## 5. Numerical resultsIn this section we consider the numerical solutions of dispersion
relation (21) for an illustrative example of the isothermal atmosphere
(Gruzinov 1998). Henceforth, we assume that the turbulent flow is
time-independent. This assumption is valid for the wave period
As an illustrative case we take where is the flow penetration factor. Consequently, the vertical flow disappears at . The correlation function is assumed to be Gaussian where is the correlation length and is the variance. For a description of the solar turbulence the reader is referred to Canuto & Christensen-Dalsgaard (1998). The Fourier transform of this correlation function is equal to We use the following values for the surface gravity
m/s First, we illustrate a dependence of the numerically obtained results on the equilibrium parameters such as the density contrast , the photospheric pressure scale height , and the temperature ratio which determines the coronal pressure scale height, . Then, we show our results as functions of the flow parameters: the variance , the correlation length , and the penetration factor . Fig. 2 presents the computed frequency
and the line-width
as functions of the angular degree
Fig. 3 shows the frequency
and the line-width for
km (solid line) and
(broken line). The latter case
corresponds to a weakly stratified atmosphere and convection zone. In
particular, the case of is
associated with a homogeneous atmosphere and convection zone. This
case was discussed by Medrek et al. (1999). The broken frequency line
lies below the solid line which fits better the SOHO/MDI data. It is
interesting to note that for the case of
km the
Fig. 4 presents and
for the variance
km/s (solid line) and
km/s (broken line). The effect of a
stronger turbulent flow is to reduce more the frequency of the
Fig. 5 shows the frequency and the line-width for the correlation length km (solid line) and km (broken line). A better fit to the SOHO/MDI data is obtained for the case of km. The line-width exhibits the minimum and maximum at and , respectively. Consequently, smaller granules decrease more the frequency and broaden the line-width. An averaging of the results over different leads to a better agreement between the theoretical and observational results (Medrek & Murawski 2000).
Fig. 6 illustrates and for the flow penetration factor (solid line) and (broken line). A larger value of corresponds to the flow which is more confined to the unperturbed interface . While both the solid and broken lines are close to the observational data, a better fit to the SOHO/MDI line-width is obtained for . A larger value of reduces more the frequency but essentially it increases the line-width. These results are in an agreement with the work by Murawski & Roberts (1993b).
Fig. 7 shows imaginary parts of
(solid line) and
(dotted line) for the turbulent
© European Southern Observatory (ESO) 2000 Online publication: August 17, 2000 |