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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 κ-mechanism (Gastine & Dintrans 2008a,b), the ionisation region is represented by a temperature-dependent radiative conductivity profile that mimics an opacity bump:

(A.1)

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,

(A.3)

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 ω₀₀ of the fundamental unstable radial mode excited by the κ-mechanism, 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

(A.4)

where L_conv is the width of the convective zone, χ = K₀/ρ₀c_p the radiative diffusivity, and s the entropy.

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:

(A.5)

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) rate-of-strain tensor given by

(A.6)

Finally, we impose the condition that all fields are periodic in the horizontal direction, while stress-free 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 ω₀₀ ∈  [3 − 4]  (see Table A.1), this corresponds approximately to 1800 periods of oscillation.

© ESO, 2011