A&A 387, 335-338 (2002)
DOI: 10.1051/0004-6361:20020393
K. Murawski^{1} - L. Nocera^{2} - E. N. Pelinovsky^{3}
1 - Department of Complex Physical Systems, Institute of Physics,
UMCS, ul. Radziszewskiego 10, 20-031 Lublin, Poland
2 -
Institute of Atomic and Molecular Physics, National
Research Council, Via Moruzzi 1, 56124 Pisa, Italy
3 -
Institute of Applied Physics, 46 Ul'anova Street 603600,
Nizhny Novgorod, Russia
Received 16 November 2001 / Accepted 12 March 2002
Abstract
The influence of space- and time-dependent random mass density field,
associated with granules, on frequencies and amplitudes of
the solar p-modes is examined in the limit of weak random
fields and small amplitude oscillations. The p-modes are
approximated by the sound waves which propagate in the gravity-free
medium. Using a perturbative method, we derive a dispersion
relation which is solved for the case of wave noise for which the
spectrum
,
where
is the Dirac's delta-function and
is the random
phase speed. We find that at
a resonance
occurs at which the cyclic frequency
tends to infinity. For
values of
which are close to the resonance point, the
frequency shift may be both negative or positive and the imaginary
part of the frequency attains the negative (positive) sign for
(
).
Key words: convection - Sun: oscillations - turbulence
The solar p-modes are essentially sound waves, for which the
dominant restoring force is pressure. The p-modes propagate if
their cyclic frequency
is higher than the acoustic cutoff
frequency
,
where
is
the specific heats ratio, g is the gravity, and h is the
isothermal density scale-height. These modes satisfy the approximate
dispersion relation
A p-mode is trapped between an upper reflecting layer and a lower turning point. Outside this region, the p-mode is evanescent. The ray paths of the sound waves that do not propagate vertically are bent by the increase with depth in sound speed until they reach the turning point where they undergo total internal refraction. The turning point is located the closer to the solar surface the higher is the spherical degree l. At the solar surface the sound waves are reflected back by the rapid decrease of mass density, where the wavelength becomes comparable to the local density scale height.
The Sun is certainly host to many dynamical phenomena, of which granulation is the most obvious, which are usually neglected in solar modeling. For the recent discussion of the effect of granulation on the solar oscillations see Nocera et al. (2001) and Murawski et al. (2001b). Granules, which exhibit random flows and consequently random mass densities (Stein & Nordlund 1998), might therefore produce observable shifts in the frequencies of sound waves propagating across them. This effect is likely to be particularly pronounced when the waves have a wavelength that is short compared with the spatial scale of the cells. Helioseismology provides a possibility of studying the manifestation of these phenomena in the oscillation frequencies.
The paper is organized as follows. In Sect. 2 we present the dispersion relation for the random p-modes. In this section, we investigate the influence of the random mass densities on frequencies and amplitudes of the solar modes. This paper is concluded by the presentation of the main results in Sect. 3.
The solar p-modes are described by the equations of hydrodynamics, with the gravity and radiative transfer terms included (Gough 1994). These general equations appear to be quite complicated even for numerical treatment. Consequently a fruitful approach to approximate these equations to a point where they can be easily discussed is proposed in this paper. It offers an insight into the behavior of the modes which are approximated locally by the sound waves (Swisdak & Zweibel 1999; Dziembowski 2000). As a consequence of this assumption, the present analysis is performed for a uniform, not a stratified, unperturbed state. Obviously, such a state is not quite realistic and there are no velocity fluctuations. However, taking into account the realistic state is a formidable task and therefore we limit our discussion to the simpler case of the random mass density field which is driven by velocity fluctuations.
Using the standard method of random wave theory for small amplitude sound waves and weak random mass density field (e.g. Nocera et al. 2001; Murawski et al. 2001a) we derive the following dispersion relation:
From the dispersion relation of Eq. (2) it follows that the random sound waves are no longer dispersionless. We will see in the forthcoming part of this paper that sound waves experience not only frequency shift but also amplitude alteration due to a presence of the random field.
Special limits of Eq. (2) were already considered in the
literature. The case of a "frozen" (space-dependent) random field
was discussed by
Nocera et al. (2001). The main conclusion was that the sound waves were
attenuated by any random field. For the Gaussian statistics,
The case of the time-dependent random field was discussed by Murawski et al. (2001a) who showed that such random field leads to energy transfer from the random field to the sound waves and consequently to wave amplification. Moreover, for the Gaussian random field the frequencies of the sound waves were lifted up, similarly as in the case of the space-dependent random field.
It is natural then to inquire about the properties of the sound waves
which propagate through the medium of space-and time-dependent random
mass density field. The study of the doubly singular Cauchy integral
associated with a fully space and time dependent random density
spectrum in Eq. (2) will be left to a forthcoming paper.
In the the following part of this work, we concentrate on the simpler
case of wave noise.
Figure 1: Real (top panel) and imaginary (bottom panel) parts of . | |
Open with DEXTER |
Figure 2: Real (solid lines) and imaginary (dashed lines) parts of versus K for (top panel) and (bottom panel). | |
Open with DEXTER |
Figure 3: Real (solid line) and imaginary (dashed line) parts of for K=2. | |
Open with DEXTER |
We define wave noise through the spectrum
(6) |
We now consider a sound wave whose wavenumber is K. In view of the
smallness of
,
its frequency
given by the
dispersion relation in Eq. (2), can be expanded as
(9) |
For
Eq. (8) can be rewritten as follows:
Real and imaginary parts of which follow from Eq. (12) are displayed in Fig. 1 versus the wavenumber K and the random phase speed . As everywhere except in the region close to , we claim that the sound waves are speeded up by the wave noise there. Indeed, the top panel of Fig. 2 shows that for the real part of the frequency shift is positive for all values of K. The imaginary part of the frequency shift is negative for all values of K. As a consequence of that we claim that for the sound waves are accelerated and attenuated for all K. The bottom panel of Fig. 2 illustrates that for the real and imaginary parts of the frequency shifts are positive and the sound waves are accelerated and amplified by the wave noise.
At the place where the phase speed of the wave noise equals the sound wave speed a resonance occurs. Figure 3 shows this resonance for K=2. Note that the resonance is of the -type; for ( ) the real and imaginary parts of the frequency shift are negative (positive) and the sound waves are decelerated and attenuated (accelerated and amplified) there. The wave deceleration and attenuation can be explained on physical grounds as for the sound wave interacts with slower propagating wave noise. This process is accompanied with an energy transfer from the sound wave into the wave noise, leading to the sound wave deceleration and attenuation. On the other hand, in the regime the wave noise moves quicker than the sound wave and the energy is transferred into the latter one. As a consequence of that the sound wave is amplified and speeded-up.
We draw the attention to the fact that, in the framework of our perturbative approach, the solutions we provide in Eqs. (10)-(12) and in Figs. 2-3, are consistent only if remains . In Fig. 2, this holds true only if K<5 and, in Fig. 3, only if .
Acknowledgements
This work was financially supported by the State Committee for Scientific Research in Poland, KBN grant No. 2 PO3D 017 17, the Italian Research Council, and INTAS 97-31931. The authors express their thanks to an unknown referee for his/her comments on the earlier version of this paper.