Evolution of anisotropy in direct numerical simulations of MHD turbulence in a strong magnetic field on elongated periodic domains

The response of initially isotropic turbulence to a strong magnetic field in the low magnetic Reynolds number regime has been studied using direct numerical simulations on elongated solution domains that are necessary for reliable results at long evolution times. Most results are obtained using a $16384\times 2{048}^2$ periodic domain of aspect ratio 8, without numerical forcing, after a presimulation that creates the desired initial conditions before the magnetic field is applied. At early times, velocity fluctuations parallel to the magnetic field becomes dominant as a result of Joule dissipation being weaker in this direction. However, this anisotropy is reversed after several large-eddy timescales. Statistics of the velocity gradients indicate a strong trend toward local axisymmetry and quasi-two-dimensionality, with reduced intermittency. Scale-dependent anisotropy is studied in spectral space in terms of a wave number $\left(k_1\right)$ along the magnetic field and a radial wave number $\left(k_r\right)$ in the orthogonal plane. Axisymmetric spectra show that the Joule dissipation plays a dominant role in causing kinetic energy to be concentrated in a narrow spectral region at very low values of $k_1$, which would not be captured if the domain were cubic. Simulations spanning over two orders of magnitude variation in the magnetic interaction parameter $\left(N\right)$ show that Reynolds stress anisotropy scales with the Joule time only for a short initial period. At large $N$, accelerated development of anisotropy leads to an even greater need for elongated domains which have not been employed frequently in the literature. Overall the results in this work provide both a confirmation of trends seen in simulations on cubic domains at earlier times and new observations at later times where the benefits of an elongated domain are clearly evident. A clear parametrization of Reynolds number effects still awaits larger simulations at inevitably higher cost.

Figure 1

(a) Taylor-scale Reynolds number and (b) flatness factor of longitudinal velocity gradients in presimulations where $n_y=n_z=256$, 512, 1024, 2048 (from bottom to top). With $n_x=\mathrm{\Lambda }{n}_y$, results for $\mathrm{\Lambda }=1$ and $\mathrm{\Lambda }=8$ are indicated by solid lines and dashed lines, respectively. The time axis $\left(t^{\prime }\right)$ is normalized by initial values of $K$ and $\langle \epsilon \rangle$ in the presimulation. The meanings of the circles on each curve are addressed in the text.

Figure 2

(a) Turbulence kinetic energy $\left(K\right)$, (b) integral length scales $L_{11}$ and $L_{22}$ of velocity components $u_1$ and $u_2$, and (c) Reynolds stress anisotropy tensor elements $(b_{11}$ (red), $b_{22}$ and $b_{33}$ (blue)) in $8192\times {1024}^2$ simulation on a $16\pi \times {\left(2\pi \right)}^2$ domain with $N=1$. Time $t$ is measured from the beginning of the presimulation, and the magnetic field is turned on at $t=t_0$. In (a) and (b) solid lines represent the presimulation (if extended), while dashed line represents MHD results. In (b) $L_{11},L_{22}$ under the magnetic field are indicated by long and short dashed lines, respectively.

Figure 3

(a) $L_{11}/\pi$, (b) $\langle J\rangle /{\langle J\rangle }_0$, (c) $\langle \epsilon \rangle /{\langle \epsilon \rangle }_0$ versus normalized time $t^{*}=(t-t_0)/{\left(T_E\right)}_0$ since when the magnetic field is turned on. Domain aspect ratio $\mathrm{\Lambda }$ increases from 1 to 64 in the direction of the arrow. Short horizontal bars in (a) mark maximum possible values for each $\mathrm{\Lambda }$.

Figure 4

Contour plots of axisymmetric spectra of Joule dissipation (left frames) and viscous dissipation (right frames), from late-time data in simulations with three different aspect ratios, corresponding to grid resolutions ${256}^3,2048\times {256}^2$, and $16384\times {256}^2$, with $N=1$ and ${\left(R_{\lambda }\right)}_0=21$.

Figure 5

Development of anisotropy of velocity gradient and vorticity variances, for the same simulation as in Table 3. (a) $\langle u_{\parallel ,\parallel }^2\rangle (•)$, $\langle u_{\parallel ,\perp }^2\rangle \left(\blacksquare \right)$, $\langle u_{\perp ,\parallel }^2\rangle (△)$, $\langle {\left(u_{\perp ,\perp }^L\right)}^2\rangle (\square )$, $\langle {\left(u_{\perp ,\perp }^T\right)}^2\rangle (\circ )$, all normalized by $\langle \epsilon \rangle /15\nu$; (b) Ratios between variance of velocity gradients: $\langle u_{\parallel ,\parallel }^2\rangle /\langle {\left(u_{\perp ,\perp }^L\right)}^2\rangle (•)$, $\langle u_{\perp ,\parallel }^2\rangle /\langle u_{\parallel ,\perp }^2\rangle \left(\blacksquare \right)$, for $2\langle u_{\parallel ,\parallel }^2\rangle /\langle u_{\parallel ,\perp }^2\rangle (△)$, $\langle u_{\parallel ,\perp }^2\rangle /\langle {\left(u_{\perp ,\perp }^T\right)}^2\rangle (\square )$, $2\langle {\left(u_{\perp ,\perp }^L\right)}^2\rangle /\langle {\left(u_{\perp ,\perp }^T\right)}^2\rangle (\circ )$; (c) $\langle {\omega }_{\parallel }^2\rangle$ (solid lines) and $\langle {\omega }_{\perp }^2\rangle$ (dashed lines), both normalized by $\langle {\omega }_i{\omega }_i\rangle$.

Figure 6

Development of anisotropy of velocity gradient and vorticity variances under a magnetic field, on domains with aspect ratio $\mathrm{\Lambda }=1$ (red), 8 (green), and 64 (blue) with the shortest dimension having 256 grid points. (a) $\langle u_{\perp ,\parallel }^2\rangle (△)$, $\langle {\left(u_{\perp ,\perp }^T\right)}^2\rangle (\circ )$, all normalized by $\langle \epsilon \rangle /15\nu$; (b) ratios between variance of velocity gradients: $2\langle u_{\parallel ,\parallel }^2\rangle /\langle u_{\parallel ,\perp }^2\rangle (△)$, $\langle u_{\parallel ,\perp }^2\rangle /\langle {\left(u_{\parallel ,\perp }^T\right)}^2\rangle (\square )$, $2\langle {\left(u_{\perp ,\perp }^L\right)}^2\rangle /\langle {\left(u_{\perp ,\perp }^T\right)}^2\rangle (\circ )$; (c) $\langle {\omega }_{\parallel }^2\rangle$ (solid lines) and $\langle {\omega }_{\perp }^2\rangle$ (dashed lines), both normalized by $\langle {\omega }_i{\omega }_i\rangle$.

Figure 7

Visualizations of normalized enstrophy $\mathrm{\Omega }/\langle \mathrm{\Omega }\rangle$ showing development of coherent vortical structures in MHD turbulence in domains of different aspect ratios. The brightest red and darkest blue represent $\mathrm{\Omega }/\langle \mathrm{\Omega }\rangle >10$ and $<0.05$, respectively. Each frame is a pair of 2 images in $y-z$ (square, on left) and $x-z$ planes (rectangle, on right.) Frames (a) and (b) are from ${2048}^3$ grid with $\mathrm{\Lambda }=1$, with $N=1$, at $t^{*}=12.58$ and 41.52, respectively; while frames (c), (d), (e) are from $16384\times 2{048}^2$ grid with $\mathrm{\Lambda }=8$, with $N=1$, at $t^{*}=0$, 12.58 and 41.52. Frames (f) and (g) are similar to (d) and (e), but from simulation at $N=8$.

Figure 8

(a) Evolution of ${\epsilon }_{\parallel }/\langle {\omega }_{\perp }^2\rangle (△)$, ${\epsilon }_{\perp }/\langle {\omega }_{\parallel }^2\rangle (◯)$ and $({\epsilon }_{\parallel }/{\epsilon }_{\perp })(\langle {\omega }_{\parallel }^2\rangle /\langle {\omega }_{\perp }^2\rangle )(\square )$. Horizontal dashed lines are at values 0.5 and 2.0. (b) Anisotropy tensor elements $d_{\parallel }={\epsilon }_{\parallel }/\left(2\epsilon \right)-1/3$ for dissipation $\left(▴\right)$, $v_{\parallel }=\langle {\mathrm{\Omega }}_{\parallel }\rangle /\langle \mathrm{\Omega }\rangle -1/3$ for vorticity covariance $(•)$, and their sum $d_{\parallel }+v_{\parallel }$ (dashed line).

Figure 9

Compensated 1D spectra $k_1{E}_{\alpha \alpha }\left(k_1\right)$: solid and dashed lines for $\alpha =1$ and $\alpha =2$, respectively. From (a) to (c): at $t^{*}=0$, 2.06, 26.25, normalized by the instantaneous kinetic energy.

Figure 10

Terms contributing to the evolution of 1D compensated spectra: (a), (c), and (e) for $d\left(k_1{E}_{11}\left(k_1\right)\right)/dt$; (b), (d), and (f) for $d\left(k_1{E}_{22}\left(k_1\right)\right)/dt$, at normalized times (from top to bottom) $t^{*}=0$, 2.06, 26.25. Different curves denote time rate of change (black), viscous dissipation (red), Joule dissipation (blue), nonlinear transfer (green) and pressure strain correlation (magenta). All are normalized by the instantaneous viscous dissipation.

Figure 11

Evolution of (left) compensated 1D transfer spectra of (a) $k_1{T}_{11}\left(k_1\right)$, (c) $k_1{T}_{22}\left(k_1\right)$, and (right) compensated radial transfer spectra of (b) $k_r{T}_{11}\left(k_r\right)$, (d) $k_r{T}_{22}\left(k_r\right)$. All the spectra shown are normalized by instantaneous viscous dissipation. Lines in red, green, blue, and black denote times $t^{*}=0$, 2.06, 6.86, and 26.25, respectively.

Figure 12

Contours of axisymmetric energy spectrum $E_A(k_1,k_r)$ at times (from left to right) $t^{*}=2.06$, 6.86, 26.25. Contour levels are set at logarithmically spaced intervals, decreasing by successive factors of 10 outwards from near the origin. (Note the differences among different frames in the upper limits of the coordinate axes shown. In frame (a) maximum values of both $k_1$ and $k_r$ are both 960.)

Figure 13

Axisymmetric spectra of TKE. From left to right, $t^{*}=0$, 2.06, 6.86, 26.25.

Figure 14

Axisymmetric spectra of the terms in the energy budget equation. From left to right: (negative of) rate of change, viscous dissipation, Joule dissipation, and positive values of spectral transfer. From top to bottom $t^{*}=0$, 2.06, 6.86, 26.25.

Figure 15

Evolution of (a) normalized integral length scale and (b) Reynolds stress anisotropy tensor element, in the direction of the magnetic field, with arrows pointing in the direction of increasing $N$ (0.5, 1, 2, 4, 8, $...$, 256). In (a), sloping dashed line has slope of 0.5, horizontal dashed lines are at heights 0.25 and 0.5. In (b), solid circles indicate time instants when $L_{11}$ exceeds 1/4 of ${\mathcal{L}}_{0x}$ as seen in (a). The upper dashed line is at height 1/6.

Figure 16

Evolution of (a) $\langle u_{\parallel ,\parallel }^2\rangle /\langle {\left(u_{\perp ,\perp }^L\right)}^2\rangle$, (b) $\langle {\left(u_{\perp ,\perp }^L\right)}^2\rangle /\langle {\left(u_{\perp ,\perp }^T\right)}^2\rangle$. Arrows point in the direction of increasing $N\left(1,2,4,8\right)$, for the same $16384\times {2048}^2$ domain of $\mathrm{\Lambda }=8$. In (b) the horizontal dashed line denotes $1/3$, which is the value in 2D isotropic turbulence.

Figure 17

Terms contributing to the evolution of 1D compensated spectra of: $d\left(k_1{E}_{11}\left(k_1\right)\right)/dt$: (a) $t^{*}=0$, with $N=1$; (b) $t^{*}=3.29$, with $N=1$; (c) $t^{*}=0$, with $N=8$; (d) $t^{*}=3.29$, with $N=8$. Different curves denote time rate of change (black), viscous dissipation (red), Joule dissipation (blue), nonlinear transfer (green), and pressure strain correlation (magenta). All curves are normalized by the instantaneous viscous dissipation.

Figure 18

Axisymmetric spectra of the terms in the energy budget equation at $N=8$. From left to right: (negative of) rate of change, viscous dissipation, Joule dissipation, and positive values of spectral transfer. Top row for results at $t^{*}=0$, bottom row for $t^{*}=26.25$.