## 2. Model atmospheresTo 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 models2-D model atmospheres were computed using a system of equations of radiation hydrodynamics describing a compressible, radiatively coupled, gravitationally stratified medium (Gadun 1995): where is a velocity deformation tensor, is the specific kinetic energy of small-scale (sub-grid) turbulence, and is a kinematic coefficient of turbulent viscosity. The turbulent viscosity was treated with a simple local gradient model of Smagorinsky (1963): then . is a dissipation of the kinetic energy of averaged movement at sub-grid level. is the spatial step, , and . 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 () as: where is the monochromatic Plank function, and the variable Eddington factors. Then , where are the weights. The boundary conditions were used in the form: and where 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 . 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 (). 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
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
and equal geometrical height ( b) Non-stationary multiscale models (MS) with dynamic interaction of several granules in the model region. The model region size, 3930 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 () 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
. Basic results concerning MS model
computations, parameters of 2-D granules, and
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 () and
geometrical height ( 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
## 2.2. 1-D modelsWe 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 = 1.25, and the SAM92 model atmosphere
was computed ## 2.3. Comparison of temperature structures of model atmospheresR.M.S. vertical velocities and fluctuations of temperature and gas pressure in OG and MS models averaged over space and modeling time at equal 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.
Temperature distributions in photospheric and in subphotospheric
layers of 1-D models are shown in Fig. 2. We point out a few
items:
© European Southern Observatory (ESO) 1997 Online publication: May 26, 1998 |