346 Chapter 11. Magnetic helicity interpreted and Woltjer-Taylor relaxationaxis of helix
helical flux tube ,
volume Vtu belinked external flux ext
Figure 11.4: Subvolume #1 is aflux tube with a possibly helical axis and volumeVtube.
Subvolume #2 is all space external to subvolume #1. All field lines in subvolume #2 which
link subvolume #1 are represented by the verticalflux tube in figure labeledψext.
- The second subvolume in this decomposition is the remaining volume ofVand is
labeled asVext.There might be field lines in this second subvolume which link the
flux tubeVtubeand theflux due to all these linkages is labeledψext.This externalflux
linkage is represented by the verticalflux tube labeledψextin Fig. 11.4.
If the dynamics within the volumeVare governed by ideal MHD, then the helicityKV
inVmust be conserved and, sinceKVis a volume integral, the volume of integration can
be subdivided into theflux tube volume and the volume external to theflux tube,
KV=KVtube+KVext.