3. Random dispersion relation
where and k are the frequency and the x-component of the wavevector, respectively.
We assume now that the speed is random. As a consequence of that all perturbed quantities have to be expanded as
Here, the symbol denotes the ensemble average and the prime corresponds to the random field which is assumed to be centered, viz.
Using the perturbative method of Howe (1971) in which a weak random field approximation is applied, we obtain a turbulent dispersion relation
Here, , , is the Fourier transform of the correlation functions:
It is noteworthy that the right hand side of Eq. (7) contains a random correction to the acoustic dispersion relation. In the case of stationary plasma () this equation reduces to Eq. (4) which is valid for a coherent acoustic wave. The random flow modifies wave frequencies as a result of scattering which can be considered as a nonlinear interaction between the incident wave with , the scattering wave with , and the "inhomogeneous wave" with . As in any nonlinear process, the interaction is effective if wavevectors and frequencies satisfy the resonance conditions (e.g., Rabinovich & Trubetskov 1989):
In the case of a space-dependent random field, the frequency of the inhomogeneous wave is zero. Therefore, the scattered wave should have the same frequency as the incident wave. Then, from dispersion relation (4) it follows that . As a consequence, from Eq. (11) we have: a) forward scattering with , ; b) backward scattering with , . Every set of defines the interacting triad. This is the Bragg condition for wave scattering. This scattering scenario is similar in the case of spatial or temporal random fields. The corresponding wavenumbers and frequencies of the scattered and inhomogeneous waves can be found from Eqs. (11) and (12).
The energetics of the wave interactions in media for which equilibrium conditions are satisfied can be described with a use of the Manley-Rowe relations (e.g., Pelinovsky 1979, Rabinovich & Trubetskov 1989). In this case, if the incident wave interacts with the inhomogeneous wave of low frequency a part of its energy is transformed into energy of the scattered wave. As a consequence of that, the energy of the incident wave is decreased. On the other hand, if the incident wave interacts with the inhomogeneous wave of high frequency its energy and the energy of the scattered wave is increased due to parametric instability of the high-frequency noise. A global picture of the interaction depends on the number of triads for which the incident wave frequency is increased or decreased. This scheme was used by Pelinovsky (1979) for analysis of the wave propagation in a random inhomogeneous ocean. Unfortunately, the case of a non-equilibrium medium, such as one with a random flow, is more complicated. In this case, the Manley-Rowe relations are not satisfied and the above scheme cannot be applied. Thus, the energetics of the process should be computed directly from Eq. (7).
Assuming the Gaussian process we have
Here, and are the correlation length and the correlation time, respectively. The variance is approximated by as follows:
The correlation length , the correlation time , and the variance correspond to the size, life-time, and the flow of the granules, respectively. A typical size of the granules is about km although smaller (larger) granules of the size 200 km ( km) are also observed. Observations show that granules have lifetimes of min. Larger convective motions such as mesogranules live for hour and supergranules of day lifetimes have been detected. Moreover, giant cells persisting for a solar rotation period may also exist.
As dispersion relation (7) is complex we consider first the case of a space-dependent random flow only.
3.1. Space-dependent random flow
In the case of the space dependent flow, . This implies and
We now introduce the dimensionless wave vector K and the dimensionless frequency ,
Using these quantities the dispersion relation (7) reads
These integrals can be expressed by the plasma dispersion function which is tabulated by Fried & Conte (1961):
The asterisk * in Eq. (21) denotes the complex conjugate.
While presenting all numerical results we choose a fixed sound speed km s-1.
Fig. 1a shows the frequency difference as a function of the spherical harmonic degree l for km s-1, Mm (solid line), Mm (dotted line), and Mm (broken line). Here, is the cyclic frequency that is obtained numerically from Eq. (16) and pertains to the motionless equilibrium (). The spherical harmonic degree l is related to the wave vector k through the following relation:
where is the radius of the Sun. As a result of the random velocity field the frequency of the random acoustic mode is lower for than the frequency of the coherent wave. For lower values of l the frequency is higher than . This effect is greatest for Mm (solid line) for which attains its maximum at . Such a frequency increase was already reported in the other contexts by Collin (1969), Razin (1995), Pelinovsky et al. (1998), and Murawski (2000a).
Fig. 1b presents the imaginary part of the random frequency, , for the same values of the parameters as in Fig. 1a. As in the case of Mm (solid line), for the overall range of displayed l, the acoustic wave is unstable. The waves corresponding to Mm (dotted line) and Mm (broken line) are unstable for and , respectively. These waves are amplified by the random flow as their amplitudes grow in time. On the other hand, waves for which are damped by the random flow as their amplitudes decrease in time. As a consequence of that we claim that granules of different size influence differently the sound waves; small granules (of the size of Mm) amplify the sound waves but large granules (of Mm) essentially damp the sound waves.
Fig. 2 displays the dependence of the acoustic wave spectrum on the variance for the fixed value of the correlation length Mm. As a consequence of stronger random flow the frequency falls more below (Fig. 2a) in agreement with former results (e.g., Murawski 2000a,b). Acoustic waves of a negative value of exist for for km s-1 (solid line of Fig. 2b) and km s-1 (dotted line). For these waves possess a positive imaginary part of the frequency and consequently, they are amplified by the random flow. The wave that corresponds to km s-1 is amplified by the random flow for the overall values of the displayed range of l.
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000