11.4 Kinking and magnetic helicity 355Case where axis is helical andb>a
Because the helix minor radiusbexceeds theflux tube minor radiusa, the entireflux
tube revolves around the axis of the helix as shown in Fig.11.8. Becausethe helical axis
closes upon itself, it must have an integral number of periods and we letNbe the number
of helix periods. However, becauseBaxisis everywhere parallel to theflux tube axis, the
flux tube axis may again be slipped through theBaxisfield lines and in this case moved
towards the helix axis until it coincides with the helix axisChelix. The writhe helicity can
thus be expressed as
Kwrithe=Φ∫
ChelixAaxis·dlaxis. (11.64)However, theflux tube links the helix axisNtimes and so
∫ChelixAaxis·dlaxis=NΦ+ψext. (11.65)Thus,
Kwrithe=NΦ^2 +Φψextifb>a. (11.66)Case whereflux tube axis is helical buta>b
In this situation theflux tube may be subdivided into an inner core with minor radius
r<b and associatedfluxΦb^2 /a^2 and an outer annular region withb<r<awith
the remainder ofΦ. We again slip theflux tube axis through the parallelBaxisfield lines
to be coincident with the helix axis. However, now only the inner core of theflux tube
(r<baxis) rotates around the helix axis. The outer annular region of theflux tube merely
wobbles around the helix axis and does not link the helix axis. Thus, the linkedflux is
∫ChelixAaxis·dlaxis=NΦb^2 /a^2 +ψext (11.67)and so the writhe helicity is
Kwrithe=NΦ^2 b^2 /a^2 +Φψextifa>b. (11.68)Hence, the helicity of theflux tube is
KVtube = Ktwist+Kwrithe= 2∫
Φ
dψ
dΦdΦ+NΦ^2 b^2 /a^2 +Φψextifa>b. (11.69)The distinction between thea<banda>bcases is sketched in Fig.11.9.