Ellipticity conditions for the extended MHD Grad-Shafranov-Bernoulli equilibrium equations

TL;DR

This paper investigates the conditions under which the extended MHD equilibrium equations transition between elliptic and hyperbolic regimes, revealing that electron inertia can cause static equilibria to become hyperbolic, which is unusual.

Contribution

It derives the ellipticity condition for extended MHD equilibrium equations, accounting for electron inertia and quasineutrality, and analyzes the transition points between elliptic and hyperbolic regimes.

Findings

Ellipticity condition expressed via a single inequality.

Electron inertia can cause static equilibria to become hyperbolic.

Transition points between elliptic and hyperbolic regimes identified.

Abstract

In this study, we find the points of transition between elliptic and hyperbolic regimes for the axisymmetric extended magnetohydrodynamic (MHD) equilibrium equations. The ellipticity condition is expressed via a single inequality but is more involved than the corresponding two-fluid ones due to the imposition of the quasineutrality condition and is also more complicated than the Hall MHD ellipticity condition, due to electron inertia. In fact, the inclusion of electron inertia is responsible for peculiar results; namely, even the static equilibrium equations can become hyperbolic.