Fundamentals of Plasma Physics

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

axial segment of helical

flux tube surface

helical axis

of flux tube

cut ribbon surface

from helical axis

to flux tube surface

axis of helix

linked externalflux 
ext




 0


 2 

Figure 11.5: Ribbon surface extending from helical axis of helicalflux tube to surface of
flux tube with normal toˆφ׈κ

Because any magnetic field line in theflux tube lies in some magnetic surface labeled

byΦ, any field line in theflux tube has a component parallel to the axis and possibly also

a component perpendicular to the axis, but never a component in the∇Φdirection. This

suggests introduction of an azimuthal coordinateθwhich is defined as the angular distance

around the axis in the direction given by∇φ×∇Φ,i.e.,θis defined such that

∇θ

|∇θ|

=

∇φ×∇Φ

|∇φ×∇Φ|

. (11.44)

The local direction of∇θdepends on bothφandθ;furthermore, forθto denote some

definite position,θmust be measured with respect to some unambiguous origin. An unam-

biguousθorigin can be defined by using as a reference the negative of the direction of the

local radius of curvature vectorκof theflux tube axis. Theθ=0surface is then a ribbon

surface, as shown in Fig.11.5, that extends from theflux tube axis to theflux tube outer sur-

face so thatˆφ×κis normal to the ribbon surface. The ribbon is considered to have a cut

atφ=0so that the ribbon ends atφ=0andφ=2πare distinct.