Large-scale dynamics of magnetic helicity

In this paper we investigate the dynamics of magnetic helicity in magnetohydrodynamic (MHD) turbulent flows focusing at scales larger than the forcing scale. Our results show a nonlocal inverse cascade of magnetic helicity, which occurs directly from the forcing scale into the largest scales of the magnetic field. We also observe that no magnetic helicity and no energy is transferred to an intermediate range of scales sufficiently smaller than the container size and larger than the forcing scale. Thus, the statistical properties of this range of scales, which increases with scale separation, is shown to be described to a large extent by the zero flux solutions of the absolute statistical equilibrium theory exhibited by the truncated ideal MHD equations.

Figure 1

Fluxes of total energy and magnetic helicity normalized with the corresponding dissipation rates for the run with $k_{f}L=40$. Black: normalized total energy flux ${\mathrm{\Pi }}_E\left(k\right)/\varepsilon$. Blue (dark gray): normalized magnetic helicity flux ${\mathrm{\Pi }}_{H_b}\left(k\right)/{\varepsilon }_{H_b}$. The inset presents the scaling ${\varepsilon }^{-}/\varepsilon \propto {\varepsilon }_{H_b}/\left(\varepsilon L\right)\propto {\left(k_{f}L\right)}^{-1}$. Black: ${\varepsilon }^{-}/\varepsilon$. Red (light gray): ${\varepsilon }_b^{-}/\varepsilon$. Blue (dark gray): ${\varepsilon }_{H_b}/\left(\varepsilon L\right)$.

Figure 2

Transfer spectra of magnetic helicity normalized with ${\varepsilon }_{H_b}L$ for the flow with $k_{f}L=40$. The time-averaged transfer $T_{H_b}\left(k\right)$ is indicated by the green (dark gray) curve while the gray curves indicate the instantaneous transfers $T_{H_b}(k,t)$.

Figure 3

Instantaneous normalized magnetic helicity transfer $T_{H_b}(K,Q)/\left({\varepsilon }_{H_b}L\right)$ from shell $Q$ to shell $K$ rescaled with $k_f$.

Figure 4

(a) Magnetic and kinetic energy spectra compensated by $k^{-2}$, where $E_u\left(k\right)$ has been shifted by a factor of ${10}^{-3}$. (b) Absolute value of magnetic helicity spectra. The spectra are collapsed by rescaling with $k/k_f$. The dashed lines correspond to the predictions from absolute equilibria, i.e., Eqs. (6). The green (light gray) curves refers to the simulation with $k_{f}L=40$, the blue (dark gray) to $k_{f}L=20$, and the red (gray) to $k_{f}L=10$. The inset shows the absolute value of the relative magnetic helicity ${\rho }_b\left(k\right)=H_b\left(k\right)/(\langle |{\mathit{a}}_{\mathit{k}}{|}^2{\rangle }^{1/2}\langle |{\mathit{b}}_{\mathit{k}}{|}^2{\rangle }^{1/2})$.