Alpha-Viscosity Effects in Slender Tori

Abstract

We explore effects of the Shakura–Sunyaev $ \alpha$-viscosity on the dynamics and oscillations of slender tori. We start with a slow secular evolution of the torus. We show that the angular-momentum profile approaches the Keplerian one on a timescale longer than a dynamical one by a factor on the order of 1$ /\alpha$ . We then focus our attention on the oscillations of the torus. We discuss the 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 viscous growth rates for the lowest-order modes, and show that already the lowest-order inertial mode is unstable for less-steep angular-momentum profiles, or very close to the central gravitating object.