10.3 The MHD energy principle 317and magnetofluid fields at the perturbed surface. If the equilibrium force balance is inte-
grated over the volume of the small Gaussian pillbox located at the perturbed magnetofluid
surface in Fig.10.5, it is seen that
0=
∫
Vpillbox,psd^3 r∇·[(
P+
B^2
2 μ 0)
I−
1
μ 0BB
]
=
∫
S,psds[
P+
B^2
2 μ 0]vacmagnetofluid
(10.102)
sinceds·B=0at the perturbed surface. The subscriptspsindicate that the volume and
surface integrals are at the perturbed surface.
Quantities evaluated at the perturbed surface are of the form
fpertsfc=f 0 +f 1 +ξ·∇f 0 ; (10.103)i.e., both absolute and convective first-order terms must be included. Since the pillbox
extent was arbitrary in Eq.(10.102), the integrand
[
P+B^2 / 2 μ 0]vac
magnetofluidmust vanish
at each point on the perturbed surface, giving the relation
(
P 1 +B 0 ·B 1
μ 0)
magneto
fluid+ξ·∇(
P+
B^2
2 μ 0)
magneto
fluid=
(
B 0 ·B 1
μ 0)
vac+ξ·∇(
B^2
2 μ 0)
vac
(10.104)
which can be rewritten as
(
P 1 +
B 0 ·B 1
μ 0)
magnetofluid=ξ·∇[
P+
B^2
2 μ 0]vacmagnetofluid+
(
B 0 ·B 1
μ 0)
vac.
(10.105)
Thus the surface integral Eq.(10.101) can be rewritten as
δWS =
1
2
∫
Smfds·ξ⊥ξ⊥·∇[
P+
B^2
2 μ 0]vacmagnetofluid+
1
2 μ 0∫
Smfds·ξ⊥(B 0 ·B 1 )vac.(10.106)
The volume integral in Eq.(10.100) is over the magnetofluid volume and the direction of
dsin Eq.(10.106) is outwards from the magnetofluid volume. The energy stored in the
vacuum region between the magnetofluid and wall is
δWvac =1
2 μ 0∫
Vvacd^3 rB 12 v=
1
2 μ 0∫
Vvacd^3 rB 1 v·∇×A 1 v=
1
2 μ 0∫
Vvacd^3 r∇·(A 1 v×B 1 v)=
1
2 μ 0∫
Svacds·(A 1 v×B 1 v) (10.107)wheredspointsoutfrom the vacuum region and∇×B 1 v=0has been used when inte-
grating by parts. At the conducting wall (which might be at infinity) the integrandvanishes
since it contains the factords×A 1 vwhich is proportional to the tangential electric field
and the tangential electric field vanishes at a conductor. Thus, only the surface integral over