2. Surface convection simulations
The 3D model atmospheres of the solar granulation which form the basis for the spectral line calculations presented here have been obtained with a 3D, time-dependent, compressible, radiative-hydrodynamics code developed to study solar and stellar surface convection (e.g. Nordlund & Stein 1990; Stein & Nordlund 1989, 1998; Asplund et al. 1999). The hydrodynamical equations of mass, momentum and energy conservation:
coupled to the equation of radiative transfer along a ray (in total eight inclined rays):
are solved on a non-staggered Eulerian mesh with 200 x 200 x 82 gridpoints; simulations with resolutions 100 x 100 x 82, 50 x 50 x 82 and 50 x 50 x 41 have also been computed but have not been utilized here since they are hampered by less good agreement between predicted and observed line profiles (Asplund et al. 2000a). In these equations, denotes the density, the velocity, the gravitational acceleration, P the pressure, e the internal energy, the viscous stress tensor, the viscous dissipation, the radiative heating/cooling rate, the monochromatic intensity, the corresponding source function, the optical depth and the absorption coefficient.
The physical dimension of the simulation box corresponds to 6.0 x 6.0 x 3.8 Mm of which about 1.0 Mm is located above continuum optical depth unity. Again, initial solar simulations extending only to heigths of 0.6 Mm suffered from problems in the predicted line asymmetries for strong lines which was traced to the influence of the outer boundary and that non-negligible optical depths were present in the line cores already at the outermost layers. Though not completely removed with the current more extended simulations, the problems have largely disappeared as discussed further in Sect. 6 and 7. The depth scale has been optimized to provide the best resolution where it is most needed, i.e. in those layers with the steepest gradients in terms of dT/dz and d/dz2, which for the Sun occurs around the visible surface (which is defined to have a geometrical depth Mm). Through this procedure, sufficient depth coverage was also obtained in the important optically thin layers. The simulation box covers about 13 pressure scale heights and 11 density scale heights in depth, while the horizontal dimension is sufficient to include granules at any time of the simulation. The spatial derivatives are computed using third order splines and the time-integration is a third-order leapfrog predictor-corrector scheme (Hyman 1979; Nordlund & Stein 1990). The code was stabilized using a hyper-viscosity diffusion algorithm (Stein & Nordlund 1998) with the parameters determined from standard hydrodynamical test cases, like the shock tube. In this respect these parameters are not freely adjustable parameters for the surface convection simulations. It is important to realize that changing the numerical resolution in effect also varies the effective viscosity, which shows that the convective efficiency and temperature structure are independent on the adopted viscosity description (Asplund et al. 2000a). Periodic horizontal boundary conditions and open transmitting top and bottom boundaries were used in an identical fashion to the simulations described by Stein & Nordlund (1998). The influence of the boundary conditions on the results are negligible. In particular the upper boundary has been placed at sufficiently great heights not to disturb the line formation process except for the cores of very strong lines, and the lower boundary is located at large depths to ensure that the inflowing gas is isentropic and featureless. No magnetic fields or rotation were included in the present calculation.
In order to obtain a realistic atmospheric structure it is crucial to correctly describe the internal energy of the gas and the energy exchange between radiation and gas. For this purpose a state-of-the-art equation-of-state (Mihalas et al. 1988), which includes the effects of ionization, excitation and dissociation, has been used. Since the solar photosphere is located in the layers where the convective energy flux from the interior is transferred to radiation, a proper treatment of the 3D radiative transfer is necessary, which has been included under the assumption of LTE and using detailed continuous (Gustafsson et al. 1975 and subsequent updates) and line (Kurucz 1993) opacities. The 3D equation of radiative transfer was solved at each timestep of the convection simulation for eight inclined rays (2 µ-angles and 4 -angles) using the opacity binning technique (Nordlund 1982). The four opacity bins correspond to continuum, weak lines, intermediate strong lines and strong lines; shorter tests with eight instead of four opacity bins revealed only very minor differences in the resulting atmospheric structures. The accuracy of the binning procedure was verified throughout the simulation at regular intervals by solving the full monochromatic radiative transfer (2748 wavelength points) in the 1.5D approximation (i.e. treating each vertical column separately in the flux calculations without considering the influence from neighboring columns) and found to always agree within 1% in emergent flux.
It is noteworthy that the convection simulations contain no adjustable free parameters besides those used to characterize the stars: the effective temperature (or equivalently, as adopted here, the entropy of the inflowing material at the bottom boundary), the surface acceleration of gravity log g and the chemical composition. The entropy at the lower boundary was carefully adjusted prior to the simulation started in order to reproduce the total solar luminosity. The solar surface gravity (log [cgs]) was employed. The chemical composition of the gas were taken from Grevesse & Sauval (1998), in particular the Fe abundance used for the equation-of-state and opacity calculations was 7.50 1. The line opacities were interpolated to the adopted Fe abundance using the standard Kurucz ODFs with log and non-standard ODFs computed with log and without any He (Kurucz 1997, private communication, cf. Trampedach 1997 for details of the procedure).
© European Southern Observatory (ESO) 2000
Online publication: July 7, 2000