Astron. Nachr./AN 323 (2002) 3/4, 417–423

Modulation of solar and stellar dynamos

S.M. TOBIAS

Department of Applied Mathematics, University of Leeds, Leeds, LS2 9JT, UK

Received 2002 May 10; accepted 2002 July 3

Abstract. In this paper I will review the progress that has been made in understanding modulation of solar and stellardynamos. I will briefly discuss the observational evidence for modulation of both the solar and stellar activity cycles. Iwill then introduce the physical processes that are likely to be significant in producing modulation, before discussing fourcompeting theories for the origin of the modulation.

Key words: dynamos – modulation – Sun – stars: activity

1. Introduction

The origin of magnetic fields in the Sun and other stars isa problem of fundamental importance. These fields, whichare believed to be generated deep within the star, are ulti-mately responsible for the formation of active regions andhence sunspots and starspots (Thomas & Weiss 1992). Themagnetic fields are highly dynamic, varying on a wide rangeof spatial and temporal scales. It is now the consensus thatthis magnetic field generation is due to the action of a hydro-magnetic dynamo in which induction by the motions of theelectrically conducting plasma is large enough to overcomethe effects of ohmic dissipation within the plasma. Solar andstellar dynamo theory is primarily concerned with the prob-lem of generating a cyclically-varying large-scale magneticfield (see e.g. Brandenburg this volume), but here I shall con-centrate on the complementary problem of understanding thedynamics of this field once it has been generated. In particularI shall focus on mechanisms that can lead to the modulationof the basic dynamo cycle.

The paper is organised as follows. In the next section Ishall briefly outline the observational evidence that modula-tion is an inherent aspect of stellar magnetic activity. I shallthen give a brief introduction to mean field dynamo theoryand the coupling of the dynamo equations to the momentumequation. In section 3 I shall review four proposed modula-tional mechanisms and focus in detail on nonlinear aspects ofmodulation before outlining my conclusions in section 4.

Correspondence to: smt@amsta.leeds.ac.uk

2. Observations: solar and stellar

2.1. Observations: solar

In this section I will summarise both the direct and proxyobservational evidence for modulation of the solar activitycycle.

2.1.1. Direct observations of sunspots

Sunspots, which are the surface manifestation of the dynamo-generated large scale toroidal field, have been systematicallyobserved in the West since the invention of the telescope inthe early seventeenth century. For a few hundred years de-tailed records of the number and location of active regionshave been kept. These data show that the basic solar magneticcycle, in which activity belts for sunspots migrate (largelysymmetrically about the equator) from mid-latitudes to theequator and reverse sign over a period of approximately 22years (see e.g. Stix this volume), is not strictly periodic. Theamplitude, and to a lesser extent the period, of the oscilla-tion changes over a longer timescale. This modulation is vis-ible in the Butterfly diagram for sunspots (with the range ofactive latitudes increasing as the amplitude of the solar cy-cle increases) but is most clearly shown in the timeseries forsunspot group number in Figure 1 (a). This figure clearlyshows the rise and fall in the amplitude of the dynamo os-cillation, which is believed to occur over a timescale of ap-proximately 80 years and is known as the Gleissberg cycle.More striking still is the period in the late seventeenth centurywhen virtually no sunspots were observed – a period knownas the Maunder Minimum. This period of reduced activityis not an observational artifact; systematic observations werebeing carried out. Another interesting feature of the sunspot

c� WILEY-VCH Verlag Berlin GmbH, 13086 Berlin, Germany 2002 0004-6337/02/3-408-0417 $ 17.50+.50/0

418 Astron. Nachr./AN 323 (2002) 3/4

Fig. 1. (a) Sunspot group number for the past 400 years. (b) Dis-tribution of spots as the Sun emerged from the Maunder Minimum(after Ribes & Nesme-Ribes 1993)

data is the distribution of sunspots as the Sun emerged fromthe Maunder minimum shown in Figure 1(b). For approxi-mately three magnetic cycles sunspots were largely confinedto the southern hemisphere of the Sun and the Butterfly dia-gram was asymmetric (Ribes & Nesme-Ribes 1993). As ac-tivity increased symmetry was re-established and the Sun’sfield has been largely dipolar since. Modulation is thereforeassociated with changes in parity (symmetry) when the fieldis weak, but can occur with no change in symmetry when thefield is strong.

2.1.2. Proxy data: � � Be and � � C

Four hundred years of direct observational data, althoughsuggestive, are not enough to construct a theory. Fortunatelyit is possible to utilise proxy data of solar magnetic variationsby analysing the variability of terrestrial isotopes such as� � Be and � � C. High energy cosmic rays entering the Earth’satmosphere are responsible for the production of these iso-topes and the cosmic ray flux is modulated by the solar mag-netic field via the solar wind. The � � Be, after approximately2 years in the atmosphere, is stored in ice cores (see e.g.Beer 2000), whilst the � � C is stored in tree rings after about30 years in the atmosphere (Stuiver 1994). The effect of so-lar variability can be analysed once the modulation due tochanges in the geomagnetic field have been carefully sub-tracted from the data. The � � Be data clearly shows the pres-ence of an eleven year cycle that is anti-correlated with so-lar activity. Close analysis of the data shows many other re-markable features (Beer et al 1998, Beer 2000, Wagner etal 2001). Most clear is that the Maunder Minimum is notan isolated event and that other minima in solar activity ap-pear in the record. These “Grand Minima” have been found

in data stretching back as long as � � � � � years (Wagner et al2001) and recur with a well-defined mean period of approxi-mately 205 years (see Figure 2(b)). It should be stressed thatthis is the result of very careful analysis of the data and iscertainly a statistically significant effect. Spectral analysis iswell known to be able to pull out well-defined frequenciesfrom chaotic time-series (see e.g. Tobias et al 1995). Anotherimportant result from analysis of the � � Be record during theMaunder Minimum is the persistence of cyclic behaviour (seeFigure 2(a)). This would seem to indicate that a cyclic dy-namo continues throughout this period of reduced activity,even if sunspots were largely not visible at the solar surface.The analysis of the � � C record, which stretches back approx-imately 11000 years, provides supporting evidence for mod-ulation of the basic activity (Stuiver 1994). Recurrent GrandMinima with a mean period of approximately 200 years, aswell as a modulation with a period of about 80 years, arefound in these data.

It should be stressed that care must be (and has been)taken in ascribing the recurrent periodicity of modulation tochanges in the solar magnetic field as opposed to terrestrialvariability. It is possible that changes in the climate couldlead to some modulation, but it is unclear whether climaticvariability could produce similar effects in both the ice-coreand the tree ring data. Taken together, these proxy data pro-vide strong support for the thesis that the solar dynamo cy-cle is modulated, with recurrent Grand Minima, and that thismodulation, although aperiodic, occurs with a well-definedaverage period.

2.2. Observations: stellar

Most information about the modulation of stellar activity cy-cles derives from proxy data which can provide informa-tion on the time-variation of the stellar activity, but not onthe spatial distribution of the generated magnetic fields (seeRosner (2000) for an excellent review of stellar observationsand discussion of the theory). Stellar activity can be inferredfrom chromospheric Ca H and K emission, which is knownto be linked to the level of coronal magnetic activity. TheMount Wilson survey, established in 1966, analyses the levelof emission for a number of solar-type (moderately rotating)stars. There is now over thirty years worth of activity data onmany of these stars (see e.g. Baliunas et al 1995). This is suf-ficient in some cases to cover four or five magnetic cycles andfor long-term patterns of behaviour to emerge. These can besummarised as follows.

The stars fall into largely three categories, those withcyclic behaviour, those showing doubly periodic (quasiperi-odic) activity and those for which the basic cycle is chaoti-cally modulated (with Grand Minima appearing). As a gen-eral rule, for stars of a fixed spectral type, activity increaseswith rotation rate, as does the range of modulation of activ-ity. (For stars exhibiting a cyclic variation, there is also a sys-tematic change in period with rotation rate and spectral type(see e.g. Saar & Brandenburg (1999)), although relating thischange to properties of the stellar dynamo may be non-trivialTobias (1998).) What is clear, however, is that modulation is a

S.M. Tobias: Modulation of solar and stellar dynamos 419

Fig. 2. � � Be data. (a) Persistence of the activity cycle through theMaunder Minimum – the Maunder Minimum region (shaded) (fromBeer et al 1998). (b) Power spectrum of � � Be data showing presenceof statistically significant 205 year peak (from Wagner et al 2001)

generic property of stellar dynamos and that this modulationincreases with the amplitude of activity. It is the explanationof the modulation of both the solar and stellar dynamo cyclesthat forms the basis for the rest of this paper.

3. Basics of dynamo theory

Dynamo theory is the theory of the generation of magneticfield by the inductive motions of an electrically conductingplasma against the dissipative action of ohmic dissipation(see e.g. Moffatt 1978). For complete understanding of thedynamo problem we are required to solve self-consistentlythe coupled set of partial differential equations for the evolu-tion of magnetic field and momentum, which in their simplestform are given by

� �

� �

� � � � � � � � � � �

�

� � (1)

�

� �

� �

� � � � � � �

�

� � � � � � � � (2)

where B is the magnetic field, u is the velocity of the plasmaand � � � �

�

� � � is the current. Here � includes the effectsof rotation, gravity and any other body forces acting on the

fluid, whilst � is the magnetic diffusivity and � is the viscosityof the fluid. The self-consistent solution of these equations isnon-trivial (to say the least) largely due to the existence ofseveral anti-dynamo theorems (e.g. Cowling 1934) that limitthe allowed form of solutions.

For this reason attention has largely turned to mean fieldtheory (Steenbeck et al 1966), in which the magnetic fieldand flow are separated into large and small-scale componentsand evolution equations for the mean fields are derived. Clo-sure approximations are used for the terms in the mean fieldequations that derive from small-scale interactions. Typicallythis procedure yields equations of the following form

� �

� �

� � � � � � � � � � � � � � � � �

�

� � (3)

�

�

� �

� �

� �

�

� � �

�

�

� � � � � � �

�

�

� � � � � �

�

� �

�

� � � (4)Here an overbar denotes a mean quantity whilst the primedvariables are small-scale fluctuations. The presence of twoimportant additions to the induction equation should benoted. There is a mean generation term � � � � � whichis vital for the regeneration of poloidal field from toroidalfield (this is often referred to as the -effect) and this is aug-mented by the inclusion of an extra contribution ( � ) to thediffusion (the turbulent diffusivity). Similar regeneration anddiffusion terms also appear in equation (4) (see below). Theprocedure outlined to derive equations (3) and (4) requiresmany assumptions (some of which are certainly inappropriatefor conditions in solar and stellar interiors) and a discussionof the limitations of the theory are beyond the scope of thispaper (though see Vainshtein & Cattaneo 1992, Weiss 1994for such a discussion) Nevertheless, the solution of equation(3) in isolation is well known to lead to exponentially grow-ing magnetic fields for large enough rotation rates, and, fora large class of problems, the dynamo instability sets in ata Hopf bifurcation leading to solutions with a well-definedperiod. In this paper I am concerned with describing mech-anisms that produce modulation of the basic cycle and notwith the difficulties of generating a large-scale field in thefirst place. For this reason I shall assume that it is possible togenerate a large-scale cyclic magnetic field and concentrateon the interaction of this magnetic field with the flow (in thenonlinear regime) via equation (4), which is believed to beresponsible for the modulation.

Equation (4) describes the evolution of the mean velocityfield. Clearly the mean forces � are responsible for drivingflows in the absence of a magnetic field. In addition contri-butions from the small-scale flows can arise via the Reynoldsstress term on the left hand side of equation (4). The redis-tribution of angular momentum by this term can in generalonly be determined self-consistently by solving an evolutionequation for the small-scale flow ( �

� ). However a measureof this effect can be ascertained by using a suitable closureapproximation or turbulence model (Kitchatinov & Rudiger1993). In this approximation angular momentum is redis-tributed via the � -effect, where � is a function of rotationrate and stratification. Again a formal discussion of the � -effect is beyond the scope of this paper, but it is important

420 Astron. Nachr./AN 323 (2002) 3/4

Fig. 3. Modulation of basic cycles. Magnetic energy versus timefrom three models that produce modulation of the basic cycle. (a)The stochastic model of Ossendrijver & Hoyng (1996) (b) The dy-namic � -effect model of Schmitt et al (1996). (c) The two layermodel of Charbonneau & Dikpati (2000).

to note that the mean magnetic field will have a dynamic ef-fect on the redistribution of the angular momentum via a pro-cess known as � -quenching (see e.g. Kitchatinov et al 1994).This dynamic effect arises via the nonlinear interaction ofthe Lorentz force with the small-scale turbulence. Anotherconsequence of the presence of a mean magnetic field is the� � � term on the right hand side of equation (4). This compo-nent of the Lorentz force (which drives flows via the Malkus-Proctor mechanism, Malkus & Proctor 1975) arises due to theinteraction of the mean magnetic field with the mean current.Both of these nonlinearities may be important in modulat-ing dynamo activities via the self-consistent driving of meanflows. These mean flows are now observed to extend through-out the convection zone as torsional oscillations (Vorontsov etal 2002)

4. Modulational mechanisms

In this section I will review four possible modulational mech-anisms that have been proposed in the literature. I will arguethat the observational data and the inferred strength of the dy-namo generated magnetic field favour a deterministic originfor the modulation and that this arises naturally through theback-reaction of the Lorentz Force. Moreover, this modula-tion can be understood by examining the underlying mathe-matical structure of the nonlinear dynamo equations. IndeedI will argue that, given the structure of the dynamo equations,modulation and chaotic variability are to be expected.

4.1. Stochastic fluctuation about the marginal state

The first modulational mechanism to be discussed arises dueto the assumptions made in averaging the induction equationto derive the mean-field equations. Hoyng (1988) argues thatthe fluctuating magnetic field may become large compared tothe mean field and that a stochastically varying component tothe -effect should be included. That is �

� � � �

�

�

where

� � � �

arises naturally in mean field theory and

�

is a fluctuating component due to the unsatisfactory natureof the averaging procedure. With this prescription for ina mean field model, and with a prescribed differential rota-tion ( -effect), Ossendrijver & Hoyng (1996) investigated theconsequences of such stochastic variations in a global meanfield solar dynamo model.

They demonstrated that if

� � � �

is sufficiently close tothe critical value for dynamo action (as measured by the non-dimensional dynamo number � � �

�

�

�

� �

�

�

� ) thenthe stochastic perturbations could be significant in determin-ing the dynamics. Fluctuations in the -effect could causethe dynamo to stochastically flip between being supercritical(growing fields) and subcritical (decaying fields) and lead tomodulation as shown in Figure 3(a). The basic cycle is mod-ulated, but that there is no regular period for the modulation(see also Platt et al (1993)). As the dynamo number is in-creased the relative importance of stochasticity to nonlinear-ity in the model decreases and so the relative efficiency of themodulation due to stochastic effects also decreases. Recallthat modulation in stars appears to increase with increasedlevels of activity.

4.2. Modulation due to a dynamic -effect

A second possible mechanism that can lead to modulationof the solar dynamo was proposed by Schmitt et al 1996.They consider a mean field model with two components tothe -effect, the classical -effect due to the interaction ofsmall-scale convection with small-scale magnetic fields anda ‘dynamic -effect’ that arises due to the instability of fluxtubes at the base of the convection zone. For this dynamic -effect to work, the magnetic field at the base of the convec-tion zone must be strong enough to become unstable to tubes,which twist as they rise due to their interaction with the Cori-olis Force. It is this twist that leads to the net generation of apoloidal component of the field and hence to an -effect. Be-cause this only switches on for fields greater than a thresholdstrength � � �

� � � �

it is termed a dynamic -effect. Schmittet al 1996 combined the two -effects in a nonlinear mean-field model. When the field is strong both -effects work intandem and the dynamo can work efficiently. However non-linear and stochastic effects can act so as to push the fieldstrength below the threshold for the dynamic -effect and itswitches off. In this Minimum state magnetic field is regen-erated stochastically by the turbulent -effect until it is abovethreshold again. An example of the time-series from such adynamo is shown in Figure 3(b). It is clear that the basic cycleis modulated and that there are large periods of reduced activ-ity (minima). Although it is likely that both dynamic and tur-bulent -effects operate in solar and stellar interiors the rel-

S.M. Tobias: Modulation of solar and stellar dynamos 421

ative importance of each effect remains uncertain. Moreover,it is not clear whether it is possible to reproduce recurrentGrand Minima with a well-defined mean period using such astochastic fluctuation model. It is more likely that stochasticeffects will be important during the periods of reduced ac-tivity, but that the period of the modulation is determined bydeterministic nonlinear effects.

4.3. Two-layer dynamos with meridional circulation

Another mechanism proposed (Charbonneau & Dikpati2000) to explain the modulation of the solar cycle relieson meridional circulation. Helioseismology indicates (e.g.Schou et al 1998) the presence of a region of strong shear atthe base of the solar convection zone known as the tachocline.This shear region, which is stably stratified, is an ideal sitefor the generation and storage of strong toroidal field. In thetwo-layer dynamo scenario the -effect arises due to the de-cay of sunspots and sunspot groups at the solar surface (Bab-cock 1961, Leighton 1969). The two layers for generationof poloidal and toroidal fields are therefore separated by theconvection zone but are coupled by a meridional flow. Thismeridional flow, which is observed to be largely polewardsat and just below the solar surface, is conjectured to continuein one large cellular motion down to the tachocline. The flowacts as a conveyor belt, transporting poloidal flux down fromthe surface to the shear layer and toroidal flux back to thesurface. Hence these two coupled layers are capable of pro-ducing dynamo action. It is then conjectured that the merid-ional flow is stochastically perturbed by convective motionsand this causes fluctuations in the efficiency of the dynamoand hence modulation of the basic cycle (as in Figure 3(c)).Again modulation is apparent but, because of the stochasticnature of the fluctuations, it is hard to see how such a modelcan reproduce the occurrence of Minima with a well-definedmean period. A more serious problem with the model is thatit ignores the possibility of generation of poloidal field closerto the tachocline (i.e. at the base of the solar convection zone)either via a classical -effect due to the cyclonic convectionor a dynamic -effect as described earlier. Recently Masonet al (2002) have shown that the presence of a layer of -effect close to the interface between the convection zone andthe tachocline will overcome the effect of an -effect at thesurface even if it is much weaker than the surface -effect.

4.4. Nonlinear dynamics via the lorentz force

The final mechanism that I shall discuss for producing modu-lation is more generic. No complicated physical scenarios arenecessary and it is not restricted to dynamos with a particu-lar structure. In this picture, the modulation arises naturallydue to the nonlinear interaction of the large-scale magneticfield with the large-scale flows. As explained in section 3,this can occur either via the direct driving of large-scale flows(the Malkus-Proctor mechanism) or the modification of theturbulent transport of angular momentum ( � -quenching). Ofcourse the large-scale magnetic field has other roles to playsuch as modifying the turbulent diffusion and turbulent -effect (Vainshtein & Cattaneo 1992), but these are thought to

be less important in producing modulation although they areof course crucial issues for the generation of the cyclic fieldin the first place (see Brandenburg this volume).

There are many mean-field models that include directlythis back-reaction of the Lorentz force on the large-scale ve-locity (see e.g. Belvedere et al (1990), Brandenburg et al(1989), Tobias (1996,1997), Kuker et al (1996), Pipin (1999)and the references therein) and a consistent picture is begin-ning to emerge. The coupling of a dynamic equation for theevolution of the velocity to the dynamo equations introducesanother timescale into the dynamo problem and leads nat-urally to quasiperiodic and chaotic modulation. Indeed twotypes of modulation have been found and these have beencharacterised as follows (Tobias 1997; Knobloch et al 1998).It is well known that dynamo instability generically setsin as a Hopf bifurcation leading to oscillatory nonlinear so-lutions with either dipole symmetry (antisymmetric toroidalfield, symmetric vector potential for the poloidal field) orquadrupole symmetry (symmetric toroidal field, antisymmet-ric vector potential for poloidal field). Type 1 modulationarises due to the nonlinear interaction between these modes,mediated via the velocity perturbations driven by the non-linear Lorentz force. Energy is exchanged between modes ofdifferent parity via the velocity perturbations although no sig-nificant changes in the energy of the velocity is observed. Theinteraction of these modes can lead to quasiperiodic modula-tion with the solution asymmetric about the equator (a mixedmode solution). This modulation is associated with changesin the parity of the solution, and the frequency of the modula-tion can be linked to a beating between the frequencies of thedipole and quadrupole modes. This type of modulation is notthat observed for the solar magnetic field over the past threehundred years (as the solar magnetic field has stayed largelydipolar), but it may be a significant effect in the modulationof other stars.

Type 2 modulation does not require the interaction be-tween modes of different parity. Here the large-scale mag-netic field of one parity (dipole or quadrupole) drives velocityperturbations and energy is exchanged between the magneticfield and the velocity. For this type of modulation the systemacts as a relaxation oscillator and the modulational timescalehas a well-defined average period that is determined by theratio of the turbulent diffusivities of magnetic field and mo-mentum (Tobias 1997). This is the type of modulation (withchanges in amplitude yet little change in parity) that has beenobserved on the Sun over the last three hundred years.

In reality either type of modulation may be important inany given star, and indeed both may act in tandem. Figure 4shows the rich nonlinear behaviour that can be found in amean-field solar dynamo model (Beer et al 1998). This isan example of an interface dynamo model (Parker 1993) inwhich the shear is confined to a layer deep within the star andthe -effect operates throughout the convection zone. Bothbutterfly diagrams in Figure 4 arise from the same model, butfor different parameters. In both figures the magnetic activityis characterised by periods of strong activity interrupted byGrand Minima. In Figure 4(a) the solution is dipolar (asym-metric toroidal field) when the field is strong. Type 2 modu-

422 Astron. Nachr./AN 323 (2002) 3/4

Fig. 4. Butterfly Diagrams showing modulation due to nonlinearinteractions (after Beer et al 1998). (a) Dynamo activity is inter-rupted by the presence of recurrent minima. Note how as the so-lution emerges from a minimum that activity is only found in onehemisphere. (b) Flipping – the minimum acts as a trigger to flip thesolution between dipole and quadrupole parity.

lation modifies the solution which is pushed towards a min-imum state. As the solution emerges from a minimum Type1 modulation occurs and the toroidal field emerges asymmet-rically from the minimum. Recall that this is exactly the be-haviour observed on the Sun as it emerged from the Maun-der Minimum. Figure 4(b) shows that even more complicatedtime dependence is possible. Here the solution enters the min-imum as a dipole but emerges as a quadrupole. The minimumhas acted as a trigger to flip the parity of the field. There-fore there is no reason to assume that the Sun’s field has al-ways been dipolar. Indeed previous episodes of quadrupoleand mixed mode solutions are possible.

These dynamic interactions between the magnetic fieldof different parities (Type 1 modulation) or between mag-netic field and velocity (Type 2 modulation) can be under-stood by analysing the underlying mathematical structure ofthe coupled dynamo system (Equations (3-4)). The cleanestand most robust way to do this is to study sets of OrdinaryDifferential Equations (ODEs), which are derived using anormal form analysis. This analysis is based on the bifurca-tion sequence expected from stellar dynamos and the symme-try properties of the underlying system. It should be stressedthat this is very different from deriving ODEs by arbitrar-ily truncating the Partial Differential Equations (e.g. Weisset al 1984). The normal form ODEs are then analysed usingthe techniques of bifurcation theory and nonlinear dynamics.Tobias et al (1995) constructed a minimal model and anal-ysed the bifurcation sequence that leads to Type 2 modula-tion. They considered modes of a fixed parity (either dipoleor quadrupole) and showed how interaction with the velocity

could produce quasiperiodic and chaotic modulation. The in-teractions between dipole and quadrupole modes that lead toType 1 modulation were analysed by Knobloch & Landsberg(1996) who demonstrated that the appearance of quasiperi-odic mixed modes was a natural consequence of the underly-ing structure of the equations. The two approaches were uni-fied by Knobloch et al (1998) who considered the interactionbetween Type 1 and Type 2 modulation. They derived a sixthorder set of ODEs and showed that all the complicated dy-namics found in the PDE models was a natural consequenceof the symmetries of the equations. Figure 5 shows a compar-ison between the flipping solution found in the PDEs shownin Figure 4(b) and a corresponding solution of the ODEs.The figure shows a projection of the solution into a three-dimensional subspace defined by the energy in the dipolemode ( � -axis), the energy in the quadrupole mode ( � -axis)and some measure of the strength of the perturbation veloc-ity ( � -axis). Both solutions show considerable modulation ofthe total energy in the field and flipping between dipole andquadrupole modes as the solution emerges from minima inactivity. Hence the interactions leading to flipping (a combi-nation of Type 1 and Type 2 modulation) can be understoodwith reference to the symmetries of the system and the non-linear modulation of solar and stellar dynamos can be ex-plained solely by the action of the Lorentz force in drivingperturbations in the differential rotation.

5. Conclusions

In this paper I have reviewed the mechanisms that have beenproposed to explain the modulation of solar and stellar mag-netic cycles. Many scenarios are possible, yet it is importantto stress that modulation appears to be an intrinsic propertyof dynamos. Indeed one would be surprised to find a strictlyperiodic dynamo cycle in a star. The two main mechanismsarise either due to stochastic fluctuations in a generation ortransport term in the dynamo equations or simply via the non-linear interaction of the Lorentz force. Given that, in orderfor active regions to appear at the correct latitudes in the Sunand stars (see e.g. Schussler this volume), a field strength ofapproximately � �

G is necessary, it is certain that the mag-netic field is strong enough for nonlinear dynamical effectsto be important. The models that include nonlinear interac-tions in a dynamic manner are capable of explaining the re-currence of minima with a well-defined timescale and eventhe properties of the Butterfly diagram as the Sun emergedfrom the Maunder minimum. Moreover one would expectnonlinear (deterministic) effects to become more dominantin increasing modulation as the amplitude of activity in a staris increased. Of course there are parameter regimes in whichfluctuations become important, for example when the field isweak, and it is vital to gain an understanding of the interac-tion between the nonlinear deterministic modulation and thestochastic perturbations that are often capable of producingsurprising effects.

Acknowledgements. I would like to acknowledge Nigel Weiss forhelpful discussions and an anonymous referee for improving the pa-per.

S.M. Tobias: Modulation of solar and stellar dynamos 423

Fig. 5. Comparison of PDE mean field model with ODE model ofKnobloch et al (1998). Three-dimensional phase portraits showingflipping between different parities for both the PDEs and ODEs (seeFigure 5 of Knobloch et al for details. In both examples the solutionspends most of the time exhibiting Type 2 modulation in the dipolesubspace – but occasionally the solution flips to a quadrupole mode

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