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# On the Origin of Magnetic Perturbations associated with the FIP Effect

M. Murabito  M. Stangalini 11 D. Baker 22 G. Valori 33 D. B. Jess 44 S.
Jafarzadeh 5566 D. H. Brooks 7788 I. Ermolli 99 F. Giorgi 11 S. D. T. Grant 11
D. M. Long 55 L. van Driel-Gesztelyi 3333 10 10 11 11

In Stangalini et al. (2021) and Baker et al. (2021), magnetic oscillations
were detected in the chromosphere of a large sunspot and found to be linked to
the coronal locations where a First Ionization Potential (FIP) effect was
observed. In an attempt to shed light onto the possible excitation mechanisms
of these localized waves, we further investigate the same data by focussing on
the relation between the spatial distribution of the magnetic wave power and
the overall field geometry and plasma parameters obtained from multi-height
spectropolarimetric non-local thermodynamic equilibrium (NLTE) inversions of
IBIS data. We find that, in correspondence with the locations where the
magnetic wave energy is observed at chromospheric heights, the magnetic fields
have smaller scale heights, meaning faster expansions of the field lines,
which ultimately results in stronger vertical density stratification and wave
steepening. In addition, the acoustic spectrum of the oscillations at the
locations where magnetic perturbations are observed is broader than that
observed at other locations, which suggests an additional forcing driver to
the $p$-modes. Analysis of the photospheric oscillations in the sunspot
surroundings also reveals a broader spectrum in between the two opposite
polarities of the active region (the leading spot and the trailing opposite
polarity plage), and on the same side where magnetic perturbations are
observed in the umbra. We suggest that strong photospheric perturbations in
between the two polarities are responsible for this broader spectrum of
oscillations, with respect to the $p$-mode spectrum, resulting in locally-
excited acoustic waves that, after crossing the equipartition layer, located
close to the umbra-penumbra boundary at photopheric heights, are converted
into magnetic-like waves and steepen due to the strong density gradient.

###### Key Words.:

Sun: atmosphere – Sun: oscillations – Sun: magnetic fields – Sun: abundances

Figure 1: SDO/HMI magnetogram (panel a), alongside SDO/AIA 304 Å (panel b),
171 Å (panel d), and 335 Å (panel c) filtergrams at the time of the IBIS
observations at 14:00 UT on 2016 May 20. The white/red boxes in panels a–d
indicate the IBIS FOV shown in the right panels. The HMI magnetogram is
saturated at $\pm 500$ G for better visibility. Panels e and f show a
photospheric continuum intensity map derived from the Fe i 617.3 nm line and a
chromospheric Ca ii 854.2 nm line core intensity map, respectively. The white
contours represent the umbra-penumbra boundary as derived from the continuum
intensity. The blue dots indicate the locations where magnetic pertubations
are detected (Stangalini et al., 2021). Box ‘A’ in the chromospheric Ca ii
854.2 nm line core image (panel f) indicates where the probability density
functions (PDFs) shown in Fig. 4 have been computed.

A movie of HMI magnetograms is available in the online material.

##  1 Introduction

Although one would expect the solar corona to have the same elemental
abundances of the solar photosphere, this is not always the case (Pottasch,
1963; Meyer, 1985a, b; Widing & Feldman, 1989, 1995; Sheeley, 1995, 1996). The
abundance variation observed in the corona depends on the first ionization
potential (FIP) of an element. Elements with FIP less than approximately 10 eV
are enhanced in the corona by a factor of 3–4 compared to the photosphere
whereas those elements with FIP greater than 10 eV tend to maintain their
photospheric abundances. This FIP effect is measured using the FIP bias which
is the ratio of an element’s abundance in the solar atmosphere to its
abundance in the photosphere. Interestingly, the FIP effect is also observed
in the solar wind, where it was suggested as a means to link components back
to their source regions in the solar atmosphere (e.g. Brooks & Warren, 2011;
Brooks et al., 2015; Hinode Review Team et al., 2019).

It is argued that the FIP effect can be due to the ponderomotive force linked
to the magnetic oscillations associated with magnetohydrodynamic (MHD) waves
(Laming, 2015). The ponderomotive force arises from the reflection/refraction
of the magnetic-like waves in the chromosphere and acts only on the low FIP
ions while leaving the mainly neutral high FIP elements unaffected. Ions are
separated from neutral elements in the chromosphere and then only the ions are
transported to the corona where they may be observed with enhanced abundances
compared to those of the photosphere. However, no observational evidence of
this scenario was available until very recently when, by exploiting a unique
combination of high resolution observations in the chromosphere and corona
with magnetic modelling, it was possible to detect magnetic perturbations in a
sunspot chromosphere and find a link with the high FIP bias locations in the
corona above the same sunspot (Stangalini et al., 2021; Baker et al., 2021,
hereafter referred to as papers A and B, respectively). These results were
also in agreement with previous studies of the same magnetic structure, where
the presence of intermediate (Alfvén) shocks were reported at the same
locations (Houston et al., 2020). However, although providing observational
support to link the FIP effect to magnetic-like waves (Laming, 2015, 2017), in
papers A and B only a few possibilities were put forward to explain the
surprising localised presence of magnetic perturbations only at particular
locations within the sunspot umbra.

Paper A reported that the magnetic perturbations were only detected on one
side of the sunspot, thus suggesting a possible role of the magnetic field
geometry or the connectivity with surrounding diffuse magnetic fields. The
authors suggested MHD mode conversion at the Alfvén-acoustic equipartition
layer (i.e, where the Alfvén and acoustic speeds nearly coincide;
$v_{A}=c_{s}$) as a possible cause, in agreement with Houston et al. (2020),
who detected intermediate shocks in the equipartition layer that was estimated
to reside between the upper photosphere and lower chromosphere.
In general, waves entering the region where the Alfvén and acoustic speeds
nearly coincide undergo a mode conversion or mode transmission process from
one form (e.g. acoustic-like to magnetic-like wave) to another (Crouch &
Cally, 2005; Suzuki & Inutsuka, 2005; Cally & Khomenko, 2015). The term ‘mode
conversion’ refers to the situation in which a wave retains its original
character (i.e., fast-to-fast or slow-to-slow), yet converts its general
nature in the form of acoustic-to-magnetic or magnetic-to-acoustic. On the
other hand, with ‘mode transmission’ one generally refers to the situation in
which the wave maintains its general nature (i.e.,‘magnetic-like’ wave or
‘acoustic-like’ mode), but changes character from fast-to-slow or slow-to-
fast. In all cases, the attack angle, that is the angle between the wavevector
and the field lines, is the dominant factor in determining both the conversion
($C$) and transmission ($T$) coefficients (Cally, 2001; Cally & Goossens,
2008), with $T+|C|=1$. In particular, the fraction of incident wave energy
flux transmitted from fast to slow acoustic waves is:

$T=e^{-\pi kh_{s}sin^{2}(\alpha)}\ $ (1)

where $k$ is the wavenumber, $h_{s}$ the thickness of the conversion layer,
and $\alpha$ the attack angle. The coefficient $C$ is a complex energy
fraction to take into account possible phase changes during the process of
mode conversion (Hansen & Cally, 2009). It was estimated that the thickness of
the conversion layer can be of the order of $200-250$ km (Stangalini et al.,
2011). From the above equation, it is clear that the conversion $C$ is larger
when the attack angle is larger. This implies that the field geometry plays a
significant role in the mode conversion and therefore should be carefully
taken into account as postulated by paper B.
In this work, in an attempt to shed light on the different mechanisms
generating the FIP effect, we investigate the wave propagation across
different heights above the sunspot as a function of the plasma and magnetic
field parameters, as inferred from multi-height spectropolarimetric
inversions. For this purpose, we make use of a combination of high-resolution
spectropolarimetric observations acquired by IBIS in the photosphere and
chromosphere, SDO/HMI line-of-sight (LOS) Dopplergrams, and SDO/AIA data to
determine the wave flux across different layers of the solar atmosphere and
analyze its relation to the global parameters such as inclination angles,
vertical gradients of the magnetic field, and density ratios of the magnetic
region.
This study can be preparatory for the scientific exploitation of future space
missions such as Solar-C EUVST and Solar Orbiter.

##  2 Observational Data

The dataset used in this work was acquired with the Interferometric
BIdimensional Spectrometer (IBIS; Cavallini, 2006) instrument at the Dunn
Solar Telescope (DST) on 2016 May 20 under excellent seeing conditions for
more than two hours, between 13:40 – 15:30 UT. This dataset has been the main
focus of other studies (see, e.g., Stangalini et al., 2018; Murabito et al.,
2019; Houston et al., 2020; Murabito et al., 2020; Stangalini et al., 2021;
Baker et al., 2021), due to the quality of the data and the large-scale nature
of the observed sunspot, which was the leading spot of AR 12546.

The observations were carried out using the Fe i 617.3 nm and Ca ii 854.2 nm
lines with a sampling of 20 mÅ and 60 mÅ, respectively. Both lines were
acquired in spectropolarimetric mode with 21 spectral points and a cadence of
48 s. A standard calibration procedure (flat field, dark subtraction,
polarimetric calibration) was first applied. In order to remove the residuals
of atmospheric aberrations, the dataset was processed with the Multi-Object
Multi-Frame Blind Deconvolution (MOMFBD; van Noort et al., 2005) technique.
From the final IBIS cubes, the circular polarisation (CP) signals (for both
the photospheric and chromospheric lines) were calculated pixel-by-pixel
following the definition given in Stangalini et al. (2021), using the maximum
amplitude of the Stokes-V spectral profile.

To complement the IBIS data and better study the wave power, we use full-disk
Dopplergrams acquired by the Helioseismic and Magnetic Imager (HMI; Schou et
al., 2012) on board the Solar Dynamics Observatory (SDO; Pesnell et al., 2012)
satellite in the interval between 13:00 – 16:00 UT, with a cadence of 45 s.
The pixel scale of these data is 0.5″. We also analyzed simultaneous
Atmospheric Imaging Assembly (AIA; Lemen et al., 2012) EUV filtergrams taken
in the 304 Å, 171 Å, and 335 Å passbands. The pixel scale of the SDO/AIA data
is 0.6″and the cadence is 12 s.

The combined IBIS, SDO/HMI and SDO/AIA data are used to investigate the
spatial distribution of the wave power penetrating the higher layers of the
sunspot atmosphere, in order to obtain a tomographic view of the embedded MHD
processes. Figure 1 shows an overview of AR 12546 as observed by the SDO/HMI
and SDO/AIA EUV (304 Å, 171 Å, and 335 Å) instruments (panels a, b, c and d)
and by the IBIS instrument (panels e and f) on 2016 May 20. The IBIS field-of-
view (FOV) captured one of the biggest sunspot of cycle , manifesting as a
strong coherent leading positive polarity sunspot, as displayed in the SDO/HMI
magnetogram and the photospheric IBIS continuum intensity maps in Fig. 1
(panels a and e). The AR at the time of IBIS observations was located near the
disc center, at X=35″ and Y=-90″. The magnetogram also shows an asymmetric
flux distribution between the trailing and leading side of the moat region
around the biggest sunspot. Indeed, moving magnetic features (MMF) activity is
observed, which is asymmetric, being more extended and vigorous on the left
(east) side of the umbra, coinciding with the segment where the blue dots are
observed in Fig. 1f.

Figure 2: Three-dimensional view of the AR 12546. From bottom to top: SDO/HMI
magnetogram, IBIS Fe i core with blue dots overplotted, and Hinode/EIS FIP
bias map. The purple surface represents the equipartition layer
($0.8<c_{s}/v_{A}<1.2$) as inferred from the spectropolarimetric inversions.
Selected field lines from a PFSS extrapolation of the coronal field link the
blue dots with regions of high FIP bias on the eastern and southern edges of
the sunspot i.e. in the penumbra. See Baker et al. (2021) for more details.

##  3 Methods and Results

###  3.1 Magnetic perturbations and local properties of the sunspot

Figure 3: Inclination angle of the magnetic field at $\log_{10}(\tau)=-1.0$
(photosphere; panel a). Expansion factor of the magnetic field between
photosphere and chromosphere (i.e. at $\log_{10}(\tau)=-4.6$) (panel b).
Density ratio between chromosphere and photosphere (panel c). Total magnetic
field gradient (panel d). The white contour in all maps represents the umbra-
penumbra boundary as seen in the continuum intensity. The hatched area
indicates the central region of the umbra, where saturation effects and low
photon flux are detected in the photosphere (Stangalini et al., 2018). The
magnetic field, the inclination angles, and the density ratios were derived
from the NICOLE inversions.

In paper B, the authors used observations obtained with the EUV Imaging
Spectrometer (EIS; Culhane et al., 2007) on board the _Hinode_ (Kosugi et al.,
2007) satellite to make a spatially resolved map of coronal composition, or
FIP bias, in the region of the sunspot (see Fig. 2). Highly fractionated
plasma with FIP bias of 3+ is observed in loops rooted in the penumbra on the
eastern and southern edges of the sunspot whereas the coronal field above the
umbra contains unfractionated plasma (FIP bias of 1–1.5). On the west, FIP
bias is approximately 2-2.5.

In order to investigate the role of the wave dynamics on the FIP effect
observed at higher layers, we studied the spatial distribution of the wave
power and compared it to the magnetic field and plasma parameters (namely, the
field inclination, density ratio, and vertical gradient of the magnetic field)
as inferred from spectropolarimetric inversions. The location of magnetic
oscillations detected in Papers A and B are shown in Fig. 1 (panel f) and Fig.
2 as blue dots. These blue dots are not uniformly distributed within the umbra
of the sunspot, but are only located towards the left side of it, close to the
umbra-penumbra (UP) boundary, therefore on the same side as the trailing
negative polarity of the AR (as shown by the magnetogram at the bottom of the
three-dimensional view of Fig. 2). Using a Potential Field Source Surface
(PFSS) extrapolation to model the magnetic field of the corona, the locations
of the blue dots were magnetically linked to regions of high FIP bias at
coronal heights as shown in Fig. 2 (see paper B for more details).

The magnetic field geometry is examined by using the non-local thermodynamic
equilibrium (NLTE) inversions already presented in Murabito et al. (2019),
which were carried out using the NICOLE code (Socas-Navarro et al., 2015) on
the same data (i.e. the best spectral scan of the data series, in terms of
contrast). Both the photospheric and chromospheric lines are inverted
simultaneously, thus providing a three-dimensional stratification of the most
relevant atmospheric parameters in the range of explored heights. More details
on the inversion procedure can be found in Murabito et al. (2019), although
here we summarize the key points for completeness. As a first step, we
investigated the atmospheric parameters obtained from the spectropolarimetric
inversions at the location of the blue dots (Fig. 1f and Fig. 2), focusing our
attention on the two atmospheric heights corresponding to the maxima of the
response functions of the two spectral lines (to magnetic field
perturbations), i.e., $\log_{10}(\tau)\approx-1.0$ for the photospheric Fe i
line and $\log_{10}(\tau)\approx-4.6$ for the chromospheric Ca ii line as
reported in Murabito et al. (2019), and in agreement with the previous study
by Quintero Noda et al. (2016).

Figure 4: (Panel a) PDF of the optical depths, $\log_{10}(\tau)$,
corresponding to the equipartition layer ($v_{A}=c_{s}$) associated with the
locations of the blue dots and the area inside box A shown in Fig. 1f. (Panel
b) PDFs of the magnetic field inclination angles at photospheric
($\log_{10}(\tau)=-1.0$, red and blue histograms) and chromospheric
($\log_{10}(\tau)=-4.6$; dashed histograms) heights. (Panel c and d) PDF
distribution of the magnetic field gradient and density ratio for the selected
region (i.e. the blue dots and the box A).

In Fig. 3 we show the photospheric (panel a) magnetic field inclination maps.
Given the location of the AR during the observations, i.e., at solar disc
center, we neglect any projection effects. Hence, the displayed maps are
consistent with the LOS reference frame. Here, we see that at photospheric
heights there is no significant difference in the inclination angle (on
average) between the left and right sides of the umbra. However, at
chromospheric heights we note a slightly larger magnetic field inclination
corresponding to the location of the blue dots (see Fig. 1f and Fig. 2), and
thus with the locations where the magnetic perturbations linked to the coronal
FIP effect are detected (see the FIP bias map in Fig 2). In order to examine
the potential role of the mode conversion process, that occurs at the
equipartition layer, we calculated the probability density function (PDF) of
the optical depths (in $\log_{10}\tau$) corresponding to this layer (i.e.,
$c_{s}=v_{A}$), which is displayed in Fig. 4a. This layer can play a
significant role in the wave energy conversion (Grant et al., 2018), and as
can be seen in Fig. 4a, it is predominantly located very close to the low
photosphere (i.e., $-1\geq\log_{10}(\tau)\geq 0$). In particular, we can note
that the right side of the umbra (i.e. blue PDF in Fig. 4a) has the
equipartition layer at much lower geometric heights, while on the opposite
side (i.e., at the location of the blue dots, green PDF in Fig. 4a) this
region extends more into the mid-photosphere.

To better investigate if the magnetic field geometry could be responsible for
a possible mode conversion process, we plot the distribution of the magnetic
field inclinations at photospheric heights, close to the equipartition layer,
at the locations corresponding to the blue dots and an area on the opposite
side (see the box A in Fig. 1f). The PDFs of the photospheric magnetic field
inclinations (Fig. 4b) show a similar distribution for both opposite locations
in the umbra, with a field inclination centered around 25∘ – 30∘. In any case
the small differences in the two distributions, if statistically significant,
can not play a role in the mode conversion process (see Eq. 1) and justify the
presence of magnetic-like disturbances on only one side. This is true for
waves travelling in all directions, that would experience a similar conversion
efficiency. However, we can not rule out the possibility of an asymmetry in
the acoustic driver itself. This aspect will be better addressed in Sect. 3.2.
For comparison, Fig. 4b displays the chromospheric inclination angles at the
two considered sides (dashed histograms).

To better highlight the differences between the chromosphere and photosphere,
we report the chromospheric magnetic field inclinations in relation to their
photospheric counterparts, i.e., a measure of the expansion factor, in Fig.
3b. This map shows that at both sides (i.e., left and right sides of the
umbra) the chromospheric magnetic field is more inclined (by about a factor
1.5) but at the left side, at the location of the blue dots, this factor
reaches 3 – 4 times larger. This finding indicates that the left side of the
umbra experiences an expansion of the field lines about 2 times faster than
the opposite side.
The difference in the magnetic field inclinations at chromospheric heights is
further supported by the different magnetic field gradients obtained from the
inversions. Figure 3d displays larger (up to 400 G / $\log_{10}\tau$) negative
values in the locations of the blue dots (i.e. the blue histogram in Fig. 4c),
meaning more rapid decreases of the magnetic flux with atmospheric height. In
fact, the opposite side of the umbra displays lower negative values (see the
red histograms in Fig. 4c). It is worth noting that there are pixels in the
right side of the umbra with a positive gradient of the magnetic field, which
means that the chromospheric magnetic field is stronger than the photosphere.
Although this finding may appear surprising, it can be explained by
considering this area as being part of the central region where the magnetic
sensitivity of the photospheric spectral line saturates due to the large
magnetic fields, resulting in an underestimation of the photospheric magnetic
field. On the other hand, the chromospheric Ca ii line is not saturated, hence
the chromospheric magnetic field inferred seems to be larger than that of the
underestimated photospheric one.

The higher negative value of the magnetic field gradient found at the location
of the blue dots further supports the idea of faster magnetic field expansion
in the chromosphere for the regions associated with the coronal FIP effect as
already seen in Fig. 3b (blue dots in Fig. 1f and Fig. 2). The density ratio
between the chromosphere and photosphere is shown in Fig. 3c. In agreement
with the above findings, this panel and its related histogram (Fig. 4d),
illustrate that a significant density drop between the two heights occurs at
the same location as the blue dots.

Figure 5: Chromospheric IBIS CP (upper-left) and $I_{\mathrm{core}}$ (upper-
right) power maps within the $4-10$ mHz range. The two boxes in the CP map
indicate where the spectral averaging has been performed for deriving the
results shown in the middle panels. The black contour in the
$I_{\mathrm{core}}$ map represents the CP power contour where strong magnetic
wave power is detected. Plots of the CP (middle-top) and $I_{\mathrm{core}}$
(middle-bottom) wave power averaged across the two boxes drawn in the upper-
left panel. For completeness, we also report the spectral averaging for the
blue dots locations as green plots. The SDO/HMI velocity $5.5-7.5$ mHz
amplification map with respect to the quiet Sun velocity power is displayed in
the lower panel. The contour represents the penumbra boundary. The dotted
white box in the SDO/HMI velocity amplification map (for $5.5-7.5$ mHz)
indicates the IBIS FOV.  Figure 6: Acoustic power in the $5$ mHz band ($1$ mHz
width) for the three AIA channel spanning from upper chromosphere to the lower
corona (upper row). Maps of $5$ mHz to $3$ mHz power ratio for the three AIA
channels (lower row). The box in all panels indicates the IBIS FOV.

###  3.2 Wave power as a function of height

In addition to the analysis of the photospheric and chromospheric magnetic
field geometries and their relation with the spatial distribution of the blue
dots (Fig. 1f and Fig. 2), associated with enhanced coronal FIP effects (see
Fig. 2), we have examined the spatial distribution of wave power as a function
of atmospheric height. In particular, by employing chromospheric IBIS Ca ii
circular polarisation (CP) and line core Doppler compensated intensity
($I_{\mathrm{core}}$) measurements, we computed spatially-resolved power maps
averaged within the frequencies range of $4-10$ mHz, which are shown in the
top panels of Fig. 5. This frequency range is chosen in such a way as to
include the maximum of the power spectrum which is dominated by frequencies in
the range of $4-4.5$ mHz in the chromosphere. These maps display unique
changes in Fourier power that are cospatial with the blue dots depicted in
Fig. 1f, which have been previously linked to the coronal FIP effect (Baker et
al., 2021; Stangalini et al., 2021).

In particular, at the locations of the blue dots, we find an excess of
magnetic wave power (upper-left panel of Fig. 5) and a power deficit of
magneto-acoustic wave power (upper-right panel of Fig. 5). Also, after
averaging the whole spectra in two isolated boxes (towards the left and right
sides of the umbra; see the boxes drawn in the upper-left panel of Fig. 5), it
is possible to see a broader overall magnetic and magneto-acoustic power
spectra (i.e., relatively large power over a wider range of frequencies)
towards the left side of the umbra (middle panels of Fig. 5) compared to those
on the right side. For completeness, we also plot the spectra at the location
of the blue dots (see the green curves in the two middle panels).
Many authors (see for instance Brown et al., 1992) have reported a high
frequency power enhancement in the $5.5-7.5$ mHz band around active regions at
both photospheric and chromospheric heights. This effect is also referred to
as acoustic halos, and is characterized by a power enhancement up to $40-60\%$
with respect to nearby quiet Sun (Hindman & Brown, 1998). It was suggested
that this acoustic enhancement could be due to fast wave refraction due to
inclined magnetic fields, in proximity of the equipartition layer (Khomenko &
Collados, 2009).
In order to check whether the observed differences in the power spectrum
between the left and right side of the umbra are accompanied by a similar
asymmetry of the acoustic halo, we computed the amplification map of the
Doppler velocity power at high frequency ($5.5-7.5$ mHz), with respect to the
quiet Sun, using SDO/HMI Dopplergrams from a larger FOV centered around the
sunspot. From the lower panel of Fig. 5, a high frequency acoustic flux
imbalance (with respect with the quiet Sun) is detected towards the left side
of the sunspot, cospatial with the locations where enhanced magnetic wave
activity and suppressed magneto-acoustic wave activity is found in the IBIS
data sequence. The acoustic halo displays a power enhancements of the order of
$50\%$, thus in agreement with previous studies (Hindman & Brown, 1998),
although asymmetric with respect to what generally reported.
Furthermore, we investigated the wave power extracted from EUV intensity data
associated with different temperature responses in the SDO/AIA filtergrams. In
particular we have used the three SDO/AIA channels at 304 Å
($\log_{10}$T$\approx 4.7$), 171 Å ($\log_{10}$T$\approx 5.8$), and 335 Å
($\log_{10}$T$\approx 6.4$), where T is the peak temperature response of the
relevant channels (Boerner et al., 2012). As such, the selected SDO/AIA
channels sample different approximate heights in the solar atmosphere, ranging
from the lower transition region through to the upper corona. In particular,
we look for differences in the acoustic flux transmission between these layers
that can be linked to the locations where the coronal FIP effect is present
(blue dots and FIP bias map shown in Fig. 2). In the upper panels of Fig. 6 we
show the spatially-resolved power maps for the three SDO/AIA channels at a
frequency of 5 mHz, which is the same frequency that dominates the
chromospheric locations corresponding to the blue dots in Fig. 1f. In
addition, the lower panels of Fig. 6 show the ratios of oscillatory power
between the 5 mHz and 3 mHz frequencies for each channels. Once again, despite
the apparent visual symmetry of the sunspot (see, e.g., Fig. 1e), we observe
an acoustic flux imbalance between the left and right side of the magnetic
structure. Indeed, for the coronal channels (171 Å and 335 Å), we note that
there exists specific spatial locations, notably towards the opposite side of
the blue dots locations, where there is a 5 mHz power excess, while this is
not the case for the 304 Å map, which is formed lower in the solar atmosphere
and displays an almost homogeneous ring of oscillatory power (similar to that
shown in comparable upper-chromospheric circular-shaped sunspot umbrae; Jess
et al., 2013).

Similarly, the lower panels of Fig. 6 highlight a power deficit at coronal
heights cospatial with the locations displaying the enhanced FIP effect. In
particular, we observe that the high-frequency acoustic flux is not able to
penetrate the upper layers of the solar atmosphere at all locations,
manifesting as a clear deficit of power in the location of the blue dots
(shown in Fig. 1f and Fig. 2). This means that the wave energy is blocked/lost
at some point in the lower atmosphere, suggesting that the umbral magneto-
acoustic waves linked with the blue dots in Fig. 1f are unable to reach
coronal heights. It is important to keep in mind that the SDO/AIA filtergrams
are sensitive only to intensity (proxy for density in optically thin media)
fluctuations and it is possible that these fluctuations exist but they are
smaller than the detection limit of the instrument.

We summarize our main observational findings as follows:

  1. 1.

The averaged photospheric inclination angle of the magnetic field lines are
not significantly different between the left and right sides of the umbra

  2. 2.

Despite the apparent symmetry of the umbra, at the locations corresponding to
the blue dots in Fig. 1f (i.e., the locations linked with the coronal FIP
effect) we measure a faster expansion of the magnetic field lines and a
significant drop in vertical plasma density.

  3. 3.

High-frequency (i.e., $4-10$ mHz) power maps of chromospheric IBIS CP and
$I_{\mathrm{core}}$ measurements display both an excess of magnetic power and
a deficit of magneto-acoustic wave power in the locations of the blue dots in
Fig. 1f (i.e., towards the left side of the umbra). This region also displays
broadened CP and $I_{\mathrm{core}}$ power spectra.

  4. 4.

An asymmetric wave flux excess at high frequencies (i.e., acoustic halo
$5.5-7.5$ mHz) is observed in the photospheric SDO/HMI LOS velocity data.

  5. 5.

The acoustic flux (at 5 mHz) on the left side of the sunspot (i.e., cospatial
with the blue dots depicted in Fig. 1f) does not efficiently reach coronal
heights, suggesting the presence of different wave propagation mechanisms at
opposite sides of the sunspot umbra.

##  4 Discussion and Conclusions

In papers A and B, it was found that regions of high FIP bias at coronal
heights are magnetically linked to chromospheric locations where magnetic
fluctuations are detected. The link between magnetic perturbations and high
FIP bias was already proposed by Laming (2015) and Baker et al. (2021) and the
aforementioned works represent observational evidence of this. Also, in papers
A and B a surprising asymmetry in the spatial distribution of magnetic
oscillations was observed, despite the apparent circular symmetry of the
sunspot investigated. These magnetic perturbations at chromospheric heights
were found to be linked to the high FIP bias locations in the corona. It was
also noted that the magnetic perturbations were all located on the same side
of the trailing magnetic polarity of AR NOAA 12546. In addition, from the
analysis of the same target at chromospheric height, Houston et al. (2020)
observed the signatures of intermediate shocks. Although papers A and B were
not focused on the investigation of the spatially asymmetric wave power, but
only on the detection of the magnetic perturbations and their link with high
FIP bias locations in the corona, a few options were put forward as a possible
explanation for the excitation of the magnetic perturbations themselves. It
was speculated that the magnetic field geometry or connectivity with the
outside diffuse fields could play a role in the wave excitation and
propagation. Along the same line, Houston et al. (2020) also argued that the
increased level of wave activity found in the same region of the sunspot could
be due to the magnetic field geometry, through the mode conversion mechanism
which converts MHD waves in different modes with an efficiency depending on
the magnetic field inclination with respect the wavevector (Schunker & Cally,
2006). This point was also further commented in papers A and B. Indeed, the
authors in paper A concluded that a possible role between the mode conversion
and the detected magnetic disturbances could exist if one takes into account
the asymmetry of the distribution of the magnetic fluctuations. In this
regard, although this sunspot with its peculiarities is but one case study,
the results reported here suggest a likely plausible explanation.

Our results show that the magnetic field inclination at the equipartition
layer (i.e. where the mode conversion takes place), located very close to the
low photosphere (i.e., $-1\geq\log_{10}(\tau)\geq 0$), is not significantly
different on the two opposite sides of the sunspot. This means that waves
travelling in all directions (e.g. $p$-modes) would experience the same
conditions (i.e. attack angle) and therefore their conversion can not justify
the asymmetric wave power observed on the left side of the sunspot (i.e. blue
dots). While on the one hand, we can rule out the mode conversion for waves
travelling in all directions ($p$-modes), we could hypothesize the existence
of a driver that acts spatially in a different way. Indeed, the study of the
power spectrum at high frequencies for the chromospheric high-resolution CP
signal and core intensity from IBIS data suggests to us an asymmetric
photospheric driver. Furthermore, the analysis of the lower synoptic SDO/HMI
observations shows an imbalanced LOS velocity power flux detected in the left
side of the whole AR. Indeed, this map has shown a broadening of the power
spectrum, referred as acoustic halo, with respect to that of a quiet sun area
(where the contribution comes from the $p$-modes only). This broadening is
also observed inside the umbra at the locations of the blue dots.
It was suggested that the halos are due to fast wave refraction in proximity
to the equipartition layer and inclined magnetic fields (Khomenko & Collados,
2009). However, in contrast to what previously reported, in this case the halo
is asymmetric and cospatial with the locations of the blue dots inside the
umbra.
An excess of acoustic wave power in between the two polarities, due to strong
photospheric plasma fluctuations in between the two polarities, could result
in an excess of incident wave power on the left side of the sunspot itself
(see Fig. 7). In this regard, it is interesting to note that an excess moving
magnetic feature activity in between the two polarities is seen in HMI imagery
covering the same observing window and depicted in Fig. 7 (see also online
movie). This provides further evidence of an high level of small scale plasma
dynamics and perturbations that may result in a variation of the acoustic
field and power. We recall that MMFs are manifestations of sunspot decay and
result from erosion of the sunspot’s magnetic field by turbulent plasma
motions (e.g., Solanki, 2003). These waves travelling toward the umbra could
experience a largely inclined equipartition layer before entering the umbra as
shown by the purple surface representing the equipartition layer in Fig. 2. In
other words, the impact angle at the equipartition layer is particularly
large, and this is an ideal condition for their conversion into magnetic-like
waves. In Fig. 2 we also note that the blue dots correspond to regions within
the umbra which a wave travelling from the left side would reach immediately
after crossing the equipartition layer. In addition, the equipartition layer
itself, appears asymmetric with respect to the centre of the umbra (i.e. a
steeper increase on the left side with respect to the right), thus possibly
affecting the conversion coefficient.

However, it is important to note that the computation of the equipartition
layer requires the knowledge of gas density (to estimate the Alfvén speed).
The NICOLE inversion code assumes hydrostatic equilibrium for the calculation
of this parameter, and this may slightly shift the inferred position of the
equipartition layer. However, we would like to point out that the most
relevant aspect of our interpretation is that the equipartition layer crosses
and intercepts the photosphere around the umbra, not its exact position. This
is guaranteed by the extreme magnetic flux of the studied sunspot. We argue
that, if the above mode conversion scenario is true, the FIP effect would be
aligned with the regions outside the sunspot where there is a local increase
of the acoustic power. This fact could be directly checked in the Hinode/EIS
data or, in the future, in the Solar Orbiter SPICE and Solar-C observations.

Figure 7: Cartoon of the excitation mechanism of the magnetic-like waves.

However, an additional aspect should be considered. The spectropolarimetric
inversions also show a very interesting and unnoticed aspect. On the left side
of the sunspot, i.e. where the magnetic perturbations were detected, the field
lines undergo a faster expansion with height. As a natural result, this is
also accompanied by a stronger decrease of the plasma density with height. We
note that these two aspects determine the ideal conditions for the development
of magneto-acoustic shocks at low heights in the atmosphere, through which the
energy contained into upward propagating waves can be dissipated well before
reaching the temperature minimum, as one would expect. In particular, the fast
density decrease with height determines the ideal condition for a fast
steepening of the wave amplitude. The intermediate shocks reported by Houston
et al. (2020) and the intense magneto-acoustic activity and magnetic
perturbations reported in papers A and B appear in agreement with this
scenario. This is also independently corroborated by SDO/AIA observations
indicating that, compared to the right side of the sunspot, acoustic wave
energy on the left side is converted/dissipated much lower in the atmosphere.
We argue that the large observed density drop could be responsible for the
formation of shocks at very low altitudes. However, we note that the larger
field inclinations found in the chromosphere towards the left side of the
sunspot may provide the necessary conditions to support a high-frequency power
halo outside the umbra, similar to that first reported by Toner & Labonte
(1993) and Braun & Lindsey (1999). As revealed by Khomenko & Collados (2009),
such power halos may be the result of wave refraction in the vicinity of the
plasma-$\beta=1$ layer. Inspection of Figures 2 & 3(d) substantiate this
hypothesis, whereby the plasma characteristics present towards the left side
of the sunspot act to support wave refraction, mode conversion, and ultimate
dissipation of wave energy through shock formation.

Nevertheless, the question remains as to whether the faster expansion of the
field lines on the left side is a consequence of the wave energy dissipation
at low heights through the aforementioned shocks, or a consequence of the
overall magnetic connectivity of the sunspot. Previous studies would suggest
it is the latter, as the effect of shocks on magnetic field geometry is
localised and any perturbations to the field are temporary as the field
relaxes back to equilibrium (e.g., de la Cruz Rodríguez et al., 2013; Houston
et al., 2018; Bose et al., 2019). Further, only the intermediate shocks in
Houston et al. (2020) are localised to the FIP region. They detect the
intensity signatures associated with slow magneto-acoustic shocks across the
entire outer umbra, inferring that if shocks were capable of a bulk change in
the umbral field, this would be evident on the side of the sunspot unconnected
with the FIP region.

In this regard, it is worth underlining that the context magnetogram obtained
by SDO/HMI over a larger FOV shows the presence of small-scale magnetic field
of opposite polarity on the same side, which constitute the trailing polarity
of this AR. We argue that due to the magnetic connectivity the bending of the
field lines is larger with the creation of low lying loops. This is in line
with the simulations reported in Dahlburg et al. (2016), which show that the
FIP effect is stronger in short, high temperature loops.

In addition, it is also unclear whether or not the wave steepening and mode
conversion are concurrent processes. In this regard, it is worth noting that
the absence of intensity fluctuations in correspondence with the magnetic
power seems to suggest that at those locations acoustic energy is mostly
converted into magnetic-like waves. Further studies are necessary in order to
shed light onto this aspect.

Further, we would like to note that this kind of study has additional
important implications. The FIP effect could be regarded as a proxy for the
wave energy dissipation. This is a long-debated problem (Jess et al., 2015)
and still the identification of the physical processes with which energy is
transferred from waves to the plasma remains a challenge.

Finally, our results highlight the importance of studying simultaneously
different heights of the solar atmosphere, by combining simultaneous or nearly
simultaneous ground-based spectropolarimetric observations in the lower
atmosphere, with high spectral resolution data of the corona acquired from
space. This is important not only for the investigation of the FIP effect, but
also for the wave dissipation mechanisms for which the FIP effect itself could
be considered as a proxy.

It is worth noting that this possibility will be widened by the new solar
mission Solar-C EUVST (Shimizu et al., 2011; Suematsu & Solar-C Working Group,
2016), which will provide an unprecedented view of the corona at high
temporal, spatial and spectral resolution. In this regard, this case study can
be considered as a pathfinder for the full exploitation of its data in
combination with high resolution spectropolarimetric imaging of the lower
atmosphere.

###### Acknowledgements.

The authors are grateful to the anonymous referee for useful comments. This
research received funding from the European Union’s Horizon 2020 Research and
Innovation 531 program under grant agreements No 824135 (SOLARNET) and No
729500 (PRE-EST). This work was supported by the Italian MIUR-PRIN grant 2017
”Circumterrestrial environment: Impact of Sun-Earth Interaction” and by the
Istituto NAzionale di Astrofisica (INAF). DBJ and SDTG wish to thank Invest NI
and Randox Laboratories Ltd. for the award of a Research & Development Grant
(059RDEN-1), in addition to the UK Science and Technology Facilities Council
(STFC) for the consolidated grant ST/T00021X/1. SJ acknowledges support from
the European Research Council under the European Union Horizon 2020 research
and innovation program (grant agreement No. 682462) and from the Research
Council of Norway through its Centres of Excellence scheme (project No.
262622). The authors wish to acknowledge scientific discussions with the Waves
in the Lower Solar Atmosphere (WaLSA; www.WaLSA.team) team, which is supported
by the Research Council of Norway (project number 262622), and The Royal
Society through the award of funding to host the Theo Murphy Discussion
Meeting “High-resolution wave dynamics in the lower solar atmosphere” (grant
Hooke18b/SCTM). D.B. is funded under STFC consolidated grant number
ST/S000240/1 and L.v.D.G. is partially funded under the same grant. The work
of D.H.B. was performed under contract to the Naval Research Laboratory and
was funded by the NASA Hinode program. D.M.L. is grateful to the Science
Technology and Facilities Council for the award of an Ernest Rutherford
Fellowship (ST/R003246/1). The italian scientific contribution to Solar-C is
supported by the Italian Space Agency (ASI) under contract to the co-financing
National Institute for Astrophysics (INAF) Accordo ASI-INAF 2021-12-HH.0
”Missione Solar-C EUVST - Supporto scientifico di Fase B/C/D”.

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