Secular Instability in Quasi‐viscous Disk Accretion

Abstract

A first-order correction in the α-viscosity parameter of Shakura & Sunyaev has been introduced in the standard inviscid and thin accretion disc. A linearised time-dependent perturbative study of the stationary solutions of this "quasi-viscous" disc leads to the development of a secular instability on large spatial scales. This qualitative feature is equally manifest for two different types of perturbative treatment -a standing wave on subsonic scales, as well as a radially propagating wave. Stability of the flow is restored when viscosity disappears.

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

“…By analogy, exactly this kind of instability is also seen to develop in Maclaurin spheroids on the introduction of a kinematic viscosity to a first order (Chandrasekhar 1987). Similar features in the quasi‐viscous flow have also been argued for in this paper following the earlier study (Bhattacharjee & Ray 2007), but, in this instance, under the pseudo‐Schwarzschild generalization.…”

“…It shall be important to realize here that the choice of a driving potential, Newtonian or pseudo‐Newtonian has no explicit bearing on the form of the equation, and the case of α= 0 corresponds to a stable configuration (Ray 2003a; Chaudhury et al 2006). The form of is identical to the one derived by Bhattacharjee & Ray (2007) for flows driven by the Newtonian potential (except for the fact that here all α‐dependent terms are scaled by a factor of ), and so none of the conclusions regarding secular instability, drawn for the case of the Newtonian potential, will be qualified upon using any of the pseudo‐Newtonian potentials (Paczyński & Wiita 1980; Nowak & Wagoner 1991; Artemova, Björnsson & Novikov 1996) which are regularly invoked in accretion‐related literature to describe rotational flows on to a Schwarzschild black hole. This shall be especially true for flows on large length‐scales, where all pseudo‐Newtonian potentials converge to the Newtonian limit.…”

“…This shall be especially true for flows on large length‐scales, where all pseudo‐Newtonian potentials converge to the Newtonian limit. On these very length‐scales, the perturbation (treated as high‐frequency travelling waves) has been known to exhibit an exponential growth behaviour under a WKB analysis (Bhattacharjee & Ray 2007). This has disturbing implications, since all physically meaningful inflow solutions will have to pass through the large length‐scales, and connect the outer boundary of the accretion disc to the event horizon.…”

“…The flow will also be unstable when is used to treat the perturbation as a standing wave between two chosen boundaries (Bhattacharjee & Ray 2007). It need not always be true that standing and travelling waves will simultaneously exhibit the same qualitative properties as far as stability is concerned.…”

“…An earlier work (Bhattacharjee & Ray 2007) on the quasi‐viscous flow, driven by the classical Newtonian potential, has revealed an instability – secular instability – when the stationary flow solutions are subjected to small time‐dependent perturbations. By analogy, exactly this kind of instability is also seen to develop in Maclaurin spheroids on the introduction of a kinematic viscosity to a first order (Chandrasekhar 1987).…”