The role of the Lorentz force in sunspot equilibrium

A&A, 699, A149 (2025)
https://doi.org/10.1051/0004-6361/202554241
c The Authors 2025
Astronomy
&
Astrophysics
The role of the Lorentz force in sunspot equilibrium
J. M. Borrero1,? , A. Pastor Yabar2 , M. Schmassmann1 , M. Rempel3, M. van Noort4, and M. Collados5,6
1
Institut für Sonnenphysik, Georges-Köhler-Allee 401A, D-79110 Freiburg, Germany
2
Institute for Solar Physics, Department of Astronomy, Stockholm University, AlbaNova University Centre, 10691 Stockholm,
Sweden
3
High Altitude Observatory, NSF National Center for Atmospheric Research, 3080 Center Green Dr., Boulder 80301, USA
4
Max Planck Institute for Solar System Research, Justus-von-Liebig 3, D-37077 Göttingen, Germany
5
Instituto de Astrofísica de Canarias, Avd. Vía Láctea s/n, E-38205 La Laguna, Spain
6
Departamento de Astrofísica, Universidad de La Laguna, E-38205 La Laguna, Tenerife, Spain
Received 24 February 2025 / Accepted 27 May 2025
ABSTRACT
Context. Sunspots survive on the solar surface for timescales ranging from days to months. This requires them to be in an equilibrium
involving magnetic fields and hydrodynamic forces. Unfortunately, theoretical models of sunspot equilibrium are very simplified as
they assume that spots are static and possess a self-similar and axially symmetric magnetic field. These assumptions neglect the role
of small-scale variations of the magnetic field along the azimuthal direction produced by umbral dots, light bridges, penumbral fila-
ments, and so forth.
Aims. We aim to study whether sunspot equilibrium is maintained once azimuthal fluctuations in the magnetic field, produced by the
sunspot fine structure, are taken into account.
Methods. We apply the FIRTEZ Stokes inversion code to spectropolarimetric observations to infer the magnetic and thermodynamic
parameters in two sunspots located at the disk center and observed with two different instruments: one observed from the ground with
the 1.5-meter German GREGOR Telescope and another with the Japanese spacecraft Hinode. We compare our results with three-
dimensional radiative magnetohydrodynamic simulations of a sunspot carried out with the MuRAM code.
Results. We infer clear variations in the gas pressure and density of the plasma directly related to fluctuations in the Lorentz force and
associated with the filamentary structure in the penumbra. Similar results are obtained in the umbra despite its lack of an observed fil-
amentary structure. Results from the two observed sunspots are in excellent qualitative and quantitative agreement with the numerical
simulations.
Conclusions. Our results indicate that the magnetic topology of sunspots along the azimuthal direction is very close to magnetohy-
drostatic equilibrium, thereby helping to explain why sunspots are such long-lived structures capable of surviving on the solar surface
for days or even full solar rotations.
Key words. magnetohydrodynamics (MHD) – polarization – Sun: magnetic fields – Sun: photosphere – sunspots
1. Introduction
Sunspots are the most prominent manifestation of the solar
magnetic activity cycle (Solanki 2003). Their inner region, the
umbra, is cool and appears dark. The umbra can feature so-
called umbral dots and light bridges, which are intrusions of
convective flows that disrupt the otherwise homogeneous mag-
netic field. The umbra is surrounded by a hotter and brighter
halo, the penumbra, which features a very distinct filamen-
tary structure. It is characterized by the presence of Evershed
flow channels that are mostly concentrated along horizontal and
radially aligned weak magnetic field lines, and that alternate
in the azimuthal direction with a more vertical and stronger
magnetic field (Lites et al. 1993; Solanki & Montavon 1993;
Stanchfield et al. 1997).
In addition, sunspots harbor a myriad of highly dynamic
phenomena that evolve on timescales comparable to the Alvén
crossing timescales and convective timescales (i.e., minutes to
hours in the upper atmosphere). Examples of such phenomena
include umbral dots and penumbral grains (Sobotka & Jurčák
?
Corresponding author.
2009), overturning convection (Ichimoto et al. 2007a;
Scharmer et al. 2011), waves (Bloomfield et al. 2007a,b;
Löhner-Böttcher & Bello González 2015), and Evershed clouds
(Cabrera Solana et al. 2007, 2008). In spite of this, and due to
their long lifetimes, which span from days to several months,
sunspots are thought to be in an equilibrium involving magneto-
hydrodynamic (MHD) forces (Rempel & Schlichenmaier 2011).
However, theoretical descriptions of this equilibrium are very
simplified. They assume, for instance, that the magnetic field is
self-similar and axially symmetric (Low 1975, 1980a,b; Pizzo
1986, 1990; Jahn & Schmidt 1994; Khomenko & Collados
2008). This assumption completely ignores the large variations
in the magnetic field in the azimuthal direction caused by
small-scale features such as umbral dots (Ortiz et al. 2010),
light bridges (Falco et al. 2016), and penumbral filaments. Here
we address the long-neglected equilibrium along the azimuthal
direction and conclusively demonstrate, using observations and
numerical simulations, that in the photosphere both the umbra
and penumbra are very close to azimuthal magnetohydrostatic
(MHS) equilibrium despite the presence of strong inhomo-
geneities in the magnetic field. These results provide decisive
observational and theoretical support for the idea that sunspots
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A149, page 1 of 12

Borrero, J. M., et al.: A&A, 699, A149 (2025)
Table 1. Atomic parameters of the spectral lines analyzed in this work.
Ion λ0
1
χlow
1
log(g f) Elec. conf1
σ α geff Instr.
[Å] [eV]
Fe i 6301.5012 3.654 −0.7184 5
P2 − 5
D0 840 0.243 1.67 Hinode/SP
Fe i 6302.4936 3.686 −1.1655 5
P1 − 5
D0 856 0.241 2.5 Hinode/SP
Fe i 15648.515 5.426 −0.6692 7
D1 − 7
D1 9752
0.2292
3.0 GREGOR/GRIS
Fe i 15662.018 5.830 0.1903 5
F5 − 5
F4 11973
0.2403
1.5 GREGOR/GRIS
Notes. λ0 represents the central laboratory wavelength of the spectral line. σ and α represent the cross section (in units of Bohr’s radius
squared, a2
0) and velocity parameter of the atom undergoing the transition, respectively, for collisions with neutral atoms under the ABO theory
(Anstee & O’Mara 1995; Barklem & O’Mara 1997; Barklem et al. 1998). 1
Values taken from Nave et al. (1994). 2
Values taken from Borrero et al.
(2003). 3
Values taken from Bloomfield et al. (2007a). 4
Values taken from Bard et al. (1991). 5
Values provided by Brian C. Fawcett from the
Rutherford Appleton Laboratory (private communication).
slowly evolve around an equilibrium state and are, in leading
order, in MHS equilibrium, thereby helping to explain their long
lifespans.
2. Observations
The observations analyzed in this work correspond to spec-
tropolarimetric observations in several magnetically sensitive
Fe i spectral lines of two sunspots located very close to the
disk center. The first sunspot was observed from the ground
using the GRIS instrument (Collados et al. 2012) attached to the
German 1.5-meter GREGOR solar telescope (Schmidt et al.
2012), whereas the second sunspot was observed with the SP
instrument (Lites et al. 2001; Ichimoto et al. 2007b) on board the
Japanese spacecraft Hinode (Suematsu et al. 2008; Tsuneta et al.
2008; Shimizu et al. 2008; Kosugi et al. 2007).
For the first sunspot, GREGOR’s Infrared Spectrograph
(GRIS) was used to record the Stokes vector Iobs
(x, y, λ) =
(I, Q, U, V) across a 4 nm wide wavelength region centered
around 1565 nm and with a wavelength sampling of δλ ≈
40 mÅ pixel−1
. This wavelength region was therefore sampled
with 1000 spectral points, out of which we selected a 7.6 Å
wide region with Nλ = 190 spectral points that included two
Fe i spectral lines (see Table 1). The spectral line Fe i 15652.8 Å
was also recorded but it is not included in our analysis because
in the umbra it appears blended with two molecular lines
(Mathew et al. 2003) and the FIRTEZ inversion code employed
in this work (see Section 3) cannot model them. During data
acquisition, five accumulations of 30 ms each were recorded for
each modulation state. This results in a total exposure time of
0.6 seconds per slit, yielding a noise level in the polarization sig-
nals of σq ≈ σu ≈ σv ≈ 10−3
in units of the quiet Sun continuum
intensity.
In the second sunspot, Hinode’s spectropolarimeter (SP)
recorded the Stokes vector Iobs
(x, y, λ) = (I, Q, U, V) across a
≈0.24 nm wide wavelength region around 630 nm and with a
wavelength sampling of δλ ≈ 21.5 mÅ pixel−1
. This wavelength
region was sampled with a total of 112 spectral points includ-
ing two Fe i spectral lines (see Table 1). In these data the total
exposure time per slit position is 4.8 seconds, resulting in similar
noise levels in the polarization signals to the ones in GREGOR
data: σq ≈ σu ≈ σv ≈ 10−3
in units of the quiet Sun continuum
intensity.
The first sunspot is NOAA AR 12049 and was observed on
May 3, 2015. Different studies of the same sunspot have been
presented elsewhere (Franz et al. 2016; Borrero et al. 2016). By
correlating our images with simultaneous HMI/SDO full-disk
continuum images, we estimate that the sunspot center was
located at coordinates (x, y) = (7300
, −8300
) (measured from the
disk center). These values correspond a to heliocentric angle of
Θ = 6.5◦
(µ = 0.993). The image scale is δx = 0.13500
pixel−1
and δy = 0.13600
pixels−1
along the x and y axes, respectively.
The scanned region contains Nx = 411 and Ny = 341 pixels
along each spatial dimension, resulting in a total field of view
of 55.500
× 46.400
. The width of the spectrograph’s slit was set
to 0.2700
(i.e., twice the scanning step). Details about the data
calibration in terms of flat-fielding, spectral stray-light correc-
tion, fringe removal, and wavelength calibration can be found in
Borrero et al. (2016).
The second sunspot is NOAA AR 10933 and was observed
on January 6, 2007. This sunspot has already been analyzed
in previous works (Franz & Schlichenmaier 2013). The sunspot
center was located at coordinates (x, y) = (7200
, −2100
) (mea-
sured from the disk center). These values correspond to a helio-
centric angle of Θ = 4.4◦
(µ = 0.997). The image scale is
δx = 0.1500
pixel−1
and δy = 0.1600
pixels−1
along the x and y
axes, respectively. The scanned region contains Nx = 350 and
Ny = 300 pixels along each spatial dimension, resulting in a
total field of view of 52.500
× 48.000
.
3. Stokes inversion
In this work we have performed two distinct inversions of each of
the datasets described in Sect. 2. The first one was a pixel-wise
inversion whereby each spatial pixel on the solar surface (x, y)
was treated independently. The second one was a point spread
function (PSF)-coupled inversion employing the PSFs described
in Sect. 4. In some aspects both work identically. Let us describe
those first.
The inversion of the observed Stokes vector, Iobs(λ, x, y), was
performed by using an initial atmospheric model described by
the following physical parameters: temperature, T(x, y, z), mag-
netic field, B(x, y, z), and line-of-sight velocity, vz(x, y, z). On the
XY plane we used the same number of (x, y) grid cells and spac-
ing as the observations. In the vertical z direction the atmospheric
model was discretized in 128 points with a vertical spacing of
∆z = 12 km. The solution to the radiative transfer equation for
polarized light from this initial model in the vertical direction
yields the synthetic Stokes vector Isyn(λ, x, y), which was com-
pared to the observed ones via a χ2
merit function. The FIRTEZ
code also provides the analytical derivatives of the Stokes vector
with respect to the aforementioned physical parameters.
It is at this point that the pixel-wise and PSF-coupled inver-
sions differ. On the pixel-wise inversion, the derivatives at
A149, page 2 of 12

Table 2. Number of nodes employed in the Stokes inversion.
Inversion type T vz Bx By Bz
Hydrostatic 4 2 2 2 2
Magnetohydrostatic 2 4 4 4 4
location (x, y) enter a Levenberg-Marquardt algorithm
(Press et al. 1986), which combined with the singular value
decomposition method (Golub & Kahan 1965) provides the
perturbations in temperature, magnetic field, and so on, to
be applied to the initial physical parameters so that the next
solution to the polarized radiative transfer equation will yield
a Isyn(λ, x, y) that is closer to the observed Iobs(λ, x, y) (i.e., χ2
minimization). The perturbations were only applied at discrete
z locations called nodes, with the final z stratification being
obtained via interpolation across the 128 vertical grid points.
The number of nodes employed in this work is provided in
Table 2. The matrices to be inverted, at a given spatial location,
are squared matrices with the following number of elements:
[N(T) × N(Bx) × N(By) × N(Bz) × N(vz)]2
, where N(T) refers
to the number of nodes allowed for the temperature, N(Bx) the
number of nodes allowed in the x component of the magnetic
field, and so forth. In this way the initial model at location (x, y)
was iteratively perturbed until χ2
reaches a minimum, at which
point we assumed that the resulting atmospheric model at that
location is the correct one.
In the PSF-coupled inversion, FIRTEZ closely follows the
method originally developed by van Noort (2012). It proceeds
in a similar fashion as in the pixel-wise inversion up until the
χ2
calculation. The merit function was not obtained by com-
paring Iobs(λ, x, y) with Isyn(λ, x, y) but rather with Ĩsyn(λ, x, y)
which results from the spatial convolution of Isyn(λ, x, y) with
the instrumental PSF F (x, y). Details about the PSFs used in
this work are given in Section 4. Now, the derivatives of the
Stokes vector with respect to the physical parameters are cou-
pled horizontally, so that, a perturbation of a physical parameter
at location (x0
, y0
) will affect the outgoing Stokes vector at (x, y).
Therefore the χ2
minimization cannot be performed individually
for each (x, y) pixel. Instead the minimization must be done at
once considering all pixels on the observed region simultane-
ously. Because of this the square matrices to be inverted are now
much larger than in the pixel-wise inversion. The number of ele-
ments is now [N(T)×N(Bx)×N(By)×N(Bz)×N(vz)×Nx ×Ny]2
.
For the coupled inversion FIRTEZ does not employ the singular
value decomposition method. Instead the coupled matrix is con-
sidered to be sparse and its inverse is found via the biconjugate
stabilized gradient method (BiCGTAB; van der Vorst 1992).
Each of the two described inversions, pixel-wise and PSF-
coupled, were first run under the assumption of vertical hydro-
static equilibrium (HE) for the gas pressure. Once this was done,
the magnetic field from the hydrostatic inversion was modified
so as to remove the 180◦
ambiguity in Bx and By (Metcalf et al.
2006) via the non-potential field calculation method (Georgoulis
2005). The resulting atmospheric model was then employed,
as an initial guess of an inversion where the gas pressure was
obtained under MHS equilibrium. A more detailed description
on the approach followed can be found in Borrero et al. (2021).
From all inferred physical parameters by the Stokes inver-
sion, in our study we focus mainly on the following ones: the
module of the magnetic field, kBk, inclination of the magnetic
field with respect to the normal direction to the solar surface,
−1 0 1
x [Mm]
−1
0
1
y
[Mm]
GREGOR/GRIS
−1 0 1
x [Mm]
−1
0
1
Hinode/SP
−0.4 0.0 0.4
x [Mm]
−0.4
0.0
0.4
MHD simulations
−5 −4 −3 −2 −1 0
log[F(x, y)]
Fig. 1. Point spread functions employed in the data analysis. Left panel:
PSF (diffraction-limited plus seeing) used in the coupled-inversion of
GREGOR/GRIS data. Middle panel: PSF (diffraction-limited) used in
the coupled-inversion of Hinode/SP data. For both GREGOR and Hin-
ode data, the full PSF (±1.6 Mm) was used in the synthesis, whereas
in the inversion only the square box was considered. Right panel: Ideal
PSF (Airy function) used to degrade the MHD simulations.
γ = cos−1
(Bz/kBk), gas pressure, Pg, density, ρ, and finally also
the Lorentz force, L = (4π)−1
(∇ × B) × B. Spatial derivatives of
the magnetic field were determined by employing fourth-order
centered finite differences over an angular array with a constant
∆φ step such that R∆φ (R being the arc radius) was approxi-
mately equal to the grid size. We note that using second- or even
sixth-order derivatives does not impact results.
4. Point spread functions
The PSF-coupled inversions (described in Section 3) were car-
ried out with FIRTEZ employing the PSFs F (x, y) shown in
Figure 1 for GREGOR/GRIS data (left panel) and Hinode/SP
data (middle panel). In the latter case, F (x, y) corresponds to
a diffraction-limited PSF for the SOT telescope on-board Hin-
ode at 630 nm, including a 50 cm primary mirror with a 17.2 cm
central obscuration caused by the secondary mirror and three
4 cm wide spiders and a 1.5 mm de-focus (Danilovic et al. 2008;
Ebert et al. 2024). In the former case, GREGOR being a ground-
based telescope subject to seeing effects, the PSF was modeled
as the sum of two different contributions:
Fgregor(x, y) = (1 − α)Fdiff(x, y) + αFseeing(x, y), (1)
where Fdiff(x, y) is the diffraction-limited PSF for a telescope
operating at 1565 nm, including a 144 cm primary mirror with
a 45 cm central obscuration caused by the secondary mirror and
four 2.2 cm wide spiders. The second contribution, Fseeing(x, y),
is meant to approximately represent the seeing, and it was mod-
eled as a Gaussian function with σ = 0.500
(or ≈375 km at
disk center) and carrying 43% of the energy on the detector
(α = 0.43; Orozco Suárez et al. 2017). While originally we had
taken a much broader Gaussian with σ = 500
(≈3750 km at the
disk center), the contribution from such a Gaussian across the
employed area of ≈9 Mm2
(see Fig. 1) is essentially constant,
leading to a poor convergence of the coupled inversion. A nar-
rower Gaussian allowed the algorithm to converge much better,
while still mimicking the effects of the seeing. It is worth not-
ing that, while the actual seeing conditions change during the
observations, Fseeing is considered to be time-independent.
According to the Rayleigh criterion, if we consider only
the diffraction-limited contribution to the PSFs, both GREGOR
and Hinode data feature a very similar spatial resolution of
about 0.300
because, while Hinode’s primary mirror is three
times smaller, it also operates at wavelengths approximately
A149, page 3 of 12

three times shorter. Even though the full PSFs in Figure 1 were
employed during the synthesis and to compute the merit func-
tion χ2
between the observed and synthetic Stokes profiles (Iobs
and Isyn
, respectively), during the inversion (i.e., to determine
the derivatives of χ2
with respect to the physical parameters)
only the red squares were considered. In both cases, Hinode
and GREGOR, the region enclosed by these red squares con-
tains more than 75% of the total amount of light received by a
given pixel on the detector. In the case of Hinode, the square box
covers ±4 pixels along each spatial dimension around the central
pixel, whereas for GREGOR the square box covers ±6 pixels.
For completeness, Fig. 1 (right panel) also includes the PSF
employed to degrade the MHD simulations to a spatial resolu-
tion comparable to that of GREGOR and Hinode observations
(see Section 5).
5. Magnetohydrodynamic simulations
The MHD simulations employed in this work were produced
with the MuRAM radiative MHD code (Vögler et al. 2005)
with the specific adaptations for sunspot simulations that are
described in Rempel et al. (2009a,b) and Rempel (2011, 2012).
The code solves the fully compressible MHD equations using a
tabulated equation of state that accounts for partial ionization in
the solar atmosphere assuming equilibrium ionization (based on
the OPAL package by Rogers et al. 1996). The radiation trans-
port module computes rays in 24 directions using a short charac-
teristics approach as described in Vögler et al. (2005). We used
four-band opacities that group spectral lines according to their
contribution height in the solar atmosphere.
We used a sunspot simulation run started from Rempel
(2012), whereby the magnetic field at the top boundary was
forced to be more horizontal with the parameter α = 2 (see
Appendix B). We continued the simulation at a horizontal res-
olution of 32 × 32 km and with a 16 km vertical resolution while
using non-gray radiative transfer with four opacity bins. The
very same simulation, but using a different timestamp, was
employed in Jurčák et al. (2020). The data (Pg, ρ, and B) with
the original 32 × 32 km resolution was convolved with the PSF
presented in Sect. 4 (see also rightmost panel in Fig. 1). This
PSF consists of an Airy function where the first minimum is
located at a distance of 232 km from the peak. This is equiva-
lent to about 0.300
at the disk center, and therefore very similar to
the ideal (i.e., diffraction-limited) spatial resolution in GREGOR
and Hinode data (see Section 4). We note that, after the applica-
tion of this PSF, the data from the MHD simulation were further
binned from 32 × 32 down to 128 × 128 km so as to also have a
comparable spatial sampling as in the observed data. From there,
γ, kBk, and L were calculated. The Lorentz force in the MHD
simulations was also calculated employing fourth-order centered
finite differences to determine the spatial derivatives of the mag-
netic field (see Section 3). A summary of the physical properties
in the simulations is displayed in Figure 2 (third column). Fur-
ther details about the initial and boundary conditions are given
in Appendix B.
6. Results
Figure 2 illustrates some of the physical parameters inferred
from the Stokes inversions and from the MHD simulations.
As expected (see van Noort 2012), the PSF-coupled inversion
(right-side panels on the first two columns) display a larger vari-
ation in the physical parameters than the pixel-wise inversion
(left-side panels in the first two columns). It is worth mentioning
that this is the first time that this method has been successfully
applied to ground-based spectropolarimetric data.
Two azimuthal cuts or arcs (i.e along the coordinate φ) are
also indicated in Figure 2. The outermost and innermost arcs
are located in the penumbra and umbra, respectively. In the
azimuthal direction in the penumbra (outermost arc indicated
by the φ coordinate), there are alternating regions of large incli-
nation (γ ≈ 90◦
; field lines contained within the solar surface)
and regions of much lower inclination (γ ≈ 40◦
; field lines
more perpendicular to the solar surface). These large variations
in the inclination of the magnetic field are produced by the
filamentary structure of the penumbra (Stanchfield et al. 1997;
Martinez Pillet 1997; Mathew et al. 2003; Borrero & Solanki
2008). Across the umbra (innermost arc indicated by the φ coor-
dinate) the inclination, γ, features a much more homogeneous
distribution.
Figure 3 shows scatter plots of the following pairs of physical
quantities in the azimuthal direction (φ coordinate) in the penum-
bra (i.e., outermost arc in Fig. 2): (a) γ − kBk (upper left), (b)
γ−Pg (upper right), (c) γ−ρ (lower left), and (d) Lφ −r−1
∂Pg/∂φ
(lower right). In this figure correlations are indicated by c and
are provided separately for each of the two observed sunspots
as well as for the degraded MHD simulations. We note that, for
the original (i.e., undegraded) simulation data, the correlations
are even higher than the ones presented here in spite of featuring
variations in the physical parameters that are unresolved in the
degraded simulation data. Below each scatter plot, three mini-
panels display the physical quantities in the azimuthal direction.
All physical parameters are taken at a constant height, z∗
, equal
to the average height along φ where the continuum optical depth
is unity: τc = 1. A similar plot to Fig. 3 but for the umbra (i.e.,
innermost arc in Fig. 2) is provided in Section 6.3 and Fig. 5.
Panel a in Fig. 3 shows a clear anticorrelation between
γ and kBk. It highlights the well-known uncombed structure
of the penumbral magnetic field (Solanki & Montavon 1993),
where regions of strong and vertical magnetic fields (γ low;
kBk large) called spines alternate along the φ coordinate with
regions of weaker and more horizontal magnetic fields (γ large;
kBk low) called intraspines (Title et al. 1993; Lites et al. 1993;
Martinez Pillet 1997). Panels b and c show a clear correlation
in regions where the inclination of the magnetic field, γ, is
larger (i.e., intraspines) and regions of enhanced gas pressure,
Pg, and density, ρ. These curves indicate that the penumbral
intraspines possess an excess gas pressure and density com-
pared to the spines. This decade-long theoretical prediction
(Spruit & Scharmer 2006; Scharmer & Spruit 2006; Borrero
2007) is observationally confirmed here for the first time.
Panel d in Fig. 3 shows that the gas pressure and density fluc-
tuations, associated with the uncombed structure of the penum-
bral magnetic field and highlighted in panel a, are directly related
to fluctuations in the azimuthal component of the Lorentz force.
This is demonstrated not only by the high degree of correlation
between Lφ and r−1
∂Pg/∂φ, but also by the fact that the slope of
the scatter plot is unity, indicating that
Lφ = r−1
∂Pg/∂φ, (2)
which corresponds to the φ component (in cylindrical coordi-
nates) of the MHS equilibrium equation,
∇Pg = ρg + L, (3)
where L has already been defined as the Lorentz force.
Our combined results prove that, in the azimuthal φ direc-
tion, penumbral spines and intraspines are in almost perfect
A149, page 4 of 12

0 10 20 30 40
x [Mm]
0
10
20
30
y
[Mm]
φ
0.6 0.7 0.8 0.9 1.0 1.1
AR 12049 ; Ic/Ic,qs at 1565 nm
0 10 20 30 40
x [Mm]
0
10
20
30
y
[Mm]
φ
0.2 0.4 0.6 0.8 1.0 1.2
AR 10933 ; Ic/Ic,qs at 630 nm
0 10 20 30 40
x [Mm]
0
10
20
30
40
y
[Mm]
φ
0.2 0.4 0.6 0.8 1.0 1.2
MHD simulation ; Ic/Ic,qs
0 10 20 30 40
x [Mm]
0
10
20
30
y
[Mm]
φ
4500 5000 5500 6000 6500 7000
AR 12049 ; T(τc = 1) [K]
0 10 20 30 40
x [Mm]
0
10
20
30
y
[Mm]
φ
4500 5000 5500 6000 6500 7000
AR 10933 ; T(τc = 1) [K]
0 10 20 30 40
x [Mm]
0
10
20
30
40
y
[Mm]
φ
4500 5000 5500 6000 6500
MHD simulation ; T(τc = 1) [K]
0 10 20 30 40
x [Mm]
0
10
20
30
y
[Mm]
φ
0 20 40 60 80 100 120
AR 12049 ; γ(τc = 1) [deg]
0 10 20 30 40
x [Mm]
0
10
20
30
y
[Mm]
φ
0 20 40 60 80 100 120
AR 10933 ; γ(τc = 1) [deg]
0 10 20 30 40
x [Mm]
0
10
20
30
40
y
[Mm]
φ
0 20 40 60 80 100 120
MHD simulation ; γ(τc = 1) [deg]
Fig. 2. Summary of analyzed data and inversion results. Top row: Continuum intensity normalized to the average quiet Sun intensity, Ic/Ic,qs.
Middle row: Temperature, T. Bottom row: Inclination of the magnetic field, γ, with respect to the normal vector to the solar surface. All values
were taken at the continuum optical depth, τc = 1. Left column: Sunspot observed with the ground telescope GREGOR and data inverted with
the FIRTEZ code. Middle column: Same as before but for the sunspot recorded with the Hinode satellite. Right column: Sunspot simulated
with the MuRAM code. Each panel is split on two: the left side of the first two columns display the results of the pixel-wise Stokes inversion,
whereas the right side presents the results of the PSF-coupled inversion. In the third column, the left side shows the simulation with the original
horizontal resolution of 32 km × 32 km, while the right side shows the simulation convolved and resampled to 128 km × 128 in order to match the
approximate resolution and sampling of the GREGOR and Hinode data. Each figure contains the direction of the center of the solar disk (blue
arrow) and displays two azimuthal paths with a varying angle, φ, at a constant radial distance from the spots’ center: the outermost arc is in the
penumbra, whereas the innermost arc lies in the umbra.
MHS equilibrium and that neither the time derivative of the
velocity, ∂v/∂t, the velocity advective term, (v · ∇)v, nor the
viscosity term, ∇τ̂, where τ̂ is the viscous stress tensor, play a
significant role in the force balance. Interestingly, in the numer-
ical simulations, the velocity advective term associated with
the Evershed flow does indeed play an important role in the
force balance in the radial direction (Rempel 2012), meaning
that magnetohydrostationary equilibrium is more adequate in the
radial coordinate.
It should be noted that the almost perfect MHS equilibrium
in the azimuthal direction (Eq. 2) is found in the observations
at short and intermediate radial distances from the center of
the sunspots: r/Rspot ∈ [0.1, 0.7] (where Rspot is the radius of
the sunspot) and in a height region restricted around ±150 km
A149, page 5 of 12

1.2 1.8 2.4 3.0 3.6
B [103
g1/2
cm−1/2
s−1
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(a)
GREGOR: c = -0.607 Hinode: c = -0.737 Simulations: c = -0.757
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
B
[10
3
]
30
60
90
γ
[deg]
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
B
[10
3
]
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
3.0
3.6
B
[10
3
]
30
60
90
γ
0.5 1.5 2.5 3.5 4.5
Pg [105
dyn cm−2
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(b)
GREGOR: c = 0.791 Hinode: c = 0.913 Simulations: c = 0.870
−40 −20 0 20 40
φ [deg]
0.5
1.5
2.5
P
g
[10
5
]
30
60
90
γ
−40 −20 0 20 40
φ [deg]
0.5
1.5
2.5
P
g
[10
5
]
−40 −20 0 20 40
φ [deg]
0.5
2.5
4.5
P
g
[10
5
]
30
60
90
γ
0.5 2.5 4.5 6.5
ρ [10−7
g cm−3
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(c)
−40 −20 0 20 40
φ [deg]
2.2
3.2
4.2
ρ
[10
−7
]
30
60
90
γ
[deg]
−40 −20 0 20 40
φ [deg]
2.2
3.2
4.2
rho
[10
−7
]
−40 −20 0 20 40
φ [deg]
0.5
2.5
4.5
6.5
rho
[10
−7
]
30
60
90
γ
−5.0 −2.5 0.0 2.5 5.0
r−1
(∂Pg/∂φ) [10−3
g cm−2
s−2
]
−5.0
−2.5
0.0
2.5
5.0
L
φ
[10
−3
g
cm
−2
s
−2
]
(d)
GREGOR: c = 0.927 Hinode: c = 0.873 Simulations (/5): c =0.974
−40 −20 0 20 40
φ [deg]
−4
−2
0
2
4
L
φ
[10
−3
]
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
L
φ
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
Fig. 3. Scatter plots of the physical quantities along the outermost azimuthal arc (i.e., penumbra) for GREGOR data (red squares), Hinode data
(blue circles), and the MHD simulation (black triangles). Pearson correlations coefficients, c, for each of these pieces of data are provided in the
legend. Linear fits to the data are shown with thick solid color lines. Upper left: γ − kBk. Upper right: γ − Pg. Lower left: γ − ρ. Lower right:
Lφ − r−1
∂Pg/∂φ. Each panel also includes three smaller subpanels in which the variations in the physical quantities along φ are displayed in purple
and green colors. These subpanels are ordered as: GREGOR (left), Hinode (middle), and MHD simulations (right). This is also indicated by the
arrows under the correlation coefficients for each data source. We note that the values of the φ components of the Lorentz force and of the gas
pressure gradient in the simulations have been divided by five.
around the continuum τc = 1 (see Appendix A for more details).
In the MHD simulations the MHS equilibrium is verified over a
similar range of radial distances but over a much broader vertical
region of about 400 km above τc = 1.
6.1. Contribution of the magnetic tension and magnetic
pressure gradient in the Lorentz force
We have already established that the azimuthal component of the
Lorentz force, Lφ, has a one-to-one relationship to the pressure
and density fluctuations in this direction seen both in observa-
tions and simulations (i.e., Eq. 2 is verified). In order to study
this in more detail, we have decomposed Lφ into the contribution
from the magnetic tension, Tφ, and from the magnetic pressure
gradient, Gφ:
Lφ = (4π)−1
(B · ∇)Bφ
| {z }
Tφ
−
1
8πr
∂kBk2
∂φ
| {z }
Gφ
, (4)
where eφ is the unit vector in the azimuthal direction. The result-
ing azimuthal variations in the magnetic tension and magnetic
pressure gradient are shown in Figure 4. As can be seen, the
similarities between the GREGOR and Hinode observations and
the MHD simulations are remarkable. In all cases, the magnetic
pressure gradient, Gφ, dominates over the magnetic tension, Tφ,
with the former reaching its maximum right at the boundary
between spines and intraspines, and the latter peaking at the cen-
ter of the spines and intraspines, respectively.
6.2. Gas pressure fluctuations: Observations versus MHD
simulations
In Figure 3 (panel d) we show the results of the three-
dimensional MHD numerical simulations but dividing Lφ and
r−1
∂Pg/∂φ by a factor of five so that the points of the scatter
plot appear on the same scale as the ones from the GREGOR
and Hinode observations. The same factor has been applied to
the decomposition of the Lorentz force into the magnetic ten-
sion and gas pressure gradient contributions in Figure 4 (bottom
panel). The reason for the larger values of the Lorentz force and
of the gas pressure variations in the azimuthal direction in the
MHD simulations is to be found in Figure 3 (panel a), in which it
is shown that the MHD simulations have a much larger range of
variations in the magnetic field, kBk ∈ [2000, 3500] Gauss, com-
pared to the observations in GREGOR and Hinode data, where
A149, page 6 of 12

−40 −20 0 20 40
−4
−2
0
2
4
GREGOR
−40 −20 0 20 40
−4
−2
0
2
4
T
φ
,G
φ
[10
−3
g
cm
−2
s
−2
]
Hinode
−40 −20 0 20 40
φ [deg]
−4
−2
0
2
4
MHD/5
Tφ
Gφ
Fig. 4. Decomposition of the Lorentz force into magnetic pressure gra-
dient and magnetic tension. Variations in the magnetic pressure gra-
dient, Gφ, (blue) and magnetic tension, Tφ, (red) along the azimuthal
coordinate, φ, are indicated in Fig. 2 (outermost arc) and were inferred
from GREGOR (top) and Hinode (middle) observations, as well as from
3D MHD simulations (bottom). Values for the latter have been divided
by a factor of five.
kBk ∈ [1300, 1900]. In order to demonstrate this, we take the
force balance equation along the azimuthal direction (Eq. 2) and
consider, as is seen in Sect. 6.1, that the main contributor to the
φ component (azimuthal direction) of the Lorentz force is the
magnetic pressure gradient in the same direction. With this, we
can write
1
r
∂Pg
∂φ
' Gφ = −
1
8π
(∇kBk2
)eφ = −
1
8πr
∂B2
∂φ
· (5)
By transforming the derivatives into finite differences we can
write
∆Pg,mhd
∆Pg,obs
'
∆B2
mhd
∆B2
obs
≈
35002
− 20002
19002 − 13002
≈ 4.3. (6)
This demonstrates that the reason the φ component of the
Lorentz force, and hence also the variation in the gas pressure in
the azimuthal direction in the MHD simulations, is larger than in
the observations is because of the larger difference in the mag-
netic field of the spines and intraspines in the MHD simulations.
In the original (i.e., undegraded) simulation data (see left-side
panels in the leftmost column in Fig. 2), the mean azimuthal dis-
tance between consecutive peaks in Pg decreases with respect
to the degraded simulation data, thereby further increasing the
values of Lφ and r−1
∂Pg/∂φ.
6.3. Results for the umbra
So far we have mostly focused on the azimuthal equilibrium
of the penumbra. Following the same procedure previously
described, we can also study whether the umbra satisfies the
MHS equilibrium given by Lφ = r−1
∂Pg/∂φ. To this end, we
present in Figure 5 the same study as in Figure 3 but along the
innermost arc (i.e., umbral arc; Fig. 2). In panel a of Figure 5 we
notice that, in the observations (red squares for Hinode and blue
circles for GREGOR), the strong anticorrelation between γ−kBk
is not present anymore. This was to be expected because the
uncombed structure (i.e. spines and intraspines) of the penum-
bral magnetic field does not appear in the umbra. This is further
seen in panels b and c, where positive correlations between γ−Pg
and γ − ρ, respectively, are not inferred in either the Hinode or
GREGOR observations.
It is important to notice that while in the penumbra the physi-
cal properties inferred from GREGOR and Hinode data are quite
similar (see Fig. 3), in the umbra there are large differences
between the two observed sunspots. This is particularly the case
for the field strength obtained from the Stokes inversion (see a-
panel in Fig. 5), with Hinode showing a larger variation in the
field strength. This is likely caused by the presence of umbral
dots along the innermost arc in Hinode data. The larger varia-
tion in the field strength along the φ coordinate then translates
into a larger variation in the gas pressure and density once MHS
equilibrium is taken into account.
In the MHD simulations, correlations and anticorrela-
tions between the aforementioned physical parameters are still
present, hinting at a clear presence of spines and intrapines in
the simulated umbra. This was also to be expected if we attend
to the innermost arc in Fig. 2 (third column). It is clear from con-
tinuum intensity and inclination maps of the MHD simulations
that penumbral filaments still exist within the umbra almost up
to the very center of the sunspot. This effect appears because
of the time stamp, during the evolution of the sunspot, cho-
sen for our analysis. Initially, the convection in the outer umbra
shows a typical coffee-bean pattern such as the one reported in
Schüssler & Vögler (2006). This pattern closely resembles the
observations. However, as time passes and the cooling front goes
deeper, the convection within the outer umbra no longer shows
coffee beans but is more penumbra-like, with radially aligned
filaments, albeit less strong and transporting less energy.
Regardless of the differences between the observed and sim-
ulated umbra, the most important fact to notice is panel d in
Fig. 5, where it is still the case that in both observed and sim-
ulated umbra, the MHS equilibrium given by Eq. (2) is strongly
satisfied. This demonstrates that in the azimuthal direction, the
umbra is also in MHS equilibrium.
6.4. Results with vertical hydrostatic equilibrium
As is described in Section 3, as part of the inversion proce-
dure and before the inversion with MHS constraints was applied,
a first inversion under vertical HE was carried out. To study
whether the inversion under vertical HE yields similar results
to those under MHS equilibrium, we present in Figure C.1 the
same results as in Fig. 3 but using HE instead of MHS. As can be
seen, panel a in Figure C.1 still features the widely known struc-
ture of the penumbral magnetic field characterized by alternating
spines and intraspines, whereby more inclined magnetic field
lines (γ → 90◦
) are weaker than the more vertical (γ → 30◦
)
magnetic field lines. However, the correlations between the mag-
netic field inclination, γ, with the gas pressure and density (pan-
els b and c, respectively), as well as the correlations between
the gas pressure variation and the Lorentz force along φ, are
completely lost under the assumption of vertical HE. This result
highlights that the conclusions presented in this paper would not
have been possible unless MHS was considered.
6.5. Results without coupled inversion
In Figure C.2 we show the same results as the ones presented
in Figure 3 but performing the MHS inversion using the pixel-
wise inversion instead of the PSF-coupled inversion (see Sect. 3).
A149, page 7 of 12

1.8 2.4 3.0 3.6
B [103
g1/2
cm−1/2
s−1
]
10
20
30
40
γ
=
arccos[B
z
/B]
[deg]
(a)
GREGOR: c = 0.117 Hinode: c = 0.029 Simulations: c = -0.867
−40 −20 0 20 40
φ [deg]
2.0
2.4
2.8
B
[10
3
]
20
30
40
γ
[deg]
−40 −20 0 20 40
φ [deg]
1.8
2.6
3.2
B
[10
3
]
−40 −20 0 20 40
φ [deg]
2.0
2.6
3.2
3.8
B
[10
3
]
20
30
40
γ
0.5 1.5 2.5 3.5 4.5
Pg [105
dyn cm−2
]
10
25
40
γ
=
arccos[B
z
/B]
[deg]
(b)
GREGOR: c = 0.611 Hinode: c = -0.267 Simulations: c = 0.856
−40 −20 0 20 40
φ [deg]
2.2
2.8
3.4
P
g
[10
5
]
20
30
40
γ
−40 −20 0 20 40
φ [deg]
2.5
3.5
4.5
P
g
[10
5
]
−40 −20 0 20 40
φ [deg]
0.5
2.5
4.5
P
g
[10
5
]
20
30
40
γ
0.2 0.6 1.0 1.4
ρ [10−6
g cm−3
]
10
25
40
γ
=
arccos[B
z
/B]
[deg]
(c)
−40 −20 0 20 40
φ [deg]
0.6
0.7
0.8
ρ
[10
−6
]
20
30
40
γ
[deg]
−40 −20 0 20 40
φ [deg]
0.9
1.1
1.3
rho
[10
−6
]
−40 −20 0 20 40
φ [deg]
0.2
0.6
1.0
1.4
rho
[10
−6
]
20
30
40
γ
−8 −4 0 4 8
r−1
(∂Pg/∂φ) [10−3
g cm−2
s−2
]
−8
−4
0
4
8
L
φ
[10
−3
g
cm
−2
s
−2
]
(d)
−40 −20 0 20 40
φ [deg]
−4
−2
0
2
4
L
φ
[10
−3
]
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
L
φ
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
Fig. 5. Same as Figure 3 but in the umbra (i.e., innermost arc in Fig. 2).
By comparing these two figures we conclude that, although the
PSF-coupled inversion is not strictly compulsory to obtain a cor-
relation between the azimuthal component of the Lorentz force,
Lφ, and the derivative of the gas pressure in the azimuthal direc-
tion (panel d) it certainly helps to improve the correlations of
the field strength, kBk (panel a), gas pressure, Pg (panel b), and
density, ρ (panel c), with the inclination of the magnetic field,
γ, with respect to the normal vector to the solar surface. The
effects of including the PSF in the inversion are also more notice-
able in the GREGOR data (ground-based) than in the Hinode
data (satellite). This occurs in spite of our limited knowledge of
GREGOR’s PSF, in particular the lack of a detailed model for
the seeing (see Sect. 4).
7. Conclusions
Although simplified theoretical MHS models of sunspots have
been proposed in the past, those models have so far ignored the
strong variations in the magnetic field in the azimuthal direction.
These variations are caused not only by the filamentary structure
of the penumbra but also by inhomogeneities in the umbral mag-
netic field caused by umbral dots and light bridges. Here we have
demonstrated, using spectropolarimetric observations from both
the ground and space as well as MHD simulations, that MHS
equilibrium is maintained even if these azimuthal variations in
the magnetic field are taken into account.
It is important to bear in mind that not all dynamic phenom-
ena occurring at small scales in sunspots (see i.e., Section 1) are
resolved at the spatial resolution of our observations, with many
of them being highly relevant for the energy transport within
the sunspot (Rempel & Borrero 2021). Consequently, the MHS
equilibrium in the azimuthal direction unearthed by the present
study might not apply at spatial scales smaller than the ones
of our observations (i.e., ≤100 km Schlichenmaier et al. 2016).
What can be stated, however, is that at those spatial scales these
aforementioned dynamic phenomena must reach a stationary
state whereby the overarching magnetic structure of the sunspot
remains close to MHS (Rempel & Schlichenmaier 2011). On the
other hand, we note that the undegraded MHD simulations, fea-
turing a horizontal resolution of 32 km, verify the MHS equi-
librium in the azimuthal direction with an even higher accuracy
than the same simulations after degradation to the spatial resolu-
tion of the observations.
Notwithstanding their almost perfect MHS equilibrium, per-
turbations from this equilibrium, such as fluting instability
(Meyer et al. 1977) or advective transport of the magnetic flux
via moving magnetic features (Martínez Pillet 2002; Kubo et al.
2008a,b), will eventually break apart or make the sunspot decay
till its eventual disappearance over long timescales.
The results presented in this work also highlight the capa-
bilities of the FIRTEZ Stokes inversion code to infer physical
parameters such as gas pressure, density, electric currents, and
Lorentz force that, until now, could not be accurately deter-
mined. Gaining access to these physical parameters opens a new
path to investigate the thermodynamic state and magnetic struc-
ture of the plasma in the solar atmosphere.
In this work we have not studied the equilibrium of sunspots
in either the radial or vertical directions (r, z). There are several
A149, page 8 of 12

important reasons for this. On the one hand, these have already
been addressed by theoretical models (Low 1975, 1980a,b;
Pizzo 1986, 1990). On the other hand, the magnetic field in
these two directions varies much more slowly than along φ
(Borrero & Ichimoto 2011), and therefore the Lorentz force will
likely play a less important role there. Finally, as was shown by
Rempel (2012, see Fig. 22), in the radial direction in the penum-
bra, the velocity advective term, (v · ∇)v, associated with the
Evershed flow contributes significantly to the equilibrium. This
particular term cannot be considered by the FIRTEZ inversion
code so far, and therefore it is not possible at this point to address
the equilibrium in the radial direction employing spectropolari-
metric observations.
Acknowledgements. The development of the FIRTEZ inversion code is funded
by two grants from the Deutsche Forschung Gemeinschaft (DFG): projects
321818926 and 538773352. Adur Pastor acknowledges support from the
Swedish Research Council (grant 2023-03313). This project has been funded by
the European Union through the European Research Council (ERC) under the
Horizon Europe program (MAGHEAT, grant agreement 101088184). Markus
Schmassmann is supported by the Czech-German common grant, funded by the
Czech Science Foundation under the project 23-07633K and by the Deutsche
Forschung Gemeinschaft (DFG) under the project 511508209. Manuel Collados
acknowledges financial support from Ministerio de Ciencia e Innovación and
the European Regional Development Fund through grant PID2021-127487NB-
I00. Matthias Rempel would like to acknowledge high-performance computing
support from Cheyenne (doi:10.5065/D6RX99HX) provided by NSF NCAR’s
Computational and Information Systems Laboratory (CISL), sponsored by the
U.S. National Science Foundation. This material is based upon work supported
by the NSF National Center for Atmospheric Research, which is a major facility
sponsored by the U.S. National Science Foundation under Cooperative Agree-
ment No. 1852977. The 1.5-m GREGOR solar telescope was built by a German
consortium under the leadership of the Institut für Sonnenphysik in Freiburg
with the Leibniz-Institut für Astrophysik Potsdam, the Institut für Astrophysik
Göttingen, and the Max-Planck-Institut für Sonnensystemforschung in Göttin-
gen as partners, and with contributions by the Instituto de Astrofísica de Canarias
and the Astronomical Institute of the Academy of Sciences of the Czech Repub-
lic. Hinode is a Japanese mission developed and launched by ISAS/JAXA, col-
laborating with NAOJ as a domestic partner, NASA and STFC (UK) as inter-
national partners. Scientific operation of the Hinode mission is conducted by
the Hinode science team organized at ISAS/JAXA. This team mainly consists
of scientists from institutes in the partner countries. Support for the post-launch
operation is provided by JAXA and NAOJ (Japan), STFC (U.K.), NASA, ESA,
and NSC (Norway). This research has made use of NASA’s Astrophysics Data
System. We thank an anonymous referee for suggestions that lead to important
improvements on the readability of the manuscript.
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A149, page 9 of 12

Appendix A: Do MHS inversions enforce ∇Pg = L?
Since the MHS inversions carried out with the FIRTEZ Stokes
inversion code take into account the Lorentz force, it is natural to
wonder whether our observational results verify r−1
∂Pg/∂φ = Lφ
as a consequence of being internally imposed. In order to answer
this question, it is convenient to take a step back and first study
the implications of having HE. Full (i.e., in 3D) HE is obtained
by neglecting the Lorentz force (L = 0 in Eq. 3). The resulting
equation can be decomposed in:
∂Pg
∂x
=
∂Pg
∂y
= 0 (A.1)
∂Pg
∂z
= −ρg (A.2)
With the exception of the FIRTEZ code, all Stokes inver-
sion codes such as SIR (Ruiz Cobo & del Toro Iniesta 1992),
SPINOR (Frutiger et al. 1999), STiC (de la Cruz Rodríguez et al.
2019), NICOLE (Socas-Navarro et al. 2015), etc. consider only
Eq. A.2 (i.e., vertical HE), but they neglect that there is also hor-
izontal HE (Eqs. A.1) that implies that the gas pressure cannot
vary in planes of constant z:
Pg , f(x, y) (A.3)
Consequently, two pixels on the solar surface, sitting next to
each other (on the XY plane) must have the same Pg(z). This
must be satisfied ∀z’s. Next, by applying Eq. A.2, we can also
conclude that all pixels on the XY plane must have the same
density ρ(z), and therefore also the same temperature T(z). In
other words, neither ρ, Pg nor T can depend on (x, y). Now, we
must take into account that typically from the inversion of the
Stokes vector we obtain T(x, y, z). This immediately implies that
horizontal HE cannot be maintained in the sense that Eqs. A.1
and A.2 cannot be simultaneously satisfied. Hence Eq. A.1 is
usually ignored and only Eq. A.2 is considered, hence the term
vertical HE.
Now, as explained in Borrero et al. (2019), FIRTEZ does not
directly solve the momentum equation in MHS (Eq. 3). Instead,
it takes the divergence of that equation and iteratively solves the
following second-order Poisson equation:
∇2
(ln Pg) = −
mag
Kb
∂
∂z
u
T

+ ∇ ·
L0
Pg
!
(A.4)
where L0
is a modified Lorentz force. Let us leave L0
aside
for the moment and ask ourselves what does FIRTEZ do when
the Lorentz force is neglected (i.e., HE). Under this assumption
FIRTEZ solves:
∇2
(ln Pg) = −
mag
Kb
∂
∂z
u
T

(A.5)
But let us now remember that typically the inversion will
yield T(x, y, z) and hence also a variable mean molecular weight
u(x, y, z), so the that the term:
∂
∂z
u
T

= f(x, y, z) (A.6)
And consequently, the resulting Pg from solving Eq. A.5 will
also be a function of (x, y, z). But this is contradictory with HE
as we just saw above because full 3D HE implies Pg = f(z) only.
So, it is clear that the method FIRTEZ employs to retrieve
the gas pressure will result in Pg = f(x, y, z) even if L0
= 0 and
thus, the fact that Pg changes horizontally (i.e., along φ), is not
necessarily due to the Lorentz force. In fact, there is no guarantee
that the following equation:
Lφ = r−1 ∂Pg
∂φ
(A.7)
must be satisfied always at all points of the three-dimensional
domain. In order to prove this particular point, the top panels on
Figure A.1 show the correlations between the right-hand side and
left-hand side of Eq. A.7 as a function of (r, z) in the observed
and simulated sunspots.
Figure A.1 shows that although the correlations are large in
the observations (left and middle panels for GREGOR and Hin-
ode, respectively), this is certainly not always the case. In fact
there are regions where the obtained correlation is actually neg-
ative. Such is the case of the high photosphere at all radial dis-
tances, and of the mid-photosphere at short radial distances (i.e.,
in the umbra). For comparison the simulated sunspot is also dis-
played in the rightmost panel. Here we can see that the MHD
simulations show a high degree of correlation over a wider range
of radial distances r/Rspot and heights z.
On the other hand, a high correlation between the rhs and
lhs side of Eq. A.7 does not necessarily mean that this equation
is strictly fulfilled. It can occur that they are correlated but not
necessarily equal. We have evaluated this by performing a linear
fit to the scatter plot of Lφ and r−1
∂Pg/∂φ (as done in panel d
of Fig. 3) and plotting on the bottom panels of Figure A.1, the
slope m of the linear fit between the rhs and the lhs of Eq. A.7 as
a function of (r, z).
Figure A.1 (bottom-right and bottom-middle panels) shows
that even though there might be very high correlations between
the two terms on either side of Eq. A.7 the actual equation is
strictly fulfilled only in the region where the slope is close to
1.0, that is, in a narrow region of the observed sunspots around
the τc = 1 level. This is in contrast with the sunspot simulated
with the MuRAM code, where Eq. A.7 is verified to a large
degree (i.e., slope 1.0) over a much wider region, both vertically
and radially, than in the observations analyzed with the FIRTEZ
code. Because of these results it is possible to state that Eq. A.7
is not strictly imposed by FIRTEZ.
The question now arises to what extent we can expect
Eq. A.7 to be strictly fulfilled. As explained above, T(x, y, z)
implies that full 3D HE cannot be fulfilled by FIRTEZ even if
the Lorentz force is zero. This can be also interpreted in the fol-
lowing way: since under HE (i.e., L0
= 0) T and Pg can only
a function of z, the fact that FIRTEZ receives T(x, y, z) already
implies the presence of a residual force not given by (∇ × B) × B
(i.e., the Lorentz force) so that the resulting gas pressure also
depends on (x, y, z).
The nature of the aforementioned residual force can be
understood by looking at the right-hand-side of Eq. A.4. Here,
after defining F = L0
/Pg and projecting onto its Hodge’s com-
ponents: F = ∇Ψ + ∇ × W, we see that Eq. A.4 becomes:
∇2
(ln Pg) = −
mag
Kb
∂
∂z
u
T

+ ∇2
Ψ (A.8)
where it is clear now that in the presence of a magnetic field,
FIRTEZ infers a gas pressure Pg that balances the potential part
Ψ of the force F = L0
/Pg with the residual forces being identified
with the non-potential part W of this force.
Therefore, one can expect the results from the FIRTEZ code
to fulfill Eq. A.7 when the potential part of the L0
/Pg actu-
ally dominates over its non-potential component. Otherwise it
is conceivable that the potential component will still be capable
A149, page 10 of 12

Borrero, J. M., et al.: AA, 699, A149 (2025)
0.2 0.4 0.6 0.8
0
500
1000
1500
z
[km]
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
GREGOR: corr(r−1
∂Pg/∂φ, Lφ)
0.2 0.4 0.6 0.8
0
250
500
750
1000
1250
1500
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
Hinode: corr(r−1
∂Pg/∂φ, Lφ)
0.0 0.2 0.4 0.6 0.8 1.0
0
500
1000
1500
−1.00 −0.75 −0.50 −0.25 0.00 0.25 0.50 0.75 1.00
MHD: corr(r−1
∂Pg/∂φ, Lφ)
0.2 0.4 0.6 0.8
r/Rspot
0
500
1000
1500
z
[km]
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
GREGOR: slope(r−1
∂Pg/∂φ, Lφ)
0.2 0.4 0.6 0.8
r/Rspot
0
250
500
750
1000
1250
1500
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
Hinode: slope(r−1
∂Pg/∂φ, Lφ)
0.0 0.2 0.4 0.6 0.8 1.0
r/Rspot
0
500
1000
1500
−1.0 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0
MHD: slope(r−1
∂Pg/∂φ, Lφ)
Fig. A.1. Study of the MHS equilibrium along the azimuthal direction for the whole sunspots. Top panels: Pearson’s correlation coefficient between
the rhs and lhs of Eq. A.7 as a function of (r, z). Bottom panels: same as on the top but for the slope of the linear regression between the rhs and
lhs of Eq. A.7 as a function of (r, z). Rs represents the radius of the sunspots. Solid black lines show the φ-average location of the τc = 1, 10−2
.
Observational results are displayed on the left (GREGOR) and middle (Hinode) panels. Results from the MHD simulations are presented in the
rightmost panels.
of introducing a correlation between Lφ and r−1
∂Pg/∂φ but the
slope of the linear fit between these two quantities might be quite
different from 1.0 (see Fig. A.1).
It is now time to remember that the modified Lorentz force
L0
used by FIRTEZ to determine the gas pressure via the solu-
tion of Eq. A.4 is not necessarily the Lorentz force obtained
via the inferred magnetic field. Instead, FIRTEZ takes the cal-
culated electric current j = c(4π)−1
∇ × B and dampens it by a
factor ∝ β2
(with β being the ratio between magnetic and gas
pressure) whenever the module kjk exceeds a maximum allowed
value. This is done with the aim that, in the upper atmosphere
where the β 1 and where spectral lines do not convey accu-
rate values of magnetic field B, electric currents determined
from these inaccurate magnetic fields do not dominate the force
balance. This explains why the agreement of the observational
results with Eq. A.7 worsens progressively from the lower atmo-
sphere towards the upper atmosphere (see Fig. A.1). This effect
reinforces the previous conclusion that the MHS equilibrium
imposed by FIRTEZ does not strictly impose Eq. A.7.
Appendix B: Simulation details
The initial magnetic field of the simulation uses the self-similar
approach used in Schlüter Temesváry (1958) and others:
Bz(r, z) = B0 f(ξ)g(z), (B.1)
Br(r, z) = −B0
r
2
f(ξ)g0
(z), (B.2)
ξ = r
p
g(z), (B.3)
whereby Bz is the vertical component of the magnetic field,
Br the radial component, r the distance from the spot axis, and z
the distance from the bottom of the box. For this simulation
f(ξ) = exp

−(ξ/R0)4

, (B.4)
g(z) = exp

−(z/z0)2

. (B.5)
The parameters are B0 = 6.4 kG, the initial field strength
on the spot axis at the bottom of the box, R0 = 8.2 Mm
and z0 = 6.4185 Mm, which determines how far away from
the axis and the bottom Bz reduces by a factor e. These
parameters lead to a sunspot with an initial flux of F0 =
1.2 × 1022
Mx and a field strength at the top of 2.56 kG.
A more detailed description of the initial state of the simu-
lation is in Rempel (2012). Other self-similar sunspot mod-
els use for f(ξ) a Gaussian (e.g. Schlüter Temesváry 1958;
Schüssler Rempel 2005) or the boxcar function (many ana-
lytical models and e.g. Panja et al. 2020) and g(z) is sometimes
related to the pressure difference between the spot center and the
quiet Sun (e.g. Schlüter Temesváry 1958).
The parameter α = 2 means that at the top of the box, the
horizontal field is twice what it would be if potential extrapola-
tion were employed:
Bz = F −1

F (Bz,0)e−z α |k|

(B.6)
and
Bx = αF −1
F (Bz,0)
−ikx
|k|
e−z α |k|
!
, (B.7)
whereby Bz,0 is the vertical field at the top of the box, z
the height above the top of the box (in the ghost cells), F the
horizontal Fourier transform, kx the Fourier modes, and |k| =
q
k2
x + k2
y. For By holds the same equation as Eq. B.7 but per-
forming the substitution kx → ky. α 1 is responsible for
Br being larger than in the simulated penumbra compared to
observations.
For a comparison to observed sunspots, see Jurčák et al.
(2020), which concludes that in sunspots with α 1
have a too low Bz at the umbral boundary. Furthermore,
Schmassmann et al. (2021) conclude, that a sufficiently strong
Bz suppresses penumbral-type overturning convection.
A149, page 11 of 12

Borrero, J. M., et al.: AA, 699, A149 (2025)
Appendix C: Additional results
1.2 1.8 2.4 3.0 3.6
B [103
g1/2
cm−1/2
s−1
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(a)
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
B
[10
3
]
30
60
90
γ
[deg]
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
B
[10
3
]
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
3.0
3.6
B
[10
3
]
30
60
90
γ
0.5 1.5 2.5 3.5 4.5
Pg [105
dyn cm−2
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(b)
−40 −20 0 20 40
φ [deg]
0.5
1.5
2.5
P
g
[10
5
]
30
60
90
γ
−40 −20 0 20 40
φ [deg]
0.5
1.5
2.5
P
g
[10
5
]
−40 −20 0 20 40
φ [deg]
0.5
2.5
4.5
P
g
[10
5
]
30
60
90
γ
0.5 2.5 4.5 6.5
ρ [10−7
g cm−3
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(c)
−40 −20 0 20 40
φ [deg]
2.2
3.2
4.2
ρ
[10
−7
]
30
60
90
γ
[deg]
−40 −20 0 20 40
φ [deg]
2.2
3.2
4.2
rho
[10
−7
]
−40 −20 0 20 40
φ [deg]
0.5
2.5
4.5
6.5
rho
[10
−7
]
30
60
90
γ
−5.0 −2.5 0.0 2.5 5.0
r−1
(∂Pg/∂φ) [10−3
g cm−2
s−2
]
−5.0
−2.5
0.0
2.5
5.0
L
φ
[10
−3
g
cm
−2
s
−2
]
(d)
GREGOR: c = -0.272 Hinode: c = 0.324 Simulations (/5): c =0.974
−40 −20 0 20 40
φ [deg]
−4
−2
0
2
4
L
φ
[10
−3
]
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
L
φ
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
Fig. C.1. Same as Figure 3 but obtained under the assumption of vertical HE (see Sect. 6.4). Results from MHD simulations (black colors and
rightmost minipanels) are identical to Fig. 3.
1.2 1.8 2.4 3.0 3.6
B [103
g1/2
cm−1/2
s−1
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(a)
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
B
[10
3
]
30
60
90
γ
[deg]
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
B
[10
3
]
−40 −20 0 20 40
φ [deg]
1.2
1.8
2.4
3.0
3.6
B
[10
3
]
30
60
90
γ
0.5 1.5 2.5 3.5 4.5
Pg [105
dyn cm−2
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(b)
−40 −20 0 20 40
φ [deg]
0.5
1.5
2.5
P
g
[10
5
]
30
60
90
γ
−40 −20 0 20 40
φ [deg]
0.5
1.5
2.5
P
g
[10
5
]
−40 −20 0 20 40
φ [deg]
0.5
2.5
4.5
P
g
[10
5
]
30
60
90
γ
0.5 2.5 4.5 6.5
ρ [10−7
g cm−3
]
30
60
90
γ
=
arccos[B
z
/B]
[deg]
(c)
−40 −20 0 20 40
φ [deg]
2.2
3.2
4.2
ρ
[10
−7
]
30
60
90
γ
[deg]
−40 −20 0 20 40
φ [deg]
2.2
3.2
4.2
rho
[10
−7
]
−40 −20 0 20 40
φ [deg]
0.5
2.5
4.5
6.5
rho
[10
−7
]
30
60
90
γ
−5.0 −2.5 0.0 2.5 5.0
r−1
(∂Pg/∂φ) [10−3
g cm−2
s−2
]
−5.0
−2.5
0.0
2.5
5.0
L
φ
[10
−3
g
cm
−2
s
−2
]
(d)
−40 −20 0 20 40
φ [deg]
−4
−2
0
2
4
L
φ
[10
−3
]
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
L
φ
[10
−3
]
−40 −20 0 20 40
φ [deg]
−2.5
0.0
2.5
−4
0
4
r
−1
(∂P
g
/∂φ)
[10
−3
]
Fig. C.2. Same as Figure 3 but obtained without horizontal coupling between the pixels due to the telescope’s PSF (see Sect. 6.5). Results from
MHD simulations (black colors and rightmost minipanels) are identical to Fig. 3.
A149, page 12 of 12

The role of the Lorentz force in sunspot equilibrium

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