Corotational Instability of Inertial-Acoustic Modes in Black-Hole Accretion Discs: Non-Barotropic Flows

TL;DR

This paper investigates how corotation resonance affects inertial-acoustic modes in non-barotropic black-hole accretion discs, revealing that entropy gradients modify mode growth and potentially explain observed high-frequency QPOs.

Contribution

It extends the analysis of corotation resonance effects from barotropic to non-barotropic flows, deriving new conditions for mode instability influenced by entropy gradients.

Findings

Mode growth depends on the gradient of effective vortensity in non-barotropic discs.

Lowest-order p-modes with azimuthal numbers 2, 3, 4 have the largest growth rates.

Frequencies of these modes are near simple integer ratios, relevant for QPOs.

Abstract

We study the effect of corotation resonance on the inertial-acoustic oscillations (p-modes) of black-hole accretion discs. Previous works have shown that for barotropic flows (where the pressure depends only on the density), wave absorption at the corotation resonance can lead to mode growth when the disc vortensity, $\zeta=\kappa^2/(2\Omega\Sigma)$ (where $\Omega, \kappa, \Sigma$ are the rotation rate, radial epicyclic frequency and surface density of the disc, respectively), has a positive gradient at the corotation radius. Here we generalize the analysis of the corotation resonance effect to non-barotropic fluids. We show that the mode instability criterion is modified by the finite radial Brunt-V\"as\"al\"a frequency of the disc. We derive an analytic expression for the reflectivity when a density wave impinges upon the corotation barrier, and calculate the frequencies and growth rates of global p-modes for disc models with various $\alpha$-viscosity parameterizations. We find that for disc fluids with constant adiabatic index $\Gamma$, super-reflection and mode growth depend on the gradient of the effective vortensity, $\zeta_{\rm eff} = \zeta/S^{2/\Gamma}$ (where $S \equiv P/\Sigma^{\Gamma}$ measures the entropy): when $d\zeta_{\rm eff}/dr > 0$ at the corotation radius, wave absorption leads to amplification of the p-mode. Our calculations show that the lowest-order p-modes with azimuthal wave number $m=2, 3, 4,...$ have the largest growth rates, with the frequencies approximately in (but distinct from) the $2:3:4...$ commensurate ratios. We discuss the implications of our results for the high-frequency quasi-periodic oscillations observed in accreting black-hole systems.