Vortex disruption by magnetohydrodynamic feedback

In an electrically conducting fluid, vortices stretch out a weak, large-scale magnetic field to form strong current sheets on their edges. Associated with these current sheets are magnetic stresses, which are subsequently released through reconnection, leading to vortex disruption, and possibly even destruction. This disruption phenomenon is investigated here in the context of two-dimensional, homogeneous, incompressible magnetohydrodynamics. We derive a simple order of magnitude estimate for the magnetic stresses—and thus the degree of disruption—that depends on the strength of the background magnetic field (measured by the parameter $M$, a ratio between the Alfvén speed and a typical flow speed) and on the magnetic diffusivity (measured by the magnetic Reynolds number $\mathrm{Rm})$. The resulting estimate suggests that significant disruption occurs when $M^2\mathrm{Rm}=O\left(1\right)$. To test our prediction, we analyze direct numerical simulations of vortices generated by the breakup of unstable shear flows with an initially weak background magnetic field. Using the Okubo-Weiss vortex coherence criterion, we introduce a vortex disruption measure, and show that it is consistent with our predicted scaling, for vortices generated by instabilities of both a shear layer and a jet.

Figure 1

Snapshots of (a) vorticity and (b) magnetic field lines for the shear layer at different field strengths (for $\mathrm{Rm}=\mathrm{Re}=500)$, shown for the central half of the channel $\left(-L_y/2\le y\le L_y/2\right)$.

Figure 2

Time series of the energies (blue $=$ kinetic; red $=$ magnetic; solid $=$ perturbation state; dashed $=$ mean state; gray solid line $=$ total energy) for the shear layer at different field strengths (for $\mathrm{Rm}=\mathrm{Re}=500)$. The curve for mean kinetic energy largely lies on top of the curve for the total energy.

Figure 3

Okubo-Weiss field for the shear layer corresponding to the $t=150$ cases shown in Fig. 1. (a) The full $W$ field given in Eq. (22); (b) the field keeping only $W<-0.2\sigma$; (c) a further filtered field keeping only the largest connected region originating from the center of the parent vortex.

Figure 4

$\mathrm{\Delta }$ vs. $M^2\mathrm{Rm}$ for the shear layer (colors, varying $\mathrm{Rm}$; markers, varying $M$; see Table 1 for marker and color values). The dotted line is obtained from a regression of the data. For display purposes, data beyond $M^2\mathrm{Rm}=5$ are omitted.

Figure 5

Snapshots of (a) vorticity and (b) magnetic field lines for the jet at different field strengths (for $\mathrm{Rm}=\mathrm{Re}=500)$, shown for the central half of the channel $\left(-L_y/2\le y\le L_y/2\right)$.

Figure 6

Time series of the energies (blue $=$ kinetic; red $=$ magnetic; solid $=$ perturbation state; dashed $=$ mean state; gray solid line $=$ total energy) for the jet at different field strengths (for $\mathrm{Rm}=\mathrm{Re}=500)$.

Figure 7

Okubo-Weiss field for the jet corresponding to the $t=150$ cases shown in Fig. 5. (a) The full $W$ field given in Eq. (22); (b) the field keeping only $W<-0.2\sigma$; (c) a further filtered field keeping only the four largest connected components, accounting for periodicity. The color scale is saturated to show the small values when $M=0.025$.

Figure 8

$\mathrm{\Delta }$ vs. $M^2\mathrm{Rm}$ for the jet (colors, varying $\mathrm{Rm}$; markers, varying $M$; see Table 2 for marker and color values). The dotted line is obtained from a regression of the data. For display purposes, data beyond $M^2\mathrm{Rm}=1$ are omitted.

Figure 9

Decomposition of the components in $J(A,{\nabla }^{2}A)$, for the shear layer simulation $M=0.05,\mathrm{Rm}=500$, at $t=55$ [cf. Fig. 1], shown for the central half of the channel $\left(-L_y/2\le y\le L_y/2\right)$.

Figure 10

(a) Plot of $\delta$ as defined in Eq. (28) against $\ell /L_v$, for the $t=55$ simulation data of the shear layer across all values of $M$ and $\mathrm{Rm}$ (colors, varying $\mathrm{Rm}$; markers, varying $M$; see Table 1 for marker and color values); the $\mathrm{Rm}=50$ data are not plotted. (b) Plot of the mean value of $maxJ(A,{\nabla }^{2}A)$ against $\mathrm{Rm}$, where the average is taken over the ten values of $M$. Both dotted lines are obtained from a regression of the data with the $\mathrm{Rm}=50$ data omitted.

Figure 11

Snapshots of $\overline{u}$ for the shear layer (a, b, c) and jet (d, e, f) at different field strengths (for $\mathrm{Rm}=\mathrm{Re}=500)$. Some of the curves lie on top of each other and are indistinguishable.