Ideal Magnetohydrodynamic Equations on a Sphere and Elliptic-Hyperbolic Property

TL;DR

This paper derives and analyzes the conical ideal magnetohydrodynamic equations projected onto a sphere, revealing their mixed elliptic-hyperbolic nature and providing new insights into electrically conducting conical flows.

Contribution

It is the first derivation and analysis of conical MHD equations, showing their existence and mixed type behavior, extending prior work on inviscid flows.

Findings

Conical MHD equations are mathematically well-defined.

The system exhibits both elliptic and hyperbolic regions.

Conical flows of electrically conducting gases are possible.

Abstract

This work contains the derivation and type analysis of the conical Ideal Magnetohydrodynamic equations. The 3D Ideal MHD equations with Powell source terms, subject to the assumption that the solution is conically invariant, are projected onto a unit sphere using tools from tensor calculus. Conical flows provide valuable insight into supersonic and hypersonic flow past bodies, but are simpler to analyze and solve numerically. Previously, work has been done on conical inviscid flows governed by the Euler equations with great success. It is known that some flight regimes involve flows of ionized gases, and thus there is motivation to extend the study of conical flows to the case where the gas is electrically conducting. To the authors' knowledge, the conical magnetohydrodynamic equations have never been derived and so this paper is the first invesitgation of that system. Among the results, we show that conical flows for this case do exist mathematically and that the governing system of partial differential equations is of mixed type. Throughout the domain it can be either hyperbolic or elliptic depending on the solution.