The Physics of Chromospheric Plasmas
ASP Conference Series, Vol. 368, 2007
Petr Heinzel, Ivan Dorotovič and Robert J. Rutten, eds.
Modeling the Solar Chromosphere
Mats Carlsson
Institute of Theoretical Astrophysics, University of Oslo, Norway
Abstract. Spectral diagnostic features formed in the solar chromosphere are
few and difficult to interpret — they are neither formed in the optically thin
regime nor in local thermodynamic equilibrium (LTE). To probe the state of the
chromosphere, both from observations and theory, it is therefore necessary with
modeling. I discuss both traditional semi-empirical modeling, numerical exper-
iments illustrating important ingredients necessary for a self-consistent theoret-
ical modeling of the solar chromosphere and the first results of such models.
1. Introduction
My keynote talk was similar in content to a recent talk at a Sacramento Peak
workshop celebrating the 70th birthday of Robert F. Stein. This written version
builds to a large extent on that writeup (Carlsson 2006), but it is updated and
some sections have been expanded.
Before discussing models of the solar chromosphere it is worthwhile dis-
cussing the very definition of the term “chromosphere”. The name comes from
the Greek words “χρωµα” (color) and “σϕαιρα” (ball) alluding to the colored
thin rim seen above the lunar limb at a solar eclipse. The color comes mainly
from emission in the Balmer Hα line. This is thus one possible definition — the
chromosphere is where this radiation originates. At an eclipse this region has
a sharp lower edge, the visible limb, but a fuzzy upper end with prominences
protruding into the corona. The nature of this region is difficult to deduce from
eclipse observations since we see this region edge on during a very short time
span and we have no way of telling whether it is homogeneous along the line of
sight or very inhomogeneous in space and time. It was early clear that the emis-
sion in Hα must mean an atmosphere out of radiative equilibrium — without
extra heating the temperature will not be high enough to have enough hydrogen
atoms excited to the lower or upper levels of the transition. Early models were
constructed to explain observations in Hα and in resonance lines from other
abundant elements with opacity high enough to place the formation in these
regions even in center-of-disk observations (lines like the H and K resonance
lines from singly ionized calcium). These early models were constructed assum-
ing one dimensional plane-parallel geometry and they resulted in a temperature
falling to a minimum around 4000K about 500 km above the visible surface, a
temperature rise to 8000K at a height of about 2000 km and then a very rapid
temperature rise to a million degree corona. These plane-parallel models have
also at Center of Mathematics for Applications, University of Oslo, Norway
http://arxiv.org/abs/0704.1509v1
50 Carlsson
led to a common notion that there is a more or less homogeneous, plane-parallel
region between these heights that is hotter than the temperature minimum. In
such a picture the chromosphere may be defined as a region occupying a given
height range (e.g. between 500 and 2000 km height over the visible surface) or a
given temperature range. We may also use physical processes for our definition:
the chromosphere is the region above the photosphere where radiative equilib-
rium breaks down and hydrogen is predominantly neutral (the latter condition
giving the transition to the corona). This discussion shows that there is no
unique definition of the term “chromosphere”, not even in a one-dimensional,
static world. It is even more difficult to agree on a definition of the “chromo-
sphere” that also encompasses an inhomogeneous, dynamic atmosphere.
As mentioned above, the first models of the chromosphere were constructed
with a large number of free parameters to match a set of observational con-
straints. Since some equations are used to restrict the number of free parameters
(not all hydrodynamical variables at all points in space and time are determined
empirically) we call this class of models semi-empirical models. Typically one
assumes hydrostatic equilibrium and charge conservation but no energy equa-
tion. The temperature as function of height is treated as a free function to
be determined from observations. In the other main class of models one tries
to minimize the number of free parameters by including an energy equation.
Such theoretical models have been very successful in explaining radiation from
stellar photospheres with only the effective temperature, acceleration of gravity
and abundances as free parameters. In the chromosphere, an additional term is
needed in the energy equation — e.g. energy deposition by acoustic shocks or
energy input in connection with magnetic fields (e.g. currents or reconnection).
It is thus clear from observations that the chromosphere is not in radiative
equilibrium — there is a net radiative loss. This loss has to be balanced by an
energy deposition, at least averaged over a long enough time span, if the atmo-
sphere is to be in equilibrium. This is often called the problem of chromospheric
“heating”. It is important to bear in mind, though, that the radiative losses may
be balanced by a non-radiative energy input without an increase in the average
temperature. The term “chromospheric heating” may thus be misleading since
it may be interpreted as implying that the average temperature is higher than
what is the case in a radiative equilibrium atmosphere. In the following we will
use the term “heating” in a more general sense: a source term in the energy
equation, not necessarily leading to an increased temperature.
Chromospheric heating is needed not only for the quiet or average Sun but
also in active regions, sunspots and in the outer atmospheres of many other
stars. I will in the following mainly discuss the quiet Sun case.
The outline of this paper is as follows: In Section 2 we discuss semi-empirical
models of the chromosphere. In Section 3 we discuss theoretical models; first
we elaborate on 1D hydrodynamical models, then we discuss the role of high
frequency acoustic waves for the heating of the chromosphere and finally we
describe recent attempts to model the chromosphere in 3D including the effects
of magnetic fields.
Modeling the Solar Chromosphere 51
2. Semi-empirical Models
Semi-empirical models can be characterized by the set of observations used to
constrain the model, the set of physical approximations employed and the set
of free parameters to be determined. Spectral diagnostics used to constrain
chromospheric models must have high enough opacity to place the formation
above the photosphere. The continuum in the optical part of the spectrum is
formed in the photosphere so the only hope for chromospheric diagnostics lies
in strong spectral lines in this region of the spectrum. Candidates are resonance
lines of dominant ionization states of abundant elements and lines from excited
levels of the most abundant elements (hydrogen and helium). Most resonance
lines are in the UV but the resonance lines of singly ionized calcium (Ca II), called
the H and K lines, fulfill our criteria. These lines originate from the ground state
of Ca II, the dominant ionization stage under solar chromospheric conditions,
and the opacity is therefore given by the density directly and the optical depth is
directly proportional to the column mass (i.e. to the total pressure in hydrostatic
equilibrium). Also the source function has some coupling to local conditions
even at quite low densities (in contrast to the strongly scattering resonance
lines of neutral sodium). Other chromospheric diagnostic lines in the optical
region are the hydrogen Balmer lines and the helium 1083 nm line. They all
originate from highly excited levels and thus have very temperature sensitive
opacity. The population of He 1083 is also set by recombination such that its
diagnostic potential is very difficult to exploit. With the advent of space based
observatories, the full UV spectral range was opened up. Continua shortward of
the opacity edge from the ground state of neutral silicon at 152 nm are formed
above the photosphere and can be used to constrain chromospheric models.
Together with observations in Ly-α, such UV continuum observations were used
by Vernazza et al. (1973, 1976, 1981) in their seminal series of papers on the solar
chromosphere. The VAL3 paper (Vernazza et al. 1981) is one of the most cited
papers in solar physics (1072 citations in ADS at the time of writing) and the
abstract gives a very concise description of the models and the principles behind
their construction: “The described investigation is concerned with the solution of
the non-LTE optically thick transfer equations for hydrogen, carbon, and other
constituents to determine semi-empirical models for six components of the quiet
solar chromosphere. For a given temperature-height distribution, the solution is
obtained of the equations of statistical equilibrium, radiative transfer for lines
and continua, and hydrostatic equilibrium to find the ionization and excitation
conditions for each atomic constituent. The emergent spectrum is calculated,
and a trial and error approach is used to adjust the temperature distribution
so that the emergent spectrum is in best agreement with the observed one.
The relationship between semi-empirical models determined in this way and
theoretical models based on radiative equilibrium is discussed by Avrett (1977).
Harvard Skylab EUV observations are used to determine models for a number
of quiet-sun regions.”
The VAL3 models are thus characterized by them using Ly-α and UV-
continuum observations for observational constraint, hydrostatic equilibrium
and non-LTE statistical equilibrium in 1D as physical description and temper-
ature as function of height as free function. To get a match with observed
line-strengths, a depth-dependent microturbulence was also determined and a
52 Carlsson
corresponding turbulent pressure was added. The number of free parameters
to be determined by observations is thus large — in principle the number of
depth-points per depth-dependent free function (temperature and microturbu-
lence). In practice the fitting was made by trial and error and only rather
smooth functions of depth were tried thus decreasing the degrees of freedom in
the optimization procedure.
The models have a minimum temperature around 500 km above the visible
surface (optical depth unity at 500 nm), a rapid temperature rise outwards to
about 6000K at 1000 km height and thereafter a gradual temperature increase
to 7000K at 2000 km height with a very rapid increase from there to coronal
temperatures.
The Ca II lines were not used in constraining the VAL3 models and the
agreement between the model representing the average quiet Sun, VAL3C, and
observations of these lines was not good. An updated model with a different
structure in the temperature minimum region was published in Maltby et al.
(1986) (where the main emphasis was on similarly constructed semi-empirical
models for sunspot atmospheres).
A peculiar feature with the VAL models was a temperature plateau intro-
duced between 20000 and 30000K in order to reproduce the total flux in the
Lyman lines. This plateau was no longer necessary in the FAL models where the
semi-empirical description of the transition region temperature rise was replaced
by the balance between energy flowing down from the corona (conduction and
ambipolar diffusion) and radiative losses (Fontenla et al. 1990, 1991, 1993).
One goal of semi-empirical models is to obtain clues as to the non-radiative
heating process. From the models it is possible to calculate the amount of non-
radiative heating that is needed to sustain the model structure. For the VAL3C
model this number is 4.2 kWm−2 with the dominant radiative losses in lines
from Ca II and Mg II, with Ly-α taking over in the topmost part.
The models described so far do not take into account the effect of the
very many iron lines. This was done in modeling by Anderson & Athay (1989).
Instead of using the temperature as a free parameter and observations as the
constraints, they adjusted the non-radiative heating function until they obtained
the same temperature structure as in the VAL3C model (arguing that they would
then have an equally good fit to the observational constraints as the VAL3C
model). The difference in the physical approximations is that they included line
blanketing in non-LTE from millions of spectral lines. The radiation losses are
dominated by Fe II, with Ca II, Mg II, and H playing important, but secondary,
roles. The total non-radiative input needed to balance the radiative losses is
three times higher than in the VAL3C model, 14 kWm−2.
The VAL3 and FAL models show a good fit to the average (spatial and
temporal) UV spectrum but fail to reproduce the strong lines from CO. These
lines show very low intensities in the line center when observed close to the
solar limb, the radiation temperature is as low as 3700K (Noyes & Hall 1972;
Ayres & Testerman 1981; Ayres et al. 1986; Ayres & Wiedemann 1989; Ayres & Brault
1990). If the formation is in LTE this translates directly to a temperature
of 3700K in layers where the inner wings of the H and K lines indicate a
temperature of 4400K. The obvious solution to the problem is that the CO
lines are formed in non-LTE with scattering giving a source function below the
Modeling the Solar Chromosphere 53
Planck-function. Several studies have shown that this is not the solution — the
CO lines are formed in LTE (e.g. Ayres & Wiedemann 1989; Uitenbroek 2000).
The model M CO constructed to fit the CO-lines (Avrett 1995) give too low
UV intensities. One way out is to increase the number of free parameters by
abandoning the 1D, one-component, framework and construct a two component
semi-empirical atmosphere. The COOLC and FLUXT atmospheric models of
Ayres et al. (1986) was such an attempt where a filling factor of 7.5% of the
hot flux tube atmosphere FLUXT and 92.5% of the COOLC atmosphere repro-
duced both the H and K lines and the CO-lines. The UV continua, however,
are overestimated by a factor of 20 (Avrett 1995). A combination of 60% of a
slightly cooler model than M CO and 40% of a hot F model provides a better
fit (Avrett 1995). Another way of providing enough free parameters for a better
fit is to introduce an extra force in the hydrostatic equilibrium equation provid-
ing additional support making possible a more extended atmosphere. With this
extra free parameter it is possible to construct a 1D temperature structure with
a low temperature in the right place to reproduce the near-limb observations of
the CO lines and a sharp temperature increase to give enough intensity in the
UV continua (Fontenla 2007).
A word of caution is needed here. Semi-empirical models are often im-
pressive in how well they can reproduce observations. This is, however, not a
proper test of the realism of the models since the observations have been used
to constrain the free parameters. The large number of free parameters (e.g.,
temperature as function of height, microturbulence as function of height and
angle, non-gravitational forces) may hide fundamental shortcomings of the un-
derlying assumptions (e.g., ionization equilibrium, lateral homogeneity, static
solution). It is not obvious that the energy input required to sustain a model
that reproduces time-averaged intensities is the same as the mean energy input
needed in a model that reproduces the time-dependent intensities in a dynamic
atmosphere. Semi-empirical modeling may give clues as to what processes may
be important but we also need to study these underlying physical processes with
fewer free parameters. This is the focus of theoretical models.
3. Theoretical Models
In contrast to semi-empirical models theoretical models include an energy equa-
tion. To model the full 3D system with all physical ingredients we know are
important for chromospheric conditions is still computationally prohibitive —
various approximations have to be made. In one class of modeling one tries to
illustrate basic physical processes without the ambition of being realistic enough
to allow detailed comparison with observations. Instead the aim is to fashion
a basic physical foundation upon which to build our understanding. The other
approach is to start with as much realism as can be afforded. Once the mod-
els compare favourably with observations, the system is simplified in order to
enable an understanding of the most important processes. I here comment on
both types of approaches.
54 Carlsson
3.1. 1D radiation hydrodynamic simulations
Acoustic waves were suggested to be the agent of non-radiative energy input al-
ready by Biermann (1948) and Schwarzschild (1948). Such waves are inevitably
excited by the turbulent motions in the convection zone and propagate outwards,
transporting mechanical energy through the photospheric layers into the chromo-
sphere and corona. Due to the exponential decrease of density with height, the
amplitude of the waves increases and they steepen into shocks. The theory that
the dissipation of shocks heats the outer atmosphere was further investigated by
various authors, see reviews by Schrijver (1995); Narain & Ulmschneider (1996).
In a series of papers, Carlsson & Stein (1992, 1994, 1995, 1997, 2002a) have
explored the effect of acoustic waves on chromospheric structure and dynamics.
The emphasis of this modeling was on a very detailed description of the radia-
tive processes and on the direct comparison with observations. The full non-LTE
rate equations for the most important species in the energy balance (hydrogen,
helium and calcium) were included thus including the effects of non-equilibrium
ionization, excitation, and radiative energy exchange on fluid motions and the
effect of motion on the emitted radiation from these species. To make the cal-
culations computationally tractable, the simulations were performed in 1D and
magnetic fields were neglected. To enable a direct comparison with observations,
acoustic waves were sent in through the bottom boundary with amplitudes and
phases that matched observations of Doppler shifts in a photospheric iron line.
These numerical simulations of the response of the chromosphere to acous-
tic waves show that the Ca II profiles can be explained by acoustic waves close
to the acoustic cut-off period of the atmosphere. The simulations of the be-
haviour of the Ca IIH line reproduce the observed features to remarkable detail.
The simulations show that the three minute waves are already present at pho-
tospheric heights and the dominant photospheric disturbances of five minute
period only play a minor modulating role (Carlsson & Stein 1997). The waves
grow to large amplitude already at 0.5 Mm height and have a profound effect
on the atmosphere. The simulations show that in such a dynamic situation it is
misleading to construct a mean static model (Carlsson & Stein 1994, 1995). It
was even questioned whether the Sun has an average temperature rise at chro-
mospheric heights in non-magnetic regions (Carlsson & Stein 1995). The simu-
lations also confirmed the result of Kneer (1980) that ionization/recombination
timescales in hydrogen are longer than typical hydrodynamical timescales under
solar chromospheric conditions. The hydrogen ionization balance is therefore out
of equilibrium and depends on the previous history of the atmosphere. Since the
hydrogen ionization energy is an important part of the internal energy equation,
this non-equilibrium ionization balance also has a very important effect on the
energetics and temperature profile of the shocks (Carlsson & Stein 1992, 2002a).
Kneer (1980) formulated this result as strongly as “Unless confirmed by consis-
tent dynamical calculations, chromospheric models based on the assumption of
statistical steady state should be taken as rough estimates of chromospheric
structure.”
Are observations in other chromospheric diagnostics than the Ca II lines
consistent with the above mentioned radiation-hydrodynamic simulations of the
propagation of acoustic waves? The answer is “No”. The continuum observa-
tions around 130 nm are well matched by the simulations (Judge et al. 2003)
Modeling the Solar Chromosphere 55
but continua formed higher in the chromosphere have higher intensity in the
observations than in the simulations (Carlsson & Stein 2002c). The chromo-
spheric lines from neutral elements in the UV range are formed in the mid to
upper chromosphere. They are in emission at all times and at all positions in
the observations, and they show stronger emission than in the simulations.
The failure of the simulations to reproduce diagnostics formed in the mid-
dle to upper chromosphere gives us information on the energy balance of these
regions. The main candidates for an explanation are the absence of magnetic
fields in the simulations and the fact that the acoustic waves fed into the compu-
tational domain at the bottom boundary do not include waves with frequencies
above 20mHz.
The reason for the latter shortcoming is that the bottom boundary is deter-
mined by an observed wave-field and high frequency waves are not well deter-
mined observationally. I first explore the possibility that high frequency acoustic
waves may account for the increased input and address the issue of magnetic
fields in the next section.
3.2. High frequency waves
Observationally it is difficult to detect high frequency acoustic waves for two
reasons: First, the seeing blurs the ground based observations and makes these
waves hard to observe. Second, for both ground based and space based observa-
tions the signal we get from high frequency waves is weakened by the width of
the response function. Wunnenberg et al. (2002) have summarized the various
attempts at detecting high frequency waves, and we refer to them for further
background.
Theoretically it is also non-trivial to determine the spectrum of generated
acoustic waves from convective motions. Analytic studies indicate that there is
a peak in the acoustic spectrum around periods of 50 s (Musielak et al. 1994;
Fawzy et al. 2002) while results from high-resolution numerical simulations of
convection indicate decreasing power as a function of frequency (Goldreich et al.
1994; Stein & Nordlund 2001).
Recently, Fossum & Carlsson (2005b) & Fossum & Carlsson (2006) ana-
lyzed observations from the Transition Region And Coronal Explorer (TRACE)
satellite in the 1600 Å passband. Simulations were used to get the width of the
response function (Fossum & Carlsson 2005a) and to calibrate the observed in-
tensity fluctuations in terms of acoustic energy flux as function of frequency at
the response height (about 430 km). It was found that the acoustic energy flux
at 430 km is dominated by waves close to the acoustic cut-off frequency and the
high frequency waves do not contribute enough to be a significant contributor to
the heating of the chromosphere. Waves are detected up to 28mHz frequency,
and even assuming that all the signal at higher frequencies is signal rather than
noise, still gives an integrated energy flux of less than 500 Wm−2, too small by
a factor of ten to account for the losses in the VAL3C model. For the field free
internetwork regions used in the TRACE observations it is more appropriate to
use the VAL3A model that was constructed to fit the lowest intensities observed
with Skylab. It has about 2.2 times lower radiative losses than VAL3C (Avrett
1981) so there is still a major discrepancy. One should also remember that
Anderson & Athay (1989) found three times higher energy requirement than in
56 Carlsson
the VAL3C model when they included the radiative losses in millions of spec-
tral lines, dominated by lines from Fe II. As pointed out by Fossum & Carlsson
(2006), the main uncertainty in the results is the limited spatial resolution of the
TRACE instrument (0.5′′ pixels corresponding to 1′′ resolution with a possible
additional smearing from the little known instrument PDF): “There is possibly
undetected wave power because of the limited spatial resolution of the TRACE
instrument. The wavelength of a 40 mHz acoustic wave is 180 km and the hor-
izontal extent may be smaller than the TRACE resolution of 700 km. Several
arguments can be made as to why this effect is probably not drastic. Firstly, 5
minute waves are typically 10–20′′ in coherence, 3 minute waves 5–10′′. In both
cases 3–6 times the vertical wavelength. This would correspond to close to the
resolution element for a 40mHz wave. Secondly, even a point source excitation
will give a spherical wave that will travel faster in the deeper parts (because of
the higher temperature) and therefore the spherical wavefront will be refracted
to a more planar wave. With a distance of at least 500 km from the excitation
level it is hard to imagine waves of much smaller extent than that at a height of
400 km. There is likely hidden power in the subresolution scales, especially at
high frequencies. Given the dominance of the low frequencies in the integrated
power, the effect on the total power should be small. It is possible to quantify
the missing power by making artificial observations of 3D hydrodynamical sim-
ulations with different resolution. This is not trivial since the results will be
dependent on how well the simuation describes the excitation of high frequency
waves and their subsequent propagation. Preliminary tests in a 3D hydrody-
namical simulation extending from the convection zone to the corona (Hansteen
2004) indicate that the effect of the limited spatial resolution of TRACE on
the total derived acoustic power is below a factor of two. Although it is thus
unlikely that there is enough hidden subresolution acoustic power to provide the
heating for the chromosphere, the effect of limited spatial resolution is the major
uncertainty in the determination of the shape of the acoustic spectrum at high
frequencies.”
Another effect that goes in the opposite direction is that the analysis as-
sumes that all observed power above 5 mHz corresponds to propagating acous-
tic waves. Especially at lower frequencies we will also have a signal from the
temporal evolotion of small scale structures that in this analysis is mistakenly
attributed to wave power.
In a restrictive interpretation the result of Fossum & Carlsson (2005b) is
that acoustic heating can not sustain a temperature structure like that in static,
semi-empirical models of the Sun. Whether a dynamic model of the chromo-
sphere can explain the observations with acoustic heating alone has to be an-
swered by comparing observables from the hydrodynamic simulation with ob-
servations. This was done by Wedemeyer-Böhm et al. (2007). They come to the
conclusion that their dynamic model (Wedemeyer et al. 2004) is compatible with
the TRACE observations (the limited spatial resolution of the TRACE instru-
ment severly affects the synthetic observations) and that acoustic waves could
provide enough heating of the chromosphere. The synthetic TRACE images do
not take into account non-LTE effects or line opacities. It is also worth noting
that the model of Wedemeyer et al. (2004) does not have an average tempera-
ture rise in the chromosphere and the dominant wave power is at low frequencies
close to the acoustic cut-off and not in the high frequency part of the spectrum.
Modeling the Solar Chromosphere 57
Their study, however, is a major step forward — the question will have to be
resolved by more realistic modeling and synthesis of observations paired with
high quality observations.
The results by Fossum & Carlsson also show that the neglect of high-
frequency waves in the simulations by Carlsson & Stein is not an important
omission. Comparing their simulations with observations shows that the agree-
ment is good in the lower chromosphere (Carlsson & Stein 1997; Judge et al.
2003) but lines and continua formed above about 0.8Mm height have much too
low mean intensities. This is probably also true for the Wedemeyer et al. (2004)
model (since it has similar mean temperature) but this needs to be checked by
proper calculations. It thus seems inevitable that the energy balance in the
middle and upper chromosphere is dominated by processes related to the mag-
netic field. This is consistent with the fact that the concept of a non-magnetic
chromosphere is at best valid in the low chromosphere — in the middle to upper
chromosphere, the magnetic fields have spread and fill the volume. Even in the
photosphere, most of the area may be filled with weak fields or with stronger
fields with smaller filling factor (Sanchez Almeida 2005; Trujillo Bueno et al.
2004).
3.3. Comprehensive models in 3D
3D hydrodynamic simulations of solar convection have been very successful in re-
producing observations (e.g. Nordlund 1982; Stein & Nordlund 1998; Asplund et al.
2000; Vögler et al. 2005). It would be very natural to extend these simulations
to chromospheric layers to study the effect of acoustic waves on the structure,
dynamics and energetics of the chromosphere. This approach would then include
both the excitation of the waves by the turbulent motions in the convection zone
and their subsequent damping and dissipation in chromospheric shocks. For a
realistic treatment there are several complications. First, the approximation of
LTE that works nicely in the photosphere will overestimate the local coupling
in chromospheric layers. The strong lines that dominate the radiative coupling
have a source function that is dominated by scattering. Second, shock forma-
tion in the chromosphere makes it necessary to have a fine grid or describe
sub-grid physics with some shock capturing scheme. Third, it is important to
take into account the long timescales for hydrogen ionization/recombination for
the proper evaluation of the energy balance in the chromosphere (Kneer 1980;
Carlsson & Stein 1992, 2002a).
Skartlien (2000) addressed the first issue by extending the multi-group opac-
ity scheme of Nordlund to include the effects of coherent scattering. This mod-
ification made it possible to make the first consistent 3D hydrodynamic simula-
tions extending from the convection zone to the chromosphere (Skartlien et al.
2000). Due to the limited spatial resolution, the emphasis was on the excitation
of chromospheric wave transients by collapsing granules and not on the detailed
structure and dynamics of the chromosphere.
3D hydrodynamic simulations extending into the chromosphere with higher
spatial resolution were performed by Wedemeyer et al. (2004). They employed
a much more schematic description of the radiation (gray radiation) and did not
include the effect of scattering. This shortcoming will surely affect the amount
of radiative damping the waves undergo in the photosphere. The neglect of
58 Carlsson
strong lines avoids the problem of too strong coupling with the local conditions
induced by the LTE approximation so in a way two shortcomings partly balance
out. The chromosphere in their simulations is very dynamic and filamentary.
Hot gas coexists with cool gas at all heights and the gas is in the cool state a
large fraction of the time. As was the case in Carlsson & Stein (1995) they find
that the average gas temperature shows very little increase with height while
the radiation temperature does have a chromospheric rise similar to the VAL3C
model. The temperature variations are very large, with temperatures as low
as 2000K and as high as 7000K at a height of 800 km. It is likely that the
approximate treatment of the radiation underestimates the amount of radiative
damping thus leading to too large an amplitude.
The low temperatures in the simulations allow for a large amount of CO to
be present at chromospheric heights, consistent with observations. For a proper
calculation of CO concentrations it is important to take into account the detailed
chemistry of CO formation, including the timescales of the reactions. This was
done in 1D radiation hydrodynamic models by Asensio Ramos et al. (2003) and
in 2D models by Wedemeyer-Böhm et al. (2005). The dynamic formation of CO
was also included in the 3D models and it was shown that CO-cooling does not
play an important role for the dynamic energy balance at chromospheric heights
(Wedemeyer-Böhm & Steffen 2007).
Including long timescales for hydrogen ionization/recombination is non-
trivial. In a 1D simulation it is still computationally feasible to treat the full
non-LTE problem in an implicit scheme (avoiding the problem of stiff equations)
as was shown by Carlsson & Stein. The same approach is not possible at present
in 3D; the non-local coupling is too expensive to calculate. Fortunately, the hy-
drogen ionization is dominated by collisional excitation to the first excited level
(local process) followed by photo-ionization in the Balmer continuum. Since the
radiation field in the Balmer continuum is set in the photosphere, it is possi-
ble to describe the photoionization in the chromospheric problem with a fixed
radiation field (thus non-local but as a given rate that does not change with
the solution). This was shown to work nicely in a 1D setting by Sollum (1999).
Leenaarts & Wedemeyer-Böhm (2006) implemented the rate equations in 3D
but without the coupling back to the energy equation. The non-equilibrium ion-
ization of hydrogen has a dramatic effect on the ionization balance of hydrogen
in the chromosphere in their simulation.
Magnetic fields start to dominate over the plasma somehere in the chromo-
sphere. Chromospheric plasma as seen in the center Hα (e.g., Rutten (2007),
De Pontieu et al. (2007)) is very clearly organized along the magnetic struc-
tures. It is very likely that acoustic heating alone is not sufficient to account for
the radiative losses in the chromosphere. It is thus of paramount importance
to include magnetic fields in chromospheric modeling but unfortunetely the in-
clusion of magnetic fields increase the level of complexity enormously. As was
the case with acoustic waves, it is necessary to perform numerical experiments
and modeling in simplified cases in order to fashion a basic physical founda-
tion upon which to build our understanding. A number of authors have studied
various magnetic wave modes and how they couple, see Bogdan et al. (2003)
and Khomenko & Collados (2006) for references. Rosenthal et al. (2002) and
Bogdan et al. (2003) reported on 2D simulations in various magnetic field con-
Modeling the Solar Chromosphere 59
figurations in a gravitationally stratified isothermal atmosphere, assuming an
adiabatic equation of state. Carlsson & Stein (2002b) and Carlsson & Bogdan
(2006) reported on similar calculations in the same isothermal atmosphere but
this time in 3D and also studying the effect of radiative damping of the shocks.
Hasan et al. (2005) studied the dynamics of the solar magnetic network in two
dimensions and Khomenko & Collados (2006) studied the propagation of waves
in and close to a structure similar to a small sunspot.
The picture that emerges from these studies is that waves undergo mode
conversion, refraction and reflection at the height where the sound speed equals
the Alfvén speed (which is typically some place in the chromosphere). The crit-
ical quantity for mode conversion is the angle between the magnetic field and
the k-vector: the attack angle. At angles smaller than 30 degrees much of the
acoustic, fast mode from the photosphere is transmitted as an acoustic, slow
mode propagating along the field lines. At larger angles, most of the energy is
refracted/reflected and returns as a fast mode creating an interference pattern
between the upward and downward propagating waves. When damping from
shock dissipation and radiation is taken into account, the waves in the low-mid
chromosphere have mostly the character of upward propagating acoustic waves
and it is only close to the reflecting layer we get similar amplitudes for the up-
ward propagating and refracted/reflected waves. It is clear that even simple
magnetic field geometries and simple incident waves create very intricate inter-
ference patterns. In the chromosphere, where the wave amplitude is expected to
be large, it is crucial to include the effects of the magnetic fields to understand
the structure, dynamics and energetics of the atmosphere. This is true even
in areas comparably free of magnetic field (such regions may exist in the lower
chromosphere).
The fact that wave propagation is much affected by the magnetic field
topology in the chromosphere can be used for “seismology” of the chromo-
sphere. Observational clues have been obtained by McIntosch and co-workers:
McIntosh et al. (2001) & McIntosh & Judge (2001) find a clear correlation be-
tween observations of wave power in SOHO/SUMER observations and the mag-
netic field topology as extrapolated from SOHO/MDI observations. These re-
sults were extended to the finding of a direct correlation between reduced oscilla-
tory power in the 2D TRACE UV continuum observations and the height of the
magnetic canopy (McIntosh et al. 2003) and the authors suggest using TRACE
time-series data as a diagnostic of the plasma topography and conditions in
the mid-chromosphere through the signatures of the wave modes present. Such
helioseismic mapping of the magnetic canopy in the solar chromosphere was
performed by Finsterle et al. (2004) and in a coronal hole by McIntosh et al.
(2004).
The chromosphere is very inhomogeneous and dynamic. There is no simple
way of inverting the above observations to a consistent picture of the chromo-
spheric conditions. One will have to rely on comparisons with full 3D Radiation-
Magneto-Hydrodynamic forward modeling. Steiner et al. (2007) tracked a plane-
parallel, monochromatic wave propagating through a non-stationary, realistic
atmosphere, from the convection-zone through the photosphere into the mag-
netically dominated chromosphere. They find that a travel time analysis, like the
ones mentioned above, indeed is correlated with the magnetic topography and
60 Carlsson
Figure 1. Temperature structure in a 3D simulation box. The bottom plane
shows the temperature at 1.5 Mm below τ500=1 ranging from 15 700K in
down-flowing plumes to 16 500 in the gas flowing into the simulation domain.
The next plane is in the photosphere and shows hot granules and cool inter-
granular lanes. In the chromosphere the isothermal surfaces show pronounced
small scall structures and corrugated shock fronts. The upper 8Mm is filled
with plasma at transition region and coronal temperatures up to 1MK.
that high frequency waves can be used to extract information on the magnetic
canopy.
Including not only 3D hydrodynamics but in addition the magnetic field,
and extending the computational domain to include the corona is a daunting
task. However, the development of modern codes and computational power is
such that it is a task that is within reach of fulfillment. Hansteen (2004) reported
on the first results from such comprehensive modeling. The 3D computational
box is 16 × 8 × 12Mm in size extending 2Mm below and 10Mm above the
photosphere. Radiation is treated in detail, using multi-group opacities including
the effect of scattering (Skartlien 2000), conduction along field-lines is solved for
implicitly and optically thin losses are included in the transition region and
corona. For a snapshot of such a simulation, see Fig.1.
After a relaxation phase from the initial conditions, coronal temperatures
are maintained self-consistently by the injection of Pointing flux from the con-
Modeling the Solar Chromosphere 61
vective buffeting of the magnetic field, much as in the seminal simulations by
Gudiksen & Nordlund (2002, 2005a, 2005b).
It is clear from these simulations that the presence of the magnetic field
has fundamental importance for chromospheric dynamics and the propagation
of waves through the chromosphere. It is also clear that magnetic fields play a
role in the heating of the chromosphere (Hansteen et al. 2007).
There are several hotly debated topics in chromospheric modeling today:
Is the internetwork chromosphere wholly dynamic in nature or are the dynamic
variations only minor perturbations on a semi-static state similar to the state in
semi-empirical models (e.g., Kalkofen et al. 1999)? Is there a semi-permanent
cold chromosphere (where CO lines originate) or is the CO just formed in the
cool phases of a dynamic atmosphere? Is there enough chromospheric heating in
high frequency waves of small enough spatial extent that they are not detected
by the limited spatial resolution of TRACE? What is the role of the magnetic
field (mode conversion of waves, reconection, currents, channeling of waves)? A
reason for conflicting results is the incompleteness of the physical description in
the modeling and the lack of details (spatial and temporal resolution) in the ob-
servations. We are rapidly progressing towards the resolution of this situation.
New exciting observations at high temporal and spatial resolution (especially
from the Swedish 1-m Solar Telescope on La Palma) are changing our view of
the chromosphere. New observing facilities are on the verge of coming on line
(GREGOR, Hinode). On the modeling side, several groups have developed codes
that start to include the most important ingredients for a comprehensive mod-
eling of the dynamic chromosphere (e.g., Hansteen 2004; Schaffenberger et al.
2006). There is still more work to do with simulations of idealized cases to
build up a foundation for our understanding and this is also a field with several
groups active at present. The future for chromospheric modeling thus looks both
promising and exciting.
Acknowledgments. This work was supported by the Research Council of Norway
grant 146467/420 and a grant of computing time from the Program for Supercomputing.
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  Spectral diagnostic features formed in the solar chromosphere are few and
difficult to interpret -- they are neither formed in the optically thin regime
nor in local thermodynamic equilibrium (LTE). To probe the state of the
chromosphere, both from observations and theory, it is therefore necessary with
modeling. I discuss both traditional semi-empirical modeling, numerical
experiments illustrating important ingredients necessary for a self-consistent
theoretical modeling of the solar chromosphere and the first results of such
models.

Introduction
My keynote talk was similar in content to a recent talk at a Sacramento Peak
workshop celebrating the 70th birthday of Robert F. Stein. This written version
builds to a large extent on that writeup (Carlsson 2006), but it is updated and
some sections have been expanded.
Before discussing models of the solar chromosphere it is worthwhile dis-
cussing the very definition of the term “chromosphere”. The name comes from
the Greek words “χρωµα” (color) and “σϕαιρα” (ball) alluding to the colored
thin rim seen above the lunar limb at a solar eclipse. The color comes mainly
from emission in the Balmer Hα line. This is thus one possible definition — the
chromosphere is where this radiation originates. At an eclipse this region has
a sharp lower edge, the visible limb, but a fuzzy upper end with prominences
protruding into the corona. The nature of this region is difficult to deduce from
eclipse observations since we see this region edge on during a very short time
span and we have no way of telling whether it is homogeneous along the line of
sight or very inhomogeneous in space and time. It was early clear that the emis-
sion in Hα must mean an atmosphere out of radiative equilibrium — without
extra heating the temperature will not be high enough to have enough hydrogen
atoms excited to the lower or upper levels of the transition. Early models were
constructed to explain observations in Hα and in resonance lines from other
abundant elements with opacity high enough to place the formation in these
regions even in center-of-disk observations (lines like the H and K resonance
lines from singly ionized calcium). These early models were constructed assum-
ing one dimensional plane-parallel geometry and they resulted in a temperature
falling to a minimum around 4000K about 500 km above the visible surface, a
temperature rise to 8000K at a height of about 2000 km and then a very rapid
temperature rise to a million degree corona. These plane-parallel models have
also at Center of Mathematics for Applications, University of Oslo, Norway
http://arxiv.org/abs/0704.1509v1
50 Carlsson
led to a common notion that there is a more or less homogeneous, plane-parallel
region between these heights that is hotter than the temperature minimum. In
such a picture the chromosphere may be defined as a region occupying a given
height range (e.g. between 500 and 2000 km height over the visible surface) or a
given temperature range. We may also use physical processes for our definition:
the chromosphere is the region above the photosphere where radiative equilib-
rium breaks down and hydrogen is predominantly neutral (the latter condition
giving the transition to the corona). This discussion shows that there is no
unique definition of the term “chromosphere”, not even in a one-dimensional,
static world. It is even more difficult to agree on a definition of the “chromo-
sphere” that also encompasses an inhomogeneous, dynamic atmosphere.
As mentioned above, the first models of the chromosphere were constructed
with a large number of free parameters to match a set of observational con-
straints. Since some equations are used to restrict the number of free parameters
(not all hydrodynamical variables at all points in space and time are determined
empirically) we call this class of models semi-empirical models. Typically one
assumes hydrostatic equilibrium and charge conservation but no energy equa-
tion. The temperature as function of height is treated as a free function to
be determined from observations. In the other main class of models one tries
to minimize the number of free parameters by including an energy equation.
Such theoretical models have been very successful in explaining radiation from
stellar photospheres with only the effective temperature, acceleration of gravity
and abundances as free parameters. In the chromosphere, an additional term is
needed in the energy equation — e.g. energy deposition by acoustic shocks or
energy input in connection with magnetic fields (e.g. currents or reconnection).
It is thus clear from observations that the chromosphere is not in radiative
equilibrium — there is a net radiative loss. This loss has to be balanced by an
energy deposition, at least averaged over a long enough time span, if the atmo-
sphere is to be in equilibrium. This is often called the problem of chromospheric
“heating”. It is important to bear in mind, though, that the radiative losses may
be balanced by a non-radiative energy input without an increase in the average
temperature. The term “chromospheric heating” may thus be misleading since
it may be interpreted as implying that the average temperature is higher than
what is the case in a radiative equilibrium atmosphere. In the following we will
use the term “heating” in a more general sense: a source term in the energy
equation, not necessarily leading to an increased temperature.
Chromospheric heating is needed not only for the quiet or average Sun but
also in active regions, sunspots and in the outer atmospheres of many other
stars. I will in the following mainly discuss the quiet Sun case.
The outline of this paper is as follows: In Section 2 we discuss semi-empirical
models of the chromosphere. In Section 3 we discuss theoretical models; first
we elaborate on 1D hydrodynamical models, then we discuss the role of high
frequency acoustic waves for the heating of the chromosphere and finally we
describe recent attempts to model the chromosphere in 3D including the effects
of magnetic fields.
Modeling the Solar Chromosphere 51
2. Semi-empirical Models
Semi-empirical models can be characterized by the set of observations used to
constrain the model, the set of physical approximations employed and the set
of free parameters to be determined. Spectral diagnostics used to constrain
chromospheric models must have high enough opacity to place the formation
above the photosphere. The continuum in the optical part of the spectrum is
formed in the photosphere so the only hope for chromospheric diagnostics lies
in strong spectral lines in this region of the spectrum. Candidates are resonance
lines of dominant ionization states of abundant elements and lines from excited
levels of the most abundant elements (hydrogen and helium). Most resonance
lines are in the UV but the resonance lines of singly ionized calcium (Ca II), called
the H and K lines, fulfill our criteria. These lines originate from the ground state
of Ca II, the dominant ionization stage under solar chromospheric conditions,
and the opacity is therefore given by the density directly and the optical depth is
directly proportional to the column mass (i.e. to the total pressure in hydrostatic
equilibrium). Also the source function has some coupling to local conditions
even at quite low densities (in contrast to the strongly scattering resonance
lines of neutral sodium). Other chromospheric diagnostic lines in the optical
region are the hydrogen Balmer lines and the helium 1083 nm line. They all
originate from highly excited levels and thus have very temperature sensitive
opacity. The population of He 1083 is also set by recombination such that its
diagnostic potential is very difficult to exploit. With the advent of space based
observatories, the full UV spectral range was opened up. Continua shortward of
the opacity edge from the ground state of neutral silicon at 152 nm are formed
above the photosphere and can be used to constrain chromospheric models.
Together with observations in Ly-α, such UV continuum observations were used
by Vernazza et al. (1973, 1976, 1981) in their seminal series of papers on the solar
chromosphere. The VAL3 paper (Vernazza et al. 1981) is one of the most cited
papers in solar physics (1072 citations in ADS at the time of writing) and the
abstract gives a very concise description of the models and the principles behind
their construction: “The described investigation is concerned with the solution of
the non-LTE optically thick transfer equations for hydrogen, carbon, and other
constituents to determine semi-empirical models for six components of the quiet
solar chromosphere. For a given temperature-height distribution, the solution is
obtained of the equations of statistical equilibrium, radiative transfer for lines
and continua, and hydrostatic equilibrium to find the ionization and excitation
conditions for each atomic constituent. The emergent spectrum is calculated,
and a trial and error approach is used to adjust the temperature distribution
so that the emergent spectrum is in best agreement with the observed one.
The relationship between semi-empirical models determined in this way and
theoretical models based on radiative equilibrium is discussed by Avrett (1977).
Harvard Skylab EUV observations are used to determine models for a number
of quiet-sun regions.”
The VAL3 models are thus characterized by them using Ly-α and UV-
continuum observations for observational constraint, hydrostatic equilibrium
and non-LTE statistical equilibrium in 1D as physical description and temper-
ature as function of height as free function. To get a match with observed
line-strengths, a depth-dependent microturbulence was also determined and a
52 Carlsson
corresponding turbulent pressure was added. The number of free parameters
to be determined by observations is thus large — in principle the number of
depth-points per depth-dependent free function (temperature and microturbu-
lence). In practice the fitting was made by trial and error and only rather
smooth functions of depth were tried thus decreasing the degrees of freedom in
the optimization procedure.
The models have a minimum temperature around 500 km above the visible
surface (optical depth unity at 500 nm), a rapid temperature rise outwards to
about 6000K at 1000 km height and thereafter a gradual temperature increase
to 7000K at 2000 km height with a very rapid increase from there to coronal
temperatures.
The Ca II lines were not used in constraining the VAL3 models and the
agreement between the model representing the average quiet Sun, VAL3C, and
observations of these lines was not good. An updated model with a different
structure in the temperature minimum region was published in Maltby et al.
(1986) (where the main emphasis was on similarly constructed semi-empirical
models for sunspot atmospheres).
A peculiar feature with the VAL models was a temperature plateau intro-
duced between 20000 and 30000K in order to reproduce the total flux in the
Lyman lines. This plateau was no longer necessary in the FAL models where the
semi-empirical description of the transition region temperature rise was replaced
by the balance between energy flowing down from the corona (conduction and
ambipolar diffusion) and radiative losses (Fontenla et al. 1990, 1991, 1993).
One goal of semi-empirical models is to obtain clues as to the non-radiative
heating process. From the models it is possible to calculate the amount of non-
radiative heating that is needed to sustain the model structure. For the VAL3C
model this number is 4.2 kWm−2 with the dominant radiative losses in lines
from Ca II and Mg II, with Ly-α taking over in the topmost part.
The models described so far do not take into account the effect of the
very many iron lines. This was done in modeling by Anderson & Athay (1989).
Instead of using the temperature as a free parameter and observations as the
constraints, they adjusted the non-radiative heating function until they obtained
the same temperature structure as in the VAL3C model (arguing that they would
then have an equally good fit to the observational constraints as the VAL3C
model). The difference in the physical approximations is that they included line
blanketing in non-LTE from millions of spectral lines. The radiation losses are
dominated by Fe II, with Ca II, Mg II, and H playing important, but secondary,
roles. The total non-radiative input needed to balance the radiative losses is
three times higher than in the VAL3C model, 14 kWm−2.
The VAL3 and FAL models show a good fit to the average (spatial and
temporal) UV spectrum but fail to reproduce the strong lines from CO. These
lines show very low intensities in the line center when observed close to the
solar limb, the radiation temperature is as low as 3700K (Noyes & Hall 1972;
Ayres & Testerman 1981; Ayres et al. 1986; Ayres & Wiedemann 1989; Ayres & Brault
1990). If the formation is in LTE this translates directly to a temperature
of 3700K in layers where the inner wings of the H and K lines indicate a
temperature of 4400K. The obvious solution to the problem is that the CO
lines are formed in non-LTE with scattering giving a source function below the
Modeling the Solar Chromosphere 53
Planck-function. Several studies have shown that this is not the solution — the
CO lines are formed in LTE (e.g. Ayres & Wiedemann 1989; Uitenbroek 2000).
The model M CO constructed to fit the CO-lines (Avrett 1995) give too low
UV intensities. One way out is to increase the number of free parameters by
abandoning the 1D, one-component, framework and construct a two component
semi-empirical atmosphere. The COOLC and FLUXT atmospheric models of
Ayres et al. (1986) was such an attempt where a filling factor of 7.5% of the
hot flux tube atmosphere FLUXT and 92.5% of the COOLC atmosphere repro-
duced both the H and K lines and the CO-lines. The UV continua, however,
are overestimated by a factor of 20 (Avrett 1995). A combination of 60% of a
slightly cooler model than M CO and 40% of a hot F model provides a better
fit (Avrett 1995). Another way of providing enough free parameters for a better
fit is to introduce an extra force in the hydrostatic equilibrium equation provid-
ing additional support making possible a more extended atmosphere. With this
extra free parameter it is possible to construct a 1D temperature structure with
a low temperature in the right place to reproduce the near-limb observations of
the CO lines and a sharp temperature increase to give enough intensity in the
UV continua (Fontenla 2007).
A word of caution is needed here. Semi-empirical models are often im-
pressive in how well they can reproduce observations. This is, however, not a
proper test of the realism of the models since the observations have been used
to constrain the free parameters. The large number of free parameters (e.g.,
temperature as function of height, microturbulence as function of height and
angle, non-gravitational forces) may hide fundamental shortcomings of the un-
derlying assumptions (e.g., ionization equilibrium, lateral homogeneity, static
solution). It is not obvious that the energy input required to sustain a model
that reproduces time-averaged intensities is the same as the mean energy input
needed in a model that reproduces the time-dependent intensities in a dynamic
atmosphere. Semi-empirical modeling may give clues as to what processes may
be important but we also need to study these underlying physical processes with
fewer free parameters. This is the focus of theoretical models.
3. Theoretical Models
In contrast to semi-empirical models theoretical models include an energy equa-
tion. To model the full 3D system with all physical ingredients we know are
important for chromospheric conditions is still computationally prohibitive —
various approximations have to be made. In one class of modeling one tries to
illustrate basic physical processes without the ambition of being realistic enough
to allow detailed comparison with observations. Instead the aim is to fashion
a basic physical foundation upon which to build our understanding. The other
approach is to start with as much realism as can be afforded. Once the mod-
els compare favourably with observations, the system is simplified in order to
enable an understanding of the most important processes. I here comment on
both types of approaches.
54 Carlsson
3.1. 1D radiation hydrodynamic simulations
Acoustic waves were suggested to be the agent of non-radiative energy input al-
ready by Biermann (1948) and Schwarzschild (1948). Such waves are inevitably
excited by the turbulent motions in the convection zone and propagate outwards,
transporting mechanical energy through the photospheric layers into the chromo-
sphere and corona. Due to the exponential decrease of density with height, the
amplitude of the waves increases and they steepen into shocks. The theory that
the dissipation of shocks heats the outer atmosphere was further investigated by
various authors, see reviews by Schrijver (1995); Narain & Ulmschneider (1996).
In a series of papers, Carlsson & Stein (1992, 1994, 1995, 1997, 2002a) have
explored the effect of acoustic waves on chromospheric structure and dynamics.
The emphasis of this modeling was on a very detailed description of the radia-
tive processes and on the direct comparison with observations. The full non-LTE
rate equations for the most important species in the energy balance (hydrogen,
helium and calcium) were included thus including the effects of non-equilibrium
ionization, excitation, and radiative energy exchange on fluid motions and the
effect of motion on the emitted radiation from these species. To make the cal-
culations computationally tractable, the simulations were performed in 1D and
magnetic fields were neglected. To enable a direct comparison with observations,
acoustic waves were sent in through the bottom boundary with amplitudes and
phases that matched observations of Doppler shifts in a photospheric iron line.
These numerical simulations of the response of the chromosphere to acous-
tic waves show that the Ca II profiles can be explained by acoustic waves close
to the acoustic cut-off period of the atmosphere. The simulations of the be-
haviour of the Ca IIH line reproduce the observed features to remarkable detail.
The simulations show that the three minute waves are already present at pho-
tospheric heights and the dominant photospheric disturbances of five minute
period only play a minor modulating role (Carlsson & Stein 1997). The waves
grow to large amplitude already at 0.5 Mm height and have a profound effect
on the atmosphere. The simulations show that in such a dynamic situation it is
misleading to construct a mean static model (Carlsson & Stein 1994, 1995). It
was even questioned whether the Sun has an average temperature rise at chro-
mospheric heights in non-magnetic regions (Carlsson & Stein 1995). The simu-
lations also confirmed the result of Kneer (1980) that ionization/recombination
timescales in hydrogen are longer than typical hydrodynamical timescales under
solar chromospheric conditions. The hydrogen ionization balance is therefore out
of equilibrium and depends on the previous history of the atmosphere. Since the
hydrogen ionization energy is an important part of the internal energy equation,
this non-equilibrium ionization balance also has a very important effect on the
energetics and temperature profile of the shocks (Carlsson & Stein 1992, 2002a).
Kneer (1980) formulated this result as strongly as “Unless confirmed by consis-
tent dynamical calculations, chromospheric models based on the assumption of
statistical steady state should be taken as rough estimates of chromospheric
structure.”
Are observations in other chromospheric diagnostics than the Ca II lines
consistent with the above mentioned radiation-hydrodynamic simulations of the
propagation of acoustic waves? The answer is “No”. The continuum observa-
tions around 130 nm are well matched by the simulations (Judge et al. 2003)
Modeling the Solar Chromosphere 55
but continua formed higher in the chromosphere have higher intensity in the
observations than in the simulations (Carlsson & Stein 2002c). The chromo-
spheric lines from neutral elements in the UV range are formed in the mid to
upper chromosphere. They are in emission at all times and at all positions in
the observations, and they show stronger emission than in the simulations.
The failure of the simulations to reproduce diagnostics formed in the mid-
dle to upper chromosphere gives us information on the energy balance of these
regions. The main candidates for an explanation are the absence of magnetic
fields in the simulations and the fact that the acoustic waves fed into the compu-
tational domain at the bottom boundary do not include waves with frequencies
above 20mHz.
The reason for the latter shortcoming is that the bottom boundary is deter-
mined by an observed wave-field and high frequency waves are not well deter-
mined observationally. I first explore the possibility that high frequency acoustic
waves may account for the increased input and address the issue of magnetic
fields in the next section.
3.2. High frequency waves
Observationally it is difficult to detect high frequency acoustic waves for two
reasons: First, the seeing blurs the ground based observations and makes these
waves hard to observe. Second, for both ground based and space based observa-
tions the signal we get from high frequency waves is weakened by the width of
the response function. Wunnenberg et al. (2002) have summarized the various
attempts at detecting high frequency waves, and we refer to them for further
background.
Theoretically it is also non-trivial to determine the spectrum of generated
acoustic waves from convective motions. Analytic studies indicate that there is
a peak in the acoustic spectrum around periods of 50 s (Musielak et al. 1994;
Fawzy et al. 2002) while results from high-resolution numerical simulations of
convection indicate decreasing power as a function of frequency (Goldreich et al.
1994; Stein & Nordlund 2001).
Recently, Fossum & Carlsson (2005b) & Fossum & Carlsson (2006) ana-
lyzed observations from the Transition Region And Coronal Explorer (TRACE)
satellite in the 1600 Å passband. Simulations were used to get the width of the
response function (Fossum & Carlsson 2005a) and to calibrate the observed in-
tensity fluctuations in terms of acoustic energy flux as function of frequency at
the response height (about 430 km). It was found that the acoustic energy flux
at 430 km is dominated by waves close to the acoustic cut-off frequency and the
high frequency waves do not contribute enough to be a significant contributor to
the heating of the chromosphere. Waves are detected up to 28mHz frequency,
and even assuming that all the signal at higher frequencies is signal rather than
noise, still gives an integrated energy flux of less than 500 Wm−2, too small by
a factor of ten to account for the losses in the VAL3C model. For the field free
internetwork regions used in the TRACE observations it is more appropriate to
use the VAL3A model that was constructed to fit the lowest intensities observed
with Skylab. It has about 2.2 times lower radiative losses than VAL3C (Avrett
1981) so there is still a major discrepancy. One should also remember that
Anderson & Athay (1989) found three times higher energy requirement than in
56 Carlsson
the VAL3C model when they included the radiative losses in millions of spec-
tral lines, dominated by lines from Fe II. As pointed out by Fossum & Carlsson
(2006), the main uncertainty in the results is the limited spatial resolution of the
TRACE instrument (0.5′′ pixels corresponding to 1′′ resolution with a possible
additional smearing from the little known instrument PDF): “There is possibly
undetected wave power because of the limited spatial resolution of the TRACE
instrument. The wavelength of a 40 mHz acoustic wave is 180 km and the hor-
izontal extent may be smaller than the TRACE resolution of 700 km. Several
arguments can be made as to why this effect is probably not drastic. Firstly, 5
minute waves are typically 10–20′′ in coherence, 3 minute waves 5–10′′. In both
cases 3–6 times the vertical wavelength. This would correspond to close to the
resolution element for a 40mHz wave. Secondly, even a point source excitation
will give a spherical wave that will travel faster in the deeper parts (because of
the higher temperature) and therefore the spherical wavefront will be refracted
to a more planar wave. With a distance of at least 500 km from the excitation
level it is hard to imagine waves of much smaller extent than that at a height of
400 km. There is likely hidden power in the subresolution scales, especially at
high frequencies. Given the dominance of the low frequencies in the integrated
power, the effect on the total power should be small. It is possible to quantify
the missing power by making artificial observations of 3D hydrodynamical sim-
ulations with different resolution. This is not trivial since the results will be
dependent on how well the simuation describes the excitation of high frequency
waves and their subsequent propagation. Preliminary tests in a 3D hydrody-
namical simulation extending from the convection zone to the corona (Hansteen
2004) indicate that the effect of the limited spatial resolution of TRACE on
the total derived acoustic power is below a factor of two. Although it is thus
unlikely that there is enough hidden subresolution acoustic power to provide the
heating for the chromosphere, the effect of limited spatial resolution is the major
uncertainty in the determination of the shape of the acoustic spectrum at high
frequencies.”
Another effect that goes in the opposite direction is that the analysis as-
sumes that all observed power above 5 mHz corresponds to propagating acous-
tic waves. Especially at lower frequencies we will also have a signal from the
temporal evolotion of small scale structures that in this analysis is mistakenly
attributed to wave power.
In a restrictive interpretation the result of Fossum & Carlsson (2005b) is
that acoustic heating can not sustain a temperature structure like that in static,
semi-empirical models of the Sun. Whether a dynamic model of the chromo-
sphere can explain the observations with acoustic heating alone has to be an-
swered by comparing observables from the hydrodynamic simulation with ob-
servations. This was done by Wedemeyer-Böhm et al. (2007). They come to the
conclusion that their dynamic model (Wedemeyer et al. 2004) is compatible with
the TRACE observations (the limited spatial resolution of the TRACE instru-
ment severly affects the synthetic observations) and that acoustic waves could
provide enough heating of the chromosphere. The synthetic TRACE images do
not take into account non-LTE effects or line opacities. It is also worth noting
that the model of Wedemeyer et al. (2004) does not have an average tempera-
ture rise in the chromosphere and the dominant wave power is at low frequencies
close to the acoustic cut-off and not in the high frequency part of the spectrum.
Modeling the Solar Chromosphere 57
Their study, however, is a major step forward — the question will have to be
resolved by more realistic modeling and synthesis of observations paired with
high quality observations.
The results by Fossum & Carlsson also show that the neglect of high-
frequency waves in the simulations by Carlsson & Stein is not an important
omission. Comparing their simulations with observations shows that the agree-
ment is good in the lower chromosphere (Carlsson & Stein 1997; Judge et al.
2003) but lines and continua formed above about 0.8Mm height have much too
low mean intensities. This is probably also true for the Wedemeyer et al. (2004)
model (since it has similar mean temperature) but this needs to be checked by
proper calculations. It thus seems inevitable that the energy balance in the
middle and upper chromosphere is dominated by processes related to the mag-
netic field. This is consistent with the fact that the concept of a non-magnetic
chromosphere is at best valid in the low chromosphere — in the middle to upper
chromosphere, the magnetic fields have spread and fill the volume. Even in the
photosphere, most of the area may be filled with weak fields or with stronger
fields with smaller filling factor (Sanchez Almeida 2005; Trujillo Bueno et al.
2004).
3.3. Comprehensive models in 3D
3D hydrodynamic simulations of solar convection have been very successful in re-
producing observations (e.g. Nordlund 1982; Stein & Nordlund 1998; Asplund et al.
2000; Vögler et al. 2005). It would be very natural to extend these simulations
to chromospheric layers to study the effect of acoustic waves on the structure,
dynamics and energetics of the chromosphere. This approach would then include
both the excitation of the waves by the turbulent motions in the convection zone
and their subsequent damping and dissipation in chromospheric shocks. For a
realistic treatment there are several complications. First, the approximation of
LTE that works nicely in the photosphere will overestimate the local coupling
in chromospheric layers. The strong lines that dominate the radiative coupling
have a source function that is dominated by scattering. Second, shock forma-
tion in the chromosphere makes it necessary to have a fine grid or describe
sub-grid physics with some shock capturing scheme. Third, it is important to
take into account the long timescales for hydrogen ionization/recombination for
the proper evaluation of the energy balance in the chromosphere (Kneer 1980;
Carlsson & Stein 1992, 2002a).
Skartlien (2000) addressed the first issue by extending the multi-group opac-
ity scheme of Nordlund to include the effects of coherent scattering. This mod-
ification made it possible to make the first consistent 3D hydrodynamic simula-
tions extending from the convection zone to the chromosphere (Skartlien et al.
2000). Due to the limited spatial resolution, the emphasis was on the excitation
of chromospheric wave transients by collapsing granules and not on the detailed
structure and dynamics of the chromosphere.
3D hydrodynamic simulations extending into the chromosphere with higher
spatial resolution were performed by Wedemeyer et al. (2004). They employed
a much more schematic description of the radiation (gray radiation) and did not
include the effect of scattering. This shortcoming will surely affect the amount
of radiative damping the waves undergo in the photosphere. The neglect of
58 Carlsson
strong lines avoids the problem of too strong coupling with the local conditions
induced by the LTE approximation so in a way two shortcomings partly balance
out. The chromosphere in their simulations is very dynamic and filamentary.
Hot gas coexists with cool gas at all heights and the gas is in the cool state a
large fraction of the time. As was the case in Carlsson & Stein (1995) they find
that the average gas temperature shows very little increase with height while
the radiation temperature does have a chromospheric rise similar to the VAL3C
model. The temperature variations are very large, with temperatures as low
as 2000K and as high as 7000K at a height of 800 km. It is likely that the
approximate treatment of the radiation underestimates the amount of radiative
damping thus leading to too large an amplitude.
The low temperatures in the simulations allow for a large amount of CO to
be present at chromospheric heights, consistent with observations. For a proper
calculation of CO concentrations it is important to take into account the detailed
chemistry of CO formation, including the timescales of the reactions. This was
done in 1D radiation hydrodynamic models by Asensio Ramos et al. (2003) and
in 2D models by Wedemeyer-Böhm et al. (2005). The dynamic formation of CO
was also included in the 3D models and it was shown that CO-cooling does not
play an important role for the dynamic energy balance at chromospheric heights
(Wedemeyer-Böhm & Steffen 2007).
Including long timescales for hydrogen ionization/recombination is non-
trivial. In a 1D simulation it is still computationally feasible to treat the full
non-LTE problem in an implicit scheme (avoiding the problem of stiff equations)
as was shown by Carlsson & Stein. The same approach is not possible at present
in 3D; the non-local coupling is too expensive to calculate. Fortunately, the hy-
drogen ionization is dominated by collisional excitation to the first excited level
(local process) followed by photo-ionization in the Balmer continuum. Since the
radiation field in the Balmer continuum is set in the photosphere, it is possi-
ble to describe the photoionization in the chromospheric problem with a fixed
radiation field (thus non-local but as a given rate that does not change with
the solution). This was shown to work nicely in a 1D setting by Sollum (1999).
Leenaarts & Wedemeyer-Böhm (2006) implemented the rate equations in 3D
but without the coupling back to the energy equation. The non-equilibrium ion-
ization of hydrogen has a dramatic effect on the ionization balance of hydrogen
in the chromosphere in their simulation.
Magnetic fields start to dominate over the plasma somehere in the chromo-
sphere. Chromospheric plasma as seen in the center Hα (e.g., Rutten (2007),
De Pontieu et al. (2007)) is very clearly organized along the magnetic struc-
tures. It is very likely that acoustic heating alone is not sufficient to account for
the radiative losses in the chromosphere. It is thus of paramount importance
to include magnetic fields in chromospheric modeling but unfortunetely the in-
clusion of magnetic fields increase the level of complexity enormously. As was
the case with acoustic waves, it is necessary to perform numerical experiments
and modeling in simplified cases in order to fashion a basic physical founda-
tion upon which to build our understanding. A number of authors have studied
various magnetic wave modes and how they couple, see Bogdan et al. (2003)
and Khomenko & Collados (2006) for references. Rosenthal et al. (2002) and
Bogdan et al. (2003) reported on 2D simulations in various magnetic field con-
Modeling the Solar Chromosphere 59
figurations in a gravitationally stratified isothermal atmosphere, assuming an
adiabatic equation of state. Carlsson & Stein (2002b) and Carlsson & Bogdan
(2006) reported on similar calculations in the same isothermal atmosphere but
this time in 3D and also studying the effect of radiative damping of the shocks.
Hasan et al. (2005) studied the dynamics of the solar magnetic network in two
dimensions and Khomenko & Collados (2006) studied the propagation of waves
in and close to a structure similar to a small sunspot.
The picture that emerges from these studies is that waves undergo mode
conversion, refraction and reflection at the height where the sound speed equals
the Alfvén speed (which is typically some place in the chromosphere). The crit-
ical quantity for mode conversion is the angle between the magnetic field and
the k-vector: the attack angle. At angles smaller than 30 degrees much of the
acoustic, fast mode from the photosphere is transmitted as an acoustic, slow
mode propagating along the field lines. At larger angles, most of the energy is
refracted/reflected and returns as a fast mode creating an interference pattern
between the upward and downward propagating waves. When damping from
shock dissipation and radiation is taken into account, the waves in the low-mid
chromosphere have mostly the character of upward propagating acoustic waves
and it is only close to the reflecting layer we get similar amplitudes for the up-
ward propagating and refracted/reflected waves. It is clear that even simple
magnetic field geometries and simple incident waves create very intricate inter-
ference patterns. In the chromosphere, where the wave amplitude is expected to
be large, it is crucial to include the effects of the magnetic fields to understand
the structure, dynamics and energetics of the atmosphere. This is true even
in areas comparably free of magnetic field (such regions may exist in the lower
chromosphere).
The fact that wave propagation is much affected by the magnetic field
topology in the chromosphere can be used for “seismology” of the chromo-
sphere. Observational clues have been obtained by McIntosch and co-workers:
McIntosh et al. (2001) & McIntosh & Judge (2001) find a clear correlation be-
tween observations of wave power in SOHO/SUMER observations and the mag-
netic field topology as extrapolated from SOHO/MDI observations. These re-
sults were extended to the finding of a direct correlation between reduced oscilla-
tory power in the 2D TRACE UV continuum observations and the height of the
magnetic canopy (McIntosh et al. 2003) and the authors suggest using TRACE
time-series data as a diagnostic of the plasma topography and conditions in
the mid-chromosphere through the signatures of the wave modes present. Such
helioseismic mapping of the magnetic canopy in the solar chromosphere was
performed by Finsterle et al. (2004) and in a coronal hole by McIntosh et al.
(2004).
The chromosphere is very inhomogeneous and dynamic. There is no simple
way of inverting the above observations to a consistent picture of the chromo-
spheric conditions. One will have to rely on comparisons with full 3D Radiation-
Magneto-Hydrodynamic forward modeling. Steiner et al. (2007) tracked a plane-
parallel, monochromatic wave propagating through a non-stationary, realistic
atmosphere, from the convection-zone through the photosphere into the mag-
netically dominated chromosphere. They find that a travel time analysis, like the
ones mentioned above, indeed is correlated with the magnetic topography and
60 Carlsson
Figure 1. Temperature structure in a 3D simulation box. The bottom plane
shows the temperature at 1.5 Mm below τ500=1 ranging from 15 700K in
down-flowing plumes to 16 500 in the gas flowing into the simulation domain.
The next plane is in the photosphere and shows hot granules and cool inter-
granular lanes. In the chromosphere the isothermal surfaces show pronounced
small scall structures and corrugated shock fronts. The upper 8Mm is filled
with plasma at transition region and coronal temperatures up to 1MK.
that high frequency waves can be used to extract information on the magnetic
canopy.
Including not only 3D hydrodynamics but in addition the magnetic field,
and extending the computational domain to include the corona is a daunting
task. However, the development of modern codes and computational power is
such that it is a task that is within reach of fulfillment. Hansteen (2004) reported
on the first results from such comprehensive modeling. The 3D computational
box is 16 × 8 × 12Mm in size extending 2Mm below and 10Mm above the
photosphere. Radiation is treated in detail, using multi-group opacities including
the effect of scattering (Skartlien 2000), conduction along field-lines is solved for
implicitly and optically thin losses are included in the transition region and
corona. For a snapshot of such a simulation, see Fig.1.
After a relaxation phase from the initial conditions, coronal temperatures
are maintained self-consistently by the injection of Pointing flux from the con-
Modeling the Solar Chromosphere 61
vective buffeting of the magnetic field, much as in the seminal simulations by
Gudiksen & Nordlund (2002, 2005a, 2005b).
It is clear from these simulations that the presence of the magnetic field
has fundamental importance for chromospheric dynamics and the propagation
of waves through the chromosphere. It is also clear that magnetic fields play a
role in the heating of the chromosphere (Hansteen et al. 2007).
There are several hotly debated topics in chromospheric modeling today:
Is the internetwork chromosphere wholly dynamic in nature or are the dynamic
variations only minor perturbations on a semi-static state similar to the state in
semi-empirical models (e.g., Kalkofen et al. 1999)? Is there a semi-permanent
cold chromosphere (where CO lines originate) or is the CO just formed in the
cool phases of a dynamic atmosphere? Is there enough chromospheric heating in
high frequency waves of small enough spatial extent that they are not detected
by the limited spatial resolution of TRACE? What is the role of the magnetic
field (mode conversion of waves, reconection, currents, channeling of waves)? A
reason for conflicting results is the incompleteness of the physical description in
the modeling and the lack of details (spatial and temporal resolution) in the ob-
servations. We are rapidly progressing towards the resolution of this situation.
New exciting observations at high temporal and spatial resolution (especially
from the Swedish 1-m Solar Telescope on La Palma) are changing our view of
the chromosphere. New observing facilities are on the verge of coming on line
(GREGOR, Hinode). On the modeling side, several groups have developed codes
that start to include the most important ingredients for a comprehensive mod-
eling of the dynamic chromosphere (e.g., Hansteen 2004; Schaffenberger et al.
2006). There is still more work to do with simulations of idealized cases to
build up a foundation for our understanding and this is also a field with several
groups active at present. The future for chromospheric modeling thus looks both
promising and exciting.
Acknowledgments. This work was supported by the Research Council of Norway
grant 146467/420 and a grant of computing time from the Program for Supercomputing.
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