Fundamentals of Plasma Physics

12.4 Semi-quantitative estimate of the tearing process 369

product J 1 zB′x


vorticity source

J 1 z

B′x

x

Figure 12.3: Product of symmetric perturbed current with antisymmetricequilibrium field
results in antisymmetric vorticity source localized nearx=0.

The amplitude factorλcan be expressed using Eq. (12.21) as

λ = −

1

μ 0

∫

layer

∇^2 ⊥A 1 zdx

≃−

1

μ 0

∫

layer

∂^2 A 1 z

∂x^2

dx

= −

1

μ 0

[(

∂A 1 z

∂x

)

+

−

(

∂A 1 z

∂x

)

−

]

(12.44)

where the subscripts±mean evaluated atx=±ǫ.For purposes of joining to the outer

ideal solution, the normalized jump derivative is defined as

∆′=

(

∂A 1 z

∂x

)

+

−

(

∂A 1 z

∂x

)

−

A 1 z(0)

(12.45)

so that

λ=−

∆′

μ 0

A 1 z(0). (12.46)

The velocity becomes

U 1 x∼

k^2 B∆′ǫ^2

2 γL

√

πμ 0 ρ 0

A 1 z(0)sign(x) (12.47)