330 Chapter 10. Stability of static MHD equilibria
here the circumflex means the modified Bessel function is evaluated at the wall, i.e.,
with its argument set to|k|b.Equation (10.159) can then be recast as
χ=β
[
I|m|(|k|r)Kˆ|′m|−Iˆ|′m|K|m|(|k|r)
]
(10.162)
whereβ=β 1 /Kˆ|′m|.The wall boundary condition has reduced the number of inde-
pendent coefficients by one.
- Vacuum side of plasma-vacuum interface: Here Eq.(10.147) is linearized to obtain
B 1 ·∇S 0 +B 0 ·∇S 1 =0 (10.163)
or using Eqs.(10.145) and (10.146)
B ̄ 1 vr−i(m
a
+kB ̄ 0 vz)ξ=0. (10.164)
On the vacuum side Eq.(10.162) gives
B ̄ 1 vr = |k|β
[
I|′m|Kˆ|′m|−Iˆ|′m|K′|m|
]
(10.165)
where omission of the argument means that the modified Bessel function is evaluated
atr=a.Substitution of Eq.(10.165) into Eq.(10.164) gives
β=
i
(m
a
+kB ̄ 0 z
)
|k|
[
I′|m|Kˆ|′m|−Iˆ|′m|K|′m|
]ξ (10.166)
so the complete vacuum field can now be expressed in terms ofξ.
- Plasma side of plasma-vacuum interface: The plasma-side version of Eq.(10.163)
gives
B ̄ 1 pr−ikB ̄ 0 pzξ=0 (10.167)
sinceB 0 pθvanishes inside the plasma. From Eq.(10.158) the perturbed radial mag-
netic field on the plasma side of the interface is
B ̄ 1 pr=|k|αI|′m|. (10.168)
The plasma side version of Eq.(10.164) is thus
α=
ikB ̄ 0 pz
|k|I|′m|
ξ (10.169)
and so the plasma fields can now also be expressed in terms ofξ.
The stability condition, Eq.(10.156), can be written in terms ofαandβto obtain
β
(
im
a
+ikB ̄ 0 vz
)(
I|m|Kˆ|′m|−Iˆ|′m|K|m|
)
−
ξ
a
>αB ̄ 0 pzikI|m|. (10.170)
Substituting forαandβand re-arranging the order gives
|k|aB ̄^20 pz
[
I|m|
I|′m|
]
−
(m+kaB ̄ 0 vz)^2
|k|a
[
I|m|Kˆ|′m|−Iˆ|′m|K|m|
I|′m|Kˆ|′m|−Iˆ|′m|K′|m|
]
>1=⇒stable.(10.171)