Universal behaviour of $\alpha$-viscosity in black hole accretion discs

The Shakura-Sunyaev $\alpha$-viscosity coefficient, defined as the ratio of total stress to total pressure, $\alpha= \mathbb{T}/p$, played an important role in the development of the accretion disc theory in the early 1970s. The origin of 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 - the modern general relativistic magneto-hydrodynamic simulations (GRMHD) of black hole accretion discs 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. It was found that the behaviour of $\alpha$ reflects some underlying, fundamental properties of the stress $\mathbb{T}$ itself. In particular, as argued by several authors, the stress must be zero at the black hole horizon. We notice that the stress calculated in GRMHD simulations by different authors, including us, has a maximum rather 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 handy in the construction of improved versions of thin and slim disc models.