Radiative convection with a fixed heat flux

We have determined the marginal stability curve of convective instability in the usual Rayleigh–Bénard configuration with radiative transfer and a fixed total heat flux at the boundaries instead of a fixed temperature. In the Milne–Eddington approximation, radiative transfer introduces a new length scale and breaks the invariance of the Boussinesq equations under an arbitrary temperature shift, which occurs when the heat flux is fixed at the boundaries. The convergence to the limits where the non-radiative cases are expected is studied in this approximation. Then, using a second-order perturbative calculation, we show that the presence of radiation can change qualitatively the instability pattern: there is a range of optical parameters where the Cahn–Hillard equation is not anymore the one appropriate to describe the instability near the threshold.