Newsgroups: sci.physics.plasma
From news@engin.umich.edu Mon Jan 23 13:32:55 1995
From: oliveria@down.engin.umich.edu (Roque Donizete de Oliveira)
Organization: University of Michigan Engineering, Ann Arbor
Subject: Re: plasma dispersion function

In article <3g0ch3$ian@mojo.eng.umd.edu>
Frank Crary <fcrary@rintintin.Colorado.EDU> wrote:

%% In article <3fgjj6$bma@mojo.eng.umd.edu>,
%% Roque Donizete de Oliveira <oliveria@down.engin.umich.edu> wrote:
%% >I have a question regarding the numerical solution of plasma dispersion
%% >equations...
%%
%% >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)
%%
%% I'm afraid I don't have any answers to your problem. (At least
%% not at the moment.) But I am curious how such a distribution
%% function could be produced. At first glance, I would expect
%% that this distribution would be extremely unstable (assuming
%% b and c were of the same order of magnitude.)

It is distribution function that can be used to model
the electron distribution function due to ECRH.
Typically b = 70 degrees (in our case), is the midplane pitch angle
for particles with turning points at the cyclotron resonance (due to the incident heating microwave) location.
c is the width (about 2 degrees). It is almost like a delta function
in pitch angle. As far as I know, the first (and only) use of such model
is described in
\bibitem{Smith1} G. R. Smith,
{\em Alfv\'{e}n Ion-Cyclotron Instability in Tandem-Mirror Plasmas. I},
Phys. Fluids {\bf 27}, 1499 (1984).
and
\bibitem{Smith2} G. R. Smith, W. M. Nevins and W. M. Sharp,
{\em Alfv\'{e}n Ion-Cyclotron Instability in Tandem-Mirror Plasmas. II},
Phys. Fluids {\bf 27}, 2120 (1984).

The problem I'm having is that even at low densities (5 10^10 cm^-3)
the imaginary part of k (the wavenumber) is always (for some wave
frequencies) negative (convective growth) throughout the plasma length.
Furthermore, its real part goes to zero somewhere along the plasma
(wave reflection, perhaps). While I've seen other papers where the real
part of k goes to zero, this behavior is not expected in our experiment.
Thus I'm desperately looking for ways to validate my result, even
questioning the validity of using one single dispersion equation
D(k,omega,z)= 0 no matter what the sign of Im[(omega - Omega_e)/k] is.
References to a derivation of the dispersion equation for the boundary
value problems are welcome (I have Landau's 1946 paper).

Also note that the f(v,theta) above is used to model the hot (3 keV) electron
component in our experiment. There is also a cold Maxwellian and warm
loss-cone bi-Maxwellian components. Thus even though the hot electrons
alone might yield an instability, the total plasma should be stable
(during its short duration) according to experimental data.

Unfortunately I need to solve the dispersion equation due to each electron
component to find out some optical properties (emission coefficient, absorption
coefficient and source function). If the solution I get for the hot component
(in which k_r goes to zero and k_i is negative) is indeed correct then
the radiation transport model is in serious jeopardy.

In case anyone has any clues as to what I'm talking about (I'm not sure
I do anymore) and has some suggestions, please don't hesitate to reply.

Thanks.

Roque
oliveria@engin.umich.edu