316 Chapter 10. Stability of static MHD equilibria
stituting Eq.(10.97) gives
δWF′ =
1
2
∫
Vmfd^3 r
γP 0 (∇·ξ)^2 +B 12 ⊥
μ 0+B 12 ‖
μ 0
−(ξ⊥·∇P 0 )B 1 ‖
B 0
−ξ⊥·J 0 ‖Bˆ 0 ×B 1 ⊥−ξ⊥·∇⊥(ξ⊥·∇P 0 )
=
1
2
∫
Vmfd^3 r
γP 0 (∇·ξ)^2 +B 12 ⊥
μ 0
+B 1 ‖
μ 0(
B 1 ‖−
μ 0 ξ⊥·∇P 0
B 0)
−ξ⊥·J 0 ‖Bˆ 0 ×B 1 ⊥−ξ⊥·∇⊥(ξ⊥·∇P 0 )
=
1
2
∫
Vmfd^3 r
γP 0 (∇·ξ)^2 +B 12 ⊥
μ 0+
(
B^20
μ 0[2ξ⊥·κ+∇·ξ⊥]−ξ⊥·∇P 0)
[2ξ⊥·κ+∇·ξ⊥]−ξ⊥·J 0 ‖Bˆ 0 ×B 1 ⊥+(ξ⊥·∇P 0 )∇·ξ⊥
−∇·[ξ⊥(ξ⊥·∇P 0 )]
=
1
2
∫
Vmfd^3 r
γP 0 (∇·ξ)^2 +B^21 ⊥
μ 0+
B 02
μ 0[2ξ⊥·κ+∇·ξ⊥]^2−(ξ⊥·∇P 0 )(2ξ⊥·κ) −ξ⊥·J 0 ‖Bˆ 0 ×B 1 ⊥
−∇·[ξ⊥(ξ⊥·∇P 0 )]
=
1
2 μ 0∫
Vmfd^3 r{
γμ 0 P 0 (∇·ξ)^2 +B^21 ⊥+B 02 [2ξ⊥·κ+∇·ξ⊥]^2
− 2 μ 0 (ξ⊥·∇P 0 )(ξ⊥·κ)+ξ⊥×B 1 ⊥·(
μ 0 J 0 ‖Bˆ 0)
}
−
1
2
∫
Smfds·ξ⊥(ξ⊥·∇P 0 ). (10.99)A new surface term has appeared because of an integration by parts;this new term is
absorbed into the previous surface term and thefluid and surface terms are redefined by
removing the primes;thus
δWF=1
2 μ 0∫
Vmfd^3 r{
γμ 0 P 0 (∇·ξ)^2 + B^21 ⊥+B 02 [2ξ⊥·κ+(∇·ξ⊥)]^2
− 2 μ 0 (ξ⊥·∇P 0 )(ξ⊥·κ)+ξ⊥×B 1 ⊥·(
μ 0 J 0 ‖Bˆ 0)
}
(10.100)
and
δWS =1
2 μ 0∫
Smfds·ξ⊥{B 0 ·B 1 −μ 0 (γP 0 ∇·ξ+ξ⊥·∇P 0 )}=
1
2 μ 0∫
Smfds·ξ⊥{B 0 ·B 1 +μ 0 P 1 } (10.101)whereB 0 ·ds=0and Eq.(10.68) have been used to simplify the surface integral. The sur-
face integral can be further re-arranged by considering the relationship between the vacuum