322 Chapter 10. Stability of static MHD equilibria
Since bothB·ˆn=0andEt=0at the wall, this reduces to
K=
∫
d^3 rA·B=const.; (10.134)i.e., the total helicity (integral of the helicity density) is conserved for an ideal plasma
surrounded by a rigid or conducting wall.
Now recall thatA 1 is a linear function ofξas given in Eq.(10.122) and thatξwas
assumed to be of orderǫ.Consider the implications of this result for perturbed magnetic
fields. We writeB=B 0 +B 1 andA=A 0 +A 1 so that the total magnetic helicity, a
constant, is
K=∫
d^3 rA 0 ·∇×A 0 =∫
d^3 r(A 0 +A 1 )·∇×(A 0 +A 1 )=
∫
d^3 rA 0 ·∇×A 0 +∫
d^3 rA 1 ·∇×A 0
+∫
d^3 rA 0 ·∇×A 1 +∫
d^3 rA 1 ·∇×A 1.
(10.135)
Thus, to first-order in the perturbation (i.e., to orderǫ),
∫
d^3 rA 1 ·∇×A 0 +∫
d^3 rA 0 ·∇×A 1 =0 (10.136)and to second-order (i.e., to orderǫ^2 which is what is relevant for the energy principle),
∫
d^3 rA 1 ·∇×A 1 =
∫
d^3 rA 1 ·B 1 =0. (10.137)This can be compared to Eq.(10.123);the second term is almost the same as Eq.(10.137)
except for a factor−μ 0 J 0 ‖/B 0 which, in general, is some complicated function of position.
However, in the special case whereμ 0 J 0 ‖/B 0 =const.,it is possible to factorμ 0 J 0 ‖/B 0
out of the integral and obtain, using Eq.(10.137),
δWF=1
2 μ 0∫
d^3 rB 12 (10.138)which is positive-definite and therefore gives absolute stability. Thus, equilibria which have
μ 0 J 0 ‖/B 0 =λwhereλ=const.are stable against current-driven modes. Since these
equilibria are helical or kinked, they may be considered as being the final relaxed state
associated with a kink instability –once a system attains this state no free energy remains
to drive further instability. Since pressure has been assumed to be negligible, Eq.(10.60)
implies that the equilibrium current must be parallel to the equilibriummagnetic field,
but does not specify the proportionality factor. What has been shown here is that if the
proportionality factor is spatially uniform, i.e.,
μ 0 J 0 =λB 0 (10.139)whereλdoes not depend on position, then the system is stable against any further helical
perturbations. This gives rise to the relation
∇×B 0 =λB 0 (10.140)