## 3. Dispersion relation for the static () atmosphereTo obtain the dispersion relation for the static () atmosphere we Fourier analyze the perturbation variables as: where , From linearized Eqs. (2) - (4) we obtain the equation which describes vertical profiles of the flux function , viz. Boundary conditions (5) and (6) lead then to the dispersion relation From this dispersion relation it follows that the ## 3.1. An instructive example: The case of isothermal plasmaSimilarly to Gruzinov (1998), we consider the case of uniform temperatures inside the solar corona and the solar interior. However, there is a temperature jump at the interface . Then, the density profiles can be written as where and are the isothermal pressure scale heights, . Here, and are the sound speeds. Consequently, there is a jump in the mass density () and the sound speed () at the interface . where: From these equations it follows that there are cut-offs in
below which
and
are complex and the wave is
propagating along the In the case when this condition is not satisfied, waves cease to be localized, they propagate and the corona and the convection zone act as sinks for such perturbations. Substituting the spatial profiles which are described by Eqs. (13) and (14) into Eq. (11) the dispersion relation for the isothermal atmosphere can be written as follows: where the density contrast is given by It is worth mentioning that for the case of the constant density atmosphere we have and . Consequently, the dispersion relation takes the form: © European Southern Observatory (ESO) 2000 Online publication: August 17, 2000 |