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Phil. Trans. R. Soc. A (2012) 370, 3070–3087
doi:10.1098/rsta.2011.0533
REVIEW
Magneto-convection
B Y R OBERT F. S TEIN*
Michigan State University, East Lansing, MI 48824, USA
Convection is the transport of energy by bulk mass motions. Magnetic fields alter
convection via the Lorentz force, while convection moves the fields via the curl(v × B)
term in the induction equation. Recent ground-based and satellite telescopes have
increased our knowledge of the solar magnetic fields on a wide range of spatial and
temporal scales. Magneto-convection modelling has also greatly improved recently
as computers become more powerful. Three-dimensional simulations with radiative
transfer and non-ideal equations of state are being performed. Flux emergence from
the convection zone through the visible surface (and into the chromosphere and corona)
has been modelled. Local, convectively driven dynamo action has been studied. The
alteration in the appearance of granules and the formation of pores and sunspots has
been investigated. Magneto-convection calculations have improved our ability to interpret
solar observations, especially the inversion of Stokes spectra to obtain the magnetic field
and the use of helioseismology to determine the subsurface structure of the Sun.
Keywords: Sun; convection; magnetic; modelling; simulation
1. Introduction: convection
Convection is the transport of energy by bulk mass motions. In a convection zone,
energy is transported as thermal energy, except in layers where hydrogen is
only partially ionized where most of the energy is transported as ionization
energy. Typically, the motions are slow compared with the sound speed so that
approximate horizontal pressure balance is maintained. As a result, warmer fluid
is less dense and buoyant, while cooler fluid is denser and gets pulled down
by gravity.
The topology of convection is controlled by mass conservation [1]. Warm
upflows diverge and tend to be laminar, while cool downflows converge and tend
to be turbulent. Convection has a horizontal cellular pattern, with the warm fluid
ascending in separate fountain-like cells surrounded by lanes of cool descending
fluid. In a stratified atmosphere, with density decreasing outward, most of the
ascending fluid must turn over and be entrained in the downflows within a density
scale height (ignoring gradients in velocity and filling factor). Fluid moving a
distance Dr in an atmosphere with a density gradient dln r/dr would, if its
density remained constant, be overdense compared with its surroundings by a
*[email protected]
One contribution of 11 to a Theme Issue ‘Astrophysical processes on the Sun’.
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factor Dr/r = −(dln r/dr)Dr, which is unstable and produces a pressure excess
in the upflow cell interiors that pushes the fluid to turn over into the surrounding
downflow lanes. Since the fluid velocity decreases inwards from the top of the
convection zone, its derivative has a sign opposite to that of the density; so the
length scale for entrainment is increased.
Temperature in stellar convection zones increases inwards; so the scale height
and, as a result, the size of the horizontal convective cellular pattern also increase
inwards. Think of the rising fluid as a cylinder. As described already, most of the
fluid entering at the bottom of the cylinder must leave through its sides within a
scale height. If the ratio of vertical to horizontal velocities does not change much
with depth, then the radius of the cylinder can increase in proportion to the scale
height and still maintain mass conservation [2].
2. Magneto-convection
In the presence of magnetic fields, convection is altered by the Lorentz force,
while convection influences the magnetic field via the curl(v × B) term in the
induction equation. Where the magnetic field is weak and the conductivity
high, the field is frozen into the ionized plasma. Convective motions drag
the field around. To maintain force balance, locations of higher field strength
(higher magnetic pressure) tend to have smaller plasma density and lower gas
pressure. Diverging, overturning motions quickly sweep the field (on granular
times of minutes) from the granules into the intergranular lanes [3–8]. In hours
(mesogranular times), the field tends to collect on a mesogranule scale. In days
(supergranule times), the slower, large-scale supergranule motions collect the
field in the magnetic network at the supergranule boundaries. Convective flows
produce a hierarchy of loop structures in rising magnetic flux. Slow upflows and
buoyancy raise the flux, while fast downflows pin it down, which produces Uand U-loops [9]. The different scales of convective motion produce loops on these
different scales, with smaller loops riding piggy-back in a serpentine fashion on
the larger loops [9,10]. Dynamo action occurs in the turbulent downflows where
the magnetic field lines are stretched, twisted and reconnected, increasing the
field strength [11–15].
Magnetic fields influence convection via the Lorentz force, which inhibits
motion perpendicular to the field. As a result, the overturning motions that
are essential for convection are suppressed and convective energy transport from
the interior to the surface is reduced. Radiative energy loss to space continues;
so regions of strong field cool relative to their surroundings. Being cooler,
these locations have a smaller scale height. Plasma drains out of the magnetic
field concentrations in a process called ‘convective intensification’ or ‘convective
collapse’ [16–22]. This process can continue until the magnetic pressure (plus a
small gas pressure) inside the flux concentration equals the gas pressure outside,
giving rise to a field strength much greater than the equipartition value with the
dynamic pressure of the convective motions. These magnetic flux concentrations
are cooler than their surroundings at the same geometric layer. However, because
they are evacuated, their opacity is reduced; so photons escape from deeper in
the atmosphere (Wilson depression [23]). Where the magnetic concentrations
are narrow, there is heating from their hotter side walls and they appear as
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bright points [24]. Where the concentrations are wide, the side wall heating is
not significant and the flux concentrations appear darker than the surroundings
as pores or sunspots.
Magnetic fields alter the granule properties—producing smaller, lower intensity
contrast, ‘abnormal’ granules [20,25]. Strong magnetic flux concentrations
typically form in convective downflow lanes, especially at the vertices of several
such lanes, owing to the sweeping of flux by the diverging convective upflows
[7,8]. They are surrounded by downflows that sometimes become supersonic. The
normal convective downflows are enhanced surrounding the flux concentrations by
baroclinic driving due to the influx of radiation into the concentration [7,26–28].
3. Equations
Magneto-convection is highly nonlinear, so it needs to be modelled using numerical simulations. To simulate magneto-convection, the conservation equations for
mass, momentum, energy and the induction equation for the magnetic field must
be solved, together with Ohm’s Law for the electric field and an equation of
state relating pressure to the density and energy. Two types of numerical studies
of magneto-convection have been undertaken: idealized and ‘realistic’. Both
approaches give valuable, but different, insights into the properties of magnetoconvection. Idealized simulations were pioneered by Weiss [29] and extensively
used by Tao et al. [30], Cattaneo [12], Abbett et al. [31], Hurlburt & Rucklidge
[32], Emonet & Cattaneo [4], Weiss et al. [5] and Cattaneo et al. [33]. They
are especially useful for gaining physical insights into convective properties. In
these calculations, an ideal gas equation of state is assumed and energy transport
is assumed to be only by diffusion and conduction. Similar calculations are
also appropriate for modelling convection in the solar interior [34,35]. ‘Realistic’
simulations were pioneered by Nordlund [36] and have been extensively developed
by Stein & Nordlund [2], Steiner et al. [37], Vögler et al. [7], Schaffenberger et al.
[38], Stein & Nordlund [8], Hansteen et al. [39], Abbett [40], Jacoutot et al. [41]
and Carlsson et al. [42]. For a discussion of the equations used, see Vögler et al. [7]
and Nordlund et al. [43]. Here, a tabular equation of state is used, which includes
the partial ionization of hydrogen, helium and other abundant elements, because,
below 40 000 K in the Sun, ionization energy dominates over thermal energy in
convective energy transport. The radiation transfer equation is solved to determine the radiative heating and cooling, because the optical depth is of order
unity near the visible solar surface, so that neither the diffusion nor optically thin
approximations are valid. Such detailed physics is necessary to make quantitative
comparisons with observations. No simulations approach the solar Reynolds
number. However, they do reproduce many of the observed solar features.
We will restrict ourselves to the more realistic surface simulations. Magnetoconvection and dynamo action in the deeper layers of the convection zone are
reviewed by Miesch [34,44].
4. Photospheric observations
The solar surface is covered with magnetic features with spatial scales from
smaller than can currently be resolved (approx. 70 km with the Swedish 1 m
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Solar Telescope) to active regions covering up to 100 Mm. These evolve on
a correspondingly wide range of time scales, from seconds for the smallest
observed features to months for some active regions. If one counts as a single
feature any contiguous collection of the same polarity with magnitude above
some cut-off, then the magnetic flux distribution is a power law of slope −1.85.
Alternatively, if one identifies features as individual flux peaks, then the
distribution is lognormal [45]. These power laws are featureless, they have no
peaks or valleys.
The large-scale magnetic structures, sunspots and active regions possess
some well-defined global properties: Hale’s polarity law (same polarities of
leading/following spots in a given hemisphere, but reversed between the Northern
and Southern Hemispheres); polarities reverse in a semi-periodic 22 year cycle; in
each cycle spots first appear at mid-latitudes and then their appearances migrate
towards the equator; Joy’s Law (that active regions are tilted with the leading
spot closer to the equator); and sunspots tend to reappear at certain active
longitudes. These properties imply the existence of a global dynamo process.
There is an excellent review of small-scale solar magnetic field observations
(network and internetwork quiet Sun) by de Wijn et al. [46]. The main
observational results are summarized hereafter: strong fields tend to be vertical
and weaker fields horizontal. Vertical kilogauss fields (in pressure equilibrium
with their surroundings) are found in the magnetic network and as isolated,
intermittent concentrations in intergranular lanes. Horizontal magnetic fields are
found all over the Sun, predominantly inside and near the edges of granules.
They are transient, intermittent and have granule-scale sizes and lifetimes
and strengths in the hectogauss range (generally less than equipartition with
the convective dynamic pressure). Weaker horizontal fields have no preferred
orientation. Stronger ones tend to align with the active regions. The horizontal
field properties are similar in the quiet Sun, plage and polar regions [47].
The spatially averaged horizontal magnetic field strength is 50–60 G, while
the spatially averaged vertical field strength is only 11 G [48]. This may be
due to the larger area covered by horizontal fields compared with the isolated
vertical field concentrations. There is no characteristic size or lifetime for the
horizontal fields (they have an exponential distribution in both size and lifetime)
[49]. Bright points in the G-band have been used as proxies for the magnetic
field. In simulations, all the bright points correspond to locations of large field
magnitude, but not all large field locations correspond to bright points [7,8].
Further, the field has a longer lifetime than the bright points.
Four orders of magnitude more magnetic flux is observed to emerge as smallscale loops in the quiet Sun than emerges in active regions. This new flux is
first seen as horizontal field (linear polarization in Stokes spectra) inside granules
followed by the appearance of vertical field at the ends of the horizontal field
(circular polarized Stokes spectra) [50–52]. These U-loop footpoints get quickly
swept into the intergranular lanes and the horizontal field to the edges of the
granules. They do not show a helical structure. Transient horizontal fields also
appear briefly where new downflow lanes form [49]. The flux in these emerging
bipoles is small, 1016 − few × 1017 Mx, but their rate of appearance is large,
around a few ×10−10 km−2 s−1 , hence their dominant contribution to the emerging
flux of the Sun [47,51,52]. Most of these small loops are low lying, with only about
a quarter reaching up to chromospheric heights.
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5. Simulations
Three different classes of magneto-convection simulations have been carried out
by various groups: the first starts from an initial convective state whose volume
is filled with magnetic field. Often this is a vertical, uniform field, but some
have studied inclined fields and some have studied mixed polarity fields. The
second starts from a localized field—a coherent flux tube near the bottom of the
computational domain, a uniform horizontal field in the inflows at the bottom
boundary, or a coherent vertical flux concentration. The first two study how flux
emerges and active regions form, while the last is used to study sunspots. The
third starts from a tiny seed field in the initial convective state and is used to
study dynamo action, sometimes in a small localized setting, sometimes on the
global scale of the entire Sun. The solar dynamo is discussed by Miesch [44] and
sunspots by Rempel [53] in this series. Here, we review flux emergence and quiet
Sun magneto-convection simulations.
(a) Magnetic flux emergence
As mentioned already, there are two types of flux emergence simulations: those
that start with a coherent, twisted horizontal flux tube [9,54–57] and those
that start from a minimally structured, uniform, untwisted, horizontal field
advected by inflows into the domain at the bottom boundary [10,58]. These
two very different approaches show many robust, common features. The fields
first appear at the surface in localized regions as small bipoles with a smallscale, mixed pepper and salt polarity. The emergence region spreads in time.
Granules become larger and darker as the field first emerges (owing to the
suppression of vertical motions by the expanding horizontal section of the
bipoles) and elongate in the direction of the horizontal component of the field.
As the bipoles begin to emerge, horizontal and vertical fields have similar
strengths, but horizontal fields are more common (cover more area) than vertical
fields, except for the strongest fields. The bipole footpoints separate with time
and the field collects into unipolar regions accompanied by flux cancellation
where opposite polarities come in contact (figure 1). As the magnetic flux
rises, it expands (figure 2).√The horizontal expansion is much larger than the
vertical expansion, so B ∝ r [57]. Convective upflows and magnetic buoyancy
carry fields towards the surface and the fast turbulent downflows push them
down. The different scales of motions produce a hierarchy of magnetic loops,
with small loops riding piggy-back on larger loops in a serpentine structure
[9,10,54,59] (figure 3),
The main differences in these two approaches are that a coherent initial flux
tube leads to a more coherent symmetrical structure when it emerges through
the surface and field line connections below the surface are more localized. In the
minimally structured approach, organized magnetic field concentrations develop
spontaneously when sufficient flux is present, instead of being imposed as initial
and boundary conditions.
The effects of emerging flux on the chromosphere and corona have been
modelled by Schaffenberger et al. [38], Abbett [40], Hansteen et al. [60] and
Martínez-Sykora et al. [55,56,61].
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(a)
(b)
(c)
Y (Mm)
40
30
20
10
0
17.00 hours
Y (Mm)
40
30
20
10
0
20.00 hours
Y (Mm)
40
30
20
10
0
23.00 hours
Y (Mm)
40
30
20
10
0
10 20 30
26.00 hours
X (Mm)
40
0
10
20 30
X (Mm)
40
0
10
20 30
X (Mm)
40
Figure 1. (a) Time sequence of emergent continuum intensity, (b) vertical magnetic field at tcont =
0.01 and (c) horizontal magnetic field at the same optical depth. The range of intensities is 0.22–
1.35I . The range of magnetic field (both vertical and horizontal) is ±2 kG. Uniform, untwisted,
horizontal field was advected into the domain at the 20 Mm depth. Time is since horizontal field
started entering from the bottom.
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time = 0 min
18
z (Mm)
0
–0.5
14
–1.0
3
9
time = 4.4 min
T (1000 K)
–1.5
10
z (Mm)
0
–0.5
6
–1.0
–1.5
3
2
9
time = 8.7 min
3000
z (Mm)
0
–0.5
2550
–1.0
–1.5
3
9
time = 12.6 min
z (Mm)
0
–0.5
|B| (G)
2100
1650
1200
–1.0
750
–1.5
3
4
5
6
y (Mm)
7
8
9
300
Figure 2. Time sequence of vertical cross sections perpendicular to an initial coherent twisted flux
tube. The grey scale is temperature and the grey coding is magnetic field strength |B|. The grey
line is the t500 = 1 (adapted from Cheung et al. [9]).
(b) Magnetic effects on convection
Diverging convective upflows quickly sweep the magnetic field into the
intergranular lanes (granular time scale of minutes) and on a longer
(supergranule) time scale into the magnetic network on the supergranule
boundaries (a day), in the process concentrating the field into sheets and at
the vertices of the lanes into ‘tubes’ of magnetic flux [7,8,38]. However, the tubes
are leaky. The coherent structures at the surface connect in complex, incoherent
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48
38
29
19
9.6
0
48
38
29
19
9.6
5
10
15
20
Figure 3. Several magnetic field lines showing large-scale loops with smaller serpentine loops riding
piggy-back on them. Shading shows downflows.
ways below the surface (figure 4; [7,8,38]). Magnetic fields react back on the
convection, distorting granules and producing ‘abnormal granulation’, where the
granules become smaller and elongated [9,25,62].
Magneto-convection simulations have been very useful in understanding and
interpreting observations. Sánchez Almeida et al. [63], Khomenko et al. [64],
Shelyag et al. [65] and Bello González et al. [66] have used simulations to calibrate
the procedures for analysing and interpreting Stokes spectra in order to determine
the solar vector magnetic field. Fabbian et al. [67] have shown that magnetic
fields alter line equivalent widths by altering the temperature stratification and
by Zeeman broadening. These two effects act in opposite directions, but still leave
a net result and hence alter abundance determinations. Zhao et al. [68], Braun
et al. [69] and Kitiashvili et al. [70] have used convection and magneto-convection
simulation results to analyse local helioseismic inversion methods.
Strong magnetic flux concentrations inhibit cross-field motions so that
convective energy transport (which requires overturning) is reduced (but not
eliminated). Magnetic concentrations also have lower plasma density than their
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1000
2.5
500
2.0
0
1.5
–500
1.0
log |B| (G)
z (km)
3.0
0.5
–1000
0
1000
2000
3000
x (km)
4000
Figure 4. Magnetic flux concentration at the solar surface and magnetic field lines showing the
complex field line connections below the surface. The ‘flux tube’ is a local surface phenomenon
(adapted from Schaffenberger et al. [38]).
surroundings because of horizontal force balance. As a result, their opacity
is smaller and optical depth surfaces are depressed into the interior (Wilson
depression). Radiatively, they are holes in the surface. The temperature structure
in these concentrations is nearly in radiative equilibrium with radiative heating
from fluid flowing down along their sides and cooling from emission in vertical
rays [28] (figure 5). Where the flux concentration is narrow, heating from the side
walls raises the internal temperature at optical depth unity and the concentration
appears bright [24]. Small magnetic flux concentrations may appear especially
bright in the continuum [28,71–73]. This enhanced brightness extends for all
the sight lines that pass through the low-density, optically thinner, magnetic
concentration where photons escape from deeper, hotter layers. In the G-band,
this contrast is enhanced by the destruction of the CH radical in the lowdensity magnetic concentrations, reducing the opacity even more [71,74–76]
(figure 6). These small bright points have been used as a proxy for magnetic
concentrations. Where the flux concentration is broader, the side wall heating
does not penetrate through the volume and the reduced convective heating makes
the concentration appear darker, as pores or micropores. As a result, although
all especially bright points correspond to strong fields, not all strong fields
are bright [7].
Figure 7 shows a pore that formed spontaneously in a simulation where
horizontal magnetic field with 5 kG strength was advected through the bottom
of a 20 Mm deep domain. It has existed for over 3 h so far. It maintains its
coherent structure down to 10 Mm below the surface. Most magnetic field lines
in the pore connect to the end of a large-scale loop rising from the bottom of
the domain, although some connect to various other structures. The pore is
surrounded by downflows and is being shoved sideways by a large-scale horizontal
flow. Additional flux is being transported into the pore by horizontal flows
along the intergranular lanes. These flows feeding the pore extend to depths of
several megametres.
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net radiative heating/cooling with 1000 K temperature contours
(Mm)
(a)
–0.5
0
0.5
1.0
net radiative heating/cooling with 250 G B-field contours
(Mm)
(b)
0
–0.5
0
0.5
1.0
–5
–10
radiative heating/cooling m = ± 1
(Mm)
(c)
–20
radiative heating/cooling m = ± 0.5
(Mm)
(d)
–0.5
0
0.5
1.0
radiative heating/cooling m = ± 0.05
(e)
(Mm)
–15
–0.5
0
0.5
1.0
–0.5
0
0.5
1.0
0
1
2
3
4
5
6
(Mm)
7
8
9
10
11
12
Figure 5. Radiative heating and cooling (1010 erg g−1 s−1 ) in a vertical slice through a magnetic
flux concentration. (a,b) Net heating (yellow and red)/cooling (green and blue) with superimposed
contours of temperature (a) and magnetic field (b). (c–e) Net heating/cooling for vertical
(cos qray,vertical = m = 1), slanted (m = 0.5) and nearly horizontal rays (m = 0.05) (adapted from
Bercik [28]).
Although the strong magnetic fields in sunspots inhibit convection, they do
not shut it down entirely. Umbral convection is observed as umbral dots and
has been simulated by Schüssler & Vögler [77]. In such strong fields, convection
manifests itself as very narrow upflow plumes of hot plasma with adjacent narrow
cool downflows. As in normal convection, the upflows are braked rapidly near the
surface where the plasma loses buoyancy owing to radiative cooling. The plasma
piles up, the gas pressure increases and makes the plasma expand latterly, which
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(a)
R. F. Stein
log T
0.4
0.2
0
–0.2
–0.4
(b)
log r
0.4
0.2
0
–0.2
–0.4
(c)
|B|
0.4
0.2
0
–0.2
–0.4
(d)
log T
–2
–1
0
1
1
2
3
4
5
Figure 6. (a–c) Temperature, density and magnetic field strength in a vertical slice through
magnetic and non-magnetic regions with the average formation height for the G-band intensity
(black line along a vertical ray and white line along a ray at cos q = 0.6). The axes are distances
in Mm. Red indicates large and purple indicates small values. (d) Temperature as a function of
log t500 . The G-band has its mean formation height at log t500 = −0.38 (mean height 54 km above
t500 = 1). The large effect is the side wall heating. A smaller effect is the destruction of the CH
radical shown by the small dips in the height of formation in the bottom panel. (From Carlsson
et al. [71].)
reduces the magnetic field strength. As a result of the enhanced density, the
optical depth increases and photons can escape only from higher cooler layers
producing a dark lane through the bright umbral dot (figure 8).
(c) Dynamo action
Meneguzzi et al. [78] and Cattaneo [12] were the first to demonstrate, via
a magneto-convection simulation, that dynamo action will occur in turbulent
convection even in the absence of rotation. This calculation was for a closed,
Boussinesq system. Questions were raised whether local dynamo action is
possible in the highly stratified solar convection zone [79] because in a stratified
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–0.5
15
Mm
–6.4
10
13.2
5
20.0
0
8
16
24
Mm
32
40
48 0
Figure 7. Magnetic field strength and vectors in a vertical plane through a pore (to the left of
centre). Field strength is in kilogauss. This pore appeared spontaneously in a simulation where
5 kG horizontal magnetic field was advected into the bottom of a 20 Mm deep domain. It has
existed so far for a little over 3 h.
density fluctuation (107 g cm–3)
1.6
–6
–4
–2
0
2
4
6
4.6
4.7
4.8
horizontal position (Mm)
4.9
8
1.5
height (Mm)
1.4
1.3
1.2
1.1
1.0
4.5
5.0
Figure 8. Vertical slice through an umbral dot. Image is density fluctuation with respect to the
surroundings. The solid line is Rossland optical depth unity. The dotted lines are isotherms. The
arrows are velocity (longest is 2.7 km s−1 ) (adapted from Schüssler & Vögler [77]).
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Figure 9. Photospheric magnetic field lines showing many low-lying, horizontally directed magnetic
structures from a simulation from the upper convection zone to the corona [40].
atmosphere with much stronger downflows than upflows magnetic flux is pumped
down [80,81]. Vögler & Schüssler [13] and Pietarila Graham et al. [15] showed that
a local surface dynamo was indeed possible in a shallow, very-high-resolution,
magneto-convection simulation with no Poynting flux in or out of the domain,
but with a high magnetic diffusivity in the bottom boundary layer to mimic the
loss of magnetic flux to the deeper convection zone. Abbett [40] showed that such
small-scale dynamo action produces many low-lying loops with large amounts of
horizontal field overlying the granules (figure 9; see also [14]). Steiner [73] argues
that the preponderance of horizontal over vertical field is an inherent consequence
of the fact that granules are wider than a scale height. Consider an area of length
L and height h. The horizontal (FH ) and vertical (FV ) fluxes for a loop are the
same, so that FH = BH Lh = FV = BV L2 , where BH is the horizontal and BV is
the vertical field and L and h are the horizontal and vertical extents of the field.
Hence BH /BV ≈ L/h and low-lying loops connecting opposite sides of granules
must have larger average horizontal than vertical field. Pietarila Graham et al.
[15] showed that the primary dynamo process was the stretching of magnetic
field lines against the magnetic tension component of the Lorentz force, with a
much smaller contribution from the work against magnetic pressure, although
the latter is important for the cascade of larger scale into smaller scale fields.
Global dynamos have been simulated by Miesch [34], Brun et al. [82], Dobler
et al. [83], Browning et al. [84] and Brown et al. [85,86]. See also the review
by Miesch [44]. Note that the fact that both the rate of magnetic flux emergence
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3083
and the probability distribution of magnetic flux magnitudes are featureless power
laws from 1016 to 1023 Mx suggests that the solar dynamo has no preferred scale,
and that it acts throughout the convection zone with each scale of convective
motions generating new flux on that scale [45,87]. That is, all the surface magnetic
features are produced by a common process (which cannot be all dominated by
surface effects because the sunspots and active regions clearly are not).
6. Future directions
The increasing computational power that is continually becoming available will
allow us to improve on current results in several directions in the future.
We will be able to explore the effects of using different initial and boundary
conditions. We will be able to self-consistently include larger regions with a
wider range of temporal and spatial scales. We will be able to increase the
resolution and reduce the diffusivity of the simulations. Modelling of sub-grid
scales needs to be further explored. Use may be made of adaptive mesh refinement.
We will be able to improve the physics included in the simulations: non-local
thermodynamic equilibrium (LTE) hydrogen ionization and non-LTE radiation
are especially important in the chromosphere as we explore the coupling between
the convection zone and upper atmosphere. It may be important to include
non-ideal magnetohydrodynamic effects. The generalized Ohm’s Law includes
the additional terms: hall, electron pressure and electron inertia. These can
all become important in thin reconnection layers. These physical effects make
the chromosphere an especially complex region. We need to study the coupling
between the convection zone, chromosphere and corona better. We need to study
the coupling between the surface where radiation cools the plasma and the
deeper layers of the convection zone more realistically. We need to explore the
coupling between the radiative interior, tachocline and the convection zone more
realistically. We need to investigate the processes of magnetic flux disappearance.
The advent of the Advanced Technology Solar Telescope will improve the
resolution of what we can see as well as increase the amount of light collected,
which will increase our sensitivity to small-scale magnetic structures.
The author was supported by NSF grant AST 0605738 and NASA grants NNX07AH79G and
NNX08AH44G. R.F.S.’s calculations were performed on the Pleiades supercomputer of the NASA
Advanced Supercomputing Division.
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