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Astron. Astrophys. 346, 633-640 (1999)

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6. Models

The interpretation of our results depends on the correct understanding of the projection effects of the solar surface velocity field. We address each of the data fields with a position angle, [FORMULA], which we assume constant within each field. In a local cartesian coordinate system the velocity vector v has the components [FORMULA]. The z-axis coincides with the local surface normal, and the x-axis is in the plane defined by the z-axis and the line of sight. The z-axis is inclined with respect to the line of sight by the angle [FORMULA]. If we introduce spherical coordinates with the inclination angle, [FORMULA], measured from the z-axis and the azimuth angle [FORMULA] in the [FORMULA]-plane, with [FORMULA] in x-direction, the velocity components read


The line-of-sight velocity [FORMULA] is then given by


The velocity power spectrum contains the time average of [FORMULA]:


We assume that there is no correlation between the angles [FORMULA] and [FORMULA], and that the power distribution in azimuth [FORMULA] is isotropic. In this case the last term of Eq. (7) makes no contribution; we obtain


Thus, for purely vertical motion one would obtain a velocity power proportional to [FORMULA].

For the diverse wave modes in the solar atmosphere the angle [FORMULA] can be determined from a theoretical model. The simplest model describes adiabatic waves of small amplitude in an isothermal plane-parallel atmosphere. In this case the variation with t and z is of the form


where [FORMULA] is the angular frequency, H is the pressure scale height, and [FORMULA] is the vertical wave number. In the present paper we do not solve the oscillation equations for the entire Sun, which would yield the frequencies [FORMULA] as eigenvalues. Instead we consider [FORMULA] as given, although we pay special attention to those values that are known as solar f- and p-mode frequencies. The vertical wave number is then given by the dispersion relation


(e.g., Stix 1989). Here [FORMULA] is the acoustic cut-off frequency, [FORMULA] is the adiabatic sound velocity, and [FORMULA] is the Brunt-Väisälä frequency. For frequencies below the acoustic cut-off, which is for the greater part of the photospheric diagnostic diagram, we have [FORMULA]; we consider evanescent waves in an atmosphere of infinite extent, and therefore take the positive sign of [FORMULA]. For frequencies above the cut-off [FORMULA] is real; in this case either sign of [FORMULA] yields the same result with Eq. (11) below. We eliminate the perturbations of pressure and density from the wave equations and obtain the ratio of the horizontal and vertical velocity components, [FORMULA] and [FORMULA], and hence the angle [FORMULA]:


In order to avoid confusion we denote the ratio of the specific heats by [FORMULA] (we take [FORMULA]).

We have calculated the inclination [FORMULA] for the modes presented in the preceding section. The Ni I line at 676.8 nm is formed around the temperature minimum in the solar atmosphere, cf. Fig. 9 below, hence we take [FORMULA] K in our model; for the mean molecular weight we take [FORMULA]. With these specifications we have [FORMULA] km, [FORMULA] km/s, [FORMULA] s-1, and [FORMULA] s-1.

The dispersion relation for the f mode is [FORMULA]. Hence expression (10) for the vertical wave number can be simplified:


and [FORMULA]. With the scale height as specified, we have [FORMULA] in the range of horizontal wave numbers considered here, and therefore take the upper sign of [FORMULA]. Eq. (11) then yields


That is, the angle [FORMULA] is exactly [FORMULA] for the f mode, independently of the horizontal wave number.

The angles [FORMULA] listed in Table 1 are calculated for the central horizontal wave numbers of the two [FORMULA]-ranges; within each of these ranges there is only a very small variation of [FORMULA]. With increasing p-mode number [FORMULA] decreases; at high frequency, in particular at and above the acoustic cut-off near 5 mHz, the wave vector becomes almost vertical. A minimum occurs near the cutoff itself, which is approximately at p7 for [FORMULA] Mm-1, and at p4 for [FORMULA] Mm-1. At a given frequency [FORMULA] increases with increasing horizontal wave number [FORMULA].


Table 1. Inclination [FORMULA] of the wave vector with respect to the local vertical direction, for adiabatic waves in an isothermal atmosphere

Let [FORMULA] be the squared Fourier amplitude for a pair [FORMULA]. For this pair, and for the corresponding angle [FORMULA] calculated with (11), we may then calculate the line-of-sight power [FORMULA] according to Eq. (8), as a function of [FORMULA]. For all values [FORMULA] of interest in the present context this function is monotonously decreasing as [FORMULA] increases. Two examples which, according to Table 1, represent the modes f and p1, are shown in Fig. 6.

[FIGURE] Fig. 6. Calculated center-to-limb variation of [FORMULA] for inclination angles [FORMULA] and [FORMULA], for isotropic and non-isotropic distributions of azimuth angles [FORMULA]

A maximum such as found for the modes f and p1 around [FORMULA] (Figs. 4b and 5b) cannot be explained in this way. However, if we allow for an anisotropy in the azimuthal distribution and calculate [FORMULA] according to Eq. (7), such a maximum is possible, as also illustrated in Fig. 6 for two values of the inclination [FORMULA] and an azimuthal distribution that distinguishes the direction towards the observer.

In order to compare the center-to-limb variation of the modes with the global behavior of the velocity power in our data we have integrated the total power and the power of the f mode and the seven lowest p modes in the range [FORMULA] Mm-1 and [FORMULA] mHz. Fig. 7 displays the result for the total power, the power contained in the eight ridges, and the f-mode power as a function of [FORMULA]. The f-mode power deviates from the [FORMULA] curve only for values of [FORMULA] below 0.4., whereas the total p-mode power decreases faster towards the limb. The inter-ridge power was measured exactly in the same way as the mode power, in a 0.18 mHz wide frequency band below the f-mode.

[FIGURE] Fig. 7. Center-to-limb variation of the total power, ridge power, and f-mode power, in comparison to the curve [FORMULA], and the inter-ridge power measured close to the f-mode.

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© European Southern Observatory (ESO) 1999

Online publication: May 21, 1999