Conservation of magnetic-helicity fluctuations due to spatial decorrelation of fluxes in decaying MHD turbulence

Abstract

Hosking & Schekochihin (2021, Phys. Rev. X 11, 041005) have proposed that statistically isotropic decaying MHD turbulence without net magnetic helicity conserves the mean square fluctuation level of magnetic helicity in large volumes -- or, equivalently, the integral over space of the two-point correlation function of the magnetic-helicity density, denoted $I_H$. Formally, the conservation and gauge invariance of $I_H$ require the vanishing of certain boundary terms related to the strength of long-range spatial correlations. These boundary terms represent the ability (or otherwise) of the turbulence to organise fluxes over arbitrarily large distances to deplete or enhance fluctuations of magnetic helicity. In this work, we present a theory of these boundary terms, employing a methodology analogous to that of Batchelor & Proudman (1956, Philos. Trans. R. Soc. A 248, 369) to determine the relevant asymptotic forms of correlation functions. We find that long-range correlations of sufficient strength to violate the conservation of $I_H$ cannot develop dynamically if the evolution equation for the magnetic vector potential is chosen to be local in space. Likewise, we find that such correlations cannot develop for a wide class of gauge choices that make this equation non-local (including the Coulomb gauge). Nonetheless, we also identify a class of non-local gauge choices for which correlations that are sufficiently strong to violate the conservation of $I_H$ do appear possible. We verify our theoretical predictions for the case of the Coulomb gauge with measurements of correlation functions in a high-resolution numerical simulation.

Figures (8)

• Figure 1: Two-dimensional slices of the current density squared $J^2 = |\bnabla \times \bm{B}|^2$, normalised to its root-mean-square value, at various times during the decay. After $t \sim t_{A0}$, we see growth of magnetic structures resulting from the merging of smaller structures.
• Figure 2: Temporal and spectral statistics of decaying MHD turbulence in the simulations with $2304^3$ grid cells. Panel (a): magnetic and kinetic energy, $E_M$ and $E_K$, as functions of time in units of the Alfvén box-crossing time $t_{A0}$. The dotted vertical lines indicate times in which the correlation functions are plotted in figures \ref{['fig:unscaled_correlators']} and \ref{['fig:scaled_correlators']}. Panel (b): magnetic- and kinetic-energy spectra. The darkest lines correspond to the initial condition, and the other three to the times indicated in panel (a), with lighter lines indicating later times. We observe a $k^4$ scaling until the peak scale, after which the spectrum is proportional to $k^{-2}$, which may be an artifact of current-sheet discontinuities. Panel (c): magnetic-helicity-variance spectrum, $\mathcal{E}_H$. The vertical axis of each panel uses the code units defined in Section \ref{['sec:initconds']}.
• Figure 3: The evolution of $p(t)$ and $q(t)$\ref{['pq']} as a function of time in the simulation. Different values of $\beta$ correspond to different scaling relations between $E_{M}$ and $\xi_M$. The simulation evolves somewhat along the line $\beta = 3/2$, which corresponds to self-similar decay that conserves $I_H$ [see \ref{['nonhelicallaws']}].
• Figure 4: Evolution of the Mach number $\mathrm{Ma}$ and Lundquist number $S$. The vertical lines indicate the times at which the correlation functions are shown in figures \ref{['fig:unscaled_correlators']} and \ref{['fig:scaled_correlators']}.
• Figure 5: The three correlation functions that contribute to $C_{\infty}$\ref{['Cinfty']} measured at ${t/t_{A0} = 0.112}$ (blue), $t/t_{A0} = 0.447$ (red) and $t/t_{A0} = 1.780$ (green) as a function of $r/L$. Each vertical axis is measured in the code units defined in Section \ref{['sec:initconds']}. Positive values are plotted with solid circles, negative ones with crosses. Dashed vertical lines indicate the integral scale $\xi_M$\ref{['xi_M']} at each time. The correlation functions decay in amplitude and shift to larger spatial scales over time. We indicate a number of power laws to guide the eye, but observe that none of these fit any given curve over more than half a decade in $r/L$.