12.7 Assignments 379(b) Define privateflux to be aflux surface that links only one of the current loops
(examples are theflux surfaces labeled 1 and 2 in Fig.12.6(a)). Define public
flux to be aflux surface that links both of the current loops (examples areflux
surfaces 3,4,5 in Fig.12.6(a)). Define the X-point to be the location in thez=0
plane where there is a field null as shown in Fig.12.6(a);letr 0 be the radius
of the X-point. Show by sketching that as the two current loops approach each
other in vacuum, a privateflux surface above the midplane will merge with a
privateflux surface below to form a publicflux surface.
(c) Show that theflux linked by a circle in thez=0plane with radiusr 0 (i.e.,
the circle follows the locus of the X-point) is the publicflux. Argue that if the
current loops approach each other, this publicflux will increase at the rate that
privateflux is converted into publicflux. By integrating Faraday’s law over
the surface of this circle in thezplane with radiusr 0 ,show that there exists a
toroidal electric field at the X-point,
Eφ=−1
2 πr 0∂ψpublic
∂t
where∂ψpublic/∂tis the rate of increase of publicflux.(d) Now suppose that the vacuum is replaced by perfectly conducting plasma sothat
plasma is frozen to field lines. Show from symmetry that the plasma velocity at
thexpoint must vanish. What isEφat the X-point if the two current loops are
immersed in a plasma which satisfies the ideal Ohm’s law,
E+U×B=0.
Can there be any conversion of privateflux into publicflux in an ideal plasma?
Sketch contours of private and publicflux when two current loops approach each
other in an ideal plasma.(e) Now suppose the two current loops are immersed in a non-ideal plasma which
satisfies the resistive Ohm’s law
E+U×B=ηJ.
Can privateflux be converted into publicflux in this situation, and if so, what is
the relationship between the rate of increase of publicflux and the resistivity?(f) By writingEφ,Br,Bz,andJφin terms of the poloidalflux function show that
the toroidal component of the resistive Ohm’s law can be expressed as
∂ψ
∂t+U·∇ψ=r^2 ∇·(
η
μ 0 r^2∇ψ)
.
Show that this equation implies thatflux convects with the plasma ifη=0,but
diffuses across the plasma ifU=0.Show that the diffusion increases ifψdevel-
ops a steep gradient at the location whereU=0.By considering your response
to (d) above, discuss how theflux gradient in thezdirection might steepen in the
vicinity of the X-point. Taking into account the relationship betweenJandψ,
show that this corresponds to developing a thin sheet of current in the midplane