3D MHD Coronal Oscillations About a Magnetic Null Point: Application of WKB Theory

TL;DR

This paper demonstrates how the WKB approximation can be applied to solve 3D MHD equations near a magnetic null point, revealing wave refraction, accumulation, and heating effects dependent on magnetic topology.

Contribution

It introduces a novel application of WKB theory to 3D MHD null points, analyzing wave behavior and current buildup for fast and Alfvén waves in a potential magnetic field.

Findings

Fast waves refocus at the null point, leading to exponential current buildup.

Alfvén waves propagate along field lines, accumulating at specific structures depending on parameters.

Wave energy accumulation results in preferential heating at magnetic nulls.

Abstract

This paper is a demonstration of how the WKB approximation can be used to help solve the linearised 3D MHD equations. Using Charpit's Method and a Runge-Kutta numerical scheme, we have demonstrated this technique for a potential 3D magnetic null point, ${\bf{B}}=(x,\epsilon y -(\epsilon +1)z)$. Under our cold plasma assumption, we have considered two types of wave propagation: fast magnetoacoustic and Alfv\'en waves. We find that the fast magnetoacoustic wave experiences refraction towards the magnetic null point, and that the effect of this refraction depends upon the Alfv\'en speed profile. The wave, and thus the wave energy, accumulates at the null point. We have found that current build up is exponential and the exponent is dependent upon $\epsilon$. Thus, for the fast wave there is preferential heating at the null point. For the Alfv\'en wave, we find that the wave propagates along the fieldlines. For an Alfv\'en wave generated along the fan-plane, the wave accumulates along the spine. For an Alfv\'en wave generated across the spine, the value of $\epsilon$ determines where the wave accumulation will occur: fan-plane ($\epsilon=1$), along the $x-$axis ($0<\epsilon <1$) or along the $y-$axis ($\epsilon>1$). We have shown analytically that currents build up exponentially, leading to preferential heating in these areas. The work described here highlights the importance of understanding the magnetic topology of the coronal magnetic field for the location of wave heating.