Fundamentals of Plasma Physics

362 Chapter 12. Magnetic reconnection

width until circular. Because only modest frictional dragging of water across the substrate

is required to do this, the process can be energetically favorable, and so result in water

beading.

12.3 Qualitative description of sheet current instability

Suppose an infinite-extent plasma has a thin sheet of currentflowing in thezdirection

as shown in Fig.12.2(a). The sheet current is centered atx=0 and extends fromy=

−∞toy=+∞.The current is uniform in both theyand thezdirections and roughly

corresponds to the initial long thin water drop. This somewhat artificial situation can be

considered as the Cartesian analog of a cylindrical shell currentflowing in thezdirection,

localized at some radiusr=r 0 and azimuthally symmetric. Thus, we identifyrwithx

andywithθ.

(a) (b) (c)

sheet

current

By 0

y

gap

typicalbar

selffield

filamentation

of current

into bar

cross-sections

detail of bars

show ing fluid

flow patterns

center of

typical

fluid

vortex

fluid

flow

vector

Figure 12.2: (a) Initial sheet current;(b) breaking up into bar cross-section filaments;(c)
detail showingfluidflow consists of set of small vortices which are antisymmetric inx.

Since all quantities depend onxonly,Ampere’s law reduces to

μ 0 Jz(x)=

∂By

∂x

(12.1)