# DKIST unveils the serpentine topology of quiet Sun magnetism in the
photosphere

Ryan J. Campbell Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University, Belfast, BT7 1NN, UK  P. H. Keys Astrophysics
Research Centre, School of Mathematics and Physics, Queen’s University,
Belfast, BT7 1NN, UK  M. Mathioudakis Astrophysics Research Centre, School of
Mathematics and Physics, Queen’s University, Belfast, BT7 1NN, UK  F. Wöger
National Solar Observatory, 3665 Discovery Drive, Boulder, CO 80303, USA  T.
Schad National Solar Observatory, 22 ´Ōhi´a Kū Street, Pukalani, HI, 96768,
USA  A. Tritschler National Solar Observatory, 3665 Discovery Drive, Boulder,
CO 80303, USA  A. G. de Wijn High Altitude Observatory, National Center for
Atmospheric Research, P.O. Box 3000, Boulder, CO 80307-3000, USA  H. N. Smitha
Max-Planck-Institut für Sonnensystemforschung, Justus-von-Liebig-Weg 3, 37077
Göttingen, Germany  C. Beck National Solar Observatory, 22 ´Ōhi´a Kū Street,
Pukalani, HI, 96768, USA  D. J. Christian Department of Physics and Astronomy,
California State University Northridge, 18111 Nordhoff Street, Northridge, CA
91330, USA  D. B. Jess Astrophysics Research Centre, School of Mathematics and
Physics, Queen’s University, Belfast, BT7 1NN, UK  Department of Physics and
Astronomy, California State University Northridge, 18111 Nordhoff Street,
Northridge, CA 91330, USA  R. Erdélyi Solar Physics and Space Plasma Research
Centre, School of Mathematics and Statistics, University of Sheffield,
Sheffield S3 7RH, UK  Department of Astronomy, Eötvös Loránd University,
Pázmány Péter sétány 1/A, H-1117 Budapest, Hungary

###### Abstract

We present the first quiet Sun spectropolarimetric observations obtained with
the Visible SpectroPolarimeter (ViSP) at the $4-$m Daniel K. Inouye Solar
Telescope (DKIST). We recorded observations in a wavelength range that
includes the magnetically sensitive Fe I $6301.5/6302.5$ $\mathrm{\AA}$
doublet. With an estimated spatial resolution of $0{\,}.{\\!\\!}{\arcsec}08$,
this represents the highest spatial resolution full-vector spectropolarimetric
observations ever obtained of the quiet Sun. We identified $53$ small-scale
magnetic elements, including $47$ magnetic loops and $4$ unipolar magnetic
patches, with linear and circular polarisation detected in all of them. Of
particular interest is a magnetic element in which the polarity of the
magnetic vector appears to change three times in only $400$ km and which has
linear polarisation signals throughout. We find complex Stokes $V$ profiles at
the polarity inversion lines of magnetic loops and discover degenerate
solutions, as we are unable to conclusively determine whether these arise due
to gradients in the atmospheric parameters or smearing of opposite polarity
signals. We analyse a granule which notably has linear and circular
polarisation signals throughout, providing an opportunity to explore its
magnetic properties. On this small scale we see the magnetic field strength
range from $25$ G at the granular boundary to $2$ kG in the intergranular lane
(IGL), and sanity check the values with the weak and strong field
approximations. A value of $2$ kG in the IGL is among the highest measurements
ever recorded for the internetwork.

Sun: photosphere — Sun: magnetic fields — Sun: granulation

††facilities: DKIST (Rimmele et al., 2020).††software: SIR explorer (Campbell,
2023), Astropy (Astropy Collaboration et al., 2022), Matplotlib (Hunter,
2007), Numpy (Harris et al., 2020).

##  1 Introduction

The discovery of transient, transverse magnetism in the quiet Sun by Lites et
al. (1996) provided observational evidence suggesting that the “magnetic
carpet” (Title & Schrijver, 1998) may not be predominantly longitudinal in
nature. The short-lived linear polarisation, observed with the Zeeman effect,
was found to be located at the edge of granules, between opposite polarity
circular polarisation signals, but in the absence of any accompanying
measurable Stokes $V$ signal at the polarity inversion line (PIL). Later,
Lites et al. (2008) used the spatial resolution afforded by the Hinode
spacecraft and the magnetically sensitive $6302.5$ $\mathrm{\AA}$ line to
unveil the spatially averaged transverse magnetic flux density ($55$ Mx cm-2)
as five times greater than the longitudinal equivalent ($11$ Mx cm-2).
However, circular polarisation remains much more prevalent than linear
polarisation in this data. Long integrations with the
Hinode/SpectroPolarimeter (SP) in sit-and-stare mode have revealed linear
polarisation in as much as half of the field of view (FOV), but with
compromised spatiotemporal resolution due to an integration time of $6.1$
minutes (Bellot Rubio & Orozco Suárez, 2012). Whether the vector magnetic
field is predominantly horizontal (transverse) or vertical (longitudinal) in
the solar photosphere is still debated, with the impact of photon noise on the
retrieval of the magnetic inclination angle a source of controversy (Borrero &
Kobel, 2011; Danilovic et al., 2016). Therefore, geometric approaches were
also sought to circumvent this problem (e.g., Stenflo, 2010, 2013; Jafarzadeh
et al., 2014; Lites et al., 2017, to name but a few). The reader is directed
to Bellot Rubio & Orozco Suárez (2019) for a review.

The conversion and transport of quiet Sun magnetic energy in the photosphere
to kinetic energy in the chromosphere and corona could have an important role
to play in explaining the high temperature of the upper solar atmosphere
(Schrijver et al., 1998; Trujillo Bueno et al., 2004). Statistical analyses of
this phenomenon with the Zeeman effect suggest that magnetic bi-poles do not
appear uniformly on the solar surface and there are regions of the very quiet
Sun where no bi-poles seem to emerge (Martínez González et al., 2012).
Martínez González & Bellot Rubio (2009) showed that $23\%$ of magnetic loops
studied reached the chromosphere, perhaps driven by convective up-flows and
magnetic buoyancy (Steiner et al., 2008), on a timescale of $8$ minutes. Gošić
et al. (2021) demonstrated that small magnetic bi-poles that emerge in the
photosphere with field strengths between $400$ G and $850$ G can reach the
chromosphere, but on a much longer timescale of up to $1$ hour. Wiegelmann et
al. (2013) extrapolated photospheric magnetic field line measurements into the
chromosphere and corona, and concluded that the energy released by magnetic
reconnection could not fully account for the heating of these layers. However,
the $110-130$ km resolution of their observations means this analysis does not
account for small-scale braiding of magnetic field lines and their potential
to drive heating via reconnection and dissipation, a mechanism which could
explain the relative weakness of the small-scale magnetic field in the solar
photosphere (Parker, 1972).

Near infrared spectropolarimetric observations have a demonstrated ability to
measure linear polarisation more effectively compared to photospheric
diagnostics in the visible (Martínez González et al., 2008a; Lagg et al.,
2016; Campbell et al., 2021a). Recently, Campbell et al. (2023) revealed an
internetwork region with a relatively large fraction of linear and circular
polarisation ($23\%$ versus $60\%$) such that a majority of magnetised pixels
displayed a clear transverse component of the magnetic field, despite having
set Stokes profiles with signal $<5\sigma_{n}$ to zero before the inversion,
where $\sigma_{n}$ is the noise level as determined by the standard deviation
in the continuum. Observations of sub-granular, small-scale magnetic loops,
with lifetimes of a few minutes and sizes of $1-2\arcsec$, with unambiguous
transverse components at the PIL, were made possible with use of an integral
field unit such that the spatiotemporal resolution was not compromised.

Simultaneous observations with visible and near infrared Zeeman diagnostics
have also been demonstrated to produce conflicting results, due to a different
in formation heights or magnetic substructure (Socas-Navarro & Sánchez
Almeida, 2003; Martínez González et al., 2008b). Indeed, Martínez González et
al. (2006) have shown that the different height of formation of the
$6301.5/6302.5$ $\mathrm{\AA}$ lines, and their sensitivities to temperature
and magnetic field strengths, causes degenerate solutions from inversions of
the Stokes vector or difficulty with the line ratio technique. This is
predicted to persist even at the $20$ km resolution of numerical simulations
(Khomenko & Collados, 2007).

The unprecedented spatial resolution made possible by the 4-m Daniel K. Inouye
Solar Telescope (DKIST; Rimmele et al., 2020), combined with the high
polarimetric sensitivity and spectral resolution of the Visible
SpectroPolarimeter (ViSP; de Wijn et al., 2022), affords the opportunity to
observe the fine-structure of the magnetic flux elements in the photosphere.
In this study, we investigate the spectropolarimetric properties of small-
scale magnetic features of the solar internetwork, as observed by DKIST, with
inversions utilised to recover the physical properties from the observed
Stokes vector. Studying the small-scale photospheric magnetic field is one of
the key research aims of the DKIST Critical Science Plan (Rast et al., 2021).
In Section 2 we describe the dataset, its reduction and provide an estimate of
the spatial resolution. In Section 3 we present the analysis of the data,
beginning with a description of the inversions. We define three case studies
and examine the spectropolarimetric properties and spatial structure of the
small-scale magnetic features in detail. Finally we discuss the results in
Section 4 before drawing our conclusions in Section 5, with a look to future
observations.

##  2 Observations & Data Reduction

###  2.1 Observations

On 26 May 2022 between 17:45 and 19:31 UT observations of a quiet Sun region
were acquired with the ViSP at the DKIST during its first cycle of
observations from the operations and commissioning phase (OCP1). The observing
sequence was designed to obtain narrow, repeated scans of the solar surface at
disk centre. The selected slit width was $0{\,}.{\\!\\!}{\arcsec}041$ with a
slit step of $0{\,}.{\\!\\!}{\arcsec}041$ and $104$ slit positions, giving a
map of $4{\,}.{\\!\\!}{\arcsec}26\ \times 76{\,}.{\\!\\!}{\arcsec}14$. The
cadence between the commencement of a given frame to the start of the next was
$13$ minutes and $17$ seconds, and $8$ frames were recorded in total. The
first arm of the ViSP captured a spectral region which includes the
magnetically sensitive photospheric Fe I line pair at $6301.5$ $\mathrm{\AA}$
and $6302.5$ $\mathrm{\AA}$ (with effective Landé g-factors of $1.67$ and
$2.5$, respectively), while the third arm captured a spectral region which
includes the chromospheric Ca II $8542$ $\mathrm{\AA}$ line. The second arm of
the ViSP did not record observations. In this paper we focus our analysis on
the Fe I doublet. The spatial sampling along the slit is
$0{\,}.{\\!\\!}{\arcsec}0298$/pixel for the arm which recorded the Fe I $6300$
$\mathrm{\AA}$ spectral region. At each slit step, 24 modulation cycles of 10
modulation states were acquired. The exposure time was $6$ ms, yielding a
corresponding $1.44$ seconds total integration time per step. The seeing
conditions during the scan were very good, with the adaptive optics (AO)
locked during $99.2\%$ of the scan step positions and a mean continuum
intensity contrast of $6.2\%$. However, the maximum continuum intensity
contrast is closer to $9\%$. The mean Fried parameter reported by the DKIST
wavefront correction system was $12$ cm during these observations. The mean
standard deviation across all pixels (except those which did not have the AO
locked) at a continuum wavelength in Stokes $Q$, $U$, and $V$ in the spectral
region adjacent to the Fe I $6302.5$ $\mathrm{\AA}$ line is $7.5\times
10^{-4}$ $I_{\mathrm{c}}$, $7.4\times 10^{-4}$ $I_{\mathrm{c}}$, and
$7.5\times 10^{-4}$ $I_{\mathrm{c}}$, respectively. The linear dispersion in
the $6300$ $\mathrm{\AA}$ region is $12.85$ $\mathrm{m\AA}$/pixel and the
spectral resolution is twice this value.

Figure 1: Summed power spectrum of the intensity, computed at a wavelength of
$6302.47$ $\mathrm{\AA}$ for each step, summed across step positions with
greater than the mean continuum intensity contrast of $6.2\%$. The dotted,
blue line shows the summed power spectrum windowed by a Hann filter before
computing the FFT. The solid, red line shows a median filter applied to the
Hann windowed power spectrum. The dashed, red vertical line indicates the
frequency that equates to a spatial resolution of $0{\,}.{\\!\\!}{\arcsec}08$.
Figure 2: Three case studies of small-scale magnetism, including a serpentine-
like magnetic configuration (upper row), two magnetic loops (middle rows), and
a magnetic granule (bottom row). From left to right is Stokes I at $6303.17$
$\mathrm{\AA}$, $L_{\mathrm{tot}}$, Stokes $V$ at $6302.45$ $\mathrm{\AA}$,
$v_{\mathrm{LOS}}$, $\gamma$, $\phi$, and $\alpha_{m}B$. The last four
parameters are derived from SIR inversions with one magnetic and one non-
magnetic model atmosphere (1C configuration). The scanning direction (i.e.
solar $X$) is shown on the $x$-axis and the slit direction (i.e. solar $Y$) is
shown on the $y$-axis. The labelled (A.1, B.1-B.2, C.1-C.4) markers highlight
the spatial location of the sample pixels: the full Stokes vector from pixel
A.1 is shown in Fig. 3, B.1 is shown in Fig. 5, B.2 is shown in Fig. 6, C.1 is
shown in Fig. 7, C.2 is shown in Fig. 8, and the circular polarisation
profiles for C.3 and C.4 are shown in Fig. 9. The region outlined by the cyan
box in the upper row is shown in Fig. 4. Figure 3: Sample observed Stokes
vector with the best-fit inverted profiles (left panels) and model associated
atmospheres (right panels), whose spatial location is labelled in the
serpentine magnetic element in the upper row of Fig. 2 (pixel A.1). The
horizontal (dot-dashed) lines show the noise thresholds ($\pm 4\sigma_{n}$),
while the vertical (dotted) lines indicate the rest wavelengths of the Fe I
doublet. The retrieved atmospheric parameters are shown as a function of
optical depth. The $\phi$, $\alpha_{m}$, and $v_{\mathrm{mac}}$ values were
$107^{\circ}$, $0.47$, and $0.015$ km sec-1, respectively. This inversion is a
1C configuration. Figure 4: Maps of $\gamma$ (upper left) and $\phi$ (upper
right) in the serpentine structure shown in the first case study of Fig. 2.
The $\gamma$ map is shown at $\mathrm{log}(\tau_{5000\mathrm{\AA}})=-1.0$. The
direction of the $\phi$ is marked with arrows whose value is given after
binning by a $3\times 3$ kernel for visual clarity. The azimuth is defined
such that a value of 0 is aligned along the line from Solar East to West,
increasing anti-clockwise. The variation in $\gamma$ and $\phi$ is also shown
across slices labelled $1$ (lower left) and $2$ (lower right), whose initial
and final locations are indicated by the blue, dashed lines. The dotted lines
shows the variation in $\gamma$ at optical depths between
$\mathrm{log}(\tau_{5000\mathrm{\AA}})=-2.0$ and
$\mathrm{log}(\tau_{5000\mathrm{\AA}})=0.0$, while the solid, red lines show
the variation in $\phi$, across the slices. The dashed, gray horizontal lines
mark the $90^{\circ}$ polarity inversion point for $\gamma$.

###  2.2 Data reduction

The ViSP calibration and reduction pipeline was applied to the data for dark
current removal, flat fielding, and polarimetric calibration 111Version 2.0.2
of the ViSP level 0 to level 1 pipeline was used.. The polarisation amplitudes
in the continuum should be negligible. Stokes $Q$, $U$, or $V$ signal at
continuum wavelengths at disk centre can be generated by cross-talk
(henceforth referred to as environmental polarisation) with Stokes $I$ when
the atmosphere is not “frozen” during the modulation process (Collados, 1999).
We followed the method of Sanchez Almeida & Lites (1992) to remove
environmental polarisation (Stokes $I\rightarrow Q,U,V$ cross-talk only). The
two $\mathrm{O}_{2}$ telluric lines close to the Fe I $6300$ $\mathrm{\AA}$
doublet provide a method for validating that this correction is appropriate,
as there should be no residual signal in $Q$, $U$, and $V$ at the wavelengths
of these tellurics if the correction is successful. The mean correction
amplitude across all pixels in Stokes $Q$, $U$, and $V$, as determined in the
continuum adjacent to the Fe I $6302.5$ $\mathrm{\AA}$ line, was $2.8\times
10^{-3}$ $I_{\mathrm{c}}$, $1.8\times 10^{-3}$ $I_{\mathrm{c}}$, and
$4.3\times 10^{-4}$ $I_{\mathrm{c}}$, respectively. There remains some small-
amplitude artefacts and residual cross-talk in the data, specifically the
continuum does not always have net-zero polarisation across the full spectrum.
In order to correct this, a simple linear least-squares regression was
performed individually to every Stokes $Q$, $U$, and $V$ profile, with the
wavelengths of the spectral and telluric lines masked from the function. The
linear fit is then subtracted from each profile, resulting in a continuum that
is consistently zero at every wavelength, with standard deviations due to
photon noise permitted.

###  2.3 Estimating the spatial resolution

To estimate the maximal spatial resolution in the data, we chose a spectral
point in the blue wing of the line at $6302.47$ $\mathrm{\AA}$. For this
wavelength, the contrast is increased relative to the continuum. We selected
those slit positions for which the contrast is above the mean value of
approximately $6.2\%$, and computed the sum of their one-dimensional spatial
power at each spatial frequency point. We also applied a Hann filter to each
slice before computing the one-dimensional fast fourier transform (FFT), to
ensure we do not introduce high spatial frequencies at the edges of each
slice. The result is shown in Fig. 1. Assuming that the noise in the data is
additive white Gaussian noise, the frequency at which the spatial power levels
can be used as an upper estimate for the noise floor at all spatial frequency
points. Increased power above this level clearly indicates the presence of
real signal in the data at that spatial frequency.

We estimate the maximal spatial resolution as $0{\,}.{\\!\\!}{\arcsec}08$ for
this scan. There is significant power at the corresponding spatial frequency
even when the Hann window is applied. We also point out that the effective
spatial resolution is expected to vary across the scan, perhaps significantly,
as a function of the quality of the atmospheric seeing conditions and the
performance of the DKIST adaptive optics correcting it. From an inspection of
the smallest structures visible in Stokes $I$, we estimate a spatial
resolution of at least $0{\,}.{\\!\\!}{\arcsec}1$ (about $72$ km).

##  3 Data analysis $\&$ Results

###  3.1 Inversion strategy

We employed the Stokes Inversion based on Response Functions (SIR) code (Ruiz
Cobo & del Toro Iniesta, 1992) with its Python-based parallelized wrapper
(Gafeira et al., 2021) to invert the full Stokes vector in the Fe I line pair
for every pixel in the dataset. The first inversion setup we describe here
allows us to produce approximate maps of the full dataset. To achieve this we
used a two-component inversion, with one magnetic and one non-magnetic model
atmosphere per pixel element, with the contribution of each given by their
respective filling factors, $\alpha$. We inverted every pixel $30$ times, with
randomised initial values in the magnetic field strength, $B$, line-of-sight
(LOS) velocity, $v_{\mathrm{LOS}}$, magnetic inclination, $\gamma$, and
magnetic azimuth, $\phi$. The $\gamma$ is defined as the angle between the
magnetic vector and the observer’s LOS (or solar normal, at disk centre),
while the $\phi$ is the angle of the magnetic field vector in the plane
perpendicular to the observer’s LOS (i.e. in the plane of the solar surface).
For each pixel we selected the solution which had the minimum $\chi^{2}$.
Apart from temperature, $T$, which had four nodes, all other parameters,
including $B$, had only one node, and thus were forced to be constant in
optical depth. In this case, where only one component is magnetic, we
explicitly refer to the filling factor of the magnetic component,
$\alpha_{m}$, and the magnetic flux density, $\alpha_{m}B$. The $T$ in both
components was always forced to be the same assuming lateral radiative
equilibrium. The microturbulent velocity, $v_{\mathrm{mic}}$ was included as a
free parameter independently in each component, and although the
macroturbulent velocity, $v_{\mathrm{mac}}$, was also included as a free
parameter, it was forced to be the same in both components, assuming it to
primarily account for the spectral resolution of the spectrograph.

We do not explicitly include an unpolarised stray light component to the
inversions. Instead, we adopt an inversion with two model atmospheres per
pixel element, allowing the filling factor of a non-magnetic model to
encapsulate the effect of an unpolarised stray light contribution. Unlike with
GREGOR, we do not have an estimate of the unpolarised stray light fraction
(Borrero et al., 2016). Nevertheless, Campbell et al. (2021b) investigated the
statistical impact a varying unpolarised stray light fraction has on the
retrieval of $T$, $B$, $v_{\mathrm{LOS}}$, and $\phi$ values using synthetic
observations and found only the $T$ contrast was impacted, with the other
parameters invariant.

In select cases the inversion will require additional free parameters in order
to fit Stokes profiles with asymmetries or abnormal or complex shapes (e.g. a
Stokes $V$ profile which does not only have one positive and one negative lobe
of equal amplitude). Further, the number of free parameters required may
increase significantly when two magnetic components are required to reproduce
the observed Stokes vector. We approached these cases with a principle of
minimising the number of free parameters as much as possible. In order to
reproduce a Stokes vector that is not adequately fit by the initial inversion,
we enabled SIR to choose the optimum number of nodes in $B$,
$v_{\mathrm{LOS}}$, and $\gamma$ up to a maximum of 10 and up to 3 in $\phi$.
We then experimented by systematically reducing the number of nodes in each of
these parameters until SIR was unable to fit the profiles with a minimum
number of free parameters. In these pixel-specific inversions, we repeated the
inversion up to $1000$ times with randomised initial model atmospheres. For
clarity, in later sections, we label inversions with two magnetic model
atmospheres as 2-component (2C) inversions. Inversions with one magnetic and
one non-magnetic model atmosphere are labelled as 1-component (1C) inversions.
In 2C inversions the non-magnetic model atmosphere is replaced by a magnetic
one.

Although in forthcoming sections we show the atmospheric parameters at a range
of optical depths, we stress that the Fe I $6300$ $\mathrm{\AA}$ doublet is
only expected to be sensitive to perturbations in $\gamma$, $\phi$, $B$, and
$v_{\mathrm{LOS}}$ in the range between
$\mathrm{log}(\tau_{5000\mathrm{\AA}})=-2.0$ and
$\mathrm{log}(\tau_{5000\mathrm{\AA}})=0.0$ in a significant way. A detailed
analysis of the diagnostic potential of this line, in the context of other
photospheric Fe I lines and their response functions, is provided by Cabrera
Solana et al. (2005) and Quintero Noda et al. (2021).

###  3.2 Case studies

An inspection of the scans revealed an abundance of circular polarisation of
both positive and negative polarities, as well as small patches of linear
polarisation. It is also clear that there are some particularly strong, large
patches of circular polarisation with much weaker magnetic flux patches in
their immediate vicinity. In the current work we focus on the weaker, smaller-
scale magnetic structures, and curate a few case studies. We used the SIR
Explorer (SIRE) tool (Campbell, 2023; Campbell et al., 2023) to locate and
analyse these case studies in the full-scan inversions. We define a magnetic
loop as two patches of opposite polarity circular polarisation in close
contact (i.e. within only a few pixels of each other) with linear polarisation
connecting them. This definition indicates that we have two vertical fields
(identified by the two opposite polarity circular polarisation signals) which
are connected with a horizontal field (identified by the linear polarisation
signals) to complete the loop. These structures could be $\Omega-$ or
$U-$shaped loops. In the absence of linear polarisation signals, we regard the
structure as a bi-pole. In addition to magnetic loops, we also identify
unipolar magnetic flux patches with linear polarisation signals but only one
polarity of circular polarisation. Finally we define a serpentine structure as
one with more than one PIL and clear linear polarisation signals at each PIL.
We further classify serpentine magnetic configurations by the number of PILs
they have.

In total we identified $53$ small-scale magnetic elements of interest,
including at least $47$ magnetic loops and $4$ unipolar magnetic patches, with
linear and circular polarisation detected in all of them. We located several
candidate serpentine structures but by examining the azimuth we were able to
discern that they were in fact two magnetic loops in close contact due to a
sharp discontinuity in azimuth values. In two of the remaining serpentine-like
magnetic elements, the polarity of the magnetic field appeared to change three
times across the structures, and in one changed only twice. However only one
serpentine structure had linear polarisation throughout and at all three PILs.
In the following sections we define three case studies from this analysis. We
unveil the potential detection of a serpentine magnetic structure in section
3.2.1, discuss the problem of degenerate solutions at the PILs of magnetic
loops in section 3.2.2, and analyse a magnetic granule in section 3.2.3.
Overview maps of these case studies are shown in Fig. 2. The upper row shows
the serpentine magnetic structure, the two middle rows show two magnetic loops
with transverse magnetism at the PIL, and the bottom row shows the magnetic
granule. Defined here is the total linear polarisation,

${L_{\mathrm{tot}}}=\frac{\int_{\lambda_{b}}^{\lambda_{r}}[Q^{2}(\lambda)+U^{2}(\lambda)]^{\frac{1}{2}}d\lambda}{I_{c}\int_{\lambda_{b}}^{\lambda_{r}}d\lambda},$
(1)

where $\lambda_{r}$ and $\lambda_{b}$ are the red ($6302.7$ $\mathrm{\AA}$)
and blue ($6302.3$ $\mathrm{\AA}$) limits of integration in wavelength across
the $6302.5$ $\mathrm{\AA}$ line, while $I_{c}$ is the Stokes $I$ signal
measured in the neighbouring continuum and spatially averaged along the slit
direction for each scan step. Here we also show maps of the azimuth, which we
have post-processed to force the values to vary between $0^{\circ}$ and
$180^{\circ}$. We stress that we have not disambiguated the data.

Figure 5: Two solutions to the Stokes vector belonging to pixel B.1 based on
two magnetic models (2C) and one magnetic model (1C). The Stokes $V$ signal
has a complex shape. In the left column the red, solid lines show the
synthetic Stokes vector for 2C, and the blue, dotted lines show the same for
1C. The model parameters for 2C and 1C are shown in the middle column and
right column, respectively. The location of this pixel is indicated in Fig. 2.
For 2C, the $\alpha$ values of model 1 and 2 were $0.93$ and $0.07$,
respectively, the $\phi$ values were $27^{\circ}$ and $191^{\circ}$,
respectively, and the $v_{\mathrm{mac}}$ was $0.77$ km sec-1 for both models.
For 1C, the $\alpha$ values of model 1 and 2 were $0.17$ and $0.83$,
respectively, the $\phi$ value was $22^{\circ}$, and the $v_{\mathrm{mac}}$
was $0.94$ km sec-1 for both models. Figure 6: As in Fig. 5, but for the
Stokes vector belonging to pixel B.2. For 2C, the $\alpha$ values of model 1
and 2 were $0.83$ and $0.17$, respectively, the $\phi$ values were
$158^{\circ}$ and $23^{\circ}$, respectively, and the $v_{\mathrm{mac}}$ was
$0.71$ km sec-1 for both models. For 1C, the $\alpha$ values of model 1 and 2
were $0.93$ and $0.07$, respectively, the $\gamma$ and $\phi$ values for model
1 were $90^{\circ}$ and $203^{\circ}$, respectively and the $v_{\mathrm{mac}}$
was $0.94$ km sec-1 for both models. There is no significant Stokes $V$
signal.

####  3.2.1 Case study I: Small-scale serpentine magnetism

The first case study is shown in the upper row of Fig. 2. Upon initial
examination at a wavelength of $6302.39$ $\mathrm{\AA}$, this appeared to be a
magnetic loop, where two patches of circular polarisation are in close contact
and linear polarisation bridges the two patches. Upon closer inspection, a
small adjustment in the wavelength position in SIRE reveals a circular
polarisation pattern that is more complex than a single magnetic loop. When
inspected at a wavelength of $6302.45$ $\mathrm{\AA}$, closer to the rest
wavelength of the line, as shown in Fig. 2, the complexity of the magnetic
topology in this structure became apparent.

If one examines the $\gamma$ map, it is clear that the polarity of the
magnetic field changes sign at least three times across the structure (as
opposed to once, as would be expected in a simple U- or $\Omega$-shaped loop).
When the pixels in this structure have reasonably symmetrical, two-lobed
Stokes $V$ profiles, and linear polarisation is abundant, the change in
$\gamma$ is unambiguous and SIR produces good fits to the observed Stokes
vectors. However, there is a narrow location at which complex Stokes $V$
profiles are found and the initial inversion is unable to fit these profiles.
A sample Stokes vector from this location, at the edge of the granule, whose
spatial location is highlighted by the marker in Fig. 2, is shown in Fig. 3.
This Stokes vector has all three polarisation parameters with signal greater
than the $4\sigma_{n}$ level, but the Stokes $V$ profile has two negative
lobes. We therefore increased the number of free parameters permitted in $B$,
$\gamma$, and $v_{\mathrm{LOS}}$ to $10$ each, and in $\phi$ to $3$, and
repeated the inversion with $200$ randomised initialisations per pixel, and
systemically reduced the number of nodes in each parameter. We found that the
best fits could be achieved with $1$ node in $B$, but $3$ each in $\gamma$ and
$v_{\mathrm{LOS}}$ for the magnetic component. For the non-magnetic component
we found only $2$ nodes in $v_{\mathrm{LOS}}$ were required. The synthetic
vector shown in Fig. 3 resulted from this process. We then ran the inversion
with this configuration in a small area, shown in Fig. 4, so we could
investigate how $\gamma$ changed as a function of optical depth.

Shown in Fig. 4 is the variation in $\gamma$ across two
$1{\,}.{\\!\\!}{\arcsec}22$ ($882$ km) slices generated by cubic interpolation
between two points, as a function of optical depth between
$\mathrm{log}(\tau_{5000\mathrm{\AA}})=-2.0$ and
$\mathrm{log}(\tau_{5000\mathrm{\AA}})=0.0$, where the spectral line is
expected to be responsive to changes in $\gamma$ according to the response
functions. Regardless of the optical depth considered, the polarity of the
magnetic field changes three times across the slices. However, the spatial
position at which this polarity inversion occurs can depend on the optical
depth. We emphasise that this is occurring on a small, sub-granular scale, as
from the point where the first polarity inversion occurs to the last is a
distance of about $400$ km in slice $1$.

To truly characterise this magnetic element as serpentine we must also examine
the azimuth. If the azimuth showed a sharp, discontinuous change between two
distinct values this could indicate that we have actually observed two
separate magnetic loops that are not connected. Figure 4 also shows the
variation in $\phi$ across the slices. In the upper (lower) slice, $\phi$
varies smoothly from a minimum of $86^{\circ}$ ($84^{\circ}$) to a maximum of
$151^{\circ}$ ($153^{\circ}$). To confidently constrain the azimuth one needs
to measure both Stokes $Q$ and $U$. However, the amplitude of the noise in the
weaker linear polarisation parameter places a limit on the value that $\phi$
could take when only one is confidently measured. Therefore, in these cases,
and especially when the amplitude of the stronger linear polarisation
parameter is large, the $\phi$ value can still be constrained. When heavily
noise contaminated, maps of $\phi$ appear random. The map of $\phi$ in Fig. 4
is smooth-varying across the magnetic element, including in those pixels where
only one linear polarisation parameter has a maximum amplitude greater than
$4\sigma_{n}$. This is in contrast to areas surrounding the magnetic element,
where the map of $\phi$ has a salt-and-pepper pattern. The spatial coherence
of this map indicates that there is enough linear polarisation signal
available in the magnetic element to constrain the azimuth and deduce that it
varies across the slices. While a change of $69^{\circ}$ in $\phi$ is not an
insignificant variation, we note that the change occurs gradually. This
suggests the magnetic field vector is changing direction not just along the
LOS, but in the plane of the solar surface across the slices, but not in a
sharp, discontinuous way that would indicate the existence of two distinct
magnetic elements. Indeed, after the $\phi$ value reaches its peak in each
slice, it continues to vary gradually.

We also repeated the inversion in the small area in Fig. 4 where the maximum
number of nodes in $B$ in the magnetic component was increased from $1$ to
$3$, and the maximum number of nodes in $v_{\mathrm{LOS}}$ in the non-magnetic
component was increased from $2$ to $3$. We found that the increased number of
free parameters made a very marginal improvement in the quality of the fits in
a small number of pixels, but made no significant difference to the $\gamma$
or $\phi$ variations in this magnetic element; most importantly, the polarity
of the magnetic field changed three times across both slices. We found no
evidence that SIR made use of the additional free parameter in
$v_{\mathrm{LOS}}$.

####  3.2.2 Case study II: Degenerate solutions at polarity inversion lines

Figure 7: As in Fig. 3, but for vector C.1. The filling factors of model 1 and
2 were $0.79$ and $0.21$, respectively, while the $\phi$ value was
$146^{\circ}$. The $v_{\mathrm{mac}}$ was $0.73$ km sec-1 for both models.
This inversion is a 1C configuration. The Stokes $V$ signal has a single lobe.
Figure 8: As in Fig. 3, but for vector C.2. The filling factors of model 1 and
2 were $0.31$ and $0.69$, respectively, while the $\gamma$ and $\phi$ values
were $78^{\circ}$ and $152^{\circ}$. The $v_{\mathrm{mac}}$ was $0.95$ km
sec-1 for both models. This inversion is a 1C configuration. The Stokes $V$
signal exhibits a regular double-lobed, anti-symmetric shape. Figure 9: In the
lower panels the Stokes V profiles from pixels C.3 and C.4 are shown on the
left and the right, respectively. Profile C.3 has over-plotted the WFA fit
obtained from the derivative of Stokes $I$ (blue, dashed line). Both profiles
have the synthetic SIR profile over-plotted (red, solid line). The horizontal,
grey, dashed-dotted lines show the $4\sigma_{n}$ threshold and the vertical,
blue, dotted lines show the locations of the blue and red lobes used for the
SFA estimate for $B$.

One of the most likely places to find complex, abnormal Stokes profiles (i.e.
a larger or smaller number of lobes than expected, or with significant
asymmetries, or both) in the quiet Sun is at, or near, the PIL in magnetic
loops. In the middle rows of Fig. 2 we group together two simple magnetic
loops for the second case study. These structures have in common that linear
polarisation is found at the PIL, however, they differ significantly in terms
of their circular polarisation signals at the same location. The magnetic loop
shown in the second row in Fig. 2 is found in the intergranular lane (IGL)
next to a tiny granule. From the azimuth map, we can immediately determine
that this magnetic loop is a distinct structure compared to its surroundings.
The sample Stokes vector B.1 shown in Fig. 5 demonstrates that the linear
polarisation signals are strong, with small asymmetries, while the circular
polarisation signal has a three-lobed shape, and SIR is unable to reproduce it
without an increase in the number of free parameters. The middle panel in Fig.
5 shows the atmospheric parameters which resulted in a good fit to the
observed Stokes vector with two magnetic model atmospheres (2C), including $1$
node in $B$, $2$ in $v_{\mathrm{LOS}}$, $2$ in $\gamma$, and $1$ in $\phi$ per
model. SIR has selected one magnetic component with a larger filling factor
($\alpha=0.93$) that is weaker ($B=133$ G) to co-exist with another magnetic
component with a much smaller filling factor ($\alpha=0.07$) and much stronger
magnetic field ($B=902$ G). The superposition of the opposing linear
stratifications in $v_{\mathrm{LOS}}$ and $\gamma$ is essential to produce the
synthetic vector, in particular to produce the circular polarisation profile
which could result both from the mixing of opposite polarity Stokes $V$
signals within the spatiotemporal resolution element and from gradients in
$v_{\mathrm{LOS}}$. Reducing the number of nodes in either of these parameters
significantly degraded the fit.

This sample pixel is not unique in this magnetic loop, as many similar Stokes
$V$ profiles are found along the PIL. Furthermore, the solution provided by
the inversion is not unique, and the mixing of two magnetic components in the
spatiotemporal resolution element is not the only physical scenario that could
be responsible for this Stokes vector. We encounter a degeneracy problem,
where very different physical solutions provide equally plausible fits to the
observed Stokes vector, when trying to fit complex profiles. In Fig. 5 we also
present an alternative, degenerate solution that does not require a 2C
configuration. In this case, a 1C configuration with a $638$ G magnetic field
that is constant in optical depth was sufficient, but both the magnetic and
non-magnetic model atmospheres required strong linear gradients in
$v_{\mathrm{LOS}}$ achieved with $2$ nodes each, and crucially the magnetic
model atmosphere required a polynomial stratification in $\gamma$ that was
achieved with $5$ nodes.

The second magnetic loop shown in the third row in Fig. 2 is a narrow patch of
magnetic flux along the granule-IGL boundary. The structure is about
$1{\,}.{\\!\\!}{\arcsec}5$ long but only $0{\,}.{\\!\\!}{\arcsec}5$ wide. The
Stokes vector from pixel B.2 is taken from the PIL, but in contrast to the
former magnetic loop (pixel B.1) there is a complete absence of a Stokes $V$
signal. However, despite this, we once again encounter degenerate solutions as
there are large asymmetries in the amplitudes of the $\sigma$-lobes of the
linear polarisation profiles. Figure 6 show solutions based on 1C and 2C
configurations. These solutions have in common that SIR selected an almost
perfectly transverse magnetic field. Indeed, as we have no circular
polarisation we also find there is no benefit to including a gradient in
$\gamma$. In the 1C solution, we required $3$ nodes each in $B$ and
$v_{\mathrm{LOS}}$ for the magnetic model atmosphere. The $\alpha_{m}$ for
this solution was very large ($\alpha_{m}=0.93$) and the magnetic model
atmosphere is nearly stationary at optical depths where the response functions
peak, while the non-magnetic model atmosphere has a strong down-flow
($v_{\mathrm{LOS}}=3.8$ km sec-1). We find that the stratification in $B$,
decreasing with height until reaching zero near where the response functions
peak, is essential. Additionally, a linear gradient in $v_{\mathrm{LOS}}$ was
insufficient to fully achieve the asymmetric amplitude of Stokes $Q$ and $U$.
As for the 2C solution, we found it to be essential that the model atmospheres
have different $v_{\mathrm{LOS}}$ stratifications and different $\phi$ values.
The primary model atmosphere, with the larger filling factor ($\alpha=0.83$),
has a very weak magnetic field strength ($B=28.0$ G) and is nearly stationary
in the LOS ($v_{\mathrm{LOS}}$ = -0.1 km sec-1), while the secondary model
atmosphere has a polynomial stratification in $v_{\mathrm{LOS}}$ and has a
much larger magnetic field strength ($B=676$ G). We were able to reduce the
number of nodes in $B$, $\gamma$, $\phi$, and $v_{\mathrm{LOS}}$ to 1 in both
model atmospheres with the exception of the $v_{\mathrm{LOS}}$ of the
secondary model atmosphere, which had $3$ nodes and could not be reduced
without losing the required asymmetries in the linear polarisation profiles.

We tested the inversions for pixels B.1 and B.1 under 1C and 2C configurations
where we allowed the temperature of both model atmospheres to vary
independently. In all cases the increase in the number of free parameters did
not improve the quality of fit, and further the key features of the parameter
stratifications in $B$, $\gamma$, and $v_{\mathrm{LOS}}$ were invariant.
Ultimately we found no evidence that increasing the number of free parameters
provided either a significantly different solution or an improvement to the
quality of the fit to the observed Stokes vector, so we favour the simpler
solutions.

####  3.2.3 Case study III: Magnetic anatomy of a granule

The bottom row in Fig. 2 shows our final case study. Notably, we find there is
significant linear and circular polarisation signal throughout almost the
entirety of the granule. We take this opportunity to explore the different
spectropolarimetric and magnetic properties of the granule and surrounding
IGLs, which is the convective building block of the solar surface, for the
first time at $70$ km resolution. To this end we select four representative
pixels. To begin, in the lower, left-most patch of circular polarisation,
linear polarisation is pervasive, while both the linear and circular
polarisation profiles are highly asymmetric and the circular polarisation is
even single-lobed. Fig. 7 shows a sample Stokes vector (pixel C.1) from this
patch, with its location at the granule-IGL boundary indicated by the blue
marker in Fig. 2. The initial 1C inversion was unable to accurately fit the
asymmetric Stokes profiles, and so the number of free parameters was
increased. The very good fit shown in Fig. 7 is the result, which was achieved
with $3$ nodes in $B$, $3$ in $v_{\mathrm{LOS}}$, $3$ in $\gamma$, and $1$ in
$\phi$ for the magnetic model atmosphere, and only $1$ node in
$v_{\mathrm{LOS}}$ for the non-magnetic model atmosphere. We found that a 1C
configuration was sufficient and did not require a 2C configuration. The $B$
stratification required to fit this profile is a gradient which decreases with
increased height in the atmosphere, from very strong, kG magnetic field with
$B>2000$ G in the deepest layers. However, we stress that SIR is simply
extrapolating to $\mathrm{log}(\tau_{5000\mathrm{\AA}})=1.0$. At
$\mathrm{log}(\tau_{5000\mathrm{\AA}})=0.0$ the field strength is $851$ G. The
magnetic model atmosphere ($\alpha_{m}=0.79$) is highly transverse in the
optical depth range the lines are responsive to but has a gradient which
changes polarity and becomes more vertical higher in the atmosphere. In order
to produce the asymmetries in all four Stokes parameters, the magnetic model
atmosphere also required a strong linear gradient in $v_{\mathrm{LOS}}$ while
the non-magnetic component has a single, positive $v_{\mathrm{LOS}}$ value
throughout the atmosphere.

Deep in the granule, in the central patch of circular polarisation, linear
polarisation is again pervasive but the circular polarisation is much more
symmetric and two-lobed. Fig. 8 shows a sample Stokes vector (pixel C.2) whose
location is indicated by the cyan marker in Fig. 2. The synthetic vector
provided by the initial 1C inversion is a good fit in this case, with the
exception of small asymmetries in the linear polarisation profiles. By adding
a gradient to the $v_{\mathrm{LOS}}$ of the magnetic model atmosphere, SIR was
able to obtain better fits. The $B$ of this pixel is moderate but the pixel
had a large $\alpha_{m}$ such that the $\alpha_{m}B$ is relatively large
($B=380$ G, $\alpha_{m}=0.31$, $\alpha_{m}B=118$ Mx cm-2). In the lower, right
patch of circular polarisation, most of the profiles do not have significant
linear polarisation and the circular polarisation profiles are typically
asymmetric. While the profiles may be asymmetric or not, the change in
polarity is nevertheless unambiguous between these three patches of small-
scale magnetism. It is perhaps not surprising that the profiles are
asymmetric, and therefore indicative of velocity gradients, in pixel C.1 given
that it is located at the granule-IGL boundary. Pixel C.2, on the other hand,
is more firmly located in the granule.

The Stokes vectors found in the IGLs surrounding the examined granule were
also inspected. Stokes $V$ profiles with varying degrees of asymmetries are
found. We find that the field strengths at these locations differ by two
orders of magnitude, and at the high end of these extremes the $B$ exceeds $2$
kG. We are motivated to sanity check these values with the weak field
approximation (WFA) and strong field approximation (SFA) in order to ensure
the inversions are well calibrated (Campbell et al., 2021a, 2023), by
estimating the longitudinal magnetic flux density with the WFA and magnetic
field strength with the SFA.

In the weak regime, as Landi Degl’Innocenti & Landolfi (2004) demonstrate, the
amplitude of Stokes $V$ is proportional to the longitudinal magnetic flux
density (where $B_{\parallel}=B\cos\gamma$),

$V(\lambda)\approx-\Delta\lambda_{B}\cos\gamma\frac{dI(\lambda)}{d\lambda},$
(2)

where,

$\lambda_{B}\approx 4.6686\times 10^{-13}\lambda_{0}^{2}g_{\mathrm{eff}}B,$
(3)

where $\lambda_{0}$ is the rest wavelength and $g_{\mathrm{eff}}$ is the
effective Landé g-factor of the spectral line. Equation 2 is only valid if the
magnetic Zeeman splitting is negligible relative to the Doppler width of a
line (i.e. $\Delta\lambda_{B}/\Delta\lambda_{d}\ll 1$) and when $\gamma$,
$\phi$, $B$, $v_{\mathrm{LOS}}$, the Doppler width, and any broadening
mechanisms are invariant along the LOS in the formation region of the line. On
the other hand, by measuring $2\Delta\lambda_{B}$ as the separation of the
lobes of Stokes $V$ when the field is strong enough and when
$\Delta\lambda_{B}/\Delta\lambda_{d}\gg 1$, it is possible to obtain $B$
directly through Eqn. 3 (Khomenko et al., 2003; Nelson et al., 2021; Campbell
et al., 2023). As in Asensio Ramos (2011), writing the expressions for
$\Delta\lambda_{B}$ and $\Delta\lambda_{d}$ allows us to say a line is in the
weak field regime when the magnetic field strength fulfills:

$B\ll\frac{4\pi mc}{g_{\text{eff}}\lambda_{0}e}\sqrt{\frac{2kT}{M}+v_{mic}},$
(4)

where $m$ is the mass of an electron, $c$ is the speed of light, $\lambda_{0}$
is the rest wavelength of the spectral line, $e$ is the charge of an electron,
$k$ is the Boltzmann constant, and $M$ is the mass of the species. For the
$6302.5$ $\mathrm{\AA}$ line, assuming $v_{\text{mic}}=1$ km sec-1 and
$T=5800$ K, we estimate a limit of $760$ G. Raising the $T$ to $7000$ K would
increase the limit to $809$ G. Eliminating the $v_{\mathrm{mic}}$ would
decrease the limit to $610$ G.

We sampled pixels at the upper-most granular boundary with low-amplitude,
symmetrical Stokes $V$ profiles, and applied the WFA. The circular
polarisation profile from the pixel labelled C.3 in Fig. 2 is shown in Fig 9.
In the initial 1C inversion, SIR fit this profile with a model atmosphere with
$B=73.4$ G, $\gamma=57.0^{\circ}$, $\alpha_{m}=0.5$, and
$v_{\mathrm{LOS}}=0.1$ km/sec. Therefore, according to SIR,
$\alpha_{m}B\cos\gamma=21.2$ Mx cm-2. The $B$ value is an order of magnitude
smaller than the limit we estimated using Eqn. 4, so we are satisfied that
using the WFA on this pixel is appropriate. Meanwhile, the WFA estimate is
obtained from the derivative of Stokes $I$ and through Eqn. 2 as $20.0$ Mx
cm-2. The pixel with the weakest $B$ that we found which satisfied the
$4\sigma_{n}$ amplitude threshold, and which had symmetrical lobes, was $25$
G, or $\alpha_{m}B\cos\gamma=12.8$ Mx cm-2. For the same pixel, the WFA
estimate was $11.7$ Mx cm-2.

We sampled pixels in the right-most IGL which has an extremely strong magnetic
field and applied the SFA by measuring the separation of the lobes of Stokes
$V$. The circular polarisation profile from the pixel labelled C.4 in Fig. 2
is shown in Fig. 9. In this case, the initial 1C inversion fit this profile
with a very strong magnetic field ($B=2048$ G, $\alpha_{m}=0.07$,
$\alpha_{m}B=139.3$ Mx cm-2) and the positive and negative lobes of Stokes $V$
are separated by $15$ increments in wavelength. The SFA estimate is thus
$2078$ G as a single increment in wavelength is equivalent to $138.55$ G. We
are both satisfied that using the SFA is appropriate and that the magnetic
field strength is accurate as there are many pixels in this IGL with field
strengths greater than $1.8$ kG.

##  4 Discussion

We have presented the first spectropolarimetric observations of the quiet Sun
with DKIST, the first $4-$m class and largest solar optical telescope ever
built. With an estimated spatial resolution of $0{\,}.{\\!\\!}{\arcsec}08$,
these observations represent the highest spatial resolution full-vector
spectropolarimetric observations ever obtained of the quiet Sun; previous high
resolution observations were achieved by the Swedish Solar Telescope
($0{\,}.{\\!\\!}{\arcsec}16$, e.g. Schnerr & Spruit, 2011), the Sunrise
balloon-borne experiment
($0{\,}.{\\!\\!}{\arcsec}14-0{\,}.{\\!\\!}{\arcsec}16$, e.g. Lagg et al.,
2010; Wiegelmann et al., 2013), the GREGOR telescope
($0{\,}.{\\!\\!}{\arcsec}3-0{\,}.{\\!\\!}{\arcsec}4$, e.g. Lagg et al., 2016;
Campbell et al., 2023), and the Hinode spacecraft
($0{\,}.{\\!\\!}{\arcsec}32$, e.g. Lites et al., 2008; Bellot Rubio & Orozco
Suárez, 2012). After locating at least $53$ magnetic elements of interest, we
curated three case studies and examined their physical properties in detail to
showcase how the small-scale magnetic fields are spatially organised at this
resolution.

We examined a particular magnetic element where the polarity inverts three
times across the structure (see Fig 4). In addition to a complex topology in
terms of the inclination angle, we also observe the azimuthal angle changing
significantly across its structure. We believe that the evidence indicates
that it is plausible that the magnetic field lines of this small-scale
magnetic structure are serpentine with significant, coherent variation in the
plane of the solar surface. We are not aware of any previous studies that have
observed these complex magnetic structures in the quiet Sun on a sub-granular
scale. Hinode observations have demonstrated that in simple magnetic loops the
azimuth can vary significantly in a few minutes (Martínez González et al.,
2010) and simulations of the twisting of magnetic field lines during small-
scale reconnection events show that gradual azimuthal changes in the
photosphere are plausible in more complex magnetic topologies (Danilović,
2009; Hansteen et al., 2017). Perhaps it is possible that these complex
topologies have not been observed at facilities with lower diffraction limited
resolutions than DKIST precisely because they lack the required angular and
spectral resolution. Similarly, the complex nature of the serpentine feature
could be lost with poorer seeing conditions. However, we also emphasise that
this is the only example of serpentine magnetism that we discovered in the
dataset which had three PILs and linear polarisation signals throughout the
structure, and in the other magnetic loops we see a much more simple variation
in $\gamma$, and typically insignificant variation in $\phi$.

An alternative interpretation would be that we observed two magnetic loops in
very close proximity. The fact that the azimuth increases to reach a maximum
in one part of the slice but then decreases somewhat less significantly in the
remainder of the slice could support this argument. However, the evidence does
not conclusively support this interpretation either because there is no
discontinuity in the variation of the azimuth like can be observed in the
magnetic loop shown in the second row of Fig. 2. We note that without
disambiguation of the azimuth we cannot be certain that the gradual, coherent
variation we measure is correct. If this alternative explanation was true we
argue that we should not observe linear polarisation signals at the middle,
apparent PIL between the magnetic loops. In other words, we should not observe
linear polarisation signals at all three apparent PILs. We note that linear
polarisation signals are pervasive across the entire slice, and thus we are
able to observe two crests in the magnetic field lines. However, we also note
that we did discover a second serpentine structure with three PILs in our
analysis, and the middle PILs did not have linear polarisation signals.

Ultimately, only repeated observations with a high-cadence time series would
allow us to conclusively determine the correct scenario. Finally we point out
that the temporal resolution may play a role due to the scanning speed of the
observations. The step cadence of these scans is about $7.4$ seconds. A
horizontal flow would have to be on the order of $4$ km sec-1 to be similar to
the stepping cadence. However, it would be premature to say this is a factor
which makes either scenario less or more plausible; for instance, the gradual
variation in the azimuth that we measured could be explained by supposing that
we observed the temporal evolution of a serpentine structure.

A physical scenario that demands an increased number of free parameters
commonly occurs at the PIL of magnetic elements, and detecting linear
polarisation at these locations has been a major challenge (Kubo et al., 2010,
2014). We examined two magnetic loops which had very different properties at
the PIL in terms of circular polarisation, but which both had strong linear
polarisation signals. In one case, we uncovered a Stokes vector which had a
complex Stokes $V$ profile. Three-lobed Stokes $V$ profiles were also observed
in the quiet Sun by Viticchié & Sánchez Almeida (2011) with Hinode/SP and
Martínez González et al. (2016); Kiess et al. (2018); Campbell et al. (2021a)
with GREGOR. Initially we attempted a 2C inversion, but with gradients in
$\gamma$ as well as in $B$ and $v_{\mathrm{LOS}}$ (see Fig. 5). We encountered
a problem of degeneracy as we found this profile could also be explained with
a more extreme gradient in $\gamma$ but without two magnetic model atmospheres
(see Fig. 5). The second interpretation is consistent with Khomenko et al.
(2005), who showed using magnetoconvection simulations that gradients in
$\gamma$ and $v_{\mathrm{LOS}}$ can generate irregular Stokes $V$ profiles. In
this case it is not as simple as saying the approach with the lower number of
free parameters is the correct one - the fact that this profile was located
along the PIL means it is plausible that the 3-lobed Stokes $V$ profile is the
physical manifestation of a smearing of signals of opposite polarities, or
even mixing within the spatiotemporal resolution element of the observations
that remains unresolved. We note that in the 1C case $\gamma$ changes polarity
along the LOS in the optical depth range in which these lines are sensitive to
perturbations in $\gamma$, and there are other examples in this study where
this occurs (see Fig. 3 and Fig. 7). Previously Rezaei et al. (2007) had shown
this was possible when the Fe I $6301.5/6302.5$ $\mathrm{\AA}$ doublet showed
two-lobed Stokes $V$ profiles with different polarities in a single spectrum,
but in our observations we find the Stokes $V$ profiles of both lines are
compatible.

In another case, we observe a long, narrow magnetic loop at the granule-IGL
boundary which has an absence of Stokes $V$ at the PIL. It is most plausible
that in this case the magnetic field at the PIL is completely transverse,
however, like in the former example, opposite polarity Stokes $V$ signals may
have mixed such that their signals have completely cancelled each other.
However, the absence of a Stokes $V$ signal cannot be presented as evidence
for the latter scenario. We also find degenerate solutions to this Stokes
vector, based on a single magnetic model and two magnetic models (see Fig. 6),
showing that degeneracy is an issue even in the absence of a Stokes $V$
profile. This degeneracy issue is a serious problem which calls into question
how accurately we can infer physical parameters from inversions, but we argue
this issue could be significantly improved by observing more photospheric
spectral lines. This is distinct from the degeneracy problem described by
Martínez González et al. (2006) because in all the inversions presented in
this study we forced the temperature of each component to be the same.
Ultimately, if the solutions requiring two magnetic models are correct it
suggests that, even at the spatiotemporal resolution of DKIST/ViSP, magnetic
flux is yet hidden to spectropolarimetric observations that use the Zeeman
effect in magnetic loops (or bi-poles).

Significant effort was made to try and balance the need for an increased
number of free parameters to fit complex Stokes profiles with the risk of
producing unphysical or unrealistic solutions from the inversions. In
particular, velocity gradients in the LOS often manifest in Stokes $Q$, $U$,
and $V$ as an amplitude asymmetry that cannot be modelled with a
$v_{\mathrm{LOS}}$ stratification that is constant in optical depth. In the
final case study we examined a magnetic feature spanning a granule which had
linear polarisation signals throughout most of its structure. The magnetic
nature of granules is important because it has implications for constraining
magnetohydrodynamic simulations, particularly in terms of circulation models,
emergence of magnetic flux, and the formation of IGLs (Rempel, 2018). It is
notable that this granule has pervasive linear polarisation, in addition to
circular polarisation, because magnetic flux emergence in granules was
observed in early Hinode/SP observations to be longitudinal in nature (Orozco
Suárez et al., 2008), but the emergence of horizontal magnetic flux has been
observed in a granule (Gömöry et al., 2010). In the granule, we found
symmetric Stokes profiles (see Fig. 8), but at the granule-IGL boundary we
sampled a Stokes vector from a pixel which had significant asymmetries in the
polarisation parameters (see Fig. 7). We inverted this Stokes vector with an
approach which sought to minimise the number of free parameters as much as
possible, but were only adequately able to reproduce it using gradients in
$B$, $\gamma$, and $v_{\mathrm{LOS}}$. Single-lobed Stokes $V$ profiles are
common in quiet Sun Hinode observations, accounting for up to $34\%$ of
profiles (Viticchié & Sánchez Almeida, 2011). They cover about $2\%$ of the
surface, are associated with inclined magnetic fields and flux emergence and
submergence processes, and are also reproduced by simulations (Sainz Dalda et
al., 2012). There is an expectation that the existence of canopy fields mean
that the magnetic field becomes statistically more horizontal with height
(Stenflo, 2013; Danilovic et al., 2016). However, in our analysis we found the
opposite in individual cases (see Fig. 3 and 7). Although it is well
understood that depth-averaged parameters can be well retrieved from
inversions of photospheric diagnostics without the inclusion of gradients in
the magnetic or kinetic parameters (see, for instance, Campbell et al., 2021b;
Quintero Noda et al., 2023), we offer a word of caution that at DKIST
resolutions it is clear that any future statistical analysis of this data will
be inadequate unless the need for gradients in $v_{\mathrm{LOS}}$ (and other
parameters) are accounted for in the inversions due to their prevalence in
these small-scale magnetic structures.

We sampled relatively symmetric circular polarisation profiles at the granular
boundaries of the magnetic granule - with one side of the granule harbouring
smaller amplitude profiles, indicative of a weak magnetic field, and the other
with amplitudes twice their size and much larger lobe separations - and
applied the weak and strong field approximations, to retrieve manual estimates
of the $\alpha_{m}B$ and $B$. This allowed us to sanity check our measurements
of vastly different magnetic field strengths spanning two orders of magnitude.
However, we note that the difference in $\alpha_{m}B$ is not as large as the
difference in $B$, indicating that the determination of $\alpha_{m}$ plays a
role. However, we expect that inversions of these lines are able to constrain
both the $\alpha_{m}$ and $B$ (Asensio Ramos, 2009) and expect to be able to
make distinctions between weak and strong fields in these lines when
$\alpha_{m}$ is a free parameter (del Toro Iniesta et al., 2010). We emphasise
that such symmetrical Stokes $V$ profiles are in the minority in this
structure, especially in the IGLs, and so we do not recommend attempting to
apply these approximations more broadly to these data except in limited
specific cases. The extremely large field strengths we recorded are at the
high end of what is expected to result from magnetic field amplification in
bright points based on high resolution spectropolarimetric observations (see,
for instance, Beck et al., 2007; Utz et al., 2013; Keys et al., 2019).

The analysis presented in this study is conducted under the assumption of
local thermodynamic equilibrium (LTE). Smitha et al. (2020) have shown that
for the Fe I $6300$ $\mathrm{\AA}$ doublet there can be departures from LTE
due to resonance line scattering and over-ionisation of Fe atoms by
ultraviolet photons. The former process makes the line stronger, while the
latter makes the line weaker, which means for the Fe I $6300$ $\mathrm{\AA}$
doublet these processes compensate each other to an extent. Nevertheless,
departures from LTE can introduce errors or uncertainties in the derived
atmospheric parameters.

##  5 Conclusions

We have analysed spectropolarimetric observations of the quiet Sun with the
first $4-$m class optical telescope in service to solar physics. This allowed
us to reveal, for the first time, the serpentine nature of the sub-granular,
small-scale magnetic field, as we demonstrated that the polarity of the
magnetic field changed at least three times in only $400$ km. The three
polarity reversals are evident from an inspection of the Stokes $V$ profiles
and confirmed by the inversions. From the inversions we were also able to
deduce that the azimuth changed in a significant but coherent way. While
beyond the scope of this study, it is clear that a statistical analysis based
on an non-LTE inversion, not just of the Ca II $8542$ $\mathrm{\AA}$ line but
also the Fe I doublet, is the next logical step and will be the subject of
future work. We were unable to trace the temporal evolution of these
structures. Observations of these structures at high-cadence ($30-90$ seconds)
should be a priority for DKIST in the future with the Diffraction Limited Near
Infrared Spectropolarimeter (DL-NIRSP) instrument (Jaeggli et al., 2022).

We found that, particularly at PILs of magnetic loops, degenerate solutions in
inversions remains a significant issue even at the spatial resolution provided
by a 4-m telescope, and this is true even in the absence of a Stokes $V$
signal. Although previous work involving simulations and synthetic
observations have shown that atmospheric parameters can be well recovered on
average (see, for instance, Campbell et al. (2021b); Quintero Noda et al.
(2023)), in our case studies we found that to truly characterise the plasma on
small scales we had to permit a larger number of free parameters to produce
the required asymmetries in the Stokes profiles and abnormal Stokes $V$
profiles. We are unable to distinguish between gradients in atmospheric
parameters along the LOS and smearing of signals in the plane of the solar
surface due to the spatial point spread function of the telescope. We concur
with the advice provided by Cabrera Solana et al. (2005); Quintero Noda et al.
(2021) and suggest that observations of the Fe I $6302.5$ $\mathrm{\AA}$ line
with the deep photospheric Fe I $15648.5$ $\mathrm{\AA}$ line should be
prioritised, as observations of diagnostics which sample a range of heights in
the photosphere could provide a method for distinguishing between these two
otherwise plausible scenarios when simultaneously inverted. These observations
would additionally benefit from the demonstrated ability of the Fe I $15648.5$
$\mathrm{\AA}$ line to observe higher linear polarisation fractions (Martínez
González et al., 2008b; Lagg et al., 2016; Campbell et al., 2023).

We thank the anonymous referee for their suggestions which improved this
manuscript. RJC, DBJ and MM acknowledge support from the Science and
Technology Facilities Council (STFC) under grant No. ST$/$P000304$/$1,
ST$/$T001437$/$1, ST$/$T00021X$/$1 and ST$/$X000923$/$1\. DBJ acknowledges
support from the UK Space Agency for a National Space Technology Programme
(NSTP) Technology for Space Science award (SSc 009). DBJ and MM acknowlege
support from The Leverhulme Trust grant RPG-2019-371. This research has
received financial support from the European Union’s Horizon 2020 research and
innovation program under grant agreement No. 824135 (SOLARNET). The research
reported herein is based in part on data collected with the Daniel K. Inouye
Solar Telescope (DKIST), a facility of the National Solar Observatory (NSO).
NSO is managed by the Association of Universities for Research in Astronomy,
Inc., and is funded by the National Science Foundation. Any opinions, findings
and conclusions or recommendations expressed in this publication are those of
the authors and do not necessarily reflect the views of the National Science
Foundation or the Association of Universities for Research in Astronomy, Inc.
DKIST is located on land of spiritual and cultural significance to Native
Hawaiian people. The use of this important site to further scientific
knowledge is done so with appreciation and respect. This material is based
upon work supported by the National Center for Atmospheric Research, which is
a major facility sponsored by the National Science Foundation under
Cooperative Agreement No. 1852977. The observational data used during this
research is openly available from the DKIST Data Centre Archive under proposal
identifier pid$\\_1\\_36$.

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