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

It has recently been recognised that local fluctuations in magnetic helicity are key to constraining the turbulent decay of magnetic fields that are statistically homogeneous, isotropic and non-helical on average. Mathematically, the constraint manifests as conservation of the so-called Saffman helicity integral, also known as the Hosking integral. In this study, we construct von-K\'arm\'an-Howarth-Monin relations for the Hosking integral and thus formulate the formal condition on relevant fourth-order two-point correlation functions for the integral to be conserved. Specifically, we show that that conservation of the Hosking integral is guaranteed if the longitudinal correlation function of the helicity density and its flux decays faster than $r^{-3}$ at large separations $r$. With high-resolution numerical simulations ($2304^3$ cells), we demonstrate that this is indeed the case in decaying turbulence: the correlation function instead decays as $r^{-4}$. We generalise Batchelor \& Proudman's theory of correlations in hydrodynamic turbulence to MHD and show that it predicts correctly the $r^{-4}$ scaling that we measure.