## 6. ModelsThe 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
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 where is the angular frequency,
(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 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 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 p
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 p
A maximum such as found for the modes f and p 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
© European Southern Observatory (ESO) 1999 Online publication: May 21, 1999 |