In our axisymmetric simulations without magnetohydrodynamical (MHD) effects, angular momentum transport in the accreting flow is given by imposing explicit viscosity. The viscous stress tensor is given by
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σij=ρν˜∂vj∂xi+∂vi∂xj−23(∇·v)δij,

where 



ν˜

 is the shear viscosity, and the bulk viscosity is neglected. To mimic angular momentum transport associated with MHD turbulence driven by the magnetorotational instability (MRI) in a sufficiently ionized disk (e.g., Balbus & Hawley 1998; Stone & Pringle 2001; McKinney & Gammie 2004; Bai 2011; Narayan et al. 2012), we assume the azimuthal components of the shear tensor are nonzero and, in spherical polar coordinates, are given by
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σrϕ=ρν˜∂∂rvϕr,



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σθϕ=ρν˜sinθr∂∂θvϕsinθ

(e.g., Stone et al. 1999; Fernández & Metzger 2013; Inayoshi et al. 2019). The strength of anomalous shear viscosity is calculated with the α-prescription (Shakura & Sunyaev 1973),
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ν˜=αcs2ΩK·exp−∣z∣H,

where α is the viscous parameter, c
s is the sound speed, 



ΩK≡(GM•/r3)1/2

, and H( ≡ c
s/ΩK) is the disk scale height. Note that the exponential factor imposes that the viscous process is active near the mid-plane. The strength of viscosity is set to
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α=α0+αmaxexp−ρcritρ2.

The first term corresponds to the strength of MRI turbulence, and the value is set to α
0 = 0.01 (e.g., Zhu & Stone 2018; Takasao et al. 2018). The second term characterizes the torque caused by non-axisymmetric structure (e.g., spiral arms) excited in a marginally unstable disk against its self-gravity. The density threshold, above which viscosity turns on, is assumed to be 



ρcrit≡ΩK2/(πG)≃2.2×10−21






(M•/105M⊙)(r/10pc)−3gcm−3

. This choice is motivated by the following reasons. The local gravitational instability of a rotating disk is described by Toomre’s Q parameter (Toomre 1964) defined by
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Q≡csκΩπGΣ,

where 



κΩ2=4Ω2+dΩ2/dlnr

 is the epicyclic frequency, and Σ is the disk surface density. For a geometrically thin cold disk around a point mass, the Q-value is approximated to 



Q≃ΩK2/(2πGρ)=0.5(ρcrit/ρ)

. Thus, the second term of the right-hand side of Equation (15) is written as 



∝exp(−4Q2)

, which is a commonly used parameterization of the effective viscosity adopted in semi-analytical models of a self-gravitating disk; 



∝exp(−QA/B)

 where A > 0 and B > 0 (Zhu et al. 2009; Takahashi et al. 2013; see also Kratter & Lodato 2016). The value of 



αmax

 depends on the level of non-axisymmetric structures in a disk. A previous three-dimensional RHD simulation study of a dusty circumnuclear disk around a BH shows that the mass inflow velocity is as high as a substantial fraction of the freefall velocity due to strong torque caused by spiral arms in the disk, indicating 



αmax≃O(1)

 (Toyouchi et al.2021). We here adopt 



αmax=2

 (Hirano et al. 2014; Fukushima et al. 2020). The critical Q-value for the onset of gravitational torque caused by spiral arms in a disk is considered to be Q ∼ 1, but the exact value depends on various properties of the disk (e.g., cooling, heating, and disk irradiation). We note that in our viscous model, the second term in Equation (15) becomes larger than α
0 when Q ≤ 1.15.
Citation: Shakura & Sunyaev 1973
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