Zonal flow evolution and overstability in accretion discs

TL;DR

This paper analyzes the stability and evolution of a viscous, self-gravitating Keplerian disc with thermal diffusion and cooling, revealing conditions for overstability and instabilities in density waves and slow modes.

Contribution

It extends classical stability analysis by including cooling and diffusion effects, providing analytical insights into the overstability and instability regions in accretion discs.

Findings

Overstability occurs for intermediate and long wavelengths at Toomre Q less than approximately 2.

A non-self-gravitating instability appears at wavelengths greater than 18H for certain adiabatic indices.

Cooling efficiency and physical parameters like Prandtl number significantly influence the growth of instabilities.

Abstract

This work presents a linear analytical calculation on the stability and evolution of a compressible, viscous self-gravitating (SG) Keplerian disc with both horizontal thermal diffusion and a constant cooling timescale when an axisymmetric structure is present and freely evolving. The calculation makes use of the shearing sheet model and is carried out for a range of cooling times. Although the solutions to the inviscid problem with no cooling or diffusion are well known, it is non-trivial to predict the effect caused by the introduction of cooling and of small diffusivities; this work focuses on perturbations of intermediate wavelengths, therefore representing an extension to the classical stability analysis on thermal and viscous instabilities. For density wave modes the analysis can be simplified by means of a regular perturbation analysis; considering both shear and thermal diffusivities, the system is found to be overstable for intermediate and long wavelengths for values of the Toomre parameter $Q \lesssim 2$; a non-SG instability is also detected for wavelengths $\gtrsim 18H$, where $H$ is the disc scale height, as long as $\gamma \lesssim 1.305$. The regular perturbation analysis does not however hold for the entropy and potential vorticity slow modes as their ideal growth rates are degenerate. To understand their evolution, equations for the axisymmetric structure's amplitudes in these two quantities are analytically derived and their instability regions obtained. The instability appears boosted by increasing the value of the adiabatic index and of the Prandtl number, while it is quenched by efficient cooling.