# The Effects of Gas Angular Momentum on Forming Magnetically Arrested Disks
and Launching Powerful Jets

Tom M. Kwan Department of Physics, The University of Hong Kong, Pokfulam Road,
Hong Kong  Lixin Dai Department of Physics, The University of Hong Kong,
Pokfulam Road, Hong Kong  Alexander Tchekhovskoy Department of Physics &
Astronomy, Northwestern University, Evanston, IL 60208, USA  Center for
Interdisciplinary Exploration & Research in Astrophysics (CIERA), Evanston, IL
60208, USA

###### Abstract

In this Letter, we investigate jet-launching abilities of Bondi-like accretion
flows with zero or low specific angular momentum by performing 3D general
relativistic magnetohydrodynamic simulations. In order to check if
relativistic jets can be launched magnetically, we thread the accretion flow
with large-scale poloidal magnetic field, and choose a rapidly spinning black
hole. We demonstrate that the magnitude of the initial gas specific angular
momentum primarily controls whether the disk can reach and sustain the
magnetically arrested disk (MAD) state that launches very powerful jets, at
$\gtrsim 100\%$ energy efficiency. We find that MAD forms in the presence of
even a very small amount of specific angular momentum, and episodic jets with
an average energy efficiency of $\sim 10\%$ can still form even when the gas
has zero initial angular momentum. Our results give plausible explanations to
why jets can be produced from various astrophysical systems that lack large
gas specific angular momenta, such as Sgr A*, wind-fed X-ray binaries, tidal
disruption events, and long-duration gamma-ray bursts.

Accretion (14) — Black holes (162) — Galactic center (565) — Gamma-ray
bursts(629) — Massive stars (732) — High mass x-ray binary stars(733) — Jets
(870) — MHD (1964), method: numerical

##  1 Introduction

Gas accretion onto black holes (BHs) is common in many astrophysical systems.
The accreting gas is usually assumed to carry some angular momentum and form a
disk. Magnetorotational instability (MRI) transports the angular momentum
outwards (Balbus & Hawley, 1998), allowing gas to accrete. In this process,
the gas mass-energy is converted to energy power relativistic jets, winds and
radiation, as observed from active galactic nuclei (AGNs), BH X-ray binaries
(XRBs), tidal disruption events (TDEs), etc. Some of the popular accretion
disk models include the advection-dominated accretion flow (ADAF, e.g.,
Narayan & Yi, 1994, 1995; Yuan & Narayan, 2014), standard $\alpha$-disk
(Shakura & Sunyaev, 1973; Novikov & Thorne, 1973) and slim disk (Abramowicz et
al., 1988) models, which apply to disks at different accretion rates.

While all disk models described above assume the disk has a relatively large
specific angular momentum, some astrophysical systems can carry very low gas
angular momentum, such as Sgr A* (Melia, 1992; Narayan et al., 1995; Ressler
et al., 2018), long gamma-ray bursts (GRBs, Fryer, 1999), tidal disruption
events (TDEs, Rees, 1988) and wind-fed high-mass X-ray binaries (HMXBs, Smith
et al., 2002; Tauris & van den Heuvel, 2006). Accretion flows with low gas
angular momentum are less studied. Yet it is important to understand whether
such accretion flows can produce powerful emissions, winds, and jets.

The simplest model for accretion flows with zero angular momentum is the Bondi
accretion model (Bondi, 1952), in which gas accretes steadily in a spherically
symmetric fashion. Bondi-like accretion flows with low angular momentum have
been extensively studied with hydrodynamic simulations, which examine the
behavior of the accretion flow on large scales (Suková & Janiuk, 2015; Suková
et al., 2017), and show that a thick equatorial torus structure can form near
the equator as the flow hits the centrifugal barrier. As a result, part of the
inflow can form an outflow, which can lead to energy feedback on the
surroundings (e.g. Proga & Begelman, 2003; Janiuk et al., 2008; Lee & Ramirez-
Ruiz, 2006; Murguia-Berthier et al., 2020). However, these studies have
neglected the magnetic or general relativistic effects, which play an
essential role in determining the behavior of inner accretion flows and in
particular the prospect of launching relativistic jets (Blandford & Znajek,
1977).

The dynamical evolution of magnetized plasma around black holes and the
production of jets involves nonlinear processes and generally lacks obvious
symmetries. This makes general relativistic magnetohydrodynamic (GRMHD)
simulations particularly attractive for modeling these systems (e.g., Gammie
et al., 2003; De Villiers & Hawley, 2003; Tchekhovskoy et al., 2011; McKinney
et al., 2012; White et al., 2016; Porth et al., 2017; The Event Horizon
Telescope Collaboration et. al., 2019). These GRMHD simulations usually start
from gas in the configuration of a torus or disk with its motion-dominated
rotation (see reviews by Tchekhovskoy et al., 2012; Davis & Tchekhovskoy,
2020). They show that if large-scale ordered magnetic fields are supplied to
such disks, the accretion flow would drag sufficient magnetic fluxes to the
vicinity of the black hole, and magnetically arrested disks (MADs, Narayan et
al. 2003, see also Bisnovatyi-Kogan & Ruzmaikin 1974, 1976) will form. In
MADs, magnetic field significantly impacts the inner disk dynamics, and very
powerful jets can be produced if the black hole spins rapidly (Blandford &
Znajek, 1977): In fact, jet power can even exceed the accretion power,
resulting in net energy extraction from the black hole (Tchekhovskoy et al.,
2011).

In contrast, the behavior of low angular momentum accretion flows is much less
clear and has not been extensively explored with GRMHD simulations. In
particular, the role of gas angular momentum for launching powerful jets
remains poorly understood. Komissarov & Barkov (2009) studied the spherical
accretion of magnetized gas with two-dimensional GRMHD simulations and showed
that such accretion flow can become highly magnetized near the event horizon
and launch outflows. More recently, several studies employed 3D GRMHD
simulations to investigate Bondi-like accretion flow and the resulting
feedback processes. Ressler et al. (2021) studied spherical accretion flow
threaded with large-scale magnetic field at different inclinations with
respect to the spin axis of the BH and explored how the jet strength depends
on the inclination. Kaaz et al. (2022) studied Bondi-Hoyle-Lyttleton (BHL)
accretion onto a spinning BH traversing through a magnetized gaseous medium
and showed such accretion could power relativistic jets. However, how the jet
power depends on the gas angular momentum has not been systematically studied.

In this work, we focus on investigating the link between the magnitude of the
gas specific angular momentum and the power of the relativistic jets. We carry
out three fully 3D GRMHD simulations of magnetized Bondi-like accretion flows
with initially no or very low angular momentum around rapidly spinning black
holes. We study the properties of the quasi-steady state accretion flow, and
investigate how the magnetic flux and gas coexist near the black hole. We
demonstrate that both non-rotating and slowly rotating accretion flows can
produce relativistic jets. However, the initial angular momentum of the gas
needs to reach a small critical value to robustly sustain the MAD state and
its powerful jets. We introduce our methodology and simulation setup in
Section 2. We present the results in Section 3, and discuss and summarize them
in Section 4.

##  2 Methodology

###  2.1 Governing equations

We carry out 3D time-dependent GRMHD simulations using the HARM code (Gammie
et al., 2003; McKinney & Gammie, 2004; Tchekhovskoy et al., 2011; McKinney et
al., 2012). The code solves the following conservation equations:

$\displaystyle\nabla_{\mu}\left(\rho u^{\mu}\right)$ $\displaystyle=0$ (1)
$\displaystyle\nabla_{\mu}T^{\mu}_{\nu}$ $\displaystyle=0$ (2)

where $\rho$ is the rest-mass density of the gas in the fluid frame, $u^{\mu}$
is the gas 4-velocity, $T^{\mu}_{\nu}$ is the energy stress tensor, which
includes matter (MA) and electromagnetic (EM) terms,

$\begin{split}T^{\mu}_{\nu}&={T^{\rm EM}}^{\mu}_{\nu}+{T^{\rm
MA}}^{\mu}_{\nu}\\\ {T^{\rm MA}}^{\mu}_{\nu}&=(\rho_{0}+u_{g}+p_{g})\
u^{\mu}u_{\nu}+p_{g}\delta^{\mu}_{\nu}\\\ {T^{\rm
EM}}^{\mu}_{\nu}&=b^{2}u^{\mu}u_{\nu}+p_{b}\delta^{\mu}_{\nu}-b^{\mu}b_{\nu}\end{split}$
(3)

where $u_{g}$ is the gas internal energy density with the adiabatic index of
$\gamma=4/3$, $p_{g}=(\gamma-1)u_{g}$ is the gas pressure,
$p_{b}=b^{\mu}b_{\nu}/2=b^{2}/2$ is the magnetic pressure, $b^{\mu}$ and
$b_{\nu}$ are the contravariant and covariant fluid-frame magnetic field
4-vectors, respectively, $\delta^{\mu}_{\nu}$ is the Kronecker delta tensor,
and we use the Lorentz-Heaviside units. The magnetic field is evolved using
the induction equation,

$\partial_{t}(\sqrt{-g}B^{i})=-\partial_{j}\left(\sqrt{-g}(b^{j}u^{i}-b^{i}u^{j})\right)$
(4)

where $\sqrt{-g}$ is the metric determinant and $B^{i}$ is the magnetic field
three-vector.

###  2.2 Simulation setup and initial conditions

We consider a rapidly rotating BH with dimensionless spin parameter $a=0.9$
and conduct three simulations with different initial gas angular momenta. The
simulations employ modified spherical-polar coordinates centered on the BH and
with a minimum resolution of $N_{r}\times N_{\theta}\times N_{\phi}=240\times
128\times 64$ cells in r-, $\theta$-, and $\phi$-directions, respectively (see
Table A.1). The grid in the $r$-direction is logarithmically-spaced. The grid
in the $\theta$-direction has higher resolution in the disk (equatorial) and
jet (polar) regions in order to resolve the turbulence in the equatorial disk
and the collimated polar jets. The grid in $\phi$-direction is uniform. The
simulation domain extends from an inner boundary at
$R_{\rm{}in}=0.8r_{H}\approx 1.15r_{g}$ to an outer boundary at
$R_{\rm{}out}=10^{5}r_{g}$. Here $r_{g}=GM_{\rm{}BH}/c^{2}$ is the BH
gravitational radius, where $G$ is the gravitational constant, $c$ is the
speed of light, and $M_{\rm{}BH}$ is the mass of the black hole.
$r_{H}=(1+\sqrt{1-a^{2}})r_{g}$ is the black hole event horizon radius. Non-
radiative ideal GRMHD simulations are scale-free in BH mass, so we use
geometric units with $G=c=M_{\rm{}BH}=1$ hereafter.

For standard Bondi accretion, the free-falling gas has a spherically symmetric
configuration. At infinity, the gas particle has specific energy $e=1$, which
stays constant during the free fall. Therefore, the self-similar solution for
the gas density $\rho$ and four-velocity $u^{\mu}$ is:

$\displaystyle\rho$ $\displaystyle\propto r^{-3/2}$ (5) $\displaystyle u_{t}$
$\displaystyle=-e=-1$ (6) $\displaystyle u^{\theta}$ $\displaystyle=0$ (7)
$\displaystyle u^{\phi}$ $\displaystyle=0$ (8) $\displaystyle u^{r}$
$\displaystyle=-\frac{\sqrt{2r(r^{2}+a^{2})}}{r^{2}+a^{2}\cos\theta}$ (9)

For the simulated gas flow with zero or low angular momentum, while we still
use Eqs. (5)-(7) to initialize $\rho$, $u_{t}$, and $u^{\theta}$, we modify
the initial $u^{\phi}$ and $u^{r}$ according to the desired disk specific
angular momentum. The disk specific angular momentum is characterized by the
circularization radius $R_{c}$, at which the centrifugal force is large enough
to counterbalance the inward gravitational force. The three simulations have
maximum specific angular momentum corresponding to $R_{c}=0$ (model a09rc0),
$10r_{g}$ (model a09rc10), and $50r_{g}$ (model a09rc50), respectively. The
specific angular momentum of a Keplerian, equatorial circular orbit is:

$l_{K}(r)=\frac{\sqrt{r}(r^{2}-2a\sqrt{r}+a^{2})}{r\sqrt{r^{2}-3r+2a\sqrt{r}}}.$
(10)

Hence we set the gas initial maximum specific angular momentum to be:

$l_{\rm{}max}=\begin{cases}0&\text{for}\ R_{c}=0\\\ l_{K}(r=R_{c})&\text{for}\
R_{c}=10r_{g}\ {\rm or}\ 50r_{g}\\\ \end{cases}$ (11)

More details of the initial specific angular momentum distribution across the
gas flow are given in Appendix A.

We thread the accretion flow with a relatively weak poloidal magnetic field.
The magnetic flux is described by the magnetic flux function
$\Psi(r,\theta)=r(1-|\cos\theta|)$, which describes parabolic magnetic flux
distribution (Tchekhovskoy et al., 2008). This description ensures that the
plasma beta $\beta\equiv p_{g}/p_{b}$ is asymptotically constant everywhere.
$\beta$ is set to have a minimum value of 100 along the equator, so that the
initial magnetic field in the disk is subdominant.

###  2.3 Diagnostics

We categorize the gas flow into three different regions: 1) the jet region,
where $\sigma=b^{2}/\rho>1$; 2) the wind region, where the conditions
$u^{r}>0$ and $\sigma<1$ are both satisfied; 3) the disk region, where the
conditions $u^{r}<0$ and $\sigma<1$ are both satisfied. (We note that a small
inflow region very close the black hole can have $\sigma>1$, but excluding
this region from the disk only makes minor changes to our results.) The above
conditions define the boundaries between these regions, i.e., the jet-wind
boundary has $\sigma=1$ and the disk-wind boundary has $u^{r}=0$. Below we
provide a summary of the definitions used for computing the various
quantities.

The rest-mass accretion rate flowing inwards through a sphere of radius $r$ is

$\dot{M}=-\int\rho u^{r}\rm dA_{\rm\theta\phi},$ (12)

where $dA_{\rm\theta\phi}=\sqrt{-g}\rm d\theta\rm d\phi$. This gives us the
accretion rate onto the black hole $\dot{M}_{H}$ when evaluated at the event
horizon $r_{H}$.

The absolute magnetic flux through the black hole event horizon is

$\Phi_{H}=\frac{1}{2}\int|B^{r}|\,\rm dA_{\theta\phi}$ (13)

which is integrated over the event horizon. One can define a dimensionless
magnetization parameter, the “MADness parameter”, by normalizing $\Phi_{BH}$
with $\dot{M}_{H}$ (Tchekhovskoy et al., 2011)

$\phi_{H}=\frac{\Phi_{H}}{\sqrt{\dot{M}_{H}r^{2}_{g}\ c}}.$ (14)

The disk enters the MAD state when $\phi_{H}\gtrsim 40$. In practice, when we
compute dimensionless quantities like $\phi$ and various efficiencies by
normalizing some quantity $Q(t)$ with $\dot{M}_{H}(t)$, we smooth
$\dot{M}_{H}(t)$ with a Gaussian function over a moving window centered at
time $t$ and with a standard deviation of $1000~{}r_{g}/c$ in order to reduce
fast fluctuations.

The energy flux flowing out through a sphere of radius $r$ is

$\dot{E}=-\int T^{r}_{t}\rm dA_{\rm\theta\phi}.$ (15)

Using this, we can calculate the net outflow efficiency as

$\eta(r)\equiv\frac{\dot{E}(r)+\dot{M}(r)}{\dot{M}_{H}}.$ (16)

The accretion efficiency $\eta_{H}$ is defined as $\eta(r)$ evaluated at the
event horizon. When $\eta_{H}>100\%$, the accretion process results in a net
energy extraction out of the BH, typically caused by the energy extraction
rate in electromagnetic form through the BZ process around a fastly spinning
BH exceeding the disk energy supply rate in the form of rest-mass energy. In
addition, we can evaluate the jet energy flux and the jet energy efficiency
as:

$\dot{E}_{\rm{}jet}=-\int T^{r}_{t}(\sigma>1)\rm dA_{\rm\theta\phi},$ (17)
$\eta_{\rm jet}\equiv\frac{\dot{E}_{\rm jet}(r)}{\dot{M}_{H}}.$ (18)

In order to obtain the radial profile of a quantity $Q(r)$, we average it over
spherical shells and weigh it by the gas density $\rho$:

$\langle Q(r)\rangle_{\rho}\equiv\frac{\int\rm
dA_{\rm\theta\phi}\rho\textit{Q}}{\int\rm dA_{\rm\theta\phi}\rho}.$ (19)

We can define the geometric half-angular thickness, ($H/R$), of the flow as:

$\frac{H}{R}\equiv\left(\frac{\int\rho(\theta-\theta_{\rm{}mid})^{2}\;\rm
dA_{\rm\theta\phi}}{\int\rho\;\rm dA_{\theta\phi}}\right)^{1/2}$ (20)

where $\theta_{\rm{}mid}\equiv\langle\theta\rangle_{\rho}$ is the polar angle
of the disk mid-plane.

##  3 Results

###  3.1 Time evolution of the accretion flow

Figure 1: We show snapshots of a vertical slice from the inner regions of the
3D GRMHD simulations with a09rc0 ($R_{c}=0$, top), a09rc10 ($R_{c}=10r_{g}$,
middle), or a09rc50 ($R_{c}=50r_{g}$, bottom), at different times $t=$ (0,
3000, 35000, 48000) $r_{g}/c$ (left to right). The color shows gas the rest
mass density $\rho_{0}$. The black lines indicate the magnetic field lines,
for which the thickness represents the magnetic field strength. The pink
contours show the boundary of the jet with $\sigma=1$. The blue contours show
the disk-wind (i.e., inflow-outflow) boundary, which has $u^{r}=0$. In all
simulations, geometrically thick disk structures (shaded with a light blue
color) form close to the black holes. In addition, wide-angle winds and
relativistic jets are launched.

We run all three simulations out to at least $t=50000\ r_{g}/c$. The flows
reach a inflow equilibrium out to $r\gtrsim 200\ r_{g}$ in all runs. We show
the evolution of the accretion flows in the three runs with different $R_{c}$
in Fig. 1. Initially, the outer gas falls in radially without sufficient
support from angular momentum. At the same time, a centrifugally supported
structure starts to form close to the black hole, owing to two factors. 1) The
small initial gas specific angular momentum (as in our model a09rc10 and
a09rc50) can result in the formation of a centrifugal barrier, which has been
demonstrated in previous analytical calculations and hydrodynamical
simulations of Bondi-like accretion (e.g., Proga & Begelman, 2003; Lee &
Ramirez-Ruiz, 2006; Suková et al., 2017). 2) Due to the GR frame-dragging
effect, the gas near black hole can gain some angular momentum. Therefore,
even in our a09rc0 model, the spherical symmetry is still broken due to the
fast spin of the black hole and the vertical configuration of the magnetic
flux near the black hole. After the formation of the centrifugal barrier, the
infalling gas hits on it and develops a shock, which propagates outwards. A
high-density bubble expands with the shock (e.g., the regions confined by two
blue $u^{r}=0$ contours in the snapshots at $t=3000r_{g}/c$). By $t\gtrsim
10000r_{g}/c$, the accretion flows have become stable in three simulations,
and winds and relativistic jets are launched in all three simulations. The
jets drive blast waves into the ambient medium, vacant the gas and prevent
accretion onto the black hole along the polar region.

Fig. 2 shows the evolution of the mass accretion rate onto the black hole
$\dot{M}_{H}$ (Eqn. 12), the MADness parameter $\phi_{H}$ (Eqn. 14), the
outflow energy efficiency $\eta_{H}$ (Eqn. 16) and the jet efficiency
$\eta_{\rm jet}$ (Eqn. 18) as functions of time.111In practice, here we
measure the BH mass $\dot{M}_{H}$ and energy $\dot{E}_{H}$ fluxes at
$r=6r_{g}$ to avoid the potential contamination of $\dot{M}$ and $\dot{E}$ by
the numerical density floors near the horizon, since the true time-averaged
$\dot{M}_{H}$ and $\dot{E}_{H}$ (if uncontaminated by the density floors)
should remain constant near the horizon when the flow has reached a quasi-
steady state. The rate that gas falls into the black hole initially starts out
high. Dragged by the infalling gas, magnetic flux builds up near the black
hole, so $\phi_{H}$ quickly grows and MRI is triggered in the accretion disk.
It is apparent that the model a09rc0 has $\phi_{H}$ growing the fastest, since
the gas in this model falls in faster with the lowest initial gas angular
momentum. After the disk forms, $\dot{M}_{H}$ drops with time, until a later
phase at $t\sim 20000r_{g}/c$ when the accretion becomes quasi-steady.

Figure 2: Time evolution of the mass accretion rate, $\dot{M}_{H}$ (top
panel), normalised magnetic flux, $\phi_{H}$ (middle panel), and the outflow
energy efficiency, $\eta_{H}$ (bottom panel, dotted line), and jet efficiency,
$\eta_{\rm{}jet}$ (bottom panel, solid line) of the three models a09rc0 (red),
a09rc10 (orange) and a09rc50 (blue). The jet efficiency is measured at
$r=100r_{g}$; the other quantities are measured close to the horizon. The
three vertical blue lines indicate the time at which the snapshots in Fig. 1
are taken. The light yellow highlight indicates the time interval over which
we compute time-average values of various quantities. The horizontal dashed
line in the middle panel indicates the critical value of $\phi_{H}\sim 40$ for
the disk to enter the MAD state. The smaller the value of $R_{c}$ and hence of
the initial specific angular momentum, the faster the magnetic fluxes
accumulate initially. However, the MAD state and very powerful jet are only
sustained in the model a09rc50 with the highest initial specific angular
momentum .

Interestingly, though magnetic fluxes accumulate faster in model a09rc0 and
a09rc10 in the initial phase, after $t\sim 20000r_{g}/c$, $\phi_{H}$ in these
two models decrease and eventually to below the MAD threshold. In contrast, we
see $\phi_{H}$ grows the slowest in model a09rc50 which has the largest
initial specific angular momentum. However, it can reach and sustain the MAD
state and demonstrate characteristic MAD behavior: high variability in
$\Phi_{H}$ and $\eta$ and the production of powerful jets with $\eta_{\rm
jet}>100\%$ from the Blandford-Znajek process ($\eta_{\rm jet}$ is plotted
using solid line in the bottom panel). We will give more analysis of the
behavior of magnetic flux in Section 3.3. In addition, there is a significant
correlation between $\eta_{H}$ and $\eta_{\rm jet}$, meaning most of the
energy produced in the accretion process is carried away by jet.

In the following sections, we compute the average values of a few physical
quantities in the window of $t=20000$–$50000r_{g}/c$. In this time window,
both gas accretion and $\phi_{H}$ have become quasi-steady in the inner
regions of the accretion flow.

###  3.2 The radial and angular structure of the flow

Figure 3:  The time-angle-integrated radial profiles of (a) density $\rho$,
(b) lab-frame radial velocity $v^{r}$ ($\rho$-weighted) over the free-fall
velocity $v_{\rm ff}$, (c) lab-frame angular velocity $\Omega$
($\rho$-weighted) over the relativistic Keplerian angular velocity
$\Omega_{\rm K}$, (d) geometric half-angular disk thickness $H/R$, and (e)
Lorentz factor of the jet $\Gamma$ ($\sigma$-weighted). The three different
colors denote different models: a09rc0 (red), a09rc010 (yellow) and a09rc50
(blue). The quantities in the inflow and outflow regions in (a)-(c) are
plotted using the solid (inflow) and dot-dashed (outflow) lines, respectively.
All models produce sub-Keplerian, sub-free-fall, and geometrically thick
disks, though the model a09r50 produces the thinnest and fastest rotating disk
among the three. The relativistic jets in the models have the peak Lorentz
factor $\Gamma=3-4$, at $r\sim 1000r_{g}$.

In this section we investigate the radial and inclination profiles of
accretion flow in the window of $t=20000$–$50000\ r_{g}/c$ when the inner
accretion flow becomes quasi-steady. Fig 3 shows the radial profiles of: (a)
the gas density, $\rho$, (b) the ratio of the lab-frame radial velocity,
$v^{r}=u^{r}/u^{t}$, to the free-fall velocity $v_{\rm ff}$ (in practice
$v^{r}/v_{\rm ff}=-u^{r}/u^{r}_{\rm ff}$ with $u^{r}_{\rm ff}$ given by Eqn.
9) , (c) the ratio of the lab-frame angular velocity, $\Omega=u^{\phi}/u^{t}$,
to the Keplerian velocity, $\Omega_{\rm K}=1/(r_{p}^{1.5}+a)$, where
$r_{p}=r\sin\theta$ is the cylindrical radius. (d) the geometric half-angular
thickness of the inflow, $H/R$, and (e) the Lorentz factor, $\Gamma=\alpha
u^{t}$, where $\alpha=1/\sqrt{-g^{tt}}$ is the time lapse. All quantities are
time-$\theta$-$\phi$-averaged. In addition, quantities in (b)-(c) are further
density-weighted to focus on the disk, while $\Gamma$ is further
magnetization-weighted to focus on the jet. We plot these quantities out to
$1000r_{g}$, where the shocks have propagated to, so $\rho$, $v^{r}$ and
$\Omega$ of the inflows decline rapidly around that radius.

One can see that the gas density and radial velocity structures of the
accretion flows in the three models are similar in the $r\lesssim 200r_{g}/c$
inflow equilibrium region. All models produce geometrically thick disks, with
model a09rc50 with the largest specific angular momentum producing the least
thick disk. The disks completely extend to the event horizon, and their
densities still approximately follow the free-fall profile $\rho\propto
r^{-1.5}$. The radial inflow velocities are 30-40% of the free-fall velocity
in the quasi-steady region. Moreover, despite the different initial
conditions, all three accretion inflows acquire some angular momentum and
rotate at the frequency of few $\times 0.1\Omega_{\rm{K}}$ near the horizon.
However, only the model a09rc50 can maintain a near-Keplerian rotation until
$r\sim 700r_{g}$, while the rotation speed in the other two models quickly
declines at larger radii. Furthermore, in all models winds are launched close
to the black hole and rapidly accelerated. They reach the maximum density
within $r=5\ r_{g}$ and terminal velocity within $r=100\ r_{g}$ . Lastly, the
jets launched in three models have similar acceleration profiles and
eventually reach $\Gamma=3-4$. The jet in the model a09rc50 is more steady and
slightly faster.

Fig. 4 shows the $\theta$-profiles of (a) $\rho$, (b) $v^{r}/v_{\rm ff}$ and
(c) $\Omega/\Omega_{\rm K}$, all evaluated on the surface at $r=100r_{g}$.
These quantities are $t$-$\phi$-$\rho$ averaged. Consistent with the $H/R$
result, the density profile of disk in model a09rc50 is the most concentrated
toward the mid plane. Indeed the a09rc50 disk most closely resembles the disks
produced in typical GRMHD simulations starting from torus set-ups, though
being geometrically thicker. The $v^{r}$ plot also clearly shows the inflow-
outflow boundary moves towards the mid-plane as the gas specific angular
momentum increases. The winds reach maximum radial velocities of $v_{r}\approx
0.1c$ right outside the jet in model a09rc0 and a09rc10, while in a09rc50
model the wind reaches a faster velocity of $v_{r}\approx 0.2c$. The wind in
a09rc50 also has the fastest angular velocity of $\Omega\approx
0.3-0.4\Omega_{\rm K}$, while the winds in model a09rc0 and a09rc10 rotate
slower with $\Omega\approx 0.1-0.2\Omega_{\rm K}$. Most interestingly, the
behavior of the angular velocity $\Omega$ in model a09rc50 and the other two
models are drastly different. Only in model a09rc50 the accretion inflow
develops a coherent rotation at all inclinations, while the gas in a09rc0 and
a09rc10 still orbit in different directions at different inclinations.

Figure 4: The time-$\phi$-averaged $\theta$-profiles of (a) $\rho$, (b)
$v^{r}/v_{\rm ff}$ ($\rho$-weighted), and (c) $\Omega/\Omega_{\rm K}$
($\rho$-weighted) for the surface at $r=100r_{g}$. The colors indicate the
three models are the same as in Fig. 3. In panel (b), solid lines indicate
inflow, and dot-dashed lines indicate outflow. In panel (c), solid lines
indicate clockwise rotation, and the dotted lines indicate counter-clockwise
rotation. The vertical dashed lines indicate the jet boundary with $\sigma=1$
in the three models. Only the accretion flow in model a09rc50 can develop
coherent rotation among the three. Model a09rc50 also produces the highest
radial and angular speed of the wind.

###  3.3 Retainment of magnetic flux in the accretion flow

We have shown that an accretion flow needs a minimum specific angular momentum
to sustain the MAD state, when it can possibly launch a powerful jet. In the
final part of the study, we focus on how the magnetic flux accumulates (or
leaks out) in these Bondi-like accretion flows.

We show the snapshots of the gas density and electromagnetic energy flux at
$t=35000r_{g}/c$ for the a09rc0 and a09rc50 models side to side in Fig. 5.
These snapshots demonstrate the characteristic behavior of the magnetic fluxes
during the quasi-equilibrium phase. It is apparent that model a09rc0 has a
highly non-axisymmetric accretion flow structure. Most strikingly, one can see
from panels (b1) and (b2) that even along the mid-plane close to the black
hole gas still does not flow inwards along all directions as seen in typical
disks, and there is a large azimuthal angle range persistently open for the
gas flowing all the way out. The electromagnetic energy fluxes in these out-
flowing regions are large, indicating that such regions provide a channel for
magnetic flux to leak out from the vicinity of the black hole. We suspect that
these “magnetic bubbles” are carried out by the wind or leak out due to
buoyancy, which leads to a low level of magnetic flux present near the black
hole. Model a09rc10 exhibits a similar behavior as model a09rc0 (not shown).
In contrast, model a09rc50 produces a more steady and axisymmetric disk
structure. There are still out-flowing regions at high latitudes; however,
along the mid-plane the outflows are patchy and not strong enough to disrupt
the inflow. It is very probable that some magnetic bubbles still form and flow
out for a short range, but later mix with the inflow and get shredded due to
gas rotation. The magnetic flux is then brought back by the inflow and
supplied to the vicinity of the black hole, which allows the magnetic flux to
accumulate and sustain the MAD state. More careful analysis of the interaction
of the magnetic flux with gas, which is out of the scope of this paper, will
be needed to fully understand the physic here.

Figure 5: Leakage of magnetic flux: The panels show snapshots at
$t=35000r_{g}/c$ in the a09rc0 model (top four panels) and a09rc50 model
(lower four panels) in the vertical plane (panels a and c) and equatorial
plane (panels b and d). The left column shows the gas density, and the right
column shows the electromagnetic energy flux. The inflow regions are shaded
with a light blue color, with blue contours marking the inflow-outflow
boundaries. The pink contours indicate the jet boundaries with $\sigma=1$. The
black lines in the left column are the density-weighted velocity streamlines,
and the white lines in the right column are the magnetic field lines. One can
see that in the a09rc0 model gas around the mid-plane can persistently flow
out at a large range of angles and carry away large magnetic fluxes. In
contrast, in the $R_{c}=50r_{g}$ run, more steady inflow structure can form,
which likely plays an important role in retaining the high level magnetic flux
in the inner accretion flow. (Full movies of these models can be seen at
https://tomkm.com/madbondi.html. )

##  4 Discussion

###  4.1 Comparison with previous GRMHD simulations of Bondi-like accretion
flows

It is worth mentioning that a few GRMHD simulations of Bondi-like accretion
flow have been performed recently, and we compare our results to theirs as
below. For example, Ressler et al. (2021) has conducted various simulations of
Bondi accretion flow focusing on understanding how the orientation between the
magnetic field and black hole spin axis plays a role in determining the
behavior of the accretion flow and the launching of the jet. Their simulation
with magnetic field orientation aligned with the black hole spin axis has the
configuration resembling our a09rc0 model the most. For this simulation, they
report that a powerful and time-varying jet is produced, while the jet power
in our a09rc0 model is relatively weak in the quasi-steady phase. We note that
the two models have different initial gas and magnetic field configurations.
More importantly, the accretion flows in the two models have been evolved for
significantly different periods, which likely has led to the discrepancy
between the results. The Ressler et al. (2021) simulation has run up to only
$t\sim 20000\ r_{g}/c$, during which period the accretion flow in our a09rc0
model is also in the MAD state, while the magnetic flux decreases to below MAD
level in later phases. More analysis is shown in Appendix B.

Furthermore, Kaaz et al. (2022) have studied the BHL accretion of magnetized
accretion onto a BH traveling through a uniform magnetized medium. Their
B100R53 model has the most analogous configuration to our a09rc0 model with
similar black hole spin parameters. In addition, this run has a similar
magnetic field strength in the wind compared to our initial disk. Their
simulation produces a sporadic jet, which is in good accordance with our
results. While their simulations with stronger initial magnetic field strength
in the wind can lead to sustained MAD accretion flow, there are some
fundamental differences between their setting and ours: First, there is a
steady supply of magnetic flux from the medium that the black hole is
traveling through in their simulations. Second, the relative motion between
the black hole (and its accretion disk) and the surrounding medium might help
retain the magnetic flux in a similar way as in our a09rc50 model.

###  4.2 Applications to various black hole accretion systems

In this section, we will briefly discuss how our results are relevant for a
few black hole accretion systems in which the accretion flows are believed to
have very low specific angular momenta.

Sgr A*: Sgr A*, the SMBH in the nuclei of our Galaxy, has a very low
luminosity $L_{\rm bol}\lesssim\ 10^{36}\text{erg}\ \text{s}^{-1}$ (Narayan et
al., 1998; Bower et al., 2019), indicating that it hosts a radiatively
inefficient accretion flow (see a review by Yuan & Narayan, 2014). Linear
polarization measurements of Sgr A* also constrains the accretion rate to be
as low as $10^{-9}-10^{-7}M_{\odot}\text{yr}^{-1}$ near the horizon (e.g.
Marrone et al., 2006). The accreting gas is likely supplied by the stellar
wind from the neighboured WR stars, which gives low cumulative angular
momentum (e.g., Quataert, 2004). Continuous accretion of stellar winds
together with their magnetic fields is sufficient to form a MAD flow as shown
by numerical simulations (Ressler et al., 2020). Furthermore, the recent Event
Horizon Telescope observation of Sgr A* favors the model that Sgr A* is
rapidly spinning and surrounded by a prograde accretion disk (The Event
Horizon Telescope Collaboration et. al., 2022).

The findings above imply that our simulations of magnetized, Bondi-like
accretion flow are very relevant for interpreting the properties of Sgr A*.
For example, suppose we take the accretion rate of
$10^{-9}-10^{-7}M_{\odot}\text{yr}^{-1}$ constrained from observation and use
the minimum jet efficiency of about $10\%$ from our simulated accretion flow
with zero angular momentum, we obtain a total jet power of $10^{36}-10^{38}\
\rm{erg\ s^{-1}}$, which is large enough to power Sgr A* radio and X-ray
emission, which has the isotropic luminosity of at most $\sim\ 10^{35}\
\text{erg}\ \text{s}^{-1}$ at radio and X-ray wavelengths (Morris & Serabyn,
1996; Baganoff et al., 2003). Future higher-resolution observations and
simulations/modeling of this type can allow us to better constrain the
accretion flow structure and jet properties in Sgr A*.

Long GRBs: The emergence of long GRBs is usually explained using the collapsar
model. In a collapsar, the core of a massive star has collapsed and formed a
spinning BH, while the stellar envelope material is still accreting onto the
BH, which launches a powerful, relativistic jet (Woosley, 1993). However,
there is a debate whether the envelope of the progenitor star could have
insufficient angular momentum to form a normal accretion disk (e.g. MacFadyen
& Woosley, 1999). Interestingly, our simulations show that even non-rotating
accretion flows can still launch a jet that is powerful enough to be the
central engine of a long GRB, which alleviates the problem, although the black
hole still needs to acquire enough angular momentum during the initial
collapsing process to gain a large spin.

TDEs: In TDEs, a star is tidally disrupted by a supermassive black hole (SMBH)
along a parabolic orbit. Interestingly, the circularization radius of the
stellar debris is usually a few tens of $r_{g}$ for typical stellar and black
hole parameters (Rees, 1988). Therefore, our work shows that the TDEs
accretion flows have the critical angular momentum needed to form MADs, if
sufficient magnetic fields are supplied to the gas. Indeed, three TDEs have
been observed to produce highly-beamed X-rays, which are believed to be
powered by jets (e.g., Bloom et al., 2011; Zauderer et al., 2013; Tchekhovskoy
et al., 2014), which is consistent with our prediction that accretion flow
with such angular momentum can still reach the MAD condition and launch
strong, relativistic jets. The majority of TDEs, however, produce thermal-like
emissions likely associated with the disks or winds (Dai et al., 2018; Gezari,
2021). This suggests that either most dormant SMBHs do not possess large spins
or most TDEs lack sufficient magnetic fluxes to form MADs.

Wind-fed BH HMXBs: High-mass X-ray binaries (HMXBs) are often wind-fed systems
instead of going through Roche-lobe overflow. In such a system, the accretion
flow around the compact object forms from the capture of the wind from the
donor giant star and therefore likely has a low specific angular momentum
(Shapiro & Lightman, 1976). Interestingly, all dynamically confirmed BH-HMXBs
have BH spin values near the maximum (e.g. Liu et al., 2008; Gou et al.,
2009). In addition, collimated, relativistic jets have been observed from some
BH-HMXBs (e.g., Cyg X-1, Stirling et al., 2001). While the full hydrodynamics
of wind-fed accretion can be complicated, our simulations of simplified Bondi-
like accretion show that indeed powerful jets can be formed from these systems
if there has sufficient magnetic fluxes. Future observations of the jet power
might even allow us to constrain the spins of BHs in HMXBs.

##  5 Summary

In this work, we have performed novel 3D GRMHD simulations of a magnetized
Bondi-like accretion flow around a rapidly spinning black hole. The three
models have different initial specific angular momenta ranging from zero to a
small value corresponding to that of a Keplerian disk circularized at
$50r_{g}$. The simulations results highlight the role that initial gas angular
momentum plays in forming MADs and producing powerful jets. We summarize the
main findings as follows:

  1. 1.

Even a Bondi accretion flow with zero gas angular momentum around a rapidly
spinning BH can still launch a relativistic jet magnetically. However, the jet
tends to be intermittent, wobbly, and relatively weak. The jet power can reach
about ten percent of the accretion power.

  2. 2.

For forming a very powerful and stable jet, the accretion flow has to achieve
and sustain the MAD state, which requires a critical specific angular momentum
of the accretion flow. However, the threshold is very small, which corresponds
to a circularization radius between $10r_{g}$ and $50r_{g}$. Under such
circumstances, the jet efficiency can approach and sometimes exceed 100%.

  3. 3.

When the angular momentum of the accretion flow does not reach this critical
value, the accretion inflow is unsteady even close to the black hole. Gas can
have different rotation directions at different inclinations. Furthermore, gas
can flow in along certain azimuthal angles around the mid-plane but
persistently flow out to large radii at other azimuthal angles. These outflow
regions contain a large amount of magnetic fluxes, providing a channel for
them to leak out. As a result, the accretion flow cannot sustain the MAD
state.

  4. 4.

In contrast, when the threshold of specific angular momentum has been reached,
a more steady accretion inflow structure can form around the black hole with
coherent rotation and persistent inflow along the mid-plane in the inner
region. Therefore, less magnetic fluxes leak out, and we suspect that the
rotating gas inflow can further shred any outflowing magnetic bubbles and
bring back some of the magnetic fluxes contained in them. This allows the MAD
condition to be reached and sustained.

We emphasize that we have only tested the simplified scenario where the gas
angular momentum, black hole spin and magnetic field axes are all aligned,
while the realistic configurations can be more complicated. Furthermore, the
treatment of cooling and radiative processes can be improved by conducting
simulations using codes incorporating radiative transfer physics, which is
particularly important in the super-Eddington accretion regime. Nonetheless,
this work has disclosed that accretion flows around spinning black holes only
need to possess small specific angular momenta to become MADs and launch very
powerful jets. The results have important consequences for understanding
various black hole accretion systems with low gas angular momentum, such as
Sgr A* and other low-luminosity AGNs, TDEs, GRBs and certain HMXBs, and in
particular, what sets the jet power in such systems.

We thank S. Woosley for useful discussions. TK and LD acknowledge the support
from the Hong Kong Research Grants Council (HKU27305119, HKU17305920) and the
National Natural Science Foundation of China (HKU12122309). We acknowledge the
computational support from NSF via XSEDE resources and from the supercluster
and the HPC computing facilities offered by ITS at HKU and the Tianhe-2
supercluster.

##  Appendix A Simulation initial setup and parameters

We conduct three simulations with different initial gas specific angular
momenta. In each simulation, the maximum specific angular momentum of the gas
is set by the characteristic specific angular momentum at the circularization
radius calculated using Eqn. 11, with $R_{c}$ = 0, 10, or 50 $r_{g}$
respectively. For our chosen magnetic flux distribution, a magnetic field line
passing through a point $(r_{0},\theta_{0})$, with
$\Psi=\Psi_{0}\equiv\Psi\left(r_{0},\theta_{0}\right)$, will cross the
equatorial plane at the “footpoint” radius of

$R_{\rm fp}=r_{0}(1-|\cos\theta_{0}|).$ (A1)

The initial specific angular momentum distribution is set as:

$l=\begin{cases}l_{\rm max}&\text{for}\ R_{\rm{}fp}>R_{\rm{}solid}\\\ l_{\rm
max}\times R_{\rm fp}^{2}/R_{\rm{}solid}^{2}&\text{for}\ R_{\rm{}fp}\leq
R_{\rm{}solid}\end{cases}$ (A2)

where $R_{\rm{}solid}\equiv 100r_{g}$ and $R_{\rm fp}$ is given by Eqn. A1.
Using this setup, close to the black hole, gas rotates approximately as a
solid body (i.e., $u^{\phi}$ is roughly constant), so we avoid having a
nonphysically high specific angular momentum. Also, outside the equator, the
value of $l$ is propagated along the magnetic field lines. We plot the initial
specific angular momentum, together the the magnetic field lines, in Fig. A.1.

A few numerical and physical parameters of the simulations are listed in table
A.1. Besides various parameters discussed in the main text, we further show
the value of $s$, the BH dimensionless spin-up parameter, which is defined as:

$s\equiv\frac{{\rm d}j}{{\rm
d}t}\frac{M_{\rm{}BH}}{\langle\dot{M}_{H}\rangle_{t}}=-j-2a(1-\eta)$ (A3)

where $j=\frac{\int T^{r}_{\phi}\rm
dA_{\rm\theta\phi}}{\langle\dot{M}_{H}\rangle_{t}}$ is the specific angular
momentum flux of the BH. Positive values indicates the spin-up of the BH. We
can see in all three runs the jet has extracted rotation energy from the BH.

Figure A.1: The initial conditions of the gas specific angular momentum $l$ and magnetic field of the three runs (a) a09rc0; (b) a09rc10; (c) a09rc50, respectively. The color shows the initial specific angular momentum. The black lines indicate the initial magnetic field lines. Table A.1: Parameter Summary of the simulations  Model |  $R_{c}$ ($r_{g}$) | $N_{r}$ | $N_{\theta}$ | $N_{\phi}$ | $\phi_{\rm{}H}$ | $\eta_{\rm{}H}(\%)$ | $\eta_{\rm{}j}(\%)$ | $\eta^{\rm{}EM}_{\rm{}j}(\%)$ | $s$
---|---|---|---|---|---|---|---|---|---
a09rc0 | 0 | 288 | 128 | 64 | 26.8 | 19.9 | 17.1 | 13.0 | -12.0
a09rc10 | 10 | 240 | 128 | 64 | 23.2 | 10.8 | 9.8 | 6.8 | -10.5
a09rc50 | 50 | 280 | 128 | 64 | 45.1 | 76.1 | 61.7 | 52.6 | -35.2

##  Appendix B Histogram of MAD parameter

Fig. B.1 shows the histograms of the instant dimensionless magnetic flux
$\phi_{\rm{}H}$ in all snapshots of the three runs, either over the early
phase of $t=(0-20000)r_{g}/c$ or over the late phase of
$t=(20000-50000)r_{g}/c$. The accretion flow has established inflow
equilibrium out to $r\gtrsim 200r_{g}$ only in the late phase, during which
period we have evaluated the results.

One can see that the accretion flow has distinct properties at early and late
phases. For model a09rc0 and a09rc10, the inner accretion flow are MAD or
nearly MAD before $t\sim 20000$ with $\phi_{H}$ varying around 40. However,
the magnetic flux decays with time and the accretion flow is no longer MAD in
the quasi-equilibrium phase. Model a09rc50 shows an opposite behavior. The MAD
state is slowly developed and then maintained in the quasi-equilibrium phase.

These results demonstrate the importance of conducting GRMHD simulations for a
sufficiently long duration and probably explains some differences in the
results obtained by Ressler et al. (2021) and this work.

Figure B.1: The histogram of the dimensionless magnetic flux $\phi_{\rm{}H}$
around the horizon $\phi_{\rm{}H}$ of the three models: (a) a09rc0, a09rc10
and a09rc50. We sample $\phi_{\rm{}H}$ in the two different time ranges: 1)
The early phase $t=(0-20000)r_{g}/c$, plotted in red, and 2) the late, quasi-
equilibrium phase $t=(20000-50000)r_{g}/c$ (when we computed the averaged
values of the quantities), plotted in blue. The orange and green dashed-dot
lines are the log-normal distribution functions fitting on the profiles of
$\phi_{H}$ in the early and late phases respectively.

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