Corotational instability of inertial-acoustic modes in black hole accretion discs and quasi-periodic oscillations

Abstract

We study the global stability of non‐axisymmetric p modes (also called inertial‐acoustic modes) trapped in the innermost regions of accretion discs around black holes. We show that the lowest‐order (highest‐frequency) p modes, with frequencies ω= (0.5–0.7) mΩISCO[where m= 1, 2, 3, … is the azimuthal wavenumber, ΩISCO is the disc rotation frequency at the Innermost Stable Circular Orbit (ISCO)], can be overstable due to general relativistic effects, according to which the radial epicyclic frequency κ is a non‐monotonic function of radius near the black hole. The mode is trapped inside the corotation resonance radius rc (where the wave pattern rotation speed ω/m equals the disc rotation rate Ω) and carries a negative energy. The mode growth arises primarily from wave absorption at the corotation resonance, and the sign of the wave absorption depends on the gradient of the disc vortensity, ζ=κ2/(2ΩΣ) (where Σ is the surface density). When the mode frequency ω is sufficiently high, such that dζ/r > 0 at rc, positive wave energy is absorbed at the corotation, leading to the growth of mode amplitude. The mode growth is further enhanced by wave transmission beyond the corotation barrier. We also study how the rapid radial inflow at the inner edge of the disc affects the mode trapping and growth. Our analysis of the behaviour of the fluid perturbations in the transonic flow near the ISCO indicates that, while the inflow tends to damp the mode, the damping effect is sufficiently small under some conditions (e.g. when the disc density decreases rapidly with decreasing radius at the sonic point) so that net mode growth can still be achieved. We further clarify the role of the Rossby wave instability and show that it does not operate for black hole accretion discs with smooth‐varying vortensity profiles. Overstable non‐axisymmetric p modes driven by the corotational instability provide a plausible explanation for the high‐frequency (≳100 Hz) quasi‐periodic oscillations observed from a number of black‐hole X‐ray binaries in the very high state. The absence of high‐frequency quasi‐periodic oscillations in the soft (thermal) state may result from mode damping due to the radial infall at the ISCO.

Inertial-acoustic oscillations of black hole accretion discs with large-scale poloidal magnetic fields

“…Their full GR results of the mode frequencies are qualitatively in agreement with the pseudo-Newtonian results (Lai & Tsang 2009), but indicate that the theoretical frequencies are too high compared to observations. The discrepancy is most severe for rapidly spinning BHs (such as GRS 1905+105).…”

“…Lai & Tsang (2009) and Tsang & Lai (2009b) showed that these modes, which consist of nearly horizontal oscillations with no vertical structure, can grow in amplitude due to wave absorption at the corotation resonance (where the wave pattern speed ω/m matches the disc rotation rate Ω). This overstability requires that the disc vortensity, ζ ≡ κ 2 /(2ΩΣ), where κ is the radial epicyclic frequency and Σ is the surface density (see Horak & Lai 2013 for the full general relativistic version of vortensity), have a positive gradient at the corotation radius rc (Tsang & Lai 2008; see also Narayan et al 1987).…”

Two Transient X-Ray Quasi-periodic Oscillations Separated by an Intermediate State in 1H 0707-495

“…Nevertheless some models do suggest frequency ratios of 1 : 2 : 3 and so on. For example, Lai & Tsang (2009) studied the global stability of non-axisymmetric p modes (also called inertial-acoustic modes) trapped in the innermost regions of accretion discs around black holes and showed that the lowest-order p modes, with frequencies ω ≈ 0.6mΩ ISCO can be overstable due to general relativistic effects, where m = 1, 2, 3, ... is the azimuthal wavenumber and Ω ISCO is the disc rotation frequency at the so-called innermost stable circular orbit (ISCO). They also suggested that overstable non-axisymmetric p modes driven by the corotational instability may account for the high-frequency QPOs observed from a number of BH X-ray binaries in the very high state while the absence of such signals in the soft (thermal) state may result from mode damping due to the radial infall at the ISCO.…”

Incompressible Modes Excited by Supersonic Shear in Boundary Layers: Acoustic CFS Instability

“…We start with the Papaloizou-Pringle instability (Papaloizou & Pringle 1987) and its plane-parallel analog (Glatzel 1988;. The PPI can operate by over-reflection of trapped p-modes via tunneling through the forbidden region around corotation (Narayan et al 1987) or by interaction with a Rossby-like mode trapped at corotation (Tsang & Lai 2008, 2009aLai & Tsang 2009). Li et al (2003) found that the presence of vertical nodes in the wave has a strong stabilizing effect due to absorption of the wave at corotation.…”