Magnetic helicity and the evolution of decaying magnetohydrodynamic turbulence

Ensemble-averaged high resolution direct numerical simulations of reverse spectral transfer are presented, extending on the many single realization numerical studies done up to now. This identifies this type of spectral transfer as a statistical property of magnetohydrodynamic turbulence and thus permits reliable numerical exploration of its dynamics. The magnetic energy decay exponent from these ensemble runs has been determined to be $n_E=(0.47\pm 0.03)+(13.9\pm 0.8)/R_{\lambda }$ for initially helical magnetic fields. We show that even after removing the Lorentz force term in the momentum equation, thus decoupling it from the induction equation, reverse spectral transfer still persists. The induction equation is now linear with an externally imposed velocity field, thus amenable to numerous analysis techniques. A new door has opened for analyzing reverse spectral transfer, with various ideas discussed.

Figure 1

Magnetic energy and helicity spectra of run H9 showing reverse spectral transfer. The black (upper) lines refer to $E_{\text{mag}}\left(k\right)$, and red (lower) lines to $k{H}_{\text{mag}}\left(k\right)$. Solid lines indicate one initial turnover time $t_0$; dotted and dash-dotted lines refer to $2t_0$, $5t_0$, and $10t_0$. The inset shows the flux of magnetic helicity at $t_0$ and $5t_0$.

Figure 2

Reynolds number dependence of decay exponents of $E_{\text{mag}}\left(t\right)$ for maximally helical initial magnetic fields. The inset shows the decay of $\Gamma \left(t\right)$ for runs H3-H11.

Figure 3

Magnetic energy spectra showing reverse spectral transfer for the decoupled system (run Hd5). Note the absence of RST for the magnetic helicity. The black (upper) lines refer to $E_{\text{mag}}(k,t)$, the red (lower) lines to $k{H}_{\text{mag}}(k,t)$, and the solid lines are earlier in time than the dotted lines. The inset shows the flux of magnetic helicity.

Figure 4

Magnetic energy and helicity spectra for the decoupled case Hd4 showing RST for an initially helical velocity field. The black (upper) lines refer to $E_{\text{mag}}(k,t)$, the red (lower) lines to $-k{H}_{\text{mag}}(k,t)$, and the solid lines are earlier in time than the dotted lines. Since $H_{\text{mag}}(k,t)$ is positive at larger $k$, it does not show on logarithmic scales.