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
fv
vmin v
fvmin
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