Evolution of Magnetic Fields in Freely Decaying Magnetohydrodynamic Turbulence

TL;DR

By quasilinearizing the Navier-Stokes equation, the induction equation in the quasinormal approximation is solved analytically and it is found that, if the magnetic field is not helical, the magnetic energy and correlation length evolve in time, respectively, as E(B) proportional to t(-2(1+p)/(3+p)) and xi(B), where p is the index of initial power-law spectrum.

Abstract

We study the evolution of magnetic fields in freely decaying magnetohydrodynamic turbulence. By quasi-linearizing the Navier-Stokes equation, we solve analytically the induction equation in quasi-normal approximation. We find that, if the magnetic field is not helical, the magnetic energy and correlation length evolve in time respectively as E_B \propto t^{-2(1+p)/(3+p)} and ξ_B \propto t^{2/(3+p)}, where p is the index of initial power-law spectrum. In the helical case, the magnetic helicity is an almost conserved quantity and forces the magnetic energy and correlation length to scale as E_B \propto (log t)^{1/3} t^{-2/3} and ξ_B \propto (log t)^{-1/3} t^{2/3}.

Figures (2)

• Figure 1: Magnetic energy spectrum in the non-helical case for $p=4$, with $\gamma = 1$. The dotted line corresponds to the initial spectrum, while continuous lines correspond, from left to right, to $t/\tau_{\rm eddy}(0) = 1,10,10^2,...,10^7$.
• Figure 2: Result of a numerical integration of Eqs. (\ref{['differential1']})-(\ref{['differential2']}) for ${\text{Re}}_B(0) = 10^{15}$, $p=4$, $h=10^{-3}$, with $\gamma = 1$. Upper panel: magnetic energy; middle panel: correlation length; Dotted lines correspond to analytical expansions. Lower panel: magnetic energy spectrum; the dotted line corresponds to the initial spectrum, while continuous lines correspond, from left to right, to $t/\tau_{\rm eddy}(0) = 1,10,10^2,...,10^7$.