  Fragmentation of rotating gaseous systems via gravitational instability is
believed to be a crucial mechanism in several astrophysical processes, such as
formation of planets in protostellar discs, of molecular clouds in galactic
discs, and of stars in molecular clouds. Gravitational instability is fairly
well understood for infinitesimally thin discs. However, the thin-disc
approximation is not justified in many cases, and it is of general interest to
study the gravitational instability of rotating fluids with different degrees
of rotation support and stratification. We derive dispersion relations for
axisymmetric perturbations, which can be used to study the local gravitational
stability at any point of a rotating axisymmetric gaseous system with either
barotropic or baroclinic distribution. 3D stability criteria are obtained,
which generalize previous results and can be used to determine whether and
where a rotating system of given 3D structure is prone to clump formation. For
a vertically stratified gaseous disc of thickness $h_z$ (defined as containing
$\approx$70% of the mass per unit surface), a sufficient condition for local
gravitational instability is $Q_{\rm 3D}\equiv\left(\sqrt{\kappa^2+\nu^2}+c_s
h_z^{-1}\right)/{\sqrt{4\pi G\rho}}<1$, where $\rho$ is the gas volume density,
$\kappa$ the epicycle frequency, $c_s$ the sound speed, and $\nu^2\equiv\rho_z
p_z/\rho^2$, where $\rho_z$ and $p_z$ are the vertical gradients of,
respectively, gas density and pressure. The combined stabilizing effects of
rotation ($\kappa^2$) and stratification ($\nu^2$) are apparent. In unstable
discs, the conditions for instability are typically met close to the midplane,
where the perturbations that are expected to grow have characteristic radial
extent of a few $h_z$.
