Fundamentals of Plasma Physics

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

and

Ktwist=

∫

Vtube

d^3 r

(

ψ(Φ)

2 π

∇φ·Baxis+Aaxis·

1

2 π

∇ψ×∇φ

)

. (11.52)

As will be shown below,Ktwistdepends onψbeing finite whereasKwrithedepends on the
extent to which theflux tube axis is helical. Theflux tube helicity can thus be entirely due
toKwrithe,entirely due toKtwist,or due to some combination of these types of helicity.

.

11.4.2Evaluation ofKtwist

Since theKtwistintegral is avolumeintegral, theflux tube may be cut atφ=0without
affecting this integral. Making such a cut means thatφis restricted to the range 0 ≤φ<
2 π and therefore does not make a complete circuit around the axis of theflux tube. The
evaluation of this integral is insensitive to the connectivity of the axis because connectivity
is a concept which makes sense only after making a complete circuit of theaxis. The axial
magnetic field within the cut volume may be expressed as

Baxis=

1

2 π

∇×Φ∇θ (11.53)

whereθis the angular distance on a contourCθencircling theflux tube axis and lies in a
magnetic surface. This representation for the axial magnetic field is appropriate here since
at eachφfor 0 ≤φ< 2 πwe may write

Φ =

∫

ds·Baxis

=

1

2 π

∫

ds·∇×Φ∇θ

=

1

2 π

∮

Cθ

dl·Φ∇θ

=

Φ

2 π

∮

Cθ

dl·∇θ

=

Φ

2 π

∮

Cθ

dθ. (11.54)

By uncurling Eq.(11.53) it is seen that vector potential associated with theaxial magnetic
field may be represented in the cutflux tube as

Aaxis=

Φ

2 π

∇θ, 0 ≤φ< 2 π (11.55)

so that

Ktwist =

1

4 π^2

∫

Vtube

d^3 r(ψ∇φ·∇Φ×∇θ+Φ∇θ·∇ψ×∇φ)

=

1

4 π^2

∫

Vtube

d^3 r

(

ψ∇φ·∇Φ×∇θ+Φψ′∇θ·∇Φ×∇φ

)

=

1

4 π^2

∫

Vtube

d^3 r

(

−ψ+Φψ′

)

∇θ·∇Φ×∇φ. (11.56)