10.8 Analysis of free-boundary instabilities 333so the stability condition becomes
x^2 +(m+x)^2 >|m|=⇒stable. (10.185)Without loss of generalityxcan assumed to be positive, in which case instability occurs
only whenmis negative. Thus, Eq.(10.185) can be written as
2 x^2 − 2 |m|x+m^2 −|m|>0=⇒stable. (10.186)The threshold for instability occurs when the left hand side of Eq. (10.186) vanishes, i.e.,
at the roots of the left hand side. These roots are
x=|m|±√
2 |m|−m^2
2. (10.187)
Since the left hand side of Eq.(10.186) goes to positive infinity for|x|→∞,the left hand
side is negative only in the region between the two roots. Thus, the plasmais unstable only
if|x|lies between the two roots. The stability condition is that|x|must lie outside the
region between the two roots, i.e., for stability we must have
x>|m|+√
2 |m|−m^2
2orx<|m|−√
2 |m|−m^2
2.
(10.188)
Form=− 1 modes this gives the stability condition
x> 1. (10.189)In un-normalized quantities and usingk=2π/LwhereLis the length of the system, the
stability condition is
2 πa
L
B 0 z
B 0 θ>1;
this is known as the Kruskal-Shafranov stability criterion (Kruskal,Johnson, Gottlieb and
Goldman 1958, Shafranov 1958).
Form=− 2 modes, the two roots coalesce atx=1and so Eq.(10.189) also gives
stability. Form≥ 3 the argument of the square root in Eq.(10.187) is negative so there is
no region of instability.
In toroidal devices such as tokamaks, the axial wavenumber corresponds to the toroidal
wavenumber since the dominant magnetic field is in the toroidal direction,i.e.B 0 z→B 0 φ
whereφ is the toroidal angle. The axial wavenumberkbecomesn/Rwherenis the
toroidal mode number. Since we saw that long axial wavelengths are the most unstable we
taken=1.The Kruskal-Shafranov kink stability condition becomes
q=
aB 0 φ
RB 0 θ> 1 (10.190)
whereqis called the safety factor. Typical tokamaks operate withq∼ 3 at the wall;this
qcondition is one of the most important design criteria for tokamaks since it dictates the
magnitude of the large and expensive toroidal field system.