*Astron. Astrophys. 359, 759-765 (2000)*
## 2. Linear equations
We consider a gravity-free plasma which is described by ideal
hydrodynamic equations. At a uniform equilibrium,
the plasma can flow with the speed
which is assumed to be small, viz.
where is the sound speed.
The present model strictly applies only to waves propagating in one
dimension with parallel random flows. Although this is an important
class to consider, it has to be admitted that the solar case is
substantially different. In particular, solar *p* -modes
propagate in three dimensions and convective flows are characterized
by strong up- and down-flows with relatively weak horizontal motions.
However, if any inhomogeneity of the solar plasma exhibits large
spatial scales in comparison to the wavelength of the *p* -mode,
then the *p* -modes can be well approximated by acoustic waves
(Swisdak & Zweibel 1999). Using this approximation is justified by
the fact that we are more concerned with the physical insight we can
derive from this model than its self-consistency.
As the amplitude of the flow (1 m s^{-1}) that is
associated with the *p* -modes is small in comparison to the
sound speed km s^{-1} we are
allowed to apply the linear theory. Consequently, small perturbations
to this equilibrium can be described by the linearized equations:
where *V* is the perturbed flow speed in the
*x*-direction, *p* is the perturbed pressure, and
is the specific heat ratio. The
indices *x* and *t* denote the space and time derivatives,
respectively.
Eqs. (2) and (3) describe the coupling between the flow *V*
and the pressure *p*; perturbations in *p* excite
oscillations in *V* and vice versa, a signal in *V* drives
oscillations in *p*. The terms
and correspond to damping
(amplification) of perturbations if
(). The terms
and
are responsible for the Doppler
shift of the frequency.
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000
helpdesk.link@springer.de |