Fundamentals of Plasma Physics

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

The twist of the magnetic field is defined as the number of times a field line goesaround
in theθdirection for one circuit in theφdirection. Ifdlφis a displacement in theφdirection
then

dφ=|∇φ|dlφ (11.15)

and similarly

dθ=|∇θ|dlθ. (11.16)

The trajectory of a magnetic field line is parallel to the magnetic field so ifdlis an increment
along a magnetic field line thenB×dl=0or

dlθ

Bθ

=

dlφ

Bφ

. (11.17)

This means that
dθ
|∇θ|Bθ

=

dφ

|∇φ|Bφ

. (11.18)

HoweverBθ=|∇ψ||∇φ|/ 2 πandBφ=|∇Φ||∇θ|/ 2 πso

dθ

dφ

=

|∇θ|Bθ

|∇φ|Bφ

=

|∇θ||∇ψ||∇φ|/ 2 π

|∇φ||∇Φ||∇θ|/ 2 π

=

|∇ψ|

|∇Φ|

. (11.19)

Finally, ifψ=ψ(Φ)then∇ψ=ψ′∇Φwhereψ′=dψ/dΦand so

dθ

dφ

=ψ′. (11.20)

Thus,θincreasesψ′times faster than doesφand so ifφmakes one complete circuit (i.e.,
goes from 0 to 2 π),thenθmakesψ′circuits (i.e.,θgoes from 0 to 2 πψ′). The number
of times the field line goes around in theθdirection for each time it goes around in theφ
direction is called the twist

T(Φ)=ψ′. (11.21)

Hence Eq. (11.11), which gave the helicity when theψflux is embedded in theΦflux, can
be expressed in terms of the twist as

K=2

∫

ΦT(Φ)dΦ. (11.22)

and if the twist is a constant (i.e.,T′=0) then

K=TΦ^2. (11.23)