Newsgroups: sci.physics.plasma
From news@engin.umich.edu Sat Jan 14 12:14:52 1995
From: oliveria@down.engin.umich.edu (Roque Donizete de Oliveira)
Organization: University of Michigan Engineering, Ann Arbor
Subject: plasma dispersion function

I have a question regarding the numerical solution of plasma dispersion
equations. When you numericaly solve for k complex (for a given omega real)
the dispersion equation for electromagnetic waves propagating
along the external magnetic field

D(k,omega) = 0 = function of Z(zeta)

where Z(zeta) is the usual plasma dispersion function defined for
Im(zeta) > 0 and analytically continued for Im(zeta) <= 0, and

omega - Omega_e
zeta = -----------------
k V

do you always use the same dispersion equation for Im(zeta) > 0
and for Im(zeta) < 0 ?

Or do you change your definition of Z(zeta), to say Z_(zeta), defined for
Im(zeta) < 0 and analytically continued for Im(zeta) >= 0 ?

I have a code that solves the local dispersion equation D(k,omega,z) = 0
where z is the axial position, for an electron distribution that
is made up of a Maxwellian in speed and a Gaussian pitch angle, like

f(v,theta) ~ exp(-v^2/a^2) * exp( -(theta - b)^2/c^2)

The resulting local dispersion equation is very complicated (in terms
of a pitch angle integral). I need to solve it for a fixed frequency omega
at several axial positions. I start from near vacuum where we know
the solution (index of refraction = 1) and move along z (towards resonance).
The problem is that the real part of k goes to 0 pretty fast and its imaginary
part is negative (growing wave) as we approach resonance (omega = Omega_e).
We don't expect this behavior. Unfortunately I can't compute its
analytical solution anywhere. I would expect the real part of k to have a
smooth transition past cyclotron resonance.

I'm trying to find a possible error I'm making. I always use the
same dispersion equation and plasma dispersion function Z(zeta)
(the latter works fine, for all zeta (by analytic continuation)).

Stix's book
(T. H. Stix, Waves in Plasmas American Institute of Physics, New York, 1992)
in the derivation of dispersion equation defines
a function Zo(zeta) that is quite similar to Fried&Conte's.
The difference is that Stix multiplied the exponential
term by sign(k), where k was assumed real, to account for waves
with k > 0 and k < 0 (the Landau contours for k > 0 and k < 0 are different).

He then points out what to do if k is complex (see page 203).
The problem is that Im( (omega - Omega_e)/k ) = 0 is a branch
cut in the complex k-plane for the dispersion relation.

Bers's paper (page 474)
(A. Bers, Space-Time Evolution of Plasma Instabilities - Absolute and
Convective}, Chapter 3.2 in Handbook of Plasma Physics, M. N. Rosenbluth
and R. Z. Sagdeev, eds., North-Holland, Amsterdan, 1983)
also talks about this and the fact that we get not 1 but 2 dispersion relations.

If k is real and > 0 Stix's Zo and Fried&Conte's Z function agree.

I wondered which plasma dispersion function I should use in a
problem where omega is real and k is complex.

Any suggestions would be appreciated.

Roque
oliveria@engin.umich.edu