332 Chapter 10. Stability of static MHD equilibria
Sausage modes
Them=0modes are the sausage instabilities and here Eq. (10.174) reduces to
[
1+B ̄^20 vz−P ̄ 0
]
[
I 0
I 0 ′
]
+B ̄^20 vz
[
−I 0 Kˆ 0 ′+Iˆ 0 ′K 0
I′ 0 Kˆ′ 0 −Iˆ 0 ′K 0 ′
]
>
1
|k|a
=⇒stable. (10.177)
For a given normalized plasma pressure and normalized wall radiusb/a, this expression
can be used to make a stability plot ofB ̄^20 vzversus|k|a.Since the wall always provides
stabilization if brought in close enough, let us consider situations where there is no wall
(i.e.,b→∞) in which case the stability condition reduces to
[
1+B ̄ 02 vz−P ̄ 0
]
[
I 0
I 0 ′
]
+B ̄ 02 vz
[
−K 0
K 0 ′
]
>
1
|k|a
=⇒stable. (10.178)
For small arguments, the modified Bessel functions of order zero have the asymptotic val-
ues
I 0 (s)≃1+
s^2
4
; K 0 (s)≃−lns (10.179)
so the stability criterion becomes
B ̄^20 vz
[
1 −k^2 a^2 ln(|k|a)
]
>P ̄ 0 =⇒stable. (10.180)
This gives a simple criterion for how muchB ̄ 02 vzis required to stabilize a given plasma
pressure against sausage instabilities. The logarithmic term is stabilizing for|k|a < 1
but is destabilizing for|k|a>1;however this region of instability is limited because we
showed that very large|k|ais stable.
Kink modes
The finitemmodes are the kink modes. It was shown earlier that large|k|ais stable so
again we confine attention to small|k|a.In addition, we again let the wall location go to
infinity to simplify the analysis. The stability condition now reduces to
|k|a
[
1+B ̄^20 vz−P ̄ 0
]
[
I|m|
I′|m|
]
+
(m+kaB ̄ 0 vz)^2
|k|a
[
−K|m|
K|′m|
]
>1=⇒stable. (10.181)
Form =0the small argument asymptotic asymptotic limits of the modified Besselfunc-
tions are
I|m|(s)≃
1
|m|!
(s
2
)|m|
; K|m|(s)≃
|m− 1 |!
2
(s
2
)−|m|
(10.182)
so the stability condition becomes
k^2 a^2
[
1+B ̄^20 vz−P ̄ 0
]
+(m+kaB ̄ 0 vz)^2 >|m|=⇒stable (10.183)
which is a quadratic equation inka. Let us consider plasmas whereB ̄^20 vz >> 1 and
B ̄ 02 vz>>P ̄ 0 ;this corresponds to a low beta plasma where the externally imposed axial
field is much stronger than the field generated by the internal plasma currents (tokamaks
are in this category). Let us define
x=kaB ̄ 0 vz (10.184)