## 3. Random dispersion relationEqs. (2) and (3) describe the sound wave which propagates in a flowing plasma. In a motionless () plasma the dispersion relation for the acoustic wave reads where and 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 flowIn the case of the space dependent flow, . This implies and We now introduce the dimensionless wave vector Using these quantities the dispersion relation (7) reads where: These integrals can be expressed by the plasma dispersion function which is tabulated by Fried & Conte (1961): where: The asterisk * in Eq. (21) denotes the complex conjugate. ## 3.1.1. ResultsWhile presenting all numerical results we choose a fixed sound
speed km s Fig. 1a shows the frequency difference
as a function of the spherical
harmonic degree 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
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
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
© European Southern Observatory (ESO) 2000 Online publication: July 7, 2000 |