Fundamentals of Plasma Physics

320 Chapter 10. Stability of static MHD equilibria

magnetic field reduces to

B 1 ‖=−B 0 [2ξ⊥·κ+∇·ξ⊥]. (10.119)

Thus, we may identify

B^21 =B 12 ⊥+B^21 ‖=B^21 ⊥+B^20 [2ξ⊥·κ+∇·ξ⊥]^2 (10.120)

and so the perturbed potential energy of the magnetofluid volume reduces to

δWF=

1

2 μ 0

∫

d^3 r

{

B^21 +ξ⊥×B 1 ⊥·B 0

μ 0 J 0 ‖

B 0

}

. (10.121)

Equation (10.67) shows that the perturbed vector potential can be identified as

A 1 =ξ×B 0 (10.122)

so that Eq.(10.121) can be recast as

δWF=

1

2 μ 0

∫

d^3 r

{

B^21 −A 1 ·B 1

μ 0 J 0 ‖

B 0

}

. (10.123)

We now show that finite A 1 ·B 1 corresponds to a helical perturbation. Consider the
simplest situation whereA 1 ·B 1 is simply a constant and define a local Cartesian coordinate
system withzaxis parallel to the localB 0 .Equation (10.122) shows thatA 1 =A 1 xxˆ+
A 1 yˆyso

A 1 ·B 1 =−A 1 x

∂A 1 y

∂z

+A 1 y

∂A 1 x

∂z

. (10.124)

Suppose both components ofA 1 are non-trivial functions ofzand, in particular, assume
A 1 x=ReA 1 xexp(ikz)andA 1 y=ReA 1 yexp(ikz).In this case

A 1 ·B 1 =

1

2

Re

[

−A∗ 1 x

∂A 1 y

∂z

+A∗ 1 y

∂A 1 x

∂z

]

=−

k

2

Re

[

i

(

A∗ 1 xA 1 y−A∗ 1 yA 1 x

)]

(10.125)

which can be finite only ifA∗ 1 xA 1 yis not pure real. The simplest such case is where
A 1 y=iA 1 xso
A 1 ·B 1 =k|A 1 x|^2 (10.126)
and
A 1 =Re[A 1 x(ˆx+iˆy)exp(ikz)] (10.127)
which is a helically polarized field sinceA 1 x∼coskzandA 1 y∼sinkz.

10.6 Magnetic helicity

Since finiteA 1 ·B 1 corresponds to the local helical polarization of the perturbed fields,
it is reasonable to defineA·Bas the density of magnetic helicity and to define the total
magnetic helicity in a volume as

K=

∫

V

d^3 rA·B. (10.128)