LAUSR

Hydrodynamic, non-magnetic instabilities can provide turbulent stress in the regions of protoplanetary discs, where the MRI can not develop. The induced motions influence the grain growth, from which formation of planetesimals begins. Thermal relaxation of the gas constrains origins of the identified hydrodynamic sources of turbulence in discs. We estimate the radiative relaxation timescale of temperature perturbations and study the dependence of this timescale on the perturbation wavelength, the location within the disc, the disc mass, and the dust-to-gas mass ratio. We then apply thermal relaxation criteria to localise modes of the convective overstability, the vertical shear instability, and the zombie vortex instability. Our calculations employed the latest tabulated dust and gas mean opacities and we account for the collisional coupling to the emitting species. The relaxation criterion defines the bulk of a typical T Tauri disc as unstable to the development of linear hydrodynamic instabilities. The midplane is unstable to the convective overstability from at most $2\mbox{ au}$ and up to $40\mbox{ au}$, as well as beyond $140\mbox{ au}$. The vertical shear instability can develop between $15\mbox{ au}$ and $180\mbox{ au}$. The successive generation of (zombie) vortices from a seeded noise can work within the inner $0{.}8\mbox{ au}$. Dynamic disc modelling with the evolution of dust and gas opacities is required to clearly localise the hydrodynamic turbulence, and especially its non-linear phase.