Velocity-Space Instabilities

Consider the dispersion relation (8.23) for an electrostatic plasma wave in an unmagnetized quasi-neutral plasma with stationary ions. This relation can be written

(8.125) |

or

where

(8.127) |

and . Taking to be real and positive, the question of whether the system is stable or not is equivalent to asking whether Equation (8.126) is satisfied for any value of lying in the upper half of the complex plane.

To answer the previous question, we employ a standard result in complex variable theory which states that
the number of zeros minus the number of poles of
in a given region of the complex
plane
is
times the increase in the argument of
as
moves once counter-clockwise
around the boundary of this region (Flanigan 2010). To determine the latter quantity, we construct
what is known as a *Nyquist diagram* (Nyquist 1932). Because the region in which we are interested is the
upper-half complex plane, we let
follow the semi-circular path shown in Figure 8.10(a), and
plot the corresponding path followed in the complex plane by
, as illustrated in Figure 8.10(b).
Now,
as
. Hence, if the radius of the semicircle in Figure 8.10(a)
tends to infinity, then only that part of the contour running along the real axis is important, and the
contour
starts and finishes at the origin. Because the function
is analytic in the upper-half
plane, by virtue of the way
in which it is defined, the number of zeros of
is equal to the change in argument (divided by
) of this
quantity as the path shown in Figure 8.10(b) is followed. However, this is just the number of times that the path
encircles the point
. Hence, the criterion for instability is that the path should encircle part of the positive real axis.
Thus, in Figure 8.10(b), the system is unstable for the indicated values of
(Cairns 1985).

In an unstable system, there must exist a point such as in Figure 8.10(b) where the contour crosses the real axis going from negative to positive imaginary part. Now, as moves along the real axis [cf., Equation (8.26)],

(8.128) |

Thus, at point , corresponding to (say), it must be the case that . Furthermore, must go from being negative to being positive as passes through from below. This implies that attains a minimum at . In other words, a necessary condition for the distribution function to be unstable is that it should attain a minimum value at some finite value of . A further condition to be satisfied is that the real part of be positive at . In other words,

(8.129) |

Note that the principal part need not be taken in the previous integral, because the numerator vanishes at the same point as the denominator. Integration by parts yields the equivalent condition

Here, has been chosen as the constant of integration in order to again make it unnecessary to take the principal part. The previous relation is called the

The previous discussion implies that a single-humped velocity distribution function, such as a Maxwellian, is absolutely stable to velocity-space instabilities (Gardner 1963). This follows because there is no finite value of at which such a distribution function attains a minimum value. In fact, assuming that the distribution function, , is such that as , we deduce that an unstable distribution function must possess at least one minimum and two maxima for in the range .