Universal behaviour of $α$-viscosity in black hole accretion discs

Abstract

The Shakura-Sunyaev $α$-viscosity coefficient, defined as the ratio of total stress to total pressure, $α= \mathbb{T}/p$, 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 $α=$ 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 $α$ 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 $α$ 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 $α$ 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 $α$-formula, deduced from the GRMHD simulations, may be handy in the construction of improved versions of thin and slim disc models.

Figures (3)

• Figure 1: The $\alpha$-viscosity coefficient calculated in GRMHD simulations by 2019ApJ...884L..37L, 2013MNRAS.428.2255P, denoted by 2013MNRAS.428.2255P, and 2025MNRAS.542..377R, denoted by 2025MNRAS.542..377R, all for non-rotating black hole accretion discs, fitted to our $\alpha$-prescription formula (\ref{['eqn:whole-domain']}). Three characteristic features are visible in all of the simulations: [1] $\alpha=0$ at the horizon $r=r_{\mathrm H}$, [2] $\alpha$ has a maximum very close to the location of the circular light ray at $r_{\mathrm{ph}}=(3/2) \, r_{\mathrm H}$, [3] for large radii $r \gg r_{\mathrm H}$, $\alpha$ is much smaller than in the plunging region. The plunging region is located between the sonic radius $r_{\mathrm S}$ and the horizon $r_{\mathrm H}$ and is highlighted by the grey colour ($r_{\mathrm S}$ is taken from 2019ApJ...884L..37L). All lines are normalised by their respective maxima: $\alpha_{\mathrm{max}} = 0.8$ for 2019ApJ...884L..37L, $0.3$ for 2013MNRAS.428.2255P, and $0.3$ for 2025MNRAS.542..377R. Since 2025MNRAS.542..377R uses only the Maxwell stress in the local fluid frame (see Section \ref{['sec:Stress-simulations']}), their $\alpha$ definition is mostly valid in the highly magnetised plunging region and we therefore only show R25 data up to $r_\mathrm{ISCO}$.
• Figure 2: The structure of a puffy disc consists of three regions: C: dense core, P: puffy region, and F: funnel. Left: The colour map of gas density with the magnetic field lines. Right: Viscous $\alpha$ colour map, and the magnetosonic surface (yellow contour). In both panels, the white dashed line indicates the photosphere, and the full line indicates the density scale height.
• Figure 3: Behaviour of the total stress $\mathbb{T} = \mathbb{T}^{\mathrm{Max}} + \mathbb{T}^{\mathrm{Rey}}$ and its Reynolds and Maxwell components (the lines shown are taken from 2019ApJ...884L..37L): the stress is zero at the horizon, it has a maximum at the circular photon orbit and is much smaller far away than in the plunging region. The behaviour of the stress is remarkably similar to the behaviour of the inverse of the square of the "gyration radius" $\tilde{r}$.