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)