# Relativistic global solutions of neutrino-dominated accretion flows with
magnetic coupling

Jiao-Zhen She, Tong Liu, Li Xue
Department of Astronomy, Xiamen University, Xiamen, Fujian 361005, China
tongliu<EMAIL_ADDRESS>
(Accepted XXX. Received YYY; in original form ZZZ)

###### Abstract

A Kerr black hole (BH) surrounded by a neutrino-dominated accretion flow
(NDAF) is one of plausible candidates of the central engine in gamma-ray
bursts. The accretion material might inherit and restructure strong magnetic
fields from the compact object mergers or massive collapsars. The magnetic
coupling (MC) process between a rapid rotating BH and an accretion disc is one
of possible magnetic configurations that transfers the energy and angular
momentum from the BH to the disc. In this paper, we investigate one-
dimensional global solutions of NDAFs with MC (MCNDAFs), taking into account
general relativistic effects, detailed neutrino physics, different MC
geometries, and reasonable nucleosynthesis processes. Six cases with different
accretion rates and power-law indices of magnetic fields are presented and
compared with NDAFs without MC. Our results indict that the MC process can
prominently impact the structure, thermal properties, and microphysics of
MCNDAFs, increase luminosities of neutrinos and their annihilations, result in
the changing of radial distributions of nucleons, and push the region of heavy
nuclei synthesis to a larger radius than counterparts in NDAFs.

###### keywords:

accretion, accretion discs - black hole physics - gamma-ray burst: general -
magnetic fields - neutrinos - nuclear reactions, nucleosynthesis, abundances

††pubyear: 2022

##  1 Introduction

Extraction of energy and angular momentum from a spinning black hole (BH) or
an accretion disc through magnetic braking is a long-standing topic. There are
several possible types of magnetic field geometries associated with this issue
(e.g. Blandford, 2002; Hirose et al., 2004; McKinney, 2005). For example,
small-scale tangled fields embedded in a disc is related to the turbulent
viscosity of accretion flow through the Balbus-Hawley instability (Balbus &
Hawley, 1991), large-scale loop fields connecting different regions of a disc
will result in the long-distance interactions between these regions (e.g.
Blandford, 2002; Hirose et al., 2004; McKinney, 2005), and large-scale open
fields emerging from a disc will lead to winds through Blandford-Payne (BP)
process (Blandford & Payne, 1982).

However, none of above geometries is directly related to BHs. Blandford &
Znajek (1977) proposed the first field geometry directly linked to BHs, which
consists of large-scale open fields emerging from the BH and ending at the
remote astrophysical load, and by which the energy and angular momentum can be
extracted from the BH and transported to distance via the Poynting flux. This
mechanism is named as the Blandford-Znajek (BZ) mechanism. BZ and BP
mechanisms had been widely considered as origins of jets or outflows (winds)
from BH binaries, active galactic nuclei, and gamma-ray bursts (GRBs, e.g.
Blandford & Znajek, 1977; Macdonald & Thorne, 1982; Rees, 1984; Cao & Spruit,
1994, 2013; Lee, Brown, & Wijers, 2000; Lee, Wijers, & Brown, 2000; Piran,
2004; Remillard & McClintock, 2006; Barkov & Komissarov, 2008; Cao, 2011). In
order to investigate interactions between a BH and a disc, the field lines
connecting the BH and disc through the plunging region were considered in a
series of works (e.g. Krolik, 1999; Gammie, 1999; Agol & Krolik, 2000; Hawley,
2000; Hawley & Krolik, 2001). Nevertheless, large-scale fields through the
disc corona rather than the unstable plunging region will lead to more intense
interactions between the BH and disc, which is generally called magnetic
coupling (MC) process (e.g. Li & Paczyński, 2000; Li, 2002). This process has
been generally adopted in researches of different classic accretion flows
around BHs, such as the thin disc (e.g. Li, 2002; Wang, Xiao, & Lei, 2002;
Wang et al., 2003; Wang, Lei, & Ma, 2003; Uzdensky, 2005; Kluzniak &
Rappaport, 2007), the slim disc (e.g. Lee, 1999), and the advection-dominated
accretion flow (e.g. Ma, Yuan, & Wang, 2007). A common conclusion amount these
works is that the MC process will largely enhance the radiation luminosity by
the additional energy extracted from BHs, meanwhile the disc structure will be
significantly changed by the additional angular momenta from BHs.

As one of candidates of GRB central engines, the neutrino-dominated accretion
flow (NDAF) has been used to explain many characteristics of GRBs in past two
decades, such as GRB jet luminosities powered by the annihilations of
neutrinos escaping from the discs (e.g. Popham, Woosley, & Fryer, 1999; Fryer
et al., 1999; Lee, Ramirez-Ruiz, & Page, 2005; Gu, Liu, & Lu, 2006; Liu et
al., 2007; Chen & Beloborodov, 2007; Zalamea & Beloborodov, 2011; Lei, Zhang,
& Liang, 2013; Xue et al., 2013; Xie, Lei, & Wang, 2016; Song, Liu, & Wei,
2020), variabilities in GRB prompt emissions driven by the instabilities of
the discs (e.g. Reynoso, Romero, & Sampayo, 2006; Janiuk et al., 2007;
Kawanaka & Mineshige, 2007; Lei et al., 2009, 2017; Liu et al., 2010;
Carballido & Lee, 2011; Kawanaka & Kohri, 2012; Kawanaka, Piran, & Krolik,
2013; Hou et al., 2014; Kimura, Mineshige, & Kawanaka, 2015; Lin et al., 2016;
Kawanaka & Masada, 2019; Shahamat, Abbassi, & Liu, 2021), extended emissions
following short-duration GRBs induced by the redistribution of the angular
momenta of the discs (e.g. Liu et al., 2012a; Cao, Liang, & Yuan, 2014), X-ray
flares originated from the instability or magnetic barriers in the discs (e.g.
Perna, Armitage, & Zhang, 2006; Proga & Zhang, 2006; Lazzati, Perna, &
Begelman, 2008; Liu et al., 2008; Lee, Ramirez-Ruiz, & López-Cámara, 2009; Luo
et al., 2013; Mu et al., 2016, 2018; Yi, Du, & Liu, 2022), brightly optical
bumps of the core-collapse supernovae (CCSNe) associated with GRBs supplied by
the disc outflows (e.g. Surman, McLaughlin, & Hix, 2006; Surman, McLaughlin, &
Sabbatino, 2011; Song & Liu, 2019; Liu et al., 2021), long-lived CCSNe with at
least two peaks produced by the disc outflow feedback (Liu et al., 2019),
luminous kilonovae driven by outflows (or the gamma-ray radiation induced by
the vertical advection) in binary compact object mergers (e.g. Yi et al.,
2017, 2018; Song, Liu, & Li, 2018; Qi et al., 2022), and possibly detectable
MeV neutrinos and gravitational waves (GWs, e.g. Romero, Reynoso, &
Christiansen, 2010; Sun et al., 2012; Pan & Yuan, 2012; Caballero et al.,
2016; Xie, Lei, & Wang, 2016; Liu et al., 2017; Wei, Liu, & Song, 2019; Wei &
Liu, 2020; Wei, Liu, & Xue, 2021). For the recent reviews, see Liu, Gu, &
Zhang (2017) and Zhang (2018).

The MC process is one of practicable mechanisms in accretion discs including
NDAFs. Lei et al. (2009) investigated the properties of magnetic coupling
NDAFs (MCNDAFs) and discussed their thermal-viscous instabilities to interpret
variabilities on GRB light curves. Luo et al. (2013) studied MCNDAFs to
investigate the observed late-time X-ray flares of GRBs. Moreover, Song, Liu,
& Wei (2020) discussed MeV neutrinos and corresponding GWs from a magnetized
NDAF including the MC process. The results all confirmed that the MC process
can significantly enhance the neutrino and neutrino annihilation luminosities.
However, these works are all simplified local algebraic solutions, and the
detailed dynamics and neutrino physics are excluded. Hence relativistic global
solutions of MCNDAFs should be visited.

In this paper, the relativistic global solutions of the axisymmetric MCNDAF in
Kerr metric are calculated. In order to highlight the MC process, we ignore
any possible BZ and BP processes also related to large-scale magnetic fields.
In section 2, we establish the basic model by introducing the fundamental
hydrodynamic and thermodynamic equations with the detailed neutrino physics
and nuclear statistical equilibrium (NSE, e.g. Seitenzahl et al., 2008; Liu et
al., 2013; Xue et al., 2013). In section 3, the numerical method and results
are exhibited. Conclusions and relevant discussions are made in section 4.

##  2 Model

Catastrophic events from high-angular-momentum compact sources such as massive
collapsars or BH-neutron star mergers are expected to form a torus. However, a
thick torus is generally unstable (Papaloizou & Pringle, 1984) and it will
undergo the self-gravity instability. If the torus with non-axisymmetric
distortions reaches an appreciable mass fraction of the central BH, the
instability may remove the non-axisymmetry from the accretion flow (Woodward,
Tohline, & Hachisu, 1994). This makes the situation possible for a stable
axisymmetric inner disc plus a instable non-axisymmetric outer thick torus. In
this paper, we only focus on the possible MC process of axisymmetric inner
disc, which is widely considered as one of plausible GRB central engine
candidates (for a review see Liu, Gu, & Zhang, 2017). The research on
interactions between an outer torus and an inner disc has gone beyond the
scope of this paper, but it is an interesting topic worthy of being involved
in future research. We take account of general relativity effects, neutrino
physics, magnetic field geometries, and nucleosynthesis processes in NDAF
model based on our previous works (Xue et al., 2013; Liu, Gu, & Zhang, 2017).
In order to make this paper reader friendly, we revisit most of these physics
in details here.

###  2.1 Relativistic hydrodynamics

The relativistic hydrodynamics in our model is refered to the ones in the ADAF
(Abramowicz et al., 1996), NDAF (Popham, Woosley, & Fryer, 1999), and slim
disc (Sądowski, 2009) models, which are all the one-dimensional global
solutions in the Kerr metric. To simplify, the hydrodynamics in our model is
described in units of $G=c=M_{\rm BH}=1$ ($M_{\rm BH}$ is the BH mass), but
the neutrino physics and thermodynamics is described in cgs units. The five
basic hydrodynamics equations are as follows.

First, the mass conservation equation is

$\displaystyle\dot{M}=-4\pi\rho
H\Delta^{1/2}\frac{V_{r}}{\sqrt{1-{V_{r}}^{2}}},$ (1)

where $\dot{M}$, $\rho$, $H$, and $V_{r}$ are the rest-mass accretion rate,
the rest-mass density, the half-thickness of disc, and the radial velocity
measured in the corotating frame, respectively. Note that
$\Delta=r^{2}-2r+a^{2}$ is a function of the Boyer-Lindquist radial coordinate
$r$, and $a$ is the total specific angular momentum of the BH.

Second, the gas energy equation is written as

$\displaystyle-\frac{\dot{M}}{2\pi r^{2}}\left(\frac{u}{\rho}\frac{d\ln
u}{d\ln r}-\frac{p}{\rho}\frac{d\ln\rho}{d\ln r}\right)=-\frac{2\alpha
pHA\gamma^{2}}{r^{3}}\frac{d\Omega}{dr}$ $\displaystyle-Q^{-}+Q_{\rm MC},$ (2)

where $u$, $p$, and $\alpha$ is the specific internal energy, pressure, and
viscosity parameter on the disc (see e.g. Kato, Fukue, & Mineshige, 2008),
respectively. The variable-$\alpha$ effect could incur the instability of the
disc (e.g. Simon & Hawley, 2009; Potter & Balbus, 2014; Kawanaka & Masada,
2019). Moreover, $A\equiv r^{4}+r^{2}a^{2}+2ra^{2}$, $\gamma$ is the Lorentz
factor, and $\Omega\equiv u^{\phi}/u^{t}$ is the angular velocity with respect
to the stationary observer. It is noted that the l.h.s of this equation
corresponds to the advection cooling rate $Q_{\rm adv}$ and the first term on
the r.h.s is the viscous heating rate $Q_{\rm vis}$. The terms of $Q^{-}$ and
$Q_{\rm MC}$ are respectively the total cooling rate and the energy injection
through MC process, which are both described in Section 2.4. Thus this
equation represents the balance between the coolings and heatings of the gas
internal energy.

Third, the equation of angular momentum conservation is

$\displaystyle\dot{M}(\mathcal{L}-\mathcal{L}_{\rm{in}})+T_{\rm MC}=\frac{4\pi
pHA^{1/2}\Delta^{1/2}\gamma}{r},$ (3)

where $\mathcal{L}\equiv u_{\phi}$ and $\mathcal{L}_{\rm in}$ are the specific
angular momentum of the accreting gas and specific angular momentum at the
inner boundary of the disc, respectively. $T_{\rm MC}$ is the MC torque, which
is described in Section 2.4.

Fourth, the equation of radial momentum conservation is

$\displaystyle\frac{V_{r}}{1-{V_{r}}^{2}}\frac{d{V_{r}}}{dr}=\frac{\mathcal{A}}{r}-(1-{V_{r}}^{2})\frac{1}{\lambda\rho}\frac{dp}{dr},$
(4)

where

$\displaystyle\mathcal{A}\equiv-\frac{A}{r^{3}\Delta\Omega^{+}_{\rm
K}\Omega^{-}_{\rm K}}\frac{(\Omega-\Omega^{+}_{\rm K})(\Omega-\Omega^{-}_{\rm
K})}{1-\tilde{\Omega}^{2}\tilde{R}^{2}}.$ (5)

The effects of gravity and rotation are combined in $\mathcal{A}$, where
$\lambda\equiv(\rho+p+u)/\rho$ is the relativistic enthalpy,
$\tilde{\Omega}\equiv(\Omega-2ar/A)$ is the angular velocity relative to the
local inertial observer, $\Omega^{\pm}_{\rm K}\equiv\pm(r^{3/2}\pm a)^{-1}$
are the angular frequencies of the corotating and counterrotating Keplerian
orbits, and $\tilde{R}\equiv A/(r^{2}\Delta^{1/2})$ is the radius of gyration.

Fifth, the equation of vertical mechanical equilibrium is (e.g. Abramowicz,
Lanza, & Percival, 1997; Xue et al., 2013; Liu, Gu, & Zhang, 2017)

$\displaystyle\frac{p}{\lambda\rho
H^{2}}=\frac{\mathcal{L}^{2}-a^{2}(\epsilon^{2}-1)}{r^{4}},$ (6)

where $\epsilon\equiv u_{t}$ is the energy at infinity along geodesics. In our
calculation, the formulae of $\epsilon$ and $\gamma$ are respectively defined
as

$\displaystyle\epsilon=-\gamma\frac{r\Delta^{1/2}}{A^{1/2}}-\frac{2ar}{A}\mathcal{L}$
(7)

and

$\displaystyle\gamma=\sqrt{\frac{1}{1-{V_{r}}^{2}}+\frac{\mathcal{L}^{2}r^{2}}{A}}.$
(8)

###  2.2 Neutrino physics

The cooling mechanism is the main difference between NDAFs and classic
accretion discs. Neutrino radiation becomes dominant in NDAFs as well as
MCNDAFs. The precise neutrino physics is considered in our calculation and the
details are described in following.

####  2.2.1 Neutrino optical depth

The total optical depth of neutrinos is

$\displaystyle\tau_{\nu_{i}}=\tau_{s,\nu_{i}}+\tau_{a,\nu_{i}},$ (9)

which consists of two parts from scattering and absorption, denoted by
subscript “$s$” and “$a$”, respectively. The symbol $\nu_{i}$ represents the
three different species of neutrinos $\nu_{\rm e}$, $\nu_{\rm\mu}$, and
$\nu_{\rm\tau}$, as well as for the antineutrinos.

The optical depth for neutrinos through scattering off electrons and nucleons
$\tau_{s,\nu_{i}}$ is given by

$\displaystyle\tau_{s,\nu_{i}}\approx H\displaystyle(\sigma_{{\rm
e^{-}},\nu_{i}}n_{\rm e^{-}}+\sigma_{{\rm e^{+}},\nu_{i}}n_{\rm
e^{+}}+{\sum_{j}}\sigma_{j,\nu_{i}}n_{j}),$ (10)

where $\sigma_{\rm e^{\mp},\nu_{i}}$ and $\sigma_{j,\nu_{i}}$ are the cross
sections of electron (positron) and nucleons (e.g. Kohri, Narayan, & Piran,
2005; Chen & Beloborodov, 2007; Xue et al., 2013; Liu, Gu, & Zhang, 2017).
$n_{\rm e^{\mp}}$ is the number density of electron (positron); based on the
Fermi-Dirac integration (see, e.g. Kohri, Narayan, & Piran, 2005; Kawanaka &
Mineshige, 2007; Liu et al., 2007; Liu, Gu, & Zhang, 2017), the number
densities of electrons and positron $n_{\rm e^{-}}$ and $n_{\rm e^{+}}$ can be
obtained, i.e.,

$\displaystyle n_{\rm
e^{\mp}}=\frac{1}{\pi^{2}\hbar^{3}}\int_{0}^{\infty}dp\frac{p^{2}}{{\rm
e}^{({\sqrt{p^{2}c^{2}+{m_{\rm e}}^{2}c^{4}}\mp{\mu_{\rm e}})/k_{\rm
B}T}}+1},$ (11)

where $\mu_{\rm e}=\eta_{\rm e}k_{\rm B}T$ is the chemical potential of
electrons, $\eta_{\rm e}$ is the electron degeneracy, and $T$ is the
temperature of the disc. $n_{j}$ ($j=$ 1, 2, …) are the number densities of
nucleons ($j=1$ for free proton, $j=2$ for free neutron, and $j\geq 3$ for
other heavy nucleons), which are obtained by the NSE as shown in Section 2.3.
From a given $n_{j}$, the mass fraction of the $j$th nucleons can be derived.

Moreover, the absorption depth for neutrinos $\tau_{a,\nu_{i}}$ is defined as

$\displaystyle\tau_{a,\nu_{i}}=\frac{q_{\nu_{i}}H}{4(7/8)\sigma T^{4}},$ (12)

where $q_{\nu_{i}}$ is the total neutrino cooling rate per unit volume, which
consists of four parts, i.e., the Urca process, the electron-positron pair
annihilation rate, the nucleon-nucleon bremsstrahlung rate, and the plasmon
decay rate,

$\displaystyle q_{\nu_{i}}=q_{\rm Urca}+q_{{\rm
e^{-}+e^{+}}\rightarrow\nu_{i}+\overline{\nu}_{i}}+q_{{\rm n+n\rightarrow
n+n+}\nu_{i}+\overline{\nu}_{i}}$
$\displaystyle+q_{\tilde{\gamma}\rightarrow\nu_{i}+\overline{\nu}_{i}},$ (13)

and more detailed physical descriptions see Xue et al. (2013) and Liu, Gu, &
Zhang (2017).

Figure 1: Profiles of (a) density $\rho$, (b) temperature $T$, (c) surface
density $\Sigma$, (d) absolute value of radial velocity $\mid V_{r}\mid$, (e)
angular velocity $\Omega$, (f) specific angular momentum $\mathcal{L}$ and (g)
half-thickness $H$ of MCNDAFs and the counterparts of NDAFs. The blue, red,
and green lines correspond to $\dot{m}=0.1$, 0.5 and 1, respectively. The
solid and dashed lines correspond to MCNDAFs with $n=4$ and $5$ as well as the
dot-dashed lines denote NDAFs.

####  2.2.2 Electron Fraction

The definition of the electron fraction is given by (e.g. Liu, Gu, & Zhang,
2017)

$\displaystyle Y_{\rm
e}=\frac{\sum\limits_{j}n_{j}Z_{j}}{\sum\limits_{j}n_{j}(Z_{j}+N_{j})},$ (14)

where $Z_{j}$ and $N_{j}$ are the number of the protons and neutrons of a
nucleus. Here we employ the strict NSE equation (see Section 2.3) to calculate
the electron fraction. Meanwhile, the condition of electrical neutrality
remain valid, which can be written as (e.g. Liu et al., 2007, 2013)

$\displaystyle\sum\limits_{j}{n_{j}}{Z_{j}}=\frac{\rho Y_{\rm
e}}{m_{u}}=n_{\rm e^{-}}-n_{\rm e^{+}},$ (15)

where $m_{u}$ is the mean mass of baryons.

Furthermore, the other relation on the number density is a bridging formula.
For the extremely neutrino optically thin and thick regions, the chemical
potentials are satisfied with $\mu_{\rm e}=\mu_{\rm n}-\mu_{\rm p}$ and
$2\mu_{\rm e}=\mu_{\rm n}-\mu_{\rm p}$ (see Yuan, 2005), respectively, where
$\mu_{\rm n}$ and $\mu_{\rm p}$ are the chemical potentials of neutrons and
protons, and the chemical potential of neutrinos is ignored because it is much
lower than the chemical potential of electron $\mu_{\rm e}$. Then the ratio of
the number density of free neutrons $n_{2}$ to that of free protons $n_{1}$
transits from the optically thin to optically thick regimes,

$\displaystyle\lg{\frac{n_{\rm 2}}{n_{\rm 1}}}={\rm e}^{-\tau_{\nu_{\rm
e}}}\frac{2\mu_{\rm e}-Q}{k_{\rm B}T}+(1-{\rm e}^{-\tau_{\nu_{\rm
e}}})\frac{\mu_{\rm e}-Q}{k_{\rm B}T},$ (16)

where $Q=(m_{\rm n}-m_{\rm p})c^{2}$, and $m_{\rm n}$ and $m_{\rm p}$ are the
masses of the neutron and proton, respectively (e.g. Liu et al., 2007; Xue et
al., 2013; Liu, Gu, & Zhang, 2017). For the more detailed descriptions on the
modification of the $\beta$-equilibrium condition from neutrino optically thin
to thick, see Li & Liu (2013).

###  2.3 Nucleosynthesis

NSE is established when all nuclear reactions achieve chemical equilibrium. We
adopt the NSE state proposed by Seitenzahl et al. (2008), which is satisfied
with the entire range of the electron
fraction111http://cococubed.asu.edu/code_pages/nse.shtml. The number density
of nucleus $n_{j}$ can be expressed as

$\displaystyle n_{j}=$ $\displaystyle g_{j}(\frac{m_{j}k_{\rm
B}T}{\hbar^{2}})^{3/2}$ $\displaystyle\times{\rm exp}[\frac{Z_{j}(\mu^{\rm
kin}_{\rm p}+{\mu^{\rm C}_{\rm p}})+N_{j}{\mu^{\rm kin}_{\rm n}}-{\mu^{\rm
C}_{\rm j}}+Q_{j}}{k_{\rm B}T}],$

where $\mu^{\rm kin}_{\rm p}$ and $\mu^{\rm kin}_{\rm n}$ are the kinetic
chemical potentials of protons and neutrons, $\mu^{\rm C}_{\rm p}$ and
$\mu^{\rm C}_{\rm j}$ are the Coulomb chemical potentials of protons and
nucleons, and $g_{j}$ is the nuclear partition functions.

###  2.4 Thermodynamics

The equation of state in our model can be written as

$\displaystyle p=p_{\rm gas}+p_{\rm rad}+p_{\rm e}+p_{\nu}+p_{\rm mag}.$ (18)

The gas pressure from free nucleons $p_{\rm gas}$ is estimated by

$\displaystyle p_{\rm gas}=\sum\limits_{j}n_{j}k_{\rm B}T.$ (19)

The photon radiation pressure $p_{\rm rad}$ is

$\displaystyle p_{\rm rad}=a_{\rm rad}T^{4}/3,$ (20)

which form is caused by the totally optical thick for photon in the disc, and
$a_{\rm rad}$ is the radiation constant.

The electron pressure $p_{\rm e}$ is composed of the contributions from
electrons and positrons. Since there is no asymptotic expansion valid for
electrons with moderate degeneracy and relativity, it has to be calculated by
the exact Fermi-Dirac distribution (e.g. Chen & Beloborodov, 2007; Liu et al.,
2007; Liu, Gu, & Zhang, 2017). It reads

$\displaystyle p_{\rm e}=p_{\rm e^{-}}+p_{\rm e^{+}},$ (21)

with

$\displaystyle p_{\rm e^{\mp}}=\frac{1}{3{\pi}^{2}{\hbar}^{3}c^{3}}$
$\displaystyle\times\int_{0}^{\infty}dp\frac{p^{4}}{\sqrt{p^{2}c^{2}+{m_{\rm
e}}^{2}c^{4}}}\frac{1}{{\rm e}^{({\sqrt{p^{2}c^{2}+{m_{\rm
e}}^{2}c^{4}}\mp{\mu_{\rm e}})/k_{\rm B}T}}+1}.$

The neutrino pressure $p_{\nu}$ is

$\displaystyle p_{\nu}=u_{\nu}/3,$ (23)

where ${u}_{\nu}$ is the energy density of neutrinos, for which we adopt the
bridging formula (e.g. Kohri, Narayan, & Piran, 2005; Liu et al., 2007; Liu,
Gu, & Zhang, 2017), i.e.,

$\displaystyle u_{\nu}=\sum_{i}\frac{(7/8)a_{\rm
rad}T^{4}(\tau_{{\nu}_{i}}/2+1/\sqrt{3})}{\tau_{{\nu}_{i}}/2+1/\sqrt{3}+1/(3\tau_{a,{\nu}_{i}})}.$
(24)

It is valid in the regimes from the optically thin to thick for neutrinos.

The magnetic pressure contributed by the tangled magnetic fields in the disc
is

$\displaystyle p_{\rm mag}=\beta_{t}p,$ (25)

and $\beta_{t}$ is the ratio of the magnetic pressure to the total pressure,
we adopt $\beta_{t}=0.1$ in our calculation.

The cooling rate $Q^{-}$ appearing in Equation (2.1) can be defined as

$\displaystyle Q^{-}=Q_{\rm ph}+Q_{\nu}+Q_{\rm rad},$ (26)

which consists of the cooling rates of photodisintegration $Q_{\rm ph}$, the
neutrino emitting $Q_{\nu}$ and the photon radiation $Q_{\rm rad}$. However,
due to photon trapping by dense material, $Q_{\rm{rad}}$ is always much
smaller than the other two, so we ignored this term in our calculation (e.g.
Liu, Gu, & Zhang, 2017).

The cooling rate of photodisintegration $Q_{\rm ph}$ is mainly contributed by
the decomposition of $\alpha$-particles, which can be written as

$\displaystyle Q_{\rm ph}=6.8\times 10^{28}\rho_{10}V_{r}H\frac{dX_{\rm
nuc}}{dr}~{}{(\mathrm{cgs~{}units})},$ (27)

where ${\rho}_{10}\equiv{\rho}/(10^{10}{\rm g~{}cm^{-3}})$, and $X_{\rm nuc}$
is the mass fraction of $\alpha$-particles (e.g. Kohri, Narayan, & Piran,
2005; Liu et al., 2007; Liu, Gu, & Zhang, 2017).

According to Equation (24), the cooling rate due to the neutrino loss
$Q_{\nu}$ can be obtain,

$\displaystyle
Q_{\nu}=\sum_{i}\frac{(7/8){\sigma}T^{4}}{(3/4)[\tau_{{\nu}_{i}}/2+1/\sqrt{3}+1/(3\tau_{a,{\nu}_{i}})]}.$
(28)

Figure 2: Profiles of (a) total pressure $p$, (b) optical depth of electron
neutrinos $\tau_{\nu_{e}}$, (c) electron degeneracy $\eta_{e}$, and (d)
electron fraction $Y_{\rm e}$ with the same denotations of colours and line-
styles as Figure 1.

Figure 3: Radial distributions of (a) neutrino cooling rates $Q_{\nu}$, (b)
viscous heating rates $Q_{\rm{vis}}$, (c) MC heating rates $Q_{\rm{MC}}$, and
(d-f) absolute values of advection cooling rates $\mid Q_{\rm{adv}}\mid$ for
$\dot{m}=0.1$, 0.5 and 1. _Panels_ (a-c): the blue, red, and green lines
correspond to $\dot{m}=$ 0.1, 0.5 and 1, respectively, and the solid and
dashed lines correspond to MCNDAFs with $n=4$ and $5$ as well as the dot-
dashed lines denote NDAFs. _Panels_ (d-f): the red and blue lines correspond
to MCNDAFs with $n=4$ and $5$ as well as the green lines denote NDAFs. The
solid and dashed parts of each line denote the positive and negative values of
$Q_{\rm{adv}}$.

The MC torque $T_{\rm MC}$ and the energy transferred from the BH to the disc
by the MC process $Q_{\rm MC}$ are given by (e.g. Wang, Xiao, & Lei, 2002; Lei
et al., 2009; Luo et al., 2013; Song, Liu, & Wei, 2020),

$\displaystyle T_{\rm
MC}=4T_{0}a_{*}(1+q)\int^{\pi/2}_{\theta_{0}}\frac{(1-\Omega/\Omega_{\rm
H})\sin^{3}\theta d\theta}{2-(1-q)\sin^{2}\theta},$ (29)

and

$\displaystyle Q_{\rm MC}=\frac{T_{\rm MC}}{4\pi r}\frac{d\Omega}{dr},$ (30)

respectively, where $q=a_{*}/(1+\sqrt{1-a_{*}^{2}})$ is a dimensionless
parameter relating to the dimensionless BH spin $a_{*}$, $T_{0}=3.26\times
10^{45}(B_{\rm H}/10^{15}~{}{\rm G})^{2}(M_{\rm
BH}/M_{\odot})^{3}~{}\rm{g~{}cm^{2}~{}s^{-2}}$ is a characteristic torque,
$\Omega_{\rm H}=(a_{*}c/2r_{\rm g})(1+q)$ is the angular velocity at the BH
horizon. $B_{\rm H}^{2}=8\pi c\dot{M}/r_{\rm g}^{2}$ is the field strength at
the BH horizon (e.g. McKinney, 2005; Lei et al., 2009), where $r_{\rm
g}=2GM_{\rm BH}/c^{2}$ is the Schwarzschild radius, while magnetic field of
inner region vary as $B_{\rm D}\propto\xi^{-n}$ described in Blandford (1976),
where $\xi=r/r_{\rm ms}$, $r_{\rm ms}$ is the marginally stable orbital
radius, and $n$ is the index measuring the concentration of the magnetic
fields on the equatorial plane. Moreover, $\theta_{0}$ is the azimuth
corresponding to the boundary between the open and closed field lines, which
will tend to zero when the MC effect vanishes and can be defined as (e.g.
Wang, Xiao, & Lei, 2002)

$\displaystyle\cos\theta_{0}=\int^{\xi_{\rm out}}_{1}d\xi$
$\displaystyle\times\frac{\xi^{1-n}\chi_{\rm
ms}^{2}\sqrt{1+a_{*}^{2}\chi^{-4}_{\rm ms}\xi^{-2}+2a_{*}^{2}\chi^{-6}_{\rm
ms}\xi^{-3}}}{2\sqrt{(1+a_{*}^{2}\chi^{-4}_{\rm ms}+2a^{2}_{*}\chi^{-6}_{\rm
ms})(1-2\chi^{-2}_{\rm ms}\xi^{-1}+a^{2}_{*}\chi^{-4}_{\rm ms}\xi^{-2})}},$

where $\chi_{\rm ms}=\sqrt{r_{\rm ms}/r_{\rm g}}$ is a dimensional parameter
relating to the marginally stable orbital radius and $\xi_{\rm out}$ is the
the dimensionless outer boundary of the MC dominated region.

###  2.5 Neutrino and annihilation luminosities

Figure 4: (a) Neutrino luminosity and (b) neutrino annihilation luminosity as
functions of the accretion rates. The black, red, and blue markers correspond
to $\dot{m}$ = 0.1, 0.5, and 1, respectively. The pluses and asterisks
correspond to MCNDAFs with $n=4$ and $5$ as well as the crosses denote NDAFs.

Due to the neutrino physics in MCNDAFs, the electron neutrinos and
antineutrinos are the main products to annihilate in the space out of the disc
(e.g. Chen & Beloborodov, 2007; Xue et al., 2013; Liu, Gu, & Zhang, 2017).
After obtaining the radial distribution of the neutrino cooling rate
$Q_{\nu}$, the neutrino luminosity can be calculated by integrating along
radial direction, i.e.,

$\displaystyle L_{\nu}=4\pi\int_{{\rm max}(r_{\rm ms},r_{\rm tr})}^{r_{\rm
out}}Q_{\nu}rdr,$ (32)

where the outer boundary of the disc $r_{\rm out}=500~{}r_{\rm g}$ is set in
our calculations. The inner edge of integration is set to be the larger one of
the neutrino trapping radius $r_{\rm tr}$ and $r_{\rm ms}$ to take into
account of the effect of neutrino trapping.

The neutrino trapping radius can be obtained by comparing the neutrino
diffusion timescale $t_{\rm diff}$ with the accretion timescale $t_{\rm acc}$.
In other words, the criterion should be given to judge that the neutrinos
produced in the equatorial planes of the disc can escape from the disc surface
instead of falling into the BH. The photon diffusion timescale in classical
accretion discs can be estimated as $\sim H/(c/3\tau)$ (e.g. Ohsuga et al.,
2002; Liu, Gu, & Zhang, 2017). Thus, for MCNDAFs, we estimate $t_{\rm
diff}\sim H/(V_{\rm n}/3\tau_{\nu_{\rm e}})$, where the neutrino speed $V_{\rm
n}\sim[1-(0.07{\rm eV}/3.7k_{\rm B}T)]^{1/2}~{}c$ (e.g. Liu et al., 2012b;
Liu, Gu, & Zhang, 2017) and the photon optical depth $\tau$ is substituted by
the neutrino optical depth $\tau_{\nu_{\rm e}}$. The other one, $t_{\rm
acc}\sim-r/V_{r}$. The lower limit of neutrino mass $0.07~{}{\rm eV}$ is
adopted here. According to the condition $t_{\rm diff}>t_{\rm acc}$, the
neutrino trapping radius can be roughly estimated by

$\displaystyle r_{\rm tr}\approx-\frac{3\tau_{\nu_{\rm e}}HV_{r}}{V_{\rm n}}.$
(33)

Obviously, the effect of neutrino trapping will greatly affect the
annihilation luminosity if it exists generally for the high accretion rates.

We adopted the Newtonian approximation method (e.g. Ruffert et al., 1997;
Popham, Woosley, & Fryer, 1999; Rosswog, Ramirez-Ruiz, & Davies, 2003; Liu et
al., 2007; Liu, Gu, & Zhang, 2017; Xue et al., 2013) to calculate the neutrino
annihilation luminosity, the value of which has the same order of magnitude to
the relativity method (Zalamea & Beloborodov, 2011). In this approach, the
disc is modeled as a grid of cells in the equatorial plane. Every cell $k$ has
its mean neutrino energy $\varepsilon_{\nu_{i}}^{k}$, neutrino radiation
luminosity $l_{\nu_{i}}^{k}$, and distance to a space point in the outside
space of the disc $d_{k}$. $l_{\nu_{i}}^{k}$ can be calculated by the surface
integral of cooling rate of each flavor of neutrino in the cell $k$. The angle
at which neutrinos from cell $k$ encounter antineutrinos from another cell
$k^{\prime}$ at that point is denoted as $\theta_{kk^{\prime}}$. Then the
neutrino annihilation luminosity at that point is described by the summation
over all pairs of cells,

$\displaystyle
l_{\nu\overline{\nu}}=\sum_{i}A_{1,i}\sum_{k}\frac{l_{\nu_{i}}^{k}}{d_{k}^{2}}\sum_{k^{\prime}}\frac{l_{\overline{\nu}_{i}}^{k^{\prime}}}{d_{k^{\prime}}^{2}}(\varepsilon_{\nu_{i}}^{k}+\varepsilon_{\overline{\nu}_{i}}^{k^{\prime}}){(1-\cos{\theta_{kk^{\prime}}})}^{2}$
$\displaystyle+\sum_{i}A_{2,i}\sum_{k}\frac{l_{\nu_{i}}^{k}}{d_{k}^{2}}\sum_{k^{\prime}}\frac{l_{\overline{\nu}_{i}}^{k^{\prime}}}{d_{k^{\prime}}^{2}}\frac{\varepsilon_{\nu_{i}}^{k}+\varepsilon_{\overline{\nu}_{i}}^{k^{\prime}}}{\varepsilon_{\nu_{i}}^{k}\varepsilon_{\overline{\nu}_{i}}^{k^{\prime}}}{(1-\cos{\theta_{kk^{\prime}}})},$

with $A_{1,i}=(1/12\pi^{2})[\sigma_{0}/c{(m_{\rm
e}c^{2})}^{2}][{(C_{V,\nu_{i}}-C_{A,\nu_{i}})}^{2}+{(C_{V,\nu_{i}}+C_{A,\nu_{i}})}^{2}]$
and $A_{2,i}=(1/6\pi^{2})(\sigma_{0}/c)$
$(2C_{V,\nu_{i}}^{2}-C_{A,\nu_{i}}^{2})$. The parameters $C_{V,\nu_{i}}$ and
$C_{A,\nu_{i}}$ can be found in Liu, Gu, & Zhang (2017), which are included in
the descriptions of the scattering and absorption of neutrinos.

Then the total annihilation luminosity can be obtained by the integration over
the whole space outside of the BH and the disc,

$\displaystyle L_{\nu\overline{\nu}}=4\pi\int_{{\rm max}(r_{\rm ms},r_{\rm
tr})}^{\infty}\int_{H}^{\infty}l_{\nu\overline{\nu}}rdrdz,$ (35)

where the inner edge is depended on the status of neutrino trapping (e.g. Xue
et al., 2013; Liu, Gu, & Zhang, 2017).

Figure 5: Pair-annihilation luminosities per $r_{\rm g}$ (a) vs. radius and
(b) vs. distance along the angular momentum axis. The blue, red, and green
lines correspond to $\dot{m}$ = 0.1, 0.5 and 1, respectively, and the solid
and dashed lines correspond to MCNDAFs with $n=4$ and $5$ as well as the dot-
dashed lines denote NDAFs.

##  3 Results

In this paper, we use a shooting method (Matsumoto et al., 1984) for the
solving of above equations, which has been proved to be effective in dealing
with the boundary value problem of NDAFs by our previous works (e.g. Xue et
al., 2013; Liu, Gu, & Zhang, 2017). Of course, the convergence of numerical
calculations in the MCNDAF model is harder than that in the NDAF model,
especially for the inner disc.

In our model, there are five parameters, i.e., the viscous parameter $\alpha$,
BH mass $M_{\rm BH}$, dimensionless BH spin $a_{*}$, dimensionless accretion
rate $\dot{m}$ [$\equiv\dot{M}/(M_{\odot}~{}\rm s^{-1})$], and power-law index
of magnetic fields $n$. Here we concentrate on influences of $\dot{m}$ and
$n$, so we fix the viscous parameter, BH mass, and BH spin with the typical
values of $\alpha=0.1$, $M_{\rm BH}=3~{}M_{\odot}$, and $a_{*}=0.9$,
respectively. The BH spin $a_{*}$ at an extreme high value is selected because
the MC effect is relatively strong in this condition (e.g. Lei et al., 2009;
Luo et al., 2013; Song, Liu, & Wei, 2020). We set the accretion rates
$\dot{m}=0.1$, 0.5, and 1, corresponding to the low, medium, and high
accretion states, respectively; and set the power-law indices $n=4$ and 5222As
shown in Figure 2(a) of Song, Liu, & Wei (2020), in the case of $n$ = 3 with
$a_{*}$ = 0.9, one can notice that $\theta_{0}$ = 0 and the outer boundary of
the MC region was too small to significantly effect on the disc, so we only
discussed the cases of $n$ = 4 and 5, which correspond the infinite MC
regions., denoting the relatively incompact and compact magnetic field
geometries, respectively.

###  3.1 Structure

Figure 6: Radial distributions of mass fractions of top seven elements, $\rm
n$, 1H, 4He, 52Cr, 54Cr, 56Fe, and 58Fe for $\dot{m}=0.1$, $0.5$, and $1$,
respectively. The solid and dashed lines correspond to MCNDAFs with $n=4$ and
$5$ as well as the dot-dashed lines denote NDAFs.

Profiles of the density $\rho$, temperature $T$, the surface density $\Sigma$,
the absolut value of the radial velocity $V_{r}$, the angular velocity
$\Omega$, the specific angular momentum $\mathcal{L}$, and the half-thickness
of disc $H$ of MCNDAFs and the counterparts of NDAFs are all shown in Figure
1. The blue, red, and green lines correspond to different $\dot{m}=0.1$, 0.5
and 1, respectively. The solid and dashed lines correspond to MCNDAFs with
$n=4$ and $5$ respectively as well as the dot-dashed lines correspond to
NDAFs. The density $\rho$ (or surface density $\Sigma$) and temperature $T$ of
MCNDAFs are obviously higher than those of NDAFs, and reach the corresponding
maximums of $\sim 10^{13}~{}\rm g~{}cm^{-3}$ and $\sim 2\times 10^{11}~{}\rm
K$ in the inner region of the disc when $\dot{m}=1$. Especially, the blue
solid line marginally exceeds the red solid one at $r\approx 1.5~{}r_{\rm g}$
in Figure 2(c), which implies $d\dot{M}/d\Sigma<0$ and indicates the viscous
instability possible there. It might be an available origin of GRB
variabilities, which is consistent with Lei et al. (2009). The inflow speed
$|V_{r}|$ of MCNDAFs are obviously slower than those of NDAFs until $r\lesssim
2~{}r_{\rm g}$. While there are only slight differences between MCNDAFs and
NDAFs on the angular velocity $\Omega$ in the range of $2~{}r_{\rm g}\lesssim
r\lesssim 30~{}r_{\rm g}$. In the region $r/r_{\rm g}<30$, the effects of the
viscosity and MC coexist and compete with each other on the transfer of
angular momentum. As it is transferred from the BH to the disc by MC effects,
the specific angular momentum is increased as shown in Figure 1(f). Meanwhile,
the temperatures of MCNDAFs at $r/r_{\rm g}<30$ are higher than those of NDAFs
as shown in Figure 1(b) due to the injection of additional energy through MC
effects. It implies that the strength of the viscosity in the inner region of
MCNDAFs is larger than that of NDAFs. Therefore, the specific angular momenta
of MCNDAFs are lower than these of NDAFs. The values of the half-thickness $H$
of MCNDAFs have some fluctuations compared to these of NDAFs. All of these are
caused by the additional energy and angular momentum brought by the MC
process, by which the accreting gas is hindered and deposited in the inner
region of the disc to result in the increasing of the temperature and density.
The specific angular momentum of the deposited gas is also increased to almost
the same as that in NDAFs before passing toward to BH, which can accommodate
much more angular momentum with denser gas than NDAFs in the same accretion
rates.

Figure 2 shows the profiles of the total pressure $p$, optical depth of
electron neutrinos $\tau_{\nu_{e}}$, electron degeneracy $\eta_{e}$, and
electron fraction $Y_{\rm e}$ of MCNDAFs. The total pressure $p$ and the total
neutrino optical depth $\tau_{\nu_{\rm e}}$ are both larger than those of
NDAFs, which are consistent with the higher gas temperature and density of
MCNDAFs and both result from the MC process. Especially, $\tau_{\nu_{\rm e}}$
is larger than 1 until $r\gtrsim 40~{}r_{\rm g}$ for $\dot{m}=1$, which means
that the inner region of MCNDAFs are totally neutrino optical thick and
thermal MeV neutrinos can not escape directly from the depths of the disc in
this region.

In addition, the electron degeneracy $\eta_{\rm e}$ is an important physical
parameter that affects the electron fraction $Y_{\rm e}$, degeneracy pressure,
and neutrino cooling (e.g. Liu, Gu, & Zhang, 2017). In Figures 2(c) and 2(d),
$\eta_{\rm e}$ and $Y_{\rm e}$ of MCNDAFs are very different to those of
NDAFs, which indicate the MC process has very remarkable influence on the disc
microphysics. Specifically, $\eta_{\rm e}$ increases for the MCNDAF cases and
its peak lies in tens of $r_{\rm g}$. On the contrary, $Y_{\rm e}$ decreases
around $\sim 10~{}r_{\rm g}$ and tends to about 0.46 in the outer boundary.
These properties will reflect in the nucleosynthesis process and related
components of the disc.

###  3.2 Cooling and luminosity

Figures 3(a-c) display the radial distributions of the neutrino cooling rates
$Q_{\nu}$, the viscous heating rates $Q_{\rm{vis}}$ and the heating rates
$Q_{\rm{MC}}$, respectively. The blue, red, and green lines correspond to
$\dot{m}=$ 0.1, 0.5 and 1, respectively, and the solid and dashed lines
correspond to MCNDAFs with $n=4$ and $5$ as well as the dot-dashed lines
denote NDAFs. One can find the neutrino cooling rates $Q_{\nu}$ of MCNDAF are
higher than NDAF and increase with the increasing accretion rates. The viscous
rates $Q_{\rm vis}$ rise more rapidly when approach to the BH in the MCNDAF
cases. The heating rates $Q_{\rm{MC}}$ in the cases of $n$ = 4 and 5 are still
remarkable in the outer regions of the discs, thus the solutions of MCNDAFs
and NDAFs cannot converge there as shown in Figures 1 and 2.

Figures 3(d-f) show the absolute values of the advection cooling rates $\mid
Q_{\rm{adv}}\mid$ in the cases of $\dot{m}=0.1$, 0.5 and 1, the red and blue
lines correspond to MCNDAFs with $n=4$ and $5$ as well as the green lines
denote NDAFs, and the dash lines denote the negative values. The advection
fractions change their signs to become negative due to the photodisintegration
or the neutrino cooling dominated there.

In Figure 4, we show the luminosities of the neutrino radiation and
annihilation both for MCNDAFs and NDAFs. Obviously, there is little difference
on the luminosities between the MCNDAFs with different magnetic indices but
with the same accretion rates. Regardless of accretion rates, the neutrino
radiation luminosities of MCNDAFs ($\sim 10^{54}~{}\rm erg~{}s^{-1}$) are
always higher about one magnitude order than those of NDAFs ($\sim
10^{53}~{}\rm erg~{}s^{-1}$), meanwhile there is always about two magnitude
order gain on the corresponding annihilation luminosities ($\sim
10^{52}-10^{54}~{}\rm erg~{}s^{-1}$ for MCNDAFs, $\sim 10^{50}-10^{52}~{}\rm
erg~{}s^{-1}$ for NDAFs). This is the multiplier effect, i.e., the incident
neutrino luminosities increased by 10 times, and the annihilation luminosity
is proportional to the product of the incident luminosities at the colliding
point, which shows the significance of MC process on the enhancement of
annihilation luminosity. We also consider the neutrino trapping in our
calculations because of the large increase of neutrino optical depths [see
Figure 2(b)], but there is no significant effect observed in our results.

We show the spatial distribution of neutrino annihilation luminosity in Figure
5. _Panels_ (a) and (b) plot the annihilation luminosity at different radii
$r$ by integrating over the distance $z$ above the plane and at different
distances above the plane $z$ by integrating the radius $r$, respectively. The
line type representation is consistent with Figure 1. One can note that the
annihilation luminosity varies rapidly along the $z$-coordinate and the
$r$-coordinate, which indicates that most of the annihilation energy escapes
outward close to the inner region and along the angular momentum axis of the
disc to power the relativistic jets. As mentioned before Equation (34), we
adopt the Newtoniam approximation method to calculate the annihilation
luminosities of NDAFs and MCNDAFs. Since the annihilation luminosities of
MCNDAFs are typically higher than these of NDAFs, and neutrinos are coming
from more inner parts of the disc, the relativistic effects, i.e., bending of
null geodesics and gravitational redshift, should be further considered in the
MCNDAF cases, which lead to the lower annihilation luminosities (e.g. Zalamea
& Beloborodov, 2011).

###  3.3 Nucleosynthesis

In nucleosynthesis calculations, we consider more than 40 elements including
free baryons, which contribute $99\%$ of accretion mass in the inner region.
Figure 6 shows the radial distributions of mass fractions for the top seven
elements, $\rm n$, $\rm{}^{1}H$, $\rm{}^{4}He$, $\rm{}^{52}Cr$,
$\rm{}^{54}Cr$, $\rm{}^{56}Fe$, and $\rm{}^{58}Fe$ with $\dot{m}=0.1$, 0.5,
and 1 respectively. For all cases, ${}^{56}\rm Fe$ is dominated in the outer
region, $\rm{}^{4}He$ become dominated in the middle region, and free neutrons
and protons become the final dominators in the hot and dense inner region. For
different accretion rates, the ratios of free neutrons to free protons of
NDAFs change significantly but this is not the case for MCNDAFs, because the
MC process results in the inner region gas hotter and denser than those of
NDAFs, which prevents the number of free protons from exceeding the number of
free neutrons. As the same reason, the inner regions dominated by free baryons
in MCNDAFs generally are larger than those in NDAFs, which results in the
regions of heavy nuclei synthesis in MCNDAFs farther from BH than NDAFs.

In our conclusions, there are very rare 56Ni produced in MCNDAFs and NDAFs.
According to the Urca and other adjusting processes, free protons are at a
disadvantage on the numbers throughout the disc, and the synthesis condition
of 56Ni is stricter than other elements of iron group, so there is hard to
mass-produce 56Ni, which decay is widely considered as the origin of CCSNe
(e.g. Liu et al., 2021). Nevertheless, its yield might be plentiful during the
adiabatic cooling process of the high-velocity disc outflows (e.g. Surman,
McLaughlin, & Hix, 2006; Liu et al., 2013, 2021; Janiuk, 2014; Song & Liu,
2019) with a rigorous and stable condition of $Y_{\rm e}\sim 0.49-0.60$,
$\rho\gtrsim 10^{6}~{}\rm g~{}cm^{-3}$, and $T\lesssim 5\times 10^{9}~{}\rm K$
(Seitenzahl et al., 2008).

##  4 Conclusions and discussion

In this paper, we calculate the one-dimensional global solutions of
axisymmetric MCNDAFs, with the strict Kerr metric, detailed neutrino physics,
different magnetic field geometries, and reasonable nucleosynthesis processes.
The structures, luminosities, and elemental abundances of six cases with the
different accretion rates $\dot{m}=0.1$, $0.5$, and $1$ as well as the power-
law indices of magnetic fields $n=4$ and $5$ are exhibited and compared with
the counterparts of NDAFs. The main results are summarized as follows.

(i) In contrast to NDAFs, the structure, thermal properties, and microphysics
of MCNDAFs have been prominently impacted by the MC process especially for the
inner region of the disc. There should exist the viscous instability in the
inner region of MCNDAFs, which leads to the GRB variabilities.

(ii) The MC process has the significant effects on the heating mode and can
increase the luminosities of neutrinos and their annihilations. The cases with
the indices $n=4$ and 5 have little difference on the structure, energy
equilibrium, luminosity, and components.

(iii) The mass fractions of 56Fe, 4He, and free neutrons are dominated in turn
in the outer, middle, and inner regions of MCNDAFs. The MC process will drive
off the heavy nuclei synthesis to the outside of MCNDAFs farther than that of
NDAFs. Almost no 56Ni appears in the outer disc.

Therefore, the MC effect plays an important role to rebuild the structure,
luminosity, and components of MCNDAFs. Moreover, although the NDAF is one of
the popular central engine model of GRBs, only a small fraction of neutrinos
can annihilate in the outside space of the disc to power GRBs and other
following radiation processes, so there still leaves over some issues related
to the much higher energy requirements from some observed luminous GRBs. The
MC process is one of the priority options to dramatically enhance the neutrino
annihilation luminosity, and might attach some other advantageous effects,
such as arising the possible instability in the inner disc (e.g. Janiuk et
al., 2007; Lei et al., 2009), redistributing the angular momentum of the disc
(e.g. Janiuk & Proga, 2008; Liu et al., 2012a), and launching the episodic
Poynting fluxes via magnetic reconnections (e.g. Yuan & Zhang, 2012; Liu, Gu,
& Zhang, 2017).

Since the drastic hyperaccretion process appears in the BH-NDAF systems, the
evolutions of the central BH mass and spin should be further considered (e.g.
Janiuk & Proga, 2008; Janiuk & Yuan, 2010; Song et al., 2015; Liu et al.,
2021). Of course, the strong disc outflows and MC process might greatly
relieve this situation. Moreover, the electron fraction $Y_{\rm e}$ of MCNDAFs
tends to the relative proton-rich phase as well as in NDAF cases (e.g. Xue et
al., 2013), which is naturally applicable to describe the initial fallback
hyperaccretion in the massive collapsar scenario. As mentioned above, the 56Ni
yield in the disc is very scarcity also caused by the self-consistent $Y_{\rm
e}$ condition and the following “missing” protons via Urca process; but the
disc outflows will change the structure and components of the disc and rewrite
the equations related to $Y_{\rm e}$, then could be the rich 56Ni mines as
well as their host CCSNe, which will prominently contribute on the chemical
evolution in the Universe (e.g. Liu et al., 2021). In future work, a
compulsive boundary condition should be included to investigate the properties
of the time-dependent NDAFs with magnetic fields and outflows lying in the
various circumstances with different element abundances.

## Acknowledgements

We thank the anonymous referee for helpful suggestions. This work was
supported by the National Natural Science Foundation of China under grants
12173031, 11822304, and 11373002, and the science research grants from the
China Manned Space Project with No. CMS-CSST-2021-B11.

## Data availability

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

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