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Astron. Astrophys. 332, 314-324 (1998)

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6. Discussion

The main distinguishing feature of turbulent convection is a random pattern of the temperature and velocity fluctuations. The size of the turbulent elements varies from the largest, defined by forcing, to the smallest, defined by viscous dissipation. However, the effect of the fluctuations on waves declines with the decrease of the size of the turbulent elements. Thus, the main effect of the fluctuations could be assigned to the largest turbulent elements. Unfortunately, the largest convective cells (for example supergranulation on the Sun) do not in general form a perfect lattice. The question arises of whether the effects of a regular convective pattern occur as well in the case of slow turbulent convection.

The effect of sound dispersion appears due to finite scale [FORMULA] in the model. In the turbulent atmosphere the sound dispersion is governed mostly by the largest turbulent elements. The method developed here allows the exact calculation of this dispersion. It probably overestimates the dispersion due to the use of a regular pattern, but underestimates it as the effect of the smaller elements is neglected. In any case there seems to be no other method to calculate the effect.

The present model gives the properties of the vibrational waves, which are nothing else but sound turbulence. Solution (42) shows that these waves are not simple acoustic plane waves, as assumed so far. The physics is clear: the wave functions of the vibrational waves follow the pattern of the temperature and velocity fluctuations. They will have this property also in the case of random fluctuations. The question is whether the banded structure of the frequency spectrumof the vibrational waves survives when the wavenumber spectrum of the fluctuations has a finite lower bound.

The advantage of the Brillouin zone diagram is that it demonstrates the coupling between the acoustic and vibrational waves at [FORMULA]. An indication of the Brillouin zones appeared in the treatment of the f mode in an atmosphere with random density fluctuations (Murawski & Goossens 1993). The wave branch in the [FORMULA] diagram (Fig. 5 of that paper) shows a peculiar behaviour for a wavelength about the correlation length, which can be considered as the size of the largest turbulent element. This indication of Brillouin zones appeared in spite of the use of the Born approximation, which is not valid at the border of the Brillouin zones. Thus, there are reasons to believe that the effect of Brillouin zones will appear in the case of random fluctuations as well.

The model with periodic fluctuations seems to be a necessary step towards the more general treatment of random fluctuations (Zhugzhda, in progress). The periodic model gives a clear physical picture of the wave phenomena in an atmosphere with fluctuations and allows to calculate the effects of convection on waves.

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© European Southern Observatory (ESO) 1998

Online publication: March 10, 1998