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 mesh-points in the horizontal and vertical directions, respectively, and a spatial step of 35 km. Thus the model domain has 17.850 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 11458 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.

N | Grid-points, | Step, | Size, | Note |

hor.ver. | km | km | ||

1. | 20164 | 10 | 1801620 | stable |

2. | 2468 | 24 | 5281584 | stable |

3. | 3860 | 28 | 10081624 | stable |

4. | 5060 | 28 | 13441624 | stable |

5. | 6260 | 28 | 16801624 | unstable |

6. | 7560 | 28 | 20441624 | unstable |

7. | 8760 | 28 | 23801624 | 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).

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