Fundamentals of Plasma Physics

11.3 Woltjer-Taylor relaxation 343

The relaxation process involves minimizing the magnetic energy

W=

1

2 μ 0

∫

B^2 d^3 r (11.24)

subject to the constraint that the magnetic helicity

K=

∫

A·Bd^3 r (11.25)

is conserved (Woltjer 1958, Taylor 1974). The minimum-energy magnetic field for the
given helicity is calledBMEand its associated magnetic energy is

WME=

1

2 μ 0

∫

BME^2 d^3 r. (11.26)

In order to determineBME, we consider an arbitrary variation B=BME+δBsatis-
fying the same boundary conditions;by assumption, this variation has a higher associ-
ated magnetic energy thanBME. The magnetic fieldBhas an associated vector potential
A=AME+δAand because the tangential electric field must vanish at the wall to prevent
helicityflux across the wall, the tangential component ofδAmust also vanish at the wall.
A naive solution for minimizingWMEwould be to setBMEto zero, but this is forbidden
because it would makeKvanish and violate the helicity conservation requirement. Thus
Wmust be minimized subject to the constraint thatKremains constant. The variation of
the magnetic energy about the minimum energy state is

δW =

1

2 μ 0

∫[

(BME+δB)^2 −BME^2

]

d^3 r

=

1

μ 0

∫

BME·δBd^3 r

=

1

μ 0

∫

BME·∇×δAd^3 r

=

1

μ 0

∫

[∇·(BME×δA)+δA·∇×BME]d^3 r

=

1

μ 0

∫

ds·BME×δA+

∫

δA·∇×BMEd^3 r

=

∫

δA·∇×BMEd^3 r (11.27)

sinceds·BME×δA=0everywhere on the wall. The variation of the helicity is

δK =

∫

(δA·BME+AME·δB)d^3 r

=

∫

(δA·BME+AME·∇×δA)d^3 r

=

∫

(δA·BME+δA·∇×AME)+∇·(δA×AME)d^3 r

= 2

∫

δA·BMEd^3 r. (11.28)