Fundamentals of Plasma Physics

5.3 The Penrose criterion 173

Q-plane origin and ending at this same origin as shown by the various possible mappings
sketched in Figs.5.6(b), (c) and (d).

complex zplane

(a)

(b)

(c)

(d)

(e)

complex Qplane

complex Qplane

complex Qplane

upper halfzplane

corresponds to

instability

Im z

Im Q

Im Q

Im Q

Re Q

Re Q

Re Q

mapping ofupper

half ofzplane

intoQplane

Re z

Cz

CQ

ReQ
 0 ,Im Q 0

positive contribution to

instability criterion

negativecontribution negativecontribution

f
v

vmin v

f
vmin

distribution function

Figure 5.6: Penrose criterion: (a)-(d) mappings, (e) instability criterion.

The upper halfzplane maps to the area inside the curveCQ. IfCQis of the form
shown in Fig.5.6(b), thenQ(z)never takes on a positive real value forzbeing in the upper
halfzplane;thus a curve of this form cannot give a solution to Eq.(5.95) corresponding