Fundamentals of Plasma Physics

380 Chapter 12. Magnetic reconnection

(current-sheet) as shown in Fig.12.6(c). Show that in order to have an X-point

geometry and also satisfy Faraday’s law, theflux function must have the form

ψ(r,z,t)=

(

1+

z^2

2 δ^2

−

(r−r 0 )^2

2 L^2

)

ψ 0 − 2 πrEφt

in the vicinity of the X-point;hereψ 0 is theflux at the X-point att=0andr 0

is the radial location of the X-point. What is the relationship betweenδandLin

vacuum and in a resistive plasma? Isδlarger or smaller thanLif the two current

loops are approaching each other? Ifδ<Lis there a current in the vicinity of

the X-point? What is the direction of this current with respect to the currents in

the two current loops? Will the force on the current loops from the current in

the current sheet accelerate or retard the motion of the two current loops towards

each other? What is the direction ofEφ? Isψincreasing or decreasing at the

X-point?

(g) Suppose that the current sheet has a nominal thickness ofδin thezdirection

and a nominal width ofLin therdirection as shown in Fig.12.6(c). LetUin

be the nominal vertical velocity with which the plasma approaches thexpoint

and letUoutbe the nominal horizontal velocity with which the plasma leaves the

xpoint region as shown in Fig.12.6(c). If the plasma motion is incompressible

show that

UinL≃Uoutδ. (12.99)

(h) Consider the transition from the ideal MHD form of Ohm’s law to the form at

the X-point. Argue that in the region of this transition the termsU×BandηJ

should have the same order of magnitude (this is essentially the same argument

that was used to analyze Eq.(12.49)). Use this result and Ampere’s law to find a

relationship betweenηJ,Binandδ.Use this to show

Uin∼η/μ 0 δ (12.100)

and explain this result in terms of the convective velocity for motion offlux

surfaces outside the current sheet and the "diffusive velocity" for motion offlux

surfaces inside the current sheet.

(i) Using Eq.(9.42) show that the MHD force acting on the plasma is

FMHD=−

∇ψ

(2π)^2 μ 0

∇·

(

1

r^2

∇ψ

)

Sketch the direction of this force and indicate where this force is finite (hint:

consider whereJis finite). Estimate the work

∫

F·dldone on plasma accelerated

through the current sheet by using relationships involvingBin,Bout,δ,andL.

Use this estimate to show that

Uout≃vA,in (12.101)

wherevA,in=Bin/

√

μ 0 ρis the Alfvén velocity at the input. (Hint: Let∆ψbe

the jump influx experienced by afluid element as it moves across the current