Magnetoconvection
1: G-band image of an
active region taken at the
Swedish Solar Telescope,
La Palma. Plage regions
and small sunspots are
visible. Note the similarity
of the granular flows to
the simulations shown in
figure 7. (Courtesy T E
Berger/Royal Swedish
Academy of Sciences.)
Solar convection and
magnetic fields
Abstract
Thermal convection is the most
significant driver of time-dependent
patterns of motion within the Sun.
Observations of magnetic and convection
phenomena in the Sun, together with an
understanding of the basic physical
processes involved, provide a basis for
large numerical simulations. Models of
solar convection produced in this way
and using considerable computer power
can investigate such problems as the role
of stratification within the Sun, in two or
three dimensions. The models are now
sufficiently complex to produce results
that can be compared with observations,
in some cases. In future this technique
offers the chance to replicate surface solar
phenomena such as sunspots, while
uncovering the complex magnetoconvection beneath.
4.14
Mike Proctor looks at the interplay between convection and magnetism
in the Sun’s photosphere, using powerful numerical simulations.
H
owever featureless the Sun may appear,
careful observation at the correct
wavelengths in space and time reveals
a wide variety of time-dependent patterns of
motion. Some of these are due to acoustic waves
in the solar plasma, comparable to air vibrations in an organ pipe (discussed by Thompson
on pages 4.21–4.25 of this issue), but the most
significant structures are those due to thermal
convection, the mechanism by which the Sun
transports heat to the surface in its outer envelope. We can see this convection in the form of
the solar granulation and supergranulation,
which are similar in appearance to the disordered cellular flows seen in vigorous laboratory
convection. Associated with these convective
structures there are magnetic fields. These are
mostly generated at the base of the convection
zone, are affected by the local velocity field, and
have a significant dynamical effect on the convection. They exist on all scales – from the
largest sunspots down to small scale “filigree”
magnetic fields that nestle between and are buffeted about by the convection. Understanding
both of the convective envelope of the Sun and
of the magnetic field structures has advanced
rapidly in recent years with the advent of largescale numerical simulations. In this article I
describe the most recent ideas on the detailed
nature of the convection and the latest attempts
to model the magnetic field behaviour.
Deep-seated convection
The Sun, in common with many late-type stars,
has a convective envelope, with inner radius
rc ~ 0.7r . Energy, produced by nuclear reactions in the region r < rc , is carried outwards by
radiation initially. But for r > rc the temperature
August 2004 Vol 45
Magnetoconvection
Convection in the photosphere
Visible manifestations of convection in the photosphere occur on several different scales. Most
easily observed is the granulation, with the
instantaneous appearance of an irregular cellular pattern of bright (hotter) and dark (cooler)
regions, with a typical horizontal extent of
about 2 arcsec, about 1500 km. Doppler measurements show that the bright regions have
outwards radial velocities of about 1 km s–1.
Thus the cells take the form of plumes of hot
material rising in the cell centre, moving horiAugust 2004 Vol 45
Magnetic fields and convection
The convective structures in the photosphere
and at greater depths are threaded by magnetic
fields on a wide variety of scales. At the smallest scale are the bright points observed in highresolution images of the granulation. They are
associated with vertical magnetic fields of the
order of 1500 gauss, close to the magnetic pressure balance limit (discussed below). They are
thus nearly evacuated and the opacity is
reduced; their brightness is due to lateral heating. The bright points are located in the downdrafts in the intergranular lanes, and are
buffeted around as the granules evolve. On a
larger scale are the network fields, which lie at
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zontally across the cell wall and falling at the
cell periphery. To some extent this resembles the
velocity field of the hexagon cell that is seen at
the onset of convection in stratified fluids.
However, the granulation cells are transient,
with a lifetime of the order of 10 minutes, comparable with the time taken for fluid elements
to traverse the cell. The pattern changes either
due to a new cell pushing through from below,
by fragments of old granules growing, or else by
the development of a dark spot in the centre of
a granule, followed by its fragmentation; these
are called “exploding granules”. As discussed
below, the intensity of convection is significantly affected by magnetic fields, but it is possible to observe granular type flows even in the
middle of sunspot umbrae.
On a much larger scale there is the so-called
supergranulation. This forms a cellular network
similar to the granulation, but on much larger
horizontal scales of the order of 10 × 106 m, with
vertical extents of at least 8 × 106 m. Horizontal
velocities are comparable with those in the granulation. Cells have lifetimes of about one day.
The likely interpretation of these cells is that they
are convective in origin, but more deep-seated
than the granulation, where the scale heights are
larger. The downdrafts at the edges of the supergranular cells are closely associated with the distribution of magnetic field in the chromosphere.
Intermediate scales are less easy to distinguish,
but there is significant evidence for mesogranulation, recognized principally from systematic
motions of granules and by secondary measurements of chromospheric emissions. These
structures are of the order of five granule diameters across. These may be some sort of overshooting effect of deeper seated convection, or
they may arise dynamically as a part of the granulation process. There is as yet no theory to
account for the size of the mesoscale, but it may
be associated with the stabilizing effect of vortical flows which, like bathplug vortices, tend to
congregate at the edges of the mesogranules.
Even simulations of incompressible convection
show the phenomenon, so it is not due to stratification (see for example figure 7).
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gradient is “superadiabatic”, meaning that if a
fluid parcel is raised by a small amount, keeping its entropy fixed, its density and temperature will change in such as way that it will find
itself buoyant with respect to its surroundings,
and so will go on rising. The inner core is stable because it is fully ionized, and the heavier
elements, principally He, are found preferentially nearer the centre. However, any attempt
to construct a self-consistent radiative regime
throughout the atmosphere leads to difficulties
near the surface, where entropy has to be carried by convection. In this case the only consistent model has turbulent motions that transport
heat upwards in the outer third of the atmosphere. There is very little direct evidence for the
forms of convection well below the surface.
Some simulations support the idea that the vertical correlations in these flows are significant
over distances comparable with the local pressure scale height HP (the distance over which
the pressure changes by a factor e), and that
horizontal correlations scale similarly. Since HP
increases with depth, we expect larger scales of
motion in deeper regions, with smaller superficial scales at the surface. Numerical studies of
the whole height and pressure range of the solar
envelope are not possible at present, but simulations covering a more limited range of scale
heights have produced some evidence in support of these predictions (e.g Stein and
Nordlund 1998) Some of these simulations use
ordinary ionized gas, while others include the
effects of radiative cooling, which are highly
important near the surface. In both cases there
is a large asymmetry between upwards and
downwards moving flows, with the vertical
transport of heat being dominated by narrow
downflows of denser cooler fluid. The simulations suggest that as the pressure scale height
increases with depth, the scales of the convection may also increase, with plumes amalgamating to form the edges of larger structures,
though this has not been directly demonstrated. In the great bulk of the convective
region, radiative transport of heat is not important compared with convection and the temperature gradient is essentially adiabatic.
Conditions near the region at the bottom of the
convection zone are not yet well understood.
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2: (a) Image of a small active region observed
at disk centre.
(b) Dopplergram of the same active region.
Dark denotes downflows, bright denotes
upflows.
(c) Magnetogram of active region.
Note the correlation between magnetic fields
and bright points shown in (a).
The arrows refer to the following features:
(1) rudimentary penumbra;
(2) light bridge, or LB, with strong, narrow
downflows;
(3) examples of bright points;
(4) a region of opposite polarity between
umbral fragments;
(5) micropore.
(From Rimmele 2004.)
4.15
Magnetoconvection
Kinematics and dynamics of magnetic fields
Induction processes
The magnetic fields in the convection zone
are both affected by the flow, and interact
dynamically with it due to the Lorentz force.
On large length scales and short times the
kinematic evolution of the field in a given
flow field can be understood by thinking of a
perfect conductor. Then Faraday’s Law shows
that the amount of magnetic flux through any
material surface cannot change. This result
leads to Alfvén’s Theorem, that magnetic field
lines move “with the fluid”, i.e. just like
material lines. In consequence there can be
both field line stretching in extensional flows,
which acts to increase magnetic energy, and
advection of magnetic field lines into regions
of converging flow. Diffusive processes,
conveniently modelled by ohmic diffusion,
lead to slippage between the field lines and
the flow, and concentration of field due to
convergence will be halted when scales are
the edges of the supergranules. Their structure
parallels that of the small bright points – indeed,
detailed observation suggests that the bright
points evolve from small bipolar regions that
arise within supergranules and are fragmented
by the granulation before finally merging with
the network (the “magnetic carpet”). Elsewhere
in the photosphere there are so-called active
regions with more extensive field structures.
These have evolved from bipolar regions that
occur when loops of flux arrive at the surface,
as part of the solar cycle. Within these areas
there are bright regions called plages where
there is enhanced background field; there is evidence that the granulation in these regions has
a smaller horizontal scale than in the ordinary
network. Also in the active regions are the dark
pores and sunspots, which are darker (because
they are cooler) than the surrounding photosphere, discussed further by Tobias and Weiss
in this issue, pages 4.28–4.33. The normal granular flows are much distorted in the neighbourhood of spots and pores, and there is
typically an annular region surrounding the
spot, with predominantly outwards flow nearer
the spot. Net flux of the same sign as the spot
is carried away from the spot in this moat flow.
There is cellular convection within the central
umbra (common to spots and pores), but it has
quite different properties from that of normal
granulation. This is principally observed as
umbral dots, which have the form of small
bright features. Their horizontal scales are
smaller than that of the granulation (200–
500 km) but with similar or longer lifetimes.
These dots may be signatures of modified overturning convection due to umbral magnetic
4.16
sufficiently short that diffusion becomes comparable with advection. Relying on molecular
values of the diffusivity would lead to scales
of field on the order of 1 km, far below present visual resolution, but the effects of smallscale turbulence and dynamical processes as
described below will in general lead to larger
transverse scales, which can be observed.
The traditional picture of the effects of
cellular flows such as the granulation on
magnetic fields is that any field will be
advected into local regions of convergence,
but is stretched out along the diverging
directions that must also emanate from such
regions. These loops develop high curvature
and so diffusion becomes important locally,
leading to reconnection of field lines, where
the topology of the field lines changes and
closed loops are formed. These eventually
decay by diffusion. In a persistent cellular
flow pattern, fluid particles that converge on
fields, or due to wave motions, as discussed
below. In many sunspots there are narrow
bright bands known as light bridges. These represent gaps in the regular umbral structure, and
either are a remnant of the formation of the
spot from smaller magnetic structures, or else
are a precursor of the eventual fragmentation of
the spot. The magnetic field is much reduced in
the light bridges, and almost normal granular
motion can be seen in the larger ones. Sunspots
are distinguished from pores by having penumbrae, which are manifested as light and dark
radial stripes. Recent high-resolution observations show bright features (penumbral grains)
running along these penumbral structures,
inwards close to the umbra and outwards further away. The inner grains seem to merge
smoothly with the outer region of umbral dots.
All these phenomena demand an explanation
in terms of the interaction between magnetic
fields and convection. I discuss the relevant
basic physical processes in the box “Physical
processes in the Sun”. In the remainder of this
article, I shall show how the above structures
may be understood, and discuss analysis and
simulations that aim to provide more details of
how convection and fields interact.
Fundamentals of magnetoconvection
Much understanding of the effects of magnetic
fields on convection can be obtained by the
analysis of simplified models. Recent reviews
can be found in Proctor (2004) and Weiss
(2001, 2003). If we consider convection in an
incompressible fluid permeated by a vertical
magnetic field (this is of course a far cry from
solar conditions, but is easy to analyse), then for
individual fixed points have diverged from
other fixed points, and so the notion of an
isolated flux tube needs modification. This
can be seen in simulations of the induction
equation for a cellular velocity field with
hexagonal symmetry, in which at large times
a significant proportion of flux appears near
the top of the cell in a region of divergence.
Disconnected flux in a chaotic flow such as
the granulation behaves very differently.
Coherent recirculation cannot happen and so
the strongest fields remain in the downdrafts.
In addition, in chaotic flows the remaining
magnetic field is continually stretched and so
magnetic energy can rise rapidly at first. If
there is significant folding of the field lines
then diffusion will shut off this growth –
otherwise there is dynamo action, as
described below, and elsewhere in this issue
(Bushby and Mason, pages 4.7–4.13).
Another important kinematic effect of
motion is flux pumping. This occurs
when the topology of the upwellings and
weak fields convection onsets as steady overturning motion on a scale comparable with the
layer depth. As the strength of the imposed
magnetic field is increased, the favoured horizontal scale for convection is reduced, finally
having a magnetic field dependence B–1/3, where
B is a measure of the field strength. If B is large
enough, and the ratio ζ of magnetic to thermal
diffusivities is less than unity, the first manifestation of convection is in the form of oscillations that are essentially slow magnetoacoustic
waves driven by the buoyancy forces. These
give way to overturning motions in a complicated transition as the driving becomes more
vigorous. In a compressible atmosphere such as
a perfect gas, for which many simulations are
performed, the physics is considerably more
complicated, but these basic ideas remain valid.
When the convection reaches finite amplitude,
the simple plan forms seen at onset in numerical experiments give way to complicated aperiodic patterns. More importantly, if the
induction effects of convection are powerful
enough compared with diffusion (i.e. if the
magnetic Reynolds number Rm = UL/η is large,
where U, L are velocity and length scales and
η is the magnetic diffusivity) then the magnetic
fields can be advected to the edges of the cells,
where they lodge in downdrafts and so have less
dynamical influence on the motion. The strong
fields can become evacuated due to the magnetic pressure, and the resulting buoyancy interferes with the downflows. In addition the
retarding effects of the curvature force act to
slow the flow locally, so that for strong fields
the stretching and hence enhancement of the
magnetic field strength is inhibited.
August 2004 Vol 45
Magnetoconvection
downdrafts are distinct, as in the granulation
where the upwards flows are disconnected
while the downwards flow is primarily in
sheets. An initial imposed horizontal field will
be preferentially drawn downwards and
become concentrated near the base of the cell.
This transport mechanism is of great current
interest in connection with the structure of
sunspots and the transport of magnetic fields
at the base of the convection zone.
The Lorentz force
The dynamical effects of the magnetic field are
expressed in two ways. Magnetic energy dissipated by resistance turns into heat and so
appears in the energy equation. While this
term is essential for total energy conservation,
it is not usually important for driving convective flow. More significantly, there is a direct
body force on the fluid, due to electric currents flowing in the presence of a magnetic
field (the Lorentz force). This force can be
divided into two parts. The first acts like a
If Rm is large enough, and the field weak
enough, to be swept aside by the flow, its
dynamical effect on the bulk of the convection
will be reduced. Thus it is possible for finite
amplitude convection to occur in circumstances
in which the layer is stable to small disturbances.
When the ambient field increases to very
strong values, there is a generalized interference
with the efficacy of convection. The amplitude
drops and finally Rm becomes so small that flux
can no longer be contained in the cell interstices.
Then either all convection ceases, or else the
amplitude of the convection becomes nonuniform, with some regions of vigorous convection that acts to expel the flux and others,
now with a greater ambient field, which have
much lower amplitudes. This phenomenon is
known as flux separation.
Most investigations have centred on an
imposed vertical field, so that the convection
remains isotropic with respect to the horizontal
directions. In the penumbra of a sunspot, however, the field lines make a significant angle with
the vertical. This breaks the isotropy, and the
system will favour motions elongated along the
horizontal field component. In addition there
will typically be a breaking of reflection symmetry, so that the two directions parallel to the
tilt direction will not be equivalent. This will
promote travelling convective structures, though
the direction of travelling is not easy to predict
as it appears to depend sensitively on the precise
boundary conditions on the magnetic field.
Finally, there is the difficult subject of dynamo
action. It can be shown that in essentially any
sufficiently vigorous turbulent flow of a conducting fluid (so that Rm is larger than some
August 2004 Vol 45
pressure field Pm proportional to the square of
the field strength (the magnetic pressure). This
takes the place of gas pressure where the field
is strong, and so can lead to the evacuation of
regions of strong magnetic field. This phenomenon can be seen in numerical simulations
as described below. There is an upper bound
to the magnetic field strength in concentrated
field structures, given by equating Pm and Pe ,
the external pressure. If magnetic flux elements have a horizontal component, then they
are buoyant, and tend to rise through the
plasma. Secondary effects can then lead to
arching of the field lines. This mechanism is
thought to account in broad terms for the rise,
through the convection zone to the surface, of
large bipolar flux regions.
The remaining part of the Lorentz force (the
curvature force) acts on curved field lines, in
the direction of their normal, and is proportional to the product of the square of the field
and the line curvature: the field line resists
bending. Thus magnetic field lines can be
value of order 50–100 depending on geometry)
the inductive effects of the flow are sufficient to
overcome the effects of folding and diffusion, so
that any small seed field will grow to large
amplitude. For a discussion see Cattaneo and
Hughes 2001. This is the (small-scale) dynamo
mechanism, which is to be distinguished from
the large-scale dynamo, responsible for the solar
cycle, that is described elsewhere in this issue
(Bushby and Mason, pages 4.7 –4.13). The consequence of such dynamo action is that fields
grow to a strength at which they have a dynamic
influence on the motion, unless they are swept
away by larger scale advective processes. The
small-scale magnetic flux elements in the network fields are possible examples of this
process. There has been little study of the effect
of an ambient field on this process, though one
revealing calculation is described below.
Simulations of magnetoconvection
All the ideas above can be recognized in a variety of different numerical simulations.
“Idealized” or “fundamental” studies seek to
focus on the basic physics of the field-flow interaction in the context of a simplified model of
convection. Typically the problem solved is that
of a layer of perfect gas heated from below, with
an imposed uniform magnetic field. Some simulations use an incompressible fluid, which
makes for simpler and faster calculations. Early
work was confined to one horizontal dimension,
but even for such 2-D incompressible models it
is possible to show the effects of flux expulsion
in strong vertical fields, and flux separation
between vigorous almost field-free convection
and feeble convection in a strong field. Early
thought of as strings under tension, and in
strong fields motion tends to become elongated in the direction of the field. This
inhibits freely overturning flows, so the
efficiency of convection is reduced.
A further important effect of the curvature
force is that the plasma can support waves,
analogous to waves on a string, where the
restoring force is provided by the magnetic
curvature force. These waves can be both
transverse and longitudinal, and if the sound
speed exceeds the Alfvén speed (proportional
to the magnetic field strength) divide into
“fast” (mainly compressional like sound
waves), “slow” (mainly transverse) and
“Alfvén” (torsional) types. The slow modes
are the most important ones that can be
driven by convection. In fact, if the effective
magnetic diffusivity (inversely proportional to
electrical conductivity) is sufficiently small
compared with the thermal diffusivity then
convection in a magnetic field will occur first
as growing oscillations.
compressible calculations in two dimensions,
following the pioneering work of Nordlund, and
Hurlburt and Toomre, were able to show
clearly the effects of the magnetic pressure on
the evacuation of strong field regions and production of internal buoyancy within these
regions. There have also been simulations of
both incompressible and compressible models in
an axisymmetric configuration, as more accurately depicting isolated flux tubes. The magnetic field, initially uniform, matches to a
potential field at infinity. For narrow cylinders
there is a single annular cell, inwards at the top
of the layer, while for larger aspect ratios there
is a more complex pattern, with a second outer
cell. Almost all the original flux is in the central
flux tube (as in the incompressible case), while
convection occupies the rest of the cell and tries
to have a cellular aspect ratio of order unity.
Systematic 3-D studies of compressible convection in extended domains have only been
carried out in the past few years (Weiss et al.
2002, see also Stein and Nordlund 2003). A
careful study has been performed for a polytrope of index 1 with an 11-fold increase of density across the layer depth. The horizontal
boundaries of the domain are stress-free and
either at uniform temperature or else (at the
top), they obey Stefan’s Law, so as to model
(crudely) the actual radiative transfer at the top
of the photosphere. Early simulations concentrated on small aspect ratio boxes with an
imposed vertical field. Near onset, steady
hexagonal cells were found, in accordance with
theories of symmetric bifurcation, and as
expected the favoured horizontal length scale
was reduced as B was increased.
4.17
Magnetoconvection
3: Compressible
magnetoconvection in a
polytrope (see also Weiss
et al. 2002). The lefthand series of plots
show magnetic field
strength for Q
(proportional to B 2 where
B is the imposed vertical
field) having values
2500, 1600, 1400, 500.
The right-hand series
show the temperature
perturbation for the
same values. The
progression from almost
steady, small-scale
motion through the fluxseparated state to the
weak-field state can be
clearly seen. Note that
the large “cells”
apparent in the magnetic
field pictures are seen to
be aggregates of smaller
cells in the temperature
pictures.
The behaviour of the convection near onset is
governed by the value of ζ. In ionized plasma
this quantity depends on the local temperature
and pressure, and in sunspots it is less than
unity at the top of the photosphere, reflecting
the enhanced effective thermal conductivity due
to radiation. At depths between 2000 and
20 000 km it becomes larger than unity due to
the effects of ionization. In the polytropic model
the stratification implies that ζ increases with
depth. If we choose values such that, as in
sunspots, ζ passes through unity at some midlevel, then it is found that in the nonlinear
regime, convection can take the form of spatially modulated oscillations, with reversing
flows near the top of the layer but smaller scale
steady motion deeper down. This form of
motion presents a plausible explanation for
umbral dots. The irregularity of these may be
accounted for by the fact that the surface
4.18
regions of the umbra are stably stratified, so
that only the strongest plumes are visible.
More recent computations have concentrated
on large aspect-ratio boxes, to minimize the
constraints provided by the periodic boundary
conditions. Two-dimensional simulations in
very wide boxes show that long horizontal
scales of motion develop in the nonlinear
regime, with magnetic flux being pushed into
thick and almost stagnant flux sheets. In three
dimensions the maximum convenient aspect
ratio is 8 × 8, and this seems sufficient to remove
any artificialities due to the boundaries. Most
of the calculations have formed a sequence in
which the imposed field B, represented by the
dimensionless parameter Q ∝ B2, is varied for a
fixed superadiabatic hydrostatic temperature
profile. For very small Q there is aperiodic convection in cells several times the layer depth.
These cellular structures have good qualitative
resemblance to the granulation, though the calculations are much more diffusive than the
actual flows. Although these cells have reasonably uniform properties, there are indications of
heterogeneity in the temperature field, suggesting that the cells can be thought of as clusters
of plumes, and may be modelling mesogranular scales rather than granular ones. The field is
tightly compressed into linear structures in the
downdrafts at cell edges. These are buffeted
about by the motion, and slide from place to
place along the interstices being strongest (giving a local tube-like appearance) at local regions
of convergence, at the instantaneous cell corners. Except for the very weakest fields there is
significant evacuation of the magnetic structures
due to the magnetic pressure. As Q is increased,
the flux structures grow thicker and become
more sheet-like; they now have a significant
local effect on the convection. Finally the
Lorentz forces are so strong that the motion is
unable to expel the field effectively from the
large cells, and the convection has a much
smaller aspect ratio, with large, almost stagnant, strong-field regions. The ultimate fate of
the convection can take several different forms.
Either all convection ceases, or there is a narrow range where convection can occur only in
isolated regions, and these regions become further and further separated. These “anti-fluxtubes”, known as “convectons”, are very hard
to find numerically in 3-D geometries, though
they can be seen easily in 2-D simulations.
During this process of gradual suppression of
convection, there is an interesting intermediate
state called flux separation, characterized by the
splitting in two of the convective region. In one,
magnetic field is expelled from regions of vigorous convection. In the other, the field (now
locally enhanced by its expulsion from the other
regions) is large enough to ensure weak, small
aspect-ratio motion. This coexistence can occur
for a range of values of Q, and it is possible for
the same parameters to have either flux separation or relatively uniform small scales of
motion. This phenomenon has been carefully
surveyed, with emphasis on the quantitative features of the solutions. By estimating the size of
the region from which flux is expelled, it can be
seen that an upper limit to the extent of this
region is given by an effective Q that is just less
than that which would be sufficient to suppress
convection entirely. It is also shown that the
transition is hysteretic, in that there can be two
different quasi-stationary configurations (as
given by the proportion of large cells) for the
same value of Q. The observed transitions have
some of the character of a phase transition. The
flux separation transition has some affinities
with the phenomenon of convectons discussed
above. It may be possible to identify these states
with the dichotomy observed in umbral convection, where for large spots there are dark
August 2004 Vol 45
Magnetoconvection
nuclei that seem quite free of umbral dots, surrounded by regions in which dots appear and
convection is more vigorous.
A possible drawback of all these models is the
velocity boundary condition at the lower
boundary, which constrains flow to return
upwards and cannot model extensive downdrafts. In consequence it cannot represent the
change in the horizontal scale with depth as
downdrafts merge. In other simulations, such as
those of Nordlund and collaborators, the lower
boundary condition allows normal flux of fluid,
and this may give a better guide to the real situation. However, in spite of this the morphology of the surface regions seem to be
convincingly represented.
There have been attempts to model the radial
striations in sunspot penumbrae by looking at
3-D instabilities of axisymmetric or 2-D tilted
field structures. Calculations have been performed using the basic axisymmetric solution
described above but now permitting azimuthal
variation within a 30° wedge. It is found that
with sufficient flux in the cylinder the axisymmetric configuration breaks down into a fluted
structure. The non-axisymmetric parts are
insignificant near the centre for geometrical reasons, and the picture bears a suggestive resemblance to the transition between umbrae and
penumbrae. Similar results have been obtained
in an incompressible Cartesian model.
Oblique field models
When the imposed field is neither vertical nor
horizontal, the reflection symmetry associated
with vertical (or horizontal fields) is broken in
a stratified layer, and steady convection becomes
impossible. Convection then sets in as travelling
waves; for small tilt there is motion varying
along the tilt, while for larger tilts the Lorentz
force favours cells with axes parallel to the tilt,
as in the penumbra. The linear 2-D problem, in
which the field angle was specified at the horizontal boundaries, showed that the direction of
the travelling waves could change sign as the tilt
angle increased. Fully nonlinear simulations
have been performed using various boundary
conditions (Hurlburt et al. 1996, Hurlburt et al.
2000). It is found that there is typically a velocity at the top of the layer in the direction of the
field tilt. For weak imposed fields the travelling
waves have a phase velocity in the opposite
direction to this flow, while for strong fields the
phase velocity and the flow velocity are in the
same sense. These 2-D solutions are relevant for
sufficiently small tilt angles. Preliminary 3-D
computations in a stratified layer show that for
small tilt the initially 3-D pattern drifts without
much change of form while roll-like structures
appear at larger angles. This transition could
well be relevant to the rather abrupt change
from umbral-type to penumbral-type morphology as one moves towards the edge of a sunspot.
August 2004 Vol 45
4: Travelling wave
motion in
compressible
convection with an
oblique imposed
field. The field points
up and to the right,
but the phase
velocity of the wave
is to the left. (From
Hurlburt et al. 1996.)
0
z
450
1
540
t
630
0
x
2
5: Numerically generated intensity image from a 3-D simulation of solar granulation. Note the
similarity with the observations of figures 1 and 2. (Courtesy Å Nordlund [to appear in The Solar-B
Mission and the Forefront of Solar Physics – Proceedings of the Fifth Solar-B Science Meeting, Tokyo,
November 12–14, 2003 ASP conference series].)
The fact that the direction of the phase velocity
of the convective cells can depend on the tilt
angle may be the explanation of the differing
movements of the penumbral grains in the inner
and outer parts of the penumbra.
field in downwards moving regions, making a
smaller angle with the horizontal, can be seized
by the convection as it passes through the photosphere, counteracting the buoyancy (Thomas
et al. 2002, Tobias and Weiss, this issue 4.28).
Flux pumping
“Realistic” simulations
While the basic ideas of flux pumping (see
above) were derived from simple analytical
models (and some simplified kinematic calculations), recent fully nonlinear simulations have
thrown more light on the mechanism. A major
effect of stratification and compressibility, leading to an enhancement of the pumping process,
is the presence of strong downdrafts of relatively dense fluid; these are much more intense
than in 3-D structures in weakly stratified
flows. In the presence of a cellular flow, horizontal field elements are displaced into these
downdrafts, which can then act effectively to
push fields downwards, in spite of their natural
buoyancy. Flux pumping is thought to be
important in pushing magnetic fields below the
convective zone into the stably stratified overshoot region, where it is available to be amplified as part of the large-scale solar dynamo
(Tobias et al. 2001). Recently the same mechanism has been invoked to account for the persistence of the comb-like structure in sunspot
penumbrae. While fields in updrafts are free to
connect to other distant flux concentrations, the
An alternative to simulating idealized models is
to attempt to include all the relevant physics.
Recent work has been reviewed by Schüssler
and Knölker (2001). This approach, pioneered
by Nordlund and his collaborators (see for
example Stein and Nordlund 1998, Stein et al.
2003) aims to include effects of radiative transfer and ionization so as to produce an accurate
model of the solar atmosphere, with diagnostic
information in terms of line profiles etc that can
be directly compared with observation. An
important difference from the simulations
described above is that the boundaries are
“open”: fluid can leave and enter the domain.
This has the potential to give a more realistic
picture of the fluxes etc in the upper layers, with
the disadvantage that the results for deeper
seated fields may depend sensitively on the prescription for the in- and out-flows. Fortunately
the surface regions do not depend sensitively
on the lower boundary condition due to the
strong stratification, and very convincing quantitative comparisons have been made with
observations. More recent work has shown the
4.19
Magnetoconvection
Convection and small-scale dynamos
Finally we have to confront the question of what
happens to small-scale dynamo generation in the
presence of a strong ambient field. This issue is
seldom directly addressed by simulations
(though the results may depend implicitly upon
it). It is known from a large body of literature
that the generation of magnetic field, on scales
comparable with those scales of the velocity field
that are comparable with those of flux tubes and
sheets, is quite different from the large-scale,
mean-field dynamo process discussed elsewhere
in this issue (Bushby and Mason pages 4.7–
4.13). Rotation and broken mirror-symmetry
are not essential and indeed, if there is no ambient field fast dynamo action (on the turnover
timescale), may be expected provided that Rm is
sufficiently large (this can mean of the order of
1000) (see e.g. Cattaneo and Hughes 2001).
4.20
6: Two-dimensional magnetoconvection with
radiative transfer. An intense flux tube can be
seen, together with the depression of the
surface (shown by the black line) due to
evacuation and cooling. (Courtesy O Steiner;
see also Grossmann-Doerth et al. 1998.)
3500
6000
10 000
14000
600
400
z (km)
existence of mesogranular scales in a stratified
model, showing that earlier 3-D incompressible
studies give a useful qualitative picture of flows
near the surface. It is also possible to observe
the merging of downdrafts to form larger structures. While it is not yet possible to extend a
numerical model through the whole depth of
the convection zone, there is no doubt that in
due course – with improvements in computer
power – such a model will be possible, and will
prove an essential tool in the task of unravelling the effects of fields at great depths.
The alternatives to moderate-resolution 3-D
calculations are extremely well-resolved 2-D
ones; these have been carried out by Steiner and
collaborators using an adaptive code (see for
example Grossmann-Doerth et al. 1998). Smallscale flux structures can be accurately resolved
to show accurately how the concentration of
flux leads to evacuation of the the flux sheet,
consequent falling plasma and the enhancement
of concentration (“convective collapse”). For
strong fields there is a strong rebound from this
evacuation, leading possibly to spicule-like outflows above intense flux sheets.
Recent 3-D work has looked at the interaction
of magnetic fields and mesoscale granular
flows. In one study a vertical magnetic field is
allowed to be swept into a downdraft; the
results are similar to the 2-D case. More interestingly, other simulations follow the fate of a
horizontal flux distribution released at the base
of the convection zone. It is shown that individual fluid elements spend very little time in the
upper convection zone, and so the scope for
local flux stretching and amplification is limited. Since this simulation has a non-zero
imposed field, it cannot shed light on local
dynamo generation, but does pose the interesting question of whether surface dynamos can
exist in the presence of strong downwards
pumping, and the effect of imposed fields on the
dynamo process as a whole.
0
–400
–800
0
800
1600
2400
x (km)
1
2
1
2
3
4
3
4
7: The effect of increasing an imposed field in incompressible magnetoconvection. In the left panels,
bright regions indicate areas of high magnetic field strength, while the right panels show temperature
contrast. The scale of the motion decreases as the imposed magnetic field increases (top left to
bottom right). The first case is almost indistinguishable from the case of no imposed field, i.e. a
dynamo. It can also be seen that the field outlines a “mesogranular” network larger than the basic cell
size. (After Cattaneo et al. 2003.)
This value is certainly exceeded even for the
granules. We can then expect that small-scale
magnetic fields will spring up everywhere, until
they significantly affect the flow field. Only
quite recently has computing power been sufficient to reach sufficiently large values of Rm on
appropriate scales for incompressible convection. Consider a vigorous convective flow in a
layer. A small seed field, with no net flux in any
direction, can grow if Rm is sufficiently large. If
a vertical magnetic field B is imposed on this system, there is a smooth transition from a dynamo
state in which the magnetic energy finds its own
level to a state of magnetoconvection in which
the magnetic energy increases with B2 (Cattaneo
et al. 2003). The precise mechanism of this transition is the subject of active current research.
Conclusion
The study of magnetoconvection, after a long
period of steady progress, has reached an exciting stage. Detailed new observations are revealing ever more intricate details of photospheric
fields, while at the same time rapid advances in
computing power are making it possible to simulate many of these effects. We can now expect
observers and theorists to make rapid progress
due to this mutual stimulus. A comprehensive
unified theory of magnetic field evolution on all
photospheric scales now looks a real possibility within the next decade. ●
M R E Proctor, Dept of Applied Mathematics and
Theoretical Physics, University of Cambridge
Centre for Mathematical Sciences, Wilberforce
Road, Cambridge CB3 0WA, UK.
I am happy to acknowledge fruitful discussions
over many years, especially with Nigel Weiss, and
also with David Galloway, David Hughes, Fausto
Cattaneo and Steven Tobias.
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