Influence of tearing instability on magnetohydrodynamic turbulence

It was proposed recently by Loureiro and Boldyrev [Phys. Rev. Lett. 118, 245101 (2017)] and Mallet et al. [Mon. Not. R. Astron. Soc. 468, 4862 (2017)] that strongly anisotropic current sheets formed in the inertial range of magnetohydrodynamic turbulence become affected by the tearing instability at scales smaller than a critical scale ${\lambda }_{\mathrm{c}}$, and larger than the dissipation scale of turbulence. If true, this process can modify the nature of energy cascade at smaller scales, leading to a new, tearing-mediated regime of magnetohydrodynamic (MHD) turbulence. In this work, we present a numerical study of strongly anisotropic, two-dimensional turbulent eddies, and we demonstrate that the tearing instability can indeed compete with their nonlinear evolution. The results, therefore, provide direct numerical support for the picture that a new regime of MHD turbulence can exist below ${\lambda }_{\mathrm{c}}$.

Figure 1

Time history of energy components. Top panel: $S=64000$, middle: $S=16000$, bottom: $S=4000$. The fluctuating $v_y$ and $b_y$ fields are initially generated by the driving force at the level corresponding to $1/64$ of their $x$-components. They grow due to nonlinear energy redistribution and/or tearing instability until they reach the magnitude of the $x$-components, at which point the anisotropic eddy is destroyed.

Figure 2

Compensated energy spectrum for the setup with $S=64000$, shown at several instances of the eddy evolution. At the latest time interval, the eddy has been destroyed by the nonlinear interaction. The spectrum at this stage is close to the spectrum of steady-state Alfvénic MHD turbulence.

Figure 3

Typical contours of the current $j_z$ for $S=64000$. The top panel shows a section of the domain at $t=41$, while the bottom one shows a different section at $t=45$. The plasmoid-like structures are not common in the flow; when present, they do not fully develop due to Alfvénic shearing flows.

Figure 4

Alignment angle as a function of the short coordinate $y$ for the setup $S=64000$ averaged over different periods of the eddy's evolution.

Figure 5

Compensated energy spectrum for setup $S=16000$ for several intermediate moments during the eddy evolution. The spectrum broadens in $k$-space and seems to approach the slope of $-19/9$, consistent with the prediction for the tearing-mediated turbulence, before the eddy is destroyed at late times.

Figure 6

Typical contours of the current $j_z$ for $S=16000$. The top panel shows a section of the domain at $t=40$, while the bottom one shows a different section at $t=45$. The tendency of turbulence to create plasmoid-like structures is more pronounced as compared to the case depicted in Fig. 3.

Figure 7

Alignment angle for setup $S=16000$ averaged over different periods of the eddy's evolution.