Fundamentals of Plasma Physics

10.3 The MHD energy principle 315

and

δWS′=

1

2 μ 0

∫

Smf

ds·ξ⊥(B 0 ·B 1 −μ 0 γP 0 ∇·ξ). (10.94)

At this point it is useful to examine the parallel and perpendicular componentsofB 1.
FiniteB 1 ⊥corresponds to changing the curvature of field lines (twanging or plucking)
whereas finiteB 1 ‖corresponds to compressing or rarefying the density of field lines. The
equilibrium force balance can be written as

μ 0 ∇P 0 =−∇⊥

B 02

2

+κB 02 (10.95)

whereκ=Bˆ 0 ·∇Bˆ 0 is related to the local curvature of the equilibrium magnetic field.
Applying the vector identity∇(C·D)=C·∇D+D·∇C+D×∇×C+C×∇×D
gives

κ=−Bˆ 0 ×∇×Bˆ 0 , (10.96)

a relation which will be used when evaluatingB 1 ‖. From Eq.(10.67) it is seen that the
parallel component of the perturbed magnetic field is

B 1 ‖ = Bˆ 0 ·∇×(ξ⊥×B 0 )

=(ξ⊥×B 0 )·∇×Bˆ 0 +∇·

[

(ξ⊥×B 0 )×Bˆ 0

]

= −B 0 ξ⊥·κ−∇·(B 0 ξ⊥)

= −B 0 ξ⊥·κ−B 0 (∇·ξ⊥)−

ξ⊥

B 0

·∇

B^20

2

= −B 0 [2ξ⊥·κ+∇·ξ⊥]+

μ 0

B 0

ξ⊥·∇P 0. (10.97)

The term involvingJ 0 in Eq.(10.93) can be expanded to give

ξ⊥·J 0 ×B 1 = ξ⊥·J 0 ⊥×Bˆ 0 B 1 ‖+ξ⊥·J 0 ‖Bˆ 0 ×B 1 ⊥

=(ξ⊥·∇P 0 )

B 1 ‖

B 0

+ξ⊥·J 0 ‖Bˆ 0 ×B 1 ⊥. (10.98)

Substituting for this term in Eq.(10.93), factoring out one power ofB 1 ‖,and then sub-