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Using the anelastic approximation of linearized hydrodynamic equations, weinvestigate the development of axially symmetric small perturbations in thinKeplerian discs. Dispersion relation is found as a solution of generalSturm-Liouville eigenvalue problem for different values of relevant physicalparameters (viscosity, heat conductivity, disc semithickness). The analysisreveals the appearance of overstable mode for Prandtl parameter higher thansome critical value. These modes have a viscous-convective nature and can serveas a seed for turbulence in astrophysical discs even in the absence of magneticfields.

A viscous-convective instability in laminar Keplerian thin discs