Fundamentals of Plasma Physics

12.6 Magnetic islands 377

It is seen that

B⊥·∇Az 0 =0 (12.93)

whereB⊥is the component perpendicular toz.Thus, the surfacesAz=const.give the
projection of the field lines in the plane perpendicular toz;these projections correspond to
the poloidalflux surfaces in toroidal geometry.
If it is assumed thatkz<<ky,so thatζis very nearly parallel toz,then it is seen that
the tearing instability adds a perturbation toAzso that now

Az(x,y)=

x^2 B∗′

2

+Az 1 coskyy. (12.94)

A sketch of a set of surfaces of constantAz(x,y)is shown in Fig.12.5. These surfaces
consist of (i) closed curves called islands, (ii) a separatrix which passes through thexpoint,
and (iii) open outer surfaces. The widthwof the separatrix can be calculated, by noting
that at thex−point

Asep=0+Az 1 (12.95)

while at the point of maximum width,x=w/ 2 ,

Asep=

(w/2)^2 B′∗

2

−Az 1. (12.96)

Equating these gives

w=4

√

Az 1

B′∗

=4

√∣

∣

∣

∣

Bx 1

kyB′∗

∣

∣

∣

∣ (12.97)

usingBx 1 =ikyAz.In tokamak terminology we identifyBx 1 →Br 1 whereris the minor
radius, and using Eq.(12.84), (12.87) andky→m/athis becomes

w=4

√∣

∣

∣

∣

Rq

nq′

Br 1

Bφ

∣

∣

∣

∣. (12.98)

It is important to realize that the width of the island is much larger than the width of the
tearing layer. Since particles tend to be attached to magneticflux surfaces, the formation
of islands means that particles can circulate around the island, therebycausing aflattening
of the pressure gradient because the pressure is constant along a magnetic field line.