# The dynamical structure of the outflows driven by a large-scale magnetic
field

Jia-Wen Li Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University,
38 Zheda Road, Hangzhou 310027, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Xinwu Cao Zhejiang Institute of Modern Physics, Department of Physics,
Zhejiang University,
38 Zheda Road, Hangzhou 310027, China<EMAIL_ADDRESS><EMAIL_ADDRESS>Shanghai Astronomical Observatory, Chinese Academy of Sciences, 80 Nandan
Road, Shanghai, 200030, China  Key Laboratory of Radio Astronomy, Chinese
Academy of Sciences, 210008 Nanjing, China

###### Abstract

Large-scale magnetic field is crucial in launching and collimating the
jets/outflows. It is found that the magnetic flux can be efficiently
transported inward by the fast moving corona above a thin disk. In this work
we investigate the dynamical structure of the outflows driven by the large-
scale magnetic field advected by the hot corona. With derived large-scale
magnetic field, the outflow solution along every field line is obtained by
solving a set of magneto-hydrodynamical (MHD) Equations self-consistently with
boundary conditions at the upper surface of the corona. We find that the
terminal speeds of the outflows driven from the inner region of the disk are
$\sim 0.01c-0.1c$. The temperatures of the outflows at the large distance from
the black hole are still as high as several ten keV. The properties of the
magnetic outflows derived in this work are roughly consistent with the fast
outflows detected in some luminous quasars and X-ray binaries. The total mass
loss rate in the outflows from the corona is about $7\%-12\%$ of the mass
accretion rate of the disk. The three dimension field geometry, the velocity,
temperature and density of the outflows derived in this work can be used for
calculating the emergent spectra and their polarization of the accretion
disk/corona/outflow systems. Our results may help understand the features of
the observed spectra of X-ray binaries and active galactic nuclei.

Galaxy jets (601); Galaxy accretion disks (562); Accretion (14); Magnetic
fields (994)

##  1 Introduction

Outflows/winds are ubiquitous phenomenon observed in accreting systems with a
large dynamical range in black hole masses, ranging from supermassive black
hole (SMBH) as the engine of active galactic nuclei (AGNs) to the stellar mass
black hole hiding in the galactic binary systems. These outflows/winds often
manifest themselves by the blueshifted absorption features in the observed
spectra (e.g., Reeves et al., 2009; Tombesi et al., 2010a, 2013, 2015; Gofford
et al., 2015; Parker et al., 2017; Reeves et al., 2020), which may be one of
the important feedback mechanisms influencing the host galaxies’ dynamics,
star formation, or even the growth of their central black holes (Springel et
al., 2005; McNamara & Nulsen, 2007; Fabian, 2012; Beckmann et al., 2017). The
launching mechanisms of these outflows, however, are still debated. Plausible
outflow launching mechanisms include those driven by the radiation force
(e.g., Proga et al., 2000; Proga & Kallman, 2004; Higginbottom et al., 2014;
Yang et al., 2018), the centrifugal magnetic force (e.g., Blandford & Payne,
1982; Cao & Spruit, 1994; Contopoulos & Lovelace, 1994; Ferreira, 1997;
Fukumura et al., 2010; Cao & Spruit, 2013; Fukumura et al., 2015; Li & Cao,
2019; Chen & Zhang, 2021; Yang et al., 2021; Yang, 2021), and/or the force due
to the thermal pressure gradient of the gas (e.g., Begelman et al., 1983; Bu &
Yang, 2018). The thermally driven winds are often employed to account for a
kind of weak and slow outflows launched from very large radius.

The ultra-fast outflows (UFOs) are flowing outward at near-relativistic speeds
of $v/c\gtrsim 0.03$, which can be very powerful. Most of them have been
observed with the features of highly ionized metallic element in their X-ray
spectra, such as $\rm Fe_{XXV}/Fe_{XXVI}$. The ionization parameter of the gas
is usually very high $(\rm log\xi\gtrsim 4)$ with H-equivalent column density
of $N_{\rm H}\gtrsim 10^{23}\rm cm^{2}$. The UFOs seem to be ubiquitous across
different types of AGNs (e.g., Reeves et al., 2009; Tombesi et al., 2010a, b,
2011, 2013, 2015; Gofford et al., 2015; Parker et al., 2017; Reeves et al.,
2020). The highly ionized outflows are also observed in some X-ray binaries
(XRBs) (Miller et al., 2006, 2008, 2012; King et al., 2014; Kosec et al.,
2018; Wang et al., 2021). This kind of outflows are quite unlikely driven by
the radiation force since the central engines in these systems have strong
X-ray emission which ionizes the gas around the disk, and then suppresses the
UV line force (Stevens & Kallman, 1990; Proga et al., 2000; Proga & Kallman,
2004; Higginbottom et al., 2014; Hagino et al., 2015). The magnetic driven
outflow mechanism is most probably responsible for the UFOs observed in
luminous AGNs or XRBs. Miller et al. (2006) reported an X-ray-absorbing disk
wind discovered in the stellar-mass black hole binary GRO J16552-40. Fukumura
et al. (2018a) modeled the variations of the Fe K properties of the UFOs in a
famous quasar PDS 456. Both of them concluded that the outflows/winds must be
driven by the magnetic field. The presence of powerful disk wind/outflows in
XRBs and AGNs may suggest strong poloidal magnetic fields threading the
accretion disks that accelerate the outflows (e.g., Miller et al., 2006;
Fender et al., 2004; Miller et al., 2015; Fukumura et al., 2015; You et al.,
2016; Fukumura et al., 2018b; Kraemer et al., 2018; Miller et al., 2020; Kong
et al., 2021).

The launching and collimating mechanisms of outflows/jets require a large-
scale magnetic field. A series of magnetohydro-dynamic simulations
investigated the time dependent evolution of the resistive accretion disk and
the accompanying Outflows/jets (e.g., Casse & Keppens, 2002, 2004; Zanni,
Ferrari, Rosner, Bodo, & Massaglia, 2007; Sheikhnezami, Fendt, Porth, Vaidya,
& Ghanbari, 2012; Bai & Stone, 2013; Stepanovs, Fendt, & Sheikhnezami, 2014;
Sheikhnezami & Fendt, 2015; Stepanovs & Fendt, 2016; Sheikhnezami & Fendt,
2018; Zhu & Stone, 2018; Mishra, Begelman, Armitage, & Simon, 2020). In the
quasi-steady state, a robust outflow observed in the simulation domain and a
large-scale magnetic field is established. The torque exerted by the magnetic
field on the disk due to the outflow leading to a tight couple between the
outflow and inflow, which haven been verified by the simulations with a
prescribed background magnetic field (see the review by Pudritz et al., 2007,
and the references therein).

It has been suggested that the magnetic field generated through the dynamo
processes in the disk can launch the outflows from the disk (Pudritz, 1981a,
b; Tout & Pringle, 1992; Brandenburg et al., 1995; Tout & Pringle, 1996;
Romanova et al., 1998; Stepanovs et al., 2014; Liska et al., 2020). However,
the detailed physics of dynamo generation of a large-scale magnetic field is
still quite unclear, though there is evidence of the winds driven by the
dynamo generated field in XRBs (Cao et al., 2021).

Alternatively, a large-scale poloidal field can be naturally formed through
advection of a weak ordered field threading the ambient gas (van Ballegooijen,
1989; Lubow et al., 1994; Spruit & Uzdensky, 2005), which has been verified by
numerical simulations (Igumenshchev et al., 2003; Bai & Stone, 2013; Suzuki &
Inutsuka, 2014; Zhu & Stone, 2018; Mishra et al., 2020). The external weak
magnetic field threading the outer region of the disk may probably be fed by a
companion star or the interstellar medium in AGNs, provided they are somewhat
magnetized (Bisnovatyi-Kogan & Ruzmaikin, 1974, 1976; van Ballegooijen, 1989;
Spruit & Uzdensky, 2005; Zhou et al., 2019; Cao & Lai, 2019; Cao & Zdziarski,
2020). The magnetic diffusivity, however, is roughly proportional to the
turbulent viscosity, i.e., the Prandtl number $\cal{P}_{\rm m}$ is a constant
around unity (Parker, 1979; Fromang & Stone, 2009; Guan & Gammie, 2009), which
implies a large-scale magnetic field threading a turbulent thin disk must
diffuse away rapidly, i.e., the transport of the magnetic flux in a turbulent
thin disk is very ineﬀicient (Lubow, Papaloizou, & Pringle, 1994). Therefore,
the key to generate a strong poloidal magnetic field threading a thin disk is
to suppress the outward diffusion of the flux when the field is accumulated
inward.

Different attempts have been made to solve this issue. One of these scenarios
suggested that, the structure of a thin disk is significantly altered due to
the angular momentum loss through the magnetic outflow, and therefore the
radial velocity of the gas in the thin disk is significantly increased. Thus,
the external field is sufficiently dragged inward by the fast moving gas in
the disk, and a strong large-scale magnetic field can be formed due to the
counterbalance of outward flux diffuse by the efficient inward field transport
(Cao & Spruit, 2013; Li & Begelman, 2014; Li, 2014; Li & Cao, 2019; Scepi et
al., 2020). Another possibility is the presence of highly conducting fast
inward moving layers above a thin disk that substantially suppress the
diffusion of the magnetic flux, which may also lead to a rather strong large-
scale magnetic field (Lovelace et al., 2009; Beckwith et al., 2009; Guilet &
Ogilvie, 2012, 2013), though the strength of the field is not very strong
(limited by the gas pressure of the layers above the disk) (Cao, 2018).
Recently, the time-dependent global numerical simulations suggested that for
the disk-corona system the mass accretion occurs mainly in the corona region,
of which the magnetic flux is efficiently transported inward, and a quasi-
static large-scale magnetic field with field lines strongly inclined towards
the disk surface is formed. This field accelerates a fraction of the hot
coronal gas into the outflows (e.g., Zhu & Stone, 2018; Mishra et al., 2020).

Fukumura et al. (2010) constructed a semi-analytical model of the magnetized
outflows launched from the accretion disk, and it is successfully applied to
model the X-ray absorber properties of XRBs and AGNs (Fukumura et al., 2017,
2018b). They employed a self-similar prescription of the outflows in the
calculations, while the boundary conditions of the outflows have not been
properly considered. However, as pointed in Bai et al. (2016), the physical
properties of the wind base (e.g., the field configuration, the gas
temperature and the gas-to-magnetic ratio) are crucial in calculating the wind
properties. Recently, we derived a global magnetic field configuration of an
accretion disk-corona system (Li & Cao, 2021). We find that the weak external
magnetic field can be efficiently transported to the inner region by the fast
moving corona above the disk. The derived field configurations are consistent
with those exhibited in the numerical simulations (e.g., Zhu & Stone, 2018;
Mishra et al., 2020). In this work, we study the global dynamics of the
magnetic outflows/winds driven by the magnetic field based on the field
configuration derived in Li & Cao (2021). The outflow solution is self
consistently derived with suitable boundary conditions at the upper surface of
the corona. This paper is organized as follows, Section 2 contains the model
of field advection and the dynamics of the magnetically driven outflows. The
numerical set-up is described in Section 3. The results and discussion of the
model calculations are given in Sections 4 and 5 respectively.

##  2 model

A fraction of the gas in the disk or the layer above the disk can be
accelerated into the outflows by the rotating large-scale magnetic field
threading the disk (Blandford & Payne, 1982). The dynamics of the outflow is
described by a set of MHD Equations, and the outflow solution is available by
solving these Equations when the field configuration and the suitable boundary
conditions at the base of the outflow are specified (e.g., Blandford & Payne,
1982; Cao & Spruit, 1994; Cao, 2014).

As introduced in Section 1, the large-scale magnetic field can be formed
through advection of the external weak field inward by the gas in the disk or
the upper layers (e.g., hot corona). In this paper, we explore how the gas is
driven by the large-scale magnetic field advected inward by the hot corona.
The global configuration of such a magnetic field has been calculated by Li &
Cao (2021), and we adopt the field configuration/strength and the corona
properties derived in their paper to calculate the dynamics of the outflows
driven from the corona of the disk. In principle, the disk and outflows are
tightly coupled through mass and angular momentum transfers between them by
the magnetic field (e.g., Ferreira et al., 2006; Cao & Spruit, 2013; Li & Cao,
2019). In this work, for simplicity, we neglect the impact of the outflows on
the dynamics of the disk, which is a reasonable approximation for weak or
moderately strong outflows. As pointed by Cao (2018), the strength of the
large-scale field dragged inward by the corona is always limited by the gas
pressure of the corona, and therefore only outflows with low or moderate power
can be formed in this case.

In this work, We consider steady axisymmetric outflows driven from the hot
corona above the disk, and we shall work in cylindrical coordinates
($R,\varphi,z$), i.e., $\partial/\partial t=\partial/\partial\varphi=0$. With
the derived global configuration/strength of the field dragged inward by the
corona of the disk, we solve a set of MHD Equations, and the dynamics of the
outflows is then derived with the properties of the gas (temperature and
density) at the surface of the corona as boundary conditions.

###  2.1 Disk structure and magnetic field configuration

The large-scale field advection in a turbulent thin accretion disk covered by
a layer of tenuous gas or a hot corona was studied in Li & Cao (2021). Here we
briefly summarize the model calculations as follows. As discussed, we have not
included the mass loss of the outflows in the calculations of the disk
structure. The mass accretion rate is assumed to be a constant radially, i.e.,
$\dot{M}_{\rm acc}=-2\pi R\Sigma v_{R}=\text{const}$, which implies the mass
loss rate in the outflows is negligible compared with the accretion rate in
the disk. Here $\Sigma\sim 2\rho H$ is the disk surface density, and $\rho,H$
are the mean disk density and the scale height of the disk, respectively. For
such a steady accretion disk, the momentum equation reads

$\frac{d}{dR}\left(2\pi R\Sigma v_{R}R^{2}\Omega\right)=\frac{d}{dR}\left(2\pi
R\nu\Sigma R^{2}\frac{d\Omega}{dR}\right).$ (1)

The radial velocity of the gas in the disk can be calculated by integrating
Equation (1), which reads

$v_{R}(R,z)=-\frac{3\nu(z)}{2R}=-\frac{\alpha c_{\rm s}^{2}(z)}{R\Omega_{\rm
k}},$ (2)

where $c_{\rm s}$ is the isothermal sound speed, $\Omega_{\rm k}$ is the
Keplerian angular velocity, and the the $\alpha$ viscosity $\nu=\alpha c_{\rm
s}H$ is adopted. The inner boundary condition of the disk is rather uncertain,
and the standard thin disk is artificially cut off at the inner stable
circular orbit, which leads to singularities of the disk quantities there
(Shakura & Sunyaev, 1973). To avoid the complexity of this issue, we adopt
Equation (2) in our calculations, which is a good approximation except in the
region very close the inner edge of the disk.

The temperature of the disks in AGNs is too low to emit hard X-rays observed
in AGNs. The power-law hard X-ray spectra of AGN are most likely due to the
inverse Compton scattering of soft photons on a population of hot electrons,
most probably in the corona above the disk (Galeev et al., 1979; Haardt &
Maraschi, 1991; Wandel & Liang, 1991; Haardt & Maraschi, 1993). Such a
sandwich disk-corona model has been well developed, and it can successfully
reproduce main features of the X-ray spectra of accretion disk-corona systems
in different scales, i.e., AGNs, X-ray binaries or even ultrluminous X-ray
sources (e.g. Di Matteo, 1998; Di Matteo et al., 1999; Wu & Gu, 2008; Cao,
2009; Merloni & Fabian, 2001, 2002; You et al., 2012; Liu et al., 2016; Lyu &
Rieke, 2017; Qiao & Liu, 2018; Arcodia et al., 2019; Yang et al., 2019; Liu et
al., 2019; Cheng et al., 2020; Zou et al., 2020; Mondal et al., 2020; You et
al., 2021; Liu et al., 2021a, b). The disk-corona system is usually described
as a two-phase accretion flow with its temperature changing sharply from the
cold disk to the hot corona in the vertical direction (Haardt & Maraschi,
1991, 1993), which is confirmed by the elaborate calculation of the vertical
disk-coronal structure with detailed vertical energy transportation included
(e.g., Liu, Mineshige, Meyer, Meyer-Hofmeister, & Kawaguchi, 2002). As done in
Li & Cao (2021), we choose the following function to describe the gas
temperature $\Theta(z)$ along $z$ direction,

$\Theta(z)=\left(\Theta_{z_{\rm
h}}-\Theta_{0}\right)\frac{e^{b\left(1-a\right)}+1}{e^{b\left(1-a\frac{z}{z_{\rm
h}}\right)}+1}+\Theta_{0},$ (3)

where the dimensionless gas temperature is defined as

$\Theta(z)\equiv\frac{c_{\rm s}^{2}(z)}{R^{2}\Omega_{\rm k}^{2}},$ (4)

$\Theta_{0}$ and $\Theta_{z_{\rm h}}$ are the dimensionless gas temperature at
the disk mid-plane and the surface of the corona (i.e., $z=z_{\rm h}$),
respectively. We use two parameters $a$ and $b$ to describe the vertical
temperature distribution of the disk-corona. Although the vertical temperature
distribution of the disk-corona system is described by an artificial function
(3), we believe it indeed reflects the basic feature of the two-phase disk-
corona system, i.e., the temperature changes sharply in $z$-direction (cf.,
Figure 1 in Liu, Mineshige, Meyer, Meyer-Hofmeister, & Kawaguchi, 2002).

The vertical density profile of the disk-corona system at radius $R$ is
obtained by integrating the Equation of hydrostatic equilibrium from $z=0$ to
$z=z_{\rm h}$, i.e.,

${\rm ln}\frac{\rho(R,z_{\rm h})}{\rho_{0}(R,0)}=-{\rm ln}\frac{\Theta_{z_{\rm
h}}}{\Theta_{0}}-\int_{0}^{z_{\rm h}}\frac{z}{R^{2}\Theta(z)}dz.$ (5)

The gas density decreases with the increase of $z$ from disk mid-plane, thus
the corona upper surface at radius $R$ (i.e., $z=z_{\rm h}$, hereafter we
refer $z=\pm z_{\rm h}$ as the corona surfaces) is defined as the gas density
goes down to a certain critical value $\epsilon\rho_{0}$, i.e.,

$\frac{\rho(R,z_{\rm h})}{\rho_{0}(R,0)}=\epsilon,$ (6)

where $\rho_{0}(R,0)$ is the gas density at the disk mid-plane. Combining
Equations (3), (5) and (6), we obtain the vertical structure of the disk-
corona system, i.e., $z_{\rm h}$ and $\rho(R,z)$, when the parameters, $a$,
$b$, $\Theta_{0}$, and $\Theta_{z_{\rm h}}$ are specified.

Based on above disk structure, the advection of the magnetic flux by the fast
moving corona can be solved with the induction Equation, i.e.,

$\frac{\partial\boldsymbol{B}}{\partial
t}=\nabla\times(\boldsymbol{v}\times\boldsymbol{B}-\eta\nabla\times\boldsymbol{B}),$
(7)

where $\boldsymbol{v}$ is the velocity of the plasma and $\eta$ is the
magnetic diffusivity caused by the plasma turbulent resistivity. We study the
outflow magnetically launched from a diffusive viscous disk-corona system. The
magnetic diffusivity $\eta$ is related to the magnetic Prandtl number as
${\cal P}_{\rm m}=\eta/\nu_{0}$, where $\nu_{0}$ is the effective turbulent
viscosity at the disk mid-plane (i.e., $z=0$) and ${\cal P}_{\rm m}$ is the
Prandtl number. Under the axisymmetric assumption, the magnetic field can be
described by a stream function $R\psi(R,z)$, and the radial component of the
induction equation reduces to (see Equation (10) and also the detailed
derivation procedures in Lubow et al. (1994))

$\frac{\partial}{\partial
t}\left[R\psi(R,z)\right]=-v_{R}(R,z)\frac{\partial}{\partial
R}\left[R\psi(R,z)\right]-\frac{4\pi\eta}{c}RJ_{\phi}(R,z),$ (8)

where $v_{R}$ is the radial velocity of the accreting gas and $J_{\varphi}$ is
the azimuthal current density within the disk and corona. In Equation (8),
$R\psi$ is proportional to the magnetic flux threading the disk within radius
$R$, the first term on the right-hand side is the flux advected inward by the
accretion flow (both disk and corona), while the second term is the outward
flux diffusion due to the turbulent resistivity. As summarized in Guan &
Gammie (2009), the net rate on which the poloidal magnetic field being
transported inward is governed by the relative intensity of the turbulent
viscosity and resistivity, if the former is dominant the magnetic field will
be dragged inward efficiently, otherwise, the field will diffuse away, i.e.,
the field advection is rather inefficient.

The stream function $R\psi(R,z)$ is proportional to the magnetic flux within a
circle with radius $R$ located at vertical position $z$. The azimuthal current
density $J_{\varphi}(R,z)$ is related to the stream function $R\psi(R,z)$ via
the Biot-Savart law,

$\psi_{\rm d}(R,z)=\psi(R,z)-\psi_{\infty}=\frac{1}{c}\int_{R_{\rm
in}}^{R_{\rm out}}\int_{0}^{2\pi}\int_{-z_{\rm h}}^{z_{\rm
h}}\frac{J_{\varphi}(R^{\prime},z^{\prime})\cos\varphi^{\prime}d\varphi^{\prime}R^{\prime}dR^{\prime}dz^{\prime}}{\left[R^{2}+R^{\prime
2}+\left(z-z^{\prime}\right)^{2}-2RR^{\prime}\cos\varphi^{\prime}\right]^{\frac{1}{2}}},$
(9)

where $c$ is the light speed and $\psi_{\rm d}(R,z)$ is contributed by the
current $J_{\varphi}(R^{\prime},z^{\prime})$ inside the disk and corona. We
assume an external uniform vertical weak magnetic field $B_{\rm ext}$ to be
advected inward by the disk-corona system, and therefore $\psi_{\infty}=B_{\rm
ext}R/2$.

For a steady disk, i.e., $\partial/\partial t=0$, the field advection is in
balance with the outward diffusion of the magnetic flux, which is described by

$-\frac{\partial}{\partial R}\left[R\psi_{\rm
d}(R,z)\right]-\frac{4\pi\eta}{c}\frac{R}{v_{R}(R,z)}J_{\phi}(R,z)=B_{\rm
ext}R.$ (10)

Differentiating Equation (9), we have

$\frac{\partial}{\partial R}\left[R\psi_{\rm
d}(R,z)\right]=\frac{1}{c}\int_{R_{\rm in}}^{R_{\rm
out}}\int_{0}^{2\pi}\int_{-z_{\rm h}}^{z_{\rm h}}\frac{\left[R^{\prime
2}+(z-z^{\prime})^{2}-RR^{\prime}\cos\varphi^{\prime}\right]J_{\varphi}(R^{\prime},z^{\prime})\cos\varphi^{\prime}d\varphi^{\prime}R^{\prime}dR^{\prime}dz^{\prime}}{\left[R^{2}+R^{\prime
2}+\left(z-z^{\prime}\right)^{2}-2RR^{\prime}\cos\varphi^{\prime}\right]^{\frac{3}{2}}},$
(11)

and substituting Equation (11) into Equation (10), it can be re-written as a
set of linear Equations

$-\sum_{j=1}^{\rm n}\sum_{l=1}^{\rm
m}J_{\varphi}(R_{j},z_{l})P_{i,j,k,l}\Delta R_{j}\Delta
z_{l}-\frac{4\pi\eta}{c}\frac{R_{i}}{v_{R}(R_{i},z_{k})}J_{\varphi}(R_{i},z_{k})=B_{{\rm
ext},k}R_{i},$ (12)

where

$P_{i,j,k,l}={\frac{1}{c}}\int_{0}^{2\pi}{\frac{\left[R_{j}^{2}+\left(z_{k}-z_{l}\right)^{2}-R_{i}R_{j}{\cos}\varphi^{\prime}\right]R_{j}}{\left[{R_{i}}^{2}+{R_{j}}^{2}+\left(z_{k}-z_{l}\right)^{2}-2R_{i}{R_{j}}\cos{\varphi^{\prime}}\right]^{\frac{3}{2}}}}\cos\varphi^{\prime}{d}\varphi^{\prime},$
(13)

and the subscripts $``_{i,j,k,l}"$ are labeled for the values of each
variables at radius $R_{i},R_{j}$ and vertical position $z_{k},z_{l}$.
$J_{\varphi}(R_{j},z_{l})$ is the current density at position $(R_{j},z_{l})$,
and $B_{{\rm ext},k}R_{i}=B_{\rm ext}R_{i}$ ($B_{\rm ext}$ is the strength of
the external uniform vertical magnetic field). Solving a set of the linear
Equations (12) with given disk structure, i.e., the radial velocity profile
$v_{R}(R,z)$ and the disk thickness $z_{\rm h}$, one can obtain the
distribution of the current density within the disk, and then the spatial
distribution of the magnetic stream function $\psi(R,z)$. The strength and
configuration of the large-scale poloidal magnetic field are then calculated
by

$\displaystyle B_{R}(R,z)=$ $\displaystyle-\frac{\partial}{\partial
z}\psi(R,z),$ (14) $\displaystyle B_{z}(R,z)=$
$\displaystyle\frac{1}{R}\frac{\partial}{\partial R}\left[R\psi(R,z)\right].$

###  2.2 Dynamics of the outflows

A set of MHD Equations describing a steady magnetically driven outflows are
(Spruit, 1996)

$\displaystyle\nabla\times\left(\boldsymbol{v}\times\boldsymbol{B}\right)=$
$\displaystyle 0,\quad(\text{induction equation})$ (15) $\displaystyle-\nabla
P-\rho\nabla\Psi_{\rm
g}+\frac{1}{4\pi}\left(\nabla\times\boldsymbol{B}\right)\times\boldsymbol{B}=$
$\displaystyle\rho\left(\boldsymbol{v}\cdot\nabla\right)\boldsymbol{v},\quad\text{(equation
of motion)}$ $\displaystyle\nabla\cdot\left(\rho\boldsymbol{v}\right)=$
$\displaystyle 0,\quad\text{(mass conservation)}$
$\displaystyle\nabla\cdot\boldsymbol{B}=$ $\displaystyle 0,$

where $\Psi_{\rm g}$ is the gravitational potential. Under the axisymmetric
assumption the magnetic field and the outflow velocity can be decomposed into
poloidal (i.e., $B_{\rm p},v_{\rm p}$) and toroidal (i.e.,
$B_{\varphi},v_{\varphi}$) components. The dynamical structure of such a
magnetized outflow is then described by a set of conservation Equations
derived from Equation (15), which reads (see also Cao & Spruit, 1994; Romero &
Vila, 2014, for the details )

$v_{\varphi}-R\Omega=\frac{v_{\rm p}B_{\varphi}}{B_{\rm p}},$ (16)

$Rv_{\varphi}-\frac{RB_{\varphi}B_{\rm p}}{4\pi\rho v_{\rm p}}=R_{\rm
A}^{2}\Omega,$ (17)

$\kappa=\frac{\rho v_{\rm p}}{B_{\rm p}},$ (18)

$\frac{1}{2}v_{\rm p}^{2}+\frac{1}{2}(v_{\varphi}-R\Omega)^{2}+h+\Psi_{\rm
eff}\equiv E.$ (19)

Here, $\kappa$ and $E$ are constants along each field line (i.e., the
streamline of the outflow), $R_{\rm A}$ is the Alfvén radius and $\Omega$ is
the angular velocity of the field line at the foot-point $R_{z_{\rm h}}$.
Equation (16) is derived from the poloidal component of the induction Equation
of the outflow, which is usually interpreted as the gas in the outflow always
moving along the field line. Equations (17) and (18) are the conservation of
the angular momentum and mass along each field line, respectively. Equation
(19) is the Bernoulli integral. The specific enthalpy $dh=dP/\rho$ and with
the polytropic law, $P=K\rho^{\gamma}$, we have

$h=\frac{\gamma}{\gamma-1}K\rho^{\gamma-1},$ (20)

where $\gamma$ is the adiabatic index and $K$ is a constant along each field
line. The effective potential $\Psi_{\rm eff}$ along the field line that co-
rotates with the angular velocity $\Omega$ at its foot-point, i.e.,

$\Psi_{\rm eff}(R,z)=-\frac{GM_{\rm
BH}}{(R^{2}+z^{2})^{\frac{1}{2}}}-\frac{1}{2}R^{2}\Omega^{2},$ (21)

where $M_{\rm BH}$ is the mass of the central balck hole. Substituting
Equations (16), (17), (18), (20) and (21) into Equation (19), we obtain

$\frac{B_{\rm p}^{2}\rho_{\rm A}}{8\pi\rho^{2}}+\frac{\Omega^{2}\rho_{\rm
A}^{2}(R^{2}-R_{\rm A}^{2})^{2}}{2R^{2}(\rho-\rho_{\rm
A})^{2}}+\frac{\gamma}{\gamma-1}K\rho^{\gamma-1}+\Psi_{\rm eff}=E,$ (22)

which can be re-written in the dimensionless form,

$\frac{\xi}{\tilde{\rho}^{2}}\frac{B_{\rm p}^{2}}{B_{\rm p,z_{\rm
h}}^{2}}+\frac{\omega}{2}\left[\frac{\tilde{R}^{2}-1}{\tilde{R}(\tilde{\rho}-1)}\right]^{2}+\theta\tilde{\rho}^{\gamma-1}-\frac{1}{(\tilde{R}^{2}+\tilde{z}^{2})^{\frac{1}{2}}}-\frac{\omega}{2}\tilde{R}=\tilde{H}(\tilde{R},\tilde{\rho},\tilde{R}_{z_{\rm
h}},\tilde{\rho}_{z_{\rm h}})=\tilde{E},$ (23)

where the dimensionless parameters are defined as

$\displaystyle\tilde{R}=\frac{R}{R_{\rm A}},\tilde{\rho}=$
$\displaystyle\frac{\rho}{\rho_{\rm A}},\tilde{z}=\frac{z}{R_{\rm
A}},\tilde{E}=E/\frac{GM_{\rm BH}}{R_{\rm A}},$ (24)
$\displaystyle\xi=\frac{\Theta_{z_{\rm h}}}{\beta_{z_{\rm
h}}}\frac{\tilde{\rho}_{z_{\rm h}}}{\tilde{R}_{z_{\rm h}}},\omega=$
$\displaystyle\frac{\Omega^{2}}{\Omega_{\rm
k}^{2}}\frac{1}{\tilde{R}^{3}_{z_{\rm h}}},\theta=\Theta_{z_{\rm
h}}\frac{\gamma}{\gamma-1}\frac{\tilde{\rho}_{z_{\rm
h}}^{1-\gamma}}{\tilde{R}_{z_{\rm h}}}.$

The subscripts $``_{\rm A}"$ and $``_{z_{\rm h}}"$ indicates the values of
each quantities at the Alfvén point and the bottom of the outflow (i.e., the
upper surface of the corona, since we assume the magnetized outflow is
launched from the upper surface of the corona), respectively. The angular
velocity $\Omega$ of the disk at the field line foot-point can be calculated
by

$R_{z_{\rm h}}(\Omega_{\rm k}^{2}-\Omega^{2})=\frac{B_{R}(R_{z_{\rm
h}})B_{z}(R_{z_{\rm h}})}{2\pi\Sigma},$ (25)

where the magnetic force against the gravity is considered in the calculations
of rotation velocity of the gas in the corona. The ratio of gas to magnetic
pressure at the surface of the corona $\beta_{z_{\rm h}}=\rho_{z_{\rm
h}}c_{{\rm s},z_{\rm h}}^{2}/\frac{B_{{\rm p},z{\rm h}}^{2}}{8\pi}$. The mass
conservation within the disk requires the gas density at disk mid-plane
$\rho(R,0)\propto R^{-3/2}$, and then the dependence of $B_{\rm p}$ on radius
$R$ is derived by Equation (8), which is indeed determined by the dynamical
structure of the disk, thus $\beta_{z_{\rm h}}$ is a function of radius $R$.
In this work the magnetic flux is mainly transported by the corona, thus in
this region gas pressure must dominate over the magnetic pressure, i.e.,
$\beta_{z_{\rm h}}\gtrsim 1$.

For a given magnetic field configuration, a physically plausible magnetic
outflow should smoothly pass through three critical points along the magnetic
field lines, i.e., the slow sonic point, Alfvén point and the fast sonic point
(Cao & Spruit, 1994). At the sonic points the Bernoulli function (23)
satisfies

$\frac{\partial\tilde{H}}{\partial\tilde{R}}\left.\right|_{\rm
s,f}=\frac{\partial\tilde{H}}{\partial\tilde{\rho}}\left.\right|_{\rm s,f}=0.$
(26)

Along any field line, $\tilde{E}$ is a constant, we have

$\tilde{H}(\tilde{R}_{\rm s},\tilde{\rho}_{\rm s},\tilde{R}_{z_{\rm
h}},\tilde{\rho}_{z_{\rm h}})=\tilde{H}(\tilde{R}_{\rm f},\tilde{\rho}_{\rm
f},\tilde{R}_{z_{\rm h}},\tilde{\rho}_{z_{\rm h}})=\tilde{H}(\tilde{R}_{z_{\rm
h}},\tilde{\rho}_{z_{\rm h}},\tilde{R}_{z_{\rm h}},\tilde{\rho}_{z_{\rm
h}})=\tilde{E},$ (27)

where the subscripts $``_{\rm s}"$ and $``_{\rm f}"$ refer to the slow and
fast sonic points respectively. Thus, we have six Equations, i.e., Equations
(26) and (27), to describe the dynamics of the outflow driven by the magnetic
field if the field configuration/strength is given.

Thus these six Equations can be solved for six variables ($\tilde{R}_{\rm
s},\tilde{\rho}_{\rm s},\tilde{R}_{\rm f},\tilde{\rho}_{\rm
f},\tilde{R}_{z_{\rm h}},\tilde{\rho}_{z_{\rm h}}$) with the derived large-
scale magnetic field strength and configuration, when the temperature and
density at the bottom of the outflow are specified as boundary conditions.

##  3 numerical setup

The configuration/strength of the large-scale magnetic field advected in an
accretion disk-corona system is available by solving the linear Equation (12),
when the values of the model parameters ${\cal P}_{\rm m}$, $\Theta_{0}$,
$\Theta_{z_{\rm h}}$ and $\epsilon$ are specified. The disk-corona is set to
extend radially between $R_{\rm in}$ and $R_{\rm out}=1000R_{\rm in}$, and
$R_{\rm in}=6R_{\rm g}$ ($R_{\rm g}=GM/c^{2}$) is assumed to be the inner
boundary of the disk. We adopt $a=1.5$, and $b=30$ in Equation (3) to describe
a sharp temperature increase from the cold disk to the hot corona in all the
calculations. $J_{\varphi}(R,z)$ is discretized on the center of the cell area
$\Delta R_{j}\times\Delta z_{l}$. In radial direction we set $n$ grid cells
distributed logarithmically within the disk area between $R_{\rm in}$ and
$R_{\rm out}$, while the $m$ grid cells in $z$-direction within the upper
($z=z_{\rm h}$) and lower $z=-z_{\rm h}$ surface of the corona is adopted. In
all of the calculation we adopt $n=100$ and $m=80$, which can achieve a good
performance on accuracy. The distribution of the current density
$J_{\varphi}(R,z)$ within the disk-corona (i.e., the region $-z_{\rm
h}<z<z_{\rm h}$) is available by solving a set of linear Equations (12), then
the spacial distribution of the magnetic potential can be calculated similarly
to Lubow et al. (1994). Equation (9) can be written in a matrix form, which
reads

$\left(R\psi_{d}\right)_{i,k}=\sum_{j=1}^{n}\sum_{l=1}^{m}Q_{i,j,k,l}J_{\varphi}(R_{j},z_{l})\Delta
R_{j}\Delta z_{l},$ (28)

$\left(R\psi_{d}\right)_{i,k}$ is related to the magnetic potential at
position $(R_{i},z_{k})$ contributed by all currents within the disk and
corona. Matrix $Q$ is defined as

$Q_{i,j,k,l}=\frac{1}{c}\frac{4R_{i}R_{j}}{\left[\left(R_{i}+R_{j}\right)^{2}+\left(z_{k}-z_{l}\right)^{2}\right]^{\frac{1}{2}}}\left[\frac{\left(2-{\rm
k}\right)K(\rm k)-2\emph{E}({\rm k})}{\rm k}\right],$ (29)

and

${\rm
k}=\frac{4R_{i}R_{j}}{\left(R_{i}+R_{j}\right)^{2}+\left(z_{k}-z_{l}\right)^{2}}.$
(30)

Here $K\left({\rm k}\right)=\int_{0}^{\frac{\pi}{2}}\left(1-{\rm
k}\sin^{2}\theta\right)^{-\frac{1}{2}}d\theta$ and $E\left({\rm
k}\right)=\int_{0}^{\frac{\pi}{2}}\left(1-{\rm
k}\sin^{2}\theta\right)^{\frac{1}{2}}d\theta$ are the complete elliptic
integrals of of the first and second kind, respectively. A singularity problem
has been encountered when evaluate the diagonal elements of matrix $Q$, i.e.,
at grid cells of $R_{i}=R_{j}$ and $z_{k}=z_{l}$, thus a smooth procedure is
applied, i.e.,

$Q_{j,j,l,l}=\frac{\lambda}{4}\left[Q_{j,j+\frac{\lambda}{2},l,l}+Q_{j,j+\lambda,l,l}+Q_{j,j-\frac{\lambda}{2},l,l}+Q_{j,j-\lambda,l,l}+Q_{j,j,l,l+\frac{\lambda}{2}}+Q_{j,j,l,l+\lambda}+Q_{j,j,l,l-\frac{\lambda}{2}}+Q_{j,j,l,l-\lambda}\right].$
(31)

here, $R_{j+\lambda}$ are the inner and outer boundaries of grid cell
$(R_{j},z_{l})$, $R_{j-\lambda}$, and $z_{l+\lambda}$, $z_{l-\lambda}$ are the
upper and bottom boundaries, respectively. and $\lambda=1/2$ is adopted in all
the calculations. We find that the final results are almost independent of the
value of $\lambda$.

The stream function $R\psi$ is proportional to the magnetic flux in the area
within radius $R$, which is a constant along the magnetic field line, thus the
field configuration is available with derived $\psi(R,z)$. Finally, the
distribution of the field strength is calculated with Equations (14).

With the derived large-scale field configuration/strength, the dynamical
properties of the outflow along a specific field line are available by solving
a set of six Equations (i.e. Equations 26 and 27) numerically when the
parameter $\gamma$ and $\beta_{z_{\rm h}}$ are specified. The obtained outflow
solution passes smoothly through the sonic and Alfvén points. The
dimensionless density of the outflow along the field line can be calculated
with the Bernoulli Equation (23). Then, we can calculate other physical
quantities (e.g., poloidal/azimuthal velocities, and temperature) of the
outflow with Equations (16), (17) and (18). Repeat the calculations along
every field line, we obtain the global outflow dynamics.

Table 1: model parameters in the calculation of the magnetic
field configurations and outflow solution

Model | a | b | $\epsilon$ | $\alpha$ | $\cal{P}_{\rm m}$ | $\beta_{z_{\rm h}}(R_{\rm in})$ | $\Theta_{0}$ | $\Theta_{z_{\rm h}}$ | $\gamma$
---|---|---|---|---|---|---|---|---|---
|  |  |  |  |  |  |  |  | 1.4
Case1: | 1.5 | 30 | $10^{-5}$ | 0.1 | 1 | 1 | 0.005 | 0.05 |
|  |  |  |  |  |  |  |  | 5/3
Case2: | 1.5 | 30 | $10^{-5}$ | 0.1 | 1 | 1 | 0.0025 | 0.025 | 1.4

Note. — $a$ and $b$ are adopted to describe a sharp temperature increase from
the cold disk to the hot corona in Equation (3). Under the vertical hydro
static equilibrium approximation, $\epsilon$ is used in Equation (6) to define
the location of the corona surface. $\beta_{z_{\rm h}}(R_{\rm in})$ is the
plasma $\beta$ parameter at the inner boundary of the disk-corona system.
$\alpha$ and $\cal{P}_{\rm m}$ are the viscosity parameter and the magnetic
Prandtl number, respectively. $\Theta_{0}$ and $\Theta_{z_{\rm h}}$ are the
dimensionless temperature at the disk mid-plane ($z=0$) and the surface of the
corona ($z=z_{\rm h}$), which are defined in Equation(3). Basically we adopt
$\gamma=1.4$ in the calculations, for comparison we recalculate case1 with
$\gamma=5/3$

##  4 Results

The vertical temperature structure of the disk-corona system is described by
Equation (3), and the upper surface of the corona is defined at $z=z_{\rm h}$
in Equation (6). We adopt ${\cal P}_{\rm m}=1$ and $\epsilon=10^{-5}$,
$\alpha=0.1$ and $\beta_{z_{\rm h}}(R_{\rm in})=1$ in all the calculations. We
carry out the calculations with the temperature of the gas at the disk mid-
plane, $\Theta_{0}=0.005$ ($H/R\simeq 0.07$) and $\Theta_{0}=0.0025$
($H/R\simeq 0.05$), respectively. All the constants and parameters adopted in
the calculations are listed in Table 1.

###  4.1 magnetic field configurations

Figure 1: Large-scale poloidal magnetic field configurations. The solid curves
are the poloidal magnetic field lines, while the dotted lines show the
location of the upper corona surface $z=\pm z_{\rm h}$ (which are derived by
Equation (6)) and the disk mid-plane, respectively. Left panel is calculated
for case 1: $\Theta_{0}=0.005,\Theta_{z_{\rm h}}=0.05$ and the derived aspect
ratio $z_{\rm h}/R=0.456$. Right panel is calculated for case 2:
$\Theta_{0}=0.0025,\Theta_{z_{\rm h}}=0.025$, and the derived aspect ratio
$z_{\rm h}/R=0.323$. $R_{\rm in}=6R_{\rm g}$ is the inner radius of the disk,
and the adiabatic index $\gamma=1.4$ is adopted.  Figure 2: The strength of
the vertical component of the large-scale magnetic field, while $B_{\rm ext}$
is the external magnetic field strength. The red line is calculated for case
1: $\Theta_{0}=0.005,\Theta_{z_{\rm h}}=0.05$, while blue line is for case 2:
$\Theta_{0}=0.0025,\Theta_{z_{\rm h}}=0.025$

The configurations of the large-scale poloidal magnetic field advected by the
disk-corona are plotted in Figure 1, in which two sets of the parameters are
adopted, i.e., case 1: $\Theta_{0}=0.005,\Theta_{z_{\rm h}}=0.05$, and case 2:
$\Theta_{0}=0.0025,\Theta_{z_{\rm h}}=0.025$), with an adiabatic index
$\gamma=1.4$. The inclination angle of the field line with respect to the
upper surface of the corona (i.e., $z=z_{\rm h}$) decreases with increasing
field line foot point radius. In this case, the azimuthal field component
would be substantially higher than poloidal component not far from the upper
surface of the corona, which implies that our assumption of a potential field
above the corona is no longer valid. Thus, our calculations will be terminated
at a certain radius, and we refer to the region within this radius as the
outflow region hereafter. We will discuss this issue further in Section 5.

In the region close to the disk mid-plane, the radial velocity is relatively
small (see the light blue region of Figure 3), the resultant field lines are
almost perpendicular to the disk mid-plane (see Figure 1). It indicates that
the magnetic diffusion is substantially suppressed in the cold disk even for a
strong magnetic field due to a small value of $\nabla\times\boldsymbol{B}$.

The radial profiles of the vertical field component are depicted in Figure 2.
The vertical magnetic field strength in the inner region of the disk is
increased by several orders of magnitude of the external field strength, which
indicates that the magnetic flux is efficiently transported to the inner
region of the disk by the hot corona. Due to the presence of the fast moving
corona above the disk, the magnetic flux is efficiently transported to the
inner region of the disk, which results in a steep slope of the magnetic field
strength as a function of radius (see Figure 2), while it is almost flat for a
traditional viscous thin disk, i.e., the external field is almost not
amplified in a conventional thin disk (see the black line in Figure 6 of Li &
Cao, 2021, for comparison).

###  4.2 outflow dynamics

The radial velocities of the disk-corona and the poloidal velocities of the
outflows are plotted in Figure 3. The poloidal velocity $v_{\rm p}$, parallel
to the poloidal field line (solid curves), of the outflow is scaled by the
local Keplerian velocity of the field line foot point $R_{z_{\rm
h}}\Omega_{\rm k}$ and the light speed $c$, respectively. The fast inward
moving corona is responsible for the radial magnetic flux transport. We assume
that the outflows are launched from the surface of the corona (i.e., $z=z_{\rm
h}$), which serves as a reservoir of the gas to feed the outflows. In this
case, the magnetic flux is mainly dragged inward by the corona, thus the gas-
to-magnetic ratio at the corona surface should be greater than unity, which
means the outflows are driven by a relatively weak magnetic field (compared
with gas pressure) (Cao, 2018). The derived slow sonic points of the outflows
are therefore very close to the surface of the corona (see red-dashed line in
Figure 3).

The outflows accelerated by the magnetic field pass through the fast
magnetosonic surface (see the green dashed lines in Figure 3), the terminal
poloidal velocities of the outflows can be as high as $\sim 0.01c-0.1c$, which
may account for some fast outflows observed in the XRBs and AGNs. The terminal
speeds of the outflows for case 2: $\Theta_{0}=0.0025,\Theta_{z_{\rm
h}}=0.025$ is slightly larger than that of the case 1: $\Theta_{z_{\rm
h}}=0.005,\Theta_{z_{\rm h}}=0.05$ (Figure 3, see also the solid lines in the
left panel of Figure 8). This is caused by the fact that the field line
inclination angle of case 2 is slightly larger than that of case 1 and the
resultant a deeper effective potential barrier along the field line, and
correspondingly a smaller mass loss rate [$\dot{M}_{\rm w}=\dot{M}_{\rm
out}-\dot{M}_{\rm acc}=(1-\dot{M}_{\rm acc}/\dot{M}_{\rm
out})\times\dot{M}_{\rm out}$].

The densities of the disks and outflows are plotted in Figure 4. The central
high density cold disks are covered with low density hot coronas, which feed a
fraction of the gas into the outflows. The toroidal velocities are plotted in
Figures 5. In the region close to Alfvén surface the toroidal velocity of the
outflow reach the maximum value, beyond the Alfvén surface the toroidal
velocity decrease very fast due to the field line being unable to enforce the
gas co-rotating with the field anymore.

The temperatures of the outflows are plotted in figure 6, which is calculated
with a polytropic Equation of state, and we adopt an adiabatic index
$\gamma=1.4$ in most of our calculations as that adopted in the numerical
simulation work done by Zhu & Stone (2018). The temperatures of the outflows
drop quickly (about several orders of magnitude) along the filed line (see the
left panel of Figure 6 and right panel of Figure 10), but it can still reach
several ten keV at a large distance from the disk (see Figure 6), which means
that the gas in the outflows accelerated from the surface of the corona may
naturally reach a high ionization state. However, we should be cautious,
because the temperature of the outflows depends on the cooling and heating
processes in the outflows, which have not been considered in this work. We
also give the results calculated with $\gamma=5/3$ for comparison, and find
that the results are not sensitive to the value of $\gamma$. The slow sonic
points and the Alfvén points are almost remain unchanged see the left panel of
Figures 4, 10 and 11. The accretion rate of the disk-corona does not show a
discernible variation (see right panel of Figure 11), while only slightly
shifts to large radii for the fast sonic points of the outflows.

The spatial structures of the field lines and its projection onto the
equatorial plane are plotted in Figure 7. The emergent spectra (especially
their polarization) of the disk-corona systems can be calculated when the
field configuration/strength derived here is properly included. The poloidal
and toroidal velocities of the outflows along a single field line
(corresponding to the field line plotted in the upper panel of Figure 7) are
plotted in the left panel of Figure 8. In the right panel of Figure 8, the
mass accretion rates within the disk are plotted, which shows only a small
fraction ($\sim 7\%-12\%$) of the accreting gas in the disk has been driven
into the outflows.

Figure 3: The velocity profiles of the disks and the outflows. The upper
panels are calculated for case 1: $\Theta_{0}=0.005,\Theta_{z_{\rm h}}=0.05$,
while the lower panels are calculated for case 2:
$\Theta_{0}=0.0025,\Theta_{z_{\rm h}}=0.025$. The blue region show the radial
velocities of the disks $v_{R}$, while the red areas show the derived poloidal
velocities of the outflows $v_{\rm p}$ ($c$ is the light speed and $R_{z_{\rm
h}}\Omega_{\rm k}$ is the local Keplerian velocity at the footpoint of the
magnetic field line). The dashed lines correspond to the position of the slow
sonic point (red), Alfvén point (blue) and the fast magnetosonic point
(green), respectively. The solid curves indicate the poloidal magnetic field
lines. Figure 4: The same as Figure 1 but for the density profiles of the
disks and the outflows, and $\rho(R_{{\rm in}},z_{\rm h})$ is the gas
densities of the gas at the upper corona surface (i.e., $z=z_{\rm h}$) at
radius $R=R_{\rm in}$. Figure 5: The same as Figure 1 but for the toroidal
velocity profiles of the outflows. Figure 6: The same as Figure 1 but for the
gas temperatures (in units of electron-volt) of the outflows. Figure 7: The
same as Figure 1, but for the configurations of the field lines. Upper panels
are the spatial structure of the field lines, and the radius of the field line
foot point is about $R/R_{\rm in}\sim 2$. Lower panels show the projection of
the field line onto the disk mid-plane. Two adjacent lines have a toroidal
angle difference of $\pi/4$. Figure 8: The same as Figure 2. Left panel: the
poloidal and toroidal velocity of the outflows along the magnetic field lines
shown in the upper panels of Figure 7. The red lines correspond to the field
line in upper-left panel of Figure 7, while the blue lines corresponds to the
field line in upper-right panel of Figure 7. The solid lines indicate the
poloidal velocities $v_{\rm p}$ of the outflows, while the dotted lines are
the toroidal velocities $v_{\varphi}$. The solid points, from left to right,
on each curves are the slow, Alfvén and fast points, respectively. Right
panel: the mass accretion rates of the disks as functions of radius, where
$\dot{M}_{\rm out}$ is the accretion rate at radius $R=R_{\rm out}$. The
adiabatic index $\gamma=1.4$ is adopted in the calculations. Figure 9:
Velocity profiles of the disks and the outflows. The parameters
$\Theta_{0}=0.005\;\text{and}\;\Theta_{z_{\rm h}}=0.05$ are adopted, which is
the same as the upper panel of Figure 3 but $\gamma=5/3$ is adopted in the
calculations.  Figure 10: Left panel: density profile, the same as the left
panel of Figure 4 but for $\gamma=5/3$. Central panel: toroidal velocity
profile, the same as the left panel of Figure 5 but for $\gamma=5/3$. Right
panel: temperature profile, the same as the left panel of Figure 6 but for
$\gamma=5/3$.  Figure 11: The same as Figure 9, the red line shows the result
for $\gamma=1.4$ while the green line is for $\gamma=5/3$. Left panel: solid
line is labeled for poloidal velocity of the outflow $v_{\rm p}$, dotted line
is for toroidal velocity $v_{\varphi}$. Right panel is the mass accretion
rates in the disks.

##  5 Discussion

As we know that the flux transport within a thin disk is very inefficient, a
fast inward moving corona above a thin disk is expected to alleviate this
dilemma (Guilet & Ogilvie, 2012; Lovelace et al., 2009; Li & Cao, 2021). The
configurations of the large-scale poloidal magnetic field advected by a disk-
corona system are shown in Figure 1 of Li & Cao (2021), and we find that the
field lines are significantly inclined to the surface of the corona, which
implies that the gas in the corona will be easily driven into outflows by the
field lines co-rotating with the gas in the disk-corona (Blandford & Payne,
1982). The field configurations derived in our work are qualitatively
consistent with those obtained in some previous numerical simulations (Zhu &
Stone, 2018; Mishra et al., 2020). Based on the configuration/strength of a
large-scale magnetic field advected by a disk-corona system (Li & Cao, 2021),
we calculate the dynamical properties of the outflows.

We find that the inclination angle of the field line with respect to the
surface of the corona decreases with increasing radius. At the large radius,
the effective potential barrier along the field line becomes shallow, which
makes the slow sonic points shifting close to the surface of the corona. It
leads to azimuthal field component being much stronger than the poloidal one,
which may suffer non-axis-symmetric instabilities. The assumption of a
potential field above the corona in this work is no longer valid, and the
winds may probably be magnetically propelled circulation along the corona
surface. A very similar case of the outflows magnetically driven from the
outer region of the disk has been discussed in Cao & Spruit (1994). As we are
more interested in the fast moving outflows from the inner region of the disk-
corona, we will not be involved in such complexity of the winds driven from
the outer region of the disk-corona.

Due to a relatively larger mass loss rate in the outflow (i.e., a heavier
outflow), it is therefore within a smaller radius the field line can enforce
the gas in heavier outflow co-rotating with it, which leads to a smaller
terminal velocity of the outflow. The toroidal field component is rapidly
developed within a short distance from the field footpoint, as shown in the
upper panel of Figure 7. Beyond the Alfvén surface the outflow is mainly
driven by the magnetic gradient force due to the strong toroidal field. As
mentioned above, all these two cases are calculated under the context of weak
magnetic field, thus the field line can not enforce the gas in the outflow to
co-rotate with the field line to a large radius where the toroidal component
of the field dominates over the poloidal component. As a result, the position
of the maximum toroidal velocity of the outflow along the field line is
located slightly above the surface of the corona, and it’s value is close to
local Keplerian velocity (see Figure 5), which is near the Alfvén surface (see
the dotted lines in left panel of Figure 8). The properties of the
magnetically driven outflows may also be affected by their radiation, which is
beyond the scope of this work. Fortunately, the adiabatic index could describe
its impact on the outflow dynamics to some extent. We re-calculate the whole
problem with a different value of $\gamma=5/3$ for comparison, which are
plotted in Figures 9-11, as mentioned in Section 4.2 we find that the results
are not sensitive to the value of $\gamma$.

As we know that the dynamical structure of the disk, the large-scale field
configuration of the disk, and the outflows are tightly coupled. To derive the
dynamical structure of the disk, for instance, one needs to know the outflow
properties first, while the calculations of the outflow solution require the
specific field configuration/strength above the disk, which in turn decided by
the whole structure of the disk itself (see Li & Cao, 2019, for the details).
In this work, the field advected by the corona is relative weak, which leads
to outflows with low or moderate power. As we focus on the dynamical
properties of the outflows, we neglect the impact of the outflows on the disk-
corona structure, which is justified in part by the fact that only a small
fraction of gas is driven into the outflows (see right panel of Figure 8).

The plasma $\beta$ parameter is important in deriving the properties of the
outflow. In the field advection scenario of this work, the gas-to-magnetic
ratio at the upper surface of the corona should be greater than unity, the
field line is easily sheared into toroidal component in the outflow, and the
resultant Alfvén surface is close to the surface of the corona. Thus, the
outflows driven by the large-scale magnetic field advected by the corona are
in general not very powerful (relative low mass loss rate and/or low terminal
speed), which are consistent with the simple estimate in Cao (2018).
Sheikhnezami et al. (2012) also carried out time-dependent simulations on the
outflows/jets launched magnetically from diffusive accretion disks, and they
confirmed that the magnetocentrifugal acceleration is more efficient in a
strong magnetic case (i.e., a lower plasma $\beta$), which may imply the
outflow accelerated from a MAD (magnetically arrested disks) can reach a very
high velocity with a low mass loss rate. Such outflows may be able to account
for the UFOs with mild relativistic velocities. In their work, the forces
exerted on the outflow in the direction parallel and perpendicular to the
field line are depicted, which clearly verify the relative importance of the
forces in the acceleration process. However, we are not able to separate the
forces (i.e., the gravity, pressure gradient, Lorentz, and centrifugal)
exerted on the outflow in our calculations, as our calculations are carried
out in the coordinates of the parallel and perpendicular to the field line,
i.e., the force consists of the parallel and perpendicular components (see
left panel of Figure 8), with which, in principle, the aforementioned forces
can be derived, however, it would be very complicated. It would be worth
trying to extract these forces information in our future work.

Although it is well believed that magnetic field must play an important role
in the formation of the outflows/winds and the relativistic jets (Blandford &
Znajek, 1977; Blandford & Payne, 1982), direct observation of the magnetic
field is quite unlikely. Polarization observations of the emission from the
relevant objects provide a powerful tool to measure the field
configuration/strength, though it is usually somewhat model dependent. In this
work, we have derived three-dimensional field configuration/strength (see
upper panel of Figure 7), the density and velocity of the outflows, which can
be used to calculated not only the emergent spectra of the disk-corona system
but also the emergent spectrum of the outflows. Compared with the polarization
observations, the derived results will surely help understand the physics of
magnetic field origin in the disks and magnetically driven outflows. Recent
high resolution polarization observations of M87 by the event horizon
telescope (EHT) provided evidence of strong magnetic field near the black
hole, i.e., the accretion flow around the black hole is likely to be in the
MAD state (Event Horizon Telescope Collaboration et al., 2021). Similar
observations on the objects containing outflows magnetically driven from the
disk-corona would be an ideal test of our model calculations, or the expansion
of our calculations to the outflows driven from the MAD can be used to explain
the observations of M87, which is beyond the scope of this work.

In black hole accreting system, the hot corona is believed to be responsible
for the non-thermal radiation in the X-ray bands (Haardt & Maraschi, 1993).
Our calculations show that a fraction of the gas in the hot corona can be
tapped into the outflows, which implies an outflowing corona located in the
transition region of the corona to the outflow. Indeed, some early X-ray
observations seem to require such outflowing coronas in the XRBs and AGNs
(e.g., Beloborodov, 1999; Liu et al., 2014). Our model calculations provide a
natural origin for such outflowing coronas. The further detailed calculations
of the spectra of the disk-corona outflow system derived in this work would
help understand the features of the X-ray observations in XRBs and AGNs, which
will be reported in the future work.

We thank Xiaodong Duan and Weixiao Wang for the helpful discussions, and thank
the referee for valuable comments/suggestions. This work is supported by the
NSFC (11773050, 11833007, 12073023), the science research grants from the
China Manned Space Project with NO. CMS-CSST-2021-A06, the CAS grant QYZDJ-
SSWSYS023, and China Postdoctoral Science Foundation NO. 2021M702865.

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Generating Synthetic Satellite Imagery
With Deep-Learning Text-to-Image Models
Technical Challenges and Implications for Monitoring and Verification

Tuong Vy Nguyen,∗ Alexander Glaser,†,‡ and Felix Biessmann∗,‡

∗Berliner Hochschule für Technik

†Program on Science and Global Security, Princeton University

‡Einstein Center Digital Future, Berlin

Abstract. Novel deep-learning (DL) architectures have reached a level where
they can generate digital media, including photorealistic images, that are
difficult to distinguish from real data. These technologies have already been
used to generate training data for Machine Learning (ML) models, and large
text-to-image models like DALL$\cdot$E 2, Imagen, and Stable Diffusion are
achieving remarkable results in realistic high-resolution image generation.
Given these developments, issues of data authentication in monitoring and
verification deserve a careful and systematic analysis: How realistic are
synthetic images? How easily can they be generated? How useful are they for ML
researchers, and what is their potential for Open Science? In this work, we
use novel DL models to explore how synthetic satellite images can be created
using conditioning mechanisms. We investigate the challenges of synthetic
satellite image generation and evaluate the results based on authenticity and
state-of-the-art metrics. Furthermore, we investigate how synthetic data can
alleviate the lack of data in the context of ML methods for remote-sensing.
Finally we discuss implications of synthetic satellite imagery in the context
of monitoring and verification.

##  1 Introduction

Novel technology for generating media content with automated and scalable
machine learning (ML) technology has become a focus of attention in many
application domains. In a future where digital data from a variety of sources
are abundant and widely available to non-governmental experts and independent
analysts, and where virtually any type of digital media can be generated in
ways that can make them effectively indistinguishable from real data, issues
of data authentication in monitoring and verification deserve a careful and
systematic analysis. While the impact of these developments on security
aspects has been studied for text data in the context of misinformation in
online social networks,11endnote: 1Ben Buchanan, Andrew Lohn, Micah Musser,
Katerina Sedova, Truth, Lies, and Automation, Center for Security and Emerging
Technology, Georgetown University, Washington, DC, May 2021. the challenges as
well as the potential of synthesized data in the image domain have been
underrepresented so far.

In this study, we assess the potential and risks associated with novel image
generation methods such as in the context of monitoring and verification with
a focus on satellite imagery.

More concretely we first investigate the technological requirements to apply
and adapt state of the art technology in image generation for the application
scenario of nuclear verification with satellite imagery. In order to assess
how easily the image generation can be controlled, we then generate a
heterogeneous set of synthetic images and change a number of semantic
dimensions in the image generation, such as seasonality, image context and
time of day. We evaluate the quality of the generated images using established
metrics and metrics that were specifically adapted to the task of remote
sensing data.

##  2 Datasets

To compare the capabilities of the different approaches over a wider range of
classes, two different datasets comprised of various objects, are leveraged.

Nuclear Facilities. We use two different approaches to [fine-tune/train] the
algorithms and generate synthetic imagery of nuclear power plants. First, we
use different versions of a single input image of the Neckarwestheim nuclear
power plant in Germany by varying the zoom factor and the rotation of the
image (Figure 1, left). Second, we obtain training data from a large number of
nuclear power plants using the Google Earth Engine (EE) in combination with
web scraping tools.22endnote: 2For this imagery, we used the “Planet SkySat
Public Ortho Imagery” image collection collected in 2015 by Planet Labs’
SkySat satellites. Images of sites that were blurred or of low quality were
removed, leaving 202 images of 185 unique nuclear power plants, which were
then used for fine-tuning (Figure 1, right).

Figure 1: Shown on the left are six variations of the input image of the
Neckarwestheim nuclear power plant; on the right are eight sample images of
nuclear power plants from different regions of the world. Overall, there are
202 input images in our dataset. Source: Google Earth.

UC Merced Land-use Dataset. Another dataset used for our fine-tuning
approaches is the UC Merced (UCM) dataset,33endnote: 3Yi Yang and Shawn
Newsam, Bag-Of-Visual-Words and Spatial Extensions for Land-Use
Classification, ACM SIGSPATIAL International Conference on Advances in
Geographic Information Systems (ACM GIS), 2010. a benchmark land-use
classification dataset from 2010, containing extracted images from the USGS
National Map Urban Area Imagery collection of urban areas in the United
States. The UCM images measure 256$\times$256 pixels and have a resolution of
about 30 cm. The set contains 21 different classes with 100 images each for a
total of 2100 images. Text captions for these images complement the
dataset.44endnote: 4Bo Qu, Xuelong Li, Dacheng Tao, Xiaoqiang Lu, Deep
Semantic Understanding of High Resolution Remote Sensing Image, 2016
International Conference on Computer, Information and Telecommunication
Systems (CITS), 2016. The splitting for training and testing was done
according to the pre-defined split in the captions set, using 1680 images for
fine-tuning and 210 each for validation and testing, respectively.

##  3 Methods and Implementation

Research on generative models for images has made dramatic progress over the
past few years. For a long time, the state of the art in this novel domain
have been Generative Adversarial Networks (GAN),55endnote: 5Ian J. Goodfellow
et al., “Generative Adversarial Nets,” arXiv:1406.2661, June 2014. which have
been explored thoroughly and applied in various domains. More recently, novel
approaches to image generation based on Diffusion Models (DM) overcame some of
the challenges associated with GAN training, resulting in a number of models
that achieved higher image quality.66endnote: 6Jascha Sohl-Dickstein, Eric A.
Weiss, Niru Maheswaranathan, Surya Ganguli, “Deep Unsupervised Learning using
Nonequilibrium Thermodynamics,” arXiv:1503.03585, March 2015; Jonathan Ho,
Ajay Jain, Pieter Abbeel, “Denoising Diffusion Probabilistic Models,”
arXiv:2006.11239, June 2020. Google’s Imagen and OpenAI’s DALL$\cdot$E 2 are
two well-known examples.77endnote: 7Chitwan Saharia et al.,“Photorealistic
Text-to-Image Diffusion Models with Deep Language Understanding,”
arXiv:2205.11487, May 2022; Aditya Ramesh, Prafulla Dhariwal, Alex Nichol,
Casey Chu, Mark Chen, “Hierarchical Text-Conditional Image Generation with
CLIP Latents,” arXiv:2204.06125, April 2022. For our work, we have opted for
Stable Diffusion as the underlying pre-trained text-to-image model for our
purposes, as the source code and model weights have been made publicly
available unlike with the other two models, and additional libraries and tools
have been published as well, making the implementation fairly easy. Stable
Diffusion is based on Latent Diffusion Models (LDM).88endnote: 8Robin Rombach,
Andreas Blattmann, Dominik Lorenz, Patrick Esser, Björn Ommer, “High-
Resolution Image Synthesis with Latent Diffusion Models,” arXiv:2112.10752,
December 2021.

Diffusion Models consist of two processes illustrated in Figure 2. Moreover,
LDMs make use of an Autoencoder architecture, transforming the images into a
lower-dimensional latent space first before inputting this into the DM
component. This allows for reduction of computational needs while still
retaining image quality on similar level. For conditioning, a pre-trained text
encoder is leveraged to transform the text prompts into embeddings and use
them as conditioning inputs for the model. These and the perturbed latent
representations are then used to train the denoising text-to-image U-Net
component of Stable Diffusion. The obtained latents are then put through the
Autoencoder’s decoder to retrieve a sample back in pixel space.

Figure 2: Basic principle of Diffusion Models. During the first stage, the
forward noising process, images are gradually perturbed by adding Gaussian
noise. In the second stage, the reverse denoising process, a neural network is
then learned to remove the noise iteratively to retrieve the original image.

To test the generalizability and the model’s zero-shot capabilities, we use
the unmodified pre-trained model as a baseline to examine how well the prior
knowledge can be leveraged to generate satellite and aerial imagery. We then
apply several fine-tuning approaches to further train the model on the
datasets described above. For the case of nuclear facilities, we then
investigate what and how well conditionings can be applied to those newly
learned concepts by adding keywords to the text prompts for variations
regarding the location, seasonality and the time of day, for example,
generating images of a nuclear power plant in the desert or in the winter. We
also apply the fine-tuning methods on our aerial imagery dataset, UCM, which
is comprised of more general known objects.

For implementation, we use three different approaches: DreamBooth
(DB),99endnote: 9Nataniel Ruiz, Yuanzhen Li, Varun Jampani, Yael Pritch,
Michael Rubinstein, Kfir Aberman, “DreamBooth: Fine Tuning Text-to-Image
Diffusion Models for Subject-Driven Generation,” arXiv:2208.12242, August
2022. Textual Inversion (TI),1010endnote: 10Rinon Gal, Yuval Alaluf, Yuval
Atzmon, Or Patashnik, Amit H. Bermano, Gal Chechik, Daniel Cohen-Or, “An Image
is Worth One Word: Personalizing Text-to-Image Generation using Textual
Inversion,” arXiv:2208.01618, August 2022. and Text-to-Image (Text2Img) fine-
tuning.1111endnote: 11huggingface.co/docs/diffusers/training/text2image.
Results are then compared to our baseline, vanilla Stable Diffusion
(SDiff).1212endnote: 12As opposed to fine-tuned models, vanilla means our pre-
trained base model has not been further modified and is used off-the-shelf.
DreamBooth and Text2Img train the U-Net component of the model, while Textual
Inversion trains in the text embedding space of the model’s text encoder. For
the nuclear facilities, we train on both the single nuclear plant in
Neckarwestheim and on all facilities, separately. For UCM, we train a model
for every of the single 21 land-use classes, using five randomly selected
input images each. Furthermore, Huggingface’s Diffusers Library is heavily
used for the training and generative processes.

##  4 Results

As part of this project, we have generated a large number of synthetic
imagery. Ultimately, these images need to be characterized and assessed in
order to draw conclusions about their usefulness for training purposes or
other applications. Qualitatively, we have found that each method has their
own strengths and weaknesses; these are discussed in greater detail in a
forthcoming report.1313endnote: 13Tuong Vy Nguyen, “Machine Learning for
Synthetic Satellite Images: Conditional Image Generation using a Vision-
Language Model,” Master’s Thesis, Berliner Hochschule für Technik, Berlin, May
15, 2023. As an example, Figure 3 shows some sample images produced with
DreamBooth for the Neckarwestheim reactor sample; in general, DreamBooth is
particularly strong at preserving image fidelity and reproducing distinct
features. Furthermore, the fewer images are used as input, the better this
seems to work for these subject-driven fine-tuning methods. This is plausible
as with more input images, there are more specifics and differences to
consider and preserving fine-grained details becomes more difficult when
binding this to a single identifier or token.

Figure 3: DreamBooth Neckarwestheim. Ten selected samples of synthetic images
with their respective text prompts. Depicted are variations regarding
seasonality, location, and time of day, all based on the original
Neckarwestheim imagery from Figure 1.

For the UCM case, we evaluate the different methods based on a qualitative and
quantitative assessment. Furthermore, a downstream task and user study are
conducted. Looking at the generated samples for each applied method, we can
observe that fine-tuning Stable Diffusion, our baseline and pre-trained text-
to-image model, leads to imagery more suitable to the remote sensing domain.
Furthermore, with the Text2Img model, the perspective from above, so our
desired viewing angle, often seems to be retained as well, even for other
objects or concepts, which are not present in the UCM dataset.

Figure 4: UCM, Text2Img. Shown are 36 randomly selected samples of synthetic
imagery. The data was generated with the Text2Img model, based on the text
captions provided in the UCM dataset.

Ultimately, however, a systematic assessment requires quantitive metrics for
an evaluation of generative models and their synthesized data; the two main
ones currently in use are the Inception Score (IS) and the Fréchet Inception
Distance (FID).1414endnote: 14Tim Salimans, Ian Goodfellow, Wojciech Zaremba,
Vicki Cheung, Alec Radford, Xi Chen, “Improved Techniques for Training GANs,”
arXiv:1606.03498, June 2016; Martin Heusel, Hubert Ramsauer, Thomas
Unterthiner, Bernhard Nessler, Sepp Hochreiter, “GANs Trained by a Two Time-
Scale Update Rule Converge to a Local Nash Equilibrium,” arXiv:1706.08500,
June 2017. The IS only assesses the synthetic images and evaluates image
quality and diversity. The higher the score, the better. The FID compares the
data distribution of real and fake images with each other and measures the
dissimilarity, thus, the lower, the better. Both metrics rely on the feature
space of a pre-trained Inception model, hence the name. To adapt these metrics
to our use cases, we exchange this underlying model with a pre-trained
classifier model, which we further fine-tune on real UCM data, resulting in
IS∗ and FID∗. The Fréchet Clip Distance (FCD) is another modification of the
original FID, where the model is replaced by a pre-trained CLIP
model.1515endnote: 15Eyal Betzalel, Coby Penso, Aviv Navon, Ethan Fetaya, A
Study on the Evaluation of Generative Models; betzalel-study-2022 and radford-
clip-2021.

Nuclear facilities. Due to lack of sufficient real data for comparison, only
the IS and IS∗ are calculated for nuclear reactors considered in this study.
Using different prompts where variations regarding the different dimensions
seasonality, location, time of day and plain (no additional conditioning) are
added to the text prompt. This is done for sample sizes $n=202$ and $n=6000$,
respectively, to make it somewhat comparable to the real training images. The
results are found in Table 1.

Model/Data |  $\text{IS}_{202}$ $\uparrow$ |  $\text{IS}^{*}_{202}$ $\uparrow$ |  $\text{IS}_{6000}$ $\uparrow$ |  $\text{IS}^{*}_{6000}$ $\uparrow$
---|---|---|---|---
Real train images | 3.12 ± 0.44 | 3.80 ± 0.43 | — | —
DB Neckar | 2.84 ± 0.38 | 2.13 ± 0.21 | 3.10 ± 0.08 | 2.26 ± 0.08
TI Neckar | 4.06 ± 0.42 | 3.52 ± 0.93 | 5.53 ± 0.11 | 4.13 ± 0.11
DB All | 2.30 ± 0.26 | 2.49 ± 0.28 | 2.60 ± 0.07 | 2.76 ± 0.05
TI All | 3.36 ± 0.36 | 4.26 ± 0.66 | 4.97 ± 0.13 | 5.39 ± 0.15
Van. SDiff | 2.99 ± 0.32 | 3.25 ± 0.59 | 3.75 ± 0.09 | 4.03 ± 0.12

Table 1: Quantitative evaluation results. Scores for the nuclear power plant
models have been calculated on the sample sizes 202 and 6000 each. $n=202$ was
chosen so the scores are comparable to the IS of our real data (the images of
nuclear power plants used for training). Larger values ($\uparrow$) are
better.

While the results appear to suggest that the TI approach yields better scores
than the pre-trained Vanilla Stable Diffusion model, it is difficult to draw
conclusions from the IS and the modified IS∗ alone. Surprisingly, we find that
the real satellite images used for training do not even achieve the best
scores. This highlights the challenges associated with automated evaluation of
generative methods, as well as the need to develop improved metrics. In
addition, the two metrics do not reflect how well the generated images align
with the text prompts or how accurately the semantic conditions and given
variations can be displayed. Thus, a user study should be conducted to further
investigate how the images are perceived by human evaluators.

UC Merced Land-use Dataset. For the quantitative assessment in the UCM case,
we generate 420 samples for each method and compare their obtained
quantitative results. It can be observed that for most of the metrics, the
ranking is as follows: Text2Img outperforms the other models, followed by DB,
then TI and, lastly, SDiff. This only deviates with the original IS, Recall
and FID∗. Randomly selected samples of each approach are shown in the
Appendix. These results indicate that if enough image data are available in a
given domain to fine-tune the models used in the empirical evaluation of
synthetic data as well as to calibrate the evaluation methods, the metrics we
investigated are consistent in that they yield robust rankings of image
generation methods.

Model/Data |  IS $\uparrow$ |  IS∗ $\uparrow$ |  FID $\downarrow$ |  FID∗$\downarrow$ |  FCD $\downarrow$ |  KID $\downarrow$ |  Precision $\uparrow$ |  Recall $\uparrow$
---|---|---|---|---|---|---|---|---
UCM Val+Test | 5.85±0.68 | 13.78±1.30 | — | — | — | — | — | —
Text2Img | 6.86±0.80 | 13.01±0.85 | 139.65 | 23.32 | 12.62 | 0.01±0.02 | 0.56 | 0.38
DB | 5.99±0.44 | 10.40±0.71 | 171.69 | 35.67 | 15.99 | 0.01±0.02 | 0.34 | 0.27
TI | 6.08±0.73 | 8.10±0.70 | 177.61 | 22.13 | 16.75 | 0.02±0.02 | 0.33 | 0.17
SDiff | 7.55±1.23 | 7.27±0.71 | 207.41 | 65.55 | 41.15 | 0.05±0.02 | 0.02 | 0.48

Table 2: Quantitative evaluation results on UCM. The real UCM test and
validation sets were used as one test set, resulting in $n=420$ real samples
to compare against. Arrows indicate whether larger ($\uparrow$) or lower
($\downarrow$) values are better.

UCM User Study. For the user study, we used 210 images of each approach,
resulting in 840 images. These were annotated by in total four users, with
each image receiving one label. For four of the tasks the annotation was
missing due to technical error, hence they were left out, leaving 836
annotated images to evaluate. The success rate is defined as the number of
images that were classified as real for each method. The results of the user
study are found in Table 3.

Method | Real images | Text2Img | DB | TI
---|---|---|---|---
Predicted as “Real” | 158 | 119 | 115 |  069
Predicted as “Fake” |  052 |  089 |  094 | 140
Success Rate |  75% |  57% |  55% |  33%

Table 3: User study results. Synthetic and real images were annotated as real
or fake during an experiment with human evaluators. The success rate is shown
for each approach. For example, out of the 210 real UCM images, 158 were
classified as real, resulting in a success rate of 75%; the other 52 images
were labeled as fake, even though they were real.

Overall, Text2img achieves the best results, followed by DB and TI. This
ranking matches most of the applied quantitative evaluation metrics.
Especially our adapted IS∗ aligns better with this than the original IS.
Surprisingly, this is not the case for the adapted FID∗ though, where the
original FID and FCD fit better. The specific reasons for this require further
research, one assumption is that the similarity of real and synthetic images
in the feature space captured by the neural networks doesn’t necessarily say
something about how real or authentic an image is actually perceived as and
the used feature space might not be suitable either. Furthermore, the FID∗
scores still fluctuate for the used sample sizes, thus, don’t seem to be as
robust as the original FID, and the ranking might change with increasing
sample size.

The results of the user study allow for a first comparison between the
different evaluation metrics and how they correlate with human visual
perception. Most importantly, the results demonstrate that all of the tested
approaches can generate imagery that is, partly, able to fool the untrained
human eye. Especially using DB or the Text2Img approach, where over 50% of the
synthetic imagery from both methods was labeled as real by human evaluators.
25% of the real images were also classified as fake, meaning the generated
images managed to mislead the users in that regard as well.

##  5 Conclusion and Outlook

Overall, it can be observed that, though not on par with real data, synthetic
data can obtain evaluation scores of the same order of magnitude and we are
able to, partly, generate realistic-looking images that can fool the human
eye. Furthermore, fine-tuning the text-to-image model itself, so the U-Net
component, seems to be the most effective space to further train, as this
gives the best results (Text2Img and DB both do this).

The development of suitable evaluation metrics for synthetic image quality is
still an open field of research. Our results show that while in some cases,
especially when not enough data is available to calibrate evaluation methods,
existing evaluation metrics do not yield robust and reliable estimates of the
synthetic image quality. At the same time our experiments demonstrate that if
enough data is provided for calibrating quality metrics for synthetic imagery,
existing metrics can work well: When adapted to the remote sensing domain, the
metrics investigated yield robust rankings of image generation methods that
aligns well with human perception. These findings underline the need for more
research on quantitative evaluation metrics for synthetic data. We believe
that this development requires a transdisciplinary effort including experts in
deep learning and the specific domain in which data is generated.

While the generation of artificial images is very interesting from a technical
viewpoint, there are ethical and societal aspects to consider: Our results
demonstrate that with open access to powerful models, technologies, and tools,
low resource requirements and a feasible computing time, virtually anyone has
the means to generate synthetic data at a larger scale with seemingly the
touch of a button. Consequently, since image synthesis and also spoofing have
become fairly easy to execute, the threat of malicious manipulations rises,
with implementation becoming even accessible to non-experts and laymen. Using
current methods, misinformation can quickly be spread through social media and
various other outlets, and viewers are mislead by generated content.
Recognizing deepfakes becomes increasingly difficult with the emerging
generative technologies and non-technical solutions could also be considered,
such as watermarking.

Considering our domain, artificially generated satellite imagery, especially
at a large scale, can alleviate the issue of inaccessible and unannotated
data. However, this provides grounds for malicious purposes as synthetic
satellite images, which contain real and fabricated features, are a notable
threat regarding fake geography1616endnote: 16Bo Zhao, Shaozeng Zhang, Chunxue
Xu, Sun Yifan, Chengbin Deng, ”Deep fake geography? When geospatial data
encounter Artificial Intelligence”, 10.1080/15230406.2021.1910075, April
2021.. We argue that the development of new generative approaches should be
accompanied by better methods to evaluate synthetic image quality and
detection of synthetic imagery, particularly in the remote sensing
domain1717endnote: 17Lydia Abady, Edoardo Daniele Cannas, Paolo Bestagini,
Benedetta Tondi, Stefano Tubaro, and Mauro Barni, ”An Overview on the
Generation and Detection of Synthetic and Manipulated Satellite Images”,
arXiv.2209.08984 September 2022..

Concluding, deepfakes can be a serious threat if containing compromising
content and are circulated in public. With emerging technologies and tools,
even non-experts have the means to generate large-scale synthetic datasets
without proper safeguards set in place. The detection of fake data and a
thorough investigation of the impact such images could have when widely
distributed is an important aspect and should be further investigated in the
future.

###### Notes and References

  1. 1 Ben Buchanan, Andrew Lohn, Micah Musser, Katerina Sedova, Truth, Lies, and Automation, Center for Security and Emerging Technology, Georgetown University, Washington, DC, May 2021.
  2. 2 For this imagery, we used the “Planet SkySat Public Ortho Imagery” image collection collected in 2015 by Planet Labs’ SkySat satellites.
  3. 3 Yi Yang and Shawn Newsam, Bag-Of-Visual-Words and Spatial Extensions for Land-Use Classification, ACM SIGSPATIAL International Conference on Advances in Geographic Information Systems (ACM GIS), 2010.
  4. 4 Bo Qu, Xuelong Li, Dacheng Tao, Xiaoqiang Lu, Deep Semantic Understanding of High Resolution Remote Sensing Image, 2016 International Conference on Computer, Information and Telecommunication Systems (CITS), 2016.
  5. 5 Ian J. Goodfellow et al., “Generative Adversarial Nets,” arXiv:1406.2661, June 2014.
  6. 6 Jascha Sohl-Dickstein, Eric A. Weiss, Niru Maheswaranathan, Surya Ganguli, “Deep Unsupervised Learning using Nonequilibrium Thermodynamics,” arXiv:1503.03585, March 2015; Jonathan Ho, Ajay Jain, Pieter Abbeel, “Denoising Diffusion Probabilistic Models,” arXiv:2006.11239, June 2020.
  7. 7 Chitwan Saharia et al.,“Photorealistic Text-to-Image Diffusion Models with Deep Language Understanding,” arXiv:2205.11487, May 2022; Aditya Ramesh, Prafulla Dhariwal, Alex Nichol, Casey Chu, Mark Chen, “Hierarchical Text-Conditional Image Generation with CLIP Latents,” arXiv:2204.06125, April 2022.
  8. 8 Robin Rombach, Andreas Blattmann, Dominik Lorenz, Patrick Esser, Björn Ommer, “High-Resolution Image Synthesis with Latent Diffusion Models,” arXiv:2112.10752, December 2021.
  9. 9 Nataniel Ruiz, Yuanzhen Li, Varun Jampani, Yael Pritch, Michael Rubinstein, Kfir Aberman, “DreamBooth: Fine Tuning Text-to-Image Diffusion Models for Subject-Driven Generation,” arXiv:2208.12242, August 2022.
  10. 10 Rinon Gal, Yuval Alaluf, Yuval Atzmon, Or Patashnik, Amit H. Bermano, Gal Chechik, Daniel Cohen-Or, “An Image is Worth One Word: Personalizing Text-to-Image Generation using Textual Inversion,” arXiv:2208.01618, August 2022.
  11. 11 huggingface.co/docs/diffusers/training/text2image.
  12. 12 As opposed to fine-tuned models, vanilla means our pre-trained base model has not been further modified and is used off-the-shelf.
  13. 13 Tuong Vy Nguyen, “Machine Learning for Synthetic Satellite Images: Conditional Image Generation using a Vision-Language Model,” Master’s Thesis, Berliner Hochschule für Technik, Berlin, May 15, 2023.
  14. 14 Tim Salimans, Ian Goodfellow, Wojciech Zaremba, Vicki Cheung, Alec Radford, Xi Chen, “Improved Techniques for Training GANs,” arXiv:1606.03498, June 2016; Martin Heusel, Hubert Ramsauer, Thomas Unterthiner, Bernhard Nessler, Sepp Hochreiter, “GANs Trained by a Two Time-Scale Update Rule Converge to a Local Nash Equilibrium,” arXiv:1706.08500, June 2017.
  15. 15 Eyal Betzalel, Coby Penso, Aviv Navon, Ethan Fetaya, A Study on the Evaluation of Generative Models; betzalel-study-2022 and radford-clip-2021.
  16. 16 Bo Zhao, Shaozeng Zhang, Chunxue Xu, Sun Yifan, Chengbin Deng, ”Deep fake geography? When geospatial data encounter Artificial Intelligence”, 10.1080/15230406.2021.1910075, April 2021.
  17. 17 Lydia Abady, Edoardo Daniele Cannas, Paolo Bestagini, Benedetta Tondi, Stefano Tubaro, and Mauro Barni, ”An Overview on the Generation and Detection of Synthetic and Manipulated Satellite Images”, arXiv.2209.08984 September 2022.

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