Astron. Astrophys. 346, 633-640 (1999)
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,
, 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
![[EQUATION]](img85.gif)
The line-of-sight velocity is
then given by
![[EQUATION]](img87.gif)
The velocity power spectrum contains the time average of
:
![[EQUATION]](img89.gif)
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
![[EQUATION]](img90.gif)
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
![[EQUATION]](img92.gif)
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
![[EQUATION]](img94.gif)
(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
:
![[EQUATION]](img102.gif)
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:
![[EQUATION]](img112.gif)
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
![[EQUATION]](img115.gif)
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]](img120.gif)
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.
![[FIGURE]](img132.gif) |
Fig. 6. Calculated center-to-limb variation of for inclination angles and , for isotropic and non-isotropic distributions of azimuth angles
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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.
![[FIGURE]](img139.gif) |
Fig. 7. Center-to-limb variation of the total power, ridge power, and f-mode power, in comparison to the curve , 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
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