CLUMPY ACCRETION ONTO BLACK HOLES. I. CLUMPY-ADVECTION-DOMINATED ACCRETION FLOW STRUCTURE AND RADIATION

Figure 1 shows the regimes of accretion disk models. We simply note that slim disks have $\dot{m}\gtrsim 1$, the standard model of SSD works between $1\gtrsim \dot{m}\gtrsim \dot{m}_2$, and accretion flows become advection-dominated when $\dot{m}_2=\alpha _{\rm A}^2\approx 0.1\alpha _{0.3}^2$, where α_0.3 = α_A/0.3 is the viscosity parameter of the ADAF, $\dot{m}=\dot{M}/\dot{M}_{\rm Edd}$, $\dot{M}$ is the accretion rate, $\dot{M}_{\rm Edd}=L_{\rm Edd}/\eta c^2=1.39\times 10^{18} \eta _{0.1}^{-1}\,m_{\bullet }\, {\rm g\, s^{-1}}$, c is the light speed, η_0.1 = η/0.1 is the radiative efficiency, L_Edd = 1.25 × 10³⁸m_• erg s⁻¹ is the Eddington luminosity, and m_• = M_•/M_☉ is the black hole mass. When the accretion rates are low enough ($\dot{m}<\dot{m}_1$), flows become a pure ADAF without clumps.

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The critical accretion rate $\dot{m}_1$ can be roughly estimated from the ionization parameter, which is defined by Ξ = L/4πR²cnk_BT, where k_B is the Boltzmann constant, L is the radiated luminosity, R is the distance to the ionizing source, n is the density, and T is the temperature of the cold clumps. The self-similar solution of the simple ADAF model gives the thermal pressure $P_{\rm A}=1.7\times 10^{16}\alpha ^{-1}c_1^{-1}c_3^{1/2}m_{\bullet }^{-1}\dot{m}r^{-5/2}{\rm \,g \,cm^{-1}\, s^{-2}}$, where c₁ = (5 + 2')g/3α², c₂ = [2'(5 + 2')g/9α²]^1/2, c₃ = 2(5 + 2')g/9α², g = [1 + 18α²/(5 + 2')²]^1/2 − 1, ' = (5/3 − γ)/f(γ − 1), γ is the adiabatic index, and f is the advection-dominated factor (Narayan & Yi 1994). In this paper, we use c₁ = 0.46, c₂ = 0.48, and c₃ = 0.31 for γ = 1.4 and f = 0.9 unless we note their specific values.³ The results in this paper are not very sensitive to the values of γ and f. Using the scaling relation of ADAF, we have its bolometric luminosity $L_{\rm ADAF}=0.2(\dot{m}/\alpha)^2L_{\rm Edd}$ (Mahadevan 1997). Here we only use the single temperature of the ADAF model. The ionization parameter defined by Ξ = L/4πR²cP_cl is used to describe the two-phase medium, where P_cl is the internal pressure of the clumps. We assume that the clumps hold a pressure balance with the ADAF (P_cl = P_A),⁴ yielding the critical accretion rate as

$$\begin{equation} \dot{m}_1=0.02 \alpha _{0.2}\Xi _{0.1}r_{1000}^{-1/2}, \end{equation} \tag{ 1 }$$

where Ξ_0.1 = Ξ/0.1 and r₁₀₀₀ = R/1000R_Sch (R_Sch = 2.95 × 10⁵m_• cm being the Schwarzschild radius). For cases with 10 > Ξ > 0.1 (Field 1965; Krolik 1998), the ionized gas holds a two-phase state with two different temperatures and a pressure balance between the hot and cold mediums. For gas with Ξ > 10 only a hot phase exists, whereas with Ξ < 0.1, only a cold phase exists. The timescale of the thermal instability is generally given by the line-cooling process, which determines the formation timescale of the clumps. With the cooling function (Böhringer & Hensler 1989), we have Δt_cl ∼ n_ek_BT/n²Λ_line = 7 × 10⁵n⁻¹₆T^2.3₆ s for a medium with one solar abundance. For an ADAF of 1 M_☉ black hole, clumps could be produced at the interacting regions between the ADAF and the SSD, namely the evaporation region, where Δt_cl will be much shorter than that of the Keplerian rotation. A detailed analysis is needed to show the production of clumps through thermal instability. Figure 2 shows a schematic of the clumpy-ADAF model.

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We would like to point out that Kuncic et al. (1996) and Celotti & Rees (1999) present further arguments to support the general existence of cold clumps in the accretion disk. This lower limit (Equation (1)) for the clumpy-ADAF is only based on the thermal instability and is regarded as a characteristic critical value. The limit of the critical accretion rate could become lower if the magnetorotational instability (MRI) is included in the ADAF. Furthermore, we presume that the ADAF part of the global clumpy-ADAF can be described by the self-similar solution.⁵ We do not consider the effects of ADAF-driven outflow on the clumps known as advection-dominated inflow and outflows (ADIOs; Blandford & Begelman 1999). However, it would be very interesting to postulate a situation in which the clumps could dynamically follow the outflows, forming clumpy outflows. ADIOs with clumps as a potential scenario will be considered in the future (we acknowledge the referee for bringing this point to our attention).

2.1.1. Clumps in the Clumpy-ADAF

2.1.2. Interaction Between Clumps and the ADAF

2.1.3. Inner Edge of the Clumpy-ADAF: Tidal Disruption

The orbiting clumps are suffering from the tidal disruption governed by the black hole. Self-gravitation of clumps is negligible; however, clumps still survive by keeping a pressure balance with the surroundings if the tidal distortion can be overcome by the thermal pressure of the ADAF. The tidal force reads F_tidal ≈ GM_•m_clR_cl/R³ for a cloud, where G is the gravitational constant and R is the distance to the hole. This tidal force is balanced by the thermal pressure of the ADAF, namely F_tidal = 4πP_AR²_cl. Since the clumps keep a pressure balance with the ADAF, we have P_A = n_clk_BT_cl, where n_cl = m_cl/(4π/3)R³_cl is the mass density of the clumps and k_B is the Boltzmann constant. Tidal disruption happens when F_tidal ⩾ 4πn_clk_BT_clR²_cl, yielding a natural limit of the inner boundary radius

$$\begin{eqnarray} &&{\cal R}_{\rm in}=\left(\frac{GM_{\bullet }m_pR_{\rm cl}^2}{4\pi k_{\rm B}T_{\rm cl}}\right)^{1/3} \approx \left\lbrace \begin{array}{@{}l}8.0 M_8^{-2/3}R_{11}^{2/3}T_4^{-1/3}R_{\rm Sch}\\ \\ 8.0 M_1^{-2/3}R_{3}^{2/3}T_4^{-1/3}R_{\rm Sch},\end{array}\right.\nonumber\\ \end{eqnarray} \tag{ 2 }$$

where M₈ = M_•/10⁸ M_☉, M₁ = M_•/1 M_☉, R₁₁ = R_cl/10¹¹ cm, R₃ = R_cl/10³ cm, and T₄ = T_cl/10⁴ K. Here we set the temperature of clumps T_cl = 10⁴ K. Within the tidal radius, clumps are destroyed to form a disk of debris or are mixed with the ADAF. Some papers have studied the formation of the accretion disk after the tidal disruption of stars by supermassive black holes (e.g., Cannizzo et al. 1990; Strubbe & Quataert 2009). Investigating the detailed processes of destroyed clumps to form an accretion disk is not the main goal of this paper; however, we focus on its influence on the ADAF. Following popular treatment, we assume that the tidal radius is the location of the disk after the captured debris' orbit has been circularized.

We should stress that we use the parameters of the clumps at the tidal radius throughout the entire ADAF even though the clumps are undergoing contraction. Collisions are neglected for the clumpy-ADAF; thus, the contraction is not important in the present situation. We use the values of the parameters listed in Table 1 throughout the ADAF. Clumps could vary in size and density in the outer parts of the ADAF, in particular, in a standard disk with clumps. In such a case, depending on the size of the clumps, they merge and fragment through collisions. This is, however, beyond the scope of this paper.

By defining the distribution function as ${\hbox{\myscr F}}= \Delta {\cal N}/R\Delta R\Delta z\Delta \phi \Delta \textbf {v}$, the dynamics of the clumps can be generally described using the Boltzmann equation. Unlike the normal stellar system, the clumps are moving in the supermassive black hole potential, but they are also being dragged by the ADAF. We start from the origin of collisionless Boltzmann equation (4-11) in Binney & Tremaine (1987)

$$\begin{equation} \frac{\partial {\hbox{\myscr F}}}{\partial t}+\sum _{\alpha =1}^6\frac{\partial \left({\hbox{\myscr F}}\dot{w}_{\alpha }\right)}{\partial w_{\alpha }}=0, \end{equation} \tag{ 3 }$$

where (x, v)≡ w and ${\dot{\textbf w}}\equiv (\dot{\textbf x},\,\dot{\textbf v}$) are the coordinates in the phase space. Generally, for a stellar system, $\sum _{\alpha =1}^6\partial \dot{w}_{\alpha }/\partial w_{\alpha }=0$ holds. The Boltzmann equation reduces to $\partial {\hbox{\myscr F}}/\partial t+\sum _{\alpha =1}^6\dot{w}_{\alpha }\partial {\hbox{\myscr F}}/\partial w_{\alpha }=0$. The drag force on the clumps depends on the velocity, making the Boltzmann equation complicated. Considering the dependence of acceleration of the clumps on their velocity, we recast the Boltzmann equation, so that

$$\begin{equation} \frac{\partial {\hbox{\myscr F}}}{\partial t}+\sum _{i=1}^3\left(v_i\frac{\partial {\hbox{\myscr F}}}{\partial x_i}+\dot{v}_i\frac{\partial {\hbox{\myscr F}}}{\partial v_i}+ {\hbox{\myscr F}}\frac{\partial \dot{v}_i}{\partial v_i}\right)=0. \end{equation} \tag{ 4 }$$

The third term in the bracket arises from the drag force, which disappears in a conservative system as described by Equation (4-13a) in Binney & Tremaine (1987). In a cylindrical coordinate, we have

$$\begin{eqnarray} &&\frac{\partial {\hbox{\myscr F}}}{\partial t}+\dot{R}\frac{\partial {\hbox{\myscr F}}}{\partial R}+\dot{\phi }\frac{\partial {\hbox{\myscr F}}}{\partial \phi }+\dot{z}\frac{\partial {\hbox{\myscr F}}}{\partial z}+ \dot{v}_R\frac{\partial {\hbox{\myscr F}}}{\partial v_R}+\dot{v}_{\phi }\frac{\partial {\hbox{\myscr F}}}{\partial v_{\phi }}+\dot{v}_z\frac{\partial {\hbox{\myscr F}}}{\partial v_z}\nonumber\\ &&+ {\hbox{\myscr F}}\left(\frac{\partial \dot{v}_R}{\partial v_R}+\frac{\partial \dot{v}_{\phi }}{\partial v_{\phi }}+ \frac{\partial \dot{v}_z}{\partial v_z}\right)=0, \end{eqnarray} \tag{ 5 }$$

where $\dot{R}=v_R$, $\dot{\phi }=v_{\phi }/R$, and $\dot{z}=v_z$. Motion equations of an individual cloud read

$$\begin{eqnarray} \dot{v}_R&=&{-}\frac{\partial \Phi }{\partial R}+\frac{v_{\phi }^2}{R}+F_{_R}; \dot{v}_{\phi }=-\frac{1}{R}\frac{\partial \Phi }{\partial \phi }-\frac{v_Rv_{\phi }}{R}+F_{\phi }; \nonumber\\ \dot{v}_z&=&{-}\frac{\partial \Phi }{\partial z}, \end{eqnarray} \tag{ 6 }$$

where Φ = GM_•/(R² + z²)^1/2, $F_{_R}=f_R(v_{_R}-V_{_R})^2$, and F_ϕ = f_ϕ(v_ϕ − V_ϕ)² are the drag forces per unit mass in the R- and ϕ-directions, respectively, and $V_{_R}$ and V_ϕ are the radial and rotational velocities of the ADAF. It should be noted that f_R and f_ϕ might be functions of the cloud's parameter, such as radius and density. Here we neglect the drag forces in the z-direction. We have $\sum _{\alpha =1}^6\partial \dot{w}_{\alpha }/\partial w_{\alpha }=2f_{\phi }(v_{\phi }-V_{\phi })+2f_R(v_{_R}-V_{_R})$ in the cylindrical coordinate frame. Considering the ϕ-symmetry of the clumpy disk, we have

$$\begin{eqnarray} &&\frac{\partial {\hbox{\myscr F}}}{\partial t}+v_R\frac{\partial {\hbox{\myscr F}}}{\partial R}+{v_z}\frac{\partial {\hbox{\myscr F}}}{\partial z}+ \left(\frac{v_\phi ^2}{R}-\frac{\partial \Phi }{\partial R}+F_{_R}\right)\frac{\partial {\hbox{\myscr F}}}{\partial v_R}\nonumber\\ &&+ \left(F_{\phi }-\frac{v_Rv_{\phi }}{R}\right)\frac{\partial {\hbox{\myscr F}}}{\partial v_{\phi }}- \frac{\partial \Phi }{\partial z}\frac{\partial {\hbox{\myscr F}}}{\partial v_z} + 2{\hbox{\myscr F}}\,\big[f_\phi (v_{\phi }-V_{\phi })\nonumber\\ &&+f_R(v_{_R}-V_{_R})\big] = 0. \end{eqnarray} \tag{ 7 }$$

This is the final version of the Boltzmann equation used for the dynamics of the clumps in the following sections.

Following the popular treatment, we solve the moment equations of the Boltzmann equation. We define the averaged parameter in velocity-space as $\langle X\rangle ={\hbox{\myscr N}\,\,}^{-1}\int X{\hbox{\myscr F}}d{\bm v}$, where X is a parameter. Here we use ${\hbox{\myscr N}}=\int {\cal N}d{\bm v}$. The zeroth-order moment equation can be obtained by integrating Equation (7):

$$\begin{equation} \frac{\partial {\hbox{\myscr N}}}{\partial t}+\frac{1}{R}\frac{\partial }{\partial R} (R{\hbox{\myscr N}}\,\langle v_{_R}\rangle) +\frac{\partial }{\partial z}({\hbox{\myscr N}}\,\langle v_z\rangle)=0. \end{equation} \tag{ 8 }$$

The first-order moment equations can be gained through multiplying Equation (7) by v_R, v_ϕ, and v_z, respectively, and integrating the equations in velocity-space. The first moment equation is given by

$$\begin{eqnarray} &&\frac{\partial }{\partial t}({\hbox{\myscr N}}\,\langle v_{_R}\rangle)+\frac{\partial }{\partial R}\left({\hbox{\myscr N}}\big\langle v_R^2\big\rangle\right)+ \frac{\partial }{\partial z}({\hbox{\myscr N}}\,\langle v_{_R}v_z\rangle)\nonumber\\ &&+{\hbox{\myscr N}}\,\left(\frac{\left\langle v_R^2\right\rangle -\left\langle v_{\phi }^2\right\rangle }{R}+ \frac{\partial \Phi }{\partial R}\right)- {\hbox{\myscr N}}\,f_R\big(\left\langle v_R^2\right\rangle \nonumber\\ &&-2\langle v_{_R}\rangle V_{_R}+V_{_R}^2\big)=0, \end{eqnarray} \tag{ 9 }$$

the second as

$$\begin{eqnarray} &&\frac{\partial }{\partial t}({\hbox{\myscr N}}\,\langle v_{\phi }\rangle)+\frac{1}{R^2}\frac{\partial }{\partial R}(R^2{\hbox{\myscr N}}\,\langle v_{_R}v_{\phi }\rangle)+ \frac{\partial }{\partial z}({\hbox{\myscr N}}\,\langle v_{\phi }v_z\rangle)\nonumber\\ &&-{\hbox{\myscr N}}\,f_{\phi }\left(\left\langle v_{\phi }^2\right\rangle -2\langle v_{\phi }\rangle V_{\phi }+V_{\phi }^2\right)=0, \end{eqnarray} \tag{ 10 }$$

and the third as

$$\begin{eqnarray} &&\frac{\partial }{\partial t}({\hbox{\myscr N}}\,\langle v_z\rangle)+\frac{\partial }{\partial R}({\hbox{\myscr N}}\,\langle v_{_R}v_z\rangle)+ \frac{\partial }{\partial z}\left({\hbox{\myscr N}}\,\left\langle v_z^2\right\rangle \right)+\frac{1}{R}{\hbox{\myscr N}}\,\langle v_{_R}v_z\rangle\nonumber\\ &&+\frac{\partial \Phi }{\partial z}{\hbox{\myscr N}}\,=0. \end{eqnarray} \tag{ 11 }$$

We will use these moment equations to discuss the dynamics of the clumpy disk.

For the ϕ- and z-direction symmetric clumpy-ADAF, we have 〈v_z〉 = 0, $\langle v_{_R}v_z\rangle =0$, and 〈v_ϕv_z〉 = 0, and we recast the continuity equation as

$$\begin{equation} \frac{1}{R}\frac{\partial }{\partial R}(R{\hbox{\myscr N}}\,\langle v_{_R}\rangle)=0, \end{equation} \tag{ 12 }$$

yielding the accretion rates of the clumps as $\dot{M}_{\rm cl}=-2\pi R H_{\rm A}{\hbox{\myscr N}}\,m_{\rm cl}\langle v_{_R}\rangle$. The first moment equation is reduced to

$$\begin{eqnarray} &&\frac{\partial }{\partial R}\left({\hbox{\myscr N}}\,\left\langle v_R^2\right\rangle \right)+ {\hbox{\myscr N}}\,\left(\frac{\left\langle v_R^2\right\rangle -\left\langle v_{\phi }^2\right\rangle }{R}+ \frac{\partial \Phi }{\partial R}\right)-{\hbox{\myscr N}}\,f_R\big(\big\langle v_R^2\big\rangle\nonumber\\ && -2\langle v_{_R}\rangle V_{_R}+V_{_R}^2\big)=0, \end{eqnarray} \tag{ 13 }$$

the second to

$$\begin{eqnarray} &&\frac{1}{R^2}\frac{\partial }{\partial R}\left(R^2{\hbox{\myscr N}}\,\langle v_{_R}v_{\phi }\rangle \right)- {\hbox{\myscr N}}\,f_{\phi }\left(\left\langle v_{\phi }^2\right\rangle -2\left\langle v_{\phi }\right\rangle V_{\phi }+V_{\phi }^2\right)=0,\nonumber\\ \end{eqnarray} \tag{ 14 }$$

and the third to

$$\begin{equation} \frac{\partial }{\partial z}\left({\hbox{\myscr N}}\,\left\langle v_z^2\right\rangle\right)+ \frac{\partial \Phi }{\partial z}{\hbox{\myscr N}}\,=0. \end{equation} \tag{ 15 }$$

Table 2 gives a summary of input and output parameters used in the present model. The above moment equations describe the dynamics of clumps; however, these are not close. We must supplement additional physical considerations to proceed. We distinguish two classes of the dynamics in light of the strength of coupling between the clumps and the ADAF. When f_R and f_ϕ are large enough, the clumps are strongly coupled with the ADAF so that the average dynamics of the clumps follows the ADAF. For small f_R and f_ϕ, the clumps are weakly coupled with the ADAF, showing weak dependence on the ADAF.

Table 2. Parameters of the Present Model

Parameter	Physical Meanings
Input Parameters
M•	Black hole mass
$\dot{M}_{\rm A}$	Accretion rate of the continuous flow
$\dot{M}_{\rm cl}$	Average accretion rate of the clumps
mcl	Average mass of individual clumps
Rin	Tidal radius as the inner radius of the clumpy-ADAF
Rout	Outer radius of the clumpy-ADAF
Rcl	Average radius of individual clumps
ncl	Hydrogen number density of clumps
${\mathfrak {M}}$	Ratio of the ADAF and clumps mass ($=\dot{M}_{\rm cl}/\dot{M}_{\rm A}$)
α	Viscosity parameter of the ADAF
ΓR	Coefficient of the R-direction drag force
Γϕ	Coefficient of the ϕ-direction drag force
Output Parameters
${\hbox{\myscr N}}\,$	Number density of the clumps
$\langle v_{_R}\rangle$	Average R-direction velocity
〈v2R〉	Average v2R
〈vϕ〉	Average ϕ-direction velocity
〈v2ϕ〉	Average values of v2ϕ
$\dot{\hbox{\myscr R}}$	Capture rates of clumps

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In the strong-coupling case, the drag force is so strong that the average dynamics of the clumps follow the ADAF, namely $\langle v_{_R}\rangle =V_{_R}$ and 〈v_ϕ〉 = RΩ_A, where Ω_A is the rotational velocity of the ADAF. So, the factor f_ϕ should be larger than a critical one. This can be understood by the fact that both the specific angular momentum and the kinetic energy of the clumps are about the same as the gas of the ADAF since they are born in the ADAF. Therefore, we have

$$\begin{equation} \langle v_{_R}\rangle =-2.12\times 10^{10}\alpha c_1r^{-1/2} {\rm \,cm\, s^{-1}}, \end{equation} \tag{ 16 }$$

$$\begin{equation} \langle v_{\phi }\rangle =2.12\times 10^{10}c_2r^{-1/2} {\rm \,cm\, s^{-1}}, \end{equation} \tag{ 17 }$$

$\langle v_{_R}\rangle /\langle v_{\phi }\rangle =-\alpha c_1/c_2$, and the height of the clumpy disk

$$\begin{equation} H_{\rm A}=2.95\times 10^5c_3^{1/2}c_2^{-1}mr {\rm \,cm}. \end{equation} \tag{ 18 }$$

From the continuity equation, we have

$$\begin{equation} {\hbox{\myscr N}}\,=-\frac{\dot{M}_{\rm cl}}{4\pi RH_{\rm A}\langle v_{_R}\rangle m_{\rm cl}}, \end{equation} \tag{ 19 }$$

where $\dot{M}_{\rm cl}$ is the averaged accretion rate of clumps. From ϕ-motion equation, we have

$$\begin{eqnarray} \left\langle v_{\phi }^2\right\rangle &=&\frac{1}{{\hbox{\myscr N}}\,f_{\phi }}\frac{1}{R^2}\frac{d}{dR}(R^2{\hbox{\myscr N}}\,\langle v_{_R}v_{\phi }\rangle)+2\langle v_{\phi }\rangle V_{\phi }-V_{\phi }^2 \nonumber\\ &=&{-}\frac{1}{2}\frac{\langle v_{_R}v_{\phi }\rangle }{f_{\phi }R}+2\langle v_{\phi }\rangle V_{\phi }-V_{\phi }^2 \end{eqnarray} \tag{ 20 }$$

and 〈v²_ϕ〉/〈v_ϕ〉² = 1 + αc₁/2c₂f_ϕR. The radial motion is rewritten by

$$\begin{eqnarray} &&\frac{d\left\langle v_R^2\right\rangle }{dR}+\frac{1}{R}\left(1+\frac{d\ln {\hbox{\myscr N}}\,}{d\ln R}-f_RR\right)\left\langle v_R^2\right\rangle \nonumber\\ &&+ \left(\left\langle \frac{\partial \Phi }{\partial R}\right\rangle -\frac{1}{R}\left\langle v_{\phi }^2\right\rangle +f_RV_{_R}^2\right)=0, \end{eqnarray} \tag{ 21 }$$

where

$$$ \left\langle \frac{\partial \Phi }{\partial R}\right\rangle =\frac{1}{H_{\rm A}}\int _0^{H_{\rm A}}\frac{\partial \Phi }{\partial R}dz \approx \frac{GM_{\bullet }}{R^2}. $$$

Considering $d\ln {\hbox{\myscr N}}\,/d\ln R=-3/2$ from Equation (19), we have

$$\begin{eqnarray} &&\frac{d\left\langle v_R^2\right\rangle }{dR}-\frac{1}{R}\left(f_RR+\frac{1}{2}\right)\left\langle v_R^2\right\rangle +R\left(\Omega _{\rm K}^2- \frac{\left\langle v_{\phi }^2\right\rangle }{\left\langle v_{\phi }\right\rangle ^2}\Omega _{\rm A}^2\right)\nonumber\\ &&+f_RV_{_R}^2=0. \end{eqnarray} \tag{ 22 }$$

Clearly, the ϕ-motion has a strong influence on the radial motion. Inserting 〈v²_ϕ〉 and 〈v_ϕ〉, we have

$$\begin{eqnarray} &&\frac{d\left\langle v_R^2\right\rangle }{dR}-\frac{1}{R}\left(f_RR+\frac{1}{2}\right)\left\langle v_R^2\right\rangle +\bigg[\left(1-c_2^2\right)R \nonumber\\ &&- \frac{1}{2}\alpha c_1c_2f_{\phi }^{-1}\bigg]\Omega _{\rm K}^2+f_RV_{_R}^2=0. \end{eqnarray} \tag{ 23 }$$

With the outer boundary condition of 〈v²_R〉 = V_out² at R = R_out, we have the solution as

$$\begin{eqnarray} \left\langle v_R^2\right\rangle &=&c^2\bigg\lbrace \frac{1}{2}\bigg[\alpha ^2c_1^2\Gamma _R\Lambda _{\frac{3}{2}}(\Gamma _R,r) +\left(1-c_2^2\right)\Lambda _{\frac{5}{2}}(\Gamma _R,r) \nonumber\\ &&-\frac{\alpha c_1c_2}{2\Gamma _{\phi }}\Lambda _{\frac{7}{2}}(\Gamma _R,r)\bigg] +\frac{V_{\rm out}^2}{c^2r_{\rm out}^{1/2}}e^{-\Gamma _Rr_{\rm out}}\bigg\rbrace r^{1/2}e^{\Gamma _R r},\nonumber\\ \end{eqnarray} \tag{ 24 }$$

where Γ_R = f_RR_Sch, Γ_ϕ = f_ϕR_Sch, and the function

$$$ \Lambda _{q}(\Gamma _R,r)=\int _r^{r_{\rm out}}x^{-q}e^{-\Gamma _Rx}dx, $$$

and q = 3/2, 5/2, 7/2. Figure 3 shows the properties of the function Λ_q. Equation (24) gives the solution of the clumps, which deviates from the ADAF.

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Physical meanings of each term in Equation (24) can be examined under some extreme cases. Generally, clumps are controlled by two factors: (1) black hole potential, and (2) ϕ- and R-direction drag forces. A drag-free cloud's, orbit is determined purely by the black hole. When f_R tends to zero, namely Γ_R = 0, clumps are orbiting around the black hole with ϕ-drag. The angular momentum of the clumps is transferred by the ADAF, in which the popular α-prescription works toward an outward transportation of the angular momentum of the ADAF, giving rise to a fast spiral down to the black hole. In such a case, we have

$$\begin{eqnarray} \left\langle v_R^2\right\rangle &=& \frac{1}{3}\left(1-c_2^2\right)\frac{c^2}{r}\left[1-\left(\frac{r}{r_{\rm out}}\right)^{3/2}\right]- \frac{\alpha c_1c_2}{20\Gamma _{\phi }}\frac{c^2}{r^2}\nonumber\\ &&\times \left[1-\left(\frac{r}{r_{\rm out}}\right)^{5/2}\right]+ \left(\frac{r}{r_{\rm out}}\right)^{1/2}V_{\rm out}^2. \end{eqnarray} \tag{ 25 }$$

The first term is the orbital motion around the black hole. The second term results in the accordance of cloud motion with the ADAF. The ϕ-drag decreases the velocity dispersion between the clumps and the ADAF. When the ϕ-drag is strong enough, we have the critical value

$$\begin{equation} \Gamma _{\phi }^c\approx \frac{3\alpha c_1c_2}{10\left(2-2c_2^2-3\alpha ^2 c_1^2\right)r_{\rm in}} \approx 8.7\times 10^{-4} \alpha _{0.2}r_{10}^{-1}. \end{equation} \tag{ 26 }$$

Here we neglect the terms of (r_in/r_out). When Γ_ϕ = Γ^c_ϕ, coupling with the ADAF is so strong that the velocity dispersion of the clumps with the ADAF is zero at the tidal capture radius (R_in), namely 〈v²_R〉^1/2 = V_in at R = R_in, where V_in is the radial velocity of the ADAF. So, the strong coupling is referred to the case with Γ_ϕ > Γ^c_ϕ. Models with Γ_ϕ < Γ^c_ϕ are the weak coupling, so the strong coupling approximation does not work. We will discuss the case below.

For an extremely strong coupling, namely Γ_ϕ → ∞, clumps tend to have

$$\begin{eqnarray} \big\langle v_R^2\big\rangle &=& c^2\bigg\lbrace \frac{1}{2}\bigg[\alpha ^2c_1^2\Gamma _R\Lambda _{\frac{3}{2}}(\Gamma _R,r) +\big(1-c_2^2\big)\Lambda _{\frac{5}{2}}(\Gamma _R,r)\bigg] \nonumber\\ &&+\frac{V_{\rm out}^2}{c^2r_{\rm out}^{1/2}}e^{-\Gamma _Rr_{\rm out}}\bigg\rbrace r^{1/2}e^{\Gamma _R r}\approx \frac{1}{3}\big(1-c_2^2\big)\frac{c^2}{r}\nonumber\\ &&\times\bigg[1- \bigg(\frac{r}{r_{\rm out}}\bigg)^{3/2}\bigg]+\bigg(\frac{r}{r_{\rm out}}\bigg)^{1/2}V_{\rm out}^2, \end{eqnarray} \tag{ 27 }$$

reaching their maximum. Here, the first term with $\Lambda _{\frac{3}{2}}$ is always smaller than the others.

Figure 4 shows the solutions of the clumpy disk for the different parameters of the drag forces. For a fixed Γ_ϕ case, the radial drag strongly influences the velocity dispersion of the clumps at large radii. On the other hand, for fixed Γ_R cases, 〈v²_R〉 is mainly determined by the Γ_ϕ and reaches their maximum as shown in Figure 4. We find that the term involving Λ_3/2 is always smaller than the other two in Equation (24). This is due to the radial velocity of the ADAF, which is smaller than the rotational, and it is represented by multiplying the factor α. Although the influence of the radial drag can be neglected, the present treatments are complete. The most important point is that 〈v²_R〉^1/2 ∼ 10〈v_R〉 for the extremely strong coupling case from Figure 4. This means that the accretion rates of the clumps are actually enhanced.

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When the birth of the clumps is not very tightly linked with the ADAF in dynamics, the strong coupling between the clumps and the ADAF is relaxed. For cases of weak coupling with the ADAF, the assumptions of $\langle v_{_R}\rangle =V_{_R}$ and 〈v_ϕ〉 = V_ϕ do not work, and cloud dynamics resembles the stellar system. We introduce the velocity dispersions $\sigma _{_R}^2=\langle v_R^2\rangle -\langle v_{_R}\rangle ^2$, σ²_ϕ = 〈v_ϕ²〉 − 〈v_ϕ〉², and σ²_z = 〈v_z²〉 − 〈v_z〉² to solve the moment equations. The moment equations should be closed up by physical considerations. Similar to stellar dynamics, we assume σ²_R = k_Rσ²_z and σ²_ϕ = k_ϕσ²_z, where k_R and k_ϕ are two constants. Since the clumps are produced by the ADAF in light of thermal instability, the velocity dispersion of the clumps should be less than the sound speed of the ADAF, and the vertical height of the clumpy disk does not exceed the ADAF height, we assume H = H_A. Employing 〈v_z〉 = 0 and σ²_z = 〈v_z²〉, we have

$$\begin{equation} \sigma _z=H_{\rm A}\Omega _{\rm K}. \end{equation} \tag{ 28 }$$

After some algebraic manipulations, we recast Equations (17) and (18) as

$$\begin{equation} \left(\frac{\langle v_{_R}\rangle ^2-\sigma _R^2}{\langle v_{_R}\rangle }\right)\frac{d\langle v_{_R}\rangle }{dR}- \frac{\langle v_{\phi }\rangle ^2+\sigma _{\phi }^2}{R}+\frac{d\sigma _R^2}{dR}+ \left\langle \frac{\partial \Phi }{\partial R}\right\rangle =0 \end{equation} \tag{ 29 }$$

and

$$\begin{equation} \langle v_{_R}\rangle \frac{d\langle v_{\phi }\rangle }{dR}+\frac{\langle v_{_R}\rangle }{R}\langle v_{\phi }\rangle -f_{\phi }\big[\sigma _{\phi }^2+ (\langle v_{\phi }\rangle -V_{\phi })^2\big]=0, \end{equation} \tag{ 30 }$$

where Equation (19) is used.

For a weak-coupling case, the clumps are mainly orbiting around the black hole. The first term with $d\langle v_{_R}\rangle /dR$, σ²_ϕ/R and dσ²_R/dR are of the same order, but the term is much smaller than 〈v_ϕ〉²/R and ∂Φ/∂R. So, we have

$$\begin{equation} \langle v_{\phi }\rangle \approx -\left\langle \frac{\partial \Phi }{\partial R}\right\rangle =V_{\rm K}, \end{equation} \tag{ 31 }$$

where V_K is the Keplerian velocity. Since 〈v_ϕ〉 ≈ V_K, we have σ_ϕ ≈ 0 and dv_ϕ/dR ≈ −v_ϕ/2R, and Equation (30) yields

$$\begin{equation} \langle v_{_R}\rangle =\frac{2f_{\phi }R}{V_{\rm K}}(V_{\rm K}-V_{\phi })^2 =7.0\times 10^4 \Gamma _{-5}r^{1/2} {\rm \,cm\, s^{-1}}, \end{equation} \tag{ 32 }$$

where Γ₋₅ = Γ_ϕ/10⁻⁵. The mean velocity of the clumps is linearly proportional to the factor f_ϕ. The velocity dispersions $\langle v_R^2\rangle =\sigma _R^2+\langle v_{_R}\rangle ^2$ and 〈v²_ϕ〉 = σ_ϕ² + 〈v_ϕ〉² are known only if k_R and k_ϕ are also known; however, the present model cannot determine k_R and k_ϕ in a self-consistent way. The two parameters should be constrained by observations. The weak-coupling case does not have significant observable effects; therefore, we do not use it further.

As a brief summary, the dynamics of the clumps are very different from the ADAF both for strong- and weak-coupling cases between the clumps and the ADAF, but the dynamical properties of the clumps depend on the ADAF. The prominent properties are the velocity dispersion of clumps is one order higher or much smaller than the radial velocity of the ADAF in strong- and weak-coupling cases, respectively. For weak-coupling cases, solutions with the clumps weakly coupled with the ADAF show that their properties are similar to stars in the black hole potential field. These properties of the strong-coupled clumpy-ADAF determine the variabilities of the accretion disk of black holes.

Finally, we would like to point out the roles of magnetic fields discussed by Kuncic et al. (1996). For magnetized clumps, magnetic fields might be in equipartition with thermal pressure. The magnetic field plays a role in resisting the MRI-turbulence, making the existence of the clumps more persistent. Furthermore, magnetic fields lower the thermal conductivity (Spitzer 1962); thus, they create a more robust clump. In reality, this may be much more complicated. Future numerical simulations may be able to uncover more details into the roles of magnetic fields in the clumpy-ADAF.

Considering the mass of clumps $\Delta M_{\rm cl}=4\pi RH_{\rm A}{\hbox{\myscr N}}\,m_c \Delta R$ within R to R + ΔR, whereas the gas mass of the ADAF is ΔM_A = 4πRH_Aρ_AΔR, we find that the mass fraction of the clumps to the ADAF is

$$\begin{equation} {\mathfrak {M}}=\frac{\Delta M_{\rm cl}}{\Delta M_{\rm A}}=\frac{\dot{M}_{\rm cl}}{\dot{M}_{\rm A}}, \end{equation} \tag{ 33 }$$

where $\dot{M}_{\rm A}$ is the accretion rate of the ADAF. In principle, the parameter ${\mathfrak {M}}$ should be self-consistently determined using the cloud formation model, but this is beyond the scope of this paper. Instead, we treat it as a free parameter in the model.

The capture rates are given by the number of clumps that enter the tidal radius per unit time. Considering a time interval of Δt, we include the number of captured clumps through the surface of the tidal radius as $\Delta {\hbox{\myscr N}}\,=2\pi R_{\rm in}H_{\rm in}{\hbox{\myscr N}}\,_{\rm in}\langle v_R^2\rangle ^{1/2}\Delta t$, where H_in and ${\hbox{\myscr N}}\,_{\rm in}$ are the height of the clumpy disk and the clump number density at the tidal radius, respectively. Therefore, the capture rates are $\dot{\hbox{\myscr R}}=\lim _{\Delta t\rightarrow 0}(\Delta {\hbox{\myscr N}}\,/\Delta t)$

$$\begin{eqnarray} \dot{\hbox{\myscr R}}&=&\delta {\mathfrak {M}}\dot{m}_{\rm A}\frac{\dot{M}_{\rm Edd}}{m_{\rm cl}}\nonumber\\ &\approx& \left\lbrace \begin{array}{@{}l}1.4\,{\times}\, 10^7 \eta _{0.1}^{-1}\delta _1{\mathfrak {M}}\,\dot{m}_{-2}m_{10}^{-1}M_1 {\rm \,s^{-1}},\\ \\ 3.6\,{\times}\, 10^9 \eta _{0.1}^{-1}\delta _1{\mathfrak {M}}\,\dot{m}_{-2}m_{22}^{-1}M_8 {\rm \,month^{-1}}, \end{array}\right. \end{eqnarray} \tag{ 34 }$$

where δ = 〈v²_R〉^1/2/〈v_R〉, δ₁ = δ/10, and $\dot{m}_{-2}=\dot{M}_{\rm A}/10^{-2}\dot{M}_{\rm Edd}$. We note that the capture rates are very high since 〈v²_R〉^1/2 ∼ 10〈v_R〉.

Since the capture timescale $\dot{\hbox{\myscr R}}^{-1}$ is much smaller than the accretion of the debris onto black holes, the captured clumps monotonically accumulate with time. For a single capture event, the tidally disrupted debris has complicated fates, which may resemble the captured stars (e.g., Rees 1988). However, the difference of the present case from the captured star should be noted: the debris of disrupted clumps will interact with the ADAF. The circularization timescale could be a multiple Keplerian timescale orbiting the black hole, possibly mixing with the local ADAF. On the other hand, after a series of captured clumps, a tiny disk of debris will be formed within the timescale of Δt_cir if the clumps take enough kinetic energy. The interaction among the debris of the captured clumps is very complicated; however, it is reasonable to assume that the timescale used to form the debris disk is of the order of Δt_cir.

Fates of the disrupted clumps depend on the cloud's properties as well as the density of the ADAF. Similar to the case of the captured star, the debris of a disrupted cloud has a specific kinetic energy ε ∼ 3(GM_•/R_P)(R_cl/R_P) for clumps approaching the black hole with a parabolic orbit, where R_P is the pericenter distance of the parabolic orbit (Lacy et al. 1982; Evans & Kochanek 1989). After multiple cycles of the parabolic orbits, the gas flow will be circularized and form a disk. The fallback timescale is given by

$$\begin{equation} \Delta t_{\rm cir}^{\rm min}\approx \left\lbrace \begin{array}{@{}l}0.7 M_1^{5/2}R_{_{P,3}}^3R_{\rm cl, 3}^{-3/2} {\rm \,s},\\ \\ 2.2 M_8^{5/2}R_{_{P,3}}^3R_{\rm cl, 11}^{-3/2} {\rm \,yr}, \end{array}\right. \end{equation} \tag{ 35 }$$

where $R_{_{P,3}}=R_{_P}/3R_{\rm Sch}$ (e.g., Strubbe & Quataert 2009). Since the debris of the clumps is embedded in the ADAF, the interaction between them is unavoidable. This makes it possible for the debris to mix with the ADAF when the interaction timescale is shorter than that of the circularization; otherwise, the debris will form a disk of its own, called a debris disk. This timescale is actually for the accumulation of the debris disk (Δt_acc ∼ Δt^min_cir).

4.2.1. Mixed with the ADAF

The swept mass rates of the debris are $\dot{\hbox{\myscr M}}\sim \pi R_{\rm cl}^2 n_{\rm A}\langle v_R^2\rangle ^{1/2}m_p$ and the lost energy rates are $\dot{E}_{\rm K}\sim \dot{\hbox{\myscr M}}c_s^2$, where c_s is the sound speed of the ADAF. The timescale of dissipating the kinetic energy of the debris is given by

$$\begin{eqnarray} &&\Delta t_{\rm diss}=\frac{E_{\rm K}}{\dot{E}_{\rm K}}\sim \frac{m_p\big\langle v_R^2\big\rangle R_{\rm cl}}{k_{\rm B}T_{\rm cl}\langle v_{_R}\rangle } \approx \left\lbrace \begin{array}{@{}l}1.2 \delta _1\big\langle v_R^2\big\rangle _{8}^{1/2}R_3T_4^{-1} {\rm \,s},\\ \\ 3.8 \delta _1\big\langle v_R^2\big\rangle _{8}^{1/2}R_{11}T_4^{-1} {\rm \,yr}, \end{array}\right.\nonumber\\ \end{eqnarray} \tag{ 36 }$$

where 〈v²_R〉₈^1/2 = 〈v²_R〉^1/2/10⁸ cm s⁻¹, and the approximation of the pressure balance between clumps and the ADAF is used. We find that Δt_diss ∼ Δt^min_cir generally holds for small 〈v²_R〉 cases, but Δt_diss is significantly longer than Δt^min_cir for large 〈v²_R〉 cases. This indicates the formation of a debris disk, especially for the very strong coupling case.

When the debris is mixed with the ADAF, the density of the ADAF within the tidal radius will be generally enhanced, so that the cooling of the ADAF increases. Cooling enhancement could be moderate by mixing the ADAF with the captured clumps. We do not focus on this case; instead, we focus on the formation of a debris disk.

4.2.2. Formation of a Debris Disk

As we have shown above, capturing the clumps is much faster than the accretion onto black holes, yielding an accumulation of debris around the hole. Since the orbits of the clumps are in the ADAF, the new disk of the accumulated debris will be inside the ADAF. Detailed formation of the debris disk could be very complicated (more so than the case of the tidally disrupted stars; see Rees 1988; Strubbe & Quataert 2009). Here we assume that the formed disk holds the approximation of radiation-pressure-dominated regions of the SSD since most of the gravitational energy will be released in this region. The accretion timescale driven by the viscosity is given by t_debris ≈ R²_debris/ν = α⁻¹(H_debris/R)⁻² t_K, where t_K = 1/Ω_K is the Keplerian rotation timescale, R_debris and H_debris are the typical radius and scale height of the debris disk, respectively, and ν is the kinetic viscosity. Using the SSD solution, we have the accretion timescale

$$\begin{equation} \Delta t_{\rm debris}=\left\lbrace \begin{array}{@{}l}1.56 \alpha _{0.2}^{-1}M_1\dot{m}_{0.1}^{-2}r_{10}^{7/2} {\rm \,s},\\ \\ 4.96 \alpha _{0.2}^{-1}M_8\dot{m}_{0.1}^{-2}r_{10}^{7/2} {\rm \,yr}, \end{array}\right. \end{equation} \tag{ 37 }$$

where r₁₀ = R_in/10R_Sch and α_0.2 = α_SSD/0.2. We find that $\dot{\hbox{\myscr R}}^{-1}\ll \Delta t_{\rm debris}$. The accumulation of the debris around the black hole follows the capture of the clumps.

For an interval Δt, the accumulated mass of the captured clumps is $\Delta M_{\rm cl}=m_{\rm cl}\dot{\hbox{\myscr R}}\Delta t$, where we neglect the swallowed clumps by the black hole during accumulation. The accretion rate of the debris disk is $\dot{M}_{\rm debris}=\Delta M_{\rm cl}/\Delta t_{\rm debris}$, and the dimensionless rate is

$$\begin{equation} \dot{m}_{\rm debris} =\delta \mathfrak {M}\dot{m}_{\rm A}\left(\frac{\Delta t}{\Delta t_{\rm debris}}\right) =0.1 \delta _1{\mathfrak {M}}\,\dot{m}_{-2}\Delta t_1, \end{equation} \tag{ 38 }$$

where Δt₁ = Δt/1.0Δt_debris. We stress that the accretion rates of the debris disk are one order higher than the undergoing ADAF, which arises from the fast radial velocity of clumps (δ ∼ 10) for the strong-coupling case. The storage of the debris is undergoing through capturing the clumps since it carries too much kinetic energy to be directly accreted or mixed with the local ADAF.

It should be pointed out that the above estimation is based on the radiation-pressure-dominated solution of the Shakura–Sunyaev model. The transition radius from gas to radiation-pressure-dominated regions is $R_{\rm tr}/R_{\rm Sch}\approx 104.1(\alpha m_{\bullet })^{2/21}(\dot{m}/\eta _{0.1})^{16/21}$. For $\dot{m}=0.1$, we find that $R_{\rm tr}\approx 18.0(\alpha m_{\bullet })^{2/21}R_{\rm Sch}>{\cal R}_{\rm in}$ generally holds. This means that the debris disk should be generally radiation-pressure-dominated. Debris disks dominated by gas pressure could still be in the ADAF regime until they reach the SSD regimes. The present estimations are valid.

4.3.1. Compton Cooling as Feedback to the ADAF

We show that a debris disk forms within Δt ∼ Δt_debris in the regime of the Shakura–Sunyaev model. The disk is radiating at a quite large luminosity, $L_{\rm debris}=\eta \dot{M}_{\rm debris}c^2\sim 10^{45}\dot{m}_{0.1}M_8$ erg s⁻¹, where $\dot{m}_{0.1}=\dot{m}_{\rm debris}/0.1$. Photons from the debris disk spanning from optics to UV for supermassive black holes and from UV to ≲ 1.0 keV for a few solar mass black holes provide extra sources to cool the hot electrons in the ADAF through Compton cooling with a timescale

$$\begin{equation} \Delta t_{\rm Comp}=\frac{n_ek_{\rm B}T}{\Lambda _{\rm Comp}}=\left\lbrace \begin{array}{@{}l}54.1 M_1\dot{m}_{0.1}^{-1}r_{1000}^2 {\rm \,ms}\\ \\ 2.1 M_8\dot{m}_{0.1}^{-1}r_{1000}^2 {\rm \,month}, \end{array}\right. \end{equation} \tag{ 39 }$$

where Λ_Comp = 2n_ek_BTσ_TF_debris/m_ec² is the Compton cooling rate, the energy flux from the debris disk is F_debris = L_debris/4πR², and $L_{\rm debris}=\dot{m}_{\rm debris}L_{\rm Edd}$. Using t_Comp = Δt_debris, we have the Compton radius, within which the ADAF is driven by the emergent photons from the debris disk to collapse through Compton cooling

$$\begin{equation} R_{\rm Comp}=5447.0 \alpha _{0.2}^{-1/2}r_{10}^{7/4}\dot{m}_{0.1}^{-1/2} R_{\rm Sch}. \end{equation} \tag{ 40 }$$

We find that R_Comp > (R_evap, R_out), suggesting that the global ADAF will be cooled through Compton cooling. With such a strong feedback,⁶ the ADAF collapses into a geometrically thin disk since it has angular momentum. The collapse timescale is mainly controlled by the vertical gravity of the black hole. The collapsing velocity is given by v_ff = H_CompΩ_K and the timescale is Δt_collapse = Ω⁻¹_K = 0.44r^3/2₁₀₀₀M₁ s = 1.7r^3/2₁₀₀₀M₈ yr. This timescale is much shorter than the dynamical; thus, shocks could be formed during the collapse and could heat the cADAF. In this paper, we neglect this heating, which could be balanced by the cooling of the condensed gas. The collapse stops until the cADAF reaches a new dynamical equilibrium of the SSD with a scale height of $H_{\rm cADAF}/R=4\times 10^{-3}\dot{m}_{-2}r^{-1}$. In such a case, clumps are then orbiting around the black holes without the ADAF-driven drag if they are bound by the magnetic field. It is also plausible for clumps to collide with the cold disk and then captured by the disk. They could also undergo a fast expansion and totally disappear since the pressure balance is broken. This feedback gives rise to a quenching of the clumpy accretions. The debris disk is acting as a switch during these processes.

4.3.2. cADAF and Revived Clumpy-ADAF

The cADAF is undergoing two processes: (1) it proceeds the accretion onto black holes at a viscosity timescale, and (2) it may be evaporated by the hot corona from the outer to inner regions. Although the formation of the hot corona on the SSD remains open, we presume here that the two competing processes determine the post-appearance of the cADAF. For the cADAF as an SSD, it is still radiation-pressure-dominated and has a timescale of

$$\begin{equation} \Delta t_{\rm cADAF}=\left\lbrace \begin{array}{@{}l}0.5 \alpha _{0.2}^{-1}M_1\dot{m}_{0.1}^{-2}r_{1000}^{7/2} {\rm \,yr},\\ \\ 5.0\times 10^7 \alpha _{0.2}^{-1}M_8\dot{m}_{0.1}^{-2}r_{1000}^{7/2} {\rm \,yr}, \end{array}\right. \end{equation} \tag{ 41 }$$

which is much longer than Δt_collapse. This indicates that the cADAF has the same accretion rates as the previous ADAF, and it radiates at $L_{\rm cADAF}=\dot{m}_{\rm A}L_{\rm Edd}\approx 1.25\times 10^{44}\eta _{0.1}\dot{m}_{-2}M_8$  erg s⁻¹. The disk enters a relatively brighter state than the ADAF. The rising timescale is about Δt_debris + Δt_Comp from the ADAF state. However, the fate of the cADAF is determined by the competition between the fueling black hole and the evaporating cADAF.

According to numerical calculations (Liu & Taam 2009), the evaporation timescale is given by

$$\begin{equation} \Delta t_{\rm evap}=\frac{M_{\rm cADAF}}{\dot{M}_{\rm evap}} \approx 1.45 r_{1000}^{3/2}\dot{m}_{-2}\dot{m}_{\rm evap,-2}^{-1} {\rm \,yr}, \end{equation} \tag{ 42 }$$

where M_cADAF is the mass of the cADAF, which is roughly equal to the total mass of the ADAF within R_out, and $\dot{M}_{\rm evap}$ is the evaporation rate. The estimation simply follows from $\dot{M}_{\rm evap}\sim 10^{-2}\dot{M}_{\rm Edd}$, which somehow depends on viscosity. We should stress here that the evaporation timescale does not depend on the mass of the black hole. Comparing Δt_cADAF with Δt_evap, we have Δt_tran = min (Δt_cADAF, Δt_evap), indicating that evaporation could govern the postappearance of the cADAF in the AGNs, whereas the viscosity of the accretion governs it in X-ray binaries. We point out that accretion onto black holes in the AGNs and X-ray binaries are different in this way. After the interval of Δt_tran, the clumpy-ADAF is revived and a new cycle starts.

Figure 5 shows a schematic of the state transition cycle in a black hole clumpy-ADAF. After the time Δt_tran, a new ADAF develops, and the object enters a low/hard state, switching on the clumpy accretion. Clumpy-ADAF is at low/hard state, but the cADAF corresponds to the high/soft state since it is an SSD. State transition happens through the feedback of the transient disk of the debris. The cADAF that powers the high state will be brighter than the ADAF at the low/hard state. These processes correspond to the transition of states in black hole X-ray binaries. For LLAGNs, there could be a component seen as a big blue bump in some low ionization nuclear emission regions (LINERs), especially in some LINERs with broad components of emission lines (Ho 2008; Younes et al. 2011, 2012). Furthermore, BL Lac objects have ADAF and some of them show light curves with quasiperiodical modulations, which could be explained by the present model. We stress that the transient disk plays a key role in feedback to the ADAF. The present model predicts a transition of accretion flows. There are two processes of shorter bursts: (1) emission from the debris disk, and (2) cooling of the ADAF through the inverse Compton scattering. The first outburst proceeds to the second, especially it is in the soft band, and the second is a burst in the hard X-ray band. A detailed comparison with observations of the X-ray binaries and AGNs would determine the size of the cADAF and the disk of the debris cloud.

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When an accretion flow has low enough rates, it turns into an ADAF through evaporation (Meyer & Meyer-Hofmeister 1994) and will become a clumpy-ADAF with a critical accretion rate of $\dot{m}\gtrsim 0.02\alpha _{0.1}$ as shown in Figure 1. Clumps are finally disrupted by the tidal force of the black hole, forming a transient tiny disk. Liu et al. (2007) and Meyer-Hofmeister & Meyer (2011) suggest that an inner disk may be formed through condensations of the ADAF if the accretion rates are in a reasonable range. In principle, this condensation model results from thermal instability, and it is physically equivalent to the clumpy-ADAF model suggested in this paper. However, their model is different from the one presented in this paper for three reasons: (1) the tiny disk is persistent in Liu et al. (2007), but it is transient in the present model; (2) there are only two components (the cold disk and ADAF) in Liu et al. (2007) and many clumps in the present model; and (3) the dynamics are different. As we have shown above, the present model can conveniently explain more observations, besides the observed iron Kα lines in the AGNs (Meyer-Hofmeister & Meyer 2011), such as variabilities of X-ray binaries and radio-loud AGNs.

In the present model, we assume that clumps in the ADAF keep a constant mass and radius before being tidally disrupted. Despite these assumptions, it holds the main features of the clumpy accretion. We may relax some of these assumptions in future studies. Here, we do not discuss the weak-coupling case of the clumpy-ADAF because it may give results without any significant differences from the pure ADAF. On the other hand, if the accretion rates are in the SSD regime, the dynamics of the clumps could be changed into a phase driven by collisions among the clumps as well as the drag force in the R- and ϕ-directions. Observations show that the X-ray regions are partially covered by clumps (Gallo et al. 2004; Ballantyne et al. 2004; Ricci et al. 2010; see a review of Turner & Miller 2009), implying that the clumps in the clumpy-SSD will be much larger than those predicted by the present Clumpy-ADAF model. These contents will be discussed as the main goals of second paper. We treat the outer boundary as the evaporation radius, where the production of clumps is likely happening due to the thermal instability there. This should be issued in more detail in a future paper.

Finally, numerical simulations of the dynamics and radiation of the clumpy-ADAF are worth performing to provide more details like in a dusty torus (Stalevski et al. 2011). We briefly discuss the radiation from the clumpy-ADAF in light of timescales of the undergoing processes. The collapse of the ADAF deals with the release of gravitational energy in the vertical direction, although this energy is smaller than that of the debris disk. The radiated cADAF is simplified as an SSD. This is valid when the collapsed timescale is much shorter than the accretion timescale. Future numerical simulations will uncover the full processes and radiation from the cADAF.

The AGNs well known in the ADAF regime include LINERs (e.g., Ho 2008) and some BL Lac objects (e.g., Wang et al. 2002, 2003). Since the timescale of the transient disk is much longer than that seen in X-ray binaries, the component of the transient disk can be easily observed, but it varies at a timescale of years.

LINERs. Evidence has been found for the presence of the significant component of the big blue bump observed in normal AGNs and quasars (Maoz 2007). Pure reprocessing emission from clumps (Figure 1 in Celotti & Rees 1999) is not enough to explain the component since the reprocessing emission achieves a peak at 10^14.5 Hz. The transient disk originated from the captured clumps definitely contributes to its emission to 10¹⁴–10¹⁶ Hz in light of the SSD model. It is trivial to test this model in LINERs since the component of the debris disk has a variability with a timescale of years depending on the black hole masses and accretion rates of the ADAF. It should be noted that the selected LINERs to test the model should have relatively higher accretion rates in the clumpy-ADAF regime rather than in the pure ADAF mode. These LINERs have broad Hα components that are 25% LINERs (Ho 2008). They could have relatively higher accretion rates (Elitzur & Ho 2009) and thus contain clumps in the ADAF. These LINERs are expected to be monitored for variabilities to test the accretion processes.

BL Lac objects. They have lower accretion rates, likely in the ADAF mode (Wang et al. 2002; 2003; Barth et al. 2003). It is expected that some of them could have clumpy-ADAF. The observed emission is overwhelmed by the boosted emission of a relativistic jet, and the emission from the debris disk or cADAF is not directly visible. It is generally postulated that the ADAF will somehow produce a relativistic jet (e.g., Meier 2001), which is evidenced by X-ray binaries (e.g., Fender et al. 2010), e.g., the radio galaxy 3C 120 (Marscher et al. 2002) and 3C 111 (Tombesi et al. 2011). As we have shown, the presence of the debris disk drives the disappearance of the ADAF through efficient Compton cooling, resulting in quenching jet formation. In such a case, thermal components (L_debris, L_Comp, and L_cADAF) could be observed if the relativistic boosting jet emission underwhelms this component. More interestingly, the model predicts a periodical presence of the clumpy-ADAF, which could lead to light curves with quasiperiodical modulation in light of the intermittent production of the jet in the clumpy-ADAF mode. Jet production quenches in the period of Δt_debris + Δt_Comp + Δt_cADAF. Radio light curves from the Web site (http://www.astro.lsa.umich.edu/obs/radiotel/umrao.php) with quasiperiodical modulations of a few years observed in some BL Lac objects and radio galaxies could be explained by the present model, e.g., 0235+164, 3C 120, 0607−157, 0727−115, 1127−145, 1156+295, 1308+326, 1335−127, OT 129, BL Lac, 3C 446, and 3C 345. These objects have supermassive black holes with masses of 10⁸–10⁹ M_☉ (Ghisellini & Celotti 2001; Barth et al. 2003; Wang et al. 2002, 2003) and Eddington ratios of 10⁻³–10⁻² (Wang et al. 2003).

We note that some sources from this Web site, such as 3C 273, are different from the sources mentioned above. These sources may have higher accretion rates than the ADAF in BL Lac type sources. This implies the necessity of the standard clumpy accretion disk discussed in previous sections.

Extensive reviews of observed properties of black hole X-ray binaries and related theoretical explanations have been given by Remillard & McClintock (2006), McClintock & Remillard (2006), Done et al. (2007), and Belloni et al. (2011). Some black hole X-ray binaries show a repeat transition from low to high states and vice versa. The present model predicts repeat transitions. From the long-term light curves of 4U 1630-47, XTE J1650-500, XTE J1720-318, H 1743-322, SLX 1746-331, XTE J1859, and Cyg X-1, they usually have an Eddington ratio of L/L_Edd ∼ 10⁻², but they show variabilities with two orders (Done et al. 2007). They show bursts with very steep rising and slow declining luminosity. For a simple estimation at the hard state, the hard X-ray luminosity (at ∼100 keV) from the ADAF is $L_{\rm HX}\sim 2.5\times 10^{35}(M_{\bullet }/M_\odot)\dot{m}_{-2}$ erg s⁻¹ (Mahadevan 1997). Once the debris disk is formed, it radiates at a luminosity of $L_{\rm SX}\sim 1.3\times 10^{37}(M_{\bullet }/M_\odot)(\dot{m}_{\rm debris}/0.1)$, and the accreting black hole transits to a high state with around a two-order change (L_HX/L_SX ∼ 10⁻²). Detailed calculations of the state transition and variability will be carried out in a forthcoming paper.

On the other hand, the presence of clumps in the ADAF will increase the radiative efficiency of the ADAF, driving the states of the ADAF into hard states with high luminosity (e.g., Figure 5 in the most recent review of Belloni et al. 2011). After the ADAF collapses, jet production quenches and the accretion flows then transit to the high state. We note that the collapse of the ADAF will squeeze itself. The squeezed gas could be blown away by the radiation, yielding outflows. This happens during the transition of states, especially from low to high states. Detailed comparisons with observations will be given in a future paper.