The behaviour of magnetoacoustic waves in the neighbourhood of a two-dimensional null point: initially cylindrically-symmetric perturbations

TL;DR

This paper investigates the behavior of magnetoacoustic waves near a 2D null point in plasmas, deriving analytical and numerical solutions that reveal wave focusing, deformation, and the influence of plasma beta on wave dynamics.

Contribution

It provides the first analytical solution for fast magnetoacoustic waves in a 2D null point for zero beta plasma and compares wave behavior in zero and non-zero beta conditions.

Findings

Fast waves focus on the null but never reach it in finite time.

Wave shape deforms into a quasi-diamond pattern along separatrices.

Pressure gradients cause wave deformation near the null.

Abstract

The propagation of magnetoacoustic waves in the neighbourhood of a 2D null point is investigated for both $\beta=0$ and $\beta \neq 0$ plasmas. Previous work has shown that the Alfv\'en speed, here $v_A \propto r$, plays a vital role in such systems and so a natural choice is to switch to polar coordinates. For the $\beta=0$ plasma, we derive an analytical solution for the behaviour of the fast magnetoacoustic wave in terms of the Klein-Gordon equation. We also solve the system with a semi-analytical WKB approximation which shows that the $\beta=0$ wave focuses on the null and contracts around it but, due to exponential decay, never reaches the null in a finite time. For the $\beta \neq 0$ plasma, we solve the system numerically and find the behaviour to be similar to that of the $\beta=0$ system at large radii, but completely different close to the null. We show that for an initially cylindrically-symmetric fast magnetoacoustic wave perturbation, there is a decrease in wave speed along the separatrices and so the perturbation starts to take on a quasi-diamond shape; with the corners located along the separatrices. This is due to the growth in pressure gradients that reach a maximum along the separatrices, which in turn reduces the acceleration of the fast wave along the separatrices leading to a deformation of the wave morphology.