3 General characteristics of the models

The analysis of every set of single-scale models was based on about 1 hour of time-sequence computed, using the detailed description of radiative transfer. The temporal step between the analysed models was also 30 s. Note, however, that three models from our list of ss models (Table 1), with computational domain sizes larger than or equal to 1680 km, gave unstable solutions. For this reason, all characteristics of these unstable models were determined during their stable phase, prior to the onset of the instability. Thus, for these models the analyzed time was 30-40 min. The analysis of the ss models is described in Sect. 3.1. In Sect. 3.2 we describe the analysis of 5 hours of the time-dependent multi-scale simulation, with a 30 s time step between the studied models.

Note that in the classical sense such terms as granule or intergranular lane can be used only to define "bright'' and "dark'' elements of the brightness field at the solar surface and cannot be invoked to separate convective flows (below the surface, for instance) according to their properties. However, we can prescribe a number of additional characteristics (temperature, pressure, velocity, etc.) for granules and intergranular lanes, if they are defined inside such locations on the surface where we observe the corresponding brightness patterns and if these additional quantities are found for the level where emergent intensity forms. We define the (solar) surface as the mean continuum level, i.e. at $\log \overline{\tau_{\rm R}} = 0$, where $\overline{\tau_{\rm R}}$ is the Rosseland optical depth averaged over a horizontal level and over the whole time of simulation.

In the model analysis we shall discern between the so called "full'' averages of model parameters (which correspond to an average over the whole horizontal extent of the simulation) and means computed separately over up- and downflows. Both are determined for a given geometrical level by averaging model data over the total simulation time. The mean parameters of up- and downflows are computed separately for those horizontal positions which are occupied by rising or sinking gas. We have also computed extreme values of the physical parameters of up- and downflows at a given horizontal level.

We additionally introduce two classes of quantities to characterize the spatial variations within up- and downflows. They will be called 1) excess or deficit (of temperature, density, etc.), determined as the difference between mean values in up- or downflows and the corresponding "full'' average; and 2) the maximum excess or deficit of a parameter if we study the difference between its extreme value and the "full'' average.

Figure 1: Development of an instability in the central upflow of a convection cell with a size of 2380 km. The velocity field, isotherms, and regions with pressure and density excesses are shown at five instances during the cell's evolution (from top to bottom). The horizontal lines denote isotherms (from bottom to top): 14 000, 12 000, 10 000, 8000, 7000, 6000, 5000, and 4500 K. Relative pressure ( $P/\overline {P}$) and density ( $\rho /\overline {\rho }$) excesses above 1.001, 1.01, 1.1, and 1.3 at any horizontal level are indicated by shading

3.1 Stable convection: Single-scale models

Single-scale models describing steady-state thermal convection are presented here. They can be briefly characterized as a treatment of granular convection in which

• convective flows are quasi-stationary in time. Moreover, convection cells do not change their horizontal size in the course of their evolution;
• convection rolls (overturning cells) of any studied scales always extend to the largest depth allowed by the computational box (about 1100 km below the surface in the present models). This means that granular convection is not a surface (shallow) phenomenon in this approach. Rather, its properties are determined by upflowing thermal plumes. Partial exceptions are the unstable ss models (Table 1), which are discussed in detail later.

In spite, or possibly because of their simplicity, the ss models can help us to understand some key results obtained from more realistic multi-scale models mostly because their basic size-dependent properties are less affected by wave and oscillations. Namely, they permit us to

• determine scale-dependent properties of synthetic granulation in a more direct manner;
• study the problem of the instability of the upflow in connection with the sizes of the convection cells;
• estimate contributions of surface effects (i.e., formation of new granules at the surface due to fragmentation of large-scale thermal flows) into observed granular pattern.

Steady-state solutions like the ss models presented here were earlier obtained also by Nelson & Musman (1978) and Steffen et al. (1989).

3.1.1 General evolution

Our model grid consists of 7 ss models with horizontal sizes of the computational domain ranging from 180 km to 2380 km (Table 1). Three models, having horizontal sizes 1680, 2040, and 2380 km, are unstable. Their convection cells fragment in a similar way as that shown in Fig. 1. Figure 1 illustrates the velocity field, isotherms and regions with pressure and density excesses at various instances during the evolution of model with horizontal size 2380 km. Fragmentation starts due to a weakening of the central flow which occurs as a result of the force exerted by the pressure gradient there (buoyancy braking, Spruit et al. 1990 and references therein). Based on these calculations we can note that it has the largest impact in layers lying deeper than the zone of high convective instability (it is discussed later). Figure 1 shows gas can still move upwards inside the high instability zone, although in deeper layers a downflow has already developed. Among our ss models we distinguish between those of the small cells, which have horizontal sizes smaller than about 800-900 km, the large cells, consisting of ss models larger than $\sim$1500 km, which exhibit unstable solutions, and intermediate cells having, sizes in the range 900-1500 km. The reasons of such a division will become clear in the next section.

Figure 2: a) rms horizontal velocities, b) rms vertical velocities, and c) their ratio, shown for five ss models: dotted lines are models with horizontal size 180 km, dash-dotted lines represent models with size 528 km, dashed lines are convection cells with a width of 1008 km, long dashes are those of 1680 km, solid lines are the largest models, having a width of 2380 km

3.1.2 Velocity field

In Fig. 2 the rms horizontal and vertical velocities as well as their ratio are plotted versus height for five ss models. Peak velocities and other quantities are plotted in Fig. 3 as a function of convection cell size. Figure 3a displays maximum values of rms horizontal and rms vertical velocities and Figs. 3b and c show the peak up- and downflows, respectively. We also represent the minimum fractions covered by downflows (Fig. 3d - "area'' filling factor).

Figure 4 reflects the heights at which the peak velocities are found: given in Fig. 4a are photospheric heights at which horizontal velocities reach their maximum values (cf. Fig. 3a). The depths at which the up- and downflow maximum, plotted in Figs. 3b and c, are located are represented in Figs. 4b and c.

Figure 3: Dependence of extreme properties of vertical and horizontal velocities on convection cell size. a) Maximum of photospheric rms horizontal velocities (squares) and maximum of rms vertical velocities (dots) in subsurface layers. b) Horizontally averaged (solid line) and the largest (dashed line) vertical velocities in upflows. For each model the velocities are plotted for the layer at which they are the largest. c) Same as frame b, but for downflows. d) Minimum values of "area'' filling factors of downflows

As follows from Figs. 2a and 3a the horizontal velocities are highly sensitive to the horizontal size of convection cells L (see Steffen et al. 1989). For instance, maximal photospheric rms horizontal velocities are strongly dependent on L: $U_{\rm rms, max} \sim L^{0.86}$and are systematically shifted into higher photospheric layers for larger flows (Fig. 4a), from 20 km to 180 km above the surface for the cell sizes studied here. This behaviour is explained due to mass balance. These results are compared with those of Steffen et al. (1989) in Sect. 5.

Figure 4: Heights and depths at which the largest up- and downflow velocities are reached as functions of cell size. a) Height of maximum photospheric horizontal velocity. b) Depths (below the surface) at which the maximum vertical velocities in upflows are located: for horizontally averaged flows (solid line) and for the largest speeds (dashed line). c) The same as b) but for downflows

The vertical velocities are less sensitive to the cell size, except for small cells. The general tendencies in the behaviour of vertical velocities are:

• in convectively unstable subphotospheric layers smaller cells exhibit larger rms vertical velocities (Figs. 2b and 3b);
• in the photosphere three layers can be recognized (Fig. 2b):

  i)

  a region in which the subphotospheric velocity distribution still dominates (up to $\sim$50 km above the surface);

  ii)

  a region (from 50 to 100 km) where the vertical velocities are nearly cell-size independent;

  iii)

  a region in which the vertical velocities produced by small-scale convective flows become several times smaller than the velocities of cells larger than 1000 km. These possess very similar vertical velocities there.

Granular convection is characterized by geometrical asymmetry between up- and downflows (Cattaneo et al. 1989; Spruit et al. 1990; Hanslmeier et al. 1991). Below we examine differences in properties between up- and downflows, again concentrating on their dependence on cell size.

According to Fig. 3b the largest upflow velocities are nearly constant over a wide range of cell sizes from 500 to 2400 km and they reach their maximal values near the surface (Fig. 4b): the larger the cell size the closer to the surface this happens (at depths of 60 km to 20 km below the surface).

It also follows from Fig. 3b that the mean velocities of horizontally averaged upflows possess a more obvious dependence on the cell size - larger flows have smaller averaged upflows. Moreover, they attain their maximum values (Fig. 4b) at deeper levels for larger flows (from 70 km to 100 ,km below the surface). Hence, the general tendency for upflows is: small-scale flows have larger mean vertical velocities.

Figure 3c shows that the largest downflow grows rapidly with increasing cell size and (Fig. 4c) is reached at increasing depth: from 400 km to 550 km below the surface.

However, horizontally averaged downflow velocities (also shown in Fig. 3c) are relatively independent on cell size. Also, these velocities are largest roughly at the same depth $\sim200-300$ km below the surface.

The asymmetry between down- and upflows becomes less pronounced for small-scale cells. This is seen from the comparison between their absolute up- and downward directed velocities (Figs. 3b and c), which lie closer to each other for small cells, as well as from the filling factors of sinking flows (Fig. 3d). In Fig. 3d we plot the minimum fractions of downflows as a function of cell size. As expected for more symmetrical flows the filling factor goes to 0.5 for smaller cells.

Figure 5: Spatially and temporally averaged stratification of temperature a), pressure b), and density c) in the five simulations plotted in Fig. 2. Designations of the models correspond to Fig. 2. Pressure and density are given in CGS

3.1.3 Thermodynamic quantities

Modeled stratifications of temperature, pressure, and density are shown in Fig. 5. In the lower photosphere (heights from 0 to +200 km) larger cells (long dashed and solid line, cf. Fig. 2) have lower temperature, pressure, and density. However, below the surface, at a depth range from 50 to 150 km, the larger cells are denser than small convection cells due to a density inversion which is particularly pronounced in the latter.

The density inversion occurs in almost all the models (the density at the surface is larger or roughly equal to the density at a depth of 150-170 km).

Figure 6: rms fluctuations of temperature a), pressure b), and density c) of the five ss models as a function of height. Models and designations are the same as in Fig. 2

Figure 7: Temperature stratifications of the five ss models (designations correspond to Fig. 2). a) The hottest fragments in the upflows of convection cells. b) Mean temperatures of upflows. c) The coolest parts of downflows. d) Mean temperatures of the downflows

Plotted in Fig. 6 are rms fluctuations of model temperature, pressure, and density. As expected to be, in each model the behaviour of temperature fluctuations and density variations in the convectively unstable regions are in accordance with the distribution of the rms vertical velocities (Fig. 2b). In the atmospheric layers the spatial pressure and density fluctuations correspond to horizontal velocities.

Figure 8: Size dependence of extreme values of temperature fluctuations. a) Maxima of rms temperature fluctuations in the envelope, i.e. below the surface. b) Maxima of photospheric rms fluctuations. c) Heights at which sign reversals of the temperature fluctuations are found

Temperature. Some statistical aspects of behaviour of temperature inside computational domains are given by series of figures. Figure 7 shows temperature stratifications of upward- and downward-directed flows, separately. Figure 8 represents dependence of maxima of rms temperature fluctuations on cell size. We give rms temperature fluctuations in the envelope (Fig. 8a) and in the photosphere (Fig. 8b). In Fig. 8c we show those photospheric heights at which atmospheric temperature fluctuations change their sign. Figure 9 exhibits relative temperature excess/deficit in upflow. In Fig. 10a we display peak values of the temperature excess sampled over depth as a function of convection cell size. We give two values: maximum temperature excess of the horizontally averaged hot upflow and the largest temperature excess due to the hottest part of the upflow. Both excesses are relative to the "full'' average temperature at the same geometrical level. The temperatures in upflows are plotted in Figs. 7a and b. The peak values of temperature deficits in downflows are given respectively in Fig. 10b. Their temperature stratifications can be found in Figs. 7c and d. And depths at which these peak values reach their maximum values are shown in Fig. 11a for upflows and in Fig. 11b for downflows.

Figure 9: Relative temperature excess of upflows. Designations of the models are the same as given in Fig. 2

Figure 10: Size dependence of extreme excesses a) or deficits b) of temperature in up- and downflows. Solid lines are peak values in horizontally averaged up- and downflows relative to the mean temperature; dashed lines are peak values in the hottest and coolest parts of these flows compared with the mean temperature

Figure 11: Depths at which peaks of temperature excess or deficit are located, a) for upflows, b) for downflows vs. convection cell size. Designations are found in Fig. 10

The main tendencies in size-dependent properties of temperature stratification and temperature spatial variations can be outlined as the follows:

• in small-scale convective cells the temperature gradient below the surface is steeper, on average, than in larger cells (Fig. 7b);
• However, the hottest upflow fragments display the opposite dependence of temperature on cell size: in larger cells the hottest subflows also have the largest temperature gradients (Fig. 7a) which have the tendency to occur at greater heights;
• On average, downflows of larger cells are cooler than those of smaller cells to depths of about 300 km below the surface. In deeper layers the temperatures of downflows seem to be similar in all the models (Fig. 7d);
• The coolest fragments always show lower temperature for larger cells (Fig. 7c);
• In subphotospheric layers the rms temperature fluctuations are higher for larger convection cells (Figs. 6a and 8a). However, this relation is not monotonous. It differs for cells larger and smaller than about 1000 km. Both groups show this tendency but in cells smaller than 1000 km the temperature inhomogeneities dissipate faster with decreasing horizontal size (Fig. 8a) than they do for the larger cells;
• As can be seen from Fig. 10a the maximal temperature excess in the horizontally averaged upflow is roughly scale-independent, while its value for the hottest part of the upflow is greater in larger cells. For cells with sizes of about 2400 km it is 4 times greater than that for the averaged upflow. This suggests that inside large upflows there are hotter portions with properties considerably different from those of the mean flow;
• Below the surface, in downflows the temperature deficit increases with cell size. Moreover, the mean downflows and their parts show similar deficits in dependence on cell size, suggesting that downflows are horizontally more homogeneous;
• In the photospheric layers the larger convection cells show larger temperature fluctuations (Fig. 8b). These values are close to semiempirical estimates made by Altrock (1976), Keil & Canfield (1978), and Kneer et al. (1980). Moreover, the temperature fluctuations also display a sign-reversal in the photosphere, in agreement with earlier investigations (e.g. Nelson & Musman 1977; Nordlund 1984; Steffen 1989; Steffen et al. 1989; Stein & Nordlund 1998; Spruit et al. 1990). This is seen in Fig. 9 where the relative temperature excess/deficit in upflows relative to the horizontally averaged temperature is plotted for various ss models. Note how above a certain height the temperature excess becomes a deficit. The geometrical heights at which this occurs are given in Fig. 8c. They vary from 70 to 170 km depending on cell size.

The depths at which $\Delta T$of horizontally averaged upflows and those at which the $\Delta T$ their horizontally hottest parts reach their maximum values are shown in Fig. 11a. The depths seen in Fig. 11a coincide with the depth at which the upflows are largest (Fig. 4b). The coincidence between the depths at which downflow velocities and $\Delta T$ in sinking flows reach their extremes is much poorer (Figs. 4c and 11b).

Density. Density stratifications in up- and downflows are given in Fig. 12. For upflows (Fig. 12a) we notice a density inversion for all the cells studied. In contrast, the density inversion, being a result of dynamic compression of the gas, occurs only in downflows associated with the largest convection cells (Fig. 12b).

Figure 12: Density stratification in the models. a) Mean density of upflows; b) mean density of downflows. Densities are given in g cm-3. Designations of the models are the same as in Fig. 2

Figure 13: Amplitudes of density excess (or deficit) in up- and downflows relative to the mean density at a given geometrical level: a) upflows, b) downflows

In Fig. 13 relative amplitudes of horizontal density fluctuations are plotted versus height. The layer, lying just below the surface, in which the convective instability is most pronounced, is clearly discernable (in both up- and downflows, Figs. 13a, b). As expected, the density fluctuations have the opposite sign of the temperature variations in the convectively unstable layer and their depth dependence is similar to those found for stratifications of temperature fluctuations.

For instance, the mean densities of upflows, shown in Fig. 13a, exhibit a sharp drop just below the surface, relative to the average over the whole cell. The magnitude of this drop is roughly scale-independent for cells larger than 1000 km (Fig. 13a). The peak deficits occur at the same depths, at which the peak of $\Delta T$ is localized (Figs. 9, 11a, and 13a). The density deficit of the least dense parts of upflows increases with cell size in agreement with the growth of the maximum temperature, given in Fig. 10a.

In subphotospheric layers the maximum density excess in downflows, and hence the buoyancy driving in them, grows with increasing cell size. The temperature deficit in downflows exhibit a similar dependence (Fig. 10b). Depths at which the $\Delta \rho$ reach their peaks also agree with the depth at which $\Delta T$ reaches its peak in downflows.

In the low and middle photosphere the density in upflows becomes larger than the average as a result of radiative cooling of hot gas and its compression by the rising flow. This is shown in Figs. 1 and 13a. In contrast, downflows are relatively less dense there (Fig. 13b). Maxima of $\Delta \rho_{\rm rms}$ in the photosphere depend almost linearly on cell size. The photospheric heights at which $\Delta \rho_{\rm rms, max}$ is located decreases slightly for larger cells and lie in the range of 50 to 100 km above the surface. The height of $\Delta \rho_{\rm rms, max}/\overline {\rho}$ increases from $\sim$100 to 300 km with growing convection cell size due to a wider peak of $\Delta \rho_{\rm rms}$ belonging to the largest cells and decreasing density in the photosphere.

Figure 14: Height dependence of pressure excesses in up- and downflows. a) Excesses of horizontal mean pressure in upflows are shown relative to the minimal pressure at the same geometrical height, which is located where velocities are nearly horizontal. b) Excesses of mean pressure in downflows relative to the minimal pressure. c) Relative difference between the horizontally averaged pressure in up- and downflows. Designations are the same as in Fig. 2

Pressure. In compressible convection the pressure fluctuations play a significant role in the evolution of structures (Hart 1973; Massaguer & Zahn 1980; Hurlburt et al. 1984; Rast 1998). In our simulations $H_{\rm P}$ (the pressure scale height) changes from about 100 km at the top of the model, over $\sim$130 km near the surface, to 450 km at the bottom of the computational domain. Thus non-local pressure fluctuations cannot be neglected (see Nesis et al. 1999).

Moreover, pressure fluctuations define geometrical topology of convective flows in stratified medium (Massaguer & Zahn 1980; Hurlburt et al. 1984). Positive pressure fluctuations are significant attribute of both: downward- and upward-directed flows. But to do work over long return path of convective flows the pressure inside upflows must be larger than in downflows. Then pressure perturbations increase buoyancy driving of downflows (making them faster and narrower) and build up buoyancy braking in upflows (converting them to broader and gentler structure).

In the ss models the pressure fluctuations are essentially different in layers near the surface, deeper zone of high convective instability located at 100-200 km below the surface, and in the quasi-adiabatic envelope below it. This can be seen from Figs. 1 and 14.

In deep layers (starting from 400-500 km below the surface), on average, the pressure in upflows is larger than in downflows (Figs. 1 and 14c) and the regions of maximum pressure are located in rapid, hot, and rarefied currents (Fig. 1). On an absolute scale the pressure fluctuations increase with depth (in the deep layers) in contrast to the relative values plotted in Fig. 14. From around -150 km below the surface through at least the lower photosphere the average pressure is considerably larger than in downflows. This is particularly true for the largest convection cells. Note that minimum pressure at a certain horizontal level usually occurs in a region close to the horizontal edge of the upflow where the velocity field is nearly horizontal. In downflows the pressure is always higher than this minimal pressure, with the exception of small-scale cells (L < 600 km). However, relative to the pressure averaged over the whole horizontal extent of the convection cell, the pressure averaged over downflows is often smaller.

Just below the zone of maximum convective instability (see the next section) the steady-state models exhibit an reversal of the pressure fluctuations: on average, the pressure in upflows is lower (Fig. 14c). In this series of ss simulations we have postulated closed lateral boundary conditions. The closed lateral "walls'' of the model domain do not permit the hot rising flows to expand and so to reduce their high pressure. Thus, the downward-directed streams are squeezed much more strongly than can be expected for a periodical treatment of the lateral boundaries. This increases the speed of the sinking gas, reducing their "area'' filling fraction, and, finally, leads to the development of inversion of pressure fluctuations.

As can be seen from Figs. 1 and 14a, at the surface very high pressure fluctuations develop above upflows due to effective cooling there (making the surface a thermal boundary for adiabatic convective flows: Spruit et al. 1990; Schüssler 1992; Rast et al. 1993). This excess pressure produces horizontal velocities in the photosphere.

Figure 15: Height of the largest pressure fluctuations, $P_{\rm rms, max}$, in the upper part of the models vs. size of the convection cells. Squares denote the absolute values of the fluctuations; triangles are fluctuations relative to the mean pressure at that height: $P_{\rm rms, max}/\overline {P}$

The height at which this $\Delta P_{\rm rms, max}$is reached in the photosphere are shown in Fig. 15. We plot two relations: $\Delta P_{\rm rms, max}$ (squares) and its value normalized to the horizontally averaged pressure (given by triangles). The maximum pressure fluctuations occur at the surface, between the peaks of photospheric density fluctuations and those of temperature.

A pressure excess (at a fixed horizontal level) is also seen near downflows in the photosphere (Figs. 1 and 14b). It directs horizontal velocities downward. But averaged over downflow it is lower than that above upflows.

3.1.4 Zone of convective instability

In general, the properties of compressible solar convection are controlled by stratification, radiative heating and cooling (thermal conduction), turbulence, and ionization effects. In this section we focus on layer containing of partially ionized hydrogen in the context of its role in granular convection.

Figure 16: Stratification of the ratio of specific heat at constant pressure ($C_{\rm P}$) to that at constant volume ($C_{\rm V}$) in up- a) and downflows b), as well as $\log C_{\rm V}$ in rising c) and sinking d) gas. The curves represent convection cells of different sizes. $C_{\rm V}$ is in units of ergg-1K-1. Designations are those given in Fig. 2

The partial ionization of hydrogen leads to the development of a pronounced convective instability in the corresponding. It produces clear signatures in the horizontal fluctuations of thermodynamic quantities as well as in the velocities.

Efficient ways in which ionization can influence the basic thermodynamic properties of a gas are: changes of specific heat, opacity, internal energy, and pressure (Rast et al. 1993).

Figure 17: Fraction hydrogen that is ionized relative to the total hydrogen number density in a) up- and ( b) downflows. Designations are the same as in Fig. 2

Due to the ionization of hydrogen both the specific heat at constant volume ($C_{\rm V}$) and that at constant pressure ($C_{\rm P}$) grow by an order of magnitude in the depth range from the surface to about 200 km below it in upflows and down to 400 km in downflows. This process is illustrated by Fig. 16, where the height-dependence of the ratio $\gamma = C_{\rm P} / C_{\rm V}$and $\log C_{\rm V}$are shown for up- and downflows.

In upflows changes in $\gamma$ and the specific heat in upflows depend only weakly on cell size and occur within a narrow depth range. In downflows, however, they change over a broader depth range.

These values can be compared with the fraction of ionized hydrogen, plotted in Fig. 17. It is obvious that the largest changes in $\gamma$ and the specific heat are observed at the top of the region of partial ionization of hydrogen, where about $5-10\%$ of hydrogen is ionized.

Figure 18: Height dependence of the work done by buoyancy forces in five ss models plotted separately for a) up- and b) downflows. Designations are the same as in Fig. 2

Figure 19: Stratification of the photon mean-free path and Rosseland opacity for up- and downflows in the models. Designations of the models correspond to those shown in Fig. 2. Horizontal solid lines in a) and b) correspond to a photon mean-free path of 50 km. The photon mean-free path is computed using the Rosseland mean opacity and is given in cm. The Rosseland opacity is in units of cm2g-1

The local pressure gradient is also changed due to changes in particle number density. As a result of recombination, for instance, the pressure is reduced and this accelerates gas to move upwards. The extremes of $\Delta T$, $\Delta \rho$, and $W_{\rm rms}$ coincide in averaged upflows and are seen at depths from 60 km to 100 km below the surface. They do not depend strongly on cell size.

Figure 20: Temperature sensitivity of the monochromatic opacity coefficient at $\lambda$ 500 nm averaged over up- a) and downflow b). Designations are the same as in Fig. 2

The buoyancy force ($W_{\rm b}$) shown in Fig. 18 was calculated as:

$$W_{\rm b} = g W \Delta \rho,$$ (1)

where W and $\Delta \rho$ are the vertical velocity and the horizontal density fluctuations, respectively, and g is the gravitational acceleration. For thermal upflows the mean buoyancy force work also depends weekly on their horizontal size, more precisely, for cells larger than 1000 km it remains roughly unchanged.

We note also the important role of opacity in these layers. The Rosseland mean opacity increases in upflows by a factor of more than 100 from the surface to 100 km below it (Fig. 19c). The absorption coefficient is most sensitive to the temperature exactly at the same depths, i.e. 50-100 km below the surface (Fig. 20a). This additionally protects hot flows from radiative cooling. The reason of this strong temperature sensitivity of opacity lies in the high dependence of $\rm {H^-}$ absorption on electron density. Above a depth of 50 km the photon mean free path in large-scale upflows (displayed in Fig. 19a) becomes larger than 50 km and they are cooled actively.

The extremes of internal energy variations are located at the same depth as those of the temperature and density fluctuations. However, these relative fluctuations are larger, for instance, by a factor of 2 with respect to relative temperature fluctuations due to ionization effects.

In summary, the zone of high instability can be easily localized in upflows: its location corresponds to that at which the most pronounced peaks of $\partial C_{\rm V}/\partial z$, T-excess and $\rho$-deficit occur. Moreover, several effects work together so as to increase the convective instability of upflows in the zone of partial ionization of hydrogen. They lead to a significant growth of the buoyancy force there, so that just below the surface there is a narrow layer which increases convective instability.

The dynamics of downflows differ in essential aspects from those of upflows. The growth of the fraction of ionized hydrogen takes place in deeper layers (Fig. 17b) as compared with the levels where the maxima of T-deficit and $\rho$-excess are observed (Figs. 11b and 13b, respectively). The layers at which the largest downward-directed velocities are seen (Fig. 4c), are also shifted with respect to the $\Delta T$- and $\Delta \rho$-extremes. At last, all downflow parameters (including $W_{\rm b}$, Fig. 18b) demonstrate sharp and well pronounced dependence on horizontal size of downflows (convective cells). This strong size dependence is explained by sensitivity of their buoyancy driving to horizontal radiative heating (Steffen et al. 1989): as broader they are as less effective is their radiative heating.

It is important, for understanding of dynamics of downdraughts, that the increase of thermal damping in downflows due to horizontal radiative heating near the surface occurs before ionization effects change thermodynamic properties of the gas. The ionization effects inhibit this process and maintain energetic downflow plumes. This follows from the stratification of the Rosseland opacity (Fig. 19d), photon mean freepath (Fig. 19b), temperature sensitivity of the monochromatic absorption coefficient at 500 nm (Fig. 20b) and is in contrast to upflows where ionization effects increase the instability of upflows before they are influenced by radiative cooling.

3.1.5 Continuum intensity fluctuations

In Fig. 21 we show relative rms fluctuations of emergent monochromatic intensity at $\lambda$ 500 nm. $\Delta I_{\rm rms}/\overline {I}$ exhibits a similar dependence on convection cell size as $\Delta T_{\rm rms}$ (discussed above and plotted in Fig. 8a): around $L \sim$ 1000 km (this corresponds to "granules'' in the size range of 500-600 km) it is seen that the dependence on L changes. We also plot the $\Delta I_{\rm rms}/\overline {I}$ resulting from large-scale models during the unstable phase of their evolution. Such values are represented in Fig. 21 by triangles and show that large cells exhibit only marginally smaller intensity fluctuations in this case.

Figure 21: Relative rms fluctuations of continuum intensity at $\lambda$ 500 nm plotted vs. convection cell size. For large-scale cells two values are shown: those corresponding to stable (squares) and unstable (triangles) solutions

Based on the scale dependence of the intensity fluctuations, these convection cells can be divided into two groups, those with sizes smaller and larger than $\sim$1000 km, respectively. For the first group radiative damping plays a very active role and intensity fluctuations decrease rapidly with decreasing cell size. For the second group $\Delta I_{\rm rms}/\overline {I}$ is virtually independent of L.

These results are in good qualitative agreement with the detailed study of Steffen et al. (1989). Quantitatively, our models exhibit higher intensity fluctuations by about a factor of 2.

3.2 Non-stationary convection: Multi-scale models

The multi-scale models are free from most of the main assumptions underlying the ss models, in particular those outlined in Sect. 3.1.

Figure 22: A representative snapshot of 2-D multi-scale models. Arrows represent the velocity field. Horizontal lines are isotherms (from bottom to top): 12 000, 10 000, 9000, 8000, 7000, 6000, 5000, and 4000 K. Mach numbers above 0.9, 1.2, and 2.0 are denoted by different strengths of the shading in the top frame. Relative pressure ( $P/\overline {P}$) and density ( $\rho /\overline {\rho }$) excesses greater than 1.02, 1.2, and 1.4 are similarly marked in the middle and bottom frames. The average $\overline {P}$ and $\overline {\rho }$ are determined over horizontal layers

A typical snapshot of the velocity field, isotherms, areas with supersonic flows, and regions having pressure and density excesses in the model are displayed in Fig. 22. The regions with large Mach number (associated with shocks) can be divided into the following 3 categories, examples of each of which are found in Fig. 22:

• supersonic oscillations of the vertical velocity localized in the upper part of the model atmosphere, near and above the traditional temperature minimum (e.g., between horizontal locations $x=(6-9)\ 10^3$ km in Fig. 22);
• supersonic horizontal granular velocities situated in the middle photosphere (e.g. near x=10⁴ km);
• supersonic downflows near and under the visible surface (at x=2000 km in Fig. 22).

Often the downflows of oscillating motions exceed the local sound speed in the upper atmosphere. Moreover, they interact with overshooting convection by increasing the granular horizontal velocities. As a consequence, the maximum Mach numbers of vertical oscillations and the maximum Mach numbers of horizontal velocities of overshooting convection oscillate fairly in phase. Subphotospheric supersonic downflows are often caused by the sharp increase of downflows after being pulsed by the impact of horizontal flows at the intergranular lane (Cattaneo et al. 1989). Their Mach number fluctuates approximately in antiphase.

The relative fluctuations of pressure and density reach their maxima in the upper atmospheric layers of the model, where they are caused by a combination of the buoyancy braking mechanism acting on convective flows and the oscillatory motions. Inhomogeneities in these layers are influenced mainly by oscillations. Consequently, their boundaries do not coincide with the spatial distribution of the granular brightness field (Fig. 22).

The temporal evolution of 2-D granules in the current set of ms models has been studied by Ploner et al. (1998,1999). They show that the evolution of individual synthetic granules ends either through fragmentation or dissolution, in the most cases, depending on their mean sizes. Granules smaller than 1000-1400 km mainly dissolve whereas the evolution of larger structures is in general terminated by their fragmentation. This result agrees well with the behaviour of the ss models if we take into account that convection cells with a horizontal sizes of 1500-1600 km produce bright "granules'' with sizes of about 1000 km. It is also in good agreement with the observations of Kawaguchi (1980) and Karpinskiy & Pravdjuk (1998).

3.2.1 General characteristics of ms models

The height dependence of temporally and spatially averaged rms vertical and horizontal velocities are shown in Fig. 23, together with filling factors of up- and downflows.

We have to note that these models are a case of direct numerical experiment. This means that we did not correct any model quantity (including velocity components) inside computational domain or at the model boundaries. Hence, large values of vertical velocities seen in Fig. 23a are resulted by our treatment of physical processes and/or details of numerical calculations. In particular, extremely large velocities are demonstrated by less dense downflows. Their behaviour depends in large degree on treatment of radiative effects. In contrast to the steady-state models we did not include the contribution of molecular lines to the opacity. This enhanced the dynamics in the ms simulation. There is also fundamental reason in producing large model oscillations because in simulations the very limited computational domain is normally used compared with the real solar case (Stein et al. 1989). They can be in principal reduced with special boundary conditions (Stein & Nordlund 1998; Grossmann-Doerth et al. 1998; Gadun et al. 1999).

Figure 24 displays the stratification of averaged rms temperature fluctuations and amplitudes of $\Delta T$-variations, where $\Delta T$ is the temperature difference between horizontally averaged up- and downflows.

Figure 23: a) rms vertical (solid line) and horizontal velocities (dots) averaged horizontally and over time versus height and b) "area'' filling factors for up- (solid line) and downflows (dot-dashed line) also as a function of height. The plotted data refer to ms models

In convectively unstable layers rms vertical velocities, horizontal velocities, and rms temperature fluctuations are close to those found for large-scale ss models. In the upper model atmosphere vertical velocities are higher in the ms models, due to the influence of oscillations. The atmospheric temperature fluctuations are also larger for the same reason.

Figure 24: a) Horizontally averaged rms temperature fluctuations of the ms models versus height and b) difference between averaged temperatures in up- and downflows. In b) the dashed line represents the zero-level of the temperature variations

The reversal of the temperature difference between up- and downflows is seen at $\sim$170 km above the surface in Fig. 24b. Near the traditional temperature minimum we note a second reversal of the temperature difference: upflows become hotter again. This is due to the increasingly large contribution from oscillations in the upper photosphere. Upflows of ms models also demonstrate an inversion of density in subphotospheric layers (Fig. 25), in agreement with the ss models.

Figure 25: Height-distribution of ms model density averaged horizontally and over time: mean density (solid), averaged density in upflows (dashed), mean density in downflows (dot-dashed). The density is in g cm-3

Vertical velocities and excesses and deficits of T, P, and $\rho$ are given in Fig. 26 for up- and downflows separately. Values averaged horizontally over up- respectively downflows are considered. These results are reasonably close to the corresponding values for the steady-state solutions, although there are also important differences. In upflows, the largest velocity is observed $\sim$120 km below the surface, at the depth of the maxima of T-excess and $\rho$-deficit (see Banos & Nesis 1990). In contrast, in downflows, the largest vertical velocities in subphotospheric layers are shifted to greater depth with respect to the extremes of temperature and density fluctuations (300-350 km vs. 200 km). The velocity maximum instead roughly coincides with the smallest area occupied by downflows (Figs. 23b and 26d).

Figure 26: Stratification of various physical parameters in the ms simulations. a) Stratification of the unsigned vertical velocities averaged horizontally over up- and downflows, b) the difference between the mean temperature in and the average temperature, c) relative excess of density and pressure in up- and downflows, d) pressure excesses in up- and downflows with respect to the minimal pressure inside upflow and a fraction of downflows (squares); all the quantities are given in percents, e) relative difference between mean pressures in up- and downflows, a measure of the amplitude of pressure fluctuations. In a- d) the solid lines represent upflows, dot-dashed ones downflows

In contrast to ss models, in these simulations we do not see a pressure inversion. However, at a depth of 270-350 km the amplitude of pressure fluctuations decreases (Fig. 26e). It is caused by a strong growth of pressure in downflows (Fig. 26d) and is close in depth to the minimal filling factor of downflows (Fig. 26d). Therefore, at these depths the rising thermal plumes exhibit the largest horizontal sizes.

Figure 27: Maximum a) and mean b) vertical velocities for granules, and minimum c) and averaged d) vertical velocities for intergranular lanes as a function of the size of the granule ( a and b), respectively of intergranular lanes ( c and d). Circles are stable convection cells, small crosses mark unstable models (fragmenting cells), and solid lines are the mean relation computed only for the stable phase of cell evolution. Plotted are the results at the visible surface $\log \tau_{\rm R} = 0$ of the steady-state models of thermal convection

Figure 28: Scale dependence of maximum temperature a), and average temperature b) of granules, as well as minimum c) and mean d) temperatures of intergranular lanes. The plotted values refer to the surface $\log \tau_{\rm R} = 0$ of steady-state simulations. Designations are the same as in Fig. 27