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Phil. Trans. R. Soc. A (2012) 370, 3049–3069
doi:10.1098/rsta.2011.0507
R EVIEW
The solar dynamo
B Y M ARK S. M IESCH*
High Altitude Observatory, National Center for Atmospheric Research,
3080 Center Green Drive, Boulder, CO 80301-2252, USA
The origins of solar magnetism lie below the visible surface of the Sun, in the highly
turbulent convection zone. Turbulent convection operates in conjunction with rotational
shear, global circulations and intricate boundary layers to produce the rich diversity of
magnetic activity we observe. Here, we review recent insights into the operation of the
solar dynamo obtained from solar and stellar observations and numerical models.
Keywords: Sun interior; Sun activity; solar dynamo; magnetohydrodynamics
1. Introduction
The year 2008 marked the 100th anniversary of George Ellery Hale’s monumental
detection of magnetic fields in sunspots by means of the Zeeman effect [1].
It was the first demonstration of magnetism in an astronomical object and
the discovery transformed not only solar physics but all of astrophysics. The
11 year sunspot cycle, discovered 65 years earlier by Heinreich Schwabe [2], was
soon seen in a new light (although not without controversy), as a remarkable
example of hydromagnetic self-organization on a colossal scale, whereby bulk
motions of a turbulent plasma produce organized, global patterns of magnetic
activity. Hale himself appreciated the implications, drawing parallels between
sunspots and laboratory experiments of magnetic fields generated by rotating
electrically charged discs. Astrophysical dynamo theory was born (or at least
reborn, because prior links between geomagnetic storms and solar activity had
already provided inklings of solar magnetism).
Modern observations reveal the handiwork of the solar dynamo with marked
clarity. Filamentary magnetic structures such as coronal loops and streamers
permeate the solar atmosphere, and the release of magnetic energy accelerates
the solar wind and powers eruptive events such as flares and coronal mass
ejections. These in turn shape the heliosphere and regulate the space environment
of planets. Solar and stellar variability and rotational evolution are intimately
linked with magnetism. Yet, the sunspot cycle still stands as one of the most
fundamental and enigmatic challenges of solar physics.
In this paper, we provide a brief overview of solar dynamo theory as it now
stands, focusing on fundamental building blocks and active frontiers. In order to
*[email protected]
One contribution of 11 to a Theme Issue ‘Astrophysical processes on the Sun’.
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M. S. Miesch
set the stage for the subsequent discussion, we give a brief overview of solar
magnetism in §2. We then address the nature of convective dynamos in §3,
highlighting the distinction between small-scale and large-scale dynamo action
and its implications for stars and the Sun in particular. In §4, we focus on the solar
activity cycle, highlighting current theoretical paradigms and open questions.
Section 5 is a summary and a look to the future.
2. Solar magnetism
(a) The solar cycle
The cyclic nature of the global solar magnetic field is most clearly demonstrated
in the structure and evolution of the mean field threading through the solar
surface. Figure 1a shows the radial component of the magnetic field in the
solar photosphere inferred from Zeeman spectroscopy as a function of latitude
and time. A longitudinal average is approximated by combining measurements
near the centre of the disc spanning one mean rotation period, known as a
Carrington rotation.
The low-latitude behaviour seen in figure 1a reflects the well-known solar
butterfly diagram, prominent also in sunspot observations [4–6]. At the beginning
of a cycle, bipolar active regions appear at latitudes of roughly ±30◦ , signifying
the emergence of coherent subsurface flux structures. As the cycle proceeds, these
active regions disperse and new flux emerges at progressively lower latitudes,
reaching the equator in approximately 11 years.1 As the active bands approach
the equator, the emergence of new active regions largely ceases, as signified
for example by a minimum in the fraction of the surface covered by spots
(figure 1b). This is known as solar minimum, when the large-scale structure of
the magnetic field in the photosphere and overlying corona is relatively simple,2
dominated by polar caps of opposite polarity (e.g. 1996–1997 in figure 1a). Much
of the magnetic energy is in the dipole component, but higher order multi-poles
also contribute significantly [7]. This was particularly evident during the recent
extended minimum in 2008–2009 when the dipole contribution dropped unusually
low relative to higher order contributions (J. Luhmann 2010, unpublished data).
The appearance of active regions at mid-latitudes signifies the beginning of a
new 11 year cycle. The polarity of the polar fields during solar minimum reverses
each 11 year sunspot cycle; so the net period of the magnetic activity cycle is
approximately 22 years.
Regions of positive and negative polarity within an active region are oriented
approximately along an east–west axis, although the trailing spots (westward
edge, following the sense of rotation) are systematically displaced poleward
1 Individual
active regions do not exhibit latitudinal propagation until after they fragment
and disperse.
2 The photospheric magnetic field at solar minimum exhibits many small-scale, short-lived features
that cover most of the surface and dominate the magnetic energy. Thus, the magnetic topology
may actually be more complex at solar minimum relative to solar maximum in the sense of
more photospheric null points, separators and separatrix surfaces. Still, the large-scale magnetic
structure is relatively simple in terms of the relative prominence of low-order, nearly axisymmetric
multi-pole moments.
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Review. The solar dynamo
(a) 90 N
30 N
EQ
30 S
90 S
Mg II index
spot area
(b) 8000
6000
4000
2000
0
(c)
0.29
0.28
0.27
0.26
1975
1980
1985
1990
1995
2000
2005
2010
date
Figure 1. (a) Vertical magnetic flux through the solar photosphere as a function of latitude and
time, averaged over longitude (courtesy of David Hathaway). Blue and yellow regions denote inward
and outward polarities. The field strength is inferred from Zeeman measurements near the centre of
the disc assuming a radial field orientation, and longitudinal averages are constructed from synoptic
maps spanning one Carrington rotation (data compiled from the Vacuum Tower Telescope and
Synoptic Optical Long-term Investigations of the Sun (SOLIS) instruments at NSO/Kitt Peak and
the Michelson Doppler Imager (MDI) instrument onboard the SOlar and Heliospheric Observatory
(SOHO)). (b) The area of the solar disc covered by sunspots as a function of time, in units of
millionths of a hemisphere (courtesy of David Hathaway; data compiled from Royal Greenwich
Observatory, US Air Force and NOAA facilities). (c) Mg II index (non-dimensional; see text) as
a function of time (courtesy of Marty Snow; data compiled from the UARS, ERS-2, SORCE and
additional NOAA missions; for further information, see Viereck et al. [3]).
relative to the leading (eastward edge) polarity. The tilt of the axis increases
with latitude, known as Joy’s Law, and is thought to arise through the influence
of the Coriolis force on buoyantly rising toroidal flux structures [8]. Furthermore,
the sense of the leading and trailing polarities in active regions reverses sign
across the equator, known as Hale’s polarity law, implying antisymmetry in the
subsurface toroidal fields that give rise to them.
The emergent flux structures that form photospheric active regions apparently
decouple from deeper seated dynamics within a few days [9], fragmenting and
dispersing over the course of several weeks. The dispersal and subsequent
transport of the residual flux from active regions is well captured by quasitwo-dimensional models that incorporate turbulent diffusion from convective
motions as well as advection by a time-varying, axisymmetric, poleward flow
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M. S. Miesch
of order 10–30 m s−1 [10,11]. Such a flow has been detected in the solar surface
layers by multiple independent techniques, including Doppler measurements, local
helioseismic inversions and correlation tracking [12].
The combined effect of Joy’s Law, active region fragmentation and surface
transport by convection and by meridional flow induces a net global, mean
poloidal field by means of a phenomenon known as the Babcock–Leighton (BL)
mechanism [13,14]. Trailing residual flux from active regions is transported
poleward, where it interacts with pre-existing polar fields of the opposite polarity
(figure 1a). This process apparently accounts for the cyclic polarity reversals in
the polar caps, although other mechanisms may also contribute to global poloidal
field generation (§4).
Although sunspots and active regions are the most familiar manifestations
of cyclic solar activity, corresponding variability is observed in many other
diagnostics, such as solar irradiance, the frequency of eruptive events (coronal
mass ejections and flares) and geomagnetic activity induced by solar forcing
[15,16]. An example is shown in figure 1c, which traces what is known as the
Mg II index, quantifying the relative emission in the core and wing of the Mg
II spectral line. This Mg II emission originates in the chromosphere, and the
core-to-wing ratio is observed to peak above active regions and supergranular
downflow lanes where the photospheric magnetic flux is predominantly vertical.
The Mg II index is commonly taken as a proxy for solar and stellar magnetic
activity [17].
Although continuous multi-wavelength monitoring of solar irradiance
variability and related magnetic activity is limited to the past three to five
decades, available data suggest that similar variations in solar magnetic activity
have been operating unabated for at least 10 000 years, with extended periods
of relatively low and high activity known as grand minima and grand maxima
[5,18]. Continuous sunspot records date back to the early seventeenth century,
and earlier variations in solar activity can be inferred from cosmogenic chemical
isotopes found in tree rings and ice cores. The data are consistent with the
notion that the 22 year activity cycle has been operating continually for at least
several millennia, although the most ancient isotope data often have insufficient
temporal resolution to demonstrate this conclusively. Longer term variations such
as grand minima and maxima appear relatively random. There is some possible
evidence for quasi-periodic behaviour on time scales of 80–2000 years, but none
as prominent or regular as the primary 11 year sunspot cycle [18].
(b) Order amid chaos
Although cyclic activity has long been regarded as the most important and
formidable challenge of solar dynamo theory, a closer look at solar magnetism
reveals that this striking regularity exists amid a turbulent maelstrom of smallscale magnetic fluctuations. High-resolution Zeeman measurements of the solar
photosphere reveal rapidly varying magnetic structure on the smallest observed
scales of order 100 km, while more sophisticated measurements based on full
Stokes polarimetry and the Hanle effect suggest that there is yet more magnetic
energy on even smaller scales that are beyond current resolution limits [19–22].
This small-scale flux component replenishes itself daily and has been referred to
as the Sun’s magnetic carpet [23,24].
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Review. The solar dynamo
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Photospheric magnetograms (vertical field maps inferred from Zeeman
measurements) reveal patches of magnetic flux spanning a vast range of scales
(figure 2a,b). By applying clumping algorithms to cross-calibrated data from the
SOlar and Heliospheric Observatory (SOHO)/Michelson Doppler Imager (MDI)
and Hinode/Solar Optical Telescope (SOT), Parnell et al. [25] have demonstrated
that the flux distribution is self-similar, following a power law extending from the
smallest resolved features (area ∼1014 cm2 , flux ∼1016 Mx) all the way to sunspots
and active regions (area ∼2 × 1019 cm2 , flux ∼1022 –1023 Mx). The distribution
persists at solar minimum and solar maximum, the main difference being the
presence of active regions in the latter, which extends the distribution to higher
flux values (figure 2c). The best-fit power-law exponent for the data in figure 2c
is −1.85. Careful scrutiny of time series suggests that the distribution of emerging
flux is even steeper, following a power law with a best-fit exponent of −2.74 [26].
Presumably, it is this emergent flux spectrum that characterizes the dynamo,
while the shallower spectrum shown in figure 2c may reflect the coalescence of flux
elements through passive advection by the granular flow or through dynamical
feedback mechanisms such as convective collapse [27]. Indeed, reprocessing of
surface flux through coalescence and/or fragmentation can readily yield power
laws similar to observational inferences [28]. Still, the power-law behaviour of
emergent bipoles and the persistence of power-law behaviour throughout the solar
cycle suggests that the scale similarity may be inherent in the field generation
and not just due to surface processing of emergent flux.
The distribution function f (F) shown in figure 2c is defined such that the
total flux per unit area from flux concentrations with values between F1 and F2
is given by
F (F1 , F2 ) =
F2
Ff (F) dF Mx cm−2 .
(2.1)
F1
The emergent flux spectrum obtained by Thornton & Parnell [26] is defined
similarly, but the integrated flux F (F1 , F2 ) now has units of Mx cm−2 d−1 . Thus,
a power-law distribution f (F) ∝ F−a with a > 2 indicates that the unsigned
flux emerging on small scales dominates over that emerging on large scales.
In particular, the rate of flux emergence in structures smaller than 1020 Mx
exceeds that in active regions by nearly three orders of magnitude [26]. This
estimate must be regarded as a lower limit, since much of the small-scale flux is
likely to be unresolved, as suggested by Hanle measurements [19]. Furthermore,
if the mean-field strength in small and large flux concentrations is comparable
(justified as a first approximation by the common saturation level of figure 2a,b),
this implies that most of the magnetic energy produced by the dynamo is likewise
in small-scale, turbulent fluctuations.
Thus, the solar dynamo does not just produce the solar cycle. Rather, it
produces a continuous spectrum of dynamically significant fields extending from
the global dipole moment to sunspots to turbulent fluctuations on the smallest
sub-granule scales we can currently resolve. The common power law across
this vast range of scales suggests that the sunspot cycle and the magnetic
carpet are dynamically linked. Although this does not necessarily rule out
meaningful models targeted at specific aspects of the solar dynamo, it is wise
to keep in mind that the multi-faceted manifestations of solar magnetism have a
common origin.
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M. S. Miesch
(b)
(c)
frequency
(× 10–46 Mx–1 cm–2)
(a)
1012
1010
SOD/NFI 06/2007
MDI/FD 12/2001
MDI/FD 12/2007
108
106
104
102
100
1016 1017 1018 1019 1020 1021 1022 1023
flux (Mx)
Figure 2. The distribution of vertical magnetic flux in the solar photosphere (from Parnell et al.
[25]). Full-disc (FD) observations from (a) the Michelson Doppler Imager (MDI) instrument on the
SOlar and Heliospheric Observatory (SOHO) combined with higher resolution observations from
(b) the Solar Optical Telescope (SOT) instrument on Hinode reveal a power-law distribution (c)
spanning five decades of flux. White and black in (a,b) indicate outward and inward fluxes with
saturation levels of 200 Mx cm−2 . The Hinode/SOT field of view is indicated by the smaller of the
two squares in (a). (c) Includes SOT and MDI FD measurements at solar minimum (2007) as well
as MDI FD measurements at solar maximum (2001), with colours as indicated. The dashed line
indicates a power law with an index of −1.85.
(c) Convection, rotation and magnetic activity
It has been realized since the formative years of solar dynamo theory a half
century ago that the key ingredient in producing organized patterns of magnetic
activity from disorganized convective motions is rotation [29–31]. Rotation
induces anisotropic momentum and energy transport, which in turn establishes
differential rotation and global meridional circulations (MCs) that amplify and
transport magnetic flux [32]. Rotation also gives rise to helical flows and fields
that can promote hydromagnetic self-organization by coupling large and small
scales. There are many subtleties in how such processes proceed in practice
(reviewed in §3b) and there are still many uncertainties as to what physical
mechanisms are responsible for establishing cyclic solar activity (reviewed in
§4). Yet, it is still fair to say that, to the best of our current understanding,
if the Sun were not rotating, it would have a magnetic carpet but not a
magnetic cycle.
In the past few decades, helioseismology has revolutionized solar dynamo
theory by mapping out the solar internal rotation profile throughout most of
the convection zone (figure 3a). The solar envelope exhibits prominent rotational
shear, with the angular velocity U decreasing monotonically by about 30 per cent
from the equator to latitudes of 60–70◦ (higher latitude inversions are unreliable
owing to the insensitivity of global acoustic modes to rotation near the rotation
axis). This latitudinal shear persists throughout the convection zone, with radial
shear largely confined to upper and lower boundary layers. The lower boundary
layer is known as the tachocline and represents a transition from latitudinal shear
in the convection zone to nearly uniform rotation in the radiative zone. At the top
of the convection zone is the near-surface shear layer (NSSL), where the rotation
rate increases inward by about 2–3% from the photosphere to r ∼ 0.95R. In this
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450
(b)
400
375
350
325
300
latitude
425
2
60
40
20
0
–20
–40
–60
dW/2p (nHz)
(a)
1
0
–1
–2
1996 1998 2000 2002 2004 2006 2008
date (yr)
Figure 3. The solar internal rotation. (a) Angular velocity (nHz) as a function of latitude and
radius inferred from SOHO/MDI observations using a subtractive optimally localized averages
(SOLA) inversion technique [33]. The dotted line indicates the base of the convection zone and
white regions near the rotation axis indicate a lack of reliable data. (b) Angular velocity variations
relative to a temporal mean versus latitude and time at a depth of 0.99 R, based on MDI and
Global Oscillations Network Group (GONG) data (from Howe et al. [34]). Solid contours indicate
photospheric magnetic activity based on magnetograms from the SOLIS instrument on Kitt Peak.
paper, we focus on the dynamo implications of this differential rotation profile,
which will be addressed in §§3b and 4. For a discussion of how mean flows are
established and maintained, see Miesch & Toomre [32].
It is often assumed that the solar dynamo operates in a kinematic regime
such that fields are passively amplified and transported by the flow. Because
this assumption helps us to justify the concept of the a-effect (§3b), it underlies
virtually all dynamo models of the solar cycle. Yet, it is also widely realized
that the kinematic assumption cannot be strictly valid for the Sun. In the
kinematic limit, the magnetohydrodynamic (MHD) induction equation is linear
and dynamo action can be regarded as a linear instability with an exponential
growth rate. Because the amplitude of the solar cycle does not appear to be
growing exponentially, this implies that there must be some nonlinear feedback
between fields and flows that limits it.
Dynamo saturation, such as magnetic self-organization, is a subtle issue that
we will address briefly in §3b. Here, we merely highlight the torsional oscillations,
which represent the most compelling evidence we have that systematic feedbacks
between large-scale fields and flows do occur. These are alternating bands of
faster and slower rotation that appear and propagate in latitude in conjunction
with the solar activity cycle (figure 3b). In particular, there is both a low-latitude
branch that propagates equatorwards along with the bands of magnetic activity
as well as a high-latitude branch that propagates poleward on a comparable
time scale.
The torsional oscillations are clearly associated with magnetism but their
significance, if any, with regard to the operation or saturation of the dynamo
is unclear. With an amplitude of less than 0.5 per cent, they do not appear to be
large enough to reflect dynamo saturation through the suppression of rotational
shear (although MHD convection simulations and non-kinematic mean-field
models do exhibit a reduction in rotational shear relative to non-magnetic models,
providing some evidence for dynamo saturation via the so-called U-quenching;
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M. S. Miesch
[35–37]). Current models generally regard them as a passive by-product of cyclic
activity, although it is worth noting that they appear each cycle before the
emergence of sunspots on the surface (figure 3c). The poleward branch is typically
attributed to the mean Lorentz force or turbulent diffusion responding to cyclic
lower latitude forcing, but the meridional structure of the low-latitude branch
suggests that it may be thermally driven, induced by enhanced radiative losses
in active bands3 [38].
It is clear from stellar observations that convection and rotation play a
central role in magnetic activity. Late-type stars with convective envelopes such
as the Sun generally exhibit similar signatures of magnetic activity, including
chromospheric and coronal emission and photometric variations indicative of
spots [17]. Although unresolved, the most active stars also exhibit detectable
Zeeman splittings [39]. Inferred field strengths scale roughly as the square of the
rotation rate, B̃ ∝ U2 , although precise estimates for the power-law index range
widely from 1.2 to 2.8 [40–42]. The increase in B̃ is likely to be due in part
to an increase in rotational shear DU, which also increases with U, although
precisely how is still being debated. Various studies have suggested power-law
relationships such that DU ∝ Un but estimates for the index n range from 0.1
to 0.7 [43–45]. Such scaling uncertainties for B̃ and DU are most probably
owing to heterogeneous sampling but they may also reflect intrinsic variability
owing to sensitivities in the underlying nonlinear dynamics [42]. Scaling
relationships for both B̃ and DU are made somewhat tighter when U is replaced
by the inverse Rossby number tc U (particularly for heterogeneous samples),
where tc is a convective turnover time obtained from theoretical models or
empirical fitting.
Scaling relationships for both B̃ and DU saturate beyond a threshold value
that depends weakly on stellar type [41,42]. For solar-like stars, the threshold
is about 10U , beyond which B̃ becomes insensitive to U and DU appears
to be suppressed, presumably by the Lorentz force (thus providing evidence
for U-quenching). In this saturated regime, the dynamo appears to operate
at peak efficiency, with the mean-field strength B̃ determined by the stellar
luminosity [46].
Many late-type stars certainly exhibit magnetic activity cycles but, despite
much literature on the subject, there are few well-established results. Data are
scarce, largely because observations spanning multiple years or even decades
are needed to verify cycle periods and amplitudes. There is some evidence that
cycle periods for solar-type stars with moderate rotation rates tend to decrease
somewhat with U but there is much scatter and there may be multiple branches
signifying distinct dynamo regimes. We refer the interested reader to the reviews
by Rempel [47], Saar [42] and Lanza [48]. Still, it is clear that the interplay
between convection and rotation lies at the root of solar and stellar magnetic
activity, so we turn now to discuss the nature of convective dynamos.
3 Sunspots
are famously dark relative to their surroundings but their appearance on the solar disc
is accompanied by bright, more weakly magnetized regions known as faculae. Facular brightenings
generally overwhelm sunspot darkenings; so the Sun is roughly 0.1 per cent brighter at solar
maximum [15].
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3. Convective dynamos
The starting point for any solar dynamo model is the magnetic induction equation
under the MHD approximation
vB
= V(v × B − hV × B),
(3.1)
vt
where v and B are the fluid velocity and magnetic field and h is the magnetic
diffusivity of the plasma. The (global-scale) ratio of the advection term to
the diffusive term on the right-hand side (RHS) is the magnetic Reynolds
number Rm = UL/h, where U and L are characteristic velocity and length scales,
respectively.
(a) Small-scale dynamos
The best understood class of turbulent dynamos are isotropic, homogeneous,
incompressible flows with large magnetic Prandtl numbers Pm = n/h 1, where n
is the kinematic viscosity. Here, the kinetic energy spectrum scales with the spatial
wavenumber k as E(k) ∝ k −p within an inertial range that extends approximately
from the energy injection scale k0 to the viscous dissipation scale kn = Req
k0 k0 . Here, Re = UL/n is the Reynolds number and q = 2/(p + 1). For
Kolmogorov turbulence, p = 5/3 and q = 3/4.
For such a flow, the magnetic field is most efficiently generated on small
scales, below the viscous dissipation scale k > kn , where the velocity field is
spatially smooth but temporally random [49,50]. Quasi-laminar (yet chaotic)
stretching generates folded field topologies such that variations in the direction
of B occur on a length scale comparable to the size of eddies e , whereas
−1/2
e [49]. The
variations perpendicular to B occur on a resistive scale r ∼ Rm
−1/2
e .
resulting magnetic energy peaks near the ohmic dissipation scale r ∼ Rm
From equation (3.1), we anticipate that the kinematic dynamo growth rate should
scale with the strength of the shear kuk ∼ k (3−p)/2 , where uk2 = kE(k). For a general
velocity field with p < 3, this implies that the smallest eddies are most efficient
at amplifying field because they have the shortest turnover times. Thus, e ∼ n
and r ∼ Pm1/2 n where n = 2p/kn .
This picture breaks down in stars where Pm < 1. Here, the ohmic dissipation
scale lies within the inertial range such that r > n . There are no scales that
appear spatially smooth from the perspective of the magnetic field. Small eddies
still have the fastest turnover times but ohmic diffusion will suppress field
amplification on scales smaller than r . Thus, we might still expect magnetic
energy generation to be most efficient near the ohmic dissipation scale r but
now quasi-laminar stretching by larger eddies > r must compete against the
disruptive effects of turbulent mixing from smaller eddies < r . Dynamo action
is more difficult to sustain and depends sensitively on the slope of the spectrum
p near r ; steeper slopes (larger p) facilitate field generation [50]. This is most
readily demonstrated in numerical simulations where Rm can be held fixed while
Pm is varied by changing the value of n. As n is decreased below h such that Pm
passes through unity, there is a sharp rise in the critical value of Rm needed to
sustain dynamo action4 [51,52].
4 The
value of Rm in stars is sufficiently large that supercriticality is ensured, despite the small
value of Pm .
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M. S. Miesch
Despite the subtleties at low Pm , it is clear that isotropic, homogeneous,
turbulent dynamos tend to generate intermittent magnetic fields on scales
much smaller than the typical scale of the velocity field LB LV . In short,
turbulent flows beget turbulent fields. Such dynamos have been referred to as
small-scale dynamos in order to distinguish them from large-scale dynamos,
which may generate magnetic fields on scales much larger than the velocity
field LB LV .
Small-scale dynamo action has been proposed to account for the intricate
flux distribution of the magnetic carpet [21,53,54]. This would then imply that
the magnetic energy in the photosphere may well peak on scales that are
currently unresolved, possibly extending down to ohmic dissipation scales of
order 100 m.
(b) Large-scale dynamos: from fluctuations to flux
The principal agents of self-organization in turbulent dynamos are helicity
and shear. Although the significance of these agents was originally appreciated
within the context of the kinematic mean-field theory, they continue to take centre
stage as three-dimensional MHD simulations become increasingly important tools
at the leading edge of dynamo research. Chaotic stretching is still required to
generate magnetic energy but helicity and shear—coupled with the geometry,
boundary conditions and other sources of anisotropy and instability such as
density stratification and buoyancy—play a dominant role in shaping the
large-scale magnetic topology.
The pioneers of mean-field dynamo theory emphasized the significance of the
kinetic helicity, defined as Hk = u · vV , where u = V × v is the fluid vorticity
and V denotes a volume integral [29–31]. The combined influence of rotation
and density stratification induces a helicity density u · v, which reflects the
presence of spiralling flows that break anti-dynamo constraints associated with
two dimensionality and mirror symmetry. More recently, the focus has shifted to
the magnetic helicity Hm = A · BV , where A is the vector magnetic potential,
defined such that B = V × A.
Magnetic helicity is produced by means of kinetic helicity through magnetic
induction but has its own particular significance, largely because Hm , unlike Hk ,
is an ideal invariant of the MHD equations. However, because A is undefined
within the addition of an arbitrary scalar gradient VL (the gauge), Hm is not,
in general, uniquely defined for a finite spatial domain. This ambiguity can be
resolved by defining a gauge-invariant relative helicity with respect to a reference
field Bp as follows [55]:
HR = (A + Ap ) · (B − Bp ) dV ,
(3.2)
V
where V is the volume of the domain in question and Bp = V × Ap . The reference
field is generally taken to be potential (V × Bp = 0) with a Coulomb gauge (V ·
Ap = 0). Furthermore, it is defined such that the component normal to the surface
S that encloses V is the same as the actual field: Bp · n̂ = B · n̂ on S, which
specifies Bp and Ap uniquely. If no field lines leave the domain, i.e. B · n̂ = 0 on
S, then Ap = Bp = 0 and HR = Hm .
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3059
If we take S to be the spherical surface of the Sun, or a single hemisphere, then
the evolution equation for HR , obtained from equations (3.2) and (3.1), is given
by [55]
dHR
8p
4ph
=−
J · dS,
(3.3)
hJ · B dV − Ap × v × B +
dt
c V
c
S
where c is the speed of light, J = c(4p)−1 V × B is the current density and we have
used cgs units throughout. Equation (3.3) is gauge invariant (with respect to A).
If h is constant in V then the first term on the RHS is equal to −8phc −1 Hc , where
Hc = J · BV is the current helicity. Thus, the rate of change of HR is determined
by ohmic dissipation, ∝ Hc , and the helicity flux through the surface S, given
by the integrand of the second term on the RHS of (3.3). If the field is entirely
contained within V (B · n̂ = 0 on S) and if the fluid is perfectly conducting (h =
0), then the magnetic helicity is conserved (HR = Hm , dHm /dt = 0).
To demonstrate the significance of helicity and shear to the generation of largescale fields, we consider the longitudinally averaged induction equation (3.1)
vB
(3.4)
= lBm · VU + V[vm × B + E − hV × B],
vt
where l = r sin q is the cylindrical radius and E = v
× B
. Throughout this
paper, angular brackets denote averages over longitude (not to be confused
with V , which indicates a volume integral) and primes indicate fluctuations
about the mean, e.g. v
= v − v. The subscript m denotes components in the
meridional plane, e.g. Bm = Br r̂ + Bq q̂.
With regard to the solar activity cycle, the primary challenge of solar dynamo
theory is to determine the nature of the turbulent electromotive force (emf)
E. Correlations between fluctuating fields and flows are generally attributed
to convective motions but they may also arise from other phenomena, such as
magnetic buoyancy (MB) or magneto-shear instabilities [4,6]. Explicit expressions
for E can only be obtained for idealized scenarios, the best known being kinematic
mean-field dynamo theory in which the field is passively advected by the flow
(the Lorentz force is neglected) and it is assumed that characteristic length
and time scales of the fluctuations are small relative to the mean. Under these
conditions, the turbulent emf is linear in B and its components can be expressed
(in Cartesian coordinates) as a converging infinite series [4,6]
vBj + ··· .
(3.5)
Ei = aij Bj + bijk
vxk
Under the additional assumptions of quasi-isotropy (antisymmetric under mirror
reflections but otherwise isotropic), and homogeneity, the a and b tensors reduce
to scalar coefficients proportional to the kinetic helicity and the kinetic energy of
the fluctuating motions, respectively.
The a-effect is inherently helical. The generation of magnetic energy is
associated with a transfer of magnetic helicity from fluctuating to mean
fields [56]. In strictly kinematic models, this transfer is implicit (fluctuating
fields never appear explicitly), but many mean-field models incorporate Lorentzforce feedbacks through slight adjustments to the basic mean-field paradigm [57].
To lowest order, Lorentz force feedbacks modify the a-effect by contributing an
extra term such that a = ak + am = −(t/3)(Hk
− (rc)−1 Hc
), where Hk
and Hc
are
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3060
M. S. Miesch
the kinetic and current helicity associated with the fluctuating motions. A meanfield analogue of equation (3.3), accounting for the conservation of magnetic
helicity in the absence of ohmic dissipation, can be used to derive an expression
for am , the result being [32,57,58]
ak
,
(3.6)
a=
2 )
1 + Rm B2 V /(Beq
2
= 4pru 2 and Rm = u 2 (3th)−1 are, respectively, the equipartition field
where Beq
strength and the magnetic Reynolds number associated with the turbulent flow.
In short, Lorentz force feedbacks lead to a marked suppression of the turbulent
a-effect for large Rm that can be attributed to the near conservation of magnetic
helicity. The generation of helical mean flows gives rise to a build-up of small-scale
helicity of the opposite sense that can efficiently quench the dynamo, unless
it is either dissipated or removed from the domain by the helicity flux in
equation (3.3).
This conclusion is not restricted to the mean-field theory. In fact, the
first derivation of am was based on an eddy-damped, quasi-normal Markovian
(EDQNM) turbulence model that did not adopt the kinematic and scaleseparation assumptions required by traditional mean-field theory [59]. Results
exhibited a self-similar inverse cascade of magnetic helicity from small to large
scales,5 which was later demonstrated in idealized numerical simulations of MHD
turbulence [60,61]. However, MHD convection simulations have demonstrated
that rotation alone is not sufficient to promote the generation of large-scale fields
by means of an inverse cascade of magnetic helicity; rather, results are sensitive
to boundary conditions, geometry and the strength and orientation of rotational
shear (reviewed by Tobias et al. [62] and Brandenburg [63]).
Whatever the nature of the initial saturation on a dynamical time scale,
the helicity will continue to evolve on a diffusive time scale according to
equation (3.3). In a closed system, a steady state can only be reached if the
current helicity vanishes J · B = 0. A helical mean field can still persist, provided
the helicity of the opposite sign prevails at small scales such that J · B =
−J
· B
. If J ∼ kB, where k is a characteristic wavenumber (k0 for the mean and
k for the fluctuations), then saturation occurs when B2 ∼ k /k0 (B )2 . In other
words, energy in the mean fields will grow slowly until it exceeds small-scale fields
by the ratio of length scales. This was first derived and demonstrated in numerical
simulations of MHD turbulence by Brandenburg [60].
The first term in equation (3.4) is the well-known U-effect whereby mean
toroidal field Bf is generated from mean poloidal field Bm and amplified by
means of rotational shear. The prominence of sunspots and the large amount
of flux contained in toroidal relative to poloidal fields suggests that the U-effect
must play an essential role in cyclic solar activity (§4). Rotational shear can
also influence the topology of large-scale fields by inducing a flux of smallscale magnetic helicity along isorotation surfaces that can potentially alleviate
dynamical a-quenching [57,64].
Global numerical simulations of convective dynamos verify that rotation
promotes and regulates dynamo action by means of helicity and shear. Figure 4
illustrates a simulation of a young solar-type star rotating at five times the solar
5 As
opposed to the forward cascade of energy from large to small scales.
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Review. The solar dynamo
3061
rate [65]. Enhanced convective Reynolds stresses produce prominent differential
rotation, which in turn generate strong toroidal fields (approx. 10 kG) in confined
bands with opposite polarity in the Northern and Southern Hemispheres, as
inferred from solar observations (figure 1). These toroidal flux structures account
for over half of the total magnetic energy of the dynamo and persist amid the
intense turbulence of the convection zone. This is in contrast to the simulation
at the solar rotation rate where the kinetic energy of the convection exceeds
that in the differential rotation and turbulent pumping effectively confines such
structures to the base of the convection zone and tachocline [66–68]. Simulations
such as this often exhibit quasi-cyclic polarity reversals, as demonstrated in
figure 4d for a fast rotator. By using sophisticated numerical methods featuring
implicit time stepping and minimal numerical dissipation, Ghizaru et al. [68] (see
[37]) have recently achieved the first global MHD simulation of solar convection to
exhibit cyclic polarity reversals on a decadal time scale (figure 5). Understanding
the origin of such behaviour requires a closer look at the fundamental physical
mechanisms that give rise to magnetic activity cycles in simulations and stars,
which we now explore (see §4).
4. Origins of cyclic solar activity
Dynamo models of the solar cycle are generally based on the longitudinally
averaged induction equation (3.4), with some theoretically or empirically
motivated parametrization for the turbulent emf E. Nearly all recent models
invoke an emf of the form
E = P(r, q : B) + g × B + ht J.
(4.1)
The first term on the RHS is the source term for the mean poloidal field. The
simplest representation is a quasi-isotropic, homogeneous, kinematic a-effect,6
as described in §3b: P(r, q : B) = ak Bf f̂. However, models often employ a
nonlinear quenching formulation such as in equation (3.6) in order to limit
the dynamo amplitude. Furthermore, many current models employ a nonlocal source term such that the value of P(r, q : B) near the solar surface
is related to the integrated toroidal flux near the base of the convection
zone [35,69,70]. This is intended to implicitly capture the mean magnetic
induction arising from the buoyant destabilization, rise, emergence and dispersal
of the toroidal flux structures that give rise to active regions; namely, the BL
mechanism discussed in §2a. Although the BL mechanism is phenomenologically
distinct from the turbulent a-effect, it is functionally similar in that Ef is
linearly proportional to the mean toroidal field Bf , apart from quenching and
non-locality.
The second and third terms in equation (4.1) are transport terms and may
be related directly to the kinematic expansion in equation (3.5). The g term
arises from the antisymmetric off-diagonal components of the a tensor and reflects
magnetic pumping by convective motions. Again, in terms of magnetic induction,
this is phenomenologically distinct but functionally equivalent to a mean MC vm ;
6 Strictly speaking, the a appearing here is the symmetric part of the full a tensor appearing in
k
equation (3.5). The antisymmetric part reflects transport and is expressed here separately as the
magnetic pumping vector g, which may be regarded as a pseudo-flow.
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3062
(b)
(c)
(d)
latitude (°)
(a)
M. S. Miesch
50
0
–50
6
8 10 12 14 16 18
time (yr)
Figure 4. Convective dynamo simulation of a young, rapidly rotating, solar-like star (rotation period
5.6 days; from Brown et al. [65]). (a) Radial velocity near the top of the convection zone in a
Molleweide projection (yellow upflow, blue/black downflow; saturation range ±70 m s−1 ). (b) Timeaveraged angular velocity (white faster, blue slower; range 1980–2100 nHz) and (c) toroidal
magnetic field (red eastward, blue westward; range ±10 kG), both averaged over longitude.
(d) Mean toroidal field in the lower convection zone as a function of latitude and time, illustrating
polarity reversals on a time scale of roughly 4 years (colour table as in c). Poleward propagating
magnetic features are accompanied by torsional oscillations (cf. §2c).
(b) 90
60
30
0
–30
–60
–90
latitude (°)
(a)
0
50
100
150
200
time (yr)
250
300
Figure 5. Magnetic cycles in a solar convection simulation (from Racine et al. [37]; see also Ghizaru
et al. [68]). (a) Snapshot of longitudinal field Bf near the base of the convection zone and
(b) longitudinally averaged toroidal field Bf as a function of latitude and time. Yellow/orange and
green/blue regions indicate eastward and westward fields, respectively, with a saturation range of
(a) ±2 kG and (b) ±4 kG. Note the regularity of reversals and the slight equatorwards propagation.
cf. equation (3.4). There is also a zonal component to g which can transport
poloidal magnetic flux in longitude [37,71]. The turbulent diffusion ht is usually
assumed to be isotropic, although this is not strictly consistent with a non-zero g.
As with the a-effect, mean-field models are increasingly incorporating implicit
Lorentz-force feedbacks by means of h-quenching [72,73].
In most solar dynamo models, the mean toroidal field is generated by the
U-effect (see §3b) near the base of the convection zone—region III in figure 6a
(for a notable exception focusing on the NSSL, see [74]). Although one may
expect that the strong radial shear in the tachocline is responsible, many recent
mean-field models find that the latitudinal shear in the lower convection zone
is more efficient at generating the global, equatorially antisymmetric toroidal
bands that give rise to sunspots [35,75,76]. By contrast, the principal mechanisms
and even the location of the poloidal source term P are currently a matter
of debate.
Photospheric measurements suggest that the flux emerging in active regions
is more than sufficient to account for the magnetic polarity reversal in the polar
caps, as suggested by figure 1a. The flux in a single active region is of order
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3063
Review. The solar dynamo
(a)
(b)
MC
80
latitude
CT
MB
60
40
20
0
III
II I
5
10
time (yr)
15
Figure 6. (a) Schematic illustrating distinct physical regimes in the solar envelope including (I)
the near-surface shear layer in the upper convection zone, (II) the bulk of the convection zone
and (III) the tachocline, overshoot region and lower convection zone (from Miesch & Toomre [32]).
Coupling mechanisms between the regions are also highlighted, including the meridional circulation
(MC), convective transport (CT) and magnetic buoyancy (MB). (b) A representative example of
a Babcock–Leighton flux-transport solar dynamo model from Rempel [35]. Contours indicate the
mean toroidal field near the base of the convection zone (solid lines eastward, dotted lines westward;
range ±12.8 kG) and the image represents the mean radial field Br near the surface (red outward,
blue inward; range ±100 G), both plotted versus latitude and time.
1022 –1023 Mx, with about 1024 –1025 Mx emerging over the course of an 11 year
sunspot cycle [77]. If the average magnetic field strength poleward of ±70◦
latitude is roughly 10 G, then a polarity reversal would require a flux of order
4 × 1022 Mx. Thus, even if much of the flux emerging in active regions is lost
through magnetic reconnection or vertical transport of magnetic loops, the BL
mechanism can account for reversals of the surface dipole moment—threading
region I in figure 6a.
Yet, can the redistribution of active region flux in the photospheric boundary
layer really determine the global dipole moment of the entire solar envelope? The
flux in photospheric active regions (region I) and the associated magnetic energy is
likely to be a small fraction of the total unsigned flux and energy in the remainder
of the convective zone (regions II and III), where most of the mass and kinetic
energy of the solar envelope reside. Photospheric observations suggest that active
regions make up only a small fraction of the magnetic flux and energy in the
photosphere, let alone the entire convection zone (§2b). Furthermore, cosmogenic
isotope measurements suggest that the 22 year magnetic activity cycle persisted
throughout the Maunder minimum, a 70 year interval in the seventeenth century
when there were very few sunspots [78]. Convection simulations exhibit cyclic
magnetic activity without capturing the dynamics of flux emergence and dispersal
that underlie the BL mechanism (figures 4 and 5; although aspects of these
simulations do not match solar observations). Amplitude and parity issues have
also motivated several authors to advocate for additional deep-seated sources
of poloidal field in BL dynamo models, including convection and tachocline
instabilities [79]. Still, the BL mechanism is undeniably operating and plays a
prominent role in most recent solar dynamo models.
Cyclic behaviour in the Sun is intimately linked with propagation; activity
bands migrate equatorwards, and residual vertical flux migrates poleward during
the course of each cycle, as evident in figure 1a. Within the MHD framework, there
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3064
M. S. Miesch
are essentially three physical mechanisms that can give rise to the propagation
of mean fields and can thus establish cyclic variability: (A) a spatial offset
between poloidal and toroidal sources (i.e. a dynamo wave), (B) advection by the
mean MC, and (C) transport by convection or other non-axisymmetric motions
(magnetic pumping, turbulent diffusion, etc.). These are not mutually exclusive;
multiple mechanisms can operate simultaneously and some phenomena, such as
MB instabilities, can contribute to all three.
The earliest solar dynamo models were of type (A). Cyclic behaviour is
attributed to a phase shift between the a-effect and the U-effect that admits
travelling wave solutions to the linear, kinematic induction equation (3.4) [6,30].
The propagation speed and the growth rate of the eigenmodes are proportional to
(a|VU|)1/2 , with waves travelling along isorotation surfaces. The natural location
for such a dynamo wave is the low-latitude tachocline, where the nearly radial
orientation of VU and the sign of a (∝ −Hk ) are favourable for equatorwards
propagation. This, together with the need to avoid dynamical a-quenching,
motivated interface dynamo models [80,81] whereby the a-effect occurs in the
convection zone (region II) while the U-effect occurs in the tachocline (region III).
Cyclic behaviour still hinges on a dynamo wave but now the cycle period depends
also on the coupling mechanism between regions II and III, usually taken to be
turbulent diffusion.
Most recent solar dynamo models advocate mechanism (B), whereby an
equatorwards circulation of 1–3 m s−1 near the base of the convection zone
(region III) accounts for the equatorwards migration of active bands as
manifested in the solar butterfly diagram and largely determines the period
of the solar cycle [35,69,73,75,76,79]. These are known as flux-transport (FT)
dynamo models, most of which attribute poloidal field generation to the
BL mechanism. Furthermore, many BL–FT dynamo models operate in a socalled advection-dominated regime whereby the coupling between regions I
and III is achieved by the MC. Such models generally employ single-celled
circulation profiles as illustrated in figure 6a, but recent results by Dikpati
et al. [82] based on modelling and photospheric observations suggest that
the presence of a high-latitude counter-cell may account for why cycles are
typically shorter (approx. 11 years) than the most recent one (approx. 13 years).
An equatorwards circulation of several m s−1 at the base of the convection
zone is consistent with global convection simulations, but estimates of the
efficiency of convective transport (CT) are about an order of magnitude larger
than typical circulation speeds, calling into question the advection-dominated
regime [67,83,84].
Magnetic pumping or anisotropic magnetic diffusivity, mechanism (C), can in
principle induce cyclic activity in mean-field models but in practice it generally
operates in conjunction with other processes. For example, BL models can be
constructed in which turbulent diffusion or magnetic pumping account for the
coupling between regions I and III [85–87]. The challenge here is to account for
the relative inefficiency of CT; if coupling were to occur on a turnover time scale
(approx. 1 month), cycles would be too short.
The picture emerging from global convection simulations is only now beginning
to take shape. For the simulation shown in figure 4, cyclic activity appears to be
primarily owing to a nonlinear dynamo wave (A), even though E is not well
represented by a simple a-effect. Poleward circulations associated with polar
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Review. The solar dynamo
3065
slip instabilities, and torsional oscillations are also present and may play a
role in polarity reversals. The simulation shown in figure 5 appears to be more
amenable to a mean-field treatment and shows evidence for all three mechanisms
for cyclic variability: (A) spatial offsets between toroidal and poloidal sources,
(B) an equatorwards circulation of approximately 0.2 m s−1 near the base of the
convection zone, and (C) equatorwards turbulent pumping of a similar amplitude,
approximately 0.2 m s−1 [37].
5. Summary and outlook
Kinematic, axisymmetric mean-field models are still the workhorse of solar
dynamo theory, but verifying their underlying assumptions and accounting for
the full richness of solar magnetic activity will require more sophisticated threedimensional MHD simulations. A major challenge for the future will be to link the
cyclic global dynamo and the magnetic carpet into a unified theoretical paradigm.
Three-dimensional dynamo simulations must also be linked to simulations and
observations of flux emergence. Current dynamo models take the strength of
the mean toroidal field near the base of the convection zone as a proxy for the
sunspot number (figure 6b). However, there is no guarantee that this relationship
is linear and the details of how it unfolds may have important implications for
the operation of the dynamo as well as solar cycle predictability and the space
weather implications of emergent flux.
The solar cycle itself still harbours many mysteries. What processes dominate
the generation of large-scale poloidal fields as encapsulated in the essential but
enigmatic turbulent emf? Can the BL mechanism overwhelm the turbulent
a-effect? Might MHD instabilities contribute to mean-field generation? What
regulates the cycle period and amplitude? Can a weak but persistent meridional
flow prevail over turbulent transport? Does the spectral and spatial flux of
magnetic helicity play a crucial role in determining the large-scale magnetic
topology and dynamo saturation? Is flux emergence merely a by-product of
dynamo action or is it essential for the establishment of cyclic activity? How
does the Sun fit within the context of other stars? Inevitably, answers can
only come from diligent observation and innovative modelling. Thanks to
continuing advances in instrumentation and high-performance computing, the
future is bright.
We are grateful to Paul Charbonneau, Guiliana De Toma, David Hathaway, Rachael Howe, Clare
Parnell, Étienne Racine, Matthias Rempel and Marty Snow for help in preparing and interpreting
figures. We also thank Paul Charbonneau, Matthias Rempel, Janet Luhmann and the two
anonymous referees for discussions (referees excluded) and comments on the manuscript. Research
support is provided through NASA grants NNH09AK14I and NNX08AI57G and computing
resources were provided by NASA HEC (Pleiades) and NSF PACI (NCSA, TACC, PSC and SDSC).
The National Center for Atmospheric Research is sponsored by the National Science Foundation.
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