Issue 
A&A
Volume 530, June 2011



Article Number  L7  
Number of page(s)  6  
Section  Letters  
DOI  https://doi.org/10.1051/00046361/201116766  
Published online  13 May 2011 
Online material
Appendix A: Setup of the simulations
A.1. The equilibrium model
As in GD2011, our system represents a zoom around an ionisation region. Since we are computing local simulations, the vertical gravity g = −ge_{z} and the kinematic viscosity ν are assumed to be constant. Following our purely radiative model of the κmechanism (Gastine & Dintrans 2008a,b), the ionisation region is represented by a temperaturedependent radiative conductivity profile that mimics an opacity bump:
with
(A.2)where T_{bump} is the position of the hollow in temperature and σ, e, and denote its slope, width, and relative amplitude, respectively. We assume both radiative and hydrostatic equilibria; that is,
where F_{bot} is the imposed bottom flux. Following GD2011, we chose L_{z} as the length scale, i.e. [x] = L_{z}, top density ρ_{top} and top temperature T_{top} as density and temperature scales, respectively. The velocity scale is then , while time is given in units of .
Table A.1 then summarises the parameters of the numerical simulations presented in this study in these dimensionless units. The penultimate column of this table contains the value of the frequency ω_{00} of the fundamental unstable radial mode excited by the κmechanism, which lies between 3 and 4 for every DNS. The last column gives the value of the Rayleigh number, which quantifies the strength of the convective motions. It is given by
where L_{conv} is the width of the convective zone, χ = K_{0}/ρ_{0}c_{p} the radiative diffusivity, and s the entropy.
Dimensionless parameters of the numerical simulations.
A.2. The nonlinear equations
With the parallel version of the alternate direction implicit (ADI) solver for the radiative diffusion implemented in the pencil code (see GD2011), we advance the following hydrodynamic equations in time:
where ρ, u, and T denote density, velocity, and temperature, respectively, while K(T) is given by Eq. (A.1). The operator D/Dt = ∂/∂t + u·∇ is the usual total derivative, while S is the (traceless) rateofstrain tensor given by
Finally, we impose the condition that all fields are periodic in the horizontal direction, while stressfree boundary conditions (i.e., u_{z} = 0 and du_{x}/dz = 0) are assumed for the velocity in the vertical one. Concerning the temperature, a perfect conductor at the bottom (i.e., flux imposed) and a perfect insulator at the top (i.e., temperature imposed) are applied.
To ensure that both the nonlinear saturation and thermal relaxation are achieved, the simulations were computed over very long times, typically t ≳ 3000. As the eigenfrequency of the unstable acoustic mode ω_{00} ∈ [3 − 4] (see Table A.1), this corresponds approximately to 1800 periods of oscillation.
© ESO, 2011
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