Fundamentals of Plasma Physics

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

two ends atφ=0andφ=2π.This definition implies thatψ=0on the outer surface of
theflux tube because the sub-ribbon area will be zero in this case.

We define the vector potential

Aazimuthal=

ψ(Φ)

2 π

∇φ (11.46)

and note that this definition is valid only in the range 0 ≤φ< 2 π.The appropriateness of
this definition is established by integrating following the contourCaround the subribbon
to obtain

∮

C

dl·Aazimuthal =

∮

C

dl·

ψ(Φ)

2 π

∇φ

=

ψ(Φ)

2 π

∫

cΦ

dl·∇φ

= ψ(Φ) (11.47)

wheredl·∇φ=dφ. As seen in Fig.11.7, the contourCconsists of four segments, namely a
segment following the outside edge of the sub-ribbon (this segment lies in theouter surface
of theflux tube whereψ=0), the segment labeledCΦon the inside edge of the sub-ribbon,
a segment at theφ=0end, and an oppositely directed segment at theφ=2π−end. Only
theCΦsegment contributes to the integral because (i)dl·∇φ=0on the endsφ=0and
φ=2π−sinceφis constant on both these ends and (ii)ψ=0on the outside edge of the
sub-ribbon.

The magnetic field inside theflux tube can thus be written as

B=Baxis+∇×

ψ(Φ)

2 π

∇φ (11.48)

whereBaxisis the magnetic field component parallel to theflux tube axis. This decom-
position of the magnetic field is valid even for a twisted or knotted axis provided it is used
only for situations where 0 ≤φ< 2 π.