372 Chapter 12. Magnetic reconnection
which in turn can be thought of as the straight cylindrical approximationof a toroid withz
corresponding to the toroidal angle.
k̂
x 0 plane only
B
̂
ẑ
ŷ.
Figure 12.4: Tilted coordinate system for general sheared field.Figure 12.4 shows the magnetic field given by Eq.(12.60) as viewed in thex=0plane.
In the sheet current analysis discussed in the previous section, the perturbed vector potential
pointed in thezdirection and was periodic in theydirection. This corresponded to having
k·A 1 =0so that the wavevector was orthogonal to the vector potential and both were
orthogonal tox.Other important properties were thatk·Bvanished at the reconnection
layer, thefluid vorticity vector was pointed in thezdirection, the perturbed currents and
perturbed magnetic fields were such thatJ 1 /J 0 >>B 1 /B 0 in the reconnection layer, and
J 1 /J 0 ∼B 1 /B 0 in the exterior region.
These relationships and approximations are generalized here and, in particular, it is
assumed that all perturbed quantities have functional dependence∼g(x)exp(ikyy+ikzz+
γt).As before, the vorticity equation is the curl of the linearized equationof motion, i.e.
ρ∂Ω 1
∂t=∇×(J 1 ×B 0 +J 0 ×B 1 ) (12.62)
and in the reconnection layer whereJ 1 /J 0 >>B 1 /B 0 this becomes
ρ∂Ω 1
∂t= ∇×(J 1 ×B 0 )
= B 0 ·∇J 1 −J 1 ·∇B 0 (12.63)
since∇·B 0 =0and∇·J 1 =0.
An essential feature of the reconnection topology is that the vorticity must be anti-
symmetric about the reconnection layer, as can be seen from examination of thefluidflow
vectors in Fig.12.2(c). Since the vorticity is created by the torque (i.e., the curl of the force),
it is clear that the torque must be antisymmetric about the reconnectionlayer. As will be
seen in the next paragraphs, the condition that the torque is antisymmetric doesnot imply
that eitherB 0 orJ 1 are antisymmetric, but rather implies some more subtle conditions.