Quasi-viscous accretion flow -- I: Equilibrium conditions and asymptotic behaviour

TL;DR

This paper introduces a novel perturbative approach to viscous accretion flows, simplifying the analysis of equilibrium and stability by reducing complex equations and revealing new dynamical behaviors and instability mechanisms.

Contribution

It presents a first-order correction method for viscous effects in accretion flows, analyzing equilibrium conditions, critical points, and stability, with a novel analogy to Schrödinger's equation.

Findings

Viscosity breaks equilibrium invariance, differentiating accretion from wind.

Critical points are saddle points and spirals, indicating specific flow behaviors.

Large-scale perturbations lead to secular instability in the disc.

Abstract

In a novel approach to studying viscous accretion flows, viscosity has been introduced as a perturbative effect, involving a first-order correction in the $\alpha$-viscosity parameter. This method reduces the problem of solving a second-order nonlinear differential equation (Navier-Stokes equation) to that of an effective first-order equation. Viscosity breaks down the invariance of the equilibrium conditions for stationary inflow and outflow solutions, and distinguishes accretion from wind. Under a dynamical systems classification, the only feasible critical points of this "quasi-viscous" flow are saddle points and spirals. A linearised and radially propagating time-dependent perturbation gives rise to secular instability on large spatial scales of the disc. Further, on these same length scales, the velocity evolution equation of the quasi-viscous flow has been transformed to bear a formal closeness with Schr\"odinger's equation with a repulsive potential. Compatible with the transport of angular momentum to the outer regions of the disc, a viscosity-limited length scale has been defined for the full spatial extent over which the accretion process would be viable.