## Appendix AWe consider streams of neutral and ionized gas components that flow
in parallel directions on either side of the KH interface with
different bulk velocities and mass densities. In the unperturbed state
the interface, say , is horizontal.
Let be ,
,
, and
for
and
,
,
, and
for
, where
and
denote the mass density and the bulk velocity. The index where where we made use of Eq. (A7) and here and in the following we
omitted the hat indicating the amplitudes of the perturbed quantities.
With the help of Eqs. (A1) and (A2) we find for the The Cartesian components of the neutral gas momentum balance equation (A6) give accordingly where denotes the Doppler
shifted frequency . We concentrate
on surface waves that decay exponentially with increasing distance
from the boundary layer and
where the minus signs holds for
and the plus sign for
. We assume that the boundary layer
is not a discontinuity but that the physical quantities vary
continuously across the small finite width of the interface. It is
reasonable to assume that the tangential velocity perturbations, the
pressure perturbations as well as
and are odd functions of where the bar denotes the mean values across the interface , and Eqs. (A14) and (A15) hold accordingly for the neutral gas
quantities. After integrating Eqs. (A9) and (A12) over Eq. (A16) represent the generalized dispersion relation for the KH modes in partially ionized plasmas. For is reduces to the well-known dispersion relation for fully ionized plasmas (see e.g. Woods 1987) © European Southern Observatory (ESO) 2000 Online publication: December 8, 1999 |