The Astron Astrophys Rev (2003) 11: 287–367
Digital Object Identifier (DOI) 10.1007/s00159-003-0019-3
THE
ASTRONOMY
AND
ASTROPHYSICS
REVIEW
The solar dynamo
Mathieu Ossendrijver
Kiepenheuer-Institut für Sonnenphysik, Schöneckstrasse 6, 79104 Freiburg, Germany
(e-mail: [email protected])
Received 5 May 2003 / Published online 15 July 2003 – © Springer-Verlag 2003
Abstract. The solar dynamo continues to pose a challenge to observers and theoreticians. Observations of the solar surface reveal a magnetic field with a complex, hierarchical structure consisting of widely different scales. Systematic features such as
the solar cycle, the butterfly diagram, and Hale’s polarity laws point to the existence
of a deep-rooted large-scale magnetic field. At the other end of the scale are magnetic
elements and small-scale mixed-polarity magnetic fields. In order to explain these phenomena, dynamo theory provides all the necessary ingredients including the α effect,
magnetic field amplification by differential rotation, magnetic pumping, turbulent diffusion, magnetic buoyancy, flux storage, stochastic variations and nonlinear dynamics.
Due to advances in helioseismology, observations of stellar magnetic fields and computer capabilities, significant progress has been made in our understanding of these and
other aspects such as the role of the tachocline, convective plumes and magnetic helicity conservation. However, remaining uncertainties about the nature of the deep-seated
toroidal magnetic field and the α effect, and the forbidding range of length scales of the
magnetic field and the flow have thus far prevented the formulation of a coherent model
for the solar dynamo. A preliminary evaluation of the various dynamo models that have
been proposed seems to favor a buoyancy-driven or distributed scenario. The viewpoint
proposed here is that progress in understanding the solar dynamo and explaining the
observations can be achieved only through a combination of approaches including local
numerical experiments and global mean-field modeling.
Key words: Sun: magnetic fields – Magnetohydrodynamics (MHD) – convection –
stars: magnetic fields
1. Introduction
The Sun displays a stunning variety of magnetic-field related phenomena across a wide
range of spatial, temporal and energy scales. Ultimately, a theory of the solar dynamo
should explain the origin of all magnetic fields observed on the Sun, their properties, how
they are related to one another, and how they change during the course of a solar cycle.
Progress in solar dynamo theory is desired also for explaining magnetic fields of stars and
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other cosmical objects ranging from planets to galaxies. For instance, magnetic torques
control the rotation of stars, and the presence of magnetic fields can have a large influence
on stellar structure and evolution. It is hoped that one day dynamo theory will produce
accurate models of stellar magnetic fields for a wide range of basic stellar parameters.
Solar dynamo theory has the advantage of allowing a very detailed comparison with
observations, thus providing the best possible test case for stellar dynamo theory. These
tasks have not yet been accomplished, and they may not be in the foreseeable future. But
dynamo theory has succeeded in presenting various avenues along which a successful
solution may one day be found, by formulating simplified models for the large-scale
solar magnetic field. Furthermore, magnetohydrodynamic computations have clarified
aspects of the solar dynamo by considering in isolation physical processes that are
thought to be relevant, and by focussing on local, small-scale phenomena that are more
easily accessible to simulation.
For these reasons, it seems justified to present a review of current ideas in solar
dynamo theory, even though the subject might appear to be overburdened with a great
many reviews already (Cowling 1981; Gilman 1986; Hoyng 1992; Parker 1970, 1987;
Rosner & Weiss 1992; Rüdiger & Arlt 2003; Schüssler 1983; Stix 1976, 1991, 2001;
Weiss 1981, 1994). Other treatments of the solar dynamo problem can be found in several
monographs (Cowling 1976; Krause & Rädler 1980; Moffatt 1978; Parker 1979a; Priest
1982; Roberts 1967; Schrijver & Zwaan 2000; Stix 2002; Zel’dovich et al. 1983). As
will become apparent, a coherent theory cannot yet be presented. Instead, it seems
appropriate to consider a rather broad selection of solar and stellar observational data
and theoretical aspects, and to sketch various global models that have been proposed. A
historical account of the development of solar dynamo theory is not intended.
2. The solar magnetic field
Solar dynamo theory usually focusses on the large-scale magnetic field and the solar
cycle, although the magnetic field assumes a smooth form with a dipolar symmetry
only at some distance from the Sun. At the solar surface, the global features are rather
well hidden from the casual observer in a sea of complex, small-scale features (Zwaan
1987). Theoretical arguments support the conclusion that the magnetic field is spatially
intermittent throughout the solar convection zone (§ 5.5.4). This has led to the concept
of the fibril state of the solar magnetic field (Schüssler 1984), and it suggests that solar dynamo theory cannot be considered complete if it deals only with the large-scale
magnetic field. Nevertheless, the most important challenge of solar dynamo theory is
without doubt posed by the solar cycle.
2.1. Solar cycle
2.1.1. Sunspot cycle
Our knowledge of the solar magnetic field is largely derived from observations of the
photosphere and the regions above it. The main diagnostic of the large-scale magnetic
field is offered by sunspots (Solanki 2003). Their number exhibits a cyclic variation
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with a period of about 11 years, commonly illustrated by the annual mean of the Zürich
sunspot number R (Fig. 1; Waldmeier 1955; Hoyt & Schatten 1998). The sunspot number
is roughly a linear measure of the surface coverage of sunspots. Sunspots are the sites
of strong magnetic fields, and they have a rather invariant field strength of 0.1–0.3 T.
This uniformity is attributed to the convective collapse mechanism which results in
approximate equipartition between magnetic pressure and ambient thermal pressure
(Parker 1978; Spruit & Zweibel 1979). Thus the magnetic field strength of sunspots has
no connection to the dynamo mechanism, being a local property of the photosphere.Also,
the sunspot number is not a linear measure of the magnetic field intensity, but of the total
magnetic flux contained in sunspots. A direct comparison of the sunspot number with
magnetic field intensities from dynamo calculations is therefore not possible without
some assumption about the relation between magnetic field intensity within the dynamo
layer and the total flux or surface filling factor of sunspots.
Fig. 1. The Zürich annual-mean sunspot number (courtesy National Geophysical Data Center,
USA)
Frequently, sunspots form bipolar pairs, consisting of a leading spot and a trailing
spot with respect to the solar rotation. Virtually all bipolar pairs obey the Hale-Nicholson
polarity rules: (1) the magnetic polarities of leading and trailing spots are opposite, and
those of the leading spots in one hemisphere are opposite to those of the leading spots
in the other hemisphere; (2) the predominant sunspot polarities reverse after the solar
minimum. Taking into account this polarity reversal, the solar magnetic field has a period
of about 22 years, also known as the Hale cycle. The axes of bipolar sunspot pairs are
slightly tilted by about 4◦ with respect to the equator (Howard 1991), leading spots being
closest to the equator (Joy’s rule). Tilt angles and the relative size distribution of active
regions are almost invariant throughout the cycle (Harvey & Zwaan 1993). The only
exception is a period of 2–3 years before solar minimum, during which small oppositepolarity ephemeral regions emerge at high latitudes (Harvey 1993). Ephemeral regions
have a typical lifetime of a few days, and they have a weak but significant preference
for the same orientation as that of sunspots. They follow the Hale cycle with a minimum
preceding the solar minimum by about 1 year. One may conclude from these features
that ephemeral regions partly originate as recycled flux from active regions, and are
partly generated locally in the convection zone (Schrijver et al. 1997).
A typical sunspot cycle is characterized by a sharp rise from minimum to maximum,
lasting 3–6 years (on average 4.8). The duration of the rise phase is anticorrelated with
the height of the maximum (Waldmeier’s rule). The maximum is followed by a gradual
decline lasting 5–8 years (on average 6.2). Individual sunspot cycles since 1710 lasted
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between 7 and 14 years (on average 11.0) and had amplitudes in the range 38 ≤ R̄ ≤ 201
(on average 105); the issue of solar-cycle variability is treated more fully in § 2.9.
Conventionally, the sunspot cycle that began in 1755 is referred to as cycle 1; 1997
saw the beginning of cycle 23.As can be seen in Fig. 3, the leading polarity on the northern
hemisphere during cycle 22 was negative; from this and the Hale-Nicholson rules one
can infer the predominant polarities for both hemispheres in every cycle. For nearly all
known cycles the amplitude of any odd-numbered cycle exceeds that of the preceding
even-numbered cycle (Gnevyshev & Ohl 1948). This might be seen as evidence for the
existence of a dipolar relic magnetic field in the radiative core of the Sun (§ 5.1).
2.1.2. Other solar-cycle indices
The solar cycle is visible not only in magnetic features but in many of the Sun’s observables including irradiance (Fröhlich & Lean 1998), surface flows (§ 2.7), coronal shape
(Bravo et al. 1998), and oscillation frequencies (Woodard & Libbrecht 1993; Elsworth et
al. 1994; Jiménez-Reyes et al. 1998). The modulation amplitude varies widely between
different indices. It is minute in visible light, and this explains why the solar cycle is
not obvious to a casual naked-eye observer. In the far-ultraviolet and X-ray range the
modulation amplitude is large (Feminella & Storini 1997). Most indices vary roughly
in phase with the sunspot number.
Solar-cycle modulations can also be measured at the Earth. The AA and AP indices
of the geomagnetic field are correlated with the solar cycle (Gonzalez & Schatten 1987).
This is attributed to compression of the Earth’s magnetopause due to coronal mass
ejections, the number of which varies in phase with the solar cycle (Hildner et al. 1976).
The magnetopause acts as a shield for cosmic rays, as a result of which the cosmic-ray flux
at the Earth exhibits a solar-cycle modulation of typically 50%. This subsequently affects
the rate at which radio isotopes are produced in the upper atmosphere. Measurements
of the concentration of 10 Be in arctic ice cores (Beer et al. 1990) and of 14 C in tree rings
thus enable us to study the history of solar activity (§ 2.9). Due to the long atmospheric
storage time, the solar-cycle variations in the 14 C data are attenuated by about two orders
of magnitude; those in the 10 Be data are only weakly attenuated (Beer 2000).
2.1.3. Butterfly diagram
The latitudinal distribution of sunspots as a function of time can be viewed from the
butterfly diagram (Fig. 2). Sunspots form two belts parallel to the equator, whose midpoints migrate equatorward from about ±27◦ to about ±8◦ during the course of a cycle
(Spörer’s law), and whose widths attain a maximum of about 36◦ during the sunspot maximum. During the Maunder minimum however, the activity belts did not extend beyond
about 20◦ of latitude, and the northern belt was almost absent (Sokoloff & Nesme-Ribes
1994). The polarities of the northern and southern wings of the butterfly diagram are
opposite, and they alternate from one sunspot cycle to the next. Sunspots exist for at
most a few months, and they hardly migrate themselves, but in the course of a cycle,
every new sunspot appears on average at a lower latitude.
The butterfly diagram is an important diagnostic of the solar dynamo, but a naive
comparison with magnetic field intensities obtained from dynamo calculations can be
misleading. Since sunspots are the product of flux tubes rising from the magnetic layer
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291
Fig. 2. Butterfly diagram of sunspot activity (courtesy D.H. Hathaway)
at the bottom of the convection zone (§ 5.3), the solar butterfly diagram reflects dynamo
action in that location. Secondly, the distribution of the deep-seated magnetic field is
masked by the process of flux emergence. The stability of the flux tubes depends on the
magnetic field intensity, latitude and other parameters. Hence the latitudinal distribution
of sunspots does not directly reflect that of the magnetic field. Paradoxically, it appears
that the differential rotation and the α effect provide more favorable conditions for
dynamo action at high latitudes than at the latitudes of sunspot emergence (§ 6.3.1).
Thirdly, flux emergence occurs as a series of random discrete events. This gives rise to
additional irregularity in the sunspot cycle (Ruzmaikin 1997) that must be entangled
from any intrinsic variability.
2.2. Deviations from symmetry
The large-scale solar magnetic field is predominantly axisymmetric and dipolar, which
is readily explained in terms of α-type dynamo action (§ 3.4.4). Sunspots and active
regions have a tendency to emerge near existing active regions. These loci of flux emergence, also referred to as active nests or longitudes, may live up to 6 months (Gaizauskas
et al. 1983; Brouwer & Zwaan 1990). They amount to small nonaxisymmetric contributions to the large-scale magnetic field that can be interpreted as non-axisymmetric
dynamo modes (§ 3.4.4). Such modes can be excited by stochastic or nonlinear effects
(§ 5.6; § 5.7).
There have been small but significant asymmetries between sunspot activity in the
northern and southern hemispheres (Newton & Milsom 1955; Howard 1974; Vizoso
& Ballester 1990; Temmer et al. 2002). One such event concerns the years 1955-1965
(Fig. 2). North-south asymmetries may be seen as evidence for a phase difference between the magnetic activity in both hemispheres (Waldmeier 1971; Swinson et al. 1986).
The effect is larger than average during solar minima (Carbonell et al. 1993), and was
particularly strong during the Maunder minimum, when the few observed sunspots were
concentrated on the southern hemisphere (Sokoloff & Nesme-Ribes 1994). Such features can be explained by interference between the dominant dipolar mode of the solar
dynamo and modes with quadrupolar symmetry with respect to the equator. Quadrupolar
modes can also be excited through nonlinear or stochastic effects. Dynamo calculations
suggest that in extreme cases, this may result in the near vanishing of the magnetic field
in one hemisphere (§ 6.2).
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2.3. Magnetic network
Magnetograms reveal an extended pattern of magnetic fields forming the magnetic network (Fig. 3). Near the solar maximum, the magnetic network at low latitudes consists
mainly of large unipolar regions in the vicinity of the bipolar active regions. Around the
solar minimum, the network is much less pronounced, and has mainly mixed polarities.
Its geometry can be described by a fractal dimension (Balke et al. 1993; Tao et al. 1995).
The magnetic network is predominantly made up of vertically oriented flux elements.
The formation of the magnetic network is attributed to diverging convective motions
that sweep up the magnetic elements originating from active regions and ephemeral regions into the granular and supergranular lanes. This explains why the magnetic network
roughly coincides with the supergranular network. The field strength of the magnetic
elements is a rather uniform 0.15 T, comparable to that in sunspot plages. This value has
no connection with the dynamo mechanism, but the number and polarity imbalance of
the magnetic network elements do vary with the solar cycle.
Fig. 3. Magnetograms taken respectively at 23-8-1990, near the maximum of cycle 22, and at
15-10-1996, near the subsequent minimum. Black (white) indicates magnetic regions of negative
(positive) polarity (courtesy NSF’s National Solar Observatory, USA)
The magnetic elements perform a random motion across the network that can be
described approximately by scalar diffusion in two dimensions, with r 2 = Dt/4.
Schrijver et al. (1996) have pointed out that the dispersal rate increases with decreasing
flux of the element, suggesting the use of an effective diffusion coefficient weighted with
the flux distribution function. This leads to D ≈ 6 · 108 m2 s−1 , similar to the result of
Simon et al. (1995), and compatible with what is required in flux transport models in
order to reproduce the magnetic flux distribution at the solar surface (§ 2.5). A smaller
value of about 2 · 108 m2 s−1 is inferred from local flux dispersal in network and plages
(Schrijver & Zwaan 2000: § 6). This difference probably reflects in situ disappearance of
flux, not included in the latter value. Turbulent diffusion of magnetic fields plays a crucial
role in the solar dynamo (§ 3.4.4), and the coefficient D is prominent in the Babcock-
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Leighton dynamo model (§ 6.3.4). However, the value of D is not representative for the
turbulent magnetic diffusivity (ηt ) in the convection zone, because magnetic diffusion
is a threedimensional process involving a vector rather than a scalar, and ηt depends on
the convective velocity, which decreases downward. Mean-field dynamo models provide
several further arguments for ηt being smaller than D by at least an order of magnitude
(§ 6.3.1).
2.4. Intranetwork magnetic fields
In between the network there is a weak, mixed-polarity magnetic field with random
orientations (Martin 1988). Its intensity has not yet been reliably established, because
the result depends on how well resolved this intranetwork magnetic field is assumed to
be (Stenflo 1982). Stenflo & Lindegren (1977) provide un upper limit for the apparent
field strength of about 9 mT. Keller et al. (1994), Lin (1995), and Lin & Rimmele (1999)
conclude from infrared observations that the intranetwork field is dominated by discrete
magnetic elements with a diameter of about 70 km and an intrinsic field strength of the
order 50 mT. Sánchez-Almeida & Lites (2000) and Socas-Navarro & Sánchez-Almeida
(2002) explain the measurements in terms of unresolved magnetic fields with a strength
in the 0.1 T range. However, Lites (2002) has argued that the analysis of the Hanle
depolarization effect does not justify the inference of unresolved strong-field elements
with a small filling factor. Rather, the measurements suggest that the filling factor of
the intranetwork field cannot be less than about 0.3–0.5, so that the intranetwork must
contain predominantly space-filling, intrinsically weak magnetic fields with a broad
distribution of field intensities centered around a value of at most a few times 10 mT
(Lin 1995; Collados 2001).
Apart from the spatial distribution, field strength and random orientation, also the
temporal behaviour and polarity imbalance serve to distinguish intranetwork from network magnetic fields. The dynamical time scale of the intranetwork flux is a few days,
similar to that of the ephemeral regions, which is short compared to the evolution time
of the network (weeks to months). The flux imbalance, defined as the ratio of the difference between the fluxes of both polarities to their total, is typically much smaller for
the intranetwork than for the network (Lites 2002). This indicates that the intranetwork
magnetic field has mainly mixed polarities and does not contribute to the Sun’s global
magnetic field. Although no systematic study of the solar-cycle dependence of the intranetwork flux has been done, the available evidence suggests that there is none. Harvey
(1993) found that the solar-cycle modulation of the magnetic flux for a given range of
apparent field strengths decreases with decreasing field strength from a factor 30–40 for
regions with Bapp 2.5 mT to a factor 1.5–2 for regions with Bapp 2.5 mT.
Altogether, the observations of intranetwork fields and ephemeral regions suggest
that small-scale dynamo action is taking place in the upper layers of the convection zone,
and presumably in the entire convection zone, to some degree independently of the solar
cycle (§ 3.8). A similar conclusion was reached by Lawrence et al. (1993) on the basis
of the scaling behaviour of quiet-Sun magnetic fields.
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Fig. 4. Butterfly diagram of the longitudinally averaged surface magnetic field (courtesy D.H.
Hathaway)
2.5. Flux transport and the polar field reversal
2.5.1. Poleward migration of magnetic fields
All magnetic structures at the solar surface, except those very close to the equator,
exhibit a slow poleward migration, with a maximum velocity of about 10 m s−1 (Bumba
& Howard 1965; Duvall 1979; Howard & Labonte 1981; Ulrich et al. 1988; Snodgrass
& Dailey 1996; Wöhl & Brajša 2001). Even though the poleward migration is detected
in all magnetic features, sunspots form an equatorward branch in the butterfly diagram
because their mean locus of emergence migrates towards the equator during the course of
a solar cycle. At high latitudes, where there are no sunspots, the magnetograms do exhibit
a poleward branch (Fig. 4). Doppler measurements suggest that the poleward migration
corresponds to a large-scale meridional flow at the solar surface. It therefore appears that
the physical origin of the poleward branch in the solar butterfly diagram is different from
that of the equatorward branch. Whereas the equatorward branch reflects circumstances
in the deep-seated dynamo layer, the poleward branch is a surface phenomenon. Hence
there is no reason to bring the poleward branch into connection with a sign change
of the radial differential rotation in the tachocline at mid latitudes (§ 4.1). Dynamo
theory suggests that this sign change can result in a reversal of the propagation of the
dynamo wave, provided that the dynamo is in a regime where meridional circulation
has a negligible influence (§ 3.4.4). However, it seems likely that meridional circulation
is important (§ 6.1), and it is possible that the equatorward and the poleward branches
both reflect the meridional circulation, but evaluated at different depths in the convection
zone.
The weak polar magnetic field has mainly one polarity at each pole, and the two poles
have opposite polarities. The polar magnetic field follows the solar cycle but, consistent
with the polar branch, it reverses during the solar maximum. Its polarity is such that
between solar maximum and solar minimum it agrees on each hemisphere with that of
the following spots of bipolar sunspot pairs (Fig. 4). At lower latitudes there is a weak
net radial surface field that, if averaged in a suitable way, is roughly in antiphase with the
solar cycle, i.e. Br Bφ < 0 (Schlichenmaier & Stix 1995). Unlike the polar magnetic
field, this may reflect a property of the deep-seated large-scale magnetic field.
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2.5.2. Flux transport models
Confirmation of the superficial nature of the poleward branch is gained from flux transport models. They are capable of reproducing rather well the surface distribution of
magnetic flux by incorporating sources in the form of active regions, diffusion across
the supergranular network and advection by the poleward meridional flow (DeVore et
al. 1984; Wang et al. 1989; Wang & Sheeley 1994). The main result of such models is
that the flux distribution at high latitudes (Fig. 4) can be explained by dispersal of flux
originating within the activity belts. Due to the tilt angle between leading and following spots, a surplus of following-polarity flux accumulates at the poles, thus leading to
the polar reversal close to the solar maximum. Schrijver (2001) has formulated a more
sophisticated flux transport model that aims to reproduce the measured flux distribution
at all scales, taking into account the flux dependence of the diffusion coefficient. He
obtains good agreement with magnetograms for the surface-integrated flux distribution,
irrespective of the values for the meridional circulation and the differential rotation.
Although the surface transport models yield an adequate description of the flux
distribution, they suffer from a number of difficulties. First, the vectorial nature of the
magnetic field is not taken into account. It seems questionable whether the dispersal of
flux from active regions can be treated as scalar diffusion at all scales. This approximation
must become increasingly problematic at large distances from the parent active region,
unless the magnetic elements originating from it become detached from the common
deep-seated footpoint to which they are initially linked (Wilson et al. 1990; Wilson &
McIntosh 1991; Wilson 1992). It is still unclear which physical mechanism is responsible
for overcoming the tension forces; perhaps reconnection or flux elimination in small
subducted loops (Schrijver & Zwaan 2000: § 6). Martens & Zwaan (2001) have proposed
a detailed mechanism for the flux dispersal from active regions based on reconnection
events in tilted prominences. In the vicinity of plages the diffusion is retarded, such that
the mean square distance grows as r 2 ∝ t 2/d , where d = 2.3 (Lawrence & Schrijver
1993). This might be caused by the subsurface connections with the parent active region.
Dikpati & Choudhuri (1994) modeled the advection of the poloidal magnetic field
using an equatorward propagating dynamo wave as the lower boundary condition. Their
results confirm that a poleward meridional flow in the upper part of the convection zone
results in a poleward branch in the butterfly diagram. This raises the question how much
the active regions and the deep-seated magnetic field each contribute to the poloidal
magnetic field at the solar surface.
A second problematic feature might be that the magnetic flux at high solar latitudes
appears to be replenished on a time scale of several days, much more rapidly than the
time scale for transport by surface diffusion and meridional flow (Stenflo 1992; Petrovay
& Szakály 1993). This would speak against the assumption that active regions are responsible for maintaining the global surface field. On the other hand, the inclusion of
even a large number of randomly oriented bipolar ephemeral regions does not significantly affect the flux transport (Wang & Sheeley 1991), and one may argue that a high
emergence rate of mixed-polarity magnetic fields could mask a long time scale, giving
the impression of a rapid renewal of the polar field.
The fact that the polar field reversal takes place before the solar minimum may suggest
that the polar magnetic field has a causal relation with the toroidal magnetic field of the
next cycle. This interpretation lies at the foundation of the Babcock-Leighton dynamo
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scenario, which can be viewed as a three-dimensional extension of the surface flux
transport models (§ 6.3.4). However, the polar magnetic field is connected to the global
poloidal magnetic field, and this acts as a source for the toroidal magnetic field in any
α-type scenario (§ 3.4.4).
2.6. Current helicity, magnetic helicity and kinetic helicity
Rotation imparts flows and magnetic fields in the solar convection zone with a systematic
twist that plays an important role in the dynamo (§ 3.4.3). The current helicity, J · B,
and the kinetic helicity, u · curl u, are measures of twist that can be estimated from
observations of the solar surface. Inasmuch as the surface values are representative also
for the interior of the convection zone, they contain clues about the solar dynamo. First,
the kinetic and current helicities can be directly compared with numerical simulations
of dynamo action in rotating convection. Secondly, they provide information about the
sign and magnitude of the α effect, which is an essential ingredient of the solar dynamo
(§ 3.4; § 5.5).
The radial component of the current helicity, Jr Br , has been estimated from vector
magnetograms of active regions. For a given latitude, the measured values form a broad
distribution centered around a mean value that is negative on the northern hemisphere,
and positive on the southern hemisphere (Seehafer 1990; Pevtsov et al. 1994; Abramenko
et al. 1996; Bao & Zhang 1998; Longcope et al. 1998; Pevtsov & Latushko 2000; Pevtsov
et al. 2001). The sign rule is not significantly different for different classes and sizes
of sunspots (Pevtsov et al. 1995). The findings for the current helicity are consistent
with the long-established observation that Hα structures have a preferred orientation
corresponding to a left-handed screw on the northern hemisphere and an opposite one
on the southern hemisphere (Hale 1927; Richardson 1941; McIntosh 1981; Martin et
al. 1992; Zirker et al. 1997). X-ray observations of coronal loops exhibit the same
hemispherical rule (Rust & Kumar 1996; Canfield & Pevtsov 1999), as is the case for
coronal mass ejections (Low 1996). The latter may be an important agent in expelling
magnetic helicity (Low 2001).
Observationally, the magnetic helicity, A · B (§ 3.3.2), is a difficult quantity, since
it cannot be measured directly. All the available indirect evidence suggests that it is
negative on the northern hemisphere of the Sun (Berger & Ruzmaikin 2000; Chae 2000).
On the other hand, MHD simulations and mean-field calculations indicate that the sign
of the large-scale magnetic helicity should be positive on the northern hemisphere if the
dynamo coefficient αφφ is also positive there (§ 3.4.5). These results may be reconciled
if the measured magnetic helicity corresponds to an intermediate scale, and if there is a
sign change at some larger scale, as is found to be the case in numerical simulations of
helical MHD turbulence (§ 3.3).
Duvall & Gizon (2000) measured the quantity (curl u)z /div u on the solar surface,
which is a proxy of the kinetic helicity; it is negative on the northern hemisphere, and
peaks at the poles. This is expected for the bulk of the convection zone, and agrees
with results from MHD simulations (Brandenburg et al. 1990; Ossendrijver et al. 2001).
From the measurements of current and kinetic helicities one can infer that the dynamo
coefficient αφφ is positive on the northern hemisphere (Kuzanyan et al. 2000).
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2.7. Torsional oscillations
Doppler measurements reveal a pattern of parallel belts of faster and slower rotation that
is correlated with the butterfly diagram (LaBonte & Howard 1982). Their amplitude is
5–10 m s−1 , which corresponds to 0.25–0.5% of the surface rotation, and the regions
of maximal shear coincide more or less with sunspot latitudes (Fig. 5). Helioseismic
observations have revealed that the disturbances extend downward to at least 60 Mm
(Howe et al. 2000b). It has been claimed that the disturbances are delayed with respect to
the sunspot cycle by 20 yrs (Yoshimura & Kambry 1993). Whereas these belts were once
thought of as being uninterrupted during the 16–20 years of their migration from pole
to equator, persistent gaps at mid latitudes now suggest that their pattern is in fact more
complex (Snodgrass 1992). Most likely, they can be interpreted as torsional oscillations
caused by Lorentz forces (§ 5.7). If this explanation is correct, then their continuation to
high latitudes may be interpreted as evidence for dynamo action outside of the sunspot
belts.
Fig. 5. Contours of the longitudinally and temporally averaged deviations of the surface rotation rate in a latitude-time diagram. Dark
(light) signifies slower (faster) rotation with
respect to the mean rate for the corresponding
latitude (courtesy NSF’s National Solar Observatory, USA)
2.8. Global magnetic modes
Stenflo & Vogel (1986) carried out a decomposition of the Sun’s surface field in terms
of spherical harmonics, and found that the temporal behaviour of axisymmetric (m = 0)
modes with odd (i.e. antisymmetric with respect to the equator) differs from that of
the axisymmetric modes with even . Whereas the odd- modes are dominated by the
22-year cycle, the even- modes form a power ridge that extends to periods as short
as ≈ 1.4 yr. No such difference between even and odd- modes is observed for nonaxisymmetric modes (m = 0), which have a main peak at the 22-year cycle, and smaller
peaks at its higher harmonics, almost irrespective of (Stenflo & Güdel 1988). A 155-day
periodicity in sunspot areas has also been reported (Carbonell & Ballester 1992). These
observations can be interpreted as evidence for the excitation of multiple axisymmetric
dynamo modes with periods shorter than the 22 year period of the fundamental mode.
Gokhale & Javaraiah (1990, 1992) and Gokhale et al. (1992) could however not confirm
the existence of the power ridge for axisymmetric odd- modes. Altogether, the evidence
for solar dynamo modes with periods shorter than 22 years is therefore inconclusive.
Possible explanations for the presence of higher dynamo modes are stochastic excitation
by convection (§ 5.6) or nonlinear effects (§ 5.7).
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2.9. Long-term modulations and solar variability
The sunspot cycle exhibits modulations on several time scales, the main one being
the 80–90 year Gleissberg cycle (de Meyer 1998). The Gleissberg cycle has also been
identified in variations of the magnetic equator of the Sun (Pulkkinen et al. 1999). From
auroral observations the occurrence of a phase shift in the Gleissberg cycle of about
35 years during the Maunder minimum has been inferred (Feynman & Gabriel 1990).
There is evidence for the Gleissberg cycle and for a 205-year periodicity in the records
of 14 C and 10 Be (Beer et al. 1994; Beer 2000), but the periodic nature of the modulations
remains somewhat uncertain.
During the Maunder minimum (1645–1715) sunspots were very few and, prior to
about 1670 no cycles are apparent (Sokoloff & Nesme-Ribes 1994). Coverage of the
sunspot record in this period is estimated to be about two thirds, and the reality of
the Maunder minimum is undisputed (Eddy 1983; Ribes & Nesme-Ribes 1993; Hoyt &
Schatten 1996). This and earlier grand minima such as the Spörer minimum (1420–1530)
are clearly visible in the 14 C and 10 Be data (Eddy 1988; Stuiver & Braziunas 1988; Beer
2000). Timing and duration of the known grand minima are irregular, but the Sun and
solar-type stars may spend as much as a third of their time in grand minima (Baliunas et
al. 1995). Surprisingly, the cosmogenic indicators suggest that the solar cycle continued
throughout the entire Maunder minimum at a reduced level (Beer et al. 1998). This might
speak against the solar dynamo being driven by magnetic buoyancy (§ 6.3.3). However,
a thorough statistical analysis has shown that the 10 Be data are not easy to interpret,
especially during grand minima (Fligge et al. 1999). Further confirmation is needed in
order to establish beyond doubt the possible continuation of the sunspot cycle during
grand minima. More recent but less impressive prolonged minima, visible in Fig. 1, are
the Dalton minimum (1800–1830) and the Modern minimum (1880–1910).
Solar variability is also apparent in the length and amplitude of sunspot cycles. There
is a tendency for long cycles to be followed by short cycles such that the phase of the
cycle is maintained within a certain bandwidth (Dicke 1978). However, the statistical
significance of the evidence for phase locking is still marginal (Gough 1987). If existent,
phase locking could be used to test solar dynamo models because it would be incompatible with the prediction of, for instance, an unrestricted random walk (§ 5.6). The
question whether solar variability should be ascribed to (quasi-)periodic or chaotic nonlinear behaviour on the one hand, or to stochastic processes on the other hand remains
as yet unresolved due to the lack of a sufficiently long and accurate dataset. In order
to answer such questions, it may be helpful to extract continuous phase and amplitude
functions from the sunspot record (Mininni et al. 2002).
2.10. Solar-cycle prediction
From the socio-economic point of view, the main relevance of studying the solar dynamo
lies in producing a tool for predicting future solar magnetic activity. As of yet, there is
no model of the solar dynamo that is able to serve that goal, and one should consider
the possibility, suggested by the theory of nonlinear dynamics, that no such model will
ever be able to predict solar activity by more than a few years in advance. From a
more practical point of view though, the prediction of solar activity can be pursued
The solar dynamo
299
as a heuristic discipline, without any recourse to dynamo theory. Along these lines,
much effort has been put into the analysis of time series and the identification of solar
diagnostics with a predictive value. Nevertheless, the findings may give clues about the
dynamo mechanism.
Prediction of future magnetic activity within the current cycle can be done with some
success by a curve-fitting method, as is suggested by Waldmeier’s rule (Hathaway et al.
1994). Results for long-term predictions are much less convincing (Layden et al. 1991).
By analyzing time series of sunspot numbers or other indicators it is possible to derive
some form of decomposition that can be used as a basis for extrapolating into the future
(de Meyer 1998). The results are doubtful, because not all relevant periodicities may be
identifiable in the limited available data. In addition, the solar cycle may not be strictly
periodic but chaotic. Even in that case some form of extrapolation may be possible if
the relevant properties of the nonlinear oscillator can be derived (Kurths & Ruzmaikin
1990; Kremliovsky 1994; Zhang 1996). The nonlinear method of Sello (2001) is able
to predict quite accurately solar activity within the current cycle by several years. Other
successful techniques are based on neural networks (Calvo et al. 1995; MacPherson et
al. 1995; Conway et al. 1998).
Alternatively one may try to identify phenomena that act as precursors to future
solar activity. Layden et al. (1991) conclude that the best predictive diagnostic for the
amplitude of the following solar maximum is provided by the geomagnetic AA and AP
indices. Some methods combine timeseries analysis with geomagnetic data (Hanslmeier
et al. 1999). The predictive value of the geomagnetic indices can be seen from the fact that
they are strongly correlated with the sunspot number, while their maximum precedes the
solar maximum by about 3–4 years. Hence the geomagnetic field is roughly in phase with
the Sun’s polar magnetic field, which is connected to the Sun’s global poloidal magnetic
field. The predictive nature of the geomagnetic indices may therefore be a consequence
of the fact that the Sun’s poloidal magnetic field acts as a source for the deep-seated
toroidal magnetic field. This is consistent with an α-type dynamo mechanism (§ 3.4.4).
3. Solar dynamo theory
3.1. Formulation of the dynamo problem
The magnetic field of the Sun is generated by dynamo action in its conducting interior.
Alternative, non-dynamo explanations of the solar magnetic field have been proposed,
but they can be virtually ruled out (§ 5.1). Dynamo action turns out to be a rather
common phenomenon in the cosmos, and has been invoked to explain magnetic fields in
widely different objects ranging from planets to interstellar clouds and galaxies. Before
discussing the specifics of the solar dynamo, it is necessary to provide a more general
theoretical framework. The dynamo problem can be formulated mathematically as a
quest for solutions with a non-decaying total magnetic energy of an appropriate set of
equations (Moffatt 1978: § 6; Krause & Rädler 1980: § 11). Certain plausible restrictions
must be added in order to exclude pathological cases (Roberts 1967: § 3; Roberts 1994),
i.e. (1) the relevant induction effects are confined to a compact volume V with boundary
S, so that the magnetic field, B, is generated within V . (2) The flow, U , is regular; e.g.
∂Ui /∂xj and V dVρU 2 /2 are finite for all t, and n · U = 0 on S.
300
M. Ossendrijver
Depending on the physical circumstances of the dynamo, the governing equations
can be different. The conditions in the bulk of the solar convection zone are such that
(1) the plasma is highly ionized and the mean free path of photons is very short, so
that radiation can be treated in the diffusive approximation. (2) The fluid velocities are
subrelativistic so that the Galilean transformations can be used and the displacement
current can be ignored in the Maxwell equations. (3) The collision frequency is very
high and the plasma is quasineutral, so that Ohm’s law is valid in its simplest form, i.e.
J = σ (E + U × B), where σ is the (Spitzer) electrical conductivity. The appropriate
description of dynamo action in the Sun under these conditions is provided by the
equations of magnetohydrodynamics (MHD) in a rotating frame. They consist of the
MHD induction equation plus the equations for mass continuity, momentum (NavierStokes) and internal energy,
∂B
∂t
dρ
dt
dU
ρ
dt
de
ρ
dt
= ∇×(U ×B − ηµ0 J ) ,
(1)
= −ρ ∇·U ,
(2)
= −∇p + ρg + J ×B − 2ρ ×U + 2∇·νρS ,
(3)
= −p∇·U + ∇·
λrad
∇e + 2νρ S2 + ηµ0 J 2 ,
CV
(4)
where J = ∇ × B/µ0 , d/dt ≡ ∂/∂t + U · ∇, and Sij ≡ 21 (∂Ui /∂xj + ∂Uj /∂xi ) is the
kinetic stress tensor. Eq. (1) is supplemented by the auxiliary relation ∇ · B = 0. The
governing parameters are the magnetic diffusivity η ≡ (µ0 σ )−1 , the vacuum permeability µ0 , the kinetic viscosity ν, and the radiative conductivity λrad = 16σSB T 3 /(3κρ),
where σSB is the Stefan-Boltzmann constant and κ is the opacity. One may also employ
the thermometric conductivity, which is defined as χ ≡ λrad /(ρCp ). Frequently used
dimensionless parameters characterizing convection and dynamo action are summed up
in Table 1. Representative values for the Sun are given in Table 2.
A few remarks should suffice to sketch some important aspects of the parameter
regime of the solar dynamo. In the convection zone, the superadiabaticity parameter ∇
is positive, which signifies instability according to the Schwarzschild criterium; in the
radiative core ∇ is negative. In reality, viscous effects cause the convective instability
to set in only if Ra exceeds a positive critical value. The solar convection zone is highly
stratified, and compressibility has a large effect on the flow (§ 4). Incompressible or
Boussinesq computations are inadequate for solar convection, although they can serve
to increase our understanding of magnetoconvection or illustrate elementary dynamo
mechanisms (§ 3.8.1). The smallness of ∇ and Ma in the bulk of the convection
zone allows one to use the anelastic approximation, which is briefly addressed in § 4.2.
Possible effects of the smallness of Pr and Pm are briefly mentioned in § 4.3.
In the solar photosphere the mean free photon path is not short, so that radiation
cannot be treated in the diffusive approximation and one has to incorporate radiative
transfer and effects of incomplete ionization. This is known to influence the distribution
of magnetic flux in the photosphere, but it seems reasonable to assume that the effect on
the solar dynamo is negligible. This assumption is not uncontested because downflowing
The solar dynamo
301
Table 1. Dimensionless parameters characterizing convection and dynamo action
parameter
measure of
∇ ≡ ∇ − ∇ad
Ra ≡ g∇d 4 /(νχHp )
Re ≡ U L/ν
Rm ≡ U L/η
Pr ≡ ν/χ
Pm ≡ ν/η
Co ≡ 2L/U (a)
Ta ≡ (2d 2 )2 /ν 2
S ≡ U τc /L
Ma ≡ U/cs
β ≡ 2µ0 p/B 2
superadiabaticity
Rayleigh nr.
Reynolds nr.
magn. Reynolds nr.
Prandtl nr.
magn. Prandtl nr.
Coriolis nr.
Taylor nr.
Strouhal nr.
Mach nr.
plasma β
Schwarzschild instability
thermal instability
hydrodynamic turbulence
ratio of adv. to diff. of B
ratio of smallest therm. to kin. scales
ratio of smallest magn. to kin. scales
rotational influence on flow
(de)stabilising effect of rotation
ratio of corr. time to turnover time
ratio of flow speed to sound speed
ratio of gas to magn. pressure
(a) Identical to the inverse Rossby number. However, in some publications on stellar activity
the Rossby number is defined as Prot /τc = 4π/Co
Table 2. Representative values of dimensionless parameters in the Sun
parameter (a)
base of convection zone
photosphere
∇
Ra
Re
Rm
Pr
Pm
Co
Ta
Ma
β
10−6
1020
0.5
1016
1012
106
10−7
10−6
2 · 10−3 · · · 0.4 (b)
1019
1
1 (d)
1013
1010
10−7
10−3
15
1027
10−4
105 · · · 107 (c)
(a) Unless stated otherwise, estimated by setting L ≈ H
p
(b) Lower value: granulation; upper value: supergranulation
(c) Magnetized plasma with 1 B 10 T
(d) Sunspots and magnetic elements
plumes, which play an important role in solar convection, originate in the photosphere
(§ 4.3).
Alfvén’s theorem asserts that for Rm 1 the magnetic field is frozen into the fluid to
good approximation, except on very small scales. Nevertheless, it will become apparent
in § 3.3 and elsewhere that Ohmic dissipation cannot in general be ignored in dynamo
theory. The correlation time of solar convection is not short (S ≈ 1), and this causes
serious though not necessarily insurmountable difficulties for mean-field theories of the
solar dynamo (§ 3.4). Unless stated otherwise, it will be tacitly assumed that S ≈ 1, so
that τc refers to the correlation time as well as the turnover time (L/U ).
302
M. Ossendrijver
The MHD equations are to be supplemented by boundary conditions (Roberts 1967:
§ 1; Jackson 1975: § 1; DeLuca & Gilman 1986). This is less trivial than apparent at first
sight, and the correct treatment for the solar dynamo is not always easily established,
especially at the external boundary.
3.2. Restrictions and conditions on dynamo action
Dynamo action faces several obstructions in the form of conditions and anti-dynamo
theorems. Magnetic field generation can be seen as an instability of conducting fluids
leading to growth of the magnetic field. A necessary but insufficient condition for the
dynamo instability is that advection dominates over diffusion, i.e. Rm 1. Backus
(1958) obtained a rigorous necessary condition for dynamo action in a sphere of incompressible fluid with constant η subject to the boundary condition U = 0 at the
surface (Moffatt 1978: § 6). The condition amounts to a lower bound on the small-scale
magnetic Reynolds number, Rms ≡ Smax /η ≥ π 2 , where Smax is the largest eigenvalue of Sij . Cowling (1934) proposed a famous theorem suggesting the impossibility
of magnetic field generation in axisymmetric dynamos. Cowling assumed stationarity
and presented an argument based on the impossibility of induction (U × B) to overcome
diffusion (ηµ0 J ) in the vicinity of a neutral line. Although his original argument was
incomplete, Cowling’s theorem has been proven since for an incompressible fluid with
constant η (Braginskii 1964a; Moffatt 1978: § 6; James et al. 1980). Hide & Palmer
(1982) generalized the neutral-line argument to the non-steady case, but their proof relies on additional assumptions. As of yet, no proof exists for more general situations
allowing for compressibility or variable resistivity (Ivers 1984).
Bullard & Gellman (1954) proved a theorem conjectured by Elsasser (1946) stating
the impossibility of dynamo action by purely toroidal motions for the case of an incompressible fluid with constant η in a sphere. This was generalized to compressible flows
and non-uniform η by Ivers & James (1988). Zel’dovich (1957) and Moffatt (1978)
proved the impossibility of dynamo action by motions in flat planes. Ivers & James
(1986) proved that a spherically symmetric radial flow cannot sustain a magnetic field.
The conclusion from these and similar theorems is that dynamo action is essentially
three-dimensional and requires complex, asymmetric flow fields, conditions that are
readily fulfilled in stellar dynamos.
Bondi & Gold (1950) have shown that dynamo action in a volume of highly conducting fluid would be largely invisible to an outside observer. This is because the magnetic
flux contained in any comoving surface patch of the dynamo is conserved for a perfect
conductor. Consequently the external magnetic dipole moment can change only through
a rearrangement of the magnetic patches, the result being optimal if flux of opposite
signs is swept onto opposite poles. As shown by Rädler & Geppert (1999), the theorem
of Bondi and Gold also applies to mean-field dynamo models. Since the magnetic dipole
moment of the Sun is known to be variable, this immediately tells us that the solar surface
cannot be a good conductor.
The solar dynamo
303
3.3. MHD turbulence
Due to the high value of Re, the flow in the solar convection zone is turbulent, and the
same holds for the magnetic field since it is well frozen into the plasma. This suggests
that the problem of the solar dynamo should be approached within the more general
framework of MHD turbulence. The complexities of Eqs. (1-4) are such that there is no
selfconsistent, comprehensive turbulence theory. Nevertheless, a number of basic mechanisms and properties are well-established, and they provide a phenomenological view
of turbulent dynamo action. Thus, in order to gain insight into the fundamental mechanisms of dynamo action in astrophysical plasmas, simplified MHD turbulence models
with controllable and well-defined properties are investigated using various analytical
and numerical tools. This is achieved by replacing the gravity force in the Navier-Stokes
equation, which drives convection, by a specified external force. Of course, the relevance
of such an analysis for the Sun must be established, which may not be easy.
3.3.1. Spectra and cascades
A typical feature of dynamo action in a rotating convecting fluid is that a large-scale
magnetic field is generated even though convection can be viewed as a predominantly
small-scale phenomenon. This suggests an approach to MHD turbulence based on transforming the relevant quantities to Fourier space, and studying the properties and evolution
of their spectra using closure models. The appropriate quantities are the invariants of
ideal MHD,
1
E≡
dV (ρ|U |2 + |B|2 /µ0 ), HM ≡ dV A · B, HC ≡ dV U · B, (5)
2
i.e. the total energy E, the magnetic helicity HM , and the cross helicity HC (Biskamp
1993). By applying the EDQNM closure method, Frisch et al. (1975) found that the
injection of kinetic helicity results in an inverse cascade of magnetic helicity leading to
a buildup of magnetic helicity at a large scale (Pouquet et al. 1976; De Young 1980).
There is an associated growth of magnetic energy at large scales that continues until the
magnetic field reaches a saturated state. This has been confirmed in numerical simulations of forced MHD turbulence (Pouquet & Patterson 1978; Meneguzzi et al. 1981;
Brandenburg 2001) and elementary helical flows (Gilbert & Sulem 1990).
The spectra of MHD turbulence can be schematically divided in an injection range, an
inertial range, where the injected flux is cascaded without significant dissipative losses,
and a range where dissipation becomes important. Each of these ranges is characterized
by spectral indices that depend on the details of the interaction between adjacent scales,
but their correct values are not easily established, due to theoretical and numerical
difficulties (Müller & Biskamp 2003). This is especially true for the solar plasma, which
is only weakly magnetized and highly intermittent.
Numerical simulations indicate that dynamo action occurs in helical flows if Rm
exceeds a critical value. Conversely, for a given value of Rm, dynamo action occurs
if the kinetic helicity exceeds a threshold value. The latter may merely reflect that the
critical value of Rm for dynamo action increases with decreasing kinetic helicity; it
may also be interpreted as suggesting that the inverse cascade sets in only if the flow is
sufficiently helical (Maron & Blackman 2002).
304
M. Ossendrijver
The inverse cascade can proceed from small scales to the scale of the mean magnetic
field while skipping an intermediate range of scales (Brandenburg 2001). In the nonlinear
regime of MHD turbulence, the inverse cascade can lead to complex magnetic energy
spectra with dominant peaks at intermediate length scales associated with intermittence
(Meneguzzi et al. 1981). Spatial intermittence also occurs in hydrodynamical turbulence,
without the presence of magnetic fields (She et al. 1990).
The dynamo behaviour depends strongly on the Prandtl number; for Pm 1 magnetic energy tends to pile up at small scales, thereby preventing the onset of an inverse
cascade. While this makes it difficult to understand how large-scale magnetic fields are
generated in galaxies (Kulsrud & Anderson 1992), dynamo action in the Sun proceeds
at small values of Pm, and here an inverse cascade is possible.
3.3.2. Magnetic helicity
Usually, the dynamo is thought of as a mechanism for generating magnetic fields. But
most dynamo mechanisms also produce magnetic helicity. Since magnetic helicity is an
ideal invariant, this can have severe consequences for the evolution of the magnetic field.
This is particularly true for dynamo action in rotating stars, because rotation enables the
generation of magnetic fields and magnetic helicity at a large scale. In the absence of
rotation, magnetic helicity conservation is not expected to play as large a role (§ 3.7;
§ 3.8).
Since HM is gauge dependent except in closed systems or if the boundary conditions
are periodic in all spatial directions, one usually resorts to the gauge independent relative
magnetic helicity (Berger & Field 1984), here also denoted by HM for convenience. The
evolution of the relative magnetic helicity, say for the northern hemisphere of the Sun,
is governed by a conservation law, derived from the MHD induction equation,
dHM
= QH − F H ,
dt
(6)
where QH ≡ −2 dV ηµ0 J · B represents Ohmic dissipation, which is controlled by
the microscopic resistivity η, and FH is the flux of magnetic helicity across the boundary
of V , not reproduced here (Brandenburg & Dobler 2001; Brandenburg et al. 2002).
One may distinguish two extreme cases. The boundary of the dynamo may be such
that a flux of helicity is not permitted. In that case HM can change only resistively, so
that for a sufficiently small η, dynamo action becomes impossible. In order to judge how
restrictive this would be in the Sun, where η is very small, one should determine the
scaling of QH with η. This is achieved by taking into account the approximate balance
between
the work done by the Lorentz force and Ohmic dissipation, i.e. WM = QJ ≡
dV ηµ0 |J |2 . Since this balance should exist irrespective of the value of η, and if one
assumes that WM and Brms are independent of η for η → 0, it follows that QH ∝ η1/2 .
Given the smallness of η in the Sun, it seems likely that a dynamo that generates magnetic
helicity on large scales, as is true for any mechanism based on an α effect, cannot comply
with the helicity constraint by relying only on Ohmic dissipation, although it cannot yet
be completely ruled out for a shallow layer near the surface, where η is large.
Alternatively, magnetic helicity may be disposed of by losses through the boundary
of the dynamo (Blackman & Field 2000b). Numerical simulations of helical MHD
The solar dynamo
305
turbulence but without allowing for such losses (Brandenburg et al. 2002) indicate that (1)
the small-scale contribution and the large-scale contribution to the magnetic helicity have
opposite signs (Seehafer 1994, 1996). They can each grow rapidly, i.e. on a convective
timescale, but not their total, which is controlled by the much longer timescale for Ohmic
dissipation. (2) The rapid growth of the magnetic helicity and the magnetic energy at
the large scale continue up to the point where the small-scale magnetic field reaches
saturation, after which their evolution is slaved by Ohmic dissipation, and therefore slow.
Based on these two tendencies it would appear that the removal of small-scale magnetic
helicity may allow the rapid evolution of the large-scale magnetic helicity and energy to
continue, thereby enabling the dynamo to work. This has indeed been shown to be true
in numerical experiments where the small-scale magnetic helicity is removed artificially
from within the volume of the dynamo by applying a Fourier filter. However, attempts
to demonstrate that a loss of small-scale magnetic helicity through a boundary of the
dynamo is able to achieve the same result have not been successful so far (Brandenburg
& Dobler 2001; Brandenburg et al. 2002). Chiueh (2000) has proposed a mechanism for
magnetic helicity losses based on rising flux tubes. Observational evidence for helicity
losses is provided by coronal mass ejections (Low 2001). The implications of magnetic
helicity conservation for mean-field dynamo theory are considered in § 3.4.5.
3.4. Mean-field theory of the solar dynamo
3.4.1. Equations and FOSA
The flow field in the solar convection zone is so complex that exact analytical solution
of the induction equation (1) is out of reach, even in the kinematic case. The main
concern of solar dynamo theory, however, is not to exactly reproduce the small-scale
structure, but to account for the large-scale magnetic field. In order to achieve this, one
may take an average, so that the small scales are washed out and knowledge of the
flow field is required only through its statistical properties. This has led to mean-field
electrodynamics, the foundations of which were developed by Steenbeck, Krause &
Rädler (1966). Extensive treatments are given by Krause & Rädler (1980) and Moffatt
(1978). It is advantageous to formulate mean-field dynamo theory in the framework of
stochastic differential equations (van Kampen 1976, 1992), as was done by Knobloch
(1978a,b) and Hoyng (1985, 1992), and write the induction equation in the general form
∂B
= DB = (D0 + D1 )B,
(7)
∂t
where D0 B = ∇ × (U 0 × B − η∇ × B) and D1 B = ∇ × (U 1 × B). The indices 0 and
1 refer to mean and fluctuating parts, e.g. B 0 ≡ B and B 1 ≡ B − B. By averaging
this equation, and subtracting the mean from the original, one obtains
∂B 0
= D0 B 0 + E,
∂t
∂B 1
= D0 B 1 + D1 B 0 + G,
∂t
(8)
(9)
where
E ≡ U 1 × B 1
(10)
306
M. Ossendrijver
is the turbulent electromotive force (EMF), and G ≡ D1 B 1 −D1 B 1 . The EMF results in
a contribution to the mean current given by σ E. On most occasions, the more convenient
notation u ≡ U 1 , b ≡ B 1 , and j ≡ J 1 will be used. In order to derive a closed equation
for B 0 , mean-field dynamo theory proceeds by assuming G ≈ 0, so that Eq. (9) can
be easily solved. This first-order smoothing approximation (FOSA) or second-order
correlation approximation (SOCA) can be justified if at least one of three conditions
is satisfied: (1) |G| |D1 B 0 | ⇔ |B 1 | |B 0 |; (2) |G| |D0 B 1 | ⇔ Rm 1 or
U1 U0 ; (3) |G| |∂B 1 /∂t| ⇔ S 1, which asserts that the correlation time must
be short compared to the turnover time. In the Sun, the first two conditions are far from
satisfied and, as long as S < 1, the third condition may be marginally satisfied, although
it may be necessary to consider higher-order corrections to FOSA (§ 3.4.6). If S ≥ 1, it is
not possible to compute the EMF using FOSA or any higher-order cumulant expansion,
but other methods may be successful (§ 3.4.6).
3.4.2. Interpretation of averages
Mean-field electrodynamics is a statistical theory and therefore a correct interpretation
of its results requires a careful examination of the averaging procedure that is adopted. A
consistent derivation of mean equations is possible only if for arbitrary functions f (r, t),
g(r, t) the averaging procedure satisfies the Reynolds rules (Krause & Rädler 1980: § 1):
f = f + f1 ;
f + g = f + g;
f g = f g.
(11)
From these rules it follows that f1 = 0 and f g = f g + f1 g1 . Also, · should
commute with differentiations and integrations with respect to t and r. The averaging
procedure may be temporal, spatial or based on an ensemble approach. Often, mean-field
theory is thought of as requiring a two-scale approximation in the spatial or temporal
domain. But this is incorrect, and if such a procedure is used to justify using a spatial
or temporal average, it must be kept in mind that it does not satisfy the Reynolds rules
exactly. Because solar convection is organized in a hierarchical structure with different
length scales and coherence times, any spatial or temporal average over an intermediate
scale contains only a finite number of independent realisations. If the Reynolds rules are
violated, this formally results in mean equations with rapidly fluctuating additive terms,
the exact form of which cannot be easily established (Hoyng 1987b, 1988).
A spatial average that does satisfy the Reynolds rules exactly is obtained if one or
more of the spatial coordinates is integrated out. For stellar dynamos, the most obvious
definition is the azimuthal average, i.e.
1
B ≡
dφ B,
(12)
2π
a procedure that goes back to Braginskii (1964a,b). The resulting mean-field dynamo
equation does not acquire additional terms, but the mean quantities are subject to fluctuations due to the finite correlation length of convection. These fluctuations may be
modeled by allowing the dynamo coefficients to have a random component (§ 5.6). By
averaging over such fluctuations, Silant’ev (2000) obtained a modified dynamo equation that can yield non-decaying solutions for the mean magnetic field for flows with
vanishing mean α.
The solar dynamo
307
The ensemble average, defined as
N
1 B i (r, t),
N→∞ N
B ≡ lim
(13)
i=1
also obeys the Reynolds rules and has the advantage that mean quantities do not acquire
fluctuating components, as long as the ensemble is conceived of as being infinitely large.
But the interpretation of mean-field theory changes in a subtle way, because the ensemble
average leads to a probabilistic description of the magnetic field (Hoyng 1987a,b, 1988,
2003). This can be understood with the help of a thought experiment. The solar cycle
is known to have frequency variations of the order δcyc / cyc ≈ 0.1 (Hoyng 1988),
and we may assume for the moment that δcyc varies in a random way. If the phases
of the ensemble members are synchronized at t = 0, their distribution is described by a
delta-function initially.As the ensemble members evolve with time, the phase distribution
broadens because in each dynamo the realisation of δcyc is different. This phase mixing
leads to increasing cancellations, and the conclusion is that the mean magnetic field
decays, even if the actual field does not. The decay time represents the coherence time
of the magnetic cycle, which is about 10 dynamo periods for the Sun. The mean-field
dynamo coefficients must be interpreted accordingly. For instance, turbulent diffusion
becomes a probabilistic effect that does not necessarily correspond to enhanced magnetic
diffusion within each ensemble member. Therefore, the existence of molecular diffusion
or magnetic reconnection is not a precondition, so that in the ensemble interpretation the
objections to turbulent diffusion raised by Piddington (1972) and Layzer et al. (1979)
do not apply already from first principles.
The question is whether in the nonlinear case a straightforward physical interpretation
of the ensemble-averaged magnetic field and the dynamo equation is still possible. If we
carry the thought experiment one step further it becomes clear that this may be difficult.
Suppose that all ensemble members are in a highly supercritical nonlinear saturated
state such that the magnetic cycles exhibit phase variations with a finite coherence time
leading to phase mixing. Irrespective of whether the variations are stochastic or nonlinear
in nature, the resulting exponentially decaying mean magnetic field is not reproduced by
a nonlinear supercritical dynamo model but by a subcritical model. Hence the physical
regime of the ensemble-averaged magnetic field is fundamentally different from that
of the ensemble members. Whatever the nonlinear mechanisms are that control the
behaviour of individual ensemble members, they are masked by probabilistic effects
in the ensemble average, and this makes it difficult to interpret the mean-field dynamo
equation. Of course, one may resolve this by adopting a less stringent definition of the
ensemble average, and ignore the phase mixing, so that the solutions of the mean-field
dynamo equation are no longer required to decay.
This leaves only three options. Either one adopts the ensemble interpretation, so that
the mean-field equations are well-defined, without the need for a two-scale approximation. In that case the mean-field equations give only a probabilistic description, unless
one chooses to ignore the phase mixing. The second option is to adopt a spatial average
that satisfies the Reynolds rules, such as the azimuthal average. This also leads to welldefined equations, while avoiding the probabilistic effects. Formally all mean quantities
acquire fluctuating components, but one may choose to ignore this aspect, as is tacitly
308
M. Ossendrijver
done in most mean-field dynamo calculations employing the azimuthal average. The
third option is to adopt a two-scale approximation and ignore the consequence that the
Reynolds rules are not satisfied, thereby accepting that the resulting mean-field equations
might not be well-defined. This is tacitly done in three-dimensional mean-field dynamo
calculations.
3.4.3. General form of the EMF
With the adoption of FOSA one obtains a linear expansion for the EMF in terms of
spatial derivatives of the mean magnetic field and, in principle, analytical expressions
for all the expansion coefficients. As long as the correlation time is short compared to
the evolutionary time scale of the mean magnetic field (|τc D0 | 1 in Eq. 7), it is
possible to do this for arbitrary mean flows (Hoyng 1985). This condition appears to be
easily satisfied in the Sun, where one may estimate τc ≈ 30 d and |D0 |−1 ≈ 1 yr. A
two-scale approximation in the spatial domain is not strictly necessary, but it simplifies
the EMF by allowing the dynamo coefficients to assume the form of local tensors instead
of integral kernels. Independently of FOSA, it is possible to take the expansion as the
starting point, and write down its general form for inhomogeneous anisotropic nonmirrorsymmetric turbulence on the basis of symmetry considerations (Krause & Rädler
1980: § 15). For convection in a rotating sphere, with two obvious preferred directions
given by stratification and rotation, i.e. er and ez , this leads to
E = α ◦ B 0 + β ◦ curl B 0 + · · ·,
(14)
where α is a tensor that describes the famous α effect. It is convenient to separate α in
symmetric and antisymmetric parts according to α ◦ B 0 ≡ α S ◦ B 0 + γ × B 0 , where
γ is the magnetic pumping vector (§ 5.5.4). The tensor β is one of several coefficients
parametrizing anisotropic turbulent magnetic diffusion. This leads to
α S ◦ B 0 = α1 (er ·ez ) B 0 + · · ·,
γ = γ1 er + γ2 (er ·ez ) ez + γ3 er ×ez ,
β ◦ curl B 0 = β1 curl B 0 + · · ·.
(15)
(16)
(17)
On most other occasions, the notation ηt ≡ β1 will be used for the scalar turbulent
diffusivity. One obtains the symmetry property with respect to the Sun’s equator for
all dynamo coefficients from the fact that E is an ordinary vector while B is a pseudo
vector. This immediately tells us for instance that α S is antisymmetric with respect to
the equator, so that α1 (r) is symmetric. The appearance of ez in the expression for α S
indicates the need for rotation; the radial pumping effect (γ1 ) does not require rotation.
Apart from such symmetry considerations, the dependence of α and β on r, B0 , and other parameters is still arbitrary at this point. The magnitudes and functional dependencies of the dynamo coefficients are notoriously difficult to establish, but they can
be estimated using various analytical and numerical methods. A more detailed account
of the physics behind the α effect and other dynamo coefficients is given in § 5.5. At
this point, it suffice to mention that FOSA yields the following well-known relations for
homogeneous isotropic turbulence, in which case α S = αI :
1
α ≈ − τc u · ω,
3
(18)
The solar dynamo
309
β1 ≈
1
τc | u2 |.
3
(19)
The dependence of α on the mean kinetic helicity points to the importance of cyclonic
motions, as was first realized by Parker (1955b). Of course, the situation in the solar
convection zone is far removed from a state of isotropic turbulence. Also, it will be
argued in § 3.4.5 that Eq. (14) is in general incompatible with the conservation of
magnetic helicity. This can be corrected by replacing the kinetic helicity in Eq. (18) by
the residual helicity and considering the effect of the magnetic helicity cascade.
3.4.4. Basic features of the mean-field dynamo equation
The essential features of the large-scale magnetic field of the Sun are already reproduced
by the simplest axisymmetric mean-field dynamo model incorporating only differential
rotation, U 0 = (r, θ ) r sin θ eφ , one α effect, α1 er ·ez = α1 cos θ , and scalar turbulent
diffusion (ηt ),
∂B 0
= ∇ × (U 0 × B 0 + α1 cos θ B 0 − ηt ∇ × B 0 ) ,
∂t
(20)
where θ is colatitude. The physical meaning of this equation can be grasped most easily by
decomposing the mean magnetic field in toroidal and poloidal components, according to
B 0 = B0t eφ + ∇ × A0p eφ , where A0p is the vector potential of B 0p , the mean poloidal
magnetic field (Krause & Rädler 1980: § 13). Differential rotation converts B 0p into
B 0t , and the α effect generates B 0p from B 0t and vice versa. Turbulent diffusion leads
to enhanced decay and diffusive transport; the associated time scale is τd ≡ L2 /ηt .
The linear growth rate of the mean magnetic field depends on a dimensionless combination of the parameters occurring in Eq. (20), the dynamo number D, which may be
defined as
α1 L L2
D = Dα D ≡
·
.
(21)
ηt
ηt
Here L denotes a typical length scale of the dynamo, is the typical difference
in rotation rate within the dynamo region, and α1 and ηt are also to be understood as
typical values. The numbers Dα and D represent turbulent magnetic Reynolds numbers
for the α effect and the differential rotation, respectively. Dynamo action, defined as
exponential growth of B 0 , occurs if |D| exceeds a geometry-dependent critical value
|Dcrit |. If D = Dcrit , then the dynamo is marginally stable.
The values of α1 and ηt in the Sun are badly known. Evidence from observations
and numerical simulations (§ 5.5) and plausible arguments suggests that α1 is positive
in the bulk of the convection zone and that α1 ≈ 0.1 m s−1 . All indications are that
ηt is similar to the value derived from the decay of sunspots, 2 · 107 m2 s−1 (Meyer
et al. 1974), but considerably smaller than the diffusion coefficient inferred from the
dispersal of magnetic flux at the solar surface (§ 2.5). The microscopic diffusivity η is at
least 5 orders of magnitude smaller than ηt and can be neglected in Eq. (20). Adopting
L ≈ 2 · 108 m, one finds τd ≈ 102 yr, and this allows the large-scale magnetic field to
vary on a time scale of years. Setting ≈ 2.5 · 10−7 s−1 for the radial differential
rotation at low latitudes, one obtains |Dα | ≈ 2 and D ≈ 103 . The fact that |Dα | D
justifies the frequently made α-approximation, following which the α effect is ignored
310
M. Ossendrijver
in the equation for the toroidal mean magnetic field. This amounts to neglecting all
components of α S except αφφ .
If |Dα | is not small compared to |D |, the dynamo is said to be of the α 2 -type;
if |D | |Dα |, the dynamo is of the α 2 -type. These regimes are relevant for rapidly
rotating solar-type stars and fully convective stars (§ 7.5). For an α dynamo, the ratio
|B 0p |/|B 0t | is of the order |Dα /D |1/2 , which explains why |B 0t | |B 0p | in the Sun.
For an α 2 -dynamo, the ratio is of the order 1.
In general, Eq. (20) has oscillating, wave-like solutions in the α regime. If
α1 ∂/∂r < 0, the waves tend to propagate in the correct equatorward direction
(Yoshimura 1975). For a linear model, the frequency of the dominant mode is of the
order cyc ≈ K|D|1/2 /τd , where K is a model-dependent constant. The aforementioned
values of α and ηt typically result in a dynamo period of the correct order of magnitude
(Köhler 1973; Choudhuri 1990). However, boundary conditions can have a large effect
on the value of K. If closed boundaries are used instead of vacuum boundary conditions
cyc can decrease by an order of magnitude (Choudhuri 1984; Kitchatinov et al. 2000).
This is because closed field lines prevent the loss of magnetic flux, which increases the
effective decay time. Nonlinear effects can modify the scaling of cyc , or change cyc
in more drastic ways (§ 5.7.7).
Boundary conditions can have a profound influence on the solutions of Eq. (20).
Ideally, they are obtained by considering the proper limit of the boundary conditions
of electrodynamics (Roberts 1967: § 1; Jackson 1975: § 1). Unfortunately the correct
description is not always easily established, especially for the external boundary. In the
case of discontinuities in the dynamo coefficients the proper matching condition for B 0
is obtained by imposing continuity of B 0 and of the tangential components of E 0 , while
allowing E0r to be discontinuous. Boundary conditions for several limiting cases with
a discontinuity of ηt were derived by Schubert & Zhang (2000). A more approximate
treatment for the interface between the radiative interior and the convection zone is
obtained by allowing the magnetic field to penetrate a given depth, as suggested by
the skin effect (Moss et al. 1990c; Brandenburg et al. 1992; Tavakol et al. 1995). The
effects of different boundary conditions were investigated by Kitchatinov et al. (2000)
and Tavakol et al. (2002).
The large-scale solar magnetic field is predominantly axisymmetric, with active
longitudes representing a small deviation from axisymmetry (§ 2.2). Strong differential
rotation tends to destroy azimuthal disturbances of the magnetic field. Weak but nonvanishing differential rotation (Roberts & Stix 1972; Moss et al. 1991; Barker & Moss
1994) and the anisotropy of α (Rüdiger 1978, 1980; Rädler et al. 1990) are known to
favor the excitation of non-axisymmetric dynamo modes. The latter is related to the
rotational quenching of the vertical α effect (§ 5.5.2). Rüdiger & Elstner (1994) found
that there is a critical value for the differential rotation that depends, among other factors,
on the degree of anisotropy of α, above which the solar dynamo favors axisymmetric
modes. Apart from active longitudes on the Sun, non-axisymmetric dynamo modes may
be important for explaining magnetic fields of rapidly rotating stars (§ 7.5).
It can be useful to investigate Eq. (20) under the most elementary conditions by
considering waves in an infinite homogeneous medium, without the need for boundary conditions. This is achieved by making a Fourier ansatz and solving the resulting
dispersion relation. For an α-type dynamo one obtains traveling waves that can be in-
The solar dynamo
311
terpreted as describing dynamo action locally at one point in the solar convection zone.
Such local results must be interpreted with care because the spherical geometry of the
Sun introduces unavoidable boundary effects at the poles and the equator. This typically
results in solutions with a spatial structure very different from that of the plane waves.
In spherical geometry, the solutions tend to have long wavelengths and large amplitudes
at low latitudes, and short wavelengths and small amplitudes near the poles, and the
critical dynamo number is usually increased (Tobias et al. 1997; Tobias 1998).
A realistic mean-field description of the solar dynamo is likely to require a more
elaborate formulation of each of the terms in Eq. (20). For instance, the mean flow
should include a term for meridional circulation, U 0m (r, θ ). This introduces an additional
dynamo number Dm ≡ U0m L/ηt . If Dm is sufficiently large, the solutions of Eq. (20)
can have equatorward propagation irrespective of the sign of α1 ∂/∂r. Furthermore,
the tensorial form of the dynamo coefficients should be used in order to account for the
anisotropy of convection (§ 5.5), and magnetic quenching should be included to account
for dynamo saturation and other nonlinear effects (§ 5.7). This is likely to require a
formulation based on magnetic-helicity conservation (§ 3.4.5).
3.4.5. Magnetic helicity conservation in mean-field dynamo theory
Even though magnetic helicity conservation is not a priori satisfied in mean-field dynamo
theory, its effect can in principle be incorporated through the magnetic-field dependence
of the α effect and other dynamo coefficients. The starting point for a discussion of the
role of magnetic helicity in α quenching is the following approximative relation between
α and the residual helicity for isotropic homogeneous turbulence (Pouquet et al. 1976),
1
1
α ≈ − τc Hres ≡ − τc u · ω − b · j /ρ ≡ αK + αM .
3
3
(22)
This relation, including the sign, has been confirmed in simulations of forced helical
MHD turbulence (Brandenburg & Subramanian 2000); derivations are given by Vainshtein & Kichatinov (1983), Blackman & Chou (1997) and Field et al. (1999). Suppose
that initially the magnetic field is weak, and that the kinetic helicity is negative, as is
the case in the bulk of the solar convection zone on the northern hemisphere. The positive α effect generates positive magnetic helicity at large scales and, due to magnetic
helicity conservation, a similar negative amount at small scales. Associated with the
magnetic helicity at each scale is a current helicity of the same sign. Due to the extra
spatial derivatives the total current helicity is dominated by small scales, and therefore
negative. Hence αM counteracts the initial α effect, leading to α quenching. This can
also be understood in terms of the Alfvén effect, which increases the alignment of u and
b (Pouquet et al. 1976; Biskamp 1993: § 7).
Unlike the magnetic backreaction on αK , which is a direct consequence of the Lorentz
force, the response on αM depends on the evolution of the magnetic helicity spectrum.
Therefore magnetic quenching is a dynamic process requiring a description based on a
differential equation for α. For isotropic conditions this equation can be written as
2
αM
∂αM
2 αB0 − ηt µ0 J 0 ·B 0
+
+ ∇ · F = −ηt kc
,
(23)
2
∂t
Beq
Rm
312
M. Ossendrijver
where F is the flux of magnetic helicity, not reproduced here, and kc is the wavenumber
at which the turbulence is forced (Kleeorin & Ruzmaikin 1982; Zel’dovich et al. 1983;
Vainshtein & Kichatinov 1983; Kleeorin et al. 1995). Recall that F may be crucial for
enabling dynamo action in the Sun. Equations of this type have been incorporated in
several mean-field dynamo models (§ 5.7.2).
Algebraic quenching of α is compatible with magnetic helicity conservation only if
the dynamo has closed boundaries and if the mean magnetic field has reached a forcefree saturated state (Kleeorin et al. 1995; Brandenburg 2001; Blackman & Brandenburg
2002). A well-studied ideal case where algebraic quenching applies is α 2 -type dynamo
action in homogeneous isotropic helical turbulence with periodic boundary conditions.
Even though not directly applicable to the Sun, a study of this elementary dynamo mechanism provides useful insights in the effects of magnetic helicity conservation before
considering more difficult cases. Once the mean magnetic field in the α 2 simulations
has reached a saturated state, the EMF can be described by
EA ≡
αK
ηt0
B0 −
µ0 J 0 .
2
2
2
1 + Rm B0 /Beq
1 + Rm B02 /Beq
(24)
By identifying the first term with αB 0 and the second with −ηt µ0 J 0 , one obtains
expressions for α and ηt that describe what is known as catastrophic quenching, because
α and ηt (and therefore E A ) are strongly reduced if Rm 1. Catastrophic quenching
is well-known from earlier dynamo simulations based on forcing by helical toy flows,
both of α (Vainshtein & Cattaneo 1992; Tao et al. 1993; Cattaneo & Hughes 1996) and
of ηt (Cattaneo & Vainshtein 1991; Vainshtein & Rosner 1991; Vainshtein & Cattaneo
1992; Cattaneo 1994). The adjective catastrophic is somewhat misplaced in the case of
ηt , where quenching enhances the efficiency of the dynamo. Saturation is achieved in
the α 2 simulations only because molecular diffusion (η) is not quenched, so that it can
overcome the α effect for a sufficiently strong magnetic field (Brandenburg 2001). Note
that a decrease of ηt means an increase of the diffusive time scale, which slows down the
dynamo. For this reason, strong quenching of ηt would be incompatible with the solar
cycle.
The results of the α 2 simulations are now understood in terms of magnetic helicity
conservation (Blackman & Field 2000a). However, it turns out that the quenching of
α and ηt may be intricately linked. The simulations are also compatible with an EMF
given by
2
αK + Rm ηt0 µ0 J 0 ·B 0 /Beq
EB ≡
B 0 − ηt0 µ0 J 0 .
(25)
2
1 + Rm B02 /Beq
This is because E A = E B if J 0 B 0 , which is true if the mean magnetic field is forcefree. The first term of E B now yields an expression for α that is identical to the stationary
solution of Eq. (23) with F = 0 and ηt = ηt0 (Gruzinov & Diamond 1994, 1995, 1996;
Blackman & Brandenburg 2002). In this interpretation, α is not subject to catastrophic
quenching, unless J 0 · B 0 = 0, and ηt is not quenched at all, even though E B is strongly
quenched. Irrespective of these different interpretations, the simulations do exclude the
possibility of having for instance B03 instead of B02 in the nominator (Brandenburg 2001).
Blackman & Brandenburg (2002) have extended the homogeneous isotropic turbulence
model by including shear in an effort to establish how α and ηt each are quenched.
The solar dynamo
313
This is possible because there the large-scale magnetic field can assume the form of
traveling waves, with a frequency that depends on ηt . The simulations indicate that ηt
is less severely quenched than α, which suggests that E B may be more general than
E A . More definite results could be established by studying how the cycle period in such
simulations varies with Rm.
In any case, the conditions for catastrophic quenching are apparently not satisfied
in the Sun, even though Rm 1. One reason for this may be that there is a significant
flux of magnetic helicity. Also, the mean magnetic field of the Sun is oscillating, so
that the above considerations based on stationarity may not apply. Even in the α 2 simulations the EMF is strongly quenched only in the final state, whereas there is no
quenching in the initial kinematic stage, which lasts many turnover times (Field &
Blackman 2002). Furthermore, Eq. (22) should be generalized to anisotropic conditions;
this leads to tensorial expressions for α and β (Rogachevskii & Kleeorin 2000, 2001;
Kleeorin & Rogachevskii 1999, 2003). The correct description of dynamical quenching
under sufficiently general conditions (cyclic α-type dynamo action, inhomogeneous
anisotropic turbulence, non-vanishing flux of magnetic helicity, spherical geometry)
remains to be established. Finally, Eq. (22) presupposes that dynamo action is the result
of helical turbulence. If the α effect has a different origin, such as magnetic buoyancy
(§ 5.5.3), the relation between α and the kinetic and current helicities is expected to
be different, although magnetic helicity conservation must also be satisfied. Thus, the
investigation of the role of magnetic helicity conservation in the Sun has hardly begun yet.
It seems unavoidable that the EMF is controlled by magnetic helicity conservation, and
that the dynamo coefficients must satisfy a dynamical equation, albeit a more elaborate
one than suggested by Eqs. (22) and (23).
3.4.6. Higher-order corrections to FOSA
It seems likely that the correlation time of solar convection is not short, but rather S ≈ 1,
so that the use of FOSA may not be justified. Hence it may be necessary to consider
a higher-order approximation. As long as S < 1, it is possible to obtain a converging
cumulant expansion, of which FOSA represents the first truncation (Knobloch 1977,
1978b; Hoyng 1985; Nicklaus & Stix 1988). In general, the resulting dynamo equation
has spatial derivatives up to the order of truncation, but for special choices of the turbulent
flow there might remain only first and second derivatives of B 0 , with modified dynamo
coefficients.
The higher-order corrections have a large effect if the correlation time of the turbulence is long (S ≈ 1). In strongly helical turbulent flows this may even result in a sign
reversal of ηt (Kraichnan 1976a,b; Knobloch 1977, 1978b). By itself, negative diffusion
leads to an instability of the mean magnetic field that would invalidate the dynamo equation. But the relevance of this is uncertain because for S ≈ 1 the cumulant expansion
converges slowly if at all, so that it might be impossible to prove the correctness of
any high-order truncation. If S ≥ 1 it is no longer possible to work with the cumulant
expansion. There are alternative analytical methods that do not require the correlation
time to be short (Dolginov & Silant’ev 1992; Carvalho 1992). Moreover, FOSA and
higher-order approximations represent formal techniques for deriving mean-field dynamo equations and analytical expressions for the dynamo coefficients. Even if S > 1,
314
M. Ossendrijver
it does not follow that a dynamo equation based on some truncation similar to Eq. (14)
cannot provide a correct description. Therefore, one may also adopt a heuristic approach
because the expansions based on FOSA do seem to capture the essential features of
dynamo action in the Sun.
3.4.7. Vishniac-Cho mechanism
Vishniac & Cho (2001) have proposed an alternative scenario for astrophysical dynamos
based on the internal flux of magnetic helicity, which results in a contribution to the EMF
given by
B0
E = −
∇ · F,
(26)
2|B 0 |2
where F is the local flux of magnetic helicity. The advantage of Eq. (26) is that it merely
describes transport of magnetic helicity through the volume of the dynamo. No largescale magnetic helicity is generated, so that the difficulties posed by magnetic helicity
conservation can be avoided. For isotropic turbulence they derive
F ≈ −2c τc (B · ω) (B · ∇) u).
(27)
which is nonvanishing if there is a correlation between the velocity gradient along field
lines and the kinetic vorticity along field lines, a condition that is reminiscent of the
stretch-twist-fold mechanism. Hence this dynamo mechanism does not require any mean
kinetic helicity. Vishniac et al. (2003) have proposed how to generalize their result to
non-isotropic conditions. A related mechanism based on the flux of magnetic helicity,
but with a different expression for F , was proposed earlier by Boozer (1986) and Bhattacharjee and Hameiri (1986). There is no indication yet that the mechanism is of great
importance in the Sun, where kinetic and magnetic helicity are both clearly present (Arlt
& Brandenburg 2001).
3.5. Modal decompositions of the magnetic field
Elsasser (1946) developed a theoretical approach to stellar and planetary dynamos based
on a decomposition of the magnetic field in terms of orthogonal decay modes, i.e.
solutions of Eq. (1) with U = 0. This amounts to replacing Eq. (1) by an infinite set
of coupled equations for the expansion coefficients, that can in principle be solved if a
suitable truncation is applied (Cowling 1976: § 5; Roberts 1967: § 3). Such an attempt was
undertaken by Bullard & Gellman (1954) in connection with the geodynamo. For certain
non-axisymmetric flows dynamo action occurs, and this was confirmed subsequently by
Kumar & Roberts (1975).
Dynamo theory as well as observations of stellar magnetic fields suggest that the 22year axisymmetric dipolar mode is but one of the possible modes of the solar magnetic
field, although the evidence for the existence of other modes is inconclusive (§ 2.8).
As pointed out by Hoyng (1987b), convection provides a natural source of random
fluctuations that could excite all eigenmodes of the solar dynamo, resulting in a spectrum
of global magnetic resonances. This idea led Hoyng (1988) to propose a statistical
approach to solar dynamo theory based on an expansion of the global magnetic field
The solar dynamo
315
in terms of mean-field dynamo modes. The method yields a set of coupled ordinary
differential equations for the expansion coefficients that is formally equivalent with
the full MHD dynamo problem. Application of the ensemble average provides coupled
equations for the mean coefficients. Due to the finite correlation time of the excitation
mechanism, the mean-field dynamo modes have a finite coherence time that should
decrease with increasing complexity of the mode. In principle, this approach allows one
to compute the complete spectrum of solar dynamo modes, and make a comparison with
observations (Hoyng & Van Geffen 1993; Hoyng & Schutgens 1995).
3.6. Local dynamo simulations in stratified convection
A numerical approach complementary to the idealized turbulence simulations (§ 3.3) is
to include all physical aspects of solar convection that are judged to be necessary for a
realistic description of the dynamo process, such as compressibility, stratification, differential rotation, and the presence of an overshoot layer. In order to capture the smallest
possible scales for a given grid size, such simulations are carried out preferentially in a
local Cartesian geometry. In some respects, the results exhibit crucial differences with
homogeneous isotropic MHD turbulence models. Typically, convection is less efficient
than helical turbulence in generating a large-scale magnetic field because convection is
only partially helical. Furthermore, the gravity force introduces effects of buoyancy and
magnetic pumping by which magnetic flux can be expelled from the dynamo region.
The problem of storage of the magnetic field becomes very important if the magnetic
field is allowed to be amplified by shear. Compressibility has a large effect on stratified
convection because it leads to a strong asymmetry between up- and downflows, which
also affects the dynamo process. Realistic MHD simulations of the solar photosphere
require the inclusion of radiative transfer and incomplete ionisation in order to allow
a detailed comparison with observations (Grossmann-Doerth et al. 1988; Steiner et al.
1998).
Dynamo action in rotating stratified compressible convection without shear was
investigated by Nordlund et al. (1992) and Brandenburg et al. (1996). The total magnetic
energy saturates at a small fraction of the total kinetic energy, and the plasma-β is large,
as is true in the solar convection zone (Table 2), but locally the magnetic field strength
can reach equipartition with the convective flow. Under such conditions, the magnetic
field assumes an intermittent structure consisting of irregular flux cigars (Nordlund et
al. 1994). In the simulations, these flux tubes are probably still too strongly controlled
by drag forces. As the magnetic field reaches a saturated state, the work done against
the Lorentz force is in approximate balance with Ohmic dissipation, and the spectra of
the magnetic energy and the magnetic helicity show evidence of a cascade. During the
initial growth phase and the saturated state the spectra are different, but the resolution
of the simulations does not yet enable a definite identification of an inertial range and
the corresponding spectral index.
Dynamo action is found if Pm exceeds a threshold value, which is of the order 1 in
the simulations. This might seem worrying because in the Sun Pm is very small. But it is
expected that for sufficiently small values of η, dynamo action occurs also for small Pm.
The dynamo-generated magnetic fields accumulate in the stably stratified overshoot layer
due to magnetic pumping (§ 5.5.4). In order to address questions such as the topology
316
M. Ossendrijver
and strength of the deep-seated global magnetic field the simulations should incorporate
differential rotation. Some of these aspects are investigated in isolation by considering
the stability and break-up of a magnetic layer (§ 5.3).
3.7. Fast dynamos
The concept of fast dynamo action is motivated by the question how the evolution of
the large-scale magnetic field of the Sun on convective time scales can be explained
2 /η ≈ 1010 yr). Usually
given the much longer time scale for magnetic diffusion (R
fast dynamo investigations address only the initial instability of the magnetic field for a
prescribed flow, such that a dynamo is said to be fast if there is exponential growth of
the magnetic field in the limit η → 0 (i.e. Rm → ∞). As will be argued, the kinematic
nature of fast dynamos thus identified may limit their relevance for explaining dynamo
action in the Sun.
The mathematical analysis of fast dynamos is difficult because for η = 0 the eigenfunctions of Eq. (1) develop structure on ever smaller scales (Moffatt & Proctor 1985),
requiring a desciption in terms of generalized functions (Bayly 1994). A well-known
candidate for fast dynamo action is the stretch-twist-fold mechanism of Vainshtein &
Zel’dovich (1972). In general, fast dynamo action is expected whenever a flow has
the property of exponentially stretching and constructively folding the magnetic field
lines. Such flows produce chaotic fluid trajectories characterized by positive Lyapunov
exponents. The chaotic trajectories can be viewed as being produced by a continuous
Lagrangian map that operates on the magnetic field (Childress & Gilbert 1995). Taking
the idealization one step further, these maps may be discretized and abstracted from the
underlying flow in order to identify elementary types of fast dynamos (Bayly 1994).
Numerical simulations using flow fields with a sufficient lack of symmetry illustrate the
existence of fast dynamos (Soward 1987, 1990, 1994; Gilbert & Bayly 1992).
In the light of magnetic helicity conservation, it might appear impossible to achieve
fast dynamo action in helical flows without having significant magnetic helicity losses.
It turns out that fast dynamo action is accompanied by a near perfect cancellation of
magnetic helicity between adjacent magnetic features in the spatial or spectral domain,
and that the mean magnetic helicity is typically very small even if the flow is helical
(Hughes et al. 1996). As a result, the magnetic energy is able to grow on a fast time
scale, but this may be an artefact of the kinematic approach, which precludes the onset
of a cascade that would lead to an accumulation of magnetic helicity at a large scale.
Numerical simulations of nonlinear MHD indicate that helical flows are fast dynamos
only in the initial kinematic phase (Field & Blackman 2002). It therefore seems that fast
dynamos with prescribed helical flows might be rather academic because they will likely
turn out to be slow in the nonlinear regime. Perhaps non-helical flows provide a more
relevant type of fast dynamos, since they do not generate large-scale magnetic helicity
even in the nonlinear regime, and so remain fast. The magnetic fields produced by such
a dynamo, being entirely small-scale and vanishing in the mean, would not contribute
to the large-scale solar magnetic field, but could be relevant for explaining small-scale
mixed-polarity fields in the solar photosphere (§ 2.4; § 3.8).
The solar dynamo
317
Fig. 6. Small-scale dynamo action in a local Boussinesq simulation of a convectively unstable
layer with zero rotation. Left: temperature (light = hot); middle and right: vertical magnetic field.
The left and middle panels correspond to a horizontal plane near the upper boundary; the right
panel represents a plane in the middle of the layer (from Cattaneo 1999)
3.8. Small-scale magnetic fields
3.8.1. Small-scale dynamo action in the solar convection zone
Observations of intranetwork magnetic fields reveal the existence of a background magnetic flux residing in small scales and characterized by mixed polarities, a seemingly
random spatial distribution, and no solar cycle dependence (§ 2.4). These properties
point to small-scale dynamo action in the solar convection zone (Spruit et al. 1987). Dynamo theory suggests that such magnetic fields can be generated by sufficiently complex
motions if the mean kinetic helicity is negligible, which is the case if rotation is slow.
This is relevant for the upper layers of the solar convection zone, where rotation has a
negligible effect at spatial scales smaller than that of supergranulation (Table 2).
Numerical simulations based on the Boussinesq approximation indicate that turbulent convection without rotation is capable of generating a highly intermittent, spacefilling magnetic field with mixed polarities and a dynamical time scale comparable to
the turnover time (Meneguzzi & Pouquet 1989; Cattaneo 1999). The calculations reveal a broad distribution of magnetic field intensities with values predominantly below
equipartition with the kinetic energy, the strongest fields being located near downflowing
channels (Fig. 6). An observer would detect only a small fraction of the magnetic flux
due to cancellation of mixed polarities below the resolution of the instrument (Emonet &
Cattaneo 2001). Dynamo simulations using such idealized set-ups serve to illustrate the
magnetic patterns that can be expected in quiet-Sun regions. Due to its small-scale nature, this type of dynamo action may be rather insensitive to boundary conditions, except
near the boundaries themselves (Theelen & Cattaneo 2000). However, for a quantitative comparison with observations, it is necessary to include compressibility, radiative
transfer and realistic boundary conditions (§ 3.6).
3.8.2. Mean magnetic energy of the solar dynamo
Small-scale mixed-polarity magnetic fields do not contribute to the mean magnetic field,
but their energy is likely to exceed that of the mean magnetic field in the Sun. Bräuer &
Krause (1973, 1974) have shown that in a highly conducting turbulent fluid and ignoring
318
M. Ossendrijver
nonlinear effects, the presence of a seed magnetic field leads to magnetic fluctuations
with an associated energy of the order |b|2 ≈ Rm B02 , thereby confirming a result of
Parker (1963a,b).Although nonlinear effects are expected to constrain the ratio |b|2 /B02
to a value smaller than Rm, this relation points to the importance of small-scale magnetic
fields in the Sun, where Rm is very large. Furthermore, the large-scale magnetic field
of the Sun does not assume the form of a homogeneous field, except perhaps in the
magnetic layer (§ 5.3). Also for this reason, the magnetic fluctuations are likely to be
large.
This does not invalidate the concept of a mean magnetic field, but it does suggest
that the mean-field induction equation should be complemented by an equation for the
mean magnetic energy, which retains contributions from all length scales. No exact
closed equation exists for the mean magnetic energy, and it is necessary to consider the
two-point correlation function of the magnetic field, Rij (r, r ; t) ≡ Bi (r, t)Bj (r , t)
(Kraichnan & Nagarajan 1967; Kazantsev 1968; Bräuer & Krause 1973; Vainshtein
1982; Kleeorin et al. 1986; Kim 1999; Kleeorin & Rogachevskii 2002). From this, the
mean magnetic energy is obtained by contraction and setting r = r .
Durney et al. (1993) investigated the growth of the magnetic energy for isotropic homogeneous non-helical turbulence, using the EDQNM spectral closure method (§ 3.3.1).
In their calculations the magnetic energy approaches equipartition with the kinetic energy on a typical timescale of 50 turnover times. In the solar convection zone, the growth
of the magnetic field is counteracted by magnetic buoyancy on a timescale of several
turnover times, an effect that is not included in the calculation. Nevertheless, the results
of such calculations confirm that the magnetic energy of small-scale magnetic fields
dominates over that of the cyclic large-scale magnetic field.
A simpler mean-field description of the magnetic energy is obtained if Ohmic dissipation is ignored. Adopting the formalism of stochastic differential equations (Eqs. 8-9) and
assuming FOSA, Knobloch (1978a) derived an equation for the tensor Tij ≡ Bi Bj /µ0
for homogeneous isotropic turbulence, that was generalized by Hoyng (1987b) to include
the effect of a mean flow. After contraction of Tij one obtains
∂U0i
∂eM
+ (U 0 · ∇) eM = 2γ eM +
Tij + ∇ · ηt ∇eM .
(28)
∂t
∂xj
ij
where eM ≡ B 2 /2µ0 is the mean magnetic energy density. The first term on the right
hand side involves the rms magnitude of the turbulent vorticity ω ≡ ∇ × u,
γ ≈ 13 τc |ω|2 ,
(29)
not to be confused with the pumping effect. Vorticity gives rise to random stretching
and winding up of magnetic field lines, causing growth of the magnetic energy. The
magnitude of γ is not well-known; a rough estimate suggests that γ ≈ ηt /2t ≈ 10−8 s−1
in the solar convection zone. Closer inspection of the equation for Tij reveals that the α
effect plays no role in the generation of magnetic energy (Ossendrijver & Hoyng 1997).
However, from the presence of an α effect one may infer the existence of small-scale
√
dynamo action because of the realizability condition, ηt γ > |α|. The second term
on the right hand side accounts for the effect of shear, which enhances the growth of
the magnetic energy, but is not a condition for growth. Finally, turbulent diffusion (ηt )
describes enhanced transport, but not dissipation, of the magnetic energy.
The solar dynamo
319
Applications of Eq. (28) to the solar dynamo were developed by Hoyng (1987b),
Van Geffen & Hoyng (1990), Van Geffen (1993) and Ossendrijver & Hoyng (1997).
Unlike the oscillating dipolar solutions of the mean-field induction equation, the fastest
growing mode of Tij is non-oscillatory. For typical parameters as they are used in the
mean-field induction equation, the mean magnetic energy exhibits exponential growth
on a rapid time scale of about one month. Since the α effect plays no role, these features
can be explained as being the result of small-scale dynamo action by vorticity and shear,
unrelated to the solar cycle. Since Eq. (28) is kinematic, it describes only the initial
instability of the magnetic energy density. Saturation would require incorporation of the
Lorentz force and Ohmic dissipation, but this results in a difficult closure problem. The
effect of Ohmic dissipation was accounted for in a heuristic way by Ossendrijver &
2 e on the right hand side.
Hoyng (1997) by adding a term −ηkdiss
M
The solutions of Eq. (28) describe the distribution of unsigned magnetic flux in the
convection zone. Typically, eM exhibits an exponential increase with depth, confirming
the result of Petrovay & Szakály (1993). They conclude that only the presence of local
sources (i.e. γ ) in the convection zone can prevent the increase with depth from being
unphysically rapid. This is another indication that small-scale dynamo action is taking
place in the solar convection zone.
4. Convection and differential rotation
Strictly speaking, the solar dynamo problem can be tackled only by solving for both the
magnetic field and the flow in a selfconsistent manner. This has not been successful yet, at
least in global models (§ 6.2). On the other hand, the ratio of the total magnetic energy to
the total kinetic energy of the convection zone is small, so that magnetic fields represent
only a small perturbation of the Sun’s global structure. The situation becomes more
complex if one takes into account the intermittence of convection, and the fact that this
carries over to the magnetic field, due to flux expulsion and other effects (§ 5.5.4). Hence
Lorentz forces can be significant locally and influence dynamo coefficients such as α,
while leaving the bulk of the flow and the hydrodynamic turbulent transport coefficients
largely unaffected. A possible exception could be the magnetic layer at the base of the
convection zone, where the filling factor of strong magnetic fields is larger (§ 5.3), but
the bottom line appears to be that convection and differential rotation control the solar
magnetic field and not vice versa. It follows that there is some justification in attacking
the problem of the solar dynamo and that of solar convection parallel but separately, and
this is the usual procedure in most models and simulations.
4.1. Observations of convective patterns and differential rotation
From observations it is inferred that convection is organized in a hierarchical pattern of
various scales (Spruit et al. 1990). On the smallest scale the solar disk is covered with
granulation, which has a typical length scale L ≈ 106 m and a life time τc ≈ 5 minutes
(Co ≈ 2 · 10−3 ). In some studies an intermediate scale of mesogranulation has been
identified with L ≈ 5 · 106 m and τc ≈ 2 h, but it seems doubtful whether this represents
a distinct feature. In order of increasing size one further distinguishes supergranulation,
320
M. Ossendrijver
Fig. 7. Rotation rate as a function of fractional
radius from helioseismic inversions obtained
with the Global Oscillation Network Group
(courtesy NSF’s National Solar Observatory,
USA)
with L ≈ 3 · 107 m, τc ≈ 1 d (Co ≈ 0.4). The existence of giant cells with L ≈ 108 m
and τc ≈ 1 month (Co ≈ 15) is suggested by numerical simulations (§ 4.2), but the
observational evidence is meager (Stix 2002: § 6).
The convective motions are superposed on a mean flow consisting of differential
rotation and meridional circulation. Solar surface rotation depends on latitude, such that
rotation is faster at the equator than at higher latitudes. The Sun’s internal rotation can
be established by means of helioseismology. This technique uses the frequency splitting
of solar oscillations, which depends on the rotation rate. By analysing oscillations that
are reflected at different depths, the angular velocity can be probed as a function of
depth (Fig. 7). These measurements reveal that depends only weakly on depth in
the bulk of the convection zone, but near the bottom there is a transition layer also
known as tachocline, in which the rotation rate changes from being almost uniform in
the radiative interior to being latitude dependent in the convection zone (Goode 1995;
Elsworth et al. 1995; Schou et al. 1998; Howe et al. 2000a). Within the tachocline,
rotation increases with distance from the core at low latitudes, while it decreases at
high latitudes; at intermediate latitudes rotation is almost independent of depth. The
thickness of the tachocline is a matter of debate. The rotation profile obtained from the
helioseismic inversions shows evidence of a smooth tachocline located in the region
0.65 < r/R < 0.75 (Kosovichev 1996). The true thickness of the tachocline may be
smaller; Charbonneau et al. (1999) propose dtach ≈ 0.04R ; others favor a very thin
tachocline with dtach ≈ 0.01 − 0.02R (Elliott & Gough 1999). Theoretical aspects of
the tachocline are considered in § 5.2 and § 5.3.
The meridional flow is poleward at the solar surface, where it has a maximum speed
of about 20 m s−1 (Giles et al. 1997; Schou & Bogart 1998). From helioseismic data the
poleward motion is known to extend downward; it would be of great interest to know
the direction and magnitude of the meridional flow at the base of the convection zone.
Detailed knowledge of flows in the upper layers of the convection zone is also becoming
available (Lindsey & Braun 1997).
4.2. Global simulations
Stellar structure models confirm that a small deviation from adiabaticity suffices for
convection to carry the entire energy flux generated by nuclear fusion in the core.
This has facilitated simplified approaches to convection based on mixing-length theory and anelastic hydrodynamics. Numerical simulations confirm that stellar convection
The solar dynamo
321
Fig. 8. Snapshot of the radial velocity at a level near the top of
the convection zone (r/R = 0.95) from a global simulation
of solar convection based on the anelastic approximation. Light
and dark regions denote upflows and downflows, respectively
(from Miesch et al. 2000)
can to some extent be described by mixing-length theory, if allowance is made for a
depth-dependent mixing-length parameter αML (Kim et al. 1996). Mixing-length theory
provides Ma ≈ αML (∇)1/2 (Stix 2002: § 6), which tells us that convection is slow
throughout the lower half of the solar convection zone (Table 2). This is computationally expensive because of the need to resolve the rapid sound waves. In the anelastic
approximation, the term ∂ρ/∂t is ignored in Eq. (2), so that sound waves are filtered
out (Gough 1969; Glatzmaier 1984). The derivation of the anelastic approximation proceeds by defining a nearly adiabatic reference state and estimating the deviations from
this state using mixing-length relations and other plausible arguments.
Nevertheless, the parameter regime of the lower half of the convection zone remains
inaccessible to numerical computation in crucial aspects (Table 2). Although the entropy
gradient is very small (i.e. |∇| 1), Ra is very large, because the viscosities are small.
It is likely that the numerical resolution must be only as high as to allow a sufficiently
large separation of scales, but this is still very high. Such problems can be alleviated to
some degree by adopting a prescription for thermal (radiative) diffusion consisting of a
small value that acts on the mean temperature stratification and a large value that acts
on the deviations from it.
Global simulations of solar convection in spherical geometry based on the anelastic approximation were carried out by various groups (Latour et al. 1976; Toomre et
al. 1976; Van der Borght 1975, 1979; Massaguer & Zahn 1980; Gilman & Glatzmaier
1981; Glatzmaier & Gilman 1981a,b; Gilman & Miller 1986). Although these computations did achieve some success in reproducing the surface differential rotation and the
outward increase of rotation at low latitudes, other features turned out to be persistently
incompatible with observations. Recent higher-resolution simulations are beginning to
succeed better in reproducing solar differential rotation (Elliott et al. 2000; Miesch et al.
2000). Unlike the previous simulations, rotation no longer exhibits a strong cylindrical
alignment. Latitudinal differential rotation is well reproduced, but discrepancies still
exist. For instance, the surface meridional flow is typically too slow at high latitudes
(Brun & Toomre 2002) and the tachocline is not reproduced, presumably because the
simulations are still too strongly diffusive. There is evidence for the formation of large
eddies reminiscent of giant cells (Fig. 8). Agreement with solar convection appears to
increase if the convection is allowed to be more strongly turbulent. This may be related to
the appearance of narrow, rapid downflows that aid the formation of an overshoot layer
and result in a larger kinetic helicity, thereby creating favorable conditions for dynamo
action (Miesch et al. 2000).
322
M. Ossendrijver
4.3. Local simulations
Local simulations can be particularly fruitful for the upper layers of the solar convection
zone, where the parameter regime is more accessible than in deeper layers (Table 2).
Due to compressibility effects, stratified convection proceeds in a highly asymmetric
fashion, with broad and slow laminar upflows punctuated by narrow, rapid turbulent
downflows (Hurlburt et al. 1984; Hurlburt & Toomre 1988; Malagoli et al. 1990). Interaction with magnetic fields occurs preferentially near such downdrafts (Brummell et
al. 1996, 1998). A realistic treatment of photospheric convection requires the inclusion
of radiative transfer (Stein & Nordlund 1989). The results of such computations have
changed our view of solar convection. Rather than being a hierarchical phenomenon consisting of distinct eddies on various scales, convection has important non-local aspects
(Stein & Nordlund 1998). Radiative cooling in the photosphere drives the formation of
downdrafts that might extend to large depths (Spruit et al. 1990; Rast & Toomre 1993a;
Rast et al. 1993). This would suggest that a correct treatment of radiative processes in
the photosphere might be important also for global convection.
In the lower part of the solar convection zone, where motions are slow, the anelastic
approximation can be advantageous (Ginet & Sudan 1987; Lantz 1995; Lantz & Sudan
1995). Here the value of Pr is very small, and numerical simulations reveal important
changes already within the range 0.1 Pr 10 (Cattaneo et al. 1991). Rast & Toomre
(1993b) point out that the steep temperature dependence of λrad creates favorable conditions for the formation of hot rising plumes near the base of the convection zone. This
effect is rarely included in numerical investigations, most of which adopt a simplified
treatment such that λrad is independent of temperature. From such considerations it is
apparent that the true nature of the flow in the deepest layers of the solar convection
zone remains a matter of speculation, and may be rather different from what is currently
found in the numerical simulations.
4.4. Mean-field theory of rotation
As is the case with the solar magnetic field, one can argue that it suffices to achieve a
mean-field description of the flow in the solar convection zone. This idea has led to the
development of a mean-field theory of stellar convection and rotation (Rüdiger 1989).
The equation for the azimuthal average of angular momentum is given by
∂ 2
(30)
ρs + ∇ · ρs 2 U 0m + ρsuφ u − LM = 0,
∂t
where s ≡ r sin θ, and
1
LM ≡
∇ · s (B0φ B 0 + bφ b )
(31)
µ0
represents the torque exerted by the Lorentz force. By adopting an assumption equivalent
to FOSA, it is possible to derive an expansion for the turbulent Reynolds stress tensor
Qij ≡ ui uj in terms of the mean rotation, the most important coefficients being
∂
+ r sin θ,
∂r
∂
+ h sin θ,
≈ −σ νt sin θ
∂θ
Qrφ ≈ −νt s
(32)
Qθφ
(33)
The solar dynamo
323
where νt is the turbulent kinematic viscosity, and σ is a scaling factor. While the diffusive
terms drive the system to a state of uniform rotation, the terms containing the -effect
enable angular momentum transport, even if rotation is uniform, thereby establishing differential rotation. The radial -effect is responsible for establishing ∂/∂r. In principle,
meridional circulation and h both contribute to ∂/∂θ , and their relative importance
in the Sun is not clear. The existence of the -effect has been confirmed in numerical
simulations of rotating stratified convection (Brandenburg et al. 1990; Pulkkinen et al.
1993). The first term of the magnetic torque LM is also known as the Malkus-Proctor effect (§ 5.7.4). In addition to causing a torque, Lorentz forces lead to magnetic quenching
of the -effect (§ 5.7.5).
Inclusion of the -effect in Eq. (30) yields solutions for (r, θ ) that have many
similarities with solar differential rotation. By incorporating FOSA results for Qij for
arbitrary rotation rates, mean-field models calibrated to the current Sun can be extrapolated to different evolutionary stages of the Sun as well as to other solar-type stars
in order to make verifiable predictions for their differential rotation (§ 7.3). Küker &
Stix (2001) present solutions for the Sun at various evolutionary stages. For the current
Sun, their model reproduces approximately the observed surface differential rotation
and meridional flow. Discrepancies remain with regard to the radial differential rotation
and the tachocline, which are not well captured. The meridional flow in some of the
mean-field models changes its sign at some depth in the convection zone, a feature that
is also found in direct numerical simulations (Brun & Toomre 2002).
5. Physical processes in the solar dynamo
5.1. Magnetic fields in the core of the Sun
The near solid-body rotation of the radiative interior points to efficient transport of angular momentum, possibly due to the presence of a relic poloidal magnetic field (Mestel
& Weiss 1987; Charbonneau & MacGregor 1993; Elsworth et al. 1995; MacGregor &
Charbonneau 1999). Gough & McIntyre (1998) have argued that a 0.1 mT relic field
would be sufficient to achieve solid body rotation; a value of 10−8 T was derived by Rüdiger & Kitchatinov (1997). Alternatively, the necessary transport of angular momentum
may be the result of weak turbulence caused by the magneto-rotational instability (Arlt
et al. 2003). This also requires the presence of a weak magnetic field in the core of the
Sun. Various magnetic instabilities that can contribute to the core field were compared
by Spruit (1999).
Due to the skin effect, an oscillating large-scale magnetic field is not able to penetrate
into the radiative core beyond a depth given by dskin ≡ (2η/ cyc )1/2 ≈ 4 km. On the
other hand, random field components are able to diffuse into the core without restriction.
Garaud (1999) estimates that this can lead to the presence of a weak magnetic field in the
core with an intensity decreasing from about 0.01 mT near the top to less then 10−10 T
below 0.3R . From the weakness of this field one may infer that a primordial magnetic
field must also be present.
Even though it is likely that there is a relic magnetic field in the radiative core, there
are only a few rather unconvincing observational indications for this. First, there is the
Gnevyshev-Ohl rule (§ 2.1.1). Secondly, the observed inclination of the heliomagnetic
324
M. Ossendrijver
equator might be explained by an inclined relic field (Bravo & Stewart 1995). But this
requires the relic field to protrude the convection zone and avoid being shredded, which
seems highly improbable.
Relic magnetic fields have been invoked as part of a non-dynamo, oscillator-based
explanation for the solar cycle. Piddington (1972, 1976) and Layzer et al. (1979) questioned the foundations of dynamo theory and proposed an alternative theory for the
solar cycle based on torsional oscillations caused by a stationary relic field. The phase
locking of the solar cycle claimed by Dicke (1978) has been interpreted as evidence for
the oscillator model, but similar behaviour can be reproduced in dynamo models as well
(§ 5.6). Non-dynamo models face several difficulties that seem impossible to overcome.
They rely on the existence of a 22-year torsional oscillation in a layer near the top of the
radiative core, but there is no observational evidence for this, and it is unclear how the
oscillation could be maintained (Cowling 1981; Rosner & Weiss 1992).
Due to the skin effect, the presence of a conducting core can have some influence on
the global dynamo, as has been suggested for the Earth (Hollerbach & Jones 1993). Nonaxisymmetric relic fields in the cores of solar-type stars might provide an explanation
for active longitudes and non-axisymmetric stellar activity (Kitchatinov et al. 2001).
Schubert & Zhang (2000, 2001) proposed an α 2 -type dynamo model for the Sun that
incorporates magnetic coupling with the inner core. They find that for a sufficiently large
η the model produces oscillating solutions, which is surprising for an α 2 -type dynamo.
However, the model is academic because it rests on the unphysical assumption that α
is independent of latitude (§ 3.4.1). This also explains why the solution for the toroidal
magnetic field has the wrong parity even though the poloidal magnetic field has the
correct dipolar parity.
5.2. Tachocline physics
The discovery of the tachocline through helioseismic inversions (§ 4.1) has not only
changed our view about the solar dynamo, but also initialized a new field of investigations. A definitive theory of the tachocline has not emerged though, partly because its
thickness and turbulence properties are uncertain.
5.2.1. Tachocline confinement
Hydrodynamic calculations suggest that the thickness of the tachocline depends on
the efficiency of the turbulent transport of angular momentum (Spiegel & Zahn 1992;
Elliott 1997). If the tachocline is very thin (§ 4.1), the question arises whether it can
be maintained purely by hydrodynamic effects, or whether magnetic fields also play
a role (Charbonneau et al. 1999). The radiative core can be subject to the magnetorotational instability, leading to near solid-body rotation of the core (Arlt et al. 2003),
and the formation of a tachocline. Gough & McIntyre (1998) argue that the presence
of a poloidal relic magnetic field in the radiative core would result in a thin magnetic
boundary layer underneath the convection zone that would rapidly deflect convective
motions, thereby confining the tachocline. For a 0.1 mT poloidal field, they estimate
that dtach ≈ 0.02R , and that of the magnetic boundary layer about 25 times less. This
model would be consistent with the observation that the tachocline does not appear to
The solar dynamo
325
exhibit significant solar-cycle variations. Forgács-Dajka & Petrovay (2001, 2002) have
proposed a scenario for tachocline confinement based on the dynamo-generated poloidal
magnetic field. For this cyclic field to penetrate into the tachocline ηt must be at least
of the order 105 m2 s−1 , a lower limit that is roughly consistent with other estimates.
According to this mechanism, dtach should depend on latitude, and vary with the solar
cycle, but the observational evidence does not seem to support this.
5.2.2. Dynamo action based on tachocline instabilities
Several suggestions have been made for dynamo mechanisms based on hydrodynamic
or MHD instabilities that could operate entirely within the tachocline. Gilman & Fox
(1997, 1999a) have shown that latitudinal differential rotation is generally unstable to
nonradial perturbations in the presence of a toroidal magnetic field. This joint instability
of differential rotation and toroidal magnetic fields may be relevant for explaining the
structure of the tachocline and, possibly, dynamo action (Gilman & Fox 1999b; Dikpati &
Gilman 1999; Gilman & Dikpati 2000, 2002). Instability can occur also in the absence of
a magnetic field for certain values of the latitudinal differential rotation (Charbonneau et
al. 1999a). In a further generalisation using the shallow-water approximation, Dikpati &
Gilman (2001a) have shown that the inclusion of radial deformations has a destabilising
effect, such that the differential rotation in the solar tachocline should be unstable. This
leads to helical disturbances that may produce a dynamo effect (Dikpati & Gilman
2001b; § 6.3.2).
Ponty et al. (2001) investigated the dynamo effect of thermal and shear-related instabilities in the tachocline by means of a numerical kinematic Boussinesq calculation.
Latitudinal differential rotation leads to the formation of an Ekman shear layer at the
bottom of the convective layer that becomes unstable for a sufficiently large Reynolds
number. This contributes to the dynamo action, by causing the magnetic flux to be more
strongly concentrated near the bottom of the unstable layer. However, their model does
not include radial shear, which is important in the solar tachocline.
5.3. Magnetic layer at the base of the convection zone
5.3.1. Observational and theoretical evidence
Several arguments speak out for the existence of a deep-seated layer in the Sun with
strong toroidal magnetic fields (Schüssler 1980, 1983). In the convection zone proper,
magnetic buoyancy would expel magnetic fields with intensities above the equipartition
value on time scales shorter than about a month (Parker 1955a, 1975, 1979a). This would
be too short for enabling the differential rotation to produce the predominantly toroidal
orientation of the large-scale magnetic field inferred from observations. Stability analysis
and dynamical calculations of rising magnetic flux tubes indicate that stronger magnetic
fields can be stored for sufficiently long times in the stably stratified region below the
convection zone (Van Ballegooijen 1982a,b; Moreno-Insertis et al. 1992; Ferriz-Mas &
Schüssler 1993, 1995). Important properties of sunspot groups such as tilt angles and
emergence latitude can be explained on this basis if the toroidal magnetic field has an
intensity of the order B ≈ 1 − 10 T (Choudhuri & Gilman 1987; D’Silva & Choudhuri
326
M. Ossendrijver
1993; Fan et al. 1993; Schüssler et al. 1994; Caligari et al. 1995, 1998). This likely
requires the tubes to have a degree of twist (Wissink et al. 2000a). For sufficiently weak
magnetic fields, magnetic pumping is able to overcome buoyancy and contribute to the
accumulation of magnetic flux near the base of the convection zone (§ 5.5.4).
5.3.2. Location, thickness and thermal properties of the magnetic layer
Although the formation and subsequent rise of magnetic loops from a magnetic layer
near the base of the convection zone cannot be doubted, the precise location and nature
of this layer are still uncertain. It may be plausibly identified with the stably stratified
overshoot layer at the top of the radiative core. On the assumption of an abrupt transition
in the thermal structure, helioseismic inversions (Christensen-Dalsgaard et al. 1995)
and mixing-length models (Skaley & Stix 1991) provide an estimate for its thickness
of dov ≈ 0.1Hp . This does not exclude the possibility of a thicker overshoot layer, if
the transition between the radiative interior and the convection zone is smooth, as is the
case in the model of Xiong & Deng (2001), who obtain dov ≈ 0.6Hp .
Near the base of the convection zone, the heat flux changes from being fully radiative to being almost entirely convective. The resulting nonvanishing divergence of
the radiative heat flux leads to heating of the magnetic flux tubes so that they are more
buoyant then previously assumed (Fan & Fisher 1996; Moreno-Insertis et al. 2002). If
the subadiabaticity in the overshoot layer is of the order ∇ ≈ −10−6 , as is predicted
in most mixing-length models (e.g., Skaley & Stix 1991), then the tubes are expelled on
a timescale of about a month (Rempel 2003). Sufficiently long storage of magnetic flux
tubes is possible only in a region that is more strongly subadiabatic, with ∇ −10−4 .
This possibility is suggested by the overshoot model of Xiong & Deng (2001) and by
numerical simulations, although the latter must be interpreted with care (§ 4.3). A homogeneous magnetic layer, on the other hand, would suppress the convective heat flux,
which, if the suppression is not too strong, can by itself lead to a suitably stable stratification with ∇ −10−4 (Rempel 2003). A stronger suppression leads to destabilization
of the upper part of the magnetic layer. This enhanced buoyancy of the magnetic field
in the overshoot layer is of interest for the α effect.
5.3.3. Topology and equilibrium of the magnetic layer
Various instabilities can lead to the break up of the magnetic layer (Hughes 1992).
From linear stability analyses the possibility of Rayleigh-Taylor type instabilities (Chandrasekhar 1961; Cattaneo et al. 1990a) and double-diffusive type instabilities (Acheson
1979; Schmitt & Rosner 1983) are known. Numerical simulations in 2D of the linear
and nonlinear evolution of an unstable magnetic layer indicate that the Rayleigh-Taylor
instability leads to the formation of mushroom-like structures that are subject to a secondary Kelvin-Helmholtz instability, which has a further disrupting effect on the layer
(Cattaneo & Hughes 1988; Cattaneo et al. 1990a,b). In 3D, the instability is found to result in the formation of arched flux tubes (Matthews et al. 1995; Wissink et al. 2000b). It
is not clear whether the magnetic layer in the Sun consists entirely of flux tubes (DeLuca
et al. 1993), or whether it can remain partly homogeneous. Due to rotation, the buoyant
fluid acquires a twist that contributes to the α effect (§ 5.5.3). The amplification of the
The solar dynamo
327
magnetic field by differential rotation and the subsequent expulsion of flux tubes might
produce a relaxation oscillation (Schmitt & Rosner 1983; Brummell et al. 2002).
The nature of the equilibrium state of the magnetic layer and the flux tubes is a
delicate issue. For very thin toroidal tubes, an equatorward meridional flow can play a
role by providing a drag force to balance the magnetic tension force that pushes the tubes
towards the poles (Van Ballegooijen 1982b; Van Ballegooijen & Choudhuri 1988). The
drag force is not likely to be important for rising loops, because the observed properties
of sunspot groups (e.g. Hale’s law and emergence latitudes) suggest that they are not
strongly disfigured during their journey through the convection zone (Schüssler 1984,
1987; Choudhuri & D’Silva 1990). In spite of this, all magnetic surface features corotate
with the surface, even though the deep-rooted footpoints of the loops rotate at a different
rate, so drag forces must be efficient in establishing corotation at some point close to the
surface - presumably because the flux tubes assume a fibril form (Zwaan 1978; Parker
1979b; Schüssler 1984). In order for the tubes to be stored for a sufficiently long time
their initial state must be one of mechanical equilibrium. The magnetic tension force
can be balanced by a Coriolis force resulting from a mass flow along the tube, and the
buoyancy force should vanish, so that the tubes have the same density as the external
medium (Caligari et al. 1995). Due to the pressure equilibrium, this means that they
have a lower temperature. If the tubes are initially not in a mechanical equilibrium, they
will migrate polewards until the equilibrium is established, and this is hard to reconcile
with the observed emergence latitude of sunspots. It is therefore likely that the magnetic
layer from which the tubes are formed is itself already in a mechanical equilibrium.
Such a mechanical equilibrium can arise naturally in an axisymmetric homogeneous
magnetic layer (Rempel et al. 2000). If the magnetic layer is located in the weakly subadiabatic overshoot region, it evolves towards an equilibrium dominated by the Coriolis
force, such that it will rotate more rapidly than the surroundings. If the magnetic layer
is located in the more strongly subadiabatic radiative core, it evolves towards a different
equilibrium in which the magnetic tension force is balanced by a latitudinal pressure
gradient. This has consequences for the tubes that emerge from the layer, because only
the former case is compatible with the required mechanical equilibrium, so that the possibility of a magnetic layer in the radiative core seems to be excluded. In addition, the
magnetic field strength of flux tubes in the radiative core would have to be prohibitively
large for them to overcome their stability and emerge from the core in order to form
sunspots.
5.4. Amplification of the toroidal magnetic field
A major issue of solar dynamo theory concerns the generation of strong toroidal magnetic
fields near the bottom of the convection zone. In particular, the question arises whether
they are generated by the radial differential rotation alone, or whether additional mechanisms are operating. The latter seems likely because the magnetic field strength that
one infers by comparing results of flux tube computations with properties of bipolar
magnetic regions is of the order 10 T, which is about 10 times the local equipartition
value with ambient convection. It is nontrivial to explain how differential rotation alone
can achieve such field enhancement in the presence of magnetic tension forces, although
it cannot be completely ruled out (Petrovay 1991).
328
M. Ossendrijver
The formation of omega loops in the convection zone can provide additional field
intensification by stretching and partially evacuating the loop section that remains anchored at the base of the convection zone (Parker 1994a). Perhaps sufficient amplification
is achieved only through a succession of such events, which would require the omega
loops to become detached from the anchored flux tube (Parker 1994b). Thin flux-tube
calculations confirm that rising loops lead to field amplification, and they suggest that
some loops can undergo an explosion-like event that enhances the magnetic field strength
still further (Moreno-Insertis et al. 1995). This can be understood by making the reasonable assumption that rising loops are in pressure equilibrium during their evolution
and have little heat exchange with the external medium. As a result, the internal pressure in the summit of the tube must decrease with height more slowly than the external
pressure. If the initial magnetic pressure in the tube is small enough, the tube summit
undergoes a catastrophic expansion at some point within the convection zone because
the internal pressure becomes equal the external pressure. This event and the subsequent
phase are not accessible to a thin flux-tube analysis, but Rempel (2002) performed MHD
simulations of exploding flux tubes using a simplified twodimensional setup. The results
confirm that significant amplification can be achieved during the phase when the tube
evacuates from its open ends, depending on the entropy contrast between the outflowing
material and the surroundings.
5.5. Dynamo coefficients
As was argued in § 3.4.3, it is possible to write down a general form of the meanfield dynamo equation without addressing in any detail the physical mechanisms behind
the dynamo coefficients, using only symmetry considerations. By making assumptions
about the mean flow and the dynamo coefficients one may explore different models
for the solar dynamo using a selection of coefficients, and this has been a frequent
approach in dynamo investigations. A more physically motivated approach is to identify
the mechanisms that contribute to the dynamo coefficients, and to compute these using
analytical or numerical tools. To a limited extent, dynamo coefficients can be estimated
from observations (§ 2.6).
5.5.1. Methods of computation
Anisotropic inhomogeneous turbulence provides a framework for analytical computation
of dynamo coefficients using closure methods. Turbulence models are able to capture
essential properties of solar convection in a statistical sense (Moffatt 1978: § 7; Krause
& Rädler 1980: § 9). Commonly used closure methods are FOSA (§ 3.4.1), the EDQNM
method (Frisch et al. 1975) and the closely related τ -approximation (Pouquet et al. 1976;
Rogachevskii & Kleeorin 2000, 2001; Kleeorin & Rogachevskii 2003). In spite of the
theoretical difficulties of applying FOSA to convective flows, the results for the dynamo
coefficients often agree qualitatively with those obtained through MHD simulations,
although they typically overestimate their magnitude (Ossendrijver et al. 2001, 2002).
In the case of α, this is probably due to the fact that convection tends to be less helical
than is assumed in the turbulence models. Also, the spatial intermittence of the solar
The solar dynamo
329
magnetic field dilutes the interaction between the magnetic field and the flow, which
results in smaller dynamo coefficients (Childress 1979, 1981).
The EMF can also be computed by solving linearized wave equations for u and
b derived from the MHD equations and taylored for a specific physical mechanism.
Examples are magnetostrophic waves and the undular instability of toroidal magnetic
flux tubes (§ 5.5.3). Due to the linearization, such computations provide no information
on the magnitude of the dynamo coefficients.
Numerical computation of dynamo coefficients does not suffer from the restrictions
of FOSA and linearization. Furthermore, it can be a powerful diagnostic tool to isolate
from numerical simulations the various effects that play a role in the solar dynamo.
Dynamo coefficients can be determined by measuring E and B 0 , and inverting the
assumed relation between them (Eq. 14). Usually simplifications are necessary in order
to reduce the number of unknown coefficients.
The coefficients α and ηt have been studied intensively through numerical simulations of isotropic turbulence and elementary chaotic flows in two and three dimensions
with the purpose of establishing the magnetic quenching (§ 3.4.5). Due to their idealized
nature, e.g. the use of periodic boundary conditions, the resulting quenching expressions
are unlikely to be directly applicable to the Sun.
5.5.2. Convective α effect
The convective (or turbulent) α effect results from passive advection of magnetic fields
by helical convection. This works only in the absence of strong magnetic curvature
forces, so that the magnetic field intensity must be less than roughly the equipartition
value. The convective α effect is an integral part of the interface and distributed dynamo
models (§ 6.3.1).
For isotropic turbulence the α effect can be approximately described by a single
pseudoscalar (Eq. 18). For this case Keinigs (1983) also derived an expression in terms
of the current helicity, such that α ≈ −ηj ·b/B02 . Numerical simulations of convection
in rotating systems indicate that α is highly anisotropic. The most important component
in the Sun is αφφ , which is responsible for generating a poloidal mean magnetic field from
a toroidal mean magnetic field (§ 3.4.4). This can also be interpreted as the generation
of an effective toroidal current that is (anti-)parallel to the original toroidal mean field
(Fig. 9). In the bulk of the convection zone, the sign of αφφ is positive (negative) on the
northern (southern) hemisphere, and it reverses near the base, in qualitative agreement
with Eq. (18) (Brandenburg et al. 1990; Ossendrijver et al. 2001). This can be understood
in terms of Coriolis forces if one takes into account that a rising (sinking) fluid parcel
expands (contracts) in the bulk of the convection zone (Fig. 9), whereas it contracts
(expands) near the base. The sign of αφφ as found in simulations of stratified convection,
but not the magnitude, is also roughly consistent with the FOSA result
αφφ ≈ −
16 2 2
τ u · ∇ ln(ρurms ),
15 c rms
(34)
which holds for mildly anisotropic turbulence (Steenbeck & Krause 1969). The measurements of the kinetic and current helicities at the solar surface (§ 2.6) also point to
αφφ being positive in the bulk of the convection zone on the northern hemisphere, as
330
g
M. Ossendrijver
er Ω
uconv
uexp
eφ
eθ
fcor
Fig. 9. Convective α effect for the bulk of the solar convection zone on the northern hemisphere.
Due to the Coriolis force, convective motions acquire a systematic twist such that the mean kinetic
helicity is negative. Embedded weak toroidal magnetic fields are passively advected leading to
the formation of loops carrying a mean current σ αφφ B 0t that is parallel to B 0t , so that αφφ is
positive
follows by applying the FOSA relations between αφφ and the kinetic or current helicity. Further analytical results for anisotropic turbulence were derived by Rüdiger &
Kitchatinov (1993) using FOSA, and by Rogachevskii & Kleeorin (2000) and Kleeorin
& Rogachevskii (2003) using the τ -approximation.
By varying the inclination of , Ossendrijver et al. (2002) determined the full α
tensor as a function of solar latitude and other parameters in local simulations of magnetoconvection with weak imposed magnetic fields. The diagonal components of α peak
near the poles, and they vanish at the equator, or in the absence of rotation. Up to a
rotation rate of about Co ≈ 2, which is typical for the lower part of the solar convection
zone, αrr is the largest diagonal component, and its sign is opposite to that of αφφ . If
the rotation rate increases beyond this point, αrr reduces due to rotational quenching,
while αφφ continues to increase (Ossendrijver et al. 2001). These tendencies are known
from the FOSA results (Rüdiger & Kitchatinov 1993). The magnitude of αφφ in the Sun
can only be roughly estimated from the simulations, since the parameter regime is not
realistic in some respects (§ 4.3). For Co ≈ 2, the simulations yield αφφ ≈ 0.05urms at
mid latitudes in the bulk of the unstable layer. Taking into account the tendency of α to
decrease with increasing degree of turbulence (Ossendrijver et al. 2002), one may conclude that 5 m s−1 is a rough upper limit for αφφ in the lower half of the solar convection
zone. The actual value is likely to be smaller by at least an order of magnitude.
Helical disturbances resulting from a hydrodynamical instability in the tachocline
(§ 5.2) can also lead to an α effect through passive advection of the magnetic field.
Depending on which modes contribute, it is expected to have a complex latitude dependence, with two or more sign changes between pole and equator (Dikpati & Gilman
2001b). This α effect should be quenched if the magnetic field intensity approaches the
equipartition value.
5.5.3. Magnetically-driven α effect
Magnetic buoyancy leads to the formation of rising loops in the overshoot layer (§ 5.3).
Linear stability analysis and numerical simulations based on the thin flux-tube approximation show that toroidal flux tubes with field strengths up to about 1–10 T can be
The solar dynamo
g
331
er Ω
fcor
ubuoy
eφ
eθ
u//
Fig. 10. Buoyancy-driven α effect for the northern hemisphere of the Sun. A rising section of a
toroidal magnetic flux tube is twisted by Coriolis forces resulting from variations in the flow along
the tube. The mean current corresponding to the twisted loops, σ αφφ B 0t , is parallel to B 0t , so
that αφφ is positive
stably stored in the solar overshoot layer (Ferriz-Mas & Schüssler 1995; Caligari et al.
1995). Due to differential rotation and other effects (§ 5.4), the field strength of the tubes
increases steadily, up to a point where they become unstable to small displacements and
form rising loops (Fig. 10). While the tubes rise they are subject to a Coriolis force so that
they acquire a systematic twist that is equivalent to an α effect. The buoyancy-driven α
effect has been computed analytically in terms of magnetostrophic waves (Moffatt 1978:
§ 10; Schmitt 2003) and thin flux tubes (Ferriz-Mas et al. 1994; Ossendrijver 2000). Numerical results for the initial phase of the instability in a homogeneous magnetic layer
are presented by Brandenburg & Schmitt (1998) and Theelen (2000a); the subsequent
nonlinear evolution leading to the formation of flux tubes was considered by Wissink et
al. (2000b). The sign of αφφ as found in the computations is predominantly positive for
the northern hemisphere, as shown in Fig. (10). The magnitude is typically of the order
10−3 urms , which would correspond to about 0.1 m s−1 in the Sun, roughly consistent
with the estimate of Ferriz-Mas et al. (1994). The requirement of a minimum magnetic
field strength for instability means that a dynamo based on a buoyancy-friven α effect
is not self-excited (§ 6.3.3).
If the toroidal magnetic field in the overshoot layer is sufficiently weak, it is subject
to a pinch instability that can lead to an α effect (Spruit 2002). This might enable selfexcited dynamo action within the overshoot layer. Exploding flux tubes in the convection
zone (§ 5.4) might also contribute to an α effect. Finally, mention should be made of the
magneto-rotational instability (Velikhov 1959; Balbus & Hawley 1991). It is unlikely to
be important for dynamo action in solar-type stars, but may be responsible for generating
weak turbulence in the radiative interior of the Sun, thereby establishing the observed
near solid-body rotation (Spruit 1999; Arlt et al. 2003).
5.5.4. Flux expulsion and pumping effects
Magnetic pumping refers to any form of transport of magnetic fields in convective layers
that does not result from bulk motion. The best known example of magnetic pumping
is the expulsion of flux away from regions of intense turbulence, an effect interpreted
as turbulent diamagnetism by Zel’dovich (1957). Flux expulsion by convective eddies
in 2D is a well-studied problem, for which Parker (1963b, 1979a) presented stationary
332
M. Ossendrijver
solutions. The dynamics were studied by Weiss (1966), who found that the time scale
for expulsion is of the order Rm1/3 τc , where Rm is the magnetic Reynolds number of
the eddies.
Flux expulsion can be important on any scale at which the turbulence is inhomogeneous, provided that the associated time scale is sufficiently short, and that the Lorentz
forces are not too strong. It therefore contributes to the intermittence of the magnetic field
in the solar convection zone (Galloway & Weiss 1981). Also, it shortens the length scales
of the magnetic field, thereby reducing the diffusive time scale and creating favorable
conditions for fast dynamo action (Childress 1979; Roberts 1987; Soward 1988).
On large scales, flux expulsion in stratified convection is directed radially inwards,
so that it counteracts magnetic buoyancy. Hence for sufficiently weak magnetic fields,
downward pumping should dominate. From analytical and numerical MHD computations using prescribed incompressible flows it is known that magnetic flux is pumped in
the direction in which the flow forms connected channels; hence the designation topological pumping (Drobyshevski & Yuferev 1974; Arter et al. 1982; Arter 1983, 1985;
Galloway & Proctor 1983). This effect may contribute to downward pumping in the Sun.
It can be quantified in terms of a drift velocity γ (Eq. 16); this requires a third-order
approximation (Moffatt 1978: § 3; Krause & Rädler 1980: § 7.2; Drobyshevski et al.
1980).
Magnetic pumping in stratified compressible convection is more likely due to the diamagnetic effect, because magnetic pumping operates throughout a convective layer also
if the connected downflowing lanes fragment into isolated plumes below a certain depth.
Petrovay & Szakály (1993) point out that on a time scale of months the mean magnetic
field in the solar convection zone can be treated as being approximately in a stationary
state defined by a balance between downward pumping and turbulent diffusion, which
leads to a monotonic increase with depth of the field strength. Thus, turbulent pumping
contributes to the accumulation of magnetic flux at the bottom of a convective layer
(Schüssler 1983, 1984; Petrovay 1991). This is well-known from numerical simulations
(Brandenburg & Tuominen 1991; Nordlund et al. 1992; Brandenburg et al. 1996; Tobias
et al. 1998, 2001; Dorch & Nordlund 2001; Ossendrijver et al. 2002; Ziegler & Rüdiger
2003). The diamagnetic effect results in a pumping velocity of the order
1
γ ≈ − τc ∇u2 ,
(35)
3
a relation that is borne out both by FOSA computations (Krause & Rädler 1980: § 9.5,
Kichatinov & Rüdiger 1992; Petrovay & Szakály 1993) and numerical simulations (Tao
et al. 1998b; Rüdiger & Ziegler 2003).
If the magnetic field is strong, flux expulsion can occur spontaneously as a consequence of magnetic quenching of the turbulence, without the need for any initial
inhomogeneity. This is because the expulsion of a small initial amount of flux leads to
more vigorous convection in that region, which enhances the expulsion, until the region
is nearly free of flux (Blanchflower et al. 1998; Tao et al. 1998a). This can be relevant for
explaining magnetic flux separation in the photosphere. Tobias (1996b) has illustrated
how the related phenomenon of magnetic quenching of ηt can contribute to accumulation
of flux in the overshoot layer.
The presence of a density gradient leads to pumping due to magnetic buoyancy.
Kitchatinov & Pipin (1993) present FOSA calculations for buoyancy-driven pumping of
The solar dynamo
333
small-scale magnetic fields. While the direction is mostly upwards (Rüdiger & Ziegler
2003), it can be downwards in special cases.
As suggested by symmetry considerations (Eq. 16), rotationally induced anisotropies
can lead to pumping effects in non-radial directions, so that Eq. (35) is no longer adequate. The relevant FOSA results are presented by Kitchatinov (1991) and Rüdiger &
Kitchatinov (1993). Kleeorin & Rogachevskii (2003) provide expressions based on a
derivation that takes into account effects of magnetic helicity conservation. In the simulations of Ossendrijver et al. (2002), the components of γ were determined numerically
through simulations of magnetoconvection with weak imposed magnetic fields. In the
latitudinal direction, there is a weak net equatorward pumping effect in the bulk of the
unstable layer that vanishes at the pole and at the equator. This could contribute to the
equatorward motion of the magnetic belts in the Sun. The azimuthal pumping effect is
predominantly in the retrograde direction within the bulk of the convective layer and the
overshoot layer. Due to its depth dependence, the azimuthal pumping velocity has the
same effect on the mean magnetic field as differential rotation; this might be relevant for
explaining magnetic cycles in stars that have negligible differential rotation (Rüdiger et
al. 2003).
In addition to the general pumping effect common to all magnetic field components,
Kichatinov (1991) showed that it is possible to define pumping velocities for separate
components of the magnetic field, provided they are divergence free. There is evidence
for this from numerical simulations (Ossendrijver et al. 2002). This may contribute to
the difference in the migration of the toroidal and the poloidal magnetic fields in the
Sun.
5.5.5. Turbulent magnetic diffusion
Turbulent diffusion of magnetic fields in isotropic conditions or due to elementary chaotic
flows in two and three dimensions has been well studied in the context of investigations of
magnetic quenching (§ 3.4.5; § 5.7.3). Such simulations confirm that for weak magnetic
fields ηt is correctly estimated by Eq. (19). For anisotropic turbulence, some analytical results based on FOSA (Kitchatinov et al. 1994b; Urpin & Brandenburg 2000) and
the τ -approximation (Rogachevskii & Kleeorin 2001) are available. They indicate that
stratification, rotation and strong magnetic fields cause βij k to be anisotropic. The dependence on the magnetic field intensity can be complex, but the quenching does not
appear to be catastrophic. The anisotropy of turbulent diffusion can aid in resolving a
number of issues of the solar dynamo such as the parity problem and the migration of
the magnetic belts (§ 6.1).
5.5.6. Cross-helicity effect
Yoshizawa (1990) has generalized the EMF by including a term proportional to the
cross-helicity u · b, which is ignored in traditional mean-field dynamo theory:
E ≈ τc u · b ∇ × U 0 .
(36)
This contribution to the mean current proportional to the mean vorticity (∇ × U 0 ) is a
result of the Alfvén effect, which causes alignment of the magnetic field with the flow.
334
M. Ossendrijver
Yoshizawa et al. (2000) have proposed a dynamo scenario based on the cross-helicity
effect, the main novelty being that rotation suffices to maintain a turbulent state with
nonvanishing cross helicity, so that differential rotation is not required for dynamo action.
It is not clear how important the cross-helicity effect is in the Sun, where differential
rotation plays an important role, because no detailed model is available. Perhaps it plays
a role in producing a seed magnetic field (Brandenburg & Urpin 1998; Blackman 2000).
5.6. Stochastic behaviour
One explanation for the variability of the solar cycle is based on the observation that
convection is inherently random. Solar convection is organized in a hierarchical structure
consisting of several scales (§ 4.1). Numerical simulations show that convection is spatially and temporally intermittent, with narrow rapid downdrafts embedded in a slowly
upflowing surrounding (§ 4.3). With each of the spatial scales is associated a finite lifetime. This random renewal of convection should affect the evolution of the magnetic field
and contribute to the variability of the solar cycle. Furthermore, solar variability must be
taken into account for a correct interpretation of the averaging procedure (§ 3.4.2). In the
ensemble average it can lead to phase mixing, and if the azimuthal average is adopted,
the dynamo coefficients acquire fluctuating components. Fluctuations should arguably
be most prominent for α. Local simulations of magnetoconvection indicate that α is
subject to a high degree of cancellation, such that a long-term spatio-temporal average
yields a value much smaller than the variance of the spatial average (Ossendrijver et al.
2001, 2002). It is not straightforward to extrapolate such results to the global dynamo,
but there the fluctuations should be smaller.
The inclusion of global (i.e. spatially fixed) α fluctuations with a correlation time
that is short compared to the cycle length can result in behaviour reminiscent of solar
variability if the fluctuations are of the order of 10% (Choudhuri 1992). Such models
exhibit an anticorrelated random walk of the amplitude and phase of the dynamo cycle,
similar to what is observed (Hoyng 1993). This is readily explained in terms of α-type
dynamo action. Nonlinear effects result in a confinement of the random walk equivalent
to phase locking (Ossendrijver & Hoyng 1996), a phenomenon that was inferred by
Dicke (1978) from an analysis of the sunspot cycle. The models also illustrate that
grand minima might be the cumulative effect of α fluctuations. Since all quantities in
such models are azimuthal averages, the underlying fluctuations in individual convective
eddies must be rather strong.
If the α fluctuations are allowed to be spatially incoherent, the solutions also exhibit
parity variations resulting in north-south asymmetries and fluctuations in the polarity
dividing line (Moss et al. 1992; Hoyng et al. 1994; Ossendrijver et al. 1996; Mininni &
Gómez 2002). This can be understood in terms of the excitation of overtones of the 22year mode. If the 22-year mode is marginally excited, then all overtones are transiently
excited, with the mean energy in each mode depending on the decay rate and on a
dimensionless parameter measuring the effective strength of the fluctuations in terms of
eddy parameters,
frms τc 1/2
ξ≡
.
(37)
N c τd
The solar dynamo
335
Fig. 11. Computed butterfly diagram for a dynamo model with spatially incoherent α fluctuations
with ξ = 1.4 · 10−2 . Dark and light denote negative and positive values of the mean toroidal
magnetic field, respectively (cf. Ossendrijver et al. 1996)
Here frms is the relative strength of the α fluctuations in individual eddies, τc the turnover
time, Nc the number of eddies from pole to pole, and τd the turbulent diffusion time (Ossendrijver et al. 1996). Solar-type behaviour is found if ξ ≈ 1.4·10−2 (Fig. 11), a number
that in a best-case scenario assuming the existence of giant cells with τc /τd ≈ 10−3 and
Nc ≈ 20 would require frms ≈ 10. It is unclear whether this level of fluctuations can be
justified.
Hoyng et al. (1994) set out to explain the modal structure of the surface magnetic
field claimed by Stenflo & Vogel (1986), but they obtain no agreement. They argue that,
if existent, it is unlikely to be reproduced by any mean-field dynamo model because
the mode amplitude should decrease only slowly for subsequent overtones, whereas the
amplitude of adjacent dynamo modes always increases rapidly, since the linear decay rate
of dynamo modes increases rapidly with increasing spatial structure. But it is unclear how
relevant linear decay rates are, because the solar dynamo operates in a nonlinear saturated
state (Brandenburg et al. 1989). Therefore it cannot yet be ruled out that stochastic
effects might explain the modal structure if it turns out to be real, and if the required
amplitude of the fluctuations can be justified. But it is likely to require a more detailed
treatment of the surface flux, along the lines of the flux transport models. Models with
spatially incoherent α fluctuations are able to reproduce such features as the observed
anticorrelation between phase and amplitude variations, north-south asymmetries and
grand minima, but they typically do not exhibit phase locking (Ossendrijver et al. 1996).
The latter might come out differently if the meridional flow is important.
5.7. Nonlinear behaviour
A glance at the full set of MHD equations (1-4) makes it obvious that the dynamo problem
is highly nonlinear. The Navier-Stokes equation is nonlinear in terms of the flow, and this
gives rise to hydrodynamical turbulence and chaotic flow trajectories. The Lorentz force
renders the induction equation nonlinear in terms of the magnetic field, and this causes
saturation of the dynamo at some field intensity determined by the balance of forces.
The importance of nonlinear effects is thus already obvious from the trivial fact that the
solar magnetic field is saturated. MHD turbulence investigations suggest that dynamo
saturation in rotating stars is a consequence of magnetic helicity conservation, such that
the growth of the large-scale magnetic field due to the inverse cascade ceases when
336
M. Ossendrijver
the small-scale magnetic field reaches equipartition with the flow (§ 3.3). This process
can, in principle, be described by dynamical quenching of the dynamo coefficients
(§ 3.4.5). Other aspects of the Lorentz force may also contribute to dynamo saturation.
For example, magnetic buoyancy explains the emergence of sunspots, but its precise
role in the solar dynamo is less clear. Beyond that, there are solar-cycle features that are
indicative of nonlinear effects, among which are grand minima, torsional oscillations,
north-south asymmetries, and the presence of other dynamo modes besides the 22-year
mode.
So far it has proven difficult to establish a satisfactory and complete treatment of
nonlinear effects in the framework of mean-field dynamo theory. While some terms,
such as the Malkus-Proctor effect (§ 5.7.4), are firmly based on physical principles,
others are heuristic parametrizations that may be fundamentally flawed. The latter may
hold for the often used algebraic quenching of αφφ , which should be replaced by a
dynamical description (§ 3.4.5). Analytical results based on FOSA yield rather complex
dependencies of the dynamo coefficients on B0 , but no catastrophic quenching (§ 5.5).
For some purposes, these uncertainties are not very important, because mean-field
models yield rather similar results irrespective of the details of the nonlinearities.
Thus, the investigation of nonlinear mean-field models in the spirit of astromathematics
(Spiegel 1994) serves to illustrate behaviour that may occur in the solar dynamo. Within
this context a further idealization can be achieved by expanding the variables in terms of
Fourier modes and applying a truncation. As a result, the partial differential equations
are reduced to a set of coupled ordinary differential equations that can be investigated
using the tools of nonlinear dynamics. Such equations may be investigated in order
to identify their generic behaviour (Tobias et al. 1995; Knobloch & Landsberg 1996).
This goal might be elusive, because the results of nonlinear models depend strongly on
the structure of the dynamo (Jennings et al. 1990), and even within one model small
structural changes can lead to qualitatively different types of behaviour (Phillips et al.
2002). Therefore the main relevance for the solar dynamo of studying idealized nonlinear models lies in identifying phenomena that may explain observed features of the
solar cycle.
5.7.1. Typical phenomena in nonlinear dynamo models
Nonlinear dynamo models exhibit several types of behaviour that are reminiscent of
long-term solar variability. A change in the parameters can lead to symmetry breaking
such that a different parity of the magnetic field is preferred (Jennings 1991; Jennings
& Weiss 1991). The coexistence of modes with different parities results in mixed-parity
solutions, which may explain the enhanced asymmetry of solar activity observed during
the Maunder minimum. Nonlinear dynamos can exhibit periodic or quasiperiodic modulations with a transition to chaos for sufficiently large supercritical dynamo numbers
(Weiss et al. 1984; Jones 1984; Ruzmaikin 1984; Tobias et al. 1995). Various forms
of temporal intermittency are known, characterized by the irregular interruption of one
type of behaviour, typically periodic oscillations, by bursts of another type of behaviour
(Tavakol 1978; Platt et al. 1993; Tworkowski et al. 1998; Covas & Tavakol 1997; Covas
et al. 1997; Tavakol & Covas 1999). Often the modulations affect both amplitude and
symmetry of the solutions, with high-amplitude oscillations of dipolar parity interrupted
The solar dynamo
337
by low-amplitude intervals where the solution assumes quadrupolar parity, reminiscent
of the Maunder minimum (Knobloch & Landsberg 1996; Tobias 1997; Knobloch et
al. 1998; Brooke et al. 1998). A predominance of dipolar symmetry as is observed on
the current Sun is not always obtained in nonlinear models. Often the solutions are of
a mixed parity resulting in rather strong asymmetries. Unless one accepts that the observed predominance of dipolar parity on the Sun is a consequence of carefully tuned
parameters, it appears that the models often suffer from a parity problem (§ 6.1).
5.7.2. Quenching of α
The convective α effect requires passive advection of the magnetic field by the flow, which
becomes increasingly difficult if the magnetic field intensity approaches the equipartition
value (§ 5.5.2). This has given rise to the concept of algebraic α quenching, according
to which α = α0 f (B0 ), where f is a decreasing function of B0 . A commonly used
2 )−1 , where K is a constant
heuristic formulation is provided by f = (1 + KB02 /Beq
of the order 1 (Stix 1972; Jepps 1975; Ivanova & Ruzmaikin 1977). Analytical computations based on closure methods yield more complicated expressions for f (§ 5.5.2).
It has been suggested that K should be of the order Rm, so that the α effect would be
negligible in the Sun. Considerations based on magnetic helicity conservation indicate
that α quenching in the Sun requires a dynamical formulation and that the inference of
catastrophic quenching is premature (§ 3.4.5).
Algebraic α quenching has been used extensively in models, and is known to produce
such phenomena as chaotic modulations (Weiss et al. 1984; Tavakol et al. 1995) and parity
changes (Brandenburg et al. 1989; Jennings & Weiss 1991). Dynamical quenching has
been incorporated in several models (Kleeorin et al. 1994, 1995; Roald & Thomas 1997;
Schmalz & Stix 1991; Schlichenmaier & Stix 1995). This introduces an effective delay
in the backreaction of the magnetic field, that can by itself lead to periodic or chaotic
modulations, as was recognized by Yoshimura (1978a,b, 1979).
5.7.3. Quenching of ηt
Magnetic quenching of ηt has the effect of enhancing the diffusive time scale, which
controls cyc . Thus, with increasing intensity of the magnetic field, i.e. increasing dynamo number, cyc decreases, at least in local Cartesian models (§ 5.7.7). In spherical
geometry, the behaviour is more complex, because of changes in the spatial structure of
the solutions (Rüdiger et al. 1994; Tobias 1998).
The inclusion of ηt -quenching in the interface dynamo model (§ 6.3.1) is sufficient
to cause the formation of a layer with strong magnetic fields through flux expulsion even
without adopting any initial reduction of ηt in this layer (Tobias 1996b). As expected,
the solutions assume the form of a surface wave tied to the interface, but due to the ηt quenching, there is a sudden transition from a weak-field regime with little quenching to
a strong-field regime with strong quenching in the magnetic layer. Such a transition may
be relevant for explaining grand minima, such that the strong-field regime corresponds
to the normal state of the solar dynamo, since this is more likely to produce sunspots
than the weak-field regime.
338
M. Ossendrijver
5.7.4. Malkus-Proctor effect
The Lorentz torque exerted by the large-scale magnetic field, also known as the MalkusProctor effect (Malkus & Proctor 1975; Proctor 1977), is given by the first term on
the right hand side of Eq. (31). The torque exerted by the small-scale magnetic field is
usually ignored; analytical computations based on FOSA suggest that it weakens the
Malkus-Proctor effect (Rüdiger et al. 1986).
For a predominantly axisymmetric large-scale magnetic field like that of the Sun, the
Malkus-Proctor effect scales as the product of the toroidal and poloidal components of
the mean magnetic field. In order for the Malkus-Proctor effect to achieve a significant
modulation, the magnetic field strength must be at least of the order of the equipartition
value with respect to kinetic energy, which is unlikely to be the case within the solar
convection zone. Also taking into account the intermittent nature of the magnetic field
in the convection zone, the Malkus-Proctor effect is more likely to be important in the
overshoot layer, where the magnetic field is stronger and probably more space filling
(§ 5.3).
The Malkus-Proctor effect can cause dynamo saturation (Brandenburg et al.
1991) and produce various types of nonlinear behaviour such as torsional oscillations (Schüssler 1981; Yoshimura 1981; Rüdiger et al. 1986; Jennings 1993), nonaxisymmetric dynamo modes (Barker & Moss 1994), long-term modulations including Gleissberg-type cycles, grand minima (Weiss et al. 1984; Kleeorin & Ruzmaikin
1989; Belvedere et al. 1990; Roald & Thomas 1997; Phillips et al. 2002) and intermittent behaviour (Moss & Brooke 2000). For small dynamo numbers Tobias (1996a,
1997) identified a weak-field regime characterized by a beat phenomenon also known
from other nonlinear models, resulting from the interference of dipolar and quadrupolar modes. This leads to strong parity modulations, unlike what is observed during the
normal solar cycle. For sufficiently large dynamo numbers, the solutions also exhibit a
strong-field regime peculiar to the Malkus-Proctor effect with only small parity modulations, that can alternate with the weak-field regime in a quasiperiodic fashion. These
alternating regimes are reminiscent of grand minima and the normal solar cycle, respectively. The timescale of the long-term modulations resulting from the Malkus-Proctor
effect is controlled by the turbulent magnetic Prandtl number, Pmt ≡ νt /ηt (Tobias 1997;
Covas et al. 2000, 2001a; Moss & Brooke 2000; Phillips et al. 2002).
For a range of dynamo parameters the torsional oscillations can have a complex
structure known as spatio-temporal fragmentation characterized by a doubling of the
cycle frequency at a certain depth in the convection zone (Covas et al. 2001b; Tavakol et
al. 2002). The frequency doubling can occur repeatedly, and this might explain reported
observations of tachocline oscillations with periods in the range 1-3 years (Howe et al.
2000b; Toomre et al. 2000).
The response of the differential rotation is not instantaneous, but occurs with a certain
delay. This aspect has been emphasized by Yoshimura, who illustrated that such a delay
can by itself lead to modulations of the solar cycle (Yoshimura 1978a,b, 1979).
5.7.5. Quenching of Reynolds stresses
Instead of directly acting on the mean flow, the magnetic field can modify the transport
of angular momentum through a backreaction on the -effect, thereby changing the
The solar dynamo
339
differential rotation (§ 4.4). The magnetic quenching of scales with the magnetic
energy density, for which reason it is likely to be more important in the solar convection
zone than the Malkus-Proctor effect (Kitchatinov 1990; Rüdiger & Kitchatinov 1990).
By incorporating Eq. (30) with -quenching, mean-field dynamo models are capable
of producing a pattern of torsional oscillations similar in shape and magnitude to the
observed one (Küker et al. 1996). -quenching has also been invoked in order to explain
long-term modulations and grand minima (Kitchatinov et al. 1994; Küker et al. 1999).
The model of Pipin (1999) exhibits a periodic modulation reminiscent of the Gleissberg
cycle that affects both amplitude and parity of the magnetic field.
5.7.6. Quenching by magnetic buoyancy
Magnetic buoyancy can contribute to dynamo saturation by expelling magnetic flux. This
can be important for strong magnetic fields in the convection zone, for which buoyancy
dominates over magnetic pumping (§ 5.5.4). For the toroidal flux system in the overshoot
layer, flux expulsion is less likely to contribute to dynamo saturation, because this would
require entire toroidal flux tubes to be expelled rather than single loops. Flux loss due
to magnetic buoyancy is sometimes included in the mean-field induction equation by
adding a heuristic loss term of the form −f (B0 )B 0 /τbuoy (Leighton 1969; DeLuca &
Gilman 1986; Schmitt & Schüssler 1989), an approach that is unlikely to be adequate. A
more realistic approach is to include an effective drift velocity or pumping effect (Moss
et al. 1990a,b; Ossendrijver 2000a). Analytical results based on FOSA were derived by
Kitchatinov & Pipin (1993).
If the solar dynamo is driven by magnetic buoyancy (§ 6.3.3), saturation may be
achieved because flux loops are expelled on an increasingly rapid time scale with increasing magnetic field strength. As a result, they acquire less and less twist, so that the
buoyancy-driven α effect (§ 5.5.3) is effective only for a finite range of magnetic field
intensities above the instability threshold.
5.7.7. Cycle frequencies of nonlinear dynamos
Non-linear effects can have a profound effect on the cycle frequency. For moderately
supercritical dynamos it is possible to obtain approximate scaling laws for each type of
nonlinear feedback. For example, interface-type dynamo models with different types of
quenching (α and ηt -quenching; Malkus-Proctor effect) yield an increase of cyc with
increasing dynamo number according to
cyc ∝ |D|n ,
(38)
where n lies in the range 0.38 n 0.67 (Tobias 1998). This range is in fact rather
tightly centered around the linear value n = 0.5 (§ 3.4.4). Similar behaviour is found for
the magnetic buoyancy prescription adopted by Schmitt & Schüssler (1989) and Moss et
al. (1990a,b). The difference between cyc in the nonlinear model and that of the linear
model turns out to scale with the average energy of the mean magnetic field according
to |cyc | ≈ KB02 q , where K depends on the type of nonlinearity, and q ≈ 1 (Tobias
1998). These results may help to interpret stellar cycles (§ 7.4). In a strongly supercritical
dynamo regime however, more complex behaviour such as frequency doubling and a
transition to chaos can occur, so that a simple scaling law for cyc is no longer adequate.
340
M. Ossendrijver
6. Global models of the solar dynamo
6.1. Dilemmas and unresolved issues
The discovery that ∂/∂r is positive near the base of the convection zone at low latitudes
led Parker (1987) to formulate the dynamo dilemma, at the core of which is the issue of
the migration of the magnetic belts. The sign of αφφ is predominantly positive on the
northern hemisphere, irrespective of the physical mechanism (§ 5.5.2; § 5.5.3). This puts
a tough constraint on solar dynamo theory. Unless one includes a meridional flow that
is equatorward at the base of the convection zone or, possibly, some additional dynamo
coefficients, mean-field models produce equatorward migration of the magnetic belts
only if αφφ is negative on the northern hemisphere (§ 3.4.4). Often the problem is
sidestepped by assuming that αφφ is negative on the northern hemisphere.
Some models suffer from a parity problem, in that they prefer a toroidal magnetic
field of symmetric parity with respect to the equator rather than antisymmetric parity,
unless the meridional flow has a specific value or the α effect is rather strongly localized
near the bottom of the convection zone (Bonanno et al. 2002). Related to this is the
common feature of α-models in thin spherical shells that the critical dynamo number
for excitation of the first mode with antisymmetric parity of B 0t is comparable to that of
the first symmetric mode (Roberts 1972; Moss et al. 1990b). Even if the antisymmetric
mode is excited first according to the linear analysis, the inclusion of small random
perturbations or nonlinear effects often results in significant quadrupolar contributions,
so that the magnetic activity becomes rather asymmetric with respect to the equator. In
the Sun such asymmetries are small except perhaps during grand minima (§ 2.2). The
parity problem appears to show up frequently in cases where cancellation of oppositepolarity toroidal flux at the equator is not allowed to play a sufficiently large role in the
dynamo process. From this one can conclude that the inclusion of a meridional flow or
anisotropic turbulent magnetic diffusion might resolve the problem by enhancing the
magnetic coupling of both hemispheres (Yoshimura 1984a,b).
The thin-shell geometry of the interface and tachocline models typically results in
overlapping magnetic cycles, such that there can be one or more polarity changes of
the toroidal magnetic field between equator and pole, whereas there is none observed
on the Sun. In practice, the problem of overlapping cycles is sometimes reduced by
assuming that the α effect exists only at low latitudes, but there is no justification for
this in the case of the convective α effect, which is strongest near the poles (§ 5.5.2).
Perhaps the issue can be resolved by noting that we may be unable to see any effect of
the opposite-polarity toroidal fields at high latitudes. Alternatively, and perhaps more
likely, the problem may be reduced if the proper values for the meridional flow and
the anisotropic magnetic diffusivity are taken into account. However, the problem of
overlapping cycles does make it highly unlikely that the solar dynamo is confined to a
very thin layer (§ 5.3).
Other observations that must be addressed by global models include the latitudinal
distribution of magnetic fields (§ 2.1.3), the phase relation between poloidal and toroidal
magnetic fields (§ 2.5), variability (§ 2.9), torsional oscillations (§ 2.7), the frequency
of the solar cycle (§ 3.4.4), and phase locking (§ 2.9). The latter is potentially relevant
for establishing what dynamo mechanism operates in the Sun. If the phase performs
The solar dynamo
341
an unrestricted random walk, this speaks against the solar cycle being controlled by
meridional circulation.
6.2. Global MHD simulations
In principle one could try to explain the solar dynamo by numerically solving the MHD
equations (1-4) in spherical geometry and incorporating all relevant physics. Due to the
limitations posed by a finite resolution, the involved viscosities (ν, η) are no smaller than
the turbulent viscosities of mean-field electrodynamics, and they must be interpreted as
such. The main difference with mean-field dynamo models is that one refrains from
parametrizing the flow any further by including an α effect or other dynamo coefficients
besides ηt .
Global MHD dynamo calculations in a spherical shell were pioneered by Gilman
& Miller (1981) and Gilman (1983), who adopted the Boussinesq approximation, and
Glatzmaier (1984, 1985), who used the anelastic approximation. The latter is more
appropriate for the strong stratification that exists in the solar convection zone. Cyclic
dynamo solutions were obtained in some cases, but with the wrong poleward migration
of the magnetic fields. This can be explained in terms of an α dynamo wave (§ 3.4.4),
because αφφ in the bulk of the convection zone on the northern hemisphere and ∂/∂r
are both positive in the simulations. Although both signs are correct, the magnetic field
migration in the simulations is opposite to that in the Sun. Apparently, the equatorward
migration in the solar butterfly diagram is a subtle feature requiring simulations with a
higher resolution, in order to capture adequately such features as the tachocline and the
meridional circulation (§ 4.2). For the same reason, the global simulations were unable to
capture the intermittent nature of the solar magnetic field and the formation and buoyant
rise of flux tubes.
In the strongly nonlinear regime dynamo action in spherical shells has been found to
exhibit large north-south asymmetries such that the magnetic field can almost disappear
on one hemisphere (Grote & Busse 2000; Busse 2000). Such hemispherical dynamo
action is reminiscent of solar activity during the Maunder minimum (§ 2.9).
6.3. Global mean-field models
Mean-field models are capable of reproducing essential features of the large-scale solar
magnetic field on the basis of only a small number of ingredients (§ 3.4.4). In actuality,
this apparent success masks two sets of issues. First, there are fundamental though
not insurmountable difficulties concerning the justification of the adopted expansions
of the turbulent EMF (§ 3.4.1; § 3.4.3) and the correct treatment of magnetic helicity
conservation (§ 3.4.5). Secondly, there is a practical problem in that we still know too little
about the internal structure of the Sun (§ 5.2) and the basic properties of the deep-seated
magnetic field (§ 5.3) to allow a definite identification of those physical ingredients that
are really important for the solar dynamo, and those that are not.
For these reasons, there are several scenarios of the solar dynamo that emphasize different mechanisms. Some are unable to reproduce the equatorward motion of the activity
belts without invoking a meridional flow. Usually only the axisymmetric component of
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M. Ossendrijver
the mean magnetic field is considered, which makes good sense in the Sun; only few
examples exist of non-axisymmetric 3D mean-field models (Rädler et al. 1990; Moss et
al. 1991, 1995; Barker & Moss 1994; Moss 1999).
6.3.1. Interface dynamo and distributed dynamo
The interface model was proposed by Parker (1993) in order to accommodate a tachocline
with super-equipartition magnetic fields while allowing for a convective α effect. This
is achieved by assuming that the tachocline coincides with the overshoot layer, and that
the convective α effect exists only in the convection zone proper. Due to the convective
stability and the strong magnetic fields, overshooting convection results in a small but
nonvanishing ηt in the tachocline, and this enables the necessary exchange of magnetic
flux with the convection zone. The idea of a depth-dependent ηt was considered earlier
by Roberts & Stix (1972) and Ivanova & Ruzmaikin (1976). A detailed exploration of the
interface model is presented by Charbonneau & MacGregor (1996, 1997) and MacGregor
& Charbonneau (1997), but cf. also Markiel & Thomas (1999). The energy balance of the
interface model was investigated by Ossendrijver & Hoyng (1997); nonlinear behaviour
by Tobias (1997, 1998).
The main problem of the interface model is that the strict separation between α effect
and differential rotation is not justified, even if one interprets the helioseismic inversions
in such a way that the radial shear is confined to a thin layer below the convection zone.
The scenario of the distributed dynamo, which incorporates the convective α effect
in the convection zone, and the differential rotation as inferred from helioseismology,
seems more likely. Unlike the distributed models from the early days of solar dynamo
theory, the model should incorporate an overshoot layer, where magnetic flux is stored
and amplified. Due to the latitudinal shear in the convection zone, the magnetic field
configuration produced by the simplest distributed models tends to be dominated by
dynamo action within the convection zone proper, with high-frequency waves migrating
radially outward (Lerche & Parker 1972). The solutions may no longer assume the form
of an interface mode characterized by strong magnetic fields in the overshoot layer that
propagate towards the equator, except for special choices of the parameters (Markiel &
Thomas 1999).
The dominance of these unphysical convection zone modes in the models might be
resolved by including downward magnetic pumping (§ 5.5.4). There would still be the
problem of the magnetic field migration. Perhaps the α effect is relevant only near the
very base of the convection zone where plausible arguments and numerical simulations
indicate that the sign of αφφ is opposite to that in the bulk of the convection zone (§ 5.5.2),
resulting in equatorward migration (§ 3.4.4). This possibility cannot be discarded, but
the helioseismic inversions do not impose a conclusion that the tachocline is very thin
(§ 5.2). Although the precise location of the sign change of the convective α effect is
unclear, this makes it likely that there is a significant contribution to dynamo action from
the region where αφφ has the ‘wrong’ sign.
If so, then this leaves the possibility that a meridional flow is responsible for the
equatorward migration (Choudhuri et al. 1995; Küker et al. 2001; Rüdiger et al. 2001).
For meridional circulation to have an effect, the dynamo must be advection-dominated,
i.e. Dm must be sufficiently large (§ 3.4.4). If the meridional flow at the base of the
The solar dynamo
343
convection zone is U0m,base ≈ 1 m s−1 , this requires ηt ≈ 107 m2 s−1 . However,
U0m,base has to be within a narrow range around 1 m s−1 (Küker et al. 2001). The
inclusion of meridional circulation also yields approximately the correct phase relation
between poloidal and toroidal magnetic fields (Bonanno et al. 2002). The rotationally
induced anisotropy of turbulent magnetic diffusion (Kitchatinov 2002) and latitudinal
pumping (§ 5.5.4) may help in deflecting the magnetic fields to lower latitudes.
In numerical simulations, the convective α effect is found to be strongest near the
poles (§ 5.5.2). In combination with the differential rotation inferred from helioseismic
inversions, this should result in the magnetic field being concentrated near the poles,
which is not observed. While sunspots emerge only at low latitudes, torsional oscillations
do continue to higher latitudes (§ 2.7), and this may point to the presence of a deepseated magnetic field outside of the sunspot belts. Perhaps this issue might therefore be
resolved by noting that according to the linear stability analysis of thin magnetic flux
tubes the growth time of the buoyancy instability is much longer for tubes anchored in
the overshoot layer at high latitudes than for tubes near the equator (Caligari et al. 1995).
Thus, the idea is that the deep-seated toroidal magnetic field at high latitudes, if existent,
would be advected towards the equator before being able to emerge and form sunspots.
However, it is unclear whether meridional circulation is able to provide the necessary
equatorward advection.
One suggested explanation for long-term modulations of the solar cycle (§ 2.9) is
that they result from spatial variations in the physical parameters, such that different
periods can be assigned to different loci in the convection zone. This might result in a
superposition of spatially distinct modes, consisting of a slowly oscillating deep-seated
mode and a 22-year mode residing in the convection zone proper (Boyer & Levy 1992).
6.3.2. Dynamo action in the tachocline
The inference of a magnetic layer near the base of the solar convection zone (§ 5.3) has
led to a number of suggestions how a dynamo may operate within this layer. DeLuca
& Gilman (1986) derived a set of equations describing α 2 -type dynamo action in a
stellar overshoot layer, without any differential rotation. However, differential rotation
is needed to explain the large field strength of the deep-seated magnetic field (§ 5.3).
Durney & De Young (1990) used the EDQNM approximation in their investigation
of turbulent dynamo action in an overshoot layer resulting from kinetic helicity and
latitudinal differential rotation. It is difficult to generate sufficiently strong magnetic
fields in such a model, because the convective α effect is quenched for field strengths
comparable to the equipartition value.
The tachocline can be subject to a hydrodynamic instability of the differential rotation
that may produce an α effect (§ 5.2). If a suitable meridional flow is included, meanfield models based on this α effect are capable of producing butterfly diagrams with
the required equatorward migration of the activity belts (Dikpati & Gilman 2001b). The
models favour odd-parity solutions for the toroidal magnetic field, in agreement with the
parity of the solar toroidal magnetic field. While the tachocline instability is a promising
new explanation for the solar dynamo, a drawback might be that it relies on a kinematic α
effect in the overshoot layer. Passive advection by ambient flows may no longer function
if the magnetic field in the overshoot layer is as strong as is inferred from flux-tube
344
M. Ossendrijver
calculations. A dynamo effect in the overshoot layer based on the kink instability was
proposed by Spruit (1999, 2002).
6.3.3. Buoyancy-driven dynamo
The idea that flux tubes are fundamental building blocks of the solar magnetic field has
led to the concept of a flux-tube dynamo driven by magnetic buoyancy (Schüssler 1980,
1984; DeLuca et al. 1983; Schüssler & Ferriz-Mas 2003). Rising magnetic flux tubes
acquire a systematic twist that is equivalent to an α effect (§ 5.5.3). This provides a
source of the poloidal magnetic field within or just above the overshoot layer (§ 5.3).
Actual flux-tube dynamo models exist only in the form of illustrative mean-field
calculations, in which the average dynamo effect of rising flux tubes is parametrized
by an α coefficient. Schmitt (1987) considered an α model with a buoyancy-driven
α effect due to magnetostrophic waves. Unlike what is found in most computations of
α (§ 5.5.3) this α effect could be negative on the northern hemisphere at low latitudes,
and change sign at mid latitudes, leading to dynamo waves with the correct equatorward
migration at low latitudes without invoking meridional circulation (Prautzsch 1993). The
consequences of the requirement of a minimum magnetic field strength for the buoyancy
instability in the overshoot layer were explored by Ferriz-Mas et al. (1994), Schmitt et
al. (1996) and Ossendrijver (2000a). Nontrivial solutions of the dynamo equation exist
only if the magnetic field strength exceeds the threshold value. A flux-tube dynamo is
therefore not self-excited, and must rely on an additional dynamo mechanism in order
to explain how the magnetic field strength can reach the threshold value. This could be
a different instability within the tachocline, or dynamo action and magnetic pumping in
the convection zone. This might be seen as a problematic feature, because it calls for
an explanation as to why two coexisting dynamos in the Sun would result in only one
predominant magnetic cycle, as observed, instead of two. On the other hand, the threshold
for instability may be much lower then previously assumed, due to the additional heating
of magnetic flux tubes resulting from the nonvanishing divergence of the radiative heat
flux in the overshoot layer (§ 5.3). While the mean-field calculations cannot establish
whether the Sun harbours a flux-tube dynamo, they do succeed in illustrating that such
a model can account in a natural way for the grand minima of solar activity (Fig. 12).
Fig. 12. Computed butterfly diagram for a dynamo model with a buoyancy-driven α effect in
the overshoot layer and spatially incoherent α fluctuations in the convection zone. Dark and
light denote negative and positive values of the mean toroidal magnetic field, respectively (from
Ossendrijver 2000a)
The solar dynamo
345
6.3.4. Babcock-Leighton dynamo
In the Babcock-Leighton model, the generation of the poloidal magnetic field is explained
in terms of the decay of bipolar active regions (Babcock 1961; Leighton 1964, 1969).
As is the case with the flux tube scenario, the buoyant rise of twisted magnetic flux loops
is considered to be an integral part of the solar dynamo, the main difference being the
locus of generation of the poloidal magnetic field. The Babcock-Leighton model can be
viewed as a threedimensional extension of the surface flux transport models (§ 2.5).
Several features of the original Babcock-Leighton model have been abandoned or
modified in later versions because of our increased understanding of magnetic buoyancy,
tachocline physics and differential rotation. For instance, the toroidal magnetic field was
assumed to be generated by latitudinal shear within the convection zone (Wang et al.
1991). For the reasons indicated in § 5.3, this appears impossible. Instead, it is generated
within the overshoot layer, and the tilt of bipolar sunspot pairs is a consequence of
Coriolis forces acting on rising loops.
The Babcock-Leighton model is essentially equivalent to an α mean-field model
that contains an α effect only near the solar surface (Choudhuri et al. 1995). The sense
of the observed tilt of bipolar sunspot pairs corresponds to a positive α on the northern
hemisphere (Stix 1974). However, one should bear in mind that the α effect merely
transforms a toroidal magnetic field into a poloidal magnetic field, and that turbulent
diffusion cannot adequately account for the buoyant rise of toroidal flux, because it is
much slower and smears out the magnetic field. Therefore a description based on the
α effect is consistent with the Babcock-Leighton scenario only in combination with a
source term for toroidal magnetic flux near the surface.
Whereas in the model of Leighton (1964, 1969) flux transport was assumed to be
purely diffusive, later versions that incorporate solar differential rotation invoke a meridional flow in order to obtain equatorward migration in the butterfly diagram (Choudhuri
et al. 1995; Durney 1995, 1996, 1997; Nandy & Choudhuri 2001). By including meridional circulation, the models are capable of reproducing approximately the correct phase
relation between poloidal and toroidal fields (Dikpati & Charbonneau 1999).
Random fluctuations in the eruption rate can lead to variations in length and amplitude of the dynamo cycle as well as north-south asymmetries (Leighton 1969). Without
meridional circulation, this would cause the phase of the solar cycle to perform an
unrestricted random walk. Currently, the data are insufficient to establish the phase behaviour of the solar cycle (§ 2.1.1). If the dynamo is controlled by meridional circulation,
then phase locking would occur (Dicke 1988). This was confirmed by Charbonneau &
Dikpati (2000), who considered random fluctuations in the source term and in the meridional flow. Also, they were able to reproduce the anticorrelation between variations in
amplitude and duration of the solar cycle (§ 2.1.1).
As suggested by the stability analysis of toroidal magnetic flux tubes stored in the
overshoot layer, the Babcock-Leighton dynamo functions only if a sufficient number of
tubes have a magnetic field strength in excess of the instability threshold. This may be
seen as a problematic feature (Layzer et al. 1979), but in combination with a weak-field
dynamo mechanism it may provide an explanation for grand minima (§ 6.3.2).
Models with a shallow α effect suffer from the parity problem (Bonanno et al.
2000; Dikpati & Gilman 2001). Another problematic feature of the Babcock-Leighton
scenario is the low efficiency. Mason et al. (2002) compared the efficiency of the interface
346
M. Ossendrijver
α effect and the Babcock-Leighton α effect in a model where both are confined to
different δ-type horizontal layers, spatially separated from the differential rotation. If
the distance between the α layers is comparable to the thickness of the convection zone,
the interface mode dominates even if the deep-seated α effect is weaker by many orders
of magnitude. Meridional circulation is not expected to change the result qualitatively,
because it would not preferentially enhance the efficiency of the Babcock-Leighton
mechanism. Radial differential rotation near the solar surface (Corbard & Thompson
2002) is also unlikely to contribute much (Dikpati et al. 2002). Even though the dynamo
efficiency is underestimated by Mason et al. (2002) because the buoyant rise of magnetic
flux is modeled by turbulent diffusion, which is inadequate, it seems hard to escape the
conclusion that the large spatial separation between the tachocline and the sources of the
poloidal magnetic field renders the Babcock-Leighton model inherently less efficient
than other models.
It has been claimed that the polar magnetic field is a predictive diagnostic for solar
activity (§ 2.10). If true, then this may merely reflect that the solar dynamo is of the
α-type, and it need not be interpreted as evidence for the Babcock-Leighton model.
7. Dynamos in solar-type stars
Observations of the Sun allow us to study many aspects of the solar dynamo in great detail,
but they provide only limited information about the dependence of dynamo action on
basic solar parameters. Observations of stellar activity can complement our knowledge
of the solar magnetic field. Together, they may help us to find a general theory for stellar
dynamos, and infer the history and future of the solar dynamo. A better understanding
of stellar dynamos would also lead to more accurate models of stellar structure. For
instance, the inclusion of a magnetic layer has an effect on surface temperatures (Lydon
& Sofia 1995; D’Antona et al. 2000).
The onset of magnetic activity in late-type stars coincides more or less with the onset
of convection. X-ray emission of ZAMS stars in the Hyades cluster is found to set in for
early F stars, i.e. for M < 1.3M (Stern et al. 1995). This is confirmed by the observation
that low-mass stars are braked more rapidly than heavier stars (Wolff & Simon 1997),
and that high-mass stars are weak in coronal emission (Simon & Drake 1989). Dynamo
action appears to occur in all types of stars that have a convective envelope; this includes T
Tauri stars, late-type stars, brown dwarfs and AGB stars. Magnetic fields on stars without
a convection zone, such as peculiar A and B stars and some upper main-sequence stars,
are more likely explained as relic fields (Landstreet 1992). Also in some evolved giants
where a convection zone does exist, the presence of large, persistent spots may point to
relic fields (Strassmeier et al. 1999).
Stellar magnetic fields can be measured directly through the Zeeman effect when
they are sufficiently strong (Landstreet 1992) or they can be detected indirectly through
chromospheric and coronal emission or photometric variations. The search for correlations between magnetic activity, stellar structure and rotation rate has focussed on
four aspects, namely the surface filling factor of magnetic fields, the emission level,
differential rotation, and cycle frequencies.
The solar dynamo
347
7.1. Starspots
7.1.1. Surface filling factors
Robinson et al. (1980) devised a method based on the Zeeman effect that allows both
the magnetic field B and the surface filling factor, f , to be determined. The inferred
field strengths are in the range B ≈ 0.1 − 0.5 T, similar to values found in magnetic
elements on the Sun, and roughly consistent with approximate equipartition with the gas
pressure. The surface filling factor of magnetic regions on cool stars can be described
by a relation
f ∝ Co0.9 ,
(39)
indicating that the surface fraction covered with spots increases with increasing rotation rate (Montesinos & Jordan 1993). For very rapid rotation f must saturate. Due to
detection limits, observed filling factors of stars are not smaller than about 0.1 in the
visible spectrum. In the infrared the Zeeman effect is stronger, and filling factors of a
few percent can be measured (Valenti et al. 1995). On the Sun, f is about 0.01 during
the activity maximum, so that magnetic fields of stars with activity levels similar to that
of the Sun are below the detection limit in the visible spectrum.
If f is sufficiently large, starspots can also be detected photometrically. The amplitude of the rotational modulation of the photometric brightness is a measure of f
(Messina et al. 2001). In many cases, the photometric variability exhibits long-term
cycles that are attributed to changes in the coverage of starspots (Henry et al. 1995;
Strassmeier et al. 1997; Radick et al. 1998; Oláh et al. 2000; Messina & Guinan 2002).
There are indications that saturation of the photometric variations occurs at a rotation
rate considerably higher than Co ≈ 6 (O’Dell et al. 1995), which is where the chromospheric emission saturates (Vilhu 1984), but this has been challenged by Krishnamurthi
et al. (1998). For large values of f , starspots may undergo changes in their geometry as
a result of which the saturation behaviour of f and the chromospheric activity could be
different (Radick et al. 1990; Foukal 1998).
As a result of Doppler imaging it is known that many rapidly rotating cool stars are
covered with large spots at high latitudes (Strassmeier & Rice 1998; Rice & Strassmeier
2001). In comparison to a star with spots at low latitudes, the magnetic torque is reduced,
and this affects the evolution of angular momentum (Solanki et al. 1997; Buzasi 1997).
7.1.2. Explaining the surface distribution of starspots
One explanation for star spots is based on the assumption that they are essentially the
same phenomenon as sunspots; this requires the presence of an overshoot layer where
toroidal magnetic flux tubes can be amplified. Calculations of rising magnetic flux tubes
in solar-type stars at various evolutionary stages indicate that the tubes can be deflected
to high latitudes by the Coriolis force (Schüssler & Solanki 1992). This effect increases
with increasing rotation rate or increasing depth of the convection zone, but the tube
summit reaches the pole only if the tube is formed already at a high latitude (Schüssler
et al. 1996; Buzasi 1997; Granzer et al. 2000). Slowly rotating stars (Prot 27 days)
should not exhibit spots above 45◦ of latitude, if the tubes are formed at low latitudes
(DeLuca et al. 1997). If rising flux tubes provide the correct explanation of starspots,
then rapidly rotating stars with high-latitude spots ought to have a deep-seated magnetic
348
M. Ossendrijver
layer like the Sun has. High-latitude spots in slowly rotating stars can be explained in
terms of rising magnetic flux tubes only if the tubes originate at high latitudes. In the
Sun, sunspots exist only at low latitudes, even though the conditions for dynamo action
appear to be more favorable at high latitudes (§ 6.3.1). If low-latitude spots are observed
on rapidly rotating stars, an explanation by rising flux tubes seems to be ruled out. These
conclusions might be modified if there is a sufficiently strong meridional flow. Stars with
very deep convective envelopes, such as post main-sequence giants, may be prevented
from exhibiting spots, because rising flux tubes are deflected so strongly that they remain
trapped in the convection zone (Holzwarth & Schüssler 2001).
On the other hand, photometric variations attributed to star spots are found also on
low-mass T Tauri stars, which are fully convective (Bouvier et al. 1995). The interpretation of these and other observations of magnetic-field diagnostics on T Tauri stars in
terms of dynamo action is complicated due to the presence of magnetospheric accretion
flows (Johns-Krull et al. 1999, 2000). In any case, the absence of an overshoot layer
precludes an explanation in terms of rising flux tubes.
Schrijver & Title (2001) propose that polar spots are the result of intensive flux
transport at the stellar surface. They present model calculations for a solar-type star based
on the nonlinear flux transport model of Schrijver (2001). Using a model calibrated to
the Sun, they find that with increasing amplitude of the magnetic cycle, but leaving other
model parameters unchanged, more flux accumulates near the poles. If the amplitude of
magnetic activity is about 20 − 30 times that of the Sun, a prominent flux ring is formed
at high latitudes, with a polarity that is opposite to that of the polar cap, and equal to that
of the trailing spots of the (hypothetical) tilted bipolar pairs. If the star’s magnetic field
is oscillatory, these flux concentrations would be persistent throughout the stellar cycle,
so that the cyclic modulations would be reduced. Technically, the formation of the polar
cap and the opposite-polarity ring are explained by the model as being a consequence of
the retardation of the flux dispersal with increasing magnetic flux density at the stellar
surface.
7.2. Chromospheric emission
For some 100 lower main-sequence stars regular measurements have been made at Mount
Wilson of the chromospheric Ca II H and K emission cores, which are known to have
a magnetic origin from solar observations (Baliunas et al. 1995). The relative Ca II HK
≡ F /σ T 4 is rather accurately parametrized by
flux density RHK
HK
eff
∝ Co
RHK
(40)
(Noyes et al. 1984a). A similar parametrization holds for the relative excess flux RHK ,
the non-chromospheric basal flux (Rutten
which is obtained by subtracting from FHK
1987; Stȩpień 1994; Montesinos et al. 2001). The various indicators of chromospheric
activity are closely correlated among one another and with X-ray indicators (Schrijver
& Zwaan 2000: § 9). Any remaining scatter in the activity-Coriolis number relation
points to additional but less significant dependencies on stellar structure. Since the magnetic field intensity in star spots is a photospheric property determined by equipartition
is mainly a measure of
between thermal and magnetic pressure, it follows that RHK
The solar dynamo
349
the filling factor, and not of the magnetic field strength (Vilhu 1984; Schrijver et al.
1989; Saar 1990). This is confirmed by their similar dependencies on Co (Eq. 39). The
chromospheric emission saturates at Co ≈ 6 (Vilhu 1984), but it is not entirely clear
whether this reflects dynamo saturation (Jardine & Unruh 1999). There are indications
that saturation of the photometric variations occurs at a considerably higher rotation
rate (O’Dell et al. 1995), but this has been challenged by Krishnamurthi et al. (1998).
Broadly speaking, Eq. (40) is compatible with dynamo theory, because the efficiency of
stellar dynamos as measured by the mean-field dynamo number (§ 3.4.4) is expected to
increase with Co due to the dependence on the α effect.
There is a clear relation between age, chromospheric emission and rotation rate. Stars
are born with widely varying rotation rates, depending on their protostellar evolution.
Magnetic braking reduces the dispersion of the rotation rates and therefore also that of
the chromospheric emission, because the more rapidly rotating stars are spun down more
strongly. This is confirmed by the distribution of X-ray luminosities among stellar types
in the Hyades cluster (Stern et al. 1995).
7.3. Differential rotation
The dependence of differential rotation on the rotation rate is expected to be rather
complicated. Within a limited range of rotation rates it may be parametrized by power
laws of the form
r ∝ nr ,
θ ∝ nθ ,
(41)
where r and θ are differences of the rotation rate across a radial or latitudinal
distance respectively, and nθ and nr are the corresponding power indices. In solar-type
stars with an overshoot layer, r is likely to be the dominant factor, but in other stars
θ may be more important. The appearance and disappearance of magnetic features
on the stellar disk due to rotation can be used to infer the variance of the surface rotation
rate, and this provides a rough measure of θ . This can be used to verify theoretical
models of stellar differential rotation, from which one could infer r .
From observations of about 100 solar-type stars Donahue et al. (1996) conclude that
θ increases mildly with increasing rotation rate, with nθ ≈ 0.7 ± 0.1. Hall (1991)
obtained an even flatter dependence with nθ ≈ 0.15. These conflicting results might be
reconciled if the K stars rotate more rigidly than the G stars, while θ is only weakly
dependent on rotation within each class of stars (Collier Cameron et al. 2001). More
observations are necessary to resolve this issue. Mean-field models of stellar rotation
are capable of reproducing a weak dependence on the rotation rate, if allowance is made
for rotationally induced deviations from sphericity of the convective heat flux (Küker et
al. 1993; Kitchatinov & Rüdiger 1993, 1995, 1999; Rüdiger et al. 1998; Küker & Stix
2001; Rüdiger & Küker 2002).
7.4. Stellar cycles
Magnetic cycles are detected in solar-type stars are found predominantly in old, slowly
rotating stars of stellar types G-K, among which is the Sun (Baliunas et al. 1995). Stars
350
M. Ossendrijver
with well-defined activity cycles have cycle periods ranging from 7 to 14 years. Some
stars have long-term activity trends that may turn out to be cyclic as the observations
continue. Stars with a low and flat activity level may be in a grand minimum. Rapidly
rotating stars rarely exhibit cycles in their chromospheric activity, but many exhibit
starspot cycles.
7.4.1. Parametrizing cycle frequencies
Brandenburg et al. (1998) identified two parallel branches for active (A) and inactive (I)
old stars (age 0.1 Gyr). Cycle frequencies on the I branch are larger than those on
the A branch by a factor of about 6. Most of the stars on branch A, which includes the
Sun, are old (age 2 − 3 Gyr) and slowly rotating, while stars on branch I tend to be
younger and more rapidly rotating. Some stars exhibit multiple periodicities that can be
assigned to different branches. Saar & Brandenburg (1999) infer a relation of the form
cyc / ∝ Cop
(42)
with p ≈ 0.5 for branches A and I. A third branch consists of rapidly rotating superactive
(S) stars (Prot 3 days) of which the chromospheric activity is saturated and non-cyclic,
but that show periodic photometric variations that can be attributed to starspot cycles,
with p ≈ −0.4. Ossendrijver (1997) found that the cycles of slowly rotating stars
(Co 1) with well-defined periods are parametrized by a somewhat steeper relation
given by cyc ∝ Cop with p = 2.0 ± 0.3.
7.4.2. Explaining stellar cycles
In combination with plausible assumptions about the dynamo mechanism, measurements
of cyc can, in principle, be used to infer how dynamos parameters depend on basic stellar
parameters such as rotation rate, convective turnover time and magnetic field strength.
Noyes et al. (1984b) considered plane dynamo waves and inferred that a nonlinearity
due to magnetic buoyancy is best able to reproduce the known sample of stellar cycles.
By comparison with a linear interface-type dynamo model Ossendrijver (1997) found
that the empirical relation for slowly rotating stars can be reproduced if the differential
rotation scales as r ∝ −1.1±0.2 , and the α effect as α ∝ Co5.1±0.6 . This would
indicate that with increasing rotation rate r decreases, a feature that is also obtained
in some mean-field models of stellar differential rotation (Kitchatinov & Rüdiger 1995).
The increase of α with Co is also physically plausible (§ 5.5.2). On the other hand, such
results can only provide rough guidance, because they rely on model assumptions. For
instance, nonlinear effects can modify the scaling of cyc (§ 5.7.7).
Perhaps it is therefore not surprising that a different conclusion was reached by Saar
& Brandenburg (1999). They set out to compare the empirical relations (40) and (42),
with p ≈ 0.5 for the I and A branches and p ≈ −0.4 for the S branch, with scaling
laws suggested by nonlinear dynamo theory. They assume that α and ηt are magnetically
controlled, with α ∝ B n , and ηt ∝ B m , where B is the strength of the magnetic field in
the dynamo layer. By balancing regenerative and dissipative terms and with the help of
∝ B κ , they infer that 0.3 < n < 1.5
additional assumptions among which that RHK
surf
and m ≈ 0.75 for branches A and I, and n ≈ 0.5 and m ≈ 0.25 for branch S. The positive
The solar dynamo
351
values of n would suggest that the α effect in solar-type stars is magnetically driven.
appears to be primarily a measure of the filling factor f (Eq. 39) and
However, RHK
, might
not of the photospheric magnetic field (§ 7.1), although f , and therefore RHK
be a measure of the intensity of the deep-seated magnetic field. Thus, the inference of
a magnetically-driven α effect in solar-type stars must await further confirmation. The
origin of the different branches is a matter of speculation; perhaps they correspond to
distinct dynamo modes. The coexistence of such modes could be the result of stochastic
or nonlinear effects (§ 5.6; § 5.7).
7.5. Magnetic fields in rapidly rotating stars and fully convective stars
7.5.1. Observational characteristics
Magnetic activity in rapidly rotating stars differs qualitatively from that of slowly rotating
stars. First, rapidly rotating stars have a higher level of chromospheric emission, such
that they are separated from the slowly rotating stars by a clear gap (Vaughan & Preston
1980). Secondly, they only rarely exhibit cycles in the chromospheric emission (Baliunas
et al. 1995). Solar X-ray emission is characterized by a very strong cycle dependence,
the ratio between solar maximum and minimum being of the order 100 for soft X-rays.
No such variability, and no magnetic cycles have been detected in the X-ray emission
of, for instance, Hyades stars, which are young rapidly rotating ZAMS stars (Stern et
al. 1995). On the other hand, a significant fraction of the rapidly rotating stars exhibit
starspot cycles. Perhaps these inferences can be reconciled because the chromospheric
emission may be saturated (Vilhu 1984). There is evidence for solar-like features such as
a hot corona and prominences, but the magnetic-field topology of rapidly rotating stars
is more complex and non-axisymmetric than that of the Sun. For instance, Donati et al.
(1999) have inferred the presence of large-scale toroidal surface structures on one such
star, a phenomenon that is unknown from slowly-rotating stars.
M dwarfs beyond spectral types M3-4 are predicted to be fully convective; some
of them are magnetically active (Stern et al. 1995; Drake et al. 1996). X-ray emission
from brown dwarfs also points to magnetic activity in fully convective stars (Neuhäuser
et al. 1999). This is consistent with the observation that the radio emission of rapidly
rotating late-type stars in the Pleiades is comparable to that of the most active T-Tauri
stars, which are fully convective (Lim & White 1995).
7.5.2. Suggestions for an explanation in terms of dynamo action
From the observations one can infer that dynamo action in rapidly rotating stars does
not rely as much on differential rotation as is the case in the Sun. Rather, their dynamos
may be of the α 2 or α 2 -type, because the α effect is expected to be strong, whereas
differential rotation may be not too different in magnitude from that of slowly rotating
stars. This is confirmed by the rarity of cycles in the chromospheric activity of rapidly rotating stars and fully convective stars, because α 2 -dynamos are typically non-oscillatory
(§ 3.4.4). One may hypothesize that cyclic variations should occur only on stars that
have a tachocline and a spot distribution that is consistent with the flux tube paradigm,
but not on fully convective stars, and not if the spot distribution is inconsistent with the
352
M. Ossendrijver
flux tube paradigm. In any case, rapidly rotating stars are expected to generate a largescale magnetic field due to the α effect, so that there is no justification in characterizing
dynamo action in such stars as being predominantly small-scale.
The α 2 -hypothesis is consistent with the inference from star-spot observations that
rapidly rotating stars can exhibit multiple, non-axisymmetric dynamo modes (Meinel &
Brandenburg 1990). One peculiar example is provided by the young active dwarf LQ
Hya, which is a solar-type star that has probably just arrived at the main sequence, having
an age of about 60 Myr. From its light curve, Berdyugina et al. (2002) infer the existence
of two active longitudes. Its magnetic activity appears to be governed by three cycles:
a 5.5-year cycle governing the relative strength of both active longitudes, a 7.7-year
cycle for the brightness modulation, and a superimposed 15-year cycle that is tentatively
identified as the analogue of the 11-year solar cycle. Starspot cycles may be seen as
evidence for α 2 -type dynamo action. Similar phenomena have been reproduced in
non-axisymmetric mean field models with weak differential rotation (Moss et al. 1991,
1995; Moss 1999; Küker & Rüdiger 1999).
7.5.3. Fully convective stars
In the Hayashi phase preceding the main sequence, the Sun was a rapidly rotating, fully
convective star. Dynamo action in fully convective stars is expected to be distinct to
some extent because without an overshoot layer long-term storage of the magnetic field
is not possible, so that the magnetic field cannot be amplified by differential rotation as
much as in the Sun (Fig. 13).
In the beginning of the Hayashi and T-Tauri stages, solar-type stars are expected
to have a strong relic magnetic field. After the onset of convection, this magnetic field
is subject to enhanced decay due to turbulent diffusion. The relic magnetic field might
survive for some time in T-Tauri stars (Tayler 1987), but it is unlikely to be present later in
the T-Tauri stage. Küker & Rüdiger (1999) have carried out 3D mean-field simulations
of α 2 -type dynamo action in a rapidly, rigidly rotating fully convective T-Tauri star,
using an anisotropic formulation of the α effect and the turbulent diffusivity. They find
that the magnetic field is always non-axisymmetric and stationary. Kitchatinov et al.
(2001) present mean-field dynamo models for the Sun in different pre main-sequence
stages incorporating a small radiative core. The model includes a non-axisymmetric
relic magnetic field in the radiative core. Perhaps this provides an explanation for active
longitudes, as well as the flip-flop phenomenon observed on LQ Hya.
Fig. 13. Isosurface of the magnetic field strength from a numerical simulation of dynamo action in a fully convective star
(courtesy W. Dobler)
The solar dynamo
353
8. Conclusion
The solar magnetic field poses a formidable research topic for observers and theoreticians
alike. Ultimately, all magnetic phenomena observed on the Sun are consequences of
dynamo action in the solar interior. Dynamo theory provides all the necessary ingredients
for achieving the goal of finding a satisfactory and consistent explanation for the solar
magnetic field.
Many aspects of the solar dynamo have been clarified considerably as a result of
recent theoretical investigations, numerical simulations, mean-field modeling, helioseismology and observations of stellar activity. These include the role of magnetic helicity,
anisotropic dynamo coefficients, the tachocline, magnetic pumping, meridional circulation and the parametrization of stellar activity. They must be taken into accounted in
any model of the solar dynamo.
The main outstanding issue is arguably the nature of the α effect, which has not
been finally resolved. Toroidal magnetic fields with a broad range of field intensities are
expelled into the convection zone as a result of various instabilities in the deep-seated
magnetic layer (§ 5.3). The relative importance of the resulting buoyancy-driven α effect
and the convective α effect remains to be established.
In any case, several considerations speak against a scenario in which the dynamo
is confined to a very thin shell. Also, the Babcock-Leighton model faces a number of
serious difficulties. From a preliminary evaluation of the dilemmas and issues facing
the solar dynamo one may therefore conclude that the buoyancy-driven or distributed
dynamo scenarios appear to suffer from the smallest number of difficulties. However,
the fact that similar solar-type stars have widely different cycle periods should warn us
that the solar cycle is a rather delicate phenomenon.
In order to obtain a more definitive explanation, a number of critical issues must
be addressed, and it seems appropriate to conclude with a number of key questions:
Which mechanism is responsible for the α effect, and where is it located? What is the
topology of the deep-seated toroidal magnetic field and which mechanisms play a role
in amplifying it? How important is the flux of magnetic helicity for allowing the solar
dynamo to comply with the magnetic helicity constraint, and what is its magnitude?
What is the direction and magnitude of the meridional flow at the base of the convection
zone, and does it explain the equatorward migration in the butterfly diagram? What
is the thickness of the tachocline and how is it confined? Is there dynamo action near
the poles? What is the role of anisotropic turbulent magnetic diffusion and nonradial
pumping effects?
Due to its proximity, the Sun will continue to offer the best possibility for testing
models and ideas that may emerge as a result of these questions. In addition, more
accurate knowledge of the dependence of stellar cycles, activity levels and differential
rotation on basic parameters will help in further constraining stellar dynamo theory. At
the same time, the steady increase of computational power allows us to slowly approach
the parameter regime of the solar dynamo, even though we may never actually reach it.
But perhaps that is neither necessary nor desirable. It may suffice to achieve a reasonable
separation of scales in the numerical simulations. Furthermore, in order to provide an
explanation that is both correct and relevant, any theory of the solar dynamo should
allow a formulation in terms of elementary mean effects as they appear in mean-field
electrodynamics.
354
M. Ossendrijver
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