5. MHD wave propagation and magnetic topology

James McLaughlin is a lecturer in Applied Mathematics at Northumbria University.

Introduction

Magnetohydrodynamic (MHD) wave behaviour in the neighbourhood of coronal null points is a fundamental plasma process. As well as being fascinating in its own right, results so far have also lead to critical insights into reconnection, mode-coupling, quasi-periodic pulsations and phase-mixing. This research topic exists at the overlap of two important areas of solar physics: MHD wave and magnetic null-point theories. This nugget will give a brief overview of this research topic (see also [1]), explain why this area of study is interesting and important, and outline the big questions that remain unanswered.

MHD waves and magnetic topology

MHD wave motions are ubiquitous throughout the solar corona [2,3,4,5] and several different types of wave motion have now been observed. In this nugget, we consider three MHD wave types: the Alfvén wave and the fast and slow magnetoacoustic waves. Table 1 lists the main properties of each wave type depending upon their plasma-β environment (plasma-β is the ratio of the gas pressure to the magnetic pressure):

	plasma-β >> 1 (high-β)	plasma-β << 1 (low-β)
Alfvén Wave	Transverse wave propagating at speed v_A
Fast Magnetoacoustic Wave	Behaves like sound wave (speed c_s)	Propagates roughly isotropically and across magnetic fieldlines (speed v_A)
Slow Magnetoacoustic Wave	Guided along B (speed v_A)	Guided along B . Longitudinal wave propagating at speed c_s.

Table 1 : Properties of MHD waves depending on plasma−β.

It is clear that the coronal magnetic field plays a fundamental role in the propagation and properties of these MHD waves, and to begin to understand this inhomogeneous, magnetised environment, it is useful to look at the topology of the magnetic field itself. A general magnetic configuration contains separatrices: which split the magnetic plane into separate regions of magnetic flux connectivity (i.e. topologically distinct regions) and null points: locations where the magnetic field is zero (B = 0) and hence the Alfvén speed is zero.

Figure 1: (a) 2D X-type potential null point with separatrices (red lines) intersecting at an angle of π/2. Null is located at the origin. (b) 3D proper radial null point, described by B=(x,y,−2z). The z−axis indicates the spine and the xy−plane at z=0 denotes the fan-plane. The red fieldlines have been tracked from the z=1 plane, the blue from z=−1.

Since magnetic pressure generally dominates thermal pressure in the solar corona, it is usual to assume plasma-β << 1. However, near null points the magnetic field vanishes and so the plasma-β can become very large. In fact, the plasma-β is infinite at the null point. Around the null point there will be a surface where β=1 and, more importantly, a layer where the Alfvén speed equals the sound speed (v_A = c_S ) which is located at a similar position. It is at this layer that mode-conversion can and will occur!

But how common are null points in the corona?

Null points are an inevitable consequence of the distributed, isolated magnetic flux sources at the photospheric surface. Using photospheric magnetograms to provide the field distribution on the lower boundary, field extrapolations suggest that there are always likely to be null points in the corona, and their number will depend upon the magnetic complexity of the photospheric flux distribution. The statistics of coronal null points have been investigated using two methodologies: direct measurement from potential field extrapolations [5,6] and as an estimate from the Fourier spectrum of magnetograms [7]. We can estimate the total number of coronal null points by multiplying these authors’ results by the surface area of the Sun (i.e. to provide a crude estimate, where we assume the Sun is free of active regions and coronal holes). The answer is between 10,000 and 40,000 coronal nulls [5,6,7].

Why is this area of study interesting and important?

The motivation for this topic can be summarised as follows:

MHD wave propagation through inhomogeneous media is a fundamental plasma process, and the study of MHD wave behaviour in the neighbourhood of magnetic null points directly contributes to this.
We now know that MHD wave perturbations are omnipresent in the corona. We also know that null points are an inevitable consequence of the distributed magnetic flux sources at the photospheric surface. So at some point, MHD waves will propagate into the neighbourhood of coronal null points – for example blast waves from a flare in a complex topology.
The topic has also provided critical insights into other areas of plasma physics, including: mode-coupling and mode-conversion, reconnection, quasi-periodic pulsations and phase-mixing [1].

Figure 2 : (a) Colour contour of v_perpendicular at t=2 for a linear fast wave driven from the upper boundary. Black lines are the WKB solution, where the three lines represent the leading, middle and trailing edges of the wave pulse. Blue cross denotes the null point (here located at origin). The fast wave is guided towards the null and refracts around it. (b) Ray paths of the WKB solution for an Alfvén wave driven at the upper boundary, after a time t=2, for starting points of x = 0, 0.025, ..., 0.5. The linear Alfvén wave propagates along the magnetic fieldlines, and accumulates along the separatrices.

Unanswered questions

Several big questions still remain unanswered in this research area:

The nature of the coupling of the modes in 3D needs to be addressed, and the importance of coupling due to the magnetic geometry verses nonlinear coupling should also be investigated.
The theory of nonlinear fast waves driving oscillatory reconnection should be extended to study more general disturbances and to investigate how robust initial findings are [8].
The key results for the both the linear fast and Alfvén waves make clear predictions as to where preferential heating should occur. It would be interesting to see the theoretical models further developed with forward modelling to provide tell-tale observational signatures, and for these synthetic results to be compared with observational data.

Looking to the future…

There is as yet no clear observational evidence for MHD wave behaviour in the neighbourhood of coronal null points. The successful detection of MHD oscillations around coronal null points will require high-spatial/temporal resolution imaging data and potential/non-potential extrapolations from co-temporal magnetograms. Two of the instruments onboard the recently launched Solar Dynamics Observatory (SDO) may satisfy these requirements: the Atmospheric Imaging Assembly (which provides high-quality imaging data) and the Helioseismic and Magnetic Imager (which provides vector magnetograms).

Thus, the first detection of MHD waves in the neighbourhood of coronal null points may be reported in the very near future.

References

[1] McLaughlin, J.A., Hood, A.W. & De Moortel, I., Space Science Reviews, accepted, DOI: 10.1007/s11214-010-9654-y
[2] Roberts, B. 2004, SOHO 13: Waves, Oscillations and Small-Scale Transient Events in the Solar Atmosphere: a Joint View from SOHO and TRACE, ESA SP-547, 1
[3] Nakariakov, V.M. & Verwichte, E. 2005, Living Reviews in Solar Physics, 2, http://www.livingreviews.org/lrsp-2005-3
[4] De Moortel, I. 2005, Phil. Trans. Roy. Soc. A, 363, 2743
[5] Tomczyk, S., McIntosh, S. W., Keil, S. L., Judge, P. G., Schad, T., Seeley, D. H. & Edmondson, J. 2007, Science, 317, 1192
[6] Régnier, S., Parnell, C. E. & Haynes, A. L. 2008, A&A, 484, L47
[7] Longcope, D. W. & Parnell, C. E. 2009, Solar Phys., 254, 51
[8] McLaughlin, J.A., De Moortel, I., Hood, A.W. & Brady, C.S. 2009, A&A, 493, 227