The vertical structure of an accretion disk has been investigated in detail by Abramowicz et al. (1997). A general accurate expression for the vertical hydrodynamical equilibrium that is valid both for a thin and a slim disk has been derived in their work. The hydrostatic balance in the vertical direction of the accretion disk is described by
19







1


ρ






∂
p


∂
z



+



∂
ψ


∂
z



+


v


r





∂


v


z




∂
r



+


v


z





∂


v


z




∂
z



=
0
,


where p is the pressure, ρ is the density of the gas, and ψ is the gravitational potential (Abramowicz et al. 1997). As we focus on how the general properties of the disk structure is affected by the radiation-driven outflows, we simplify the expression of the hydrostatic balance in the vertical direction as
20







1


ρ






∂
p


∂
z



+



∂
ψ


∂
z



=
0
,


where we assume the terms of ∂vz/∂r and ∂vz/∂z to be negligible. It has been pointed out that the widely used approximation for the vertical component of gravity, GMz/R3, is only valid for the thin disk with H/R ≪ 1, and a more accurate expression of vertical gravity should be adopted for a slim disk (McClintock et al. 2006; Gu & Lu 2007; Jiao et al. 2009; Gu 2012; Cao & Gu 2015). As discussed in Cao & Gu (2015), there are upper limits on the radiation flux, frad, and half-thickness, H, for a slim disk, above which the radiation force will overwhelm the vertical gravity. The maximal flux can be calculated with
21






f


rad


=
q
(
H
)
=



GM
ρ
H






r


2


+


H


2












r


2


+


H


2




−


r


s






2





.


How the gas is blown away from the disk surface by the radiation force is still quite unclear, and, therefore, it is assumed that the gas at the disk surface will be accelerated into outflows by the radiation force when the radiation force overwhelms the vertical gravity. Then, the mass accretion rate in the disk decreases due to outflows. This self-adjustment mechanism leads to a maximal radiation flux/thickness of the disk (see the detailed discussion in Cao & Gu 2015). Thus, we assume that the outflows are triggered in the disk where the condition
22






Q


rad


−


≥
2


f


rad


max


,


is satisfied, where 






f


rad


max




 is the maximal radiation flux and then the mass accretion rate is self-adjusted by the outflows to maintain
23






Q


rad


−


≡
2


f


rad


max


.


This relation is adopted in the energy of Equation (15) for the region in the disk where outflows are driven. Otherwise, the outflows are suppressed, and the disk is the same as the conventional slim disk without outflows, which is described by a set of equations for a slim disk (see Chapter 7 in Kato et al. 2008), i.e., the continuity equation:
24






M


˙


=
−
2
π
r
Σ


v


r


,


where 






M


˙




 remains constant radially, and the angular momentum equation is
25






v


r


(
Ω


r


2


−


j


in


)
=
−
α


rc


s


2


.


Except for Equations (23)–(25), the radial-component of the momentum equation, the equation of state, and energy equation are the same as the disk with outflows (see Equations (4) and (14)–(18)).
Citation: McClintock et al. 2006
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