Fundamentals of Plasma Physics

318 Chapter 10. Stability of static MHD equilibria

the magnetofluid-vacuum interface remains. Since an element of surfacedspointing out
of the vacuum points into the magnetofluid, usingdsto mean out of the magnetofluid (as
before), this becomes

δWvac=−

1

2 μ 0

∫

Smf

ds·(A 1 v×B 1 v) (10.108)

whereSmfhas been used instead ofSvacto indicate thatdspoints out of the magnetofluid.
The tangential component of the electric field must be continuous at the magnetofluid-
vacuum interface. This field is not necessarily zero. The electric field inside the mag-
netofluid is determined by Ohm’s law, Eq.(10.37). Thus the electric field on the mag-
netofluid side of the magnetofluid-vacuum interface is

E 1 p=−v 1 ×B 0 (10.109)

while on the vacuum side of this interface the electric field is simply

E 1 v=−

∂A 1 v

∂t

(10.110)

whereA 1 vis the vacuum vector potential. Defining the surface normal unit vectorˆnand
integrating both electric fields with respect to time, the condition that the tangential electric
field is continuous is seen to be

nˆ×A 1 v=ˆn×(ξ×B 0 )=−nˆ·ξB 0 (10.111)

wherenˆ·B 0 =0has been used. Thus, the second surface integral in Eq.(10.106) can be
written as

1

2 μ 0

∫

S

dsnˆ·ξ⊥B 0 ·B 1 v.=−

1

2 μ 0

∫

S

dsˆn×A 1 v·B 1 v=δWvac. (10.112)

These results are now summarized. The perturbed potential energy can be expressed
as
δW=δWF+δWint+δWvac (10.113)
where the contribution from thefluid volume interior is

δWF=

1

2 μ 0

∫

Vmf

d^3 r

{

γμ 0 P 0 (∇·ξ)^2 +B^21 ⊥+B 02 [2ξ⊥·κ+(∇·ξ⊥)]^2

+ξ⊥×B 1 ⊥·Bˆ 0 μ 0 J 0 ‖− 2 μ 0 (ξ⊥·κ)(ξ⊥·∇P 0 )

}

(10.114)

the contribution from the vacuum-magnetofluid interface is

δWint=

1

2 μ 0

∫

Smf

ds·ξ⊥ξ⊥·∇

[

μ 0 P+

B^2

2

]vac

magnetofluid

(10.115)

and the contribution from the vacuum region is

δWvac=

1

2 μ 0

∫

vac

d^3 rB 12 v. (10.116)