# Modelling the propagation of slow magneto-acoustic waves in a multi-stranded
coronal loop

S. Krishna Prasad Aryabhatta Research Institute of Observational Sciences,
Nainital-263001, India  T. Van Doorsselaere  Centre for mathematical Plasma
Astrophysics, Mathematics department, KU Leuven, Celestijnenlaan 200B, 3001
Leuven, Belgium

###### Abstract

We study the propagation properties of slow magneto-acoustic waves in a multi-
thermal coronal loop using a 3D MHD model, for the first time. A bundle of 33
vertical cylinders, each of 100 km radius, randomly distributed over a
circular region of radius 1 Mm is considered to represent the coronal loop.
The slow waves are driven by perturbing the vertical velocity ($v_{z}$) at the
base of the loop. We apply forward modelling to the simulation results to
generate synthetic images in the coronal channels of SDO/AIA. Furthermore, we
add appropriate data noise to enable direct comparison with the real
observations. It is found that the synthetic images at the instrument
resolution show non-cospatial features in different temperature channels in
agreement with previous observations. Time-distance maps are constructed from
the synthetic data to study the propagation properties. The results indicate
that the oscillations are only visible in specific channels depending on the
temperature range of plasma existing within the loop. Additionally, the
propagation speed of slow waves is also found to be sensitive to the available
temperature range. Overall, we propose that the cross-field thermal properties
of coronal structures can be inferred using a combination of numerical
simulations and observations of slow magneto-acoustic waves.

Magnetohydrodynamical simulations (1966), Solar coronal heating (1989), Solar
coronal loops (1485), Solar coronal waves (1995)

##  1 Introduction

There is ample evidence in the literature suggesting that a sizable fraction
of the solar coronal loops possess multithermal cross-sections. Because the
cross-field thermal conduction is heavily suppressed in the solar corona, such
structures can exist as multi-stranded loops. The volume filling factor for
the plasma within the coronal loops is estimated to be a few to several tens
of %, indicating the presence of fine structure within the coronal loops (Di
Matteo et al., 1999; Warren et al., 2008; Tripathi et al., 2009; Gupta et al.,
2015). Similar characteristics were found in active region fan loops (Young et
al., 2012).

It has been observed that coronal structures appear diffuse (or fuzzy) at hot
temperatures and sharp at warm temperatures (Dudok de Wit & Auchère, 2007;
Tripathi et al., 2009). Eclipse observations in the past also displayed
similar behaviour (see for e.g., Fig. 2 of Singh et al., 2006). An explanation
was provided in terms of sub-resolution strands within coronal loops (Guarrasi
et al., 2010), but it necessitated the reemergence of sharper structures at
very hot ($\sim$6 MK) temperatures. Later, Reale et al. (2011) showed that
thin, discrete structures indeed reappear at hotter temperatures and thus
confirmed the existence of sub-resolution structures within coronal loops.

Using high-resolution spectroscopic observations in H$\alpha$, Antolin &
Rouppe van der Voort (2012) studied the dynamics and morphological properties
of coronal rain. The authors find that the downfalling plasma condensation
blobs roughly trace the coronal loops but have much narrower widths (on an
average $\sim$ 310 km) suggesting the presence of fine structure within those
loops. Ji et al. (2012) employ high-resolution imaging observations in He i
10830 Å line, to identify ultrafine magnetic structures of 100 km diameter
cospatial to an active region loop system. The observed magnetic braiding
(Cirtain et al., 2013) and nano jets (Antolin et al., 2021; Patel & Pant,
2022) also support sub-resolution structuring within coronal loops. Other
studies investigated the distributions of cross-sectional widths of coronal
loops and additionally compared them across different instruments to identify
if the structures are completely resolved (Aschwanden & Peter, 2017; Williams
et al., 2020).

The cross-field thermal structure of a coronal loop is generally obtained from
multi-wavelength data using techniques such as Emission Measure (EM) loci
(e.g., Del Zanna et al., 2002) and Differential Emission Measure (DEM) (e.g.,
Hannah & Kontar, 2012). Early studies produced contradicting results with some
researchers suggesting that coronal loops have an isothermal cross-section
(Del Zanna & Mason, 2003; Cirtain et al., 2007; Noglik et al., 2008; Tripathi
et al., 2009) while some others indicate that they instead have a multi-
thermal cross-section (Martens et al., 2002; Schmelz et al., 2005; Schmelz &
Martens, 2006; Patsourakos & Klimchuk, 2007). Additionally, Warren et al.
(2008) found coronal loops which are not entirely isothermal but have a narrow
temperature distribution over their cross-sections. Later studies involving
the analysis of multiple loop structures show the existence of both isothermal
and multi-thermal loops (Aschwanden & Boerner, 2011; Brooks et al., 2012;
Schmelz et al., 2016).

The multi-stranded and multithermal nature of coronal loops is also evident
from the observations of waves and oscillations. For instance, the
observations of transverse decayless oscillations often show a coherent wavy
pattern across the multiple threads of a loop (Wang et al., 2012; Nisticò et
al., 2013; Mandal et al., 2021). Furthermore, the different dynamical
behaviour of these oscillations across multiple wavelengths also suggests a
multi-thermal nature of the loop. It is believed that the differential
propagation of slow magnetoacoustic waves, as observed using multiple
wavelength channels, indicates a multithermal structure within a loop (King et
al., 2003). In fact, Krishna Prasad et al. (2017) extracted two individual
components of a multithermal loop using the observations of propagating slow
waves. Additionally, Krishna Prasad & Van Doorsselaere (2021) have shown that
the multi-wavelength behaviour of standing slow waves also suggests the
presence of multithermal structure within a coronal loop.

Coronal loops with multiple strands were also considered in numerical models
by a number of authors (e.g., Ofman, 2009; Klimchuk et al., 2010; Luna et al.,
2010; Magyar & Van Doorsselaere, 2016; Guo et al., 2019). In this article, we
study the propagation of slow magnetoacoustic waves in a coronal loop using a
multi-stranded and multithermal loop model for the first time. In such a
model, because the slow waves propagate at different speeds in different
strands owing to their differences in temperatures, they gradually go out of
phase, resulting in the cumulative oscillation amplitude decreasing over a
distance. This implies the slow waves propagating in a multithermal loop will
appear to be damping even though there is no real dissipation of energy. As we
will see, this damping indeed occurs in our model. We call this the
Multithermal Apparent Damping (MAD) which is analysed separately in detail in
Van Doorsselaere et al. (2024). Concurrently, a similar damping effect is seen
by Fedenev et al. (2024), who modelled the propagation of slow magnetoacoustic
waves in a plasma slab with smoothly varying temperature and density profiles
transverse to the magnetic field. We describe our numerical setup, including
the driver details, in Section 2, followed by a discussion of obtained results
in Section 3, and at last we present our conclusions in Section 4.

##  2 Numerical simulations

We perform 3D MHD simulations to study the propagation of slow magneto-
acoustic waves in a multi-thermal coronal loop. Considering ideal conditions
(i.e., no dissipation terms), the MHD equations are numerically solved using
the publicly available MPI-AMRVAC code (version 2.0 Porth et al., 2014; Xia et
al., 2018). We use a second-order predictor-corrector type two-step scheme for
the time integration and the ‘hlld’ flux scheme for the spatial
discretization. The errors on the divergence of the magnetic field are treated
using a Generalized Lagrange Multiplier (GLM) method.

Figure 1: A 3D rendering of the multi-stranded loop model employed. The bottom
panel shows the cross-sectional density profile highlighting the distribution
of strands across the loop cross section.

The numerical setup involves a bundle of vertical cylinders randomly
distributed within a circular region (see Fig. 1) representing a multi-
stranded loop. The radius of each strand is 100 km and that of the loop is 1
Mm. A total of 33 strands are considered resulting in a filling factor of 0.33
for the loop (Katsukawa & Tsuneta, 2005; Ugarte-Urra et al., 2005; Van
Doorsselaere et al., 2014; Gupta et al., 2015). The physical size of the
simulation domain is considered to be 100 Mm along the loop length ($z$-axis)
and 2.5$\times$2.5 Mm2 in the perpendicular direction ($x$–$y$ plane). The
corresponding number of grid cells is 128 along the $z$-axis and
256$\times$256 in the $x$–$y$ plane. Consequently, the spatial scale is
$\approx$0.78 Mm along the loop length and $\approx$9.76 km in the cross-
sectional plane. The plasma temperature within each strand is obtained from a
random normal distribution whose peak is at log $T=6.0$, and has a standard
width of 0.15 in the logarithmic scale. The corresponding plasma (number)
densities are also obtained from an uncorrelated random normal distribution
with a peak value of log $n=9.2$ and a standard width of 0.10. The resultant
plasma temperature in the loop strands varies from 0.41 MK to 1.73 MK.
Accordingly, the number densities vary from 1.2$\times$109 cm-3 to
2.5$\times$109 cm-3. These values fall in the typically observed range for
coronal loops. The external (including those locations within the loop but
outside the strands) plasma temperature and number densities are kept fixed at
log $T=6.0$ and log $n\approx 8.7$ (1/3rd of the internal peak value),
respectively. The gravitational stratification is ignored, meaning the plasma
parameters are the same all along the loop length. Also, following the
definition of the strand, the plasma parameters are kept constant across each
strand. The magnetic field $B$ is considered to be vertical
($B_{x}$,$B_{y}$=0), parallel to the loop axis, with a magnitude of 50 G
($B_{z}$=50). We understand that the coronal magnetic field strength values
typically inferred from the direct (Lin et al., 2000, 2004; Kuridze et al.,
2019; Brooks et al., 2021) and indirect (Nakariakov & Ofman, 2001; Tomczyk et
al., 2007; Jess et al., 2016; Yang et al., 2020) observational techniques
range from a few Gauss to a few tens of Gauss, but considering the complicated
setup a strong field is required to keep the loop stable, so we consider a
value on the higher side in this model. Moreover, since we are interested in
only slow magneto-acoustic waves here, the effect of the magnetic field on
their propagation is going to be negligible. The magnetic field is largely
uniform across the domain except within the loop strands. In order to
compensate for the high gas pressure within the hot and dense strands, the
magnetic field is reduced so that the total pressure is uniform keeping the
loop stable. However, the strong magnetic field also ensures the plasma beta
is very small, and consequently, the reduction in the magnetic field required
is marginal. The $B_{z}$ value across the domain varies from 49.7 – 50 G.

All the boundaries are kept open with zero gradients for all variables except
for the vertical velocity ($v_{z}$) at the bottom boundary which we perturb to
generate slow magneto-acoustic waves in the loop. The $v_{z}$ at the bottom of
the loop is continuously perturbed using a sinusoidal driver of the form

$v_{z}=v_{z0}{\,}\mathrm{sin}(2\pi t/P).$ (1)

Here, $t$ is time, $P$ is the oscillation period, and $v_{z0}$ is the
oscillation amplitude. The oscillation period is chosen as 180 s, a typical
value for propagating slow magneto-acoustic waves observed in active region
loops. The oscillation amplitude is set to $\approx$7.6 km s-1 within the loop
(0 outside the loop) which is about 5 % of the sound speed at the peak plasma
temperature (log $T=6.0$). This small amplitude keeps the generated waves in
the linear regime and is in line with that found in observations (for e.g.,
Krishna Prasad et al., 2012; Sharma et al., 2017; Krishna Prasad et al., 2019;
Nakariakov & Kolotkov, 2020). The simulation is run for 1200 seconds in
physical time which should approximately capture about seven oscillation
cycles near the base of the loop.

##  3 Results and Discussion

The evolution of plasma density and velocity within the simulation clearly
shows fluctuations corresponding to slow magneto-acoustic waves propagating
along the individual strands within the loop. In order to compare the
propagation properties of these waves with those from observations, we
generate synthetic images using the forward modelling code FoMo (Van
Doorsselaere et al., 2016). This code takes the plasma temperature, density,
and velocity at each voxel (volume element) along a desired line-of-sight
direction, and estimates the expected emissivities (taking solar abundances
into account), which are subsequently convolved with the telescope
characteristics (e.g., spectral transmission, detector sensitivity, etc.) and
integrated along the line of sight, to synthesise intensity images for a
particular wavelength channel/filter. Alternatively, one could also use the
code to generate synthetic spectral lines following a similar procedure. Here,
we employ the code to generate synthetic images for all the six coronal
channels (94 Å, 131 Å, 171 Å, 193 Å, 211 Å, and 335 Å) of the SDO/AIA (Lemen
et al., 2012; Pesnell et al., 2012) assuming the line of sight is along the
$y$-axis (i.e., perpendicular to the loop length).

Figure 2: Sample synthetic images of the loop in AIA 171 Å channel generated
assuming the line-of-sight direction to be perpendicular to the loop axis
(i.e., along $y$-axis). The simulation resolution is still preserved in these
images. The left, center, and right panels display the first, 45th and 90th
frames corresponding to 0 s, 450 s, and 900 s, from the beginning of the
simulation. The center and right panels show the compressive signature of
propagating slow waves within individual strands. A movie associated with this
figure showing the propagation of slow waves across the entire simulation
duration (1200 s) is available online. Figure 3: Sample synthetic images of
the loop in AIA 193 Å channel generated assuming the line-of-sight direction
to be perpendicular to the loop axis (i.e., along $y$-axis). The simulation
resolution is still preserved in these images. The left, center, and right
panels display the first, 45th and 90th frames corresponding to 0 s, 450 s,
and 900 s, from the beginning of the simulation. The center and right panels
show the compressive signature of propagating slow waves within individual
strands. A movie associated with this figure showing the propagation of slow
waves across the entire simulation duration (1200 s) is available online.

Sample images thus produced in the AIA 171 Å and 193 Å channels are shown in
Figs. 2 & 3, respectively. The associated movies (available online) reveal how
the simulated slow waves are manifested in these two wavelength channels.
Notably, the compressive signature of slow waves is also visible in the centre
and right panels of the images. Importantly, the cross-field structure of the
loop appears different in these two channels. This is because these two
channels are meant to observe different temperature plasma. Note that the
resolution of the simulation is still preserved in these images. More
particularly, the spatial scale in the transverse direction ($\approx$9.76 km
per pixel) is much lower than that is actually achievable in SDO/AIA
observations ($\approx$430 km). Therefore, the intensities were rebinned to
match the pixel scale to that of AIA. Along the $z$-axis, this would require a
bit of interpolation, as the pixel scale in this direction is slightly larger
($\approx$780 km) than that of AIA. Additionally, we also add a random noise
component to these images that is generally expected in observations. The
amplitude of the noise within each AIA channel is estimated by incorporating
various components including the photon noise, readout noise, digit noise,
compression noise, dark noise, subtraction noise, and the despiking noise
(Yuan & Nakariakov, 2012; Boerner et al., 2012). The resultant images from AIA
171 Å and 193 Å channels are shown in Figs. 4 & 5 respectively.

Figure 4: Sample synthetic images of the loop in AIA 171 Å channel after
degrading the pixel scale to match with that of the AIA and adding
corresponding data noise. The images in individual panels correspond to the
same time frame as that shown in Fig. 2. Note that these images are quite a
bit stretched in the $x$-direction to highlight the key differences. Figure 5:
Sample synthetic images of the loop in AIA 193 Å channel after degrading the
pixel scale to match with that of the AIA and adding corresponding data noise.
The images in individual panels correspond to the same time frame as that
shown in Fig. 3. Note that these images are quite a bit stretched in the
$x$-direction to highlight the key differences.

Note that, after scaling them to AIA resolution, there are hardly 6 pixels in
the transverse direction, so the images look quite stretched along the
$x$-axis. Also, to avoid a jagged appearance due to the availability of very
few pixels, the images in Fig. 3 are smoothed using a bilinear interpolation.
While this did not have any impact on the overall structure, it brings out a
noisy appearance in the horizontal direction. Nevertheless, what appeared as
multiple strands in Figs. 2 & 3 seems to be flattened out in these images.
Importantly, in the AIA 171 Å channel (see Fig. 4), the bundle of strands
emerge as a single isolated structure which is what we regularly see in
observations. The AIA 193 Å channel on the other hand, displays a bifurcated
structure (see Fig. 5) with two strands which are seemingly located adjacent
to the structure visible in the 171 Å channel. This is again because the two
channels are capturing different temperature plasma. It may be noted that in
the solar coronal observations, loops captured in different temperature
channels are occasionally found next to each other rather than being exactly
cospatial (Nagata et al., 2003; Schmieder et al., 2004; Habbal et al., 2011;
Nisticò et al., 2014). Those loops are generally interpreted as multi-thermal
loops. The natural reproduction of this behaviour in our simulations implies
that one could indeed use this trait to infer the underlying multi-thermal
nature of coronal loops.

Subsequently, we collapse the loop in the transverse direction and stack the
intensity profiles from each time step next to each other to build a time-
distance map. Such maps for all the AIA channels are displayed in Fig. 6.

Figure 6: Time-distance maps constructed from the synthetic data for all the
six coronal channels of AIA. The slanted ridges of alternating brightness
visible in some of the channels highlight the propagation of slow waves along
the loop.

As can be seen, slanted ridges of alternating brightness, a signature of the
propagating slow magneto-acoustic waves, are visible in the AIA 171 Å, 193 Å,
and 211 Å maps. The temporal spacing between the ridges is about 3 minutes
which is the same as the period of the driver. Additionally, in these maps,
the backward slanted ridges may be seen near the top of the domain (forming a
crisscross pattern) indicating some unwanted reflection from the top boundary.
But, we have data covering enough spatial extent to study the propagation
properties so the reflection part can be ignored for the present study. More
importantly, the ridges are very faint in the AIA 131 Å channel and not at all
visible in the 94 Å and 335 Å channels. This behaviour is also seen in
observations and the specific channels the ridges are visible will apparently
vary from loop to loop. For instance, the active region loop studied by Jess
et al. (2012) shows propagating waves in all channels except the 94 Å while
that by Sharma et al. (2020) does not show any oscillations in the 335 Å and
94 Å channels. A few other studies report oscillations in fewer channels often
because of the lower amplitudes at remaining wavelengths (Kiddie et al., 2012;
Uritsky et al., 2013; Krishna Prasad et al., 2017). In the current case, the
absence of oscillations in some of the channels is because the loop does not
possess plasma at their respective temperatures.

The characteristic peak response temperatures for the AIA 94 Å, 131 Å, 171 Å,
193 Å, 211 Å, and 335 Å channels are 7.08 MK, 0.56 MK, 0.79 MK, 1.41 MK, 1.78
MK, and 0.22 MK respectively. The 94 Å, 131 Å, and 335 Å channels have
additional peaks at 1.58 MK, 11.2 MK, and 2.5 MK respectively. These values
are obtained from peak locations in the AIA temperature response curves
(version 10; Boerner et al., 2014). Since our simulated loop has plasma
temperature varying between 0.41 MK to 1.73 MK, the oscillations are expected
to be visible in the 131 Å, 171 Å, and 193 Å channels. However, the
oscillations are nearly invisible in the 131 Å channel and on the other hand
we see oscillations in the 211 Å channel albeit at a lower amplitude. This is
perhaps due to the differences in the number of strands and densities at the
extreme temperatures and could also be partly related to the sensitivities of
the respective detectors. Nevertheless, this behaviour highlights the
importance of forward modelling taking all the instrumental effects into
account and clearly indicates that one cannot simply conjecture the cross-
field thermal information based on the peak response temperatures. The lower
amplitudes for slow waves, observed in the hot 211 Å channel, are generally
attributed to high thermal conduction in hotter plasma. However, in the
current setup, there are no dissipative mechanisms and there is no plasma
corresponding to the peak response temperatures of 211 Å. Therefore, one
should exercise caution in directly attributing the low (or high) oscillation
amplitudes to the strength of dissipative processes.

Lastly, we compute the propagation speeds of slow waves observed in the 171 Å,
193 Å, and 211 Å channels. To achieve this, the light curve at each spatial
location is cross-correlated with that at the adjacent position and the
corresponding time lags are measured as a function of distance along the loop.
For this purpose, distances between 4 – 48 Mm are only considered to avoid
boundary effects at the bottom (see 171 Å image) and reflections at the top.

Figure 7: Time lags between the oscillations measured as a function of
distance along the loop. Different symbols/colors denote data for different
wavelength channels. The overplotted solid lines represent linear fits to the
corresponding data. The propagation speeds estimated from the measured slopes
are listed in the plot.

The resulting time lags, as a function of distance, are shown for all three
channels in Fig. 7. The inverse of the slope derived from a linear fit to the
time lag values will give us the propagation speed. As listed in the figure,
the speed values obtained are 138.2$\pm$0.5 km s-1, 181.1$\pm$0.9 km s-1, and
182.6$\pm$1.0 km s-1, respectively, for the 171 Å, 193 Å, and 211 Å channels.
The expected propagation speeds according to their characteristic peak
response temperatures are 135.1 km s-1, 180.5 km s-1, and 202.8 km s-1.
Evidently, the obtained propagation speeds match pretty well with the expected
speeds for the 171 Å and 193 Å channels but that in the 211 Å channel,
surprisingly, does not match. Instead, the propagation speed of the 211 Å
channel is nearly the same as that found in the 193 Å, possibly suggesting
that the emission seen at this wavelength is close to the 193 Å peak
temperature. Note that the line of sight angle considered here is
perpendicular to the loop axis, so there is no projection effect unlike that
found in observations.

##  4 Summary and Conclusions

Employing a 3D MHD model, we study the propagation of slow magneto-acoustic
waves in a multi-stranded, multi-thermal loop for the first time. The loop is
modelled as a vertical cylinder composed of a bundle of 33 strands with
randomly varying temperatures and densities. The gravitational stratification
and other non-ideal effects are ignored. Slow waves are generated in the
system by perturbing the vertical velocity at the bottom of the loop. The
simulation results are then forward-modelled to produce synthetic images in
all 6 coronal channels of SDO/AIA. Additionally, we estimate and add the
corresponding data noise as would have been present during real observations.
It is observed that the emission in the 193 Å channel is adjacent to that
found in the 171 Å channel rather than being co-spatial. Although this
behaviour is dependent on the distribution of plasma and the line of sight
direction, we do note that the observed loops are known to exhibit similar
traits. Time-distance maps are constructed from the synthetic data to study
the propagation properties of slow waves. It is observed that the oscillations
are only visible in certain channels and not visible in others and their
appearance is related to the available plasma temperatures within the loop.
The propagation speed of slow waves is also found to be sensitive to this
information. Indeed, Van Doorsselaere et al. (2024) have shown that the
bandpass and the peak response temperature of an imaging filter can influence
the wave propagation speeds that may be observed in a multi-thermal loop, in
good agreement with the results found here.

Observations in the past (for e.g., Jess et al., 2012; Kiddie et al., 2012;
Uritsky et al., 2013; Krishna Prasad et al., 2017; Sharma et al., 2020) have
often shown that slow magneto-acoustic waves are visible only in certain
wavelength channels. Additionally, their amplitudes and propagation speeds are
found to vary from one channel to the other. While this differential
propagation behaviour has been previously suggested to be due to a multi-
thermal nature of the coronal loops (King et al., 2003; Krishna Prasad et al.,
2017), our results demonstrate that we can take this to the next step and
possibly infer the range of plasma temperatures within a coronal loop. This
can be achieved, for example, by adjusting our loop model with an appropriate
range of plasma temperatures that may reproduce the multi-wavelength behaviour
of slow magneto-acoustic waves observed in a particular loop. The cross-field
thermal information provides an important constraint for the coronal heating
models so we believe this could be a significant step forward. The
Differential Emission Measure (DEM) technique as well, in principle, provides
this information but it suffers from line-of-sight integration effects.
Importantly, it is highly non-trivial to discern whether the cross-field
thermal characteristics obtained solely belong to the loop plasma or some
background/foreground structures. On the other hand, because we use the
properties of waves propagating within the loop here, one can stay assured
that the thermal information obtained purely belongs to the loop.

In summary, via a multi-stranded loop model, we demonstrate that the
seismological capabilities of slow magneto-acoustic waves can be further
extended to obtain additional information on the cross-field thermal structure
of coronal loops. We anticipate future studies to build on this and develop
different plasma distribution models for comparison with the observation
characteristics of slow waves. Moreover, because the slow waves are regularly
observed in a variety of structures including active region fan loops, polar
plumes, on-disk plumes, hot flare loops, etc., this technique is widely
applicable in the solar corona and thus can be very useful once properly
established.

SKP is grateful to FWO Vlaanderen for a senior postdoctoral fellowship (No.
12ZF420N). The resources and services used in this work were provided by the
VSC (Flemish Supercomputer Center), funded by the Research Foundation -
Flanders (FWO) and the Flemish Government. SKP would like to thank Norbert
Magyar for valuable discussions on AMRVAC simulations. TVD was supported by
the European Research Council (ERC) under the European Union’s Horizon 2020
research and innovation programme (grant agreement No 724326), the C1 grant
TRACEspace of Internal Funds KU Leuven, and a Senior Research Project
(G088021N) of the FWO Vlaanderen. TVD would like to thank Dipankar Banerjee,
Vaibhav Pant and Krishna Prasad for their hospitality when visiting ARIES,
Nainital in spring 2023, where this work was discussed. TVD received financial
support from the Flemish Government under the long-term structural Methusalem
funding program, project SOUL: Stellar evolution in full glory, grant
METH/24/012 at KU Leuven, and also the DynaSun project (number 101131534 of
HORIZON-MSCA-2022-SE-01).

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