3D MHD coronal oscillations about a magnetic null point: Application of WKB theory

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, B=[x, epsilon y, -(epsilon + 1)z]. Under our cold-plasma assumption, we have considered two types of wave propagation: fast magnetoacoustic and Alfven 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 Alfven speed profile. The wave and thus the wave energy accumulate at the null point. We have found that current buildup is exponential and the exponent is dependent upon epsilon. Thus, for the fast wave there is preferential heating at the null point. For the Alfven wave, we find that the wave propagates along the field lines. For an Alfven wave generated along the fan plane, the wave accumulates along the spine. For an Alfven 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.