Fundamentals of Plasma Physics

10.8 Analysis of free-boundary instabilities 329

so that, when expressed as a relationship between normalized fields, Eq.(10.154) becomes

B ̄ 1 vθ+B ̄ 0 vzB ̄ 1 vz−ξ

a

>B ̄ 0 pzB ̄ 1 pz=⇒stable. (10.156)

What remains to be done is express all first-order magnetic fields in terms ofξ.This is
relatively easy because the current is confined to the surface layer sothe magnetic field is
a vacuum magnetic field everywhere but exactly on the surface.
SinceBis a vacuum magnetic field for both 0 <r<aanda<r<b, it must be of the
formB=∇χwhere∇^2 χ=0in both regions, but there could be a jump discontinuity at
r=a.Because the perturbed fields were assumed to have anexp(imθ+ikz)dependence,
the Laplacian becomes

∇^2 χ=

d^2 χ(r)

dr^2

+

1

r

dχ(r)

dr

−

(

m^2

r^2

+k^2

)

χ(r)=0 (10.157)

which is just the equation for modified Bessel functions,I|m|(|k|r)andK|m|(|k|r).Ab-
solute value signs have been used to avoid possible confusion later for situationswheremor
kcould be negative. TheI|m|(|k|r)function is finite atr=0but diverges atr=∞while
the opposite is true forK|m|(|k|r).
The surface current causes a discontinuity in the tangential components of the perturbed
magnetic field. Hence, different solutions must be used in the respective plasma and vac-
uum regions and then these solutions must be related to each other in such a way as to
satisfy the requirements of this discontinuity. The forms of the plasma and vacuum solu-
tions are, in addition, constrained by the respective boundary conditions atr=0and at
r=b. In particular, theK|m|(|k|r)solution is not allowed in the plasma because of the
constraint that the magnetic field must be finite atr=0;thus inside the plasma the solution
is of the form
χ=αI|m|(|k|r) inplasmaregion 0 <r<a−. (10.158)
On the other hand, both theI|m|(|k|r)andK|m|(|k|r)solutions are permissible in the
vacuum region and so the vacuum region solution is of the general form

χ=β 1 I|m|(|k|r)+β 2 K|m|(|k|r) in vacuum regiona+<r<b. (10.159)

The objective now is to express the coefficientsα,β 1 ,andβ 2 in terms ofξso that all
components of the perturbed magnetic field can be expressed as a function ofξ, both in the
plasma and in the vacuum.
The functional dependence of these coefficients is determined by considering boundary
conditions at the wall and then at the plasma-vacuum interface:

1. Wall: Since the wall is perfectly conducting it is aflux conserver. There is no radial
   magnetic field initially and so there is noflux linking each small patch of the wall.
   ThusB ̄ 1 vr(b)must vanish at the wall in order to maintain zeroflux at each patch of
   the wall. Using Eq.(10.159) to calculate the perturbed radial magnetic field at the wall,
   this condition means that
   β 1 I|′m|(|k|b)+β 2 K|′m|(|k|b)=0 (10.160)
   which may be solved forβ 2 to give

β 2 =−β 1

Iˆ|′m|

Kˆ′

|m|

; (10.161)