The convective stability of fully stratified baroclinic discs
We examine the convective stability of hydrodynamic discs with full stratification in the local approximation and in the presence of thermal diffusion (or relaxation). Various branches of the relevant axisymmetric dispersion relation derived by Urpin (2003) are discussed. We find that when the vertical Richardson number is larger than or equal to the radial one (i.e. $|Ri_z|\geq|Ri_x|$) and wavenumbers are comparable (i.e. $|k_x|\sim|k_z|$) the disc becomes unstable, even in the presence of radial and vertical stratifications with $Ri_x>0$ and $Ri_z>0$. The origin of this resides in an hybrid radial-vertical Richardson number. We propose an equilibrium profile with temperature depending on the radial and vertical coordinates and with $Ri_z>0$ for which this destabilization mechanism occurs. We notice as well that the dispersion relation of the "convective overstability" is the branch of the one here discussed in the limit $|k_z|\gg|k_x|$ (i. e. two-dimensional disc).