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*):

(2) |

where

- 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
*U*are 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 (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) 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
.
The vertical velocities
are different at these heights.
For instance, below the surface
,
shown in Fig. 3b, can be approximated
by
;
in the lower photosphere (at a
fixed geometrical height of about 50 km above the surface)
is scale-independent:
;
while in the middle photosphere (150-200 km)
is roughly *L*^{0.25}.
If we assume scale-independent
then from
(6) we get
,
in good agreement
with the relation for
that we found
directly from the simulations.
Note, however, that the maxima of
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 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.

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