Fundamentals of Plasma Physics

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

It should be noted that the ability forCto be continuously deformed intoC′imposes the

requirement that a surfaceSexists betweenCandC′;this is only true ifCdoes not link

C′.

We now consider some limiting cases forKwritheand to aid in visualization it should

be recalled that theflux tube axis is assumed to have a helical trajectory with helix major

radiusRand helix minor radiusbas shown in Fig.11.6.

Limiting case whereflux tube axis not helical

This situation corresponds to havingb= 0 in which case theflux tube axis lies in

a plane. The contourCaxisin Eq.(11.60) can thus be slipped from its original position

through theBaxisfield lines to the surface of theflux tube without crossing anyBaxis

field lines. This is topologically possible because by definition, allBaxisfield lines in the

flux tube are parallel to theflux tube axis. Thus, here

Kwrithe=Φ

∫

C′

Aaxis·dlaxis (11.62)

whereC′lies on theflux tube surface. The line integralC′encircles the externalflux linked

by theflux tube and so

Kwrithe=Φψext, when theflux tube axis is not helical (11.63)

helical axis

offlux tube

slit cut in flux tube

parallel to flux tube

axis, faces helicalaxis

of flux tube

axis ofhelix

Figure 11.8: Helical axis offlux tube may be slipped throughBaxisfield lines offlux tubes
to coincide of axis of helix (note that open slit influx tube always faces axis of helix).