Fundamentals of Plasma Physics

10.3 The MHD energy principle 313

which shows thatω^2 must be pure real. Negativeω^2 corresponds to instability.

Equation (10.78) can be considered as an eigenvalue problem whereω^2 is the eigen-
value andξis the eigenvector. As in the usual linear algebra sense, eigenvectors having
different eigenvalues are orthogonal and so, in principle, a basis set ofnormalized orthog-
onal eigenvectors{ζm}can be constructed where

F 1 (ζm)=−ω^2 mρζm. (10.81)

Thus any arbitrary displacement can be expressed as a suitably weighted sum of eigenvec-
tors,
ξ=

∑

m

αmζm. (10.82)

The orthogonality of the basis set gives the relation
∫
d^3 rζm·F(ζn)=−ω^2 m

∫

d^3 rρζm·ζn=0ifm =n. (10.83)

Instability will result if there exists some perturbation that makesδWnegative. Con-
versely, if all possible perturbations result in positiveδWfor a given equilibrium, then the
equilibrium is MHD stable.

10.3.4Evaluation ofδW

SinceF 1 consists of three terms involvingJ 0 ,B 1 andP 1 respectively,δWcan be decom-
posed into three terms,
δW=δWJ 0 +δWB 1 +δWP 1. (10.84)
Using Eqs. (10.72), (10.67), (10.68) in Eq.(10.76) shows that these terms are

δWJ 0 =−

1

2

∫

d^3 rξ·J 0 ×B 1 , (10.85)

δWB 1 =−

1

2 μ 0

∫

d^3 rξ·(∇×B 1 )×B 0 , (10.86)

δWP 1 =

1

2

∫

d^3 rξ·∇P 1. (10.87)

Although the above three right hand sides are in principle integrated over the volumes of
both the magnetofluid and the vacuum regions, in fact, the integrands vanish in the vacuum
region for all three cases and so the integration is effectively overthe magnetofluid volume
only. The second two integrals can be simplified using the vector identities∇·(C×D)=
D·∇×C−C·∇×Dand∇·(fa)=a·∇f+f∇·ato obtain

δWB 1 =

1

2 μ 0

∫

Vmf

d^3 r(ξ×B 0 )·∇×B 1

=

1

2 μ 0

∫

Vmf

d^3 rB 12 +

1

2 μ 0

∫

Smf

ds·B 1 ×(ξ×B 0 ) (10.88)