On the convexity of Relativistic Ideal Magnetohydrodynamics

TL;DR

This paper investigates how magnetic fields influence the convexity properties of relativistic magnetohydrodynamics equations, revealing that magnetic fields tend to make the system more convex and reduce non-convex states.

Contribution

It introduces a generalized fundamental derivative for relativistic MHD that explicitly accounts for magnetic effects, showing magnetic fields always promote convexity.

Findings

Magnetic fields always positively contribute to the fundamental derivative.

Magnetic fields reduce the domain of non-convex thermodynamical states.

Material and Alfvén waves are unaffected by convexity issues.

Abstract

We analyze the influence of the magnetic field in the convexity properties of the relativistic magnetohydrodynamics system of equations. To this purpose we use the approach of Lax, based on the analysis of the linearly degenerate/genuinely non-linear nature of the characteristic fields. Degenerate and non-degenerate states are discussed separately and the non-relativistic, unmagnetized limits are properly recovered. The characteristic fields corresponding to the material and Alfv\'en waves are linearly degenerate and, then, not affected by the convexity issue. The analysis of the characteristic fields associated with the magnetosonic waves reveals, however, a dependence of the convexity condition on the magnetic field. The result is expressed in the form of a generalized fundamental derivative written as the sum of two terms. The first one is the generalized fundamental derivative in the case of purely hydrodynamical (relativistic) flow. The second one contains the effects of the magnetic field. The analysis of this term shows that it is always positive leading to the remarkable result that the presence of a magnetic field in the fluid reduces the domain of thermodynamical states for which the EOS is non-convex.