Scaling of the Hosking integral in decaying magnetically-dominated turbulence

TL;DR

This paper investigates the scaling behavior of the Hosking integral in decaying magnetically dominated turbulence, confirming its gauge invariance and near-conservation, and exploring its role in self-similar decay processes.

Contribution

The study demonstrates the gauge invariance and near-conservation of the Hosking integral in turbulence decay and links it to self-similar evolution of magnetic energy spectra.

Findings

Hosking integral is gauge-invariant and nearly conserved at high Lundquist numbers.

Turbulence decay follows a self-similar pattern influenced by the Hosking integral.

Decay behavior varies with magnetic diffusivity, indicating different governing time scales.

Abstract

The Saffman helicity invariant of Hosking and Schekochihin (2021, PRX 11, 041005), which we here call the Hosking integral, has emerged as an important quantity that may govern the decay properties of magnetically dominated nonhelical turbulence. Using a range of different computational methods, we confirm that this quantity is indeed gauge-invariant and nearly perfectly conserved in the limit of large Lundquist numbers. For direct numerical simulations with ordinary viscosity and magnetic diffusivity operators, we find that the solution develops in a nearly self-similar fashion. In a diagram quantifying the instantaneous decay coefficients of magnetic energy and integral scale, we find that the solution evolves along a line that is indeed suggestive of the governing role of the Hosking integral. The solution settles near a line in this diagram that is expected for a self-similar evolution of the magnetic energy spectrum. The solution will settle in a slightly different position when the magnetic diffusivity decreases with time, which would be compatible with the decay being governed by the reconnection time scale rather than the Alfv\'en time.