10.8 Analysis of free-boundary instabilities 327can be written as
0=−μ 0 ∇P 0 −∇(
B 02
2
)
+B 0 ·∇B 0. (10.141)
Because∇·B 0 =r−^1 ∂/∂r (rB 0 r)=0for an azimuthally symmetric, axially uni-
form system and becauseBrcannot be singular,B 0 rmust be zero everywhere. Since all
equilibrium quantities depend only onr,the third term in Eq.(10.141) can be expanded
B 0 ·∇B 0 =(
B 0 θ(r)ˆθ+B 0 z(r)ˆz)
·∇
(
B 0 θ(r)ˆθ+B 0 z(r)ˆz)
=B^20 θˆθ·∇ˆθ=−B 02 θ
rˆr.(10.142)
Thus, Eq.(10.141) has onlyrcomponents and reduces to the relation
0=−μ 0∂P 0
∂r−
∂
∂r(
B^20
2
)
−
B 02 θ
r. (10.143)
On integrating across the surface layer, this becomes
P 0 +
B^20 pz
2 μ 0=
B^20 vθ+B 02 vz
2 μ 0(10.144)
where the subscriptprefers to being at the inner radius (plasma side) of the surface layer and
the subscriptvrefers to the outer radius (vacuum side). The reason there is noB^20 pθ
term on the left hand side is that there are no interior plasma currents (from Ampere’s
law 2 πaB 0 pθ=μ 0 Iz=0).
The equilibrium surface can be described by the functionr=aor equivalently, the
surface-defining equation
S 0 (r)=r−a=0. (10.145)
The incompressible perturbation is characterized by a harmonic deformation of the plasma
surface such thatr=a+ξeimθ+ikz. The surface equation for the perturbed surface is
thus
S(r,θ,z)=r−a−ξeimθ+ikz=0 (10.146)
where without loss of generality it is assumed thatξ > 0. The equilibrium magnetic field
hadB 0 r=0,i.e., had no component normal to the surface, and so the equilibrium magnetic
field lines at the surface lie in the surface. This is equivalent to stating thatB 0 ·∇S 0 =0
which implies that the equilibrium magnetic field is tangential to the surface. Since the
magnetic field is assumed frozen into the plasma, the magnetic field mustcontinueto be
tangential to the surface even when the surface becomes deformed from its equilibrium
shape. Thus, the condition
B·∇S=0 (10.147)
must be satisfied at all times where∇S is in the direction normal to the surface;this is
essentially a statement that no magnetic field line penetrates the surface.
A quantityfat the perturbed surface (denoted by the subscript ps) can be expressed
as
fps=f 0 +f 1 +ξ·∇f 0 (10.148)
where the the middle term is the absolute first-order change and the last termis the convec-
tive term due to the motion of the surface.