On the self-similarity of nonhelical magnetohydrodynamic turbulence

Abstract

We re-analyze the Olesen arguments on the selfsimilarity 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.

The decay of isotropic magnetohydrodynamics turbulence and the effects of cross-helicity

“…(1997). Note that our prediction (3.6) is in agreement with the self-similar analysis of Campanelli (2016).…”

“…These theoretical decay rates span from −1 for σ = 1 to −1.4 for σ = 4: these values are closer to the three-dimensional simulations of Banerjee & Jedamzik (2004) than the two-dimensional ones of Galtier et al (1997). Note that our prediction (3.6) is in agreement with the self-similar analysis of Campanelli (2016). These new predictions for K and L are successfully assessed numerically in figure 4 for σ B = 2 and σ B = 4.…”

Turbulent Magnetic Relaxation in Pulsar Wind Nebulae

“…A potentially significant effect, that was not accounted for in earlier studies, is evolution of the eddy scale. In particular, growth of over time is now understood to be a general feature of turbulent magnetic relaxation (Zrake 2014;Brandenburg et al 2015;Campanelli 2016). Previously, the "inverse energy transfer" was thought to operate only when the field had a significantly non-zero magnetic helicity measure (see e.g.…”

Classes of Hydrodynamic and Magnetohydrodynamic Turbulent Decay

“…(2) that α = −3 + 2/q. He argues that for a given subinertial range spectral exponent α, the exponent q is given by [12,[15][16][17] q = 2/(3 + α)…”

“…In the latter case, cosmological magnetic fields generated in the early Universe provide the initial source of turbulence, which leads to a growth of the correlation length by an inverse cascade mechanism [11], in addition to the general cosmological expansion of the Universe. In the last two decades, this topic has gained significant attention [12]. The time span since the initial magnetic field generation is enormous, but it is still uncertain whether it is long enough to produce fields at sufficiently large length scales to explain the possibility of contemporary magnetic fields in the space between clusters of galaxies [13].…”