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

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.