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Phil. Trans. R. Soc. A (2012) 370, 3114–3128
doi:10.1098/rsta.2011.0556
REVIEW
Numerical models of sunspot formation and
fine structure
B Y M ATTHIAS R EMPEL*
High Altitude Observatory, National Center for Atmospheric Research,
PO Box 3000, Boulder, CO 80307, USA
Sunspots are central to our understanding of solar (and stellar) magnetism in many
respects. On the large scale, they link the magnetic field observable in the photosphere to
the dynamo processes operating in the solar interior. Properly interpreting the constraints
that sunspots impose on the dynamo process requires a detailed understanding of the
processes involved in their formation, dynamical evolution and decay. On the small
scale, they give an insight into how convective energy transport interacts with the
magnetic field over a wide range of field strengths and inclination angles, leading to
sunspot fine structure observed in the form of umbral dots and penumbral filaments.
Over the past decade, substantial progress has been made on both observational and
theoretical sides. Advanced ground- and space-based observations have resolved, for
the first time, the details of umbral dots and penumbral filaments and discovered
similarities in their substructures. Numerical models have advanced to the degree that
simulations of entire sunspots with sufficient resolution to resolve sunspot fine structure
are feasible. A combination of improved helioseismic inversion techniques with seismic
forward modelling provides new views on the subsurface structure of sunspots. In this
review, we summarize recent progress, with particular focus on numerical modelling.
Keywords: magnetohydrodynamic; solar magnetic field; sunspots
1. Introduction
Sunspots are the most prominent manifestation of large-scale magnetic field on
the visible solar surface. Understanding how sunspots form, evolve and decay is
central to our understanding of solar magnetism, solar variability and its impact
on the heliosphere. The solar magnetic field is produced in the convection zone
(including the underlying overshoot region) through a magnetohydrodynamics
(MHD) dynamo process [1]. Most of the observational constraints on the magnetic
field that the dynamo produces come from photospheric observations, mostly
in the form of active regions (sunspot groups) and their temporal behaviour
throughout the solar magnetic cycle [2]. Unravelling the detailed implications
for the magnetic field within the solar convection zone requires a detailed
*[email protected]
One contribution of 11 to a Theme Issue ‘Astrophysical processes on the Sun’.
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understanding of the processes involved in the transport of the magnetic field
towards the visible solar surface and formation of active regions and sunspots
in the photosphere [3,4]. While the process of flux emergence is formally a
part of the dynamo itself, it is typically modelled independently owing to the
vast difference in the time and length scales involved. Substantial progress in
modelling capabilities over the past decade has enabled a transition towards
three-dimensional models incorporating the effects of convection down to scales of
granulation in the solar photosphere. With these improvements, a self-consistent
description of flux emergence across scales from the base of the convection zone
into the solar photosphere and beyond is in reach within the next decade.
While magnetic field and associated flows in the bulk of the solar convection
zone are difficult to infer from surface observations, this task is much more
promising for the uppermost 10–20 Mm of the convection zone where substantial
progress was made through the application of local helioseismology [5–7]. Progress
in this area thrives from a combination of stable, high-resolution space-based
observations, advanced inversion methods and most recently also numerical
forward modelling of wave propagation through sunspots. In particular, the latter
promises to resolve many of the challenges that currently restrict the reliable use
of helioseismic inversions within and near sunspots.
The fine structure of sunspots present at the smallest currently observable
scales has puzzled observers for more than a century (see recent reviews [8–12]).
In the umbra of sunspots, fine structure is manifest as dot-like plumes with a
diameter of several hundred kilometres (umbral dots), and in the penumbra as
elongated filaments in which strong horizontal outflows (first discovered about a
century ago [13]) along almost horizontal fields are embedded in a background of
more vertically inclined field (see also figure 1 for fine structure as it is present in a
numerical sunspot model). Sunspot fine structure is the consequence of magnetoconvective energy transport in strongly magnetized plasma, and therefore related
to a fundamental astrophysical process. Over the past decade, the physical
understanding of magneto-convection in sunspots was advanced forward by a
combination of new instrumentation (both ground- and space-based) and realistic
MHD models, which finally progressed to the point at which entire sunspots can
be modelled with a resolution sufficient for capturing the essence of sunspot fine
structure.
In this review, we will present a selective summary of these developments with a
special emphasis on recent advances in MHD modelling capabilities. After a brief
review of recent developments in ground- and space-based observation in §2, we
focus on MHD models of sunspot fine structure and of sunspot formation and
flux emergence in §3. Progress in our understanding of the subsurface structure
of sunspots from local helioseismology is presented in §4. Concluding remarks are
given in §5.
2. Recent advances and future directions in observations
Over the past decade, several advancements in ground- and space-based
observations have pushed our understanding of sunspot fine structure forward.
Substantial improvements in ground-based observations resulted from the
development of advanced image correction (adaptive optics) and reconstruction
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M. Rempel
I
Bz
Vz
VR
Figure 1. Sunspot fine structure at the t = 1 level. Shown are (in clockwise order) the intensity
(I ), vertical magnetic field (Bz ), radial flow (with respect to centre of sunspot) (VR ) and vertical
flow velocity (Vz ). The range shown is 0.3 . . . 1.5I for I (with I being the quiet sun intensity),
±2.5 kG (1 kG = 0.1 T) for Bz , ±8 km s−1 for VR and ±2 km s−1 for Vz .
methods that pushed existing telescopes to their diffraction limit. Examples for
ground-based observations of sunspot fine structure under optimal conditions
have been described in previous studies [14–26].
While the filamentary structure of the penumbra was already known from
lower-resolution observations, recent high-resolution data revealed an additional
substructure of penumbral filaments in terms of dark cores embedded in bright
filaments [14], which has also been confirmed for umbral dots [21,27] and light
bridges and provides strong observational evidence for a common origin of these
structures. Ground-based instrumentation is advanced even further with the
recently deployed 1.6 m New Solar Telescope telescope at the Big Bear Solar
Observatory as well as the 1.5 m Gregor telescope on Tenerife, which will become
operational soon. New 4 m class telescopes such as the Advanced Technology
Solar Telescope as well as the European Solar Telescope will become available
over the next decade. A major step forward in space-based instrumentation is
the Hinode satellite—in particular, the spectro-polarimeter of the Solar Optical
Telescope [28]. Examples of recent results concerning sunspot fine structure are
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those of Ichimoto et al. [29,30], Franz & Schlichenmaier [31] and Bharti et al.
[32]. The recently launched Solar Dynamics Observatory (SDO) will further
advance the study of sunspots and active regions through full-disc high-resolution
magnetograms and Doppler data for helioseismic inversions.
3. Recent advances in magnetohydrodynamics modelling
(a) General considerations
Challenges for MHD models of the solar convection zone, including the
photosphere, arise primarily from the vast separation in length and time scales
that are the consequence of the strong stratification of the system, while the
physical ingredients are well known. In the bulk of the convection zone, singlefluid MHD, combined with an ideal gas equation of state, is sufficient—when
approaching the photosphere, the equation of state has to take into consideration
partial ionization. In the photosphere, full three-dimensional radiative transfer is
essential, but remains manageable in complexity owing to the validity of local
thermal equilibrium, and the fact that the frequency dependence of opacities
can be captured sufficiently well in most cases by either grey or multi-band
approaches. Many of these approximations were introduced by Nordlund [33]
and have been reviewed recently in the study of Nordlund et al. [34], see also
the contribution by Stein [35]. In that sense, the solar convection zone and lower
photosphere differ substantially from the solar chromosphere and corona, where,
in addition to a large-scale separation, more complicated physics such as non-local
thermodynamic equilibrium radiative transfer, multi-fluid MHD, non-equilibrium
ionization, and kinetic effects matter.
Nevertheless, the density variation by about six orders of magnitude within the
convection zone implies a transition from strongly subsonic motions near the base
of the convection zone (Mach numbers ∼10−4 ) to supersonic motions in the
photosphere and a variation of the pressure scale height from about 50 Mm to
100 km. As a consequence, a model of the whole convection zone up into the
photosphere is currently still out of reach.
(b) Modelling of sunspot fine structure
Most aspects of sunspot fine structure can be addressed by models that are
restricted to the uppermost few megametres of the convection zone. Nevertheless,
substantial challenges remain owing to the massive scale separation between the
typical size of a sunspot (greater than 10 Mm) and the observed scale of fine
structure smaller than a few hundred kilometres, which requires grid resolutions
as small as a few 10 km. As a consequence, magneto-convection in strong
magnetic fields was addressed initially mainly through idealized MHD simulations
neglecting effects from partial ionization and radiative transfer. ‘Realistic’ MHD
simulations considering those effects were initially applied to small sections of
sunspots, and only in the past few years have computing resources been allowed
for simulations of complete sunspots. In addition to computing resources, these
simulations also require robust numerical schemes that can equally well handle
low and high Mach number flows, as well as high- and low-plasma beta regimes.
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The primary goal of magneto-convective sunspot models is to provide a
consistent description that explains the brightness of umbra and penumbra,
the fine structure (umbral dots and penumbral filaments) and the origin of the
Evershed flow. Apart from an initial setup and boundary conditions that define
the overall field geometry of the sunspot, such models should reproduce all the
other aspects that are self-consistently based on MHD and radiative transfer,
provided that a sufficient numerical resolution is used.
(i) Idealized models
The first two- and three-dimensional MHD models addressed magnetoconvection in idealized setups wherein the researchers studied the influence of
field strength and inclination on convection patterns. A central parameter in
these studies is the ratio of magnetic to thermal diffusivity, z, which characterizes
the mode of convection. Of particular interest is the situation with z < 1, which
is realized in the upper-most parts of the convection zone. Here, it was found
that the energy is transported by oscillatory columnar convection [36–39], which
transitions to travelling waves in the presence of inclined magnetic fields [40,41].
(ii) Magnetohydrodynamics models of umbral dots
A substantial step forward in realism was presented by Schüssler & Vögler [42],
who modelled a sunspot umbra using MHD simulations with three-dimensional
radiative transfer and a realistic equation of state. Here, it was found that
umbral dots are quasi-stationary plumes (lifetimes of about half an hour)
with overturning convection and substantially reduced field strength that are
embedded in the vertical background field of the umbra. Interestingly, the
oscillatory convection mode found in previous idealized investigations was not
present. These simulations also predicted a substructure in terms of dark lanes
within umbral dots similar to those that were identified previously in penumbral
filaments by Scharmer et al. [14]. Observational confirmations for dark lanes in
umbral dots were given by Bharti et al. [27] and Rimmele [21].
(iii) Magnetohydrodynamics models of penumbral filaments
The first MHD simulations with radiative transfer addressing the structure
of the penumbra were introduced by Heinemann et al. [43] and later refined
by Rempel et al. [44]. To keep the computational expense moderate, these
simulations were using a ‘slab-geometry’, in which the computational domain
is restricted to a narrow cut through the centre of a sunspot and horizontal
periodicity is assumed. Relaxing these constraints, Rempel et al. [45] presented
a numerical MHD simulation of a pair of full sunspots, which provided, for the
first time, extended penumbrae with strong horizontal outflows (Evershed flow).
Results from a very recent high-resolution simulation [46,47] are presented in
figure 1. The computational domain with an extent of 49 × 49 × 6.1 Mm3 is
resolved with a total of 4.8 billion grid points (3072 × 3072 × 512). In clockwise
order, figure 1 presents (bolometric) intensity, vertical magnetic field strength,
radial and vertical flow velocity. Here, radial refers to the cylindrical radius
with respect to the centre of the sunspot. The simulation shows umbral dots
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in the centre and the filamentary structure of the penumbra with dominantly
radially aligned structures present in all four quantities shown. Throughout
most of the penumbra, bright filaments are associated with radially elongated
convection cells having upflows in their centre and downflows in their periphery.
Unlike granulation, where downflows have typically a stronger velocity amplitude,
upflows dominate in the inner penumbra and possibly lead to an observational
bias towards upflows. This might explain the lack of detection of a convective flow
pattern in some observations [23,31]. A recent analysis of a numerical sunspot
model by Bharti et al. [48] confirmed that most of the downflows remain hidden
after the data is degraded to the resolution of typical observations.
The horizontal divergence of upflowing material strongly reduces the vertical
magnetic field, while the horizontal field (not shown in figure 1) is moderately
enhanced, leading to a complex magnetic structure consisting of filaments with
an almost horizontal field embedded in a more vertically inclined background
field. Fast horizontal outflows with flow velocities close to the Alfvén velocity
are found in the horizontal field component, consistent with the observationally
inferred structure (see reviews [8–12]). The three-dimensional structure available
from the numerical simulations allows us to further understand processes that
are intrinsically hidden from photospheric observations. We highlight here
three aspects.
Mass and energy transport in penumbra. The brightness of sunspot penumbrae
is typically more than 70 per cent of the brightness of granulation, indicating
that a substantial amount of overturning convective motions should be present.
Interestingly, the observational evidence for overturning motions in a sunspot
penumbra is not that clear. While there is some evidence for overturning
convection [16,20,21,30,32,49], other studies such as [23,29,31,50] see mostly
support for a larger-scale flow pattern with upflows in the inner and downflows in
the outer penumbra. Recent MHD models [51] clearly
show vigorous convection in
the penumbra and a relationship of the form I ∝ vz r.m.s. (t = 1), i.e. a brightness
of 0.7I requires about 1 km s−1 vertical r.m.s. velocity, which is about half the
value found in granulation. Direct evidence for overturning convection was found
recently by Joshi et al. [24] and Scharmer et al. [25] using observations in the
rather deep forming CI 5380 line.
Penumbral fine structure. The complicated fine structure of the penumbra is a
direct consequence of overturning convective motions. The vertical magnetic field
component is strongly weakened in upflows owing to the horizontal divergence of
fluid motions. At the same time, the horizontal field component shows a moderate
enhancement within filaments near an optical depth of unity, which results from
a combination of induction from the vertical shear profile of the Evershed flow
with upward advection that concentrates horizontal flux near the optical depth of
unity, where most of the mass flux turns over [51]. The result is a fine structure
consisting of essentially horizontal fields within flow channels, embedded in a
background of more vertically inclined fields. The combination of a reduction of
vertical and moderate enhancement of horizontal field components leads to an
overall reduction of field strength within flow channels in the inner penumbra
and an increase in the outer penumbra.
Origin of the Evershed flow. Since its discovery by Evershed [13], the origin
of fast outflows in sunspot penumbrae has been central to almost all theoretical
and observational investigations of sunspot fine structure. While many simplified
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models considered flows channelled by magnetic field in magnetic flux tubes, the
large degree of overturning motions found in simulations limits the meaningful
applicability of the flux tube concept in the case of penumbral flow channels.
Based on the work of Heinemann et al. [43], it was proposed by Scharmer et al.
[52] that the Evershed flow is essentially the horizontal convective flow component
along the field, enhanced in amplitude due to anisotropy. A recent analysis by
Rempel [51] showed that the Evershed flow originates from a narrow boundary
layer forming in upflow regions just beneath the t = 1 level. Here, the horizontal
magnetic field component increases strongly with height, leading to a sharp kink
in field lines. Pressure-driven upflows provide the energy source for the horizontal
acceleration of the Evershed flow, with the Lorentz force facilitating the energy
exchange between the vertical and horizontal direction. The consequence is a
magnetized horizontal outflow in the deep photosphere, with a flow amplitude
close to the Alfvén velocity (5–10 km s−1 ).
(iv) Future challenges
MHD simulations with radiative transfer and realistic equations of state
have evolved to the point where they include all essential building blocks of
sunspot fine structure. All aspects of sunspot fine structure from umbral dots to
penumbral filaments and light-bridges are explained through a common magnetoconvective process that is modulated by the background field strength and
inclination angle. Large-scale properties, such as the flux and overall extent
of penumbrae, is, in most simulations, a consequence of initial and boundary
conditions; however, details of fine structure are the consequence of (mostly)
resolved magneto-convection processes that are found to be robust within the
currently accessible resolution range. While the overall flow and field structure at
photospheric levels agrees well with the observationally inferred fine structure, a
detailed comparison based on forward modelling of spectral line signatures is
still in an exploratory state. That and resolving the apparent discrepancy with
regard to the different amounts of overturning motions (§3b(iii)) present in
simulations and observations are currently the prime challenges that require
further investigation.
(c) Modelling of flux emergence and sunspot formation
The numerical models described in §3b all have in common that they are
initialized with a mostly monolithic sunspot-like field structure and then evolved
for a time span of typically a few hours, which is sufficient as far as the
development of sunspot fine structure is concerned, but processes that lead to
the formation of sunspots in the first place are outside their realm.
The formation of sunspots is the part of the larger-scale flux emergence process
that transports magnetic fields from within the solar convection zone into the
photosphere and beyond into the chromosphere and corona. Currently, there is
no single model that can capture the whole process—typically, three regions are
modelled independently: (i) the deep convection starting at its base up to about
20 Mm beneath the solar photosphere, (ii) the top of the convection zone starting
at about 10–20 Mm depth upwards into the photosphere, and (iii) the outer solar
atmosphere starting in the photosphere upwards into the solar corona. Here, we
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focus entirely on the second region and refer to the review by Fan [3] and the
contribution by Archontis [4] for further details regarding modelling the other
regions.
The flux emergence process in the solar convection zone faces two challenges:
the interaction with vigorous convection and the strong density contrast of 106
between the bottom of the convection zone and the photosphere. The strong
expansion due to the density drop implies a substantial weakening of magnetic
field during the flux emergence process. It was found by Cheung et al. [53] in nearsurface flux emergence
simulations that the magnetic field scales with density
√
roughly as B ∝ 9, which is primarily a consequence of preferred expansion in
the horizontal direction (the expansion along the magnetic field direction does
not weaken the field due to the additional stretching term). Assuming that this
relation holds over a larger depth range implies that even a field of 100 kG (1 kG =
0.1 T) strength at the base of the convection zone (which is a factor of 10 larger
than the equipartition value) is weakened to about 100 G in the photosphere,
while a typical sunspot field strength is about 3 kG. Because the pressure scale
height decreases from about 50 Mm at the base of the convection zone to about
100 km in the photosphere, most of the density drop is found near the top of the
convection zone. As a consequence, a model that focuses on the top 15 Mm of
the convection zone includes a density drop of about 104 out of the 106 , while
covering only 7.5 per cent of the total convection zone depth. Many effects related
to the strong expansion and weakening of rising magnetic fields and the required
re-amplification in the photosphere can be studied in numerical models of the
near-surface layers.
In addition, substantial progress relied heavily on advanced computing
resources, as numerical simulations on the scale of the active region (100 Mm)
are demanding, even if only a very moderate resolution of about 100 km is
adopted. While idealized simulations (many of them neglecting convection) have
been used in the past, recently the emergence and interaction of magnetic flux
with photospheric granulation was addressed by several groups [54–61]. Flux
emergence simulations on the scale of supergranulation and beyond were recently
performed by Cheung et al. [53] and Stein et al. [62]. While Stein et al. [62]
followed the approach of Stein & Nordlund [54], in which the horizontal field
was advected across the bottom boundary condition in upflow regions, Cheung
et al. [53] kinematically advected a half torus of flux into the domain, following
the approach of Fan & Gibson [63]. The latter approach allows one to introduce
a substantially larger amount of flux into the domain, and as a result, Cheung
et al. [53] were successful in simulating the formation of a small active region in
a 92 × 49 Mm2 wide and 8 Mm deep domain (figure 2). In the early stages, the
evolution was dominated by the marked expansion due to the density contrast
of about 103 , at later stages the flux was re-amplified in a pair of small sunspots
with about 3 × 1021 Mx (108 Mx = 1 Wb) flux, which was about 40 per cent
of the flux initially transported into the domain across the bottom boundary.
While this simulation showed, for the first time, that the strong weakening of
the magnetic field as a consequence of the flux emergence process is consistent
with the formation of sunspots at later stages, the overall extent of the domain
was not sufficient to clearly quantify the role played by boundary conditions.
This work is currently being extended to 150 × 75 Mm2 wide and 16 Mm deep
domains. It is found that flux emergence and sunspot formation processes are
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M. Rempel
(a)
2
1
Bz (kG)
y (Mm)
10
0
–1
log10 |B| (kG)
–10
(b)
z (Mm)
0
0
–2
–4
–6
–30
–20
–10
0
x (Mm)
10
20
–2
4
2
0
30
Figure 2. The formation of a small sunspot pair with about 3 × 1021 Mx flux. (a) A photospheric
magnetogram and (b) the subsurface field strength in the vertical plane running through the
simulation domain along the horizontal dotted line in (a). The simulation was performed in a
92 × 49 Mm2 wide and 8 Mm deep domain. This figure is reproduced from Cheung et al. [53] by
permission of the American Astronomical Society. (Online version in colour.)
only weakly dependent on the initial twist and field strength (10 and 20 kG give
essentially similar results) of the rising flux imposed at the bottom boundary. The
dynamics are strongly driven by changes in the stratification that are a result of
the rising plasma, regardless of field strength. Most importantly, these simulations
demonstrate that sunspots can form from fields with about 10 kG field strength
(peak value, average value of emerging flux is 5 kG) in 16 Mm depth, which is
consistent with the expectations from flux emergence simulations that start at
the base of the convection zone with about 50–100 kG field strength. Future
improvements will also require an upward extension of the simulation domain
into the upper solar atmosphere (chromosphere, corona) to reduce the potential
influence from the top boundary condition on the flux emergence process.
Overall, these results indicate that a coherent and consistent picture of the flux
emergence process in the solar convection zone and photosphere is developing.
Over the coming decade, coupled models that describe the whole process from
the base of the convection zone into the photosphere and beyond are within reach.
4. Local helioseismology
Understanding the subsurface magnetic field and flow structure of sunspots is a
central goal of helioseismic studies. Although substantial progress has been made
over the past decade (see reviews [5,6,64]), many results remain controversial.
Most challenges in applying helioseismic inversion techniques to sunspots have
their root in the fact that sunspots cannot be treated as small perturbations—at
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least not in the upper-most few megametres—which limits the applicability of
linear inversion methods. In addition, the effect of the magnetic field cannot
be captured by an isotropic wave speed perturbation. Direct applications of
time distance helioseismology revealed a two-layer structure underneath sunspots,
which is manifest in thermal and velocity structure [64–66]. It was found that the
uppermost 4 Mm show a reduction in wave propagation speed and converging
flows, while deeper layers show an enhancement of wave speed combined with
outflows. The application of ring diagram analysis [67,68] provided results that
are comparable on a qualitative level, although the transition from reduced to
enhanced wave speed is found in a greater depth of about 7 Mm. It was recently
pointed out by Gizon et al. [69,70] and Moradi et al. [71] that many of the observed
signatures originate from the upper-most 1 or 2 Mm, where the sunspot imposes
a rather large perturbation. Only after properly addressing these near-surface
effects are reliable deeper inversions possible.
A new promising approach that has been followed recently in parallel by several
groups is the forward modelling of wave propagation through stationary sunspot
models [72–76]. Using this approach, Cameron et al. [77] demonstrated that the
observed helioseismic signatures of the f, p1 and p2 modes can be captured by
a rather shallow sunspot model. The strong role of near-surface effects was also
deduced by Braun & Birch [78] from the observed frequency dependence of travel
time shifts.
In addition to the forward modelling of wave propagation through stationary
sunspot models, the three-dimensional radiative MHD simulations of sunspots
described in §3 can also be used for the study of helioseismic signatures.
In these simulations, solar-like oscillations are naturally excited by convection
to the extent that the modes are allowed for by the finite size simulation
domain. Helioseismic studies require simulations in wider and deeper domains
that have a reduced grid resolution to allow for evolution over time scales of
at least a few days. Additionally, the larger domain size permits the study
of large-scale flow systems such as the moat flow, which was originally found
through tracking of magnetic fields surrounding sunspots [79–82] and more
recently through helioseismology [69,70,83]. Flow structures on the scale of active
regions [84] are currently still out of reach for numerical simulations. With
regard to helioseismic signatures, three-dimensional radiative MHD simulations of
sunspots also point towards rather shallow wave speed perturbations. The thermal
structure (discussed in [71]) as well as large-scale flow structure (figure 3 and
[85]) do not show a two-layer structure, as indicated in some of the helioseismic
inversions mentioned already.
The combination of high-resolution data available today through the SDO, in
combination with several forward modelling methods that account for the full
complexity of sunspots, makes substantial progress likely in the near future.
5. Summary
Over the past decade, substantial progress in our understanding of sunspots
resulted from a combination of high-resolution observations and numerical
simulations from first principles. The latter benefited substantially from a
combination of robust numerical methods with some ‘innovative’ computing
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radial velocity
z – zphot (Mm)
0
–2
–4
–6
–8
0
10
30
20
R (Mm)
–0.5
0
VR (Vr.m.s.)
0.5
Figure 3. Azimuthally averaged radial flow velocity as function of depth and distance from centre
of sunspot. This numerical simulation was evolved for 24 h in a 75 Mm wide and 9 Mm deep
computation domain. The velocity amplitude is relative to the convective r.m.s. velocity of
undisturbed solar convection at the respective depth level. Positive values indicate outflows. The
sunspot is surrounded by an outflow cell at all depth levels. The fast flow concentrated around the
indicated t = 1 level is the Evershed flow, which merges in deeper layers with outflows reaching
about 50% of the convective r.m.s. velocity over a wide depth range. Further out (R > 22 Mm), a
rather superficial photospheric outflow of about 300 m s−1 is present, with indication of an inflow
below 4 Mm depth. (Online version in colour.)
approaches that resolve the most relevant processes with highest precision, but
that also relax other aspects (such as high Alfvén velocities) where possible to
keep the computational expense manageable. In addition, the steady increase
in computing capabilities allowed these models to pass a threshold at which
computations on the scale of entire sunspots with sufficient resolution to capture
sunspot fine structure became possible. The simultaneous focus on large and small
scales substantially reduces (albeit does not eliminate) uncertainties introduced
through initial and boundary conditions and makes the entire simulation more
self-consistent with less (hidden) parameters free to adjust. This substantial
increase in realism leads to models that simultaneously explain several aspects
of sunspot structure, such as magneto-convective energy transport in umbra
and penumbra, filamentation and large-scale flow systems without requiring fine
tuning, apart from setting up a large-scale magnetic field structure through initial
and boundary conditions that resembles a typical sunspot and using a sufficiently
high resolution. While these models agree very well with the observationally
inferred thermal, magnetic and flow structure of sunspots, a detailed comparison
with spectro-polarimetric data in terms of line shapes and polarization signals is
still at an exploratory stage. While the currently affordable numerical resolution
appears to be sufficient to capture the essence of the underlying magnetoconvection processes, it is still marginal for detailed forward modelling of spectral
signatures [47].
Phil. Trans. R. Soc. A (2012)
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On the larger scale, photospheric MHD simulations now approach the scale of
active regions and time scales of several days. Work in progress by several groups
focuses on the processes involved in the formation and decay of active regions
as well as large-scale flow systems, such as the moat flow surrounding sunspots.
These simulations provide, as a by-product, very realistic oscillation modes that
can be used for an in-depth comparison with helioseismic data from missions such
as the SDO or used as a testbed for helioseismic inversion procedures. The latter
is, in particular, beneficial for inversions in the proximity of sunspots, which are
most relevant for our understanding of solar magnetism.
The ultimate goal of studying sun (star) spots is making the connection to
the MHD processes in the solar (stellar) interiors responsible for the magnetic
field generation in the first place. On the largest scales, the relevant quantities
are the overall flux and its distribution over the solar (stellar) surface. This
is tied most strongly to the dependence of the underlying dynamo process
on rotation rates, stellar luminosities and convection zone depths, and has to
be studied in the context of global dynamo models [1]. Details of the surface
appearance of spots (including the fine structure) are governed by the properties
of the photosphere, most importantly, the gas pressure and Mach number of
the photospheric convection. While sunspot models could be generalized to
explore a wider (stellar) photospheric parameter range, currently no observational
constraints exist for this level of detail.
M.R. is partially supported through NASA grant NNH09AK02I (SDO Science Center) to the
National Center for Atmospheric Research. The National Center for Atmospheric Research is
sponsored by the National Science Foundation. M.R. thanks M. Miesch for very helpful comments
on the manuscript.
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