Solar dynamo theory
Solar dynamo theory:
a new look at the origin of
small-scale magnetic fields
Fausto Cattaneo and David W
Hughes delve beneath the surface
of the Sun with numerical models
of turbulent convection.
O
bservations of the solar magnetic field
reveal a bewildering variety of structures and activities. Phenomena vary
in scale from sizes comparable to the solar
radius to the limit of present resolution, and in
duration from tens of years to minutes (see, for
instance, Schrijver and Zwann 2000). At the
large end of the scale are active regions: highly
structured complexes, with a characteristic lifetime of one month, with fluxes of the order of
1022 Mx and typically containing sunspots.
The number of active regions and their latitude
of emergence have a cyclic variation with a
period of approximately 11 years – the solar
cycle. The large-scale spatial organization and
longevity of active regions suggest strongly
that they are surface manifestations of a
strong, deeply-seated, coherent magnetic field.
At low resolution, the solar surface away
from active regions – the quiet photosphere –
appears magnetically uninteresting. However,
recent high-resolution observations have
revealed the quiet photosphere to be the site of
highly dynamic magnetic activity. It has long
been observed that turbulent convective
motions at the solar surface have two dominant characteristic scales, granulation with a
typical size of 1000 km and a lifetime of
5–10 minutes, and supergranulation with characteristic size of 15–20 000 km and lifetime of
10–20 hours. It now appears that with each
scale of convection there is an associated magnetic component evolving on a comparable
timescale. Magnetic field emerges at the solar
surface mostly in the form of ephemeral
regions, small bipolar regions with (unsigned)
fluxes of the order of 1019 Mx. These appear
preferentially in the centres of supergranules,
the two polarities then drifting apart before
breaking up into smaller magnetic elements.
These are then swept to the supergranular
3.18
lthough magnetic dynamo action
is traditionally associated with
rotation, fast dynamo theory shows
that chaotic flows, even without
rotation, can act as efficient small-
A
boundaries where they form the magnetic network. Recent estimates indicate that all of the
quiet network flux is replaced within a 40 hour
period, a timescale of the same order as the lifetime of supergranules (Schrijver et al. 1997).
In addition, weaker field concentrations are
found away from the magnetic network. These
granular fields comprise tiny magnetic elements of both polarities, with sizes at the limit
of current resolution and observed field intensities between 200 and 1000 G (Lin and Rimmele 1999). The pattern formed by these small
elements overlies that of the granulation and
appears to evolve on a comparable timescale.
Given that the quiet photosphere appears to
be somewhat more lively than previously
thought, it is natural to speculate on the origin
of magnetic fields therein. Two distinct possibilities immediately spring to mind. One is that
the mechanism responsible for the generation
of active regions and the origin of the solar
cycle – the so-called solar dynamo – is responsible also for the magnetic activity of the quiet
Sun. In this scenario, magnetic fields in the
quiet photosphere could arise as the by-product of the decay of active regions; alternatively,
they could be seen as the small-scale magnetic
noise generated by the (large-scale) solar
dynamo. The other possibility is that magnetic
fields in quiet photospheric regions are generated by local dynamo action driven by the
granular and supergranular flows. In reality,
both are likely to contribute to some extent;
the problem is then to determine which is the
dominant mechanism. Recent observations
and numerical studies indicate that local
dynamo action might actually be the most
important.
Is rotation important?
Traditionally, solar dynamo theory has concen-
scale dynamos. Indeed, numerical
simulations suggest that granular and
supergranular convection may
generate locally a substantial part of
the field in the quiet photosphere.
trated on the problem of the origin of active
regions and of the solar cycle, with the aim of
reproducing the features of the large-scale field
(Parker 1979). Practically all models of the
large-scale solar magnetic field have been
derived within the framework of mean field
electrodynamics. One of the great successes of
this theory has been to clarify the relationship
between magnetic field generation and rotation.
In simple terms the dynamo mechanism can
be viewed as the generation of toroidal field
from poloidal field, occurring concurrently
with the opposite process of creating poloidal
from toroidal field. One half of the cycle is
readily envisaged. Differential rotation, with
either radius or latitude, is an extremely effective way of dragging out toroidal field from an
existing poloidal field. Formally this is contained in the ∇ × (〈U〉 × 〈B〉) term in equation
(6) in the box “Mean field theory”. Helioseismology now suggests that most of the differential rotation is confined to a thin layer at
the bottom of the convection zone – the
tachocline (Schou et al. 1998); this is commonly identified with the seat of the solar dynamo
(see, for example, Weiss 1994). Closing the
dynamo loop, namely regenerating the largescale poloidal field from the large-scale
toroidal field, is ascribed to the α-effect occurring in flows that lack reflectional symmetry
(see Box). In most physical circumstances,
flows of this type are closely associated with
the presence of helicity. This quantity, which is
possibly the simplest measure of the lack of
reflectional symmetry, is non-zero in flows in
which, for instance, there is a correlation
between rising and clockwise twisting motions.
In the solar context, helical turbulence is generated by the interaction between convection
and the Coriolis force (Parker 1955). It is
important to note that both aspects of the
June 2001 Vol 42
Solar dynamo theory
Mean field theory
The evolution of a magnetic field B in an
electrically conducting fluid moving with
velocity U is described by the magnetic
induction equation
∂B
= ∇ × (U × B) + η∇2B
(1)
∂t
where η is the magnetic diffusivity (inversely
proportional to the electrical conductivity).
The starting point for the derivation of
mean field electrodynamics is the assumption that the quantities U and B can be
meaningfully divided into mean and fluctuating parts, where the characteristic scale of
variation of the mean greatly exceeds that of
the fluctuations. Thus one can effect the
decomposition
U = 〈U〉 + u
(2)
B = 〈B〉 + b
(3)
where the angle brackets denote a spatial
average such that 〈u〉 and 〈b〉 vanish. The
objective of mean field theory is to derive an
equation for the evolution of the mean magnetic field 〈B〉 (comprehensive reviews of the
theory can be found in Moffatt 1978, Krause
and Rädler 1980). This is achieved by substituting the decomposition (3) into the induction equation (1) and averaging, to give
∂〈B〉
= ∇ × (〈U〉 × 〈B〉 + ε) + η∇2〈B〉 (4)
∂t
where ε = 〈u × b〉 is the mean electromotive
force. Comparison of equations (1) and (4)
shows how the mean field is subject not
only to diffusion, and advection by the
mean velocity, as one might expect, but also
to the influence of the mean electromotive
force. In order to obtain a closed system of
equations it is necessary to express ε, which
is defined as the product of fluctuating
quantities, in terms of mean quantities
alone. This is achieved by exploiting the linearity of the induction equation to write
∂〈B〉j
εi = αij 〈B〉j + βijk +…
(5)
∂xk
The contributions from the higher derivatives in (5) are expected to decrease rapidly
provided that 〈B〉 varies on a sufficiently
large scale. The tensors αij and βijk are determined by η and by the fluctuating velocity
solely. In particular, if u is a random velocity, as in a turbulent flow, then αij and βijk
are statistical quantities determined by the
statistics of u. Their physical interpretation
is most clearly appreciated in the case where
the underlying turbulence is homogeneous
and isotropic so that αij and βijk are themselves isotropic tensors, i.e. αij = αδij and
βijk = βijk. In this case, equation (4) reduces
to
∂〈B〉
= ∇ × (〈U〉 × 〈B〉 + α〈B〉) + (η + β)∇2〈B〉 (6)
∂t
The quantity β is seen to be an enhancement of the magnetic diffusivity due to the
underlying turbulence, while α is a source of
mean field due to interactions between fluctuations. It is now apparent why the α-term
plays such an important role in the model-
ling of astrophysical dynamos, since it provides a mechanism to generate large-scale
fields from the underlying turbulent velocity.
It is therefore important to determine the
circumstances under which it is non-zero. In
this regard, we note that αij establishes a
(linear) relationship between the mean electromotive force (a polar vector) and the
mean magnetic field (an axial vector). Under
parity transformations, polar and axial vectors behave differently; the former remain
invariant while the latter change sign. In
order for the relationship between these two
vectors to remain valid under parity transformations, as indeed it must, the quantity
αij itself must change sign. This property
identifies αij as a pseudo-tensor, and correspondingly, α as a pseudo-scalar, to distinguish them from regular tensors and scalars
that do not change sign under parity transformations. Since, as mentioned above, αij is
determined solely by η and by the velocity
field, it follows that in order for αij to be
non-zero, the velocity itself must not be
invariant under parity transformations – in
other words the velocity must lack reflectional symmetry. In most physical situations,
lack of reflectional symmetry is associated
with motions in rotating bodies. It is worth
noting that, unlike α, the β term defines a
relationship between two polar vectors, and
is thus a regular tensor. It can therefore be
non-zero both in reflectionally- and nonreflectionally-symmetric turbulence.
dynamo process, namely the α-effect and the
differential rotation, are the consequences of
motions in a rotating frame. This has led to the
widespread perception that dynamo action is
impossible in the absence of rotation. However, this is not quite the case; the above arguments imply that rotation is necessary only for
the generation of large-scale magnetic fields.
Application of mean-field arguments leads to
important consequences in bodies, such as the
Sun, with motions extending over a wide range
of spatial and temporal scales; in particular,
only those motions that “feel” the rotation can
contribute to the (large-scale) dynamo process.
In the Sun, where the rotation period is approximately one month, neither the granulation nor
the supergranulation are significantly rotationally constrained. Therefore, their contribution
to the generation process (the α-effect) is
expected to be negligible. On the other hand,
their contribution to the turbulent diffusion
(the β term) is expected to be significant. Thus,
application of mean field theory to the granulation and supergranulation predicts that no
dynamo action of mean-field type could be
associated with these convective scales. How-
ever, this leaves open the intriguing possibility
of dynamo action that generates small-scale
fields without the generation of large-scale
fields – a so-called small-scale dynamo.
The study of small-scale dynamo action has
received a lot of attention recently in the context of fast dynamo theory, the study of
dynamo action in fluids with very high electrical conductivity. One of the interesting results
to emerge from fast dynamo theory is that any
highly conducting turbulent fluid is likely to
act as a dynamo, the essential ingredient being
the chaotic properties of the flow, but not necessarily its helicity or lack of reflectional symmetry (see the box “Fast dynamo action”).
Thus, arguments from fast dynamo theory are
certainly suggestive that the solar granulation
and supergranulation, by virtue of their turbulent nature, together with the high electrical
conductivity of the solar plasma, could act as
small-scale dynamos. However, to make the
idea convincing, there are two aspects of this
problem that need further discussion. One is
that although fast dynamo theory makes it
plausible that sufficiently complicated chaotic
flows are capable of field generation, it does
not make it certain. Thus, whether convectively driven turbulence can indeed act as a
dynamo needs to be verified. The other, more
important, aspect is that the ideas of fast
dynamo theory are fundamentally kinematic.
In a kinematic approach, the back-reaction of
the magnetic field on the flow (the Lorentz
force) is neglected, the field therefore being
treated as purely passive. Kinematic theory
determines the rate at which a weak magnetic
field will be amplified, but not the amount by
which it will be amplified. This last point can
only be settled by taking the Lorentz force into
account. From a physical point of view, kinematic theory describes the initial phases of
amplification from a state of weak magnetization. As the magnetic field is amplified, the
Lorentz force grows until it becomes comparable to the other forces acting on the fluid. The
flow itself is then modified so as to prevent further amplification of the field. It is the magnetic field in this equilibrated state, which results
from a dynamical balance of magnetic and
hydrodynamic forces, that should be compared
with observations. In a fully dynamical
approach to dynamo theory, the equations
June 2001 Vol 42
3.19
Solar dynamo theory
describing the evolution of the magnetic and
velocity fields must be solved self-consistently.
The resulting mathematical problem is highly
nonlinear and really can only be tackled by a
numerical approach.
Is turbulent convection a small-scale
dynamo?
1: Snapshot of the temperature fluctuations in
a horizontal plane near the surface of the
layer. Light tones correspond to hot (rising)
fluid, dark tones to cold (sinking) fluid. The
computational domain is 20 times wider in
each horizontal direction than it is deep.
2: Six snapshots of the temperature
fluctuations in a square subdomain of width
4, showing the evolution of a typical
convective cell. The snapshots are spaced at
equal intervals and cover approximately one
turnover time. The position of the initial
snapshot is shown in figure 1. The colour
table is the same as in figure 1.
1.2
1.0
0.8
0.6
0.4
0.2
0.0
0
10
20
time
30
40
3: Time evolution of the kinetic (green) and magnetic (blue) energy densities. Both quantities are
measured in terms of the initial value of the kinetic energy (before dynamo action sets in), and the
time is in units of the turnover time. The magnetic energy density has been scaled up by a factor of 5
to facilitate the comparison with the kinetic energy. The end of the kinematic regime (at t ≈ 7)
corresponds to the earliest epoch at which departures from an exponential behaviour are observed.
3.20
The fundamental questions that must be
addressed are whether non-helical but otherwise turbulent convection can act as an efficient dynamo and, if so, whether the resulting
magnetic field resembles, at least qualitatively,
that of the solar photosphere. Although convection in the Sun involves many complex
physical processes such as compressibility, ionization, strong stratification and radiative
transfer, it is conceivable that they may not be
crucial to the process of field generation and,
as such, could be neglected in an initial study.
On the other hand, theories of high conductivity dynamos suggest that the essential ingredients are large regions of chaotic streamlines
together with a lack of symmetry in the underlying flow, both of which can be achieved by
ensuring that the convection is well into the
turbulent regime. The simplest model of thermally driven convection consists of a layer of
incompressible fluid heated from below and
described by the Boussinesq approximation; it
neglects all of the effects outlined above, but it
correctly captures the interplay between temperature fluctuations and buoyancy forces. The
vigour of the convection is controlled by the
Rayleigh number, a dimensionless measure of
the temperature drop across the layer. Provided the horizontal extent of the layer is sufficiently large, a turbulent state of convection is
achieved for sufficiently high Rayleigh numbers. Such flows can be simulated efficiently on
modern supercomputers (Cattaneo 1999).
The structure of the convection for one such
realization is captured by figure 1, which shows
the temperature fluctuations in a horizontal
plane near the top of the layer. The flow consists of a pattern of convective cells with hot
fluid rising in the cellular centres and cold fluid
sinking at the cellular boundaries. The Rayleigh
number for this case is 500 000, roughly 760
times the critical value at which convection sets
in. In such a strongly nonlinear regime, the pattern of convection is both highly irregular and
strongly time-dependent, with the lifetime of
the individual cells being comparable to the
turnover time – i.e. the time for a fluid element
to traverse the depth of the layer. The evolution
of a typical cell is shown in figure 2.
It is reasonable to expect that such a complicated flow will have chaotic particle paths
everywhere; thus we anticipate that for large
magnetic Reynolds number this velocity is a
strong candidate for (kinematic) dynamo
action. In the present case, Rm ≈ 1000 which,
June 2001 Vol 42
Solar dynamo theory
though not huge in the astrophysical sense, far
exceeds the critical value for the onset of field
generation, believed to be O(10) for generic
chaotic systems. Figure 3 shows the time evolution of the kinetic and magnetic energy densities following the introduction of a weak seed
field into a state of fully developed convection.
The field is initially amplified kinematically at
an exponential rate with growth rate comparable to the turnover frequency, as is to be
expected from fast dynamo considerations. At
the end of the kinematic regime the Lorentz
force becomes significant, leading to a modification of the convection and to the eventual
saturation of the dynamo growth. This modification is subtle, probably involving the chaotic
structure of the fluid trajectories, but not at all
evident in the overall appearance of the convection, except for a reduction in its vigour,
visible as a decrease in the kinetic energy density. In the present case the magnetic energy in
the dynamical regime is approximately 20% of
the kinetic energy, indicating that this type of
convection is an efficient nonlinear dynamo.
Figure 4 shows the magnetic field distribution in a horizontal plane near the top of the
layer. Magnetic field of both polarities is concentrated at the cellular boundaries and corners into thin structures separated by regions
of nearly field-free fluid. The magnetic field in
these concentrations is intense, with an energy
density exceeding the kinetic energy of the
flow. The magnetic field is highly dynamical,
evolving on the same timescale as the velocity;
figure 5 shows the evolution of the field over a
small area. It is interesting that the magnetic
field described by such a simple model nevertheless shares so many features with that
observed in the quiet solar photosphere.
In contrast with solar observations, which
are by nature restricted to the surface, numerical simulations permit the measurement of the
field throughout the interior. Remarkably, it
emerges that the surface is not entirely representative of the magnetic field structure over
the bulk of the fluid. In the interior the magnetic field becomes more pervasive, with moderately strong magnetic fluctuations occurring
almost everywhere. This property is illustrated
in figure 6, which shows the magnetic field distribution in a horizontal plane in the middle of
the layer (cf. figure 4), and figure 7, a volumerendered image of the magnetic field intensity.
This shows that where, as here, the flow is an
efficient nonlinear dynamo, the magnetic field
becomes an important part of the flow dynamics, and the turbulence must be regarded as
essentially hydromagnetic in nature.
The way ahead
The considerations above support the idea that
a substantial fraction of the magnetic field in
the quiet photosphere is generated locally by
June 2001 Vol 42
4: Snapshot of the vertical component of the
magnetic field in a horizontal plane near the
surface of the layer. Orange tones correspond
to nearly field-free regions, yellow and blue
tones correspond to strong magnetic fields
with opposite polarity.
5: Six snapshots of the magnetic field
intensity in a square subdomain of width 4,
showing the evolution of a typical magnetic
structure. The snapshots are spaced at equal
intervals and cover approximately one
turnover time. The location of the initial
snapshot is shown in figure 4. The colour
table is the same as in figure 4.
6: Snapshot of the vertical component of the
magnetic field in a horizontal plane in the middle of
the layer. The colour table is as in figure 4.
3.21
Solar dynamo theory
Fast dynamo action
In most astrophysical systems the characteristic timescale for magnetic diffusion greatly
exceeds that for advection of the field; i.e.
the magnetic Reynolds number Rm , the ratio
of these timescales, is enormous. In order for
dynamo processes to be astrophysically relevant, they must therefore operate in the large
Rm regime. This idea has been formalized as
fast dynamo theory, which addresses the
problem of magnetic field generation in the
limit of infinite Rm (vanishing magnetic diffusivity). The rigorous treatment of the fast
dynamo problem is mathematically very
involved (for an appraisal see the monograph
by Childress and Gilbert 1995); however, the
underlying ideas can be discussed in reasonably straightforward terms.
In general, dynamo action succeeds if, on
average, the rate of field amplification exceeds
the rate of field destruction. In a turbulent,
highly conducting fluid, the former is due to
the stretching of magnetic field lines whereas
the latter is due to enhanced diffusion. The
problem of quantifying these two processes in
terms of the properties of the advecting velocity is the very essence of fast dynamo theory.
small-scale dynamo action. The present model
is too simplified to allow a direct quantitative
comparison with observations. For that kind
of analysis some of the missing physics would
definitely have to be included such as, for
instance, compressibility and radiative transfer,
both of which are known to be important in
the uppermost layers of the solar convection
zone.
The ability of turbulent convection to support small-scale dynamo action also raises an
interesting question related to the process of
generation of large-scale magnetic fields. Mean
field theory predicts that, in the presence of
helicity, large-scale fields may become unstable
to dynamo amplification. This result is based
in part on the important, though not always
explicitly stated, assumption that the smallscale fields do not experience dynamo growth.
This, however, appears to be at odds with the
predictions of fast dynamo theory suggesting
that small-scale dynamo action will occur in
most turbulent situations. The resolution of
this apparent inconsistency is one of the key
issues in the application of mean field theory to
astrophysical dynamos. ●
Fausto Cattaneo is Assistant Professor in the
Department of Mathematics, University of Chicago
and David W Hughes FRAS is Professor and Head
of the Department of Applied Mathematics at the
University of Leeds.
3.22
As a starting point it is instructive to consider the case when the magnetic diffusivity
actually vanishes (i.e. perfect conductivity)
and for which, by Alfvén’s theorem, magnetic field lines move with the fluid. The analogy of the evolution equations for magnetic
field lines and material lines, namely
D
D
B = B . ∇u and δx = δx . ∇u
(1)
Dt
Dt
thus suggests that useful information about
the magnetic field may be extracted from the
average properties of fluid trajectories.
The evolution of small fluid elements can
be described by a displacement, a rotation
and a deformation. The latter is of particular
importance and typically involves stretching
along at least one direction and contraction
along at least one of the others. If these two
processes proceed, on average, at an exponential rate then the flow is said to be chaotic, with the rates of stretching and contraction being known as the Lyapunov
exponents (a readable account of chaotic systems is contained in Ott 1993).
Since chaotic flows stretch material lines at
an exponential rate, naively one might
expect that any chaotic flow should lead to
fast dynamo action. Crucially, however, gra-
dients also increase exponentially; thus the
diffusion term will inevitably become significant, no matter how small the diffusivity,
provided it is non-zero. Thus the success or
failure of fast dynamo action depends on the
competition between two exponential
processes; line stretching, measured locally
by the largest Lyapunov exponent, and gradient growth, measured locally by the smallest (i.e. the most negative).
An interesting complication, and one which
gives this problem its unique flavour, arises
from the vectorial nature of the magnetic
field. This implies that magnetic field decay
is controlled not only by the growth of gradients, but also by the way in which the field
lines are oriented relative to each other. Field
lines with like polarity are less affected by
diffusion when brought together than are
field lines of opposite polarity. In a finite volume, the relative orientation of neighbouring
field lines depends on global properties of
the flow; consequently the way in which the
flow “packs” the magnetic fields becomes
important. Thus the rate at which a fast
dynamo operates, if at all, depends on a
complicated combination of local and global
properties of the flow.
7: Volume rendering of magnetic field intensity over a 10 ×10 subdomain at one instant in time. Bright,
opaque regions correspond to intense fields; dark, transparent regions correspond to nearly field-free fluid.
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