Abstract
Understanding the phenomenology captured in direct numerical simulation (DNS) of magnetohydrodynamic(MHD)turbulence rests upon models and assumptions concerning the scaling of field variables and dissipation. Here compressible MHDturbulence is simulated in two spatial dimensions by solving the isothermal equations of resistive MHD on a periodic square grid. In these simulations it is found that the energy spectrum decreases more slowly with , and the viscous cutoff length is larger, than would be expected from the 1941 phenomenology of Kolmogorov (K41). Both these effects suggest that the cascade time is modified by the presence of Alfvén waves as in the phenomenology of Iroshnikov and Kraichnan (IK). Motivated by this, these scaling exponents are compared with those of the IKbased model of Politano and Pouquet [Phys. Rev. E52, 636 (1995)], which is an extension of the model of She and Leveque [Phys. Rev. Lett.72, 336 (1994)]. However, the scaling exponents from these simulations are not consistent with the model of Politano and Pouquet, so that neither IK nor K41 models would appear to describe the simulations. The spatial intermittency of turbulent activity in such simulations is central to the observed phenomenology and relates to the geometry of structures that dissipate most intensely via the scaling of the local rate of dissipation. The framework of She and Leveque implies a scaling relation that links the scaling of the local rate of dissipation to the scaling exponents of the pure Elsässer field variables (). This scaling relation is conditioned by the distinct phenomenology of K41 and IK. These distinct scaling relations are directly tested using these simulations and it is found that neither holds. This deviation suggests that additional measures of the character of the dissipation may be required to fully capture the turbulent scaling, for example, pointing towards a refinement of the phenomenological models. It may also explain why previous attempts to predict the scaling exponents of the pure Elsässer fields in twodimensional magnetohydrodynamicturbulence by extending the theory of She and Leveque have proved unsuccessful.
This work was supported in part by the Engineering and Physical Sciences Research Council through a CASE scholarship with the United Kingdom Atomic Energy Authority. The computing facilities were provided by the Centre for Scientific Computing of the University of Warwick with support from a Science Research Investment Fund grant.
I. INTRODUCTION
II. STATISTICAL MEASURES
III. NUMERICAL PROCEDURE
IV. RESULTS
V. CONCLUSIONS
Key Topics
 Turbulent flows
 42.0
 Magnetohydrodynamics
 35.0
 Intermittency
 12.0
 Turbulence simulations
 12.0
 Reynolds stress modeling
 10.0
Figures
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 steadystate 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 steadystate averages are calculated.
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 steadystate 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 steadystate 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.
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 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 .
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.
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.
Timeaveraged energy spectra [compare Eq. (3)] for highresolution runs 3 (lower) and 4 (upper). The vertical axis is normalized by (a) and (b). The bold lines indicate the powerlaw index as estimated by a powerlaw 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.
Timeaveraged energy spectra [compare Eq. (3)] for highresolution runs 3 (lower) and 4 (upper). The vertical axis is normalized by (a) and (b). The bold lines indicate the powerlaw index as estimated by a powerlaw 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 ESSconsistent 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.
Scaling of the ESSconsistent 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 highorder code (a) and the Lagrangian remap code that solves an equation of state (b). The vertical axis is normalized by .
Scatter plots of normalized energy spectra for decaying turbulence runs for the isothermal highorder code (a) and the Lagrangian remap code that solves an equation of state (b). The vertical axis is normalized by .
Tables
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 steadystate 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 steadystate rms fluctuations in density are for all runs.
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 steadystate 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 steadystate 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).
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 selfsimilarity. 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.
Test of the refined similarity hypothesis as modified for consistency with extended selfsimilarity. 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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