Fundamentals of Plasma Physics

11.4 Kinking and magnetic helicity 349

Figure 11.6 shows a cross-section of this system. Theflux tube has minor radiusaand

the helical axis of theflux tube traces out a trajectory with minor radiusbabout an axis

which has major radiusR.Representative magnetic surfaces (surfaces of constantΦ), the

ribbon surface, and the angleθare also shown.

axis of flux tube

2 b

2 a

R

flux tube

surface

ribbon

surface

magnetic surface,

constant 

axis of helix

Figure 11.6: Cross-section showingflux tube of minor radiusa, with its axis making a helix
of radiusb. Interiorflux surfaces withΦ=const.and the ribbon surface definingθ=0
are also shown.

ThefluxΦwas a generalization of the toroidalflux of an axisymmetric system. We

now define a corresponding generalization of the poloidalfluxψ(Φ)as the magneticflux

penetrating a sub-ribbon extending inwards from the outer surface of the possibly helical

flux tube to some given interior magnetic surfaceΦ.This subribbon is shown in Fig.11.7

and the definition is such that

ψ =

∫

subribbon

ds·B

=

∫

subribbon

ds·∇×A

=

∮

C

dl·A (11.45)

where the contourCfollows the perimeter of the subribbon and specifically follows the