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Astron. Astrophys. 324, 281-288 (1997)

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2. Model atmospheres

To minimize the effect of possible errors in oscillator strengths and damping constants, we studied the differential dependence of Li and Fe abundances on model atmospheres of the Sun computed with different convection treatments.

2.1. 2-D models

2-D model atmospheres were computed using a system of equations of radiation hydrodynamics describing a compressible, radiatively coupled, gravitationally stratified medium (Gadun 1995):




where i, j = 1, 3, [FORMULA] is the total specific energy (equal to the sum of kinetic energy [FORMULA] and internal energy e), P is the total pressure, [FORMULA] is the Reynolds stress tensor [FORMULA]:


[FORMULA] is a velocity deformation tensor, [FORMULA] is the specific kinetic energy of small-scale (sub-grid) turbulence, and [FORMULA] is a kinematic coefficient of turbulent viscosity. The turbulent viscosity was treated with a simple local gradient model of Smagorinsky (1963):


then [FORMULA]. [FORMULA] is a dissipation of the kinetic energy of averaged movement at sub-grid level. [FORMULA] is the spatial step, [FORMULA], and [FORMULA]. [FORMULA] is a divergence of the radiative energy flux (value of radiative heating/cooling).

The system of equations was integrated using the conservative large particle method (Belozerkovskiy & Davidov 1982). Usually this method is used with schemes of the first order of accuracy in space and time. In this work the order of accuracy in an approach of convective terms depends on the smoothness of the solution and varied up to second order in regions with the smooth flow.

The radiative energy transfer was treated by a momentum method (a differential approach) with the variable Eddington factors and LTE. Within the frame of this approach the complete momentum relation may be written for a given value of the radiative heating/cooling ([FORMULA]) as:


where [FORMULA] is the monochromatic Plank function, and [FORMULA] the variable Eddington factors. Then [FORMULA], where [FORMULA] are the weights. The boundary conditions were used in the form:




where [FORMULA] is the vertical component of radiative flux at the upper boundary. The final equation with its boundary conditions can be solved using a standard difference technique. In the optically deep layers the diffusion approximation was taken for finding [FORMULA]. The detailed frequency dependence of the monochromatic continuum opacity as well as the radiation transfer in spectral lines were taken into account. The line opacities were considered using directly the earlier Kurucz's opacity distribution function (ODF, Kurucz 1979) for standard solar abundances without molecular lines. The system of equations and the numerical techniques were described in detail by Gadun (1995).

In this work we consider two kinds of 2-D model atmospheres:

a) Quasi-stationary time-dependent models which describe only a single scale of the solar granulation (single scale or "one granule" -OG- approach, Gadun 1995). Thermal convection is treated in these models as quasi-stationary, cellular, and laminar. Topology of flows is not changed during the modeling process. These models have low Reynolds numbers ([FORMULA]). Due to the large influence of numerical viscosity and the absence of non-stationary interaction with neighbouring granules, the secondary motions in the upper layers are non-active, and photospheric velocity fields are governed by overshooting convection. Using these models it is possible to estimate the impact of the inhomogeneities on the lithium and iron abundances in the idealized approach, when these inhomogeneities are caused by the laminar, stationary and plane convective flows.

The size of the model region of OG models was chosen as 1295 [FORMULA] 2030 km in horizontal and vertical direction with a spatial step of 35 km. The atmosphere region occupies about 900 km. Although the flow topology in these models is time-independent they are affected by oscillations, and to take them into account we carried out the line computations over a short sample of time-dependent models. To consider a dependence of the synthesized line profiles on the wave component, computations were carried out for 11 models with a temporal step of 30 s between them. We also used two models obtained by the average of all 11 OG models at equal levels of [FORMULA] and equal geometrical height (h). Here [FORMULA] is a monochromatic optical depth at 5000 Å. Such models are labelled as OGA models.

b) Non-stationary multiscale models (MS) with dynamic interaction of several granules in the model region. The model region size, 3930 [FORMULA] 2030 km, was sampled with the same spatial step as in OG models. These models treat convection as a non-stationary process. The Reynolds number is higher ([FORMULA]) than for OG models. There are several evolved granulation flows with the interaction between them in the model regions. In these models, secondary motions in the middle and upper photosphere are very active. They have a significant impact on the radiative-dynamic state of photospheric layers, and the structure of these layers can be described more properly.

We note that the flow topology and the radiative-dynamic situation in the photosphere layers are strongly time-dependent. To exclude selection effects, we performed the synthesis of spectral lines over the whole sample of time-dependent MS models. Afterwards we carried out the temporal averaging procedure over 529 models with 30 s step. This corresponds to the integration time of [FORMULA]. Basic results concerning MS model computations, parameters of 2-D granules, and [FORMULA] power spectra for these models are given in Gadun & Vorob'yov (1995) and Gadun & Pikalov (1996). Lines were also computed for the "averaged MS" (MSA) models which were obtained by averaging all 529 MS models over equal optical depths ([FORMULA]) and geometrical height (h) levels.

A few important points are noted:

- by definition, OG and MS models are self-consistent. They define both the temperature structure and the velocity field in the modeling region.

- In fact, OGA and MSA models are "1-D-like" models. Strictly speaking, they are not self-consistent. In such averaged models the thermodynamic quantities are not consistent with the relation of hydrostatic equilibrium and the equation of state. To solve this problem, in Paper I the authors computed a new 1-D model where the temperature gradient was taken from the averaged 3-D model, and P and [FORMULA] were found from the equations of state and hydrostatic equilibrium. In this paper we are interested in the study of the impact of inhomogeneities on the abundance determination, and we consider only averaged models. Of course, the information about the velocity field is absent in the averaged OGA and MSA models. To compute the line profiles for these models we should define micro- and macroturbulent velocity field parameters. These parameters can be directly obtained from 2-D models by the filtering of velocities over large- and small-scale structures. But, in principle, each line has its own filtering parameters. For this reason, we decided to work with 1-D models in the frame of the classical spectral line analysis. Namely, we used for them the micro- and macroturbulent velocities similar to those which were found for 1-D classical solar model atmospheres to fit weak FeI and FeII lines (see Section 4.1).

2.2. 1-D models

We obtained iron and lithium abundances in the solar atmosphere using also several 1-D model atmospheres:

- semiempirical HOLMU model atmosphere (Holweger & Müller 1974),

- theoretical Kurucz (1993) model atmosphere computed with convective overshooting (K93),

- theoretical model atmosphere (PK79) computed by the code SAM92. This program is a modification of ATLAS9 (Kurucz 1993). Opacity was considered in the frame of the opacity sampling approach, and the subroutine XLINOPOS was taken from SAM71 (Pavlenko 1991). To compute the blocking effect due to the absorption by lines of atoms and ions we used the list of Kurucz (1992). The mixing length theory parameter used was [FORMULA] = 1.25, and the SAM92 model atmosphere was computed without convective overshooting. The temperature structure of that model is similar to that present in the Kurucz (1979) solar model.

2.3. Comparison of temperature structures of model atmospheres

R.M.S. vertical velocities and fluctuations of temperature and gas pressure in OG and MS models averaged over space and modeling time at equal [FORMULA] level are shown in the Fig. 1. The different behavior of the velocity field in OG and MS models, as well as the more pronounced fluctuations of thermodynamic values in MS models, may be explained by the presence of convective flows (inhomogeneities) of larger scales in these models and the interaction between them. The growth of the vertical velocity in the upper photosphere is the result of increasing oscillatory motions.

[FIGURE] Fig. 1. R.M.S. vertical velocities (a), fluctuations of the temperature (b), and gas pressure (c) at equal [FORMULA] levels in the sequences of MS and OG 2-D models. The data were obtained by averaging over the space and modeling time. Solid and dashed lines represent MS and OG models, respectively

Temperature distributions in photospheric and in subphotospheric layers of 1-D models are shown in Fig. 2. We point out a few items:
- [FORMULA] and [FORMULA] in the forming region of weak and intermediate Fe I lines with low excitation potential. The model K93 has higher temperatures in these layers in comparison with our 1-D models (PK79, OGA, and MSA) due to differences in opacity computation procedure and opacity source lists.
- MSA and OGA models have the largest temperature gradients in the lower photosphere.
- Significant differences between averaged OG and MS models were found in the lower photosphere and in sub-photospheric layers. Their structures depend on the averaging procedure over equal [FORMULA] or h layers: models averaged over h are hotter near irradiating layers but they have cooler subphotospheric layers. Indeed, hot and cold structures in high temperature-dependent opacity regions provide different contributions due to the averaging procedure.
- The effect of the convective overshooting in the model K93 is obvious: temperatures in low photosphere become higher.

[FIGURE] Fig. 2. Temperature distributions in the models used: in atmospheric layers of OGA a, MSA b, and PK79 and HOLMU c models in comparison with 1-D hydrostatic models (K93) of Kurucz (1993)
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© European Southern Observatory (ESO) 1997

Online publication: May 26, 1998