Nonlinear variations in axisymmetric accretion

TL;DR

This paper investigates the nonlinear dynamics of axisymmetric accretion flows, revealing potential instabilities through a dynamical systems approach and analyzing wave perturbations with WKB methods.

Contribution

It introduces a nonlinear perturbation framework for accretion flows, linking the equations to analogue black hole metrics and analyzing stability via dynamical systems.

Findings

Nonlinear perturbations lead to saddle points indicating potential instabilities.

High-frequency wave analysis shows growth in amplitude and energy flux due to nonlinearity.

The acoustic horizon delineates stable and unstable regions in the flow.

Abstract

We subject the stationary solutions of inviscid and axially symmetric rotational accretion to a time-dependent radial perturbation, which includes nonlinearity to any arbitrary order. Regardless of the order of nonlinearity, the equation of the perturbation bears a form that is similar to the metric equation of an analogue acoustic black hole. We bring out the time dependence of the perturbation in the form of a Li\'enard system, by requiring the perturbation to be a standing wave under the second order of nonlinearity. We perform a dynamical systems analysis of the Li\'enard system to reveal a saddle point in real time, whose implication is that instabilities will develop in the accreting system when the perturbation is extended into the nonlinear regime. We also model the perturbation as a high-frequency travelling wave, and carry out a Wentzel-Kramers-Brillouin analysis, treating nonlinearity iteratively as a very feeble effect. Under this approach both the amplitude and the energy flux of the perturbation exhibit growth, with the acoustic horizon segregating the regions of stability and instability.