1 Introduction

A large number of fully nonlinear 2-D and 3-D simulations of compressible convection has been carried out to better understand basic features of solar (stellar) thermal convection. They can be roughly divided into two basic groups according to their aims and the detail way they treat physical processes (Nesis et al. 1992).

The models of the first group were developed to study the major physical mechanisms underlying convectively unstable solar or stellar envelopes. This includes, among others, the simulations of Hurlburt et al. (1984, 1986), Cattaneo et al. (1989, 1990, 1991), Malagoli et al. (1990), Rast & Toomre (1993a, 1993b), Rast (1998). Although they cannot be used for direct comparisons with observations, their simplicity allows individual physical processes to be isolated and investigated in detail (compressibility, turbulence, ionization effects, topology of convective flows, their transport properties, etc.).

The basic aim of the second group of models is the numerical reconstruction of the complicated structure of solar or stellar atmospheres and their envelopes, including a more or less detailed description of radiative transfer and ionization effects in a compressible, stratified, turbulent medium. These models can be applied to the simulation of spectral lines and reproduce numerous observational properties of solar granulation such as spectral line widths, line shifts, and bisector shapes (Dravins et al. 1981, 1986; Nordlund 1984; Steffen 1989, 1991; Steffen & Freytag 1991; Gadun 1986; Atroshchenko & Gadun 1994; Gadun et al. 1999; Asplund et al. 2000b); the distribution of granule sizes (Wöhl & Nordlund 1985; Gadun & Vorob'yov 1996); the evolution of granules (Rast et al. 1993; Rast 1995; Ploner et al. 1998,1999); the reversal of the thermal contrast between granules and intergranular lanes in the photosphere (Nordlund 1984; Steffen 1989; Gadun et al. 1999); correlations between various atmospheric and spectral line parameters (Gadun et al. 1997), and localized anomalies probably related to shocks (Solanki et al. 1996; Gadun et al. 1997). They were used to determine abundance of chemical elements on the Sun (Atroshchenko & Gadun 1994; Gadun & Pavlenko 1997; Asplund et al. 2000a; Asplund 2000).

The basic aim of the present project is a numerical study of the size-dependent properties of 2-D thermal convection carried out for the conditions of the solar photosphere and the underlying superadiabatic zone. The main advantage of 2-D simulations is a rapid gain and thus a thorough insight into the effects of varying free parameters and boundary conditions.

For this purpose, we have computed two sets of 2-D models. The first treats thermal convection as roll-like quasi-stationary flows, which are similar to those constructed by Nelson & Musman (1978) and Steffen et al. (1989). These models are relatively free for problems of large influence of wave-oscillating phenomena on their size-dependence properties, particularly for small-scale cells. The second treats solar convection as a fully non-stationary phenomenon and is a case of more realistic modeling. We have simulated convective and oscillatory motions in a domain with a horizontal size of about 18000 km. Using both sets of models we examine the roles of ionization, radiative transport effects, stability of the flows and their transport properties, concentrating mainly on their size dependence. Finally, we compare the theoretical and observed brightness fields seen as granulation to interpret the observed size dependence of granular brightness. We show here that large scatter in size-dependent properties of small granules can be real because they reflect their various evolutionary history.