Universal behaviour of α-viscosity in black hole accretion discs

The Shakura-Sunyaev $\alpha$-viscosity coefficient, defined as the ratio of total stress to total pressure, $\alpha= \mathbb{T}/p$, began to play an important role in the development of accretion disc theory in the early 1970s. The origin of the turbulence that causes the stress $\mathbb{T}$ was unknown at that time; Shakura and Sunyaev assumed $\alpha=$\,const. Today we know that this was not quite realistic -- modern general relativistic magneto-hydrodynamic simulations (GRMHD) of black hole accretion discs have revealed that $\alpha$ changes by about an order of magnitude within the disc, being smaller far away from the black hole and larger in the plunging region close in, and it has been found that the behaviour of $\alpha$ reflects some underlying, fundamental properties of the stress $\mathbb{T}$. In particular, it has been argued by several authors, that $\mathbb{T}$ must be zero at the black hole horizon. We note that the stress calculated in three independent GRMHD simulations of accretion discs around non-rotating black holes, made by a variety of authors (including ourselves), each has its prominent maximum close to the location of the circular photon orbit. We propose a formula that accurately describes this ``universal'' behaviour of $\alpha$ in terms of the ``gyration radius'', a physical characteristic of rotation well known in Newtonian dynamics and in the black hole case uniquely defined by the Kerr space-time geometry. Analytic and semi-analytic models of black hole accretion discs provide an invaluable insight into fundamental physics, and the GRMHD simulations do not aspire to replace them. Rather, simulations could help to improve analytic models by making them more realistic. For example, our $\alpha$-formula, deduced from the GRMHD simulations, may be useful in the construction of improved versions of thin and slim disc models.