368 Chapter 12. Magnetic reconnection

The stream-functionf 1 is a solution of the Poisson-like system Eq.(12.25) and Eq.(12.33),

∂^2 f 1

∂x^2

−k^2 f 1 =

1

γρ 0

dAz 0

dx

ikJz 1. (12.35)

Since the perturbed current peaks atx=0and has a width of the order ofǫ, it may be
characterized by the Gaussian profile

Jz 1 ≃

λ

ǫ

√

π

e−x

(^2) /ǫ 2
(12.36)
where
λ=

∫

layer

Jz 1 dx (12.37)

is the total perturbed current in the tearing layer. The gradient of the vectorpotential can
be written as

dAz 0

dx

=−By 0 (x)≃−

x

L

By′ 0. (12.38)

Assuming that the tearing layer is very narrow gives

∂^2 f 1

∂x^2

>>k^2 f 1 (12.39)

so that Eq.(12.35) becomes

∂^2 f 1

∂x^2

=−

B′y 0

γLρ 0

ik

λ

ǫ

√

π

xe−x

(^2) /ǫ 2

ikB′y 0
2 γLρ 0
λǫ
√
π
d
dx
e−x
(^2) /ǫ 2

. (12.40)

The profiles ofJz 1 ,By 0 (x)and their product (right hand side of Eq.(12.40)) are shown in
Fig.12.3.
Integrating Eq.(12.40) with respect toxgives

∂f 1

∂x

=

ikBy′ 0

2 γLρ 0

λǫ

√

π

e−x

(^2) /ǫ 2
(12.41)
which incidentally givesU 1 y=−∂f 1 /∂x.Since it is desired to find the magnitude ofU 1 x
in the regionx∼ǫ,a rough ‘order of magnitude’ integration of Eq.(12.41) in this region
gives
f 1 ∼
ikByo′ λǫ^2
2 γL

√

πρ 0

sign(x)forx∼ǫ (12.42)

and so

U 1 x∼−

k^2 Bλǫ^2

2 γL

√

πρ 0

sign(x)forx∼ǫ. (12.43)