Fundamentals of Plasma Physics

356 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxation

a
b

a b

flux tube

surface

flux tube

surface

a

b

b

a

axis of helix

core region 0 ≤r
b

outerregion b≤r≤a

Figure 11.9: Top: case whereflux tube radiusais less than radiusbof helix traced out by
flux tube axis. Bottom: Case wherea>b;the core region 0 ≤r<brotates around the
axis of the helix whereas the outer region withb<r<awobbles about the axis of the
helix.

11.4.4Twist and writhe exchange in a kink instability

A kink instability is governed by ideal MHD and so must be helicity-conserving. The

number of turns of the kink is defined by the initial twist conditionk·B=kφBφ+kθBθ=

0 so ifkθ=1/aandkφ=n/Rthen the kink will result in a helix withN=n.Since

helicity is conserved, it is seen that as the amplitude of the kink increases(i.e.,bincreases),

theflux tube twistdψ/dΦwill have to decrease. In particular if the twist is uniform so that

ψ=TΦwhere twistTis independent ofΦ,then as in Section 11.2.2 we may write

2

∫

Φ

dψ

dΦ

dΦ=TΦ^2 (11.70)

and so from conservation of helicity we may conclude that

T+Nb^2 /a^2 =const.forb<a; (11.71)

this is an example of the Calugareanu theorem (Calugareanu 1959).