Prospects and challenges of numerical modelling of the Sun at millimetre wavelengths

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Prospects and challenges of numerical

modelling of the Sun at millimetre

wavelengths

Sven Wedemeyer 1∗, Gregory Fleishman 2, Jaime de la Cruz Rodr´ıguez3,

Stanislav Gun´

ar4, Jo˜

ao M. da Silva Santos5, Patrick Antolin6, Juan Camilo

Guevara G ´

omez 1, Mikolaj Szydlarski 1, Henrik Eklund 1,3

1Rosseland Centre for Solar Physics, Institute of Theoretical Astrophysics,

University of Oslo, Postboks 1029 Blindern, N-0315 Oslo, Norway

Center For Solar-Terrestrial Research, New Jersey Institute of Technology, Newark,

NJ 07102, USA

3Institute for Solar Physics, Dept. of Astronomy, Stockholm University, AlbaNova

University Centre, SE-10691 Stockholm, Sweden

4Astronomical Institute, The Czech Academy of Sciences, 25165 Ondˇ

rejov, Czech

Republic

5National Solar Observatory, 3665 Discovery Drive, Boulder, CO, USA

6Department of Mathematics, Physics and Electrical Engineering, Northumbria

University, Newcastle upon Tyne, NE1 8ST, UK

Correspondence*:

Sven Wedemeyer

sven.wedemeyer@astro.uio.no

ABSTRACT

The Atacama Large Millimeter/submillimeter Array (ALMA) offers new diagnostic possibilities

that complement other commonly used diagnostics for the study of our Sun. In particular, ALMA’s

ability to serve as an essentially linear thermometer of the chromospheric gas at unprecedented

spatial resolution at millimetre wavelengths and future polarisation measurements have great

diagnostic potential. Solar ALMA observations are therefore expected to contribute signiﬁcantly to

answering long-standing questions about the structure, dynamics and energy balance of the outer

layers of the solar atmosphere. In this regard, current and future ALMA data are also important

for constraining and further developing numerical models of the solar atmosphere, which in

turn are often vital for the interpretation of observations. The latter is particularly important

given the Sun’s highly intermittent and dynamic nature that involves a plethora of processes

occurring over extended ranges in spatial and temporal scales. Realistic forward modelling of the

Sun therefore requires time-dependent three-dimensional radiation magnetohydrodynamics that

account for non-equilibrium effects and, typically as a separate step, detailed radiative transfer

calculations, resulting in synthetic observables that can be compared to observations. Such

artiﬁcial observations sometimes also account for instrumental and seeing effects, which, in

addition to aiding the interpretation of observations, provide instructive tools for designing and

optimising ALMA’s solar observing modes. In the other direction, ALMA data in combination with

other simultaneous observations enables the reconstruction of the solar atmospheric structure

arXiv:2210.13894v1 [astro-ph.SR] 25 Oct 2022

Wedemeyer et al. Numerical modelling of the Sun at millimetre wavelengths

via data inversion techniques. This article highlights central aspects of the impact of ALMA for

numerical modelling for the Sun, their potential and challenges, together with selected examples.

Keywords: Sun: radio radiation, Sun: atmosphere, Sun: magnetic ﬁelds, Radiative transfer

1 INTRODUCTION

When pointed at the Sun, the Atacama Large Millimeter/submillimeter Array (ALMA, Wootten and

Thompson, 2009) mostly observes radiation that originates from the solar chromosphere. This atmospheric

layer, which is situated between the photosphere below and the transition region and corona above, is

highly dynamic and intermittent and shows variations on a large range of spatial and temporal scales.

Plasma with chromospheric conditions can also be found in the corona in the form of prominences and

coronal rain. These structures are integral components of the solar corona in the sense that they do not only

reﬂect speciﬁc physical processes of the corona but also inﬂuence its evolution (Vial and Engvold, 2015;

Antolin and Froment, 2022). The investigation of the thermodynamic conditions and morphology of these

dense and cool structures supported by the magnetic ﬁeld is therefore also a major ﬁeld in which ALMA

can make a major contribution.

Despite very active research regarding the chromosphere, which involves observations at many different

wavelength ranges supported by numerical simulations, yet many fundamental questions concerning this

layer remain open. The main reason is that the chromosphere is notoriously difﬁcult to observe. Only a

small number of spectral lines and continua are formed in the chromosphere, usually across extended

height ranges. The formation of chromospheric spectral lines involves non-equilibrium effects such as,

e.g., non-local thermodynamic equilibrium (NLTE

, see, e.g., Uns

old, 1955; Mihalas, 1978; Carlsson and

Stein, 1992, and references therein) and time-dependent hydrogen ionisation (Carlsson and Stein, 2002).

Consequently, the few currently available diagnostics like the spectral lines of singly ionised calcium

and magnesium are difﬁcult to interpret, in particular in combination with instrumental limitations. As a

result, the physical properties of the observed atmospheric region can only be derived with rather large

uncertainties, hampering the progress in understanding this important part of the solar atmosphere.

Observations of the solar continuum radiation with ALMA, as offered on a regular basis since 2016,

provide unprecedented diagnostic possibilities that are complementary to other chromospheric diagnostics

(Bastian, 2002; Karlick

y et al., 2011; Benz et al., 2012; Wedemeyer et al., 2016; Bastian et al., 2018). The

radiation continuum at sub-millimetre/millimetre ((sub-)mm) wavelengths, including the range accessed by

ALMA, forms essentially under conditions of local thermodynamic equilibrium (LTE) so that the observed

brightness temperature,

, is closely related to the actual (electron) temperature of the chromospheric gas

in a (corrugated) layer whose average height roughly increases with the selected observing wavelength.

Unfortunately, observations at (sub-)mm wavelengths prior to ALMA had too low spatial and temporal

resolution for resolving the small spatial and short temporal scales on which the intricate chromospheric

dynamics occur. For instance, the Berkeley-Illinois-Maryland Array (BIMA) had a spatial resolution

corresponding to a restored beam size of

∼10

” at a wavelength of

λ= 3.5

mm (White et al., 2006;

Loukitcheva et al., 2009). The full diagnostic potential of millimetre wavelengths has therefore only

been unlocked by ALMA thanks to its high temporal and (comparatively high) spatial resolution, which

signiﬁcantly exceeds the resolution achieved by previous millimetre and radio observatories. It should

In this context, NLTE or non-LTE describes deviations from LTE conditions for the atomic level populations (which can then be calculated under the

assumption of statistical equilibrium), the electron density (mostly due to non-equilibrium hydrogen ionisation), and the the radiative source function that is no

longer given by the Boltzmann function.

This is a provisional ﬁle, not the ﬁnal typeset article 2

Wedemeyer et al. Numerical modelling of the Sun at millimetre wavelengths

be noted that the millimetre wavelengths addressed in this article are also referred to in terms of their

corresponding frequencies (a few 10s GHz up to

∼

1 THz) and as (ALMA) receiver bands

that covered the

discussed wavelength range. In particular, ALMA Bands 3 and 6, which have been used most frequently

for solar observations so far, refer to (central) wavelengths of 3.0 mm and 1.3 mm and corresponding

frequencies of 100 GHz and 230-240 GHz, respectively.

Because ALMA is relatively new as a diagnostic tool for the solar chromosphere, still many aspects are not

understood well yet. For instance, the exact formation heights and thus the layers sampled by the different

receiver bands of ALMA and likewise the oscillatory behaviour seen in the ALMA observations are still

debated (see, e.g., Jafarzadeh et al., 2021; Patsourakos et al., 2020; Narang et al., 2022; Nindos et al., 2021).

On the other hand, as solar observing with ALMA is still in its infancy, its capabilities will continue to

improve in the near future. However, any new ability to obtain unprecedented observations in any part of the

spectrum always brings its own challenges in the interpretation of the resulting data. Fortunately, diagnostics

and understanding of the ALMA observations and their relationship to the coordinated observations in

other spectral domains can beneﬁt from dedicated numerical modelling.

Like in many other ﬁelds of astrophysics, numerical simulations have developed into an essential tool

in solar physics. Also in the context of solar observations with ALMA, simulations help to interpret

observational data but can also be used to develop and optimise new observing strategies (see, e.g.,

Wedemeyer-B

ohm et al., 2007; Loukitcheva et al., 2015; Fleishman et al., 2021a). In return, comparison

with observations provide crucial tests for the veracity of existing models of the solar atmosphere. In

this brief overview article, the potential value of numerical simulations for solar science with ALMA is

explored, ranging from forward modelling of thermal and non-thermal mm continuum radiation and the

impact of magnetic ﬁelds (Sect. 2) to data inversion techniques and modelling of instrumental and seeing

effects (Sect. 3). Examples of scientiﬁc applications are presented in Sect. 4, followed by a summary and

outlook in Sect. 5.

2 FORWARD MODELLING AND ARTIFICIAL OBSERVATIONS

Forward modelling of the solar atmosphere is typically split into the following steps: (i) Radiation

(magneto)hydrodynamics simulations (Sect. 2.1), (ii) synthesis of observables via radiative transfer

calculations (Sect. 2.2), and (iii, optionally) application of simulated observational effects (e.g., limited

angular resolution, see Sect. 2.3).

2.1 Radiation magnetohydrodynamic simulations

Semi-empirical models of the solar atmosphere like those by Vernazza et al. (1981), Fontenla et al.

(1993), Avrett and Loeser (2008) and others have been an important milestone and still are widely used

for reference. Millimetre continuum observations were also used for the construction of these models,

which therefore give a ﬁrst idea of where and under which conditions the radiation continuum at different

millimetre wavelenghts is formed – on average. While the employed radiative transfer modelling, which

even accounts for NLTE, is elaborate, this class of models can by nature not account for the pronounced

temporal and spatial variations seen at the much increased resolution of modern observations.

The next step in the development towards realistic models was therefore to account for temporal variations

in the chromosphere. The time-dependent one-dimensional simulations by Carlsson and Stein (1995) and

variations therefore capture well the dynamics introduced by shock waves that propagate through the solar

2https://www.eso.org/public/teles-instr/alma/receiver-bands/

Frontiers 3

Wedemeyer et al. Numerical modelling of the Sun at millimetre wavelengths

Figure 1.

Forward modelling of synthetic brightness temperatures at millimetre wavelengths starting from

a 3D model snapshot (or a time series of such) from a Bifrost simulation (top left). The model features

an enhanced network region in the centre surrounded by Quiet Sun. The radiative transfer calculations

with ART produce a brightness temperature map for the selected frequencies (shown here: 239 GHz, top

right) to which then a simple point spread function (PSF) equivalent to an idealised synthesised beam for

ALMA can be applied (lower right). The PSF is shown in the top left corner of the panel. A more realistic

simulation of atmospheric seeing and instrumental effects as done with SASim (and then reconstructed

with SoAP) results in a stronger degradation (lower left) of the original image than compared to the simple

PSF. Please note that only a moderate example is shown that corresponds to good observing conditions.

The synthesised beam is shown in the top left corner of the reconstructed image.

atmosphere and the resulting implications for the chromospheric plasma properties, i.e. the ionisation

degree of hydrogen and thus the (non-equilibrium) electron density (Carlsson and Stein, 2002). These

simulations have been used for the synthesis of the millimetre continuum and thus provided ﬁrst predictions

This is a provisional ﬁle, not the ﬁnal typeset article 4

Wedemeyer et al. Numerical modelling of the Sun at millimetre wavelengths

of the brightness temperature variations that a telescope with sufﬁcient resolution would be able to observe

(Loukitcheva et al., 2008).

However, the solar atmosphere and in particular the highly dynamic and intermittent chromosphere is

a truly time-dependent three-dimensional phenomenon, which poses signiﬁcant challenges for realistic

modelling capable of reproducing observational ﬁndings. Also, temporal variations on short timescales

are typically connected to spatial variations across short length scales. Consequently, accounting for the

full time-dependence and multi-dimensionality of the solar chromosphere is a substantial step forward

from one-dimensional approaches. In view of limited computational resources, early 3D simulations were

restricted in the overall number of grid cells, seeking a compromise between required resolution and

extent of the computational domain, and the physical processes that could be numerically treated (see, e.g.,

Skartlien et al., 2000; Wedemeyer et al., 2004). The enormous increase in computational power over the last

decades now enable simulations with much higher numbers of grid cells and thus a better representation of

the chromospheric small-scale structure and larger extents of the modelled region. However, self-consistent

numerical simulations of whole Active Regions are still at the modelling frontier (Rempel et al., 2009).

Numerical two-dimensional (2D) and and three-dimensional (3D) models produced with the radiation

magnetohydrodynamics (rMHD) simulation codes Bifrost (Gudiksen et al., 2011) and

CO5BOLD

(Freytag

et al., 2012) have already been used as basis for the synthesis of mm continuum radiation (see, e.g.,

Wedemeyer-B

ohm et al., 2007; Loukitcheva et al., 2015) but an increasing number of codes is developing

the necessary functionality, e.g., MURaM (Przybylski et al., 2022). Both Bifrost and

CO5BOLD

solve

the equations of magnetohydrodynamics and radiative energy transfer together with a realistic equation

of state and realistic opacities and further relevant physics. A typical model includes a small part of

the solar atmosphere (from a few Mm to a few 10 Mm, cf. Wedemeyer et al., 2016, and references

therein) and extends from the upper convection zone into the chromosphere and/or low corona (see

Fig. 1, upper left). This way the dynamics in the model are driven self-consistently and all layers mapped

by ALMA are included. A simulation typically starts with an evolved model snapshot (or any other

initial condition) and is evolved in time step by step, where the computational time steps are of the

order of 1 ms to 100 ms, depending on the magnetic ﬁeld strength in the model. Simulation snapshots

of the physical parameters can be output at freely selectable cadence. Modelling the layers of the solar

atmosphere above the temperature minimum in a realistic way requires the inclusion of additional physical

processes and deviations from equilibrium conditions that are usually computationally expensive. As

discussed in Sect. 2.2.1, the detailed treatment of time-dependent non-equilibrium hydrogen ionisation,

like it is implemented in Bifrost, is of particular importance for the continuum radiation at millimetre

wavelengths. Adding also non-equilibrium ionisation of helium and ion-neutral interactions (ambipolar

diffusion) signiﬁcantly increases the computational costs. Consequently, only a small number of models so

far can account for these additional ingredients and are necessarily limited to 2.5D in order to render such

modelling computationally feasible (Mart

ınez-Sykora et al., 2020). These models suggest that the effective

formation heights of the millimetre continuum in both ALMA Band 3 and 6 is similar in active regions

(ARs) and network regions, which contradicts results from previous simulations (see Wedemeyer et al.,

2016, and references therein) and actual ALMA observations (e.g., Hofmann et al., 2022). Clearly, the

inclusion of more physical processes relevant under chromospheric conditions as implemented in the 2.5D

simulations by Mart

ınez-Sykora et al. (2020) is an essential step in the right direction. However, given

the complicated small-scale dynamics of the chromosphere, modelling this layer in full 3D at sufﬁcient

resolution is a critical requirement that comes with high computational costs.

Frontiers 5

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ProspectsandchallengesofnumericalmodellingoftheSunatmillimetrewavelengthsSvenWedemeyer1,GregoryFleishman2,JaimedelaCruzRodr´guez3,StanislavGun´ar4,JoaoM.daSilvaSantos5,PatrickAntolin6,JuanCamiloGuevaraG´omez1,MikolajSzydlarski1,HenrikEklund1;31RosselandCentreforSolarPhysics,InstituteofTheoretical...

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