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Astron. Astrophys. 359, 759-765 (2000)

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3. Random dispersion relation

Eqs. (2) and (3) describe the sound wave which propagates in a flowing plasma. In a motionless ([FORMULA]) plasma the dispersion relation for the acoustic wave reads

[EQUATION]

where [FORMULA] and k are the frequency and the x-component of the wavevector, respectively.

We assume now that the speed [FORMULA] is random. As a consequence of that all perturbed quantities have to be expanded as

[EQUATION]

Here, the symbol [FORMULA] denotes the ensemble average and the prime corresponds to the random field which is assumed to be centered, viz.

[EQUATION]

Using the perturbative method of Howe (1971) in which a weak random field approximation is applied, we obtain a turbulent dispersion relation

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

Here, [FORMULA], [FORMULA], is the Fourier transform of the correlation functions:

[EQUATION]

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 ([FORMULA]) 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 [FORMULA], the scattering wave with [FORMULA], and the "inhomogeneous wave" with [FORMULA]. As in any nonlinear process, the interaction is effective if wavevectors and frequencies satisfy the resonance conditions (e.g., Rabinovich & Trubetskov 1989):

[EQUATION]

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 [FORMULA]. As a consequence, from Eq. (11) we have: a) forward scattering with [FORMULA], [FORMULA]; b) backward scattering with [FORMULA], [FORMULA]. Every set of [FORMULA] 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

[EQUATION]

[EQUATION]

Here, [FORMULA] and [FORMULA] are the correlation length and the correlation time, respectively. The variance [FORMULA] is approximated by [FORMULA] as follows:

[EQUATION]

The correlation length [FORMULA], the correlation time [FORMULA], and the variance [FORMULA] correspond to the size, life-time, and the flow of the granules, respectively. A typical size of the granules is about [FORMULA] km although smaller (larger) granules of the size 200 km ([FORMULA] km) are also observed. Observations show that granules have lifetimes of [FORMULA] min. Larger convective motions such as mesogranules live for [FORMULA] hour and supergranules of [FORMULA] 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, [FORMULA]. This implies [FORMULA] and

[EQUATION]

We now introduce the dimensionless wave vector K and the dimensionless frequency [FORMULA],

[EQUATION]

Using these quantities the dispersion relation (7) reads

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

where:

[EQUATION]

These integrals can be expressed by the plasma dispersion function [FORMULA] which is tabulated by Fried & Conte (1961):

[EQUATION]

where:

[EQUATION]

[EQUATION]

[EQUATION]

[EQUATION]

The asterisk * in Eq. (21) denotes the complex conjugate.

3.1.1. Results

While presenting all numerical results we choose a fixed sound speed [FORMULA] km s-1.

Fig. 1a shows the frequency difference [FORMULA] as a function of the spherical harmonic degree l for [FORMULA] km s-1, [FORMULA] Mm (solid line), [FORMULA] Mm (dotted line), and [FORMULA] Mm (broken line). Here, [FORMULA] is the cyclic frequency that is obtained numerically from Eq. (16) and [FORMULA] pertains to the motionless equilibrium ([FORMULA]). The spherical harmonic degree l is related to the wave vector k through the following relation:

[EQUATION]

where [FORMULA] is the radius of the Sun. As a result of the random velocity field the frequency [FORMULA] of the random acoustic mode is lower for [FORMULA] than the frequency [FORMULA] of the coherent wave. For lower values of l the frequency [FORMULA] is higher than [FORMULA]. This effect is greatest for [FORMULA] Mm (solid line) for which [FORMULA] attains its maximum at [FORMULA]. Such a frequency increase was already reported in the other contexts by Collin (1969), Razin (1995), Pelinovsky et al. (1998), and Murawski (2000a).

[FIGURE] Fig. 1a and b. The frequency difference [FORMULA] and the imaginary part of the frequency [FORMULA] as functions of the spherical harmonic degree l for the case of the space-dependent random flow with variance [FORMULA] km s-1. The solid, dotted and broken lines represent the data obtained from Eq. (16) for [FORMULA] Mm, [FORMULA] Mm, and [FORMULA] Mm, respectively. For high values of l, the effect of the random flow is to reduce frequencies and damp the acoustic waves. For low values of l the acoustic waves can be amplified and their frequencies can be shifted up by the random flow.

Fig. 1b presents the imaginary part of the random frequency, [FORMULA], for the same values of the parameters as in Fig. 1a. As in the case of [FORMULA] Mm (solid line), [FORMULA] for the overall range of displayed l, the acoustic wave is unstable. The waves corresponding to [FORMULA] Mm (dotted line) and [FORMULA] Mm (broken line) are unstable for [FORMULA] and [FORMULA], respectively. These waves are amplified by the random flow as their amplitudes grow in time. On the other hand, waves for which [FORMULA] 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 [FORMULA] Mm) amplify the sound waves but large granules (of [FORMULA] Mm) essentially damp the sound waves.

Fig. 2 displays the dependence of the acoustic wave spectrum on the variance [FORMULA] for the fixed value of the correlation length [FORMULA] Mm. As a consequence of stronger random flow the frequency [FORMULA] falls more below [FORMULA] (Fig. 2a) in agreement with former results (e.g., Murawski 2000a,b). Acoustic waves of a negative value of [FORMULA] exist for [FORMULA] for [FORMULA] km s-1 (solid line of Fig. 2b) and [FORMULA] km s-1 (dotted line). For [FORMULA] these waves possess a positive imaginary part of the frequency [FORMULA] and consequently, they are amplified by the random flow. The wave that corresponds to [FORMULA] km s-1 is amplified by the random flow for the overall values of the displayed range of l.

[FIGURE] Fig. 2a and b. As in Fig. 1 but for a random flow with correlation length [FORMULA] Mm and variance [FORMULA] km s-1 (solid line), [FORMULA] km s-1 (dotted line), and [FORMULA] km s-1 (broken line). A stronger random flow reduces the frequencies of the acoustic waves more strongly as well as amplifying them further. The frequency-reduction effect is more pronounced for higher values of l.

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© European Southern Observatory (ESO) 2000

Online publication: July 7, 2000
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