376 Chapter 12. Magnetic reconnection
Thus, Eq.(12.80) becomes
γ=0.55(∆′a)^4 /^5 τ−R^3 /^5 τ−A^2 /^5
(
na^2
R
q′
q
) 2 / 5
(12.89)
where
τ−R^1 =
η
a^2 μ 0
. (12.90)
In the form given by Eq.(12.89), it is seen that the essential source for thetearing instability
isq′.From this point of view, the ‘free energy’ is in the gradient ofqand so as the tearing
mode uses up this free energy, theqprofile will beflattened.
x
y
W
separatrix
Figure 12.5: Surfaces of constantAzshowing magnetic islands and separatrix with width
w.
12.6 Magnetic islands
The tearing instability changes the topology of the magnetic field and causes the formation
of magnetic ‘islands’ (Bateman 1978). The equilibrium magnetic field has the form
B=B ̄+B∗(x)ˆy=B ̄+xB∗′yˆ=B ̄+∇Az 0 (x)×zˆ (12.91)
where
Az 0 (x)=
x^2 B′∗
2