4. Dispersion relation for the turbulent () plasma
The solar plasma below the visible layers is a dynamic environment, supporting convection which reveals itself principally on two spatial scales of motion: a large scale supergranulation with horizontal scale of km and flows of 0.1-0.4 km/s and much smaller scale granulation with horizontal scale of km and flows of 1-3 km/s (Simon et al. 1991). Temporal scales associated with these motions range from 30 minutes for granule overturn times to weeks for giant cells. Such dynamic medium is expected to influence the f -mode.
We consider a model in which weak turbulent field is settled in the convection zone. This assumption is valid since the turbulence reveals speeds km/s which are small in comparison to the sound speed km/s. Consequently, we can use the following expansion:
where and represent coherent and random fields, respectively. The symbol denotes ensemble averaging (e.g., Ishimaru 1978). Substituting this expansion into linearized Eqs. (2)-(6) after lengthy algebra in which the binary collision approximation (Howe 1971) has been used, we obtain the dispersion relation
The functions and , , are solutions of Eq. (10). The formulae which determine the right hand side of this equation are presented in Appendix A. Here, denotes the Fourier transform operator evaluated at and .
From dispersion relation (21) it follows that the dependence of the cyclic frequency on the wavevector k differs from non-turbulent dispersion relation (11). The turbulent field changes the f -mode frequency. This change is described by the real part of . As a consequence of scattering by turbulent flow, the energy of the f -mode is partially transformed into the turbulent field (Pelinovsky et al. 1998). This phenomenon is associated with the imaginary part of the frequency, .
© European Southern Observatory (ESO) 2000
Online publication: August 17, 2000