Hosking integral in nonhelical Hall cascade

TL;DR

This paper demonstrates that in electron-controlled magnetic field evolution, the Hosking integral still governs turbulence decay but with different scaling laws, leading to slower decay and different correlation length growth compared to MHD.

Contribution

It extends the understanding of the Hosking integral's role to nonhelical Hall cascade, revealing altered decay laws and correlation growth in electron-dominated regimes.

Findings

Correlation length grows as t^{4/13}

Magnetic energy density decays as t^{-10/13}

Results agree with numerical simulations of nonhelical Hall cascade

Abstract

The Hosking integral, which characterizes magnetic helicity fluctuations in subvolumes, is known to govern the decay of magnetically dominated turbulence. Here we show that, when the evolution of the magnetic field is controlled by the motion of electrons only, as in neutron star crusts, the decay of the magnetic field is still controlled by the Hosking integral, but now it has effectively different dimensions than in ordinary magnetohydrodynamic (MHD) turbulence. This causes the correlation length to increase with time $t$ like $t^{4/13}$ instead of $t^{4/9}$ in MHD. The magnetic energy density decreases like $t^{-10/13}$, which is slower than in MHD, where it decays like $t^{-10/9}$. These new analytic results agree with earlier numerical simulations for the nonhelical Hall cascade.