6 Conclusion

We have analyzed time-dependent sets of 2-D HD models to obtain size-dependent properties of simulated thermal convection and granulation under solar conditions. The two sets of models treat thermal convection as steady-state convection rolls or as highly dynamic, non-stationary convection. Our basic results can be summarized as follows.

Single-scale convection cells can be divided into 3 groups, depending on their sizes: small-scale (up to $\sim$900 km), intermediate-scale (1000-1500 km, these correspond to sizes of granules in the range of $\sim$600-1000 km) and large-scale (larger than 1500 km) convection cells. Thermal damping due to radiative transfer in horizontal direction determines the properties of the first group to a large extent: their temperature and brightness fluctuations decrease very rapidly with decreasing size. Upflows in the large-scale convection cells are retarded by pressure excess (buoyancy braking), which determines their evolution: large computed granules end basically by fragmentation. In a narrow size range between these two extremes are cells inside which both these effects have a minimum impact on the upflows. This can be deduced from the dependence of granular properties on their size.

We have studied in detail the importance of the zone of high convective instability associated with the partial ionization of hydrogen: ionization effects increase the convective instability of thermal plumes and protect them from thermal damping. There is an asymmetry between hot rising and cool sinking flow in the instability and thermal damping mechanisms (in upflows the ionization effects increase the instability of upflows before at the solar surface radiative damping reduces it again. In downflows the opposite sequence of process takes place). As a consequence for upflows the work of buoyancy force in this zone is roughly independent of their horizontal size, while cool downflows, influenced by radiative heating, show a very strong dependence on their size.

Both kinds of models show a density inversion in subphotospheric layers. It is more pronounced in small convection cells and inside upflows.

From an observational point of view our investigation demonstrate that we can distinguish between non-stationary multi-scale convection and quasi-steady convection from the size dependence of the vertical velocities of small granules at the solar surface. However, they cannot be distinguished in the dependence of temperature or emergent intensity of granules or lanes.

We have compared brightness properties of 2-D simulated granulation with real observations of Hirzberger et al. (1997) and found them to be in qualitative agreement as far as granules are concerned.

For intergranular lanes the theoretical prediction that the brightness of small-scale inhomogeneities decreases linearly with increasing size finds observational confirmation only for the darkest lanes. We expect that the quantitative disagreement between simulations and observations for the range of small-scale lanes may result, at least partially, from problems faced by the observations due to spatial smoothing (produced by seeing and finite telescope resolution) and possible shortcomings from the granule finding algorithm. Simple tests confirm the significant influence of spatial smearing. Hence, based on the simulations presented here we conclude that determining the true size dependence of granular brightness field is beyond the spatial resolution of current observations.

On the other hand, the different dimensions of the studied theoretical (1-D) and observed (2-D) intensity fields are almost certainly responsible for a part of the discrepancy.

Acknowledgements

A. Gadun and S.R.O. Ploner are grateful to the Swiss National Science Foundation for supporting this work. A. Hanslmeier, K. Puschmann, A. Gadun, and K. Pikalov thank the Austrian and the Ukrainian Academies of Sciences for financing the exchange of scientists. A. Hanslmeier acknowledges the financial support from the Austrian Fonds zur Förderung der wissenschaftlichen Forschung. The authors thank Prof. A. Getling for useful comments.