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, , which we assume constant within each field. In a local cartesian coordinate system the velocity vector v has the components . 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 . If we introduce spherical coordinates with the inclination angle, , measured from the z-axis and the azimuth angle in the -plane, with in x-direction, the velocity components read
The line-of-sight velocity is then given by
The velocity power spectrum contains the time average of :
We assume that there is no correlation between the angles and , and that the power distribution in azimuth 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 .
For the diverse wave modes in the solar atmosphere the angle 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 is the angular frequency, H is the pressure scale height, and 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 as eigenvalues. Instead we consider 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 is the acoustic cut-off frequency, is the adiabatic sound velocity, and 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 ; we consider evanescent waves in an atmosphere of infinite extent, and therefore take the positive sign of . For frequencies above the cut-off is real; in this case either sign of 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, and , and hence the angle :
In order to avoid confusion we denote the ratio of the specific heats by (we take ).
We have calculated the inclination 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 K in our model; for the mean molecular weight we take . With these specifications we have km, km/s, s-1, and s-1.
The dispersion relation for the f mode is . Hence expression (10) for the vertical wave number can be simplified:
and . With the scale height as specified, we have in the range of horizontal wave numbers considered here, and therefore take the upper sign of . Eq. (11) then yields
That is, the angle is exactly for the f mode, independently of the horizontal wave number.
The angles listed in Table 1 are calculated for the central horizontal wave numbers of the two -ranges; within each of these ranges there is only a very small variation of . With increasing p-mode number 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 Mm-1, and at p4 for Mm-1. At a given frequency increases with increasing horizontal wave number .
Table 1. Inclination of the wave vector with respect to the local vertical direction, for adiabatic waves in an isothermal atmosphere
Let be the squared Fourier amplitude for a pair . For this pair, and for the corresponding angle calculated with (11), we may then calculate the line-of-sight power according to Eq. (8), as a function of . For all values of interest in the present context this function is monotonously decreasing as increases. Two examples which, according to Table 1, represent the modes f and p1, are shown in Fig. 6.
A maximum such as found for the modes f and p1 around (Figs. 4b and 5b) cannot be explained in this way. However, if we allow for an anisotropy in the azimuthal distribution and calculate according to Eq. (7), such a maximum is possible, as also illustrated in Fig. 6 for two values of the inclination 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 Mm-1 and 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 . The f-mode power deviates from the curve only for values of 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.
© European Southern Observatory (ESO) 1999
Online publication: May 21, 1999