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