Fundamentals of Plasma Physics

11.4 Kinking and magnetic helicity 345

11.4 Kinking and magnetic helicity

Magnetic helicity can be manifested in various forms and plasma dynamics cause this form
to be altered. We now discuss an important example of helicity-conserving, morphology-
altering dynamics, namely the situation where kink instability causes thetwist of a straight-
axisflux tube to be transformed into the writhe of the axis of an untwistedflux tube (Berger
and Field 1984, Moffatt and Ricca 1992).
Consider a volumeVbounded by a surfaceSwhere no magnetic field lines penetrate
the surfaceS so thatB·ds=0over all ofS.It is assumed here thatVextends to infinity,
but the analysis would equally apply for a finite-extent volumeVso long as no magnetic
field lines penetrate its bounding surfaceS. The helicity contained inVis

KV=

∫

V

A·Bd^3 r. (11.38)

Equation (10.129) showed thatKVis gauge-independent because no field lines penetrate
S.
The volumeVis now decomposed into two subvolumes which haveB·ds=0at all
locations on their mutual interface. Thus, any specific field line inVis entirely in one or
the other of the two subvolumes.

1. The first subvolume, calledVtube,is a closedflux tube of minor radiusawith a pos-
   sibly helical axis as shown in Fig. 11.4. The length of the axis of thisflux tube is
   denotedlaxisand the variableξdenotes the distance along this axis from some fixed
   reference pointx 0 on the axis. Incrementingξfrom 0 tolaxisthus corresponds to
   going once around theflux tube axis. A pseudo-angular coordinateφis defined as

φ=2π

ξ

laxis

(11.39)

so that going once around the axis corresponds to incrementingφfrom 0 to 2 πandˆφ

defines the local direction of the axis. Because the axis of theflux tube has the complex

curvature of a helix, everywhere along the axis there exists a radius of curvature vector

κ=φˆ·∇ˆφwhich is at right angles toˆφ. The local radius of curvature of the axis

isrcurve =

∣

∣

∣φˆ·∇ˆφ

∣

∣

∣

− 1

(see discussion of Eq.(3.85)). From the point of view of

an observer inside theflux tube, theflux tube appears as a long curved tunnel that

eventually closes upon itself. Theflux tube minor radiusamust always be smaller

thanrcurvewhich corresponds locally to the condition that by definition the major

radius of a torus must exceed the minor radius. Theflux tube interior volume is now

imagined to be filled with fiduciary lines all of which are parallel to theflux tube axis;

the lengths of these fiduciary lines will vary according to their location relative to the

axis. We defineξ′as the distance along a fiduciary line of lengthl′from a plane

perpendicular to theflux tube axis at the reference pointx 0. The meaning ofφcan

then be extended to indicate the distance along any fiduciary line using the relation

ξ′=φl′/ 2 π.Incrementingφfrom 0 to 2 πthus corresponds to going once around any

or all of these fiduciary lines and soφprovides an unambiguous measure of fractional

distance along theflux tube for any point within theflux tube even though theflux

tube may be curved, twisted, or helical.