Criticality and turbulence in a resistive magnetohydrodynamic current sheet

Scaling properties of a two-dimensional (2d) plasma physical current-sheet simulation model involving a full set of magnetohydrodynamic (MHD) equations with current-dependent resistivity are investigated. The current sheet supports a spatial magnetic field reversal that is forced through loading of magnetic flux containing plasma at boundaries of the simulation domain. A balance is reached between loading and annihilation of the magnetic flux through reconnection at the current sheet; the transport of magnetic flux from boundaries to current sheet is realized in the form of spatiotemporal avalanches exhibiting power-law statistics of lifetimes and sizes. We identify this dynamics as self-organized criticality (SOC) by verifying an extended set of scaling laws related to both global and local properties of the current sheet (critical susceptibility, finite-size scaling of probability distributions, geometric exponents). The critical exponents obtained from this analysis suggest that the model operates in a slowly driven SOC state similar to the mean-field state of the directed stochastic sandpile model. We also investigate multiscale correlations in the velocity field and find them numerically indistinguishable from certain intermittent turbulence (IT) theories. The results provide clues on physical conditions for SOC behavior in a broad class of plasma systems with propagating instabilities, and suggest that SOC and IT may coexist in driven current sheets which occur ubiquitously in astrophysical and space plasmas.

Figure 1

Time series of the number of unstable grid sites with $Q=D_{max}$ through several cycles, and the corresponding time series of the total magnetic energy dissipation. Statistics shown in Figs. 4, 6 are obtained from the unloading cycles. Those separate quiet, laminar periods with no energy dissipation where the velocity field is smooth. Note that large-scale loading and unloading do not occur for cellular automata models, such as the BTW sandpile, or for many models of IT, but do happen, e.g., in magnetospheric dynamics [12].

Figure 2

Numerical simulation of a current sheet and its qualitative behavior (see Ref. [16] for details). The model is driven by a steady, uniform plasma inflow at the top and bottom boundaries as shown by arrows. The left boundary is closed, while the right is open. Plasma energized through annihilation in the spatial magnetic field reversal leaves the region at the right. Inset: (Top) A snapshot of regions where the diffusive Poynting flux (see text) exceeds a threshold, used to define avalanches. (Bottom) At the same time, the corresponding velocity field ($v_x$) in the system. Note that while the magnetic field lines appear smooth at this scale, the plasma velocity field is highly intermittent.

Figure 3

For several $J_c$ values ranging over a factor of ${10}^2$, the quantity $\xi$ (triangles, right axis) measured at different driving rates $h_f$, the susceptibility $\chi$ (stars, left axis), and the power-law fit to the susceptibility values providing the exponent $\gamma$. Inset shows an example of stochastic evolution of the number of unstable grid sites during a single global unloading.

Figure 4

Data collapse using Eq. (12) for avalanches with different maximum area $s_{max}$ with ${\tau }_T=1.95,{\lambda }_T=0.68,{\tau }_E=1.48$, and ${\lambda }_E=1.47$. Insets: anisotropic scaling of $E$ and $T$ with the maximum avalanche extent $l$ in $x$ and $z$ directions.

Figure 5

The neighborhood of an initial site of instability shortly after the initiation of an unloading event. (a) Poynting flux due to slipping magnetic flux in the interior of the outward expanding current wave. (b) Transition from laminar to turbulent velocity field due to $\mathit{J}\times \mathit{B}$ acceleration at the current wave as it propagates through the magnetic field.

Figure 6

Left: ESS plots of velocity structure functions with $l$ parallel to the $z$ axis. The inset shows the same functions with subtracted average slopes. Right: Dependence of $\zeta \left(q\right)/\zeta \left(3\right)$ on the order $q$ for our current sheet model (CSM, with error bars), hierarchical models of IT mentioned in the text, Müller and Biskamp (MB) [34] model ($g=3,\gamma =2/3,C=1$), Kolmogorov model (K41) [35], as well as the exponents from 2d ideal MHD simulations [36].