Fundamentals of Plasma Physics

11.4 Kinking and magnetic helicity 353

The three direction gradients∇θ,∇Φ,and∇φform an orthonormal coordinate system
and an element of volume in this system is given by

d^3 r=dlΦdlθdlφ=

dΦdθdφ

∇θ·∇Φ×∇φ

(11.57)

sincedθ=dlθ|∇θ|,dφ=dlφ|∇φ|,anddΦ=dlΦ|∇Φ|. Thus,

Ktwist =

1

4 π^2

∫Φ

0

dΦ

∫ 2 π

θ=0

dθ

∫(2π)−

φ=0

dφ

(

−ψ+Φψ′

)

= −

∫Φ

0

ψdΦ+

∫Φ

0

ψ′ΦdΦ

= 2

∫Φ

0

Φ

dψ

dΦ

dΦ. (11.58)

where the integration limit(2π)−corresponds to being infinitesimally less than 2 π.The last

line has been obtained using the relationship

∫

d(ψΦ)=[ψΦ]surfaceaxis =

∫

Φdψ+

∫

ψdΦ
and noting that the integrated term vanishes sinceψ=0in theflux tube surface andΦ=0
on theflux tube axis.

11.4.3Evaluation ofKwrithe

Connectivity of theflux tube is now the important issue. In order to evaluate theKwrithe
integral the volume element is expressed as

d^3 r=dl·ds (11.59)

wheredlis an increment of length along the axis anddsis an element of surface in the plane
perpendicular to the axis so thatBaxis·ds=dΦ.Because the line integral will involve a
complete circuit of theflux tube axis, in contrast to the earlier evaluation ofKtwist, we now
avoid using the gradient of a scalar to denote distance along the axis. Using Eq.(11.59) the
writhe helicity may therefore be expressed as

Kwrithe =

∫

Caxis

Aaxis·dlaxis

∫

dΦ

= Φ

∫

Caxis

Aaxis·dlaxis. (11.60)

This integral differs topologically from the integrals of the previous section, because here
the contour is acomplete circuit, i.e.,φvaries from 0 to 2 π,and not from 0 to 2 π−.
A contour pathCof an integral

∫

CA·dlmay be continuously deformed into a new
contour pathC′without changing the value of the integral provided no magneticflux is
linked by the surfaceSbounded byCandC′;this behavior is a three dimensional analog
to the concept of analyticity for a contour integral in the complex plane. The validity of
this assertion is established by the relation

0=

∫

S

B·ds=

∮

A·dl=

∫

C

A·dl−

∫

C′

A·dl. (11.61)