Can Magnetic Field Lines Break and Reconnect?
Kirk T. McDonald
Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544
(December 29, 2015; updated June 18, 2017)
1
Problem
Following discussion by Giovanelli [1] and Hoyle [2] of solar flares near neutral points in
the solar magnetic field, Dungey (1953) [3] argued that in a region where time-dependent
magnetic fields “collide,” field lines near the “neutral point” can be said to “break” and
“reconnect.”
Although the magnetic field energy is negligible very close to a neutral point, Dungey
argued that is the separatrices do not intersect at a right angle, as illustrated in the above
left figure, the field energy density is greater in the region between where the angle between
the separatrices is obtuse, rather than acute, such that “breaking and reconnection” as in
the right figures above is associated with the release of magnetic field energy.
This theme was pursued by Sweet [4, 5], and then by Parker [6, 7], who discussed the
figure below in [6].
Discuss whether “breaking” and “reconnecting” actually occurs in Parker’s example,
taking this to be a pair of collinear, “point” magnetic dipoles, m = m ẑ located at (x, y, z) =
(0, 0, ±a).
1
2
2.1
Solution
Collinear Point Dipoles
The magnetic field of the two “point” dipoles can be written (in Gaussian units) as
3((r + a) · z)(r + a)
B(r)
3((r − a) · z)(r − a)
ẑ
ẑ
−
−
=
5
3 +
5
3 ,
m
|r − a|
|r − a|
|r + a|
|r + a|
(1)
where a = (0, 0, a). At a point r = (x, 0, 0) on the midplane between the two dipoles, the
magnetic field is
3a(x x̂ − a ẑ)
B(x, 0, 0)
ẑ
3a(x x̂ + a ẑ)
ẑ
= − 2
− 2
+
− 2
2
5/2
2
3/2
2
2
5/2
m
(x + a )
(x + a )
(x + a )
(x + a2)3/2
4a2 − 2x2
=
ẑ.
(2)
(x2 + a2)5/2
√
The magnetic field is zero on the ring x2 + y 2 = 2a2 of radius 2a in the midplane, z = 0.
All magnetic field lines emanating
from the dipole at z = −a within angle 0 < θ < α to
√
−1
2 ≈ 55◦ , end up on the dipole at z = a. Likewise all field
the z-axis, where α = tan
lines emanating from the dipole at z = −a within angle β < θ < 180◦ to the z-axis, where
β has a value greater than 90◦ end up on the dipole at z = a. Only field lines emanating
from the dipole at z = −a within angle α < θ < β return to this dipole. Angles α and β do
not depend on the separation 2a of the two magnetic dipoles, and the pattern of field lines
does not depend on this distance, in contrast to the two left figures above from [6]. Only
the rightmost figure is correct qualitatively.
No field√lines in this example “break” or “reconnect” as the distance 2a changes, although
the radius 2a of the “neutral point/ring” does change.
The figure below was generated by the Wolfram CDF applet at
http://demonstrations.wolfram.com/ElectricFieldLinesDueToACollectionOfPointCharges/
for four electric charges arranged as two dipoles, in which particlar example α ≈ 57◦ and β ≈ 180 − 23◦ .
One can drag the charges around in applet, and the field lines will be recalculated in real time, permitting
one to verify that no lines are broken or reconnected as the geometry of the four charges of the two coplanar
dipoles is varied.
2
2.2
When Can “Breaking and Reconnection” Occur?
In the solar model of Sweet and Parker, the magnetic field above the surface of the Sun was
described by a pair of equal and opposite (fictitious) magnetic poles that were on the surface,
supported by some kind of half-circular solenoids current below the surface. A coplanar pair
of such magnetic dipoles, with all poles on the surface of the Sun, actually lead to “breaking
and reconnection” of field lines above the surface, although if the four poles are not collinear,
there can be “breaking and reconnection” at neutral points the lie on the surface.
This configuration was discussed in Sweet’s second paper [5]. Here, the two dipoles have
(fictitious) magnetic charges that are well separated, and the second dipole is aligned along
the midplane of the first, as sketched below.
As the positive charge of the “horizontal” dipole is moved from left to right, with the
other three charged kept fixed, the field lines in the plane of the dipoles evolve as illustrated
below.
In the upper two figures, no field lines go “downwards” from the upper, positive charge
3
to the lower, negative charge, while in the lower two figures there are field lines that do so.
In the portion of the upper-left figure reproduced below, the four field line segments a, b,
c and d are the separatrices between lines the veer off in opposite directions close to the blue
+ symbol, at which location the magnetic field is zero. It is ambiguous whether segment a
connects to segment b or to segment d, etc.
The argument of Sweet is that as the charge labeled 3 moves to the left, with the other
charges kept fixed, the field line that initially combined segments a and d “breaks” and
“reconnects” to become a combination of segments a and b, while the field line that initially
combined segments b and c “breaks” and “reconnects” to become a combination of segments
c and d.
This “breaking” and “reconnecting” occurs at the common point of line segments z, b, c
and c, where the field strength is zero. As such, there would be no field-energy cost to the
process of “breaking/reconnecting,” so this process is consistent with conservation of (field)
energy.1
2.3
Comment
While the notion of “breaking” and “reconnection” of field lines is popular in the astrophysical literature, it is often overdramatized. A possibly better view is that these terms serve
as jargon for interesting phenomena which deserve more complete description.2
References
[1] R.G. Giovanelli, A Theory of Chromospheric Flares, Nature 158, 81 (1946),
http://physics.princeton.edu/~mcdonald/examples/astro/giovanelli_nature_158_81_46.pdf
[2] F. Hoyle, Some Recent Researches in Solar Flares (Cambridge U. Press, 1949), p. 103,
http://physics.princeton.edu/~mcdonald/examples/astro/hoyle_flares_49.pdf
[3] J.W. Dungey, Conditions for the Occurrence of Electrical Discharges in Astrophysical
Systems, Phil. Mag. 44, 725 (1953),
http://physics.princeton.edu/~mcdonald/examples/EM/dungey_pm_44_725_53.pdf
1
For another discussion by the author of knotted field lines, see [8].
For a discussion of “breaking and reconnection” of magnetic fields lines when a magnet is tied in a knot,
see [8].
2
4
[4] P.A. Sweet, The Neutral Point Theory of Solar Flares, p. 123 of IAU Symposium 6,
Electromagnetic Phenomena in Cosmical Physics, ed. B. Lehnert (Kluwer, 1958),
http://physics.princeton.edu/~mcdonald/examples/EM/sweet_piua_6_123_56.pdf
[5] P.A. Sweet, The Production of High Energy Particles In Solar Flares, Nuovo Cim. Suppl.
8, 188 (1958), http://physics.princeton.edu/~mcdonald/examples/EM/sweet_ncs_8_188_58.pdf
[6] E.N. Parker, Sweet’s Mechanism for Merging Magnetic Fields in Conducting Fluids, J.
Geophys. Res. 62, 509 (1957),
http://physics.princeton.edu/~mcdonald/examples/EM/parker_jgr_62_509_57.pdf
[7] E.N. Parker, The Solar Flare Phenomenon and the Theory of Reconnection and Annihilation of Magnetic Fields, Ap. J. Suppl. 8, 177 (1963),
http://physics.princeton.edu/~mcdonald/examples/EM/parker_apjs_8_177_63.pdf
[8] K.T. McDonald, Can the Field Lines of a Permanent Magnet Be Tied in Knots? (June
12, 2017), http://physics.princeton.edu/~mcdonald/examples/knot.pdf
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