Presidential Address
Turbulent magnet
Nigel Weiss recounts his
Presidential Address 2001, given to
the RAS A&G Ordinary Meeting
on 9 February 2001.
T
he resurgence of interest in cosmical
magnetic fields has been driven by a
combination of observational advances
with rapid progress in theory and interpretation. My chosen subject of magnetic fields in
stars like the Sun not only reflects my own
interests but also provides an opportunity to
show how numerical models, coupled with a
deeper understanding of nonlinear processes,
help us to explain the features that are being
seen. Here I shall focus on small-scale local
structures near the surface of the Sun, leaving
global magnetic fields – activity cycles,
dynamos and so forth – for next year. Fortunately, these topics can largely be decoupled. I
shall start by recalling the key properties of our
local star, the Sun, and go on to describe some
recent observations from space, with Yohkoh,
SOHO and TRACE, and from the ground, at
the Swedish Vacuum Solar Telescope (SVST)
on La Palma. Then I shall discuss theoretical
results, starting with simple kinematic models,
proceeding to fully nonlinear dynamic models
of stratified magnetoconvection and continuing to small-scale turbulent dynamos. Finally, I
shall confront these model calculations with
the actual observations.
The Sun as our star
The Sun is the only star on which fine-scale
structures can actually be resolved. The image
Early sunspot observations
Modern observational astronomy began with
a technological advance: the invention of the
telescope in about 1608. The news spread
rapidly and reached Galileo at Padua in June
1609. He used his improved instrument to
observe the Moon and the Milky Way and
he discovered the satellites of Jupiter. There
is no reason to doubt his statement in May
1612 that he had been observing sunspots for
18 months, i.e. since November 1610, or his
later claim that he had noticed them earlier
that year, before moving from Padua to Florence. Fortunately solar activity was near a
maximum and there were plenty of spots to
3.10
ecent high-resolution observations, from the ground and from space, have
revealed the fine structure of magnetic features at the surface of the Sun. At
the same time, advances in computing power have at last made it possible to
develop models of turbulent magnetoconvection that can be related to these
observations. The key features of flux emergence and annihilation, as observed by
the MDI experiment on SOHO, are reproduced in kinematic calculations, while
three-dimensional numerical experiments reveal the dynamical processes that are
involved. The pattern of convection depends on the strength of the magnetic field:
as the mean field decreases, slender rising plumes give way to a regime where
magnetic flux is separated from the motion and then to one where locally intense
magnetic fields nestle between broad and vigorously convecting plumes. Moreover,
turbulent convection is itself able to act as a small-scale dynamo, generating
disordered fields near the solar surface.
R
from TRACE (the Transition Region and
Coronal Explorer) in figure 1 shows slender
loops, only a few hundred km across, that follow field lines in the solar atmosphere and are
linked to magnetic flux emerging through the
solar photosphere. Such features make the Sun
a unique laboratory for plasma physics, in
parameter regimes that cannot be matched in
the laboratory. Observations keep throwing
up phenomena that would never have been
anticipated had they not been seen. Sunspots
themselves, which are dark because they are
the sites of strong magnetic fields, remain a
classic example. The first telescopic observations were made almost 400 years ago and led
immediately to unseemly arguments about
priority (see “Early sunspot observations”
below). Between then and now there have
been enormous increases in resolution, as can
be seen by comparing the images in figures 2
and 3, yet we still do not properly understand
the origin and stability of isolated flux tubes,
or the nature and cause of a sunspot’s filamentary penumbra (Thomas and Weiss 1992).
As Eugene Parker puts it, “Nature is cleverer
than we are” or, in the words of Blaise Pascal,
who was born in Galileo’s lifetime: “Imagination tires sooner of conceiving than Nature
does of providing”, (“L’imagination se lassera
plutôt de concevoir que la nature de fournir”).
At this point we should recall the structure
of the Sun (Stix 1989). Energy is generated by
thermonuclear processes in its central core,
with a radius of 0.2 R , where the solar
radius R = 700 000 km. This energy is transported outwards by photons through the
radiative zone, to a radius of 0.7 R , where
radiative transport gives way to convection.
be seen, so other independent discoveries followed rapidly. The first recorded observations are due to Thomas Harriot in England
– but his manuscripts, with drawings made
in December 1610, were only rediscovered at
Alnwick Castle in 1786. So the credit for the
first published account (printed in the
autumn of 1611) goes to Johann Fabricius
(or Goudsmid) from East Friesland, who
observed several dark spots in March 1611
and noted that they appeared to rotate with
the Sun. Christoph Scheiner, at the Jesuit
university in Ingolstadt, had noticed a similar
spot just three days earlier, and observed
continuously from October to December
1611. His account, published anonymously
in January 1612, drew a response from
Galileo, who carried out a systematic series
of observations from June to August 1612
and published his results the following year,
arguing conclusively that the spots were on
the Sun and that they rotated with it. From
then on there was a fierce dispute between
Galileo and Scheiner (neither of whom ever
mentioned Fabricius) as to who had first discovered sunspots. Their arguments were
aggravated by disagreement over Aristotelian
cosmology and the Copernican world system. Scheiner later moved to Rome, where
he continued his observations and eventually
described them in a sumptuous volume,
Rosa Ursina sive Sol, which appeared in
June 2001 Vol 42
Presidential Address
ic fields in the Sun
1: Fine loops above an active region on the
Sun are revealed in this image from TRACE,
taken in the extreme ultraviolet. These loops,
only a few hundred kilometres across, follow
magnetic field lines that emerge from the
photosphere and extend upwards into the
corona before returning to the solar surface.
What we see at the solar photosphere is the
upper boundary of this convection zone. We
have an enormous wealth of observational
results relating to the Sun’s atmosphere (which
contains only 10–10 of the solar mass). By contrast, there is still relatively little direct information about the solar interior, though we can
exploit helioseismology to probe deeper into
the convection zone. Yet the structure of the
atmosphere is determined by magnetic fields
that are generated within the convection zone
and emerge from it through the photosphere.
Turbulent convection near the photosphere
gives rise to various cellular patterns (Schrijver and Zwaan 2000). The remarkable image
in figure 2, obtained at the SVST, shows not
only sunspots but also the photospheric granulation, caused by small-scale convective
plumes, about 1000 km across, rising and
1630. His dispute with Galileo probably
affected the subsequent trial.
It took 300 years before a scientific
advance revealed the origin of sunspots. The
development of spectroscopy and the discovery of the Zeeman effect in 1894 made it
possible for George Ellery Hale to measure
magnetic fields of several thousand gauss in
sunspots. Thus began the study of solar
magnetism, which was eventually extended
to other stars by Harold and Horace Babcock in the 1950s. The explanation of the
darkness of sunspots as being caused by
magnetic suppression of convection was put
forward by Ludwig Biermann and Thomas
Cowling in the 1940s.
June 2001 Vol 42
2: This remarkable image of the solar photosphere shows two sunspots and several dark pores. The
solar granulation, caused by cellular convection just below the photosphere, is clearly visible and the
tiny, bright points are sites of intense magnetic fields. The field shown is about 120 000 km2 and the
smallest features are a few hundred km across. The image, in the CH G-band at 4305 Å, was obtained
on the Swedish Vacuum Solar Telescope at La Palma by T Berger and G Scharmer on 12 May 1998 and
processed at the Lockheed-Martin Advanced Technology Center in Palo Alto. (Courtesy of T Berger.)
3: Early sunspot observations.
This figure, from Scheiner’s
Rosa Ursina, completed in 1630,
shows several sunspots on
successive days, as they are
carried round by the Sun’s
rotation. The distinction
between the umbra and
penumbra of a sunspot had
already been recognized.
(Courtesy of the RAS.)
3.11
Presidential Address
4: This image of the
Sun, from the
Michelson Doppler
Imager on SOHO, shows
motion towards and
away from the Earth as
blue- and red-shifted
patches, after the
axisymmetric
differential rotation has
been removed. The
patches are produced
by horizontal motion at
the solar surface,
corresponding to
supergranular cells
with characteristic
diameters of about
30 000 km. (Courtesy of
D Hathaway.)
(a)
(b)
5: X-ray emission from the hot corona of the Sun maps magnetic fields at the solar surface. (a) A soft
X-ray image of the Sun, obtained with the Normal Incidence X-ray Telescope on a rocket flight on 11
September 1989. (b) The corresponding magnetogram, from Kitt Peak National Observatory, shows
oppositely directed fields as blue and red against a lighter neutral background. (Courtesy of L Golub.)
3.12
impinging on a stable layer, so that bright, hot
features are enclosed by a network of cooler,
sinking gas (Berger et al. 1998). The pattern is
constantly changing and individual cells have
lifetimes of about 10 minutes.
Superimposed on this is the supergranulation, a much larger pattern of cells with diameters between 20 000 and 30 000 km (Hagenaar et al. 1997). This is demonstrated by the
Doppler image in figure 4, from the MDI
instrument on SOHO (the Solar and Heliospheric Orbiter), which shows gas spreading
outwards from cell centres and moving
towards or away from the observer (Hathaway et al. 2000). Associated with this motion
is a magnetic network surrounding the supergranules and made visible by Ca+ emission.
There is also evidence for an intermediate cellular scale, the mesogranulation (Shine et al.
2000), which is probably produced by collective interactions between the granular convection cells (Cattaneo et al. 2001a).
Magnetic fields on the Sun
The outermost part of the solar atmosphere,
the corona, is heated magnetically to temperatures of several million degrees and therefore
radiates in the ultraviolet and X-ray regions of
the spectrum. Its rich and complicated structure is determined by the fields that emerge
from the photosphere, as can be seen by comparing the soft X-ray image in figure 5 with
the accompanying magnetogram.
The magnetogram shows features on several
different scales. First, there are the prominent
active regions within which sunspots are
located, which correspond to bright loops and
flaring activity in the corona. Then there is the
irregular magnetic network that outlines the
supergranules and, finally, there are tiny features distributed across the face of the Sun.
The close-up image of magnetic fields in figure
6, obtained from the MDI instrument on
SOHO, shows many such features, scattered
like pepper and salt across the field of view.
The evolution of these fields can be followed
from time-sequences of the MDI results
(Schrijver et al. 1997, 1998). Small-scale magnetic flux emerges within supergranules as
bipolar features (ephemeral active regions),
whose footpoints spread apart, while the
fields are fragmented to yet finer scales by the
granulation. These small flux elements
migrate towards the boundaries of the supergranules and merge into the magnetic network, where fields with opposite polarities
collide, reconnect and cancel out. The total
rate at which unsigned magnetic flux emerges
on these scales is significantly greater than
that associated with solar activity and
sunspots. The ratio of the total unsigned flux
in the network to the rate at which it emerges
implies that individual flux elements have a
June 2001 Vol 42
Presidential Address
lifetime of less than a day before they are
annihilated (Hagenaar 2001).
The interaction between magnetic flux elements and granular convection can be followed in images obtained with the SVST on
La Palma and processed digitally at the Lockheed-Martin Solar and Astrophysics Laboratory in Palo Alto. When the solar p-modes –
which are so important for helioseismology –
are filtered out, using techniques originally
developed for observations from space, the
individual granules are plainly visible, as can
be seen in figures 3 and 7. These images were
both taken in the CH G-band at 4305 Å and
the tiny, bright features are the sites of intense
magnetic fields, with strengths of up to
1500 gauss, so that the magnetic pressure in a
flux tube would be close to the external gas
pressure, and the flux tube should therefore be
almost completely evacuated. It is apparent
that these flux elements nestle between the
individual granules and then move along the
dark intergranular lanes as the cellular pattern
evolves (Berger et al. 1995, 1998; Berger and
Title 1996).
6: A composite image of intense small-scale magnetic fields that form a “magnetic carpet” at the solar
photosphere. Bipolar regions constantly emerge and split into tiny flux elements that are swept into
the magnetic network that partially encloses supergranules. Black and white pixels denote oppositely
directed fields at high resolution. Extreme ultraviolet emission is represented by the green-shaded
background. The field lines correspond to a potential field in the solar atmosphere. (N E Hurlburt.)
7: Close-up of bright points
between granules in the photosphere. They correspond to
intense magnetic fields in the
intergranular lanes, where
cooler gas is sinking (but not
all strong fields are bright).
This G-band image of a region
about 60 000 km across was
obtained by Göran Scharmer at
the SVST on 10 July 1997. (G
Scharmer and D Hathaway.)
Kinematic modelling of emerging flux
The simplest numerical models are twodimensional and purely kinematic, without
any dynamical effects. In a perfectly conducting plasma the magnetic field B satisfies the
induction equation
∂B
= curl u × B
(1)
∂t
where u is the fluid velocity, and field lines are
frozen into the plasma. So if B is vertical at the
surface, where the vertical velocity vanishes,
∂B
z = –uH . ∇Bz – Bz∇ . uH
(2)
∂t
where uH is the horizontal velocity and the
z-axis points upwards. Thus individual flux
elements will move with the flow like passive
test particles (or “corks”) and accumulate in
regions where the horizontal velocity converges.
To model the emergence and fragmentation
of ephemeral active regions we need to introduce two families of corks (labelled + and –)
to represent oppositely directed fields and
then to arrange for these corks to annihilate
each other when they collide. Then we can
represent the supergranules by axisymmetric
sources, thereby generating a Voronoi tessellation of the plane with a network that encloses
every source, and allow each supergranule to
have a finite lifetime, after which a new source
appears (Simon et al. 1995).
The results in figure 8 show that such a simple calculation already reproduces the essential features of a statistically steady network,
with a balance between flux emergence and
cancellation, and a lifetime for flux elements
that is compatible with that derived from
June 2001 Vol 42
30
40
×103 km
50
60
256
224
192
160
128
96
64
32
0
8: 2-D kinematic modelling of bipolar flux emergence and the formation of a statistically steady magnetic network. Circles mark the positions of supergranules and boxes and crosses represent small flux
elements with oppositely directed magnetic fields. Flux emerges near the centres of supergranules
and is swept to their boundaries where it merges with the network. In the network, flux tubes with
oppositely directed fields collide and reconnect, to reach a steady state. (Simon et al. 2001.)
3.13
Presidential Address
9: Dynamical modelling of three-dimensional nonlinear magnetoconvection in a wide box, with aspect ratio λ = 8. Narrow plumes in a fully compressible and
strongly stratified layer undergo spatially modulated oscillations in the strong field regime (R = 105, Q = 1600). The left-hand panel shows |B| 2 at the top and
(reflected) bottom of the layer, with the colour table running from violet to red as the field strength increases. Magnetic flux is swept aside by rising plumes at
the upper boundary and drawn into them at the base. The right-hand panel shows the vertical temperature gradient, proportional to the intensity at the upper
boundary. Temperature fluctuations are shown on the sidewalls and the arrows represent the tangential components of the velocity on the bounding surfaces.
This pattern corresponds to behaviour in the dark nuclei of sunspot umbrae. (See Weiss et al. 2001.)
observations (Simon et al. 2001).
Despite its success, the limitations of this
kinematic approach are, however, obvious.
The real magnetic field is a solenoidal vector
and three-dimensional effects cannot be
ignored. Moreover, passive flux concentration
produces fields that are strong enough to
affect the motion so that the Lorentz force
(j × B) can no longer be neglected. Thus it
becomes necessary to carry out fully dynamical, three-dimensional computations – and
they are inevitably much more complicated.
Modelling 3-D compressible
magnetoconvection
There are two approaches to modelling astrophysical convection. One is to include as
many properties of the real atmosphere as
possible – chemical composition, ionization,
radiative transfer and so forth – with the aim
of comparing the simulation directly with
observable features. This has been achieved
successfully for photospheric granulation in
the absence of magnetic fields (Stein and
Nordlund 1998). The alternative method,
which I shall pursue, is to investigate complex
nonlinear behaviour in an idealized configuration. This approach yields qualitative results
that can be used to interpret behaviour that is
seen on the Sun but cannot supply a quantitative comparison with observations.
The induction equation has to be augmented
by adding an equation of motion,
3.14
Du
ρ = –∇p + ρg + j × B + Fvisc
(3)
Dt
where Fvisc is the viscous force and other symbols have their usual meanings, together with
the continuity equation
∂ρ
= –∇ . (ρu)
(4)
∂t
the perfect gas equation of state, p = RTρ, and
an equation for the entropy S:
∂S
(5)
ρT = –∇ . (k∇T) + Qdiss
∂t
where radiative transfer is treated in the diffusive limit and the last term represents the dissipative sources. Moreover, the induction
equation has to be extended to include the
effects of ohmic dissipation, with a magnetic
diffusivity η, so that
∂B
= curl u × B + η∇2B
(6)
∂t
The magnetic Reynolds number Rm = UL/η,
where L and U are typical measures of length
and velocity respectively, measures the ratio
of advection to diffusion. In a star, Rm >> 1:
Rm ≈ 106 in the photosphere and Rm ≈ 109 in
the deep convective zone, but the currently
attainable limit in numerical experiments lies
in the range Rm ≈ 100–1000.
We have systematically explored the effect of
imposing an initially uniform vertical magnetic magnetic field B0 on convection in a strongly stratified layer of fluid contained in a box
with square cross-section subject to standard
illustrative boundary conditions (Weiss et al.
1996; Rucklidge et al. 2000).
To obtain meaningful results it is essential
that the aspect ratio λ of the box (its normalized width) should be sufficiently large: the
runs presented here have λ = 4 or 8 and consequently require massive computing power.
Fortunately, with support from PPARC, we
have had access to the Hitachi SR-2201 at the
University of Cambridge High Performance
Computing Centre. The runs to be described
here have required a resolution of up to
256 × 256 × 100 mesh points.
The governing equations are solved within
the domain, with idealized boundary conditions at the top and bottom of the box and
periodic lateral boundary conditions. The
thermal forcing is measured by a Rayleigh
number R which is proportional to the mean
superadiabatic temperature gradient. For the
runs illustrated here we fix R = 105, a value
high enough to generate turbulent convection
in the absence of any magnetic field, and vary
the strength of the imposed field B0 , which is
measured by a Chandrasekhar number
Q ∝ |B0|2 (Weiss et al. 2001). When Q is very
large, convection is completely suppressed,
but with our choice of parameters convection
sets in at Q ≈ 4200. In the magnetically dominated regime, with 4000 > Q > 2000, we
obtain steadily (but feebly) convecting solutions with very narrow plumes disposed in a
hexagonal pattern. As Q is further decreased
the motion gradually grows more vigorous,
June 2001 Vol 42
Presidential Address
10: Flux separation in nonlinear magnetoconvection, for exactly the same parameters as the solution in figure 9 but started with different initial conditions. The
cluster of broad and vigorous plumes is able locally to expel nearly all the magnetic flux, so that convection is unimpeded. In the rest of the region, where the
average field strength is increased, only small-scale convection can occur. The whole pattern, which evolves chaotically, is related to the appearance of bright
umbral dots in sunspots (cf. Weiss et al. 2001).
11: Vigorous convection with flux separation (Q = 1000). The box is split into two regions, one with vigorous convection and the other with strong fields and
weaker, small-scale convective plumes. The pattern keeps changing but the separation persists indefinitely (cf. Weiss et al. 2001).
until the plumes eventually become timedependent, undergoing irregular spatially
modulated oscillations, in which adjacent
plumes wax and wane aperiodically in
strength. This pattern is illustrated in figure 9.
The right-hand panel shows a snapshot of
temperature fluctuations at the upper boundJune 2001 Vol 42
ary, with an array of slender plumes that vary
in strength, as can be seen from the temperature fluctuations on the sidewalls. The lefthand panel shows the field strength, measured
by |B|2, at the top and bottom of the layer. As
a plume impinges upon the upper boundary
there is a radial flow that carries field lines
outwards, just as corks were transported in
the kinematic calculation. Thus magnetic flux
is compressed into a network that encloses the
rising plumes, and the field strength is anticorrelated with the temperature. (Conversely,
at the base of the layer, the field is concentrated into the centres of the plumes – but that is
3.15
Presidential Address
12: Formation of strong transient magnetic features in intergranular lanes for Q = 200 in a box with λ = 4. Magnetic flux moves rapidly along the intergranular
lanes and the most intense magnetic fields appear at corners. This pattern is similar to that suggested by observations of intergranular magnetic fields, as
shown in figure 7 (cf. Weiss et al. 2001).
a consequence of introducing a rigid boundary, which is not present in a star.)
The pattern in figure 9 corresponds to an
attractor that is stable but is not unique. With
different initial conditions but the same parameter values (Q = 1600) an entirely different
pattern appears. Figure 10 illustrates this new
phenomenon of flux separation. There is a
cluster of broad and vigorous convective
plumes from which most of the magnetic flux
has been expelled, thereby strengthening magnetic fields in the rest of the region. Outside
this cluster the field is therefore strong enough
to limit motion into narrow plumes that are
relatively ineffectual. The whole pattern is
chaotic and the strong plumes migrate slowly
through the region. This pattern, a form of
phase separation with fronts that separate vigorous convection from strong fields, is characteristic of a strong field regime that exists in
our model for 2000 ≥ Q ≥ 1000. For Q ≤ 1400
the narrow plumes become unstable and give
way to larger scale convection. Figure 11
shows flux-separated behaviour for Q = 1000
(Tao et al. 1998). There is a constantly changing pattern, with several clusters of broad and
vigorous plumes surrounded by magnetically
dominated regions of small-scale weak convection. Note also that, although the active
rising plumes are enclosed by cooler sinking
fluid, the downward motion is rapidly focused
into narrow plumes that plunge towards the
lower boundary. This is a characteristic feature of convection in a strongly stratified layer
3.16
and has appeared in many turbulent simulations (cf. Spruit et al. 1990).
When the imposed field strength is decreased
yet further, the motion squeezes magnetic flux
into lanes where the field is locally too strong
to allow any small-scale motion to occur. Figure 12 shows a close-up (in a box with λ = 4)
of a solution for Q = 200. Now there are
strong magnetic fields in lanes enclosing clusters of active plumes. The magnetic flux
migrates rapidly along these lanes, accumulating preferentially at the corners. Note, however, that there are no individual isolated flux
tubes. Rather, magnetic flux at the surface
behaves like a fluid that moves in spurts along
the network of lanes. There are ephemeral
concentrations, giving rise to fields that are
locally intense. At such locations the magnetic
pressure rises and the gas pressure and density consequently drop to values that correspond to almost complete evacuation. This
results in numerical difficulties that set a current limit to this investigation.
Turbulent dynamos
What happens as the mean imposed field B0 is
yet further reduced? In particular, what do we
expect to find in the limit when there is no net
imposed magnetic flux (B0 = 0)? It is known
that even simple velocity fields possessing the
property of Lagrangian chaos may act as fast
dynamos in the limit as the magnetic Reynolds
number Rm → ∞ (Childress and Gilbert 1995).
In such flows the stretching of field lines, as
neighbouring fluid elements move apart at an
exponentially increasing rate, predominates
over dissipation at the ever-decreasing scale of
the intermittent fields that are produced. Figure 13 illustrates the elegant and complex
structure that appears in such a model calculation (Brummell et al. 1995). So we might
expect that highly turbulent convection is able
to maintain a similarly disordered field.
This process has indeed been demonstrated
for convection in a Boussinesq (i.e. incompressible) fluid layer. (See the article by Cattaneo and Hughes in this issue, A&G 42 pages
3.18–3.22.) In this model calculation, in a box
with λ = 10 and a very high Rayleigh number
(R = 5×105), producing a magnetic Reynolds
number Rm ≈ 1000, there is a gradual transition from magnetoconvection to dynamo
action as B0 is progressively reduced (Cattaneo et al. 2001b). In the limiting case, when
B0 = 0, a small seed field grows exponentially
until the Lorentz force is able to saturate its
growth. Thereafter the small-scale turbulent
magnetic field, with zero net flux, is maintained indefinitely (Cattaneo 1999). Figure 7
of Cattaneo and Hughes (on page 3.21 of this
issue) shows the structure of this intermittent
field, which is strongest locally in isolated
regions where the fluid flow converges; note
that there is an up–down symmetry in this
problem that was not present in the stratified
layers of figures 9 to 12. These ambitious
calculations (which required up to
1024 × 1024 × 96 gridpoints) confirm that sufJune 2001 Vol 42
Presidential Address
13: Intermittent
field structures in
fast dynamos.
Results for a linear
(kinematic) dynamo
model with a largescale, chaotic
velocity field and a
magnetic Reynolds
number Rm ≈ 105.
The convoluted
structure of the
magnetic field, with
rapidly alternating
directions
(indicated by black
and white against a
red background)
illustrates the
complex structure
of the intermittent
field. (From
Brummell et al.
1995, courtesy of F
Cattaneo.)
ficiently vigorous convection is indeed able to
generate a turbulent magnetic field that can be
maintained forever, as opposed to amplifying
pre-existing flux that must eventually decay.
Relating theory to observations
The next challenge is to relate these theoretical models to observational results. As we
have seen, the action lies in small-scale structures, and in each case these fine structures
have barely been resolved. Moreover, we have
to extrapolate from numerical experiments
with Rm ≤ 1000 to the solar atmosphere, where
Rm ≈ 106. All the same, we can use results from
model problems to interpret features that can
be observed.
We have already seen that the formation of
the magnetic network that surrounds the
supergranules can be modelled by a straightforward two-dimensional kinematic calculation. Given a total flux of 3×1022 Mx in the
network and the observed rate of flux emergence (at 2×1022 Mx hr–1), the emerging flux is
replaced after a residence time of only
12–24 hr. This flux appears over the entire
surface of the Sun, at sunspot minimum as
well as sunspot maximum. Moreover, the
ephemeral regions are essentially oriented randomly, with only a weak bias towards the orientation of fields in active regions.
That suggests that the emerging flux is not
related to that in active regions, which must
be formed near the tachocline at the base of
the convection zone. Rather, it must owe its
origin to small-scale dynamo action near the
photosphere. Since the ephemeral regions
emerge with fluxes of around 3×1018 Mx and
are then split by granular convection into flux
June 2001 Vol 42
elements with much smaller fluxes, of order
1017 Mx (Schrijver et al. 1997, 1998), this
dynamo must operate at some level below that
where the granules themselves are formed.
That in turn suggests that the photospheric
granulation may itself also be capable of acting as a turbulent dynamo, generating ultrasmall-scale fields. Is that perhaps the origin of
the weak intranetwork fields whose existence
has frequently been reported? Furthermore, if
there are really three different scales of
dynamo action, how do these scales interact,
and do the larger scales feed the smaller scale
activity? The persistence of plage regions,
which are associated with distorted granulation and the presence of a significant average
field, certainly suggests that their flux is
anchored at some deeper level in the convection zone.
There is a close correspondence between the
observed behaviour of intergranular magnetic
fields, as revealed by bright points in the continuum or in G-band images like those in figures 3 and 7, and the strong fields formed in
the numerical experiment that was illustrated
in figure 12. In each case there are flux concentrations that are locally intense but constantly changing. The bright points only indicate positions where some threshold is
exceeded, but in the computations we can follow the continuous distribution of the field.
Thus we can confirm that magnetic flux does
indeed behave like a “magnetic fluid”.
Finally, we come back to sunspots. In the
dark central umbra there is a strong magnetic
field that is only slightly inclined to the vertical. Nevertheless, the reduced flux of energy
has to be carried by convection to just below
the visible surface. In such a magnetically
dominated regime we should expect to find
small-scale time-dependent convection, with
slender plumes like those in figure 9, that are
concealed beneath a radiative blanket.
Although there are indeed dark nuclei with no
visible structure within the umbrae of
sunspots, there are also bright features known
as umbral dots. They seem to be examples of
flux separation, which allows isolated bright
plumes (like the clump in figure 10) to form
and then to penetrate the radiative blanket.
In conclusion, I should emphasize that this
lecture has been restricted to complex and
chaotic behaviour such as we see near the
solar surface — the only place where these
detailed local fields and velocities can be measured. As one moves on to study larger features it becomes necessary to introduce the
effects of rotation, which brings some order
out of all the chaos. Investigating deeper
structures and large-scale systematic behaviour then leads one to apply dynamo theory to
global fields in stars like the Sun. That, however, is a subject for my next address. ●
Nigel Weiss FRAS, DAMTP, University of
Cambridge, Cambridge CB3 9EW.
Acknowledgments: The work I have described
rests on collaborations with Sean Blanchflower,
Derek Brownjohn, Fausto Cattaneo, David Hughes, Neal Hurlburt, Paul Matthews, Michael Proctor, Alastair Rucklidge, Louis Tao and Steven
Tobias (all of whom have been in Cambridge at
some stage); I have also benefited enormously
from working with George Simon and Alan Title,
both primarily observers. This research has relied
on continuing support from PPARC.
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