5 Discussion

The behaviour of the velocity in the low photosphere of the model is explained by the condition of mass balance which follows from the continuity equation. For simple models of anelastic convective flows in a stratified medium it can be shown (Steffen et al. 1989; Spruit et al. 1990; Schüssler 1992) that the horizontal velocity (U) is a function of the horizontal size of the convection cell (l):

$$U \sim W l / H,$$ (2)

where W is the vertical velocity and H the scale height of the vertical mass flux (vertical momentum density). This proportionality is valid in both the 2-D and 3-D cases. However, in real granular convection W and H are themselves functions of l and the final relation between U and l can be nonlinear. For instance, Steffen et al. (1989) found from their ss models that the maximum rms horizontal velocities could be represented as $U_{\rm rms, max} \sim \sqrt {L}$, where L is the horizontal size of the model cells. Our simulations, however, give $U_{\rm rms, max} \sim L^{0.86}$. The differences between their and our results occur for the following reasons.

• Different geometry of the models: Steffen et al. (1989) computed 2-D models in cylindrical geometry, while we employed 2-D Cartesian geometry. In the latter case the horizontal velocities Uare larger than in 2-D cylindrical flows or in fully 3-D cartesian geometry. This follows from the continuity equation and is clearly seen in the comparison between our $U_{\rm rms}$ (Fig. 2a) and those found by Steffen et al. (1989 - their Fig. 2a) or between the present 2-D results and 3-D computations done by Atroshchenko & Gadun (1994 - their Fig. 1b);

• Differences in the velocity may also result from different upper boundary conditions for velocities. We do not reduce the velocities at the top of the model in any form, while Steffen et al. (1989) postulated that model velocities had to vanish there completely. We expect that this affects the sensitivity of model properties to the convection cell sizes.

In all the 2-D models of Steffen et al. (1989) $U_{\rm rms}$ reaches its maximum approximately at the same height in the photosphere. At this height the vertical velocity is roughly scale-independent.

In our model atmospheres we find that $H \sim L^{0.14}$. The vertical velocities are different at these heights. For instance, below the surface $W_{\rm rms, max}$, shown in Fig. 3b, can be approximated by $W_{\rm rms, max} \sim L^{-0.13}$; $W_{\rm rms}$ in the lower photosphere (at a fixed geometrical height of about 50 km above the surface) is scale-independent: $W_{\rm rms} \sim L^{-0.02}$; while in the middle photosphere ($\sim$150-200 km) $W_{\rm rms}$ is roughly L^0.25. If we assume scale-independent $W_{\rm rms}$ then from (6) we get $U_{\rm rms} \sim {L^{0.86}}$, in good agreement with the relation for $U_{\rm rms, max}$ that we found directly from the simulations. Note, however, that the maxima of $U_{\rm rms}$occurs in our models at different photospheric heights for different cell size (Fig. 4a).

Consider now the density inversion exhibited by at least our ss models (Fig. 5c). A surface density inversion is also found in models of stellar convective envelopes based on mixing length theory (Böhm-Vitense 1958). For instance, for stars with masses smaller than 4 $M_{\odot}$ Ergma (1970) obtained that their density inversion was most pronounced when the effectiveness of convection was low. Under effectiveness of convection we understand here the ratio of energy transported by a convective element to the energy lost due to radiative exchange.

In the present 2-D simulations the ratio of transported energy to radiative losses depends on the size of the convection cell. The smallest convection cells represent convective transport of energy with a large loss of energy through radiation. According to Fig. 5c these are also the models with the largest density inversion. Thus 2-D simulations and predictions of mixing-length theory are in good agreement with each other in this sense.

The occurrence of a density inversion when the effectiveness of convection is low can be explained as follows: ineffective convection means a larger role of radiation in energy transfer, producing steeper temperature gradients. Since, on average, the pressure must always grow with depth to compensate for the gravity force, at the peak of the temperature gradient the density is expected to show an inversion according to the equation of state.