Thermal baroclinic instabilities in accretion disks I: Combined dispersion relation for Goldreich-Schubert-Fricke Instability and Convective Overstability in disks around young stars

This paper discusses the Goldreich-Schubert-Fricke instability (GSF) and the convective overstability (COS) in the context of baroclinic thermal instabilities in rotating disks around young stars. The vertical shear instability (VSI) is a global extension of the GSF that affects geometrically thin disks but follows the same stability criterion. The COS, on the other hand, also possesses a twin for stellar interiors, specifically, Shibahashi's vibrational stability of rotating stars. We derive a combined dispersion relation for GSF and COS with arbitrary cooling times for local perturbations and determine a new stability criterion beyond the Solberg-H{\o}iland\ criterion. The paper shows that in extension to the stability criterion for the vertically unstratified case ($N^2_R > 0$), one also needs a barotropic disk structure to ensure stability towards COS modes. We demonstrate that a baroclinic disk atmosphere always has a buoyantly unstable direction, although not necessarily in the radial nor vertical direction. The paper predicts that for cooling times longer than the critical cooling time for VSI, GSF modes will always be accompanied by COS modes of similar growth rate. The numerical companion paper II tests the predictions of growth rates from this paper.