4 Size-dependence of observable properties of 2-D granulation

In Sect. 3 we aimed to obtain a better understanding of 2-D granulation, in particular the size dependence. Now we consider how the observable physical properties of convection cells in both the ms models and the ss models, depend on their size, which is now represented by the size of the visible granule to be directly compared with the observations.

In accordance with the definition generally used by observers we consider granules to be regions of excess brightness. We have computed the monochromatic emergent intensity at $\lambda$ 500 nm at each horizontal location at each time step and have identified granules using isophotes of the monochromatic intensity: a granule is defined as a one-dimensional brightness excess above the mean intensity emerging from the simulation at the corresponding time-step. For every granule and intergranular we have also computed a number of physical parameters as described in Sect. 3.

4.1 Steady-state models

In the ss models that go unstable, i.e. models with horizontal size of 1680 km or greater, as the fragmentation starts a part of unstable models is seen to darken until it takes on the appearance of an intergranular lane. It splits up the single brightness excess (single granule) into several smaller brightness fragments which are, for a short time, actually produced by the same broad convective flow. These brightness fragments we shall call the unevolved structures in the following.

Figure 29: Dependence of maximum a) and mean b) intensities of granules and minimum c) and mean d) intensities of intergranular lanes on their sizes. Plotted are results of two sets of unstable models which have horizontal sizes 2044 and 2340 km

Shown in Figs. 27 and 28 are the velocities and temperatures of granules and intergranular lanes computed from ss models. We distinguish between stable (represented by open circles) and unstable (crosses) models.

In Fig. 27 the stable and unstable solutions differ essentially in the small-scale range. Stable small-scale flows exhibit larger velocities connected with granules and lower downward-directed velocities associated with intergranular lanes than we found for equally small, unevolved structures in unstable solutions (which are formed during the fragmentation of the original large cell).

In Fig. 28, on the other hand, the two populations of small-scale granules almost coincide (although, stable small-scale cells display slightly higher granule temperatures).

The temperatures of intergranular lanes belonging to stable convection cells exhibit the same size dependence as the intergranular lanes produced by the fragmentation of large-scale unstable cells.

In Fig. 29 we plot the intensity of two sets of scans computed for unstable "steady-state'' models only. The intensity distributions look similar to the temperature distributions (Fig. 28), with the exception of the mean intensities of intergranular lanes. The fact that this behaviour is not seen in Fig. 28d suggests that the temperature gradient is significantly different for small and large-scale intergranular lanes bordering unstable granules.

So, the nature of thermal convective flows, either as shallow (surface) convection or as a collection of deep-formed convection cells, is well defined in the dependence of vertical velocities of small-scale granules on their sizes. But it cannot be recognized in scale dependence of the temperature.

Figure 30: Histograms of the relative granule count a) and of the contribution of granules to the total "area'' covered by granules b) obtained for intensity at $\lambda$ 500 nm. In c) and d) the same quantities are plotted for intergranular lanes. The figure refers to multi-scale models. The acoustic modes were filtered out prior to analysis. The bins are 100 km wide

4.2 Multiscale simulation: Non-stationary convection

Because ms models produce oscillations with strong amplitudes we have removed the p-modes from the studied time-dependent intensity field as well as from the associated variables. The removal technique is described by Ploner et al. (1998,1999) and is based on filtering out p-modes seen in the $k-\omega$ diagram. We have also carried out the analysis described below without first filtering the p-modes. The basic results remain unchanged, but the oscillations lead to a larger scatter, which makes trends more difficult to recognize.

Figure 31: Maximum horizontal a), maximum b) and mean c) vertical velocities in granules, and minimum d) and average e) vertical velocities in intergranular lanes as a function of their sizes. Plotted are the values at the solar surface ( $\log \tau_{\rm R} = 0$) in multi-scale models after the removal of p-modes

Histograms of relative granule/intergranular-lane counts and their contribution to the total area are plotted in Fig. 30. The major "area'' contributors among the granules are those with sizes from 600 to 1300 km (Fig. 30b).

In the histogram of intergranule counts there is a sharp peak at 500 km. They are also the major contributors to the total "area'' of dark lanes (Fig. 30d).

Velocity and temperature properties of granules and intergranular lanes are represented in Figs. 31 and 32, respectively. The striking feature of Fig. 31a is the almost linear dependence of the maximum horizontal velocity on granule size, in general agreement with the mass balance condition (Nelson & Musman 1978; Nordlund 1985; Steffen et al. 1989), which was also found for ss models (Sect. 3.1.2).

The vertical velocities of granules, shown in Figs. 31b and c, increase with granular size up to $\sim$600 km and remain unchanged or decrease for larger granules. The small granules exhibit a large scatter which can be produced by granules with different evolutionary histories. We do not exclude, however, the influence of model oscillations on behaviour of their velocity field in ms simulations.

Granules are smallest either just after their birth through fragmentation of their parent granule, or shortly before their death through dissolution. The evolution of granules in these ms models has been studied in detail by Ploner et al. (1999) and they find that indeed the mean vertical velocities of small granules near the beginning of their lifetime (stronger upflows) and averaged velocities of the very smallest ones near the ends of their lifetimes (weak upflows) differ significantly. The properties of these very small "granules'' in these models are a complicated case for direct study because they are affected by model oscillations significantly and because in non-stationary simulations they additionally depend on prehistory of flow evolution (Ploner et al. 1999).

The vertical velocities of the ss models show qualitatively the same dependence (Fig. 27) on granule respectively lane size, with the exception of the smallest granules. These exhibit a much larger range of velocities, due mainly to the fact that the stable small-scale convection cell exhibit much larger upflows than the small granules formed through fragmentation of a larger granule. Not surprisingly the granules of the highly variable ms simulation exhibit a behaviour more similar to the unstable ss models.

Plotted in Figs. 31d and e are velocities in the intergranular lanes. They suggest the presence of two families of intergranular lanes: those with width above 200 km and rapid downflows and those which have only weak vertical velocity (which may even correspond to a weak upflow) and which are generally narrower than 200 km. The coexistence of these two families of intergranular lanes is especially clearly seen in Fig. 31d - in the plot of minimum vertical velocities found inside the intergranular lanes. The second family, the narrower, slowly flowing "lanes'' are usually associated with the fragmentations of a granule and are the precursors of the new lane formed by the fragmentation (see Ploner et al. 1999). This conclusion is supported by the properties of the lanes formed by the splitting of large ss convective cells. The fact that lanes with strong downward-directed velocities cannot be narrower than roughly 200 km depends on the mass balance condition and, probably, on the spatial step used, because for ss models with smaller horizontal domain size the well evolved downflows can be practically as narrow as 100 km.

Figure 32: Size dependence of maximum a) and mean temperatures b) of granules as well as minimum c) and mean d) temperatures of intergranular lanes. Shown are values at the surface, $\log \tau_{\rm R} = 0$, of multi-scale models

In Fig. 32 the temperature at $\tau_{\rm R}=1$ of granules and lanes is plotted vs. their size. The granular temperature increases with granular size up to sizes of about 400-600 km and then remains unchanged (Fig. 32a - for maximal granular temperature) or even decreases on average (Fig. 32b - mean temperature). Note that the scatter in temperature of the small granules is small compared to the scatter exhibited by the velocities in Fig. 31. Temperatures of intergranular lanes also show two dependences: a sharp decrease with size up to $\sim$700 km and a slower decrease of their minimum temperatures beyond that (Fig. 32c). Their mean temperatures actually remain unchanged for sizes above 600-800 km.

The scale dependence of temperatures of granules and intergranular lanes, shown in Fig. 32, are similar to those given in Sect. 4.1. for the ss models (Fig. 28). For those models the temperatures of newly formed small-scale granules (intergranular lanes) are close (almost coincident) to those derived from evolved structures.

The fact that many of the smallest granules are cooler than the smallest lanes, although by definition the former are brighter (compare with Fig. 32), is due to the different temperature gradient in the two types of features (the gradient is larger in the granules).

The scale size dependence of the intensity, shown in Fig. 33, roughly reflects the same behaviour as seen in Fig. 32 for the temperature. It is nevertheless interesting that the size-independence of the mean intensity of intergranular lanes larger than 600-700 km (Fig. 33d) is in contrast to the behaviour of the ss models (Fig. 29d).

4.3 Interpretation of scale-dependent relationships

In two previous sections we have ascertained that simulated small-scale granules significantly differ in their velocity features if they have different history of the evolution. If small-scale convective flows are well evolved, quasi-stable, and deep-formed then granular velocities, associated with them, decrease with increasing their sizes (circles, Figs. 27a and b). The opposite scale dependence is realized for small-scale granules which become seen at the beginning stage of fragmentation of underlying large-scale thermal upflows (Figs. 27a and b, crosses). In ss modeling they are well separated while in ms modeling the presence of these two sorts of small-scale granules leads to large scatter of granular velocities and velocities of intergranular lanes for region of small-scale structures.

Figure 33: Dependence of maximal a) and mean b) intensities of granules and minimal c) and mean d) intensities of intergranular lanes on their sizes. The intensity computed at $\lambda$ 500 nm. Acoustic oscillations were removed from the multi-scale models prior to analysis

The results described above can be explained by invoking two mechanisms, which act on small- and large-scale structures, respectively.

For small-scale inhomogeneities the smoothing influence of horizontal energy exchange through radiation becomes important (Steffen et al. 1989). This effect is most important near the $\tau = 1$ surface. This mechanism causes the temperature difference between small granules and lanes to diminish with decreasing size (see Fig. 32). This also explains the increase of intergranular lane brightness and decrease of granular brightness seen in Fig. 33 with decreasing size.

Figure 34: The data of Figs. 33a and b are replotted as a function of the "area'' of granules and intergranular lanes (see text for details)

The second mechanism acts on large-scale upflows. They evolve mainly adiabatically and could in principle have higher temperatures and vertical velocities than is exhibited by the simulations. However, they are effectively retarded by the pressure excess that builds up in granules (Schüssler 1992). This pressure excess causes buoyancy braking, leading to the reduced upflow velocities in larger granules (above a granule size of 1000 km) seen in Figs. 31b and c (see also Ploner et al. 1999). In many cases the buoyancy braking concentrates the strongest upflows into certain locations in the large granule, e.g. near its boundary, making the flow (and the brightness structure) inhomogeneous.

Granules are relatively free of both effects only over a relatively narrow range of sizes ($\sim$600-1000 km, these correspond to convection cell sizes of $\sim$900-1500 km).

4.4 Comparison with observations

The scale dependence of the brightness field of our models (Fig. 33) can be compared with the observed distribution of Hirzberger et al. (1997 - their Figs. 8 and 10). Qualitatively theoretical and observational granule data look similar - the granular brightness increases linearly with the size of small-scale granules. It becomes approximately size-independent or even decreasing slightly at larger scales. However, the largest scatter of granular intensities (at the "elbow'') is located at smaller scales ($\sim$400-800 km) in our models relative to the observations ($\sim$1500 km). The simulations also predict a linear decrease of brightness of small-scale intergranular lanes, which is hardly visible in the observations, excluding the darkest lanes. Finally, the amplitudes of the intensity fluctuations resulting from the simulations are larger than in the observations.

These differences can be explained by (1) the spatial smoothing of the observed brightness map (due to seeing), (2) different spatial dimensions of the studied intensity fields (2-D in observations and 1-D in the models), and (3) different granule finding algorithms. The high scatter of intensities seen for small granules can be explained in part by the coexistence of flows of different nature located there, i.e. small granules recently formed through fragmentation, or small granules nearing the end of their lifetime. Some of the points may even represent small-scale bright upflows inside larger granules.

To better compare the simulated data with the observations, we have replotted Figs. 33a and b in terms of granule area. To find the "area'' of 2-D synthetic granules we assumed that they have a circular shape. For intergranular lanes this simple shape presentation can be hardly assumed and is used here mostly in qualitative sense. The new dependencies are shown in Fig. 34.

Figure 35: The data of Fig. 33 are plotted, after spatial smearing to simulate the limited spatial resolution typical of observations (see text), as a function of the "area'' of granules

For granules we can now hardly see any "elbow'' at all, only a small decrease of their maximum intensity in the small-scale area range. Narrow intergranular lanes, however, still demonstrate a stronger decrease of brightness towards smaller scales. Therefore, the difference between the present 2-D models and the observational relationships found by Hirzberger et al. (1997) basically results from spatial smoothing and possibly in the differences in the granule finding algorithm.

In Fig. 35 we display the granular data as in Figs. 33 and 34, but after spatially smearing the emergent intensity. The point spread function (PSF) used consists of the sum of two Gaussians with dispersions of 180 and 540 km, respectively (Deubner & Mattig 1975). The comparison of this figure with the results of Hirzberger et al. (1997) supports the argument that spatial smearing can explain a significant part of the quantitative disagreement between the simulations and the observations. This implies that the finite spatial resolution of the current observations can in principal distort the true size dependence of granular properties substantially.

The different dimensions of the observed and simulated granular brightness fields probably also play a role. Only the comparison of the results of 3-D simulations with these observations can clarify this point.