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We examine the convective stability of hydrodynamic discs with fullstratification in the local approximation and in the presence of thermaldiffusion (or relaxation). Various branches of the relevant axisymmetricdispersion relation derived by Urpin (2003) are discussed. We find that whenthe 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. $|k_x|\sim|k_z|$) thedisc becomes unstable, even in the presence of radial and verticalstratifications with $Ri_x>0$ and $Ri_z>0$. The origin of this resides in anhybrid radial-vertical Richardson number. We propose an equilibrium profilewith temperature depending on the radial and vertical coordinates and with$Ri_z>0$ for which this destabilization mechanism occurs. We notice as wellthat the dispersion relation of the "convective overstability" is the branch ofthe one here discussed in the limit $|k_z|\gg|k_x|$ (i. e. two-dimensionaldisc).

The convective stability of fully stratified baroclinic discs