Alpha-viscosity effects in slender tori

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.

💡 Research Summary

The paper investigates how the Shakura‑Sunyaev α‑viscosity influences both the secular evolution and the oscillation spectrum of slender tori—geometrically thin, axisymmetric fluid configurations that serve as idealized models for accretion structures around compact objects. The authors begin by deriving the long‑term (secular) response of a torus whose initial specific angular‑momentum distribution follows a power law L(r)∝r^q (0 ≤ q ≤ ½). By inserting the viscous stress tensor into the angular‑momentum conservation equation, they show that the torus relaxes toward a Keplerian profile L_K∝r^{1/2} on a timescale τ_visc≈t_dyn/α, where t_dyn is the dynamical (orbital) time at the torus centre. This result demonstrates that even a modest viscosity (α≈0.01) can prolong the redistribution of angular momentum by two orders of magnitude relative to the dynamical time, while simultaneously allowing the torus radius and thickness to adjust slowly as viscous heating modifies the internal pressure balance.

Having established the background state, the authors turn to a linear perturbation analysis that includes the viscous terms to first order. Perturbations are expressed as ξ∝exp