338 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxationTheflux tubes did not actually have to be thin: a fatflux tube #2 could be decomposed
into a number of adjacent thinflux tubes in which caseK 1 would beΦ 1 times the sum of
theflux in all of the thin #2flux tubes. The helicity is thus just the sum offlux tube linkages
since if the twoflux tubes each had unitflux, there would be one linkage offlux tube #1 with
flux tube #2 and one linkage offlux tube #2 withflux tube #1. As a generalization, ifflux
tube #2 wrapped aroundflux tube #1 twice, then the contributions would beK 1 =2Φ 1 Φ 2
andK 2 =2Φ 1 Φ 2 in which case the helicity would beK=4Φ 1 Φ 2. This would correspond
to two linkages offlux tube #2 onflux tube #1 and two linkages offlux tube #1 onflux tube
#2.flux tube # 1
Bof flux tube # 2
Bflux tube # 2
Figure 11.2: Flux tube #2 deformed so that it tightly encircles flux tube #1 and its
cross-section is squeezed and stretched so as to uniformly coverflux tube #1 like a coat
of paint (shaded area).
11.2.2Twist helicity
Now suppose that the major radius offlux tube #2 is first shrunk untilflux tube #2 tightly
encirclesflux tube #1 and then as sketched in Fig.11.2, the cross-section offlux tube #2 is
both squeezed and stretched until its field lines are uniformly distributedover the length of
flux tube #1;the field lines offlux tube #2 are shown shaded in Fig.11.2. The result is that
the volume offlux tube #2 is like a thin coat of paint applied toflux tube #1 (shaded region
in figure). LetΦdenote theflux influx tube #1 anddψdenote theflux influx tube #2 so,
using Eq.(11.8), the helicity in this configuration is seen to be
dK=2Φdψ. (11.9)