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

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5. Numerical results

In 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 T which is much lower than the turnover time [FORMULA], viz. [FORMULA]. Hence, we get a corresponding condition for the frequency [FORMULA]. Consequently, our model is valid for sufficiently high frequencies.

As an illustrative case we take

[EQUATION]

where [FORMULA] is the flow penetration factor. Consequently, the vertical flow disappears at [FORMULA].

The correlation function is assumed to be Gaussian

[EQUATION]

where [FORMULA] is the correlation length and [FORMULA] 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

[EQUATION]

We use the following values for the surface gravity [FORMULA] m/s2 and the polytropic index [FORMULA]. In the following figures, the frequency [FORMULA] and the line-width [FORMULA] (e.g., Osaki 1990) are displayed as functions of the spherical harmonic l for various parameters of the equilibrium and turbulent flow. The SOHO/MDI data (Duvall et al. 1998) is shown by the dotted curve for comparison purposes.

First, we illustrate a dependence of the numerically obtained results on the equilibrium parameters such as the density contrast [FORMULA], the photospheric pressure scale height [FORMULA], and the temperature ratio

[EQUATION]

which determines the coronal pressure scale height, [FORMULA]. Then, we show our results as functions of the flow parameters: the variance [FORMULA], the correlation length [FORMULA], and the penetration factor [FORMULA].

Fig. 2 presents the computed frequency [FORMULA] and the line-width [FORMULA] as functions of the angular degree l for two values of the density contrast: [FORMULA] (solid line) and [FORMULA] (broken line). The dotted line represents the SOHO/MDI data (Duvall et al. 1998). The broken line of the frequency [FORMULA] lies above the solid line and it fits better the SOHO/MDI data. The f -mode exists only if its modal frequencies are higher than the cut-off frequencies which are described by Eq. (17). In this case it is a mode which is localized in the vertical direction z. At frequencies below these cut-offs, the solar corona and the convection zone become transparent and the mode is not localized anymore. Instead, the f -mode can propagate through these regions and it leaks away. It is not an eigen-mode that persists in time. As a consequence of that the computed frequencies do not exist for low values of l. The line-width corresponding to [FORMULA] (broken line) is higher than the line-width for [FORMULA] (solid line) for overall values of l. These lines lie close to the observational data (dotted line). These theoretical curves cross the observational line at [FORMULA].

[FIGURE] Fig. 2. The frequency [FORMULA] and the line-width [FORMULA] of the solar f -mode as a function of the angular degree l for the density ratio [FORMULA] (solid curve) and [FORMULA] (broken curve). The other equilibrium parameters are [FORMULA] and [FORMULA] km. A turbulent flow is characterized by the correlation length [FORMULA] km, the variance [FORMULA] km/s and the penetration factor [FORMULA]. The SOHO/MDI data is represented by the dotted curve.

Fig. 3 shows the frequency [FORMULA] and the line-width [FORMULA] for [FORMULA] km (solid line) and [FORMULA] (broken line). The latter case corresponds to a weakly stratified atmosphere and convection zone. In particular, the case of [FORMULA] 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 [FORMULA] km the f -mode exists for the entire range of l while at [FORMULA] the solid curve exhibits a cut-off which corresponds to unstable ([FORMULA]) f -mode. The line-width corresponding to [FORMULA] km (broken line) is higher than the line-width for [FORMULA] km (solid line). Although the solid line lies below the observational curve it is close to the SOHO/MDI data (dotted line). The broken curve exhibits a maximum at [FORMULA]. The results for large [FORMULA] are in agreement with the work by Medrek et al. (1999).

[FIGURE] Fig. 3. Dispersion curves for [FORMULA] km (solid curve) and [FORMULA] km (broken curve). The other parameters are: [FORMULA], [FORMULA], [FORMULA] km, [FORMULA] km/s and [FORMULA]. The SOHO/MDI data is represented by the dotted curve.

Fig. 4 presents [FORMULA] and [FORMULA] for the variance [FORMULA] km/s (solid line) and [FORMULA] km/s (broken line). The effect of a stronger turbulent flow is to reduce more the frequency of the f -mode and to increase the line-width (Murawski et al. 1998). As a consequence of that the broken curve lies below the solid line which is close to the observational data. The line-width corresponding to [FORMULA] km/s (broken line) is higher that the line-width for [FORMULA] km/s (solid line). For the influence of [FORMULA] on the line-width see also Murawski et al. (1998), Medrek & Murawski (2000), and Medrek et al. (1999).

[FIGURE] Fig. 4. Dispersion curves for [FORMULA] km/s (solid curve) and [FORMULA] km/s (broken curve) and [FORMULA], [FORMULA], [FORMULA] km, [FORMULA] km, and [FORMULA]. The SOHO/MDI data is represented by the dotted curve.

Fig. 5 shows the frequency [FORMULA] and the line-width [FORMULA] for the correlation length [FORMULA] km (solid line) and [FORMULA] km (broken line). A better fit to the SOHO/MDI data is obtained for the case of [FORMULA] km. The line-width exhibits the minimum and maximum at [FORMULA] and [FORMULA], respectively. Consequently, smaller granules decrease more the frequency and broaden the line-width. An averaging of the results over different [FORMULA] leads to a better agreement between the theoretical and observational results (Medrek & Murawski 2000).

[FIGURE] Fig. 5. Dispersion curves for [FORMULA] km (solid curve) and [FORMULA] km (broken curve) and [FORMULA], [FORMULA], [FORMULA] km, [FORMULA] km/s and [FORMULA]. The SOHO/MDI data is represented by the dotted curve.

Fig. 6 illustrates [FORMULA] and [FORMULA] for the flow penetration factor [FORMULA] (solid line) and [FORMULA] (broken line). A larger value of [FORMULA] corresponds to the flow which is more confined to the unperturbed interface [FORMULA]. 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 [FORMULA]. A larger value of [FORMULA] reduces more the frequency but essentially it increases the line-width. These results are in an agreement with the work by Murawski & Roberts (1993b).

[FIGURE] Fig. 6. Dispersion curves for [FORMULA] (solid curve) and [FORMULA] (broken curve). The other parameters are: [FORMULA], [FORMULA], [FORMULA] km, [FORMULA] km/s, and [FORMULA] km. The SOHO/MDI data is represented by the dotted curve.

Fig. 7 shows imaginary parts of [FORMULA] (solid line) and [FORMULA] (dotted line) for the turbulent f -mode. As the imaginary parts differ from zero the f -mode leaks into the solar corona and the solar interior. This leakage is higher into the solar interior as [FORMULA]. Consequently, the wave profiles are no longer purely exponential but are oscillatory in the z-direction. The envelopes of these oscillations in the convection zone and the solar corona are exponentials: [FORMULA] and [FORMULA], respectively.

[FIGURE] Fig. 7. The imaginary part of [FORMULA] (solid curve) and [FORMULA] (dotted curve) for the f -mode as functions of the angular degree l for [FORMULA], [FORMULA], [FORMULA] km, [FORMULA] km, [FORMULA] km/s, and [FORMULA].

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© European Southern Observatory (ESO) 2000

Online publication: August 17, 2000
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