Fundamentals of Plasma Physics

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

. (12.92)