Lack of universality in decaying magnetohydrodynamic turbulence

Using computations of three-dimensional magnetohydrodynamic (MHD) turbulence with a Taylor-Green flow, whose inherent time-independent symmetries are implemented numerically, and in the absence of either a forcing function or an imposed uniform magnetic field, we show that three different inertial ranges for the energy spectrum may emerge for three different initial magnetic fields, the selecting parameter being the ratio of nonlinear eddy to Alfvén time. Equivalent computational grids range from ${128}^3$ to ${2048}^3$ points with a unit magnetic Prandtl number and a Taylor Reynolds number of up to 1500 at the peak of dissipation. We also show a convergence of our results with Reynolds number. Our study is consistent with previous findings of a variety of energy spectra in MHD turbulence by studies performed in the presence of both a forcing term with a given correlation time and a strong, uniform magnetic field. However, in contrast to the previous studies, here the ratio of characteristic time scales can only be ascribed to the intrinsic nonlinear dynamics of the paradigmatic flows under study.

Figure 1

(a) Temporal evolution of the total dissipation for the highest Reynolds number for each type of flow, represented by runs I6 (solid), A6 (dashed), and C6 (dotted), as described in Table . (b) Ratio of total magnetic to kinetic energy $E^M/E^V$ for these same runs. Note that I6 has noticeably less dissipation and more magnetic energy.

Figure 2

Ratio of magnetic- to kinetic-energy spectra averaged over $\Delta t=0.5$ (1.5–2 turnover times) about the maximum of dissipation for run I6 (solid), A6 (dashed), and C6 (dotted); same labels as in Fig. 1. Note the dominance of the magnetic energy at large scale for run I6 and the tendency toward equipartition at small scales for all runs, with a slight excess of magnetic energy.

Figure 3

Total energy spectra (a) compensated by $k^{5/3}$ and averaged over $\Delta t=0.5$ (1.5–2 turnover times) about the maximum of dissipation and ratio of nonlinear to Alfvén time scales as a function of wave number (b) for the same runs (labels as in Fig. 1: solid line for I, dash for A, and dots for C). Slopes are given only as a reference. The three arrows indicate the magnetic Taylor scale. Note that the three spectra follow noticeably different spectral laws and possibly different scale dependence for their time scales as well (see text).

Figure 4

Variation with Reynolds number Re of the ratio of eddy turnover to Alfvén time scales computed at the Taylor scale ${\lambda }^M$ for each flow. I1–I6 $(+)$, A1–A6 (◇), and C1–C6 $(\ast )$ are plotted in order of resolution and Reynolds number, as listed in Table . All quantities are computed in an interval of $\Delta t=0.5$ about the peak of dissipation for each run. Note the rather different values of these ratios and onset of saturation for the highest Re indicative of the beginning of a convergence to a high-Re regime.

Figure 5

Scaling as a function of Reynolds number Re of the maximum value of the dissipation over time (a) and of the Taylor Reynolds number $R_{\lambda }$ computed at the instantaneous peak of dissipation (b); straight line indicates the turbulent scaling $R_{\lambda }\sim {\text{Re}}^{1/2}$. Symbols are as in Fig. 4.

Figure 6

Total energy spectra averaged over $\Delta t=0.5$ (1.5–2 turnover times) around the time of maximum dissipation for different Reynolds numbers for the following flows (see Table ): I runs compensated by $k^2$ (a), A runs compensated by $k^{5/3}$ (b), and C runs compensated by $k^{3/2}$ (c). Dash-triple dots, dash-dots, dashes, dots, and solid lines represent, respectively, the runs I2–I6, A2–A6, and C2–C6. Equivalent resolutions range from ${128}^3$ to ${2048}^3$ grid points.

Figure 7

(a) Magnetic- to kinetic-energy ratio as a function of wave number at late time $\left(t∊\left[6,6.5\right]\right)$ and averaged over $\Delta t=0.5$. Again, we plot I6 (solid), A6 (dashed), and C6 (dotted). Note that the excess of magnetic energy at low wave number in I6 has subsided to 1/15 of its former value. (b) Total energy spectra for the same runs over the same time interval. Arrows indicate the new magnetic Taylor wave number for each run. In the insert, the same spectra are compensated by $k^{3/2}$. Note the similar energy ratios and inertial range scaling for the three flows.