Generation of chiral asymmetry via helical magnetic fields

It is well known that helical magnetic fields undergo a so-called inverse cascade by which their correlation length grows due to the conservation of magnetic helicity in classical ideal magnetohydrodynamics (MHD). At high energies above approximately $10$ MeV, however, classical MHD is necessarily extended to chiral MHD and then the conserved quantity is $\langle\mathcal{H}\rangle + 2 \langle\mu_5\rangle / \lambda$ with $\langle\mathcal{H}\rangle$ being the mean magnetic helicity and $\langle\mu_5\rangle$ being the mean chiral chemical potential of charged fermions. Here, $\lambda$ is a (phenomenological) chiral feedback parameter. In this paper, we study the evolution of the chiral MHD system with the initial condition of nonzero $\langle\mathcal{H}\rangle$ and vanishing $\mu_5$. We present analytic derivations for the time evolution of $\langle\mathcal{H}\rangle$ and $\langle\mu_5\rangle$ that we compare to a series of laminar and turbulent three-dimensional direct numerical simulations. We find that the late-time evolution of $\langle\mathcal{H}\rangle$ depends on the magnetic and kinetic Reynolds numbers ${\rm Re}_{_\mathrm{M}}$ and ${\rm Re}_{_\mathrm{K}}$. For a high ${\rm Re}_{_\mathrm{M}}$ and ${\rm Re}_{_\mathrm{K}}$ where turbulence occurs, $\langle\mathcal{H}\rangle$ eventually evolves in the same way as in classical ideal MHD where the inverse correlation length of the helical magnetic field scales with time $t$ as $k_\mathrm{p} \propto t^{-2/3}$. For a low Reynolds numbers where the velocity field is negligible, the scaling is changed to $k_\mathrm{p} \propto t^{-1/2}\mathrm{ln}\left(t/t_\mathrm{log}\right)$. After being rapidly generated, $\langle\mu_5\rangle$ always decays together with $k_\mathrm{p}$, i.e. $\langle\mu_5\rangle \approx k_\mathrm{p}$, with a time evolution that depends on whether the system is in the limit of low or high Reynolds numbers.