# Evolution of MHD Torus and Mass Outflow Around Spinning AGN

Ramiz Aktar1 , Kuo-Chuan Pan123, and Toru Okuda4
1Department of Physics and Institute of Astronomy, National Tsing Hua
University, 30013 Hsinchu, Taiwan
2Center for Theory and Computation, National Tsing Hua University, Hsinchu
30013, Taiwan
3Physics Division, National Center for Theoretical Sciences, National Taiwan
University, Taipei 10617, Taiwan
4Hakodate Campus, Hokkaido University of Education, Hachiman-Cho 1-2, Hakodate
040-8567, Japan Contact e-mail<EMAIL_ADDRESS>
###### Abstract

We perform axisymmetric, two-dimensional magnetohydrodynamic (MHD) simulations
to investigate accretion flows around spinning AGN. To mimic the space-time
geometry of spinning black holes, we consider effective Kerr potential, and
the mass of the black holes is $10^{8}M_{\odot}$. We initialize the accretion
disc with a magnetized torus by adopting the toroidal component of the
magnetic vector potential. The initial magnetic field strength is set by using
the plasma beta parameter ($\beta_{0}$). We observe self-consistent turbulence
generated by magneto rotational instability (MRI) in the disc. The MRI
turbulence transports angular momentum in the disc, resulting in an angular
momentum distribution that approaches a Keplerian distribution. We investigate
the effect of the magnetic field on the dynamics of the torus and associated
mass outflow from the disc around a maximally spinning black hole
$(a_{k}=0.99)$. For the purpose of our analysis, we investigate the magnetic
state of our simulation model. The model $\beta_{0}=10$ indicates the
behaviour similar to the “magnetically arrested disk (MAD)” state, and all the
other low magnetic model remains in the SANE state. We observe that mass
outflow rates are significantly enhanced with the increased magnetic field in
the disc. We find a positive correlation between the magnetic field and mass
outflow rates. We also investigate the effect of black hole spin on the
magnetized torus evolution. However, we have not found any significant effect
of black hole spin on mass outflows in our model. Finally, we discuss the
possible astrophysical applications of our simulation results.

###### keywords:

accretion, accretion discs, black hole physics, (magnetohydrodynamics) MHD,
ISM: jets and outflows, quasars: supermassive black holes

††pubyear: 2022††pagerange: Evolution of MHD Torus and Mass Outflow Around
Spinning AGN–C

##  1 Introduction

Active galactic nuclei (AGNs) are ubiquitously observed at every galaxy’s
center. In general, accretion on to AGN may produce highly radiative luminous
events across the entire electromagnetic spectrum. However, AGN does not
always show high radiative emission, and surprisingly they spend most of the
time in a quiescence state. Based on the observed optical/UV spectral
properties, AGNs are classified as Type 1 and Type 2. Type 1 AGNs shows both
broad band ($\gtrsim 1000$ km s-1) and narrow band ($\lesssim 1000$ km s-1)
emission lines in optical/UV. On the other hand, AGNs only manifest narrow
bands are known as Type 2. In recent years, it has been observed that AGN-type
changes between Type 1 and Type 2, and these are called changing-look AGNs
(CLAGNs) (LaMassa et al., 2015; MacLeod et al., 2016; Ricci et al., 2016). In
this regard, the low luminosity AGNs (LLAGNs) are defined as luminosity
$0.1\%$ LEdd, where LEdd is the Eddington luminosity. These low luminosity
AGNs can be explained based on radiatively inefficient accretion flows (RIAFs)
onto AGNs (Narayan et al., 1995; Yuan & Narayan, 2014). RIAF models are
basically hot, optically thin, advection-dominated and include some other
effects such as outflows and convection (Yuan & Narayan, 2014). The radiation
energy is much smaller compared to the thermal energy in RIAF model.

Further, astrophysical jets and mass outflows are commonly observed in black
hole X-ray binaries (BH-XRBs) and AGNs (Mirabel et al., 1992; Fender & Gallo,
2014) and reference therein. Several theoretical studies are carried out to
investigate jets and mass outflows around black holes starting from the
Penrose process (Penrose, 1969). Penrose process first provides an explanation
of the extraction of energy from the infalling matter into the rotating black
hole. Later, the seminal paper by Blandford & Znajek (1977) (BZ) showed that
jet energy could be extracted from the rotational energy from large-scale
magnetic fields around spinning black holes. Subsequently, (Blandford & Payne,
1982) (BP) pointed out that the matter can also leave the surface of the
accretion disc due to magneto-centrifugal acceleration. Several simulation
studies confirm the role of black hole spin in generating powerful jets around
black holes (Tchekhovskoy et al., 2010, 2011; Tchekhovskoy & McKinney, 2012;
Narayan et al., 2022). There are several other models have been proposed to
address the mass outflow around black holes. In this regard, the advection-
dominated inflow-outflow solutions (ADIOS) have been proposed in the
literature. In this model, the inward decrease of mass accretion rate is due
to the mass loss as outflow at every radius of the disc $(\dot{M}_{\rm
acc}\propto r^{s};0\leq s<1)$ (Blandford & Begelman, 1999; Becker et al.,
2001; Blandford & Begelman, 2004; Xue & Wang, 2005; Begelman, 2012; Yuan et
al., 2012). Simultaneously, convection-dominated accretion flow (CDAF) has
been introduced to explain the mass outflow from hot HD and MHD accretion
flows (Narayan et al., 2000; Quataert & Gruzinov, 2000). In this model, the
accreting gas moves in and out with convective eddies, and this motion
provides the fluxes of inflowing and outflowing matter. It is also believed
that inward angular momentum is transported by convection and outward
transport by viscous stresses. Numerical simulations also investigated the
mass outflow, which is in support of the CDAF model (Igumenshchev &
Abramowicz, 1999; Stone et al., 1999; Igumenshchev et al., 2000, 2003). On the
other hand, accretion shock-driven mass outflow studies also investigated in
analytical as well simulation study and their astrophysical applications
around black hole (Chattopadhyay & Das, 2007; Kumar & Chattopadhyay, 2013; Das
et al., 2014; Okuda & Das, 2015; Aktar et al., 2015, 2017; Okuda et al., 2019;
Kim et al., 2019; Okuda et al., 2022, 2023).

It is noteworthy to mention that in recent years, an important development has
been achieved in the accretion disc theory for highly magnetized accretion
flows recognized as “magnetically arrested disk (MAD)” (Narayan et al., 2003).
Tchekhovskoy et al. (2011) first showed in their pioneering work that hot
accretion flow in MAD state can launch a very powerful jet based on GRMHD
simulation around a spinning black hole. They also showed that the jet carries
more power than the accretion energy for a highly spinning black hole. This is
because of the extraction of rotational energy from the black hole via the BZ
process. Subsequently, several numerical studies confirm the existence of the
MAD state along with the “standard and normal evolution (SANE)" state in
highly magnetized flow (Narayan et al., 2012; McKinney et al., 2012, 2015;
Dihingia et al., 2021, 2022; Chatterjee & Narayan, 2022; Dihingia et al.,
2023; Dhang et al., 2023; Jiang et al., 2023). Moreover, high angular
resolution polarization observations of M87 by the Event Horizon Telescope
observe that the accretion flow in this system is likely to be in the MAD
state (Event Horizon Telescope Collaboration et al., 2021).

One of the fundamental questions in accretion disc physics is how the angular
momentum transfer in the disc. Initially, the seminal paper by Shakura &
Sunyaev (1973) proposed the ‘$\alpha$-disc’ model. However, the origin of ‘ad-
hoc’ viscosity is still questionable in this model. Conversely, it has been
widely accepted in recent years that the mechanism of angular momentum
transport in the accretion flows is the magnetorotational instability (MRI)
(Balbus & Hawley, 1991, 1998). This instability is set to increase the initial
magnetic field exponentially in the accreting gas until the
magnetohydrodynamic (MHD) turbulence develops in the system. In this turbulent
state, the Maxwell and Reynolds stress transport angular momentum outwards and
causes inward mass accretion. To understand the nonlinear turbulence state of
MRI, numerical simulation studies of accretion flows are inevitable. Several
numerical simulations also show that the Maxwell stress always dominates over
the Reynolds stress by a factor of several in MHD flows. Over the years,
several MHD simulations have been carried out to investigate accretion flows
around black holes considering magnetized torus. In this context, Hawley
(2000) investigate global three-dimensional MHD simulations of non-radiative
accretion flows. In this work, Hawley (2000) showed that mass accretion is
primarily driven by Maxwell stress, and it is enhanced by MRI. On the other
hand, Machida et al. (2000) showed that the magnetic field is enhanced by MRI
and buoyantly escapes from the disc to form a magnetically active disc corona.
There are several other MHD simulations have been carried out considering non-
radiative as well as radiative accretion flows around black holes in the
literature considering pseudo-Newtonian potential (Paczyńsky & Wiita, 1980)
around non-spinning black hole (Kuwabara et al., 2000; Hawley & Krolik, 2001;
Stone & Pringle, 2001; Hawley & Krolik, 2002; Kuwabara et al., 2005; Ohsuga et
al., 2009; Ohsuga & Mineshige, 2011; Igarashi et al., 2020). Moreover, it has
been investigated that global MHD simulation is more realistic than global HD
simulation because MHD simulations self-consistently generate shear stress via
MRI turbulence. On the other hand, in HD simulation, one needs an ‘ad hoc’
viscosity to generate shear stress in the flow. In recent years, with the
considerable development of numerical advancement, general relativistic
simulations of MHD torus around black holes have been investigated (Gammie et
al., 2003; De Villiers & Hawley, 2003; De Villiers et al., 2003; Tchekhovskoy
et al., 2011; Narayan et al., 2012; Dihingia et al., 2021; Chatterjee &
Narayan, 2022; Dihingia et al., 2022; Narayan et al., 2022; Dhang et al.,
2023; Curd & Narayan, 2023; Jiang et al., 2023).

In this paper, we investigate the time-dependent accretion and associated mass
outflows considering MHD torus around spinning AGNs. Here, the space-time
geometry is modeled around a spinning black hole using effective Kerr
potential derived by Dihingia et al. (2018). This effective potential quite
accurately mimics space-time geometry around Kerr black hole having spin
$0\leq a_{k}<1$. Dihingia et al. (2018) also showed that the analytical
transonic solutions using this effective potential in semi-relativistic flow
are in excellent close agreement with general relativistic results even in the
maximally spinning regime. During accretion, the accreting gas moves inward
with the increase of magnetic pressure. Also, the gas expands in the vertical
direction below and above the torus via MRI turbulence and carries a
significant amount of magnetic field with it as magnetized mass outflow
(Machida et al., 2000; Hawley & Krolik, 2001; Hawley & Balbus, 2002). In
general, torus evolution around the black hole is primarily determined by
three parameters: the strength of the initial magnetic field, flow angular
momentum, and spin of the black hole. First, we investigate the magnetic state
of the accretion flow by calculating normalized magnetic flux for our
simulation model (Tchekhovskoy et al., 2011). Further, we examine the effect
of these three parameters on the accretion and associated mass outflow around
AGNs. More specifically, we investigate the correlation between the magnetic
field and spin on the mass outflow around black hole. We also estimate
bremsstrahlung luminosity under optically thin disc approximation from our
model and investigate the effect of basic parameters on luminosity variation.

We organize the paper as follows. In section 2, we present the description of
the numerical model and governing equations. In section 3, we discuss the
results of our model in detail. Finally, we draw the concluding remarks in
section 4.

##  2 Numerical Method

We perform axisymmetric two-dimensional MHD simulations using the publicaly
available numerical simulation package PLUTO111http://plutocode.ph.unito.it
(Mignone et al., 2007). We adopt the unit system as $G=M_{\rm BH}=c=1$, where
$G$, $M_{\rm BH}$ and $c$ are the gravitational constant, the mass of the
black hole and the speed of light, respectively. In this unit system, we
measure distance, velocity, and time as $r_{g}=GM_{\rm BH}/c^{2}$, $c$ and
$t_{g}=GM_{\rm BH}/c^{3}$, respectively. Here, we consider no explicit
resistivity in the flow. We also ignore any radiation transport and loss in
our model.

###  2.1 Governing Equations

In this paper, we write MHD governing equations in cylindrical coordinates
$(r,\phi,z)$. The governing equations are as follows

$\displaystyle\frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\bm{v})=0,$ (1)
$\displaystyle\frac{\partial(\rho\bm{v})}{\partial
t}+\nabla\cdot(\rho\bm{v}\bm{v}-\bm{B}\bm{B})+\nabla P_{t}=-\rho\nabla\Phi,$
(2) $\displaystyle\frac{\partial E}{\partial
t}+\nabla\cdot[(E+P_{t})\bm{v}-(\bm{v}.\bm{B})\bm{B}]=-\rho\bm{v}.\nabla\Phi,$
(3) $\displaystyle\frac{\partial\bm{B}}{\partial
t}+\nabla\cdot(\bm{v}\bm{B}-\bm{B}\bm{v})=0,$ (4)

where $\rho$, $\bm{v}$, and $\bm{B}$ are the mass density, fluid velocity, and
magnetic field, respectively. $P_{t}=P_{\rm gas}+B^{2}/2$ is the total
pressure comprised with gas pressure $(P_{\rm gas})$ and magnetic pressure
$(B^{2}/2)$. $\Phi$ represents the gravitational potential in the presence of
a black hole. The Bernoulli parameter (Be) can be obtained as

$\displaystyle Be=\frac{P_{\rm gas}}{\gamma-1}+\frac{1}{2}\rho
v^{2}+\frac{1}{2}B^{2}+\Phi^{\rm eff}-1,$ (5)

where $\gamma$ is the adiabatic index. Here, we assume the adiabatic equation
of state $P_{\rm gas}=\rho\epsilon(\gamma-1)$. Here, $\epsilon$ is the
specific internal energy. Also the adiabatic sound speed is defined as
$c_{s}=\sqrt{\frac{\gamma P_{\rm gas}}{\rho}}$. We subtract the rest mass
energy of the gaseous matter from the total energy to obtain the Bernoulli
parameter (Narayan & Yi, 1994; Narayan et al., 2012). Here, $\Phi^{\rm eff}$
is the effective Kerr potential as described in the sub-section 2.2.

###  2.2 Graviational potential

The gravitational potential around spinning black holes is modeled using
Dihingia et al. (2018) effective Kerr potential. The effective Kerr potential
is given by

$\displaystyle\Phi^{\rm
eff}\left(r,z,a_{k},\lambda\right)=\frac{1}{2}\ln\left(\frac{A(2R-\Sigma)r^{2}-4a_{k}^{2}r^{4}}{\Sigma\lambda\left(\Sigma\lambda
R^{2}+4a_{k}r^{2}R-2\lambda R^{3}\right)-A\Sigma r^{2}}\right),$ (6)

where $R=\sqrt{r^{2}+z^{2}}$ = spherical radial distance,
$\Delta=a_{k}^{2}+R^{2}-2R$, $\Sigma=\frac{a_{k}^{2}z^{2}}{R^{2}}+R^{2}$ and
$A=\left(a_{k}^{2}+R^{2}\right)^{2}-\frac{a_{k}^{2}r^{2}\Delta}{R^{2}}$. Here,
$\lambda$ and $a_{k}$ are specific flow angular momentum and black hole spin,
respectively. It is to be noted that the gravitational and centrifugal
potential is coupled in this potential. Therefore, we modify the radial
momentum conservation equations in the PLUTO code to incorporate the effective
Kerr potential. The Keplerian angular momentum can be obtained in the
equatorial plane $(z\rightarrow 0)$ as

$\displaystyle\lambda_{K}=\sqrt{r^{3}\frac{\partial\Phi_{\rm eff}}{\partial
r}|_{\lambda\rightarrow 0}}.$ (7)

The angular frequency is $\Omega=\lambda/r^{2}$ and the circular orbital
period at distance $r$ is $P_{\rm orb}=2\pi\Omega^{-1}$.

###  2.3 Accretion Torus set up

We consider an equilibrium accretion disc around a spinning black hole. The
black hole is surrounded by a hot corona (Matsumoto et al., 1996; Hawley,
2000; Hawley & Krolik, 2001). The equilibrium torus solutions can be obtained
by adopting the Newtonian analog of relativistic tori (Abramowicz et al.,
1978). The density distribution of the torus can be obtained considering
constant angular momentum flow $(\lambda=\rm constant)$ as (Matsumoto et al.,
1996; Hawley, 2000; Kuwabara et al., 2005)

$\displaystyle\Phi^{\rm
eff}\left(r,z,a_{k},\lambda\right)+\frac{\gamma}{\gamma-1}\frac{P_{\rm
gas}}{\rho}={\cal C}={\rm constant}.$ (8)

The constant ‘$\mathcal{C}$’ can be determined considering zero-gas pressure
surface $(P_{\rm gas}\rightarrow 0)$ at $r=r_{\rm min}$ at the equatorial
plane. Here, $r_{\rm min}$ represents the inner edge of the torus. Using
adiabatic equation of state $P_{\rm gas}=K\rho^{\gamma}$, the density
distribution inside the torus can be determined as

$\displaystyle\rho=\left[\frac{\gamma-1}{K\gamma}\left({\cal C}-\Phi^{\rm
eff}\left(r,z,a_{k},\lambda\right)\right)\right]^{\frac{1}{\gamma-1}},$ (9)

where $K$ can be determined considering the density maximum surface
$(\rho_{\rm max})$ at $r=r_{\rm max}$ at the equatorial plane and is given by

$\displaystyle K=\frac{\gamma-1}{\gamma}\left[\mathcal{C}-\Phi^{\rm
eff}\left(r_{\rm max},0,a_{k},\lambda\right)\right]\frac{1}{\rho_{\rm
max}^{\gamma-1}}.$ (10)

Now we assume that the density distribution outside the torus is isothermal,
non-rotating, and high-temperature halo surrounding the black holes (Matsumoto
et al., 1996; Kuwabara et al., 2005). For the halo, the density distribution
is assumed to be in hydrostatic equilibrium as

$\displaystyle c_{s}^{2}\frac{\nabla\rho}{\rho}=-\nabla\Phi^{\rm
eff}\left(r,z,a_{k},\lambda\rightarrow 0\right),$ (11)

where $c_{s}$ is the sound speed. Integrating equation (11) from maximum
density of the torus $(r_{\max},0)$ to the outside torus $(r,z)$ upto halo is
given by

$\displaystyle\rho={\eta}\rho_{\rm max}\exp\left[\left(\Phi^{\rm
eff}\left(r_{\rm max},z,a_{k},\lambda\rightarrow 0\right)-\Phi^{\rm
eff}\left(r,z,a_{k},\lambda\rightarrow 0\right)\right){\cal H}\right],$ (12)

where, $\cal{H}$ = $1/c_{s}^{2}$ and for representative case, we consider
$\cal{H}$ equal to 2 throughout the simulation. Here, we also introduce a
constant factor $\eta$ which represents the density fraction between the
density of the halo to the maximum torus density.

Figure 1: Initial equilibrium torus profile of density ($\log\rho$) and
magnetic field lines (grey lines) for $a_{k}=0.99$ and $\beta_{0}=10$. We
consider the initial maximum density at $r=50r_{g}$. See the text for details.

Figure 2: Distribution of gas density in $(r-z)$ plane for different initial
plasma-$\beta$ at various times of evolution. The first, second, third, and
fourth rows correspond to $\beta_{0}$ = 10, 50, 100, and 1000, respectively.
Similarly, the first, second, third, and fourth columns represent at times
$t_{1}=500t_{g}$, $t_{2}=5000t_{g}$, $t_{3}=10500t_{g}$, and
$t_{4}=17500t_{g}$, respectively. The grey lines represent the magnetic field
lines.

Figure 3: Distribution of Temperature ($T$), plasma-$\beta$, azimuthal
magnetic field ($B_{\phi}$) and magnetization parameter $(\sigma_{\rm M})$ for
different initial magnetic field ($\beta_{0}$). First, second, third, and
fourth columns correspond to $\beta_{0}$ = 10, 50, 100, and 1000,
respectively, at time $t_{3}=10500t_{g}$. See the text for details. Figure 4:
Temporal evolution of $(a)$ mass accretion rate $(\dot{M}_{\rm acc})$ in
Eddington unit and $(b)$: normalized magnetic flux $(\dot{\phi}_{\rm acc})$ in
code units accumulated at the black hole inner boundary with the simulation
time for different initial plasma-$\beta$ parameter $\beta_{0}$ = 10 (red), 50
(blue), 100 (magenta) and 1000 (green), respectively. Dashed and dotted
horizontal lines are for $\dot{\phi_{\rm acc}}=15$ and $50$, respectively. See
the text for details. Figure 5: Radial variations of different flow variables:
($a$): density ($\rho$), ($b$): temperature ($T$), ($c$): radial velocity
($v_{r}$), $(d)$: plasma-beta ($\beta$), $(e)$: magnetization parameter
($\sigma_{\rm M}$), $(f)$: specific angular momentum ($\lambda$), $(g)$: mass
accretion rate ($\dot{M}_{\rm acc}$), $(h)$: normalized Reynolds stress
($\alpha_{\rm gas}$), and $(i)$: normalized Maxwell stress ($\alpha_{\rm
mag}$), respectively. Here, CDAF implies Convection Dominated Accretion Flow
and $\lambda_{\rm K}$ is the Keplerian angular momentum. See the text for
details. Figure 6: Variation of $(a)$: Bernoulli parameter ($Be$) $(b)$: mass
outflow rates $(\dot{M}_{\rm out})$ and $(c)$: luminosity $(L)$ with time for
different $\beta_{0}$ =10 (red), 50 (blue), 100 (magenta) and 1000 (green),
respectively. See the text for details.

Figure 7: Distribution of density $(\rho)$, Temperature ($T$), plasma-$\beta$
($\beta$) and magnetization parameter $(\sigma_{\rm M})$ for various spin
($a_{k}$) values. The first, second, third and fourth column are for
$a_{k}=0.99,0.8,0.5$, and 0.0, respectively at time $t_{3}=10500t_{g}$. Figure
8: Variation of $(a)$: Bernoulli parameter ($\rm Be$) $(b)$: mass outflow
rates $(\dot{M}_{\rm out})$ and $(c)$: luminosity $(L)$ with time for
different spin values as $a_{k}$ = 0.99 (red), 0.80 (blue), 0.50 (magenta) and
0.0 (green), respectively.

Figure 9: Distribution of density $(\rho)$ Temperature ($T$), plasma-$\beta$
($\beta$) and magnetization parameter $(\sigma_{\rm M})$ for various angular
momentum. The first, second, third and fourth columns are for
$\lambda=\lambda_{\rm K},7.00,6.80$ and 6.60, respectively at time
$t_{3}=10500t_{g}$

.

Figure 10: Variation of specific angular momentum $(\lambda)$ for different
time of evolution $(a)$: $t=0$ s, $(b)$: $t=7.42\times 10^{5}$s, $(c)$:
$t=5.2\times 10^{6}$ s and $(d)$: $t=7.42\times 10^{6}$ s, respectively.
Figure 11: Variation of $(a)$: Bernoulli parameter $(\sigma)$, $(b)$: mass
outflow rates $(\dot{M}_{\rm out})$ and $(c)$: luminosity $(L)$ with time for
different specific angular momentum as $\lambda=\lambda_{\rm K},7.00,6.80$ and
6.60, respectively.

###  2.4 Magnetic field configuration

In this work, we consider a poloidal magnetic field configuration following
Hawley & Krolik (2002). The initial magnetic field configuration is set by
considering a toroidal component of vector potential as

$\displaystyle A_{\phi}=B_{0}[\rho(r,z)-\rho_{\rm min}],$ (13)

where, $\rho_{\rm min}$ is the minimum density in the torus. $B_{0}$ is the
normalized initial magnetic field strength. The magnetic field strength is
parameterized by the ratio of the gas pressure to the magnetic pressure, and
it is constant in the initial disc. The initial magnetic field strength is set
by using the initial plasma-$\beta$ parameter as $\beta_{0}=\frac{2P_{\rm
gas}}{B_{0}^{2}}$. The poloidal magnetic field configuration is obtained using
$\bm{B}=\nabla\times\bm{A}$. Moreover, in MHD simulation, it is usually
convenient to define a parameter known as magnetization parameter $\sigma_{\rm
M}=\frac{B^{2}}{\rho}$. The magnetization parameter represents the ratio of
magnetic energy to rest mass energy in the flow (Dihingia et al., 2021, 2022;
Dhang et al., 2023; Curd & Narayan, 2023).

###  2.5 Initial and boundary conditions

Here, we consider two-dimensional MHD flows in cylindrical coordinate
$(r,\phi,z)$ in PLUTO code. To simulate the MHD accretion flows, we employ
HLLD Riemann solver, second-order-in-space linear interpolation, and the
second-order-in-time Runge–Kutta algorithm. We also enforce the hyperbolic
divergence cleaning method for solving the induction equation
$\nabla\cdot\bm{B}=0$ (Mignone et al., 2007). In this work, we consider the
computational domain is $1.5r_{g}\leq r\leq 200r_{g}$ in the radial direction
and $-100r_{g}\leq z\leq 100r_{g}$ in the vertical direction. The number of
grid is $(n_{r},n_{z})=(896,896)$. We perform a convergence test of our
simulation model, described in Appendix A. We use uniform grid spacing in both
the radial and vertical directions. We set the inner boundary at $r_{\rm
in}=2.5r_{g}$. The inner boundary of the black hole is set to be absorbing
boundary conditions (Okuda et al., 2019, 2022, 2023). Moreover, the
axisymmetric boundary condition is set at the origin, and all the rest are set
to outflow boundary conditions. To set up the initial torus around the black
hole, we consider the inner edge of the torus at $r_{\rm min}=32r_{g}$. The
maximum pressure surface is at $r_{\rm max}=50r_{g}$. The minimum density in
the torus is set to be $\rho_{\rm min}=0.5\rho_{\rm max}$. The maximum density
is choosen as $\rho_{\rm max}=10\rho_{0}$ (see the Table 1). We initially set
the magnetic field is zero at $\rho\leq\rho_{\rm min}$ .i.e., the initial
magnetic field is embedded only inside the torus; otherwise, it is zero. Here,
the mass of the AGN is chosen as $10^{8}M_{\odot}$, and the adiabatic index is
$\gamma=4/3$. Also, the density fraction between the maximum density of the
torus to the density of the halo $\eta$ is set $10^{-4}$ (see equation 12).
The C.G.S. and code unit system used in this work is summarized in Table 1. In
the PLUTO code, we are required to specify three fundamental units, such as
unit density $(\rho_{0})$, unit length $(L_{0})$, and unit velocity $(v_{0})$.
Therefore, we first specify the physical unit (C.G.S.) system for our model
and then convert the unit system into code units for the simulation run, as
depicted in Table 1.

In numerical simulation, especially in MHD simulation, sometimes it can be
encountered that density and pressure become negative due to evacuated matter
in the supersonic and highly magnetized flows. In general, fixing pressure and
density to a particular lower value as floor values is recommended to avoid
such a situation throughout the simulation run. Additionally, in the PLUTO
code, there is an efficient way to handle this situation by implementing two
flags turned on, i.e., (i) SHOCK FLATTENING to MULTID and (ii) FAILSAFE to
YES. In this FAILSAFE scheme, PLUTO first saves the solution arrays before
attempting the step before encountering a negative density. Then, it retries
the step again while tagging the critical zone(s) to be updated using FLAT
reconstruction and the HLL Riemann solver. Therefore, we fix the floor density
and pressure in our simulation as $\rho_{\rm floor}=10^{-5}$ and $P_{\rm
floor}=10^{-8}$, respectively. Moreover, we turn on the FAILSAFE scheme in
PLUTO, as mentioned above. Moreover, our choice for the magnetic field
strength is confined by the requirement that the flow is super-Alfvénic, i.e.,
$|v_{p}|>|v_{Ap}|$ which implies $\sigma_{\rm M}<4\pi v_{p}^{2}$, where
$v_{p}$ is the poloidal fluid velocity and $v_{Ap}=[B^{2}/4\pi\rho]^{1/2}$ is
the poloidal Alfvén velocity, respectively (Proga & Begelman, 2003).
Therefore, we impose the conditions $\sigma_{\rm M}<4\pi v_{p}^{2}$ for all
the radii for the entire computational domain in our model.

Table 1: Units used in this paper  Units | C.G.S values | code units
---|---|---
Density |  $\rho_{0}=1\times 10^{-12}$ g cm-3 | $\rho_{0}=1\times 10^{-12}$
Length |  $r_{g}=1.485\times 10^{13}$ cm | $r_{g}=GM_{\rm BH}/c^{2}$
Velocity |  $c=2.998\times 10^{10}$ cm s-1 | $c$
Time |  $t_{g}=4.953\times 10^{2}$ s | $t_{g}=GM_{\rm BH}/c^{3}$
Magnetic field |  $B_{0}=1.063\times 10^{5}$ G | $B=\frac{B_{0}}{c\sqrt{4\pi\rho_{0}}}$
Mass | $M_{\rm BH}=10^{8}M_{\odot}^{*}$ |
$*$ $M_{\odot}$ is the mass of the sun. |  |

##  3 Simulation Results

First, to set up the initial torus around the black hole, we fix the black
hole spin ($a_{k}$) and supply Keplerian angular momentum ($\lambda_{\rm K}$)
(equation 7) at the pressure maximum surface. The formation of the initial
torus is mainly dependent on the three parameters such as flow angular
momentum ($\lambda$), spin ($a_{k}$) of the black hole, and location of the
inner edge of the torus ($r_{\rm min}$). The initial magnetic field is
embedded inside the torus (see sub-section 2.4), and with time evolving, MRI
grows in the disc (Balbus & Hawley, 1991). We represent the initial
equilibrium torus of the density profile with magnetic field lines in Figure
1. Here, we consider the spin, angular momentum, and plasma-$\beta$ are
$a_{k}=0.00$, $\lambda=\lambda_{\rm K}$ and $\beta_{0}=10$, respectively. MRI
transports angular momentum outwards and gradually increases the accretion of
matter toward the black hole. As a result, the edge of the disc drifts towards
the black hole horizon. The accretion flow becomes turbulence via MRI. In this
turbulent state, the gas spreads in the vertical direction. With time evolve,
the poloidal magnetic field enhances in the disc, and the magnetic pressure
drives the matter outward from the disc as outflow (Machida et al., 2000;
Hawley & Krolik, 2001; Hawley & Balbus, 2002; De Villiers & Hawley, 2003).

###  3.1 Effect of magnetic field

In this section, we first examine the effect of magnetic field on the
evolution of torus around spinning AGN. For the purpose of analysis, we
consider four initial plasma-$\beta$ parameters. In this analysis, we also fix
the maximal spin value $a_{k}=0.99$ for AGN. To set up the initial torus, we
supply Keplerian flow angular momentum $\lambda_{\rm K}=7.21$ (see equation
7). Here, we compare the effect of the strength of the magnetic field on the
torus evolution keeping all other parameters fixed. Initially, we compare
density distribution on $(r-z)$ plane at a different simulation time as
depicted in Figure 2. The first, second, third, and fourth rows are for
$\beta_{0}$ = 10, 50, 100, and 1000, respectively. Also, the different columns
are for different times of torus evolution $t_{1}=500t_{g}$,
$t_{2}=5000t_{g}$, $t_{3}=10500t_{g}$ and $t_{4}=17500t_{g}$, respectively. In
the first column, we observe that the initial torus is formed around AGN for
all the magnetic field configurations. With the decrease of plasma-$\beta$
parameters. i.e., with the increase of magnetic field in the disc, the MRI
enhances Maxwell stress more rapidly. As a result, the transport of angular
momentum amplifies with the increase of MRI and initial disc spreads
throughout the computational domain. It is found that matter from the torus
expands more easily and rapidly for a higher magnetic field compared to the
lower magnetic field, depicted in Figure 2. It implies that the magnetic field
plays an essential role in torus evolution around AGN. It is also observed
that mass is ejected from the disc as outflow. Due to the mass escape from the
disc, the torus gradually destroys and forms a uniformly dense disc with a
minimum density of torus at the end time of the simulation (see Figure 2,
first row, last column). It is to be mentioned that we check the quality
factors for resolving MRI in our simulation model (Hawley et al., 2011, 2013).
We find that both the quality factors in radial and vertical directions are
$Q_{r},Q_{z}\gtrsim 15$, see Appendix B. Therefore, our simulation model is
able to resolve MRI very efficiently.

In Figure 3, we represent the distribution of temperature ($T$),
plasma-$\beta$ $(\beta)$, azimuthal magnetic field $(B_{\phi})$ and
magnetization parameter $(\sigma_{\rm M})$ for different initial magnetic
field strength ($\beta_{0}$) at time $t_{3}=10500t_{g}$. Column first, second,
third and four are for different initial magnetic field $\beta_{0}=10,50,100$
and 1000, respectively. The disc is heated up ($T\gtrsim 10^{9}$K) by
releasing the gravitational energy and expands in the vertical direction more
efficiently for a higher magnetic field compared to a low magnetic field. The
temperature is distributed throughout the disc for a higher magnetic field
compared to the lower one. We show the plasma-$\beta$ distribution in the
second row of Figure 3. It is observed that gas pressure is dominated in the
disc region. However, the plasma-$\beta$ distribution becomes low ($\beta<1$)
in the disc corona where the magnetic field buoyantly escapes from the disc to
the disc corona for a high magnetic field compared to a low one. In fact, in a
low magnetic field case, it is impossible to escape the magnetic field from
the torus, and it remains confined within the torus. Further, we present the
distribution of the azimuthal magnetic field ($B_{\phi}$) in the third row of
Figure 3. We also observe that the azimuthal magnetic fields buoyantly escape
from the disc to the disc corona. In general, the azimuthal magnetic fields
are antisymmetric with respect to the equatorial plane, and they facilitate
magnetic reconnection. This magnetic reconnection may trigger heating in the
disc (see Figure 3, first row). Moreover, the azimuthal magnetic field
occasionally changes its polarity and escapes from the disc. Finally, we
present the distribution of the magnetization parameter ($\sigma_{\rm M}$) in
the fourth row of Figure 3. We observe that the magnetization parameter
($\sigma_{\rm M}$) becomes low in the equatorial disc region and significantly
increases ($\sigma_{\rm M}\sim 1$) away from the equatorial plane for all the
magnetized case, i.e., $\beta_{0}=10,50$ and 100. Moreover, we find that the
matters escaping from the disc are magnetically driven as depicted by the
distribution of plasma-$\beta$ and magnetization parameter in Figure 3.
However, we do not observe very high $\sigma_{\rm M}$ values for our
simulation, even in the highly magnetized case ($\beta_{0}=10$). High values
of magnetization parameters usually refer to highly relativistic jets with
high Lorentz factor, i.e., BZ jet (Dihingia et al., 2021; Narayan et al.,
2022; Dihingia et al., 2022; Jiang et al., 2023). In this work, we only find
magnetized mass outflow with $\sigma_{\rm M}\sim 1$ but not relativistic jets.

####  3.1.1 Magnetic state of the accretion flow

Here, we investigate the characteristics of the magnetic state of the
accretion flow in our model. It is usually investigated two quantities, mass
accretion rate $(\dot{M}_{\rm acc})$ and the normalized magnetic flux
threading the BH horizon ($\dot{\phi}_{\rm acc}$) (Tchekhovskoy et al., 2011;
Narayan et al., 2012). In this work, the mass accretion rate is defined as the
mass flux entering through the inner boundary ($r_{\rm in}$) towards the black
hole and is defined as

$\displaystyle\dot{M}_{\rm acc}=-2\pi\int{\rho(r,z)rv_{r}dz}.$ (14)

Here, the ‘-ve’ sign indicates the inward direction of mass flux.
Additionally, we define dimensionless normalized magnetic flux threading to
the black hole horizon $(\dot{\phi}_{\rm acc})$. The normalized magnetic flux
usually refers to the “MAD-ness" parameter. Therefore, the normalized magnetic
flux is given by (Tchekhovskoy et al., 2011; Narayan et al., 2012; Dihingia et
al., 2021; Dhang et al., 2023)

$\displaystyle\dot{\phi}_{\rm
acc}=\frac{\sqrt{4\pi}}{2}\frac{\int{|B_{r}|_{r=r_{\rm
in}}dz}}{\sqrt{\dot{M}_{\rm acc}}},$ (15)

where $r_{\rm in}$ represents the inner boundary. Here, $B_{r}$ is the radial
component of the magnetic field. We represent the temporal variation of mass
accretion rate and the normalized magnetic flux in Figure 4a and Figure 4b,
respectively. Here, we fix spin $a_{k}=0.99$ and angular momentum
$\lambda=\lambda_{\rm K}$ and vary initial magnetic field ($\beta_{0}$). We
observe that the mass accretion rate lies in the sub-Eddington limit
($\dot{M}_{\rm acc}<\dot{M}_{\rm Edd}$) throughout the simulation run for all
the cases, shown in Figure 4a. It is generally observed that the value of
saturated magnetic flux is a good indicator of the characteristic of the
magnetic state of the accretion flow. We find that magnetic flux continues to
grow and becomes saturated after $t\sim 2000,2500,3000t_{g}$ for
$\beta_{0}=10,50$ and 100, respectively. In the previous studies in GRMHD
simulation, it is observed that the criterion to enter the MAD state is when
the normalized magnetic flux reaches the critical value. Generally the
threshold value of magnetic flux is $\dot{\phi}_{\rm acc}$ $\sim$ 15 for MAD
state (Tchekhovskoy et al., 2011; Narayan et al., 2012; Dihingia et al., 2021;
Jiang et al., 2023). In this work, we find that the saturated magnetic flux
crosses the threshold value for MAD state for the high magnetic case model
($\beta_{0}=10$), shown in Figure 4b. We argue that the magnetic model
$\beta_{0}=10$ is similar to the MAD state as generally found in GRMHD
simulation. However, the saturated magnetic flux remains more or less below
the threshold value of MAD for the case of $\beta_{0}=50$ and 100. These two
models are high-magnetized SANE states. It is to be mentioned that the inner
boundary is higher compared to the event horizon for $a_{k}=0.99$,i.e.,
$r_{\rm in}>r_{\rm H}$ in our model, where $r_{\rm H}=1+\sqrt{1-a_{k}^{2}}$.
We also observe that for low-magnetic flow ($\beta_{0}=1000$), the mass
accretion rate attains saturation value after a long time ($t\geq
12500t_{g}$), and the mass accretion rate is very low compared to the high
magnetized flow. Moreover, the saturation magnetic flux value is much lower
compared to the higher magnetic case and remains in the low magnetic SANE
state.

####  3.1.2 Radial dependence of the flow variables

Now, we investigate the comparison of the overall radial variation of various
flow variables. We represent density $\rho$ in g cm-3, temperature $T$ in K
and radial velocity $v_{r}$ in units of speed of light in Figure 5a, 5b and
5c, respectively. Here all the variables are vertically space-averaged between
$-2r_{g}\leq z\leq 2r_{g}$ and time-averaged over $10500t_{g}\leq t\leq
12500t_{g}$. The red, blue, magenta and green curves are for $\beta_{0}$ = 10,
50, 100, and 1000, respectively. The radial distribution of density follows
$\rho\sim r^{1/2}$ except low magnetic flow $\beta_{0}$ = 1000 in the inner
region of the disc ($r<20r_{g})$. It shows steeper dependence on the radial
coordinate. Similarly, temperature profile follows $T\sim r^{-1}$ and radial
velocity also shows $v_{r}\sim r^{-3/2}$ dependence (see the appendix C). This
is almost similar to the non-radiative convection dominated accretion flows
(CDAF) (Narayan et al., 2000; Machida et al., 2001; Igarashi et al., 2020).
The dashed black lines represent CDAF model dependence. The density declines
inwards towards the horizon as the radial velocity increases inwards, depicted
in Figure 5a and 5c. Consequently, temperature increases towards the horizon
and forms a low-density, hot, RIAF-like flow, shown in Figure 5b. In Figure 5d
and 5e, we represent the space averaged plasma-$\beta$ $(\beta)$ and
magnetization parameter $(\sigma_{\rm M})$, respectively. The plasma-$\beta$
parameter decreases at the inner part of the disc near the black hole horizon
($\beta\sim 1$). It implies that the magnetic pressure increases near to the
horizon for all the cases. On the hand, we observe that the magnetization
parameter increases significantly near to the horizon. However, we find that
the magnetization parameter is lower than unity ($\sigma_{\rm M}<1$) in the
equatorial plane for all the cases. In MHD flows, the angular momentum
transports outward region, and gas accretes inwards toward the horizon via
MRI. Initially, we set all the torus with constant angular momentum
($\lambda$). For the representation, we supply constant Keplerian angular
momentum $(\lambda_{\rm K}=7.21)$ at the pressure maximum surface. We observe
that the specific angular momentum becomes nearly Keplerian distribution for
all the cases, depicted in Figure 5f. The black dotted curve represents the
theoretical Keplerian angular momentum distribution for $a_{k}=0.99$ (see
equation 7). Figure 5g represents mass accretion rates in Eddington units
($\dot{M}_{\rm Edd}$). The mass accretion rates are calculated using equation
(14) where the integration is carried out over the vertical direction
$-2r_{g}$ to $2r_{g}$. We observe that the mass accretion rate is always in
the sub-Eddington range ($\dot{M}<\dot{M}_{\rm Edd}$) throughout the radial
distance. Also, the mass accretion rate is more or less constant very near to
the black hole ($r<10r_{g}$). Now we calculate normalized Reynolds stress
$\alpha_{\rm gas}$ and normalized Maxwell stress $(\alpha_{\rm mag})$. The
Reynolds stress is calculated as $\alpha_{\rm gas}=\frac{<\rho v_{r}\delta
v_{\phi}>}{P_{\rm gas}}$ and the Maxwell stress is estimated as $\alpha_{\rm
mag}=-\frac{<2B_{r}B_{\phi}>}{B^{2}}$ (Hawley, 2000; Stone & Pringle, 2001;
Proga & Begelman, 2003). We observed that Maxwell stress lies $\alpha_{\rm
mag}<0.65$ and Reynolds stress is $\alpha_{\rm gas}<0.3$. It also confirms
that Maxwell’s stress is much stronger than Reynolds’s stress. It implies that
the outward transport of angular momentum is predominantly driven by Maxwell
stress in MHD flows.

####  3.1.3 Mass outflows and luminosity

Further, it is observed that the gaseous matter in the disc expands vertically
above and below the torus due to MRI turbulence, and it carries a significant
amount of magnetic field with it (see Figure 2, 3). This indicates the
possibility of a magnetized mass outflow from the disc. In this work, one of
the main purposes is to estimate mass outflow from the disc. One of the major
criteria for mass outflow is that when the Bernoulli parameter is positive
$Be\geq 0$,i.e., unbound flow (Narayan & Yi, 1994; Narayan et al., 2012; Penna
et al., 2013). In Newtonian hydrodynamics, the Bernoulli parameter $(Be)$ is
the sum of the kinetic energy, potential energy, and enthalpy as shown in
equation (5). In general, at a large distance from the black hole, the
potential energy diminishes. Here, we subtract unity from total energy to
eliminate the rest of mass-energy from the gas (see equation (5)). In Figure
6a, we show the variation of the Bernoulli parameter ($Be$) with simulation
time for $\beta_{0}$ = 10 (red), 50 (blue), 100 (magenta) and 1000 (green). We
observe that initially $Be<0$, i.e., bound or no outflows for $\beta_{0}=10$,
50, 100. With the growing MRI and increasing magnetic pressure (i.e., magnetic
energy) in the disc, the Bernoulli parameter becomes positive $Be>0$, depicted
in Figure 6a. However, for low magnetic case $(\beta_{0}=1000)$ $Be$ remains
negative, shown in the inset figure of Figure 6a. It implies that mass outflow
is not possible for low magnetic flow. We also observe a significant positive
correlation between initial plasma-$\beta$ and Bernoulli parameter.

In this paper, we calculate the mass outflow rates by calculating the rate of
outflowing matter through the outer $z$-boundary $(z=\pm z_{\rm out})$ as

$\displaystyle\dot{M}_{\rm out}=2\pi\int^{r_{\rm out}}_{r_{\rm
in}}\left[\rho(r,z_{\rm out})v_{z}(r,z_{\rm out})-\rho(r,-z_{\rm
out})v_{z}(r,-z_{\rm out})\right]rdr,$ (16)

where, $v_{z}$ is the vertical velocity as a function of $(r,z)$. Here, we
ignore the mass loss from the $r_{\rm out}$ boundary. In Figure 6b, we compare
the mass outflow rates $\dot{M}_{\rm out}$ in units of g s-1 for different
magnetic field strengths. Interestingly, we observe that mass outflow rates
significantly increase with the increase of magnetic field. For example, the
mass outflow rates roughly increase almost ten times if we decrease initial
plasma-$\beta$ from 50 to 10. It implies that the mass outflow rates are
purely magnetically driven in our model. Moreover, we find the quasi-periodic
nature of mass outflow rates in the magnetized flow (Okuda et al., 2019).

Now, we attempt to estimate the luminosity emanating from our model. If we
consider here only ion-electron bremsstrahlung emission in a single
temperature approximation, the optical thickness $\Delta\tau=\kappa\rho\Delta
r$ across the mesh size $\Delta r$ for the present model is

$\displaystyle\Delta\tau=6\times
10^{-24}\left(\frac{\rho}{10^{-12}}\right)^{2}\left(\frac{T}{10^{13}}\right)^{-3.5}\left(\frac{\Delta
r}{0.02\times 1.5\times 10^{13}}\right)<<1,$ (17)

where the opacity $\kappa$ is given by the Kramers approximation corresponding
to the bremsstrahlung emission (Kley, 1989). Accordingly, the gas is fully
optically thin to the bremsstrahlung, and then the total luminosity due to the
bremsstrahlung emission is given as

$\displaystyle L=\int e_{\rm ff}dV,$ (18)

where, $e_{\rm ff}$ is the bremsstrahlung emission rate per unit volume. The
integration is carried out all over the computational domain. However, in a
realistic accretion flow, we cannot ignore synchrotron emission in a
magnetized flow (Dihingia et al., 2022; Okuda et al., 2023; Curd & Narayan,
2023). We find that the flow in our model is optically thick to the
monochromatic synchrotron emission in some radial zones for some frequency
ranges in the radio to IR band. Then, the synchrotron luminosity is only
estimated correctly by solving the radiative transfer (Okuda et al., 2023).
Therefore, in the present formalism, we may regard bremsstrahlung luminosity
as the measure of luminosity for the correlation. In Figure 6c, we show the
variation of luminosity with time for all the magnetic flow cases. We find
that there is a completely steady nature of luminosity for low magnetic flow
$\beta_{0}$ = 1000. It can also be visualized from the density and temperature
distribution in Figure 2 and Figure 3. The magnetic field in the high
$\beta_{0}$ case is not significant enough to trigger the turbulent flow in
the torus. The fluctuation is minimal in this case. However, with the increase
of the magnetic field, turbulence is enhanced in the torus via MRI, and
consequently, the luminosity increases with time. Also, we observe quasi-
periodic variation in the luminosity for higher magnetic field cases due to
the turbulent nature of the accretion flow. This quasi-periodic nature of
luminosity variations in a magnetized flow can explain flaring events, and
QPOs originated from AGNs (Okuda et al., 2019, 2022). Moreover, we observe
that total luminosity significantly increases nearly 100 times with the
increase of magnetic field from $\beta_{0}$ = 50 to 10, similar to mass
outflow rates. The simulation run ends when there is no structure of torus at
all, i.e., minimum density ($\rho_{\rm min}$) covers the whole disc (see
Figure 2, first row, last column). At the end time of the simulation, most of
the gas escaped from the disc to create a sudden increase of mass outflow
rates and luminosity, shown in Figure 6b,c for $\beta_{0}$ = 10. Moreover, it
is to be noted that MRI is well-sustained for longer periods of evolution if
we consider large initial torus size (Jiang et al., 2023).

###  3.2 Effect of black hole spin

In this section, we investigate the effect of black hole spin on the evolution
of torus. Here, we use the same initial torus configuration for $\beta_{0}$ =
100 as mentioned in sub-section 3.1 but only vary black hole spin as
$a_{k}=0.99,0.80,0.50$, and 0.0. i.e., maximally spinning to a non-spinning
black hole. It is observed that the initial torus size is slightly decreased
with the lowering of the black hole spin (Utsumi et al., 2022). Because with
the decrease of black hole spin, the total angular momentum of the system
decreases. In Figure 7, we show density $\rho$, temperature $T$,
plasma-$\beta$ and magnetization parameter $(\sigma_{\rm M})$ distribution at
time $t_{3}=10500t_{g}$. The first, second, third, and fourth columns are for
spin $a_{k}=0.99$, 0.80, 0.50, and non-spinning black hole $a_{k}=0.0$,
respectively. We observe more or less similar trends for density, temperature,
plasma-$\beta$, and magnetization parameter distribution for spinning or non-
spinning black holes. We find that magnetized mass outflow is possible for
rotating as well as non-rotating black holes, as depicted in Figure 7.
Further, we observe no significant effect of black hole spin on the accretion
process of the torus as such as pointed out by Jiang et al. (2023).

To investigate the effect of spin on the torus evolution rigorously, we
compare the Bernoulli parameter, mass outflow rates and luminosity variation
by varying spin, depicted in Figure 8a, 8b and 8c, respectively. The red,
blue, magenta and green curves are for spin $a_{k}=0.99,0.80,0.50$, and 0.0,
respectively. We do not find any significant correlation of spin on the
Bernoulli parameter, shown in Figure 8a. Moreover, we observe absolutely no
correlation between spin ($a_{k}$) and mass outflow ($\dot{M}_{\rm out}$) as
well as between spin and luminosity $(L)$. In this regards, a very weak
correlation between mass outflow rates and spin has been observed based on
analytical investigation (Aktar et al., 2015). This is because the mass
outflow happens throughout the disc, and the effect of black hole spin
diminishes with the increase of the distances from the event horizon.

###  3.3 Effect of flow angular momentum

In the same spirit, we investigate the effect of flow angular momentum on the
torus evolution. The angular momentum plays an essential role in forming the
initial torus around the black hole. In this regard, we consider the same
initial configuration of torus set up as sub-section 3.1 and 3.2. But, we only
vary the specific angular momentum in the sub-Keplerian range
($\lambda<\lambda_{\rm K}$). Here, we fix the spin of the black hole as
$a_{k}=0.99$. Now, we vary specific angular momentum as $\lambda=\lambda_{\rm
K},7.00,6.80$ and 6.60, where $\lambda_{\rm K}=7.21$. The initial torus size
decreases significantly with the decrease of angular momentum, and beyond a
critical $\lambda$, there is no possibility of torus formation. It is obvious
that angular momentum provides a repulsive centrifugal force against
attractive gravity force to form a torus. It is found that there it is not
possible to form torus beyond $\lambda<6.60$ for this initial and boundary
conditions. In Figure 9, we represent density $(\rho)$, temperature $(T)$,
plasma-$\beta$ ($\beta$) and magnetization parameter ($\sigma_{\rm M}$)
distribution similar as Figure 7. Here, the first, second, third, and fourth
columns are for angular momentum $\lambda=\lambda_{\rm K}$, 7.00, 6.80 and
6.60, respectively. It is observed that the initial torus size is much smaller
in $\lambda=6.60$ compared to $\lambda=\lambda_{\rm K}$. Further, we represent
the variation of angular momentum at different times of evolution in Figure
10. The red, blue, magenta and green curves are for angular momentum
$\lambda=\lambda_{\rm K},7.00,6.80$, and 6.60, respectively. The dotted black
curves are theoretically calculated specific Keplerian angular momentum (see
equation 7). We observe that with time evolution, the angular momentum
distribution becomes near Keplerian distribution for all the cases, even if
for the sub-Keplerian flow. Moreover, we investigate the effect of angular
momentum on the Bernoulli parameter, mass outflow rates and luminosity,
depicted in Figure 11a, 11b and Figure 11c, respectively. We also observe no
correlation between angular momentum and mass outflow rates and luminosity
similar to the black hole spin in our model.

##  4 Discussions and Conclusions

In this paper, we present the simulation results in two-dimensional MHD
accretion flows around spinning AGN. We use PLUTO code to simulate MHD flows
(Mignone et al., 2007). To mimic the general relativistic effects of spinning
black holes, we adopt effective Kerr potential introduced by Dihingia et al.
(2018). The advantage of this Kerr potential is that we investigate accretion
flows in multi-dimension and with higher spatial resolution for highly
spinning black holes without doing expensive and complex general-relativistic
simulations. In this work, we adopt axisymmetric, two-dimensional torus
evolution around spinning AGN, considering non-resistive and non-radiative
(RIAF) approximation. The initial magnetic field in the torus is configured
following Hawley & Krolik (2002) (see sub-section 2.4). The toroidal magnetic
field is developed in the disc due to shear, and the poloidal magnetic field
is amplified via MRI (Hawley, 2000; Hawley & Krolik, 2002) (see Figure 2). MRI
ignites the non-linear turbulence in the accretion flow, and the flow becomes
in an MHD turbulence state. As a result, Maxwell’s stress transport angular
momentum outwards, and mass accretion happens. In this work, we examine the
effect of the magnetic field, the spin of the black, and the flow angular
momentum on the torus evolution.

In Figure 2 and 3, we represent the distribution of density, temperature,
plasma-$\beta$, azimuthal magnetic field, and magnetization parameter by
varying initial magnetic field strengths. We find that the magnetic field
plays a pivotal role in the torus evolution. We observe that gas and magnetic
field can easily escape from the disc for highly magnetized disc compared to
low magnetized flow, as depicted in Figure 3 of $\beta$ and $\sigma_{\rm M}$
distribution. We also investigate the magnetic state of the accretion flow for
our model. For the purpose of analysis, we calculate normalized magnetic flux
$(\dot{\phi}_{\rm acc})$ with time (Tchekhovskoy et al., 2011; Narayan et al.,
2012; Dihingia et al., 2021). We observe that the model $\beta_{0}=10$ is
similar to the MAD, shown in Figure 4b. On the other hand, model
$\beta_{0}=50$ and $\beta_{0}=100$ are highly magnetized SANE and
$\beta_{0}=1000$ is in the low magnetized SANE state. The radial distribution
of density, temperature, and radial velocity follow non-radiative CDAF
distribution in magnetized flow, depicted in Figure 5a, b, c. We observe that
the magnetic pressure and magnetic energy increases with the radial distance
towards the horizon as shown in the radial variation of $\beta$ and
$\sigma_{\rm M}$ in Figure 5d and Figure 5e, respectively. Also, we find that
the angular momentum distribution is nearly Keplerian for all the cases; see
Figure 5f. Figure 5g indicates that the accretion flow remains sub-Eddington
limit throughout the disc. Also, the variation of Reynolds’s stress and
Maxwell’s stress confirms that Maxwell’s stress is more prominent compared to
Reynolds’s stress in magnetized flow, depicted in Figure 5h and 5i. Further,
we examine the effect of the magnetic field on the mass outflow rates and
luminosity emanating from the disc. To check the unbound matter as mass
outflow as far as energy is concerned, we also calculate the Bernoulli
parameter with time as depicted in Figure 6a (Narayan & Yi, 1994; Narayan et
al., 2012; Penna et al., 2013). We find a positive correlation between initial
magnetic field and Bernoulli parameter as shown in Figure 6a. We also observe
a significant positive correlation between magnetic field strengths and mass
outflow and as well as luminosity, shown in Figure 6a,b,c. Interestingly, we
find an almost steady nature of luminosity variation for less magnetized flow.
On the other hand, the quasi-periodic nature of luminosity variation is
observed in magnetized flow. Therefore, the magnetized flow may be very useful
to explain the QPOs observed for various black hole sources (Okuda et al.,
2019, 2022).

Further, we examine the effect of black hole spin on the torus evolution in
magnetized flow. To investigate that, we plot the variation of Bernoulli
parameter, mass outflow rates and luminosity by varying black hole spin,
depicted in Figure 8. Interestingly, we have not found any correlation between
black hole spin and mass outflow rates for magnetized flow (Aktar et al.,
2015). Further, we observe that the black hole spin has no prominent role in
the accretion dynamics (Jiang et al., 2023). Similarly, we investigate the
effect of flow angular momentum on the magnetized flow. It is observed that
initial torus formation is strongly dependent on angular momentum, shown in
Figure 9. Beyond a critical angular momentum, torus formation is impossible.
We also find no correlation between angular momentum and mass outflow rates
from our MHD simulation model, depicted in Figure 11.

It is to be emphasized that our simulation model belongs to the semi-
relativistic regime by adopting effective Kerr potential to mimic the general
relativistic effects around spinning black holes. One of the limitations of
the semi-relativistic model is that it is unable to explain highly
relativistic jets, which are commonly observed around AGNs or XRBs. The
general consensus is that highly relativistic jets are originated via BZ
process (Blandford & Znajek, 1977) around spinning black holes. In this
regard, McKinney & Gammie (2004), for the first time, explored relativistic
jets generated via BZ mechanism based on GRMHD simulation. Later, a growing
number of independent simulation studies confirms that BZ jets are generic
phenomena in GRMHD simulations around highly spinning black holes to explain
the relativistic jets (Tchekhovskoy et al., 2011; Narayan et al., 2012;
Dihingia et al., 2021), and reference therein. Therefore, one needs to model
accretion flows considering GR effects around spinning black holes to address
relativistic jets. We hope to investigate the accretion-jets mechanism by
incorporating the GRMHD simulation scheme around spinning black holes in the
future.

In this work, we consider two-dimensional MHD flows in an axisymmetric
assumption. To get the complete picture, we need to investigate global three-
dimensional simulation studies. Moreover, the radiation transport mechanism
always plays a crucial role in explaining state transition for black holes.
Also, radiation flux can drive mass outflow to the relativistic limit. i.e.,
jets formation. Further, the radiation mechanism in the MHD flows may explain
the CLAGNs (Igarashi et al., 2020). We plan to investigate global three-
dimensional radiation-dominated MHD flows around spinning black holes in the
future.

## Acknowledgments

We sincerely thank the anonymous referee for the valuable suggestions and
comments that helped us to improve the manuscript. This work is supported by
the National Science and Technology Council of Taiwan through grant NSTC
111-2811-M-007-033, 111-2112-M-007-037, and by the Center for Informatics and
Computation in Astronomy (CICA) at National Tsing Hua University through a
grant from the Ministry of Education of Taiwan. The simulations and data
analysis have been carried out on the CICA Cluster at National Tsing Hua
University. We want to thank Indu Kalpa Dihingia for valuable comments and
discussions during the preparation of the manuscript.

## Data Availability

The data and code underlying this article will be shared on reasonable request
to the corresponding author.

## References

  * Abramowicz et al. (1978) Abramowicz M., Jaroszynski M., Sikora M., 1978, A&A, 63, 221
  * Aktar et al. (2015) Aktar R., Das S., Nandi A., 2015, MNRAS, 453, 3414
  * Aktar et al. (2017) Aktar R., Das S., Nandi A., Sreehari H., 2017, MNRAS, 471, 4806
  * Balbus & Hawley (1991) Balbus S. A., Hawley J. F., 1991, ApJ, 376, 214
  * Balbus & Hawley (1998) Balbus S. A., Hawley J. F., 1998, Reviews of Modern Physics, 70, 1
  * Becker et al. (2001) Becker P. A., Subramanian P., Kazanas D., 2001, ApJ, 552, 209
  * Begelman (2012) Begelman M. C., 2012, MNRAS, 420, 2912
  * Blandford & Begelman (1999) Blandford R. D., Begelman M. C., 1999, MNRAS, 303, L1
  * Blandford & Begelman (2004) Blandford R. D., Begelman M. C., 2004, MNRAS, 349, 68
  * Blandford & Payne (1982) Blandford R. D., Payne D. G., 1982, MNRAS, 199, 883
  * Blandford & Znajek (1977) Blandford R. D., Znajek R. L., 1977, MNRAS, 179, 433
  * Chatterjee & Narayan (2022) Chatterjee K., Narayan R., 2022, ApJ, 941, 30
  * Chattopadhyay & Das (2007) Chattopadhyay I., Das S., 2007, New Astron., 12, 454
  * Curd & Narayan (2023) Curd B., Narayan R., 2023, MNRAS, 518, 3441
  * Das et al. (2014) Das S., Chattopadhyay I., Nandi A., Molteni D., 2014, MNRAS, 442, 251
  * De Villiers & Hawley (2003) De Villiers J.-P., Hawley J. F., 2003, ApJ, 592, 1060
  * De Villiers et al. (2003) De Villiers J.-P., Hawley J. F., Krolik J. H., 2003, ApJ, 599, 1238
  * Dhang et al. (2023) Dhang P., Bai X.-N., White C. J., 2023, ApJ, 944, 182
  * Dihingia et al. (2018) Dihingia I. K., Das S., Maity D., Chakrabarti S., 2018, Phys. Rev. D, 98, 083004
  * Dihingia et al. (2021) Dihingia I. K., Vaidya B., Fendt C., 2021, MNRAS, 505, 3596
  * Dihingia et al. (2022) Dihingia I. K., Vaidya B., Fendt C., 2022, MNRAS, 517, 5032
  * Dihingia et al. (2023) Dihingia I. K., Mizuno Y., Fromm C. M., Rezzolla L., 2023, MNRAS, 518, 405
  * Event Horizon Telescope Collaboration et al. (2021) Event Horizon Telescope Collaboration et al., 2021, ApJ, 910, L13
  * Fender & Gallo (2014) Fender R., Gallo E., 2014, Space Sci. Rev., 183, 323
  * Gammie et al. (2003) Gammie C. F., McKinney J. C., Tóth G., 2003, ApJ, 589, 444
  * Hawley (2000) Hawley J. F., 2000, ApJ, 528, 462
  * Hawley & Balbus (2002) Hawley J. F., Balbus S. A., 2002, ApJ, 573, 738
  * Hawley & Krolik (2001) Hawley J. F., Krolik J. H., 2001, ApJ, 548, 348
  * Hawley & Krolik (2002) Hawley J. F., Krolik J. H., 2002, ApJ, 566, 164
  * Hawley et al. (2011) Hawley J. F., Guan X., Krolik J. H., 2011, ApJ, 738, 84
  * Hawley et al. (2013) Hawley J. F., Richers S. A., Guan X., Krolik J. H., 2013, ApJ, 772, 102
  * Igarashi et al. (2020) Igarashi T., Kato Y., Takahashi H. R., Ohsuga K., Matsumoto Y., Matsumoto R., 2020, ApJ, 902, 103
  * Igumenshchev & Abramowicz (1999) Igumenshchev I. V., Abramowicz M. A., 1999, MNRAS, 303, 309
  * Igumenshchev et al. (2000) Igumenshchev I. V., Abramowicz M. A., Narayan R., 2000, ApJ, 537, L27
  * Igumenshchev et al. (2003) Igumenshchev I. V., Narayan R., Abramowicz M. A., 2003, ApJ, 592, 1042
  * Jiang et al. (2023) Jiang H.-X., Mizuno Y., Fromm C. M., Nathanail A., 2023, MNRAS, 522, 2307
  * Kim et al. (2019) Kim J., Garain S. K., Chakrabarti S. K., Balsara D. S., 2019, MNRAS, 482, 3636
  * Kley (1989) Kley W., 1989, A&A, 208, 98
  * Kumar & Chattopadhyay (2013) Kumar R., Chattopadhyay I., 2013, MNRAS, 430, 386
  * Kuwabara et al. (2000) Kuwabara T., Shibata K., Kudoh T., Matsumoto R., 2000, PASJ, 52, 1109
  * Kuwabara et al. (2005) Kuwabara T., Shibata K., Kudoh T., Matsumoto R., 2005, ApJ, 621, 921
  * LaMassa et al. (2015) LaMassa S. M., et al., 2015, ApJ, 800, 144
  * MacLeod et al. (2016) MacLeod C. L., et al., 2016, MNRAS, 457, 389
  * Machida et al. (2000) Machida M., Hayashi M. R., Matsumoto R., 2000, ApJ, 532, L67
  * Machida et al. (2001) Machida M., Matsumoto R., Mineshige S., 2001, PASJ, 53, L1
  * Matsumoto et al. (1996) Matsumoto R., Uchida Y., Hirose S., Shibata K., Hayashi M. R., Ferrari A., Bodo G., Norman C., 1996, ApJ, 461, 115
  * McKinney & Gammie (2004) McKinney J. C., Gammie C. F., 2004, ApJ, 611, 977
  * McKinney et al. (2012) McKinney J. C., Tchekhovskoy A., Blandford R. D., 2012, MNRAS, 423, 3083
  * McKinney et al. (2015) McKinney J. C., Dai L., Avara M. J., 2015, MNRAS, 454, L6
  * Mignone et al. (2007) Mignone A., Bodo G., Massaglia S., Matsakos T., Tesileanu O., Zanni C., Ferrari A., 2007, ApJS, 170, 228
  * Mirabel et al. (1992) Mirabel I. F., Rodriguez L. F., Cordier B., Paul J., Lebrun F., 1992, Nature, 358, 215
  * Narayan & Yi (1994) Narayan R., Yi I., 1994, ApJ, 428, L13
  * Narayan et al. (1995) Narayan R., Yi I., Mahadevan R., 1995, Nature, 374, 623
  * Narayan et al. (2000) Narayan R., Igumenshchev I. V., Abramowicz M. A., 2000, ApJ, 539, 798
  * Narayan et al. (2003) Narayan R., Igumenshchev I. V., Abramowicz M. A., 2003, PASJ, 55, L69
  * Narayan et al. (2012) Narayan R., SÄ dowski A., Penna R. F., Kulkarni A. K., 2012, MNRAS, 426, 3241
  * Narayan et al. (2022) Narayan R., Chael A., Chatterjee K., Ricarte A., Curd B., 2022, MNRAS, 511, 3795
  * Ohsuga & Mineshige (2011) Ohsuga K., Mineshige S., 2011, ApJ, 736, 2
  * Ohsuga et al. (2009) Ohsuga K., Mineshige S., Mori M., Kato Y., 2009, PASJ, 61, L7
  * Okuda & Das (2015) Okuda T., Das S., 2015, MNRAS, 453, 147
  * Okuda et al. (2019) Okuda T., Singh C. B., Das S., Aktar R., Nandi A., Dal Pino E. M. d. G., 2019, PASJ, 71, 49
  * Okuda et al. (2022) Okuda T., Singh C. B., Aktar R., 2022, MNRAS, 514, 5074
  * Okuda et al. (2023) Okuda T., Singh C. B., Aktar R., 2023, MNRAS, 522, 1814
  * Paczyńsky & Wiita (1980) Paczyńsky B., Wiita P. J., 1980, A&A, 88, 23
  * Penna et al. (2013) Penna R. F., Kulkarni A., Narayan R., 2013, A&A, 559, A116
  * Penrose (1969) Penrose R., 1969, Nuovo Cimento Rivista Serie, 1, 252
  * Proga & Begelman (2003) Proga D., Begelman M. C., 2003, ApJ, 582, 69
  * Quataert & Gruzinov (2000) Quataert E., Gruzinov A., 2000, ApJ, 539, 809
  * Ricci et al. (2016) Ricci C., et al., 2016, ApJ, 820, 5
  * Shakura & Sunyaev (1973) Shakura N. I., Sunyaev R. A., 1973, A&A, 500, 33
  * Spruit et al. (1987) Spruit H. C., Matsuda T., Inoue M., Sawada K., 1987, MNRAS, 229, 517
  * Stone & Pringle (2001) Stone J. M., Pringle J. E., 2001, MNRAS, 322, 461
  * Stone et al. (1999) Stone J. M., Pringle J. E., Begelman M. C., 1999, MNRAS, 310, 1002
  * Tchekhovskoy & McKinney (2012) Tchekhovskoy A., McKinney J. C., 2012, MNRAS, 423, L55
  * Tchekhovskoy et al. (2010) Tchekhovskoy A., Narayan R., McKinney J. C., 2010, ApJ, 711, 50
  * Tchekhovskoy et al. (2011) Tchekhovskoy A., Narayan R., McKinney J. C., 2011, MNRAS, 418, L79
  * Utsumi et al. (2022) Utsumi A., Ohsuga K., Takahashi H. R., Asahina Y., 2022, ApJ, 935, 26
  * Xue & Wang (2005) Xue L., Wang J., 2005, ApJ, 623, 372
  * Yuan & Narayan (2014) Yuan F., Narayan R., 2014, ARA&A, 52, 529
  * Yuan et al. (2012) Yuan F., Bu D., Wu M., 2012, ApJ, 761, 130

##  Appendix A Numerical Convergence Test

Figure 12: Time evolution of mass accretion rate $(\dot{M}_{\rm acc})$ and
normalized magnetic flux $(\dot{\phi}_{\rm acc})$ for different simulation
resolution. Red (dashed-dotted), blue (dotted), black (dashed-dotted), magenta
(dashed-dotted) and green (solid) are for resolution ($348\times 348$),
($640\times 640$), ($768\times 768$), ($800\times 800$) and ($896\times 896$),
respectively. Figure 13: Time-averaged radial variation of MRI quality factor
$Q_{r}$ (solid) and $Q_{z}$ (dotted). The quality factor are space-averaged
vertically over $-5r_{g}<z<5r_{g}$. The red and blue curves are for
$\beta_{0}=10$ and 50, respectively.

Ideally, the convergence test of the numerical model is compared with the
exact analytical solution. However, there is a very limited exact solution
available for fluid equations. Also, the simulation method is based on the
discretization of the computational domain as well as various approximation
methods. Here, we perform a convergence test for our simulation model by
calculating two important parameters such as mass accretion rate
$(\dot{M}_{\rm acc})$ and normalized magnetic flux $(\dot{\phi}_{\rm acc})$ in
time using equation (14) and (15) by varying different simulation resolution,
shown in Figure 12. We vary the computational resolution as $348\times 348$
(Red dashed-dotted), $640\times 640$ (blue dotted), $768\times 768$ (black
dashed-dotted), $800\times 800$ (magenta dashed-dotted) and $896\times 896$
(green solid), respectively. We observe that with the increase of resolution,
the mass accretion rate and normalized magnetic flux are settled down in mean
values,i.e., convergence values for ($896\times 896$). Therefore, we adopt the
high resolution ($896\times 896$) for all the simulation runs in this paper,
keeping in mind the numerical accuracy and long-term simulation run.

##  Appendix B Resolution of the MRI

In magnetized accretion flow, MRI is one of the essential triggering
mechanisms to transport angular momentum. Here, we check whether the flow is
subjected to MRI in our simulation model. In order to have MRI-driven
accretion, it is necessary to resolve the fastest-growing MRI mode. To
quantify the criteria to resolve the MRI, we define the characteristic
wavelength of the fastest growing mode of MRI as $\lambda_{\rm MRI}=2\pi
V_{\rm A}/\Omega$. Here, $V_{\rm A}$ is the Alfvén speed, and $\Omega$ is the
angular velocity. The quality factor for resolving MRI is defined as (Hawley
et al., 2011, 2013)

$\displaystyle Q_{r}=\frac{2\pi V_{{\rm A}r}}{\Omega\Delta r},$ (19)
$\displaystyle Q_{z}=\frac{2\pi V_{{\rm A}z}}{\Omega\Delta z},$ (20)

where, $V_{{\rm A}r}$ and $V_{{\rm A}z}$ are the radial and vertical component
of Alfvén speed, respectively. $\Delta r$ and $\Delta z$ are the grid sizes in
the radial and vertical directions, respectively. In general, it is shown that
the quantitative values are $Q_{r}\gtrsim 10$ and $Q_{z}\gtrsim 10$ to
sufficiently well-resolved MRI (Hawley et al., 2011, 2013). In Figure 13, we
represent the time-averaged radial variation of quality factors
$(Q_{r},Q_{z})$ of MRI. Here, we consider the time average over
$t=(7000-10000)t_{g}$ and vertical average over $-5r_{g}<z<5r_{g}$. We show
the variation of the quality factor for magnetized flow $\beta_{0}=10$ (red)
and 50 (blue), depicted in Figure 13. Solid and dashed curves are for $Q_{r}$
and $Q_{z}$, respectively. We observe $Q_{r}$ $\gtrsim 15$ and $Q_{z}$
$\gtrsim 15$ for our simulation result. It clearly indicates that our
simulation model is able to resolve MRI in good agreement.

##  Appendix C Radial dependence of flow variables in CDAF Model

The self-similar solution has been introduced to investigate accretion flow
around black holes in the literature (Spruit et al., 1987; Narayan & Yi, 1994;
Narayan et al., 2000). Self-similarity aims to obtain a dimensionless system
of equations whose primary essence is to scale physical quantities with local
flow variables. In this regard, the Keplerian angular velocity and linear
velocity can be written as $\Omega_{\rm K}=\left(\frac{GM_{\rm
BH}}{r^{3}}\right)^{1/2}$ and $v_{\rm K}=r\Omega_{\rm K}=\left(\frac{GM_{\rm
BH}}{r}\right)^{1/2}$, respectively. Here, $G$, $M_{\rm BH}$, and $r$ are the
gravitational constant, the mass of the black hole, and cylindrical radius,
respectively. Therefore, the self-similar scaling of angular velocity, sound
speed, and scale height can be written as

$\displaystyle\Omega=\Omega_{0}\Omega_{\rm K}\propto r^{-3/2}$ (21)
$\displaystyle c_{s}^{2}=c_{0}^{2}v_{\rm K}^{2}\propto r^{-1},$ (22)
$\displaystyle H=c_{s}/\Omega_{\rm K}=c_{0}r,$ (23)

where $\Omega_{0}$ and $c_{0}$ are the dimensionless constant and can be
determined. Now, we assume the density variation as

$\displaystyle\rho=\rho_{0}r^{-\alpha}.$ (24)

and the pressure is scaled as

$\displaystyle P=\rho c_{s}^{2}\propto r^{-1-\alpha}.$ (25)

Therefore, the radial dependence of temperature can be written

$\displaystyle T\propto r^{-1}.$ (26)

Now, the mass accretion rate is obtained from the mass conservation equation
as

$\displaystyle\dot{M}=2\pi\rho vrH.$ (27)

To make the mass conservation dimensionless, i.e., independent of radial
distance, then the velocity scale is

$\displaystyle v\propto r^{\alpha-2}.$ (28)

Narayan & Yi (1994) indicates that there are two kinds of solutions possible:
(i) advection-dominated accretion flows (ADAF) when $\alpha=3/2$ and (ii)
convection-dominated accretion flows (CDAF) when $\alpha=1/2$. In the CDAF
model, the angular momentum transfers inward via convection with very small
viscosity (Narayan & Yi, 1994; Narayan et al., 2000).