On the self-similarity of nonhelical magnetohydrodynamic turbulence

TL;DR

This paper re-analyzes the self-similarity properties of nonhelical magnetohydrodynamic turbulence, establishing conditions on initial fields that determine whether the evolution proceeds through self-similar decay or involves inverse energy transfer.

Contribution

It clarifies the conditions under which the energy spectra evolve self-similarly in nonhelical MHD turbulence, extending previous arguments with new criteria based on initial field homogeneity.

Findings

Self-similar evolution occurs if initial fields are homogeneous functions of the same degree.

Selective decay governs the evolution when initial spectra are power laws with different slopes.

Inverse energy transfer can occur even when evolution is self-similar.

Abstract

We re-analyze the Olesen arguments on the self-similarity properties of freely evolving, nonhelical magnetohydrodynamic turbulence. We find that a necessary and sufficient condition for the kinetic and magnetic energy spectra to evolve self-similarly is that the initial velocity and magnetic field are not homogeneous functions of space of different degree, to wit, the initial energy spectra are not simple powers of the wavenumber with different slopes. If, instead, they are homogeneous functions of the same degree, the evolution is self-similar, it proceeds through selective decay, and the order of homogeneity fixes the exponents of the power laws according to which the kinetic and magnetic energies and correlation lengths evolve in time. If just one of them is homogeneous, the evolution is self-similar and such exponents are completely determined by the slope of that initial spectrum which is a power law. The latter evolves through selective decay, while the other spectrum may eventually experience an inverse transfer of energy. Finally, if the initial velocity and magnetic field are not homogeneous functions, the evolution of the energy spectra is still self-similar but, this time, the power-law exponents of energies and correlation lengths depend on a single free parameter which cannot be determined by scaling arguments. Also in this case, an inverse transfer of energy may in principle take place during the evolution of the system.