The temporal evolution of the space averaged kinetic energy (lower curve) and magnetic energy (upper curve) for runs 4 (panel a) and 3 (panel b) (see Table I). The solid part of the curves shows the period over which steady-state averages are calculated. The horizontal lines show the time averages calculated over this period. The broken part of the curves shows the temporal evolution before steady-state averages are calculated.
The temporal evolution of the alignment of the velocity and magnetic fields for runs 4 (panel a) and 3 (panel b) (see Table I). The horizontal lines show the time averages calculated over this period.
ESS scaling in the field, order 6 against order 3. The boxes surrounding the data points indicate the standard error present in the time averaging process. Marked on the plots are the approximate values of and . Benzi et al. (Refs. 22,23) found that ESS scaling should extend down to approximately . The break in scaling occurs at approximately in these plots, showing that the viscous cutoff of the cascade is higher than expected from Kolmogorov phenomenology (K41) and is closer to that of Iroshnikov and Kraichnan (IK).
ESS scaling of Elsässer field variables [compare Eq. (7) with and ] for run 4. We exclude since this necessarily gives perfect scaling. The boxes surrounding the data points indicate the standard error present in the time averaging process. See column 4 of Table II for the corresponding inferrred values of .
Relative scaling exponents obtained from the driven 2D simulation of run 4 (error bars), the driven 2D simulation of Ref. 8 (upper broken line , lower broken line ), and the 2D decaying simulation of Ref. 7 (solid line with square markers). Values reported from the 3D decaying simulation of Ref. 10 (solid line) and the MHD intermittency model of Politano and Pouquet (Ref. 12) (dotted line) are shown for comparison.
Time-averaged energy spectra [compare Eq. (3)] for high-resolution runs 3 (lower) and 4 (upper). The vertical axis is normalized by (a) and (b). The bold lines indicate the power-law index as estimated by a power-law fit over the inertial range. It can be seen that the power law has a negative index whose magnitude is less than the predicted by K41 but is greater than the predicted by IK. The scaling range is extended to higher for run 4 compared to run 3, since run 4 is performed at a higher Reynolds number.
Scaling of the ESS-consistent refined similarity hypothesis for the Kolmogorov case [compare Eq. (10a)] (top) and the IK case [compare Eq. (10b)] (bottom) for run 4, order . Here the scaling range extends to below as found in Ref. 23. However, the ideal gradient of unity is not recovered, see Table III. The boxes surrounding the data points indicate the standard error present in the time averaging process.
Scatter plots of normalized energy spectra for decaying turbulence runs for the isothermal high-order code (a) and the Lagrangian remap code that solves an equation of state (b). The vertical axis is normalized by .
Parameters of driven turbulence runs. is the number of the grid points, is the time period over which the steady state is tracked in terms of the nonlinear turnover time, is the ratio of the simulation box length to the Kolmogorov dissipation scale, and is the kinetic Reynolds number. Statistics are calculated from snapshots taken approximately every two nonlinear turnover times. The steady-state kinetic energy is approximately twice that of the magnetic energy. The viscosity is set equal to the magnetic diffusivity making the magnetic and kinetic Reynolds numbers equal and the magnetic Prandtl number equal to unity. The rms sonic Mach number is and the steady-state rms fluctuations in density are for all runs.
Ratios of scaling exponents calculated from the Elsässer field variables for different runs (see Table I). Errors are an estimate of the possible range of straight line fits that can be drawn on the ESS plots (see Fig. 4).
Test of the refined similarity hypothesis as modified for consistency with extended self-similarity. K41 or IK refers to Eqs. (10a) and (10b), respectively. The symbols or represent scaling derived from the and Elsässer field variables, respectively. Exact agreement would be indicated by a value of 1.0 across all columns.
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