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Astron. Astrophys. 360, 707-714 (2000)

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3. Dispersion relation for the static ([FORMULA]) atmosphere

To obtain the dispersion relation for the static ([FORMULA]) atmosphere we Fourier analyze the perturbation variables as:

[EQUATION]

where [FORMULA], k is the horizontal wavenumber and [FORMULA] is the frequency of the f -mode. [FORMULA] represents a fluid variable such as [FORMULA], [FORMULA] and p.

From linearized Eqs. (2) - (4) we obtain the equation which describes vertical profiles of the flux function [FORMULA], viz.

[EQUATION]

Boundary conditions (5) and (6) lead then to the dispersion relation

[EQUATION]

[EQUATION]

From this dispersion relation it follows that the f -mode frequency depends on the density and mode profiles either side of the interface, the wavevector k and the gravity g.

3.1. An instructive example: The case of isothermal plasma

Similarly 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 [FORMULA]. Then, the density profiles can be written as

[EQUATION]

where [FORMULA] and [FORMULA] are the isothermal pressure scale heights, [FORMULA]. Here, [FORMULA] and [FORMULA] are the sound speeds. Consequently, there is a jump in the mass density ([FORMULA]) and the sound speed ([FORMULA]) at the interface [FORMULA].

Eq. (10) then leads to

[EQUATION]

where:

[EQUATION]

From these equations it follows that there are cut-offs in [FORMULA] below which [FORMULA] and [FORMULA] are complex and the wave is propagating along the z-direction. We have the following conditions for [FORMULA] and [FORMULA] to be real:

[EQUATION]

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:

[EQUATION]

where the density contrast is given by

[EQUATION]

It is worth mentioning that for the case of the constant density atmosphere we have [FORMULA] and [FORMULA]. Consequently, the dispersion relation takes the form:

[EQUATION]

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© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000
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