Growth rates of the Weibel and tearing mode instabilities in a relativistic pair plasma

TL;DR

This paper introduces a spectral Galerkin algorithm to solve the linear dispersion relation in inhomogeneous, magnetized, relativistic plasmas, enabling analysis of instabilities like Weibel and tearing modes with improved accuracy.

Contribution

It extends previous algorithms to handle inhomogeneous plasmas using spectral methods, allowing for detailed analysis of relativistic plasma instabilities.

Findings

Validated the algorithm against analytical growth rates for the Weibel instability.

Applied the method to the relativistic tearing mode without assuming current sheet thickness.

Demonstrated the algorithm's effectiveness in complex, inhomogeneous plasma configurations.

Abstract

We present an algorithm for solving the linear dispersion relation in an inhomogeneous, magnetised, relativistic plasma. The method is a generalisation of a previously reported algorithm that was limited to the homogeneous case. The extension involves projecting the spatial dependence of the perturbations onto a set of basis functions that satisfy the boundary conditions (spectral Galerkin method). To test this algorithm in the homogeneous case, we derive an analytical expression for the growth rate of the Weibel instability for a relativistic Maxwellian distribution and compare it with the numerical results. In the inhomogeneous case, we present solutions of the dispersion relation for the relativistic tearing mode, making no assumption about the thickness of the current sheet, and check the numerical method against the analytical expression.