0. Introduction

Good afternoon. I'm William Simpson, a student of Theoretical Physics and Mathematics at the University of St. Andrews. I've been invited today to tell you a little bit about a project I understook with The Solar Group over a short period last Summer - a project motivated by a specific question that is current in Solar Physics at the moment: are coronal null points a necessary requirement for solar flares?

According to certain mathematical models proposed, magnetic null points - which are points in the magnetic field where the field strength vanishes - play an important role in the sudden release of vast stores of energy in the solar atmosphere. If this is true, then we should expect to find them in the vicinity of a solar flare site around the time of the solar flare. My objective was to see whether an analysis of the magnetic field in the region of a particular solar phenomenon might corroborate or falsify such assertions.

I was looking for any stable null points that might have emerged in the magnetic field of active region AR0486 between the hours of 10:00 to 12:58 UT on Oct. 28th 2003 -- it was during this interval that the X17.2 solar flare ripped through the solar atmosphere with an explosive energy comparable to several million terrestrial nuclear bombs!

We shall discuss my approach to this problem presently. For now, I should like to review some of the relevant physics behind this project, and to suggest some reasons for why answers to these kinds of questions really matter.

1. An introduction to solar flares

1.1 What is a solar flare?

Let me begin by telling you something about solar flares.

A solar flare is an explosive phenomenon that can occur in intense magnetic regions of our Sun. It involves the sudden, fast release of vast stores of energy, heating Earth-sized active regions to tens of millions of degrees and producing large-scale perturbations throughout the solar atmosphere. For a short while, the site of a solar flare can be the hottest place on the Sun, producing more X-ray and radio-wave radiation than all the other regions of the Sun collectively.

1.2 Solar flare classes

There are different ways of classifying solar flares, but since they emit the bulk of their energy in X-rays, X-ray brightness is a useful way of characterising their size, the X-class being the largest.

The X17.2 flare is classified in the top 5 of the most powerful flares ever observed since 1976.

1.3 Some interstellar effects of solar flares

For the largest solar flares, the effects of these explosions are far-reaching, moving beyond the surface of our Sun to affect the interplanetary medium. During a solar flare eruption, high energy electrons and protons are flung out into space at posing serious problems for astronauts and satellites by interfering with electronic equipment.

Vast quantities of hot coronal gases are ejected into space too some of this impinging upon the Earths magnetosphere and creating strong electric currents in the ionosphere that can, for instance, cause massive damage to power lines, plunging communities into darkness. In short, solar flares affect the Earth in significant ways. If we can understand these phenomena, we might be better able to predict them, and to protect ourselves from their more harmful effects. We might also learn some interesting Physics along the way.

2. Magnetic reconnection and null points

2.1 Magnetic reconnection

So where does the energy for a solar flare come from, and why is it suddenly released in such a dramatic way?

Almost certainly, solar flares are energised by magnetism. It is theorised that, when the magnetic fields in an active region of the Sun become contorted, there is a build up of magnetic energy in the low corona and this build up can become very large indeed. The mechanism generally believed to be responsible for the release of this energy is magnetic reconnection. This important phenomenon involves a restructuring of a magnetic field caused by a change in the connectivity of its field lines, and is considered the most important process for explaining large-scale, dynamic, fast releases of magnetic energy.

2.2 Null points

The phenomenon of reconnection has commonly been associated with the presence of magnetic null points. For instance, in the so-called magnetic breakout model it is proposed that reconnection at a null point in the corona above a sheared neutral line causes the removal of magnetic flux responsible for holding down the sheared low-lying field, allowing it to erupt explosively outward.

My concern within the narrow scope of this project has not been with the details of such models, but with finding a way of identifying and tracking stable null points in active regions, during periods of intense solar activity.

It is appropriate then, at this point, to go into some slightly more technical detail on the nature of magnetic null points.

By itself, finding a point where the field vanishes tells us little about the local magnetic structure the topology of the field in the immediate vicinity of one null point could be quite different near another. However, if we assume that the magnetic field near a null approaches zero linearly, we can approximate the components of the field in this region by means of a first order Taylor expansion about the neutral point. This is tantamount to calculating the elements of the Jacobian at that point multiplied by the position vector, each of the elements of this matrix being a partial derivative of one of the components of the magnetic field evaluated at the position of the null point.

A good deal of information about a nulls properties is encoded in the Jacobian. For our purposes, it is sufficient to note that the eigenvectors and eigenvalues of this matrix tell us something directly about the skeleton structure of the null point we are dealing with. There are, in fact, two basic components that make up the skeleton of any 3-dimensional null point :

• the fan of a null point is the surface formed by the set of field lines radiating out of or into the null point
• the spine is formed by the two field lines that enter or leave the null point (but not in the plane of the fan)

The spine is directed away from the null if the field lines in the fan are directed towards it we will call this a negative null or the field lines in the fan are directed away from the null, whereas the spine points into it we shall call this a positive null.

Now it can be shown from linear analysis of the magnetic field that the spine of a null point lies along the eigenvector of the Jacobian that relates to the single eigenvalue of opposite sign to the real parts of the remaining eigenvalues. The remaining eigenvectors define the plane of the fan. It can be shown that

1. if the real components of the eigenvalues associated with the fan are greater than zero, we have a positive neutral point, whereas
2. if they are less than zero, we have a negative neutral point.

We shall refer to this as the Null Point Nature Criteria.

Much more has been disclosed in the relevant literature on the structure of three-dimensional magnetic neutral points. But we are prevented here from exploring the topologies of our null points in much further detail by the nature of the data we are dealing with, and which we must now discuss.

3. The data for analysis

3.1 The nature of the data

One of the problems we face in this undertaking is the difficulty of measuring the coronal magnetic field directly. We must therefore resort to using potential field data (something I left to my supervisor, Dr. Regnier, to produce ;-). This is a neat trick, but there are some points to note about this kind of extrapolation:

1. it assumes the coronal magnetic configuration is in a force-free equilibrium at the time of observation (J x B = 0), and
2. it assumes, moreover, that the field is current-free (J = 0). The magnetic energy of the potential field configuration is therefore the lowest magnetic energy a magnetic configuration can reach with the same photospheric field - it does not include the free energy.

This imposes some limitations on our study, one of which is that we are prevented from exploring more sophisticated null point classificatory schema.

3.2 The structure of the data

For the purposes of this project, 175 magnetic configurations were produced, each defining the three components of the field at uniformly spaced, discrete points in a volume region approximately 300 x 200 x 200 Mm, and collectively covering in one minute intervals the period from 10:00 to 12:58 UT on Oct. 28th, 2003. (The flare itself began at 11:01).

4. Finding null points

To find null points we are, by definition, looking for points in a magnetic configuration where the magnetic field strength goes to zero.

The task of discovering the location of possible null points in a given configuration and the problem of determining their properties both require recourse to 3d interpolation theory, where our magnetic field data which defines the field strength only at discrete points is extrapolated into the space between defined points. For finding with some precision the location of zero points in the field, a code written by Andrew Haynes was employed. For computing the Jacobian at each point in which useful information about the null is encoded the technique was to take the nearest points in the configuration where the field was defined, and, by interpolation, to calculate the partial derivatives from small spatial perturbations about that point within the domain of the interpolated field. By calculating and interpreting the eigenvalues and eigenvectors of the Jacobian, tables of classified, possible null points could then be produced for all 175 time-indexed configurations. You can see a graphical representation of the first magnetic configuration, depicting the nature and location of the possible null points discovered in the region.

5. Null point stability

On examining in sequence graphical representations of each configuration, however, it is clear that the majority of the apparent null points located by this technique seem to appear and vanish at random they are likely unphysical, arising most probably from noise in the MDI magnetograms as well as theoretical constraints, such as the boundary conditions imposed on the extrapolation and the assumption of the linearity of the field. The majority of these pseudo-nulls occur towards the periphery of the configuration, where the magnetic field strength is weaker and the noise has a more pronounced effect.

Now our conclusions ought to be based on null points we consider to be physical and stable (if a null point is stable, we will assume it is physical) if indeed any exist in this region during this time interval. The next important step, then, is to begin to formulate a discriminating criteria to sort through this plethora of null points. For a stable null, we will require that

1. it persists through time,
2. the nature of the null remains unchanged, and
3. the skeleton structure of the null is geometrically stable.

We shall refer to this as the Null Stability Criteria.

Now a visual appraisal of the data suggests a small number of possible candidates satisfying criteria 1 and 2. Our concern now will be to establish null identity through time with more rigor. If some null point appears to persist through the interval, we would like to be able to track it, pull together all the information we know about its various instances in the different time frames, and analyse the accumulated data. We will then be able to see whether or not it its skeleton structure is geometrically stable. For this, we will demand that the orientation of the nulls fan plane is not changing erratically with time, and investigate by analysing the behaviour of its eigenvectors throughout the time interval.

6. Tracking null points

Our first step, then, is to see whether any of the possible null points discovered in each time frame can actually be traced from one time frame to the next. In other words, we are seeking to establish the diachronic identity of any real null points in our active region. So consider two consecutive time frames. We will regard a null point found in one time frame to be associated with a null point found in a consecutive time frame if the following criteria hold:

1. first, we will require that the spatial distance between lower-i (whose position is defined in time frame upper-I) and lower-j (whose position is defined in time frame upper-J) should be smaller than that of some null n (whose position is defined in time frame upper-I);
2. secondly, we will require that the spatial distance between lower-i and lower-j should be smaller than the distance between i and some other null m (whose position is defined in time frame upper-J)
3. thirdly, we will require that the trajectory of a null should have the semblance of continuity the time lapse between each of our configurations is sufficiently small for us to expect to be able to track the motion of a null point without anticipating large leaps in the distances between time frames; consequently, we shall place a constraint on the distance between consecutive instances of a null
4. finally, we preclude from association with a null point in one frame any null points in another frame that have a different nature.

These four considerations constitute the diachronic identity criteria, in its incipient form, that I have employed here.

Thus, by making a set of associations over all 175 configurations, we form the diachronic identity signature of a null point.

6b. Null points tracked

Apply the criteria, and this is what you get.

This movie represents our active region over the interval. Every possible null is depicted by a cross, hovering at a certain height, and you will notice that seven nulls all of them labelled are being successfully tracked through the time interval. An eighth null, which appears in the movie to persist, has proven more difficult to track with this algorithm, however. You can see the N6 arrow wandering off Part of the problem here, of course, is that in noisy regions of the field, where many null points appear to be forming and disappearing (most of them are surely unphysical), our null tracker will form false associations, and once our algorithm has been misled into making an association with a pseudo-null that has suddenly appeared close-by, subsequent associations are quite likely to be mistaken as well. I am inclined to think that further development of the identity tracking algorithm will improve its reliability.

7. Stable null points

We are now in a position to impose the full demands of our null stability criteria, which subsumes the diachronic identity criteria, with the added requirement of geometric stability, requiring us to consider the behaviour of the eigenvectors of each null over the time interval.

To put it simply, we intend to discount as unstable (and possibly unphysical) any null whose geometry exhibits erratic behaviour with time. To this end movies were produced of all the successfully tracked null points, depicting the spine and fan plane of each null for each time frame, and the behaviour of the eigenvector plots can be observed directly you can see one such movie here.

It seems clear on viewing the movies that the null points so far identified are well divided between the fairly (geometrically) stable, where the spine and the fan plane remain at more or less the same orientation, and the obviously unstable. In our case, the angle between the normal of the fan plane and the z-axis was calculated for each candidate null, in each time frame. Taking the standard deviation of the set of angles (call it the angle deviation), we can clearly see that the geometric instability of nulls N2,3,4 and N7 are (or are close to being) of a higher order of magnitude than nulls N0, 1 and N5 . We will set them aside.

7. Discussion and conclusions

Let us summarise and conclude here:

First, we enumerated the apparent null points for all 175 magnetic configurations. Second, we obtained a set of nulls that persist through time (N0-7) by imposing the diachronic identity criteria. Third, by imposing the full rigours of the null stability criteria on the diachronic identity signatures of nulls (N0-7), we were able to set aside from consideration nulls exhibiting a large angle deviation (namely N2-4,7). This reduced our set of stable, physical nulls to 3. So there does, in fact, appear to be at least 3 stable coronal nulls in AR0486 during the time of the X17.2 Solar Flare; namely, N0, N1 and N5. N1 and N5 are perhaps not particularly interesting, occurring in parts of the active region where the z component of the magnetic field is low, at some distance from the location of the solar flare.

The most significant finding has been null point N0, which occurs near the location of the solar flare itself, in an area where the magnetic field contours are dense. This is consistent with those theories that have posited one or more coronal null points as a necessary requirement for solar flares; our investigation has not falsified that claim. Of course, one confirmation is a far cry from establishing the truth of such a proposition many more active regions exhibiting similar solar activity would have to be similarly investigated before we could hope to convincingly answer our motivating question. But there is good reason to think that the work that has been done here could be extended to do just that.