[Paper Review] Three-dimensional structure of an alpha accretion disk

W. Kluźniak, D. Kita|arXiv (Cornell University)|Jun 20, 2000

Phase Equilibria and Thermodynamics3 references16 citations

TL;DR

This paper presents an analytic solution for the three-dimensional structure of a thin, axisymmetric alpha accretion disk using a systematic expansion in the small parameter ε = H̄/R̄. It reveals that for α < 0.685, a significant backflow region forms in the disk midplane due to viscous stress gradients, with up to 40% of the mass flow outward at large radii, challenging the standard one-dimensional accretion paradigm.

ABSTRACT

An analytic solution is presented to the three-dimensional problem of steady axisymmetric fluid flow through an accretion disk. The solution has been obtained through a systematic expansion in the small parameter epsilon =H/R (the ratio of disk thickness to its radial dimension) of the equations of viscous hydrodynamics. The equation of state was assumed to be polytropic. For all values alpha&lt; 0.685 of the viscosity parameter, we find significant backflow in the midplane of the disk occuring at all radii larger than a certain value; however, in the inner regions of the disk the fluid always flows toward the accreting object. The region of backflow is separated from the region of inflow by a surface flaring outwards from a circular locus of stagnation points situated in the midplane of the disk.

Motivation & Objective

• To resolve the three-dimensional structure of a viscous, geometrically thin accretion disk beyond one-dimensional height-averaged approximations.
• To investigate the dynamical origin of midplane backflow observed in numerical simulations but not captured by standard disk models.
• To determine the global structure of the velocity field and mass flux distribution in the meridional plane using a systematic asymptotic expansion.
• To quantify the fraction of mass flowing outward in the disk midplane as a function of the viscosity parameter α.
• To assess the implications of backflow for accretion-dominated flows and the interpretation of X-ray nova observations.

Proposed method

• Employed a systematic expansion in ε = H̄/R̄ (disk aspect ratio) to solve the viscous hydrodynamics equations in 3D, assuming steady, axisymmetric, and polytropic flow.
• Used a polytropic equation of state with index n = 3/2, neglecting thermal effects but including viscous torques as a perturbation.
• Applied an inner boundary condition with vanishing viscous torque at r = rₘ, introducing a natural length scale r₊ = Ωₘ²rₘ⁴/(GM*) relevant to black hole or spinning star disks.
• Solved the radial and vertical momentum equations perturbatively, deriving the radial velocity u₁(r,z) and density ρ₀(r,z) to leading order.
• Calculated the mass flux in the radial direction and defined the outflow fraction Γ(r,α) as the ratio of outward mass flux in the midplane to net accretion rate.
• Evaluated the asymptotic outflow fraction Γ∞(α) in the limit r → ∞ using integrals over the vertical structure, with γ* = z_vert/h as a key variable.

Experimental results

Research questions

• RQ1 What is the three-dimensional structure of a viscous, thin accretion disk when solved beyond the one-dimensional, height-averaged approximation?
• RQ2 Under what conditions does backflow occur in the disk midplane, and how is it sustained by the inflowing fluid?
• RQ3 How does the fraction of mass flowing outward in the midplane depend on the viscosity parameter α?
• RQ4 What is the global structure of the velocity field, particularly the location and extent of the stagnation surface separating inflow and outflow?
• RQ5 Can backflow solutions coexist with advective energy transport, and what would be the implications for interpreting X-ray nova emission deficits?

Key findings

• For all α < 0.685, a backflow region forms in the disk midplane at radii beyond a critical radius r_stag, with fluid flowing outward despite net accretion.
• The backflow is driven by viscous stress gradients and is fed by the inflowing fluid, forming a closed circulation pattern in the meridional plane.
• The asymptotic outflow fraction Γ∞(α) reaches approximately 0.4 for α ≤ 0.1, meaning up to 40% of the total mass flux is outward at large radii.
• For 0.1 < α < 0.5, the outflow fraction remains significant at ∼0.35, indicating that backflow is a robust feature for low to moderate α.
• The critical value α_cr ≈ 0.685 marks the threshold beyond which the backflow region disappears, as γ*∞ becomes unphysical.
• The solution is not factorizable (i.e., not separable as R(r)Θ(θ)), distinguishing it from prior axisymmetric numerical solutions and highlighting the role of viscous stress gradients in 3D structure.