Statistics of the inverse-cascade regime in two-dimensional magnetohydrodynamic turbulence

We present a detailed direct numerical simulation of statistically steady, homogeneous, isotropic, two-dimensional magnetohydrodynamic turbulence. Our study concentrates on the inverse cascade of the magnetic vector potential. We examine the dependence of the statistical properties of such turbulence on dissipation and friction coefficients. We extend earlier work significantly by calculating fluid and magnetic spectra, probability distribution functions (PDFs) of the velocity, magnetic, vorticity, current, stream-function, and magnetic-vector-potential fields, and their increments. We quantify the deviations of these PDFs from Gaussian ones by computing their flatnesses and hyperflatnesses. We also present PDFs of the Okubo-Weiss parameter, which distinguishes between vortical and extensional flow regions, and its magnetic analog. We show that the hyperflatnesses of PDFs of the increments of the stream function and the magnetic vector potential exhibit significant scale dependence and we examine the implication of this for the multiscaling of structure functions. We compare our results with those of earlier studies.

Figure 1

Log-log plots of spectra for run R1 with $k_{\mathrm{inj}}=500$ vs the wave number $k$: (a) kinetic energy $E^u\left(k\right)$ (red curve with filled circles) and magnetic energy $E^b\left(k\right)$ (blue curve with filled squares), (b) the compensated spectra $k^0{E}^u\left(k\right)$ (red curve with filled circles) and $k^{1/3}{E}^b\left(k\right)$ (blue curve with filled squares), (c) vorticity ${\left|\omega \left(k\right)\right|}^2$ (red curve with filled circles) and current density ${\left|j\left(k\right)\right|}^2$ (blue curve with filled squares) spectra, and (d) stream function ${\left|\varphi \left(k\right)\right|}^2$ (red curve with filled circles) and magnetic potential ${\left|\psi \left(k\right)\right|}^2$ (blue curve with filled squares) spectra. These spectra are averaged over 30 independent field configurations, which are separated from one another by ${\tau }_{\mathrm{eddy}}$ (Table 1). A clear inverse-cascade inertial range is visible in the interval $5\le k\le 110$. The inset of (a) shows a semilogarithmic plot of the total-energy flux $\Pi \left(k\right)$ vs $k$.

Figure 2

Log-log plots of spectra, for run R2, vs the wave number $k$: (a) kinetic energy $E^u\left(k\right)$ (red curve with filled circles) and magnetic energy $E^b\left(k\right)$ (blue curve with filled squares), (b) the compensated spectra $k^{-1/3}{E}^u\left(k\right)$ (red curve with filled circles) and $k^{1/3}{E}^b\left(k\right)$ (blue curve with filled squares), (c) the vorticity spectrum ${\left|\omega \left(k\right)\right|}^2$ (red curve with filled circles) and the current density spectrum ${\left|j\left(k\right)\right|}^2$ (blue curve with filled squares), and (d) the stream-function spectrum ${\left|\varphi \left(k\right)\right|}^2$ (red curve with filled circles) and magnetic potential spectrum ${\left|\psi \left(k\right)\right|}^2$ (blue curve with filled squares). The inset of (a) shows a semilogarithmic plot of the total-energy flux $\Pi \left(k\right)$ vs $k$.

Figure 3

Plots, for run R2, (a) of the total energy (green curve with filled diamonds), magnetic energy (blue curve with filled squares), and kinetic energy (red curve with filled circles) vs time [measured in units of ${\tau }_{\mathrm{eddy}}$ (Table 1)]. The origin on the horizontal axis is chosen at a time at which a statistically steady state has been obtained; data for averages are collected thereafter for $\simeq 30{\tau }_{\mathrm{eddy}}$. (b) The log-log plots of kinetic-energy spectra, at times $t=10{\tau }_{\mathrm{eddy}}$ (red curve with filled circles), $20{\tau }_{\mathrm{eddy}}$ (blue curve with filled squares), and $30{\tau }_{\mathrm{eddy}}$ (green curve with filled diamonds). (c) The log-log plots of the magnetic-energy spectra, at times $t=10{\tau }_{\mathrm{eddy}}$ (red curve with filled circles), $20{\tau }_{\mathrm{eddy}}$ (blue curve with filled squares), and $30{\tau }_{\mathrm{eddy}}$ (green curve with filled diamonds).

Figure 4

Semilogarithmic plots of the PDFs of (a) the vorticity $\omega$, (b) the current density $j$, (c) the stream function $\varphi$, and (d) the magnetic potential $\psi$. Blue dashed lines indicate Gaussian distributions. The deviations of our PDFs from Gaussian ones are small and visible only in the tails of these distributions (see text for their flatnesses).

Figure 5

Plots of the PDFs for (a) $cos\left({\beta }_{\omega ,j}\right)$, with ${\beta }_{\omega ,j}$ the angle between $\omega$ and $j$ and $cos\left({\beta }_{\omega ,j}\right)=\frac{\omega ·j}{\left|\omega \right|\left|j\right|}$, and (b) $cos\left({\beta }_{u,b}\right)$, with ${\beta }_{u,b}$ the angle between $\mathbf{u}$ and $\mathbf{b}$, i.e., $cos\left({\beta }_{u,b}\right)=\frac{\mathbf{u}·\mathbf{b}}{\left|\mathbf{u}\right|\left|\mathbf{b}\right|}$.

Figure 6

(a) The PDF of $\Lambda$ [red (gray) curve] and ${\Lambda }_b$ [blue (black) curve], and (b) the joint PDF of $\Lambda$ and ${\Lambda }_b$, for run R1, with the axes rescaled by the rms value of the quantity. The color bar is logarithmic.

Figure 7

(a) The PDF of $\Lambda$ [red (gray) curve] and ${\Lambda }_b$ [blue (black) curve], and (b) the joint PDF of $\Lambda$ and ${\Lambda }_b$, for run R2, with the axes rescaled by the rms value of the quantity. The color bar is logarithmic.

Figure 8

Left panel: Overlaid plot of the contours of $\varphi$ on a pseudocolor plot of $\Lambda$. Right panel: Overlaid plot of the contours of $\psi$ on a pseudocolor plot of ${\Lambda }_b$. These are for the run R1. The $\Lambda$ and ${\Lambda }_b$ fields have been filtered in such a way that all values of $\Lambda ,{\Lambda }_b\ge 4000$ are set to $\Lambda ,{\Lambda }_b=4000$, and $\Lambda ,{\Lambda }_b\le -2000$ are set to $\Lambda ,{\Lambda }_b=-2000$. These plots are for run R1.

Figure 9

Left panel: Overlaid plot of the contours of $\varphi$ on a pseudocolor plot of $\Lambda$. Right panel: Overlaid plot of the contours of $\psi$ on a pseudocolor plot of ${\Lambda }_b$. These plots are for run R2.

Figure 10

Pseudocolor plots (a) $\psi$ (run R1), (b) $\varphi$ (run R1), (c) $\psi$ (run R2), and (d) $\varphi$ (run R2).

Figure 11

Plots of PDFs for run R1 of the increments of (a) $\delta j$, (b) $\delta \psi$, (c) $\delta b_{\left|\right|}$, (d) $\delta \omega$, (e) $\delta \varphi$, and (f) $\delta u_{\left|\right|}$, with $l/l_{\mathrm{inj}}=6$ [red (dark gray) curves] and $l/l_{\mathrm{inj}}=12$ [green (light gray) curve]. The blue, dashed curves indicate Gaussian distributions.

Figure 12

Plots of the hyperflatnesses vs the scaled length $l/l_{\mathrm{inj}}$ for the following increments: (a) $\delta \varphi$ (red curve with filled circles), $\delta \omega$ (blue curve with filled squares), $\delta u_{\left|\right|}$ (green curve with filled diamonds) (for run R1), (b) $\delta \psi$ (red curve with filled circles), $\delta j$ (blue curve with filled squares), $\delta b_{\left|\right|}$ (green curve with filled diamonds) (for run R1), (c) $\delta \varphi$ (red curve with filled circles), $\delta \omega$ (blue curve with filled squares), $\delta u_{\left|\right|}$ (green curve with filled diamonds) (for run R2), and (d) $\delta \psi$ (red curve with filled circles), $\delta j$ (blue curve with filled squares), $\delta b_{\left|\right|}$ (green curve with filled diamonds) (for run R2).

Figure 13

Plots, for run R1, of (a) $S_2^{\omega }$ vs $l/l_{\mathrm{inj}}$ (in the inset, we zoom into the region $10l_{\mathrm{inj}}\le l\le 20l_{\mathrm{inj}}$), (b) multiscaling exponents vs $p$ for $\varphi$ [red (gray) curve] and $\psi$ [blue (black) curve], without extended self-similarity (ESS; see text), and (c) multiscaling exponents ratios vs $p$ for $\varphi$ [red (gray) curve] and $\psi$ [blue (black) curve], with ESS. For these multiscaling exponents, we use data for the structure functions in the range $10\le l/l_{\mathrm{inj}}\le 30$.