Newsgroups: sci.physics.plasma
From: ctd@mpe-garching.mpg.de (Christian T. Dum)
Organization: MPE
Subject: Re: complex group velocity
Approved: eastman@glue.umd.edu

>                         
>In article <3atoep$nuh@mojo.eng.umd.edu> oliveria@down.engin.umich.edu
>  (Roque Donizete de Oliveira) writes:
>From: oliveria@down.engin.umich.edu (Roque Donizete de Oliveira)
>Subject: Re: complex group velocity
>Date: 22 Nov 1994 21:39:37 GMT

>In article <aiqbe$m3n@mojo.eng.umd.edu>,
>Frank Crary <fcrary@benji.Colorado.EDU> wrote:

>>In article <3aiqbe$m3n@mojo.eng.umd.edu>,
>>Roque Donizete de Oliveira <oliveria@hadron.engin.umich.edu> wrote:
>>>I'm looking for recent references that discuss
>>>the problem of whether or not complex group
>>>velocity makes sense
>>
>I found a paper that promises to be interesting in this subject:

>  \bibitem{Muschietti}  L. Muschietti and C. T. Dum,
>  {\em Real Group Velocity in a Medium with Dissipation},
>  Phys. Fluids B {\bf 5}, 1383 (1993).

>>I'm not sure about the group velocity, but if the
>>real part of the wave number isn't much larger than
>>the complex part, you may have even more problems.
>>I don't know if you are deriving the wave number from
>>the fluid or kinetic equations, but in either case,
>>the derivation hits problems when the wave number
>>is largely complex. In kinetic theory, the common
>>derivation of dispersion relations has a built-in
>>assumption that the complex part is small; if this
>>isn't true, then the derivation is not valid.

I  believe that you have come across an embarrassing problem which is far
from having been resolved.  The usual assumption of negliglible imaginary
part  probably is violated more often than it is satisfied.  Take cyclotron
waves, for example. The frequency stays then relatively constant, near a
harmonic,  while the growth rate generally will have a strong wave number
dependence. Hence with the definition v_g=d\omega/dk one can  easily get
imaginary parts of v_g comparable or even much larger than the real part, even
if the growth rate remains small compared to the frequency.

I don't see anything wrong with the derivation of such dispersion relations. 
The restriction to small imaginary parts mentioned in the comments above
applies only to approximate solutions based on the separation of real and
imaginary parts, \gamma=-Imag \epsilon/{d Real \epsilon/d\omega}, rather
than a rigorous solution of \epsilon =0  for complex \omega=\omega_r+i\gamma.
(or complex k).

The next step I think is to examine the concept of  group velocity
itself.  It is always used in the context of some approximation to the problem
of wave propagation in a dispersive medium.
The traditional  way (Brillouin 1914) is to consider the space-time evolution
of a wave packet. The group velocity describes then the propagation speed
of the peak in the envelope of this packet.  In the paper mentioned above (
Muschietti and Dum, 1993) we have extended this approach to a dissipative
medium, using saddle point methods to approximately evaluate the
corresponding Fourier integral.  The basic effect is easily understood by
realizing that d\gamma/dk represents differential growth,   leading to
shifts in the dominant wave number etc. with time.  The effective
propagation speed ( group velocity) thus is a combination of the real and
imaginary  parts of d\omega/dk which depends on the width of the packet.

The group velocity also appears in the ray tracing equations for an
inhomogenous medium.  Bernstein (1975) considered first order corrections which
arise for weak dissipation. We have numerically compared his approach with
ours in the limit of a homogenous plasma. It would be importantant to extend
results to inhomogenous plasmas.  Another important extension is needed in the
kinetic  eqation for the  wave spectra of turbulent plasmas.

    C. T. Dum
    Max Planck Institute f. Extraterr. Physik
      and Center for Space Research, M. I. T.