The computational method and the boundary conditions used here to simulate 2-D granular convection were described in the papers of Gadun (1995), Gadun & Vorob'yov (1996), Gadun & Pavlenko (1997), and Gadun et al. (1999). One set of models analysed here has $512\times58$ mesh-points in the horizontal and vertical directions, respectively, and a spatial step of 35 km. Thus the model domain has 17.850 $\times$ 2.030 Mm, since the cells at the right and left model boundaries are used to treat the lateral boundary conditions. The model atmosphere occupies about 800 km at the top of the model. The time step was 0.3 s and a total of 71 600 time steps, corresponding to 5 h 58 min of solar (hydrodynamic) time, were calculated. Of these the last 5 hours were analyzed.

The calculations were carried out with open upper and lower boundary conditions and periodical conditions at the lateral boundaries. The frequency dependence of the continuum absorption coefficient was included in 97 frequency intervals when treating the radiative energy transport. Absorption due to atomic lines was also considered in the frame of the ODF (opacity distribution function) approach by Kurucz (1979). To solve the system of hydrodynamic equations we used the large particle method (Belotserkovskiy & Davydov 1982). In order to conserve computational resources we first periodically extended the fully relaxed models with a smaller spatial domain and periodic lateral conditions (they had 114 $\times$ 58 mesh-points) in the horizontal direction by introducing small variations of the input parameters into the extended model to ensure asymmetric solutions. This was used as the initial condition for the present simulations. Since the simulated solar convection is fully unstable, with interacting flows at different scales, we call them multi-scale models (ms models) in the follows.

To better examine scale-dependent properties of simulated granulation we have also computed a grid of single-scale steady-state models, each with a different horizontal size of the computational domain (hereafter called single- scale or ss models). Their scale-dependent properties are significantly less suffer from wave-oscillating effects and evolutionary history. This is particularly important for subsequent analysis of small-scale inhomogeneous. Their basic parameters are given in Table 1. The model atmospheres were roughly 500-550 km deep in all the models. Note that it is mainly the restricted horizontal extent of the domain which leads to the less turbulent, quasi-stationary flows of these models, since the treatment of the physical processes is otherwise very similar to the ms models.

**Table 1:** Computed single-scale steady-state models
N	Grid-points,	Step,	Size,	Note
	hor. $\times$ ver.	km	km
1.	20 $\times$ 164	10	180 $\times$ 1620	stable
2.	24 $\times$ 68	24	528 $\times$ 1584	stable
3.	38 $\times$ 60	28	1008 $\times$ 1624	stable
4.	50 $\times$ 60	28	1344 $\times$ 1624	stable
5.	62 $\times$ 60	28	1680 $\times$ 1624	unstable
6.	75 $\times$ 60	28	2044 $\times$ 1624	unstable
7.	87 $\times$ 60	28	2380 $\times$ 1624	unstable

We used a simple grey approach to treat radiative energy transfer for the initial duration of about 1 hour of real solar (hydrodynamic) time, i.e. in the interval before the statistically stable regime of the simulations was reached. The simulations of convective motions in each model domain were continued for another hour beyond this moment, but now including a detailed treatment of radiative transfer. This includes transfer effects in molecular lines within the framework of the ODF tables of Kurucz (1993).

2 Two-dimensional hydrodynamics