Fundamentals of Plasma Physics

100 Chapter 3. Motion of a single plasma particle

Figure 3.14: Specific example (withzdependence suppressed) showingψ andχrela-
tionship: top is plot of functionψ(r)/ψ 0 = (r/L)^2 /(1 + (r/L)^6 ), middle and bot-
tom plots show corresponding normalized effective potential for 2 πPθ/qψ 0 = +0. 2 and
2 πPθ/qψ 0 =− 0. 2 .Both middle and bottom plots have a minimum atr/L≃ 0 .45;mid-
dle plot also has a minimum atr/L≃ 1. 4 .The two minima in the middle plot occur when
χ(r)/χ 0 = 0but the single minimum in the bottom plot occurs at a finite value ofχ(r)/χ 0
indicating that an axis-encircling particle must have finite energy.

1. qψ/Pθispositive. If 2 π|Pθ|<|qψmax|there exists a location insidermaxwhere

Pθ=

q

2 π

ψ (3.161)

and there exists a location outsidermaxwhere this equality holds as well.χvanishes

at these two points which are also local minima ofχbecauseχis positive-definite.

The top plot in Fig.3.14 shows a nominalψ(r)/ψ 0 flux profile and the middle plot

shows the correspondingχ(r)/χ 0 ;thezandtdependence are suppressed from the

arguments for clarity. There exists a maximum ofχbetween the two minima. We

consider a particle initially located in one of the two minima ofχ.IfH < χmaxthe

particle will be confined to an effective potential well centered about theflux surface