A chromospheric resonance cavity in a sunspot mapped with seismology

Abstract

Sunspots are intense collections of magnetic fields that pierce through the Sun’s photosphere, with their signatures extending upwards into the outermost extremities of the solar corona¹. Cutting-edge observations and simulations are providing insights into the underlying wave generation², configuration3,4 and damping⁵ mechanisms found in sunspot atmospheres. However, the in situ amplification of magnetohydrodynamic waves⁶, rising from a few hundreds of metres per second in the photosphere to several kilometres per second in the chromosphere⁷, has, until now, proved difficult to explain. Theory predicts that the enhanced umbral wave power found at chromospheric heights may come from the existence of an acoustic resonator8,9,10, which is created due to the substantial temperature gradients experienced at photospheric and transition region heights¹¹. Here, we provide strong observational evidence of a resonance cavity existing above a highly magnetic sunspot. Through a combination of spectropolarimetric inversions and comparisons with high-resolution numerical simulations, we provide a new seismological approach to mapping the geometry of the inherent temperature stratifications across the diameter of the underlying sunspot, with the upper boundaries of the chromosphere ranging between 1,300 ± 200 km and 2,300 ± 250 km. Our findings will allow the three-dimensional structure of solar active regions to be conclusively determined from relatively commonplace two-dimensional Fourier power spectra. The techniques presented are also readily suitable for investigating temperature-dependent resonance effects in other areas of astrophysics, including the examination of Earth–ionosphere wave cavities¹².

Full text

A chromospheric resonance cavity in a sunspot mapped
with seismology
David B. Jess1,2, Ben Snow3, Scott J. Houston1, Gert J. J. Botha4, Bernhard Fleck5, S. Krishna
Prasad1, Andr´
es Asensio Ramos6,7, Richard J. Morton4, Peter H. Keys1, Shahin Jafarzadeh8,9,
Marco Stangalini10,11, Samuel D. T. Grant1& Damian J. Christian2
1Astrophysics Research Centre, School of Mathematics and Physics, Queen’s University Belfast,
Belfast BT7 1NN, U.K.
2Department of Physics and Astronomy, California State University Northridge, Northridge, CA
91330, U.S.A.
3Centre for Geophysical and Astrophysical Fluid Dynamics, University of Exeter, Exeter, EX4
4QF, U.K.
4Department of Mathematics, Physics and Electrical Engineering, Northumbria University, New-
castle upon Tyne, NE1 8ST, U.K.
5ESA Directorate of Science, Operations Department, c/o NASA/GSFC Code 671, Greenbelt, MD
20071, U.S.A.
6Instituto de Astrof´
ısica de Canarias, C/V´
ıa Lactea s/n, E-38205 La Laguna, Tenerife, Spain
7Departamento de Astrof´
ısica, Universidad de La Laguna, E-38206 La Laguna, Tenerife, Spain
8Institute of Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, NO-0315 Oslo,
Norway
9Rosseland Centre for Solar Physics, University of Oslo, P.O. Box 1029 Blindern, N-0315 Oslo,
Norway
1

10Italian Space Agency (ASI), Via del Politecnico snc, 00133 Roma, Italy
11INAF-OAR National Institute for Astrophysics, Via Frascati 33, 00078 Monte Porzio Catone
(RM), Italy
Sunspots are intense collections of magnetic fields that pierce through the Sun’s photosphere,
with their signatures extending upwards into the outermost extremities of the solar corona1.
Cutting-edge observations and simulations are providing insights into the underlying wave
generation2, configuration3, 4, and damping5mechanisms found in sunspot atmospheres. How-
ever, the in-situ amplification of magnetohydrodynamic waves6, rising from a few hundreds
of m/s in the photosphere to several km/s in the chromosphere7, has, until now, proved dif-
ficult to explain. Theory predicts that the enhanced umbral wave power found at chromo-
spheric heights may come from the existence of an acoustic resonator8–10, which is created
due to the substantial temperature gradients experienced at photospheric and transition re-
gion heights11. Here we provide strong observational evidence of a resonance cavity existing
above a highly magnetic sunspot. Through a combination of spectropolarimetric inversions
and comparisons with high-resolution numerical simulations, we provide a new seismolog-
ical approach to map the geometry of the inherent temperature stratifications across the
diameter of the underlying sunspot, with the upper boundaries of the chromosphere ranging
between 1300 ±200 km and 2300 ±250 km. Our findings will allow the three-dimensional
structure of solar active regions to be conclusively determined from relatively commonplace
two-dimensional Fourier power spectra. The techniques presented are also readily suitable
for investigating temperature-dependent resonance effects in other areas of astrophysics, in-
2

cluding the examination of Earth-ionosphere wave cavities12 .
Spectropolarimetric observations, captured in the Si I10827 ˚
A and He I10830 ˚
A lines at
high spatial (110 km per pixel), temporal (14.6 s) and spectral (0.04 ˚
A per pixel) resolution,
were acquired across the centre of a large sunspot on 14 July 2016 using the Facility Infrared
Spectropolarimeter13 (FIRS) at the Dunn Solar Telescope. Simultaneous contextual imaging is
provided by the Rapid Oscillations in the Solar Atmosphere14 (ROSA) and the Interferometric
BIdimensional Spectrometer15 (IBIS) instruments. Spatially resolved Doppler velocities are de-
rived as a function of time for the entire 86 minute data sequence, providing 35,350 individual ve-
locity measurements with amplitudes in the range of ±0.3 km/s and ±6 km/s for the photospheric
Si I10827 ˚
A and upper-chromospheric He I10830 ˚
A time series, respectively. The resulting im-
ages and spectra (Fig. 1) highlight the persistent and regular wave signatures manifesting in the
sunspot umbra.
A long-lived filamentary structure, consistent with previous observational studies16, 17 , natu-
rally segregates the sunspot into two distinct umbrae (Fig. 1). The centres of gravity (or barycen-
ters) of each isolated umbra are calculated, allowing the wave characteristics to be studied as a
function of distance from their respective umbral core. Fourier spectral energies18 are computed
for each of the 101 spatial pixels crossing the sunspot umbrae (Fig. 2), revealing distinct differ-
ences between the upper-chromospheric He I10830 ˚
A spectra and their co-spatial photospheric Si I
10827 ˚
A counterparts. Most notable is the fact that all of the He I10830 ˚
A spectral energies can be
categorised by three distinct regions: (region I;<5mHz) the evanescent regime with frequencies
3

below the acoustic cut-off, (region II;6−17 mHz) the region where propagating waves become
permissible and demonstrate broad spectral peaks and strong spectral energies that are consistent
with previous observational findings19, and (region III;18 −27 mHz) the final regime where the
spectral energy demonstrates a steep power-law relationship with gradient α. It is region III that
acts as both an indicator for the presence of a resonant layer9, as well as the ability to use the
spectral slope, α, as a diagnostic tool for estimating the thickness of the temperature structuring of
the chromospheric resonance cavity11.
Fitting the He I10830 ˚
A spectral energy gradients for region III through maximum-likelihood
statistical approaches20 reveals a strong correspondence between the steepness of the slope and the
distance subtended from the corresponding umbral barycenter (Fig. 3). While the spectral gradients
for region II remain consistent across the entire extent of the sunspot (with characteristic spectral
gradients of −2.1±0.2), the spectral slopes for region III vary as a function of distance from their
respective umbral barycenter, with gradients as shallow as −5.4±0.6at the core of the relevant
umbra, extending to gradients as steep as −7.8±0.6at maximal distances (∼3000 km) from
each barycenter. Spectral slopes of this magnitude closely resemble the strong dissipative ranges
previously documented in studies of the solar wind21.
To compare with the observational findings, the Lare2D22 numerical non-linear compress-
ible MHD code, which is employed in a 1.5D configuration, is driven by the photospheric velocity
profiles extracted from the observational Si I10827 ˚
A Doppler shifts and allowed to evolve in
time. The embedded atmospheric model is constrained by HAnle and Zeeman Light23 (HAZEL)
4

inversions applied to the spectropolarimetric data products24, with the computed velocity signals
extracted with a cadence of 14.6s (to match that of the FIRS observations) following propagation
of the wave signatures to the upper temperature gradient corresponding to the commencement of
the transition region, which is consistent with the predicted formation height of the He I10830 ˚
A
spectral line25. The velocity time series is cropped to 86 minutes in duration to match that of
the observations and converted into spectral energies (Fig. 2e,f). This process is repeated for in-
put atmospheres scaled to 80%, 90%, 110% and 120% of the original temperature stratification
height, providing resonance cavity depths (photosphere to the base of the transition region) span-
ning 1700 −2545 km. The spectral energies computed for both the modelled and observed time
series show similar trends across regions I,II and III (Fig. 2). In particular, the modelled region III
demonstrates an identical rise in spectral energy at ∼20 mHz, before dropping off very rapidly
with increasing frequency. Importantly, re-running the numerical simulations for an atmospheric
profile devoid of the steep transition region temperature gradient produces spectral energies where
the secondary ∼20 mHz spectral peak is absent. This verifies that the steep temperature gradient
intrinsic to the solar transition region, which amplifies the spectral energies at ∼20 mHz, is re-
quired for the initiation of resonance behaviour. The maximum-likelihood fitted spectral slopes for
region III reveals that shallower spectral gradients correspond to inherently deeper chromospheric
cavities (Fig. 3b), allowing the observed spectral slopes for region III to unveil the cavity depths
of the local sunspot atmosphere. Importantly, a larger cavity depth introduces a greater resonant
energy content, hence providing more energy across the frequency range, and thus reducing the
steepness of the associated spectral slope11.
5

The range of gradients measured for region III (18−27 mHz) of the sunspot spectral energies
span −5.4±0.6(close to the umbral barycenter) to −7.8±0.6(at the outermost extremities of the
umbra), suggesting that the chromospheric resonance cavity is thickest near the core of the umbra,
dropping to its thinnest depth at the penumbral boundary (Fig. 3). The upper geometric height of
the chromosphere, τchromo, which corresponds to the uppermost boundary of the resonance cavity
before the commencement of the transition region, can be defined (Fig. 3b) as,
τchromo (km) = Region III spectral slope + 26.408
0.009131 cos θ , (1)
where θis the inclination angle of the wave propagation path with respect to the normal to the
solar surface. Use of the inclination angle is important since the Lare2D numerical code simulates
the wave evolution along a given magnetic field line, which may be inclined with respect to the
solar normal. Hence, taking the magnetic field inclinations into consideration allows for the con-
version of a wave propagation distance into a true geometric height of the chromosphere for that
particular spatial location. Utilising the vector magnetic fields derived from HAZEL inversions24
yields inclination angles ranging from 0degrees at the umbral barycenters, through to approxi-
mately 50 degrees towards the outer umbral boundaries, providing geometric heights for the upper
chromosphere on the order of 2300 ±250 km and 1300 ±200 km for the umbral cores and um-
bral/penumbral boundaries, respectively (Fig. 4).
Here, we show strong evidence substantiating the presence of a chromospheric resonance
cavity above a sunspot. We reveal how high resolution spectropolarimetric observations, when
combined with cutting-edge numerical MHD simulations, provide the spectral energy sensitivity
necessary to accurately measure the high-frequency spectral gradients that are modulated by the
6

depth of the chromospheric resonance cavity. Importantly, the variable cavity depths across the
diameter of the sunspot have important implications for atmospheric seismology, since the umbral
atmosphere can no longer be considered as a homogeneous slab environment. Instead, thicknesses
of the chromospheric resonance layer will need to be incorporated into seismological estimations
in order to improve the accuracy of such techniques. Looking ahead, fiber-fed spectrographs on
the upcoming 4m Daniel K. Inouye Solar Telescope will provide two-dimensional spectral energy
maps of sunspots with unprecedented resolving power, allowing revolutionary three-dimensional
atmospheric reconstructions to be uncovered.
Furthermore, the topic of resonance cavities is fundamentally important across a wide range
of ongoing astrophysical research including, but not limited to, the examination of near-Earth
ionospheric wave cavities12. As a result, understanding the physics responsible for the creation
of resonance cavities, along with their impact on the universe around us, is of paramount impor-
tance. Our results enable the astrophysical community to benchmark, through novel seismological
approaches, what atmospheric characteristics are required to form a stable resonance cavity (e.g.,
specific temperature stratifications), what impact this has on waveforms interacting with the cavity
structure (e.g., power enhancements at well-defined frequencies), and how cutting-edge numerical
simulations can be employed alongside high-precision spectropolarimetric data products to deduce
physical parameters corresponding to the local plasma conditions (e.g., cavity depth).
7

1. Borrero, J. M. & Ichimoto, K. Magnetic Structure of Sunspots. Living Reviews in Solar
Physics 8, 4 (2011).
2. Bogdan, T. J. et al. Waves in the Magnetized Solar Atmosphere. II. Waves from Localized
Sources in Magnetic Flux Concentrations. Astrophys. J. 599, 626–660 (2003).
3. Lites, B. W., Thomas, J. H., Bogdan, T. J. & Cally, P. S. Velocity and Magnetic Field Fluctu-
ations in the Photosphere of a Sunspot. Astrophys. J. 497, 464–482 (1998).
4. Nagashima, K. et al. Observations of Sunspot Oscillations in G Band and CaII H Line with
Solar Optical Telescope on Hinode. Publ. Astron. Soc. Jpn 59, S631–S636 (2007).
5. Grant, S. D. T. et al. Alfv´
en wave dissipation in the solar chromosphere. Nature Physics 14,
480–483 (2018).
6. Kobanov, N. I. & Makarchik, D. V. Propagating waves in the sunspot umbra chromosphere.
Astron. Astrophys. 424, 671–675 (2004).
7. Tziotziou, K., Tsiropoula, G., Mein, N. & Mein, P. Dual-line spectral and phase analysis of
sunspot oscillations. Astron. Astrophys. 463, 1153–1163 (2007).
8

8. Hollweg, J. V. A new resonance in the solar atmosphere. I. Theory. Sol. Phys. 62, 227–240
(1979).
9. Botha, G. J. J., Arber, T. D., Nakariakov, V. M. & Zhugzhda, Y. D. Chromospheric Resonances
above Sunspot Umbrae. Astrophys. J. 728, 84 (2011).
10. Felipe, T. Origin of the chromospheric three-minute oscillations in sunspot umbrae. A&A 627,
A169 (2019).
11. Snow, B., Botha, G. J. J. & R´
egnier, S. Chromospheric seismology above sunspot umbrae.
Astron. Astrophys. 580, A107 (2015).
12. Toledo-Redondo, S., Salinas, A., Fornieles, J., Port´
ı, J. & Lichtenegger, H. I. M. Full 3-D
TLM simulations of the Earth-ionosphere cavity: Effect of conductivity on the Schumann
resonances. Journal of Geophysical Research (Space Physics) 121, 5579–5593 (2016).
13. Jaeggli, S. A. et al. FIRS: a new instrument for photospheric and chromospheric studies at the
DST. Mem. Soc. Astron. Italiana 81, 763 (2010).
14. Jess, D. B. et al. ROSA: A High-cadence, Synchronized Multi-camera Solar Imaging System.
Sol. Phys. 261, 363–373 (2010).
15. Cavallini, F. IBIS: A New Post-Focus Instrument for Solar Imaging Spectroscopy. Sol. Phys.
236, 415–439 (2006).
9

16. Socas-Navarro, H., McIntosh, S. W., Centeno, R., de Wijn, A. G. & Lites, B. W. Direct
Imaging of Fine Structure in the Chromosphere of a Sunspot Umbra. Astrophys. J. 696, 1683–
1688 (2009).
17. Beck, C., Choudhary, D. P. & Rezaei, R. A Three-dimensional View of the Thermal Structure
in a Super-penumbral Canopy. Astrophys. J. 788, 183 (2014).
18. Stull, R. An Introduction to Boundary Layer Meteorology. Atmospheric and Oceanographic
Sciences Library (Springer Netherlands, 2012).
19. Reznikova, V. E., Shibasaki, K., Sych, R. A. & Nakariakov, V. M. Three-minute Oscillations
above Sunspot Umbra Observed with the Solar Dynamics Observatory/Atmospheric Imaging
Assembly and Nobeyama Radioheliograph. Astrophys. J. 746, 119 (2012).
20. Barret, D. & Vaughan, S. Maximum Likelihood Fitting of X-Ray Power Density Spectra: Ap-
plication to High-frequency Quasi-periodic Oscillations from the Neutron Star X-Ray Binary
4U1608-522. Astrophys. J. 746, 131 (2012).
21. He, J., Wang, L., Tu, C., Marsch, E. & Zong, Q. Evidence of Landau and Cyclotron Resonance
between Protons and Kinetic Waves in Solar Wind Turbulence. Astrophys. J. Lett. 800, L31
(2015).
22. Arber, T. D., Longbottom, A. W., Gerrard, C. L. & Milne, A. M. A Staggered Grid,
Lagrangian-Eulerian Remap Code for 3-D MHD Simulations. Journal of Computational
Physics 171, 151–181 (2001).
10

23. Asensio Ramos, A., Trujillo Bueno, J. & Landi Degl’Innocenti, E. Advanced Forward Model-
ing and Inversion of Stokes Profiles Resulting from the Joint Action of the Hanle and Zeeman
Effects. Astrophys. J. 683, 542–565 (2008).
24. Houston, S. J. et al. The Magnetic Response of the Solar Atmosphere to Umbral Flashes.
Astrophys. J. 860, 28 (2018).
25. Avrett, E. H., Fontenla, J. M. & Loeser, R. Formation of the Solar 10830 Angstrom Line.
In Rabin, D. M., Jefferies, J. T. & Lindsey, C. (eds.) Infrared Solar Physics, vol. 154 of IAU
Symposium, 35 (1994).
11

Acknowledgements D.B.J. would like to thank the UK Science and Technology Facilities Council (STFC)
for an Ernest Rutherford Fellowship, in addition to a dedicated standard grant which allowed this project to
be undertaken. D.B.J. and S.D.T.G. also wish to thank Invest NI and Randox Laboratories Ltd. for the award
of a Research & Development Grant (059RDEN-1) that allowed the computational techniques employed to
be developed. B.S. is supported by STFC research grant ST/R000891/1. S.K.P. wishes to thank the UK
STFC for support. A.A.R. is grateful to the Spanish Ministry of Economy and Competitiveness through
project AYA2014-60476-P. S.J. acknowledges support from the European Research Council under the Eu-
ropean Unions 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). M.S.
is grateful for funding received from the European Research Council under the European Unions Horizon
2020 Framework Programme for Research and Innovation, grant agreements H2020 PRE-EST (no. 739500)
and H2020 SOLARNET (no. 824135), in addition to support from INAF Istituto Nazionale di Astrofisica
(PRIN-INAF-2014). D.J.C. would like to thank California State University Northridge for start-up funding.
The Dunn Solar Telescope at Sacramento Peak/NM was operated by the National Solar Observatory (NSO).
NSO is operated by the Association of Universities for Research in Astronomy (AURA), Inc., under coop-
erative agreement with the National Science Foundation (NSF). The SDO/AIA imaging employed in this
work are courtesy of NASA/SDO and the AIA, EVE, and HMI science teams. The authors wish to acknowl-
edge 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). Imagery pro-
duced by the Visualization and Analysis Platform for atmospheric, Oceanic and solar Research (VAPOR;
www.vapor.ucar.edu), a product of the Computational Information Systems Laboratory at the National Cen-
ter for Atmospheric Research.
12

Author Contributions D.B.J. and D.J.C. designed the observational instrumentation setup. D.B.J., S.J.H.
and S.K.P. undertook the ground-based observations. D.B.J, S.J.H., A.A.R. and S.D.T.G. performed analysis
of the observations. B.S. and G.J.J.B. designed and carried out numerical MHD simulations. D.B.J., B.S.,
S.J.H., G.J.J.B., S.K.P., P.H.K., S.J., M.S., B.F. and R.J.M. interpreted the observations and simulations.
D.B.J., B.S., S.J.H., R.J.M. and S.D.T.G. prepared and processed all data products. All authors discussed
the results and commented on the manuscript.
Competing Interests The authors declare that they have no competing financial interests.
Correspondence Correspondence and requests for materials should be addressed to David B. Jess (email:
d.jess@qub.ac.uk).
13

Methods
Observations. The sunspot at the centre of the active region NOAA 12565 on 14 July 2016 was
the primary focus of the observing campaign. The image sequence duration was approximately
86 minutes, and was obtained during excellent seeing conditions between 13:42 – 15:08 UT with
the Dunn Solar Telescope at Sacramento Peak, New Mexico. The Hydrogen-Alpha Rapid Dy-
namics camera26 (HARDcam) and Rapid Oscillations in the Solar Atmosphere14 (ROSA) imaging
systems were simultaneously used to capture active region NOAA 12565 at G-band, blue con-
tinuum (4170 ˚
A), Ca II K and Hαwavelengths, which was positioned at heliocentric co-ordinates
(−58200,3000 ), providing a heliocentric viewing angle of 38◦(µ'0.79). This location corresponds
to N05.2E38.1 in the conventional heliographic co-ordinate system. To complement the ROSA and
HARDcam data streams, the Facility Infrared Spectropolarimeter13 (FIRS) slit-based spectrograph
and Interferometric BIdimensional Spectrometer15 (IBIS) imaging spectrograph acquired contem-
poraneous observations of the same active region in the He I10830 ˚
A and Ca II 8542 ˚
A spectral
lines, respectively.
The FIRS instrument was configured to obtain diffraction-limited spectropolarimetry of the
He I10830 ˚
A upper-chromospheric absorption lines by utilising a 7500 slit length (providing a spa-
tial sampling of 0.
0015 per pixel along the slit), combined with a 0.
00225 slit width. A confined
five-step raster was obtained by moving the slit 0.
00225 after each integration, providing a narrow
7500×1.
00125 slot-type field of view that passed through the centre of the umbral core. Each spectrum
obtained consisted of 12 consecutive additions of the modulation states to increase the signal-to-
noise of the relevant Stokes profiles, providing a final cadence equal to 14.6 s. The observations
14

were reduced into science-ready data products using the publicly available National Solar Obser-
vatory FIRS pipeline27, ultimately providing a spectral sampling of 0.04 ˚
A for the He I10830 ˚
A
spectra. To assist with the determination of the FIRS slit pointing and alignment, a slit-jaw camera
(in sync with the acquisition of each spectrum) was employed alongside the ROSA, HARDcam
and IBIS image sequences to allow the precise spatial location and orientation of the FIRS slit to
be mapped.
The IBIS system was deployed in imaging mode (i.e., no spectropolarimetric information
was retrieved) to increase the field of view captured and to decrease the time taken to acquire a
spectral scan. A total of 47 discrete, non-equidistant wavelength steps were utilised across the
Ca II 8542 ˚
A line profile with a spatial sampling of 0.
00098 per pixel, providing a circular field
of view with a diameter of 9700 and a spectral coverage of 8540.82 −8543.42 ˚
A (i.e., line core
±1.3˚
A). In total, 543 imaging spectral scans were completed, each with a cadence of 9.4s. A
radial blueshift correction was performed to compensate for the classically mounted etalons28. A
whitelight camera, in sync with the IBIS narrowband sequences, was employed to de-stretch the
resulting spectral scans29, 30. Contextual ROSA 4170 ˚
A continuum and IBIS Ca II 8542 ˚
A line core
images are displayed in Fig. 1, alongside contours depicting the precise location of the FIRS slit.
Establishing the Doppler velocities. The orientation of the FIRS slit resulted in it crossing an
approximate 11 Mm (∼1500) expanse of the sunspot umbra. This extent is highlighted by a solid
green line in Fig. 1b, with pixels beyond this corresponding to penumbral or quiet Sun locations.
Each Stokes Ispectrum extracted from the sunspot umbra is normalised by its own respective
average continuum intensity, Ic(Extended Data Fig. 1a). When the time-spectral evolution of an
15

umbral pixel is examined (see, e.g., Extended Data Fig. 1b), it is clear to see regular periodic wave
signals with a characteristic period on the order of 3minutes.
It must be noted that the He I spectral window around 10830 ˚
A is actually a collection of three
independent electron transitions:31 2s3S1– 2p3P0at 10829.09 ˚
A, 2s3S1– 2p3P1at 10830.25 ˚
A, and
2s3S1– 2p3P2at 10830.34 ˚
A. Typically, under conditions where velocity signatures are signifi-
cantly subsonic,32, 33 the He Itriplet is observed as two absorption features since the 10830.25 ˚
A
and 10830.34 ˚
A profiles are fully blended together (forming the deep “red” component), while
the more shallow “blue” component at 10829.09 ˚
A remains isolated34. It is common practice to
perform spectropolarimetric inversions on the blue He Icomponent since it has a higher effective
Land´
eg-factor (geff = 2.0at 10829.09 ˚
A, versus geff = 1.75 and geff = 1.25 at 10830.25 ˚
A and
10830.34 ˚
A, respectively) and potential blends do not need to be considered. Indeed, the HAZEL
inversions performed on this dataset were applied to the blue He Icomponent24 . On the other
hand, (subsonic) Doppler velocity measurements can be derived more reliably from the red He I
component due to its significantly larger line depth and intrinsically better signal-to-noise. For the
purpose of establishing Doppler maps of the sunspot umbra, we fit the red component of the He I
spectra since the resulting profiles are fully blended, with no evidence of supersonic flows (see,
e.g., Extended Data Fig. 1a).
By fitting all 35,350 (101 pixels across the 11 Mm umbral diameter and 350 acquisitions
in time) He I10830 ˚
A absorption lines with a Voigt profile35 (a combination of Gaussian and
Lorentzian profiles due to the Doppler and pressure broadening sensitivities, respectively, of the
He I10830 ˚
A line), the intrinsic Doppler velocities were mapped. Velocity oscillations with am-
16

plitudes in the range of ±6km s−1, were found to span the entire diameter of the chromospheric
sunspot umbra and last throughout the duration of the observing period (see, e.g., Extended Data
Fig. 1b).
Segregating the umbra into two distinct regions. Within the 11 Mm umbral region, there ex-
isted a persistent brightening that was only visible in the chromospheric image sequences obtained
by IBIS (Ca II 8542 ˚
A) and HARDcam (Hα6563 ˚
A). The brightening, at approximately 5.6 Mm
(≈7.700) along the umbral portion of the FIRS slit, is highlighted with a pink cross in Fig. 1b
and in Extended Data Fig. 2a. Due to an absence of this feature in the photospheric observations
(see, e.g., the ROSA 4170 ˚
A continuum image depicted in Fig. 1a), the chromospheric umbral
brightening is likely to be a long-lived filamentary structure16,36 . Indeed, from inspection of the
time-distance map of the fitted He IDoppler velocities in Extended Data Fig. 1b, the umbral bright-
ening (highlighted using a vertical dashed pink line) exhibits preferential red-shifted Doppler ve-
locities, a characteristic that is consistent with previous observational measurements17. Extended
Data Fig. 2b displays a magnified view of the chromospheric sunspot umbra, where the persistent
brightening is better revealed (and highlighted using a pink cross).
To focus our study purely on the umbral signatures, we decompose the umbral spectra into
two distinct regions that are isolated from one another by the long-lived chromospheric filamentary
structure crossing the FIRS slit. To do this, we extract a time-averaged Stokes I/Icintensity along
the umbral portion of the FIRS slit (see, e.g., the solid green line in Extended Data Fig. 2b) that
corresponds to the He I10830 ˚
A line core. This intensity profile is plotted in Extended Data Fig. 2c,
where the vertical dashed pink line highlights the location of the persistent chromospheric umbral
17

brightening, which demonstrates intensities considerably above the quiescent umbral background.
Next, the centres of gravity (or barycenter) for the intensity profile south (i.e., <5.6Mm) and north
(i.e., >5.6Mm) of the umbral brightening are calculated to be 2.8Mm and 8.3Mm, respectively,
which are represented by the vertical dashed red lines in Extended Data Fig. 2c. Thus, the green
and blue shaded regions in Extended Data Fig. 2c represent the southern and northern umbral
regions, respectively, which are isolated from one another by the persistent chromospheric umbral
brightening. Now, any signatures extracted from the data can be related directly to a particular
umbral region for further characterisation, in addition to the distance from their respective umbral
barycenter.
Regions present in the Fourier spectra. In accordance with recent theoretical work11, we isolate
the spectral energy plots into three distinct regions:
•Region I(<5mHz) – Part of the spectral energy that is governed by the local acoustic cut-off
frequency, ωc, which only allows waves to propagate upwards providing,
ω > ωc=cs
2Hr1+2δH
δz ,
where csis the local sound speed, zis the atmospheric height and H=c2
s/(γg)with γthe
adiabatic index and gthe acceleration due to gravity37,38 . Due to the almost vertical nature
of the magnetic field lines at the core of the sunspot umbra24, we choose an upper limit for
region Iat 5 mHz, which is consistent with other sunspot observations39.
•Region II (6−17 mHz) – Portion of the spectral energy where propagating waves become
permissible (i.e., >5mHz) and demonstrate broad spectral peaks and strong spectral energies
18

that are consistent with previous observational findings19,40, 41 . Here, region II continues from
6−17 mHz (allowing for a buffer region between 5−6mHz to allow the overall spectrum
to reach a peak energy), revealing a gradual decrease in the spectral energy with frequency.
The upper boundary of region II is set at 17 mHz, since another peak at frequencies beyond
18 mHz commences the beginning of region III.
•Region III (18 −27 mHz) – The final regime of the spectral energy corresponds to the range
where steep spectral gradient declines are found. This is the region where the spectral energy
is proportional to a fαscaling, where fis the frequency and αis the linear gradient when
plotted on log–log axes11. Due to the 14.6s cadence of the FIRS observations, the resulting
Nyquist frequency is 34 mHz. From examination of the individual spectral energies, white
noise fluctuations commence around 28 −30 mHz, which was identified by the flattening
of the spectral energy beyond these frequencies (see Fig. 2d). As a result, we set the upper
boundary for region III at 27 mHz to avoid contamination from high-frequency white noise.
It is region III that acts as a diagnostic tool for estimating the thickness and temperature
structuring of the chromospheric resonance cavity.
It must be noted that the spectral frequency ranges for regions I(<5mHz), II (6−17 mHz) and III
(18 −27 mHz) remain fixed throughout the data analyses.
Features visible in the spectral energy maps. Fig. 2d displays the spectral energies for all 101
pixels across the sunspot using a graduated blue-to-pink colour scale. While there is scatter in the
spectral energy at each component frequency, the general trend remains the same across all umbral
19

pixels, reiterating the usefulness of sub-dividing the spectral energy densities into their constituent
regions (i.e., regions I,II and III). The mean spectral energy is overplotted in Fig. 2d using a solid
black line, along with the maximum-likelihood fitted lines of best fit for regions II and III using
solid red lines. It is clear to see that, on average, the maximum-likelihood fitted spectral gradient
for region III is steeper than that for region II, as predicted in recent theoretical models11.
It is possible to re-display Fig. 2d, only now preserving the information along the spatial
diameter of the chromospheric sunspot umbra. Extended Data Fig. 3 displays a two-dimensional
map of the spectral energy, plotted as a function of frequency (x-axis) and distance across the
umbra (y-axis). As with the spectral energies plotted in Fig. 2d, it is clear to see a dominant broad
band of peak power at ≈5mHz across the entirety of the umbra. From visual inspection, it is
also possible to identify wedge-shaped traces of peak power extending outwards from the north
and south umbral barycenters (identified by the horizontal dashed green lines). A black dotted
line in Extended Data Fig. 3 tracks the frequency corresponding to the weighted spectral energy
centroid (between frequencies of 3−17 mHz) across the entirety of the sunspot umbra. It can
be seen that at the north/south umbral barycenters correspond to the highest centroid frequency
(∼6.5mHz or ∼155 s), while the furthest extremities of each umbral section demonstrate the
lowest centroid frequencies (∼5.0mHz or ∼180 s). This type of behaviour is consistent with the
umbral barycenters displaying the most vertical magnetic fields, hence pushing the acoustic cut-
off frequency to higher values37. A similar phenomenon has also been observed in IBIS spectral
imaging observations of umbral oscillations39, and reiterates the appropriateness of defining the
start of region II at 6 mHz.
20

From Extended Data Fig. 3, it is also possible to identify the second enhancement of spectral
energy corresponding to region III (≈18 mHz or 55 s). Here, there appears to be more pronounced
enhancements of spectral energy close to the north/south umbral barycenters (i.e., coincident with
the horizontal dashed green lines in Extended Data Fig. 3), when compared to similar frequencies
at the very extreme edges of the sunspot umbra (e.g., distances of approximately 0Mm and 11 Mm
in Extended Data Fig. 3). The spectral energy associated with region III is much weaker than that
found in region II, often by 1−2orders of magnitude (also visible in Fig. 2d). As the spectral gra-
dient present in region III contains information related to the structuring of the underlying umbral
resonance cavity, it is important to calculate the spectral slope with a high degree of precision.
Fitting the spectral energy gradients. To calculate the spectral slopes corresponding to regions II
and III, maximum-likelihood fitted gradients are computed for each of the 101 spectral energies
across the chromospheric umbra. Often linear lines of best fit are established to determine the
spectral slopes of Fourier power spectra42. However, the weighted least-squares minimisation
process assumes that the data to be fitted are normally (Gaussian) distributed20, which may not
necessarily be the case, especially when the periodogram of a stationary, linear stochastic process
naturally follows a χ2
2distribution43, 44. As a result, we apply the maximum-likelihood approach20,
which has recently been successfully applied to solar wave studies45, to calculate the spectral
gradients for regions II and III of the 101 spectral energies across the diameter of the sunspot umbra.
The spectral gradients for regions II and III, as a function of distance from their respective umbral
barycenter, are plotted in Fig. 3a. It is clear that the spectral slopes for region I I are relatively
constant, with a slope of −2.1±0.2. Contrarily, the spectral gradients for region III are seen to
21

vary as a function of distance from their respective umbral barycenter, with gradients as shallow
as −5.4±0.6at the core of the relevant umbra, extending to gradients as steep as −7.8±0.6at
maximal distances (∼3000 km) from each barycenter.
We believe the gradients displayed for region II in Fig. 3a point towards a universal sunspot
characteristic for waves detected in chromospheric spectral lines. A recent observational exami-
nation of the spectral power slopes for an entirely different sunspot, using ground-based images
obtained in the Ca II K and Hαline cores, found similar spectral gradients (within the 6−17 mHz
spectral range) to those presented here42. These gradients are steeper than both the f−1and f−5/3
relationships that would be expected for granulation (i.e., pink) noise patterns46 and the Kolo-
mogorov inertial range47, respectively. Instead, they more closely resemble the f−2red noise that
is linked to the strong viscous dissipative regimes associated with Brownian motion48. Region III,
on the other hand, has even steeper spectral gradients that more closely resemble the strong dissi-
pative ranges previously documented in studies of the solar wind21, 49. Importantly, the steepening
of the spectral slopes as one moves away from the umbral barycenters indicates a strong depen-
dency between the value of the spectral gradient and the characteristics of the underlying resonance
cavity in which the spectral signatures were generated.
Numerical magnetohydrodynamic simulations. The numerical code employed in the current
work is based on the well-documented Lare2D22 software. Here, a velocity driver is injected
into an atmospheric model, containing realistic temperature and density structuring, allowing the
Lare2D code to evolve the idealised non-linear compressible MHD equations to compute the ve-
locity signatures as a function of distance along the computational domain. The numerical domain
22

covers the vertical range −21 Mm ≤z≤21 Mm and is resolved by 8192 grid cells. The up-
per convection zone is modelled by a polytropic temperature profile and situated at z < 0Mm
in the domain. Above z= 0 Mm the temperature profile of sunspot model50 ‘M’ is used, which
connects smoothly at the transition region with a typical coronal temperature profile51. Near the
upper boundary the temperature profile is flattened (i.e., a constant value is used) to create an
open boundary at z= 21 Mm. An initial equilibrium is obtained by solving the pressure balance
equation, and the two horizontal directions of the computational domain are invariant, making the
resulting simulation 1.5D.
In order to make the numerical outputs as realistic as possible, we employ the HAZEL in-
versions of the umbral barycenter pixels (i.e., where the magnetic field inclination angles are ap-
proximately 0degrees) of active region NOAA 12656 [ref. 24] to provide chromospheric plasma
constraints, which allow the sunspot model50 ‘M’ to be re-scaled across the rest of the computa-
tional grid. This re-sampled sunspot model formed the background atmosphere embedded within
the Lare2D code, with the temperature values plotted using a solid black line in Extended Data
Fig. 4. It must be noted that the sunspot under current investigation is slightly less magnetic and
fractionally hotter than the standard sunspot model50 ‘M’ atmosphere. Comparisons with the out-
puts of the Very Fast Inversion of the Stokes Vector52 (VFISV) algorithm, applied to co-temporal
SDO/HMI vector magnetogram data53 that have a formation height in the low photosphere, reveal
maximum umbral magnetic field strengths on the order of 2000 G, which is consistent with the
re-sampled sunspot models.
Previous theoretical work11 that studied the characteristics of sunspot resonance cavities em-
23

ployed a variety of injected photospheric velocity drivers, including those corresponding to white
(f0), pink (f−1) and red (f−2) noise signatures, which are believed to be representative of the
spectral signatures present in the Sun’s upper convection zone54. However, the spectroscopic ob-
servations of active region NOAA 12565 obtained by FIRS allows us to provide a better estimate of
the real underlying photospheric velocity signal. Extracting bisector velocities of the Si I10827 ˚
A
absorption feature at 20% of the maximum line depth provide photospheric velocities that corre-
spond to an optical depth of log(τ500nm)∼ −0.65, or ∼50 km above the photospheric layer50,55 .
The extracted Si I10827 ˚
A velocity signatures have peak amplitudes on the order of 300 m/s (see
the Fig. 2a), which is consistent with previous sunspot oscillation studies56. These photospheric
velocity perturbations are re-scaled and applied at the lower (z=−21 Mm) boundary of the
Lare2D code and allowed to evolve. The re-scaling is to ensure that the wave root mean square
(rms) amplitudes produced by the simulations at z= 0 Mm (i.e., the photosphere) are consistent
with the observed Si I10827 ˚
A profile fluctuations.
Data interpretation The velocity outputs from the Lare2D simulation were extracted at a cadence
of 14.6s (to match that of the FIRS observations) at an atmospheric height of 2120 km (vertical
dotted black line in Extended Data Fig. 4), which is consistent with the approximate formation
height of the He I10830 ˚
A spectral line25, 57. The velocity time series was then cropped to 86 min-
utes in duration (again, to match that of the observations) and converted into spectral energies by
following the same methodology applied to the FIRS He I10830 ˚
A data. There are a number
of distinct similarities between the observed and simulated velocity time series (Fig. 2c,e). First,
both the observed and simulated time series appear modulated by a long-term trend. Such long-
24

period modulation has been extensively observed in magnetoacoustic wave studies58–60, which is
normally explained as a consequence of beat phenomena created by the superposition of a number
of closely spaced frequencies61. This observed phenomenon further supports the presence of a
chromospheric resonance cavity, since it has been theoretically shown62 that long-term modulating
periods can be created in the confines of resonant filters. Secondly, the velocity amplitudes corre-
sponding to the modelled (rms = 3.2km/s) and observed (rms = 3.5km/s) upper chromosphere
are very similar, demonstrating that the wave amplification process is accurately accounted for in
the Lare2D model. Finally, the spectral energies computed for both the modelled and observed
time series show similar trends across regions I,II and III. In particular, the modelled region Ialso
displays the relatively flat spectral energy that is consistent with the presence of evanescent waves.
Next, the modelled region II reveals a similar peak wave energy at ∼5mHz, followed by a gradual
decline in spectral energy with a maximum-likelihood fitted gradient equal to −2.3±0.3. As per
the observed spectral gradients for region II (see the magenta data points in Fig. 3a), the modelled
values also closely map to the presence of red noise (i.e., f−2). Lastly, the modelled region III
demonstrates an identical rise in spectral energy at ∼20 mHz (∼50 s), before dropping off very
rapidly with increasing frequency.
Importantly, however, is the fact that the blue data points in Fig. 3a indicate that the spectral
slope associated with region III varies as a function of distance away from the umbral barycenter.
Theoretical work11 has revealed that the thickness of the chromospheric resonance cavity has im-
plications for the steepness of the measured spectral gradient, with shallower resonance cavities
demonstrating steeper spectral slopes than their thicker cavity counterparts. To investigate this
25

effect further, the thickness (i.e., atmospheric height span) of the resonance cavity was re-scaled
at 80%, 90%, 110% and 120% of the original depth (Extended Data Fig. 4), providing the at-
mospheric parameters detailed in Extended Data Fig. 5. For each resonance cavity thickness, the
numerical models were recomputed, with the spectral energies calculated and the corresponding
spectral gradients measured using an identical maximum-likelihood approach. Utilising the strati-
fied temperature profiles listed in Extended Data Fig. 5, the computed spectral slopes for region III
can be plotted as a function of the magnetoacoustic wave propagation distances (Extended Data
Fig. 6). It can be seen that larger wave propagation distances (i.e., increased cavity depths) intro-
duce a greater resonant energy content, hence providing more energy across the frequency range,
and thus reducing the steepness of the associated spectral slope.
The general trends depicted in Fig. 3 and Extended Data Fig. 6 are consistent with previ-
ous modelling efforts11, whereby deeper resonance cavities produce inherently shallower spectral
gradients in region III (18 −27 mHz). Of course, it must be noted that Fig. 3 and Extended Data
Fig. 6 depict the variations in the spectral slopes as a function of the distance over which the mag-
netoacoustic waves propagate. In an idealised case, where the magnetic fields are aligned with the
normal to the solar surface, these propagation distances will be identical to the geometric height of
the upper chromosphere. On the other hand, if the magnetic field lines are inclined to the solar nor-
mal, then this angle will need to be incorporated into the calculation to estimate the true geometric
height of the upper chromosphere in that location. With this in mind, it becomes possible to esti-
mate the depth of the chromospheric resonance cavity for each location within the sunspot umbra
simply by comparing the measured spectral slope of region III to the reference spectral energies
26

computed via the Lare2D numerical models.
Here, we have utilised the presence of a spectral energy peak at ∼20 mHz (i.e., region III;
see Fig. 2d) as an indicator of a resonance cavity existing in our observational and simulated data,
which is consistent with theoretical and numerical investigations documented in recent years9,11 .
However, to confirm that a resonance cavity is the mechanism responsible for the elevated spectral
energies in the range of 18−27 mHz, we create an independent numerical test whereby the Lare2D
simulations are re-run for a background atmosphere devoid of the steep transition region temper-
ature gradient. For this test, the temperature reaches the chromospheric plateau value (∼6000 K;
see Extended Data Fig. 4) at an atmospheric height of 0km, then remains constant through to the
upper boundary of the simulation domain, hence removing the conditions necessary for an acoustic
resonator to operate (i.e., the temperature gradient synonymous with the transition region).
The resulting spectral energies (Extended Data Fig. 7) reveal how removing the transition
region entirely from the model atmosphere produces a shallower spectral gradient following the
∼5mHz peak. This is likely a consequence of the flattened temperature profile modifying the
acoustic cut-off frequency with atmospheric height, hence resulting in a different distribution of
energies across the frequency spectrum63. Importantly, removing the steep temperature gradient
inherent to the solar transition region, which is believed to be required for the initiation of res-
onance behaviour9, 11, acts to alleviate the rise in spectral energies at ∼20 mHz. As a result, we
conclude that the heightened spectral energies contained within the observed and simulated re-
gion III (18 −27 mHz; Fig. 2) are a direct consequence of the lower (photospheric) and upper
(transition region) temperature gradients intrinsic to the solar atmosphere, hence giving rise to the
27

creation of a resonance cavity.
The range of gradients measured for region III (18−27 mHz) of the sunspot spectral energies
span −5.4±0.6(close to the umbral barycenter) to −7.8±0.6(at the outermost extremities of the
umbra). Immediately, this suggests that the chromospheric resonance cavity is thickest near the
core of the umbra, dropping to its thinnest depth at the penumbral boundary. Extended Data Fig. 6
allows the wave propagation distance, τprop, to be defined as,
τprop (km) = Region III spectral slope + 26.408
0.009131 ,
where 0.009131 is the gradient of the dashed black line and 26.408 is the intercept on the yaxis
(Extended Data Fig. 6). The extreme values of the measured spectral slopes, −7.8and −5.4,
provide wave propagation distances on the order of 2035 km and 2300 km, respectively (shaded
magenta and green regions in Extended Data Fig. 6).
However, the magnetic fields spanning the diameter of the sunspot umbral chromosphere are
not all vertical in nature (Extended Data Fig. 8). Examining the magnetic field inclination angles,
θ, derived from HAZEL inversions applied to the He I10830 ˚
A spectropolarimetric data reveals
that the umbral cores have the most vertical magnetic fields (approximately 0degrees), while the
outermost extremities of the umbrae demonstrate the most inclined magnetic fields (approximately
35 −50 degrees on average). Furthermore, the filamentary structure that segregates the sunspot
into two isolated umbrae displays increased inclination angles approaching 40 degrees (located at
approximately 5.6Mm along the FIRS slit in Extended Data Fig. 8).
As a result, the stratified wave propagation path lengths (Extended Data Fig. 4) are tilted
28

from the solar normal by the inclination angle, θ, which needs to be taken into consideration when
estimating the true atmospheric height of the upper chromospheric boundary. As such, the true
geometric height of the upper chromosphere, τchromo, can be defined as,
τchromo (km) = Region III spectral slope + 26.408
0.009131 cos θ
=τprop cos θ .
Utilising the spatially-resolved inclination angles and spectral gradients provides true geometric
heights of the upper boundary of the umbral cavity spanning 1300 ±200 km (spectral slope of
−7.8±0.6and an inclination angle of approximately 50 degrees; outer umbral edge) through to
2300 ±250 km (spectral gradient of −5.4±0.6and a vertically-orientated magnetic field; umbral
barycenter). This can be visualised graphically in Fig. 4, where the pink isocontours represent the
geometric height of the upper chromospheric boundary across the diameter of the sunspot umbra.
29

Data Availability The data used in this paper are from the observing campaign entitled “The influence of
Magnetism on Solar and Stellar Atmospheric Dynamics” (NSO-SP proposal T1081; Principle Investigator:
D. B. Jess), which employed the ground-based Dunn Solar Telescope, USA, during July 2016. Additional
supporting observations were obtained from the publicly available NASA’s Solar Dynamics Observatory
(https://sdo.gsfc.nasa.gov) data archive, which can be accessed via http://jsoc.stanford.edu/ajax/lookdata.html.
The data that support the plots within this paper and other findings of this study are available from the cor-
responding author upon reasonable request.
30

26. Jess, D. B. et al. The Source of 3 Minute Magnetoacoustic Oscillations in Coronal Fans.
Astrophys. J. 757, 160 (2012).
27. Derks, A., Beck, C. & Mart´
ınez Pillet, V. Inferring telescope polarization properties through
spectral lines without linear polarization. Astron. Astrophys. 615, A22 (2018).
28. Cauzzi, G. et al. The solar chromosphere at high resolution with IBIS. I. New insights from
the Ca II 854.2 nm line. Astron. Astrophys. 480, 515–526 (2008).
29. Jess, D. B., Mathioudakis, M., Christian, D. J., Crockett, P. J. & Keenan, F. P. A Study of
Magnetic Bright Points in the Na I D1Line. Astrophys. J. Lett. 719, L134–L139 (2010).
30. Stangalini, M., Del Moro, D., Berrilli, F. & Jefferies, S. M. MHD wave transmission in the
Sun’s atmosphere. Astron. Astrophys. 534, A65 (2011).
31. Lagg, A., Woch, J., Solanki, S. K. & Krupp, N. Supersonic downflows in the vicinity of
a growing pore. Evidence of unresolved magnetic fine structure at chromospheric heights.
Astron. Astrophys. 462, 1147–1155 (2007).
32. Aznar Cuadrado, R., Solanki, S. K. & Lagg, A. Velocity distribution of chromospheric down-
flows. In Kneer, F., Puschmann, K. G. & Wittmann, A. D. (eds.) Modern solar facilities -
advanced solar science, 173 (2007).
33. Gonz´
alez Manrique, S. J. et al. Fitting peculiar spectral profiles in He I 10830 ˚
A absorption
features. Astronomische Nachrichten 337, 1057 (2016).
31

34. Gonz´
alez Manrique, S. J. et al. Temporal evolution of arch filaments as seen in He I 10 830
˚
A. Astron. Astrophys. 617, A55 (2018).
35. Zaghloul, M. R. On the calculation of the Voigt line profile: a single proper integral with a
damped sine integrand. Mon. Not. R. Astron. Soc. 375, 1043–1048 (2007).
36. Henriques, V. M. J. et al. Stable umbral chromospheric structures. Astron. Astrophys. 574,
A131 (2015).
37. Bel, N. & Leroy, B. Analytical Study of Magnetoacoustic Gravity Waves. Astron. Astrophys.
55, 239 (1977).
38. Roberts, B. MHD Waves in the Solar Atmosphere. In Lacoste, H. (ed.) SOHO 13 Waves,
Oscillations and Small-Scale Transients Events in the Solar Atmosphere: Joint View from
SOHO and TRACE, vol. 547 of ESA Special Publication, 1 (2004).
39. L¨
ohner-B¨
ottcher, J., Bello Gonz´
alez, N. & Schmidt, W. Magnetic field reconstruction based
on sunspot oscillations. Astronomische Nachrichten 337, 1040 (2016).
40. Jess, D. B. et al. Solar coronal magnetic fields derived using seismology techniques applied
to omnipresent sunspot waves. Nature Physics 12, 179–185 (2016).
41. Jess, D. B. et al. An Inside Look at Sunspot Oscillations with Higher Azimuthal Wavenumbers.
Astrophys. J. 842, 59 (2017).
42. Krishna Prasad, S. et al. The Frequency-dependent Damping of Slow Magnetoacoustic Waves
in a Sunspot Umbral Atmosphere. Astrophys. J. 847, 5 (2017).
32

43. Groth, E. J. Probability distributions related to power spectra. Astrophys. J. Supp. 29, 285–302
(1975).
44. Papadakis, I. E. & Lawrence, A. Improved Methods for Power Spectrum Modelling of Red
Noise. Mon. Not. R. Astron. Soc. 261, 612 (1993).
45. Morton, R. J., Weberg, M. J. & McLaughlin, J. A. A basal contribution from p-modes to the
Alfv´
enic wave flux in the Sun’s corona. Nature Astronomy (2019).
46. Matthaeus, W. H. et al. Density and Magnetic Field Signatures of Interplanetary 1/f Noise.
Astrophys. J. Lett. 657, L121–L124 (2007).
47. Huang, S. Y., Hadid, L. Z., Sahraoui, F., Yuan, Z. G. & Deng, X. H. On the Existence of the
Kolmogorov Inertial Range in the Terrestrial Magnetosheath Turbulence. Astrophys. J. Lett.
836, L10 (2017).
48. Sheikholeslami, M., Abelman, S. & Ganji, D. D. Numerical simulation of mhd nanofluid flow
and heat transfer considering viscous dissipation. International Journal of Heat and Mass
Transfer 79, 212 – 222 (2014).
49. Goldstein, M. L., Roberts, D. A. & Fitch, C. A. Properties of the fluctuating magnetic helicity
in the inertial and dissipation ranges of solar wind turbulence. J. Geophys. Res. 99, 11519–
11538 (1994).
50. Maltby, P. et al. A new sunspot umbral model and its variation with the solar cycle. Astrophys.
J. 306, 284–303 (1986).
33

51. Avrett, E. H. & Loeser, R. Models of the Solar Chromosphere and Transition Region from
SUMER and HRTS Observations: Formation of the Extreme-Ultraviolet Spectrum of Hydro-
gen, Carbon, and Oxygen. Astrophys. J. Supp. 175, 229–276 (2008).
52. Borrero, J. M. et al. VFISV: Very Fast Inversion of the Stokes Vector for the Helioseismic and
Magnetic Imager. Sol. Phys. 273, 267–293 (2011).
53. Schou, J. et al. Design and Ground Calibration of the Helioseismic and Magnetic Imager
(HMI) Instrument on the Solar Dynamics Observatory (SDO). Sol. Phys. 275, 229–259 (2012).
54. Rabello-Soares, M. C., Roca Cortes, T., Jimenez, A., Andersen, B. N. & Appourchaux, T. An
estimate of the solar background irradiance power spectrum. Astron. Astrophys. 318, 970–974
(1997).
55. Gonz´
alez Manrique, S. J., Kuckein, C. & Ruiz Cobo, B. Comparison of bisectors with inver-
sions based on response functions to infer line-of-sight velocities on the Si I 10827 ˚
A. Astron.
Astrophys. (2019).
56. Felipe, T., Khomenko, E., Collados, M. & Beck, C. Multi-layer Study of Wave Propagation in
Sunspots. Astrophys. J. 722, 131–144 (2010).
57. Vernazza, J. E., Avrett, E. H. & Loeser, R. Structure of the solar chromosphere. III - Models
of the EUV brightness components of the quiet-sun. Astrophys. J. Supp. 45, 635–725 (1981).
58. Centeno, R., Collados, M. & Trujillo Bueno, J. Spectropolarimetric Investigation of the Prop-
agation of Magnetoacoustic Waves and Shock Formation in Sunspot Atmospheres. Astrophys.
J. 640, 1153–1162 (2006).
34

59. Centeno, R., Collados, M. & Trujillo Bueno, J. Wave Propagation and Shock Formation in
Different Magnetic Structures. Astrophys. J. 692, 1211–1220 (2009).
60. Krishna Prasad, S., Jess, D. B. & Khomenko, E. On the Source of Propagating Slow Magne-
toacoustic Waves in Sunspots. Astrophys. J. Lett. 812, L15 (2015).
61. Marsh, M. S. & Walsh, R. W. p-Mode Propagation through the Transition Region into the
Solar Corona. I. Observations. Astrophys. J. 643, 540–548 (2006).
62. Zhukov, V. I. Oscillations on the Sun in regions with a vertical magnetic field. II. On the
calculation of the sunspot umbral oscillations. Astron. Astrophys. 433, 1127–1132 (2005).
63. Zhugzhda, Y. D. Three-minute oscillations in sunspots: Seismology of sunspot atmospheres.
Astronomy Letters 33, 622–643 (2007).
35

Figure 1
The velocity signatures of the magnetised sunspot atmosphere observed on 14 July 2016.
(a & b) ROSA 4170 Å continuum (photosphere) and IBIS 8542 Å line core (chromosphere) images of
the sunspot atmosphere at 13:42 UT on 14 July 2016, with the axes displaying the associated
heliocentric co-ordinates. The solid red lines highlight the position of the FIRS slit, while the solid
green line indicates the portion of the slit that clearly crosses the chromospheric umbra (i.e., not
overlapping with penumbra or quiet Sun structures). The pink cross identifies a persistent umbral
brightening that segregates the umbral spectra into two distinct regions.
(c) A velocity-time image that documents the spectral and temporal evolution of the He I 10830 Å
Stokes I line profile from a single umbral pixel. The black-to-white colour scale represents the
inverted spectral intensities to assist with visual clarity, while the horizontal dashed red line
indicates the rest position of the He I 10830 Å line core.

Figure 2
Doppler velocities and spectral energies of observed and simulated time series.
(a & b) The Si I 10827 Å (photospheric) Doppler velocity signal for an umbral pixel, with its
corresponding spectral energy displayed on log–log axes.
(c) The co-spatial He I 10830 Å (upper chromospheric) Doppler velocity signal, where higher
frequency waves are more readily visible when compared to panel (a).
(d) The calculated He I 10830 Å spectral energies, where the graduated blue-to-pink coloured lines
represent spectra derived across the entire sunspot umbral diameter. The solid black line represents
the average umbral spectral energy.
(e & f) The simulated velocity time series, which is extracted from the Lare2D computational domain
at an atmospheric height that is compatible with the formation of the He I 10830 Å spectral line,
along with its corresponding spectral energy. The dashed red lines, panels (a), (c) and (e), highlight a
zero velocity for visual reference. Orange, green and blue shaded regions (bounded by black vertical
dashed lines; panels (d) and (f) isolate the spectral energies into regions I (< 5 mHz), II (6 – 17 mHz)
and III (18 – 27 mHz). The solid red lines, panels (d and (f), highlight the respective maximum-
likelihood fits spanning the frequency domains corresponding to regions II and III.

Figure 3
Spectral energy gradients of observed and simulated time series.
(a) The spectral power gradients measured for each of the umbral Fourier energy densities,
displayed as a function of distance from their associated umbral region barycenter. Here, the
magenta and blue colours correspond to regions II (6 – 17 mHz) and III (18 – 27 mHz), while the circle
and diamond symbols relate to the southward and northward locations in relation to the persistent
umbral brightening, respectively. The vertical error bars placed on each data point correspond to the
maximum-likelihood 1σ fitment uncertainties when measuring the spectral power-law gradients.
The dashed magenta and blue lines highlight the linear lines of best fit associated with regions II and
III, while the shaded magenta and blue regions identify the standard deviations for the lines of best
fit, respectively.
(b) The spectral slopes of region III (18 – 27 mHz), which are calculated from the maximum-likelihood
fitments of the Fourier spectral energies produced by Lare2D numerical simulations, as a function of
the variable resonance cavity depths imposed in the modelled atmospheres. The vertical error bars
highlight the maximum-likelihood 1 fitment uncertainties achieved when measuring the
corresponding spectral power-law gradients. The dashed black line maps the linear best-fit line
through the data points, while the blue shaded region (bounded by the black dotted lines) highlights
the 95% confidence level associated with the fitted line.

Figure 4
Three-dimensional visualisation of the geometric extent of the chromosphere above active region
NOAA 12565.
The geometric extent of the chromosphere, visualised here as the pink isocontours extending
upwards from the photospheric (ROSA 4170 Å continuum) umbra and through the chromospheric
(IBIS 8542 Å line core). It can be seen that the depth of the resonance cavity is suppressed in the
immediate vicinity of the trans-umbral filamentary structure, providing geometric heights of
approximately 1300 km, which is consistent with the depth measured at the outermost edges of the
umbra. The cores of the umbrae display the largest resonance cavity depths, often with geometric
heights on the order of 2300 km. An image of the Earth is added to provide a sense of scale.
Note: The pink resonance cavity depth contours are not to scale.