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
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