Structures in magnetohydrodynamic turbulence: Detection and scaling

We present a systematic analysis of statistical properties of turbulent current and vorticity structures at a given time using cluster analysis. The data stem from numerical simulations of decaying three-dimensional magnetohydrodynamic turbulence in the absence of an imposed uniform magnetic field; the magnetic Prandtl number is taken equal to unity, and we use a periodic box with grids of up to ${1536}^3$ points and with Taylor Reynolds numbers up to 1100. The initial conditions are either an $X$-point configuration embedded in three dimensions, the so-called Orszag-Tang vortex, or an Arn’old-Beltrami-Childress configuration with a fully helical velocity and magnetic field. In each case two snapshots are analyzed, separated by one turn-over time, starting just after the peak of dissipation. We show that the algorithm is able to select a large number of structures (in excess of 8000) for each snapshot and that the statistical properties of these clusters are remarkably similar for the two snapshots as well as for the two flows under study in terms of scaling laws for the cluster characteristics, with the structures in the vorticity and in the current behaving in the same way. We also study the effect of Reynolds number on cluster statistics, and we finally analyze the properties of these clusters in terms of their velocity–magnetic-field correlation. Self-organized criticality features have been identified in the dissipative range of scales. A different scaling arises in the inertial range, which cannot be identified for the moment with a known self-organized criticality class consistent with magnetohydrodynamics. We suggest that this range can be governed by turbulence dynamics as opposed to criticality and propose an interpretation of intermittency in terms of propagation of local instabilities.

Figure 1

(Color online) Top: global view of dissipative clusters in the $j^2$ field selected by our algorithm in the ABC run I. The largest cluster has been removed to let see the intermediate size clusters; only one tenth of the remaining clusters are shown. Bottom: zoomed-in view of two selected current clusters, showing strong curvature of the sheets; the vorticity behaves similarly. The lower left edge (direction marked with red in the color version) is parallel to the $z$ axis chosen as the scanning direction.

Figure 2

(Color online) Two-dimensional visualization of the current at 15 regularly spaced 512 by 512 (run 1) slices in the $z$ direction with $z∊\left[0,70\right]$ (in pixel units). A dissipative cluster made up of two twisted current sheets merging is shown in the middle of the box (highlighted in red in the color version).

Figure 3

(Color online) Geometric scaling of the current sheet structures detected in run I based on the $j^2$ field for several combinations of the snapshot time $t$ and of the threshold parameter $a_{th}$: $t=4$, $a_{th}=2$ (medium gray or red), and $a_{th}=3$ (black); $t=5$, $a_{th}=2$ (light gray or green), and $a_{th}=3$ (dark gray or blue). Dotted (dashed) vertical lines show the boundaries of the dissipative (inertial) scaling ranges used for computing the scaling exponents reported in Tables . The statistics are roughly insensitive to the detection threshold $a_{th}$ and are stable in time.

Figure 4

(Color online) Scaling of probability distributions of current sheet structures detected in run I. Color coding and notations are the same as in the previous figure. Tilted dotted lines show power laws in the inertial and dissipative ranges for $t=4$ and $a_{th}=2$. As in the case of the geometric scaling, the shapes of the distributions are stable with respect to the detection threshold and time.

Figure 5

(Color online) Geometric scaling of current structures in runs I and II detected by thresholding the $j^2$ field at two different levels: $N=1536$, $a_{th}=2$ (medium gray or red lines), and $a_{th}=3$ (black); $N=512$, $a_{th}=2$ (light gray or green), and $a_{th}=3$ (dark gray or blue). Dotted (dashed) vertical lines show the boundaries of the dissipative (inertial) scaling ranges. The higher Reynolds number ABC flow (run II) develops considerably thinner current sheet structures described by smaller volumes and roughly the same surface areas compared to the lower resolution run. The dissipation rate in run II is slightly higher, and the geometry of current sheets generated in this run is significantly more complex than the $j^2$ structures observed in run I.

Figure 6

(Color online) Probability distributions of current structures in runs I and II. Tilted dotted lines show log-log regression slopes for run II; the remaining notations are the same as in the previous figure. With the exception of thickness distribution, all the statistics here exhibit power-law scaling with consistent sets of inertial range $\tau$ exponents.

Figure 7

(Color online) Comparison of scaling exponents of current structures in runs I and II ($N=512$ and $N=1536$, respectively) detected by thresholding the $j^2$ field at the level of two standard deviations above the mean. The error bars are approximate confidence intervals ($\pm 3$ standard errors) for each exponent. The plots show consistent distribution exponents but greater geometric exponents ($D_L$, $D_A$, and $D_V$) at higher Reynolds number. This difference becomes less prominent at higher detection thresholds (not shown). Similar tendencies are observed for the vorticity structures.

Figure 8

(Color online) Scaling characteristics of current and vorticity structures in ABC and OT flows (runs I and III, $N=512$, $t=4$, $a_{th}=2$) addressing the role of the velocity–magnetic-field alignment. In run I, structures in $j^2$ and ${\omega }^2$ are shown with light gray (green) and dark gray (blue) lines, whereas in run III, they are given with middle gray (red) and black lines. Note a similarity in the volume and area statistics and a significant difference in the alignment patterns of the two flows ($\chi$ is the cosine of the local alignment angle between $\mathbf{v}$ and $\mathbf{b}$). In addition to a stronger normalized $H_C$, the OT flow exhibits systematically thinner structures (smaller thickness for a given linear size for both $j^2$ and ${\omega }^2$) compared to the ABC flow.

Figure 9

(Color online) Probability distributions of the dimensionless volume and area in runs I (middle gray or red), II (light gray or green), and III (dark gray or blue), demonstrating a wider range of small-scale power-law behavior in the high $R_V$ (high-resolution) simulation.