except in a number of cases where the discrepancies are attributed to shearing effects on the unstable modes.

Figure 1

Nonmagnetic (${\mathrm{Lo}}_0=0$) axisymmetric flow solutions at a Rossby number $\mathrm{Ro}=2.5\times {10}^{-4}$. (a) and (b) for an Ekman number $\mathrm{Ek}={10}^{-4}$, (c) and (d) for $\mathrm{Ek}={10}^{-5}$. (a) and (c) display the angular velocity ${\mathrm{\Omega }}^{\prime }$, and (b) and (d) the meridional circulation (counterclockwise in the northern hemisphere and clockwise in the southern hemisphere). These solutions are representative of the asymptotic regime of Proudman [10] and Stewartson [11], characterized by fast outer boundary rotations and slightly faster inner boundary rotations.

Figure 2

Comparison of the asymptotic solution of Proudman [10] with different nonmagnetic numerical flow solutions. (a) and (b) show, respectively, the angular velocity ${\mathrm{\Omega }}^{\prime }$ and the stream function $\psi$ as function of the cylindrical radial coordinate $s$. Solid gray curves display Proudman's solution as given in Eqs. (11a)–(11d). This solution is discontinuous at the tangent cylinder location. For each Ekman number case, solid and dashed curves illustrate numerical solutions at a small Rossby number $\text{Ro}=2.5\times {10}^{-4}$ and at the largest $\text{Ro}$ explored in this study, respectively [see the legend in (a)]. The numerical solutions are taken at a depth of $z=r_o/2d$, where $z$ is the dimensionless cylindrical vertical coordinate.

Figure 3

Temporal evolution of the volume averaged kinetic ($E_{\text{kin}}$) and magnetic ($E_{\text{mag}}$) energies for the axisymmetric case at $\text{Ek}={10}^{-5}$, $\text{Ro}=0.03$, $\text{Pm}=2$, and ${\mathrm{Lo}}_0=4.47\times {10}^{-3}$. (a) and (c) display the whole simulation run and (b) and (d) the initial evolution only (note the different scales of the left and right vertical axes). The thin solid line in (c) illustrates the Ohmic decay of a pure dipole field occurring on the timescale $t_d=r_o^2/{\pi }^2\eta$. The vertical dotted line indicates the time at which nonaxisymmetric perturbations are applied (run Pm2 in Sec. 5). Flow and field solutions at this perturbation time are shown in Fig. 4.

Figure 4

(a) and (b) Comparison of axisymmetric interior flow solutions between nonmagnetic and magnetic cases at $\text{Ek}={10}^{-5}$ and $\text{Ro}=0.03$. Solid black curves show the steady nonmagnetic (${\mathrm{Lo}}_0=0$) solution. Magnetic solutions at $\text{Pm}=2$ and 6 are illustrated by the gray and red curves, respectively [see the legend in (a) for their ${\mathrm{Lo}}_0$ values]. Dashed curves display solutions taken at the times $t$ (also given in the legend) when nonaxisymmetric perturbations will be applied. They represent the basic axisymmetric flow configurations of runs Pm2 (dashed gray curves) and Pm6a (dashed red curves) described in Sec. 5. Finally, dotted gray curves show the case at $\text{Pm}=2$ during the diffusive phase of the field evolution (note that the solution is identical to the nonmagnetic one). (c) Magnetic field solution for the case corresponding to the dashed gray curves in (a) and (b). This represents the perturbed axisymmetric field configuration of run Pm2 in Sec. 5. Color contours show the axisymmetric azimuthal field $B_{\phi }$ and dashed contour lines the poloidal field.

Figure 5

Scaling of the toroidal to poloidal magnetic field ratio $\langle B_{\phi }\rangle /\langle B_p\rangle$ during the diffusive phase of the field evolution with the magnetic Reynolds number $\text{Rm}$. Here angular brackets denote rms volume averages. Squares and circles indicate simulations at the Ekman numbers $\text{Ek}={10}^{-4}$ and ${10}^{-5}$, respectively. The symbol color codes the magnetic Prandtl number $\text{Pm}$ (see the legend). The solid black line illustrates the linear scaling $\langle B_{\phi }\rangle /\langle B_p\rangle \propto \text{Rm}$ expected for $\text{Rm}{\text{Ek}}^{1/2}\lesssim 1$. Capital letters A–F indicate six cases representative of the different field solutions obtained (see Fig. 6). These cases have Ekman and Rossby numbers $\text{Ek}={10}^{-5}$ and $\text{Ro}=0.03$, respectively, and different $\text{Pm}$ values [case A, $\text{Pm}=5\times {10}^{-3}$ (or $\text{Rm}=15$); case B, $\text{Pm}=0.5$ (or $\text{Rm}=1.5\times {10}^3$); case C, $\text{Pm}=1$ (or $\text{Rm}=3\times {10}^3$); case D, $\text{Pm}=1.7$ (or $\text{Rm}=5.1\times {10}^3$); case E, $\text{Pm}=3$ (or $\text{Rm}=9\times {10}^3$); case F, $\text{Pm}=6$ (or $\text{Rm}=1.8\times {10}^4$)].

Figure 6

Axisymmetric magnetic field solutions during the diffusive phase of the field evolution. Color contours show the azimuthal field $B_{\phi }$ and dashed contour lines the poloidal field. The six selected cases (A–F) have different magnetic Reynolds numbers $\text{Rm}$ and are representative of the various regimes depicted in Fig. 5. The Ekman number is $\text{Ek}={10}^{-5}$ and the Rossby number $\text{Ro}=0.03$ in all cases.

Figure 7

Force balances for the six axisymmetric field solutions A–F (from top to bottom) shown in Fig. 6. Left and right panels illustrate, respectively, the axisymmetric azimuthal field $B_{\phi }$ and the force terms in the toroidal induction equation (15b) as a function of the vertical coordinate $z$. These profiles are taken at the cylindrical radius $s$ where $|B_{\phi }|$ is maximum and shown for the northern hemisphere ($z>0$) only. The different force terms [$\mathrm{\Omega }$ effect, field advection, and magnetic diffusion; see the legend in (b)] are calculated beneath the outer Ekman boundary layer and their amplitude is normalized to the absolute maximum of the forces. The vertical dotted line in each panel indicates the position of the $|B_{\phi }|$ maximum.

Figure 8

Temporal evolution of the toroidal magnetic energy of different azimuthal modes $m$ in (a) run Pm2 and (b)–(d) the three runs at $\text{Pm}=6$. The energy is evaluated as an average over a spherical surface at depth $r/r_o$ roughly located where the axisymmetric azimuthal field is maximum ($r/r_o=0.8$ and 0.5 for run Pm2 and the three runs at $\text{Pm}=6$, respectively). Vertical dotted lines indicate the times at which the snapshots of Fig. 9 are taken.

Figure 9

Nonaxisymmetric instability during the linear phase of its growth. Runs Pm2, Pm6a, Pm6b, and Pm6c are shown from top to bottom. Left and right panels display, respectively, map projections and meridional sections of the total nonaxisymmetric azimuthal field $B_{\phi }^{\prime }$. The map projections are taken at depths $r/r_o$ indicated on top of each panel. These snapshots are taken at the times displayed in the magnetic energy evolution of Fig. 8.

Figure 10

Meridional sections (northern hemisphere only) showing the temporal evolution of the total nonaxisymmetric azimuthal field $B_{\phi }^{\prime }$ in run Pm6c. The instability drifts in the vertical direction towards the inner boundary at constant velocity and with a mostly fixed wave packet structure. The gray dot marks the position of a mode crest. The initial vertical position of this crest is indicated by the horizontal dotted line, which can be used as a reference to track the slow instability motion.

Figure 11

Comparison of the unstable regions predicted by the local dispersion relation with the numerical simulation results. Meridional sections of the total nonaxisymmetric azimuthal field $B_{\phi }^{\prime }$ during the linear phase of the instability growth are shown for runs (a) Pm2 and (b) Pm6a. Black contour lines in the northern hemisphere display the growth rates $\gamma /\mathrm{\Omega }>0$ predicted by the local dispersion relation for the most unstable azimuthal mode $m_{\text{max}}$ observed in the simulation ($m_{\max }=4$ and 5 for runs Pm2 and Pm6a, respectively).

Figure 12

Comparison of the growth rates $\gamma /{\mathrm{\Omega }}_o$ of the unstable azimuthal modes $m$ observed in the numerical simulations (data points) with those predicted by the local dispersion relation (solid curves). The theoretical predictions are obtained with the estimates of the dispersion relation parameters listed in Table 2 (see the main text for further details). (a) Runs Pm2 and Pm3 show good agreement with the theoretical growth rates, whereas (b) runs at $\text{Pm}=6$ and 8 present larger deviations.