# On the origin of magnetic fields in stars II: The effect of numerical
resolution

James Wurster1,2, Matthew R. Bate2, Daniel J. Price3, and Ian A. Bonnell1
1Scottish Universities Physics Alliance (SUPA), School of Physics and
Astronomy, University of St. Andrews, North Haugh, St Andrews, Fife KY16 9SS,
UK
2School of Physics and Astronomy, University of Exeter, Stocker Rd, Exeter EX4
4QL, UK
3Monash Centre for Astrophysics and School of Physics and Astronomy, Monash
University, Vic 3800, Australia
jhw5<EMAIL_ADDRESS>
(Submitted: Revised: Accepted: )

###### Abstract

Are the kG-strength magnetic fields observed in young stars a fossil field
left over from their formation or are they generated by a dynamo? Our previous
numerical study concluded that magnetic fields must originate by a dynamo
process. Here, we continue that investigation by performing even higher
numerical resolution calculations of the gravitational collapse of a 1 M⊙
rotating, magnetised molecular cloud core through the first and second
collapse phases until stellar densities are reached. Each model includes Ohmic
resistivity, ambipolar diffusion, and the Hall effect. We test six numerical
resolutions, using between $10^{5}$ and $3\times 10^{7}$ particles to model
the cloud. At all but the lowest resolutions, magnetic walls form in the outer
parts of the first hydrostatic core, with the maximum magnetic field strength
located within the wall rather than at the centre of the core. At high
resolution, this magnetic wall is disrupted by the Hall effect, producing a
magnetic field with a spiral-shaped distribution of intensity. As the second
collapse occurs, this field is dragged inward and grows in strength, with the
maximum field strength increasing with resolution. As the second core forms,
the maximum field strength exceeds 1 kG in our highest resolution simulations,
and the stellar core field strength exceeds this threshold at the highest
resolution. Our resolution study suggests that kG-strength magnetic fields may
be implanted in low-mass stars during their formation, and may persist over
long timescales given that the diffusion timescale for the magnetic field
exceeds the age of the Universe.

###### keywords:

magnetic fields — MHD — methods: numerical — stars: formation

††pagerange: On the origin of magnetic fields in stars II: The effect of
numerical resolution–References††pubyear: 2021

##  1 Introduction

What is the origin of magnetic fields in low-mass stars? Observations show
strong, kG-strength surface magnetic fields on low-mass stars that weaken as
they age (e.g. Yang et al., 2005; Donati & Landstreet, 2009; Lavail et al.,
2017; Donati et al., 2020; Sokal et al., 2020). Since young, low-mass stars
are fully convective, it is generally assumed that any birth magnetic fields
are quickly diffused and replaced by dynamo-generated fields (Chabrier &
Küker, 2006). Moreover, their long-term evolution is consistent with their
magnetic fields being generated by convective dynamos since stellar rotation
rates also decrease with time due to the emission of magnetised winds and
outflows (e.g. Parker, 1958; Schatzman, 1962; Weber & Davis, 1967; Skumanich,
1972; Pizzolato et al., 2003; Wright et al., 2011; Vidotto et al., 2014; See
et al., 2015). However, given the large dispersion in the observed magnetic
field strengths of young stars (e.g. Johns-Krull, 2007; Yang & Johns-Krull,
2011), there is speculation that the magnetic fields of low-mass stars may be
dominated by primordial or ‘fossil’ magnetic fields that are implanted during
the star formation process (Tayler, 1987; Moss, 2003; Tout et al., 2004; Yang
& Johns-Krull, 2011); these studies have so far failed to find any correlation
between the measured magnetic field properties and the stellar properties
thought to be important for dynamo action. Therefore, the strength and
geometry of magnetic fields implanted in protostars during the star formation
process remains unknown.

The formation of protostellar cores in a magnetised medium has been the focus
of many numerical studies (e.g. Machida et al., 2006, 2007; Tomida et al.,
2013; Tomida et al., 2015; Machida, 2014; Bate et al., 2014; Tsukamoto et al.,
2015a; Vaytet et al., 2018; Wurster et al., 2018a, d; Machida & Basu, 2019).
These studies typically focused on the formation of discs and outflows, and
how they are affected by ideal, resistive and/or non-ideal magnetic fields.
Additionally, several studies have commented on the magnetic field strength in
the stellar cores that form. In ideal magnetohydrodynamics (MHD) studies of
the formation of an isolated star, magnetic fields of $10^{4}-10^{6}$ G are
shown to be implanted in the stellar core at birth (e.g. Machida et al., 2006,
2007; Bate et al., 2014; Wurster et al., 2018d; Vaytet et al., 2018). This
magnetic field was much stronger than expected, and largely resulted because
ideal MHD was employed.

Non-ideal MHD, however, is a more realistic prescription when modelling star
formation since star forming regions are only weakly ionized (e.g. Mestel &
Spitzer, 1956; Nakano & Umebayashi, 1986; Umebayashi & Nakano, 1990). When
adding magnetic diffusion in the form of Ohmic resistivity and/or ambipolar
diffusion, the stellar core field strength was $\sim$$10^{2}$ G at birth (e.g.
Machida et al., 2007; Vaytet et al., 2018), which is much smaller than in
ideal MHD simulations and is below the observed kG field strengths. The
dispersive Hall effect has been shown to have a significant impact on the star
forming region (e.g. Tsukamoto et al., 2015b, 2017; Wurster et al., 2016,
2018c, 2021), thus it must be included when investigating the magnetic
properties during star formation. When including Ohmic resistivity, ambipolar
diffusion and the Hall effect, we (Wurster et al., 2018d, herein Paper I)
found that the maximum magnetic field strength of $B_{\text{max}}<1$ kG
initially resided in a magnetic wall (e.g. Tassis & Mouschovias, 2005, 2007a,
2007b); similar magnetic walls were presented in, e.g., Tomida et al. (2015)
and Vaytet et al. (2018). In Paper I, the location of our magnetic wall moved
inwards during the second collapse phase, but the location of the maximum
magnetic field strength never became coincident with the centre of the stellar
core. For the duration of the simulations, the central magnetic field strength
within the stellar core was $B_{\text{cen}}<300$ G. Only slightly higher field
strengths were obtained when using a cosmic ray ionisation rate ten times
higher than the canonical value of $\zeta_{\text{cr}}=10^{-17}$ s-1 (Spitzer &
Tomasko, 1968) to calculate the non-ideal MHD effects. Artificial resistivity
also affected the magnetic field strength, however, the fiducial algorithm
used (Price et al., 2018) yielded stronger field strengths than its more
resistive predecessor (Tricco & Price, 2013). Finally, Paper I modelled two
resolutions that differed by a factor of ten in mass resolution; decreasing
the resolution decreased the maximum field strength but yielded a similar
central field strength. Therefore, in Paper I, we concluded that the origin of
magnetic fields in low-mass stars was not a fossil field, but generated at a
later time by a dynamo action.

Numerical resolution plays a crucial role in astrophysical simulations.
Ideally, every numerical study would include a convergence study, where
simulations of higher and higher resolution are run until it becomes clear
that increasing the resolution no longer affects the physical results; it
would then only be the highest resolution simulations that would be analysed.
While numerical studies typically use sufficient resolution to meet the
required resolution criteria (e.g. Bate & Burkert, 1997; Truelove et al.,
1997; Nelson, 2006; Commerçon et al., 2008) of the object they are trying to
resolve, most exclude resolution studies, typically due to limited
computational resources. Many studies, however, perform their simulations at
various resolutions to understand how resolution affects their results (e.g.
Schmidt et al., 2010; Federrath, 2015; Wurster et al., 2016, 2018d, 2019;
Cunningham et al., 2018; Grudić et al., 2021), while several have shown that
their results remain unconverged, even when using the highest resolution
feasible (e.g. Mac Low et al., 1998; Meru & Bate, 2011; Joos et al., 2013;
Bate et al., 2014; Hosokawa et al., 2016; Haugbølle et al., 2018; Meyer et
al., 2018; Hennebelle, 2018), and a few have shown that convergence has been
reached (e.g. Commerçon et al., 2008; Meru & Bate, 2012; Lee & Hennebelle,
2018a, b; Yamamoto et al., 2021)111The preceding list contains examples from
cluster formation, star formation, disc and turbulence studies. Neither the
list nor the topics is exhaustive.. Although convergence studies are required
to produce robust conclusions, this is not always possible.

Increasing numerical resolution tends to increase the magnetic field strength
in a newly formed protostar (e.g. Bate et al., 2014; Wurster et al., 2018d)
since the protostar is better resolved and its properties (including gas
density and magnetic field strength) suffer less smoothing. Therefore, the
question arises as to whether or not the conclusions of Paper I will hold at
even higher resolutions. The magnetic field strength should be less dependent
on resolution when employing non-ideal MHD since the non-ideal processes
represent physical dissipation that are not directly related to resolution.
Artificial resistivity should also become less important for increasing
resolution since it is a second-order numerical term, and is generally weaker
than artificial dissipation (e.g. Wurster et al., 2016). However, even when
including non-ideal MHD, there were resolution effects at the resolutions
previously investigated (e.g. Wurster et al., 2018d; Wurster & Bate, 2019).

In this paper, we investigate the effect of resolution and build upon the
analysis of Paper I. We present six models with mass resolutions varying by a
factor of 300 between our highest and lowest resolution model; our highest
resolution model has a mass resolution 10 times higher than the fiducial
simulation presented in Paper I. In Section 2, we summarise our methods and in
Section 3 we present our initial conditions. We present our results in Section
4, discuss timescales and initial environments in Section 5, and conclude in
Section 6.

##  2 Methods

Our method is identical to that from Paper I. We solve the self-gravitating,
radiation non-ideal magnetohydrodynamics equations using sphNG, which is a
three-dimensional Lagrangian smoothed particle hydrodynamics (SPH) code that
originated from Benz (1990). Over the past 30 years, the code has been
substantially modified to include (e.g.) a consistent treatment of variable
smoothing lengths (Price & Monaghan, 2007), individual time-stepping (Bate et
al., 1995), radiation as flux limited diffusion (Whitehouse et al., 2005;
Whitehouse & Bate, 2006), magnetic fields (for a review, see Price, 2012), and
non-ideal MHD (Wurster et al., 2014, 2016). Gravitational forces are
calculated using a binary tree, where the gravitational potential is softened
using the SPH kernel such that the softening varies with the smoothing length
(Price & Monaghan, 2007). For stability of the magnetic field, we use the
source-term subtraction approach (Børve et al., 2001), constrained
hyperbolic/parabolic divergence cleaning (Tricco & Price, 2012; Tricco et al.,
2016), and artificial resistivity (as described in Price et al., 2018)222This
artificial resistivity is generally weaker than the resistivity algorithm in
Tricco & Price (2013), as compared in Paper I and Wurster et al. (2017).. For
a more detailed description, see Wurster et al. (2018a).

To self-consistently calculate the non-ideal MHD coefficients, we use Version
1.2.5 of the Nicil library (Wurster, 2016). This includes cosmic ray
ionisation of light and heavy metals, and thermal ionisation at high
temperatures ($T\gtrsim 1000$ K). There are three dust grain populations that
differ only in charge, where the charges are $\pm 1$ and 0. We include the
three non-ideal MHD terms that are important for star formation: Ohmic
resistivity, ambipolar diffusion and the Hall effect. Ohmic resistivity is
calculated implicitly, as described in the appendix of Wurster et al. (2018a),
and the remaining two terms are calculated explicitly.

There are minor differences between this version of sphNG and Nicil compared
to the versions used in Paper I. For consistency, we recomputed the models
from Paper I so that all the models presented here are calculated using the
same version of sphNG and Nicil.

##  3 Initial conditions

Our initial conditions are identical to those in Paper I, which are the same
as our previous studies (e.g. Bate et al., 2014; Wurster et al., 2018a, c, d,
2021). We initialise a spherical core of mass 1 M⊙ with radius
$R_{\text{c}}=4\times 10^{16}$ cm and a uniform density of
$\rho_{0}=7.42\times 10^{-18}$ g cm-3; the core has an initial (isothermal)
sound speed of $c_{\text{s}}=\sqrt{p/\rho}=2.2\times 10^{4}$ cm s-1, and a
solid body rotation about the $z$-axis of $\Omega=1.77\times 10^{-13}$ rad
s-1, which corresponds to a ratio of rotational-to-gravitational energy of
$\beta_{\text{r}}\simeq 0.005$. The core is placed in pressure equilibrium
with a warm, low-density medium of edge length $4R_{\text{c}}$;
magnetohydrodynamic forces are periodic across the boundary of this box but
gravitational forces are not.

The entire domain is threaded with a uniform magnetic field that is parallel
to and aligned with the rotation axis. The initial magnetic field strength is
$B_{0}=163\mu$G, which is equivalent to a mass-to-flux ratio of $\mu_{0}=5$ in
units of the critical mass-to-flux ratio (e.g. Mestel, 1999; Mac Low &
Klessen, 2004). Although this strength is weaker than generally observed in
molecular cloud cores (for reviews, see, e.g., Crutcher, 1999; Heiles &
Crutcher, 2005; Hull & Zhang, 2019), this initial mass-to-flux ratio was
chosen to match our previous studies and as a compromise since higher field
strengths would be even more computationally expensive (see Section 3.1). For
non-ideal MHD, we use the canonical cosmic ray ionisation rate of
$\zeta_{\text{cr}}=10^{-17}$ s-1 (Spitzer & Tomasko, 1968).

We intentionally choose an aligned orientation of the magnetic field and
rotation vectors, as in Paper I. This orientation will yield a small, nearly
axisymmetric protostellar disc that forms just prior to the stellar core
phase, which results in a simpler analysis than if we had to account for the
large disc with an $m=2$ instability that appears if the two vectors are
initially anti-aligned (e.g. Wurster et al., 2018c, 2021). Moreover, the model
with the aligned orientation is computationally less expensive, reaching
$\rho_{\text{max}}=10^{-2}$ g cm-3 $\sim$5 faster than the anti-aligned
orientation.

Within a given simulation, all SPH particles have an equal mass, and the
particles are initially placed on a cubic lattice.

###  3.1 Parameter space

In this study, we only investigate the effect of resolution. To resolve the
local Jeans mass throughout the collapse, we require at least $3\times 10^{4}$
particles in the sphere (e.g. Bate & Burkert, 1997), given our initial
conditions and equation of state. Here, we investigate six clouds, where the
resolutions are given in Table 1. Throughout this paper, each simulation will
be referred to by the number of particles in the cloud core.

$N_{\text{cloud}}$ | $N_{\text{medium}}$ | d$t_{\text{sc, final}}$ (yr)
---|---|---
$10^{5}$ | $5.2\times 10^{4}$ | 40
$3\times 10^{5}$ | $1.5\times 10^{5}$ | 40
$10^{6}$ | $4.8\times 10^{5}$ | 40
$3\times 10^{6}$ | $1.5\times 10^{6}$ | 21
$10^{7}$ | $4.8\times 10^{6}$ | 13
$3\times 10^{7}$ | $1.5\times 10^{7}$ | 4
Table 1: The number of particles in the cloud core of each simulation
($N_{\text{cloud}}$; first column) and the number of particles in the warm
medium ($N_{\text{medium}}$; second column). Each cloud core has a mass of 1
M⊙, thus $m_{\text{particle}}=$ M⊙$/N_{\text{cloud}}$. In each simulation,
every SPH particle has the same mass. The third column lists the number of
years the simulation was evolved after the formation of the stellar core.

Our lowest resolution model, $N=10^{5}$, matches the mass resolution in our
cluster study (Wurster et al., 2019); the model with $N=10^{6}$ matches that
in our turbulence vs non-ideal MHD studies (Wurster & Lewis, 2020a, b) and the
model with $N=3\times 10^{6}$ matches our core-collapse studies (Wurster et
al., 2018a, c, d, 2021).

Fig. 1 shows the number of CPU hours it takes to reach any given maximum
density, which is a proxy for time. Given our computational resources, the
$N=3\times 10^{7}$ model ended after approximately 2 yrs of wall-clock time
using 256 CPUs; by the end of this simulation, we were modelling the stellar
core in nearly real time. Given that runtime increases super-linearly with
resolution, even higher resolutions are currently prohibitively expensive to
run. All simulations were run using the hybrid openMP-MPI version of
sphNG333The $N=10^{5}$ model was run only using openMP. on the DiRAC2.5 Data
Intensive service at Leicester computer cluster.

Figure 1: Cumulative number of CPU hours per simulation as a function of
maximum density, which is a proxy for time. All simulations were performed on
the DiRAC2.5 Data Intensive service at Leicester computer cluster using the
hybrid openMP-MPI version of sphNG${}^{\ref{foot1}}$. The wall-clock
equivalent for the $N=3\times 10^{7}$ model is $\sim$2 yrs.

##  4 Results

As with our previous studies, we follow the gravitational collapse of the
cloud core through the first hydrostatic core phase
($10^{-12}\lesssim\rho_{\text{max}}/(\text{g cm}^{-3})\lesssim 10^{-9}$),
through the second collapse phase ($10^{-8}\lesssim\rho_{\text{max}}/(\text{g
cm}^{-3})\lesssim 10^{-4}$) and into the stellar core phase, where we continue
to evolve the models for 4 to 40 yrs, depending on the resolution (third
column of Table 1). We define the stellar core formation to occur at
d$t_{\text{sc}}=0$, which occurs when $\rho_{\text{max}}=10^{-4}$ g cm-3.

###  4.1 Evolution of the density and temperature

In addition to physical mechanisms that delay the collapse of the cloud core
(as discussed by, e.g., Bate et al., 2014; Tsukamoto et al., 2015a, b; Wurster
et al., 2016; Machida et al., 2018; Wurster et al., 2018a, b, c, d, 2021),
numerical resolution also affects the overall collapse time (e.g. Wurster &
Bate, 2019). The top panel of Fig. 2 shows the collapse times of our models,
where the difference in reaching $\rho_{\text{max}}=10^{-4}$ g cm-3 is $\sim$1
kyr, with the lowest resolution collapsing the fastest. By
$\rho_{\text{max}}\approx 10^{-13}$ g cm-3, the collapse exits the isothermal
collapse phase and gas begins to trap radiation and heat up (see Fig. 3).
Thus, physical processes in addition to gravity become increasingly important.
When we normalise the collapse time to when each model reaches
$\rho_{\text{max}}=10^{-13}$ g cm-3, we find much better agreement, with the
time between $\rho_{\text{max}}=10^{-13}$ and $10^{-4}$ g cm-3 differing by
only $\sim$60 yrs (bottom panel of Fig. 2). Therefore, the discrepancy in
collapse times primarily occurs during the isothermal collapse phase when
gravity is the dominant physical process.

Figure 2: Maximum density as a function of time for each model. The top panel
shows the absolute time, while the bottom panel shows the time normalised to
the time when $\rho_{\text{max}}=10^{-13}$ g cm-3, which is when the collapse
exits the isothermal collapse phase; see the grey reference line. There is a
small ‘bounce’ at $\rho\approx 10^{-12}$ g cm-3 as the first core forms. The
models are reasonably converged when normalised to
$\rho_{\text{max}}=10^{-13}$ g cm-3. This indicates that gravity during the
isothermal collapse phase is the primary cause for the large range of absolute
collapse times.  Figure 3: Maximum temperature as a function of maximum
density. Top panel shows the entire evolution, while the bottom panel zooms in
on the first core phase. The bounce in maximum density near the formation of
the first core ($\rho_{\text{max}}\approx 10^{-12}$ g cm-3) is visible, where
the maximum temperature continues to increase despite a slight overall cooling
of the first core. The heating at $\rho_{\text{max}}\approx 4\times 10^{-10}$
g cm-3 for $N\geq 10^{7}$ is due to the formation of the magnetic wall. The
maximum temperature during the first core phase is slightly lower at any given
$\rho_{\text{max}}$ for increasing resolution. Outside of the first core
phase, the maximum temperatures agree within 20 per cent.

As the first core forms at $\rho_{\text{max}}\approx 10^{-12}$ g cm-3, there
is a small ‘bounce’ in the maximum density (Fig. 2). During its formation, the
core is essentially optically-thick and therefore behaves adiabatically during
the bounce. The core cools slightly, although the maximum temperature
continues to slowly increase (Fig. 3). The bounce is nearly non-existent in
$N=10^{5}$, and is naturally better resolved for higher resolutions.

Fig. 4 shows the gas density at three times during the first core phase, both
perpendicular and parallel to the rotation axis. The high-density central
regions of the core are similar at all resolutions (bottom row in each panel),
however, the surrounding gas structure is resolution-dependent. At low
resolutions, radius of the first core is overestimated due to the
comparatively large SPH smoothing length. With increasing resolution, steeper
density profiles are resolved, and the first core converges to a radius of
$\approx 5$ au. The shape of the core transitions from oblate to prolate as
resolution is increased. For $N\geq 3\times 10^{6}$, a distinct pseudo-disc
forms in the mid-plane with density $\rho\gtrsim 10^{-12}$ g cm-3 around the
prolate core, leading to a more structured core and central regions.

Figure 4: Density slices through the first hydrostatic core perpendicular
(top) and parallel (bottom) to the rotation axis for three maximum densities
late in the first core phase. Density profiles are broader at lower
resolutions and the cores tend to be oblate; at higher resolutions, the
density profiles are steeper and the first cores are prolate.

The lack of a consistent trend with resolution in the normalised collapse
times (bottom panel of Fig. 2) is a result of the different gas structures in
the first core and how well-resolved the bounce is. When we consider only
cores with similar structures (i.e. $N\geq 3\times 10^{6}$), then we again see
a slight increase in collapse time with increasing resolution.

Fig. 5 shows the gas temperature at four times during the first core phase,
both perpendicular and parallel to the rotation axis. At low resolutions
($N\leq 3\times 10^{5}$) or high maximum densities
($\rho_{\text{max}}=10^{-8}$ g cm-3), the temperature profile in the
equatorial plane (top panel) follows the density profile. In the remaining
panels, the temperature profile contains sub-structure and instead follows the
magnetic field strength profile, which includes the magnetic wall (see Section
4.2.3 below). The vertical temperature profile (bottom panel) approximately
follows the density profile for $\rho_{\text{max}}\leq 10^{-9}$ g cm-3. At
higher maximum densities, the central core is more efficient at heating the
surrounding gas to create the hot, spherical envelope.

Figure 5: Temperature slices through the first hydrostatic core perpendicular
(top) and parallel (bottom) to the rotation axis for four maximum densities
late in the first core phase; maximum densities and spatial scales are chosen
to highlight our discussion of the magnetic wall in Section 4.2.3. At lower
resolutions, the cores are hotter since the slightly larger cores are less
efficient at radiating energy cool more slowly. By $\rho_{\text{max}}=10^{-9}$
g cm-3, the central region efficiently traps energy and heats up, nearly
independent of the surrounding environment or resolution.

Prior to the formation of the first core, the maximum temperatures in the
isothermal collapse phase ($\rho_{\text{max}}\lesssim 10^{-13}$ g cm-3) agree
within 20 per cent. The variation in maximum temperature with resolution
during the first core phase (Figs. 3 and 5) is again due to the fact that
steeper gradients can be resolved with higher resolution. First, as we have
seen above, at low resolution the size of the first core is over-estimated,
meaning that radiation finds it more difficult to escape. Second, in the flux-
limited diffusion approximation, the radiative flux is proportional to the
gradient of the radiation energy density, which can be steeper with higher
resolution, leading to more rapid cooling. The bounce of the first core can
also be seen in Fig. 3 at $\rho_{\text{max}}\approx 10^{-12}$ g cm-3, and the
heating due to the formation of the magnetic wall can be seen at
$\rho_{\text{max}}\approx 4\times 10^{-10}$ g cm-3 in the high resolution
calculations. After the first core phase, the maximum temperature is no longer
in the magnetic wall and becomes dependent only on the central gas that is
collapsing to ultimately form the stellar core; during this second collapse
phase, the maximum temperatures agree within five per cent amongst the
resolutions.

After the formation of the stellar core at $\rho_{\text{max}}=10^{-4}$ g cm-3,
it is more reasonable to compare the models normalised to the formation time
of the stellar core, d$t_{\text{sc}}=0$. Figs. 6 and 7 shows the gas density
around the stellar core for several times after its formation, which is
resolution-dependent. In all cases, the gas surrounding the stellar core has a
disc-like distribution immediately after stellar core formation (top two rows
of Fig. 7). However, this is quickly lost with $N\leq 3\times 10^{5}$,
probably due to rapid angular momentum transport caused by SPH artificial
viscosity, but also magnetic torques.

Figure 6: Density slices through the stellar core perpendicular to the
rotation axis as measured from the formation of the stellar core at
$\rho_{\text{max}}=10^{-4}$ g cm-3. As the systems evolves, the higher
resolution models ($N\geq 10^{6}$) form small discs that develop gravitational
instabilities (e.g. the $m=2$ instability at 10 yr for the model with
$N=3\times 10^{6}$) while the lower resolution simulations retain smooth
density profiles that steepen with time. Figure 7: Density slices through the
stellar core as in Fig. 6, but parallel to the rotation axis. The flattened
discs form and persist for resolutions of $N\geq 10^{6}$, while they dissipate
for $N\leq 3\times 10^{5}$.

At higher resolutions, small, $r\lesssim 1$ au discs form and persist until
the end of the simulations. Furthermore, the discs in $N\geq 3\times 10^{6}$
form weak spiral instabilities, with the most notable being a prominent but
transient $m=2$ instability at 10 yr for $N=3\times 10^{6}$. The stellar cores
themselves are hot ($T\sim 10^{4}-10^{5}$ K), while the surrounding gas is
much cooler ($T\sim 3000$ K).

###  4.2 Evolution of the magnetic field before stellar core formation

####  4.2.1 Growth of the magnetic field strength

Fig. 8 shows the evolution of the maximum and average magnetic field strengths
(i.e. $B_{\text{max}}$ and $\left<B_{\text{fhc}}\right>$, respectively). The
maximum magnetic field strength is approximately independent of resolution
until late in the first core phase ($\rho_{\text{max}}\approx 10^{-9}$ g
cm-3); we have previously shown that it is in the early first core phase
($\rho_{\text{max}}\approx 10^{-12}$ g cm-3) where the non-ideal processes
cause $B_{\text{max}}$ to diverge from ideal MHD simulations, with the growth
rate slower in the models employing non-ideal MHD (Wurster et al., 2018a, d,
2021). The sudden increase in $B_{\text{max}}$ at the end of the first core
phase ($\rho_{\text{max}}\approx 10^{-9}$ g cm-3) is from the formation of the
magnetic wall (see Section 4.2.3 below), where the magnetic field is ‘piling
up’; there is no corresponding increase in $\left<B_{\text{fhc}}\right>$,
indicating that the increase in $B_{\text{max}}$ is localised to the wall and
not spread throughout the first core.

By the end of the first core phase ($\rho_{\text{max}}\approx 10^{-8}$ g
cm-3), the maximum magnetic field strength differs by factor of $\sim$30
amongst the resolutions, but the average magnetic field strength is well
converged for resolutions $N\geq 3\times 10^{6}$. By the formation of the
stellar core at $\rho_{\text{max}}=10^{-4}$ g cm-3, the maximum field strength
differs by $\sim$300, although this difference is exacerbated by the slower
growth rate of $N=10^{5}$ compared to the remaining models; this difference
decreases to a factor of $\sim$65 when excluding $N=10^{5}$ and to a factor of
$\sim$10 when including only the three highest resolutions. This increasing
magnetic field strength for increasing resolution is due to both the gas being
better resolved and less numerical dissipation from artificial resistivity.
Therefore, although our models are converging, we have not yet reached
convergence in $B_{\text{max}}$.

At $\rho_{\text{max}}=10^{-4}$ g cm-3, the maximum field strength of the
models with $N\geq 10^{7}$ reaches the observed $\sim$ kG field strength of
young, low-mass stars. However, the maximum field strength resides outside the
stellar core itself (Section 4.2.3). When we consider the average magnetic
field strength of the gas comprising the first core (i.e.
$\rho_{\text{max}}\geq 10^{-12}$ g cm-3, which includes the magnetic wall),
then all the average field strengths prior to the formation of the stellar
core are below 10 G.

Figure 8: Evolution of the maximum magnetic field strength against maximum
density (top), and the evolution of the average magnetic field strength in the
first hydrostatic core (bottom). The vertical reference line represents the
formation of the stellar core, the horizontal reference line represents the
observed threshold of 1 kG, and the grey lines in the bottom panel are the
maximum field strengths copied from the top panel for reference. The average
field strength is calculated as
$\left<B_{\text{fhc}}\right>=10^{\left[\sum_{i}\log(B_{i})\right]/n}$, and
includes only gas with $\rho\geq 10^{-12}$ g cm-3 (i.e. the gas in the first
or second hydrostatic cores). The maximum magnetic field strength increases
with increasing resolution starting late in the first core phase; the maximum
field strengths in our higher resolution models surpass the 1 kG threshold
used to determine the origin of magnetic fields in low-mass stars. The average
magnetic field strength typically remains below 10 G, and is approximately
converged for $N\geq 3\times 10^{6}$.

####  4.2.2 Magnetic field in the first core

Fig. 9 shows the magnetic flux threading the equatorial plane of the first
hydrostatic core (i.e. the gas with $\rho_{\text{max}}\geq 10^{-12}$ g
cm-3)444Note that this includes the pseudo-disc for $N\geq 3\times 10^{6}$..
As a function of $\rho_{\text{max}}$, the magnetic flux is the same within 50
per cent for all models. For $10^{-12}\lesssim\rho_{\text{max}}/(\text{g
cm}^{-3})\lesssim 10^{-9}$, the magnetic flux grows more rapidly than the
radius, indicating that the magnetic field is being dragged in as the cloud
gravitationally collapses and that the magnetic field strength is increasing.
This is confirmed by plotting the magnetic flux against mass of the first core
(bottom panel of Fig. 9) during the first core phase
($10^{-12}\lesssim\rho_{\text{max}}/(\text{g cm}^{-3})\lesssim 10^{-9}$); the
magnetic flux increases as the mass of the first core increases, indicating
that as mass enters the first core it drags the magnetic field with it.

Figure 9: Evolution of the magnetic flux passing through the mid-plane of the
first hydrostatic core,
$\Phi_{\text{B}}=\int_{0}^{r_{\text{fhc}}}B_{\text{z}}\text{d}A$. Plotting
against the mass of the first core (bottom) highlights the growth of the first
core at $10^{-12}\lesssim\rho_{\text{max}}/(\text{g cm}^{-3})\lesssim
10^{-9}$. The increasing magnetic flux represents the magnetic field diffusing
into the first core and the increasing magnetic field strength.

####  4.2.3 The magnetic wall

Unlike the density profile in the equatorial plane (top panel of Fig. 4), the
magnetic field strength is not a smooth profile in the first hydrostatic core.
Instead, magnetic walls form as the magnetic field decelerates the charged
particles, but not the neutral gas (e.g. Tassis & Mouschovias, 2005, 2007a,
2007b). This prevents the magnetic flux from reaching the central regions and
instead the magnetic field ‘piles up’ in a torus of higher magnetic field
strength such that the maximum magnetic field strength lies in this torus
rather than at the centre of the core which is coincident with the maximum
density. This explains the rapid increase in $B_{\text{max}}$ but not
$\left<B_{\text{fhc}}\right>$ shown in Fig. 8 since the increase is localised
to the torus.

Fig. 10 shows the magnetic field strength in slices through the core late in
the first core phase, both perpendicular and parallel to the rotation axis.
The formation time of the magnetic wall is resolution-dependent, with the wall
forming at slightly lower maximum densities at higher resolutions; this does
not correspond to a large change in absolute or relative time (recall Fig. 2).

Figure 10: Magnetic field strength slices through the first hydrostatic core
perpendicular (top) and parallel (bottom) to the rotation axis for four
maximum densities late in the first core phase; spatial range and maximum
densities are chosen to highlight the formation and early evolution of the
magnetic wall. The magnetic wall forms earlier at higher resolutions, and is
less axisymmetric. The asymmetries are caused by the Hall effect, and the
greater asymmetries at higher resolutions are a result of resolving shorter
whistler waves.

The temperature profile (Fig. 5) is similar to the magnetic field strength
profile (Fig. 10). The magnetic walls are hotter than the surrounding gas, and
the maximum temperature lies within this wall for $N\geq 10^{7}$ between
$\rho_{\text{max}}\approx 4\times 10^{-10}$ and $10^{-8}$ g cm-3. Although
there is a temperature increase in the walls at lower resolutions compared to
the surrounding gas, the temperature in the walls is still lower than the
central temperature for $N\leq 3\times 10^{6}$. Both Ohmic resistivity and
ambipolar diffusion contribute to heating the gas (e.g. Wurster et al., 2014),
thus as the ionised gas is slowed down to form the wall, it also heats up,
accounting for the higher temperature in the magnetic walls.

Ambipolar diffusion tends to be the process that produces the strongest
magnetic walls; assuming the initial cloud is axisymmetric and that the Hall
effect is excluded, then the magnetic wall will also be axisymmetric (Wurster
et al., 2021). The magnetic field strengths are similar at all resolutions
until the formation of the magnetic wall (see Fig. 8), however, as discussed
above, the density profile is smoothed out at lower resolutions. Therefore,
the density at $r\approx 3$ au is slightly higher at lower resolutions,
meaning that the effect of ambipolar diffusion is weaker. This results in the
later formation time of the magnetic wall at lower resolutions. Once the wall
has formed, the higher resolution models have stronger magnetic field
strengths in the wall since the magnetic field is better resolved and there is
less artificial dissipation. This results in a stronger effect of ambipolar
diffusion, amplifying the wall.

The spiral structure (bottom row of the top panel in Fig. 10 for $N\geq
3\times 10^{6}$) in the magnetic field strength is caused by the Hall effect
(Wurster et al., 2021). Unlike Ohmic resistivity and ambipolar diffusion, the
Hall effect is a dispersive term that splits the Alfvén wave into left- and
right-circularly polarised waves. The right (whistler) wave propagates faster
than the Alfvén wave, and its speed increases for decreasing wavelength (e.g.
Sano & Stone, 2002; Pandey & Wardle, 2008; Wurster et al., 2016; Marchand et
al., 2018, 2019). Therefore, as numerical resolution is increased, additional
whistler wavelengths are resolved. This yields dispersion on smaller scales
and results in the spiral structure in the magnetic field shown in Fig. 10.
These additional whistler wavelengths prevent numerical convergence, at least
at the resolutions presented here.

Between $\rho_{\text{max}}=10^{-9}$ and $10^{-8}$ g cm-3, the gas in the
centre of the first core becomes oblate (bottom panel of Fig. 4), and the
magnetic field is amplified in this region (bottom panel of Fig. 10). By
$\rho_{\text{max}}=10^{-8}$ g cm-3 for $N\geq 10^{7}$, the magnetic wall has
been disrupted due to the short wavelength whistler waves, creating the highly
structured magnetic field within the first core. Despite this disruption, the
maximum magnetic field strength continues to reside outside of the centre of
the core.

Therefore, ambipolar diffusion is primarily responsible for the formation of
the magnetic wall, but the Hall effect is responsible for its asymmetric
structure, with the Hall effect’s importance increasing with increasing
resolution as whistler waves with shorter wavelengths are resolved.

####  4.2.4 The rapid second collapse

Fig. 11 shows the magnetic field strength in slices through the inner parts of
the first hydrostatic core during the rapid second collapse phase. The
collapse is so quick that the field does not evolve significantly outside
$\approx 1$ au, but there is rapid growth on scales $\ll 1$ au. In the higher
resolution models, much of the magnetic flux that ends up in the vicinity of
the stellar core comes from the highly-structured magnetic field that was
produced by the Hall effect during the disruption of the magnetic wall. This
indicates that to obtain a complete understanding of how fossil magnetic
fields are implanted into stellar cores may require accurate modelling of the
Hall effect.

Figure 11: Magnetic field strength slices through the inner parts of the first
hydrostatic core that are perpendicular to the rotation axis during and after
the rapid second collapse phase. The collapse is so quick that the field does
not evolve significantly outside $\approx 1$ au, but there is rapid growth on
scales $\ll 1$ au.

###  4.3 Evolution of the magnetic field after stellar core formation

To determine the origin of magnetic fields in low-mass stars, we investigate
the magnetic field strength at and immediately after the formation of the
stellar core. In  at a resolution of $N=3\times 10^{6}$, we concluded that the
magnetic fields in low-mass stars were generated by a dynamo later in life
since the stellar core magnetic field strengths at birth were $<1$ kG.

Fig. 12 shows the maximum magnetic field strength, $B_{\text{max}}$, and
average field strength of the stellar core, $\left<B_{\text{sc}}\right>$,
after its formation. The maximum field strength in $N=3\times 10^{7}$
surpasses the kG threshold at stellar core formation and remains above until
the end of the simulation 4 yr later. For models with $N\leq 3\times 10^{6}$,
the maximum magnetic field strength is $B_{\text{max}}\lesssim 1$ kG.
Therefore, there is a resolution-dependence on the maximum magnetic field
strength, and whether it is above or below the kG threshold.

Figure 12: Evolution of the maximum (top) and stellar core (bottom) magnetic
field strengths, as measured from the formation of the stellar core. The
average field strength is calculated as
$\left<B_{\text{sc}}\right>=\left[\sum_{i}\log(B_{i})\right]/n$, and includes
only gas with $\rho\geq 10^{-4}$ g cm-3. Vertical and horizontal solid lines
are included for reference. For increasing resolution, both field maximum and
stellar field strengths increase, where the maximum magnetic field strength
resides outside of the stellar core. Fluctuations in the stellar core strength
result from magnetic flux being advected between the core itself and the
surrounding gas. For the first six months after stellar core formation in
$N=3\times 10^{7}$, the stellar core strength surpasses the observed 1 kG
threshold.

However, as mentioned above in Section 4.2.3, the maximum magnetic field
strength lies outside the stellar core itself. The bottom panel of Fig. 12
shows the average magnetic field strength in the stellar core. The average
magnetic field strength in the stellar core in the first few years after its
formation is similar for models with $N\leq 3\times 10^{6}$, and lies well
below 1 kG. However, it increases significantly for the two highest resolution
models, and for the highest resolution simulation it exceeds 1 kG for
approximately 6 months after stellar core formation.

After stellar core formation, there is an immediate decline in magnetic field
strength, followed by an evolution where the field strengths fluctuate up to
$\sim$2 dex, with larger fluctuations in $\left<B_{\text{sc}}\right>$ than
$B_{\text{max}}$. Although the field strength decreases for $N\leq 3\times
10^{5}$, the fluctuations and short evolution time make extracting a trend
from the remaining models challenging. Nonetheless, Fig. 12 shows that the
magnetic field in the core is continuing to dynamically evolve. Figs. 13 and
14 show the magnetic field strength in a slice through the stellar core
perpendicular to the rotation axis; the two figures show the field strengths
on different spatial scales, and the defined boundary of the stellar core at
$\rho_{\text{max}}=10^{-4}$ g cm-3 is shown in the latter figure.

Figure 13: Magnetic field slices through the stellar core perpendicular to the
rotation axis. For increasing resolution, the magnetic field strength
increases and the spiral structures become finer and more intricate.  Figure
14: Magnetic field slices through the stellar core as in Fig. 13, but on a
smaller spatial scale. The contour represents the boundary of the stellar core
at $\rho_{\text{max}}=10^{-4}$ g cm-3. The magnetic field is continually
advected in and out of the core due to the continual gas distribution between
the core and its surrounds, leading to fluctuations in the magnetic field
strength of the stellar core shown in Fig. 12.

Naturally, for higher resolution, numerical dissipation555At the temperatures
in the stellar core, physical dissipation is negligible. is lower and the
magnetic structures are better resolved, as can be seen in these figures. This
leads to higher magnetic field strengths, as previously discussed. At low
resolutions ($N\leq 3\times 10^{5}$), the entire region in and surrounding the
stellar core is permeated with a weak, unstructured magnetic field. For
$N=10^{6}$, the field strength grows with time, with an increase of $\sim$1
dex at d$t_{\text{sc}}\approx{14}$ yr, which corresponds to the formation of
the spiral structure clearly seen at d$t_{\text{sc}}={20}$ yr in Fig. 13; note
that this spiral structure does not exist in the density profile, confirming
that it is produced by Alfvén waves.

Similar to the magnetic wall (Section 4.2.3), the magnetic field structures
are well-defined for $N\geq 3\times 10^{6}$, with spiral structures seen in
the strength of the magnetic field surrounding the stellar core (Figs. 13 and
14). These structures are naturally tighter and more finely structured for
increasing resolution. This complex magnetic field is continually being
advected between the core itself and the surrounding gas since, at this stage,
there is no rigid boundary at the edge of the core. Moreover, the evolution
leads to transitions between the toroidal and poloidal components of the
magnetic field within the core itself. This evolution of the magnetic field in
and near the core leads to the fluctuations shown in the bottom panel of Fig.
12.

Therefore, the magnetic field in and around the core is dynamically evolving,
leading to fluctuations in both the maximum and stellar core magnetic field
strengths. Furthermore, these fluctuations are resolution-dependent, and our
results are not numerically converged. This clearly illustrates the difficulty
of determining the strength of the fossil field that may be implanted in a
stellar core. However, in our highest resolution model, the average magnetic
field strength of the stellar core does exceed the 1 kG threshold for
d$t_{\text{sc}}\approx{6}$ months, suggesting that with even higher resolution
the observed magnetic fields of young low-mass stars may be able to be
provided by fossil fields. The temperatures within the stellar core are high
enough that artificial resistivity in the stellar core is likely responsible
for the decay of the magnetic field strength after stellar core formation.

###  4.4 First hydrostatic core outflows

First core outflows are magnetically launched from the pseudo-disc during the
first core phase. They typically expand at a few km s-1 and contain $M<0.01$
M⊙ by the formation of the stellar core (e.g. Wurster et al., 2016, 2018c,
2021); models that include all three non-ideal processes and are initialised
with the rotation and magnetic field vectors aligned as modelled here yield
the fastest outflows compared to other orientations or combination of non-
ideal processes (Wurster et al., 2021). Fig. 15 shows the radial velocity of
the first core outflows, and Fig. 16 shows the evolution of the momentum,
mass, and average velocity of the outflows. In the latter, the gas is defined
to be in the outflow if it is at least 30∘ above/below the mid-plane, its
radial velocity vector is at least 30∘ above/below the mid-plane, and
satisfies $\rho\leq 10^{-8}$ g cm-3 and $|v_{\text{r}}|/|v|>0.5$; we divide
the outflow into fast ($v_{\text{r}}>2$ km s-1) and slow ($0.5$ km
s-1$<v_{\text{r}}<2$ km s-1) components.

Figure 15: Radial velocity of the first core outflow in a slice through the
core perpendicular to the rotation axis. The top two rows are plotted at
constant maximum density late in the first core phase, while the bottom three
rows are plotted after the formation of the stellar core, as measured from its
formation. Contours are at $v_{\text{r}}=0.5$ and $2$ km s-1, which are the
boundary velocities for the slow and fast outflows shown in Fig. 16. The fast
component of the outflows initially exists in the lobes, but at late times, an
additional fast component forms above and below the core. Figure 16: The total
momentum (top row), total mass (middle row) and average velocity (bottom row)
in the slow ($0.5<v_{\text{r}}/$(km s-1)$\ <2$; left-hand column) and fast
($v_{\text{r}}>2$ km s-1; right-hand column) outflows. The vertical grey line
represents the defined formation density of the stellar core. Gas is in the
outflow if it is at least 30∘ above/below the mid-plane, its radial velocity
vector is at least 30∘ above/below the mid-plane, and satisfies $\rho\leq
10^{-8}$ g cm-3 and $|v_{\text{r}}|/|v|>0.5$. The slow outflow has converged
for $N\geq 10^{7}$, while the fast outflow has not yet converged (excluding
velocity), with mass and momentum slightly decreasing for increasing
resolution.

The first core outflow is not properly resolved with $N=10^{5}$ (Fig. 15); the
total mass of the outflow measure during the second collapse phase is
$\sim$$3\times 10^{-4}$ M⊙ (Fig. 16) which corresponds to only 30 SPH
particles. This was the resolution of our star cluster simulations (Wurster et
al., 2019), confirming that the reason they were not observed in that study
was due to the resolution limit.

A resolved outflow forms at resolutions $N\geq 3\times 10^{5}$, while the
structure qualitatively converges for $N\geq 10^{7}$ (Fig. 15). There is both
a fast and slow component to the outflow, with the average velocity of each
outflow well-converged at $\sim$1.5 and 2.5 km s-1 for the slow and fast
outflows, respectively; for $N\geq 10^{6}$, part of each outflow reaches
speeds of $v_{\text{r}}\gtrsim 3$ km s-1 at $\rho_{\text{max}}=10^{-8}$ g
cm-3.

With increasing resolution, there is increasing mass and momentum in the slow
outflows, until the values converge for $N\geq 10^{7}$ (left-hand column of
Fig. 16); the mass in $N=3\times 10^{6}$ is only $\sim$1.5 times lower than
the converged value, suggesting it also captures the outflow quite well. The
fast outflows are embedded in the lobes of the slow outflows, and there is
decreasing mass and momentum for increasing resolution (when the fast outflows
are resolved); during the second collapse, the mass in the fast outflow
differs by only a factor of $\sim$2 amongst the resolutions. Given that the
fast component is part of the first core outflows rather than a separate
outflow, there is some blurring between the fast and slow components, leading
to this difference amongst the resolutions. We conclude that first core
outflows are well modelled for resolutions of $N\geq 3\times 10^{6}$.

Near the end of the simulations ($\rho_{\text{max}}\gtrsim 10^{-2}$ g cm-3),
there is an increase in the mass and momentum of the fast outflows (right-hand
column of Fig. 16). For the higher resolution models ($N\geq 3\times 10^{6}$),
a fast, vertical component forms above and below the stellar core (fourth row
of Fig. 15), which ultimately merges with the fast lobes at late times (bottom
row of Fig. 15). For the lower resolution models ($N\leq 10^{6}$), the
increase in mass and momentum is first due to the launching radius of the
first core outflow decreasing, and then due to the velocity in the lobes and
above/below the core increasing.

In agreement with our previous work (e.g. Wurster & Lewis, 2020b; Wurster et
al., 2021), stellar core outflows are not launched, at least for as long as we
are able to follow the models.

##  5 Discussion

###  5.1 Timescales

Due to computational limitations, our simulations end 4 to 40 yrs after the
formation of the stellar core, which is much shorter than the typical observed
age of a young stellar object of $\sim$$10^{6}$ yr (e.g. Bouvier et al., 2014;
Ansdell et al., 2016; Froebrich et al., 2018; Teixeira et al., 2020); even the
extremely young stars of $\sim$$5\times 10^{3}$ yr observed by (e.g.) Furuya
et al. (2006) and Yusef-Zadeh et al. (2017) are still much older than
presented here. Our early end time is limited by our high resolution and
densities in the stellar cores, and hence short timesteps (recall Section
3.1). Therefore, our conclusions about the origin of magnetic field in stars
is based upon a protostar’s characteristics at its birth.

After reaching the maximum magnetic field strength shortly after stellar core
formation (Figs. 8 and 12), there is a rapid decrease in field strength over
the first few years; the length of this initial decrease becomes shorter with
increasing resolution. After this rapid decrease, the trend is less clear,
given the fluctuations in maximum and stellar core magnetic field strengths.
Thus, predicting how the magnetic field will evolve over the next $10^{6}$ yr
is challenging.

At the very high temperatures in and near the stellar core, the gas is highly
ionised, therefore Ohmic resistivity is the dominant non-ideal effect. The
diffusion timescale for Ohmic resistivity is given by

$\tau_{\text{OR}}=\frac{L^{2}}{\eta_{\text{OR}}},$ (1)

where $L$ is the characteristic length scale which we set to the radius of the
stellar core, and $\eta_{\text{OR}}$ is the coefficient for Ohmic resistivity,
which is dependent on the number density of the $j$ chemical species that are
present $n_{j}$, their mass $m_{j}$, their charge $eZ_{j}$, and their plasma-
neutral collisional frequency $\nu_{j\text{n}}$; the coefficient is given by

$\eta_{\text{OR}}=\frac{c^{2}}{4\pi}\left[\sum_{j}\frac{n_{j}\left(eZ_{j}\right)^{2}}{m_{j}\nu_{j\text{n}}}\right]^{-1},$
(2)

where $c$ is the speed of light; see also Wardle & Ng (1999), Wardle (2007),
Wurster et al. (2016), and Wurster (2016). At these high temperatures, the
present chemical species are likely all gas since the dust grains would have
evaporated at lower temperatures (e.g. Lenzuni et al., 1995). Unlike ambipolar
diffusion or the Hall effect, this term is independent of the magnetic field
strength. The evolution of the Ohmic timescale is plotted in Fig. 17.

Figure 17: The evolution of the Ohmic timescale in the stellar core after
stellar core formation, as given by Eqn. 1. The timescale initially increases,
before stabilising at $\tau_{\text{OR}}\sim 10^{10}$ yr. Since
$\tau_{\text{OR}}\gg 10^{6}$ yr, this suggests that the stellar magnetic field
will not decay by Ohmic resistivity by the age at which most young stellar
objects are observed.

In our models, the stellar core radius is
$0.01<r_{\text{sc}}/\text{au}<0.015$, therefore the evolution of
$\tau_{\text{OR}}$ is primarily defined by the evolution of Ohmic resistivity.
After the initial increase, the Ohmic timescale remains at
$\tau_{\text{OR}}\sim 10^{10}$ yr. Since $\tau_{\text{OR}}\gg 10^{6}$ yr, the
magnetic field in the stellar core is not expected to decrease due to Ohmic
resistivity. Therefore, based upon resistivity arguments, the magnetic field
strength in the stellar core shortly after its birth is indicative of its
magnetic field strength at later times as well. This timescale, however, is an
upper limit since other processes, such as turbulent diffusion (see Section
5.2.2), may contribute to diffusing the magnetic field out of the stellar
core. This would yield a somewhat quicker field decay than just the Ohmic
rate, however, it is expected that the strong stellar magnetic field will
persist.

As shown in Figs. 12 – 14, the magnetic field in and around the stellar core
is dynamically evolving, with magnetic flux being advected in and out of the
stellar core. Therefore, the magnetic field evolution in the entire region is
important to determine the evolution of $\left<B_{\text{sc}}\right>$. Since
$\eta_{OR}$ increases only slightly outside of the stellar core, assuming a
larger radius to encompass the dynamically evolving region would only increase
the Ohmic timescale. Therefore, the fluctuating core field strength is
dominated by a combination of magnetic advection and artificial resistivity,
and will likely continue to do so until a gap appears between the stellar core
and the inner edge of the disc. Since Ohmic resistivity is un-important on
these timescales, the vast majority of the resolution dependence is almost
certainly due to the artificial resistivity; this is consistent with higher
resolutions yielding stronger magnetic fields. Thus, our results and the Ohmic
diffusion timescale are consistent with strong fields persisting to the ages
of observed young stars.

###  5.2 Initial conditions

The results of star formation simulations are inherently dependent on the
initial conditions, such as the initial mass, density profile, rotational
profile, thermal energy, and magnetic field geometry. Thus, the stellar field
strength may also be dependent on these initial conditions. We briefly comment
on the possible effect of the initial magnetic field strength and turbulence.

####  5.2.1 Magnetic field strength

Star forming regions are observed to have normalised mass-to-flux ratios of
$0.5\lesssim\mu\lesssim 3$ (e.g. Girart et al., 2006; Stephens et al., 2013;
Koch et al., 2014; Qiu et al., 2014; Hull et al., 2017; Karoly et al., 2020),
although some regions have strengths similar to that presented here (e.g.
B335; Maury et al., 2018). Cores tend to have larger mass-to-flux ratios than
their envelopes (Li et al., 2014), thus our weaker initial magnetic field
strength may better represent a slightly evolved core. Therefore, how would
our conclusions change if we had modelled an initially stronger magnetic field
strength?

Naively, one would expect that stronger initial field strengths would lead to
stronger maximum and core strengths. However, the ideal MHD study of Bate et
al. (2014) yielded similar maximum magnetic field strengths for both
$\mu_{0}={5}$ and 10.

When non-ideal MHD processes are introduced, predicting magnetic field
strengths becomes more challenging. For example, the effect of ambipolar
diffusion depends on the magnetic field strength at low densities, therefore
the magnetic dissipation caused by ambipolar diffusion is higher in stronger
magnetic fields; moreover the magnetic wall would also be stronger since it is
caused primarily by ambipolar diffusion (e.g. Tassis & Mouschovias, 2005,
2007a, 2007b; Wurster et al., 2021); this may help to prevent the magnetic
field from entering the stellar core. In the cloud-collapse simulations of
Masson et al. (2016) which included ambipolar diffusion, prior to the first
core phase, their model with $\mu_{0}={2}$ had a larger distribution of
magnetic field strengths at any given density and generally stronger strengths
than their model with $\mu_{0}={5}$. However, in the first core itself, the
magnetic field strengths were slightly higher in the model with $\mu_{0}={5}$.

Therefore, the effect of increasing the initial magnetic field strength is not
straightforward, and additional simulations would be required to test this
parameter space (although this is out of the scope of this project). Based
upon the results of Bate et al. (2014) and Masson et al. (2016), changing the
magnetic field strength will quantitatively change our results, but likely
will not qualitatively affect our conclusions.

While predicting the effect of increasing the initial magnetic field strength
remains challenging, it is indisputable that including some or all of the non-
ideal processes decreases that magnetic field strength compared to ideal MHD
models (e.g. Machida et al., 2007; Tomida et al., 2015; Tsukamoto et al.,
2015a; Wurster et al., 2018a, d, 2021; Masson et al., 2016; Marchand et al.,
2018; Vaytet et al., 2018), at least for strong initial field strengths. How
much the field strength is decreased compared to ideal MHD will depend on the
microphysics of the non-ideal model (e.g. Zhao et al., 2020, 2021; Wurster et
al., 2018a; Wurster, 2021). Nonetheless, non-ideal MHD prevents the unobserved
$>100$ kG fields obtained by ideal MHD simulations (e.g. Machida et al., 2006;
Bate et al., 2014). This shows that non-ideal MHD effects are extremely
important to reduce the field strength, but are not so dominate as to
effectively wipe out a stellar field and seem to give field strengths similar
to those that are observed in young stars (with sufficient resolution).

####  5.2.2 Turbulent velocity fields

Turbulence is typically included in star formation simulations that are
initialised from larger cores (e.g. $5-10^{3}$ M⊙ cores), and has been
investigated as a possible solution to the so-called ‘magnetic braking
catastrophe’ (i.e. rotationally supported discs do not form in the presence of
strong, ideal magnetic fields since well-ordered magnetic fields are very
efficient at transporting angular momentum outwards; e.g. Allen et al., 2003;
Price & Bate, 2007; Mellon & Li, 2008; Hennebelle & Fromang, 2008). Turbulence
causes a misalignment between the magnetic field and rotation vectors (e.g.
Joos et al., 2013), which hinders the outward transport of angular momentum
and hence promotes disc formation. However, turbulence also can act as an
effective diffusivity, transporting magnetic flux outwards (e.g. Santos-Lima
et al., 2012, 2013; Joos et al., 2013). Thus, turbulence leads to weaker
magnetic fields in the first core phase, which may lead to weaker stellar core
strengths.

To the contrary, Seifried et al. (2012, 2013) argued that turbulence did not
act as an effective diffusivity and did not cause a magnetic flux loss, at
least on larger scales. At first core densities and lower, they found
$B\propto\rho^{0.5}$, independent of the level of turbulence. For disc
formation, they argued that discs formed simply due to the turbulent velocity
structure and not magnetic flux loss. Unfortunately, this conclusion may have
been a result of their use of the mass-to-flux ratio as a diagnostic tool
(Santos-Lima et al., 2013). Thus, it would appear that at the first core
stage, turbulence likely causes magnetic flux loss and hence lower first core
magnetic field strengths.

The above studies focused on disc formation, and their methods prevented them
from investigating the stellar core magnetic field strength. To the best of
our knowledge, Wurster & Lewis (2020b) is the only study to model the
formation of the stellar core from a turbulent magnetised molecular cloud.
This study used the same initial conditions as presented here, but the lower
mass resolution of $10^{6}$ SPH particles in the cloud. Increasing the initial
level of turbulence from Mach 0 to Mach 1 either had negligible effect on the
maximum magnetic field strength or decreased it by an order of magnitude,
depending on the other initial properties of the sphere and physical processes
included. Since lower $B_{\text{max}}$ tends to lead to lower
$\left<B_{\text{fhc}}\right>$ and $\left<B_{\text{sc}}\right>$ (Figs. 8 and
12), it is likely that including turbulence will decrease the stellar core
magnetic field strength.

##  6 Summary and conclusion

In this study, we investigated the effect of numerical resolution on the
gravitational collapse of a molecular cloud core through the first and stellar
collapse phases to the formation of a protostar. We tested six resolutions,
where each model was initialised as a 1 M⊙ spherical cloud of uniform density
and included between $N=10^{5}$ and $3\times 10^{7}$ equal-mass SPH particles.
The domain was threaded with an initially vertical magnetic field that was
aligned with the rotation axis and had a normalised mass-to-flux ratio of 5.
Our models included Ohmic resistivity, ambipolar diffusion and the Hall
effect; the aligned orientation of the magnetic field and rotation vectors
meant that the Hall effect would hinder disc formation.

Our main conclusions are as follows:

  1. 1.

Increasing resolution increases the length of time it takes to reach stellar
densities from the beginning of the simulation, primarily due to a lengthening
of the slow, isothermal collapse phase. The collapse to stellar densities
requires similar evolution times at all resolutions when measured from the end
of the isothermal collapse at $\rho_{\text{max}}=10^{-13}$ g cm-3.

  2. 2.

We obtain numerical convergence for the maximum magnetic field strength until
late in the first core phase, with the maximum field remaining below 1 G. The
average magnetic field strength in the first core is
$\left<B_{\text{fhc}}\right>\lesssim 10$ G throughout the first core and
second collapse phases.

  3. 3.

At $\rho_{\text{max}}\approx 10^{-9}$ g cm-3, the maximum magnetic field
strength diverges amongst the models due to the formation of a magnetic wall
in the outer parts (radii $\approx 3$ au) of the first core. Following its
formation, the maximum field strength resides in the magnetic wall rather than
in the centre of the first core. The wall forms earlier, is better well-
defined and less axisymmetric at higher resolutions. At high numerical
resolutions, short whistler waves created by the Hall effect are resolved,
which cause the initially axisymmetric wall to become unstable and the
magnetic field becomes highly asymmetric.

  4. 4.

After the break up of the magnetic wall, the maximum field strength is
unconverged, even with our highest resolutions. By the formation of the
stellar core at $\rho_{\text{max}}=10^{-4}$ g cm-3, the maximum field strength
differs by a factor of $\sim$300 between the highest and lowest resolution
models (but only a factor of $\sim$10 between the three highest resolution
models).

  5. 5.

The magnetic field is dynamically evolving in and around the stellar core, and
the field strength fluctuates up to two orders of magnitude after the stellar
core’s formation. With increasing resolution, the field strength in the core
also increases. In our highest resolution model, the average magnetic field
within the stellar core exceeds 1 kG for 6 months after the formation of the
stellar core. The average magnetic field strengths of the stellar cores in the
next two highest resolution models peak around 1 kG, but quickly decay to
$\left<B_{\text{sc}}\right>\lesssim 100$ G. In all models, the maximum
magnetic field strength lies within a small (radius $\approx 1$ au) disc
surrounding the stellar core rather than within the stellar core itself. For
our two highest resolution models, the maximum magnetic field strength exceeds
1 kG for as long as we are able to follow the models.

  6. 6.

First core outflows are launched in all models with $N\geq 3\times 10^{5}$ and
are converged for resolutions of $N\geq 3\times 10^{6}$. Stellar core outflows
do not form in any of our models.

Multiple numerical studies have shown that the initial star forming
environment affects the resulting star. Therefore, our results and hence our
conclusions may be affected by (e.g.) turbulence, different initial magnetic
field strengths, and magnetic field geometry. Determining exactly how each of
these processes affects the magnetic field implanted at birth would require
additional studies. In our study, the magnetic field strengths during the
second collapse phase and within the stellar core increase with increasing
resolution. Numerical convergence of the magnetic field has not been obtained
in these phases. Given the fine structure that develops in the magnetic field
due to the Hall effect late in the evolution of the first hydrostatic core,
numerical convergence of the magnetic field strength appears to be
computational prohibitive to achieve with existing computational resources.
However, given that with our highest resolution calculation we are able to
obtain an average magnetic field strength in the stellar core in excess of 1
kG that is sustained for 6 months, we cautiously conclude that substantial
magnetic fields may be implanted in low-mass stars during their formation.
Since the Ohmic diffusion timescale in the stellar core is much longer than
the age of the young stellar objects that are currently being observed, it is
probable that these birth magnetic fields persist over long timescales,
suggesting that a dynamo process later in the star’s life is not required to
generate a strong stellar magnetic field.

## Acknowledgements

We would like to thank the referee for useful comments that improved the
quality of this manuscript. JW and MRB acknowledge support from the European
Research Council under the European Community’s Seventh Framework Programme
(FP7/2007- 2013 grant agreement no. 339248). JW and IAB acknowledge support
from the University of St Andrews. DJP received funding via Australian
Research Council grants FT130100034, DP130102078 and DP180104235. This work
was performed using the DiRAC Data Intensive service at Leicester, operated by
the University of Leicester IT Services, which forms part of the STFC DiRAC
HPC Facility (www.dirac.ac.uk). The equipment was funded by BEIS capital
funding via STFC capital grants ST/K000373/1 and ST/R002363/1 and STFC DiRAC
Operations grant ST/R001014/1. DiRAC is part of the National e-Infrastructure.
Several figures were made using splash (Price, 2007).

## Data availability

The data underlying this article will be available upon reasonable request.

## References

  * Allen et al. (2003) Allen A., Li Z.-Y., Shu F. H., 2003, ApJ, 599, 363
  * Ansdell et al. (2016) Ansdell M., et al., 2016, ApJ, 828, 46
  * Bate & Burkert (1997) Bate M. R., Burkert A., 1997, MNRAS, 288, 1060
  * Bate et al. (1995) Bate M. R., Bonnell I. A., Price N. M., 1995, MNRAS, 277, 362
  * Bate et al. (2014) Bate M. R., Tricco T. S., Price D. J., 2014, MNRAS, 437, 77
  * Benz (1990) Benz W., 1990, in Buchler J. R., ed., Numerical Modelling of Nonlinear Stellar Pulsations Problems and Prospects. Kluwer, Dordrecht, p. 269
  * Børve et al. (2001) Børve S., Omang M., Trulsen J., 2001, ApJ, 561, 82
  * Bouvier et al. (2014) Bouvier J., Matt S. P., Mohanty S., Scholz A., Stassun K. G., Zanni C., 2014, Protostars and Planets VI, pp 433–450
  * Chabrier & Küker (2006) Chabrier G., Küker M., 2006, A&A, 446, 1027
  * Commerçon et al. (2008) Commerçon B., Hennebelle P., Audit E., Chabrier G., Teyssier R., 2008, A&A, 482, 371
  * Crutcher (1999) Crutcher R. M., 1999, ApJ, 520, 706
  * Cunningham et al. (2018) Cunningham A. J., Krumholz M. R., McKee C. F., Klein R. I., 2018, MNRAS, 476, 771
  * Donati & Landstreet (2009) Donati J. F., Landstreet J. D., 2009, ARA&A, 47, 333
  * Donati et al. (2020) Donati J. F., et al., 2020, MNRAS, 491, 5660
  * Federrath (2015) Federrath C., 2015, Monthly Notices of the Royal Astronomical Society, 450, 4035
  * Froebrich et al. (2018) Froebrich D., et al., 2018, MNRAS, 478, 5091
  * Furuya et al. (2006) Furuya R. S., Kitamura Y., Shinnaga H., 2006, ApJ, 653, 1369
  * Girart et al. (2006) Girart J. M., Rao R., Marrone D. P., 2006, Science, 313, 812
  * Grudić et al. (2021) Grudić M. Y., Guszejnov D., Hopkins P. F., Offner S. S. R., Faucher-Giguére C.-A., 2021, MNRAS,
  * Haugbølle et al. (2018) Haugbølle T., Padoan P., Nordlund Å., 2018, ApJ, 854, 35
  * Heiles & Crutcher (2005) Heiles C., Crutcher R., 2005, in Wielebinski R., Beck R., eds, Lecture Notes in Physics, Berlin Springer Verlag Vol. 664, Cosmic Magnetic Fields. p. 137 (arXiv:astro-ph/0501550), doi:10.1007/11369875˙7
  * Hennebelle (2018) Hennebelle P., 2018, A&A, 611, A24
  * Hennebelle & Fromang (2008) Hennebelle P., Fromang S., 2008, A&A, 477, 9
  * Hosokawa et al. (2016) Hosokawa T., Hirano S., Kuiper R., Yorke H. W., Omukai K., Yoshida N., 2016, ApJ, 824, 119
  * Hull & Zhang (2019) Hull C. L. H., Zhang Q., 2019, Frontiers in Astronomy and Space Sciences, 6, 3
  * Hull et al. (2017) Hull C. L. H., et al., 2017, ApJ, 847, 92
  * Johns-Krull (2007) Johns-Krull C. M., 2007, ApJ, 664, 975
  * Joos et al. (2013) Joos M., Hennebelle P., Ciardi A., Fromang S., 2013, A&A, 554, A17
  * Karoly et al. (2020) Karoly J., Soam A., Andersson B. G., Coudé S., Bastien P., Vaillancourt J. E., Lee C. W., 2020, ApJ, 900, 181
  * Koch et al. (2014) Koch P. M., et al., 2014, ApJ, 797, 99
  * Lavail et al. (2017) Lavail A., Kochukhov O., Hussain G. A. J., Alecian E., Herczeg G. J., Johns-Krull C., 2017, A&A, 608, A77
  * Lee & Hennebelle (2018a) Lee Y.-N., Hennebelle P., 2018a, A&A, 611, A88
  * Lee & Hennebelle (2018b) Lee Y.-N., Hennebelle P., 2018b, A&A, 611, A89
  * Lenzuni et al. (1995) Lenzuni P., Gail H.-P., Henning T., 1995, ApJ, 447, 848
  * Li et al. (2014) Li Z.-Y., Krasnopolsky R., Shang H., Zhao B., 2014, ApJ, 793, 130
  * Mac Low & Klessen (2004) Mac Low M.-M., Klessen R. S., 2004, Reviews of Modern Physics, 76, 125
  * Mac Low et al. (1998) Mac Low M.-M., Klessen R. S., Burkert A., Smith M. D., 1998, Phys. Rev. Lett., 80, 2754
  * Machida (2014) Machida M. N., 2014, ApJ, 796, L17
  * Machida & Basu (2019) Machida M. N., Basu S., 2019, ApJ, 876, 149
  * Machida et al. (2006) Machida M. N., Inutsuka S.-i., Matsumoto T., 2006, ApJ, 647, L151
  * Machida et al. (2007) Machida M. N., Inutsuka S.-i., Matsumoto T., 2007, ApJ, 670, 1198
  * Machida et al. (2018) Machida M. N., Higuchi K., Okuzumi S., 2018, MNRAS, 473, 3080
  * Marchand et al. (2018) Marchand P., Commerçon B., Chabrier G., 2018, A&A, 619, A37
  * Marchand et al. (2019) Marchand P., Tomida K., Commerçon B., Chabrier G., 2019, A&A, 631, A66
  * Masson et al. (2016) Masson J., Chabrier G., Hennebelle P., Vaytet N., Commerçon B., 2016, A&A, 587, A32
  * Maury et al. (2018) Maury A. J., et al., 2018, MNRAS, 477, 2760
  * Mellon & Li (2008) Mellon R. R., Li Z.-Y., 2008, ApJ, 681, 1356
  * Meru & Bate (2011) Meru F., Bate M. R., 2011, MNRAS, 411, L1
  * Meru & Bate (2012) Meru F., Bate M. R., 2012, MNRAS, 427, 2022
  * Mestel (1999) Mestel L., 1999, Stellar magnetism. Clarendon, Oxford
  * Mestel & Spitzer (1956) Mestel L., Spitzer Jr. L., 1956, MNRAS, 116, 503
  * Meyer et al. (2018) Meyer D. M.-A., Kuiper R., Kley W., Johnston K. G., Vorobyov E., 2018, MNRAS, 473, 3615
  * Moss (2003) Moss D., 2003, A&A, 403, 693
  * Nakano & Umebayashi (1986) Nakano T., Umebayashi T., 1986, MNRAS, 218, 663
  * Nelson (2006) Nelson A. F., 2006, MNRAS, 373, 1039
  * Pandey & Wardle (2008) Pandey B. P., Wardle M., 2008, MNRAS, 385, 2269
  * Parker (1958) Parker E. N., 1958, ApJ, 128, 664
  * Pizzolato et al. (2003) Pizzolato N., Maggio A., Micela G., Sciortino S., Ventura P., 2003, A&A, 397, 147
  * Price (2007) Price D. J., 2007, PASA, 24, 159
  * Price (2012) Price D. J., 2012, Journal of Computational Physics, 231, 759
  * Price & Bate (2007) Price D. J., Bate M. R., 2007, MNRAS, 377, 77
  * Price & Monaghan (2007) Price D. J., Monaghan J. J., 2007, MNRAS, 374, 1347
  * Price et al. (2018) Price D. J., et al., 2018, PASA, 35, e031
  * Qiu et al. (2014) Qiu K., Zhang Q., Menten K. M., Liu H. B., Tang Y.-W., Girart J. M., 2014, ApJ, 794, L18
  * Sano & Stone (2002) Sano T., Stone J. M., 2002, ApJ, 570, 314
  * Santos-Lima et al. (2012) Santos-Lima R., de Gouveia Dal Pino E. M., Lazarian A., 2012, ApJ, 747, 21
  * Santos-Lima et al. (2013) Santos-Lima R., de Gouveia Dal Pino E. M., Lazarian A., 2013, MNRAS, 429, 3371
  * Schatzman (1962) Schatzman E., 1962, Annales d’Astrophysique, 25, 18
  * Schmidt et al. (2010) Schmidt W., Kern S. A. W., Federrath C., Klessen R. S., 2010, A&A, 516, A25
  * See et al. (2015) See V., et al., 2015, MNRAS, 453, 4301
  * Seifried et al. (2012) Seifried D., Banerjee R., Pudritz R. E., Klessen R. S., 2012, MNRAS, 423, L40
  * Seifried et al. (2013) Seifried D., Banerjee R., Pudritz R. E., Klessen R. S., 2013, MNRAS, 432, 3320
  * Skumanich (1972) Skumanich A., 1972, ApJ, 171, 565
  * Sokal et al. (2020) Sokal K. R., Johns-Krull C. M., Mace G. N., Nofi L., Prato L., Lee J.-J., Jaffe D. T., 2020, ApJ, 888, 116
  * Spitzer & Tomasko (1968) Spitzer Jr. L., Tomasko M. G., 1968, ApJ, 152, 971
  * Stephens et al. (2013) Stephens I. W., et al., 2013, ApJ, 769, L15
  * Tassis & Mouschovias (2005) Tassis K., Mouschovias T. C., 2005, ApJ, 618, 783
  * Tassis & Mouschovias (2007a) Tassis K., Mouschovias T. C., 2007a, ApJ, 660, 370
  * Tassis & Mouschovias (2007b) Tassis K., Mouschovias T. C., 2007b, ApJ, 660, 388
  * Tayler (1987) Tayler R. J., 1987, MNRAS, 227, 553
  * Teixeira et al. (2020) Teixeira P. S., Scholz A., Alves J., 2020, A&A, 642, A86
  * Tomida et al. (2013) Tomida K., Tomisaka K., Matsumoto T., Hori Y., Okuzumi S., Machida M. N., Saigo K., 2013, ApJ, 763, 6
  * Tomida et al. (2015) Tomida K., Okuzumi S., Machida M. N., 2015, ApJ, 801, 117
  * Tout et al. (2004) Tout C. A., Wickramasinghe D. T., Ferrario L., 2004, MNRAS, 355, L13
  * Tricco & Price (2012) Tricco T. S., Price D. J., 2012, Journal of Computational Physics, 231, 7214
  * Tricco & Price (2013) Tricco T. S., Price D. J., 2013, MNRAS, 436, 2810
  * Tricco et al. (2016) Tricco T. S., Price D. J., Bate M. R., 2016, Journal of Computational Physics, 322, 326
  * Truelove et al. (1997) Truelove J. K., Klein R. I., McKee C. F., Holliman John H. I., Howell L. H., Greenough J. A., 1997, ApJ, 489, L179
  * Tsukamoto et al. (2015a) Tsukamoto Y., Iwasaki K., Okuzumi S., Machida M. N., Inutsuka S., 2015a, MNRAS, 452, 278
  * Tsukamoto et al. (2015b) Tsukamoto Y., Iwasaki K., Okuzumi S., Machida M. N., Inutsuka S., 2015b, ApJ, 810, L26
  * Tsukamoto et al. (2017) Tsukamoto Y., Okuzumi S., Iwasaki K., Machida M. N., Inutsuka S.-i., 2017, PASJ, 69, 95
  * Umebayashi & Nakano (1990) Umebayashi T., Nakano T., 1990, MNRAS, 243, 103
  * Vaytet et al. (2018) Vaytet N., Commerçon B., Masson J., González M., Chabrier G., 2018, A&A, 615, A5
  * Vidotto et al. (2014) Vidotto A. A., et al., 2014, MNRAS, 441, 2361
  * Wardle (2007) Wardle M., 2007, Ap&SS, 311, 35
  * Wardle & Ng (1999) Wardle M., Ng C., 1999, MNRAS, 303, 239
  * Weber & Davis (1967) Weber E. J., Davis Jr. L., 1967, ApJ, 148, 217
  * Whitehouse & Bate (2006) Whitehouse S. C., Bate M. R., 2006, MNRAS, 367, 32
  * Whitehouse et al. (2005) Whitehouse S. C., Bate M. R., Monaghan J. J., 2005, MNRAS, 364, 1367
  * Wright et al. (2011) Wright N. J., Drake J. J., Mamajek E. E., Henry G. W., 2011, ApJ, 743, 48
  * Wurster (2016) Wurster J., 2016, PASA, 33, e041
  * Wurster (2021) Wurster J., 2021, MNRAS, 501, 5873
  * Wurster & Bate (2019) Wurster J., Bate M. R., 2019, in Loren-Aguilar P., Bate M. R., Rasull H., Ali A., Wurster J., eds, Proc. SPHERIC 2019: 14th International SPHERIC Workshop. University of Exeter, United Kingdom, p. arXiv:1906.12276 (arXiv:1906.12276)
  * Wurster & Lewis (2020a) Wurster J., Lewis B. T., 2020a, MNRAS, 495, 3795
  * Wurster & Lewis (2020b) Wurster J., Lewis B. T., 2020b, MNRAS, 495, 3807
  * Wurster et al. (2014) Wurster J., Price D. J., Ayliffe B., 2014, MNRAS, 444, 1104
  * Wurster et al. (2016) Wurster J., Price D. J., Bate M. R., 2016, MNRAS, 457, 1037
  * Wurster et al. (2017) Wurster J., Bate M. R., Price D. J., Tricco T. S., 2017, in Crespo A. J. C., Gesteira M. G., Altomare C., eds, Proc. SPHERIC 2017: 12th International SPHERIC Workshop. Universidade de Vigo, Spain (arXiv:1706.07721)
  * Wurster et al. (2018a) Wurster J., Bate M. R., Price D. J., 2018a, MNRAS, 475, 1859
  * Wurster et al. (2018b) Wurster J., Bate M. R., Price D. J., 2018b, MNRAS, 476, 2063
  * Wurster et al. (2018c) Wurster J., Bate M. R., Price D. J., 2018c, MNRAS, 480, 4434
  * Wurster et al. (2018d) Wurster J., Bate M. R., Price D. J., 2018d, MNRAS, 481, 2450 (Paper I)
  * Wurster et al. (2019) Wurster J., Bate M. R., Price D. J., 2019, MNRAS, 489, 1719
  * Wurster et al. (2021) Wurster J., Bate M. R., Bonnell I. A., 2021, MNRAS, 507, 2354
  * Yamamoto et al. (2021) Yamamoto Y., Okamoto T., Saitoh T. R., 2021, MNRAS, 504, 3986
  * Yang & Johns-Krull (2011) Yang H., Johns-Krull C. M., 2011, ApJ, 729, 83
  * Yang et al. (2005) Yang H., Johns-Krull C. M., Valenti J. A., 2005, ApJ, 635, 466
  * Yusef-Zadeh et al. (2017) Yusef-Zadeh F., Wardle M., Kunneriath D., Royster M., Wootten A., Roberts D. A., 2017, ApJ, 850, L30
  * Zhao et al. (2020) Zhao B., Caselli P., Li Z.-Y., Krasnopolsky R., Shang H., Lam K. H., 2020, MNRAS, 492, 3375
  * Zhao et al. (2021) Zhao B., Caselli P., Li Z.-Y., Krasnopolsky R., Shang H., Lam K. H., 2021, MNRAS, 505, 5142


# Algebraic valuation ring extensions as limits of complete intersection
algebras

Dorin Popescu  Simion Stoilow Institute of Mathematics of the Romanian
Academy, Research unit 5, P.O. Box 1-764, Bucharest 014700, Romania,
University of Bucharest, Faculty of Mathematics and Computer Science Str.
Academiei 14, Bucharest 1, RO-010014, Romania,  Email<EMAIL_ADDRESS>
###### Abstract.

We show that an algebraic immediate valuation ring extension of characteristic
$p>0$ is a filtered union of complete intersection algebras of finite type.

Key words : immediate extensions, pseudo convergent sequences, pseudo limits,
smooth morphisms, complete intersection algebras.
2020 Mathematics Subject Classification: Primary 13F30, Secondary
13A18,13F20,13B40.

## Introduction

B. Antieau and R. Datta have recently proven a positive characteristic
analogue [1, Theorem 4.1.1] of Zariski’s theorem [17]. It says that every
perfect valuation ring of characteristic $p>0$ is a filtered union of its
smooth ${\bf F}_{p}$-subalgebras. This result is an application of [14,
Theorem 1.2.5] which relies on some results from [2]. Also E. Elmanto and M.
Hoyois proved that an absolute integrally closed valuation ring of residue
field of characteristic $p>0$ is a filtered union of its regular finitely
generated ${\bf Z}$-subalgebras (see [1, Corollary 4.2.4]). We remind that a
filtered direct limit (in other words a filtered colimit) is a limit indexed
by a small category that is filtered (see [13, 002V] or [13, 04AX]). A
filtered union is a filtered direct limit in which all objects are subobjects
of the final colimit, so that in particular all the transition arrows are
monomorphisms.

It is well known that if the fraction field extension of an immediate
extension $V\subset V^{\prime}$ is finite and $p>0$ then $V^{\prime}$ may fail
to be a filtered direct limit of smooth $V$-algebras as shows [9, Example
3.13] inspired from [6] (see also [9, Remark 6.10]). An inclusion $V\subset
V^{\prime}$ of valuation rings is an _immediate extension_ if it is local as a
map of local rings and induces isomorphisms between the value groups and the
residue fields of $V$ and $V^{\prime}$. After seeing [4, Theorem 6.2] (see
also [15], [16]) we understood that in general we should expect that
$V^{\prime}$ is a filtered union of its complete intersection $V$-subalgebras.
In the Noetherian case a morphism of rings is a filtered direct limit of
smooth algebras iff it is a regular morphism (see [7], [12]).

A complete intersection $V$-algebra essentially of finite type is a local
$V$-algebra of type $C/(P)$, where $C$ is a localization of a polynomial
$V$-algebra of finite type and $P$ is a regular sequence of elements of $C$.
Theorem 1 stated below says that $V^{\prime}$ is a filtered union of its
$V$-subalgebras of type $C/(P)$. Since $V^{\prime}$ is local it is enough to
say that $V^{\prime}$ is a filtered union of its $V$-subalgebras of type
$T_{h}/(P)$, $T$ being a polynomial $V$-algebra of finite type, $0\not=h\in T$
and $P$ is a regular sequence of elements of $T$. Clearly, $T_{h}$ is a smooth
$V$-algebra and in fact it is enough to say that $V^{\prime}$ is a filtered
union of its $V$-subalgebras of type $G/(P)$, where $G$ is a smooth
$V$-algebra of finite type and $P$ is a regular sequence of elements of $G$.
Conversely, a $V$-algebra of such type $G/(P)$ has the form $T_{h}/(P)$ for
some $T,h,P$ using [12, Theorem 2.5]. By abuse we understand by a complete
intersection  $V$-algebra of finite type a $V$-algebra of such type $G/(P)$,
or $T_{h}/(P)$ which are not assumed to be flat over $V$.

The goal of this paper is to establish the following theorem.

######  Theorem 1.

Let $V^{\prime}$ be an immediate extension of a valuation ring $V$ and
$K\subset K^{\prime}$ the fraction field extension. If $K^{\prime}/K$ is
algebraic then $V^{\prime}$ is a filtered union of its complete intersection
$V$-subalgebras of finite type.

##  1\. Algebraic immediate extensions of valuation rings

An inclusion $V\subset V^{\prime}$ of valuation rings is an _immediate
extension_ if it is local as a map of local rings and induces isomorphisms
between the value groups and the residue fields of $V$ and $V^{\prime}$.

Let $\lambda$ be a fixed limit ordinal and $v=\\{v_{i}\\}_{i<\lambda}$ a
sequence of elements in $V$ indexed by the ordinals $i$ less than $\lambda$.
Then $v$ is _pseudo convergent_ if

$\mathrm{val}(v_{i}-v_{i^{\prime\prime}})<\mathrm{val}(v_{i^{\prime}}-v_{i^{\prime\prime}})\
\ \mbox{for}\ \ i<i^{\prime}<i^{\prime\prime}<\lambda$ (see [3], [11]). A
_pseudo limit_ of $v$ is an element $w\in V$ with

$\mathrm{val}(w-v_{i})<\mathrm{val}(w-v_{i^{\prime}})\ \ \mbox{(that is,}\ \
\mathrm{val}(w-v_{i})=\mathrm{val}(v_{i}-v_{i^{\prime}}))\ \ \mbox{for}\ \
i<i^{\prime}<\lambda$. We say that $v$ is

  1. (1)

_algebraic_ if some $f\in V[T]$ satisfies
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{i^{\prime}}))$ for large enough
$i<i^{\prime}<\lambda$;

  2. (2)

_transcendental_ if each $f\in V[T]$ satisfies
$\mathrm{val}(f(v_{i}))=\mathrm{val}(f(v_{i^{\prime}}))$ for large enough
$i<i^{\prime}<\lambda$.

The following lemma is a variant of Ostrowski ([6, page 371, IV and III], see
also [11, (II,4), Lemma 8]).

######  Lemma 2.

(Ostrowski) Let $\beta_{1},\ldots,\beta_{m}$ be any elements of an ordered
abelian group $G$, $\lambda$ a limit ordinal and let
$\\{\gamma_{s}\\}_{s<\lambda}$ be a well-ordered, monotone increasing set of
elements of G, without a last element. Let $t_{1},\ldots,t_{m}$, be distinct
integers. Then there exists an ordinal $\nu<\lambda$ such that
$\beta_{i}+t_{i}\gamma_{s}$ are different for all $s>\nu$. In particular,
there exists an integer $1\leq r\leq m$ such that

$\beta_{i}+t_{i}\gamma_{s}>\beta_{r}+t_{r}\gamma_{s}$

for all $i\not=r$ and $s>\nu$.

###### Proof.

The known proof from the quoted papers gives the second statement even when
the integers $t_{i}$ are not necessarily positive. Then apply this statement
iteratively to get the first statement.

######  Lemma 3.

Let $V\subset V^{\prime}$ be an immediate extension of valuation rings,
$K\subset K^{\prime}$ its fraction field extension and $(v_{i})_{i<\lambda}$
an algebraic pseudo convergent sequence in $V$, which has a pseudo limit $x$
in $V^{\prime}$, but no pseudo limit in $K$. Set
$x_{i}=(x-v_{i})/(v_{i+1}-v_{i})$. Let $s\in{\bf N}$ be the minimal degree of
the polynomials $f\in V[Y]$ such that
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$ and
$g\in V[Y]$ a polynomial with $\operatorname{deg}g<s$. Then there exist $d\in
V\setminus\\{0\\}$ and $u\in V[x_{i}]$ for some $i<\lambda$ with $g(x)=du$ and
$\mathrm{val}(u)=0$.

###### Proof.

In the Taylor expansion111The polynomials $D^{(n)}f\in R[Y]$ for $f\in R[Y]$,
the so called Hasse-Schmidt derivatives, make sense for any ring $R$: indeed,
one constructs the Taylor expansion in the universal case $R={\bf
Z}[a_{0},\dots,a_{\operatorname{deg}f}]$ by using the equality
$n!\cdot(D^{(n)}f)=f^{(n)}$ and verifying over $\operatorname{Frac}(R)$.

$\textstyle
g(x)=\sum_{n=0}^{\operatorname{deg}g}(D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n}\quad\text{with}\quad
D^{(n)}g\in V[Y]$

the values $\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})$ are pairwise
distinct for all $n\leq\operatorname{deg}g$ and every large enough $i$ because
$\mathrm{val}(x-v_{i})=\mathrm{val}(v_{i+1}-v_{i})$ increases,
$\operatorname{deg}(D^{(n)}g)<s$ and so $\mathrm{val}((D^{(n)}g)(v_{i}))$ is
constant for all $n\leq\operatorname{deg}g$ and $i$ large (we could also apply
Lemma 2). We cannot have
$\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})<\mathrm{val}(g(v_{i}))$ for
some $n>0$ and $i$ large because otherwise we get from above that
$\mathrm{val}(g(x))=\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})$
increases, which is false. It follows that
$\mathrm{val}((D^{(n)}g)(v_{i})\cdot(x-v_{i})^{n})>\mathrm{val}(g(v_{i}))$ for
all $n>0$ and $i$ large and so

$g(x)=g(v_{i})+\sum_{n=1}^{\operatorname{deg}g}(D^{(n)}g)(v_{i})(v_{i+1}-v_{i})^{n}x_{i}^{n}\in
g(v_{i})(1+{\mathfrak{m}}^{\prime}\cap V[x_{i}]),$

which is enough.

######  Lemma 4.

Let $V\subset V^{\prime}$ be an immediate extension of valuation rings,
$K\subset K^{\prime}$ its fraction field extension and $(v_{i})_{i<\lambda}$
an algebraic pseudo convergent sequence in $K$, which has a pseudo limit $x$
in $V^{\prime}$, algebraic over $V$, but no pseudo limit in $K$. Assume that
$h=$Irr$(x,K)$ is from $V[X]$. Then

  1. (1)

$\mathrm{val}(h(v_{i}))<\mathrm{val}(h(v_{j}))$ for large $i<j<\lambda$,

  2. (2)

if $h$ has minimal degree among the polynomials $f\in V[X]$ such that
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$, then
$V^{\prime\prime}=V^{\prime}\cap K(x)$ is a filtered union of its complete
intersection $V$-subalgebras.

###### Proof.

The first part follows from [5, Corollary 5.5]. Now, assume that
$\operatorname{deg}h$ is minimal among $f$ having the property from (2). As in
Lemma 3 we consider $(x_{i})_{i<\lambda}$ and note that for a polynomial $g\in
V[X]$ of degree $<\operatorname{deg}h$ we have $g(x)=du$ for some $d\in
V\subset\\{0\\}$ and $u\in V[x_{i}]$ for some $i<\lambda$ with
$\mathrm{val}(u)=0$. An element of $V^{\prime\prime}$ has the form $g(x)/t$
for some $g\in V[X]$ with $\operatorname{deg}g<\operatorname{deg}h$ and $t\in
V^{\setminus}\\{0\\}$ such that $\mathrm{val}(g(x))\geq\mathrm{val}(t)$. It
follows that $g(x)/t=(d/t)u\in V[x_{i}]_{{\mathfrak{m}}^{\prime}\cap
V[x_{i}]}$, where ${\mathfrak{m}}^{\prime}$ is the maximal ideal of
$V^{\prime}$.

Thus $V^{\prime\prime}$ is the union of
$(V[x_{i}]_{{\mathfrak{m}}^{\prime}\cap V[x_{i}]})_{i<\lambda}$, which is
filtered increasing as in the proof of [8, Lemma 3.2], or [10, Lemma 15]. We
have $V[x_{i}]\cong V[X_{i}]/(h_{i})$, where $h_{i}$ is defined below. Set
$g_{i}=h(v_{i}+(v_{i+1}-v_{i})X_{i})\in V[X_{i}]$. By construction
$x_{i}=(x-v_{i})/(v_{i+1}-v_{i})$ and so $g_{i}(x_{i})=h(x)=0$. As $h$ is
irreducible in $K[X]$ we get $g_{i}$ irreducible in $K[X_{i}]$ too because it
is obtained from $h$ by a linear transformation. So $K[x_{i}]\cong
K[X_{i}]/(g_{i})$. We have $g_{i}=uh_{i}$ for some primitive polynomial
$h_{i}$ of $V[X_{i}]$ and a nonzero constant $u$ of $K$. Then
$V^{\prime\prime}$ is a filtered union of its complete intersections
$V$-subalgebras.

######  Remark 5.

The extension $V\subset V^{\prime\prime}$ from (2) of the above lemma is
isomorphic with the one constructed in [3, Theorem 3].

Next, we need the following elementary lemma.

######  Lemma 6.

Let $B$ be a complete intersection algebra over a ring $A$ and $C$ a complete
intersection algebra over $B$. Then $C$ is a complete intersection algebra
over $A$.

###### Proof.

Suppose that $B\cong A[X]/(f)$, where $X=(X_{1},\ldots,X_{n})$ and
$f=(f_{1},\ldots,f_{r})$ is a regular sequence of elements in $A[X]$.
Similarly, $C\cong B[Y]/(g)$, where $Y=(Y_{1},\ldots,Y_{m})$ and
$g=(g_{1},\ldots,g_{s})$ is a regular sequence of elements in $B[Y]$. Since
$A[X,Y]$ is flat over $A[X]$ we see that $f$ is a regular sequence of elements
in $A[X,Y]$. Note that $g_{i}$ is a residue class modulo $fA[X,Y]$ of a
polynomial $G_{i}\in A[X,Y]$, $1\leq i\leq s$.

We claim that $f,G$ form a regular sequence of elements of $A[X,Y]$. It is
enough to see that $G$ is a regular sequence of elements of $A[X,Y]/(f)\cong
B[Y]$, which is true by assumption. So, $C\cong A[X,Y]/(f,G)$ is a complete
intersection $A$-algebra.

######  Lemma 7.

Let $V^{\prime}$ be an immediate extension of a valuation ring $V$, $K\subset
K^{\prime}$ their fraction field extension and $K^{\prime}=K(x)$ for some
$x\in V^{\prime}$, $x\not\in K$ algebraic over $K$. Then there exists
$K_{1}\subset K^{\prime}$, $K_{1}\not=K^{\prime}$ containing $K$ such that $x$
is a pseudo limit of an algebraic pseudo-convergent sequence
$v=(v_{j})_{j<\lambda}$ in $V_{1}=V^{\prime}\cap K_{1}$ having $x$ as a pseudo
limit and for a polynomial $h\in V_{1}[Y]$ with minimal degree among the
polynomials $f\in V_{1}[Y]$ with
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$ there
exists no pseudo limit of $v$ in $K_{1}(h(x))$.

###### Proof.

By [3, Theorem 1] there exists an algebraic pseudo-convergent sequence
$v=(v_{j})_{j<\lambda}$ in $V$ having $x$ as a pseudo limit but no pseudo
limits in $K$. Let $h\in V[Y]$ be a polynomial with minimal degree among the
polynomials $f\in V[Y]$ such that
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$. Note
that $\operatorname{deg}h>1$ because $(v_{j})_{j<\lambda}$ has no pseudo
limits in $K$ (see the proof of [3, Theorem 3]) and so
$[K^{\prime}:K(h(x))]=\operatorname{deg}h>1$.

If $v$ has no pseudo limits in $K(h(x))$ then we may take $K_{1}=K$. If $v$
has a pseudo limit in $K(h(x))$ then we may complete it to a pseudo convergent
sequence ${\tilde{v}}={\tilde{v}}_{j<{\tilde{\lambda}}}$ over
${\tilde{V}}=V^{\prime}\cap K(h(x))$ with $x$ as a pseudo limit, which has no
pseudo limit in $K(h(x))$ using again [3, Theorem 1]. As above there exists a
polynomial ${\tilde{h}}\in{\tilde{V}}[Y]$ which has minimal degree among all
polynomials $f\in{\tilde{V}}[Y]$ such that
$\mathrm{val}(f({\tilde{v}}_{j}))<\mathrm{val}(f({\tilde{v}}_{j+1}))$ for
large $j<{\tilde{\lambda}}$.

For our proof it is enough to change $K$ by $K(h(x))$, $h$ by $\tilde{h}$ and
$v$ by $\tilde{v}$. If the new $v$ has a pseudo limit in the new $K(h(x))$
then we proceed as above finding a new $\tilde{v}$ and a new $\tilde{h}$ and
again we may change $K$ by $K({\tilde{h}}(x))$ and $v,h$ with
${\tilde{v}},{\tilde{h}}$. Proceeding in this way we arrive in some steps
($K^{\prime}/K$ is finite) in the case when $v$ has no pseudo limit in
$K(h(x))$. It is enough to take $K_{1}$ to be this new $K$.

######  Proposition 8.

Let $V^{\prime}$ be an immediate extension of a valuation ring $V$, $K\subset
K^{\prime}$ their fraction field extension and $K^{\prime}=K(x)$ for some
$x\in V^{\prime}$ algebraic over $K$. Then $V^{\prime}$ is a filtered union of
its complete intersection $V$-subalgebras.

###### Proof.

We may assume $K\not=K^{\prime}$. By Lemma 7 there exists $K_{1}\subset
K^{\prime}$, $K_{1}\not=K^{\prime}$ containing $K$ such that $x$ is a pseudo
limit of an algebraic pseudo-convergent sequence $v=(v_{j})_{j<\lambda}$ in
$V_{1}=V^{\prime}\cap K_{1}$ and for a polynomial $h\in V_{1}[Y]$ with minimal
degree among the polynomials $f\in V_{1}[Y]$ with
$\mathrm{val}(f(v_{i}))<\mathrm{val}(f(v_{j}))$ for large $i<j<\lambda$ there
exists no pseudo limit of $v$ in $K_{1}(h(x))$. Set $g=$Irr$(x,K_{1}(h(x)))$.
By Lemma 4 (1) we have $\operatorname{deg}h\leq\operatorname{deg}g$. But $x$
is a root of $g^{\prime}=h(Y)-h(x)\in K_{1}(h(x))[Y]$ and so $g$ divides
$g^{\prime}$. Thus $g$ is $g^{\prime}$ multiplied with a unit of
$V^{\prime\prime}=V^{\prime}\cap K_{1}(h(x))$ because
$\operatorname{deg}g^{\prime}=\operatorname{deg}h\leq\operatorname{deg}g\leq\operatorname{deg}g^{\prime}$.

Applying Lemma 4 (2), we see that $V^{\prime}$ is a filtered union of its
complete intersection $V^{\prime\prime}$-subalgebras. Use induction on
$[K^{\prime}:K]$, case $K=K^{\prime}$ being trivial. By induction hypothesis
$V^{\prime\prime}$ is a filtered union of its complete intersection
$V$-subalgebras. Therefore, $V^{\prime}$ is also a filtered union of its
complete intersection $V$-subalgebras using Lemma 6.

Proof of Theorem 1.

Firstly assume $K^{\prime}/K$ is finite, let us say
$K^{\prime}=K(x_{1},\ldots,x_{n})$. Set $V_{i}=V^{\prime}\cap
K(x_{1},\ldots,x_{i})$, $1\leq i<n$ and $V_{0}=V$, $V_{n}=V^{\prime}$. Then
$V_{i+1}$ is a filtered union of its complete intersection $V_{i}$-algebras
for all $0\leq i<n$ by Proposition 8 and Lemma 6, which is enough. In general,
express $K^{\prime}$ as a filtered union of some subfields $(K_{i})_{i}$ of
$K^{\prime}$ which are finite extensions of $K$. Then $V^{\prime}$ is a
filtered increasing union of $V^{\prime}\cap K_{i}$.

######  Remark 9.

If $V$ is a valuation ring of characteristic $\not=0$ then an algebraic
separable valuation ring extension of it may fail to be a filtered direct
limit of smooth $V$-algebras, for instance by [9, Example 3.13, Theorem 6.9,
Remark 6.10] (see [6, Sect 9, No 57]); but it is a filtered union of its
complete intersection $V$-subalgebras as in Theorem 1.

##  2\.  Conflict of Interests/Competing Interests are missing

We have no conflict of interests.

## References

  * [1] B. Antieau and R. Datta, Valuation rings are derived splinters, Math. Zeitschrift, 299, (2021, 827-851, arxiv/AG:2002.010627v1.
  * [2] A. J. de Jong, Smoothness, semi-stability and alterations, Inst. Hautes Études Sci. Publ. Math. (1996), no. 83, 51–93.
  * [3] I. Kaplansky, Maximal fields with valuations, Duke Math. J. 9 (1942), 303-321.
  * [4] K. Kato, V. Thatte, Upper Ramification Groups for Arbitrary Valuation Rings, arXiv/AC:1909.09832v1.
  * [5] F. V. Kuhlmann, I. Vlahu The relative approximation degree in valued function fields, Math. Z., 276, (2014), 203–235.
  * [6] A. Ostrowski, Untersuchungen zur arithmetischen Theorie der Körper, Math. Z. 39 (1935), no. 1, 321-404.
  * [7] D. Popescu, General Neron Desingularization and approximation, Nagoya Math. J., 104, (1986), 85-115.
  * [8] D. Popescu, On Zariski’s uniformization theorem, in Algebraic geometry, Bucharest 1982 (Bucharest, 1982), Lecture Notes in Math., vol. 1056, Springer, Berlin, 1984, 264-296.
  * [9] D. Popescu, Algebraic extensions of valued fields, J. Algebra 108, (1987), no. 2, 513-533.
  * [10] D. Popescu, Néron desingularization of extensions of valuation rings with an Appendix by Kęstutis Česnavičius, in "Transcendence in Algebra, Combinatorics, Geometry and Number Theory", Eds. Alin Bostan, Kilian Raschel, Springer Proceedings in Mathematics and Statistics 373, 275-307, arxiv/AC:1910.09123v4.
  * [11] O. F. G. Schilling, The theory of valuations, Mathematical Surveys, Number IV, American Math. Soc., (1950).
  * [12] R. Swan, Neron-Popescu desingularization, in "Algebra and Geometry", Ed. M. Kang, International Press, Cambridge, (1998), 135-192.
  * [13] A. J. de Jong et al., The Stacks Project. Available at
http://stacks.math.columbia.edu.

  * [14] M. Temkin, Tame distillation and desingularization by p-alterations, Ann. of Math. (2) 186 (2017), no. 1, 97–126.
  * [15] V. Thatte, Ramification theory for Artin–Schreier extensions of valuation rings, J. Algebra, 456, (2016), 355-389.
  * [16] V. Thatte, Ramification theory for degree p extensions of arbitrary valuation rings in mixed characteristic $(0,p)$, J. Algebra, 507, (2018), 225-248.
  * [17] O. Zariski, Local uniformization on algebraic varieties, Ann. of Math. 41 (1940), 852-896,