tokens: internal energy per unit mass, g is the gravitational acceleration of the black hole located at ( r , θ) = ( z 0 , π), and f is the radiation force per unit mass (see Section 2.2 for detail). An adiabatic equation of state p /ρ = (γ − 1) e with γ = 5/3 is assumed in our calculations. In the last term of equation ( 5 ), is the approximate formula of the net heating rate introduced by Blondin ( 1994 ) where, n is the number density, G Compton is the heating rate via the Compton scattering G X is the difference between the heating rate by the X-ray photoionization and the cooling rate via the recombination and L b,l is the cooling rate by the bremsstrahlung and line emission In equation ( 9 ), the first term represents the effect of the bremsstrahlung and the last two terms show the line cooling in an optically thin atmosphere. The temperature T is obtained by T = μ m p p/k B ρ, where μ (=0.5) is the mean molecular weight, m p is the proton mass, and k B is the Boltzmann constant. The ionization parameter, ξ, is defined as where F X is the X-ray flux. In this study, the X-ray is assumed to come from the point source at the centre with luminosity L X = f X L D , where f X is a parameter. We also assume the temperature of the X-ray radiation to be . The X-ray flux is attenuated via the absorption of which the cross-section is set to be σ X = σ e for ξ > 10 5 , or 100σ e for ξ ≤ 10 5 , where σ e is the mass-scattering coefficient for free electrons. Such a treatment is employed in previous works (Proga et al. 2000 ; Nomura et al. 2016 , N17 ). We note that the scattered and reprocessed photons are ignored in our simulations (see also Section 4.2 ). 2.2 Line force The radiation force in equations ( 2 ) and ( 3 ) is described as where F D is the radiation flux of the disc emission integrated by the wavelength throughout the entire range, and F line is the line-driving flux (UV flux), which is the same as F D but integrated across the UV transition band of 200– Å. Both fluxes are calculated by integrating the radiation from the inner hot region of the disc, where the effective temperature is larger than . We divide the disc surface into 4096 grids for radial direction and 4096 grids for azimuthal direction, and numerically calculate and . The radial components of these fluxes are attenuated due to electron scattering, of which the optical depth is measured from the origin of the coordinate. On the other hand, θ-components of and are supposed not to be diluted. The force multiplier, M , is defined by Stevens Kallman ( 1990 ) Here, t is the local optical depth parameter written as Here, the thermal speed, v th , is set to that corresponds to the thermal speed of the hydrogen gas of the temperature of . The velocity gradient, d v /d s , depends on the direction of each light-ray from the disc surface to the point of interest. However, in order to reduce the computational cost, we evaluate d v /d s along the direction of the disc line-driving flux ( F
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