 
Summary: Radiative convection with a xed heat
ux
S. Auma^tre
Departement de Modelisation Numerique, IRPHE / LABM CNRS,
IMT La Jetee, Technop^ole de Ch^ateauGombert, Universite de la Mediterranee,
38 rue Frederic JoliotCurie, 13451 Marseille Cedex 20, France
(July 1, 2001)
We have determined the marginal stability curve of convective instability in the usual Rayleigh{
Benard conguration with radiative transfer and a xed total heat
ux at the boundaries instead
of a xed 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
ux is xed 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.
Keywords : Pattern formation, Marginal stability curve, Long wavelength approximation, Radiation
absorption length, Grey media.
I. INTRODUCTION
On account to its role in atmospheric motions, the eect of radiative transfer on convective
instability has been intensively studied since the pioneer works of Goody [1] and Chandrasekhar [2].
