370 Chapter 12. Magnetic reconnection
and using Eq.(12.36), the current density becomes
J 1 z∼−∆′
ǫμ 0√
πA 1 z(0). (12.48)We now repeat the induction equation, Eq.(12.20),
−
∂A 1 z
∂t−U 1 x
∂A 0 z
∂x=ηJ 1 z (12.49)and substitute forU 1 x,J 1 zand assume that∂A 0 z/∂x=−By 0 ≃−Bǫ/Lin the tearing
layer. This gives
γ −
k^2 B^2 ∆′ǫ^3
2 γL^2√
πμ 0 ρ 0=
η∆′
ǫμ 0√
π
#1 #2 #3(12.50)
where the terms have been numbered for reference in the following discussion.
In the ideal plasma limit, terms # 1 and # 2 balance each other while term # 3 is small;
this gives the frozen-in condition. At exactlyx=0,term # 2 vanishes and so terms # 1
and # 3 must balance each other, resulting in diffusion of the magnetic field. At the edge of
the tearing layer, which is the transition from the ideal limit to the diffusive limitall three
terms are of the same size.Thus, the three terms may be equated;this gives two equations
which may be solved forγandǫwith∆′as a parameter. Equating terms # 1 and # 3 gives
γ=
η∆′
ǫμ 0√
π(12.51)
while equating terms # 2 and # 3 gives
γ=
(kB′)^2 ǫ^4
2 ηρ 0(12.52)
whereB′=B/Lis the derivative of the equilibrium field atx=0.Equating these last
two equations to eliminateγgives the width of the tearing layer to be
ǫ≃[
2 η^2 ρ 0 ∆′
μ 0√
π(kB′)^2] 1 / 5
. (12.53)
Substitutingǫback into Eq.(12.51) gives
γ=0.55(∆′)^4 /^5[
η
μ 0] 3 / 5 [
(kB′)^2
ρ 0 μ 0] 1 / 5
. (12.54)
This result can be put in a more transparent form by defining characteristic times for ideal
processes and for resistive processes. The characteristic time forideal processes is the
Alfvén timeτA, defined as the time to move the characteristic lengthLwhen traveling at
the Alfvén velocity, i.e.,
τ−A^1 =vA
L=
√
B^2 /ρ 0 μ 0
L=
√
(B′)^2
ρ 0 μ 0