Magnetic helicity, weak solutions and relaxation of ideal MHD

TL;DR

This paper investigates magnetic helicity conservation in weak solutions of ideal MHD, proposing a relaxed system that separates turbulence effects from helicity conservation, and addresses open questions about integrability conditions and turbulent solutions.

Contribution

It introduces a new relaxed MHD system decoupling turbulence effects from magnetic helicity conservation, and answers key open questions in the field.

Findings

Sharpness of the $L^3$ integrability condition for magnetic helicity conservation.

Existence of turbulent bounded solutions dissipating energy and cross helicity with constant magnetic helicity.

Abstract

We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by-products we answer two open questions in the field: We show the sharpness of the $L^3$ integrability condition for magnetic helicity conservation and provide turbulent bounded solutions for MHD dissipating energy and cross helicity but with (arbitrary) constant magnetic helicity.