# Applications of Fast Magnetic Reconnection Models to the Atmospheres of the
Sun and Protoplanetary Disks

Fulvia Pucci11affiliationmark:  , K. Alkendra P. Singh22affiliationmark:
33affiliationmark:  , Uma Gorti44affiliationmark:  55affiliationmark:  , Marco
Velli66affiliationmark:  , Neal J. Turner11affiliationmark:  , Disha
Varshney22affiliationmark:  , Maria Elena Innocenti77affiliationmark:
<EMAIL_ADDRESS>11affiliationmark:  Jet Propulsion Laboratory,
California Institute of Technology, 4800 Oak Grove Drive, Pasadena, California
91109, USA  22affiliationmark:  Plasma Astrophysics Research Laboratory,
Department of Physics, Institute of Science, BHU, Varanasi 221005, India
33affiliationmark:  Astronomical Observatory, Graduate School of Science,
Kyoto University , Yamashina, Kyoto 607-8471, Japan  44affiliationmark:
Department of Earth, Planetary, and Space Sciences, UCLA, 595 Charles Young
Drive East, Los Angeles, California 90095, USA  55affiliationmark:  NASA Ames
Research Center, MS 245-1, PO Box 1, Moffett Field, California 94035, USA
66affiliationmark:  SETI Institute, 339 Bernardo Ave Suite 200, Mountain View,
California 94043, USA  77affiliationmark:  Institut für Theoretische Physik,
Ruhr-Universität Bochum, Bochum, Germany

###### Abstract

Partially-ionized plasmas consist of charged and neutral particles whose
mutual collisions modify magnetic reconnection compared with the fully-ionized
case. The collisions alter the rate and locations of the magnetic dissipation
heating and the distribution of energies among the particles accelerated into
the non-thermal tail. We examine the collisional regimes for the onset of fast
reconnection in two environments: the partially-ionized layers of the solar
atmosphere and the protoplanetary disks that are the birthplaces for planets
around young stars. In both these environments, magnetic nulls readily develop
into resistive current sheets in the regime where the charged and neutral
particles are fully coupled by collisions, but the current sheets quickly
break down under the ideal tearing instability. The current sheets collapse
repeatedly, forming magnetic islands at successively smaller scales, till they
enter a collisionally-decoupled regime where the magnetic energy is rapidly
turned into heat and charged-particle kinetic energy. Small-scale, decoupled
fast reconnection in the solar atmosphere may lead to preferential heating and
energization of ions and electrons that escape into the corona. In
protoplanetary disks such reconnection causes localized heating in the
atmospheric layers that produce much of the infrared atomic and molecular line
emission observed with the Spitzer and James Webb Space Telescopes.

##  1\. Introduction

Magnetic reconnection plays a key role in the solar atmosphere, from the
photosphere, chromosphere to filaments, and prominences, as well as in the
interstellar medium with its star-forming molecular clouds, and the planet-
forming protoplanetary discs that orbit young stars (e.g. Ballester et al.
2018; Ni et al. 2020; Ji et al. 2022 and references therein) and many other
environments. Reconnection in stars and accretion disks contributes to the
development of their hot coronae, the dynamo regeneration of their magnetic
fields, and the launching of their supersonic winds (e.g. Zweibel & Yamada
2009; Yamada et al. 2010; Yamada 2022 and references therein). In many of
these environments, the plasma is only partially ionized. The substantial
neutral fraction alters the rates of both reconnection (Zweibel, 1989; Jara-
Almonte et al., 2019; Pucci et al., 2020) and the instabilities that can
disrupt the current sheets (CSs) where reconnection takes place (Soler &
Ballester, 2022). Thus the partially-ionized nature of the plasma must be part
of attempts to understand these systems.

Through the solar photosphere and chromosphere the plasma density varies
sharply with height and the ionization degree of hydrogen varies from about
$\chi\sim 10^{-4}$ in the photosphere to $\chi\sim 1$ at the top of the
chromosphere (Vernazza et al. 1981; Fontenla et al. 1993). A wide variety of
dynamical events which characterize these regions are attributed, at least in
part, to reconnection: chromospheric jets (Shibata et al. 2007; Liu et al.
2009; Morton 2012; Tian et al. 2014), Ellerman Bombs (Fang et al. 2006; Nelson
et al. 2015), and type II white light flares (e.g., Ding et al. 1999).
Recently, improvements in the angular resolution of solar telescopes have
allowed many small-scale events in the low solar atmosphere to be observed
(Yang et al., 2015; Xue et al., 2016). High-temperature compact bright points,
possible signatures of magnetic reconnection, have UV counterparts frequently
observed with the Interface Region Imaging Spectrograph (IRIS; Peter et al.
2014).

Protoplanetary disks (PPDs) consist of gas and dust orbiting young stars. They
form as a consequence of angular momentum conservation during the collapse
under self-gravity of dense cores within interstellar molecular gas clouds.
They therefore are a universal step in forming stars (Shu et al., 1987) and
provide the raw materials from which planets form (Evans et al., 2003).

The ionization in PPDs’ surface layers comes mainly from photons emitted by
the central star in the X-ray (Igea & Glassgold, 1999) and UV bands (Perez-
Becker & Chiang, 2011). The disk midplane is optically-thick to these photons,
but remains ionized at least weakly, thanks to cosmic rays (Padovani et al.,
2014) and the decay of radioactive isotopes (Umebayashi & Nakano, 2009). The
ionization fraction $\chi$ ranges from $<10^{-14}$ near the midplane (Wurster
& Li, 2018) to $>10^{-4}$ in the disk atmosphere, where far-UV photons ionize
most of the carbon atoms (Perez-Becker & Chiang, 2011). Midplane temperatures
are high enough for thermal ionization in the fraction of an au nearest the
star (Desch & Turner, 2015).

Magnetic fields threading the disks play fundamental roles in the evolution
and dispersal of the planet-forming material. The fields readily slip through
the gas in the weakly-ionized parts of the disk interior and are better-
coupled to the gas in the more-strongly-ionized surface layers. Reconnection
can affect many aspects of the planet-forming environment, including the
saturation amplitude of the turbulence driven by magneto-rotational
instability or MRI (Sano & Miyama, 1999; Sano & Inutsuka, 2001), the formation
of gas rings that may determine planets’ initial locations (Suriano et al.,
2018, 2019) and the localized heating needed to form the chondrules common in
primitive meteorites (McNally et al., 2013). Magnetic forces also launch winds
from the disk atmosphere near the plasma $\beta=1$ surface where the dynamics
transitions from gas-dominated to magnetically-dominated (Gressel et al.,
2015, 2020; Pascucci et al., 2023; Martel & Lesur, 2022). Flows near this
surface, dragging the magnetic field footpoints at the wind source can
generate CSs leading to magnetic reconnection. While the processes listed
above may be understood through modeling they remain difficult to observe
directly, though there is near-term potential for chemical and thermodynamic
signatures of reconnection to be detected through molecular near- and mid-
infrared spectroscopy with the James Webb Space Telescope (JWST).

The idea that magnetic reconnection operates similarly across the solar
atmosphere, protoplanetary disks, and other astrophysical contexts, comes from
the fact that any CS forming in these environments is subject to the tearing
instability (Furth et al., 1963; Loureiro et al., 2007), leading to magnetic
reconnection. Reconnection follows a longer period over which flows in the
plasma store energy locally, in macroscopic, coherent magnetic structures.
Once reconnection begins, the stored magnetic energy is quickly converted into
plasma heating and particle acceleration. The conditions for triggering such
fast reconnection can be understood using the ideal tearing model (IT, Pucci &
Velli 2014). Under this model, the CS first thins gradually as reconnection
proceeds slowly. Then, once the CS aspect ratio (length over thickness)
reaches a critical threshold, further evolution is dominated by a fast tearing
mode. This model explains numerical MHD results from Landi et al. (2015);
Tenerani et al. (2015b); Landi et al. (2017); Huang et al. (2017a) and plasma
kinetic results from Del Sarto et al. (2016); Pucci et al. (2017); Shi et al.
(2020). The variety of scales involved and the universality of the process
suggest that, in the partially ionized case (Zweibel, 1989; Pucci et al.,
2020), there are three regimes of reconnection: (1) coupled ion and neutral
dynamics, (2) weakly coupled and (3) decoupled dynamics. These regimes are
separated by the thickness of the CS being larger or smaller than
characteristic scales, defined by the parameters of the system and, in
particular, by collision frequencies of ions with neutrals. In Pucci et al.
(2020) the onset of fast magnetic reconnection generated by the tearing
instability is discussed for each regime.

In this work we apply the model to reconnection in the lower solar atmosphere
and protoplanetary disks. In Sec. 2 we summarize the processes involved in the
onset of fast magnetic reconnection. In Sec. 3 we generalize to partially-
ionized plasmas. In Sec. 4 we present analytical calculations of the
transitions between the reconnection regimes due to recursive processes. In
Sec. 5 we introduce the spatial and temporal scales involved in the
reconnection process in the solar chromosphere and PPDs. In Sec. 6 we apply
the model to the solar atmosphere, examining the length scales at which
magnetic reconnection can be observed in the lower, partially-ionized layers
and comparing them with the observable range of scales. In Sec. 7 we carry out
similar calculations for PPD atmospheres, contrast the reconnection regimes
with the magnetic diffusion regimes commonly considered in this context, and
finally estimate the local heating resulting from the dissipation of the
magnetic fields. A summary and conclusions are in Sec. 8.

##  2\. Onset of fast magnetic reconnection, a summary.

In this section, we summarize the onset of the tearing mode instability,
leading to fast magnetic reconnection in a resistive, fully-ionized plasma.
For simplicity we describe the process starting from a 2D Harris CS (Harris,
1962), though the analysis is also valid when there is an out of plane
component of the magnetic field, i.e. a guide field, that may also vary so as
to make the equilibrium force-free – see e.g. Eq. (5) and Eq. (6) in Landi et
al. 2015. The scale of the variation of the equilibrium magnetic field
determines the initial CS thickness $a$.

The linear stability (for incompressible fluctuations) does not depend on the
presence or absence of a magnetic field in the direction orthogonal to the
reconnection plane ($z$ or $\hat{k}$) and whether the equilibrium is force-
free or pressure balanced (Shi et al., 2020). At large Lundquist numbers (low
collision regimes) two regions define the solution structure of the perturbed
magnetohydrodynamics (MHD) equations: a boundary layer of thickness 2$\delta$
around the center ($y=0$) of the CS, where $\delta$ is defined so it separates
resistive inner regions $y<\delta$ from outer regions where magnetic
diffusivity $\eta_{O}$ due to resistivity and growth rate of the tearing
instability may be neglected. From here on, barred quantities are normalized
to the CS _thickness_ $a$. $\bar{\tau}_{A}=a/v_{A}$ is the Alfvén crossing
time using the Alfvén speed $v_{A}$, $\bar{k}=ka=2\pi n/\mathcal{L}$ where $n$
is the wavenumber so that the minimum wavevector available in the system is
$\bar{k}_{min}=2\pi/\mathcal{L}$, for some system lengthscale $\mathcal{L}$.
The Lundquist number
$\bar{S}=\bar{\tau}_{R}/\bar{\tau}_{A}=a\,v_{A}/\eta_{O}$,
$\bar{\tau}_{R}=a^{2}/\eta_{O}$ is the Ohmic diffusion time over the thickness
$a$. The dependence of the maximum growth rate of the tearing instability
$\gamma_{\rm max}$ on the Lundquist number (Ohmic magnetic diffusivity) can be
expressed as

$\gamma_{\rm
max}\bar{\tau}_{A}\sim\bar{S}^{-\frac{1}{2}}\,,\,\,\,\frac{\delta_{\rm
max}}{a}\sim\bar{S}^{-\frac{1}{4}}\,,\,\,\,k_{\rm
max}a\sim\bar{S}^{-\frac{1}{4}}.$ (1)

where the subscript “max” indicates the maximum growth rate available,
corresponding to the wavevector $k_{\rm max}$ (see e.g. Del Sarto et al. 2016
for the Harris CS). The expression for $\gamma_{\rm max}$ in Eq. (1) suggests
that when the Lundquist number is large, as in most astrophysical plasmas, the
growth rate becomes negligible and the process inefficient. We also notice
that the system has an intrinsic length scale $L$, limiting the length of the
CS (e.g. the length of a magnetic loop in the solar corona, the pressure scale
height in a disk atmosphere etc..). This suggested the relation for the
“ideal” tearing instability Pucci & Velli (2014), i.e. for an instability
where the growth rate survives independently of the Lundquist number in the
ideal limit $S\rightarrow\infty$. Rescaling the dispersion relation to the CS
length rather than the thickness, the maximum growth rate of the tearing
instability (corresponding to the fastest growing mode) reads:

$\gamma\tau_{A}\sim
S^{-\frac{1}{2}}\left({\frac{a}{L}}\right)^{-\frac{3}{2}}.$ (2)

where we dropped the subscript “max” to simplify the notation. For an inverse
aspect ratio varying as $a/L\sim S^{-\alpha}$, any $\alpha<1/3$ leads to
growth rates diverging in the ideal limit, while any $\alpha>1/3$ leads to
growth rates tending to zero as the Lundquist number grows (Pucci & Velli,
2014; Tenerani et al., 2015b). This result is general: additional effects such
as viscosity (Tenerani et al., 2015a) and Hall current (Pucci et al., 2017)
result in different scalings for the critical aspect ratio at which fast
reconnection is triggered. It also extends to situations where non-collisional
terms in Ohm’s law break the frozen-in conditions (e.g. Del Sarto et al.,
2016; Mallet, 2020).

##  3\. Reconnection in partially ionized plasmas.

In partially-ionized plasmas, the electron-neutral, ion-neutral and electron-
ion collisions cause an Ohmic-type diffusion of the magnetic field. With three
different species undergoing collisions, such as electrons, ions, and
neutrals, a single-fluid description leads to an appropriately modified
magnetic induction equation (Wardle, 2007, eq. 28):

$\displaystyle\dfrac{\partial B}{\partial t}=\nabla\times({\bf v}\times{\bf
B})-\nabla\times[\eta_{O}(\nabla\times{\bf B})$
$\displaystyle+\eta_{H}(\nabla\times{\bf
B})\times{\hat{b}}+\eta_{AD}(\nabla\times{\bf B})_{\perp}]$ (3)

where $\hat{b}={\bf B}/|{\bf B}|$, and $\eta_{H}$, and $\eta_{AD}$ are
respectively the Hall, and ambipolar diffusivities (AD). The velocity is here
the velocity of the neutral fluid, and the coefficients are calculated
neglecting electron pressure and other forces associated with the ionized
components that are assumed to be negligible. If the only charge carriers are
ions and electrons the Ohmic diffusivity becomes:

$\eta_{O}=\dfrac{c^{2}}{\omega^{2}_{pe}}(\nu_{ei}+\nu_{en})+\dfrac{c^{2}}{\omega^{2}_{pi}}\nu_{in}$
(4)

where $\omega_{pe}$ is the electron plasma frequency, the electron-ion, ion-
neutral and electron-neutral collision frequencies are respectively
$\nu_{ei,in,en}$ and $c$ is the speed of light. In the lower solar atmosphere
helium and hydrogen are by far the most abundant species (Fontenla et al.,
1993) and collisions between neutrals and ions are negligible so the last term
in Eq. (4) can be dropped. For protoplanetary disks, usually the electron ion-
collisions are negligible and, under this assumption, in Wardle (2007), a
prescription for the calculation of non-ideal coefficients is provided as a
function of the local magnetic field, ions and neutrals collision frequencies,
masses and abundances. Wardle (2007) describes three regimes in which each of
the terms in Eq. (4) dominates the diffusion process in protoplanetary disks,
where the main charge carriers may vary from dust grains near the midplane, to
electrons and $H^{+}$ ions in the upper atmospheric layers (see Sec. 7). The
ambipolar diffusivity can be written as (see e.g. Ni et al. 2020):

$\eta_{AD}=\dfrac{{|\bf
B}|^{2}}{|\mu_{0}|}\left(\dfrac{\rho_{n}}{\rho}\right)^{2}(\rho_{i}\nu_{in}+\rho_{e}\nu_{en})^{-1}$
(5)

where $\rho=\rho_{i}+\rho_{n}$ is the total mass density, $\rho_{i}$,
$\rho_{n}$ and $\rho_{e}$ are respectively the ion, neutral and electron mass
density; $\mu_{0}$ is the vacuum permeability. Assuming quasi neutrality
($n_{e}\sim n_{i}$), this expression is the same as in Wardle (2007), where
$\rho\sim\rho_{n}$. Usually diffusion processes are slow compared to the
advection time and the local heating they provide depends on the actual value
of the coefficients. In the IT scenario magnetic reconnection supported by
Ohmic diffusivity can occur in a time comparable to the magnetic field
advection $\tau_{rec}\sim\tau_{A}$. The Hall effect can modify the
reconnection scenario when the scale of the equilibrium magnetic field
gradient $a$, generating the CS, is $a\sim d_{i}=c/\omega_{pi}$, where $d_{i}$
is the ion inertial length and $\omega_{pi}$ the ion plasma frequency
(Terasawa, 1983). In particular for modifications of the IT model, see Pucci
et al. (2017) for the linear theory, where the critical aspect ratio depends
on the ion inertial length while reconnection still proceeds at the fastest
speed in the system, and Shi et al. (2019a) for nonlinear simulations. In the
linear analysis of reconnecting instabilities the effect of non-ideal terms is
assessed by their relative weight compared to the first advective term on the
right hand side of Eq. (3). For the AD to become relevant specifically for
reconnection dynamics,

$a\leq v^{2}_{Ai}/(v_{A}\nu_{in}).$ (6)

Note that AD does not violate the flux conservation associated with electron
and ion motion. It appears in Ohm’s law only because the advective term is
written for the frame of the neutral fluid. The assumption is that collisions
partly couple neutrals to ions so the neutrals participate indirectly in the
reconnection process. In the discussion that follows, we assume instability
sets in on an initial state where ions and neutrals move with the same
velocity. We include AD and its effect on the linear instability. However AD
may also play a role in the formation of favorable initial conditions with CSs
(Brandenburg & Zweibel, 1994; Vishniac & Lazarian, 1999; Braginskii, 1965).
AD-driven CS steepening may be inhibited by guide fields (Ni et al., 2020),
though this does not affect the stability directly, in the sense that for a
given CS thickness the presence or absence of a guide field does not change
the sheet’s stability.

The relative importance of the different non-ideal effects in the partially
ionized environments of the solar atmosphere and protoplanetary accretion
disks are discussed in detail below.

###  3.1. Reconnection: the tearing mode equations.

Recently Pucci et al. (2020) extended the instability calculations presented
by Zweibel (1989) to the full resistive tearing mode dispersion relation, and
from there obtained the trigger conditions for IT in the partially ionized
case. In the latter work, there is no drift of the ions with respect to
neutrals in the equilibrium. The current carriers in the starting equilibrium
CS, considered to be force-free, are therefore the electrons only. In the
linearized equation for incompressible perturbations the ions and neutrals are
collisionally coupled but the equations are then resolved in terms of the ion
flow. Electron-neutral collisions are not taken into account in Ohm’s law,
again written in terms of the perturbed ion flow, though we consider the
electron-neutral collisions in our calculation of the diffusive coefficients
here. Compared to Eq. (3), the Zweibel (1989) model neglects the Hall effect,
while AD does not appear explicitly because of the different choice of
variables in Ohm’s law (ion flow vs neutral flow), though the corresponding
couplings, namely ion neutral momentum transfers, are included via the
modified ion momentum equation. For the different forms of Ohm’s law we refer
to Malyshkin & Zweibel (2011); Murphy & Lukin (2015).

The Hall effect was considered by Terasawa (1983); Pucci et al. (2017); Shi et
al. (2020), from which we derive where the Hall effect plays a role in our
calculations and discuss its effects on the triggering of fast magnetic
reconnnection.

In Pucci et al. (2020); Zweibel (1989) the linearized momentum and induction
equations are written for the ion velocity and vector potentials (respectively
$\phi\,,\psi$) as (primes denote derivatives with respect to the $a$-scaled
variable $y/a$):

$\displaystyle(\gamma\bar{\tau}_{Ai})^{2}\left(1+\dfrac{\nu_{in}}{\gamma+\nu_{ni}}\right)(\phi^{\prime\prime}-\bar{k}^{2}\phi)=$
$\displaystyle-F(\psi^{\prime\prime}-\bar{k}^{2}\psi)+F^{\prime\prime}\psi$
(7)
$\displaystyle\psi=\bar{k}F\phi+\frac{1}{\bar{S}\gamma\bar{\tau}_{Ai}}(\psi^{\prime\prime}-\bar{k}^{2}\psi),$

where $\bar{S}$ is defined using the Alfvén speed calculated with the ion mass
density. We want to stress again that barred quantities will be normalized to
the CS thickness, so that, for example $\bar{\tau}_{Ai}=a/v_{Ai}$ is the
Alfvén time calculated with the ion density, $\gamma$ is the tearing growth
rate associated with a mode with wave vector $\bar{k}=ka$ along the
equilibrium magnetic field. We calculate the collision frequencies assuming
binary elastic (energy and momentum conservation) collisions between electrons
and neutrals so that $\nu_{ni}=\dfrac{n_{i}m_{i}}{n_{n}m_{n}}\
\nu_{in}\Rightarrow\nu_{ni}<\nu_{in}$ at most heights in the solar atmosphere
(see Tab. 1 in Singh et al. 2015) and in protoplanetary disks. Note that the
opposite limit $\nu_{ni}\gg\nu_{in}$ leads to the standard tearing of a
completely ionized plasma. Following Zweibel (1989) we may redefine a starred
Alfvén time and Lundquist number

$\displaystyle\bar{\tau}_{Ai}(1+\frac{\nu_{in}}{\gamma+\nu_{ni}})^{1/2}$
$\displaystyle:=$
$\displaystyle\bar{\tau}_{Ai}f_{M}^{1/2}\rightarrow\bar{\tau}_{A}^{*},$ (8)
$\displaystyle\bar{S}^{*}$ $\displaystyle:=$
$\displaystyle\bar{S}\frac{\bar{\tau}_{Ai}}{\bar{\tau}_{A}^{*}}=\bar{S}f_{M}^{-1/2}.$
(9)

Inserting $\bar{\tau}_{A}^{*}$ into Eqs. (3.1) and substituting
$\bar{S}\bar{\tau}_{Ai}$ with $\bar{S}^{*}\bar{\tau}_{A}^{*}$ and
$\gamma\bar{\tau}_{Ai}$ with $\gamma\bar{\tau}^{*}_{A}$, the tearing mode
equations regain their standard form, so that all the properties of the
dispersion relation discussed previously now apply to the starred quantities.
We are going to analyze the maximum growth rate of the tearing instability Eq.
(1), because the fastest growing mode is the most relevant in the context of
triggering fast magnetic reconnection in natural plasmas causing an efficient
energy conversion. In particular, from Eq. (1) we have that
$\gamma\bar{\tau}_{A}^{*}$ follows the same scaling with $\bar{S}^{*}$ as in
the standard tearing theory:

$\displaystyle\gamma\bar{\tau}^{*}_{A}\sim(\bar{S}^{*})^{-1/2}\Rightarrow\gamma\bar{\tau}_{Ai}\sim(\bar{S})^{-1/2}\left(\dfrac{\bar{\tau}_{Ai}}{\bar{\tau}^{*}_{A}}\right)^{1/2}.$
(10)

When the growth rate is negligible compared to both collision frequencies, the
factor $f_{M}^{1/2}$ becomes

$f_{M}^{1/2}=(1+\frac{\nu_{in}}{\nu_{ni}})^{1/2}=(1+\frac{\rho_{n}}{\rho_{i}})^{1/2}=\left(\frac{\rho}{\rho_{i}}\right)^{1/2}.$
(11)

###  3.2. Reconnection regimes.

With the two ion-neutral collision frequencies, two intrinsic length scales
are introduced into the resistive MHD equations, $a_{c1,c2}$ defined as (Pucci
et al., 2020)

$a_{c1,c2}=\left({\eta_{m}v_{A,Ai}}/{\nu_{ni,in}^{2}}\right)^{1/3}.$ (12)

If the thickness of the CS $a\gg a_{c1}$ fast reconnection onset kicks in in a
regime where the neutrals and ion dynamics is coupled. For $a_{c1}\geq a\geq
a_{c2}$ fast reconnection onset occurs in the intermediate regime. Finally
when $a<a_{c2}$ fast reconnection dynamics involves ions only, with neutrals
not really noticing. In Fig. 1 (left two columns) we summarized the regimes
(divided by colors) and the transition scales (labelled in red and orange),
showing that from the coupled to the decoupled regime we are transitioning to
thinner CS. Ultimately, the plasma parameters and dynamics determine the
regime in which fast reconnection onset occur.

Figure 1.— Diagram summarizing magnetic reconnection regimes. From top to
bottom, the CS thickness decreases and reconnection happens faster. CSs can
shrink also through the recursive operation of tearing instability. At left on
each row is the CS thickness $a$ for fast reconnection, relative to the
critical length scales $a_{c1}$ and $a_{c2}$. At right are the critical IT CS
thickness for fast reconnection and the timescale for the IT instability. In
the coupled regime, the characteristic reconnection time is the Alfveń time
including the density of all species. In the decoupled regime, the
characteristic time is the Alfveń time using the density of the ions alone.
The critical thickness then depends weakly on $\nu_{in}/\tau_{Ai}$ to the
exponent $\zeta$. This results in reconnection speeding up as we move from the
top of the diagram to the bottom. A single CS can pass through all three
regimes while thinning, until either the magnetic field is consumed or the
recursive tearing ends because the trigger conditions are no longer met.

In Fig. 1 right three columns, we summarize the results for the onset of fast
magnetic reconnection from Pucci et al. (2020). In particular, we report the
critical CS thickness $a_{c}$ for each regime, the values of the growth rate
at the transitions between regimes (red and orange colors) and the timescale
over which the instability proceeds in each regime once the critical thickness
is reached. The latter depends on the ion-neutral collision frequencies, and
in particular in the fully decoupled regime has a weak dependence on
$\nu_{in}/\tau_{Ai}$ expressed via the exponent $\zeta$ (non barred quantities
are normalized with $L$ so that $S=Lv_{Ai}/\eta_{O}$). The growth rate of the
tearing instability and so the speed at which reconnection proceeds in the
fully coupled regime will be of the order of $\tau_{A}$, where the Alfvén
speed is calculated with the total (ions+neutrals density). In particular when
$a_{c}\ll a_{c1,c2}$ the time-scale becomes the shortest ion-only Alfvén time,
$\tau_{Ai}=\bar{\tau}_{Ai}a/L$. To compare the trigger conditions found in
Pucci et al. (2020) with those arising when including explicit ambipolar, Hall
and Pedersen terms in Ohm’s law Wardle (2007) and differential equilibrium
flows between neutrals and ions, a linear study of the tearing instability
should be carried out within a three fluid framework, where the dynamics of
the positive and negative charge carriers and of the neutrals are solved for
the IT mode. However, the main AD effects on magnetic reconnection, i.e. the
consequences of ion - neutral drift on reconnecting mode growth rates are
captured by the critical scales of Pucci et al. (2020).

##  4\. Energy cascade towards smaller scales

On the basis of the above discussion, one may outline the behavior of the
tearing instability in a simple CS that is slowly thinning. We shall assume
the tearing mode develops on the global, ion-neutral coupled, Alfvénic time-
scale, since this often proves to be the case for the environments discussed
in this work. Modeling work has shown that a recursive reconnection regime may
appear (Shibata & Tanuma, 2001; Tenerani et al., 2015b; Shibayama et al.,
2015; Huang et al., 2017b), where CSs form at successively smaller scales.
This suggests a transition to the intermediate and then fully decoupled regime
as the sheets’ thicknesses decrease, accelerating the nonlinear evolution of
the tearing mode, and so of the energy conversion.

The natural thinning of the CS results from nonlinear processes driven by the
self-attraction of parallel currents and the repulsion of oppositely directed
currents. Considering a simple 2D magnetic configuration as in Tenerani et al.
(2015b), the reconnecting magnetic field generates an inflow-outflow pattern
at the x-point. These flows tend to collapse the x-point into a new sheet that
is stretched by the outflow but being kept at the initial thickness by the
inflow. Once the sheet lengthens sufficiently, it itself is unstable to
reconnection on a faster timescale, breaking the initially macroscopic CS into
ever smaller reconnecting regions. Each then becomes further unstable
accelerating the process until a turbulent regime, with heating and
acceleration, is obtained (Fig. 2 in Tenerani et al. 2015b, Fig.2 in Shibayama
et al. 2015, Fig. Huang et al. 2017b). As discussed by Shibata & Tanuma
(2001); Singh et al. (2019), different recursive reconnection models predict
different numbers of plasmoids at the $n$-th stage of the reconnection cascade
and so different number of CSs separating plasmoids (regions of close magnetic
field lines, see the figures and references mentioned above). The process
continues down to smaller scales until energy is dissipated, by relevant
microscopic processes; for the fully ionized corona, for example, these are
collective kinetic effects. In partially-ionized plasmas, if a reconnection
event is initiated at scales at which ions and neutrals are coupled, see the
blue area in the diagram in Fig. 1, as sheets keep shrinking, they will become
unstable to secondary IT instability in the intermediate or fully decoupled
regimes. Information on plasmoid scaling and energy transfer to the small
scales can be translated into spectral features (Schekochihin, 2022), and not
only can help distinguish one model prediction to another, but might also
result as an observable feature of the reconnnection layer.

###  4.1. Transition from the coupled to the Intermediate regime.

Understanding the energy cascade at smaller scales requires finding a relation
between the $n_{th}$ recursive reconnection stage properties and the
$n_{th}-1$ parent reconnecting process. As discussed in Tenerani et al.
(2016), we do not make any assumption as in e.g. Shibata & Tanuma (2001); Ji &
Daughton (2011). We follow instead Tenerani et al. (2015b) where it is
empirically observed that the characteristics of the $n_{th}$ CS thickness is
related to the $n_{th}-1$ CS length by $a_{n}/L_{n-1}\sim S^{-1/2}$, because
the upstream magnetic field is seen to be swept into the reforming x-points
and remains of the same intensity as the background at each level (see e.g.
Schekochihin 2022). For sake of simplicity let’s consider the second step of
the recursive collapse, then

$a_{1}/L_{0}\sim S^{-1/2},$ (13)

where we labelled with subscript $0$ the initial macroscopic unstable CS
length, with subscript $1$ the parameters of the secondary reconnecting
sheets, and assumed we are in the fully coupled ion-neutral regime. Later in
this work we will show this hypothesis is satisfied for a relatively large
range of CS lengths in the lower solar atmosphere as well as in protoplanetary
disks. In the partially ionized case $\delta$ corresponding to the maximum
growth rate is given by
$\delta/a\sim\bar{S}^{*-1/4}=(\tau_{D}/\bar{\tau}_{A}^{*})^{-1/4}=S^{-1/4}f_{M}^{-1/8}$,
where again $S$ is the Lunquist number calculated with Alfvén speed based on
the ion mass density only. As discussed in Pucci et al. (2020), since $f_{M}$
is invariant for the IT rescaling, the solution is the same as for the classic
IT with corrections depending on the regime, $\delta/L=S^{-1/2}f_{M}^{-1/8}$.
Based on the assumptions above, Eq. (13) becomes

$a_{1}/L_{0}\sim S^{-1/2}(\rho_{n}/\rho_{i})^{-1/8}$ (14)

This means that if the CS thickness at the $n=1$ recursive reconnection stage
$a_{1}<a_{c1}$, the triggering condition for the tearing instability at this
stage will be the one described in Fig. 1 for the intermediate regime, i.e.
$a_{c}\sim LS^{-1/3}(\nu_{in}/\tau_{Ai})^{1/3}$. This occurs when

$a_{1}<a_{c1}.$

The inequality above gives

$\displaystyle\tau_{A}/\tau_{ni}<(S\ \rho/\rho_{i})^{1/4}$ (15)

where $\tau_{ni}\sim 1/\nu_{ni}$ as the collision time neutral-ion.

As we will show in the next sections, in a partially ionized plasma with a
significant neutral population, Eq. (15) is often satisfied and secondary
reconnection will occur in the intermediate or decoupled regime, even if the
dynamics ultimately depend on the thickness and so length of the CS and the
local Lundquist number.

##  5\. Advection scales in Solar and disk atmospheres.

In this section we address the characteristic lengths and temporal scale
variations of the environments in question, in order to calculate magnetic
Reynolds numbers necessary to determine the critical thicknesses of fast
magnetic reconnection. The reason we focus on the Reynolds number is the
following. Once a CS has formed and become unstable to fast magnetic
reconnection thanks to resistivity and neutral collisions breaking flux-
freezing, the sheet’s critical thickness is the key parameter to understand
the role of other non-ideal effects in the reconnection process. The critical
thickness, estimated as shown in Fig. 1 can be then compared to the lengths
calculated in Sec. 2 to find whether the Hall effect and AD alter
reconnection’s onset. AD is included in our model and affects the reconnection
regimes in which a degree of coupling between ions and neutrals is present.
The Hall effect has scales defined by the ion inertial length $d_{i}$ and
affects reconnection at kinetic scales, where collisions between neutrals and
ions are generally negligible. In this paper we will discuss in detail when
corrections due to the Hall effect and AD are relevant, even if they are not
expected to significantly change the conclusions. The picture could be further
refined using three-fluid calculations to obtain corrections to the CS
thickness explicitly proportional to $\eta_{AD}$. However, such corrections
would be small since the neutrals’ role in the ion and electron dynamics is
already included here.

We adopt the following definition for the magnetic Reynolds number:

$R_{m}=\frac{Lv}{\eta_{O}}.$ (16)

where $L$ is the characteristic macroscopic lengthscale of the system and $v$
is the convective speed found in the first term on the right-hand side of Eq.
(3). The Lundquist number $S$ is obtained by replacing $v$ with $v_{A}$. Eq.
(2) for fully ionized plasmas and Eq. (11) for partially ionized plasmas tell
us the efficiency of reconnection due to the tearing instability depends on
the magnetic Reynolds number. For the model in Pucci et al. (2020) this
dependence reflects on the critical scale of which fast magnetic reconnection-
and so efficient magnetic energy conversion- can be achieved. In order to
discuss the reconnection dynamics it is then important to understand how to
estimate the magnetic Reynolds numbers. In this section we will discuss the
(temporal and spatial) scales involved in convecting and diffusing the
magnetic field in the solar chromosphere and photosphere, as well as in
protoplanetary disks. A similar discussion holds for any other plasma
environment.

###  5.1. Magnetic Reynolds number in the solar atmosphere.

The solar photosphere and chromosphere are gravitationally stratified. This
means the characteristic length scale $L$ of reconnection in these layers is
limited to at most the smaller of the pressure scale height $H_{p}$ and the
typical transverse convective scale, defined by granulation at $\sim 1000$ km.
Since $H_{p}$ is the smaller of the two, $L<H_{p}$. In the solar corona in
contrast, the gravitational scale height is much greater than the transverse
and magnetic field scales, the plasma $\beta$ ($\simeq c_{\rm
s}^{2}/v_{A}^{2}$ ) is very small, and gravity can often be neglected. In the
photosphere and chromosphere then, the pressure scale height defines the
maximum vertical coherent scale achievable so that the appropriate magnetic
Reynolds number becomes (c.f. Takeuchi & Shibata 2001)

$R_{\rm m}=\frac{H_{p}c_{s}}{\eta_{O}},$ (17)

where $c_{\rm s}$ is the speed of sound. Indeed in these atmospheric layers
the plasma $\beta$ may be larger than unity. This is especially important near
the boundary of an isolated flux tube, where the Alfvén velocity is comparable
to the sound speed. We can estimate the pressure scale height in the solar
atmosphere using the C7 model in Avrett & Loeser (2008). Once the Ohmic
magnetic resistivity $\eta_{O}$ has been calculated (see e.g. Kovitya & Cram
1983; Singh & Krishan 2010a) we can, for example, evaluate $R_{\rm mc}\sim
10^{\rm 5}-10^{\rm 4}$ in the solar photosphere at the solar temperature
minimum (Kovitya & Cram, 1983).

To compare with the usual definition of the Lundquist number in Eq. (16), we
parameterize

$Lv_{A}=H_{p}c_{s}L/(H_{p}\beta^{1/2})\simeq 10H_{p}c_{s}.$

We then conclude that in order to discuss the onset of fast magnetic energy
conversion through reconnection, we should consider how the choice of the
parameters affects the critical threshold for the onset of the tearing
instability, ultimately leading to the reconnection process.

###  5.2. Magnetic Reynolds number in protoplanetary disk atmospheres.

In protoplanetary disks the magnetic fields’ evolution depends on dynamical
processes that may include magneto-rotational instability (MRI) leading to
channel flows and turbulence (Balbus & Hawley, 1991; Hawley & Balbus, 1991;
Gardiner & Stone, 2005; Miller & Stone, 2000; Sano & Inutsuka, 2001),
magnetically driven winds (Suzuki & Inutsuka, 2009; Blandford & Payne, 1982;
Ferreira & Pelletier, 1995; Gressel et al., 2015, 2020; Iwasaki et al., 2024),
photoevaporative winds (Ercolano et al., 2021; Komaki et al., 2023), vertical
shear instability (Nelson et al., 2013; Umurhan et al., 2016; Latter &
Papaloizou, 2018; Schäfer et al., 2020; Hammer & Lin, 2023), magnetic buoyancy
and Parker instability (Miller & Stone, 2000; Hirose & Turner, 2011), and
magnetic reconnection (Miller & Stone, 2000; Inutsuka & Sano, 2005; McNally et
al., 2013). Depending on the physical and chemical state of the environment,
each of these processes can be of interest for local as well as global disk
evolution, accretion of disk material onto the star, and the dispersal of the
disk back into interstellar space. The dimensionless Reynolds and Elsasser
numbers define how the fields evolve on large scales, but are insufficient for
understanding how magnetic reconnection proceeds once a CS forms. In this
paper we focus on reconnection, so the magnetic Reynolds number defined above
is a key parameter governing the CS thickness and so the role of the other non
ideal effects.

Most of the definitions of magnetic Reynolds numbers in protoplanetary disks
have been provided in the context of the MRI and its saturation in MHD
simulations (see the discussion in Pucci et al. 2021). Sano et al. (1998)
defined $R_{m}=v^{2}_{A}/\Omega\eta_{O}=1$, where $\Omega$ is the angular
velocity of the disk, below which MRI driven turbulence relaxes in channel
flows. The latter define a new macroscopic lengthscale in the system, which
can be disrupted by magnetic reconnection events. In the case of a poloidal
field with zero vertical net flux, Fleming et al. (2000) argued that the MRI
can be sustained when the effective magnetic Reynolds number
$R_{m}=c^{2}_{s}/\Omega\eta_{O}\leq 10^{4}$. Nauman & Blackman (2017) defined
$R_{m}=L^{2}\Omega/\eta_{O}>10^{3}$, where $L$ is the size of the simulation
domain in the radial direction, above which the magnetic turbulence induced by
the MRI can be sustained, and energy is stored in the turbulent magnetic
field, defining a new length at the eddy scale. In general the definition of a
magnetic Reynolds/Lundquist number depends on the dynamics and, in a
protoplanetary disk, its calculation is difficult because of uncertainties on
magnetic field detection (Li et al., 2016; Vlemmings et al., 2019). Recent
measurements using the Zeeman effect provide estimates on the upper limits of
magnetic fields of $|B|\sim 10^{-2}\rm{G}$ (Vlemmings et al., 2019),
significantly smaller than the values we see in the solar atmosphere. In this
work we will vary the magnetic field in a range $B=10^{-4}-10^{2}\rm{G}$ to
cover a sufficiently large parameter space. The disk atmosphere is vertically
stratified so that a scale height can be defined as $H_{p}:=c_{s}/\Omega$
where $c_{s}$ is the sound speed and $\Omega$ the orbital frequency. The
rotational speed of the atmospheric layer provides a torque, storing energy in
the magnetic field lines, so we can think about the rotational speed as the
main driver for the so-called accumulation or “build-up” phase (for magnetic
field concentration and amplification in PPDs see Pringle 1981; Wardle 2007;
Lesur 2021). Reconnection is particularly important in the regions where the
plasma parameter $\beta\leq 1$, since most of the energy is stored in the
magnetic field, which dominates the dynamics. We will then adopt the Alfvén
speed as the main driver velocity for magnetic field convection in the regions
of interest, so the Lundquist number for the disk will be estimated here as
$S=H_{p}v_{A}/\eta_{O}$. The latter will be calculated following Wardle
(2007), see the discussion on the model for PPDs adopted in this work, Sec. 7.

##  6\. Partially ionized reconnection in the solar atmosphere

In this section, we will discuss the regimes in which fast magnetic
reconnection can be triggered in the solar photosphere and chromosphere.

Height |  $T$[K] | $n_{e}$ | $n_{n}$ | $n_{HeI}$ | $n_{HeII}$ | $\nu_{ei}$ | $\nu_{en}$ | $\nu_{in}$
---|---|---|---|---|---|---|---|---
$0$ | $6\ 583$ | $8.4\cdot 10^{13}$ | $1.19\cdot 10^{17}$ | $1.18\cdot 10^{16}$ | $11.7$ | $6.4\cdot 10^{9}$ | $7.97\cdot 10^{9}$ | $2.63\cdot 10^{9}$
$20$ | $6\ 231$ | $4.26\cdot 10^{13}$ | $1.06\cdot 10^{17}$ | $1.18\cdot 10^{16}$ | $11.7$ | $3.53\cdot 10^{9}$ | $7.23\cdot 10^{9}$ | $2.39\cdot 10^{9}$
$50$ | $5\ 826$ | $1.91\cdot 10^{13}$ | $9.68\cdot 10^{16}$ | $9.52\cdot 10^{15}$ | $0.17$ | $1.77\cdot 10^{9}$ | $6.12\cdot 10^{9}$ | $2.02\cdot 10^{9}$
$100$ | $5\ 431$ | $8.75\cdot 10^{12}$ | $7.24\cdot 10^{16}$ | $7.04\cdot 10^{15}$ | $1.02\cdot 10^{-2}$ | $9.15\cdot 10^{8}$ | $4.42\cdot 10^{9}$ | $1.46\cdot 10^{9}$
$125$ | $5\ 288$ | $6.7\cdot 10^{12}$ | $6.15\cdot 10^{16}$ | $7.04\cdot 10^{15}$ | $1.02\cdot 10^{-2}$ | $7.28\cdot 10^{8}$ | $3.71\cdot 10^{9}$ | $1.22\cdot 10^{9}$
$200$ | $5\ 010$ | $3.31\cdot 10^{12}$ | $3.55\cdot 10^{16}$ | $3.44\cdot 10^{15}$ | $2.56\cdot 10^{-4}$ | $4\cdot 10^{8}$ | $2.08\cdot 10^{9}$ | $6.87\cdot 10^{8}$
$350$ | $4\ 700$ | $9.34\cdot 10^{11}$ | $1.03\cdot 10^{16}$ | $1.55\cdot 10^{15}$ | $1.52\cdot 10^{-5}$ | $1.25\cdot 10^{8}$ | $5.87\cdot 10^{8}$ | $1.94\cdot 10^{8}$
$450$ | $4\ 485$ | $3.87\cdot 10^{11}$ | $4.32\cdot 10^{15}$ | $4.29\cdot 10^{14}$ | $0.41$ | $5.7\cdot 10^{7}$ | $2.4\cdot 10^{8}$ | $4\cdot 10^{7}$
$490$ | $4\ 435$ | $2.72\cdot 10^{11}$ | $2.99\cdot 10^{15}$ | $2.74\cdot 10^{14}$ | $2.98$ | $4.14\cdot 10^{7}$ | $1.65\cdot 10^{8}$ | $2.72\cdot 10^{7}$
$525$ | $4\ 410$ | $2\cdot 10^{11}$ | $2.16\cdot 10^{15}$ | $2.18\cdot 10^{14}$ | $5.78$ | $3.12\cdot 10^{7}$ | $1.19\cdot 10^{8}$ | $1.96\cdot 10^{7}$
$560$ | $4\ 400$ | $1.47\cdot 10^{11}$ | $1.55\cdot 10^{15}$ | $1.73\cdot 10^{14}$ | $9.62$ | $2.44\cdot 10^{7}$ | $8.5\cdot 10^{7}$ | $1.4\cdot 10^{7}$
$615$ | $4\ 435$ | $9.16\cdot 10^{10}$ | $9.06\cdot 10^{14}$ | $1.09\cdot 10^{14}$ | $18.9$ | $1.44\cdot 10^{7}$ | $5\cdot 10^{7}$ | $8.24\cdot 10^{6}$
$660$ | $4\ 510$ | $6.42\cdot 10^{10}$ | $5.83\cdot 10^{14}$ | $6.94\cdot 10^{13}$ | $32.1$ | $9.83\cdot 10^{6}$ | $3.24\cdot 10^{7}$ | $5.34\cdot 10^{6}$
$750$ | $4\ 840$ | $4.76\cdot 10^{10}$ | $2.4\cdot 10^{14}$ | $2.85\cdot 10^{13}$ | $60.9$ | $7\cdot 10^{6}$ | $1.4\cdot 10^{7}$ | $2.3\cdot 10^{6}$
$800$ | $5\ 090$ | $5.46\cdot 10^{10}$ | $1.5\cdot 10^{14}$ | $1.91\cdot 10^{13}$ | $68.7$ | $7.11\cdot 10^{6}$ | $8.84\cdot 10^{6}$ | $1.45\cdot 10^{6}$
$854$ | $5\ 430$ | $8.06\cdot 10^{10}$ | $9.09\cdot 10^{13}$ | $1.3\cdot 10^{13}$ | $80$ | $9.5\cdot 10^{6}$ | $5.55\cdot 10^{6}$ | $9.15\cdot 10^{5}$
$900$ | $5\ 720$ | $1.16\cdot 10^{11}$ | $6.09\cdot 10^{13}$ | $8.97\cdot 10^{12}$ | $93.8$ | $1.3\cdot 10^{7}$ | $3.8\cdot 10^{6}$ | $6.29\cdot 10^{5}$
$946$ | $5\ 969$ | $1.51\cdot 10^{11}$ | $4.18\cdot 10^{13}$ | $5.26\cdot 10^{12}$ | $120$ | $1.54\cdot 10^{7}$ | $2.67\cdot 10^{6}$ | $4.4\cdot 10^{5}$
$971$ | $6\ 100$ | $1.73\cdot 10^{11}$ | $3.43\cdot 10^{13}$ | $5.26\cdot 10^{12}$ | $120$ | $1.7\cdot 10^{7}$ | $2.22\cdot 10^{6}$ | $3.7\cdot 10^{5}$
$1\ 065$ | $6\ 400$ | $2.09\cdot 10^{11}$ | $1.72\cdot 10^{13}$ | $2.68\cdot 10^{12}$ | $169$ | $1.91\cdot 10^{7}$ | $1.14\cdot 10^{6}$ | $1.89\cdot 10^{5}$
$1\ 143$ | $6\ 531$ | $2\cdot 10^{11}$ | $1.01\cdot 10^{13}$ | $1.4\cdot 10^{12}$ | $686$ | $1.76\cdot 10^{7}$ | $6.8\cdot 10^{5}$ | $1.11\cdot 10^{5}$
$1\ 398$ | $6\ 610$ | $1.44\cdot 10^{11}$ | $1.88\cdot 10^{12}$ | $4.22\cdot 10^{11}$ | $563$ | $1.25\cdot 10^{7}$ | $1.3\cdot 10^{5}$ | $21\ 000$
$1\ 520$ | $6\ 623$ | $1.42\cdot 10^{11}$ | $7.96\cdot 10^{11}$ | $1.35\cdot 10^{11}$ | $959$ | $1.25\cdot 10^{7}$ | $53\ 400$ | $8\ 850$
$1\ 722$ | $6\ 643$ | $1.03\cdot 10^{11}$ | $1.92\cdot 10^{11}$ | $6.66\cdot 10^{10}$ | $2\ 890$ | $8.95\cdot 10^{6}$ | $13\ 000$ | $2\ 130$
$1\ 946$ | $6\ 667$ | $5.53\cdot 10^{10}$ | $4.37\cdot 10^{10}$ | $1.56\cdot 10^{10}$ | $13\ 500$ | $4.82\cdot 10^{6}$ | $3\ 000$ | $487$
$2\ 098$ | $6\ 700$ | $3.29\cdot 10^{10}$ | $2.26\cdot 10^{10}$ | $2\ 530$ | $6.16$ | $2.9\cdot 10^{6}$ | $1\ 520$ | $252$
Table 1Input parameters for the model vs. height in the solar atmosphere. The height has units of $10^{5}$ cm, the temperature $T$ is in Kelvin, the number densities are in cm-3. $n_{HeI}$ is the neutral helium abundance while $n_{HeII}$ is the ionized helium abundance. The electron-ion ($\nu_{ei}$), electron-neutral ($\nu_{en}$), and ion-neutral ($\nu_{in}$) collision frequencies are in $s^{-1}$, see the text for how these quantities are estimated. Height | $\eta_{O}$ | $\eta_{H}$ | $\eta_{AD}$ | $v_{A}$ | $H_{p}$ | $S$ | $R_{m}$
---|---|---|---|---|---|---|---
$0$ | $4.83\cdot 10^{7}$ | $7.1\cdot 10^{7}$ | $3.09\cdot 10^{5}$ | $6.67\cdot 10^{5}$ | $4\cdot 10^{7}$ | $1.1\cdot 10^{6}$ | $1.12\cdot 10^{6}$
$20$ | $7\cdot 10^{7}$ | $1.37\cdot 10^{8}$ | $6.53\cdot 10^{5}$ | $6.76\cdot 10^{5}$ | $3.78\cdot 10^{7}$ | $7.2\cdot 10^{5}$ | $7.07\cdot 10^{5}$
$50$ | $1.17\cdot 10^{8}$ | $2.94\cdot 10^{8}$ | $1.57\cdot 10^{6}$ | $6.88\cdot 10^{5}$ | $3.53\cdot 10^{7}$ | $3.9\cdot 10^{5}$ | $3.83\cdot 10^{5}$
$100$ | $1.72\cdot 10^{8}$ | $5.88\cdot 10^{8}$ | $3.99\cdot 10^{6}$ | $7.19\cdot 10^{5}$ | $3.3\cdot 10^{7}$ | $2.34\cdot 10^{5}$ | $2.36\cdot 10^{5}$
$125$ | $1.9\cdot 10^{8}$ | $7.42\cdot 10^{8}$ | $5.8\cdot 10^{6}$ | $7.33\cdot 10^{5}$ | $3.21\cdot 10^{7}$ | $2.1\cdot 10^{5}$ | $2.04\cdot 10^{5}$
$200$ | $2.11\cdot 10^{8}$ | $1.25\cdot 10^{9}$ | $1.47\cdot 10^{7}$ | $8\cdot 10^{5}$ | $3.04\cdot 10^{7}$ | $1.7\cdot 10^{5}$ | $1.7\cdot 10^{5}$
$350$ | $2.16\cdot 10^{8}$ | $3.07\cdot 10^{9}$ | $8.79\cdot 10^{7}$ | $9.46\cdot 10^{5}$ | $2.85\cdot 10^{7}$ | $1.5\cdot 10^{5}$ | $1.51\cdot 10^{5}$
$450$ | $2.17\cdot 10^{8}$ | $5.7\cdot 10^{9}$ | $6\cdot 10^{8}$ | $1.1\cdot 10^{6}$ | $2.72\cdot 10^{7}$ | $1.4\cdot 10^{5}$ | $1.4\cdot 10^{5}$
$490$ | $2.15\cdot 10^{8}$ | $7.28\cdot 10^{9}$ | $1.01\cdot 10^{9}$ | $1.16\cdot 10^{6}$ | $2.7\cdot 10^{7}$ | $1.4\cdot 10^{5}$ | $1.39\cdot 10^{5}$
$525$ | $2.11\cdot 10^{8}$ | $8.97\cdot 10^{9}$ | $1.57\cdot 10^{9}$ | $1.22\cdot 10^{6}$ | $2.7\cdot 10^{7}$ | $1.39\cdot 10^{5}$ | $1.41\cdot 10^{5}$
$560$ | $2.1\cdot 10^{8}$ | $1.1\cdot 10^{10}$ | $2.44\cdot 10^{9}$ | $1.27\cdot 10^{6}$ | $2.7\cdot 10^{7}$ | $1.42\cdot 10^{5}$ | $1.41\cdot 10^{5}$
$615$ | $2\cdot 10^{8}$ | $1.51\cdot 10^{10}$ | $4.85\cdot 10^{9}$ | $1.38\cdot 10^{6}$ | $2.7\cdot 10^{7}$ | $1.51\cdot 10^{5}$ | $1.5\cdot 10^{5}$
$660$ | $1.9\cdot 10^{8}$ | $1.89\cdot 10^{10}$ | $8.16\cdot 10^{9}$ | $1.47\cdot 10^{6}$ | $2.74\cdot 10^{7}$ | $1.65\cdot 10^{5}$ | $1.62\cdot 10^{5}$
$750$ | $1.21\cdot 10^{8}$ | $1.95\cdot 10^{10}$ | $1.5\cdot 10^{10}$ | $1.68\cdot 10^{6}$ | $3\cdot 10^{7}$ | $2.8\cdot 10^{5}$ | $2.88\cdot 10^{5}$
$800$ | $8.26\cdot 10^{7}$ | $1.48\cdot 10^{10}$ | $1.55\cdot 10^{10}$ | $1.8\cdot 10^{6}$ | $3.09\cdot 10^{7}$ | $4.44\cdot 10^{5}$ | $4.49\cdot 10^{5}$
$854$ | $5.28\cdot 10^{7}$ | $8.6\cdot 10^{9}$ | $1.24\cdot 10^{10}$ | $1.93\cdot 10^{6}$ | $3.3\cdot 10^{7}$ | $7.7\cdot 10^{5}$ | $7.63\cdot 10^{5}$
$900$ | $4\cdot 10^{7}$ | $5.29\cdot 10^{9}$ | $9.46\cdot 10^{9}$ | $2.04\cdot 10^{6}$ | $3.47\cdot 10^{7}$ | $1.1\cdot 10^{6}$ | $1.08\cdot 10^{6}$
$946$ | $3.38\cdot 10^{7}$ | $3.63\cdot 10^{9}$ | $8.74\cdot 10^{9}$ | $2.18\cdot 10^{6}$ | $3.62\cdot 10^{7}$ | $1.4\cdot 10^{6}$ | $1.37\cdot 10^{6}$
$971$ | $3.14\cdot 10^{7}$ | $3\cdot 10^{9}$ | $8.03\cdot 10^{9}$ | $2.22\cdot 10^{6}$ | $3.7\cdot 10^{7}$ | $1.53\cdot 10^{6}$ | $1.53\cdot 10^{6}$
$1\ 065$ | $2.73\cdot 10^{7}$ | $2.02\cdot 10^{9}$ | $8.69\cdot 10^{9}$ | $2.46\cdot 10^{6}$ | $3.88\cdot 10^{7}$ | $1.89\cdot 10^{6}$ | $1.89\cdot 10^{6}$
$1\ 143$ | $2.51\cdot 10^{7}$ | $1.81\cdot 10^{9}$ | $1.09\cdot 10^{10}$ | $2.67\cdot 10^{6}$ | $3.96\cdot 10^{7}$ | $2.12\cdot 10^{6}$ | $2.11\cdot 10^{6}$
$1\ 398$ | $2.5\cdot 10^{7}$ | $1.54\cdot 10^{9}$ | $2.88\cdot 10^{10}$ | $3.3\cdot 10^{6}$ | $4.01\cdot 10^{7}$ | $2.21\cdot 10^{6}$ | $2.17\cdot 10^{6}$
$1\ 520$ | $2.5\cdot 10^{7}$ | $1.23\cdot 10^{9}$ | $3.97\cdot 10^{10}$ | $3.68\cdot 10^{6}$ | $4.02\cdot 10^{7}$ | $2.2\cdot 10^{6}$ | $2.17\cdot 10^{6}$
$1\ 722$ | $2.5\cdot 10^{7}$ | $1.21\cdot 10^{9}$ | $9.88\cdot 10^{10}$ | $4.1\cdot 10^{6}$ | $4.03\cdot 10^{7}$ | $2.2\cdot 10^{6}$ | $2.19\cdot 10^{6}$
$1\ 946$ | $2.5\cdot 10^{7}$ | $1.62\cdot 10^{9}$ | $2.51\cdot 10^{11}$ | $4.52\cdot 10^{6}$ | $4.04\cdot 10^{7}$ | $2.23\cdot 10^{6}$ | $2.2\cdot 10^{6}$
$2\ 098$ | $2.5\cdot 10^{7}$ | $2.28\cdot 10^{9}$ | $5.19\cdot 10^{11}$ | $4.51\cdot 10^{6}$ | $4.07\cdot 10^{7}$ | $2.23\cdot 10^{6}$ | $2.28\cdot 10^{6}$
Table 2Derived physical parameters for the solar atmosphere, in c.g.s. units.
The order of magnitude for the calculation of the non ideal Hall and AD
coefficients is the same for both the values of magnetic fields $B_{0}$ at the
bottom of the photosphere and so we chose $B_{0}=1200G$. The Alfvén speed in
the table is also obtained using $B_{0}=1200G$ and the total (neutral and ion)
mass density. The scale height $H_{p}$ is defined by Eq. (19). The Lundquist
number $S$ is calculated using $v_{A}$, $H_{p}$, and $\eta_{O}$ while the
Reynolds number $R_{m}$ is calculated using the speed of sound $c_{s}$ in
place of the Alfvén speed.

###  6.1. Lower solar atmospheric structure.

The model we adopt for the solar atmosphere is the one labeled C7 in Avrett &
Loeser (2008), where the quantities are functions only of height in the solar
atmosphere. The number densities of ions and neutrals are listed in Tab. 1
assuming quasi-neutrality $n_{i}\sim n_{e}$. As the most abundant species in
the solar atmosphere are hydrogen and helium, the neutral and ionized helium
abundances $n_{\rm HeI}$ and $n_{\rm HeII}$ are also provided. The ionized
helium abundance is orders of magnitude lower than the electron abundance (and
so positive ion abundance, under quasi-neutrality), though helium contributes
noticeably to the neutral population. Henceforth we will adopt hydrogen as the
main neutral species. The interspecies collision frequencies $\nu_{ei}$,
$\nu_{en}$, $\nu_{in}$ are calculated using Zhadnov (1962), for which the
collision cross-sections are obtained from Vranjes & Krstic (2013). In Tab. 1
we also list the temperature at each height. The temperature profile plotted
in Fig. 2 shows a minimum about 500 km above the solar surface.

Magnetic field at a given height is evaluated using the relation (Leake et
al., 2005)

$B=B_{s}\left(\frac{\rho}{\rho_{s}}\right)^{\alpha},$ (18)

where $B_{s}$ is the magnetic field at the solar surface, defined as the
surface with optical depth equal to unity (bottom of the photosphere). We
choose $\alpha=0.3$ so the magnetic field weakens with height. We experiment
with $\alpha$ over the range $0.3$ to $0.6$ (Leake et al., 2005) with only
minor impacts on the results. We consider two different field strengths at the
solar surface, $B_{s}=1200$ G and $B_{s}=2200$ G (Singh & Krishan, 2010b).

In Fig. 2 we show also the total density, from which we derive the Alfvén
speed profile. The pressure scale height is

$H_{p}=\frac{\mathcal{R}T}{\mu g},$ (19)

where $\mathcal{R}$ is the gas constant, $\mu$ the mean mass per mole of
atmospheric plasma, and $g$ the local acceleration due to gravity (Sec. 5.1).

Figure 2.— Temperature $T$, Alfvén speed $v_{A}$, total density $\rho$, and
pressure scale height $H_{p}$, all in c.g.s. units, as functions of height in
the solar atmosphere using Tab. 1.

We now have all the quantities needed to calculate the critical scales in Eq.
(12).

###  6.2. Critical scales for the onset of fast magnetic reconnection in the
lower solar atmosphere.

In Tab. 2 we listed the resulting Ohmic diffusivity and, for comparison, the
Hall and AD coefficients. The Hall coefficient is the same order of magnitude
as the Ohmic magnetic resistivity. The AD depends on the magnetic field and
increases with height in the atmosphere. While these quantities tell us which
diffusion process is dominant, as discussed before, diffusion is slow compared
to advection and fast reconnection. We will come back to this later in this
section. Magnetic Reynolds numbers, necessary to calculate the critical CS for
the IT mode, inherited the contribution due to the ion-neutral collisions from
the diffusivity.

In Fig. 3 we show the results for the transition between different
reconnection regimes (color coded), as a function of height in the solar
atmosphere. The coupled regime (blue area) extends down to CSs with thickness
of $a\ll 1$ km for all the heights, both in the photospheric as well as in the
chromospheric layer.

Figure 3.— Reconnection regime versus height in the solar atmosphere and
thickness of the CS. The critical CS thicknesses $a_{c1}$ and $a_{c2}$ are the
red and orange lines, respectively. Blue shading shows the regime where ions
and neutrals are well-coupled by collisions, yellow the intermediate regime,
and green the regime where ions and neutrals are decoupled and reconnection
proceeds at the faster Alfvén speed of the ions alone. The dotted overlay
marks where the Hall effect is expected to play a role, and the barred overlay
is the where AD must be considered. The black dashed line near top indicates
the thickness of a CS having length $L=H_{p}$ in the resistive coupled regime.
A CS of length $L$ lying above the dashed line can reconnect slowly, but once
fast reconnection begins at the dashed line, thinner CSs below the dashed line
can form only through recursive (secondary) reconnection events.

The Lundquist numbers and the magnetic Reynolds numbers resulted to be of the
same order of magnitude, and so we used $S$ for the results shown in Fig. 3.

The critical length scale $a/H_{p}\sim S^{-1/3}(\rho_{n}/\rho_{i})^{1/6}$
associated with the longest available CS, i.e. limited by $L<H_{p}$ at each
height, is shown in Fig. 3 (dashed black line). This means there is a range of
CS thicknesses in the solar atmosphere $a\lesssim 100$ km for which
reconnection begins with ion-neutral coupled dynamics. When the magnetic field
dominates the dynamics ($\beta<1$), larger CSs are likely to form. Scales down
to $a\sim 100$ km could be resolved with DKIST or SOLAR-C (Watanabe, 2014;
Shimizu et al., 2020; Rimmele et al., 2020), while smaller-scale features may
be hidden since the photon mean free path $\sim 100$ km at the photosphere
(Judge et al., 2015). As explained in Sec. 4, we expect primary CSs to break
into secondary CSs that are unstable to IT in the intermediate or decoupled
regime (see also Tenerani et al. 2015a; Singh et al. 2019).

###  6.3. Hall and Ambipolar diffusion effects.

In Fig. 3 the dotted and barred areas indicate where the Hall and AD
contributions to magnetic reconnection might not be negligible. For the AD to
be important, a degree of coupling between neutrals and ions is necessary, so
ambipolar effects are not relevant in the decoupled regime. The upper edge of
the barred area is calculated using Eq. (6). If and only if there is a neutral
point, AD steepens the current profile in the forming current sheet (Ni et al.
2020; Zweibel & Brandenburg 1997; Tolman et al. 2023), once the dynamics of
reconnection are taken into account.

The upper limit of the dotted area is defined by the local ion inertial length
$d_{i}$, for $H^{+}$ ions. The Hall effect on its own does not break the
frozen-in condition, as field lines remain frozen to the electron flow, but
the Hall term destabilizes reconnection and affects tearing and the IT
instability criterion (Terasawa 1983; Pucci et al. 2017; Shi et al. 2020).
This means that in the dotted region in Fig. 3, the critical length-scale at
which reconnection becomes efficient will also depend on the ion inertial
length, and the CS will become unstable at slightly smaller aspect ratio than
the simple resistive case, with corrections provided in Pucci et al. (2017).
Still, even if both the AD and Hall diffusion are relevant to the reconnection
process, the energy conversion will proceed at a pace determined by the regime
in which the reconnection event is happening, i.e. coupled, intermediate or
decoupled, see Fig. 1.

###  6.4. Fractal reconnection scenario.

As discussed in Sec. 4, secondary reconnection is likely to occur in the
intermediate or the decoupled regime in partially-ionized plasmas. In the
lower solar chromosphere we have from Tab. 1 that $\rho/\rho_{i}\sim
n_{n}/n_{e}={10^{3}}$ since the plasma is quasi-neutral and $m_{i}\sim m_{n}$.
The density ratio falls to $\rho/\rho_{i}\sim 1$ in the fully-ionized upper
solar atmosphere. The Lundquist number $S$ ranges from $S=10^{5}-10^{6}$ (Tab.
2). Choosing $S=10^{5}$ and $\rho/\rho_{i}=10^{3}$, for Eq. (15) to be
satisfied $\tau_{A}\leq 100\tau_{ni}$. In the coupled regime (blue region of
Fig. 1), as $a_{c}$ approaches $a_{c1}$ then
$\gamma\sim\tau_{A}\sim\tau_{ni}$. In the first equality here we use the fact
that the CS has already developed into fast IT in the primary reconnection
stage. Then Eq. (15) is satisfied and secondary reconnection begins in the
intermediate regime.

That reconnection proceeds faster at smaller scales, where the ions and
neutrals decouple, leads to runaway acceleration of the ions that could
produce tails in the ion energy distribution like those needed for a
significant contribution to coronal heating from velocity filtration (Scudder,
1992; Scudder & Olbert, 1979a, b).

##  7\. Reconnection in protoplanetary disk atmospheres.

Reconnection has not been clearly identified in PPDs. Even the global angular
scales are unresolved, since the nearest PPDs are more distant than the Sun by
seven orders of magnitude. In this section we discuss where reconnection is
likely to occur in PPDs, how reconnection events can be initiated in the disk
context, and how the resulting magnetic energy dissipation may impact the disk
and its atmosphere.

We apply the reconnection picture described above in sections 3 and 4 to a
model disk orbiting a star of $M=M_{\odot}$ that is about $10^{6}$ yr old, an
age at which such disks are typically accreting onto their central stars. We
focus on conditions near 2.5 au from the star, since (1) the disks’ central
few au is the launching point for winds observed through optical line emission
(Simon et al., 2016; Banzatti et al., 2019; Pascucci et al., 2023) and (2) the
disk atmosphere in this same distance range is the source of a multitude of
mid-infrared molecular emission lines that carry information on the excitation
conditions (Salyk et al., 2011; Pontoppidan et al., 2014; Walsh et al., 2015).

At 2.5 au, the dipole component of the star’s magnetic field is millions of
times weaker than at the stellar photosphere and the higher-multipole
components are weaker still. Measured field strengths at young near-Solar-mass
stars’ photospheres are in the kilogauss range (Yang & Johns-Krull, 2011),
corresponding to fields at au distances less than a milligauss. Comparing this
to the disk fields swept in with the gas during the collapse of the parent
molecular cloud core (Masson et al., 2016) and the gauss-range fields inferred
for the asteroid belt’s location in the early Solar system (Weiss et al.,
2021), it seems likely that the stellar fields outside 1 au are negligible
compared with the fields intrinsic to the disk.

###  7.1. Turbulence as a driver for magnetic reconnection.

Though MRI is not thought to be active at the midplane of protoplanetary disks
(Gressel et al. 2015; Mori et al. 2019; Iwasaki et al. 2024), it can be
triggered in any layer where both of the Elsasser numbers exceed unity and the
magnetic pressure is less than the gas pressure (Lesur et al. 2022). The
nature of the magnetic dynamics and in particular the presence of MRI-driven
turbulence depend on the heating and cooling processes and chemical reactions
included, since these govern the non-ideal coefficients in the induction
equation (Gressel et al., 2015; Bai, 2014; Gressel et al., 2020; Iwasaki et
al., 2024). The static disk model we employ has more species and more heating
and cooling processes than can typically be treated in dynamical calculations.
The coupling of chemistry to thermodynamics leads to a disk with a hot,
extended, ionized atmosphere. As a result, we shall see below that the
criteria for MRI are met in a layer spanning heights above the midplane from
0.35 to 0.75 au.

If MRI is present some of the magnetic fields generated in the MRI turbulent
region will rise into the atmosphere due to their buoyancy (Miller & Stone
2000) and through the Parker instability (Hirose & Turner 2011). Reconnection
can develop in turbulent environments driven by the MRI (McNally et al., 2013;
Miller & Stone, 2000; Inutsuka & Sano, 2005), and MRI-driven currents and
reconnection events can cause enough ionization to sustain MRI turbulence
itself (Inutsuka & Sano, 2005). Recent work on MHD turbulence shows that
eddies develop strong anisotropies in the inertial range and that magnetic
reconnection in the ideal regime can even break the cascade leading to
dissipation (Boldyrev & Loureiro, 2020). The energy released potentially
contributes to the heating of protostellar winds and jets (Kim & Ostriker,
2000).

The role and importance of reconnection may not be readily observable in disk
simulations because limited spatial resolution leads to underestimating
Lundquist and magnetic Reynolds numbers in certain regions, and also because
time-averaging magnetic fields and assuming axisymmetry can hide the
perturbations driving reconnection.

###  7.2. Magnetic field tangling and CSs in the protoplanetary disk
atmosphere.

The magnetic topology in protoplanetary disks is driven by the field’s
coupling to flows arising from differential rotation and turbulence. CSs may
form as natural boundaries separating flux volumes rooted in different disk
regions. For example, if the disk atmosphere has a beta transition at which
gas and field are coupled, then any flows in the gas will move atmospheric
magnetic field footpoints. In a system where a form of line-tying exists,
differential rotation of footpoints leads naturally to the formation of CSs
separating field lines rooted at different radii, or in regions separating
open and closed field lines in the magnetically linked star-disk system
(Lovelace et al., 1995; Tout & Pringle, 1996; Romanova et al., 1998; Uzdensky,
2006; Takasao et al., 2017).

One large-scale CS is that formed by the hourglass magnetic configuration (Li
et al., 2014) of the disk’s magneto-centrifugal wind when acted upon by the
differential orbital rotation. This leads naturally to toroidal fields with
opposite signs in the upper and lower hemispheres. CSs can also form at the
boundary between the so-called “dead zone” and the wind launching region,
where magnetic field is wound up by differential rotation. In diffusive MHD
calculations including those by Gressel et al. (2015, 2020) and Iwasaki et al.
(2024), this CS is offset from the midplane and occurs in the atmosphere on
either the upper or lower face of the disk.

###  7.3. Disk chemistry and radiation.

We assess reconnection in protoplanetary disks in the context of a
thermochemical disk model (Ruaud & Gorti, 2019) having mass
$0.01\mathrm{M}_{\odot}$ and orbiting a 1$\mathrm{M}_{\odot}$ star whose
optical, UV, and X-ray spectrum are those of the well-studied young star TW
Hya.

As detailed in Ruaud & Gorti (2019), we determine the disk’s density structure
by solving for vertical hydrostatic balance under a surface density constraint
set by the disk mass. Gas temperature is found by balancing heating by dust
collisions, UV, and X-rays with cooling by line and continuum emission from
various ions, atoms, molecules, and grains, whose abundances are determined by
solving a chemical network. The network includes $\sim 6000$ gas-phase
reactions, photo-reactions (UV and X-rays), cosmic ray ionization, and grain
surface reactions among $\sim 800$ species. The gas density, temperature, and
chemistry, and the dust size distribution are all coupled. The grain size
distribution at each height is determined by balancing fragmentation,
coagulation, and settling. The cooling is computed using non-LTE radiative
transfer in the main spectral lines treating collisional and radiative
processes. The dust’s thermal radiation is also included.

The density structure, heating and cooling, and chemistry are computed
iteratively to mutual consistency. Fig. 4 shows the resulting variation with
height of the total (ions plus neutrals) density and temperature. Also plotted
there is the pressure scale height $H_{p}$, used below as an estimate of the
largest coherent scale for CS formation $L=H_{p}$ as discussed in Sec. 5.2.
The main charge carriers’ abundances versus height are shown in Fig. 5.

Figure 4.— Conditions in the model protoplanetary disk versus height $z$ at
distance 2.5 au from the Solar-mass star, obtained from the thermochemical
calculation: temperature $T$ in Kelvin, ratio of gas to magnetic pressure
$\beta$, total (neutral plus ion) mass density $\rho$, and scale height
$H_{p}=c_{s}/\Omega$ where $\Omega$ is the orbital frequency. The black dashed
vertical line marks where $\beta=1$. The second horizontal axis shows the
height in units of the midplane pressure scale height $H_{0}=H_{p}(z=0)$.
Since the scale height increases with $z$, the domain spans a large multiple
of the midplane scale height. Figure 5.— Main charge carriers’ abundances
versus height in the model protoplanetary disk at a distance 2.5 au from the
Solar-mass star. The plot runs from the midplane up to $z=1.2$ au, above which
the abundances are nearly uniform due to the high ionization rate. All
abundances are relative to the number density of hydrogen nuclei, $n_{H}$.
“Metals” is the sum of Mg+ and Fe+ ions.

The model disk extends from the midplane up to heights above the base of the
layer where the star’s X-ray and FUV photons can heat the gas to near $10^{4}$
K while it remains neutral. The temperature at these heights is too low for
the hot gas to readily escape, since the virial temperature at our fiducial
distance of 2.5 au from the star is $\gtrsim 20,000$ K (Gorti et al., 2015;
Pascucci et al., 2023). Photoevaporative escape from our fiducial radius is
possible at larger heights where the gas is fully ionized, but for the range
of heights $z\lesssim 1$ au we are concerned with here, the disk structure
does not significantly depart from hydrostatic equilibrium. The thermal
structure implies the base of the photoevaporative wind lies at a height 1.14
au and the launching speed is just 16% of the thermal speed, by the methods in
Adams et al. (2004) and Gorti et al. (2015). This slow outflow will have only
slight effects on the disk structure in the region of interest here (we focus
on $z\sim 0.6$ au where $\beta=1$). Examining how reconnection operates in
this hydrostatic model is thus sufficient.

###  7.4. Modeling dust grains.

Dust grains’ surfaces catalyze chemical reactions including the recombination
of free charges. Dust also enables photoelectric heating and continuum
radiation cooling. Dust thus can have major effects on the induction
equation’s non-ideal coefficients (Wardle, 2007; Tsukamoto & Okuzumi, 2022).
Our model disk has a ratio of dust to gas surface density $\Sigma_{\rm
dust}/\Sigma_{\rm gas}=0.01$, near the value in the interstellar medium. The
silicate and carbonaceous grains’ size distribution when integrated through
the disk thickness corresponds to equilibrium between collisional aggregation
and fragmentation, with number density $n_{d}(a)\propto a^{-3.5}$ over the
range $0.005\leavevmode\nobreak\ \mu{\rm m}<a<1$ cm. Each grain size’s
vertical distribution within the gas column is determined by balancing
settling with stirring by weak turbulence, so that the bigger grains are
concentrated near the midplane as observed (Bouwman et al., 2008; Pinte et
al., 2008). The grain temperature is size- and composition-dependent and the
solids are released into the vapor phase from grains whose temperature exceeds
the sublimation threshold of $\sim 1500$K.

###  7.5. Non-ideal coefficients and Elsasser numbers.

Both the AD and the reconnection heating rate depend on the magnetic field
strength. We estimate the field in our model disk annulus as follows.
Protoplanetary disks deliver material to the surfaces of their central young
stars at rates typically around ${\dot{M}}=10^{-8}$ Solar masses per year at
age one million years (Manzo-Martínez et al., 2020). If this accretion flow is
driven by the torque from a magneto-centrifugal wind, the magnetic field
threading the disk cannot be weaker than
$B>(2{\dot{M}}\Omega/{\sqrt{3}R})^{1/2}$ (Wardle 2007 and Bai & Goodman 2009
Eq. 7). For the disk annulus located $R=2.5$ au from the Solar-mass star, this
amounts to a lower limit on the field strength $B>30$ milligauss.

While measurements of magnetization in primitive meteorites suggest fields up
to about 500 milligauss at 1 to 3 au from the young Sun in the solar system’s
first million years (Weiss et al., 2021), we give our model disk annulus a
field with the minimum strength of 30 milligauss to obtain a lower bound on
the reconnection heating rate.

Empirical evidence on protoplanetary disks’ magnetic field geometries is so
limited that a wide range of possibilities remains open. We therefore simply
assume the total field is uniform in height to make exploration of the
parameter space straightforward. Fig. 4 shows that the 30-milligauss field
yields a ratio of gas to magnetic pressure $\beta\sim 10^{4}$ at the midplane.
This ratio falls with height, being less than unity above $z=0.6$ au. The two
other magnetic field properties needed below are the magnitude of the vertical
field component, which governs whether magneto-rotational turbulence is
present, and the magnitude of whichever field component is involved in
reconnection. Again seeking the simplest reasonable choice since the evidence
is too limited to support greater complexity, we assume both the field’s
vertical component and its reconnecting component are one-third of the total
field, or 10 milligauss.

We can now determine the coefficients in the induction equation’s non-ideal
terms at each height in the model disk annulus using Eqs. (25)–(31) in Wardle
(2007). Included are the contributions to the electric current from all the
charged gas-phase and grain species tracked in the thermochemistry code;
inter-species collision frequencies from Wardle & Ng (1999); and the magnetic
field described above.

The resulting non-ideal coefficients yield the Ohmic and ambipolar Elsasser
numbers $\Lambda_{O/AD}=v^{2}_{Az}/(\Omega\eta_{O/AD})$ that govern the
magnetic field dynamics away from current sheets in the disk context (Lesur et
al., 2022). These two dimensionless numbers are plotted versus height in Fig.
6. The ambipolar Elsasser number is the smaller of the two and less than
unity, hence the limiting factor for MRI, only in a thin layer around height
0.3 au. Above about 0.35 au both Elsasser numbers exceed unity, a necessary
condition for MRI to operate. A further requirement is plasma $\beta$ greater
than unity, which holds below 0.66 au. The layer in the model disk atmosphere
between 0.35 and 0.66 au thus meets the conditions for the onset of turbulence
driven by the MRI. We have checked that these results depend weakly on the
choice of field strength. For example, if the field were twice the minimum
capable of driving the typical accretion flow, the two Elsasser numbers would
both exceed unity above 0.34 au and the plasma beta exceed unity below 0.58
au, so the layer of the disk atmosphere in between would remain subject to
MRI. The stronger field would enable reconnection heating rates several times
greater than we discuss below.

Figure 6.— Dimensionless Ohmic and ambipolar Elsasser numbers versus height in
the model protoplanetary disk at a distance 2.5 au from the Solar-mass star. A
necessary condition for magneto-rotational instability leading to turbulence
is that both Elsasser numbers exceed unity. This condition is met above 0.35
au. The plot extends from the midplane to $z=1.2$ au, above which the Elsasser
numbers are almost independent of height.

###  7.6. Critical CS thickness and regimes for the onset of fast
reconnection.

Figure 7.— Reconnection regime versus height in the model protoplanetary disk
and thickness of the CS. The distance from the central star $R=2.5$ au. The
fully-coupled, intermediate, and decoupled regimes are shown in blue, yellow,
and green, respectively. A black dashed line marks the greatest CS thickness
$a$ available in the system given the maximum coherent length scale $L=H_{p}$,
calculated as $a\sim H_{p}S^{-1/3}(\rho_{n}/\rho_{i})^{1/6}$ (Fig. 1). The
dotted and barred overlays indicate where the Hall effect and AD,
respectively, are expected to play roles in determining the critical threshold
for fast magnetic reconnection.

Next we discuss the dominant effects in reconnection dynamics, which are
determined not by the Elsasser numbers but by the Alfvén speed and diffusion
coefficients as described in Sec. 3. We calculate for the model protoplanetary
disk the critical CS thicknesses $a_{c1}$ and $a_{c2}$ at which reconnection
transitions from the fully-coupled down to the intermediate regime and from
the intermediate down to the decoupled regime following Fig. 1. These two
scales vary with height $z$ in the disk atmosphere as shown in Fig. 7 by the
red and the orange line, respectively. They divide the fully-coupled regime in
blue from the intermediate regime in yellow and the decoupled regime in green.

A black dashed line near the top of Fig. 7 shows the thickness of the longest
CS available at each height in the disk. This CS has length $L\sim H_{p}$ and
thickness given by the coupled regime critical reconnection scale $a_{c}$ in
the blue row of Fig. 1. The black dashed line remains in the blue zone even
high in the disk atmosphere above 1 au, where the stellar X-rays make ionized
hydrogen the main charge carrier (Fig. 5). Thus even here, magnetic
reconnection can be triggered in the decoupled regime on the longest CSs.

The scale height $H_{p}$ increases with $z$ since the sound speed rises
alongside the temperature (Fig. 4). We note that CSs even longer than $H_{p}$
could form above the $\beta=1$ surface, where the total pressure is mostly
magnetic. The magnetic pressure dominates above some height not just in the
simplified magnetic geometry we consider here, but also in more detailed MHD
treatments of protoplanetary disks’ magnetically-launched winds (Gressel et
al., 2015, 2020; Martel & Lesur, 2022; Pascucci et al., 2023).

###  7.7. The role of AD and Hall effect in magnetic reconnection.

The barred and dotted overlays on Fig. 7 indicate where the AD and Hall
effect, respectively, can alter the critical thresholds for fast magnetic
reconnection.

AD is relevant when there is a degree of coupling between neutrals and ions,
i.e. in the coupled and intermediate regimes (blue and yellow in Fig. 1). In
the decoupled regime (green), AD does not affect reconnection since neutrals
are not involved in the reconnection process. We determine the upper boundary
of the barred area in Fig. 7 using Eq. (6). Reconnection on the longest
available current sheet, shown by the dashed black line, is unaffected by AD
at the heights plotted. AD does however act on shorter CSs throughout the disk
atmosphere. The effects of AD on the trigger conditions for IT were treated by
Zweibel (1989) and Pucci et al. (2020) neglecting the electron-neutral
collisions in Ohm’s law (Sec. 3). Extending such modeling to all three fluids
– ions, electrons, and neutrals – would not change the maximum reconnection
speed but might slightly modify $a_{c}$.

For the Hall effect, the upper boundary of the dotted area in Fig. 7 is the
ion inertial length $d_{i}$, calculated at each height using the locally most-
abundant charge carrier from Fig. 5. The Hall effect alters the onset of fast
tearing instability as discussed in Pucci et al. (2017), Shi et al. (2019b),
Jara-Almonte et al. (2021), and Sec. 6. The Hall effect is destabilizing,
making the CS unstable to fast tearing at slightly larger thicknesses, as
quantified in Pucci et al. (2017).

###  7.8. Comparing disk diffusion regimes.

Three non-ideal terms occur in the magnetic induction equation 3. Of these,
only the Ohmic diffusivity is essential to the triggering of fast magnetic
reconnection through the tearing instability. The AD and Hall effect change
the geometry of fast-reconnecting CSs. Though Ohmic dissipation produces
heating even in current sheets not undergoing tearing instability, it is much
slower there in transferring energy from the magnetic field to the gas and
depends on the local collision frequencies.

In contrast, all three non-ideal terms can be important for the magnetic
field’s global evolution in protoplanetary disks, affecting for example the
existence and behavior of MRI turbulence (Lesur et al., 2022) and the Parker
magnetic buoyancy instability (Terquem & Papaloizou, 1996). Which non-ideal
term dominates versus height in the disk atmosphere and strength of the
magnetic field was explored by Bai (2011) for a model disk with temperature
independent of height, but treating some of the same ionization and
recombination reactions we consider here. There it was often true that the
Ohmic diffusivity dominated near the midplane, while AD dominated for strong
magnetic fields and for the atmosphere’s upper layers. Our calculation with
the thermodynamics coupled to the chemistry yields a disk atmosphere much
hotter than the midplane but nevertheless qualitatively similar diffusion
regimes, as shown in the left panel of Fig. 8. There is no combination of
field strength and height for which the Hall term dominates in this model disk
annulus.

###  7.9. Fractal and recursive reconnection.

As discussed in Sec. 4, while the primary trigger occurs in the coupled
regime, the secondary CS reconnection will most probably lead to regimes in
which ions and neutrals are not fully coupled. Assuming the CS length to be
$H_{p}$ (the longest CS achievable in the system) we estimated the primary
reconnection onset occurs in the fully coupled regime. In Fig. 8 (left) we
superimposed bars when the secondary onset occurs in the intermediate regime.
Shorter CSs than $L=H_{p}$ are not prevented to form, so that primary
reconnection trigger could also occur in the intermediate or decoupled regime.
Indeed, the length of the CS – and so its thickness to achieve fast
reconnection – is affected by the local dynamics and by the CS thinning.

Figure 8.— Left panel: Largest of the magnetic diffusivities vs. height $z$
above the disk midplane and strength $B$ of the magnetic field. The Ohmic term
dominates in the blue and the ambipolar diffusion term in the green region.
Superimposed bars indicate where secondary reconnection events occur in the
intermediate regime. The colored curve across the center indicates the
volumetric heating rate $Q$ (erg cm-3 yr-1) where the plasma $\beta$ parameter
is unity at each height $z$, using the right-hand color scale. Right panel:
Total cooling rate (dotted black curve) and selected heating rates vs. height
$z$. The blue curve is UV heating and the orange curve is X-ray heating. The
main heating process in the thermochemical model above 0.6 au is the X-ray
heating. Reconnection heating is stronger than the X-ray heating in the disk
atmosphere, by the estimate shown as the solid black curve. This estimate is
for the coupled regime, assuming the efficiency of conversion of magnetic to
heat and kinetic energy is just 10%, and only one-third of the total field
participates, so the strength of the reconnecting field is $B=10$ mG. Since
reconnection exceeds all other heating processes, it can affect the disk
atmosphere’s temperature, composition, and emitted radiation.

###  7.10. Consequences for disk heating.

In this section we explore whether magnetic reconnection results in plasma
heating and particle acceleration capable of affecting the protoplanetary
disk’s temperature, chemical composition, and dynamics.

We focus on the layer where the magnetic and gas pressures are comparable.
This layer is important if MRI is present since MHD modeling of small disk
patches indicates the $\beta=1$ surface separates a turbulent interior from a
magnetized disk corona. Fields generated beneath are buoyant and rise into the
corona where they can undergo reconnection (Miller & Stone, 2000; Hirose &
Turner, 2011). The $\beta\sim 1$ layer is important too if MRI is absent,
since magnetocentrifugal winds are typically launched around this height
(Wardle & Königl, 1993; Königl et al., 2010; Bai, 2014; Gressel et al., 2015,
2020; Iwasaki et al., 2024). Reconnection heating near the launching point
could change the launching speed and thus the rates at which the winds remove
the disk’s mass and angular momentum, with consequences for disk lifetimes.

To see whether reconnection heating is significant, we compare its rate with
the other heating processes at work in the model protoplanetary disk. The
volumetric reconnection heating rate $Q\simeq B^{2}/(8\pi)\times(v_{A}/L)$ has
units erg cm-3 s-1. We assume the fast reconnection is triggered in the fully-
coupled regime, which is a reasonable assumption for the longer CSs with
$L\sim H_{p}$ (Fig. 7).

$Q$ is shown as a function of magnetic field strength in the left panel of
Fig. 8 by color variations along the curve marking where the plasma parameter
$\beta=1$. The strength of the reconnecting field is taken to be one-third of
the total field, just as for our fiducial total field strength of 30
milligauss (Sec. 7.5). The right panel of Fig. 8 is specific to the fiducial
30 milligauss total field strength and compares $Q$ against the main heating
processes in the thermochemical disk model. Above about $z=0.5$ au, the
reconnection heating (solid black curve) exceeds all other heating processes.
Reconnection thus has the potential to raise the temperature of the gas in and
around CSs.

An important question that cannot be addressed without modeling the magnetic
dynamics in detail is when, for how long and where reconnection happens in
such protoplanetary disk contexts. Even MHD simulations that do not resolve
the CSs may yield information on the distribution of $Q$ over space and time,
useful for evaluating the contribution to heating the disk atmosphere.
However, even without such modeling we can obtain an upper limit to the time-
averaged reconnection heating rate per unit disk area. This upper limit is the
rate at which the accretion flow converts the gravitational potential energy
of the inward-spiraling disk gas into other forms, such as magnetic energy.

The power released by steady-state accretion per unit disk area is
$D\sim(3/4\pi){\dot{M}}\Omega^{2}$ (Pringle, 1981; Spruit, 2010), which
amounts to 400 erg cm-2 s-1 using a typical accretion rate onto T Tauri stars
of ${\dot{M}}=10^{-8}M_{\odot}$ yr-1 and the orbital frequency $\Omega$ for
our location 2.5 au from the star. By comparison the height-integral of $Q$
from Fig. 8 right panel is $17$ erg cm-2 s-1, more than an order of magnitude
below the available accretion power. Thus reconnection can heat the disk
atmosphere at rates exceeding all other known heating processes while
consuming a power well within that available locally from the accretion flow.

###  7.11. Reconnection Onset in Global MHD simulations.

To quantify reconnection heating in global models of MRI turbulence and
magnetocentrifugal wind launching, it will be necessary to spatially resolve
the onset of fast reconnection. We explore whether this is feasible for
reconnection taking place in the layers where $\beta>1$ and starting with CS
thickness $a>a_{c1}$, so that the reconnection is in the fully-coupled regime.

From the discussion in Sec. 2, the critical thickness to trigger fast
reconnection $a_{c}\sim LS^{-1/3}(\rho_{i}/\rho_{n})^{1/6}\sim 0.008$ au,
where $S\sim 10^{10}$ using the Ohmic diffusivity from $z=1$ au in the
thermochemical model disk annulus. In two-dimensional MHD simulations, grids
of 1,000 cells are feasible over a domain extending from the midplane up to
$z=2$ au to span the disk layers where reconnection is taking place. The
vertical grid spacing $\Delta z\sim 0.002$ au is then small enough to identify
CSs capable of beginning fast reconnection. An MHD calculation on such a grid
could estimate the places and times where reconnection produces heating at the
rates $Q$ discussed above in Sec. 7.10.

##  8\. Conclusions

We have investigated the onset of fast magnetic reconnection in the partially-
ionized plasmas found in the lower atmospheric layers of the sun and in the
atmospheres of protoplanetary disks. We combined an analytic picture of the
onset of reconnection via the tearing instability with models of the plasma’s
composition and thermodynamics in the solar and disk atmospheres. The results
indicate that reconnection may contribute to heating these environments. The
main conclusions are:

  1. 1.

Even when reconnection begins with the ions, electrons, and neutrals all well-
coupled by collisions, the tearing instability breaks the initial current
sheet into secondary sheets at successively smaller spatial scales, where
magnetic dissipation occurs in the intermediate or the completely decoupled
regime. As tearing proceeds recursively, the ions fully decouple from the
neutrals, raising the effective Alfvén speed and accelerating the conversion
of magnetic energy into heat.

  2. 2.

In the solar photosphere and chromosphere, current sheets can have lengths up
to the pressure scale height. This provides a wide range of scales over which
reconnection begins in the fully-coupled regime, so that neutrals are involved
in the reconnection dynamics. These initial current sheets have thicknesses
$a=10-100$ km, the upper end of the range being near the resolution limit of
DKIST (Campbell et al., 2023). Since this is also about the photon mean free
path at the photosphere (Judge et al., 2015), much thinner current sheets
would be visible only indirectly through their heating and brightening
effects.

  3. 3.

The Hall effect and ambipolar diffusion alter the critical scales for fast
tearing instability’s onset. Corrections due to the Hall effect (Pucci et al.,
2017) and ambipolar diffusion (Pucci et al., 2020) have been determined
separately, but not together, though the Hall term likely is important only in
the decoupled regime. Ambipolar diffusion can steepen current sheets in the
presence of neutral sheets with no guide field (Brandenburg & Zweibel, 1994;
Ni et al., 2020), but not more generally, so that the results of Pucci et al.
(2020) should hold, but we defer analyses of the linear stability of AD
steepened sheets to future work.

  4. 4.

In protoplanetary disks, current sheets’ length is limited by the density
scale height when plasma $\beta>1$. In this case there is a wide range of
scales over which reconnection is triggered in the fully-coupled regime, so
neutrals are also accelerated by the process.

  5. 5.

Where the disk is well-magnetized with $\beta<1$, the largest coherent scales
available can be governed by disk winds if these are not disrupted by other
instabilities. At higher plasma $\beta$ where MRI can develop, large-scale
magnetic structures may also be created via channel flows. Channel flows’
breakup by magnetic reconnection can drive time-dependent disk winds (Suzuki
et al., 2010) that could move the disk surface up and down. We have shown that
in such large-scale reconnecting current sheets, the neutrals are accelerated
directly by the reconnection. The whole plasma can thus be lifted up as part
of the reconnection ejecta.

  6. 6.

Our work also has implications for the idea that ambipolar diffusion
steepening of protoplanetary disks’ midplane current sheet induces rapid
accretion, pinching the field into a reversing configuration, removing
magnetic flux, and yielding dense rings Suriano et al. (2018). These processes
may be greatly accelerated if the current sheet thins enough that reconnection
takes place in the decoupled regime.

  7. 7.

The layer in protoplanetary disk atmospheres where the plasma $\beta$
parameter reaches unity and magnetic winds are launched is susceptible, if
current sheets form, to reconnection at rates sufficient to increase the local
sound speed and so the wind launching speed.

  8. 8.

In our model disk atmosphere the power deposited locally by magnetic
reconnection is sufficient to raise temperatures above those established by
the stellar X-ray and UV heating, even for the minimum field strength
$B\approx 10$ milligauss consistent with magnetically-driven accretion.
Stronger fields would provide even more magnetic energy for conversion to
heat. Such heating would alter the atmosphere’s chemical composition and
radiative emissions. A related scenario was proposed by McNally et al. (2013)
with reconnection in a turbulent environment heating the parts of the
protosolar disk near the young Sun to produce the glassy, spherical chondrules
found in many primitive meteorites.

Further investigation of the magnetic dissipation rates and timescales and the
pathways taken by the deposited energy should include modeling that treats the
multi-species nature of the plasmas. Multi-fluid treatment of the partially
ionized solar plasma has been demonstrated by Singh & Krishan (2010c);
Khomenko et al. (2014); Alharbi (2022). Through multifluid, multispecies
modeling of the solar atmosphere with hydrogen and helium, Wargnier et al.
(2023) showed that the two species’ differing dynamics drove chemical
reactions involving helium and enriched the helium abundance in the solar wind
and coronal mass ejections. They also showed the heating rates for each
species at the CS location. Here we would like to point out that an important
consequence of the fractal reconnection scenario is the eventual decoupling of
the ions from the neutrals, populating energetic tails in the electrons and
ions, whose effects at large Knudsen numbers (Scudder, 1992), might be
fundamental to the subsequent coronal expansion into the solar wind. Recent
observations of ubiquitous lower coronal jetting seem to provide indirect
evidence for the role of such processes (Raouafi et al., 2023; Bale et al.,
2023). In the context of PPDs a multifluid treatment is relevant at
intermediate heights from the disk mid-plane in which heavier species than
hydrogen act like the positive charge carriers, as shown by the chemical model
adopted in this paper.

Copyright 2023. All rights reserved. We would like to thank Prof. Kazunari
Shibata for fundamental discussions and insights on the physics of the solar
atmosphere and magnetic reconnection. F.P. would like to thank Dr. Yasuhiro
Hasegawa for insights on wind launching driven by magnetic reconnection.
F.P.’s research was supported by an appointment to the NASA Postdoctoral
Program at the Jet Propulsion Laboratory, administered by Oak Ridge Associated
Universities under contract with NASA. K.A.P.S. gratefully acknowledges the
UGC Faculty Recharge Program of the Ministry of Human Resource Development
(MHRD), Govt. of India and University Grants Commission (UGC), New Delhi, the
incentive grant of the Institute of Eminence (IoE) Program, BHU, and the
Visiting Associateship Program of IUCAA, Pune. M.V. was supported by the NASA
Parker Solar Probe Observatory Scientist Grant No. NNX15AF34G. M.E.I.
acknowledges support from the German Science Foundation (Deutsche
Forschungsgemeinschaft, DFG) within the Collaborative Research Center SFB1491.
This work was carried out in part at the Jet Propulsion Laboratory, California
Institute of Technology, under contract with NASA and with the support of
NASA’s Exoplanets Research Program through grant 17-XRP17_2-0081 to N.J.T.

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