Numerical 3-D radiative hydrodynamical simulations are the main tool for the analysis of the interface between the solar convection zone and the photosphere. The equation of state is one of the necessary ingredients of these simulations. We compare two equations of state that are commonly used, one ideal and one nonideal, and quantify their differences. Using a numerical code we explore how these differences propagate with time in a 2-D convection simulation. We show that the runs with different equations of state (EOSs) and everything else identical relax to statistically steady states in which the mean temperature (in the range of the continuum optical depths typical for the solar photosphere) differs by less than 0.2 %. For most applications this difference may be considered insignificant.

Realistic time-dependent 3-D numerical simulations

While both studies clearly indicate that the results of the different realistic simulations provide a consistent qualitative and quantitative description of the solar convection, the differences between different simulation runs are still present. Moreover, these differences may become significant in certain experiments and measurements. A good example is the problem of the abundance measurement from the comparison of the synthetic and observed spectra. This problem is extremely model-dependent, and high accuracy is critical.

In this paper we isolate the influence of the EOS on the hydrodynamical convection
simulations. In Sect.

The boundaries in the

Two types of EOS are used in the codes for the solar convection simulations:
(1) the nonideal EOS including the effects of the pressure ionisation,
Coulomb interaction and electron degeneracy and (2) the ideal EOS for a
mixture including the partial ionisation effects. MHD

MHD here stands for the authors Mihalas, Hummer and Däppen (Mihalas et al., 1988).

The range of validity of the ideal EOS may be estimated for the trivial case
of partially ionised pure hydrogen. In Fig.

To compute nonideal EOS is extremely time-consuming and impossible to do on the fly in codes for 3-D numerical simulations. Instead EOS is precomputed and results are stored in lookup tables suitable for fast interpolation. The computational cost of the ideal EOS is much lower; however, when the molecules are taken into account, it is necessary to solve the equations iteratively, and, therefore, the use of the precompiled lookup tables may save considerable computing time as long as the table grid is sufficiently fine to limit the interpolation errors.

The ionisation fraction of hydrogen
in the

The mass density, the electron number density and the specific internal energy per mass computed on the temperature–pressure grid using the VMW EOS.

The VMW EOS accounts for the partial ionisation of the mixture composed of atomic and molecular
hydrogen (H, H

We evaluated the VMW EOS for a large

Hereafter we use “internal energy” for the specific internal energy per mass.

.The OPAL EOS is computed in the physical picture through an activity expansion
of the grand canonical partition function of the plasma. The results are
distributed as lookup tables

The relative difference between the VMW EOS and the OPAL EOS
in the mass density, the electron number density and the internal energy
((VMW-OPAL)/VMW). The VMW EOS is computed on
the

The relative differences between the mass density, the electron number density
and the internal energy in the VMW and the OPAL EOS in the

Note that we use overlapping biquadratics for the interpolation as in the code included in the OPAL distribution. The interpolation errors may be suppressed by a different interpolation scheme, e.g. with bicubic splines.

. The vertical bands below 3500 K in the relative difference of the electron density correspond to the area where the OPAL electron densities are unreliable.To study propagation with time of differences caused by EOS choice
in a near-surface convection simulation, we initiate and run two simulations with
all parameters identical except for the EOS lookup tables.
For this experiment we use the version of the MURaM code
described in

Blow-up of Fig. 4 in the low-pressure and low-temperature region relevant
for the near-surface convection. The fragments of the contour lines from
Fig.

Vertical velocity at fixed height (850 km above the bottom of the simulation box) over the first 24 min of the simulation with the OPAL EOS (left) and with the VMW EOS (right). The dashed line approximately marks the instance when the two simulation runs separate.

The initial model is prepared by replicating the
temperature and the density of a 1-D convectively stable model in two dimensions.
The magnetic field is equal to 0 throughout the simulation runs.
The internal energy is computed from these two quantities using both
VMW and OPAL, so that we have two initial snapshots, one consistent
with each EOS.
The specific energy in the snapshots is then perturbed with a multiplicative factor of
1

The difference between the mean temperature of the simulation runs with OPAL and VMW in the geometrical height scale (left) and in the continuum optical depth scale at 500 nm (right). Note different scales in the two panels.

In the first iteration step, the relative difference between the temperature in the
two runs is within

In MURaM the flux of the emergent radiation

The tables of thermodynamical quantities (the mass density, the specific internal energy and
the electron density) computed using two equations of state – one ideal

To check how this difference may affect the result of the hydrodynamical simulations, we
simulated the 2-D solar convection with the two EOSs with an otherwise identical set-up.
The two simulation runs separate in the parameter space as soon as the convective
cells are developed.
Nevertheless, they evolve separately to a statistically nearly identical steady state, with
the temperature difference in the optical depth scale below 10 K throughout the photosphere.
This experiment demonstrates that the two EOSs are interchangeable for simulations of near-surface solar convection.
Moreover, the electron density in the OPAL tables, corrupted at low temperatures,
may be recomputed for a large portion of the

This result is in agreement with the studies of

This work is partially supported by the Spanish Ministry of Economy and Competitiveness (MINECO) through projects AYA2011-24808, AYA2010-18029 and AYA201455078-P. This work contributes to the deliverables identified in FP7 European Research Council grant agreement 277829, “Magnetic connectivity through the Solar Partially Ionized Atmosphere”. This research has made use of NASA's Astrophysics Data System. The topical editor L. Ofman thanks two anonymous referees for help in evaluating this paper.