Alpha-viscosity effects in slender tori

TL;DR

This paper investigates how alpha-viscosity influences the dynamics and oscillations of slender tori, revealing damping of acoustic modes and potential instability of inertial modes depending on angular momentum profiles.

Contribution

It provides a detailed analysis of viscous effects on oscillation modes in slender tori, including growth rates and stability conditions, using perturbation theory.

Findings

High-order acoustic modes are damped by viscosity.

High-order inertial modes are enhanced by viscosity.

Lowest-order inertial mode can become unstable under certain conditions.

Abstract

We explore effects of the Shakura-Sunyaev alpha-viscosity on the dynamics and oscillations of slender tori. We start with a slow secular evolution of the torus. We show that the angular-momentum profile approaches the Keplerian one on the timescale longer than a dynamical one by a factor of the order of 1/\alpha. Then we focus our attention on the oscillations of the torus. We discuss effects of various angular momentum distributions. Using a perturbation theory, we have found a rather general result that the high-order acoustic modes are damped by the viscosity, while the high-order inertial modes are enhanced. We calculate a viscous growth rates for the lowest-order modes and show that already lowest-order inertial mode is unstable for less steep angular momentum profiles or very close to the central gravitating object.