11.3 Woltjer-Taylor relaxation 34111.2.3Conservation of magnetic helicity during magnetic reconnection
Ideal MHD constrains magneticflux to be frozen into the frame of the plasma. This
means that the topology (connectedness) of magnetic field lines in a perfectlyconducting
plasma cannot change. As will be shown in Chapter 12, introduction of a small amount
of resistivity allows the frozen-in condition to be violated at locations where the velocity
must vanish due to symmetry. At these locations, the approximation of the resistive MHD
Ohm’s lawE+U×B=ηJto the ideal formE+U×B= 0necessarily fails. In-
stead, Ohm’s law has the formE=ηJso that magnetic field lines can diffuse across the
plasma and reconnect with each other. Even though the reconnection is localized to a small
region in the vicinity of whereUvanishes, it nevertheless changes the overall magnetic
field topology. As will be shown below, reconnection destroys individual linkages between
flux tubes, but creates a replacement linkage for every destroyed linkage so that the total
system helicity is conserved. The result is that the ideal MHD constraint of having per-
fectflux conservation everywhere is replaced by the somewhat weaker constraint that total
magnetic helicity is conserved. Reconnection necessarily involvesenergy dissipation since
reconnection requires finite resistivity. Thus reconnection dissipates magnetic field energy
while conserving magnetic helicity.
Conservation of helicity during reconnection is demonstrated in Fig. 11.3. Two linked
untwisted ribbons of magneticflux are shown in 11.3(a). If each ribbon is imagined to be
a magnetic field line bundle having nominalfluxΦ,then according to Eq.(11.8) this initial
configuration has helicityK=2Φ^2 .The ribbons are then cut at their line of overlap as
in Fig.11.3(b) and reconnected as in Fig.11.3(c) to form one long ribbon. This long ribbon
in Fig.11.3(c) may then be continuously deformed until in the shape shown in Fig.11.3(f),
a long ribbon havingtwotwists. Thus, the twist parameter for Fig.11.3(f) isT =2and
so according to Eq.(11.23) this two-twist ribbon has a helicityK= TΦ^2 =2Φ^2 .Hence,
magnetic helicity is conserved by the reconnection. Since the magnetic equivalent of cut-
ting ribbons with scissors involves the dissipative process of resistive diffusion of magnetic
field across the plasma, it can be concluded that magnetic reconnection conserves helicity,
but dissipates magnetic energy.
11.3 Woltjer-Taylor relaxation
Woltjer (1958) provided a mathematical proof that the lowest energy state ofa magnetic
system with a fixed amount of magnetic helicity was a certain kind of force-free state, but
did not provide any detailed explanation on how the system would attain this state. Taylor
(1974) argued that because reconnection conserves helicity while dissipating magnetic en-
ergy, reconnection events will provide the mechanism by which an isolatedplasma relaxes
towards a state having the lowest magnetic energy consistent with conservation of helicity.
The discussion of Eq.(10.133) showed that the isolation requirement corresponds to having
no field lines penetrate the surface bounding the plasma and also arrangingfor this surface
to be an electrostatic equipotential. It will therefore be assumed that the plasma is bounded
by a perfectly conducting wall and that no magnetic field lines penetrate this wall. The
tangential electric field and hence the tangential component of the vector potential must
therefore vanish at the wall.