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Old 21st November 2011, 08:56 AM   #5117
W.D.Clinger
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a simple derivation of magnetic reconnection, part 5 of 5

In this final part 5, we use the magnetic fields developed in parts 1 through 4 to understand the two main results of [Dungey 1958] (see references at the bottom of this post).


Review of parts 1 through 4

My web page on magnetic reconnection contains a more graphic summary of parts 1 through 4 of this derivation. Here I'll just highlight the main facts we need for part 5.

Starting from Maxwell's equations, we derived the magnetic field that's generated by four conducting rods at the corners of a square. When all four rods carry the same current, with the current running in the same direction for rods that are on the same diagonal of the square but in the opposite direction for rods on the other diagonal, we get a magnetic field B4 that approximates Dungey's figure 1.

The center of that square is a neutral point (where B4 is zero). Even tiny variations in the magnetic field can result in magnetic reconnection of magnetic field lines at the neutral point.

All magnetic fields, including B4, satisfy Gauss's law for magnetism, which states that the divergence of the magnetic flux density is zero at every point; equivalently, the magnetic flux through every closed surface is zero. That implies there are no magnetic monopoles, sinks, or sources. All of that is easy to prove for B4, and I have proved it several times using several different methods of proof.

I have also proved that, for every value of the z coordinate, the xy plane determined by z contains two magnetic field lines that begin at the neutral point and two other magnetic field lines that end at the neutral point. B4 therefore serves as a simple counterexample to a popular myth that has been repeated by Wikipedia and other unreliable sources. (Michael Mozina and other EU folk have relied upon that myth as the technical basis for their denials of magnetic reconnection. Since the myth is untrue, their denials have no technical basis at all.)

Reducing the current through the rods in one diagonal to zero while maintaining a constant current through the other rods gives us a much more dramatic example of magnetic reconnection. When current flows through only two parallel rods, and flows in the same direction through each rod, we get a "figure 8" field B2 that reproduces Dungey's figure 2.

That figure 8 field B2 also contains a neutral point, and provides yet another counterexample to the Wikipedia-endorsed myth. It's not quite as simple, because the magnetic field lines that begin or end at the neutral point of B2 are curved instead of straight. That makes this counterexample's calculations more complicated than for B4.

The magnetic fields B4 and B2 are vacuum solutions of Maxwell's equations. Once those vacuum solutions are thoroughly understood, we can reproduce the main results of [Dungey 1958] by considering what happens when we introduce some plasma and allow a magnetic field such as B4 to act upon charged particles that are moving in the z direction.


Historical context

Dungey's 1958 paper is only 5 pages long, but it contains several minor errors that confused me on my first readings of it. (I'll point out some of those errors below.) Another problem is that Dungey's paper is a reply to a 1953 paper by T G Cowling, and I have not read Cowling's paper (which does not appear to be available online).

In the first sentence of the introduction to Dungey's 1958 paper, Dungey writes:

Originally Posted by Dungey
The suggestion that a solar flare results from an electrical discharge situated in the neighborhood of a neutral point of the magnetic field was made by Giovanelli [2].
That's puzzling, because I don't see the word "discharge" anywhere within the paper cited [Giovanelli 1947].

Originally Posted by Dungey
The defining feature of a discharge in this context is the existence of a large current density.
Aha. Giovanelli talked a lot about current density.

It must have been Dungey (or Cowling) who put the word "discharge" into Giovanelli's mouth. Evidently "discharge" is Dungey's or Cowling's idiosyncratic synonym for "large current density". In what follows, I will speak of current density instead of discharges, because current density has a mathematically precise meaning and is less misleading than the word "discharge".

In [Giovanelli 1947], Giovanelli was trying to figure out how a large current density could be associated with sunspots and solar flares. Giovanelli argued that electric fields alone can't do the job, so he turned to magnetic fields. Giovanelli then argued that magnetic fields can't produce large current densities either except in the vicinity of a neutral point.

(Giovanelli was still assuming that the large current density would be generated by induction, and was trying to figure out how induction could generate a large current density fast enough to be consistent with observation. It is my impression that modern researchers have pretty much given up on induction as the source of these particular large current densities, because induction processes can't go fast enough.)

T G Cowling apparently objected to Giovanelli's hypothesis that large current densities occur in solar flares near neutral points of the magnetic field. Cowling gave two distinct reasons for his objection:
  1. According to calculations, the large current density would occur in a sheet that's only a few meters thick, and Cowling just didn't believe something so thin could cause the observed phenomena.
  2. Cowling thought Lenz's law would limit the current density.
The first of those objections was just an argument from incredulity. Dungey answered the second of those objections by a qualitative argument using magnetic fields similar to B4 [Dungey 1958].

Before we consider Dungey's arguments in detail, let's take a moment to understand how a small plasma current can interact with B4 to form a thin sheet of increased current density near the neutral point.


Interaction with plasma current: first-order approximation

The second paragraph of Dungey's second section begins with these observations:

Originally Posted by Dungey
Consider a neutral point N, where the lines of force in one plane have the form shown in Fig. 1. The limiting lines of force through N form an X and would be perpendicular, if there were no current flowing in the z-direction (normal to the paper).
With no current, the situation is exactly what we saw in a vacuum with B4, and exactly what we saw when we zoomed in on the neutral point of B4.

Originally Posted by Dungey
For the field of Fig. 1 there is a roughly uniform current in the z-direction, which contributes a field directed clockwise.
Something's wrong here. As we saw in part 2 of this derivation, a current in the z-direction contributes a field directed counterclockwise, not clockwise.

Dungey appears to be thinking of electrons drifting in the z-direction. Because the electrons are negatively charged, their movement in the z-direction counts as a current in the opposite direction. (We have Benjamin Franklin to blame for that.)

In what follows, I will assume the plasma current is in the negative z-direction, and consists entirely of electrons moving in the positive z-direction. I will also assume the plasma is electrically neutral, as would result from the presence of stationary protons or other positively charged particles that balance the charge of the electrons. So far as I can tell, those assumptions are consistent with everything in [Dungey 1958].

With those assumptions, Dungey's figure 1 makes sense:



The magnetic force always runs perpendicular to both the current and the magnetic field. If the current were running in the positive z direction, then Dungey's magnetic force lines (labelled "f") would be pointed in the wrong direction, but Dungey's magnetic force lines are correct for a current flowing in the negative z direction.

Along the x axis, the magnetic force compresses the current toward the neutral point N. Along the y axis, the magnetic force pulls the current away from the neutral point. Those forces result in a narrow sheet of current near the neutral point. The sheet is narrow along the x axis, and long along the y axis.

(For that result, Dungey cites one of his own papers from 1953, but it's pretty obvious from Dungey's figure 1 and the argument given above.)

T G Cowling agreed that pre-existing magnetic fields similar to B4 would tend to flatten any current near the neutral point into a sheet, but Cowling thought that flattening process would be limited by Lenz's law:

Quote:
The effect of induced currents is always to oppose the changes to which they are due.

The main result of Dungey's 1958 paper is that Lenz's law does not limit the flattening of currents near the neutral points of magnetic fields such as B4 and B2:

Originally Posted by Dungey
It is found, however, that Lenz's law is reversed at a neutral point and this will now be explained.
To understand that reversal, we must look at what happens to the magnetic field when a fairly uniform current in the (minus) z direction is compressed into a small region of high current density near the neutral point.


Interaction with plasma current: second-order approximation

A small and uniform current in the z or minus-z direction has little influence near the neutral point of B4, so long as that current is centered near the neutral point and is uniform over a much larger region than the region we're examining. (Why? Because the magnetic field generated by that current at some point p is dominated by the contributions of the nearby current. When the nearby current is nearly uniform, its contributions to the magnetic field at p nearly cancel.)

As noted above, however, magnetic fields such as B4 tend to concentrate that current into a sheet near the neutral point. As that happens, the magnetic field generated by the increased current density near the neutral point begins to distort the original magnetic field, resulting in magnetic reconnection that looks something like this:



That animation was created by adding a fifth conducting rod to the four rods that generate the field B4, and placing that fifth rod at the origin and parallel to the other four rods. When no current is running through that fifth rod, we get the initial magnetic field B4:

http://www.cesura17.net/~will/Epheme...re1zoomout.png

Zooming in on the origin, we see this:



Increasing the current through the fifth rod to 2% of the current through the other four rods begins to distort the magnetic field:

http://www.cesura17.net/~will/Epheme...ness/MR/b5.png



Comparing that graph to the one above it, we see that the field line in the NE quadrant that originally ended at the origin has merged with the field line in the SE quadrant that originally began at the origin, and that merged field line is moving away from the origin toward the east. Similarly, the field line in the SW quadrant that originally ended at the origin has merged with the field line in the NW quadrant that originally began at the origin, and that merged field line is moving away from the origin toward the west.

Meanwhile, the eight field lines whose arrows were originally marked near the y axis are becoming weaker as they advance toward the origin. (Why weaker? The current at the origin is running in the negative z direction, so its contribution to the magnetic field runs clockwise, which cancels some of the original intensity of the field lines that were running in more of a counterclockwise direction.) The original neutral point has now split into two black regions of very weak (but not quite zero) magnetic field intensity on the y axis above and below the origin.

Increasing the current through the fifth rod to 8% of the current through the other four rods results in still more magnetic reconnection and an even more obviously asymmetric magnetic field:

http://www.cesura17.net/~will/Epheme...ess/MR/b5A.png



The vertical stretching and horizontal compression of the X shape is beginning to become evident in that graph, although it has not yet reached the level of distortion shown in Dungey's figure 1.

Getting back to Lenz's law, we can now use our five-rod approximation to answer a non-trivial question about plasma physics: Has the increasing current density at the origin acted to diminish the magnetic force that was causing the increase in current density?

No! The magnetic field lines that were compressing the plasma along the x axis toward the origin have become stronger, not weaker. Meanwhile, the magnetic field lines that were pulling the plasma away from the origin along the y axis have become weaker. The net result of increasing current density at the origin is to increase the forces that will increase that current density still further.

Originally Posted by Dungey
It is only necessary to consider the signs of the variables to show that they all increase in magnitude indefinitely, and hence Lenz's law is reversed.
That is the most important result of Dungey's 1958 paper.


Dungey's speculation about magnetic reconnection

In his section 4, Dungey performs a few simple calculations to show that an extremely thin sheet of very high current density could produce the observed emissions from solar flares. Those calculations dismiss Cowling's argument from incredulity.

That's the second main result of Dungey's paper.

Dungey doesn't stop there, however.

Having gone through the exercises above, Dungey obviously knew about the magnetic reconnections that result from increasing current density near the neutral point of an X-shaped magnetic field, and was impressed by how rapidly those reconnections occur and by how violently they snap the magnetic forces around.

Dungey goes on to speculate about the possible importance of those magnetic reconnections. (Recall that "discharge" is just Dungey's word for the rapid increase in current density near the neutral point.)

Originally Posted by Dungey
The effect of the discharge is to 'reconnect' the lines of force at the neutral point, and this happens quickly. The 'reconnection' upsets the mechanical equilibrium in the neighborhood in a way that can be visualized, if the lines of force are seen as strings. Then the mechanical disturbance will spread from the neutral point and may have energy comparable to the energy of the spot field in the solar atmosphere. This disturbance, characterized by a sudden onset, may account for several features: surge prominences and Doppler shifts, which are probably different aspects of the same phenomenon; the emission of a mechanical disturbance responsible for magnetic storms; the activation of prominences; the triggering of other neutral points, in whose neighborhood the field is weak, resulting in multiple flares.

I am not a physicist. To me, that looks like hand-waving. It was informed hand-waving, however. I am not qualified to evaluate Dungey's speculation, or to evaluate the role of magnetic reconnection in solar flares, but it isn't hard to understand that Dungey is speculating about the possible effects of magnetic reconnection.


Conclusion

Starting with the basics of electromagnetism as taught by first-year textbooks such as Edward M Purcell's Electricy and Magnetism, and using little math beyond basic vector calculus, we have proved that magnetic reconnection is an undeniable consequence of Maxwell's equations.

Along the way, we have refuted several common myths about magnetic fields.

We concluded this five-part series by using a variation of the experiment I've been recommending to Michael Mozina to replicate the main result of [Dungey 1958].

For easy reference, I have created a web page that links to all parts of this series (and links to some related material as well).


References

[Dungey 1958] J W Dungey. The neutral point discharge theory of solar flares. A reply to Cowling's criticism. Proceedings of Electromagnetic Phenomena in Cosmical Physics, edited by Bo Lehnert. International Astronomical Union number 6, Cambridge University Press, page 135.

[Giovanelli 1947] R G Giovanelli. Magnetic and electric phenomena in the sun's atmosphere associated with sunspots. Monthly Notices of the Royal Astronomical Society, Volume 107, 1947, pages 338-355.

Last edited by W.D.Clinger; 21st November 2011 at 09:26 AM. Reason: removed bogus line break, corrected a quotation
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