Relativistic magnetohydrodynamics in one dimension

TL;DR

This paper derives analytical solutions and equations for one-dimensional relativistic magnetized plasma dynamics, providing benchmarks for numerical codes and advancing understanding of relativistic MHD behavior.

Contribution

It introduces new analytical solutions and linear equations for relativistic magnetohydrodynamics, including simple wave solutions and hodograph equations, for the first time.

Findings

Derived self-similar solutions for hot magnetized plasma expansion.

Formulated linear hodograph and Darboux equations for relativistic plasma motion.

Reduced complex relativistic MHD equations to a single linear differential equation.

Abstract

We derive a number of solution for one-dimensional dynamics of relativistic magnetized plasma that can be used as benchmark estimates in relativistic hydrodynamic and magnetohydrodynamic numerical codes. First, we analyze the properties of simple waves of fast modes propagating orthogonally to the magnetic field in relativistically hot plasma. The magnetic and kinetic pressures obey different equations of state, so that the system behaves as a mixture of gases with different polytropic indices. We find the self-similar solutions for the expansion of hot strongly magnetized plasma into vacuum. Second, we derive linear hodograph and Darboux equations for the relativistic Khalatnikov potential, which describe arbitrary one-dimensional isentropic relativistic motion of cold magnetized plasma and find their general and particular solutions. The obtained hodograph and Darboux equations are very powerful: system of highly non-linear, relativistic, time dependent equations describing arbitrary (not necessarily self-similar) dynamics of highly magnetized plasma reduces to a single linear differential equation.