Hall cascade with fractional magnetic helicity in neutron star crusts

The ohmic decay of magnetic fields in the crusts of neutron stars is generally believed to be governed by Hall drift which leads to what is known as a Hall cascade. Here we show that helical and fractionally helical magnetic fields undergo strong inverse cascading like in magnetohydrodynamics (MHD), but the magnetic energy decays more slowly with time $t$: $\propto t^{-2/5}$ instead of $\propto t^{-2/3}$ in MHD. Even for a nonhelical magnetic field there is a certain degree of inverse cascading for sufficiently strong magnetic fields. The inertial range scaling with wavenumber $k$ is compatible with earlier findings for the forced Hall cascade, i.e., proportional to $k^{-7/3}$, but in the decaying cases, the subinertial range spectrum steepens to a novel $k^5$ slope instead of the $k^4$ slope in MHD. The energy of the large-scale magnetic field can increase quadratically in time through inverse cascading. For helical fields, the energy dissipation is found to be inversely proportional to the large-scale magnetic field and proportional to the fifth power of the root-mean square (rms) magnetic field. For neutron star conditions with an rms magnetic field of a few times $10^{14}\,$G, the large-scale magnetic field might only be $10^{11}\,$G, while still producing magnetic dissipation of $10^{33}\,$erg$\,$s$^{-1}$ for thousands of years, which could manifest itself through X-ray emission. Finally, it is shown that the conclusions from local unstratified models agree rather well with those from stratified models with boundaries.