FRACTAL LEVEL SETS AND MULTIFRACTAL FIELDS IN DIRECT SIMULATIONS OF TURBULENCE

The fractal nature of level sets and the multifractal nature of various scalar and vector fields in hydromagnetic and hydrodynamic turbulence are investigated using data of direct simulations. It turns out that fields whose evolution is governed by stretching terms (vortex stretching, magnetic-field line stretching) exhibit "near singularities" that result in a multifractal scaling. Such stretching terms can lead to a rapid increase in the local value of the field. Fields without rapid local increase have no multifractal scaling. Furthermore, the simulations support recent theoretical suggestions that the fractal properties of the level sets of various fields are quite insensitive to the existence of stretching. Indeed, all the fields under study (temperature, vorticity magnitude, magnetic-field magnitude) show rather universal behavior in the geometry of their level sets, consistent with a two-dimensional geometry at small scales, with a crossover to a universal fractal geometry at large scales. The dimension at large scales is compatible with the theoretical prediction of about 2.7. The most surprising result of the simulations is that it appears that the "near singularities" are not efficiently eliminated by viscous dissipation, but rather seem to be strongest at the Kolmogorov cutoff. The effects of the singularities do not quite penetrate into the inertial range. We offer a simple analytic model to account for this behavior. We conclude that our findings may be due to the relatively small Reynolds numbers, but may also be indicative of generic behavior at larger Reynolds numbers. We offer some thoughts about the expected scaling behavior in the inertial range in light of our findings.