# Magnetically Ejected Disks: Equatorial Outflows Near Vertically Magnetized
Black Hole

Vladimír Karas  [  ]1,a Kostas Sapountzis  [  ]2 & Agnieszka Janiuk  []2

1Astronomical Institute  Czech Academy of Sciences  Boční II 1401  [1]
CZ-14100 Prague  Czech Republic
2Center for Theoretical Physics  Polish Academy of Sciences  Al. Lotnikow
32/46  [1] P-02-668 Warsaw  Poland
<EMAIL_ADDRESS>
###### Abstract

Black holes attract gaseous material from the surrounding environment. Cosmic
plasma is largely ionized and magnetized because of electric currents flowing
in the highly conductive environment near black holes; the process of
accretion then carries the magnetic flux onto the event horizon, $r\simeq
R_{+}$. On the other hand, magnetic pressure acts against accretion. It can
not only arrest the inflow but it can even push the plasma away from the black
hole if the magnetic repulsion prevails. The black hole does not hold the
magnetic field by itself.

In this contribution we show an example of an equatorial outflow driven by a
large scale magnetic field. We initiate our computations with a spherically
symmetric distribution of gas, which flows onto the domain from a large
distance, $r\gg R_{+}$. After the flow settles in a steady (Bondi) solution,
we impose an axially symmetric configuration of a uniform (Wald) magnetic
field aligned with the rotation axis of the black hole. Then we evolve the
initial configuration numerically by employing the MHD code that approaches
the force-free limit of a perfectly conducting fluid.

We observe how the magnetic lines of force start accreting with the plasma
while an equatorial intermittent outflow develops and goes on ejecting some
material away from the black hole.

###### keywords:

black hole physics – magnetic fields – accretion

##  1 Introduction

Accretion is ubiquitous process in the Universe. By gradual accretion, various
components of highly diluted environment are gathered and brought onto cosmic
bodies – planets, stars, even galaxies (see, e.g., Fox & Davé, 2017; Shakura,
2018). Accretion is driven primarily by action of gravitational and
electromagnetic forces. Gravity of the central body acts at long range and it
attracts gas and dust, whereas the electric and magnetic components can act in
a complex, mutually interrelated manner; they can cause either attraction or
repulsion of the plasma, depending on the conditions.

Astrophysical plasmas are characterized by their high conductivity which
ensures that the force-free condition is typically an excellent approximation
to describe the cosmic environment (Beskin, 2010). Moreover, in magnetospheres
of compact objects like neutron stars and black holes there are regions where
the energy density of the electromagnetic field greatly exceeds the inertial
(kinetic, rest-mass, and thermal) energy of matter. Plasma motions follow the
evolving magnetic field lines. Force-free electrodynamics describes
magnetically dominated relativistic plasma as long as the inertial forces can
be neglected; while this assumption is correct, in very diverse circumstances
it becomes gradually violated in case of very low ionisation and low
temperature, where the dissipation effects play a role, and for ultra-
relativistic acceleration near the light cylinder, where the particle mass is
important.

In the limit of vanishing magnetic field and sufficiently high density (short
mean free path) the hydrodynamical approach is adequate. The best-known
analytical framework then describes the stationary, spherically symmetric
inflow, a.k.a. Bondi solution (Bondi, 1952), where the actual form of the flow
is determined by the boundary conditions at infinity and at the black hole
horizon. This has been generalized in several ways; in particular, Silich et
al. (2008) include the effect of additional source of energy from stars of the
Nuclear Star Cluster, which is relevant for many nuclei containing Nuclear
Star Clusters, including the Galactic center source Sgr A* (Schödel et al.,
2014). On the other hand, in the limiting case of zero density (electro-
vacuum) the solution is described by the source-free coupled Einstein-Maxwell
equations. These are tractable only under very constraining assumptions and
symmetries, however, the problem can be simplified for electro-magnetic fields
that are weak (albeit non-vanishing) in comparison with the gravitational
field (the assumption valid in the vicinity of astrophysical black holes).
Electro-magnetic test field solutions on the fixed background of Kerr metric
then provide an adequate description of magnetic fields in interaction with
the black hole gravity (Wald, 1974; King et al., 1975).

In this contribution we are interested in a gradually evolving structure of
magnetic field, as the system goes over from the initial, homogeneous solution
to the interaction with the force-free magnetosphere near an accreting Kerr
black hole (Frank et al., 2002; Kato et al., 2008). We deliberately impose
axial symmetry along the black hole rotation axis for the magnetic field and
the inflowing medium (we employ a two-dimensional scheme). Although this
constraint will have to be relaxed to describe astrophysically realistic
systems, we want to reveal the transition from the initial state of magnetic
lines running around the black hole, i.e., the magnetic flux being expelled
out of the horizon (partially in the case of non-rotating or moderately
rotating black hole, dimension-less spin parameter $|a|<1$, and completely in
the case of extreme rotation, $a=1$; see Bičák & Dvořák, 1976). As the force-
free plasma starts inflowing with spherical symmetry at the initial
configuration, the Meissner expulsion is immediately diminished and, at later
stages, the magnetic lines start to produce reconnection regions in the
equatorial plane. Subsequently, localized blobs emerge and they are eventually
ejected away from the black hole due to the magnetic pressure.

We build our study following a series of works that have been previously
published by various authors. In particular, Komissarov & McKinney (2007)
explored the magnetized, rotating black holes embedded in the plasma. By
employing the general-relativistic magneto-hydrodynamics 2D HARM code (Gammie
et al., 2003; McKinney, 2006; Sapountzis & Janiuk, 2019) they found that the
Meissner expulsion indeed disappears due to the presence of accreting medium
even in the case of (almost) maximally rotating black hole. Penna (2014) gave
arguments to understand the essence of the Meissner–type magnetic field
expulsion near black holes. Various authors (Pan & Yu, 2016; Gong et al.,
2016; Camilloni et al., 2020) studied the analytical properties of force-free
black-hole magnetospheres especially in the context of jets emerging from the
vicinity of the ergosphere. Nathanail & Contopoulos (2014) and East & Yang
(2018) noticed the formation of current sheets near a black hole immersed in a
magnetized plasma. Some results suggest that the role of ergosphere is
essential in producing the plasma structures and ejecting matter in the force-
free medium (Blandford & Znajek, 1977).

##  2 The model set-up

###  2.1 Strong gravitational and weak magnetic fields

Gravitational field is described by Kerr metric, which can be written in the
well-known Boyer-Lindquist coordinate system $(t,r,\theta,\phi)$ (Misner et
al., 1973; Chandrasekhar, 1983; Wald, 1984). This spacetime is asymptotically
flat and it obeys the axial symmetry about the rotation axis and stationarity
with respect to time; the singularity is hidden below the event horizon. The
mass $M$ of the black hole is concentrated in the origin of the coordinate
system. The spin parameter $a$ of the Kerr metric describes its rotation; the
condition about the presence of the outer event horizon at a certain radius,
$r=R_{+}$ (where the horizon encompasses the singularity) leads to the maximum
value of the dimensionless spin rate: $|a\leq 1|$. The solution can be then
written in the form of the metric element (Misner et al., 1973; Chandrasekhar,
1983)

${\rm d}s^{2}=-\frac{\Delta\Sigma}{A}\,{\rm
d}t^{2}+\frac{\Sigma}{\Delta}\,{\rm d}r^{2}+\Sigma\,{\rm
d}\theta^{2}+\frac{A\sin^{2}\theta}{\Sigma}\;\left({\rm d}\phi-\omega\,{\rm
d}t\right)^{2},$ (1)

where $\Delta(r)=r^{2}-2r+a^{2}$, $R_{+}=1+\sqrt{1-a^{2}}$,
$\Sigma(r,\theta)=r^{2}+a^{2}\cos^{2}\theta$,
$A(r,\theta)=(r^{2}+a^{2})^{2}-{\Delta}a^{2}\sin^{2}\theta$,
$\omega(r,\theta)=2ar/A(r,\theta)$. Dimension-less geometrical units are
assumed with the speed of light $c$ and gravitational constant $G$ set to
unity. In physical units the gravitational radius is thus equal to
$R_{\rm{}g}=c^{-2}GM\approx 4.8\times 10^{-7}M_{7}\,$pc; the corresponding
light-crossing time-scale $t_{\rm{}g}=c^{-3}GM\approx 49\,M_{7}\,$sec, where
$M_{7}\equiv M/(10^{7}M_{\odot})$.

Kerr metric is a solution of Einstein’s equation for the gravitational field
of a rotating black hole in vacuum. Even in the case of strongly magnetized
gaseous environment around the black hole the contribution of an
astrophysically realistic magnetic energy to the space/time curvature is
negligible. We can thus neglect its effect on the metric terms and assume a
weak-field limit on the background of Kerr black hole; the space-time metric
is not evolved in our scheme. In order to initiate the numerical code we can
employ an initially uniform magnetic field (Wald, 1974; Bičák et al., 2007),
which is fully described by two non-vanishing components of the four-
potential,

$\displaystyle A_{t}$ $\displaystyle=$ $\displaystyle
Ba\Big{[}r\Sigma^{-1}\left(1+\cos^{2}\theta\right)-1\Big{]},$ (2)
$\displaystyle A_{\phi}$ $\displaystyle=$ $\displaystyle
B\Big{[}{\textstyle\frac{1}{2}}\big{(}r^{2}+a^{2}\big{)}-a^{2}r\Sigma^{-1}\big{(}1+\cos^{2}\theta\big{)}\Big{]}\sin^{2}\theta,$
(3)

in dimension-less Boyer-Lindquist coordinates and $B$ is the magnetic
intensity of the uniform field far from the event horizon. The magnetic field
(and the associated electric component) are generated by currents flowing in
the accreted medium far from the black hole, as the latter does not support
its own magnetic field. The set of two non-vanishing four-potential vector
components defines the structure of the electromagnetic tensor,
$F_{\mu\nu}\equiv A_{[\mu,\nu]}$; by projecting onto a local observer frame
one then obtains the electric and magnetic vectors ${\bf\it E}$ and ${\bf\it
B}$.111Let us note that typical cosmic plasmas are ionized and perfectly
conducting, and so the approximation of force-free electromagnetic action is
justified at high accuracy (e.g. Somov, 2012): ${\bf\it E}+{\bf\it v\times
B}=0$, where ${\bf\it E}$ and ${\bf\it B}$ are electric and magnetic
intensities, ${\bf\it v}$ is velocity of the plasma. Magnetic field is thus
frozen in the plasma. However idealized the initial configuration may be, the
numerical solution rapidly evolves in a complex entangled structure, with
field lines turbulent within the accreting medium and more organized in the
empty funnels that develop outside the fluid structure.

###  2.2 Two limiting cases for the initial distribution of plasma

The plasma forms an accretion disk or a torus residing in the equatorial
plane, so that the axial symmetry is maintained. In a non-magnetized (purely
hydrodynamical) limiting case one can find the classical solution for the
density distribution $\rho\equiv\rho(r,z)$ and pressure $P\equiv P(r,z)$, and
the geometrical shape $H\equiv H(r)$ of a non-gravitating barytropic torus
$P=K\rho^{k}$. Introducing enthalpy of the medium, $W(P)\equiv\int dP/\rho$
and setting $P_{\rm in}=P_{\rm out}=0$ at the inner and the outer edges of the
density distribution (Abramowicz et al., 1978; Kozlowski et al., 1978)

$W_{\rm out}-W_{\rm in}=\int_{R_{\rm in}}^{R_{\rm out}}\,\frac{l(R)^{2}-l_{\rm
kep}(R)^{2}}{R^{3}}\,dR=0,$ (4)

where $l_{\rm kep}(R)$ is the radial profile of the Keplerian angular momentum
density in the equatorial plane. Several properties of this solution are worth
mentioning (Abramowicz, 1971): (i) The level surfaces of functions $P,$
$\rho,$ and $W$ coincide; (ii) If the torus boundary $P=0$ forms a closed
surface, the torus center is defined by where $dP/dR=0,$ the pressure is
maximum; (iii) The shape of the torus can be found by integrating the vertical
component of the Euler equation.

Above a certain critical value, $W>W_{c},$ the torus forms a stable
configuration (see the shaded region), while for $W<W_{c}$ matter overflows
onto the central object even if we neglect viscosity. This behaviour resembles
the Roche lobe overflow in binary systems, however, here it is a consequence
of the non-monotonic radial dependence of the Keplerian angular momentum in
the relativistic regime near the black hole (Abramowicz et al., 1980; Penna et
al., 2013). Whereas the material inflowing from atmosphere of a primary
component of the binary system concentrates near the equatorial plane and
naturally forms the torus, in the case of a single, isolated central black
hole matching the inner (toroidal) structure to the outer reservoir of matter
depends on many circumstances at the outer boundary region. This is also the
case of super-masive black holes residing in nuclei of galaxies, which are fed
by interstellar medium from a surrounding (spheroidal) nuclear star cluster
and the galaxy bulge; flattening of the structure is a parameter that can vary
from disk-type equatorial inflow up to perfectly spherical (Bondi-type)
inflow/outflow solution (Silich et al., 2008; Różańska et al., 2017).

Unlike the above-discussed toroidal configuration, let us now assume that the
angular momentum of accreted fluid is negligible and its velocity has a non-
vanishing component only in the radial direction. At large radius, in our case
suitable as a boundary condition, we are allowed to consider the problem
within the framework of spherically symmetric Newtonian inflow with
$v_{r}=v<0$ (positive $v$ would correspond to a symmetrical problem of an
outflow or a wind). The equation of continuity gives $4\pi
r^{2}\rho(r)v(r)=-\dot{M}$, where the constant on the right-hand side has a
meaning of mass accretion rate. In the Euler equation, density of the external
force ${\bf\it f}$ has only the radial component, $GM\rho/r^{2}=-f_{r}(r)$,
and so we can write

$v\,\frac{dv}{dr}+\frac{1}{\rho}\frac{dP}{dr}+\frac{GM}{r^{2}}=0.$ (5)

Introducing the sound speed by $dP=c_{\rm s}^{2}(r)\,d\rho$ the Euler equation
can be manipulated into the well known form

$\frac{1}{2}\left(1-\frac{c_{\rm
s}^{2}}{v^{2}}\right)\frac{d\,v^{2}}{dr}=-\frac{GM}{r^{2}}\left[1-\frac{2c_{\rm
s}^{2}r}{GM}\right].$ (6)

The solutions can be classified according to their behaviour at the sonic
point, where the medium flows at the speed of sound, $r_{\rm s}=GM/2c_{\rm
s}^{2}(r_{\rm s})$. For the spherical adiabatic accretion one can find six
qualitatively different solutions to the above equations in the $(v,r)$-plane.
Inflows (accretion flows) and outflows (ejection or stellar wind) are both
possible; which mode is realized in a particular situation depends on boundary
conditions (in our case, only accretion is possible at the inner boundary).

Figure 1: The accretion rate on the horizon of the black hole initially grows
and then saturates as the steady state has been reached. The curves are
normalized to an arbitrary value $\dot{M_{0}}$ and parameterized by three
values of the dimension-less spin $a$. Time is in geometrical units and scaled
with the black hole mass $M$.

Let us note that the assumptions about constancy of $l(r)$ and the vanishing
magnetization, $\beta=\infty$, are astrophysically unrealistic but they are
useful to simplify calculations and allow an analytical insight. The
magnetization $\beta$ parameter is taken here as ratio of total hydrodynamic
pressure to the magnetic pressure within the magnetized fluid, i.e.,
$\beta\equiv U_{\rm tot}/U_{\rm mag}$. Realistic models must relax the extreme
assumptions about the strict geometrical symmetry and stationarity to allow
the system to evolve in time, which is a crucial aspect of the mutual
interaction between different components.

As mentioned above, to overcome some of the limitations we adopt the numerical
scheme by the HARM code (Gammie et al., 2003; McKinney, 2006).222In this paper
we explore an axially symmetric configuration, which is obeyed by all
components of the system: the gravitational field of the rotating black hole,
the interacting electromagnetic field of external origin, and the fluid
surrounding the black hole. We thus employ the two-dimensional version of the
code. Imposing the axial symmetry allows us to examine the role of magnetic
expulsion from the horizon of extreme Kerr black hole and to observe how this
effect is reduced by the accreted plasma. At the same time we are not confused
by non-axisymmetric effects, which are known to reduce the magnetic expulsion,
too. It will be interesting to generalize our discussion to a non-axisymmetric
configuration of an oblique (inclined with respect to the rotation axis)
magnetic field. This allows us to explore the parameter space of the system. A
fraction of the material injected spherically at the outer boundary of the
computational domain remains bound and it forms an accretion torus or an
accretion disk associated with the central black hole. This occurs at a
relatively small radius where relativistic effects play a crucial role and
decide whether the gas falls onto the black hole or becomes redistributed and
ejected (we do not include radiative cooling in the present work). The
interplay between the frozen-in magnetic field and the infalling plasma that
thermalizes the mechanical energy and generates additional overpressure,
eventually determines the ratio between accretion and ejection.

Figure 2: Gradually evolving accretion rate onto the magnetize black hole with
the initial set-up of the Bondi inflow and the imposed Wald magnetic field for
three values of spin $a$. The magnetization parameter $\beta\simeq 0.1$
(magnetically dominated medium) at $t=0$. Long-term progress is shown in the
left panel; a detail of the initial phase in the right panel.

##  3 Results

###  3.1 Mass accretion rate

We set the outer boundary of the computational domain at the radius
$10^{3}R_{\rm g}$, where the inflow is purely radial at initial time. The
inner boundary is set at $\simeq 0.65R_{\rm g}$, i.e. a fraction of
gravitational radius and also inside the horizon radius for the corresponding
value of spin $a$. The grid domain has been resolved at $600\times 512$ points
in $(r,\theta)$ coordinates and the polytropic index set to $k=4/3$. We use
the magnetic intensity $B$ of the Wald field to determine the plasma parameter
$\beta$ (in our notation, $\beta$ closer to zero corresponds to a more
magnetized plasma).

In order to initialize the computation we employ the hydrodynamic (non-
magnetized), purely spherical inflow. We set $\beta\rightarrow\infty$ and
allow the inflow to build a steady-state Bondi accretion at a certain level of
$\dot{M}$ (see Fig. 1). Once the inflow stabilizes to a quasi-steady state
inflow, we impose the large-scale Wald magnetic field along the rotation axis,
which is then evolved further. Because of the perfect conductivity and the
force-free approximation (apart from the effective small-scale numerical
dissipation), the magnetic field-lines remain attached to plasma. However, the
evolution of the system can be strongly altered if the magnetic field is
strong enough, so that its repulsive tendency halts accretion. This effect is
governed by the magnetization $\beta$-parameter, which is not uniform across
the computational domain and changes in time. While the accretion is not much
influenced in the limit of negligible magnetization ($\beta\gg 1$), where the
gravitational attraction of the black hole prevails, in the case of
equipartition between the magnetic and hydrodynamic pressure ($\beta\approx
1$) near the horizon the inflow is partially diverted into an outflow and the
accretion rate is diminished (see Fig. 2). Let us note that the mass and spin
of the black hole are not updated during the simulation because the amount of
accreted material is tiny compared to the black hole mass.

Figure 3: Graph of inverse accretion rate $1/\dot{M}$ exhibits four phases of
the magnetized flow evolution. The initial configuration starts with the Bondi
spherical inflow from the outer boundary of the integration domain. The
central black hole rotates with the Kerr spin parameter $a$ and it is
magnetized by the Wald uniform magnetic field $B$. Spherical symmetry of the
inflow is quickly lost by its interaction with the magnetic field but the
axial symmetry is imposed in our 2D computations. In case of a rapidly
rotating black hole, the magnetic flux vanishes initially (Meissner effect)
but it starts growing with accretion of the plasma. Part of the inflowing
material is diverted to an outflow along the equatorial plane and accelerated
by reconnection events (they are caused by numerical resistivity); the
resulting accretion rate thus gradually drops and it exhibits some random
glitches at later stages of its temporal evolution (see the text for further
details). Figure 4: The magnetic flux $\Phi(t)$ (in arbitrary code units)
across a hemisphere on the black hole horizon. The initially frozen-in
magnetic flux grows (in absolute value) due to accretion of plasma. At later
stages the flux starts decreasing as the magnetic intensity decreases and the
field eventually escapes to radial infinity.

Figure 5: The initial phase of the flux evolution from the previous figure is
shown in the semi-logarithmic scale and the dimensionless units (scaled by the
typical values). At time zero the magnetic flux of the Wald configuration
vanishes for Kerr black hole at extreme rotation ($a\rightarrow 1$) in
agreement with the Meissner effect. The magnetic expulsion is eliminated as
soon as some plasma gets accreted, even in the purely 2D configuration with
the imposed axial symmetry.

In the strongly magnetized case the accretion rate drops because an outflow
develops in the equatorial plane, where the field lines expand radially and
carry plasma with them. The field line expansion is not uniform across the
integration domain, leading to the distortion of the field lines and
eventually to the formation of the current sheet and the reconnection events
happening in the equatorial plane (notice the associated spikes around
$t=180$). It is illustrative to plot the inverse accretion rate, where we can
clearly identify four different phases of the system evolution (see Fig. 3).
The section denoted A is where the initial Bondi accretion prevails; in part B
we notice the formation of the current sheet followed by a rarefaction phase
of the field lines in part C. Here, more matter is expelled from the black
hole and it eventually ends up in a region of much lower density. In the D
phase the reconnection events occur and the outflow accelerates near the
equator.

Figure 6: Four time steps (from top left to bottom right panels) with the
magnetic field lines and colour-coded plasma density near a rapidly rotating
($a=0.99$) black hole with $B=8$. Magnetic field lines are expelled out of the
horizon at $t=0$ (before plasma starts to be accreted), but they start
crossing the horizon and accreting with the plasma as time passes. We
intentionally select the magnetically dominated system at the initial stage,
$\beta(t=0)\ll 1$, which is expected to support the Meissner expulsion. Still,
once some plasma arrives at the event horizon, we observe rapid accretion of
the magnetic flux which cannot inhibit the radial motion. At late stages the
field lines adopt more radial configuration near horizon and they eventually
induce the ejection of plasma in the equatorial plane. The logarithmic scale
of density is shown on the colour bar in arbitrary units. Time (growing in the
three snapshots from left to right) corresponds to the code units.

Figure 7: Similar as in the previous figure but plotted here over a larger
integration domain and longer period of time. Ejection of equatorial material
is observed in the form of plasmoids, which represent a disk or a ring of
outflowing material in the adopted 2D approximation. In the vicinity of the
rotating black hole ergosphere the frame-dragging effect acts on the plasma as
well as the magnetic field lines. As a result of the frame-dragging mechanism,
the outflowing material adopts a toroidal component of the orbital motion.

###  3.2 Magnetic flux across a hemisphere

In order to reveal the changes of the magnetic field near the black hole, we
study the evolution of the magnetic flux inflowing into the hemisphere located
on the horizon. As mentioned above, the initial configuration is the vacuum
solution of the uniform flux tube oriented in the direction parallel to the
rotation axis, i.e. the Wald solution, but this configuration starts quickly
changing once the inflowing plasma arrives in the domain.333Some more
complicated configurations were explored by Kološ & Janiuk (2020).
Interestingly, these authors found that a parabolic magnetic field also
develops in the accretion torus funnel around the vertical axis, for any
initial magnetic field configuration. As also mentioned above, the initial
configuration for the plasma inflow is the Bondi solution. As soon as a field
line enters the ergosphere, it has to terminate at the event horizon.
Therefore, the magnetic field lines in a force-free magnetosphere are not
expelled by even extreme rotation of the black hole. Figure 4 shows the
magnetic flux as a function of time and Fig. 5 exhibits a more detailed view
of the brief initial period. Finally we vary the value of the Wald magnetic
intensity $B$ parameter to obtain more initially magnetized case (lower plasma
$\beta$). Notice that the code itself poses a limit $\beta>10^{-4}$ under
which it starts introducing artificial density floor in order to avoid
numerical integration problems. Figures 6–7 show several snapshots of the
magnetic field and plasma density at different resolution. An equatorial
outflow forms at late stages of the system evolution and it drives plasmoids
in the outward direction.

##  4 Conclusions

Astrophysical black holes can be detected and their parameters constrained by
observations in the electromagnetic domain only if the accretion process takes
place, lights up the system, and produces the characteristic spectral features
and variability of the emerging radiation signal. However, the properties of
the cosmic environment vary in very broad range: from magnetically dominated
(almost) vacuum filaments of the organized field lines to the force-free field
lines frozen in the accreting medium.

In the present contribution we were interested to explore the transition
between two extreme states: from the initial configuration, where the
Wald–type uniform field (aligned with the black hole rotation axis) comes to
the contact with the Bondi–type radial inflow solution. The fluid drags the
field lines onto the black hole, and simultaneously becomes influenced and
partly expelled by the evolving magnetic field until the final state is
reached after many dynamical periods. Let us note that the initial vacuum,
ordered, homogeneous, parallel to the rotation axis magnetic field is an
idealized situation. It has been frequently employed in order to define the
starting configuration and we also use it as a test bed solution that can
represent a rotating black hole embedded into a large-scale (exceeding the
size of the horizon) magnetic filament, which allows us to model the rapid
disappearance of the magnetic expulsion once the conducing medium starts to be
accreted. An interesting development emerges as the magnetic lines are bent in
the radial direction near the horizon and they start reconnecting in the
equatorial plane, thus accelerating the outflow in the direction perpendicular
to the rotation axis. In fact, the accretion of magnetic field lines onto the
black hole starts the process of their bending from the initial Wald
configuration to the split-monopole topology, which leads to the rapid
disappearance of the Meissner expulsion in our system.

We can suggest that the resulting equatorial outflow is possible thanks to the
fact that the Meissner effect does not operate in the magnetosphere filled
with plasma.

Let us note that accretion disk backflows have been observed in various
circumstances including the simulations of accreting black holes and stars
(see Kluzniak & Kita, 2000; Mishra et al., 2020). In several aspects the
system discussed in our present work is rather distinct: it does not include a
magnetic star as a source of dipole-like magnetic field (we considered a black
hole in the centre, which was magnetized by external currents), neither an
equatorial accretion disk as the initial condition (Bondi spherical inflow was
assumed as the condition at the outer boundary). Indeed, we suggest that the
backflows are rather generic features that can occur in different accreting
systems.

Abandoning the axial symmetry will be the next step towards a more realistic
description. Also, once an oblique magnetic field (inclined with respect to
the rotation axis) and a twisted (non-spherical as well as non-axisymetric)
accretion flow are considered, we can expect the outflowing plasmoids to be
scattered in a wide range of directions.

The authors thank the Czech Ministry of Education Youth and Sports Mobility
project No. 8J20PL037 to support the Czech-Polish scientific cooperation, and
the Czech Science Foundation – Deutsche Forschungsgemeinschaft collaboration
project No. 19-01137J on processes around supermassive black holes. KS and AJ
were supported by grants No. 2016/23/B/ST9/03114 and 2019/35/B/ST9/04000 from
the Polish National Science Center. We acknowledge computational resources of
the Warsaw ICM through the grant Gb79-9. The authors thank the referee for
helpful comments and a number of suggestions to improve our text.

## References

  * Abramowicz (1971) Abramowicz, M. A. 1971, Acta Astron., 21, 81
  * Abramowicz et al. (1980) Abramowicz, M. A., Calvani, M., & Nobili, L. 1980, ApJ, 242, 772
  * Abramowicz et al. (1978) Abramowicz, M., Jaroszynski, M., & Sikora, M. 1978, A&A, 63, 221
  * Beskin (2010) Beskin, V. S. 2010, MHD Flows in Compact Astrophysical Objects: Accretion, Winds and Jets, Astronomy and Astrophysics Library. Berlin, Heidelberg: Springer
  * Bičák & Dvořák (1976) Bičák, J. & Dvořák, L. 1976, General Relativity and Gravitation, 7, 959
  * Bičák et al. (2007) Bičák, J., Karas, V., & Ledvinka, T. 2007, in Black Holes from Stars to Galaxies – Across the Range of Masses, 238, 139
  * Blandford & Znajek (1977) Blandford, R. D. & Znajek, R. L. 1977, MNRAS, 179, 433
  * Bondi (1952) Bondi, H. 1952, MNRAS, 112, 195
  * Camilloni et al. (2020) Camilloni, F., Grignani, G., Harmark, T., et al. 2020, arXiv:2007.15662
  * Chandrasekhar (1983) Chandrasekhar, S. 1983, The Mathematical Theory of Black Holes. Oxford: Clarendon Press
  * East & Yang (2018) East, W. E. & Yang, H. 2018, Phys. Rev. D, 98, 023008
  * Fox & Davé (2017) Fox, A. & Davé, R. 2017, Gas Accretion onto Galaxies: Astrophysics and Space Science Library, Volume 430, Springer
  * Frank et al. (2002) Frank, J., King, A., & Raine, D. J. 2002, Accretion Power in Astrophysics. Cambridge, UK: Cambridge University Press
  * Gammie et al. (2003) Gammie, C. F., McKinney, J. C., & Tóth, G. 2003, ApJ, 589, 444
  * Gong et al. (2016) Gong, X.-B., Liao, Y., & Xu, Z.-Y. 2016, Physics Letters B, 760, 112
  * Kato et al. (2008) Kato, S., Fukue, J., & Mineshige, S. 2008, Black-Hole Accretion Disks — Towards a New Paradigm. Kyoto: Kyoto University Press
  * King et al. (1975) King, A. R., Lasota, J. P., & Kundt, W. 1975, Phys. Rev. D, 12, 3037
  * Kluzniak & Kita (2000) Kluzniak, W. & Kita, D. 2000, astro-ph/0006266
  * Kološ & Janiuk (2020) Kološ, M. & Janiuk, A. 2020, arXiv:2004.07535
  * Komissarov & McKinney (2007) Komissarov, S. S. & McKinney, J. C. 2007, MNRAS, 377, L49
  * Kozlowski et al. (1978) Kozlowski, M., Jaroszynski, M., & Abramowicz, M. A. 1978, A&A, 63, 209
  * McKinney (2006) McKinney, J. C. 2006, MNRAS, 368, 1561
  * Mishra et al. (2020) Mishra, R., Čemeljić, M., & Kluźniak, W. 2020, in Proceedings of Polish Astronomical Society Meeting (Olsztyn, September 2019), arXiv:2006.01851
  * Misner et al. (1973) Misner, C. W., Thorne, K. S., & Wheeler, J. A. 1973, Gravitation. San Francisco: W. H. Freeman
  * Nathanail & Contopoulos (2014) Nathanail, A. & Contopoulos, I. 2014, ApJ, 788, 186
  * Pan & Yu (2016) Pan, Z. & Yu, C. 2016, ApJ, 816, 77
  * Penna (2014) Penna, R. F. 2014, Phys. Rev. D, 89, 104057
  * Penna et al. (2013) Penna, R. F., Kulkarni, A., & Narayan, R. 2013, A&A, 559, A116
  * Różańska et al. (2017) Różańska, A., Kunneriath, D., Czerny, B., et al. 2017, MNRAS, 464, 2090
  * Sapountzis & Janiuk (2019) Sapountzis, K. & Janiuk, A. 2019, ApJ, 873, 12
  * Shakura (2018) Shakura, N. 2018, Accretion Flows in Astrophysics: Astrophysics and Space Science Library, Volume 454. Springer
  * Schödel et al. (2014) Schödel, R., Feldmeier, A., Kunneriath, D., et al. 2014, A&A, 566, A47
  * Silich et al. (2008) Silich, S., Tenorio-Tagle, G., & Hueyotl-Zahuantitla, F. 2008, ApJ, 686, 172
  * Somov (2012) Somov, B. V. 2012, Plasma Astrophysics, Part I: Fundamentals and Practice, Astrophysics and Space Science Library, Volume 391. New York: Springer Science
  * Wald (1974) Wald, R. M. 1974, Phys. Rev. D, 10, 1680
  * Wald (1984) Wald, R. M. 1984, General Relativity. Chicago: University of Chicago Press