Fluctuations and Energy Balance in Solar and Stellar Dynamos

160
Fluctuations and Energy Balance in Solar and Stellar Dynamos

Transcript of Fluctuations and Energy Balance in Solar and Stellar Dynamos

Page 1: Fluctuations and Energy Balance in Solar and Stellar Dynamos

Fluctuations and Energy Balance

in Solar and Stellar Dynamos

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How countless are your works! But they are concealed from sight.

King Akhenaten, Hymn to Aten (c. 1370 BC)

CIP-GEGEVENS KONINKLIJKE BIBLIOTHEEK, DEN HAAG

Ossendrijver, Antonius Joannes Hendrikus

Fluctuations and energy balance in solar and stellar dynamos/ Antonius Joannes Hendrikus Ossendrijver.-Utrecht: Universiteit Utrecht, Faculteit Natuur- enSterrenkundeProefschrift Universiteit Utrecht.- Met lit. opg. -Met samenvatting in het Nederlands.ISBN 90-393-1345-8

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Fluctuations and Energy Balance

in Solar and Stellar Dynamos

Fluctuaties en energiebalansin zonne- en sterdynamo’s

(Met een samenvatting in het Nederlands)

PROEFSCHRIFT

Ter verkrijging van de graad van doctor aan

de Universiteit Utrecht, op gezag van

de Rector Magnificus Prof. Dr. J. A. van Ginkel,

ingevolge het besluit van het College van

Decanen in het openbaar te verdedigen op

Maandag 9 September 1996 des namiddags te 4:15 uur

DOOR

Antonius Joannes Hendrikus Ossendrijver

geboren op 3 augustus 1967 te Amersfoort

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Promotores: Prof. Dr. M. KuperusSterrenkundig Instituut, Universiteit Utrecht

Prof. Dr. H. van BeijerenInstituut voor Theoretische Fysica, Universiteit Utrecht

Copromotor: Dr. P. HoyngStichting Ruimteonderzoek Nederland

This work was supported by the Netherlands Foundation for Research in Astronomy (ASTRON).

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mephistopheles. So baut man sich ein maßig Kartenhaus,Der großte Geist baut’s doch nicht vollig aus.

J.W. Goethe, Faust II, Zweiter Akt.

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Contents

1 Solar and stellar dynamos 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Solar and stellar activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.1 The sunspot cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2.2 The magnetic cycle of the Sun . . . . . . . . . . . . . . . . . . . . . . . . 5

1.2.3 Variability in the solar cycle . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.2.4 Proxy records of solar activity . . . . . . . . . . . . . . . . . . . . . . . . 7

1.2.5 Stellar activity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3 Dynamo theory of Sun and stars . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.3.1 Formulation of the dynamo problem . . . . . . . . . . . . . . . . . . . . . 12

1.3.2 Internal structure and flow fields of the Sun . . . . . . . . . . . . . . . . . 13

1.3.3 Location of the dynamo . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.3.4 Kinematic dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.3.5 Mean field electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.3.5.1 The mean magnetic field . . . . . . . . . . . . . . . . . . . . . . 17

1.3.5.2 Mean-field dynamo models of the Sun . . . . . . . . . . . . . . . 18

1.3.5.3 Effects of variability in averaging procedures . . . . . . . . . . . 19

1.3.5.4 The mean magnetic energy tensor . . . . . . . . . . . . . . . . . 20

1.3.5.5 Stochastic differential equations . . . . . . . . . . . . . . . . . . 22

1.3.5.6 Application to the induction equation . . . . . . . . . . . . . . . 23

1.3.5.7 Application to the magnetic energy tensor equation . . . . . . . 23

1.3.6 Nonlinear dynamos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.3.7 MHD turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3.8 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Stochastic and nonlinear fluctuations in a mean field dynamo 29

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.2 Model without α-quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.1 Model equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.2.2 Weak forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.2.1 Equations and solutions . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.2.2 Statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . 35

2.2.2.3 Phase-amplitude correlation . . . . . . . . . . . . . . . . . . . . 36

2.2.3 Dynamo frequency decrease . . . . . . . . . . . . . . . . . . . . . . . . . . 38

2.3 Model with α-quenching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

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2.3.2 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.3 Weak forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.3.3.1 Linearised equations and solutions . . . . . . . . . . . . . . . . . 40

2.3.3.2 Statistical properties . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.3.3.3 Phase-amplitude correlation . . . . . . . . . . . . . . . . . . . . 43

2.3.4 Dynamo frequency decrease . . . . . . . . . . . . . . . . . . . . . . . . . . 43

2.4 The sun . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.5 The nonlinear regime without α-fluctuations . . . . . . . . . . . . . . . . . . . . . 45

2.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.7 Appendix A: Statistical properties of the phase and amplitude of the toroidal field 49

2.8 Appendix B: Derivation of the dynamo frequency decrease . . . . . . . . . . . . . 51

3 Stochastic excitation and memory of the solar dynamo 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Dynamo model and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.3 Mode excitation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.3.2 Decomposition into eigenmodes . . . . . . . . . . . . . . . . . . . . . . . . 58

3.3.3 Phase-amplitude correlation of the fundamental mode . . . . . . . . . . . 61

3.3.4 Excitation of overtones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Application to the solar cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4.1 Phase-amplitude correlation . . . . . . . . . . . . . . . . . . . . . . . . . . 64

3.4.2 North-South asymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

3.6 Appendix A: Biorthonormal representation . . . . . . . . . . . . . . . . . . . . . 70

3.7 Appendix B: Phase-amplitude correlation of the fundamental mode . . . . . . . . 71

3.8 Appendix C: Derivation of r.m.s. mode coefficients . . . . . . . . . . . . . . . . . 72

4 Mean magnetic field and energy balance of Parker’s surface wave dynamo 73

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.2 The mean magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.2.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.2.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2.4 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 The mean magnetic energy tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.3.2 Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

4.3.3 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.4 Constraints on the parameters . . . . . . . . . . . . . . . . . . . . . . . . 84

4.3.5 Marginally stable solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.3.5.1 Values of the parameters . . . . . . . . . . . . . . . . . . . . . . 85

4.3.5.2 Properties of the mean magnetic energy tensor . . . . . . . . . . 85

4.3.5.3 Energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.3.5.4 Root mean square magnetic field strength . . . . . . . . . . . . . 89

4.3.5.5 Implications for the mean magnetic field . . . . . . . . . . . . . 91

4.3.6 Growing solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

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4.4 Summary and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.5 Appendix A: Dispersion relation for the mean magnetic field . . . . . . . . . . . 93

4.6 Appendix B: Expressions for the mean magnetic energy tensor . . . . . . . . . . 944.7 Appendix C: Boundary conditions for the mean magnetic energy tensor . . . . . 96

5 Energy balance and resistive dissipation in Parker’s surface wave dynamo 97

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2 Mean field dynamo theory and resistive dissipation . . . . . . . . . . . . . . . . . 99

5.3 The mean magnetic field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.4 The mean magnetic energy tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5.4.1 Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.4.2 Constraints on the parameters . . . . . . . . . . . . . . . . . . . . . . . . 105

5.4.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.4.4 Marginally stable solutions . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.4.5 Mean magnetic energy density . . . . . . . . . . . . . . . . . . . . . . . . 1075.4.6 Magnetic energy balance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4.7 Calibration of the r.m.s. magnetic field strength . . . . . . . . . . . . . . 1125.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.6 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

6 On the cycle periods of stellar dynamos 1176.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

6.2 Activity cycle length, rotation period and color . . . . . . . . . . . . . . . . . . . 1196.3 Dynamo model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.3.1 Geometry and equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.3.2 Dynamo parameters of lower main-sequence stars . . . . . . . . . . . . . . 125

6.3.3 Theoretical cycle periods . . . . . . . . . . . . . . . . . . . . . . . . . . . 1266.3.4 The solar calibration model . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6.3.5 Stellar dynamo models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.4 Summary and conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

7 Fluctuaties en energiebalans in zonne- en sterdynamo’s 1317.1 Inleiding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

7.2 De zonnevlekkencyclus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1327.3 De magnetische cyclus van de zon . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

7.4 Variabiliteit in de zonnecyclus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1347.5 Natuurlijke archieven van zonneactiviteit . . . . . . . . . . . . . . . . . . . . . . 135

7.6 Magnetische activiteit van sterren . . . . . . . . . . . . . . . . . . . . . . . . . . . 1367.7 Dynamotheorie van zon en sterren . . . . . . . . . . . . . . . . . . . . . . . . . . 137

7.7.1 Dynamowerking in zon en sterren . . . . . . . . . . . . . . . . . . . . . . . 1377.7.2 Gemiddelde-veldentheorie . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

7.7.2.1 Lengtemiddeling . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.7.2.2 Ensemblemiddeling . . . . . . . . . . . . . . . . . . . . . . . . . 140

Referenties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Dankwoord 142

Curriculum vitae 143

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References 145

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Chapter 1

Solar and stellar dynamos

faust. Sie tritt hervor! - und, leider schon geblendet,Kehr’ ich mich weg, vom Augenschmerz durchdrungen.So ist es also, wenn ein sehnend HoffenDem hochsten Wunsch sich traulich zugerungen,Erfullungspforten findet flugeloffen;Nun aber bricht aus jenen ewigen GrundenEin Flammenubermaß, wir stehn betroffen;Des Lebens Fackel wollten wir entzunden,Ein Feuermeer umschlingt uns, welch ein Feuer!

J.W. Goethe, Faust II, Erster Akt.

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SOLAR AND STELLAR DYNAMOS

1.1 Introduction

Our Sun is the most prominent object in the sky and the source of all life. On first appearanceit is a disk of unchanging perfection, so radiant that it cannot be directly observed. Only underspecial circumstances can the naked eye discern a few of the phenomena that disturb the simplepicture. For example, one may see sunspots during sunset or sunrise, or the corona during atotal eclipse. We now associate these and other surface phenomena with magnetic fields. In fact,the Sun embodies a complex magnetic field wich has a profound influence not only on the solarsurface but even on the Earth. Magnetic explosions or flares on the Sun send large streams ofparticles into the solar system, that can cause aurorae and geomagnetic storms a few days later.During such an event, radiocommunications are disturbed. In March 1989 there was a series ofstrong flares which even led to a break down of power supplies in Canada (Gorney 1990). Theconnection between eruptions on the Sun and magnetic disturbances on Earth was suspectedalready by Carrington after a similar event in 1859 (Kippenhahn 1994). The solar magneticfield also influences the production of radioactive isotopes in the upper layers of the Earth’satmosphere because it shields off cosmic rays. This mechanism allows us to reconstruct thehistory of the solar cycle for example from tree rings and ice layers (§ 1.2.4).

There have been many attempts to establish correlations between the number of sunspotsand the Earth’s climate. In 1801 William Herschel claimed that grain prices were high whenthe Sun was spot-free. Similar and more bizarre claims followed after the discovery of the 11-year sunspot cycle in 1843. At present there is no convincing proof for the existence of suchcorrelations, which must result from changes in the Sun’s spectrum or luminosity during thesunspot cycle. It is for instance unclear whether the measured irradiance variations of 0.1% arelarge enough to influence our climate, since the resulting temperature change is estimated tobe no larger than about 0.03oC (Willson & Hudson 1991). A better, but by no means perfectcorrespondence seems to exist between changes in solar activity and climate on long timescales(Wigley 1988). For example, sunspot activity was very low during the Maunder minimum(1645-1715), a period which roughly coincides with the Little Ice Age.

The study of magnetic fields began with the discovery of magnetic stones, whose strangeattractive forces have puzzled man ever since. By the first century, the compass was knownin China. In Europe, the systematic study of magnetism began with Petrus Peregrinus, whodescribed various features of magnets in his Epistola de Magnete (1269), such as the two polesand the attractive and repulsive forces between them. In 1600 William Gilbert concluded thatthe Earth itself is a magnet. Johannes Kepler was impressed by this idea and in AstronomiaNova (1609) he envokes magnetism, rather than gravity, in order to explain the motion of theplanets and of the Sun. Cosmic magnetism remained unproven, though, until 1908, when Halediscovered magnetic fields in sunspots (see the next section). Since that time magnetic fields havebeen found or are suspected to exist in other planets, stars (for a review see Landstreet 1992),compact objects, accretion disks, and galaxies (for a review see Wielebinski & Krause 1993).

1.2 Solar and stellar activity

1.2.1 The sunspot cycle

A remarkable event in the history of solar observations is the discovery of dark spots on the solardisk. One percent of the sunspots is estimated to become large enough to enable observationwith the naked eye, during sunrise or sunset. Some 150 pre-telescopic reports of sunspots are

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1.2. Solar and stellar activity

Figure 1.1: Annual mean relative sunspot number R, showing the 11-year cycle. Note theprolonged absence of sunspots during the Maunder minimum (1645-1715). (After Kippen-hahn 1994.)

known from China and Korea, the earliest dating back to the first century BC (Eddy 1980,Stephenson 1987). In classical antiquity, sunspots remained largely unnoticed, except for anisolated reference by Theophrastus of Athens (c. 370-290 BC). The Sun was thought to be aperfect body with no place for spots, and observations were regarded with low esteem. On theother hand, the astronomical diaries compiled by Mesopotamian scholars contain a wealth ofphenomena related to the Moon, the Sun, planets and comets, but, surprisingly, no reference tosunspots. In medieval Europe and the Arab world, sunspots were sporadically mentioned andsometimes mistaken for transits of Mercury (Bray & Loughhead 1964).

Yet it seems plausible that their existence had not remained entirely unnoticed, so thatwhen the telescope was invented around 1608, several observers were purposely directing theirtelescope at the Sun to study the curious dark spots. Shortly thereafter in 1611, one of thesepioneers, the medical student Johann Fabricius from East Friesland, wrote the first publicationin which sunspots are mentioned: De maculis in sole observatis et apparente earum cum soleconversione narratio.1 As the title indicates, he realised that the spots corotate with the solarsurface. By the same time Galileo Galilei and Christoph Scheiner, a German Jesuit priest, began

1Treatise on the spots observed on the Sun and (on) their apparent rotation with the Sun.

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SOLAR AND STELLAR DYNAMOS

to observe sunspots. The existence of the spots was still controversial. After a few years Galileipublished his book Istoria e Dimonstrazioni intorno alle Macchie Solari, in which he showsthat sunspots are concentrated in activity belts and estimates the solar rotation rate. InitiallyScheiner believed sunspots to be planets in transit, but after observing them for many years hereached similar conclusions as Galilei in his book Rosa Ursina sive Sol (1630). Ignoring Fabricius,both Scheiner and Galilei claimed priority for the discovery of sunspots (Berthold 1894).

In spite of the great interest that was aroused by sunspots, subsequent progress in theirunderstanding was slow. This was aggravated by the virtual absence of sunspots between 1645and 1715, a period that is now referred to as the Maunder minimum. Interest waned, andreappeared only near the end of the 18th century, when the question of the physical natureof sunspots was addressed for explicitly. In 1771 the German priest Ludwig Christoph Schulenargued, three years prior to the Scottish astronomer Alexander Wilson, that a sunspot is locatedbelow the surrounding surface (the Wilson effect). Both Wilson and William Herschel explainedthe darkness of sunspots by saying that they were holes in a hot atmosphere through whicha cool surface was visible. Herschel was the first to study the variations in the number ofsunspots on long timescales. Unfortunately, the Maunder minimum prevented him from findingany regularity (Rudiger 1989, ch. 1).

The discovery of the solar cycle was made by the German pharmacist Heinrich SamuelSchwabe, who almost daily observed the Sun during 40 years, starting from 1826. In 1843 henoticed that almost no sunspots could be observed, as had also been the case in 1833, whilemany were seen around 1838. Schwabe published his data in an article in the Astronomis-che Nachrichten (1843) in which he concludes that sunspots vary according to a cycle of 10years. His findings acquired great fame after being included in the Kosmos by Alexander vonHumboldt (1850). In 1848 the Swiss astronomer Rudolf Wolf introduced the Relative SunspotNumber R, a measure of solar activity that is still employed today. It is defined by

R = K(10g + f), (1.1)

where g is the number of sunspot groups, f is the number of individual spots and K is acoefficient that corrects for differences in the quality of the equipment and of the observer.Since R fluctuates strongly on timescales of days, usually a yearly average is employed to obtaina smooth curve (Fig. 1.1). Wolf analysed the available sunspot records and reconstructed thesolar cycle backwards to 1730, establishing a mean cycle length of 11.1 years. Wolf was thefirst to suggest that the sunspot record is modulated periodically by what is now known as theGleissberg cycle, whose period he estimated as 7 sunspot cycles, while modern estimates varybetween 80 and 90 years. There is some support for this modulation and others with periodsof about 130-150 years and 200 years from medieval auroral records (Siscoe 1980, Attolini etal. 1988).

The English astronomer Richard Christopher Carrington was the first to study the latitudedistribution and velocity of sunspots in great detail. From 1855 onwards he observed sunspotsfor seven years, during which he discovered the equatorward migration of activity belts andmeasured the differential rotation. A revealing picture of the sunspot cycle arises from thebutterfly diagram (Fig. 1.2), devised by Maunder (1922). It gives the latitude distribution ofsunspots as a function of time and clearly shows the two belts migrating toward the equator inthe course of one 11-year cycle.

In retrospect, one wonders why sunspots were largely ignored for so long, and why the solarcycle was discovered so late. Apparently, what can be observed often has to compete withpreconceived ideas on what should be observed. For a long time, the very thought of spots on

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1.2. Solar and stellar activity

Figure 1.2: Butterfly diagram. Sunspots are indicated in black. During the course of onecycle the average position at which they appear gradually shifts toward the equator, while theirnumber first rises rapidly and then gradually declines. (After Zirin 1988.)

the Sun was too absurd to be considered. Similarly perhaps, it was common belief even longafter the Maunder minimum that no regularity could be found in the appearance of sunspots.Therefore, it is not surprising that amateurs like Fabricius and Schwabe, not constrained byunshakeable knowledge, could play such an important role.

1.2.2 The magnetic cycle of the Sun

In 1908 the American astronomer George Ellery Hale (1908) established the presence of magneticfields in sunspots by measuring the Zeeman effect. It is now known that sunspots are only themost conspicuous feature of the Sun’s highly complex magnetic surface structure. The variousaspects of magnetic activity on the Sun cannot be treated here in detail, see for instance Bray& Loughhead (1964) or Zwaan (1981, 1992) for a description. In this introduction, only thosefeatures are sketched which refer to the large-scale magnetic field of the Sun.

The work of Hale marks the beginning of solar physics in the modern sense, a new branch ofastrophysics that was dominated for decades by the observatory founded by him on Mt Wilsonin California. Hale noticed that sunspots form pairs of opposite polarity. These bipolar pairsconsists of a leading spot and a following spot with respect to the solar rotation. By 1914, justafter the solar minimum, Hale had discovered a regular pattern in their behaviour: (1) in eachhemisphere the leading spot is of one polarity, the following spot of the opposite polarity, (2) thepolarity of the leading spot in one hemisphere is the opposite of that in the other hemisphere,(3) the axis of a bipolar pair is tilted toward the equator at an angle of about 10o (Joy’s law),and (4) all polarities reverse after the solar minimum. The existence of a 22 year cycle wasannounced by Hale and Nicholson (1924) only after the next solar minimum, since Hale initiallyexpected a polarity reversal to occur at the solar maximum around 1918.

As was first discovered by Hale, the field strength in a sunspot has a typical value of afew kG. Such strong fields (i.e. comparable to the equipartition value with the ambient gaspressure Beq ≈ √

8πp) are thought to inhibit convection, resulting in a temperature differencebetween the sunspot’s innermost part or umbra, and the ambient photosphere of about 1800 K.This explanation of a sunspot’s apparent darkness is due to Biermann (1941). Due to strongbuoyancy, the magnetic field in sunspots has predominantly a vertical orientation. Sunspots

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occur in larger structures called active regions, which are formed when magnetic flux emergesfrom the convection zone. Their typical lifetime ranges from about one day to several weeks.During the decay phase of a spot, its magnetic flux is gradually dispersed into the magneticnetwork, which forms an irregular pattern all over the solar disk, that can be observed inmagnetograms. This pattern correlates with the boundaries of the supergranular cells of solarconvection (§ 1.3.2). The magnetic network is thought to consist of vertical fluxtubes, discretemagnetic elements with field strengths of the order 1 kG. The filling factor f , the total fractionof the solar disk that is covered with strong field elements, is at most about 0.01 during solarmaximum. At the solar minimum, magnetic activity does not vanish completely, since a residualstructure, the quiet network, can still be observed. Some magnetic patterns have been seen topersist for many rotation periods. For example, active regions have a tendency to emerge nearexisting active regions (Gaizauskas et al. 1983), and sunspots are often born in active nests thatmay live several years (Van Driel-Gesztelyi et al. 1992). Combined with Hale’s polarity laws,these observations indicate that the strong magnetic fields observed on the surface of the Sun arenot isolated phenomena, but must be seen as signatures of a large-scale, predominantly toroidalmagnetic field inside the Sun.

Outside sunspots and the network there is a weak and diffuse magnetic field that has nopreferred orientation and is mostly of mixed polarity. It has a typical field strength of a fewGauss. At high latitudes, where there are no sunspots, there is a weak polar field which doeshave a distinct polarity. It is also governed by a 22-year cycle, but its phase is delayed byabout 90o with respect to the sunspot cycle. Thus the polarity of the polar field reverses nearthe solar maximum, as was first noticed by Babcock (1959). Between solar minimum andmaximum, the polarity of the polar field in each hemisphere agrees with that of the leadingspots, but between the solar maximum and the next solar minimum it agrees with that of thefollowing spots. From observations of short-lived ephemeral magnetic regions at high latitudes(Harvey 1992) the existence of a weak poleward branch in the butterfly diagram is inferred. Theglobal surface distribution of magnetic flux and the reversal of the polar field can be explained bya diffusion model for the magnetic network, in which the active regions are source terms, whilethe differential rotation and meridional circulation provide advection (Leighton 1964, Wang etal. 1989, Wang & Sheeley 1994). The mechanism by which flux disappears at the smallest scalesis however not explained by this model.

1.2.3 Variability in the solar cycle

As can be seen in Fig. 1.1, the solar cycle does not consist of identical oscillations like those ofan ideal pendulum, but shows large variability. Figure 1.2 shows spatial variability in the formof asymmetries between the two hemispheres. There is inherent variability in the solar cycleon short time scales up to about a month because the magnetic field is organised in discreteelements that emerge and disappear at irregular times. The apparent variability is enhancedby rotational modulation and because we observe only half of the solar surface. However, theseeffects by themselves cannot explain the observed variability on long time scales.

For example, the asymmetry between the two hemispheres can be systematic and persistent,as was the case during the period 1955-1965 (Fig. 3.7 in chapter 3). This suggests that theobserved asymmetry reflects the asymmetry of the hidden large-scale magnetic fields. Further-more, it has been shown that the degree of asymmetry is significantly larger during the solarminimum (Carbonell et al. 1993). A possible explanation is given in chapter 3.

Turning to Fig. 1.1, we notice three features of the temporal variations: (1) the length of the

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individual cycles varies, (2) the maximum sunspot number changes from cycle to cycle, and (3)during the Maunder minimum (1645-1715) sunspots were almost absent.

In order to analyse the cycle periods, one defines a phase variable ψ, which is the sum ofthe mean phase and the fluctuating phase, i.e. ψ = −ωt + δψ (the minus sign is introduced inaccordance with chapter 2). Here ω ≈ 2π/22 year is the mean frequency of the magnetic cycle.There is no simple way to reconstruct from the sunspot number a continuous time series forδψ. We may consider for example the times of solar maxima ti, and estimate the correspondingvalues of δψi as

δψi+1 − δψi = ω(ti+1 − ti) − π. (1.2)

The observations show that δψi is correlated with the maximum sunspot number logRi accordingto δψi + µ logRi = constant (Fig. 2.7 in chapter 2). This means that cycles of larger amplitudehave a shorter duration, as was first noted by Wolf in 1861. It is argued in chapters 2 and 3that mean field dynamo theory provides a simple explanation for this correlation.

The residual δψi + µ logRi denotes the part of δψi that is not correlated with Ri. Thisquantity may tell us how well the solar cycle keeps track of its phase. Two extreme cases canbe distinguished. If the residual is bounded this suggests that the solar cycle is coupled to aclock, as was claimed by Dicke (1978, 1988). If, on the other hand, the residual is unbounded,e.g. if it describes a random walk, then the solar cycle is not coupled to a clock. Unfortu-nately, the available data seem insufficient at the moment to decide between these possibilities(Gough 1987). The question whether the observed variability is of nonlinear, chaotic natureor of stochastic nature has also attracted some attention. Although techniques are available todistinguish between these possibilities (e.g. Atmanspacher et al. 1988, Wales 1991), here too theanswer remains inconclusive because of a lack of data (Jones 1984).

A striking feature in the sunspot record is the Maunder minimum. At the time that itoccurred the solar disk was not yet observed on a regular basis, but there is little doubt thatsunspots were indeed almost absent (Eddy 1977, 1983). As a result it is probably impossibleto infer whether or not the solar cycle persisted during this period. Perhaps further historicalresearch can be of help by tracing more sunspot records. It is of interest for dynamo theory toknow for example whether the cyclic dynamo was turned off and taken over by a non-periodicdynamo.

The Maunder minimum raises the question of the behaviour of the solar cycle on long timescales. How often in the past did the Sun go through similar intervals of low activity and dogrand minima recur according to a regular pattern? The only way to obtain definite answersto these questions of temporal variability is to extend the record of solar activity much furtherinto the past by using proxy data.

1.2.4 Proxy records of solar activity

The production of radioisotopes including 14C and 10Be in the Earth’s atmosphere due to cosmicrays is modulated by the combined effect of the geomagnetic field and the magnetic field frozenin the solar wind. Charged particles are deflected by these magnetic fields, so that the cosmic-ray flux in the upper atmosphere is reduced. The cosmogenic isotopes are eventually depositedin sediments or trapped in organic materials such as wood so that their concentrations can bemeasured. If the material samples can be dated, this technique provides a way to study thehistory of solar activity (Eddy 1988). The principal limitation is posed by the isotope’s half

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Figure 1.3: Fluctuations in the 14C concentration. The positive excursions at the far right arethe Maunder minimum (right) and the Sporer minimum (left). (Adapted from Weiss 1994, afterStuiver & Braziunas 1988.)

life, which determines the time horizon of the measurements. A practical problem is to separatesolar and geomagnetic contributions, especially in long-term trends.

The 14C atoms are incorporated into CO2, and are stored in living organisms such as woodthrough the carbon cycle. Carbon 14 has a half life of 5730 years, so that this method isapplicable only to wood no older than a few tens of millennia. For the American bristleconepine, a well-established chronology of annual tree rings reaching back about 7500 years into thepast allows the measurements of 14C to be dated to within a year (Damon et al. 1978). SinceCO2 spends several decades in the atmosphere before being absorbed in trees, the 11-year cyclecannot be detected in tree-ring data. Figure 1.3 shows the 14C abundance as a function of timefor the last 9,500 years. The slow trend is attributed to the changing geomagnetic field, whilethe superimposed excursions are anticorrelated with the long-term variations in solar activity.The Maunder minimum is clearly visible, as well as several earlier grand minima, such as theSporer minimum (about 1420-1530). They seem to repeat at irregular intervals, but always witha typical duration of 200-300 years. There is also some evidence of the Gleissberg cycle andother periodicities (Stuiver & Braziunas 1988).

Beryllium 10 attaches to aerosols and is removed from the atmosphere by precipitation within1-2 years. This timescale is short enough for the atmospheric content of 10Be to be modifiedby the solar cycle. Results based on dated arctic ice cores indeed show a clear modulation inphase with the solar cycle, as well as the grand minima that are known from 14C measurements(Raisbeck & Yiou 1988, Beer et al. 1990). Owing to 10Be’s long half life of about 1.5 millionyears, it is possible in principle to reconstruct the solar history for the last 100,000 years ormore, but this has not been done so far. A disadvantage of using 10Be is that dating the samplesis more difficult because the 10Be deposition depends on local weather conditions, whereas the

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14C concentration is globally homogeneous. Furthermore, the geomagnetic record is insufficientto extract the solar contribution for samples older than about 10,000 years. Nevertheless, therecord of 10Be in ice cores offers a promising tool to study variations in the cycle period.

In this connection possible periodicities in annual layers of sediment should be mentioned.However, sedimentation is a phenomenon controlled by climate, and a correlation between cli-matic changes and the solar cycle remains unproven. A well-known example are Australiansediments (varves) dating back 750 million years of which the layers are modulated with a 12layer period, suggesting a relation with the solar cycle. Later it was established that theselayers were in fact monthly tidal deposits, completely unrelated to the solar cycle (Sonnet &Williams 1987).

From what is discussed above, it is clear that a study of the history of the solar cycle couldanswer questions about the nature of grand minima, other periodicities besides the 11-year cycleand about the phase memory. Future research may also reveal long-term trends in the cycleperiod, related to evolutionary effects. This brings us to the effects of changes in stellar structureand rotation rate.

1.2.5 Stellar activity

By observing magnetic activity in stars one can study how dynamo action depends on stellarstructure and on rotation. Stellar magnetic fields can be measured directly through the Zeemaneffect when they are very strong (e.g. in Ap stars; see Landstreet 1992 for a review) or theycan be detected indirectly through chromospheric and coronal emission that is associated withmagnetic activity (see Schrijver 1991). Robinson et al. (1980) devised a direct method thatallows both the magnetic field B and the filling factor f to be determined for cool stars.2 Theresulting field strengths are in the range B ≈ 1 − 5 kG, roughly consistent with values foundin magnetic regions on the Sun. It does not increase with stellar rotation rate, but seems to berelated to the photospheric gas pressure according to B <

∼√

8πp (Saar 1990). Stars that rotatemore rapidly have a larger fraction of their surface covered with magnetic regions; the fillingfactors can be described by a relation f ∝ Ro−0.9 (Saar 1990, Montesinos & Jordan 1993a). HereRo = Prot/τc is the Rossby number, the ratio of rotation period Prot to the convective turnovertime τc. This parameter indicates how strongly the Coriolis force can influence convection. Forsmall Rossby numbers (rapid rotation) the expression for f saturates, since f ≤ 1. Due todetection limits, observed filling factors of stars are not smaller than about 0.1 in the visiblespectrum. However, in the infrared the Zeeman effect is stronger, and filling factors of a fewpercent can be measured (Valenti et al. 1995). On the Sun, f is about 0.01 during the activitymaximum, so that magnetic fields of stars with activity levels similar to that of the Sun arebelow the detection limit in the visible spectrum. At present the direct methods have yieldedmostly isolated measurements of magnetic fields and filling factors on very active stars but nocyclic variations.

Fortunately magnetic activity of cool stars can also be detected indirectly by measuring theassociated chromospheric excess emission. The possibility of detecting stellar cycles in the H andK emission cores of Ca II was exploited by Olin Wilson, who in 1966 initiated the ”HK-project”at the Mt Wilson observatory (Wilson 1978). The HK-project now monitors these Calcium linesfor over a hundred stars on or near the lower main sequence, having a spectral type roughly

2Cool stars, or late-type stars, have a spectral type F, G, K or M; i.e. they have surface temperatures belowabout 7500K.

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Figure 1.4: Stellar activity in lower main-sequence stars. (a) Low and flat activity, (b) high andvariable activity, (c,d) low and cyclic activity. The numbers indicate B−V and the cycle periodin years. (From Baliunas et al. 1995.)

between F0 and M2 (i.e. B−V in the range 0.4 <∼ B−V <

∼ 1.5).3 The spectral type F0 roughlycoincides with the onset of a convective envelope in main sequence stars. The magnetic originof the Ca II H and K lines is confirmed by observations of the Sun, where the emission in theselines varies in phase with the magnetic cycle.

Figure 1.4 shows the intensity variations of the Ca II H and K flux, relative to the continuum,versus time for several example stars including the Sun. These pictures show a variety ofbehaviour on several timescales. On a timescale of about a month, the activity index is noisydue to rotational modulation caused by the patchy magnetic surface structures. Four classes oflong-term behaviour are distinguished by Baliunas et al. (1995): stars with (1) a flat activitylevel, (2) long-term trends, (3) irregular variations and (4) cyclic variations. The approximatefractions of stars in each respective category are 13:13:24:50 according to Baliunas et al. Somestars seem to have two cycle periods. However, if one admits to the cyclic group only those starsof which the periods are determined with a high confidence level (those rated good or excellentby Baliunas et al.) their percentage drops to about 15%. These stars with well-defined cycleshave periods between 7 and 21 years, and none have two cycle periods (chapter 6). Anotherdifficulty in assessing the type of variability arises from the short time interval of nearly 30 yearscovered by the data. Stars with long-term trends may in fact have cyclic variations with periods

3The colour index B−V measures the difference in magnitude between the blue and visual spectral bands.It is an intrinsic property of a star’s spectrum and it decreases with increasing stellar mass, i.e. with increasingsurface temperature.

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that are as yet too long to be detected.

Certain trends can be identified in the dependence of chromospheric variability on stellar typeand age (Baliunas et al. 1995). Long-term changes and cyclic variations occur predominantly,flat activity levels exclusively in stars with a low activity level. Irregular variations are most oftenfound among stars with a high activity level. A flat activity level may indicate a grand minimum(Baliunas & Jastrow 1990). Stars with a low activity level with respect to the continuum are oldand rotate slowly, while stars with a high activity level tend to be young and rapidly rotating.Such a correlation between age, rotation and magnetic activity arises from the gradual spindown of magnetically active stars as a result of the torque exerted by a magnetized stellar wind(Skumanich 1972). Thus the data suggest that only old stars have magnetic cycles and grandminima, whereas young stars have irregular or multiperiodic variations. Most of the stars withwell-defined magnetic cycles are slowly rotating K-stars, some are G-stars (chapter 6). F-starsgenerally have a flat record, a slow trend or small-amplitude variations. These differences maybe related to the depth of the convection zone, which is large for K-stars and small for F-stars.

The search for correlations between magnetic activity, stellar structure and rotation rate hasfocussed on two aspects, namely the mean activity level and the cycle period. ChromosphericCa II H and K emission is believed to have two separate origins, one being magnetic activity, theother acoustic heating (Rutten 1991, Schrijver 1995). The latter is independent of stellar rotationand is therefore called the basal flux Fb. The magnetic excess flux is obtained by subtractingthe basal flux from the purely chromospheric flux F ′, i.e. ∆FHK = F ′

HK − FHK,b. On the Sun,this quantity is correlated with the magnetic flux density (Schrijver et al. 1989). It increaseswith increasing rotation rate, and also has a dependence on B−V , which differs from that of theconvective turnover time (Stepien 1994). The Ca II excess flux is found to be correlated withother chromospheric excess line fluxes as well as with the coronal emission in the extreme ultraviolet and X-ray domain (Rutten 1987, Montesinos & Jordan 1993b, Mathioudakis et al. 1995).This suggests that they all have a common magnetic origin. The sum of all the chromosphericexcess fluxes and the coronal fluxes yields the total energy flux related to magnetic heating, auseful quantity that provides a constraint on the magnetic energy balance.

Noyes et al. (1984a) found a correlation between the relative flux in the chromospheric Ca IIH and K lines, corrected for photospheric emission, R′

HK = F ′HK/σT

4eff , and the Rossby number

Ro = Prot/τc, such that R′HK increases with decreasing Ro. Here the convective turnover time

τc is a function of spectral type and must be obtained from model calculations. Noyes et al.assumed that τc is practically independent of B−V for B−V >

∼ 0.8, but according to more recentconvection models (Gilliland 1985, Kim & Demarque 1996), τc is a monotonically increasingfunction of B−V also for B−V >

∼ 0.8. Hence it seems that R′HK correlates with Ro for

B−V <∼ 0.8, whereas it correlates with only Prot for B−V >

∼ 0.8. Thus the Rossby number canbe a useful tool for parametrising some, but not all of the indicators of magnetic activity, andonly for a limited range of stellar types (Stepien 1994).

It has proven harder to establish trends in stellar cycle periods. Noyes & Weiss (1984b)found a correlation between the cycle period Pcyc and the Rossby number, Pcyc ∝ Ro−1.25, butthis result was criticized by others after more cycle periods had been measured (Soderblom 1988,Saar & Baliunas 1992). In chapter 6 of this thesis it is argued that the criticism is unjustifiedfor two reasons: (1) stars with ill-defined spurious cycle periods should not be included in theanalysis and (2) slowly and rapidly rotating stars should be considered separately, since theirdynamo mechanisms may be different.

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1.3 Dynamo theory of Sun and stars

1.3.1 Formulation of the dynamo problem

It is the goal of dynamo theory to explain the generation and maintainance of magnetic fields incosmic objects by motions in electrically conducting fluids. Hence those objects whose magneticfields are thought to be frozen-in relic fields fall outside the scope of dynamo theory. Evenin such cases though, which include compact stars and probably Ap stars, the magnetic fieldsmay originate from dynamo action in preceding evolutionary stages. The technical possibility ofgenerating currents in a self-excited dynamo (i.e. without using permanent magnets or batteries)was first demonstrated by Werner von Siemens in 1866 (see Krause 1993, p. 487). Dynamotheory became a subject of astrophysics when Larmor (1919) proposed such a mechanism toexplain the magnetic field of the Sun. In its most general form, the dynamo problem consists ofsolving a set of four differential equations, complemented by an equation of state. These includethe magnetohydrodynamic (MHD) induction equation,

∂B

∂t= ∇× (u × B− η∇× B), (1.3)

where η = c2/4πσ is the magnetic diffusivity and σ the plasma conductivity. The first termon the right of Eq. (1.3) describes the advection of the magnetic field by the fluid motions, thesecond term represents the effect of resistive dissipation, or diffusion of the field relative to thefluid. Dynamo action occurs when the velocity field u is capable of sustaining or amplifying themagnetic field in the presence of resistive dissipation. For each of these competing mechanismsthere is a typical timescale, namely the advection timescale τa and the diffusion timescale τdrespectively:

τa = l/u; τd = l2/η, (1.4)

in which l represents a typical length scale and u a typical velocity. The decay time of thelarge-scale field in the solar convection zone (l ≈ 7 × 1010 cm, η ≈ 104 cm2 s−1) is τd ≈ 1010

year. The ratio τd/τa is the magnetic Reynolds number,

Rm = ul/η. (1.5)

If Rm ≫ 1, the advection term dominates and the magnetic field is said to be frozen into thefluid (Alfven’s theorem). The Reynolds number decreases with decreasing length scale l of themagnetic field. On astrophysical length scales, Rm is usually much larger than unity, e.g. for thelarge-scale field in the solar convection zone Rm ≈ 108. On the dissipation length scale, (aboutone km in the Sun), Rm is of order unity and resistive dissipation becomes important.

A useful way of visualizing dynamo action in a star is to decompose the magnetic field intoroidal and poloidal components,

B = Bt + Bp. (1.6)

In an axisymmetric geometry, poloidal vectors lie in meridional planes, i.e. in planes throughthe rotation axis, while toroidal vectors are perpendicular to these planes, in the longitudinaldirection. Dynamo action can be defined as a mechanism that maintains a magnetic field byconverting Bp into Bt and vice versa. As shown in the following sections, two crucial ingredientsfor this mechanism are differential rotation and helicity, both of which are present in rotating

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cool stars. The effect of the former is to stretch the field lines in the longitudinal direction, whilethe latter is a more subtle effect, that is explained in § 1.3.5.

The velocity field u is determined by the equation of motion,

ρ∂u

∂t+ (u · ∇)u

= −∇p+ ρg + fL + fv. (1.7)

The forces on the r.h.s. include the pressure force, gravity, the Lorentz force fL = J×B/c, andviscous forces, denoted by fv. The mass density ρ obeys the continuity equation,

∂ρ

∂t+ ∇ · ρu = 0. (1.8)

The temperature is determined by the energy equation,

ρT∂s

∂t+ u · ∇s

= ǫ, (1.9)

where T is temperature, s is entropy per unit mass and ǫ contains the relevant sources of heat,such as thermal conduction, radiation, convective transport, viscous dissipation and resistivedissipation (J2/σ).

From these equations it is clear that the dynamo is linked to the star’s internal structure andflow fields. The next two sections summarize the relevant observations and model calculationswith respect to the Sun’s interior and the location of the solar dynamo, with occasional referencesto stellar dynamos.

1.3.2 Internal structure and flow fields of the Sun

The Sun consists of a core with a radius of about 5 × 105 km in which energy is produced bynuclear fusion and transported outwards by radiation, and an outer layer with a depth of about2 × 105 km where energy is transported by convection.

In the convection zone, hot gas elements with a density less than that of the surroundingmedium rise, while cool elements sink. From observations of the Sun it is inferred that convectionis organised in cellular patterns of various scales (for a discussion, see for instance Stix 1989,ch. 6.). On the smallest scale the whole solar disk is seen to be covered with granulation,consisting of granules having a typical size l ≈ 103 km and a lifetime τc of about 5 minutes. Inorder of increasing size one further distinguishes mesogranulation (l ≈ 5 × 103 km, τc ≈ 2 h)and supergranulation (l ≈ 3 × 104 km, τc ≈ 1 d). The existence of giant cells (l ≈ 105 km,τc ≈ 1 m) extending throughout the lower parts of the convection zone is suggested by allconvection models but the observational evidence is meagre (Gilman 1986). Giant cells may beof particular relevance for the solar dynamo, which is believed to be located near the bottomof the convection zone (§1.3.3). Recent convection models predict downward flows localised innarrow plumes that originate near the top of the convection zone and may penetrate into theovershoot layer (e.g. Nordlund et al. 1992, Rieutord & Zahn 1995).

A property of convective motions that is crucial for the dynamo (see below) is the kinetichelicity or Schraubensinn :

h = u1 · (∇× u1), (1.10)

where u1 is the convective velocity and ∇ × u1 the vorticity. Cells that rise vertically in arotating stratified medium expand and acquire a preferred sense of rotation as a result of theCoriolis force associated with the expansion velocity. The opposite sense of rotation and hence a

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helicity of the same sign is obtained by the sinking, contracting cells. The resulting net helicityshould change its sign in the equatorial plane and may have a latitude (θ) dependence roughlyequal to that of the Coriolis force, i.e. h ∝ cos θ. It is found in convection models that nearthe bottom of the convection zone h should have the opposite sign to that in the bulk, becausehere the downward flows must diverge and the upward flows must converge. The magnitudeof h depends on u1 and also on the normalized helicity H, which is the correlation coefficientbetween u1 and the vorticity:

H =〈u1 · (∇× u1)〉√

〈u21〉〈|∇ × u1|2〉

. (1.11)

In general one expects H to increase with increasing rotation rate (stronger Coriolis forces),but if the convective cells are tilted over angles larger than, say, 90o, H may decrease again.Larger structures are more strongly affected by the Coriolis force (i.e. have smaller Rossbynumbers) than small structures, which suggests that the giant cells play an important role inthe dynamo. This is confirmed by measurements of stellar activity, which can to a certain extentbe parametrized by the Rossby number Ro = Prot/τc, where τc is the convective turnover timenear the bottom of the convection zone (§1.2.3).

The convective motions are superposed on a steady mean flow u0 = u − u1, consisting ofdifferential rotation and meridional circulation :

u0 = Ω(r, θ)r sin θ eφ + um. (1.12)

It had already been noticed by Scheiner in the 17th century that solar rotation is faster at theequator than at higher latitudes, i.e. that the Sun rotates differentially. The Sun’s internalrotation may be established by means of helioseismology. This technique uses the frequencysplitting of solar oscillations, which depends on the rotation rate. By analysing oscillations thatare reflected at different depths, the angular velocity can be probed as a function of depth. Thesemeasurements reveal that Ω depends only weakly on r in the bulk of the convection zone, butnear the bottom Ω sharply decreases with depth near the equator while it sharply increases withdepth near the poles (Goode 1995). Since differential rotation provides the second importantingredient of the solar dynamo, this has implications for the location of the dynamo.

The meridional circulation um denotes a large-scale flow pattern in meridional planes, andis a feature predicted by all convection models. On the solar surface it is slow compared tothe rotation, with poleward speeds of about 20 m s−1. More relevant for the dynamo is themeridional flow at the base of the convection zone, and there um is probably only a few m s−1

equatorward, too small to be important for the dynamo (Gilman 1992), but see Choudhuri etal. (1995) for a different view.

Extrapolating the flow patterns of the Sun to other stars with different rotation rates ishardly possible at present since the theory of differential rotation has many uncertainties, seeRudiger (1989) for a review. It appears that differential rotation in solar-type stars decreaseswith increasing rotation rate (Kitchatinov & Rudiger 1995), a result that is confirmed by acomparison of observed cycle periods of solar-type stars with predictions of mean-field dynamotheory (chapter 6).

1.3.3 Location of the dynamo

Before discussing in more detail the various aspects of solar dynamo theory, a few simple argu-ments will be presented with respect to the location of the solar dynamo. The total sub-surface

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flux as inferred from flux measurements of active regions is of the order 1024 Mx (Galloway &Weiss 1981, Golub et al. 1981). If the flux is contained in a belt of width 2 × 1010 cm, centredat the equator, the field strengths are of the order 2.5 × 103 G. Such strong fields are expelledfrom the convection zone within several months as a result of magnetic buoyancy (Parker 1979,Schussler 1979, Moreno-Insertis 1986). This timescale is too short to permit amplification of thefield by differential rotation. Hence the large-scale solar magnetic field cannot have its originin the convection zone proper. It is now believed that the flux is concentrated in the overshootlayer below the convection zone (Zwaan 1978, Galloway & Weiss 1981; see Hughes 1992 for areview). This thin layer with a thickness of about 2× 104 km is penetrated from above by over-shooting giant cells and by convective downdrafts. Magnetic buoyancy in the overshoot layer issuppressed due to the subadiabatic stratification or due to the combination of a meridional flowand the magnetic tension force (Van Ballegooijen 1982). The field strength in this layer must beof the order 2.5×104 −1×105 G to explain the observed surface characteristics of the magneticfield. Calculations on the stability and rise of flux tubes show that such strong fields can indeedbe maintained in the overshoot layer on long timescales (Caligari et al. 1995). Eventually thefluxtubes become unstable and one or more loops form that rise to the solar surface. If thefield strength is about 105 G, the emergence of these loops is confined to a strip centred onthe equator, similar to the observed activity belts, and the loops have a tilt with respect to theequator that is in agreement with Joy’s law (D’Silva & Choudhuri 1993, Schussler et al. 1994).These calculations however do not explain how strong fields are generated. Presumably thisis achieved by the strong differential rotation which prevails in the overshoot layer. Hence theobservations of differential rotation that were mentioned in the previous section also point to alocation of the solar dynamo near the bottom of the convection zone. In these deep layers, thenecessary helicity may be provided by overshooting giant cell convection. The convection zoneproper may also be the site of a dynamo, but the resulting magnetic field will be weak, diffuseand non-periodic (Spruit et al. 1987, Van Geffen 1993a, Durney et al. 1993).

1.3.4 Kinematic dynamos

The full dynamo problem can be solved only through numerical integration of Eqs. (1.3–1.9), butthis is an enormous task that seems impossible for stellar dynamos in the near future (§1.3.8).A variety of approaches have been devised to reduce Eqs. (1.3–1.9) to simpler equations thatare more easily handled and hopefully still give an adequate description of the solar magneticfield, or at least demonstrate some of the mechanisms that play a role in the solar dynamo. Inthis and following sections the various schemes will be sketched briefly.

In the kinematic or laminar approach one prescribes the velocity field u and solves only theinduction equation (1.3), which is then linear. But attempts to construct dynamo models usingsimplified flow fields are faced with obstructions in the form of several non-existence theorems.The impossibility of dynamo action was proven by Cowling (1934) for the case of axisymmetricmotions, by Bullard & Gellman (1954) for purely toroidal motions and by Zel’dovich (1957) andMoffatt (1978) for motions in flat planes. The conclusion from these and similar theorems isthat dynamo action is essentially three-dimensional and requires complex, asymmetric flow fields.An elementary example of such a dynamo mechanism is the stretch-twist-fold (STF) process,conceived by Vainshtein & Zel’dovich (1972), see Fig. 1.5. A closed flux tube is stretched totwice its length, twisted to the form of an eight and then folded. After one cycle the cross sectionis unchanged but the field strength is doubled. However, the original topology can be restoredonly if diffusive reconnection takes place at the point where the loops cross. This demonstrates

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Figure 1.5: The stretch-twist-fold dynamo. (After Roberts 1994.)

the importance of ohmic diffusion in avoiding the formation of sharp field gradients.

A systematic investigation of the kinematic dynamo problem was initiated by Elsasser (1946).His approach involves a decomposition of B in terms of orthogonal decay modes (i.e. solutionsof Eq. 1.3 with u = 0). This amounts to replacing Eq. (1.3) by an infinite set of coupledequations for the expansion coefficients, that can in principle be solved if a suitable truncationis applied (Cowling 1957, ch. 5, Roberts 1994). Such an attempt was undertaken by Bullard &Gellman (1954) in connection with the Earth’s dynamo. For certain non-axisymmetric flows theyobtained dynamo action, but their conclusion was not confirmed by later numerical simulations,which demonstrates that the effects of truncation are subtle. An application of this method tothe Sun using eigenmodes of the mean-field dynamo equation (1.13) instead of decay modes wasproposed by Hoyng (1988).

A modern approach to kinematic dynamo theory is the concept of the fast dynamo, which isconcerned with the question whether a given flow field u can result in an exponentially growingmagnetic field in the limit η ↓ 0 (i.e. Rm → ∞). This is a relevant approach for the Sun, inwhich Rm ≫ 1. It represents a difficult problem, since in the absence of ohmic diffusion theeigenfunctions of Eq. (1.3) develop an ever more complicated structure on smaller scales andmust be described mathematically by generalized functions (Bayly 1994). Numerical simulationsusing flow fields with a sufficient lack of symmetry suggest the existence of fast dynamos, but amathematical proof is still absent. A highly idealized formulation of the fast dynamo problemis provided by map dynamos. In these models one studies the effect on the magnetic field ofdiscrete fluid deformations, applied at fixed time intervals (Bayly 1994). The reader interestedin fast dynamos is referred to a large body of literature, see e.g. Soward (1994) and referencestherein.

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1.3.5 Mean field electrodynamics

1.3.5.1 The mean magnetic field

The turbulent flow field in the solar convection zone is so complex that exact analytical solutionsof the induction equation (1.3) are out of reach, even in the kinematic approach. The mainconcern of solar dynamo theory, however, is not to provide an exact description of small-scalestructures, but to account for the large-scale magnetic field as described in § 1.2.2. This suggeststaking an average, so that the small scales are washed out and knowledge of the flow fieldis required only through its statistical properties. An attempt in this direction was made byParker (1955), who was the first to realise the crucial role played in dynamos by cyclonic motions,i.e. convection with a non-zero mean helicity (Eq. 1.10). Parker’s intuitive model obtained a solidbasis after the introduction of mean-field electrodynamics by Steenbeck, Krause & Radler (1966).Their analysis resulted in the following equation for the mean magnetic field B0 = 〈B〉:

∂B0

∂t= ∇×

u0 × B0 + αB0 − (β + η)∇× B0

, (1.13)

in which u0 represents the mean flow,

u0 = 〈u〉 = u− u1, (1.14)

and the turbulent transport coefficients α and β depend on statistical properties of the turbulentflow field u1, namely the mean kinetic helicity (Eq. 1.10) and the root mean square (rms) velocity:

α = −13τc〈h〉, β = 1

3τc〈u21〉. (1.15)

Here τc is the correlation time of the largest eddies in the turbulence. The interpretation ofthe average 〈·〉 is discussed further below. A derivation of Eq. (1.13), and the conditions underwhich it is valid will be presented later in this section. For detailed treatments of mean-fieldelectrodynamics see for instance Krause & Radler (1980) and Moffatt (1978).

The meaning of the terms in Eq. (1.13) is as follows. The first term on the r.h.s. describesadvection of the mean field by the mean flow. In this way differential rotation converts poloidalinto toroidal mean field. The second term on the r.h.s. in turn generates poloidal from toroidalmean field through the α-effect, which can be visualised as follows. Magnetic field lines movealong with rising and sinking convective elements but in doing so they are tilted in a systematicway because the fluid has a net helicity h (Eq. 1.10). The average effect of all these cyclonicevents is the creation of a net current ∝ αB0 parallel to the mean field itself.

The magnitude of the α-effect depends critically on the mean kinetic helicity, a quantitythat is not easily determined since it involves an unknown correlation coefficient between u1

and the vorticity ∇×u1 (Eq. 1.11). But even if the kinetic helicity in the convection zone wereknown, another ill-understood effect should be taken into account, namely the intermittency ofthe magnetic field. It is known from simulations of magnetoconvection that magnetic flux tendsto be expelled to the boundaries of convective cells (see Galloway & Weiss 1981). The magneticfield in the solar convection zone has an intermittent structure consisting of large unmagnetizedareas and small fluxtubes in which the field strength has at least equipartition values.4 Thesefluxtubes can partly resist the Coriolis forces, so that they do not experience the full effect ofthe helicity of the convective cells. In other words, the kinetic helicity of the fluxtubes may

4Within the convection zone, the equipartition field strength commonly refers to equipartition with the kineticenergy of convection, i.e. Beq ≈

4πρu21. This is not to be confused with equipartition with the gas pressure p,

or B ≈ √8πp, which approximately holds for fluxtubes at the stellar surface.

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be smaller than that of the ambient unmagnetized plasma, and, consequently the α-coefficientmay be smaller than expected from Eq. (1.15) (Gilman & Miller 1981). These two effects mayaccount for the well-known discrepancy between rough estimates of α (essentially the convectivevelocity u1 ≈ 103 cm s−1) and values of α that are required in mean-field solar dynamo models(of the order 10 cm s−1).

The third term on the r.h.s. of Eq. (1.13) includes turbulent diffusion and ohmic diffusion.The former describes in a statistical manner the effect of turbulence on the mean magneticfield, namely to enhance its decay and its diffusive transport. This enhanced transport is infact observed on the solar surface (§ 1.2.2) and from it one estimates β ≈ 6 × 1012 cm2 s−1

(Wang et al. 1989). Similar or somewhat smaller values are required in mean-field solar dynamomodels. The ordinary magnetic diffusivity η is at least 5 orders of magnitude smaller and canbe neglected in Eq. (1.13).

Equation (1.13) has wave-like solutions, that travel towards the equator if α∂Ω/∂r is neg-ative in the northern hemisphere and positive in the southern hemisphere. The direction ofpropagation becomes poleward if the sign of α∂Ω/∂r is reversed. The growth rate of the meanfield depends on a dimensionless combination of the parameters occurring in Eq. (1.13), thedynamo number C, which may be defined as

C =α∆ΩL3

2β2. (1.16)

Here L denotes a typical length scale of the dynamo, ∆Ω is the typical difference in rotation ratewithin the dynamo region and α and β are also understood as typical values. Dynamo action,loosely defined as exponential growth of B0, occurs if |C| exceeds a geometry-dependent criticalvalue |Ccrit|. If C = Ccrit, then the dynamo is marginally stable.

1.3.5.2 Mean-field dynamo models of the Sun

Several mean-field models try to incorporate the main characteristics of the solar dynamo, asdescribed in §§ 1.3.2 and 1.3.3. An essential feature in these models is the overshoot region, inwhich the strong toroidal fields (B ≈ 105 G) are produced. Such a field strength is far abovethe equipartition value, so that the turbulence in the overshoot layer must be severely inhibitedby the Lorentz force. While there is no consistent way to account for the Lorentz force in mean-field theory (see § 1.3.6), its effect can be described qualitatively in linear theory by assuming asuitable spatial dependence for the turbulent transport coefficients.

Thus Parker (1993) devised the surface wave dynamo, in which the α-effect and the differen-tial rotation are spatially separated. The former is restricted to the convection zone, while thelatter resides in the overshoot layer, in which the strong fields inhibit the α-effect (i.e. α = 0)and largely suppress turbulent diffusion (i.e. β is reduced). Turbulent transport between thetwo layers, essential for dynamo action, requires a nonzero value of β in the overshoot layer.Parker’s plane parallel geometry is also employed in chapters 4 to 6 of this thesis.

Prautzsch (1993) used a spherical geometry and incorporated recent results from helioseis-mology (Fig. 1.6). He finds periodic solutions with equatorward migration of the activity beltsonly if the α-effect in the overshoot layer is concentrated near the equator, rather than havinga cos θ-dependence as one would expect if the Coriolis force plays a role. In the latter case, thebutterfly diagram does not agree with the observations, due to the presence of strong polewardbranches, which arise because both the differential rotation ∂Ω/∂r and the α-effect are strongestnear the poles. A concentration of α near the equator may result if the α-effect is caused by

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Figure 1.6: Contour levels of the toroidal mean field at the solar surface. The α-coefficient isconcentrated near the equator and the differential rotation is based on results of helioseismology.(After Prautzsch 1993.)

magnetostrophic waves in the overshoot layer (Schmitt 1987), since fluxtubes with strong fields,crucial for the existence of these waves, can be stably stored at low latitudes whereas theybecome unstable at high latitudes (Moreno-Insertis et al. 1992).

A similar problem of strong poleward branches was found in the more elaborate model byRudiger & Brandenburg (1995). It has been suggested that a slow meridional circulation sufficesto transform poleward motion near the overshoot layer into equatorward motion at the top ofthe convection zone (Choudhuri et al. 1995).

1.3.5.3 Effects of variability in averaging procedures

Mean-field electrodynamics is a statistical theory and therefore a correct interpretation of itsresults requires a careful examination of the averaging procedure that is adopted. A con-sistent derivation of equations for mean quantities is possible only if for arbitrary functionsf(x, t), g(x, t) and constants c the averaging procedure satisfies the Reynolds rules (Krause &Radler 1980, ch. 2):

〈f + g〉 = 〈f〉 + 〈g〉, (1.17)

〈f〈g〉〉 = 〈f〉 〈g〉, (1.18)

〈c〉 = c, (1.19)

and 〈·〉 should commute with differentiations and integrations with respect to t and x. If theserules are not exactly satisfied, the resulting equation for the mean magnetic field has rapidlyfluctuating extra terms, the exact form of which cannot be easily established (Hoyng 1987).

A spatial average that satisfies the Reynolds rules exactly is obtained if one or more of thespatial coordinates is integrated out. For a star, the most obvious definition is the longitudinalaverage, a procedure that was applied for the first time by Braginskii (1965). The mean-fielddynamo equation (1.13) holds exactly in this case. Because the turbulence is organised in apattern of coherent eddies with a definite size, a longitudinal circle contains only a finite numberof them, so that α, β and u0 acquire a fluctuating component. These fluctuations may bemodelled as stochastic forcing terms of which the statistical properties are derived from those of

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the convective eddies, see for instance Choudhuri (1992) and Hoyng (1993) and chapters 2 and3 of this thesis for a treatment of fluctuations in α.

Random forcing provides one explanation of variability in the solar cycle. In particular,fluctuations in α can explain the observed anticorrelation between cycle length and amplitude(Hoyng 1992) and can cause behaviour reminiscent of the Maunder minimum (chapters 2 and3). Furthermore, random forcing excites eigenmodes of the mean-field equation, resulting inNorth-South asymmetries in the simulated butterfly diagram that have a close resemblance tothe observed asymmetries (chapter 3). An alternative explanation of solar variability is basedon nonlinear oscillations, see § 1.3.6.

The ensemble average also obeys the Reynolds rules and it has the advantage that the meanquantities defined by it do not have fluctuating components. Hence the mean-field dynamoequation (1.13) with constant coefficients α and β is also exact in the ensemble average. But theinterpretation of the mean-field concept has now changed in a subtle but drastic way, as can beunderstood with the help of a thought experiment. The solar cycle is known to have frequencyvariations of the order δω/ω ≈ 0.1, where ω = ω0 + δω is the dynamo frequency. At present, thestatistical properties of δω are not well-established due to a lack of data. But the values of δωduring consecutive cycles are probably independent - this is certainly the case in the stochasticmodel discussed in chapter 3. In the thought experiment we may schematically represent thetime dependence of the magnetic field of one ensemble member in the form B ∝ cosψ, where ψ =ψ0+ωt is the dynamo phase (we ignore the envelope modulations for the moment). Suppose thatthe phases of all ensemble members are synchronized at t = 0. In other words, the distributionof phases is a delta-function peaked at ψ0 initially. If we now let the ensemble members evolvewith time, the distribution of phases broadens because in each dynamo consecutive values ofδω are independent. This phase mixing leads to increasing cancellation, and one conclusion ofthe thought experiment is that the mean field decays. The typical timescale of phase mixing,and hence also the typical decay time of B0, is about 10 dynamo periods for the solar dynamo.Clearly, the decay of the mean field must be understood as a statistical effect, because the actualfield need not decay while the mean field does.

Phase mixing has a different effect on the magnetic energy density B2/8π ∝ cos2 ψ = 12 +

12 cos 2ψ. If we carry out the same thought experiment with B2, we see that the fundamentalmode of 〈B2〉 is constant and non-periodic, whereas the ensemble average of the periodic partof B2 decays due to phase mixing. This thought experiment lies at the heart of the combineduse of equations for the mean field and the mean magnetic energy when the ensemble averageis employed. For a stationary dynamo the fundamental mode of Eq. (1.13) must satisfy theconstraint that B0 decays on a timescale of about 10 dynamo periods, while the fundamentalmode of 〈B2〉 must be marginally stable. These requirements form the basis of the finite energymethod (Hoyng 1987, Van Geffen 1993a). In chapters 3 and 4 of this thesis, the finite energymethod is applied to the Sun.

1.3.5.4 The mean magnetic energy tensor

In the previous section it was shown how a careful interpretation of the ensemble average inmean-field dynamo theory led to the concept of the finite energy method. Another reason forstudying the mean magnetic energy is that the fluctuating component of the magnetic field,B1 = B −B0, can be much larger than the mean field:

B1 ≈ R1/2m B0 (1.20)

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(Krause & Radler 1980). This result is a well-known dilemma of mean-field dynamo theory.Although based on a linear calculation that does not take into account the back-reaction of themagnetic field on the flow, it suggests that the mean magnetic field alone cannot give a completedescription of the solar magnetic field. Expression (1.20) does not affect the validity of the mean-field equation, but it does urge us to consider also the mean magnetic energy. The fundamentaldifference between 〈B〉 and 〈B2〉 is that the mean field is essentially a large-scale phenomenon,while the mean magnetic energy retains contributions from all length scales: 〈B2〉 = B2

0 + 〈B21〉.

In general no closed equation exists for 〈B2〉, and it is necessary to consider what is henceforthcalled the mean magnetic energy tensor 〈BB〉/8π. The equation governing this tensor is obtainedby adding Bi∂tBj +Bj∂tBi and inserting the induction equation (1.3):

(∂t + u · ∇)BiBj =∑

k

(∇kui)BkBj + (∇kuj)BkBi + η∇2BiBj

−2η∑

k

(∇kBi)(∇kBj). (1.21)

The discussion on how to average this equation is postponed to the next section on stochasticdifferential equations. It turns out that the dissipative term −2η

k(∇kBi)(∇kBj) results in aclosure problem. The following expression for the mean magnetic energy tensor T = 〈BB〉/8πis not exact but is based on an approximative treatment of resistive dissipation discussed inchapter 5:

(∂t + u0 · ∇)Tij =∑

kl

∇k(αǫiklTlj + αǫjklTli) +∑

k

(∇ku0i)Tkj + (∇ku0j)Tki

+ 25γ(2

k

Tkkδij − Tij) + ∇ · β∇Tij − 2νTij . (1.22)

From the mean energy tensor T one can calculate the mean magnetic energy density 〈B2〉/8π =∑

k Tkk and the correlation coefficients of the magnetic field,

Cij =Tij

TiiTjj=

〈BiBj〉√

〈B2i 〉〈B2

j 〉. (1.23)

The meaning of the various terms in Eq. (1.22) is explained in more detail in chapters 4 and5 and can be summarized as follows. The second term on the l.h.s. denotes advection by themean flow u0. On the r.h.s. there are two terms related to the α-effect. Next come two termsthat depend on spatial derivatives of the mean flow, i.e. on differential rotation. Then there isa term involving the mean magnitude of the turbulent vorticity ∇× u1:

γ ≈ 13τc〈|∇ × u1|2〉. (1.24)

Vorticity gives rise to random stretching and winding up of magnetic field lines. Hence it causedthe mean magnetic energy to increase, and the correlation coefficients to decrease. This explainsthe plus-sign in front of the diagonal components of T and the minus-sign in front of the off-diagonal components of T. The magnitude of γ is not well-known; a rough estimate suggests thatγ ≈ β/l2 ≈ 5× 10−7 s−1 for the largest eddies in the solar convection zone. Turbulent diffusion(β) plays a somewhat different role here than it does in the mean-field equation, because it doesnot give rise to decay of the mean magnetic energy but merely to enhanced diffusive transport.The three dynamo coefficients α, β and γ can be combined to yield the normalised helicityH = −α/√βγ (Eq. 1.11). Hence they are constrained by a Schwartz-type inequality:

α√βγ

∣ ≤ 1. (1.25)

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Finally, the last term on the r.h.s. accounts for resistive dissipation in an approximate way. Inchapter 5 it is found that the magnitude of ν that is required for marginal stability of T is ofthe same order as γ.

1.3.5.5 Stochastic differential equations

In this section a few results from the theory of stochastic differential equations will be sum-marised. A clear treatment of this mathematical technique can be found in Van Kampen (1976,1992). It was applied in the context of dynamo theory for the first time by Hoyng (1988, 1992),to whom the reader is referred for a more complete discussion. The equations for the meanmagnetic field and for the mean magnetic energy can both be derived with the technique ofstochastic differential equations. We start with a differential equation of the form

∂tA = [L0 + L1]A, (1.26)

where A stands for any scalar, vector or tensor, L0 is a time independent operator and L1 isa stochastic operator with 〈L1〉 = 0 and with a correlation time τc. Similarly we may writeA = 〈A〉 + A1. The goal is to derive an equation for A0 = 〈A〉. We average Eq. (1.26), whichprovides

∂tA0 = L0A0 + 〈L1A1〉, (1.27)

∂tA1 = L0A1 + L1A0 +G, (1.28)

where G = L1A1 − 〈L1A1〉. In order to proceed, this set of equations must be closed, which canbe achieved by ignoring G. This represents the first-order smoothing approximation (FOSA),valid if the secular effect of G is smaller than that of L0A1 and L1A0, which is true if |τcL1| ≪ 1.5

On neglecting G, Eq. (1.28) can be solved in a straightforward manner. The solution A1

itself is not of much interest, but it leads to the following expression for 〈L1A1〉:

〈L1A1〉 =

∫ ∞

0ds 〈L1(t)e

sL0L1(t− s)〉A0(t− s). (1.29)

With the help of this term Eq. (1.27) can be closed, provided that A0(t − s) is replaced by anexpression for A0(t). One possibility is to approximate A0(t−s) ≈ exp(−sL0)A0(t). Substitutionin Eq. (1.29) then yields

〈L1A1〉 =

∫ ∞

0ds 〈L1(t)e

sL0L1(t− s)〉 e−sL0 A0. (1.30)

A further simplification is obtained if |τcL0| ≪ 1, i.e. if the evolutionary timescale associatedwith L0 is large with respect to τc. In that case the exponential operators can be ignored.Substituting the result into Eq. (1.27) yields the following equation for A0:

∂tA0 = [L0 +

∫ ∞

0ds 〈L1(t)L1(t− s)〉]A0. (1.31)

5FOSA is also known in the literature as the second order correlation approximation (SOCA) or the quasi-linear

approximation.

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1.3.5.6 Application to the induction equation

In the case of the induction equation (1.3), we identify A with the magnetic field B, L0 with∇× (u0×· · ·)+η∇2 and L1 with ∇× (u1×· · ·). FOSA is justified if u1τc/l ≪ 1. This conditionis not satisfied in the Sun, since the convective eddies have a lifetime that is comparable to theirturnover time, i.e. τc ≈ l/u1. However, no satisfactory derivations of the mean-field dynamoequation exist if the correlation time is long, see Van Kampen (1976, 1992). In the solar contextthe validity of Eq. (1.13) therefore remains unproven. In what follows, this problem must beleft aside. The viewpoint in this thesis is a practical one, namely that the mean-field dynamoequation is remarkably successful in describing the large-scale solar magnetic field, in spite of thedifficulties on a theoretical level. Two less problematic common assumptions are the following:

(1) |τcL0| ≪ 1, so that Eq. (1.31) holds. This condition is valid in the Sun since the timescalefor field amplification due to differential rotation is much longer than τc.

(2) Homogeneous pseudo-isotropic (i.e. isotropic but not mirror-symmetric) turbulence. Inthat case the transport coefficients reduce to constants, given by

α = −13

∫ ∞

0ds 〈u1(t) · [∇× u1(t− s)]〉; β = 1

3

∫ ∞

0ds 〈u1(t) · u1(t− s)〉. (1.32)

The dominant contribution to these integrals comes from s <∼ τc. Hence the integration maybe approximated by a factor τc, leading to Eq. (1.15). In the anisotropic case α and β aretensors, while inhomogeneous turbulence leads to extra terms involving gradients of α andβ. One of these effects is turbulent diamagnetism, which leads to expulsion of magnetic fluxout of regions with strong turbulence, so that the field acquires an intermittent structure(Zel’dovich 1957).

This concludes the summary of the assumptions made in deriving Eq. (1.13). Details of theactual calculation of expression (1.31) can be found in Hoyng (1992).

1.3.5.7 Application to the magnetic energy tensor equation

Equation (1.21) represents a stochastic differential equation, of a more difficult type than theinduction equation, since it cannot be expressed in the form ∂tBB = LBB for some differentialoperator L. The closure problem is caused by the dissipative term −2η

k(∇kBi)(∇kBj). Inchapters 4 and 5 of this thesis, two possible scenarios are considered to resolve the closureproblem. Here I shall briefly summarize them, together with a third exact method, that, to myknowledge, has not yet been tried in the context of stochastic differential equations:

(1) Resistive dissipation is neglected (η = 0). The mean and fluctuating operators are thendefined as

L(0)ijkl = (∇ku0i)δjl + (∇ku0j)δik − δikδjl u0 · ∇, (1.33)

L(1)ijkl = (∇ku1i)δjl + (∇ku1j)δik − δikδjl u1 · ∇. (1.34)

On applying Eq. (1.31), a third transport coefficient appears besides α and β, the vorticitycoefficient:

γ = 13

∫ ∞

0ds 〈[∇× u1(t)] · [∇× u1(t− s)]〉. (1.35)

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Again, the integration may be approximated by a factor τc to yield Eq. (1.24).

In this approximation, which was introduced by Knobloch (1978) and Hoyng (1987), tur-bulent transport of magnetic energy to the boundary becomes the only energy sink of thedynamo. As it turns out, a stationary mean magnetic energy can then be obtained onlyfor unphysically large values of the turbulent diffusivity β (chapter 4).

(2) The effect of resistive dissipation is approximated by a scale-independent decay term:

−2η∑

k(∇kBi)(∇kBj) ≈ −2νBiBj, which amounts to an extra term −2νδikδjl in L(0)ijkl

(Eq. 1.33). Results obtained with this primitive approximation suggest that resistivedissipation represents by and large the dominant energy sink of the dynamo (chapter 5).

(3) The mean magnetic energy tensor is obtained by evaluating the two-point correlationfunction of the magnetic field, Rij(x,x

′; t) = 〈Bi(x, t)Bj(x′, t)〉, at x = x′. Equationsfor Rij have been studied for instance by Brauer & Krause (1973), Vainshtein (1982)and by Kleeorin et al. (1986), using various approximations. The advantage of thismethod is that no closure problem is posed by the equation for Bi(x, t)Bj(x

′, t). Hencethe theory of stochastic differential equations can be applied to derive an equation forRij(x,x

′; t). The closure problem is resolved because the original term−2η

k(∇kBi)(∇kBj) in the equation for BiBj is now replaced by a corresponding term inthe equation for Bi(x)Bj(x

′) that can be expressed in the form −2η∑

k∇k∇′kBi(x)Bj(x

′).The removal of the closure problem comes at the price of having to solve a much morecomplicated partial differential equation. Furthermore, there is probably no longer a sep-aration of timescales, since on small length scales the term −2η

k∇k∇′kBi(x)Bj(x

′) canbe very large, so that the condition |τcL0| ≪ 1 is violated. Nevertheless, this methodwould in principle yield an exact expression for the mean magnetic energy tensor.

1.3.6 Nonlinear dynamos

In the kinematic approach, dynamo action can result in unlimited growth of the magnetic fieldstrength. This is impossible if the full dynamo problem (Eqs. 1.3–1.9) is solved, since the Lorentzforce provides a nonlinear feedback which opposes dynamo action (Lenz’s rule) and gives riseto saturation of the field strength. While the Lorentz force can probably be neglected for fieldstrengths much smaller than equipartition values, it is bound to be important in the overshootlayer, where B ≈ 105 G ≫ Beq.

There is no consistent way to include these nonlinear dynamic effects in mean-field models,but several heuristic approaches have been suggested. For an overview of nonlinear mean-fieldmodels and references to various calculations see for instance Schmitt & Schussler (1989) orSchmitt (1993). The Lorentz force can be expressed as the sum of two terms,

fL =1

4π(B · ∇)B − 1

8π∇B2. (1.36)

the first of which represents tension and curvature forces, while the second represents the mag-netic pressure force. The former acts along the magnetic field lines and tends to reduce curvature.This effect is modelled in mean-field theory by α-quenching, an ad-hoc nonlinear modificationof the α-coefficient that accounts schematically for the tension forces on the helical convectiveeddies. Stix (1972) was the first to introduce such a nonlinear α-effect by setting

α = α0 f(B20), (1.37)

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1.3. Dynamo theory of Sun and stars

where f is a monotonically decreasing function of the field strength. A similar effect may beinvoked to quench the turbulent diffusivity β.

If a fluxtube (index f) is in pressure equilibrium with the ambient medium (index a), thenpgas,f + B2/8π = pgas,a, and its density is less than that of the surrounding gas. This givesrise to a magnetic buoyancy force, which tends to expel fluxtubes to the stellar surface. Inmean-field models, buoyancy is often parametrized by adding to Eq. (1.13) a nonlinear termof the form −g(B0)B0/τb, where τb is the buoyancy-loss timescale and g is a monotonicallyincreasing function of B0 (Leighton 1969). The physical justification for such a local decayterm is problematic since buoyancy is essentially a transport phenomenon, that is perhaps moreadequately modelled by enhanced turbulent diffusion in the radial direction. Furthermore, fluxloss is achieved only when entire fluxtubes are expelled from the dynamo region but not whena fraction of a fluxtube emerges as an active region (Weiss 1994).

The Lorentz force may also lead to saturation of the magnetic field by counteracting dif-ferential rotation; this has been modelled in mean-field theory using paramatrisations for u0

similar to α-quenching (Yoshimura 1978, Rudiger 1989). A different approach consists of solv-ing a separate nonlinear equation for the evolution of α (e.g. Kleeorin et al. 1995) or of u0 (e.g.Tobias 1996).

None of these nonlinear mean-field models are based on selfconsistent derivations. Further-more, one may argue that the backreaction of the fluctuating component of the magnetic field,B1, is more important than that of the much weaker mean field B0 (Eq. 1.20). Thus nonlinearmean-field models are much more problematic than kinematic ones. Nevertheless, a study ofsimple nonlinear equations may be justified if it reveals some generic properties of nonlineardynamos (Weiss 1993, Tobias et al. 1995). With this in mind many investigations of chaotic dy-namos have been undertaken, employing truncated nonlinear toy-systems, similar to the Lorenzequations. They exhibit a rich variety of behaviour, including modulated oscillations and chaotic(aperiodic) oscillations (e.g. Weiss et al. 1984, Spiegel 1994), sustained intervals of low activity(e.g. Tobias 1996) and, in spherical models, solutions of mixed parity (i.e. neither symmetricnor antisymmetric with respect to the equator, see for instance Moss et al. 1991 or Jennings &Weiss 1991 and references therein).

All these features are relevant for explaining the various types of solar variability mentioned in§ 1.2.3, namely amplitude and period variations, grand minima and asymmetries with respect tothe equator. However, it appears that nonlinear toy models display chaotic, irregular variationsonly at highly supercritical dynamo numbers (Eq. 1.16), and it is unknown whether such acondition applies to the solar dynamo. At smaller dynamo numbers the variations assumethe form of periodic modulations, inconsistent with the variability of the solar cycle (Moss etal. 1992). The viewpoint adopted in this thesis is that stochastic fluctuations explain manyfeatures of the variability in a natural way while nonlinearities are important mainly for limitingthe magnetic field strength (chapters 2 and 3).

1.3.7 MHD turbulence

From the considerations in the previous two sections it is clear that in a complete theory forthe solar dynamo a number of difficulties must be resolved, namely the role of (1) small-scalemagnetic fields, (2) the Lorentz force and of (3) resistive dissipation. These issues are relatedto the large and complex subject of MHD turbulence (see for instance Zel’dovich et al. 1983, ch.8, or Biskamp 1993, ch. 7). The main goal of turbulence theory is to describe the distributionof equilibrium properties of turbulent plasmas in Fourier (k) space. In this representation,

25

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SOLAR AND STELLAR DYNAMOS

every spectral component corresponds to a certain spatial scale, which is convenient, becausethe turbulent dynamo can be considered as a transfer process, or cascade, of various quantitiesbetween different scales. A disadvantage of spectral methods is that spatial structures cannotbe discussed because the phase information of the Fourier components is lost.

The range of scales in turbulent fluids with high Reynolds numbers extends from the injectionscale lin (say, the radius of the Sun) down to the dissipation scale ld (less than 1 km in the Sun).The intermediate scales define the inertial range:

ld ≪ l ≪ lin (1.38)

Magnetic energy is injected into the spectrum at a rate ǫin and is converted to smaller scalesat the transfer rate ǫt, until it is dissipated on the smallest scale at a rate ǫd. In a stationarysituation, the cascade continues until a length scale is reached where all the injected energy canbe dissipated, and we have ǫin = ǫt = ǫd. If it is known how ǫt depends on l, the shape ofthe turbulent spectrum can be determined. Here the Lorentz force must be taken into account,because it will start to influence the magnetic field at a certain length scale once the equipar-tition field strength is reached. The turbulence then assumes the form of random MHD waves,propagating at the Alfven speed. Due to this Alfven effect the energy transfer is delayed andthe resulting spectrum is less steep (∝ k−3/2) than a Kolmogorov spectrum (∝ k−5/3).

While magnetic energy is transferred from large to small scales, magnetic helicity A ·B (Ais the vector potential) is transported in the opposite direction (Pouquet et al. (1976). Thisinverse cascade of magnetic helicity, leading to a build-up of magnetic energy on large scales, isdriven by the mean kinetic helicity (Eq. 1.10), as is the α-effect of mean-field dynamo theory.Hence, these investigations on MHD turbulence confirm the cyclonic mechanism through whicha convective dynamo generates a magnetic field.

One conclusion to be drawn from MHD-turbulence calculations is that the magnetic field ina convective dynamo is distributed over a large range of length scales, so that small-scale fieldsmay dominate the magnetic energy. Furthermore, resistive dissipation may be important forthe energy balance. Whereas definite statements concerning the importance of these effects canprobably be obtained only from numerical simulations of the full dynamo problem, an attempt ismade in this thesis to account for small-scale fields and resistive dissipation within the frameworkof mean-field theory by studying an equation for the mean magnetic energy.

1.3.8 Numerical simulations

Attempts to simulate the solar dynamo by solving the full dynamo problem (1.3–1.9) face variousdifficulties. Due to the high Reynolds number in the solar convection zone, the smallest presentfield structures have a typical length scale of about 1 km, and, secondly, the field is highlyintermittent. These features cannot be reproduced in a numerical simulation, since the totalvolume of the convection zone of about 6×1017 km3 would require an equal amount of gridpoints.At present computational speed is such that the highest achievable value of Rm is of the order 103

(Brummel et al. 1995). A lack of resolution may explain why first attempts by Gilman (1981,1983) and Glatzmaier (1985) failed to reproduce correctly the details of both the differentialrotation and of the butterfly diagram. A more feasible numerical approach to the solar dynamoconsists of solving Eqs. (1.3–1.9) in a box. Such calculations demonstrate local dynamo action(Nordlund et al. 1992) and are appropriate to study specific effects and sub-problems, but it isdifficult to draw conclusions from their results concerning the large-scale dynamo at the bottom

26

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1.3. Dynamo theory of Sun and stars

of the solar convection zone. For more details on computational aspects of the full dynamoproblem, see Brandenburg (1994).

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Chapter 2

Stochastic and nonlinear fluctuations

in a mean field dynamo

AbstractWe study the effect of rapid stochastic fluctuations in the kinetic helicity in a plane parallel meanfield dynamo model for the Sun. The α-parameter has a fluctuating component δα = α−α0, whichis modelled as a random forcing term. The fluctuations give rise to variations in the amplitude andphase of the dynamo wave, such that shorter cycles have higher amplitudes, as is observed in thesolar cycle. By making a second order expansion close to the unperturbed marginally stable dynamowave we are able to go beyond the weak forcing limit studied by Hoyng (1993). We show that withincreasing strength of the forcing the effective dynamo frequency decreases. We introduce a simplenon-linearity to model α-quenching and derive a set of linear equations for the mean field, valid inthe weak forcing case. With α-quenching, phase and amplitude fluctuations are bounded, but stillcorrelated. The strength of the α-quenching is measured by a parameter q = −(Te/α0)(∂α/∂T )|Te ,where Te is the equilibrium value of the toroidal field. We make a comparison with sunspot data,and conclude that these are well explained by the model if δα/α0 ≈ 2.2 and q ≈ 0.7. Finallywe briefly consider the alternative possibility of fluctuations caused by nonlinear dynamics, withoutexternal forcing (δα = 0). We show that the resulting phase-amplitude diagram does not agree withobservations. Although this is no proof that the phase-amplitude correlation cannot be reproducedby nonlinear chaos, we conclude that stochastic noise provides a more natural explanation.

A.J.H. Ossendrijver and P. Hoyng

Astronomy & Astrophysics (in press)

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STOCHASTIC AND NONLINEAR FLUCTUATIONS IN A MEAN FIELD DYNAMO

2.1 Introduction

Observations of solar activity that have been recorded since the beginning of the 18th centuryin the form of sunspot number counts, demonstrate the well known periodic behaviour but alsoreveal that the solar cycle has considerable variability. The half-period of the cycle may deviatefrom its average 11 years by more than two years and activity as measured by the maximumof the relative sunspot number R may vary by as much as 100% from cycle to cycle. Since theconnection between solar activity and magnetic fields was discovered, theoretical models havebeen constructed on the basis of the induction equation to explain the solar cycle. One of themost common approaches is mean field dynamo theory. It provides an equation for the meanmagnetic field B0 = 〈B〉,

∂B0

∂t= ∇× [u0 × B0 + αB0 − β∇× B0]. (2.1)

Here u0 stands for the large scale velocity field (differential rotation). If we denote the turbulentvelocity field by u1 and the typical correlation time of the turbulence by tc, we can express theturbulent diffusivity as β = 1

3tc〈|u1|2〉 and α as α = −13〈h〉tc. Here h is the kinetic helicity, i.e.

h = u1 · ∇ × u1. (2.2)

Solutions of Eq. (2.1) reproduce some of the basic features of the large scale solar magneticfield, such as its periodicity, dipolar structure and the equatorward migration of activity belts.Variability however is not accounted for: all consecutive cycles of the mean field dynamo haveidentical periods, whereas the solar cycle shows variability in the duration and amplitude of theoscillations.

In order to explain the variability that is observed in the solar cycle (as well as the possibleexcitation of non-dipole dynamo-modes) two different causes have been put forward: nonlineari-ties and stochasticity. For both processes it has been argued that they appear in the dynamo in anatural way and can produce variability (for nonlinearities see for instance Weiss et al. 1984 andSchmitt & Schussler 1989; for stochasticity see Hoyng 1987 and, applied to nonlinear systems,Crossley et al. 1986 and Moss et al. 1992). In this paper the role of stochastic fluctuations issingled out as the main source of variability, and we shall consider nonlinear effects primarily asa means to confine the magnitude of the field.

The dynamo responsible for the solar cycle is thought to be located in the overshoot layerat the base of the convection zone. A natural source of stochasticity in the convection zone isprovided by the rising and sinking convective cells. Through the effect of the Coriolis force thatacts on the expanding and contracting cells the convective fluid acquires a non-zero net helicityh. Magnetic field lines that are oriented along the toroidal direction are dragged along with thefluid, giving rise to poloidal field components, perpendicular to the original field. In qualitativeterms, this is the origin of the well-known α-effect, described by the α-term in Eq. (2.1). Weshould stress however that there are as yet no reliable estimates of the magnitude of 〈h〉 or ofα. It is common procedure in linear mean field dynamo theory to adopt the value required toobtain a mean field with given properties. In this paper we shall focus rather on the effect offluctuations in α, and our conclusions will not depend on the mean value of α.

Usually, the stochastic nature of α and other parameters is ignored. This is correct if anensemble average is used to define mean quantities. If however a spatial average is used, onemust take into account that the convection zone consists of a finite number of convective cells.Within a convective cell, physical parameters are approximately coherent. A spatial average

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2.2. Model without α-quenching

is thus equivalent to an average over a finite number of discrete convective cells, so that theaverage quantity necessarily has a fluctuating component. The approach followed by Choudhuri(1992) and Hoyng (1993) and also in this paper is to define mean quantities 〈A〉 as azimuthalaverages, i.e.

〈A〉(r, θ) =1

∫ 2π

0dφA(r, θ, φ). (2.3)

On each azimuthal circle there are a finite number of convective cells so that one must allowfor a fluctuating component in 〈A〉. Therefore α, β and u0 are not constant with respect totime but fluctuate on the timescale tc of the convective motions. Since this timescale (tc ∼1month) is much shorter than the solar dynamo period one may try to model these fluctuations asstochastic terms. In principle, all three parameters that appear in Eq. (2.1) have such a stochasticcomponent. However, fluctuations in α = −1

3〈h〉tc dominate because the kinetic helicity h mayhave opposite signs in different convective cells, giving rise to cancellation and a potentially largeinfluence of fluctuations. The turbulent diffusivity β is the mean of a positive definite quantityso that no cancellations can occur in the averaging procedure. The relative level of fluctuationsin β is therefore expected to be lower as compared to that in α. Hoyng (1993) studied the effectof stochastic fluctuations for a linear αω-dynamo in a plane geometry, in the weak forcing limit.He found a permanent correlation between phase and amplitude fluctuations such that shortercycles have higher amplitudes. Sunspot data confirmed this correlation but also indicated thatthe model could be inproved by including some (nonlinear) quenching mechanism.

The main goal of this paper is to study the effect of α-quenching on the phase-amplitudecorrelation. After introducing the model equations based on Hoyng (1993), we shall brieflyrecapitulate some of his calculations for the weak forcing case in a slightly different notation.We derive the phase-amplitude correlation and explain an effect that was previously overlooked,namely a decrease in the effective dynamo frequency. Next we introduce α-quenching and wecalculate its effect on the phase and amplitude variations, their correlation and on the dynamofrequency decrease. After that we make a comparison with the observed solar cycle record.Finally we briefly compare our results with those from the alternative explanation of variability,based on nonlinear chaos without ’external’ fluctuations in α.

2.2 Model without α-quenching

2.2.1 Model equations

We employ a plane geometry, where x, y and z denote the radial, azimuthal and latitudinaldirections respectively (Fig. 2.1). Effects of the spherical geometry are thus ignored; the modelhas only a local validity and α and β are spatially constant. All quantities are φ-averages andthus axisymmetric, i.e. ∂/∂y = 0. Differential rotation in the dynamo layer is schematicallydescribed by

u0 = u0(x)ey,∂u0

∂x= a, (2.4)

where a is a constant (see Goode et al. 1991). The mean field is separated in poloidal andtoroidal components according to

B0 = ∇×Apey +Btey, (2.5)

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Figure 2.1: Plane geometry of the dynamo model, consisting of a single layer of thickness Lextending infinitely in the azimuthal (y) and latitudinal (z) directions

where Ap is the poloidal vector potential, and Bt the toroidal component of the mean field. Forboth components a plane wave ansatz is made,

Ap = P (t) cosχ sin(πx/L), (2.6)

Bt = kT (t) cosψ sin(πx/L), (2.7)

where P and T are the amplitudes of Ap and Bt. Their phases are defined as

χ(z, t) = kzz − ωt+ δχ(t) + φ0, (2.8)

ψ(z, t) = kzz − ωt+ δψ(t). (2.9)

Here kz is the latitudinal wavenumber, δχ and δψ are the fluctuating components of the phasesand φ0 is a constant phase lag between the poloidal and toroidal field components. The wavenum-ber k =

k2z + (π/L)2 is included in Eq. (2.7) for dimensional reasons; we refer to Hoyng (1993)

for details. For δα = δχ = δψ = 0 Eqs. (2.5–2.7) represent an undisturbed dynamo wave, trav-elling in the z-direction. We introduce dimensionless time, τ = t/td, where td = 1/β0k

2 is themean field diffusion time. The dimensionless correlation time of the fluctuations is τc = tc/tdand the dimensionless dynamo frequency ω = ωtd. If td is chosen suitably, the dynamo periodis 2πtd/ω = 22 years. The fluctuations in α are modelled by

α(τ) = α0 + δα(τ) = α0[1 + α(τ)]. (2.10)

With α we denote the relative fluctuations. We assume that α0 > 0; our model is thus to besituated on the southern hemisphere.

We substitute Eqs. (2.5–2.7) in Eq. (2.1) and obtain the following equations for P , T , δχand δψ:

ddτ P = −P + dα(1 + α)T cos(φ0 + δ), (2.11)ddτ T = −T + dα(1 + α)P cos(φ0 + δ) + dωP sin(φ0 + δ), (2.12)

P ddτ δχ = ωP − dα(1 + α)T sin(φ0 + δ), (2.13)

T ddτ δψ = ωT + dα(1 + α)P sin(φ0 + δ) − dωP cos(φ0 + δ), (2.14)

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2.2. Model without α-quenching

where

δ = δχ− δψ. (2.15)

The two dimensionless dynamo numbers dα and dω are given by

dα = α0/β0k and dω = akz/β0k3. (2.16)

In the numerical simulations, after every interval 2τc the value of α is updated by drawing anew value from a stationary gaussian distribution with a prescribed width αrms.

1 During theseintervals α remains constant. As we shall see, the statistical properties of our solutions dependonly on the fluctuations in α relative to the mean value α0 and on τc, in the combination α2τc,which plays the role of a diffusion coefficient,

D = 14 α

2rmsτc. (2.17)

We can distinguish a weak forcing regime and a strong forcing regime. The boundary betweenthese regimes lies at D ≈ 0.05, and was defined in a practical way, based on the numericalsimulations (see sections 2.2.2 and 2.2.3).

The first step is to establish relations between the parameters φ0, ω, dα and dω such that theunperturbed dynamo wave (α = δ = δχ = δψ = 0) is marginally stable (P = T = 0, P = Ps,T = Ts).

2 Hoyng (1993) showed that Eqs. (2.11–2.14) yield

ω = tanφ0, (2.18)

d2α = 1 − ω2, (2.19)

dαdω = 2ω, (2.20)

Ps/Ts = dα cosφ0 =√

cos 2φ0 sign dα. (2.21)

Since we have chosen α0 > 0, i.e. dα > 0, it follows that Ps/Ts > 0, ω > 0 and 0 < φ0 < π/4.The phase lag φ0 between the poloidal vector potential and the toroidal magnetic field deter-mines the dynamo type. Two extreme cases and an intermediate regime can be distinguished,commonly denoted as the α2-dynamo (φ0 ≪ 1), αω-dynamo (π/4 − φ0 ≪ 1) and α2ω-dynamo(φ0 intermediate). We shall be concerned with the αω-limit only. In that case a further sim-plification is possible since φ0 ≈ π/4, dα ≈

√2Ps/Ts ≪ 1 and dω ≫ 1. Introducing the new

variable

ρ = (P/T )(Ps/Ts)−1, i.e. ρ = ρ

d

dτlog(P/T ), (2.22)

Eqs. (2.11–2.14) reduce to

ddτ logP = −1 +

√2(1 + α)

1

ρcos(π/4 + δ), (2.23)

ddτ log T = −1 +

√2ρ sin(π/4 + δ), (2.24)

ddτ δχ = 1 −

√2(1 + α)

1

ρsin(π/4 + δ), (2.25)

ddτ δψ = 1 −

√2ρ cos(π/4 + δ). (2.26)

1In this way the correlation function of α is 〈α(τ )α(τ + σ)〉 = α2rms(1 − |σ|/2τc) and zero for |σ| > 2τc so

that τc satisfies the definition∫

0dσ 〈α(τ )α(τ + σ)〉 = α2

rmsτc. Another model that satisfies the definition is

〈α(τ )α(τ + σ)〉 = 8Dδd(σ) where δd(σ) is the Dirac delta-function (∫

0dσ δd(σ) = 1/2). This will be used later

in analytical work.2Where convenient, we shall employ a dot to denote differentiation with respect to τ .

33

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Figure 2.2: Two simulations of Eqs. (2.23–2.26) with φ0 = 0.7853 and τc = 0.0238, i.e. tc = 1month. Plotted is the toroidal magnetic field |Bt| in arbitrary units, at a fixed radial position xand colatitude z, as a function of time. The unit of time is the dynamo period of the unperturbedsolution. a) αrms = 0.58, D = 2 × 10−3. b) αrms = 2.25, D = 0.03. Notice that in b) thedynamo period is effectively longer than in a); see section 2.2.3 for an explanation of this effect.

These are the equations on which the analysis in this paper is based. In accordance with the αω-approximation, the α-effect does not create toroidal from poloidal field and is therefore absentin Eqs. (2.24) and (2.26). From Eqs. (2.23–2.26) we can deduce a closed set of two equations interms of δ (Eq. 2.15) and ρ (Eq. 2.22):

ρδ = −√

2(1 + α) sin(π/4 + δ) +√

2ρ2 cos(π/4 + δ), (2.27)

ρ =√

2(1 + α) cos(π/4 + δ) −√

2ρ2 sin(π/4 + δ). (2.28)

Two numerical solutions of Eqs. (2.23–2.26) with weak forcing are shown in Fig. 2.2. In the nexttwo sections we shall investigate the main features and (statistical) properties of these solutions.

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2.2. Model without α-quenching

2.2.2 Weak forcing

2.2.2.1 Equations and solutions

Equations (2.23–2.26) and (2.27-2.28) are nonlinear and have multiplicative random forcingterms.3 If the fluctuations are sufficiently weak, we expect that the solutions do not deviatemuch from the stationary unperturbed dynamo wave. From numerical practice, we estimatethat this requires D <

∼ 0.05. It is then allowed to linearise the equations by making an expansionfor small δ, and ρ close to 1. Furthermore only additive forcing terms are retained. 4 Weintroduce

ǫ = ρ− 1 (2.29)

and approximate 1/ρ ≈ 1 − ǫ, cos(π/4 + δ) ≈ (1 − δ)/√

2 and sin(π/4 + δ) ≈ (1 + δ)/√

2. ThenEqs. (2.23–2.26) reduce to

ddτ logP = −δ − ǫ+ α, (2.30)ddτ log T = δ + ǫ, (2.31)

ddτ δχ = −δ + ǫ− α, (2.32)ddτ δψ = δ − ǫ. (2.33)

From Eqs. (2.27–2.28) we obtain in compact notation:

u = Mu + αf , u =

(

δǫ

)

, (2.34)

where

M =

(

−2 2−2 −2

)

and f =

(

−11

)

. (2.35)

This result can also be derived by combining Eqs. (2.15), (2.22), (2.29) and (2.30–2.33). Theeigenvalues λ1,2 = −2(1 ± i) of M have negative real parts so that the stationary unperturbedsolution u = (0, 0) is stable. If initial conditions are imposed at τ = τ0, then for τ ≫ τ0 thesolution of Eq. (2.34) can be written as

δ(τ) = −√

2

∫ ∞

0dσ α(τ − σ)e−2σ cos(π/4 + 2σ), (2.36)

ǫ(τ) =√

2

∫ ∞

0dσ α(τ − σ)e−2σ sin(π/4 + 2σ). (2.37)

2.2.2.2 Statistical properties

In this section we derive various statistical properties of the linearised system, that are neededin section 2.2.3, where we consider the effect of α-fluctuations on the dynamo frequency. Since〈α〉 = 0 we see from Eqs. (2.36) and (2.37) that 〈u〉 = 0. Averages such as 〈uu〉 and 〈uα〉 arecalculated most easily by assuming

〈α(τ)α(τ ′)〉 = 8D δd(τ − τ ′), (2.38)

3Our original equations without α-quenching (Eqs. 2.11–2.14) are linear with respect to P and T , as in lineardynamo theory, but nonlinear in δ.

4Multiplicative terms, e.g. αδ that appear in the expansion are small compared to α because (αδ)rms ≤αrmsδrms = αrmsD ≪ αrms (see section 2.2.2 for δrms).

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where δd(τ) is the Dirac delta-function. This correlation function is not identical to the oneused in the simulations.1 However, the dimensionless correlation time τc is very small comparedto the two other timescales in the problem (the diffusion timescale, which is 1 and the dynamoperiod 2π/ω ≈ 2π). In that case the exact shape of the correlation function 〈α(τ)α(τ ′)〉 isimmaterial for the outcome of our calculation and then model (2.38) is very handy in analyticalwork. We can now in a straightforward manner calculate the three independent components of〈uu〉, i.e. 〈δ2〉, 〈δǫ〉 and 〈ǫ2〉, by inserting Eqs. (2.36–2.37). These calculations are of a type thatappears several times in this paper. It is therefore useful to first derive a general expression,that we shall apply wherever needed. Consider the integrals I(τ) =

∫∞0 dσA(σ)α(τ − σ) and

J(τ) =∫∞0 dσB(σ)α(τ − σ). Applying Eq. (2.38) and then changing the integration variables

to σ1 = σ − σ′ and σ2 = (σ + σ′)/2, we find

〈IJ〉 =

∫ ∞

0

∫ ∞

0dσdσ′A(σ)B(σ′)〈α(τ − σ)α(τ − σ′)〉

= 8D

∫ ∞

0dσA(σ)B(σ) (2.39)

Applying this method to 〈uu〉, where δ and ǫ are given by Eqs. (2.36) and (2.37), we obtain

〈uu〉 = D

(

1 −1−1 3

)

. (2.40)

From Eqs. (2.36–2.37) and (2.38) we also obtain

〈αu〉 = 4Df . (2.41)

In order to clarify the physical meaning of the averages, consider again the linearised equa-tion (2.34). Random forcing leads to rapid excursions in the δǫ-plane parallel to f , i.e. on curvesδ + ǫ = constant. Therefore if by random forcing δ increases, ǫ must decrease and vice versa.This explains for example that 〈δǫ〉 = 〈u1u2〉 = −D is negative. Similarly, Eq. (2.34) showsthat if α is positive, δ must decrease and ǫ must increase; hence 〈αδ〉 = −4D is negative and〈αǫ〉 = 4D is positive (Eq. 2.41).

2.2.2.3 Phase-amplitude correlation

Figure 2.3 shows that phase and amplitude variations of the toroidal mean field are correlated.Suppose that the toroidal field |Bt| has maxima at t = ti (i = 0, 1, ..) and that ψi = ψ(ti) andTi = T (ti). The phase-amplitude correlation predicts that if log(Ti+1/Ti) > 0, then on averageδψi+1 − δψi < 0. Since ψi+1 − ψi = −π, we derive from Eq. (2.9) that the corresponding timeinterval is given by ti+1 − ti = [π + δψi+1 − δψi]/ω. The physical meaning of the correlation isthus that an increase in cycle amplitude corresponds, on average, to a decrease in cycle duration.In order to explain this feature, we add Eqs. (2.31) and (2.33):

d

dτ(log T + δψ) = 2δ. (2.42)

Note that the ǫ-terms have cancelled out; we will return to this point later. Equation (2.42)describes motion perpendicular to the line log T + δψ = 0. We integrate over time from σ = 0to σ = τ , i.e.

log(T/T0) + δψ − δψ0 = 2

∫ τ

0dσ δ(σ). (2.43)

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2.2. Model without α-quenching

and we assume that the initial conditions were imposed at a time τ0 far in the past so thatthey have already faded out at σ = 0. In that case δ is given by Eq. (2.36). The nature ofthe correlation between log T and δψ is determined by the statistical properties of the r.h.s. ofEq. (2.43). Its mean value is zero, because 〈δ〉 = 0. The next step would be to calculate ther.m.s. value of Eq. (2.43).

Suppose for the sake of argument that δ is the velocity of a particle, moving in one dimension.Then the ’velocity auto correlation function’ can be derived from Eq. (2.36) in a manner verysimilar to the derivation of Eq. (2.39):

〈δ(τ)δ(τ + σ)〉 = De−2|σ|(cos 2|σ| − sin 2|σ|). (2.44)

It has the rather unusual property (for a velocity auto correlation function) that it is period-ically positive and negative. The oscillations are caused by the periodic behaviour of δ itself(see Eq. 2.34). As will be shown next, Eq. (2.44) has another peculiarity with an importantconsequence. From the velocity auto correlation function we calculate a diffusion coefficient,using the Green-Kubo formula (Kubo 1957). In the present case, we find that it vanishes:

∫ ∞

0dσ 〈δ(τ)δ(τ + σ)〉 = 0. (2.45)

The subsequent intervals of positive and negative correlations lead to exact cancellation of theassociated diffusion coefficient. Therefore, returning now to Eqs. (2.42) and (2.43), it followsthat the motion perpendicular to the line log T + δψ = constant is non-diffusive. Indeed, weobtain from Eq. (2.44) that5

∫ ∞

0

∫ ∞

0dσ dσ′ 〈δδ′〉 = D/4. (2.46)

Consequently, the r.m.s. value of Eq. (2.43) is constant for τ → ∞:6

[log(T/T0) + δψ − δψ0]rms =√D. (2.47)

By contrast, a similar calculation for ǫ (see Eq. 2.37) yields

〈ǫ(τ)ǫ(τ + σ)〉 = De−2|σ|(3 cos 2|σ| + sin 2|σ|), (2.48)

with a corresponding diffusion coefficient

∫ ∞

0dσ 〈ǫ(τ)ǫ(τ + σ)〉 = D. (2.49)

Due to the ǫ-terms in Eqs. (2.31) and (2.33) log T and δψ perform correlated random walks, suchthat there is diffusive behaviour parallel to the line log T +δψ = constant, but not perpendicularto it. This may seem surprising with Eq. (2.42) in mind because δ fluctuates in a random fashionjust like ǫ does. We have shown however that their statistical properties are very different.

5Here δ stands for δ(σ) and δ′ for δ(σ′). This notation is adopted throughout the paper also for other variables.6Strictly speaking, this result is valid only for φ0 = π/4. If φ0 6= π/4, Hoyng (1993) showed that there is in

Eq. (2.43) an extra term ∝ √τ , which in our case (φ0 = 0.7853) becomes important at τ >

∼ [32(π/4 − φ0)2]−1 ≈

3.2 · 106. For the Sun (Ps/Ts ≈ 0.1, Eq. 2.21) it may become important at τ ≈ 1.3 × 103. We ignore this effect.

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Figure 2.3: Logarithmic amplitude log(T/T0) versus phase deviation δψ′ − δψ′0 of the toroidal

field, obtained from a numerical integration of Eqs. (2.23–2.26) between ωτ/2π = 0 and ωτ/2π =80 for φ0 = 0.7853, τc = 0.0238 and αrms = 0.58 (D = 2× 10−3). Here δψ′ is linearly detrended;see section 2.3 for an explanation. Note the correlation between phase- and amplitude variations.The solution performs a random walk parallel to the correlation line log T + δψ′ = 0. The r.m.s.displacement perpendicular to this line is constant, as indicated by the dotted lines (Eq. 2.47):the correlation persists forever.

2.2.3 Dynamo frequency decrease

A striking effect that can be observed in the solutions of Eqs. (2.23–2.26) is a decrease in theeffective dynamo frequency with increasing strength of the forcing (Fig. 2.2). This frequencychange results from a trend ∝ τ in the phase variables δψ and δχ. In the linearised equa-tions (2.30–2.33), such a trend cannot occur; for an explanation we must turn to the originalequations (2.23–2.26). In Appendix B we give a derivation of this trend for a more general modelwith α-quenching; this is achieved by making a second order expansion and estimating the meanvalue of all the terms, using results from the linearised equations. In the strong forcing case(D >

∼ 0.05), which we do not treat here, this procedure is no longer adequate. Here we applythese results to define effective detrended phase functions and an effective dynamo frequency.In Appendix B (q = 0, no α-quenching) we derive:

〈 ddτ log P 〉 = 〈 ddτ log T 〉 = 0, (2.50)

〈 ddτ δχ〉 = 〈 ddτ δψ〉 = 2D. (2.51)

It follows that, on average, the amplitude of the magnetic field does not grow but the phases ofits two components show a trend ∼ 2Dτ relative to the stationary unperturbed solution. Thistrend can be absorbed by defining effective phase variables and an effective dynamo frequency(see Eqs. 2.8–2.9), i.e.

δχ′ = δχ− 2Dτ, (2.52)

δψ′ = δψ − 2Dτ, (2.53)

ω′ = ω − 2D. (2.54)

In other words, the α-fluctuations effectively decrease the dynamo frequency. This effect maybe compared with the energy shift in the theory of scattering.

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2.3. Model with α-quenching

Figure 2.4: Sketch of the quenching function h as a function of the toroidal magnetic fieldamplitude T . For small excursions from the equilibrium value Te, the quenching function h canbe approximated by a linear function of T (the dashed line).

The existence of a trend in the phase also affects the phase-amplitude correlation that wasfound in section 2.2.2, because we now have log T + δψ = 2Dτ + constant, i.e. log T + δψ′ =constant. This modified correlation is confirmed by Fig. 2.3.

2.3 Model with α-quenching

2.3.1 Introduction

The dynamo equation investigated in the previous sections is linear with respect to the magneticfield. This is apparent from the fact that a closed set of equations was derived from it that containthe magnetic field only through P/T (Eqs. 2.27, 2.28). In our model the absence of a bound onthe magnetic field is clearly visible in the unrestricted random walk of log T . This unsatisfyingproperty results from ignoring the feedback of the magnetic field on the velocity field by theLorentz force. Being proportional to |B|2 it would introduce a nonlinearity into the dynamoequation that limits the field growth. Obviously the evaluation of this force would necessitatefull dynamic calculations of the dynamo. Instead we shall adopt a model and assume that αdepends on the toroidal field T (the strongest component as T ≫ P in the αω-approximation):

α(τ) = α0[h(T (τ)) + α(τ)], (2.55)

replacing the previous expression (2.10). It is unknown how the magnetic field precisely couplesto the α-parameter, but the essence of the mechanism should be as is indicated in Fig. 2.4. Thetoroidal magnetic field amplitude is stabilised at T = Te if α is equal to some reference valueα0. In the calculations, the equilibrium toroidal field is conveniently chosen to be Te = 1. Anassumption of the type (2.55) has been made by many authors (e.g. Stix 1972, Rudiger 1973,Brandenburg et al. 1989, Schmitt et al. 1989). For values of T close to the equilibrium value Te,the quenching function h can be approximated by a straight line

h(T ) =

1 − q(T/Te − 1) (T/Te < 1 + 1/q)0 (T/Te > 1 + 1/q)

, (2.56)

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where the constant q can be defined as q = −(Te/α0)(∂α/∂T )|Te . Even though T stays near Te,there will be occasional brief but large jumps in T and to prevent h from becoming negative,it is cut off at T = Te(1 + 1/q). We shall investigate our model with α-quenching in the weakand intermediate forcing cases, using the same approach as in section 2.2 for the model withoutquenching.

2.3.2 Equations

The equations with α-quenching are easily derived from those without quenching by means ofthe substitution α → α − qη. This quick method is applied throughout section 2.3. Instead ofEqs. (2.23-2.26) we now have

ddτ logP = −1 +

√2(h+ α)

1

ρcos(π/4 + δ), (2.57)

ddτ log T = −1 +

√2ρ sin(π/4 + δ), (2.58)

ddτ δχ = 1 −

√2(h+ α)

1

ρsin(π/4 + δ), (2.59)

ddτ δψ = 1 −

√2ρ cos(π/4 + δ). (2.60)

In Fig. 2.5 two simulations are shown for weak forcing and q = 0.7. From Eqs. (2.57–2.60),(2.15) and (2.22) we derive equations for δ and ρ, analogous to Eqs. (2.27–2.28):

ρδ = −√

2(h+ α) sin(π/4 + δ) +√

2ρ2 cos(π/4 + δ), (2.61)

ρ =√

2(h+ α) cos(π/4 + δ) −√

2ρ2 sin(π/4 + δ). (2.62)

Due to the field dependent quenching term h, a closed set of equations is obtained only ifEq. (2.58) is included. Most of the following analysis is based on these three equations andproceeds along lines similar to sections 2.2.2 and 2.2.3.

2.3.3 Weak forcing

2.3.3.1 Linearised equations and solutions

In the weak forcing case (D <∼ 0.05), Eqs. (2.58) and (2.61–2.62) can be expanded in the neigh-

bourhood of the stationary unperturbed solution ρ = 1, δ = 0 and T = Te = 1. For that purposewe introduce

T = 1 + η. (2.63)

The following analysis is very similar to that in section 2.2.2. From Eqs. (2.30–2.33) we obtain

ddτ logP = −δ − ǫ− qη + α, (2.64)ddτ log T = δ + ǫ, (2.65)

ddτ δχ = −δ + ǫ+ qη − α, (2.66)ddτ δψ = δ − ǫ, (2.67)

If we combine Eqs. (2.15), (2.22) and (2.64–2.67), and use η = (1 + η) ddτ log T , we find thefollowing equation for v:

v = Nv + αg, v =

δǫη

, (2.68)

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2.3. Model with α-quenching

Figure 2.5: Two simulations of Eqs. (2.57–2.60) with φ0 = 0.7853, τc = 0.0238 and q = 0.7.Plotted is the toroidal magnetic field |Bt| at a fixed radial position x and colatitude z in arbitraryunits as a function of time. The time unit is the dynamo period of the unperturbed solution.Apart from the quenching, all parameters are the same as in Fig. 2.2. Notice the reducedamplitude variations compared to Fig. 2.2. a) αrms = 0.58, D = 2 × 10−3. b) αrms = 2.25,D = 0.03. This figure is representative for the Sun (see section 2.4). Notice that in b) thedynamo period is effectively longer than in a); see section 2.3.4 for an explanation of this effect.

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where

N(q) =

−2 2 q−2 −2 −q

1 1 0

and g =

−110

. (2.69)

The eigenvalues of N satisfy the equation

ξ3 + 4ξ2 + 8ξ + 4q = 0. (2.70)

For 0 < q < 8 the eigenvalues ξ have negative real parts, signifying that the zero-solution v = 0is an attractor. In that case we can write the solution of Eq. (2.68) as

v(τ) =

∫ ∞

0dσ α(τ − σ)V−1eΞσV g. (2.71)

Here we have written eNσ = V−1eΞσV where Ξ = VNV−1 is a diagonal matrix containing thethree eigenvalues ξi of N; the transformation matrix V−1 has the corresponding eigenvectors asits columns. At q = 8 a bifurcation occurs and for q > 8 two eigenvalues of N have positive realparts; the zero-solution v = 0 becomes unstable and Eq. (2.71) diverges. At this point nonlinearoscillations set in and our linearised equations (2.64–2.68) break down. We shall come back tothese oscillations in section 2.5.

2.3.3.2 Statistical properties

From Eqs. (2.71), (2.38) and (2.39) we obtain the following tensor of correlation coefficients:

〈vivj〉 = −8D∑

klmn

(V−1)ik(V−1)jm

VklglgnVmnξm + ξk

. (2.72)

The explicit expressions for V, V−1, and ξi are to be inserted into Eq. (2.72). After carrying outthe summations a simple result is obtained:7

〈vv〉 =D

1 − q/8

1 −1 1/2−1 3 −1/21/2 −1/2 2/q

. (2.73)

The main effect of quenching, namely to bound the magnetic field amplitude, is clearly visiblein 〈η2〉 = 〈v2

3〉 = 2D/q(1 − q/8), which is finite for 0 < q < 8. Without quenching log T (orη) performs an unrestricted random walk (section 2.2.2), which explains the divergence of 〈η2〉for q = 0. The overall factor 1/(1 − q/8) in 〈vv〉 diverges on approaching the bifurcation atq = 8. This indicates that α-quenching slows down the relaxation towards the marginally stableunperturbed solution and thereby increases the cumulative effect of fluctuations. Finally wenote that from Eqs. (2.71) and (2.38) we obtain

〈vα〉 = 8Dg, (2.74)

independent of q.

7Since V and ξi are rather complicated functions of q this result is not obvious and could only be obtainedwith the help of computer algebra.

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2.3. Model with α-quenching

2.3.3.3 Phase-amplitude correlation

Without α-quenching, δψ and log T both performed unrestricted, correlated random walks. Withα-quenching however, T is forced to remain close to Te, thereby inhibiting the diffusive behaviourof log T . The question is now whether the quenching also inhibits the diffusive behaviour of δψand whether it affects the correlation between δψ and log T . To answer this question we integrateEqs. (2.65) and (2.67) from 0 to τ :

δψ − δψ0 =

∫ τ

0dσ (δ − ǫ), (2.75)

log(T/T0) =

∫ τ

0dσ (δ + ǫ). (2.76)

Next we calculate the r.m.s. value of these expressions, and of their sum. This envolves compli-cated manipulations (Appendix A), and we only state the final result:

[δψ − δψ0]rms =

D(4 + q)

q(1 − q/8), (2.77)

[log(T/T0)]rms =

4D

q(1 − q/8), (2.78)

[log(T/T0) + δψ − δψ0]rms =

D

1 − q/8. (2.79)

Thus for small amplitude fluctuations the analysis shows that α-quenching limits the amplitudeas well as the phase fluctuations of the toroidal field, while maintaining the phase-amplitudecorrelation (see Fig. 2.6). Equation (2.78) is equivalent to the expression for 〈η2〉 = 〈v2

3〉 fromEq. (2.73). This is proven as follows: log T ≈ η and 〈η(τ)η(0)〉 = 0 for τ ≫ 1, so that〈[log T − log T0]

2〉 ≈ 2〈η2〉. If there is no α-quenching (q = 0) then Eqs. (2.77) and (2.78) divergebecause δψ and log T are not restricted, while Eq. (2.79) becomes equivalent to Eq. (2.46) (seealso Eq. 2.94).

2.3.4 Dynamo frequency decrease

The change in the effective dynamo frequency that can be observed in Fig. 2.5 is equivalentto a trend ∝ τ in the phase variables δψ and δχ. A derivation of this trend is presented inAppendix B; here we apply these results to define effective detrended phase functions and aneffective dynamo frequency:

δψ′ = δψ − 2Dτ

1 − q/8, (2.80)

δχ′ = δχ− 2Dτ

1 − q/8, (2.81)

ω′ = ω − 2D

1 − q/8. (2.82)

The detrended phase δψ′ is correlated with log T , since log T + δψ = 2Dτ/(1− q/8)+ constant,i.e. log T + δψ′ = constant. This modified correlation is confirmed by Fig. 2.6.

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Figure 2.6: Logarithmic amplitude log(T/T0) versus phase deviation δψ′ − δψ′0 of the toroidal

field, obtained from numerical integration of Eqs. (2.57-2.60) between ωτ/2π = 0 and ωτ/2π =80 with φ0 = 0.7853, τc = 0.0238, αrms = 0.58 (D = 2 × 10−3) and q = 0.7. Here δψ′ is thedetrended phase of the toroidal field; see section 2.3.4 for an explanation. Comparison withFig. 2.3 shows the effect of α-quenching on the amplitude variations. Both log T and δψ′ arelimited by the α-quenching. Dotted lines indicate the r.m.s. values predicted by Eq. (2.79).

2.4 The sun

For a comparison between our model and the solar cycle, we consider the sunspot numbers, whichare thought to be an indicator of the dominantly toroidal magnetic field in the dynamo layer.From the sunspot maxima Ri (i = 0, 1, ..26) and their epochs ti, we estimate the logarithmicamplitude log T (ti) and phase δψ(ti) of the toroidal field. We employ the same method asHoyng (1993) but have added two more data points, for the maxima in 1979 and 1989. Here weonly repeat the essential steps. The phase lags for all the maxima are given by

δψi − δψ0 = ω′(ti − t0) − iπ (2.83)

where δψi = δψ(ti). We assume that the toroidal magnetic field and the sunspot numbers arerelated through a power law, i.e.

log(Ti/T0) = µ log(Ri/R0) (2.84)

where Ti = T (ti) and Ri = R(ti). Expressions (2.83) and (2.84) are fitted to Fi = log(Ti/T0) +δψi− δψ0 − ζ by minimizing 〈F 2〉 = (N − 3)−1∑

i F2i with respect to ω′, µ and an offset ζ. The

best fit has µ = 1.1, ω′ = 0.28 yr−1, ζ = 0.3 and 〈F 2〉 = 0.18, i.e. Frms = 0.42 (see Fig. 2.7).

In theory we can use Frms as an estimate for Eq. (2.79). A quantitative comparison is howevercomplicated by two factors. First, the number of data in our sample is too small for an accurateestimate of expression (2.79). Second, our model considers only phase and amplitude variationsin a single dynamo mode. In fact, overtones of the fundamental dynamo mode may also beexcited and thus contribute to the variability in the sunspot number. Therefore, the variabilityof the single mode in the present model is likely to be somewhat less than sunspot variability.

For these reasons we shall be satisfied to obtain qualitative agreement with the observedphase-amplitude correlation, and we shall do so in two steps. First it will be demonstrated

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2.5. The nonlinear regime without α-fluctuations

Figure 2.7: Logarithmic amplitude log(Ti/T0) = µ log(Ri/R0) versus phase deviation δψi− δψ0,derived from the maximum sunspot numbers Ri and their epochs ti (see text). The best fit(µ log(Ri/R0)+ δψi− δψ0 = ζ, dashed line) has µ = 1.1, ω′ = 0.28 yr−1 and ζ = 0.3. The pointshave been connected by straight lines as to indicate their temporal order.

that a model without α-quenching does not agree with the observations; next, that agreementis much better with α-quenching.

In Fig. 2.8 we show two timeseries from numerical simulations of Eqs. (2.57–2.60), for D =0.03 (αrms = 2.25, τc = 0.0238; see also Figs 2.2b and 2.5b). The 27 consecutive points in thesetimeseries were sampled in the same way as the solar data in Fig. 2.7. Without α-quenching(Fig. 2.8a) the phase and logarithmic amplitude of the toroidal field perform an unrestrictedrandom walk parallel to the line log T + δψ′ =constant. The r.m.s. deviation from this line isindicated by the two dotted lines (Eq. 2.79). For D = 0.03, the spread around the correlationline is comparable to that in the solar data (Fig. 2.7). It is in fact somewhat smaller, which is nota serious problem, for the reasons just mentioned, but what is striking is that the displacementparallel to the correlation line is much larger than in the solar data. This discrepancy wouldbe even greater for larger values of D, and demonstrates that α-quenching must be envoked tolimit the phase and amplitude variations.

The value of q that is required to obtain best agreement with Fig. 2.7 is estimated as follows.The toroidal field amplitude has an equilibrium value Te/T0 = (Re/R0)

µ, and we estimate Rµe =1N

iRµi = 110. With this value we then estimate: [log(T/Te)]rms = µ[

i[log(Ri/Re)]2/N ]1/2 ∼

0.44. For D = 0.03 we obtain q = 0.7 (see Eq. 2.78; we discarded q = 7.3). Figure 2.8b shows atime series with q = 0.7. The similarity with solar data is very good.

2.5 The nonlinear regime without α-fluctuations

In the previous section it was shown that a simple mean field dynamo model with stochasticfluctuations in the α-parameter and α-quenching to limit the growth of the magnetic field, iscapable of reproducing the observed phase-amplitude correlation of the solar cycle. An interest-ing question is whether the nonlinearity that was introduced through α-quenching is capable byitself, without stochastic forcing, of producing variability with a similar correlation.

To answer this question, we look for solutions to Eqs. (2.57–2.60) that exhibit nonlinearoscillatory behaviour. Such solutions can develop if the stationary solution δ = ǫ = η = 0 is

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Figure 2.8: Logarithmic amplitude log(T/T0) versus phase deviation δψ′ − δψ′0 of the toroidal

field with φ0 = 0.7853 and D = 0.03 (αrms = 2.25, τc = 0.0238), obtained from numericalsimulations of Eqs. (2.57–2.60). We included in these plots only 27 consecutive points thatcorrespond to maxima of |Bt| (see Figs. 2.2b and 2.5b), so that the resulting timeseries aresampled in the same way as Fig. 2.7. a) without quenching: q = 0; b) with quenching: q = 0.7.

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2.6. Discussion

Figure 2.9: Numerical integration of Eqs. (2.57-2.60) with α = 0, φ0 = 0.7853 and q = 12.Plotted is the toroidal magnetic field |Bt| at a fixed radial position x and colatitude z in arbitraryunits as a function of time. Random forcing has been switched off; the irregular behaviour iscaused by the nonlinearity of the quenching term.

unstable, which is the case for q > 8. Figure 2.9 shows an example for q = 12. The correspondingphase-amplitude diagram is shown in Fig. 2.10. Here we detrended the phase variable accordingto δψ′ = δψ − Bτ , where B = 0.1529, and we sampled the solution in the same manner asFig. 2.7. The points of the timeseries lay on a roughly torus-shaped object, which does notsuggest a linear correlation between δψ′ and log T , and they are ordered in a regular way, incontrast with the observations (Fig. 2.7). We have not considered other nonlinear systems, buttake this as an indication that it may be difficult to find a nonlinear dynamo equation thatexhibits a realistic phase-amplitude correlation.

2.6 Discussion

We have studied the effect of time dependent fluctuations in α on a marginally stable planedynamo wave in the αω-limit. All our numerical calculations were carried out for φ0 = 0.7853,i.e. for Ps/Ts = 0.014 (Eqs. 2.18–2.21), but in the αω-limit, the results do not depend on φ0. Ascan be seen from Eqs. (2.23–2.28), a different value for φ0 affects only the ratio P/T , not theother variables. Thus our results are generally applicable within the αω-limit, and in particularto the Sun, for which we may take Ps/Ts ≈ 0.1, i.e. π/4 − φ ≈ 5 × 10−3, dα = 0.14 anddω = 14 (Eqs. 2.18–2.21). For the remaining parameters we adopt the following values. Thedynamo layer has a thickness L = 2 × 109 cm. The latitudinal wavenumber is kz ≈ 2π/λz ≈1.3 × 10−10 cm−1. Here the latitudinal wavelength is λz ≈ 5 × 1010 cm, which correspondsto a strip of about 60o at the base of the convection zone, centered at the equator. It followsthat k =

k2z + (π/L)2 = 1.7 × 10−9 cm−1, and from Eq. (2.16) we obtain α0 = 1.7 cm/s and

β0 = 7.3 × 109 cm2/s.The main feature of our model is a correlation between the (detrended) phase δψ′ and the

logarithmic amplitude log T of the toroidal field. The existence of a phase-amplitude correlationis confirmed by observations. A qualitative comparison between our model and the sunspot datasuggests that D = 0.03 and q = 0.7. If we identify the convective cells with ’giant cells’ (theyare the most likely candidate for the origin of variability in the solar cycle, see Ossendrijver et

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Figure 2.10: Logarithmic amplitude log(T/T0) versus phase deviation δψ′ − δψ′0 of the toroidal

field, with φ0 = 0.7853 and q = 12, corresponding to the numerical solution shown in Fig. 2.9.The solution was sampled in the same manner as Figs. 2.7 and 2.8.

al. 1996b), then the correlation time of the fluctuations is τc = 0.0238, i.e. tc = 1 month. Itfollows that the required strength of the α-fluctuations is δα/α0 ≈ αrms = 2

√D/τc = 2.2. A

similar value (δα/α0 ≈ 3) was found by Ossendrijver et al. (1996b).

Since α is related to the azimuthal average of the kinetic helicity through the expressionα = −1

3〈h〉tc, large helicity variations must underly the α-fluctuations. Their required sizerelative to the mean helicity depends on the number of giant cells that take part in the averagingprocedure. This number can be expressed as 2Nc sin θ, where θ is the colatitude and Nc is thenumber of giant cells from pole to pole. A simple estimate yields Nc ≈ πR/λG ≈ 20, whereR = 6 × 105 km is distance from the origin to the middle of the convection zone, and λG ≈ 105

km is the typical size of a giant cell. It follows that δh/h0 is larger than δα/α0 by a factor√2Nc sin θ ≈ 6, i.e. δh/h0 ≈ 13 for θ = 60o. The fluctuations in the helicity thus have to be

strong relative to its mean value. It is unknown at the moment whether they can be explainedby convection models.

A value of q = 0.7 implies that the α-effect is effectively shut off (i.e. h = 0, see Eq. 2.56) ifT >

∼ (1+1/q)Te ≈ 2.4Te. This occurs when the sunspot number R exceeds the value (2.4)1/µRe ≈240 (Eq. 2.84).

Our model predicts that log T + δψ′ is confined to a small interval of O(√D), so that phase

and (logarithmic) amplitude of the toroidal field are locked to one another. This would suggestthat there is a clock hidden in the Sun, as claimed by Dicke (1978). At present, the availablesunspot data may be insufficient to confirm this with certainty. Other calculations, done in aspherical geometry, where δα is a stochastic function of time and latitude, show that phase andlogarithmic amplitude perform a random walk with a fast component parallel to the correlationline and a slow component perpendicular to it (Ossendrijver et al. 1996b). Thus the narrowconfinement of log T + δψ′ may be a consequence of the assumption that δα is a stochasticfunction of time only.

The physical mechanism behind the phase-amplitude correlation is easily demonstrated byconsidering a Fourier analysis of plane dynamo waves. These waves have a frequency and a

48

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2.7. Appendix A: Statistical properties of the phase and amplitude of the toroidal field

growth rate given by (Moffatt 1978):

Ω =√

|αks|/2; Γ = −βk2 +√

|αks|/2, (2.85)

where k is the wavevector, s = ey ·(k×a)/k and a = ∇u0. The toroidal component of the dynamowave is of the form Bt ∝ Re eΓt+iψ, where ψ = k ·r−Ωt is the phase, and log |Bt| = Γt+constantis the logarithmic amplitude of the dynamo wave. Suppose that for α = α0 the dynamo waveis marginally stable (Γ = 0) and that Ω = Ω0. If there is a constant perturbation δα = α − α0

then the resulting changes in dynamo frequency and growth rate are identical, i.e. δΩ = δΓ.It follows that log |Bt| + ψ is invariant with respect to variations in α that are slow comparedto the dynamo period. Our analysis reveals that the phase-amplitude correlation also holds forrapid fluctuations in α.

There is some evidence that the anti-correlation of cycle length and amplitude is a generalfeature of solar-type stars, see Soon et al. (1994). A quantitative comparison with our phase-amplitude correlation is difficult because these authors employ the ratio of cycle period torotation period rather than the phase deviation δψ− δψ0. Evenso, their result suggests that thesame mechanism is responsible for period and amplitude variations in the solar cycle and forthe variations from one solar type star to another.

We have found that the α-fluctuations give rise to a decrease in the effective (dimensionless)dynamo frequency. We cannot give a simple physical explanation for this effect. It may beanalogous to the energyshift in the theory of scattering. For the Sun this decrease of thedimensionless dynamo frequency amounts to 2D(1 − q/8) ≈ 7%.

Finally we have briefly studied the alternative explanation of variability, namely nonlinearoscillations. We cannot rule out the possibility that a nonlinear model exhibits a similar phase-amplitude correlation. However, our model with α-quenching but without α-fluctuations failedto do so in the nonlinear regime (section 2.5), and no other nonlinear model, succesful in thisrespect, is known to us. We suggest that the observed phase-amplitude correlation be employedto constrain nonlinear dynamo models. Our conclusion is that the most natural explanation forthe variability in the solar cycle is fluctuations in the turbulence.

Acknowledgements

The authors want to thank prof. H. van Beijeren for the numerous stimulating discussions. Thisresearch was supported by the Dutch Foundation for Astronomical Research (ASTRON).

2.7 Appendix A: Statistical properties of the phase and ampli-

tude of the toroidal field

The main purpose of this appendix is to derive the estimates (2.77–2.79). Starting point for ourcalculation are Eqs. (2.75–2.76). Their r.m.s. values are5

〈[δψ − δψ0]2〉 =

∫ τ

0

∫ τ

0dσ dσ′ 〈δδ′ − 2δǫ′ + ǫǫ′〉, (2.86)

〈[log(T/T0)]2〉 =

∫ τ

0

∫ τ

0dσ dσ′ 〈δδ′ + 2δǫ′ + ǫǫ′〉, (2.87)

〈[log(T/T0) + δψ − δψ0]2〉 = 4

∫ τ

0

∫ τ

0dσ dσ′ 〈δδ′〉. (2.88)

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STOCHASTIC AND NONLINEAR FLUCTUATIONS IN A MEAN FIELD DYNAMO

The integrands of Eqs. (2.86–2.88) are elements of the matrix of correlation coefficients 〈v(τ)v(τ+σ)〉. We first work out these, assuming σ > 0. With Eq. (2.38) we find from Eq. (2.71)

〈vi(τ)vj(τ + σ)〉 = 8D∑

kl

∫ τ

0

∫ τ+σ

0dσ′ dσ′′ Ωijkle

ξkσ′+ξlσ

′′

δd(σ + σ′ − σ′′), (2.89)

with

Ωijkl =∑

mn

(V−1)il(V−1)jkVlmgmgnVkn. (2.90)

To evaluate this integral we make a change of variables to σ1 = (σ′ + σ′′)/2 and σ2 = σ′ − σ′′.The integration over σ2 becomes trivial because of the Dirac delta function. There remains

〈vi(τ)vj(τ + σ)〉 = 8D∑

kl

Ωijkl e(ξk−ξl)σ/2

∫ τ+σ/2

σ/2dσ1 e(ξk+ξl)σ1

= −8D∑

kl

Ωijkl

ξk + ξleξkσ (σ > 0) (2.91)

where we used τ ≫ 1 to derive the last equality. Notice that for σ = 0 Eq. (2.72) is restored.Returning to Eqs. (2.86–2.88), proceed to integrate over σ and σ′. This integral splits up intotwo parts corresponding to σ < σ′ and σ > σ′ respectively, i.e.

∫ τ

0dσ

∫ τ

0dσ′ 〈viv′j〉 = −8D

kl

∫ τ

0dσ

∫ σ

0dσ′

Ωjikl

ξk + ξleξk(σ−σ′)

+ 8D∑

kl

∫ τ

0dσ

∫ τ

σdσ′

Ωijkl

ξk + ξleξk(σ′−σ) (2.92)

= 8D∑

kl

Ωijkl + Ωjikl

ξk + ξl(τ/ξk + 1/ξ2k). (2.93)

The summands are complicated functions of q (see Eqs. 2.70 and 2.90) but a simple expressionis obtained if the summation is carried out.8 The matrix elements that are relevant here are

∫ τ

0

∫ τ

0dσ dσ′ 〈δδ′〉 =

D/4

1 − q/8, (2.94)

∫ τ

0

∫ τ

0dσ dσ′ 〈δǫ′〉 = − D/4

1 − q/8, (2.95)

∫ τ

0

∫ τ

0dσ dσ′ 〈ǫǫ′〉 =

D(4 + q/4)

q(1 − q/8). (2.96)

Note that none of these contain a term proportional to τ (see Eq. 2.93); they turn out to bezero. Inserting Eqs. (2.94–2.96) into Eqs. (2.86–2.88) we finally have

〈[δψ − δψ0]2〉 =

D(4 + q)

q(1 − q/8), (2.97)

〈[log(T/T0)]2〉 =

4D

q(1 − q/8), (2.98)

〈[log(T/T0) + δψ − δψ0]2〉 =

D

1 − q/8. (2.99)

8This calculation was carried out using computer algebra.

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2.8. Appendix B: Derivation of the dynamo frequency decrease

2.8 Appendix B: Derivation of the dynamo frequency decrease

The derivation of the dynamo frequency decrease presented here is for the general case with α-quenching, but the end result is also valid for q = 0. A dynamo frequency change is equivalent toa trend ∝ τ in the phase variables δψ and δχ. To explain this trend, we expand Eqs. (2.57–2.60)and (2.27–2.28) up to quadratic order, neglecting all higher order terms. The following analysisis therefore only valid if forcing is weak, say D <

∼ 0.05. First we approximate 1/ρ ≈ 1 − ǫ+ 2ǫ2,cos(π/4 + δ) ≈ (1 − δ − 1

2δ2)/

√2 and sin(π/4 + δ) ≈ (1 + δ − 1

2δ2)/

√2. Passing by details we

obtain

ddτ logP = −δ − ǫ− qη + α− 1

2δ2 + ǫ2 + δǫ− αδ − αǫ+ qǫη + qδη, (2.100)

ddτ log T = δ + ǫ− 1

2δ2 + δǫ, (2.101)

ddτ δχ = −δ + ǫ+ qη − α+ 1

2δ2 − ǫ2 + δǫ− αδ + αǫ+ qδη − qǫη, (2.102)

ddτ δψ = δ − ǫ+ 1

2δ2 + δǫ. (2.103)

The trend in these equations is given by the mean values of the r.h.s. terms. Note that if onlyterms linear in α, δ and ǫ are retained then 〈 ddτ δχ〉 = 〈 ddτ δψ〉 = 0, i.e. no trend remains and thedynamo frequency is independent of the strength of the forcing.

As we shall see, there is also a contribution to the mean value of Eqs. (2.100–2.103) from 〈δ〉and 〈ǫ〉. For that reason we consider the following equation for v, obtained from Eqs. (2.101–2.103) using Eqs. (2.15), (2.22) and (2.29):

v = N(q)v + αg + r, v =

δǫη

, (2.104)

where

g =

−110

and r =

−ǫ2 − αδ + αǫ+ qδη − qǫη−ǫ2 − 2δǫ − αδ + qδη−1

2δ2 + δǫ+ δη + ǫη

. (2.105)

The next step is to average Eq. (2.104). Since η is stabilised at η = 0 by the α-quenching, wehave 〈η〉 = 0. It can also be proven from Eqs. (2.61–2.62) that 〈δ〉 = 〈ǫ〉 = 0, so that 〈v〉 = 0.We thus obtain N〈v〉 + 〈r〉 = 0, from which we can solve 〈v〉:

〈v〉 = −N−1〈r〉, (2.106)

Recall that in the linearised theory of section 2.2 we had 〈u〉 = 0. By taking into account thesecond order terms we find that 〈v〉 is of the same order as 〈r〉, i.e. O(D). But there is alsoour result Eq. (2.40), which says that δrms, ǫrms and ηrms are O(

√D). Apparently v has a small

nonzero average O(D) and it fluctuates with a much larger r.m.s. amplitude O(√D). To find

〈v〉 we may thus evaluate 〈r〉 in Eq. (2.105) with the help of Eqs. (2.73–2.74), derived fromlinear theory. Extracting the relevant coefficients from Eqs. (2.73–2.74) and inserting them intoEq. (2.106) we find

〈v〉 =D

1 − q/8

2−1/2

0

. (2.107)

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STOCHASTIC AND NONLINEAR FLUCTUATIONS IN A MEAN FIELD DYNAMO

Returning to Eqs. (2.100–2.103), we proceed to average the second order terms. To lowest orderin D they are provided by linear theory, i.e. by Eqs. (2.73–2.74) We thus obtain

〈 ddτ log P 〉 = 〈 ddτ log T 〉 = 0, (2.108)

〈 ddτ δχ〉 = 〈 ddτ δψ〉 =2D

1 − q/8. (2.109)

The trend in the phase variables is equivalent to a decrease in the dynamo frequency. There isno trend in the logarithmic amplitudes.

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Chapter 3

Stochastic excitation and memory of

the solar dynamo

AbstractWe consider a simple axisymmetric mean field dynamo model for the Sun in the αΩ-limit and studythe effect of rapid, latitude dependent stochastic fluctuations in α. The fluctuations excite overtonesof the fundamental mode of the mean magnetic field. We decompose the mean field into eigenmodesand derive an equation for the mode coefficients. Transient mode excitation gives rise to a mean fieldwith spatial and temporal variability, and may provide an explanation for grand minima, the observedphase-amplitude correlation, North-South asymmetries and, close to the equator, reverse polarityregions in the solar butterfly diagram. We find that the North-South asymmetry often peaks nearthe activity minimum, in agreement with the observations. The most likely candidate for the originof the fluctuations are giant cells. Sunspot data are well reproduced if α, defined as an azimuthalaverage, has fluctuations δα of the order δα/α0 ≈ 3 at colatitude θ = 60o, assuming that thereare 20 giant cells from pole to pole, with a coherence time of 1 month. The model predicts thatthe resulting phase and amplitude fluctuations of the fundamental mode are correlated for about 90dynamo periods.

A.J.H. Ossendrijver, P. Hoyng and D. Schmitt

Astronomy & Astrophysics (in press)

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STOCHASTIC EXCITATION AND MEMORY OF THE SOLAR DYNAMO

3.1 Introduction

One of the puzzling features of the solar cycle is its temporal and spatial variability. The cycleperiod varies roughly between 9 and 13 years; sunspot numbers show large variations from cycleto cycle and may even all but vanish, as happened during the Maunder minimum. Asymmetriesin the yearly averaged sunspot numbers between the two hemispheres are evidence of spatialvariability in the solar dynamo (Vizoso & Ballester 1990 and Carbonell et al. 1993).

In the simplest linear mean field dynamo theory this behaviour is not accounted for. Solutionsof the dynamo equation,

∂B0

∂t= ∇× [u0 × B0 + αB0 − β∇× B0]. (3.1)

are strictly periodic and produce a series of identical magnetic cycles. Thus the phase of thecycle can be predicted for any future moment if the initial phase is known and there is notemporal or spatial variability. In Eq. (3.1) B0 and u0 are the mean magnetic field and meanflow respectively, α represents the α-effect and β is the turbulent magnetic diffusivity.

These properties come about because α, β and u0 are tacidly assumed to be independent oftime, which is incorrect for two reasons. Firstly, the Lorentz force acts as a feedback mechanismthat makes α, β and u0 functions of B0. Hence they will implicitly depend on time at a givenposition (Weiss et al. 1984 or Schmitt & Schussler 1989).

Here we focus on the second reason, namely the stochastic behaviour of the dynamo itself.Its physical origin is the chaotic velocity field in the convection zone, which consists of risingand sinking gas elements. Within these convective cells the velocity field is coherent. Since theirnumber is finite, spacially averaged quantities retain a fluctuating component. In particular thisis true for α = −1

3〈h〉tc. Here h is the helicity of the turbulent velocity field u1, i.e.

h = u1 · ∇ × u1 (3.2)

and tc is the correlation time of the turbulent motions. We interpret all mean quantities asaverages over longitude φ and concentrate on fluctuations in α (Choudhuri 1992). They arelikely to be more important than fluctuations in β and u0, because h may have different signsin neighbouring cells, which gives rise to cancellations and a possibly large influence of fluctu-ations. These fluctuations introduce a stochastic driving term in Eq. (3.1) that can excite alleigenmodes, and thus can bring about spatial and temporal variability in the magnetic cycle(Moss et al. 1992).

An earlier investigation by Hoyng (1993) had shown that there is a correlation betweenphase and amplitude fluctuations of the magnetic cycle, such that weaker cycles last longerand stronger cycles last shorter. The correlation persists forever in the exact αΩ-limit, so thatphase and amplitude are locked. However, in this study, α was a stochastic function of timeonly, and the resulting phase locking cannot be accepted without further investigation. It couldbe an artefact of the assumption that δα is stochastic function of time, but not of the spatialcoordinates. This gave further motivation to consider a somewhat more sophisticated stochasticmodel of the α-coefficient by including a θ-dependence. In section 3.2 we present our model andequations. Section 3.3 is devoted to a study of the excitation of dynamo modes, and specialattention is paid to the fundamental mode and its phase-amplitude correlation. Finally we shallcompare our model in a qualitative manner with the observed phase-amplitude correlation andNorth-South asymmetries in sunspot data.

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3.2. Dynamo model and equations

3.2 Dynamo model and equations

The geometry of our dynamo model is that of a spherical shell, located at the base of theconvection zone, at a distance R from the origin, cf. Hoyng et al. (1994). For the mean velocityu0 we assume a radial gradient of the angular velocity at the base of the convection zone as issuggested by helioseismic measurements (Goode 1995):

u0 = Ωr sin θ eφ, dΩ/dr = Ω′ = constant. (3.3)

We introduce dimensionless time,

τ = t/td, (3.4)

where td = R2/β is the turbulent diffusion time. All quantities are averaged over φ and thusaxisymmetric. For δα we use a simple stochastic model, based on the assumption that theconvection zone consists of square cells with a surface area of (π/Nc)

2 steradians. Here Nc isthe number of cells from pole to pole. The kinetic helicity has a constant part h0 = hm cos θand a fluctuating part δh = hmf(θ, φ, τ), i.e.

h = h0 + δh = hm(cos θ + f). (3.5)

We assume that h(θ, φ, τ) is spatially constant within a cell and we model f(θ, φ, τ) as a stochasticfunction which is updated once every correlation time τc = tc/td, independently for every cell.Its r.m.s value frms is a measure of the strength of the helicity variations from one cell to another.The effective strength of the fluctuations is measured by a ’diffusion coefficient’ (see section 3.3.3for a motivation of this terminology), defined as

∆ =f2rms

N2c

τc. (3.6)

In section 3.3 we shall demonstrate that various average properties of the magnetic field areexpressed in terms of ∆. A discussion of the parameters Nc, τc and frms is presented in section3.5. We average Eq. (3.5) over φ and obtain the following expression for α = −1

3〈h〉tc:

α = α0 + δα = αm

cos θ +frmsF√2Nc sin θ

. (3.7)

Here α0 denotes the constant part and δα the fluctuating part; F (θ, τ) is a stochastic functionwith 〈F 〉 = 0 and 〈F 2〉 = 1. In our simulations, F is updated once every correlation time,independently for every latitude interval θc = π/Nc.

1 The factor 2Nc sin θ is the number of cellson a ring at colatitude θ.

The magnetic field is separated into a poloidal and a toroidal component:

B0 = ∇×Apeφ +Bteφ. (3.8)

1The corresponding autocorrelation function is 〈F (θ0, τ0)F (θ0 +θ, τ0 +τ )〉 = (1−|θ|/θc)(1−|τ |/τc) for |θ| < θcand |τ | < τc, and zero elsewhere. For analytical calculations, delta-functions are more convenient and we shalluse 〈F (θ0, τ0)F (θ0 +θ, τ0 + τ )〉 = τcθc δ(θ)δ(τ ). Both definitions satisfy

∫ π

0dθ∫

0dτ 〈F (θ0, τ0)F (θ0 +θ, τ0 + τ )〉 =

τcθc/4. The resulting error is small as long as τc is much smaller than all relevant time scales and θc is muchsmaller than all relevant spatial scales.

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The radial dependence of Ap and Bt is assumed to be that of a spherical wave, i.e.

Ap = αmtd Pcos kr

kr, (3.9)

Bt = Tcos kr

kr, (3.10)

where P (θ, τ) and T (θ, τ) are real. We have included in our definition of Ap a factor αmtd, sothat P and T have the same dimension. We substitute Eqs. (3.8–3.10) in Eq. (3.1) and obtainthe following equation in the αΩ-limit:

∂z

∂τ= (D + V) z where z =

(

PT

)

. (3.11)

Here D and V are operators given by

D =

(

L α0/αm

C ∂∂θ sin θ L

)

, V =

(

0 δα/αm

0 0

)

. (3.12)

The dynamo number C is defined as

C = αmΩ′t2d = αmR4Ω′/β2. (3.13)

The operator L contains r as a parameter, for which we choose r = R, yielding

L =1

sin θ

∂θsin θ

∂θ− 1

sin2 θ− (kR)2. (3.14)

The boundary conditions at the poles are

z(0, τ) = z(π, τ) = 0. (3.15)

3.3 Mode excitation

3.3.1 Introduction

Numerical solutions of Eq. (3.11) show that the presence of a stochastic component in α leads totemporal and spatial variability in the magnetic cycle, see Fig. 3.1. This can be understood as aconsequence of the stochastic excitation of eigenmodes. Our aim is to decompose the magneticfield in terms of eigenmodes and find an equation for the mode coefficients. We introduce thefollowing notation for the eigenfunctions and eigenvalues of D:

bi =

(

PiTi

)

, λi = γi + iωi. (3.16)

Since D is real, the complex eigenmodes of D form conjugate pairs. The general solution ofEq. (3.11) without random forcing (V = 0) is z =

i cieλiτbi, where ci are constants. The

modes are ordered according to their growth rate γi, such that γ0 ≥ γ1 ≥ γ2... Two modes thatform a complex conjugate pair have subsequent indexes, the lower index referring to the modewith ωi > 0. The two modes that grow most rapidly are referred to as the fundamental modes;the other modes as overtones of the fundamental mode; e.g. mode 2 is the first overtone, mode4 the second, etc. Our dynamo model is linear, and in order to stabilize the magnetic field, weset the dynamo number C to the critical value C = −1319, at which the fundamental mode is

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3.3. Mode excitation

Figure 3.1: Contour plots of the toroidal field T obtained through numerical integration of Eq.(3.11) with kR = 3, C = −1319, τc = 2.25 × 10−3 = 0.010 dynamo periods, and Nc = 50,for three different values of frms. The middle figure has the highest resemblance with the solarbutterfly diagram, variability being too low in the top figure and too high in the bottom figure.The value kR = 3 was chosen because the simulated butterfly diagrams then have approximatelythe same tilt angle with respect to the equator as in the solar butterfly diagram. Contours areseparated by 1/15 of the maximum of |T | in the plot; no toroidal field less than this threshold isreproduced. Drawn and dashed contours have opposite polarity. Notice the asymmetry betweenthe hemispheres and the transequatorial activity.

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Table 3.1: Dimensionless frequencies and growth rates for kR = 3 and C = −1319. The letters aand s indicate whether the toroidal field of the mode is antisymmetric or symmetric with respectto the equator. Complex conjugate modes are not shown.

i symm. γi ωi

0 a 0.00 27.92 s −2.09 27.64 s −8.94 0.005 a −17.4 26.27 s −28.4 27.49 a −42.1 31.0

11 s −56.3 34.413 a −71.9 37.515 s −89.2 40.417 a −108 42.919 s −130 45.3

marginally stable (γ0 = γ1 = 0). In that case the overtones are damped (e.g. the first overtoneis marginally stable for C = −1577). Eigenmodes have the following symmetry properties: ifTi is antisymmetric, then Pi is symmetric with respect to the equator (and vice versa), becausethe off-diagonal matrix elements of D are antisymmetric with respect to the equator.

In Table 3.1 we give the frequencies and growth rates of the fundamental mode and thefirst ten overtones with ωi ≥ 0, for kR = 3 and C = −1319.2 The fundamental mode has adimensionless period 2π/ω0 = 0.225, which corresponds to 22 years if td = 3.08 × 109 s. Noticethat mode 4 is a non-oscillatory real mode. More details on the choice of the parameters can befound in Hoyng et al. (1994).

3.3.2 Decomposition into eigenmodes

We write the general solution of Eq. (3.11) as a linear superposition of eigenmodes of the unforcedequation, i.e.

z(θ, τ) =∑

i

ai(τ)bi(θ). (3.17)

We combine Eq. (3.17) and (3.11), providing∑

i

aibi =∑

i

ai (λi + V)bi. (3.18)

Our goal is to find an equation for ai. Unfortunately D is not a self-adjoint operator, i.e.(Dbi,bj) 6= (bi,Dbj), where the inner product is defined as (bi,bj) =

∫ π0 dθ sin θ [P ∗

i Pj + T ∗i Tj ].

Consequently the eigenfunctions bi are not orthogonal. However, the adjoint operator D, definedthrough the identity

(x,Dy) = (Dx,y) (3.19)

2The finite grid size and time step used for the simulations required a dynamo number C = −1275 for marginalstability of the fundamental mode, slightly different from the exact value C = −1319.

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3.3. Mode excitation

Figure 3.2: Contour plots of Re Tieiωiτ , the toroidal component of eigenmode i, and its adjoint

counterpart Re Tieiωiτ for i = 0, 2, kR = 3 and C = −1319. Their frequencies and growth rates

are listed in Table 1. The adjoint eigenfunctions are a mathematical tool for decomposing themagnetic field in terms of dynamo modes, see the text. The toroidal components Ti are largerthan the poloidal components Pi roughly by a factor

|C|, the reverse being true for the adjointeigenfunctions.

has eigenfunctions bi that are orthogonal to bj (see Appendix A for a proof). We shall assumethat bi and bj are properly normalised, such that they form a bi-orthonormal set of functions,i.e. (bi, bj) = δij .

It is easily checked through partial integration of the left hand side of Eq. (3.19) that

D =

(

L −C ∂∂θ sin θ

α0/αm L

)

. (3.20)

The adjoint eigenfunctions satisfy

D bi = λ∗i bi. (3.21)

It turns out that bi can be found from the eigenfunctions of D with a dynamo number of theopposite sign by exchanging the poloidal and toroidal components:

bi,C =

(

0 11 0

)

bi,−C . (3.22)

For a proof of Eqs. (3.21–3.22) and more details on the adjoint eigenfunctions see Appendix Aand also Kleeorin & Ruzmaikin (1984) and Kleeorin et al. (1995).

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Figure 3.3: The absolute values of (complex) mode coefficients a0 and a2, in arbitrary units.There are no oscillations, since all information on phases is lost. The coefficients are obtainedthrough an eigenmode decomposition of a numerical solution of Eq. (3.11). The parameters arethe same as in Fig. 3.1b, which represents only the interval t = 600 to 620 dynamo periods.The first overtone is excited through a coupling with the dominant fundamental mode, whichexplains the similarity between |a0| and |a2|. There are long intervals (50 or more dynamoperiods in this example) of low activity, reminiscent of grand minima, such as the Maunderminimum observed on the Sun.

The biorthonormal set of functions provides us with a convenient tool for decomposing themagnetic field into separate modes, by forming the inner product of bi with z and insertingEq. (3.17):

ai = (bi, z). (3.23)

The solutions z of Eq. (3.11) are real at all times and all latitudes, because D and V are real,and because we choose real boundary conditions for z. It follows from Eq. (3.23), that the modecoefficients ai, like the eigenfunctions bi, form complex conjugate pairs, e.g. a1 = a∗0, a3 = a∗2etc. We can thus restrict our calculation of ai to modes with ωi ≥ 0. The mode coefficients obeya simple equation, found by taking the inner product of bi with both sides of Eq. (3.18):

ai = λiai +∑

j

aj(bi,Vbj) ≈ λiai + gia0. (3.24)

Here the coupling coefficient gi is defined as

gi = (bi,Vb0) =

∫ π

0dθ sin θ P ∗

i T0 δα/α0 (3.25)

=frms√2Nc

∫ π

0dθ

√sin θ P ∗

i T0 F (θ, τ). (3.26)

The last expression was derived by inserting Eq. (3.7) into Eq. (3.25). On the r.h.s. of Eq. (3.24)we have made the approximation that the eigenmodes are excited mainly by the fundamentalmode, because it has the largest amplitude. This is illustrated by Fig. 3.3, which shows |a0|and |a2| as a function of time. In this long time series we can see that the fundamental mode isdominant, and that, on the average, |a2| mimics the behaviour of |a0|.

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3.3. Mode excitation

Equation (3.24) is identical to Eq. (23) of Hoyng et al. (1994), who use a decompositionbased on the non-orthogonal eigenfunctions bi. Unfortunately, their expression (B.12) for gicontains a poorly converging summation over all the eigenmodes, which is unpractical. In thepresent derivation, the summation is replaced by a single integral involving the eigenfunctionb0 and the adjoint eigenfunction bi.

3.3.3 Phase-amplitude correlation of the fundamental mode

For the fundamental mode, Eq. (3.24) becomes particularly simple, and we may write

d

dτlog a0 = iω0 + g0. (3.27)

For g0 = 0, a0 is an undamped oscillation with frequency ω0. The (complex) forcing term g0superposes random fluctuations on the oscillation. What actually happens can be clarified byseparating log a0 into real and imaginary parts. The imaginary part corresponds to the phaseof the fundamental mode, which we write as ω0τ + x1:

log a0 = x2 + i(ω0τ + x1), (3.28)

where x1 is the fluctuating part of the phase and x2 = log |a0|.3 If we combine Eqs. (3.28) and(3.27), we find

d

(

x1

x2

)

=

(

Im g0Re g0

)

. (3.29)

This equation describes a random walk in the x1x2−plane. The random walk nature of thelogarithmic amplitude is illustrated by Fig. 3.3. Here we see that there are long intervals duringwhich |a0| is very small, reminiscent of the grand minima of the Sun, such as the Maunderminimum. However, in a more realistic model, nonlinear effects (α-quenching) would prohibitlarge values for |a0| and thereby make the intervals of small |a0| less pronounced than in Fig. 3.3.

The two components of the r.h.s. of Eq. (3.29) are integrals over the same stochastic functionδα (Eq. 3.25). The random walks of x1 and x2 are thus correlated; the jumps in x1 and x2 arein fact drawn from a two dimensional Gaussian distribution of elliptical shape. Its major axisis oriented along the line x2 = sx1. The slope s = tanψ depends on the model parameters. ForkR = 3 and C = −1319 the corresponding angle is ψ = 29o; see Appendix B for a derivation.Figure 3.6 shows four numerical integrations of Eq. (3.29). The correlation between x1 and x2

implies a correlation between phase and amplitude variations, such that a larger cycle amplitude(x2 > 0) corresponds, on average, to a shorter cycle duration (x1 > 0). In Appendix B we alsoderive that for kR = 3, C = −1319 and initial condition x(0) = 0:

x‖,rms = 38√

∆τ, (3.30)

x⊥,rms = 7.7√

∆τ . (3.31)

Here x⊥ and x‖ are the projections of x on the minor and the major axis of the gaussian ellipticaldistribution respectively. Thus, x performs a random walk, not only parallel to the line x2 = sx1

but also perpendicular to it, although about 5 times more slowly. Equations (3.30–3.31) justifythe name ’diffusion coefficient’ for ∆.

3Note that the fluctuating part of the phase x1 is defined here with the opposite sign to that of δψ in chapters 1and 2.

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Figure 3.4: |a0|, |a2| and |a4| as a function of time, in arbitrary units, obtained through aneigenmode decomposition of the numerical solution shown in Fig. 3.1b. There are no oscillations,since all information on phases is lost. Besides the fundamental mode, overtones are excited byfluctuations in α.

It was shown by Hoyng (1993) that, in the exact αΩ-limit, a stochastic model with α-fluctuations independent of azimuth θ leads to a similar correlation, but with x⊥,rms = constant.This would mean that phase and amplitude fluctuations are forever locked to one another, i.e.the dynamo would have an infinitely long ’memory’ so that it would always know the phase x1

that corresponds to the amplitude x2.In order to analyse the physical origin of this infinitely long memory we should consider the

limit of θ-independent fluctuations in our model. A little care is needed to define what is meantby ”θ-independent”. Presently, the fluctuations are a random function of θ for a given τ and ofτ for a given θ. Suppose now that F (θ, τ) is no longer a random function of θ, while it remainsa random function of τ . Then one may always write F (θ, τ) = F1(θ)F2(τ), where F1 is a givenfunction of θ and F2 a random function of τ .

From Eqs. (3.26) and (3.29) we then get

x =frms√2Nc

F2(τ)

∫ π

0dθ

√sin θ F1(θ)

(

Im P ∗0 T0

Re P ∗0 T0

)

. (3.32)

The vector on the right hand side of Eq. (3.32) has a fixed orientation, which indicates thatx performs a random walk on a line in the x1x2-plane (i.e x⊥ = 0). We thus confirm that ifδα is a stochastic function of time only, phase and amplitude fluctuations are forever locked toone another. A more realistic calculation that treats δα as a stochastic function of time andlatitude, removes this property and identifies it as an artefact of the model.

It may be objected that this argument is incomplete since if we take F1(θ) =constant inEq. (3.32) as did Hoyng (1993), then x = 0: if F1(θ) is symmetric with respect to the equatorthen the integral in Eq. (3.32) vanishes as P0 is symmetric and T0 antisymmetric (Appendix A).The fundamental mode is insensitive to fluctuations δα that are symmetric with respect to theequator (g0 vanishes for symmetric δα). However, this is caused by the fact that our model hasa spherical geometry with an antisymmetric α0, so that the eigenmodes have certain symmetryproperties. The model of Hoyng (1993) has a plane geometry and has translational invariancealong the direction of field migration. In this respect the two models are essentially different.

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3.3. Mode excitation

If we ask ourselves how long the phase memory is maintained, the answer is provided bythe correlation time τcorr which is associated with the phase variations. This correlation timeindicates how long it takes, on the average, for the phase fluctuations x1 to drift away from itsinitial value x1(0) = 0 a typical distance of the order 1. We therefore put x1,rms = 1 and insertEq. (3.53), which yields

τcorr = 8.9 × 10−4/∆. (3.33)

For ∆ = 2 × 10−4 (the solar value, see Fig. 3.1b) we find τcorr = 4.4, i.e. 20 dynamo periods forthe phase memory.

The implication of the phase-amplitude correlation is that we can combine the phase vari-ations x1 with the amplitude variations x2 and calculate a ’corrected’ or residual phase x1 −x2/ tanψ = x⊥/ sinψ (Appendix B), with a much longer phase memory. We can estimate theassociated correlation time by putting x⊥,rms/ sinψ = 1 and inserting Eq. (3.31), which providesfor ψ = 29o:

τ ′corr = 4.0 × 10−3/∆. (3.34)

For ∆ = 2 × 10−4 we find τ ′corr = 20, i.e. about 90 dynamo periods. This phase-amplitudecorrelation time is much longer than the correlation time that is associated with phase fluctua-tions only. The reason that we obtain a longer correlation time is that we have made use of thephase-amplitude correlation in order to compensate partly the random walk in the phase withthe random walk in the logarithmic amplitude.

3.3.4 Excitation of overtones

In this section we shall derive the average excitation level of the overtones and we shall brieflyexamine how this excitation gives rise to asymmetries in the butterfly diagram. Figure 3.4shows |a0|, |a2| and |a4| as a function of time, derived from the solution shown in Fig. 3.1b.The dominant mode is the fundamental mode but overtones are also excited. In Appendix C wederive from Eq. (3.24) the following expression for the r.m.s. excitation level |ai|rms of mode i:

ri =|ai|rms

|a0|rms=

−π∆Ji4γi

(i ≥ 2). (3.35)

The excitation levels depend on coefficients Ji,

Ji =

∫ π

0dθ sin θ |P ∗

i T0|2, (3.36)

a number of which are given in Table 3.2. There we also give the corresponding theoreticaland time averaged values for ri/

√∆. The theoretical values predicted by Eq. (3.35) seem to

be too small by a factor√

2; as of yet we have no explanation for the discrepancy. Apart fromthe unaccounted factor

√2, the agreement between theoretical and experimental values of ri is

quite good. This justifies the omission of all coupling terms from Eq. (3.24) except that withthe fundamental mode.

In the example shown in Fig. 3.4, the fundamental mode is always dominant, but for shorttime intervals, the first overtone can be almost as strong. This phenomenon becomes moreimportant as frms, the strength of the fluctuations, increases. Occasionally, the first overtone mayeven dominate the field. During such an interval, the magnetic field has a quadrupolar structure.From Eq. (3.35) and Table 3.2 we derive that |a2|rms ≈ |a0|rms if frms ≈ Nc

−4γ2/πτcJ2 ≈ 54.

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Table 3.2: Coefficients Ji and ri/√

∆, for kR = 3 and C = −1319. The last column shows timeaveraged values, obtained from the solution in Fig. 3.1b.

i Ji ri/√

Eq. (3.35) time av.

2 1.01 × 103 19.5 264 9.46 × 101 2.88 4.65 9.25 × 102 6.47 117 1.43 × 103 6.30 8.69 1.00 × 103 4.32 7.0

11 1.59 × 103 4.70 6.513 1.37 × 103 3.86 5.915 1.73 × 103 3.90 5.417 1.73 × 103 3.54 5.219 1.91 × 103 3.41 4.7

For such a large value of frms, the first overtone will dominate for a large fraction of time.This is not observed in the Sun, which suggests that frms is less than 54. Even so, our modelpredicts that after a long period of dipolar symmetry, there will be an occasional short intervalof quadrupolar symmetry.

Our contour plots (especially Figs. 3.1b and 3.1c) show a clear asymmetry between the twohemispheres. We shall study the issue of asymmetries more carefully in section 3.4.2, where acomparison is made with the observed asymmetry in the solar butterfly diagram.

3.4 Application to the solar cycle

3.4.1 Phase-amplitude correlation

Our model predicts a correlation between the phase and amplitude of the fundamental mode.A similar correlation has also been found in sunspot data (Hoyng 1993). We shall brieflyrecapitulate how the correlation is derived from the data. For that purpose we use the times ofsolar maxima tk and the corresponding maximum sunspot numbers Rk. These were providedby the National Geophysical Center in Boulder, Colorado. We assume that the magnetic fielddepends on the sunspot number through a simple power law T ∝ Rµ. We ignore for the momentthe effect of overtones. The sunspot numbers thus provide us information about the fundamentalmode, and in particular we can estimate the phase shift x1 and the logarithmic amplitude x2

as3

x1(tk) = kπ − ω(tk − t0), (3.37)

x2(tk) = log[T (tk)/T (t0)] = µ log[R(tk)/R(t0)]. (3.38)

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3.4. Application to the solar cycle

Figure 3.5: On the vertical axis: logarithmic amplitude x2 = log Tk/T0 = µ logRk/R0 at timetk of the k-th solar maximum; Rk is the mean sunspot number. On the horizontal axis: phaseshift x1 of the cycle with respect to the mean phase (see Eq. 3.37 for the definition) at time tk.The mean dynamo frequency ω = 0.28 yr−1, and µ/s = 0.90 were determined by a least squarefit, and for s we took s = 0.56. The 27 subsequent datapoints (covering 13 dynamo periods) areconnected by straight lines.

where ω is the dynamo frequency. The 27 data points are fitted to a straight line x2 = sx1 byminimizing

S(ω, µ/s) =26∑

k=0

[x1(tk) − x2(tk)/s]2 (3.39)

with respect to ω and µ/s, i.e. by solving the equations ∂S/∂ω = ∂S/∂(µ/s) = 0. The best fitis obtained for ω = 0.28 yr−1, i.e. 2π/ω = 22 yr, and µ/s = 0.90. We have adopted s = 0.56in Fig. 3.5. A fairly good correlation between phase shift x1 and logarithmic amplitude x2 isobserved.

For comparison we show in Fig. 3.6 four time series of x1 and x2, obtained from a numericalsimulation of Eq. (3.29), using the same parameters as in Fig. 3.1b. Each series covers 27consecutive maxima of the magnetic cycle. The similarity between these curves and the solardata shown in Fig. 3.5 suggests that the observed phase-amplitude correlation is reasonably wellreproduced by our model if we adopt the parameters from Fig. 3.1b, i.e. if ∆ = 2×10−4. Due tothe small number of available solar data, we are not able to accurately determine the requiredvalue of ∆ from Fig. 3.5 itself. The same is true for the parameter kR, which determines s,x⊥,rms and x‖,rms through the eigenfunctions (see Appendix B). We have chosen kR = 3 andhave not tried other values, although the agreement between Figs. 3.5 and 3.6 might be furtherimproved in this manner. A detailed comparison is further complicated by two effects. Firstly,we have ignored in Fig. 3.6 variability in the overtones of the fundamental mode. In the solardata (Fig. 3.5) we should suspect the presence of noise terms arising from the excitation ofovertones. Secondly, we have ignored the back reaction of the magnetic field on α. This backreaction limits the amplitude of the magnetic field. As a result, the random walk behaviour ofthe mode coefficients is inhibited and Eqs. (3.30–3.31) must ultimately break down. Althoughwe have not verified it for the present model, results from a plane parallel dynamo model with

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Figure 3.6: Logarithmic amplitude x2 versus phase shift x1, obtained from four different nu-merical simulations (see text). The parameters are the same as in Fig. 3.1b. As in Fig. 3.5, weincluded x1 and x2 in the plot only at 27 consecutive times at which the phase ω0τ + x1 hasadvanced by the amount π, and connected these points by straight lines.

α-fluctuations suggest that the phase-amplitude correlation survives if α-quenching is included(Ossendrijver & Hoyng 1996a).

3.4.2 North-South asymmetry

A well known feature of solar activity is the asymmetry between the two hemispheres (seeCarbonell et al. 1993 and references therein). It is measured by the asymmetry coefficient,

AS =N − S

N + S, (3.40)

where N and S are sunspot areas on the northern and southern hemispheres respectively. Theasymmetry of solar activity has been recorded for a long time. The longest dataseries is providedby sunspot area counts. Fig. 3.7 shows the asymmetry of the yearly average sunspot areas from1874 to 1976. The data were provided by the National Geophysical Center in Boulder, Colorado.

A difficulty in comparing observed asymmetries with our model is that sunspot data have,on top of the behaviour dictated by the dynamo field, an extra stochastic component due tothe random way in which spots appear at the solar surface. However, it was found that theasymmetry is statistically significant, i.e. not only due to the above mentioned stochastic process,and that it often peaks near solar minimum (Carbonell et al. 1993). The same authors alsoconclude that the asymmetry is of stochastic and not of chaotic nature.

In order to make a comparison between sunspot data and our model, some assumption mustbe made to relate sunspot area to the magnetic field strength in our model. Since it is not ouraim to reproduce the data quantitatively, we take as a measure of sunspot area the toroidal

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3.5. Discussion

Figure 3.7: North-south asymmetry coefficient of observed sunspot areas. Minima of solaractivity are indicated by vertical bars.

magnetic field, regardless of its sign, as follows:

N =

∫ π/2

0dθ sin θ |T (θ, t)|, (3.41)

S =

∫ π

π/2dθ sin θ |T (θ, t)|. (3.42)

In Fig. 3.8 the asymmetry coefficient of the solution in Fig. 3.1b is plotted. It often peaksat activity minimum, which can be explained as follows. At activity minimum, the field ofthe fundamental mode |a0b0 + a1b1| is at its minimum or close to it. In general, the firstovertone is much weaker than the fundamental mode. Interference between these two modes,i.e. asymmetry, is a large effect when both modes are equally strong, and this occurs only nearactivity minimum, provided that the first overtone is not strongly excited.

If by chance the first overtone is nearly as strongly excited as the fundamental mode, i.e.if |a2/a0| ≈ 1 (Fig. 3.4), mode interference can be strong not only near activity minimum, butduring a large fraction of the magnetic cycle. In that case, asymmetry is a strong effect and showsless preference to peak near the activity minimum. All these effects can be observed in Fig. 3.8.Also notice that the asymmetry can persist for several dynamo periods without changing its sign.The explanation for this feature lies in the the decay time of the first overtone, 1/|γ2| = 0.49 ≈ 2dynamo periods, which determines the typical coherence time of the asymmetries. Within thistime interval, the asymmetry can maintain its sign because the frequency of the first overtone,ω2 = 27.6, differs only slightly from the fundamental frequency ω0 = 27.9, giving rise to a slowbeat period 2π/(ω0 − ω2) = 27 ≈ 120 dynamo periods. This beat period depends on kR, and ittends to be longer for dynamos in shells than for dynamos in spheres (Moss et al. 1992).

3.5 Discussion

We have studied a mean field dynamo model located in a spherical shell, in which α is a stochasticfunction of time and latitude. There are theoretical and observational motivations for consideringsuch a model.

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Figure 3.8: Asymmetry coefficient AS for the solution shown in Fig. 3.1b. Minima of N +S areindicated by vertical bars. Asymmetry often peaks at activity minimum, provided that the firstovertone is only weakly excited; see the text for an explanation.

On the theoretical level, we are prompted to treat α as a stochastic function because we definethe mean field as an average over longitude φ. Following this definition, all mean quantities havea stochastic component, because the velocity field is organised in coherent elements (eddies), ofwhich there are a finite number on circles of constant latitude.

Observations of the solar cycle show a number of features related to spatial and temporalvariability: correlated phase and amplitude fluctuations, the Maunder minimum, North-Southasymmetries and transequatorial activity. The occurance of several grand minima of solar activ-ity is suggested by variations of 14C in tree rings (Stuiver & Braziunas 1988) and 10Be in arcticice cores (Raisbeck & Yiou 1988, Beer et al. 1988).

Our dynamo model with an α-coefficient that is a stochastic function of time and colatitude,tries to address all these features. Due to the fluctuations in α, the phase and logarithmicamplitude of the fundamental mode perform a random walk. It is shown that grand minima canbe obtained, but it is not intended to model details of the solar magnetic history. It should bepointed out that no nonlinearity was needed to limit the growth of the magnetic field, becausethe dynamo number was critical. For supercritical excitation nonlinearities are necessary, andthese would make the grand minima in our model less pronounced.

A different treatment of grand minima, based on the combination of a random triggeringprocess and a nonlinear dynamo is suggested by Ferriz-Mas et al. (1994). They considered adynamo effect due to instability of magnetic flux tubes in the overshoot region of the Sun. Thiseffect only works for field strengths above a critical value for the onset of instability, and belowan upper limit when instability is so fast that magnetic flux is expelled from the generationregion by magnetic buoyancy. Model calculations based on this effect by Schmitt et al. (to bepublished) show the occurance of grand minima when the field strength falls below the firstcritical value. A random triggering process (e.g. field pushed down from the convection zoneinto the dynamo region where it is wound up by differential rotation) then leads again out of aminimum.

In our model there is no phase-locking of the magnetic cycle, and therefore no ’internalclock’ in the dynamo, as was suggested by Dicke (1978). We are able to reproduce qualitativelythe observed correlation between phase and amplitude of the solar cycle. Phase and amplitude

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3.5. Discussion

fluctuations of the fundamental mode are correlated but not locked to one another.

Interference between the fundamental mode and (mainly) the first overtone leads to asymme-try in the amount of flux on the two hemispheres. In agreement with solar data, the asymmetryoften peaks during the activity minimum, and can persist for many dynamo periods.

The dividing line between the two polarities in the simulated butterfly diagram fluctuatesaround the equator. This transequatorial activity can explain reverse polarity regions close tothe equator, but cannot explain the observed latitude distribution of such regions, which doesnot peak near the equator (Harvey 1992). Since the fraction of bipolar active regions that have areverse polarity increases with decreasing region size (Harvey 1992), they may originate from theconvection zone dynamo that produces weak, small scale fields, rather than from the overshootdynamo that is thought to be reponsible for the solar cycle.

We see the reasonable agreement between our model and the solar cycle on the issues men-tioned above as a confirmation of the conclusion of Carbonell et al. (1993), namely that theunderlying physical process that causes variability in the solar magnetic field is of stochasticrather than of chaotic origin.

Apart from the parameters Nc, frms and τc, no detailed knowledge of the convective cellsis necessary to model the stochastic behaviour. A rough estimate of the strength frms of thefluctuations suggests that for Nc = 50 and τc = 2.25 × 10−3, we require frms ≈ 15. In otherwords, solar data are reproduced if local helicity fluctuations δh, i.e. from one convective cell toanother, are frms/ cos θ ≈ 30 times stronger than the mean value at a colatitude of θ = 60o. Thecorresponding fluctuations in α are relatively smaller by a factor

√2Nc sin θ, so that δα/α0 ≈ 3.

As suggested by Eqs. (3.30, 3.31) and (3.35), the effective strength of the stochastic behaviouris measured by the diffusion coefficient ∆ (Eq. 3.6), rather than by frms. Thus other combinationsof the parameters frms, Nc and τc are also compatible with the observed variability, providedthat ∆ ≈ 2 × 10−4. This constraint then yields the required value of frms, given a certain sizeand coherence time of the cells.

The typical size of a supergranulation cell is λS ≈ 3 × 104 km, so that Nc ≈ πR⊙/λS ≈ 73,and its typical coherence time is tc = 2 days, i.e. τc ≈ 5.6 × 10−5. This requires a value offrms ≈

√∆Nc/

√τc ≈ 140. For supergranulation cells located at θ = 60o, we find δh/h0 ≈ 270

and δα/α0 ≈ 24. It is unlikely that helicity fluctuations are this large for supergranulation cells.Furthermore, these cells do not extend deep enough into the convection zone to play a role inthe dynamo process. The supergranulation is therefore not likely to play a role in the stochasticbehaviour of the dynamo.

A better candidate are the giant cells, because they are larger (λG ≈ 105 km), have a longercoherence time (tc ≈ 1 month, i.e. τc ≈ 10−3), and extend sufficiently deep into the convectionzone to affect the dynamo process. For an account on observations of giant cells, see Stix (1989).Their number from pole to pole can be estimated as Nc ≈ πR/λG ≈ 20, where R = 6 × 105 kmis the distance from the origin to a point halfway the convection zone. This yields frms ≈ 8.Thus, for giant cells located at θ = 60o, we find δh/h0 ≈ 16 and δα/α0 ≈ 3. Although helicityvariations are less extreme than for supergranulation cells, they are still much larger than themean value, so that in a given cell the kinetic helicity has a chance of almost 50% of having asign opposite to the mean value. Whether such helicity variations are in fact possible in giantcells, is a question that needs to be answered through convection models.

We conclude that the most likely candidate for the origin of the stochastic behaviour of thesolar cycle are the giant convective cells.

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Acknowledgements

The authors want to thank mr. E. Erwin of the National Geophysical Data Center in Boulderfor providing us with the sunspot data, and prof. H. van Beijeren for many useful discussions.This research was supported by the Dutch Foundation for Research in Astronomy (ASTRON).

3.6 Appendix A: Biorthonormal representation

We shall first prove that the eigenfunctions of D and of its adjoint D form a biorthogonal basis.It is then shown that there exists a simple transformation which enables us to obtain the adjointeigenfunctions from the original eigenfunctions.

The inner product is defined as

(bi,bj) =

∫ π

0dθ sin θ [P ∗

i Pj + T ∗i Tj ]. (3.43)

The adjoint operator D is defined through identity (3.19) and has eigenfunctions bi and eigen-values λi, i.e.

Dbi = λibi. (3.44)

The eigenfunctions are orthogonal to bj , for

(bi,Dbj) = λj(bi,bj) = (Dbi,bj) = λ∗i (bi,bj), (3.45)

so that

(λj − λ∗i )(bi,bj) = 0. (3.46)

Consequently (bi,bj) = 0 unless λj = λ∗i , i.e. unless i = j, if the subscripts are arrangedproperly. After normalisation we have a biorthonormal set of functions, satisfying (bi,bj) = δij .

A comparison between D (Eq. 3.12) and D (Eq. 3.20) shows that there is an unexpected,simple relation between the two operators, for

DC = ED−CE, with E =

(

0 11 0

)

. (3.47)

If we use E2 = 1 and apply DC to Ebi,−C , we obtain

DC Ebi,−C = λi,−C Ebi,−C . (3.48)

We thus find the eigenfunctions and eigenvalues of DC without explicitly solving Eq. (3.44). Ifthe subscripts of bi,−C are arranged in such a way that λi,−C = λ∗i,C we can identify

bi,C = Ebi,−C and λi,C = λi,−C . (3.49)

Finally we mention that the symmetry of bi with respect to the equator is the reverse of thatof bi, e.g. T0 is symmetric and P0 is antisymmetric (see Table 3.2).

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3.7. Appendix B: Phase-amplitude correlation of the fundamental mode

3.7 Appendix B: Phase-amplitude correlation of the fundamen-

tal mode

Here we shall demonstrate that there is a correlation between phase and amplitude variations ofthe fundamental mode, and derive a number of their statistical properties. Our starting pointis Eq. (3.29), from which we derive

(

x1

x2

)

=

(

∫ τ0 dσ Im g0∫ τ0 dσRe g0

)

. (3.50)

The statistical properties of x depend on the stochastic function g0 (Eqs. 3.25–3.26), and throughg0 on F . In our analytical calculations we adopt the following simple statistical model for theautocorrelation of F , namely

〈F (θ, τ)F (θ′, τ ′)〉 = τcθc δ(θ − θ′)δ(τ − τ ′), (3.51)

where θc = π/Nc is the latitudinal extent of our schematic convective cell. In the numericalsimulations, the autocorrelation of F is different.1 Nevertheless, the calculations are hardlyaffected as long as θc = π/Nc is much smaller than other spatial scales and τc is much smallerthan other time scales. We thus approximate for instance4

〈Im g0(τ)Im g0(τ′)〉 =

π∆

2δ(τ − τ ′)

∫ π

0dθ sin θ (Im P ∗

0 T0)2. (3.52)

Similar expressions involving Re g0 and Im g0 are treated analogously. Before calculating thestatistical properties of x (Eq. 3.50), we remark that

∫ τ0

∫ τ0 dσdσ

′ δ(σ − σ′) = τ and obtain5

〈x21〉 =

π∆τ

2

∫ π

0dθ sin θ (Im P ∗

0 T0)2 = 1.12 × 103 ∆τ, (3.53)

〈x22〉 =

π∆τ

2

∫ π

0dθ sin θ (Re P ∗

0 T0)2 = 394∆τ, (3.54)

〈x1x2〉 =π∆τ

2

∫ π

0dθ sin θ (Re P ∗

0 T0)(Im P ∗0 T0) = 594∆τ. (3.55)

The right hand sides of Eqs. (3.53-3.55) were evaluated numerically for kR = 3 and C = −1319.The correlation coefficients 〈xixj〉 form a matrix, which can be diagonalized through a rotationover an angle ψ. The components of the rotated vector x′ = R(ψ)x are

x‖ = x1 cosψ + x2 sinψ, (3.56)

x⊥ = x1 sinψ − x2 cosψ, (3.57)

so that one component is parallel, and one is perpendicular to the line x2 = x1 tanψ. Withoutgiving a proof, we mention that for ψ = 29o (i.e. s = tanψ = 0.56) their correlation coefficientvanishes, i.e. 〈x‖x⊥〉 = 0, and that

x‖,rms = 38√

∆τ , (3.58)

x⊥,rms = 7.7√

∆τ. (3.59)

4Change the integration variables to θ − θ′ and (θ + θ′)/25Change the integration variables to σ − σ′ and (σ + σ′)/2.

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STOCHASTIC EXCITATION AND MEMORY OF THE SOLAR DYNAMO

3.8 Appendix C: Derivation of r.m.s. mode coefficients

In this section we give a short derivation of Eqs. (3.35) and (3.36), which replaces that by Hoynget al. (1994). Our starting point is Eq. (3.24). Its stationary solution is for i ≥ 2

ai(τ) =

∫ ∞

0dσ eλiσa0(τ − σ)gi(τ − σ). (3.60)

The term eλiσ effectively cuts off the integration at σ ≈ 1/|γi| <∼ 0.5 (see Table 3.1). To firstapproximation, we can thus ignore the stochastic behaviour of a0 from σ = 0 onwards, andsubstitute a0(τ − σ) ≈ a0(τ) e−iω0σ:

ai(τ) ≈ a0(τ)

∫ ∞

0dσ e(λi−iω0)σgi(τ − σ). (3.61)

We multiply the result with its complex conjugate and average. To first approximation, a0(τ)and gi(τ − σ) factorize, which provides

〈|ai|2〉 = 〈|a0|2〉∫ ∞

0

∫ ∞

0dσdσ′ eγi(σ+σ′)ei(ωi−ω0)(σ−σ′)〈gi(τ − σ)g∗i (τ − σ′)〉. (3.62)

The autocorrelation function 〈gi(τ)g∗i (τ ′)〉 is found by inserting Eqs. (3.25) and (3.51):

〈gi(τ)g∗i (τ ′)〉 =π∆

2δ(τ − τ ′)

∫ π

0dθ sin θ |P ∗

i T0|2. (3.63)

We substitute this result in Eq. (3.62), and obtain5

〈|ai|2〉 =π∆Ji

2〈|a0|2〉

∫ ∞

0

∫ ∞

0dσdσ′ δ(σ − σ′) eγi(σ+σ′)ei(ωi−ω0)(σ−σ′) (3.64)

=π∆Ji

2〈|a0|2〉

∫ ∞

0dσ2e

2γiσ2 = −π∆Ji4γi

〈|a0|2〉, (3.65)

where Ji is a numerical coefficient, given by

Ji =

∫ π

0dθ sin θ |P ∗

i T0|2. (3.66)

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Chapter 4

Mean magnetic field and energy

balance of Parker’s surface wave

dynamo

AbstractWe study the surface wave dynamo proposed by Parker (1993) as a model for the solar dynamo, bysolving equations for the mean magnetic field B0, as well as for the mean magnetic ’energy tensor’T = 〈BB〉/8π. This tensor provides information about the energy balance, r.m.s. field strengths andcorrelation coefficients between field components. The main goal of this paper is to check whetherthe equations for B0 and T are compatible, i.e. whether both have physically acceptable solutionsfor the same set of physically acceptable parameters. We apply the following constraints: the meanfield has a period of 22 years and, taking into account the effect of period variations, a decay timeof 10 dynamo periods, and the mean magnetic energy is marginally stable. We find that under theseconstraints, the equations for B0 and T are incompatible. Marginal stability of the mean magneticenergy requires a turbulent diffusion in the convection zone of the order β2

>∼ 3 × 1014 cm2 s−1,

whereas the conditions on the mean field require β2 ≈ 1012 cm2 s−1. We suggest that in order toremove the inconsistency an additional energy sink is required, possibly Ohmic dissipation, hithertonot accounted for in the equation for T.

A.J.H. Ossendrijver and P. Hoyng

Astronomy & Astrophysics (submitted)

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MEAN MAGNETIC FIELD AND ENERGY BALANCE OF PARKER’S SURFACE WAVE DYNAMO

4.1 Introduction

In recent years observational evidence and various calculations have resulted in a coherent setof requirements for solar dynamo models. From measurements of the emerging flux in activeregions it is inferred that the total subsurface flux is of the order 1024 Mx (Galloway & Weiss1981; Golub et al. 1981). If contained in the convection zone of depth 2× 1010 cm and within alatitudinal band of width 2×1010 cm, centered at the equator, this amount of flux would requirefield strengths of the order 2.5× 103 G. Such fields cannot be maintained in the convection zoneand are expelled by buoyancy forces on a time scale of several months, much shorter than thetime scale for the amplification of toroidal field by radial velocity shear.

Van Geffen & Hoyng (1993b) and Van Geffen (1993c, 1993d) have confirmed that a dynamooperating within the convection zone cannot be responsible for the solar cycle. These authorsfound that the field strength at the base of the convection zone is only of order Brms ≈ 200 Gand that the mean field B0 decays on a timescale of about 2 weeks due to phase mixing (thisconcept is explained below). A convection zone dynamo thus produces only a weak, aperiodicfield.

For these reasons it is believed that the flux is concentrated in the stably stratified overshootlayer between the radiative core and the convection zone (Zwaan 1978, Spiegel & Weiss 1980).In this layer, which has a thickness of about 2 × 109 cm, the field must be of order 2.5 × 104 Gto explain the measured flux. Taking into account the possibility that not all the flux in theovershoot layer emerges at the surface, the actual field may be of order 105 G. Calculations onthe stability of flux tubes show that such strong fields can indeed be maintained in the overshootlayer (Ferriz-Maz & Schussler 1994). A value of 105 G is also in agreement with the field strengthrequired for rising flux tubes to resist the Coriolis force sufficiently and to emerge at the solarsurface within the observed activity belts and having the correct tilt with respect to the equator(Choudhuri 1989, D’Silva & Choudhuri 1993).

The creation of the mean magnetic field B0 is described by the following equation:

∂tB0 = ∇× [u0 × B0 + αB0 − β∇× B0]. (4.1)

Here u0 = Ωr sin θeφ is the large scale velocity field. Helioseismological data confirm that theSun rotates differentially and that the radial velocity shear is concentrated in a thin layer nearthe base of the convection zone. Thus strong azimuthal magnetic fields can be produced bydifferential rotation in the overshoot layer. The sign of ∂Ω/∂r is positive at low latitudes andnegative at high latitudes (Goode 1995). If u1 denotes the required turbulent velocity field andif τc is the typical correlation time of the turbulence, we can write α and the turbulent diffusivityβ as

α = −1

3τc〈u1 · (∇× u1)〉, β =

1

3τc〈u2

1〉. (4.2)

Thus α is proportional to the mean helicity of the turbulent velocity field. It is non-vanishing in arotating turbulent medium such as the solar convection zone. Since in the Northern hemisphereα is believed to be positive in the bulk of the convection zone and to change its sign near thebase, α∂Ω/∂r probably has the right sign in the lower part of the convection zone for dynamowaves to travel towards the equator at low latitudes.

Due to the strong azimuthal magnetic field in the overshoot layer, well above the equipartitionvalue, α and β are greatly reduced compared to their values in the convection zone. It wassuggested by Parker (1993) that such an overshoot layer dynamo can be modelled through

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4.1. Introduction

spatial separation of the radial velocity shear and the α-effect, whose combined action providesthe mechanism to generate magnetic fields.

In this paper we approach mean field dynamo theory from a statistical point of view, em-ploying the ensemble average to define mean quantities. Every ensemble member representsa dynamo that is on average marginally stable with, in the solar case, a mean period of 22years and, on top of that, period fluctuations of the order ∆P/P ≈ 0.1 (Hoyng 1987, VanGeffen 1993c). If we ignore for the moment latitude dependence and details of the shape of amagnetic cycle, we may schematically represent the magnetic field of one ensemble member asB ∝ cos(ωt+ψ0), where ω = 2π/P is the dynamo frequency, subject to variations (δω/ω ≈ 0.1)and ψ0 is an initial phase. The frequency variations give rise to phase mixing between theensemble members, so that the resulting mean field is slightly subcritical and decays at a rate|γ| ≈ 0.1/P . The decay time is interpreted as the coherence time of the actual magnetic field.

The result of phase mixing is different, however, for the magnetic energy density |B|2/8π,since it is positive definite. This is easily demonstrated by squaring the same simplified expres-sion for the magnetic field of an ensemble member, i.e. B2 ∝ cos2(ωt+ψ0) = 1

2 + 12 cos 2(ωt+ψ0).

Thus we may separate the magnetic energy of an oscillating dynamo into a constant part, andan oscillating part (in reality there is also a modulation of the amplitude of B, but we ignore thisfor the moment). If we apply the ensemble average to B2, we see that the fundamental mode of〈B2〉 is (roughly) constant and non-periodic. The ensemble average of the periodic part of B2

corresponds to an overtone of the fundamental mode. This mode decays due to phase mixing.

Another difference between the mean magnetic field and the mean magnetic energy lies inthe treatment of small scale fields. The r.m.s. value of the small scale field may exceed the large

scale mean field by a factor of the order R1/2m (see e.g. Krause & Radler 1980, p. 42). Here

Rm is the magnetic Reynolds number, which is of the order 106 for the solar convection zone.Therefore small scale fields are important in the solar dynamo, and this further justifies a studyof the mean magnetic energy, which has contributions from all length scales.

The combined use of equations for the mean magnetic field and the mean magnetic energyhas been named the ’finite energy method’ (Hoyng 1987, Van Geffen 1993c). To summarise, itprovides us with the following constraints: the mean magnetic energy is marginally stable andnon-oscillating and, in the solar case, the mean magnetic field has a period of 22 years and adecay time of about 10 dynamo periods.

The motivation for using Parker’s model came from the intuitive feeling that here the decaytime of the mean field may be of the right order, much longer than that of a convection zonedynamo (see Van Geffen & Hoyng 1993b and Van Geffen 1993c), because the overshoot layer,where strong fields are produced, has a lower level of turbulence, hence maybe also less variability.Van Geffen (1993d) found that the mean field decay time did not increase significantly when alower turbulent diffusivity is adopted near the base of the convection zone. However, since hiscalculations were carried out on a finite grid and his spatial resolution in the overshoot layerwas insufficient, he was not able to treat large discontinuities. In the present paper we shallemploy a simplified geometry that allows us to obtain analytical expressions for the mean fieldand the mean magnetic energy on both sides of the interface between the overshoot layer andthe convection zone.

The model proposed by Parker employs a flat geometry, and, in our case, has local validityin the northern hemisphere of the Sun (see Fig. 4.1). It describes the convection zone (region2) in which α 6= 0 and velocity shear is zero, with below it a thin overshoot layer (region 1) in

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MEAN MAGNETIC FIELD AND ENERGY BALANCE OF PARKER’S SURFACE WAVE DYNAMO

Figure 4.1: Geometry of Parker’s interface wave dynamo. In our calculations we adopted d1 =2 × 109 cm, d2 = 2 × 1010 cm and a = 4 × 10−6 s−1.

which α = 0 and in which there is a constant velocity shear, i.e.

u0(x) = u0(x)ey, ∂xu0 =

a (−d1 < x < 0)0 (0 < x < d2).

(4.3)

We have to allow for a small amount of turbulent diffusion in the overshoot layer, because thedynamo requires transport between the two regions for its functioning. Thus the ratio of theturbulent diffusivities in the two regions satisfies

fβ =β1

β2≪ 1, (4.4)

Cartesian coordinates are used with x denoting the radial, y the longitudinal and z the azimuthalcoordinate; the interface between overshoot layer and convection zone is at x = 0 and theequator is at z = 0. Translational symmetry is assumed along the y-axis; only axisymmetricsolutions are considered. The geometry of the model differs from Parker’s original model in thatboundaries are specified at x = −d1 and x = d2. The introduction of boundaries is necessaryfor computing the energy balance of the dynamo, in particular the energy loss into empty space(section 4.3). For the sake of consistency, the equation for the mean magnetic field is solvedin the same geometry. The effect of the boundaries on B0 is expected to be small, since thedynamo operates at the interface between overshoot layer and convection zone.

The paper is organised as follows. First we shall look for solar-type solutions of the equationfor the mean magnetic field. Then we shall attempt to do the same with the equation for T. Ourconclusion will be that the corresponding parameter regimes are incompatible. This motivatesus finally to briefly consider growing solutions of the equation for T.

4.2 The mean magnetic field

4.2.1 Equations

We express B0 in terms of the toroidal field T and the poloidal vector potential P ,

B0 = Tey + ∇× Pey. (4.5)

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4.2. The mean magnetic field

Substitution of Eq. (4.5) into Eq. (4.1), using the index 1 for the overshoot layer (x < 0) andindex 2 for the convection zone (x > 0), provides

(∂t − β1∇2)P1 = 0, (4.6)

(∂t − β1∇2)T1 = −a∂zP1, (4.7)

(∂t − β2∇2)P2 = αT2, (4.8)

(∂t − β2∇2)T2 = 0. (4.9)

The right hand term of Eq. (4.9) is in fact not zero but equal to −α∇2P2. For the Sun this termis usually neglected (the αω-approximation), since it represents the production of toroidal fieldfrom poloidal field through the α-effect, which is of little importance relative to the productionthrough differential rotation. We consider plane wave solutions that propagate towards theequator along the z-axis:

Pi = Re pi eikzz+λtTi = Re ti eikzz+λt (i = 1, 2). (4.10)

Here kz is the latitudinal wavevector and λ is given by

λ = iω + γ. (4.11)

The dynamo equations (4.6-4.9) then reduce to a set of ordinary differential equations for thecomplex wave amplitudes pi and ti, which depend on x only:

λ+ β1(k2z − ∂2

x) p1 = 0, (4.12)

λ+ β1(k2z − ∂2

x) t1 = −iakz p1, (4.13)

λ+ β2(k2z − ∂2

x) p2 = α t2, (4.14)

λ+ β2(k2z − ∂2

x) t2 = 0. (4.15)

4.2.2 Boundary conditions

Turbulent diffusivity vanishes in the radiative core. Therefore no poloidal field or magnetic fluxmay penetrate below x = −d1 (see also Choudhuri 1990):

p1 = 0∂xt1 = 0

at x = −d1. (4.16)

At the interface between regions 1 and 2 continuity of B0 is required, so that

t1 = t2p1 = p2

∂xp1 = ∂xp2

at x = 0. (4.17)

We can integrate the toroidal component of Eq. (4.1), i.e. Eqs. (4.7) and (4.9), from x = −δ tox = δ and let δ ↓ 0. This yields another condition for t1 and t2:

β1∂xt1 = β2∂xt2 at x = 0. (4.18)

Boundary conditions at x = d2 are provided by matching the field in the convection zone to apotential field above the solar surface (Krause & Radler 1980). The potential field is describedby

∇× B0 = 0 for x > d2, (4.19)

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MEAN MAGNETIC FIELD AND ENERGY BALANCE OF PARKER’S SURFACE WAVE DYNAMO

Figure 4.2: The mean magnetic field B0 with α = −150 cm s−1, β1 = 1.72 × 1010 cm2s−1,β2 = 1.55 × 1012 cm2s−1, P = 22 years and τd = 220 years. The mean field is concentratedin ’belts’ of alternating polarity migrating toward the equator (downwards in the figure). a)Poloidal field lines of B0 as a function of x and z at t = 0. Drawn field lines have clockwise,dashed field lines anticlockwise orientation. Poloidal field is created only in the convection zone,and diffuses into the overshoot layer. This diffusion, in combination with the migration of thedynamo wave along the z-axis, results in a phase lag that increases with the distance to theinterface x = 0. b) Toroidal (azimuthal) field T as a function of x and z at t = 0.

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4.2. The mean magnetic field

so that by Eq. (4.5),

∇2P3 = 0, ∇× T3ey = 0, (4.20)

where the index 3 denotes the region above the solar surface (x > d2). Inserting for P3 and T3

a similar plane wave solution (4.10) as was done in regions 1 and 2, we obtain

(∂2x − k2

z) p3 = 0, t3 = 0. (4.21)

Continuity of B0 implies

t2 = t3p2 = p3

∂xp2 = ∂xp3

at x = d2. (4.22)

The physically relevant solution of Eq. (4.21) decays exponentially with distance to the solarsurface, i.e. p3 = constant × e−kz(x−d2). Then boundary conditions (4.22) can be expressedsolely in terms of p2 and t2:

(∂x + kz) p2 = 0t2 = 0

at x = d2. (4.23)

4.2.3 Solutions

Equations (4.12-4.15) can be solved analytically:

p1/d1 = A1eκ1x +B1e

−κ1x, (4.24)

t1 = A2eκ1x +B2e

−κ1x +iad1kzx

2β1κ1(A1e

κ1x −B1e−κ1x), (4.25)

p2/d2 = A3eκ2x +B3e

−κ2x − αx

2β2κ2d2(A4e

κ2x −B4e−κ2x), (4.26)

t2 = A4eκ2x +B4e

−κ2x, (4.27)

where

κ1 =√

k2z + λ/β1, κ2 =

k2z + λ/β2. (4.28)

It is assumed that Re κ1 > 0 and Re κ2 > 0. The poloidal vector potentials p1 and p2 are dividedby d1 and d2 respectively so that all the integration constants A1, · · ·, B4 are dimensionless.

If we apply the boundary conditions to the general solution (Eqs. 4.24-4.27), we obtain eightrelations between the integration constants. These relations are presented in Appendix A; theyform a square coefficient matrix, whose determinant should vanish. Every root of the resultingdispersion relation corresponds to a dynamo mode, but we are interested only in the fundamentalmode, i.e. the one with the largest growth rate.

Since the dispersion relation cannot be solved analytically, we resorted to a numericalmethod. A useful initial value λ0 for the iterations can be obtained from a simplified dis-persion relation. To that purpose we first ignore all terms that grow exponentially with |x|, i.e.B1 = B2 = A3 = A4 = 0. The dispersion relation then is

κ1κ2(κ1 + κ2)(fβκ1 + κ2) = −iaαkz/4β22 , (4.29)

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MEAN MAGNETIC FIELD AND ENERGY BALANCE OF PARKER’S SURFACE WAVE DYNAMO

which corresponds to Eq. (22) of Parker (1993). Second, we use fβ ≪ 1 and approximate|κ1| ≈

|λ0|/β1 ≫ |κ2| (see below for a justification). This provides

κ21κ

22 = −iC/4, (4.30)

where the dynamo number C is defined as

C =β1aα

β32k

3z

. (4.31)

Inserting κ1 and κ2 it follows that

λ0(1 + λ0/β2k2z) + iβ2k

2zC/4 = 0. (4.32)

We conclude that to first approximation, the fastest growing solution has a frequency ω0 and agrowth rate γ0 given by

ω0/β2k2z =

18

−1 +√

1 + C2, (4.33)

γ0/β2k2z = −1

2 +√

18

1 +√

1 + C2. (4.34)

These estimates are useful as starting values for solving the exact dispersion relation iteratively.

4.2.4 Parameters

From helioseismological measurements (Goode 1995) the total velocity jump at the base of theconvection zone, near the equator, is estimated to be ad1 = 8 × 103 cm s−1, so that a =4× 10−6 s−1 for d1 = 2× 109 cm. For the wavenumber in the azimuthal direction we took kz =10−10 cm−1, which corresponds to a half wavelength of π/kz ≈ 3.1 × 1010 cm or 180o/kzR⊙ ≈26o. Thus |λ|/β2 ≈ ω⊙/β2 ≈ 10−20 has the same order of magnitude as k2

z . It follows that|λ|/β1 ≫ k2

z , which justifies the approximation that we adopted for κ1 in Eq. (4.29).

The value of α is poorly known and is varied between −25 and −500 cm s−1. For thesmallest value of |α| we obtain the largest ratio of diffusivities, fβ = 8.68×10−2. We did not usesmaller values of |α|, because fβ should satisfy fβ ≪ 1 (Eq. 4.4). An upper limit on |α| is givenby lΩ, where l is a typical convective length scale and Ω ≈ 2 × 10−6 s−1 is the solar rotationrate (see Stix 1989). Near the base of the convection zone we may estimate l ≈ 109 cm, so thatlΩ ≈ 2×103 cm s−1. This value is an upper limit, because the size of α depends on the correlationbetween the turbulent velocity field u1 and the vorticity ∇×u1 (see section 4.3.4). The chaoticnature of the convection zone suggests that this correlation may be weak, i.e. |α| ≪ lΩ.

The fundamental mode of the mean field should have a period of 22 years and a decay time1/γ of about 10 dynamo periods, i.e. ω = ω⊙ = 9.05×10−9 s−1 and γ = γ⊙ = 1.44×10−10 s−1.We obtained solutions that meet these criteria as follows. Starting with a first guess of therequired values of β1 and β2, we calculated the resulting eigenvalue λ for the fundamental modeby solving the dispersion relation. We repeated this by applying successive corrections to β1

and β2, until the solar solution was reached. The result is presented in Table 4.1, and a typicalsolution is shown in Fig. 4.2.

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4.3. The mean magnetic energy tensor

Table 4.1: Combinations of α, β1 and β2 for which the dynamo period P = 22 years and themean field decay time τd = 220 years.

α β1 β2 fβ C[cm s−1] [cm2 s−1] [cm2 s−1]

−25 5.31 × 1010 6.12 × 1011 8.68 × 10−2 −23.2−50 2.80 × 1010 9.24 × 1011 3.03 × 10−2 −7.11−150 1.72 × 1010 1.55 × 1012 1.11 × 10−2 −2.75−500 1.25 × 1010 2.61 × 1012 4.80 × 10−3 −1.42

4.3 The mean magnetic energy tensor

4.3.1 Equations

For the mean magnetic ’energy tensor’ T = 〈BB〉/8π an equation was derived by Knobloch (1978)and Hoyng (1987) in the case of zero ordinary diffusivity:

(∂t + u0 · ∇)Tij =∑

kl

∇k(αǫiklTlj + αǫjklTli) +∑

k

(∇ku0i)Tkj + (∇ku0j)Tki

+25γ(2

k

Tkkδij − Tij) + ∇ · β∇Tij . (4.35)

The mean magnetic energy density is ǫ =∑

iTii. The tensor T is related to the mean magneticstress tensor, which can be expressed in terms of T as 2Tij − ǫδij . We employ the off-diagonalcomponents of T to define correlations coefficients of the field components:

Cij =Tij

TiiTjj=

〈BiBj〉√

〈B2i 〉〈B2

j 〉. (4.36)

These coefficients must satisfy |Cij | ≤ 1. A number of remarks can be made on the individualterms of Eq. (4.35).

The advection term is zero because the differential rotation u0 (Eq. 4.3) is in the y-directionand only axisymmetric solutions are considered, i.e. ∂/∂y = 0. For the gradient of u0 in theconvection zone we insert ∇iu0j = aδixδjy. There is a third dynamo coefficient, related to ther.m.s. vorticity,

γ =1

3τc〈|∇ × u1|2〉. (4.37)

It represents the creation of small scale magnetic field due to the random field line stretchingby the convective motions. The effect of vorticity is to enhance the diagonal components ofT and the mean magnetic energy, but to decrease the off-diagonal components. The α-effectplays only a minor role in the mean energy equation. Its main influence is on the off-diagonalcomponents of T, i.e. on Cij. Ohmic dissipation is not accounted for in Eq. (4.35); this pointwill be discussed further in section 4.4.

As was argued in the introduction, we focus on the fundamental mode of Eq. (4.35). Thismode corresponds to the ensemble average of the non-oscillating part of the magnetic energy

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Figure 4.3: The diagonal components of the mean energy tensor T and the correlation coefficientsCij as a function x (Tyy is set to 1 at x = d2), for a marginally stable solution with fβ = 0.1,lt/d2 = 1 and κ0 = 100 (see Table 4.3). The transition from overshoot layer to convectionzone is smooth, because the toroidal magnetic energy produced in the overshoot layer is rapidlytransported into the convection zone by turbulent diffusion.

density. It is marginally stable, axisymmetric (∂/∂y = 0) and independent of latitude (∂/∂z =0).

Due to its symmetry T has only six independent tensor elements that can be convenientlyarranged into one vector, whose components are Tµ, with µ = xx, xy, xz, yy, yz, zz. We can nowapply Eq. (4.35) to the geometry of the Parker model and under the assumptions mentionedabove, we obtain the following equations for Tµ in the overshoot layer (region 1) and T ′

µ in theconvection zone (region 2):

(∂t − β1∂2x)Tµ =

ν

(aXµν + 25γ1Γµν)Tν , (4.38)

(∂t − β2∂2x)T

′µ =

ν

(αΞµν∂x + 25γ2Γµν)T

′ν , (4.39)

Here X, Ξ and Γ are constant matrices, given in Appendix A. We separate the time dependenceof Tµ with the following ansatz:

Tµ(x, t) = Tµ(x) eΛt, (4.40)

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Figure 4.4: The diagonal components of the mean energy tensor T and the correlation coefficientsCij as a function of x (Tyy is set to 1 at x = d2), for a marginally stable solution with fβ = 10−4,lt/d2 = 1 and κ0 = 100 (see Table 4.3). By far the largest component is Tyy, the mean energyof the toroidal field, which is highly concentrated in the overshoot layer, because turbulentdiffusion is less efficient than in Fig. 4.3. Notice the strong correlation between Bx and By inthe overshoot layer.

and similarly for T ′µ. Now equations (4.38–4.39) reduce to a set of twelve ordinary second order

differential equations, see Appendix B for explicit expressions.

4.3.2 Boundary conditions

A total of 24 boundary conditions are required to specify a solution of Eqs. (4.38–4.39). Forthe details the reader is referred to Appendix C; here it suffice to give a general idea of theirderivation. We can integrate Eq. (4.35) over a small distance δ along the x-axis on either sideof one of the boundaries and let δ ↓ 0, which yields

[[β∂xTij + α∑

l

(ǫixlTlj + ǫjxlTli)]] = 0, (4.41)

where [[...]] indicates the jump at the boundary. This condition is applied at x = −d1 and x = 0.It implies that the energy flux β∂xǫ vanishes at x = −d1 and that it is continuous at x = 0.

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We also assume continuity of T at x = 0. The boundary conditions at x = d2, the convectionzone - vacuum interface, are less trivial. We assume that the magnetic energy density at thesurface is equally distributed among its components Txx, Tyy and Tzz. The idea is that due tothe reprocessing of the (predominantly azimuthal) field on its way up through the convectionzone, the field is small scale and shows no net preference for any direction at the surface. Bythe same token, the off-diagonal elements of T should then vanish near the surface:

T ′xx = T ′

yy = T ′zz

T ′xy = T ′

xz = T ′yz = 0

at x = d2. (4.42)

We assume that the energy flux through the surface is proportional to the energy density itself,

(∂x +κ0

d2) ǫ2 = 0 at x = d2. (4.43)

Here κ0 is a dimensionless constant that measures approximately the ratio of the efficiency ofenergy transport through the surface at x = d2 and through the bulk of the convection zone.The nature of the energy transport at the convection zone - vacuum interface is discussed indetail in Van Geffen & Hoyng (1993b). These authors estimate that κ0 is of the order 30−300.1

When κ0 is much larger than unity, its value affects ǫ2 only in a thin surface layer near x = d2.In fact, for κ0 ≫ 1 all the boundary conditions for T at x = d2 have a negligible effect on thesolutions in the bulk of the dynamo. This is expected, since the dynamo is seated near theinterface of the two regions, while the upper part of the convection zone plays only a passiverole.

4.3.3 Solutions

The general solution of Eqs. (4.38–4.39) is presented in Appendix B. The 24 boundary conditionsconstitute 24 relations between the integration constants, whose determinant should vanish.Every root Λ of this dispersion relation corresponds to a separate mode of Eqs. (4.38–4.39). Weare only interested in the fundamental mode which, as we argued in the introduction, should benon-periodic (Im Λ = 0). It has been observed before, that this is indeed always the case (VanGeffen & Hoyng 1993b), although no formal proof is available. By themselves, the oscillatingmodes are unphysical because they have a magnetic energy of alternating sign. They play a roleas transients in initial value problems, which we are not considering here. In the following, thefundamental mode is simply referred to as ’the solution of the energy equation’ since no othermodes are considered.

4.3.4 Constraints on the parameters

Our search in parameter space for physically acceptable solutions is made easier by a numberof constraints. The first constraint concerns the normalised helicity H in the convection zone,which is defined as

H = − α√β2γ2

=〈u1 · (∇× u1)〉

〈|u1|2〉〈|∇ × u1|2〉. (4.44)

1Van Geffen & Hoyng (1993b), who use (∂x + 1/ρ) ǫ = 0 at the upper surface of the convection zone, findR⊙/ρ ≈ 100 − 1000 where ρ = d2/κ0 is, so that κ0 ≈ 0.3R⊙/ρ ≈ 30 − 300.

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It measures the correlation between the small scale velocity field u1 and the vorticity ∇ × u1

and obeys a Schwartz-type inequality:

|H| ≤ 1. (4.45)

This inequality implies a lower bound on γ2 for given α and β2. No reliable estimate for H isknown, but it is likely that |H| ≪ 1 (Moffatt 1978, p. 284). If applied to the overshoot region,condition (4.45) is automatically satisfied because α is neglected there.

A second constraint is based on the following estimate of the typical length scale of theturbulence in the convection zone (Eqs. 4.2 and 4.37):

lt =

β2

γ2≤ d2. (4.46)

Obviously the turbulent length scale, i.e. the typical size of a convective cell, should not exceedthe thickness of the convection zone. Hence there is the condition lt ≤ d2, which implies anotherlower bound on γ2. This condition turns out to be much stronger than (4.45).

The central idea of Parker (1993) is that strong magnetic fields suppress the turbulence inthe overshoot layer. For simplicity we assume that γ is suppressed by the same factor fβ ≪ 1as β:

γ1

γ2= fβ. (4.47)

4.3.5 Marginally stable solutions

4.3.5.1 Values of the parameters

We set lt = d2 and thereby adopt the minimal value of γ2 that is compatible with Eq. (4.46).We allow a somewhat wider range of values for fβ here than we did for the mean field equation(see Table 4.1), and adopt four different values between fβ = 0.1 and fβ = 10−4. The valueof γ1 then follows from Eq. (4.47). The α-coefficient plays an insignificant role in the energyequation, and a change in α has a negligible effect on the growth rate Λ. For this reason we donot vary α in Eq. (4.39), but set α = −150 cm s−1. We adopt two values for κ0, namely 100and 300. The parameters d1, d2 and a are the same as in Fig. 4.1.

We obtain marginally stable solutions using the following method. First we make a guessof the required value of β2, solve the dispersion relation and thus obtain the growth rate of thefundamental mode. We then apply successive corrections to β2, while keeping fβ and lt constant,until Λ equals zero. The physical idea behind this method is that we need to make the turbulentdiffusion efficient enough to transport all the produced magnetic energy to the upper boundaryof the dynamo (we shall consider the energy balance with more detail in section 4.3.5). Thevalues of β2 that are required for marginal stability of T and the corresponding values of γ2 andH are shown in Table 4.2. Since in reality lt may be smaller than d2, vorticity may play a largerrole than is assumed here, so that these values of β2 must be understood as minimum values.

4.3.5.2 Properties of the mean magnetic energy tensor

Figures 4.3 and 4.4 show solutions with the smallest and the largest adopted value of fβ respec-tively. They serve to illustrate how T is affected by varying the degree of suppression of theturbulence in the overshoot layer. Some general features of the energy tensor are:

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Table 4.2: Parameters of marginally stable solutions of the equation for the mean magneticenergy tensor, with lt/d2 = 1.

κ0 fβ β2 γ2 H[cm2 s−1] [s−1]

100 10−1 3.1 × 1014 7.8 × 10−7 9.6 × 10−3

” 10−2 5.5 × 1014 1.4 × 10−6 5.4 × 10−3

” 10−3 1.6 × 1015 4.1 × 10−6 1.8 × 10−3

” 10−4 7.1 × 1015 1.8 × 10−5 4.2 × 10−4

300 10−1 3.0 × 1014 7.4 × 10−7 1.0 × 10−2

” 10−2 5.3 × 1014 1.3 × 10−6 5.7 × 10−3

” 10−3 1.6 × 1015 4.0 × 10−6 1.9 × 10−3

” 10−4 7.0 × 1015 1.8 × 10−5 4.3 × 10−4

⊙ the dominant component of T is Tyy, the magnetic energy of the toroidal field. It reachesits maximum value in the overshoot region and, the smaller fβ, the stronger it declinesnear the base of the convection zone.

⊙ A strong correlation between Bx (the radial field) and By (the toroidal field) is observed inthe overshoot region, which is understandable because of the strong velocity shear in they-direction, transforming radial into toroidal field in a systematic way. The correlationsare much weaker in the convection zone.

⊙ The larger the ratio fβ is, the smoother the transition from the overshoot layer to theconvection zone is. With a larger β1, the efficiency of turbulent diffusion is increased inthe overshoot layer. The magnetic energy of the toroidal field is then transported out ofthe overshoot layer more rapidly, and it has less opportunity to be enhanced by differentialrotation. This is clear from the correlation coefficient Cxy which is reduced for x < 0 butenhanced for x > 0, and from the reduced contrast between Txx, Tyy and Tzz.

⊙ The sharp drop of Txx, Tyy, Tzz and ǫ near the surface x = d2 has the following origin.The rate of energy transport through the dynamo is set by the slowest transport process,which for κ0 ≫ 1 is turbulent transport in the convection zone. At the surface a muchmore efficient transport process takes over so that ǫ and hence Txx, Tyy and Tzz decline.

A detailed analysis of the energy balance of marginally stable solutions is given in the nextsection. An important conclusion is that marginal stability (Re Λ = 0) is reached only if β2

>∼

3× 1014 cm2 s−1, a very high value also found by Van Geffen (1993c). The implications of suchlarge turbulent diffusivities for the mean magnetic field are considered in section 4.3.5

4.3.5.3 Energy balance

The interpretation of Eqs. (4.38-4.39) can be clarified by considering the equations for the meanmagnetic energy density, ǫ1 =

iTii in the overshoot layer and ǫ2 =∑

iT′ii in the convection

zone:

∂tǫ1 = 2aTxy + (2γ1 + β1∂2x) ǫ1, (4.48)

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4.3. The mean magnetic energy tensor

Figure 4.5: Mean magnetic energy density ǫ as a function of x (ǫ is set to 1 at x = d2) for thetwo marginally stable solutions of Figs. 4.3 (fβ = 0.1, drawn curve) and 4.4 (fβ = 10−4, dashedcurve). Energy is produced by differential rotation in the overshoot layer and by vorticity inboth layers, and it is transported by turbulent diffusion. For fβ = 0.1, the transition from theovershoot layer to the convection zone is smooth. The smaller fβ is, the less efficient turbulenttransport is in the overshoot layer, which gives rise to a high concentration of magnetic energy.

∂tǫ2 = (2γ2 + β2∂2x) ǫ2. (4.49)

These equations follow directly from Eq. (4.38–4.39) by contraction. We point out that α doesnot appear here. It can be shown that if α = 0 but γ1 and γ2 6= 0, then Txz = Tyz = 0, whilethe other tensor components are almost unaffected. Mean magnetic energy is not created bythe α-effect, but by vorticity and differential rotation. The α-effect is, however, vital for theproduction of the mean field, see section 4.2. A typical solution is shown in Fig. 4.5.

From Eqs. (4.48-4.49) we obtain the total energy balance by integrating over the volumeof the dynamo region. Since T is independent of y and z, only the integration over the radialcoordinate x remains, yielding

∂tE1 = ΛE1 = Qω +Qγ1 − F1, (4.50)

∂tE2 = ΛE2 = Qγ2 + F1 − F2. (4.51)

The physical dimension of each of the terms is erg cm−2 s−1. The total magnetic energies ofboth regions per unit of surface area are

E1 =

∫ 0

−d1dx ǫ1, E2 =

∫ d2

0dx ǫ2. (4.52)

The energy production rate due to differential rotation is

Qω = 2a

∫ 0

−d1dxTxy. (4.53)

There are two contributions from vorticity:

Qγ1 = 2γ1E1, Qγ2 = 2γ2E2. (4.54)

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Figure 4.6: Energy flow diagram for a marginally stable solution with lt/d2 = 1, fβ = 0.1 andκ0 = 100. All the energy production rates and fluxes are normalised with respect to Qtot. Mostof the magnetic energy resides in the convection zone.

Figure 4.7: Energy flow diagram for a marginally stable solution with lt/d2 = 1, fβ = 10−4 andκ0 = 100. All the energy production rates and fluxes are normalised with respect to Qtot. Mostof the magnetic energy resides in the overshoot layer.

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4.3. The mean magnetic energy tensor

The terms mentioned so far are source terms and provide a total energy production rate of

Qtot = Qω +Qγ1 +Qγ2. (4.55)

Turbulent diffusion enables the transport of mean magnetic energy out of the dynamo layers,if allowed by the boundary conditions, and constitutes the sole energy sink in our model. Asstated by boundary condition (4.89), there is no energy flux through x = −d1 into the solarinterior. The energy flux F1, from the overshoot layer into the convection zone, and F2, throughthe upper surface of the convection zone, are defined as

F1 = −β1∂xǫ1|x=0, F2 = −β2∂xǫ2|x=d2 . (4.56)

Outward fluxes are positive; boundary condition (4.43) ensures that F2 > 0. Continuity of fluxat x = 0 is guaranteed by Eqs. (4.91).

If production and loss of magnetic energy are in balance, i.e. F2 = Qtot, the growth rateΛ is zero and the dynamo is marginally stable. By virtue of boundary condition (4.43), theenergy flux F2 depends on κ0. If we adopt the maximum value, κ0 = 300 rather than κ0 = 100,we see from Table 4.3 that only slightly smaller values of β2 are required for marginal stability.The physical reason is that a change in κ0 influences only a thin layer near x = d2 and doesnot significantly alter the energy balance: the amount of flux that is transported up to x = d2

is determined in the bulk of the dynamo. When this flux reaches x = d2, the rapid decline ofǫ2 (Fig. 4.5) guarantees that, more or less independent of the magnitude of the flux, it is allrapidly lost through the boundary. Thus only a more efficient transport of energy through theconvection zone could significantly increase the flux F2.

Figures 4.6 and 4.7 give a graphic representation of the energy flow through the dynamo fortwo typical marginally stable solutions.

4.3.5.4 Root mean square magnetic field strength

Since the equation for T is linear and all the boundary conditions are linear as well, T isdetermined up to an arbitrary multiplicative factor. We can calibrate T by an external estimateof the mean energy flux or of the r.m.s. field strength, through the relation Brms =

√8πǫ. Three

such estimates are considered here.

The first is based on the following argument. Marginal stability means that all the energythat is produced in the dynamo is transported away through the upper surface of the convectionlayer. This energy flux, presumably carried by Alfven waves, can act as a heating mechanismfor the corona. The total energy flux that is required for the heating process is estimated tobe Fc ≈ 5 × 106 erg cm−2 s−1 (Withbroe & Noyes 1977, Kuperus et al. 1981). With boundarycondition (4.43) for the energy flux at the surface of the convection zone an expression for ther.m.s. surface field strength is obtained:

Brms(d2) =

8πFcd2

β2κ0(4.57)

where κ0 is a dimensionless constant that is estimated to be 30-300 (see Eq. 4.43).

The second estimate concerns the r.m.s. field strength on the solar surface due to activeregions and small flux tubes, which are known to have a field strength of about 103 G. Theycover about 1% of the surface, so that Brms ≈

√10−2 × 106 = 100 G. The third estimate concerns

the r.m.s. field strength in the overshoot layer, which is believed to be about 2.5× 104 − 105 G.

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Table 4.3: Root mean square field strength at the base of the overshoot layer and the top of theconvection zone for marginally stable solutions with α = −150 cm s−1 and lt/d2 = 1. The fieldstrengths are calibrated by assuming F2 = Fc.

κ0 fβ rB Brms(d2) Brms(−d1)[G] [G]

100 10−1 8.8 9.0 79” 10−2 12 6.8 84” 10−3 32 3.9 125” 10−4 95 1.9 178

300 10−1 15 5.3 81” 10−2 22 4.0 87” 10−3 55 2.3 127” 10−4 170 1.1 186

Ideally, the model should reproduce all three estimates. Unfortunately the first two turnout to be hard to reconcile for marginally stable solutions. The maximum r.m.s. surface fieldstrength that we obtain in our calculations by applying Eq. (4.57) is a mere 8.8 G, see Table 4.3.For this same solution we could adopt Brms = 100 G at x = d2 but then the energy flux(Eqs. 4.43 and 4.56) would be too large, namely F2 = 6.2 × 108 erg cm−2 s−1 ≈ 120Fc. Thesetwo conditions can be made compatible only by adopting much smaller values for β2 or κ0. Butβ2 is determined by the constraint of marginal stability and we cannot decrease κ0 as much aswould be required.

We can also judge the solutions by the maximum ratio of field strengths rB that is attained:

rB =Brms(−d1)

Brms(d2). (4.58)

The estimated value for the Sun is rB ≈ 250–1000, but this value is not attained by our solutions(Table 4.3). The highest ratio, rB = 170, is found with fβ = 10−4 and κ0 = 300. This ratiodepends on fβ and κ0. The smaller fβ, i.e. the less efficient turbulent diffusion in the overshootlayer is, the more magnetic energy accumulates there, and the larger rB is. We could reach avalue of rB that lies within the estimated range, by adopting a ratio fβ < 10−4, but we havenot pursued this since the constraint of marginal stability would require a turbulent diffusivityβ2 larger than 7 × 1015 cm2 s−1. This would cause a further decrease of the r.m.s. surface field(Eq. 4.57). We could also achieve larger values of rB by adopting a value of κ0 larger than 300,but this would also cause a decrease of Brms(d2).

An alternative method of fixing a reference value for T is to start at the base of the overshootlayer and adopt Brms = 2.5 × 104 G at x = −d1. For fβ = 10−4 and κ0 = 300 we obtain ar.m.s. surface field that is closest to the estimated value of 100 G, namely Brms(d2) = 2.5 ×104/170 ≈ 150 G. We had shown already that the conditions F2 = Fc and Brms(d2) = 100G are incompatible for marginally stable solutions. With Brms(d2) = 150 G, fβ = 10−4 andβ2 = 7.0 × 1015 cm2 s−1, the energy flux at the solar surface (Eqs. 4.43 and 4.56) becomes9.2 × 1010 erg cm−2 s−1 ≈ 1.8 × 104 Fc. Clearly, the model predicts a much too large energyflux, even if we adopt the minimum r.m.s. field strength of 2.5 × 104 G in the overshoot layer.

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Table 4.4: Properties of growing solutions.

α [cm s−1] 1/Λ [months] F2/Qtot lt/d2

−25 1.44 5.4 × 10−7 0.39−50 1.41 1.0 × 10−5 0.29−150 0.96 1.1 × 10−3 0.16−500 0.20 9.1 × 10−3 0.08

4.3.5.5 Implications for the mean magnetic field

The diffusivities that are required to obtain marginally stable solutions of the mean energyequation are at least a factor 102 larger than the values that were adopted for the mean fieldequation (Table 4.1). The effect of large diffusivities on the mean field can be clarified withthe help of Eq. (4.31) for the dynamo frequency. If ω = ω⊙ and β2 = 3 × 1014 cm2 s−1 thenω/β2k

2z ≈ 3 × 10−3 ≪ 1 so that |C| ≪ 1. Then Eq. (4.31) becomes

ω ≈ β2k2z |C|/4 =

β1a|α|4β2

2kz, (4.59)

which, after substitution of ω⊙, a and kz, yields the following estimate of the value of |α|,required for a solar-type mean field:

|α| ≈ 9 × 10−13β2/fβ . (4.60)

Although α does formally appear in Eq. (4.38–4.39), it has a negligible influence on the growthrate Λ. In other words, T remains almost marginally stable even if a different value of α isadopted. Hence we have some liberty to change α for a given solution of Eqs. (4.38–4.39). Theturbulent diffusivities of our marginally stable solutions in Table 4.2 imply that we need |α|to be of the order 3 × 103 − 6 × 105 cm s−1 (Eq. 4.60). But we have the following two upperlimits for |α|. First, there is condition (4.45). Since the solutions presented in Table 4.2 all havelt/d2 = 1, we may employ Eq. (4.46) to write this condition in the form |α| ≤ β2/d2. Second,|α| should not exceed the typical maximum value lΩ ≈ 2 × 103 cm s−1 in the lower part of theconvection zone (section 4.2.4). Commonly |α| is assumed to be about a factor 102 smaller.Some of the required values of α violate condition (4.45), and all of them exceed lΩ. In otherwords, α would have to be unphysically large to counteract the strong turbulent diffusion andproduce a dynamo period of 22 years.

4.3.6 Growing solutions

From the previous section it followed that the turbulent diffusivities required for marginal sta-bility of the mean magnetic energy are incompatible with the turbulent diffusivities implied bythe mean field equation. In this section the constraint of marginal stability is abandoned andwe adopt the much lower diffusivities from section 4.2. As a consequence, turbulent diffusionis no longer capable of transporting all the produced energy to the boundary of the convectionzone (see the ratio F2/Qtot in Table 4.4 and Fig. 4.8), and the magnetic energy grows exponen-tially. We have chosen to determine γ2 by adopting H = 0.3 (Eq. 4.44); the resulting turbulentlength scales lt (Table 4.4) all satisfy constraint (4.46). Since the solutions grow exponentially,

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Figure 4.8: Energy flow diagram for an exponentially growing solution with parameters as inTable 1 with α = −150 cm s−1 and H = 0.3. All the energy production rates and fluxes arenormalised with respect to Qtot. Only a small fraction of the produced energy escapes throughthe upper surface of the convection zone.

we cannot calibrate the magnetic field strength or the energy fluxes. Instead we focus on thetypical growth times 1/Λ (Table 4.4), and we note that they all are of the order of 1 month.This suggests that our equation for T is deficient because an energy loss mechanism operatingon a typical time scale of 1 month has not been included.

4.4 Summary and discussion

We have solved equations for the mean magnetic field B0 and for the mean magnetic energytensor T, in the geometry of Parker’s surface wave dynamo. The interpretation of B0 and Tin terms of an ensemble average gave rise to the following constraints: the fundamental modeof the mean field has a period of 22 years and a decay time of 10 dynamo periods, and thefundamental mode of the mean magnetic energy is non-periodic and marginally stable.

For marginal stability of T we require a turbulent diffusivity of the order β2>∼ 3×1014 cm2 s−1.

Then α has to be unphysically large for the mean field to have a period of 22 years.

Furthermore the r.m.s. field strengths and energy fluxes cannot simultaneously assume solarvalues: if the energy flux F2 is equated to Fc, the estimated flux required for heating the corona,then field strengths are one to two orders of magnitude too small. Conversely, if we adoptBrms = 100 G at the surface of the convection zone or Brms = 2.5 × 104 G at the base of theovershoot layer, then the energy flux F2 is at least a factor 120 too large.

The requirement that the maximum ratio of field strengths rB be in the range 250–1000 canbe fulfilled only if fβ <

∼ 10−4. For marginal stability of T we then need β2 > 7 × 1015 cm2 s−1,which aggravates our dilemma.

Concerning the large diffusivities that are required, we confirm the results of Van Gef-fen (1993d), who found similar values, in spite of the fact that the overshoot layer was badlyresolved in his finite grid calculation. It is evident that transport of mean magnetic energy by

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4.5. Appendix A: Dispersion relation for the mean magnetic field

turbulent diffusion cannot provide an energy sink that is efficient enough to make T marginallystable. The main conclusion of this paper is therefore that one must envoke an alternative energysink that is not accounted for in Eq. (4.35) to balance the energy budget.

One possible mechanism is magnetic buoyancy. Due to the ’undulatory instability’, fluxtubes in the overshoot layer that reach a critical field strength of about 105 G will start to riseand cross the entire convection zone within about a month (Moreno-Insertis 1992, Caligari etal. 1995). Thus the time scale for magnetic buoyancy is comparible to the typical e-folding time1/Λ of our growing solutions in section 4.3.6. But for marginal stability of the magnetic energyit is required that all the magnetic energy is transported out of the overshoot layer within abouta month. Such a high rate of flux loss is not observed on the Sun. Although buoyancy can beimportant for limiting the growth of the magnetic field, its importance for the energy balancemay therefore be small. Furthermore there exists no satisfying treatment of buoyancy in meanfield dynamo theory, since it is an essentially non-linear effect.

Another possible candidate is Ohmic dissipation, which operates through an energy cascadefrom large scales to the very small dissipative scale, where the magnetic field is dissipated andconverted into heat. The time scale on which the energy cascade transports energy to thesmallest length scale is thought to be no longer than the eddy turnover time in the convectionzone, which is on the order of one month. Thus also for this loss mechanism the time scale agreesroughly with the e-folding time of our growing solutions. Indeed, hydromagnetic calculations byNordlund et al. (1992) suggest that all the magnetic energy is destroyed by resistive dissipation ata timescale of about half a convective turnover time. Their calculations were done with boundaryconditions that do not allow escape of energy, so that they do not address the question aboutthe importance of flux loss for the energy balance. However, they demonstrate that ohmicdissipation can in principle provide a sufficiently large energy loss term. But the inclusion ofresistive dissipation into the equation for T is non-trivial. The difficulty lies in a closure problemthat arises from the scale-dependence of resistive dissipation. Due to the scale dependence, onecan no longer derive a closed equation for T, since T is a mean quantity that has contributionsfrom all length scales. This issue is studied in a subsequent paper.

Acknowledgements

This research was supported by the Dutch Foundation for Research in Astronomy (ASTRON).

4.5 Appendix A: Dispersion relation for the mean magnetic field

The eight boundary conditions for the mean field provide the following relations between theintegration constants:

e−κ1d1A1 + eκ1d1B1 = 0, (4.61)ad1kz(κ1d1 − 1)

2β1κ1A1 − κ1A2

e−κ1d1 +ad1kz(κ1d1 + 1)

2β1κ1B1 + κ1B2

eκ1d1 = 0, (4.62)

A2 +B2 −A4 −B4 = 0, (4.63)

A1 +B1 − (A3 +B3)/fd = 0, (4.64)

fdκ1(A1 −B1) − κ2(A3 −B3) +α

2β2κ2d2(A4 −B4) = 0, (4.65)

iad1kz2β2κ1

(A1 −B1) + fβκ1(A2 −B2) − κ2(A4 −B4) = 0, (4.66)

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e−κ2d2A4 + eκ2d2B4 = 0, (4.67)

(kz + κ2) eκ2d2A3 + (kz − κ2) e−κ2d2B3 −α

2β2

1 +1 + kzd2

κ2d2

eκ2d2A4

− α

2β2

1 − 1 + kzd2

κ2d2

e−κ2d2B4 = 0. (4.68)

Here fd = d1/d2. These equations can be written as an 8 by 8 matrix that operates on theintegration constants A1 · · · B4. The dispersion relation is obtained by demanding that thedeterminant of the matrix vanishes.

4.6 Appendix B: Expressions for the mean magnetic energy ten-

sor

With ansatz (4.40) Eqs. (4.38-4.39) become

(Λ − β1∂2x) Tµ =

ν(aXµν + 25γ1Γµν) Tν , (4.69)

(Λ − β2∂2x) T

′µ =

ν(αΞµν∂x + 25γ2Γµν) T

′ν . (4.70)

Here the matrices Γ, X and Ξ are

Γ =

1 0 0 2 0 20 −1 0 0 0 00 0 −1 0 0 02 0 0 1 0 20 0 0 0 −1 02 0 0 2 0 1

,X =

0 0 0 0 0 01 0 0 0 0 00 0 0 0 0 00 2 0 0 0 00 0 1 0 0 00 0 0 0 0 0

, (4.71)

Ξ =

0 0 0 0 0 00 0 −1 0 0 00 1 0 0 0 00 0 0 0 −2 00 0 0 1 0 −10 0 0 0 2 0

. (4.72)

Only Ξ and Γ can be diagonalised simultaneously. Therefore Eq. (4.70) is easily solved, but weshall start with Eq. (4.69), which is a bit more cumbersome. First we note that

Txx − Tzz = C1ek1x +D1e

−k1x, (4.73)

Txz = C3ek1x +D3e

−k1x, (4.74)

where k1 =√

(Λ + 25γ1)/β1. Next we solve Tyz, which is coupled to Txz in a resonant manner:

Tyz = C5ek1x +D5e

−k1x − ax

2β1k1(C3e

k1x −D3e−k1x). (4.75)

The following equation for w = (Txx + Tzz, Txy, Tyy) remains:

(Λ + 25γ1 − β1∂

2x)w =

85γ1 0 8

5γ1

a/2 0 045γ1 2a 4

5γ1

w + 1

2a(Txx − Tzz)

010

. (4.76)

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4.6. Appendix B: Expressions for the mean magnetic energy tensor

The matrix M on the right hand side is transformed into diagonal form UMU−1 by means of atransformation matrix U which is not reproduced here explicitly. Defining w′

µ =∑

ν Uµνwν , weobtain from Eq. (4.76)

w′µ = Cµe

kµx +Dµe−kµx − aUµ4

2mµ(Txx − Tzz) (µ = 2, 4, 6), (4.77)

where mµ are the eigenvalues of M, i.e. the solutions of

m3µ − 24

25γ21m

2µ − 8

5a2γ1 = 0. (4.78)

The wavenumbers kµ are

kµ =√

(Λ + 25γ1 −mµ)/β1 (µ = 2, 4, 6). (4.79)

The solutions of Eq. (4.76) are

Txx + TzzTxyTyy

= U−1

w′2

w′4

w′6

. (4.80)

Equations (4.73–4.75) and (4.80) provide the expressions for all tensor components in region 1.

In region 2 we proceed as follows. Ξ and Γ are transformed to diagonal forms SΞS−1 andSΓS−1 respectively, where S is the corresponding transformation matrix (not reproduced here).Solutions of Eq. (4.70) are easily found on this basis and then transformed back, so that

T ′µ =

ν(S−1)µνhν (µ = xx, xy, .., zz; ν = 7, 8, .., 12), (4.81)

where

hµ = Cµekµ+x +Dµe

kµ−x (µ = 7, 8, .., 12). (4.82)

The wavenumbers kµ± are

k7± = ±√

(Λ − 2γ2)/β2, (4.83)

k8± = −iα

2β2±√

(Λ + 25γ2)/β2 − α2/4β2

2 , (4.84)

k9± = iα

2β2±√

(Λ + 25γ2)/β2 − α2/4β2

2 , (4.85)

k10± = ±√

(Λ + 25γ2)/β2, (4.86)

k11± = −iα

β2±√

(Λ + 25γ2)/β2 − α2/β2

2 , (4.87)

k12± = iα

β2±√

(Λ + 25γ2)/β2 − α2/β2

2 . (4.88)

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MEAN MAGNETIC FIELD AND ENERGY BALANCE OF PARKER’S SURFACE WAVE DYNAMO

4.7 Appendix C: Boundary conditions for the mean magnetic

energy tensor

In the solar interior (x < −d1), there is no turbulent diffusivity, so that β in Eq. (4.35) assumesthe negligible molecular value η ≪ β1. There is no α-effect below the convection zone (α = 0for x < 0). If we then apply Eq. (4.41) to x = −d1, we find β1∂xTij|−d1 = 0, i.e.

∂xTµ = 0 at x = −d1 (µ = xx, xy, .., zz). (4.89)

Consequently the energy flux across this boundary is zero: ∂xǫ1|−d1 = 0. Continuity of T atx = 0 yields

Tµ = T ′µ at x = 0 (µ = xx, xy, .., zz). (4.90)

We apply Eq. (4.41) at x = 0, which provides

β1∂xTxx = β2∂xT′xx

β1∂xTxy = β2∂xT′xy − αT ′

xz

β1∂xTxz = β2∂xT′xz + αT ′

xy

β1∂xTyy = β2∂xT′yy − 2αT ′

yz

β1∂xTyz = β2∂xT′yz + α(T ′

yy − T ′zz)

β1∂xTzz = β2∂xT′zz + 2αT ′

yz

at x = 0. (4.91)

Note that the first, fourth and sixth of these equations together guarantee a continuous energyflux across x = 0. For the conditions at the upper surface of the convection zone, we refer tothe main text.

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Chapter 5

Energy balance and resistive

dissipation in Parker’s surface wave

dynamo

AbstractWe study the surface-wave dynamo model for the Sun, proposed by Parker (1993), by solving equa-tions for the mean field B0 and for the mean magnetic ’energy tensor’ T = 〈BB〉/8π. The followingconstraints apply: the mean field has a period of 22 years and, taking into account the effect ofsolar cycle variability, a decay time of about 10 dynamo periods. The mean magnetic energy ismarginally stable. In Ossendrijver & Hoyng (1996c = ch. 4) it was shown that an extra loss termwas required in the equation for T. Here we explore the possibility that resistive dissipation providesthe extra term. It is shown that with a heuristically modified equation for T, the inconsistencies ofch. 4 are removed. Our main results are as follows. Most of the magnetic energy is produced bydifferential rotation in the overshoot layer, a smaller fraction by vorticity. Assuming a r.m.s. surfacefield of the order 100 G, we find a maximal r.m.s. field strength in the overshoot layer of the orderBmax

rms = (0.2 − 1.0) × 105 G, in agreement with current estimates for the Sun. The energy fluxthrough the upper surface of the convection zone is about (1 − 6) × 106 erg cm2 s−1, enough forheating the solar corona. This flux amounts to only a fraction of at most 7 × 10−4 of the rate atwhich magnetic energy is produced, the rest is dissipated within the dynamo itself on a timescale ofabout 0.4− 3 months. Such a high dissipation rate calls for a very efficient build up of the magneticenergy by differential rotation, on a time scale of days.

A.J.H. Ossendrijver and P. Hoyng

Astronomy & Astrophysics (submitted)

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ENERGY BALANCE AND RESISTIVE DISSIPATION IN PARKER’S SURFACE WAVE DYNAMO

5.1 Introduction

Both theoretical and observational arguments indicate that the solar dynamo is located in arelatively thin overshoot layer near the base of the convection zone (Galloway & Weiss 1981,Hughes 1992). In this layer, which rotates differentially, the magnetic field can be stored longenough due to the subadiabatic stratification to be amplified up to values of the order 105 G,well above the equipartition value Beq ≈ 103 G (Schussler et al. 1994). As a result of Lorentzforces, these strong fields suppress the turbulence. In the convection zone, the magnetic fieldis much weaker because flux tubes with field strengths that exceed Beq are rapidly lost due tobuoyancy forces. Thus the turbulence is not suppressed here, and the Coriolis force can acton the convective motions, which in a stratified medium leads to a non-vanishing net kinetichelicity 〈u1 · (∇× u1)〉. Here u1 denotes the turbulent, convective velocity field,

u1 = u− u0, (5.1)

and u0 denotes the large scale velocity field that is responsible for the strong fields in theovershoot layer. The cyclonic motions give rise to the α-effect of mean field electrodynamics,which provides the following equation for the mean magnetic field B0:

∂tB0 = ∇× [u0 × B0 + αB0 − β∇× B0]. (5.2)

The α-parameter and the turbulent diffusivity are

α ≈ −1

3τc〈u1 · (∇× u1)〉, β ≈ 1

3τc〈u2

1〉, (5.3)

where τc is the typical correlation time of u1.The surface-wave dynamo, proposed by Parker (1993) is a model for the solar dynamo that

is based on the spatial separation of differential rotation and the α-effect. The idea is thatdue to the suppression of the turbulence in the overshoot layer, the α-effect and differentialrotation appear to be concentrated in adjacent but distinct locations. The mean magnetic fieldis generated near the interface of the convection zone and the overshoot layer.

As Parker (1993) did, we adopt a local approximation and do our calculations in a planeparallel geometry (see Fig. 5.1). Cartesian coordinates are used with x denoting the radial, ythe longitudinal and z the azimuthal coordinate. Only axisymmetric solutions are considered.The dynamo consists of an overshoot layer between x = −d1 and x = 0 and a convection zonebetween x = 0 and x = d2. The α-effect exists only in the convection zone. The turbulentdiffusivity is β1 in the overshoot layer and β2 in the convection zone, with

fβ =β1

β2≪ 1. (5.4)

Differential rotation is prescribed by

u0 = u0(x) ey , ∂xu0 =

a (−d1 < x < 0)0 (0 < x < d2).

(5.5)

The rotation rate increases outward, as is observed to be the case at low latitudes (Goode 1995).Our approach to mean field dynamo theory is statistical and is based on the ’finite energy

method’ (see Van Geffen & Hoyng 1993b, Van Geffen 1993c and ch. 4). If the parameters αand β are taken as non-fluctuating constants, as is usually the case, the dynamo equation (5.2)applies strictly only for a mean field that is defined as an ensemble average. But the magnetic

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5.2. Mean field dynamo theory and resistive dissipation

Figure 5.1: Geometry of Parker’s interface wave dynamo. In our calculations we adopted d1 =2 × 109 cm, d2 = 2 × 1010 cm and a = 4 × 10−6 s−1.

field of each ensemble member exhibits period variations of the order ∆P/P ≈ 0.1, and theconsecutive variations are probably independent (see Ossendrijver et al. 1996b). Suppose thatall the ensemble members are synchronised at t = 0, i.e. the distribution of phases is a delta-function initially. Then, as time progresses, the distribution broadens as a result of phasemixing, and the final state is a homogeneous distribution of phases over the 22 year cycle,i.e. over positive and negative polarities. Since contributions to the mean field from ensemblemembers with opposite polarities cancel out, the mean field decays, and it vanishes in the finalstate. We can estimate the typical decay time of the mean field as follows. Since the periodvariations have a relative size of about 10%, the typical time scale for the phase mixing, i.e. thecoherence time of the actual magnetic field, can be estimated as 10 dynamo periods. Thus themean field has a decay time of about 10 dynamo periods.

The result of phase mixing is different for the magnetic energy density B2/8π, because it isa positive definite quantity. Applying the ensemble average, we obtain a ’fundamental mode’with a constant, non-periodic mean magnetic energy, and a set of decaying overtones (see ch. 4for more details).

Thus we solve equations for the mean field B0 and the ’mean energy tensor’ T = 〈BB〉/8πunder the following constraints: the mean field has a period of 22 years and a decay time ofabout 10 dynamo periods, and the mean magnetic energy is stationary and non-periodic.

5.2 Mean field dynamo theory and resistive dissipation

Both the mean field equation and the mean energy equation are derived from the inductionequation

∂tB = ∇× [u× B − η∇× B]. (5.6)

The velocity field u (Eq. 5.1) has a large scale component that is more or less constant in time,and a rapidly varying turbulent component. This suggests that in the kinematic limit Eq. (5.6)can be treated as a linear stochastic differential equation with a multiplicative driving term u1.Under certain assumptions, the theory of stochastic differential equations (see for instance Van

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ENERGY BALANCE AND RESISTIVE DISSIPATION IN PARKER’S SURFACE WAVE DYNAMO

Kampen 1992) provides an equation for the mean field B0 = 〈B〉. Here it suffice to mentionthat for a stochastic differential equation of the form

∂tA = [L0 + L1]A (5.7)

where L0 is a time independent operator and L1 is a stochastic operator with 〈L1〉 = 0, oneobtains

∂t〈A〉 = [L0 +

∫ ∞

0ds 〈L1(t)L1(t− s)〉] 〈A〉. (5.8)

This is a general result, valid if the correlation time τc of the stochastic operator is sufficientlyshort (τc|L1| ≪ 1 and τc|L0| ≪ 1). It is questionable whether the former condition applies to theSun, but we ignore this well-known problem of mean field electrodynamics for the moment. Inthe case of the induction equation (5.6), the term containing u1 is identified as L1 and the otherterms (including the dissipative term) form L0. A derivation of the mean field equation (5.2)is presented e.g. by Krause & Radler (1980) or Hoyng (1992). In principle the same procedurecan be applied to what will henceforth be called the ’magnetic energy tensor’ BB/8π, whichobeys the following equation, obtained from Eq. (5.6) by adding Bi∂tBj +Bj∂tBi:

(∂t + u · ∇)BiBj =∑

k

(∇kui)BkBj + (∇kuj)BkBi

+ η∇2BiBj − 2η∑

k

(∇kBi)(∇kBj). (5.9)

Unfortunately, this equation as it stands cannot be treated in the same manner as the inductionequation itself. The reason is that it does not represent a closed equation for BB, i.e. it cannotbe written in the form ∂tBB = LBB for some differential operator L. The closure problemis caused by the term −2η

k(∇kBi)(∇kBj). One manner in which this difficulty has beenresolved is to ignore resistive dissipation, assuming that it plays no role in the energy balanceof the dynamo. In that case, a closed equation is obtained for the energy tensor and the samemethod as was used for averaging the induction equation can be applied (see Knobloch 1978and Hoyng 1987).

However, solutions of this approximative equation turn out to be problematic. The core ofthe problem is that the only energy sink available to the dynamo is the energy flux throughthe upper surface of the convection zone (see section 5.3.5). Marginal stability of the energycan then be realised only by making the energy transport through the dynamo by means ofturbulent diffusion sufficiently effective, which in practice means that turbulent diffusivity hasto be unphysically large; see ch. 4 for a discussion of this problem.

Resistive dissipation provides an additional energy sink, operating through a cascade fromthe largest scale to the small length scale where the field is dissipated. However, unlike for themean field equation where it suffices to define a total diffusivity β + η, it is clear for the reasonmentioned above that no simple recipe is available for modifying the mean energy equation totake into account resistive dissipation. This fundamental difference between the mean field andthe mean magnetic energy originates in the different treatment of small scale fields. The meanfield gives a description of only the largest scales in the dynamo, since the rapidly varying smallscale fields cancel out in the averaging procedure. The magnetic energy on the other hand,contains all length scales, as can be seen by writing B = B0 + δB, which provides

〈B2〉 = B20 + 〈(δB)2〉. (5.10)

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5.2. Mean field dynamo theory and resistive dissipation

This explains why resistive dissipation does not affect B0, but can be very important for T.

In this paper an attempt is made to treat the effect of resistivity on the mean energy tensorin a heuristic and approximative way. For turbulent fluids with high Reynolds numbers theenergy distribution over the length scales is characterised by the inertial-range spectrum. Theinertial range is defined by (see for instance Moffatt 1978, ch. 11 or Biskamp 1993, p. 196)

ld ≪ l ≪ lin. (5.11)

Three different length scales can thus be distinguished:

1. a large injection length scale lin ≈ d2 related to variations of the mean field and the meanenergy;

2. the inertial range length scales l related to the stretching and folding of the field lines dueto the turbulent velocity;

3. the dissipative length scale ld. By definition, this is the length scale at which the dissipativetime scale equals the energy transfer time scale.

The magnetic energy is transferred through a cascade from lin down to ld, where resistivedissipation becomes important. For stationary turbulence, conservation of energy implies thatthe injection rate, the transfer rate and the dissipation rate of the magnetic energy are equal.Hence, in an equilibrium situation, resistive dissipation should balance the energy production.Whatever the energy injection rate, a balance is achieved by continuing the energy cascade downto a sufficiently small value of ld. This suggests the following simple approximation: we replacethe last term of Eq. (5.9) by a scale-independent term −2ν BiBj , and we omit the term η∇2BiBj,since it will give rise to a negligible term after averaging (similar to the treatment of resistivedissipation in the equation for B0). The value of the dissipation coefficient ν is dictated by theenergy production rate. It is not expected that this term can describe the effect of resistivedissipation in detail, but it should account for its average effect on the mean magnetic energytensor T. This approximation resolves the closure problem and enables us to study if resistivedissipation can, in principle, solve the inconsistencies that we encountered when it is ignored.The resulting equation for BiBj can be averaged using the standard methods:

∂tBiBj =∑

kl

[L(0)ijkl + L

(1)ijkl]BkBl, (5.12)

with

L(0)ijkl = (∇ku0i)δjl + (∇ku0j)δik − δikδjl(u0 · ∇ + 2ν), (5.13)

L(1)ijkl = (∇ku1i)δjl + (∇ku1j)δik − δikδjl u1 · ∇. (5.14)

For the details on how to average Eq. (5.12) by applying Eq. (5.8), the reader is referred toHoyng (1987); only L(0) has been modified here. The resulting equation (5.15) for the meanenergy tensor is the subject of section 5.4. Before coming to that, some basic results of the meanfield equation are summed up in the next section.

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ENERGY BALANCE AND RESISTIVE DISSIPATION IN PARKER’S SURFACE WAVE DYNAMO

Table 5.1: Combinations of α, β1 and β2 for which the dynamo period P is 22 years and themean field decay time τd is 10 dynamo periods.

α [cm s−1] β1 [cm2 s−1] β2 [cm2 s−1] fβ

−25 5.31 × 1010 6.12 × 1011 8.68 × 10−2

−50 2.80 × 1010 9.24 × 1011 3.03 × 10−2

−150 1.72 × 1010 1.55 × 1012 1.11 × 10−2

−500 1.25 × 1010 2.61 × 1012 4.80 × 10−3

5.3 The mean magnetic field

We solve the mean magnetic field from Eq. (5.2), under the constraint that the fundamentalmode has a period of 22 years and a decay time of about 10 dynamo periods. In ch. 4 wehave presented the boundary conditions and the general solution of Eq. (5.2) in the geometry ofFig. 5.1. The constraints on the mean field are met only for certain combinations of the dynamoparameters. We found these combinations by applying successive corrections to β1 and β2, theresult of which is shown in Table 5.1.

The remaining parameters were chosen as follows. For α we have allowed a range of values.With decreasing |α|, we see an increase of the ratio of diffusivities fβ (Table 5.1). But weargued in the introduction that fβ should satisfy fβ ≪ 1, hence we have not used values for |α|below 25 cm s−1. On the other hand, |α| should not exceed lΩ, where l is a typical convectivelength scale and Ω ≈ 2 × 10−6 s−1 is the solar rotation rate (Stix 1989). Near the base of theconvection zone we have l ≈ 109 cm, i.e. lΩ ≈ 2× 103 cm s−1. We should consider this numberis an upper limit because the mean helicity 〈u1 · (∇×u1)〉 depends on the unknown correlationcoefficient between u1 and the vorticity ∇×u1. In the calculations we adopted a maximal valueα = −500 cm s−1.

For the gradient of the azimuthal velocity we took a = 4×10−6 s−1 and for the wavenumberin the azimuthal direction kz = 10−10 cm−1.

5.4 The mean magnetic energy tensor

5.4.1 Equations

The arguments presented in section 5.2 lead to the following equation for the mean magneticenergy tensor T = 〈BB〉/8π, which now includes a dissipative term −2νTij on the right handside:

(∂t + u0 · ∇)Tij =∑

kl

∇k(αǫiklTlj + αǫjklTli) +∑

k

(∇ku0i)Tkj + (∇ku0j)Tki

+25γ(2

k

Tkkδij − Tij) + ∇ · β∇Tij − 2νTij . (5.15)

The mean energy density is

ǫ =∑

i

Tii. (5.16)

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5.4. The mean magnetic energy tensor

Figure 5.2: Diagonal components of the mean energy tensor T and correlation coefficients Cij asa function of x (Tyy is set to 1 at x = d2). The solution is marginally stable with α = −50 cm s−1,H = 0.71 and κ0 = 100 (see Tables 5.1–5.2 for β1, β2, γ2 and ν). Note the large, predominantlytoroidal field By in the overshoot layer and its strong correlation with the radial field Bx. Inthe convection zone, there is a strong correlation between Bz and By.

The off-diagonal components of T provide correlations coefficients of the field components:

Cij =Tij

TiiTjj=

〈BiBj〉√

〈B2i 〉〈B2

j 〉. (5.17)

These coefficients must satisfy |Cij | ≤ 1. A number of remarks can be made on the individualterms of Eq. (5.15).

The advection term vanishes because the differential rotation u0 (Eq. 5.5) is in the y-directionand because we consider only axisymmetric solutions (∂/∂y = 0). For the gradient of u0 weinsert ∇iu0j = aδixδjy. The vorticity coefficient γ represents the small-scale dynamo action dueto random stretching and shearing of field lines; it is given by

γ =1

3τc〈|∇ × u1|2〉. (5.18)

The diagonal components of T and the mean magnetic energy are enhanced by vorticity, butthe off-diagonal components are reduced. The new dissipative term reduces all components of

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ENERGY BALANCE AND RESISTIVE DISSIPATION IN PARKER’S SURFACE WAVE DYNAMO

Figure 5.3: Diagonal components of the mean energy tensor T and correlation coefficients Cij asa function of x (Tyy is set to 1 at x = d2). The solution is marginally stable with α = −25 cm s−1,H = 0.05 and κ0 = 100 (see Tables 5.1–5.2 for β1, β2, γ2 and ν). Compared to Fig. 5.2, vorticityis stronger. As a result, the toroidal magnetic field is less predominant and Cyz is much smaller.

T at the same rate and leaves the correlation coefficients unaffected.

We focus on the fundamental mode, i.e. the non-periodic, constant part of the mean magneticenergy. In our plane parallel geometry, this mode is also independent of latitude (∂/∂z = 0).

Due to its symmetry T has only six independent tensor elements that can be convenientlyarranged into one vector, whose components are Tµ, with µ = xx, xy, xz, yy, yz, zz. We adoptthe plane parallel geometry of Parker’s model (Fig. 5.1). This provides us with two sets ofequations, for Tµ in the overshoot layer, and for T ′

µ in the convection zone:

(∂t + 2ν1 − β1∂2x)Tµ =

ν

(aXµν + 25γ1Γµν)Tν , (5.19)

(∂t + 2ν2 − β2∂2x)T

′µ =

ν

(αΞµν∂x + 25γ2Γµν)T

′ν . (5.20)

For the constant matrices X, Ξ and Γ, we refer to ch. 4. The boundary conditions for Tµ and T ′µ

can also be found in ch. 4. Here it suffice to mention the one boundary condition, that containsa free parameter. This condition concerns the energy flux escaping from the dynamo through

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5.4. The mean magnetic energy tensor

the upper boundary of the convection zone, which we assume to be proportional to the energydensity:

(∂x +κ0

d2) ǫ2 = 0 at x = d2. (5.21)

The dimensionless constant κ0 is estimated to be of the order 30−300 (Van Geffen & Hoyng 1993b).1

It measures the ratio of the efficiency of energy transport through the surface at x = d2 andthrough the bulk of the convection zone. When κ0 ≫ 1, its value affects ǫ2 in a thin layer nearx = d2, but has little influence on T elsewhere.

5.4.2 Constraints on the parameters

The parameters α, β and γ are functions of the turbulent velocity field u1. They are not entirelyindependent, but related through constraints. The normalised helicity H in the convection zoneis defined as

H = − α√β2γ2

=〈u1 · (∇× u1)〉

〈|u1|2〉〈|∇ × u1|2〉. (5.22)

For this coefficient, which measures the correlation between the turbulent velocity and thevorticity, a Schwartz-type inequality holds:

|H| ≤ 1. (5.23)

The inequality implies a lower bound on γ2, for a given value of α and β2. No reliable estimatefor H is known, but it is likely that |H| ≪ 1 (Moffatt 1978, p. 284). We return to this issue insection 5.5.

There is also a constraint on the typical length scale of the turbulence in the convectionzone, which we estimate as follows (Eqs. 5.3 and 5.18):

lt =

β2

γ2≤ d2. (5.24)

This length scale roughly corresponds to the typical size of a convective ’giant’ cell and shouldnot exceed the thickness d2 of the convection zone. For a given value of β2, this again impliesa lower bound on γ2. Condition (5.24) appears to be about equally strong as condition (5.23),see Table 5.2.

Furthermore we assume that the strong magnetic fields that suppress the turbulent velocityfield in the overshoot layer, have an equally large effect on γ and β:

γ1

γ2= fβ; fβ ≪ 1. (5.25)

We have no knowledge about the value of the dissipation coefficients in the two dynamo regions.The simplest approximation then is to assume that

ν1 = ν2 = ν. (5.26)

With this assumption, our treatment of resistive dissipation amounts to the introduction of aglobal decay factor e−2νt for a given solution of Eqs. (5.19–5.20) without resistive dissipation.Hence we can achieve marginal stability of T by adopting a suitable value for ν, while keepingthe other parameters unchanged.

1Van Geffen & Hoyng (1993b), who use (∂x + 1/ρ) ǫ = 0 at the upper surface of the convection zone, concludeR⊙/ρ ≈ 100 − 1000. Thus κ0 = d2/ρ ≈ 0.3R⊙/ρ ≈ 30 − 300.

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Table 5.2: Vorticity coefficients, turbulent length scales, dissipation coefficients and correspond-ing dissipative time scales (in months) for marginally stable solutions, with β1, β2 as in Table 5.1and 30 <

∼ κ0<∼ 300.

H α γ2 lt/d2 ν 1/2ν[cm s−1] [s−1] [s−1] [mo]

1.00 −50 2.7 × 10−9 0.92 6.1 × 10−8 3.12” −150 1.4 × 10−8 0.52 8.2 × 10−8 2.33” −500 9.6 × 10−8 0.26 1.4 × 10−7 1.33

0.71 −25 2.0 × 10−9 0.87 7.4 × 10−8 2.56” −50 5.4 × 10−9 0.65 7.8 × 10−8 2.45” −150 2.9 × 10−8 0.37 1.0 × 10−7 1.83” −500 1.9 × 10−7 0.18 2.2 × 10−7 0.89

0.32 −25 1.0 × 10−8 0.39 1.3 × 10−7 1.44” −50 2.7 × 10−8 0.29 1.4 × 10−7 1.41” −150 1.5 × 10−7 0.16 2.0 × 10−7 0.96” −500 9.6 × 10−7 0.08 9.5 × 10−7 0.20

0.10 −25 1.0 × 10−7 0.12 3.0 × 10−7 0.64” −50 2.7 × 10−7 0.09 3.2 × 10−7 0.60” −150 1.4 × 10−6 0.05 1.4 × 10−6 0.13

0.05 −25 4.1 × 10−7 0.06 4.9 × 10−7 0.39” −50 1.1 × 10−6 0.05 1.1 × 10−6 0.18

5.4.3 Parameters

For the parameters α, β1 and β2 we employ the values shown in Table 5.1. For each of thesecombinations we fix γ2 by adopting several values of |H| in the range from H = 1 to H = 0.05,provided that Eq. (5.24) is satisfied (Table 5.2). Then γ1 follows from Eq. (5.25).

Here H = 1 represents the maximal helicity, or minimal vorticity case (Eq. 5.22). We shouldstress that it is likely that |H| ≪ 1 (see section 5.5). But by decreasing |H| for fixed values of αand β2, the role of vorticity increases, so that beyond a certain point the energy production isdominated by random stretching of field lines, with only a negligible contribution of differentialrotation. Beyond this point, the solutions do not correspond to the solar dynamo, since theydo not exhibit strong toroidal fields in the overshoot layer. Hence, for a given combination of αand β2, there is a lower limit on |H|. From numerical experience we conclude that for solutionswith |H| <∼ 0.05, the dominant source of magnetic energy is random field line stretching.

It turns out that the values of ν, required for marginal stability of T are, within the givenaccuracy of Table 5.2, insensitive to changes of κ0, as long as κ0 ≫ 1. In section 5.4.5 we shallconsider for every set of parameters in Table 5.2 the resulting maximal ratio of r.m.s. fieldstrengths and compare it with solar estimates. As will be shown, this ratio depends on κ0, andwe shall adopt a value within the range 30 <

∼ κ0<∼ 300 that achieves reasonable agreement. Even

so, such agreement cannot be attained for all the parameter combinations of Table 5.2.

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5.4. The mean magnetic energy tensor

5.4.4 Marginally stable solutions

The method of finding the solutions of Eqs. (5.19–5.20) by means of the ansatz

Tµ(x, t) = Tµ(x) eΛt, (5.27)

and a similar one for T ′µ, is described in detail in ch. 4.

With ansatz (5.27) we obtain from Eqs. (5.19–5.20) twelve ordinary second order differen-tial equations. The general solution has 24 integration constants, that are determined by theboundary conditions. Non-trivial solutions are found if the corresponding dispersion relation issatisfied. We are interested only in the fundamental mode, i.e. the mode that has the largestgrowth rate Re Λ. As was mentioned in the introduction, the fundamental mode should benon-periodic (see ch. 4 for a more complete discussion). This was indeed found to be the case,although no general proof is available.

In addition, the fundamental mode must be marginally stable. If resistive dissipation isneglected (ν = 0) as in ch. 4, T in general has a positive growth rate Re Λ0. But we canadapt the solutions that were derived in ch. 4 (Appendix C) to the present case with resistivedissipation if we perform the substitution Λ0 → Λ+2ν in all expressions. Thus we can transforman exponentially growing solution into a marginally stable one by adopting ν = Re Λ0/2. InTable 5.2 the required dissipation coefficients and the corresponding dissipative time scale 1/2νare given for a range of solutions. We shall return to this time scale in section 5.5.

Two examples of a marginally stable solutions are shown in Figs. 5.2 and 5.3. These solutionsare acceptable as models of the solar dynamo, as will be argued in the next section. Their mainfeatures are as follows:

⊙ if |H| is large, then the mean magnetic energy of the toroidal field Tyy is much largerthan the other components of T everywhere. For smaller |H| (larger vorticity) this effectis reduced and occurs mainly in the overshoot layer. The dominance of Tyy is due to thestrong differential rotation. Vorticity increases all diagonal components equally, therebyreducing the dominance of Tyy.

⊙ There is a large correlation between Bx and By in the overshoot layer, due to the strongdifferential rotation which creates toroidal magnetic field (By) from radial magnetic field(Bx). Since vorticity destroys correlations, we observe a decline of Cxy in the convectionzone, which is the more rapid, the stronger vorticity is.

⊙ If |H| is large, there is a strong correlation between By and Bz in the convection zone.This correlation can be ascribed to the α-effect, which creates poloidal field (Bz) fromtoroidal field (By). The magnetic field that is produced by vorticity does not show such acorrelation, which explains why Cyz decreases if |H| decreases.

5.4.5 Mean magnetic energy density

The mean magnetic energy density (ǫ1 in the overshoot layer and ǫ2 in the convection zone)obeys the following equations, obtained by contraction of Eqs. (5.19–5.20):

∂tǫ1 = 2aTxy + (2γ1 − 2ν + β1∂2x) ǫ1, (5.28)

∂tǫ2 = (2γ2 − 2ν + β2∂2x) ǫ2. (5.29)

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Figure 5.4: Mean magnetic energy density ǫ as a function of x (ǫ is set to 1 at x = d2) for amarginally stable solution with H = 0.1, α = −50 cm s−1 and κ0 = 300. Energy is producedby differential rotation in the overshoot layer and by vorticity in both layers. It is transportedby turbulent diffusion, inward where ∂xǫ > 0 and outward where ∂xǫ < 0. Note the exponentialdecline in the convection zone.

A graphic representation is given in Fig. 5.4. With respect to the energy density there are anumber of qualitative differences between the solutions presented here and those in ch. 4.

First, the energy flux −β∂xǫ at a given point x can be directed outward but also inward.The boundary conditions (see ch. 4) do not allow an energy flux into the radiative core, so thatin ch. 4, where turbulent diffusion provided the sole energy sink, marginal stability could berealised only if −β∂xǫ > 0 everywhere. But with resistive dissipation included, there can be aninward energy flux, because the dissipation occurs everywhere in the dynamo.

Second, there is a sharp exponential decline of ǫ accross the convection zone. This is readilyunderstood from Eq. (5.29), which has the following solution (in the notation of ch. 4):

ǫ2(x) = Cek7x +De−k7x, k7 =√

2(ν − γ2)/β2. (5.30)

Since only marginally stable solutions (Λ = 0) are considered, the time dependent factor isabsent. We apply boundary condition (5.21) and obtain

ǫ2(x)

ǫ2(d2)=

1

2

( κ0

k7d2+ 1

)

e−k7(x−d2)

−1

2

( κ0

k7d2− 1

)

ek7(x−d2). (5.31)

From Table 5.2 we see that many solutions have ν > γ2, so that k7 is real, with a typical valuek7d2 ≈ 10. Thus Eq. (5.31) describes an exponentially declining energy density for the bulkof the convection zone, because this region constitutes a net sink of energy. Without resistivedissipation (ν = 0 as in ch. 4) this effect cannot occur for marginally stable solutions and k7 ispurely imaginary. The second term on the r.h.s. becomes noticeable if x − d2 ≈ d2/κ0, i.e. ina thin layer near x = d2. The mean magnetic energy density is related to the r.m.s. magneticfield:

Brms =√

8πǫ. (5.32)

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5.4. The mean magnetic energy tensor

Figure 5.5: Ratio rB of maximal r.m.s. field strength to r.m.s. surface field strength for variousmodels, plotted versus the normalised helicity H. The full circles denote solutions with κ0 =100. The drawn curves are interpolations between solutions with identical values of α, fromα = −25 cm s−1 (top curve) to α = −500 cm s−1 (bottom curve). The solutions that areacceptable as models for the solar dynamo are situated between the dotted lines. In a numberof cases we have adjusted rB by changing κ0 (open circles).

Thus, with sufficient net dissipation we can have Brms(0) ≫ Brms(d2), as is expected to be thecase for the solar dynamo. For active regions and small flux tubes at the solar surface with a fieldstrength of about 103 G and a filling factor of about 10−2, we can estimate Brms ≈

√10−2 × 106 =

100 G. In the overshoot layer the field is thought to be of the order (0.2− 1)× 105 G. Thus theratio of maximal r.m.s. field strength to r.m.s. surface field strength,

rB =Bmax

rms

Brms(d2)(5.33)

should be of the order rB ≈ (0.2 − 1.0) × 103. Only a subset of the solutions in Table 5.2obeys this criterium, as can be seen in Fig. 5.5. In particular, we may infer from Fig. 5.5 thatthere are no acceptable solutions for |α| >∼ 3 × 102 cm s−1, since here vorticity dominates theenergy production even in the minimal vorticity case (H = 1). This is due to the fact thatcondition (5.23) requires higher values of γ2 if |α| increases.

For κ0 ≫ 1 the influence of the boundary condition for ǫ2 (Eq. 5.21) is felt only in a thinlayer of the order d2/κ0. Thus, a change in κ0 leaves T unaffected in the bulk of the dynamo andmerely affects the ratio rB, see Eq. (5.31). In most of our calculations we obtained an acceptablesolution with κ0 = 100, but on a few occasions a different value within the range 30 <

∼ κ0<∼ 300

was required. These modified solutions are indicated by the open symbols in Fig. 5.5. In whatfollows, we shall consider only the solutions that are acceptable as models for the solar dynamo.

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Figure 5.6: Energy flow diagram for the solution of Fig. 5.2 (α = −50 cm s−1 and H = 0.71).The energy production rates and fluxes are normalised with respect to Qtot.

Figure 5.7: Energy flow diagram for the solution of Fig. 5.3 (α = −25 cm s−1 and H = 0.05).The energy production rates and fluxes are normalised with respect to Qtot.

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5.4. The mean magnetic energy tensor

Table 5.3: Various energetic properties of marginally stable solutions that are acceptable asmodels for the solar dynamo. The absolute numbers in the last four columns result from thecalibration Brms(d2) = 100 G.

H α κ0 E1/E F2/Qtot xmax/d1 Bmaxrms F2 AE AQtot/L⊙

[cm s−1] [kG] [*] [1038erg]

1.00 −50 100 0.84 2.3 × 10−4 −1.00 31 1.8 4.1 0.013” −150 200 0.85 6.6 × 10−4 −0.51 27 6.2 3.6 0.015

0.71 −25 100 0.82 1.9 × 10−5 −1.00 78 1.2 16 0.10” −50 ” 0.86 8.8 × 10−5 −1.00 44 1.8 8.5 0.034” −150 ” 0.84 4.3 × 10−4 −0.38 21 3.1 2.2 0.012

0.32 −50 ” 0.89 1.0 × 10−5 −1.00 98 1.8 42 0.290.10 −50 300 0.57 2.0 × 10−4 −0.20 22 5.5 2.8 0.0450.05 −25 100 0.77 2.7 × 10−6 −1.00 73 1.2 29 0.73

* 106 erg cm−2 s−1

5.4.6 Magnetic energy balance

To derive the energy balance we integrate Eqs. (5.28–5.29) over the volume of the dynamo region.Since T and hence ǫ are independent of y and z, only an integration over x remains, resulting in

∂tE1 = ΛE1 = Qω +Qγ1 − F1 −Qd1, (5.34)

∂tE2 = ΛE2 = Qγ2 + F1 − F2 −Qd2. (5.35)

These energy production rates have the dimension of erg cm−2 s−1. The total magnetic energies(in erg cm−2) of both regions are

E1 =

∫ 0

−d1dx ǫ1, E2 =

∫ d2

0dx ǫ2. (5.36)

The total magnetic energy of the dynamo is E = E1 + E2. All the acceptable solutions show ahigh concentration of magnetic energy in the overshoot layer, as indicated by the ratio E1/E inTable 5.3.

In the energy balance equation we encounter the following terms: the energy productionrates due to differential rotation and vorticity,

Qω = 2a

∫ 0

−d1dxTxy, (5.37)

Qγ1 = 2γ1E1, Qγ2 = 2γ2E2, (5.38)

the energy fluxes at x = 0 and x = d2,

F1 = −β1∂xǫ1|x=0, F2 = −β2∂xǫ2|x=d2 , (5.39)

and the dissipative terms

Qd1 = 2νE1, Qd2 = 2νE2. (5.40)

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Figure 5.8: Energy production rate due to vorticity (Qγ = Qγ1 + Qγ2) relative to the totalenergy production rate for various models, plotted versus the normalised helicity H. The drawncurves are interpolations between solutions with identical values of α, from α = −500 cm s−1

(top curve) to α = −25 cm s−1 (bottom curve).

Furthermore the total energy production rate is given by

Qtot = Qω +Qγ1 +Qγ2. (5.41)

For a marginal stable solution (Λ = 0) the total energy production is balanced by an equallylarge energy loss due to resistive dissipation and turbulent diffusion, i.e. Qtot = Qd1 +Qd2 +F2.Figure 5.6 gives a graphic representation of the energy balance for a typical marginally stablesolution; see also Table 5.3. All these rates are normalised with respect to Qtot.

Concerning the energy balance, we make the following remarks. As is indicated by the ratioQγ/Qtot, most of the energy is produced by differential rotation and not by vorticity (Fig. 5.8).The contribution of vorticity increases, when |H| decreases (with α fixed) or when |α| increases(with H fixed). The α-effect itself plays virtually no role in the energy balance: it does notappear in Eq. (5.29) for the mean energy, and has a negligible effect on the growth rate Λ. Theα-dependence of the energy balance arises only because we fix H at certain values so that αdetermines the vorticity coefficient γ2 (Eq. 5.22).

The ratio F2/Qtot tells us how much of the produced energy is transported out of the dynamoby turbulent diffusion. In every case virtually all the produced magnetic energy is dissipatedinternally. We shall return to this issue in section 5.5.

5.4.7 Calibration of the r.m.s. magnetic field strength

The energy equation and its boundary conditions are linear and therefore the absolute value ofT for a marginally stable solution must be determined externally. In section 5.4.6 we mentionedthat Brms ≈ 100 G at x = d2 and that the maximal field strength in the overshoot layershould be Bmax

rms ≈ (0.2 − 1.0) × 105 G. Our selection of acceptable solutions was based on therequirement that the corresponding ratio rB (Eq. 5.33) falls within the estimated range. Thus byadopting the calibration Brms(d2) = 100 G, the maximal r.m.s. field strengths have the correct

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5.5. Discussion

magnitude for the solar-type solutions (see Table 5.3). The flux loss through the upper surfaceof the convection zone is (Eqs. 5.21 and 5.39)

F2 =κ0β2

d2ǫ2(d2). (5.42)

With the present calibration, the fluxes F2 are of the order F2 = (1− 6)× 106 erg cm−2 s−1 (seeTable 5.3). If a different calibration were used, say Brms → fBrms, the fluxes scale with f2. Wewould also obtain slightly different fluxes if we would change κ0. Thus one should not attachtoo much significance to the exact values of F2. Nevertheless we note that they have the sameorder of magnitude as the estimated flux Fc ≈ 5× 106 erg cm−2 s−1, required for the heating ofthe solar corona (Withbroe & Noyes 1977, Kuperus et al. 1981). This agreement is possible onlybecause the turbulent diffusivities used here are of the order β2 ≈ 1012 cm2 s−1, rather than themuch higher values that are required for marginal stability if resistive dissipation is ignored.

Next we shall calculate the resulting magnetic energy in the dynamo and the energy pro-duction rates. Although our model is based on a local approximation, we can make a roughestimate of the global energy of the dynamo by multiplying E with the dynamo surface A.If we assume that the bottom of the convection zone is located at a radius of Rc = 5 × 1010

cm and that the dynamo operates within a latitudinal band of width 2 × 1010 cm, we findA ≈ 2πRc × 2 × 1010 ≈ 6.3 × 1021 cm2. We then estimate the absolute energy as AE and theabsolute total energy production rate as AQtot (Table 5.3). We stress that these numbers arerough estimates, first due to the uncertainty in A, but more importantly, due to dependence onthe calibration of Brms. If we multiply Brms by a factor f , then the total energy and all energyproduction rates and fluxes scale as f2.

5.5 Discussion

We have studied a solar dynamo model proposed by Parker (1993), the ’surface wave dynamo’,by solving equations for the mean magnetic field and for the mean magnetic energy tensor. Themean quantities are interpreted as ensemble averages, so that the dynamo parameters may betreated as constants. We chose the parameters in such a way, that the mean field has a periodof 22 years and a decay time of about 10 dynamo periods. This slight sub-criticality of the meanfield is motivated by the concept of phase mixing, which arises in the ensemble average due tovariations in the dynamo period.

Next we solved the equation for the mean magnetic energy tensor under the constraint ofmarginal stability. We argued that this constraint can be met only if, apart from turbulenttransport of energy, an additional loss term is included. This energy loss can be providedby resistive dissipation. With respect to resistive dissipation, we pointed out a fundamentaldifference between the mean field and the mean energy. Since the mean field corresponds to thelargest length scale of the dynamo, it is not affected by resistive dissipation, which operates atthe smallest length scale. The mean magnetic energy on the other hand, is an average over alllength scales of the dynamo, and we have no knowledge about the distribution of energy overthe length scales. Thus the scale-dependence of resistive dissipation poses a closure problem.Our approach has been to apply the simplest possible, scale-independent approximation, anddescribe the average effect on the mean magnetic energy tensor through a dissipation coefficientν.

With this heuristic treatment of resistive dissipation, we were able to obtain marginallystable solutions of Eqs. (5.19–5.20) for the same set of parameters that we adopted for the

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ENERGY BALANCE AND RESISTIVE DISSIPATION IN PARKER’S SURFACE WAVE DYNAMO

mean field equation. Thus the inconsistencies that arise when turbulent diffusion is assumed toprovide the only energy sink of the dynamo are removed. These inconsistencies were caused bythe abnormally high turbulent diffusivities (β2

>∼ 3 × 1014 cm2 s−1, see ch. 4) that are required

for marginal stability of T. In particular, we note that a r.m.s. magnetic field at the upperboundary of the order 100 G is consistent with an energy flux through the upper boundary ofthe order F2 ≈ 5 × 106 erg cm−2 s−1 only if β2 is of the order 1012 cm2 s−1 (Eq. 5.42).

If, in the convection zone, resistive dissipation exceeds the energy production by vorticitythen the mean magnetic energy density exhibits an exponential decline. Hence we can easilyachieve a high ratio rB of maximal r.m.s. field strength to r.m.s. surface field strength, inagreement with current estimates for the Sun. The requirement that there be a net dissipationin the convection zone implies an upper limit on vorticity. In fact, for all acceptable solutions,most of the magnetic energy is produced by differential rotation in the overshoot layer, ratherthan by vorticity (Fig. 5.8).

The normalised helicity H measures the correlation between the turbulent velocity field andits vorticity (Eq. 5.22). As the variability of the solar cycle indicates, there may be an importantrandom component in the turbulent velocity field (see Ossendrijver et al. 1996b). Accordingly,u1 and ∇×u1 should be largely uncorrelated, i.e. |H| ≪ 1. Using different arguments, the samecondition was derived by Moffatt (1978, p. 284). If we thus accept only solutions with, say,|H| <∼ 0.1, our calculations suggest a tight range for α, namely 25 <

∼ |α| <∼ 50 cm s−1 (Fig. 5.5).

From the coefficient of resistive dissipation ν, we get the typical ohmic decay time 1/2ν,averaged over the length scales. The range of values for the acceptable solutions in Table 5.3 is1/2ν ≈ 0.4 − 3 months (see Table 5.2), which is comparable to the typical turnover time in theconvection zone. In numerical simulations of hydromagnetic convection by Nordlund et al. (1992)a similar time scale for ohmic decay was found. Due to the impenetrable boundaries that wereemployed by these authors, the importance of energy transport out of the dynamo region cannotbe assessed from their calculations. However, relevant for our discussion is their result that themagnetic energy of a small scale dynamo in the convection zone can attain a stationary statethrough a balance with resistive dissipation, and that the timescale for the energy conversionis of the order of the convective turnover time. Hence, although our treatment of resistivedissipation is simplified, there is no inconsistency at this level. We find that the energy lossby turbulent transport represents only a small fraction (at most about 7 × 10−4) of the energyloss, so that resistive dissipation has to balance the combined action of vorticity (the small scaledynamo) and differential rotation (the large scale dynamo): Qd ≈ Qtot.

What may seem somewhat striking is the corresponding rate at which energy is dissipated.If we adopt a surface r.m.s. field strength Brms(d2) = 100 G, we find that the total magneticenergy of the dynamo is generated and dissipated at a rate Qtot corresponding to at least about1% of the solar luminosity.

Other authors have estimated that viscous and ohmic heating in the convection zone canindeed amount to a considerable fraction of the convective flux, i.e. of the solar luminosity(Brandenburg 1993, Hewitt et al. 1975). We stress that the dissipated heat is fed back into thedynamo. It is therefore not lost and should not necessarily have direct observable consequencesfor the solar luminosity.

The observed luminosity variations are due rather to the net change of the total magneticenergy through the course of the solar cycle, which give rise to a change in the temperaturegradient (Spiegel & Weiss 1980). The cyclic behaviour of the magnetic energy is not treatedhere, since we only consider the fundamental mode of the mean magnetic energy, which isconstant and non-periodic. But we may estimate the corresponding rate as AE/11 years, which

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5.6. Acknowledgements

amounts to (0.2−3.2)×10−3 of the solar luminosity for the solutions shown in Table 5.3. Thesenumbers are of the same order of magnitude as the observed variations (Foukal & Lean 1988,Hudson 1988). However, the effect on the luminosity caused by a change in the temperaturegradient in the overshoot layer is not well understood (Gilliland 1982, Gilman 1986).

The high rate of energy production poses a strong demand on the major energy source of thedynamo, which is the differential rotation. A simple estimate of the total kinetic energy of thedifferential rotation AEdiff in the overshoot layer yields AEdiff ≈ 1

2Ad1ρ(∆v0/2)2 ≈ 1

8ρAd31a

2 ≈2.3 × 1037 erg. Here A ≈ 6.3 × 1021 cm2 is the total dynamo surface area, ρ ≈ 0.23 g cm−3 isthe mass density and ∆v0 = ad1 ≈ 8× 103 cm s−1 is the difference in rotational velocity accrossthe overshoot layer. It follows that AEdiff is about two orders of magnitude smaller than thetotal magnetic energy of the dynamo. Therefore differential rotation can provide the energy forthe dynamo only if it is very efficiently replenished. In fact, the build-up time scale of AEdiff

must be about AEdiff/AQω <∼ 6 days, if we insert AQω ≈ AQtot

>∼ 0.01L⊙ (see Table 5.3). Such

a rapid conversion of kinetic energy to magnetic energy in the overshoot layer requires in turnan efficient coupling with the convection zone.

5.6 Acknowledgements

This research was supported by the Dutch Foundation for Research in Astronomy (ASTRON).

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Chapter 6

On the cycle periods of stellar

dynamos

AbstractWe show that the cycle periods of slowly rotating lower main-sequence stars with periodic chromo-spheric activity can be parametrized by the rotation period Prot and the convective turnover time τcthrough a relation of the form Pcyc ∝ P brotτ

cc , with b = 2.5 ± 0.5 and c = −2.1± 0.5. This suggests

a common dynamo mechanism for slowly rotating stars. Using a simple linear mean-field dynamomodel, we are able to reproduce the observed relation if ∆Ω, the total difference in angular velocityalong the radial direction, scales as ∆Ω ∝ P prot with p ≈ 0.26, and if the α-coefficient scales asα ∝ Roq with q ≈ −5.2. This would suggest that, with increasing rotation rate, differential rotationdecreases mildly while α increases rapidly.

A.J.H. Ossendrijver

Astronomy & Astrophysics (submitted)

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ON THE CYCLE PERIODS OF STELLAR DYNAMOS

6.1 Introduction

The flux emitted in the Ca II H and K lines serves as a diagnostic of stellar magnetic activity(Wilson 1978, Saar & Baliunas 1992, Baliunas et al. 1995). These emission lines are formedin the chromosphere by non-thermal heating related to magnetic fields. Direct confirmation ofthe magnetic nature of Ca II H and K emission is observed on the Sun, where the strengthof these lines is correlated with the magnitude of, and the area covered by magnetic fields(Schrijver et al. 1989). On the main sequence chromospheric activity is limited to the interval0.4 <

∼ B−V <∼ 1.6, which roughly coincides with the range of stars that have a convective

envelope. This suggests that stellar magnetic fields are produced by dynamos and raises thequestion whether different features of stellar magnetic activity can be parametrized by quantitiesrelated to dynamo theory.

The overall chromospheric activity level of lower main-sequence stars is well parametrized byan empirical Rossby number, Roe = Prot/τe (Noyes et al. 1984a, Stepien 1994). Here Prot is thestellar rotation period and τe(B−V ) is the empirical convective turnover time, whose functionaldependence on B−V is determined from the data themselves by minimizing the scatter in therelation between activity and Roe. The resulting parametrization indicates that stellar activityincreases with decreasing Roe (more rapid rotation). For B−V <

∼ 0.8 the empirical τe closelyresembles τc, the theoretical turnover time near the base of the convection zone, (Gilman 1980,Gilliland 1985, Kim & Demarque 1996). For B−V >

∼ 0.8, however, the activity level dependsonly on Prot and not, or only mildly, on B−V (Stepien 1994). Hence the empirical turnover timeτe is essentially constant for B−V >

∼ 0.8. This is not in disagreement with the results of Gilman,since his calculations did not extend beyond B−V = 0.8, but the calculations by Gilliland andKim & Demarque reveal a further increase of τc with B−V . It follows that for B−V >

∼ 0.8 theRossby number is not a useful indicator of the chromospheric activity level.

Four main categories of chromospherically active stars are identified by Baliunas et al. (1995),namely stars with a constant activity level (13%), long-term variations (13%), irregular varia-tions (24%) and periodic variations (50%). Here the percentages indicate the fraction of starswithin each category, as estimated by Baliunas et al. In this paper we focus on the stars withperiodic variations. The assignment of a star to this category rather than to that of the irregularvariations or long-term trends depends on the confidence level at which one requires the periodsto be determined. In fact, the percentage of stars that have well-defined cyclic variations isno more than 15%. Here well-defined means that the periodicity is rated good or excellent byBaliunas et al. (1995). Furthermore, within the time interval (about 25 years) spanned by theobservations, stars with cyclic variations on time scales longer than about 20 years are hard todistinguish from stars with long-term variations or a constant activity level. Only continuedobservations can resolve these issues by increasing the reliability of the period determinationsand by allowing longer periods to be detected.

As a result, the search for possible trends in the cycle length Pcyc has yielded mixed results.Noyes et al. (1984b) found a correlation between cycle length and the empirical Rossby numberfor a small sample of stars with clear periodic variations. In the mean time, the sample of starswith periodic activity variations has grown, and it has been claimed that there is no longer anyevidence of a correlation between cycle length and rotation period or Rossby number for theextended sample (Soderblom 1988, Saar & Baliunas 1992).

However, a trend may be concealed by a large spurious scatter that is caused by stars withill-defined cycle periods. Secondly, the absence of a clear trend may be due to the fact thatqualitatively different dynamo mechanisms are active in different types of stars. There may well

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6.2. Activity cycle length, rotation period and color

Figure 6.1: Logarithm of the convective turnover time τc (in days) versus B−V . The datapoints represent model calculations by Kim & Demarque (1996). The dotted line is a thirdorder polynomial fit to the data (Eq. 6.1).

exist a trend within a group of stars that is sufficiently homogeneous.

Several further arguments can be given to justify a renewed investigation of possible trendsin the cycle length. Firstly, we now have at our disposal estimates of the convective turnovertime for lower main-sequence stars up to B−V ≈ 1.6 (Kim & Demarque 1996). Most of the starsstudied by Noyes et al. (1984b) are in the range 0.8 < B−V < 1.4, for which no estimate of τcwas available at that time, apart from the empirical τe. Secondly, the accuracy of the measuredcycle periods has increased due to the longer observation interval. Thirdly, for several stars newmeasurements of the rotation periods are available, that differ from previous values, or replacevalues that were predicted by means of the observed correlation between the Ca II flux and Roe.

In section 6.2 we examine the available cycle periods and look for trends in terms of dynamoparameters. In section 6.3 we compare the observed trends with a simple linear mean fielddynamo model. Section 6.4 contains our conclusions.

6.2 Activity cycle length, rotation period and color

The sample of stars that exhibit chromospheric activity can be divided into two groups accordingto their rotation rate: rapidly rotating, young stars with a high level of activity and slowlyrotating, older stars with a moderate or low activity level (Baliunas et al. 1995). Activityvariations with well-defined periods are observed predominantly in older stars. Younger starstend to display stronger, more irregular activity variations. This may reflect the presence ofdifferent dynamo mechanisms in slowly and rapidly rotating stars. Hence we expect a singletrend in the cycle periods to exist only within either subgroup.

In Table 6.1 we summarize the properties of all known lower main-sequence stars with well-defined activity cycles. In order to obtain a homogeneous sample we exclude the rapidly rotating

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Table 6.1: Stars with periodic chromospheric activity

HD No. B−V Pcyc(a) Prot

(b) τc(c) Ro

[yr] [d] [d]

Sun 0.66 11.0 25 19 1.323651 0.85 13.8 44 30 1.484628 0.88 8.4 38 31 1.2210476 0.84 9.6 39 29 1.3316160 0.98 13.2 48 35 1.3526965 0.82 10.1 43 28 1.5232147 1.06 11.1 47 39 1.2181809 0.89d 8.2 41 32 1.30103095 0.75 7.3 31 25 1.26160346 0.96 7.0 37 35 1.07166620 0.89 15.8 43 32 1.36219834A 0.80 21.0 42 27 1.54219834B 0.91 10.0 43 32 1.329562e 0.64 > 20 29 18 1.6210700e 0.72 > 20 34 23 1.48141004e 0.60 > 20 26 15 1.72143761e 0.60 > 20 17 15 1.13115404f 0.93 12.4 18 33 0.54152391f 0.76 10.9 11 25 0.44156026f 1.16 21.0 21 43 0.42201091f 1.18 7.3 35 44 0.80201092f 1.37 11.7 38 54 0.71

a Cycle periods taken from Baliunas et al. (1995).b Rotation periods taken from Baliunas et al. (1996).c Convective turnover times taken from Kim & Demarque (1996).d This star is a binary. We estimated B−V assuming that the chromospheric activity can beattributed to a star of type K0V (cf. Baliunas et al. 1995).e These stars are possibly in a Maunder minimum phase according to Baliunas & Soon (1995).f These stars rotate rapidly (Ro <

∼ 0.9) and have a high level of activity. HD 201091 is anintermediate case.

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6.2. Activity cycle length, rotation period and color

stars from the analysis. The definition of rapid rotation used here is that Ro <∼ 0.9, see below

for a motivation. The resulting subset is similar to the group of solar-type stars examined bySoon et al. (1994) and Baliunas & Soon (1996). Our subset is somewhat more restricted inthat we do not include HD 201091 and HD 201092. For these stars, which have B−V >

∼ 1.1,the dividing line between low and high activity, is ill-defined due to a lack of stars (Baliunas etal. 1995). Thus they may in fact be stars with a high activity level. Furthermore, their Rossbynumbers are low compared to those of the other solar-type stars. We have classified them asrapidly rotating stars, putting the dividing line between rapid and slow rotation at Ro ≈ 0.9.In fact, HD 201091 is close to the dividing line and can not be assigned to either group withcertainty, see further below.

The third column gives cycle periods, compiled from Baliunas et al. (1995). We have includedonly stars with well-defined cycles (those rated ”good” or ”excellent”), as well as four stars thatmay be in the equivalent of a Maunder minimum or, alternatively, have cycle periods longer thanabout 20 years (Baliunas & Soon 1995). There are no stars with well-defined periods shorterthan 7 years. Given the large fluctuations in cycle length observed on the Sun (δPcyc/Pcyc ≈ 0.1),a coverage of at most 3 cycles implies a potentially large error in the values of Pcyc. FollowingNoyes et al. (1984b), we assume an error of 15%.

The fifth column contains the convective turnover times τc near the bottom of the convectionzone. These are based on an interpolation of values of the local turnover time (in fact, half theglobal turnover time) calculated by Kim & Demarque (1996), for lower main-sequence stars inthe mass range 0.5 ≤M/M⊙ ≤ 1.2. Their results indicate that τc is roughly independent of ageon the main sequence, depending only on stellar mass, i.e. on B−V . We chose a stellar age of2 Gyr and fitted the corresponding values of τc to a third order polynomial:

τc = −68.3 + 224.8x − 177.2x2 + 57.0x3; x ≡ B−V, (6.1)

where τc is in days. The resulting curve is shown in Fig. 6.1. We reiterate that, unlike τe, theincrease of τc with B−V does not come to a halt or slow down significantly beyond B−V = 0.8.

In trying to identify trends in the cycle periods of slowly rotating stars we proceed along linessimilar to those followed by Noyes et al. (1984b). As a first step we consider the dependenceon the rotation period. In Fig. 6.2 we plot the cycle period log Pcyc versus log Prot - the slowlyrotating stars are indicated by the full circles. For stars with similar values of B−V we canobserve an increase of Pcyc with increasing Prot. But stars of different spectral type seem tobe located on different curves, that are shifted to the left by an amount that increases withdecreasing B−V . Hence the cycle period cannot be parametrized by rotation period alone andmust also involve a color-dependent term, which may be provided by the convective turnovertime. We assume that Pcyc depends on Prot and τc through a powerlaw, i.e.

Pcyc ∝ P brotτcc , (6.2)

and we determine the corresponding exponents by performing a least square fit of the formlog Pcyc = a+ b log Prot + c log τc. In Fig. 6.3 we plot logPcyc versus logProt + (c/b) log τc. Thus,all the data points are shifted along the horizontal axis by the amount (c/b) log τc, relativeto their positions in Fig. 6.2. The best fit, indicated by the dotted line, has a = 0.2 ± 0.4,b = 2.5 ± 0.5 and c = −2.1 ± 0.5. Hence c/b ≈ −0.9, so that our result is in basic agreementwith a parametrisation of the cycle period by the Rossby number Ro = Prot/τc, as suggestedby Noyes et al. (1984b). However, we find a steeper dependence on Ro, with a power of about2.5, rather than 1.25. Although the four ”Maunder minimum stars” were not included in theleast square fit, the resulting shift puts three of them at a location in Fig. 6.3, that is roughly in

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Figure 6.2: logPcyc versus logProt, for all the stars in Table 6.1. Open symbols denote therapidly rotating stars; arrows denote the four ’Maunder minimum stars’. The labels indicateB−V .

agreement with a cycle period of about 20 years. This suggests that their chromospheric activitymay prove to be periodic in the future. The anomalous position of the fourth of these stars, HD143761, suggests that it has no periodic activity but is in a real Maunder minimum state.

The cycle periods of rapidly rotating stars, indicated in Fig. 6.3b by the open circles, aremuch longer than what would be expected on the basis of the relation for the cycle periods ofslowly rotating stars. The deviation appears to increase with decreasing Rossby number. Thissuggests the existence of a different dynamo mechanism for rapid rotators. We put the dividingline between slow and rapid rotators at Ro ≈ 0.9, but of course it cannot be drawn sharply.Thus HD 201091 (Ro = 0.8) must be seen as an intermediate case, but if we assign it to theslow rotators, the resulting least square fit would be almost the same.

6.3 Dynamo model

The existence of a correlation between Pcyc, Prot and τc points to a common dynamo mechanismfor the stars in the sample under consideration. We compare the observed correlation with thepredictions of a simple model, based on linear mean field dynamo theory. Although the validityof the linear approach is open to debate (cf. Noyes et al. 1984b, Jennings & Weiss 1991), it maybe justified by the slow rotation rate and low activity level of the selected stars.

6.3.1 Geometry and equations

The dynamo model that we employ was proposed by Parker (1993) for the Sun. It consists oftwo plane parallel layers: the overshoot layer (region 1) and the convection zone (region 2), with

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6.3. Dynamo model

Figure 6.3: logPcyc versus logProt + (c/b) log τc. The symbols have the same meaning as inFig. 6.2. The dotted line is a least square fit applied to the slowly rotating stars (excludingthe four ’Maunder minimum stars’). Labels indicate the Rossby number. Top: slowly rotatingstars. Bottom: the complete sample of stars.

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thickness d1 and d2 respectively. The main motivation for the model arises from the presenceof strong magnetic fields in a thin layer under the convection zone. The presence of strongfields (B ≈ 105 G) is supported by observations and theoretical considerations (Hughes 1992,Schussler et al. 1994). It gives rise to a suppression of the turbulence, so that α and β arereduced. Helioseismology suggests that differential rotation is concentrated near the same layer(Goode 1995). Hence differential rotation and the α-effect may be spatially separated, theformer being restricted to the overshoot layer, the latter to the convection zone. Some turbulentdiffusion is required in the overshoot layer in order to provide communication with the convectionzone.

We assume that this model applies for all the stars in our sample. We use x for the radial,y for the longitudinal and z for the latitudinal coordinates, and we consider only axisymmetricsolutions (∂/∂y = 0). The overshoot layer is located at −d1 ≤ x ≤ 0 and the convection zoneat 0 ≤ x ≤ d2.

The mean magnetic field can be written as the sum of a toroidal and a poloidal component,i.e. B0 = Tey + ∇× Pey. It is governed by the following equations (Parker 1993, Ossendrijver& Hoyng 1996c):

(∂t − β1∇2)P = 0

(∂t − β1∇2)T = −a∂zP

(−d1 ≤ x ≤ 0),

(∂t − β2∇2)P = αT

(∂t − β2∇2)T = 0

(0 ≤ x ≤ d2).

(6.3)

We model the results of helioseismology for the equatorial region in a schematic way by adoptingthe following large scale velocity field u0:

u0 = u0(x)ey, ∂xu0 =

a (−d1 ≤ x ≤ 0)0 (0 ≤ x ≤ d2).

(6.4)

Here the constant a denotes the radial velocity gradient. If u1 = u − u0 denotes the turbulentvelocity field, having a correlation time τc, then the α-coefficient and the turbulent diffusivityin the convection zone are given by

α = −13τc〈u1 · (∇× u1)〉; β2 = 1

3τc〈u21〉. (6.5)

In the overshoot layer, strong magnetic fields lead to a suppression of the turbulent diffusivityby a factor

fβ =β1

β2≪ 1. (6.6)

We seek solutions of the form P = p(x) exp(ikzz + λt), and similarly for T . Here kz is the wavevector in the latitudinal direction. The boundary conditions are as follows (cf. Ossendrijver &Hoyng 1996c):

P = ∂xT = 0 at x = −d1,

[[P ]] = [[T ]] = [[∂xP ]] = [[β∂xT ]] = 0 at x = 0,

(∂x + kz)P = T = 0 at x = d2.

(6.7)

Here [[· · ·]] denotes the jump at the boundary. In the second line, β assumes the value β1 forx < 0 and β2 for x > 0.

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6.3. Dynamo model

Table 6.2: Parameters of lower main-sequence stars

M/M⊙ 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

B−V a 1.89 1.60 1.37 1.16 0.95 0.78 0.65 0.57 0.49

Rc [1010cm]b 1.35 1.93 2.54 3.16 3.61 4.14 4.97 6.04 7.19

Hp [109cm]b 3.78 3.68 3.98 4.40 4.73 4.99 4.91 4.30 3.27

d2 [1010cm]b 1.11 1.16 1.32 1.51 1.62 1.71 1.60 1.30 0.93

a B−V taken from Kim & Demarque (1996), assuming a stellar age of 2 Gyr.For M/M⊙ = 0.4 an extrapolation was used.b Compiled from Copeland et al. (1970).

The frequency λ can be separated in complex and real parts according to λ = iω + γ,where ω = π/Pcyc is the dynamo frequency and γ the mean field growth rate. We solve thedispersion relation numerically and we identify the fundamental mode, i.e. the fastest growingmode. Then the turbulent diffusivities β1 and β2 are tuned in such a way, that the fundamentalmode is slightly subcritical, with a decay time τdec = 1/|γ| of about 10 dynamo periods. For amotivation of this requirement and for analytical solutions of Eqs. (6.3) we refer to Ossendrijver& Hoyng (1996c).

6.3.2 Dynamo parameters of lower main-sequence stars

In this section we present expressions for the parameters that occur in Eqs. (6.3) in terms of thestellar structure and the rotation rate. A number of these expressions relate stellar parametersto solar parameters, which are treated in section 6.3.4.

Apart from τc (Eq. 6.1), the parameters related to stellar structure are derived from a setof models by Copeland et al. (1970). Out of the models presented by these authors we havechosen the series with a composition given by X = 0.7, Y = 0.27, Z = 0.03 and with a mixinglength parameter αML = lML/Hp = 1.5, see Table 6.2. We employ the pressure scale height atthe bottom of the convection zone Hp = kT/µmHg to define the thickness d1 of the overshootlayer (see Skaley & Stix 1991):

d1 = 0.4Hp. (6.8)

For the Sun, this amounts to about 2× 104 km (Table 6.2). We assume that the total difference∆Ω in angular velocity accross the overshoot layer depends on the rotation rate in the followingmanner:

∆Ω =( Prot

Prot,⊙

)p∆Ω⊙. (6.9)

No reliable estimate is known for p, but there are theoretical indications that p is positive, sothat differential rotation decreases with increasing rotation rate, even for very slow rotators(Ro ≫ 1) (Kitchatinov & Rudiger 1995). The corresponding velocity gradient in the overshootlayer can be expressed as

a =Rc∆Ω

d1=Rc∆Ω⊙d1

( Prot

Prot,⊙

)p, (6.10)

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where Rc denotes the distance from the origin to the bottom of the convection zone (Table 6.2).Following Eq. (6.5), we estimate α as α ≈ −1

3Hτc〈u21〉/l ≈ −1

3HHp/τc, where l ≈ Hp ≈u1,rmsτc is the typical length scale of the convective cell and H denotes the normalised helicitycoefficient,

H =〈u1 · (∇× u1)〉√

〈u21〉〈|∇ × u1|2〉

. (6.11)

This coefficient measures the correlation between u1 and the vorticity ∇×u1. It is non-vanishingdue to the effect of rotation on the convective motions, which suggests a dependence on theRossby number. Since |H| ≤ 1, the correlation must saturate for small values of Ro (rapidrotation), but if rotation is slow, we may assume H = H(1)Roq, with q < 0. Here H(1), thenormalised helicity for Ro = 1, is taken to be the same for all the stars in our sample. Its valueis fixed by the solar calibration model (section 6.3.4). The α-coefficient now becomes

α = −13H(1)RoqHp/τc. (6.12)

Here the convective turnover time is calculated from B−V (Table 6.2) with the help of expres-sion (6.1).

We assume next that all the stars have activity belts similar to those of the Sun. Theseactivity belts originate at a latitude of about 35o during the activity minimum and migratetoward the equator in the course of one cycle period, after which new belts of opposite polarityappear at high latitude. The wave vector in the latitudinal direction that is associated with theequatorward migration of the magnetic field can be estimated as

kz =360

70

1

Rc. (6.13)

The turbulent diffusivities are determined in the following way. We determine the ratio fβ =β1/β2 by means of the solar calibration model in section 6.3.4 and we assume that it is the samefor all the stars. The value of β2 (and of β1 = fββ2) is fixed by a condition on the decay time ofthe mean magnetic field, see section 6.3.5. However, we shall demonstrate in section 6.3.4 that,to good approximation, the cycle period depends on β1 and β2 only through their ratio fβ, sothat Pcyc is not affected by this condition.

6.3.3 Theoretical cycle periods

The exact cycle periods of the stellar dynamos are found by solving the dispersion relation ofEqs. (6.3) numerically for a star with a given mass and rotation rate. In order to gain insightin the parameter dependence of Pcyc we may employ an approximative expression. This allowsus to compare in a simple manner the observed relation between Pcyc, Prot and τc with thepredictions of our model. Ossendrijver & Hoyng (1996c) derived the following expression usingan approximative dispersion relation, which is valid if fβ ≪ 1:

Pcyc ≈π√

8

β2k2z

[

− 1 +√

1 + C2]−1/2

(6.14)

where C is the dynamo number, given by

C =aαfββ2

2k3z

. (6.15)

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Table 6.3: Parameters of the solar calibration model

∆Ω⊙ 1.57 × 10−7 s−1

α −25 cm s−1

H(1) 0.099

β1 4.1 × 1010 cm2 s−1

β2 5.8 × 1011 cm2 s−1

fβ 0.071

Pcyc 11 yr

τdec 220 yr

A useful approximation is obtained for large dynamo numbers (|C| ≫ 1):

Pcyc ≈ π√

8 (a|α|kzfβ)−1/2. (6.16)

Notice that Pcyc now depends on the turbulent diffusivities only through their ratio fβ, whichwe take as a constant. Hence we do not require for the moment any further knowledge of β1 orβ2, the values of which will be determined in section 6.3.5. Since the dynamo number (Eq. 6.15)does depend on β1 and β2, we shall verify the validity of Eq. (6.16) also in section 6.3.5, andassume for the moment that |C| ≫ 1. Inserting Eqs. (6.8–6.13) we may write

Pcyc ≈4.30P 0.5p

rot,⊙√

H(1)fβ∆Ω⊙P

−0.5(p+q)rot τ0.5(q+1)

c . (6.17)

We point out that Pcyc depends on stellar structure only through τc. This allows us to compareEq. (6.17) with expression (6.2) that describes the observations, and we conclude that

b = −0.5(p + q) ≈ 2.5

c = 0.5(q + 1) ≈ −2.1⇔

p = 1 − 2(b+ c) ≈ 0.26

q = 2c− 1 ≈ −5.2.(6.18)

The positive value of p suggests that differential rotation decreases slightly with increasingrotation rate (Eq. 6.9). The rather large negative value of q indicates a strong dependence of αon rotation.

6.3.4 The solar calibration model

The calibration of the stellar dynamo models is based on the solar parameters that are shownin Table 6.3. The total difference in angular velocity across the overshoot layer near the equatoris estimated from Goode (1995). The constant H(1) is determined from Eq. (6.12) by adoptinga value for α, namely −25 cm s−1. The choice of α is somewhat arbitrary (see Ossendrijver &Hoyng 1996d for a discussion of α). We determine β1 and β2 following an iterative method, i.e.by solving Pcyc and τdec from the (exact) dispersion relation and applying corrections to β1 andβ2, until Pcyc = 11 years and τdec = 20Pcyc.

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Figure 6.4: logPcyc versus B−V . The solid points denote the observed cycle periods of slowlyrotating stars (Table 6.1). Dotted curves are theoretical cycle periods following from modelcalculations (triangles) at a constant rotation rate. The labels indicate the rotation period indays.

6.3.5 Stellar dynamo models

Having defined the solar calibration model, we now apply the scaling laws for the radial velocitygradient a and for α to lower main-sequence stars with masses in the range 0.7 ≤M/M⊙ ≤ 1.1,which provides complete overlap with all the slowly rotating stars in Table 6.1. The purposeof this calculation is to verify whether indeed the dynamo number is large (|C| ≫ 1), so thatexpression (6.17), with which the observed cycle periods were compared, is a legitimate approx-imation. We therefore determine the turbulent diffusivity β2 (or β1 = fββ2). This is achievedby applying successive corrections to β1 and β2, while keeping fβ constant, until, for a givenstellar mass and rotation rate, we obtain a solution of Eqs. (6.3) with τdec = 20Pcyc.

The result is presented in Fig. 6.4, where we plot the theoretical cycle periods of slowlyrotating stars (triangles), versus B−V for various rotation rates. Besides the model calculationswe have included all the observed cycle periods of the slowly rotating stars from Table 6.1(full circles). Some measured cycle periods show significant deviations from the correspondingtheoretical value, but this is expected because a similar scatter can be observed with respect tothe least square fit in Fig. 6.3. Nevertheless, there is reasonable agreement on the whole betweenthe theoretical curves and the measured cycle periods.

The scaling exponents p and q (cf. Eq. 6.17) that we derived are valid only if the dynamonumber is sufficiently large. The smallest dynamo number that occurs in any one of the solutionsshown in Fig. 6.4 is |C| ≈ 7, and the resulting difference between exact and approximative values

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6.4. Summary and conclusions

of Pcyc is typically a few percent. Given the large uncertainty (about 15%) in the observed cycleperiods, we ignore this effect.

6.4 Summary and conclusions

We have demonstrated that for slowly rotating lower main-sequence stars (Ro >∼ 0.9) with well-

defined periodic chromospheric activity the cycle period Pcyc can be parametrised according toPcyc ∝ P brotτ

cc , with b = 2.5 ± 0.5 and c = −2.1 ± 0.5. The existence of such a relation points to

a common dynamo mechanism for slowly rotating stars. Cycle periods of rapidly rotating stars(Ro <

∼ 0.9) do not follow the same law, which suggests a different dynamo mechanism from thatof the slow rotators. Other qualitative differences with slow rotators support this conclusion:the chromospheric activity of rapidly rotating stars is higher, more irregular and only rarelyexhibits well-defined cycles.

We used a simple linear mean field dynamo model to model the cycle periods of slowlyrotating stars. The observed correlation between Pcyc, Prot and τc is reproduced if the differentialrotation and α scale as ∆Ω ∝ P prot and α ∝ Roq respectively, with p ≈ 0.26 and q ≈ −5.1. Thepositive sign of p suggests that differential rotation would decrease slightly with increasingrotation rate. This trend is not very significant, but is supported by calculations of Kitchatinov& Rudiger (1995), who found a similar dependence on rotation, with p ≈ 0.4. The negativevalue of q indicates that the α-coefficient should increase if Ro decreases, in accordance withthe common assumption that |α| increases with increasing rotation rate. However, our resultalso implies that for constant Prot, |α| increases with increasing τc, i.e. with increasing B−V(Eq. 6.12). This is due to the fact that convective cells with longer turnover times are morestrongly influenced by rotation.

The values of the scaling exponents p and q depend somewhat on the assumptions that aremade for fβ, the ratio of the turbulent diffusivities and kz, the latitudinal wavevector. We madethe simplest possible assumption, namely that for slowly rotating lower main-sequence stars fβis a constant, independent of B−V and Prot. In estimating kz we assumed that the activitybelts extend over 35o of latitude on either side of the equator. In reality the geometry of theactivity belts may depend on the magnetic field strength and other stellar parameters, but theresulting angle is unlikely to differ by more than a factor two.

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Fluctuaties en energiebalans in

zonne- en sterdynamo’s

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FLUCTUATIES EN ENERGIEBALANS IN ZONNE- EN STERDYNAMO’S

7.1 Inleiding

De zon is het meest opvallende hemellichaam en de bron van al het leven op aarde. Op heteerste gezicht lijkt zij een volmaakte lichtgevende schijf, zo helder dat men haar meestal nietdirect kan bekijken. Maar wanneer het zonlicht voldoende wordt verzwakt, blijkt dat het simpelebeeld wordt verstoord. Zo zijn kort voor zonsondergang met het blote oog soms zwarte vlekjeszichtbaar. Zulke zonnevlekken worden al eeuwen waargenomen, maar pas sinds het begin vandeze eeuw is bekend dat zonnevlekken gebieden zijn met een sterk magneetveld. De typischeveldsterkte in een zonnevlek is ongeveer 10.000 keer groter dan die van het aardse magneetveld.De zon is dus een krachtige magneet. Maar het bestaan van zonnevlekken geeft al aan dat hetmagneetveld van de zon een ingewikkelde structuur heeft.

Behalve de zonnevlekken doen zich in de nabijheid van het zonsoppervlak nog meer verschi-jnselen voor waarin magneetvelden een rol spelen. Zo kan men tijdens een totale zonsverduister-ing een lichtgevende krans van ijl en zeer heet gas zien rondom de zon: de corona. Ook vindener op de zon enorme explosies plaats: de zonnevlammen (Van den Oord & Kuijpers 1993, VanOss 1995). Hierbij worden massa’s geladen deeltjes de ruimte in geslingerd, die na enkele dagende aarde bereiken en dan het radioverkeer verstoren en zelfs het electriciteitsnet buiten werkingkunnen stellen. Dit zijn slechts enkele voorbeelden van magnetische activiteit, waaruit blijkt datde zon meer is dan een volmaakte, saaie lichtbron.

Magneetvelden maar komen voor in tal van hemellichamen, waaronder planeten, sterren,accretieschijven rondom compacte objecten alsmede melkwegstelsels. Het beste laboratoriumdat de mens ter beschikking staat om magnetische verschijnselen te bestuderen is echter de zon.

7.2 De zonnevlekkencyclus

Wie als eerste het bestaan van zonnevlekken heeft vermeld is niet meer te achterhalen. Wel iszeker dat zonnevlekken al tijdens de 1e eeuw v.C. in China en Korea vaak zijn waargenomen. Inde klassieke oudheid is er slechts een enkele keer melding van gemaakt, zoals door Theophrastusvan Athene (c. 370-290 v.C.). Dit heeft misschien te maken met het wereldbeeld van de Grieksefilosofen. Volgens hen bevindt de aarde zich in het centrum van de wereld en beschrijven deplaneten en de zon cirkelbanen om haar heen. Zon en planeten beschouwden zij als volmaaktehemellichamen, waardoor elke gedachte aan zonnevlekken bij voorbaat werd uitgesloten. Boven-dien hadden zij slechts weinig waardering voor waarnemingen.

De Griekse filosofen, met name Aristoteles en Ptolemaios, hebben tot na de middeleeuwenons denken bepaald, vooral omdat hun denkbeelden door de kerk werden overgenomen. Pas metde komst van de Verlichting in de 17e eeuw verandert het geocentrische wereldbeeld. De aanzethiertoe gaf Nicolaus Copernicus, die in 1543 een boek uitbracht met de titel De RevolutionibusOrbium Coelestium (Over de omwentelingen der hemelse sferen). Hij ontdekte dat de bewegingenvan de planeten veel eenvoudiger kunnen worden beschreven door aan te nemen dat niet de aardemaar de zon de centrale plaats inneemt in de wereld. Deze ommezwaai was volgens hem ookesthetisch te verantwoorden:

In het midden van alle hemellichamen zetelt de Zon. Wie zou in deze prachtigetempel deze lamp op een andere of betere plaats kunnen zetten dan van waaruit zealles gelijktijdig verlichten kan? (Dijksterhuis 1975).

Na Copernicus werden de wetten van de Aristoteliaanse natuurkunde steeds meer in twijfelgetrokken, ondanks bezwaren van de kerk. Toen omstreeks 1608 de telescoop werd uitgevonden

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Figure 7.1: Zonnevlekken zoals gezien door Hevelius.

was de tijd dan ook rijp voor een herontdekking van de zonnevlekken. Het is niet duidelijkaan wie deze moet worden toegeschreven, maar de eerste publikatie over zonnevlekken stamtuit 1611 en staat op naam van de Oostfries Johann Fabricius. Ongeveer tegelijkertijd begonnenGalileo Galilei en de Duitse Jezuiet Christoph Scheiner zonnevlekken waar te nemen.

Zij allen zagen dat de vlekjes evenwijdige banen volgen over de zonneschijf en dat ze diebinnen een dag of dertien doorkruisen. Scheiner vertegenwoordigde aanvankelijk het orthodoxestandpunt dat het niet om vlekken ging maar om schaduwen van planeten die voor de zon langstrekken. Fabricius en Galilei begrepen echter dat de vlekken zich op het zonsoppervlak moetenbevinden vanwege hun onregelmatige en veranderlijke vorm. Hiermee toonden zij bovendien aandat de zon om een as draait met een periode van iets meer dan 25 dagen. De resultaten vande eerste onderzoekingen aan zonnevlekken tot halverwege de 17e eeuw zijn verschenen in eenaantal prachtig geıllustreerde boeken. Figuur 7.1 is een voorbeeld uit het boek Selenographiavan Johann Hewelke (Hevelius) uit Danzig. Toch verdwenen de zonnevlekken daarna weer voorlange tijd uit het gezichtsveld, niet alleen figuurlijk maar ook letterlijk. Tijdens de periode1645-1715, nu bekend als het Maunderminimum, waren er namelijk bijna geen zonnevlekkente zien. Dat verklaart misschien waarom pas in 1843 de Duitse apotheker Heinrich Schwabe

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ontdekte dat het aantal zonnevlekken varieert volgens een 11-jarige cyclus. In Figuur 1.1 iste zien hoe de zonnevlekkenactiviteit - voor zover bekend - sinds de 17e eeuw is verlopen. De11-jarige cyclus en het Maunderminimum zijn duidelijk te herkennen. Op dit moment (1996)klimt de zon uit het minimum van 1994 en zijn er weinig zonnevlekken, maar omstreeks 1999zal het aantal maximaal zijn, om dan weer af te nemen tot het volgende minimum dat ongeveerin 2005 zal plaatsvinden.

Een andere manier om de zonnevlekkencyclus weer te geven is in het vlinderdiagram, zieFig. 1.2. Hierin is uitgezet tegen de tijd op welke breedte de zonnevlekken voorkomen, waarbij+90 graden de noordpool, 0 graden de evenaar en -90 graden de zuidpool van de zon voorstellen.De zonnevlekken zijn geconcentreerd in twee gordels die zich bevinden binnen ongeveer 35 gradenaan weerszijden van de evenaar. Gedurende een zonnecyclus bewegen deze gordels geleidelijknaar de evenaar toe om dan tijdens het zonneminimum plaats te maken voor twee nieuwe gordelsop hogere breedtes.

7.3 De magnetische cyclus van de zon

In 1908 bewees de Amerikaanse sterrenkundige George Ellery Hale dat er in zonnevlekkensterke magneetvelden heersen. De zonnevlekkencyclus is dus de afspiegeling van een mag-netische cyclus. Wat opvalt is dat het magneetveld niet gelijkmatig over de zon is verspreidmaar geconcentreerd is in kleine magnetische gebieden. Deze magnetische gebieden vormeneen ingewikkeld patroon, waarvan de zonnevlekken het meest opvallende onderdeel uitmaken(Schrijver & Zwaan 1993). Ondanks de complexiteit zijn er ook regelmatigheden te ontdekken.Zo vond Hale dat zonnevlekken vaak paarsgewijs voorkomen en dat de verbindingslijn tussentwee vlekken van een paar onder een hoek van ongeveer 10o staat met de evenaar. Omdatde vlekken met het zonsoppervlak meedraaien spreekt men binnen zo’n paar van een vlek dievoorgaat en een vlek die volgt. Hale vond dat op elk halfrond de voorgaande en volgende vlektegengestelde polariteit hebben. Bovendien is de polariteit van een voorgaande vlek in het no-ordelijk halfrond tegengesteld aan die van een voorgaande vlek op het zuidelijk halfrond. Ookontdekte hij dat na een zonneminimum de polariteiten omwisselen. Deze ompoling houdt in datop de zon een kompasnaald in fase met de magnetische cyclus afwisselend naar het noorden enhet zuiden wijst. Een magnetische cyclus van de zon duurt dus 22 jaar want hij omvat tweezonnevlekkencycli. Het aardse magneetveld is niet periodiek maar men kan zich indenken watvoor navigatieproblemen zich zouden voordoen in de scheepvaart als dit wel zo was.

Uit deze wetmatigheden blijkt dat aan het magneetveld op de zon een samenhangend engrootschalig magneetveld binnen in de zon ten grondslag moet liggen. Dit magneetveld is nietwaarneembaar; de zonnevlekken die wij zien ontstaan wanneer af en toe delen van het mag-neetveld uit diepe lagen naar het zonsoppervlak omhoog komen drijven.

7.4 Variabiliteit in de zonnecyclus

Een opvallende eigenschap van de zonnecyclus is zijn onregelmatige verloop, ofwel zijn vari-abiliteit. Uit Fig. 1.1 blijkt dat het maximale aantal zonnevlekken sterk varieert van cyclus totcyclus en ook dat de periode meestal niet precies 11 jaar is maar enkele jaren korter of langer kanzijn. Tijdens het Maunderminimum (1645-1715) was de zonnevlekkencyclus zelfs bijna helemaalwas uitgeschakeld. Uit het vlinderdiagram (Fig. 1.2) blijkt verder dat het aantal zonnevlekkenop het ene halfrond jarenlang systematisch kan verschillen van dat op het andere halfrond -

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men noemt dit asymmetrie ten opzichte van de evenaar. Een doel van dit proefschrift is teonderzoeken wat de oorzaak van deze verschijnselen is en hoe zij in een theorie voor het onstaanvan magneetvelden kunnen worden beschreven.

De variaties in de duur van opeenvolgende cycli roepen bijvoorbeeld de vraag op of dezonnecyclus gestuurd wordt door een klok of niet. In principe is het mogelijk om door statistischeanalyse erachter te komen welke van deze twee gevallen van toepassing is door na te gaan of dezonnecyclus met het verstrijken der tijd in de pas blijft lopen met een denkbeeldige klok of niet.Helaas reiken de waarnemingen niet ver genoeg terug om hierover uitsluitsel te geven. Maarzoals in de volgende paragraaf zal blijken zijn er nog andere methodes om het verleden van dezonneactiviteit te achterhalen.

7.5 Natuurlijke archieven van zonneactiviteit

De aarde wordt continu bestookt met hoogenergetische deeltjes afkomstig uit ons melkwegstelsel.Deze kosmische straling bereikt het aardoppervlak niet maar wordt in de hogere lagen van deatmosfeer geabsorbeerd. Daarbij worden gewone atomen omgezet in radioaktieve atomen, zoalskoolstof-14 (14C) en beryllium-10 (10Be). De aangemaakte hoeveelheid van deze radioısotopenhangt af van de intensiteit van de kosmische straling die de aarde bereikt. Nu heeft een mag-neetveld de eigenschap dat het electrisch geladen deeltjes kan afbuigen. Aldus schermen demagnetische velden van zon en aarde de kosmische straling gedeeltelijk af. Een gevolg hiervanis dat het gehalte aan 14C en 10Be in de atmosfeer gemoduleerd wordt door variaties in demagnetische velden van zon en aarde.

Waarom is dit van belang? Om met 14C te beginnen; dit wordt net als gewoon koolstofopgenomen in levende organismen. Door nu het 14C-gehalte te meten in dateerbare boomringenis het mogelijk de magnetische activiteit van de zon tot duizenden jaren terug te reconstrueren.Figuur 1.3 toont de variaties in het 14C gehalte voor de afgelopen 9500 jaar, zoals gemeten injaarringen van de Amerikaanse bristlecone spar. De langzame trend wordt toegeschreven aande verandering van het aardse magneetveld en de uitschieters daar bovenop aan de magnetischeactiviteit van de zon. Voordat 14C wordt opgenomen door bomen en planten als componentvan CO2, heeft het gemiddeld zo’n veertig jaar in de atmosfeer doorgebracht. Gedurende dezetijdspanne wordt al het aangemaakte 14C goed gemengd, zodat de 11-jarige zonnecyclus nietmeer terug te vinden is in het 14C-gehalte van boomringen. Wel zijn de variaties op langetijdschalen zichtbaar, waaronder het Maunderminimum (1645-1715, de rechter pijl in Fig. 1.3) enhet zogenaamde Sporerminimum (1420-1530, de linker pijl). Merk op dat een lagere magnetischeactiviteit dus overeenkomt met een hoger 14C-gehalte, zoals verwacht. Uit dergelijk onderzoekblijkt dat de zon op onregelmatige tijden periodes van lage magnetische activiteit heeft gehadmet een typische duur van enkele eeuwen.

Beryllium-10 hecht zich aan stofdeeltjes en bereikt het aardoppervlak na 1-2 jaar door neer-slag. Deze tijdschaal is zo kort dat het 10Be-gehalte in de atmosfeer wel gemoduleerd wordtdoor de 11-jarige cyclus. Dit is bevestigd door metingen van het 10Be-gehalte in ijskernen diemen in gletsjers of poolkappen opboort. In principe is het mogelijk zo de geschiedenis van dezonnecyclus tot honderdduizenden jaren terug te reconstrueren, maar op dit moment is mennog niet zover vanwege een aantal praktische moeilijkheden. Zo is bijvoorbeeld de datering vanjaarlijkse ijslaagjes veel moeilijker dan van boomringen omdat de dikte ervan sterk van plaatstot plaats kan varieren en omdat ze meer en meer samengedrukt zijn naarmate ze dieper liggen.Ook weet men niet genoeg van het aardse magneetveld in het verre verleden om de bijdrage

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daarvan aan de variaties in het 10Be-gehalte te scheiden van de bijdrage van de zonneactiviteit.

De bovengenoemde methodes vertellen ons iets over hoe vaak de zon Maunderminimum-achtige toestanden heeft doorgemaakt en, misschien in de nabije toekomst, of de zonnecyclusdoor een klok wordt gestuurd en of de periode ervan in de loop van de geschiedenis is veranderd.Deze laatstgenoemde vraag houdt verband met mogelijke evolutionaire effecten op de zonneac-tiviteit. De zon maakt namelijk langzaam haar energievoorraad op en daarbij verandert haarinterne structuur.

7.6 Magnetische activiteit van sterren

Evolutionaire effecten in de zon zijn pas na miljoenen jaren merkbaar. Om toch te kunnenzien hoe magnetische activiteit afhangt van de interne bouw van een ster ligt het voor de handnaar andere sterren te kijken. De afstand tot de sterren is zo groot dat deze zich zelfs door degrootste telescopen nog steeds als een punt voordoen. Het is daarom niet mogelijk om zoals opde zon stervlekken te zien, behalve wanneer die een flink deel van het steroppervlak bedekkenzodat zich ten gevolge van rotatie grote helderheidsvariaties voordoen. De bedekkingsgraad vanstervlekken geeft men weer door een getal tussen 0 en 1, de vulfactor f . Op de zon varieert fvan 0.001 tijdens het zonneminimum tot een waarde niet groter dan ongeveer 0.01 tijdens hetzonnemaximum. Om dergelijke variaties in magnetische activiteit te kunnen meten zijn specialetechnieken nodig.

Als de vulfactor niet kleiner is dan ongeveer 0.1 kunnen de magnetische veldsterkte en devulfactor beide worden gemeten door gebruik te maken van het Zeeman-effect. Dit effect berustop de splitsing van spectraallijnen in de aanwezigheid van een magneetveld. Uit dergelijke metin-gen blijkt dat de vulfactor van magnetisch actieve sterren groter is naarmate de rotatiesnelheidgroter is, terwijl de veldsterkte in de stervlekken samenhangt met de gasdruk aan het steropper-vlak maar onafhankelijk is van de rotatiesnelheid. Ook bij deze methode moet de magnetischeactiviteit van de ster veel sterker zijn dan die van de zon om gedetecteerd te kunnen worden.We kunnen niet uitsluiten dat de sterren die hiervoor in aanmerking komen een heel andermagnetisch gedrag vertonen dan de zon.

Om de magnetisch activiteit van zonachtige sterren te detecteren kijkt men naar het sterlichtdat in bepaalde spectraallijnen, waaronder de zogenaamde Calcium H- en K-lijnen, wordt uit-gezonden (Schrijver 1990). Het blijkt dat de hoeveelheid licht die hierin wordt uitgezonden eengevoelige maat is voor de magnetische activiteit van de ster. Dit weet men uit waarnemingenvan de zon, waar deze spectraallijnen sterk zijn in magnetische gebieden en zwak daarbuiten.Zo wordt nu al bijna 30 jaar lang op de berg Wilson in Californie regelmatig de magnetischeactiviteit gevolgd van zo’n honderd sterren, die hiervoor speciaal zijn uitgekozen, want lang nietalle sterren zijn magnetisch actief - zie § 7.7.

Uit dit zogenaamde HK-project blijkt bijvoorbeeld dat magnetische cycli niet uniek zijn voorde zon maar bij een kwart van de zon-achtige sterren voorkomen, zie Fig. 1.4c. De lengtes vande cycli varieren van 7 tot meer dan 20 jaar en hangen af van de draaisnelheid van de ster envan zijn interne structuur (hoofdstuk 6). Ook zijn er aanwijzingen dat sommige sterren in eenMaunderminimum-achtige toestand verkeren (Fig. 1.4a). Deze twee types van gedrag komenvooral voor bij oude sterren zoals de zon. Zulke sterren draaien langzaam vanwege de remmendewerking die het magneetveld gedurende miljarden jaren op de ster heeft uitgeoefend. Jongesterren roteren veel sneller en hebben sterkere, meer onregelmatige magnetische activiteit; eenvoorbeeld hiervan is te zien in Fig. 1.4b.

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7.7 Dynamotheorie van zon en sterren

7.7.1 Dynamowerking in zon en sterren

De oorsprong van magneetvelden in zon en sterren wordt gezocht in de electrische stromendie lopen in het binnenste van deze hemellichamen, net als in een fietsdynamo. De theoriedie zich hiermee bezighoudt heet dan ook dynamotheorie. Toch zijn er belangrijke verschillentussen kosmische dynamo’s en fietsdynamo’s. Zo werkt een kosmische dynamo natuurlijk zonderpermanente magneet en zonder vastomschreven geısoleerde circuits!

Het uitgangspunt van dynamotheorie zijn de vergelijkingen van de magnetohydrodynamica(MHD), die op hun beurt zijn gebaseerd op de Maxwellvergelijkingen. De MHD-vergelijkingenbeschrijven hoe bewegingen in een gas van geladen deeltjes, ofwel een plasma, aanleiding geventot een magnetisch veld. We spreken van dynamowerking als de gasbewegingen zodanig zijn dathet magneetveld gedurende lange tijd in stand blijft of versterkt wordt.

Om verder in te gaan op dynamowerking in sterren, is het nodig eerst het begrip van demagnetische veldlijn in te voeren. Men kan een magneetveld beschouwen als een soort bundelvan onzichtbare spaghettidraden of veldlijnen. Hoe meer veldlijnen er bij elkaar liggen, hoesterker het magneetveld is. Een belangrijke eigenschap van veldlijnen in een goed geleidendplasma is ook dat deze niet zelfstandig bewegen ten opzichte van de materie maar praktisch’ingevroren’ zijn. De gasbewegingen in het binnenste van een ster sleuren de veldlijnen dus mee.

Uit het voorgaande blijkt dat kennis van de gasbewegingen in zon en sterren essentieel isom dynamowerking te begrijpen. Het loont daarom de moeite even stil te staan bij de internestructuur van de zon. De zon is een grote gasbol met een straal van 700.000 km. De temperatuuraan het zonsoppervlak is ongeveer 5800 K en deze neemt met de diepte toe tot zo’n 15 miljoengraden in het centrum. Binnen een straal van 175.000 km van het centrum zijn de temperatuuren de dichtheid hoog genoeg om kernfusie mogelijk te maken. De energie die hierbij vrijkomtwordt naar buiten getransporteerd door licht, tot op een afstand van zo’n 500.000 km van hetcentrum. Vanaf dat punt tot aan het zonsoppervlak wordt het energietransport gerealiseerddoor convectie. In de convectielaag stijgen continu hete gasbellen op en dalen koele gasbellenweer neer, als in een pan kokend water. Tijdens het opstijgen en dalen zetten de gasbellen uit enkrimpen ze respectievelijk. Omdat de zon om haar eigen as draait werkt er op de uitzettende enkrimpende gasmassa’s een Corioliskracht, die maakt dat de gasbellen een voorkeursdraairichtingkrijgen, afhankelijk van het halfrond. Een dergelijk cyclonisch effect bestaat ook bij hoge- enlage-druksystemen op aarde. Deze voorkeursdraairichting blijkt een belangrijke rol te spelen indynamowerking.

Anders dan in een vast lichaam, draaien in de zon niet alle delen even snel. Zo zag ChristophScheiner in de 17e eeuw al aan de beweging van zonnevlekken dat de evenaar van de zon snellerdraait dan gebieden op hogere breedtes. Omdat we niet in het inwendige van de zon kunnen kij-ken is het veel moeilijker om erachter te komen hoe de draaisnelheid van de diepte afhangt. Sindskort is het mogelijk door middel van helioseismologie op indirecte wijze toch in de zon te kijken(Schrijver 1991). Het zonsoppervlak voert namelijk allerlei trillingen uit, die hun oorsprong opverschillende dieptes hebben. Door deze trillingen te analyseren weet men tegenwoordig dat bijde evenaar het zonsoppervlak langzamer draait dan diepere lagen. Deze zogenaamde differentielerotatie is het tweede belangrijke ingredient voor dynamowerking in sterren.

De essentie van dynamowerking in sterren kan nu worden begrepen aan de hand van detwee genoemde ingredienten, convectie en differentiele rotatie, en uit het ingevroren zijn van demagnetische veldlijnen. Hiertoe ontbinden we de veldlijnen eerst in twee componenten, te weten

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Figure 7.2: Het principe van dynamowerking in sterren.

poloıdale veldlijnen, in vlakken door de rotatieas, en toroıdale veldlijnen, loodrecht daarop.Dynamowerking in sterren is te beschouwen als een kringproces waarbij poloıdale veldlijnenworden omgezet in toroıdale veldlijnen en andersom, zie Fig. 7.2. Differentiele rotatie wikkeltde poloıdale veldlijnen op in de toroıdale richting, zoals spaghetti op een vork. Hierdoor wordtdus poloıdaal magneetveld omgezet in toroıdaal magneetveld. Het omgekeerde mechanisme ismoeilijker voor te stellen, maar heeft alles te maken met convectie. Volgens het cyclonischeeffect krijgen de convectieve gasbellen tijdens hun verticale beweging gemiddeld een draai mee.De magnetische veldlijnen die door de gasbellen lopen krijgen dezelfde draai, omdat ze zijningevroren. Het netto effect blijkt te zijn dat er poloıdale veldlijntjes bijkomen. Deze kunnenweer door differentiele rotatie worden opgewikkeld tot toroıdale veldlijnen, waarmee de cirkel isgesloten.

Samenvattend kunnen we zeggen dat een sterdynamo wordt aangedreven door convectie endifferentiele rotatie. Hieruit volgt dat de locatie van een sterdynamo gezocht moet worden inde convectielaag. De waarnemingen (zie § 7.6) bevestigen dat magnetische activiteit vooralvoorkomt bij sterren met een convectielaag.

Het blijkt verder dat een dynamo alleen kan functioneren aan de bodem van de convectielaag.Een van de redenen hiervoor is de magnetische drijfkracht, een opwaartse kracht die werkt op allegasmassa’s waarin zich magneetvelden bevinden. Deze kracht is in de convectielaag zo effectiefdat magneetvelden te snel naar boven komen drijven om te kunnen worden opgewikkeld door dedifferentiele rotatie. Juist onder de convectielaag bevindt zich echter een dunne overgangslaagmet een dikte van ongeveer 20.000 km, waarin de magnetische drijfkracht minder effectief is.Hier kan de differentiele rotatie lang genoeg veldlijnen opwikkelen om een toroıdaal magneetveldte creeren met een sterkte van wel 100.000 Gauss - ongeveer een half miljoen maal sterker dandat van de Aarde. Uiteindelijk wint locaal de drijfkracht het toch en drijft er een lus van hetmagneetveld naar boven. Daar waar de lus door het zonsoppervlak heenbreekt zien wij tweezonnevlekken van tegengestelde polariteit.

Het oplossen van de MHD-vergelijkingen voor een ster in zijn geheel is op dit moment on-mogelijk, zelfs met de snelste computers. Er zijn verschillende benaderingen bedacht om eenbeter hanteerbaar probleem te krijgen dat wel kan worden opgelost. Het is bijvoorbeeld mogelijk

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dynamowerking in een simpele geometrie met eenvoudige plasmastromingen theoretisch te on-derzoeken. Men kan ook op de computer een klein stukje van de zon modelleren en hiervoor deMHD-vergelijkingen oplossen. Maar het is onmogelijk om zo het grootschalige magneetveld vaneen ster als de zon te beschrijven. Om de 22-jarige cyclus en het vlinderdiagram te reproducerenis het nodig de belangrijke dynamo-ingredienten realistisch te modelleren voor de gehele zon,liefst in een bolvormige geometrie.

7.7.2 Gemiddelde-veldentheorie

Een vereenvoudigde vorm van dynamotheorie waarbij het toch mogelijk is het grootschaligemagneetveld van de zon te reproduceren inclusief de 22-jarige cyclus en het vlinderdiagram isgemiddelde-veldentheorie. Deze theorie beschrijft niet hoe het magneetveld er precies uitzietmaar hoe het er ’gemiddeld’ uitziet. Het voordeel hiervan is dat je niet alle kleine details vande convectielaag hoeft te kennen om het gemiddelde magneetveld te kunnen beschrijven. Eenprobleem van deze theorieen is dat niet bewezen is of zij geldig zijn voor zon en sterren. Deafleiding van de theorie maakt namelijk gebruik van bepaalde aannames waaraan in de zon nietis voldaan.

In de gemiddelde-veldenbenadering worden alleen uitspraken gedaan over gemiddelde groothe-den. De keuze van de middelingsprocedure is hierbij van grote invloed op de uitkomst. Indit proefschrift worden twee soorten middeling toegepast, namelijke lengtemiddeling in hoofd-stukken 2 en 3, en ensemblemiddeling in hoofstukken 4 t/m 6.

7.7.2.1 Lengtemiddeling

Met lengte wordt hier bedoeld heliografische lengte, d.w.z. de hoek tussen een punt in de zonen de meridiaan, gemeten langs een cirkel evenwijdig aan de evenaar. Bij een lengtemiddelingwordt de desbetreffende grootheid, bijvoorbeeld het magneetveld, gemiddeld over een volledigelengtecirkel van 360o. In de vorige paragraaf hebben we gezien dat de convectielaag van de zonbestaat uit opstijgende en neerdalende gasmassa’s. Aan de bodem van de convectielaag hebbendeze convectieve cellen afmetingen van tienduizenden kilometers en spreekt men van reuzecellen.Een lengtemiddeling omvat dus, afhankelijk van de grootte van de cirkel waarover wordt gemid-deld, slechts een klein aantal reuzecellen - varierend van enkele tientallen aan de evenaar toteen aan de noord- en zuidpool. In elke reuzecel beweegt het gas op een iets andere manier en -wat met name relevant is voor dynamowerking - heerst een iets andere voorkeursdraairichting.Bovendien verandert dit patroon in de tijd want een reuzecel heeft een typische levensduur vaneen dag of twintig, waarna een nieuwe cel zijn plaats inneemt. Het resultaat van een lengtemid-deling in de convectielaag is op verschillende tijdstippen daarom telkens iets anders. Met anderewoorden: de gemiddelde grootheden vertonen tijdsafhankelijke variaties.

Dit is van belang in verband met de waargenomen variabiliteit van de zonnecyclus (§ 7.4). Inhoofdstukken 2 en 3 wordt aangetoond dat m.b.v. snelheidsfluctuaties in reuzecellen een aantaleigenschappen van de variabiliteit in de zonnecyclus goed kan worden verklaard. Een voorbeeldhiervan is Fig. 3.1, waarin een aantal gesimuleerde vlinderdiagrammen van de zon is te zien.De opgelegde fluctuaties maken dat de contourlijnen van het magneetveld grillig worden op eenmanier die statistisch veel overeenkomsten blijkt te hebben met het waargenomen vlinderdiagram(Fig. 1.2). Een van de conclusies van hoofdstuk 3 is dat de zonnedynamo niet wordt aangestuurddoor een klok.

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7.7.2.2 Ensemblemiddeling

Om de ensemblemiddeling te begrijpen zullen we een gedachtenexperiment uitvoeren. Stel datwe niet een zon ter beschikking hebben maar een ensemble van oneindig veel zonnen, en steldat het magneetveld van al deze zonnen zich op het tijdstip t = 0 in dezelfde fase van dezonnecyclus bevindt. Bij de ensemblemiddeling wordt de desbetreffende grootheid gemiddeldover alle zonnen in het ensemble. Omdat dit er oneindig veel zijn, worden alle tijdsafhankelijkevariaties in de convectielaag hierbij uitgemiddeld. Het gemiddelde magneetveld heeft dan ookgeen variabiliteit, zoals wel bij de lengtemiddeling het geval is.

Het effect van variabiliteit komt in de ensemblemiddeling op een andere manier tevoorschijn.Als het waar is dat de zonnedynamo niet wordt aangestuurd door een klok, dan zullen de opeen-volgende kleine variaties in de lengte van de 11-jarige cyclus maken dat alle zonnen in het ensem-ble steeds meer ten opzichte van elkaar uit fase gaan lopen. Dit verschijnsel heet fasemenging;het vindt bijvoorbeeld ook plaats bij een ensemble van slingers waarvan de slingerlengtes kleinevariaties vertonen. Als alle slingers oorspronkelijk dezelfde uitwijking, bijvoorbeeld +1 naarrechts hebben, dan zijn er na verloop van tijd evenveel slingers die een uitwijking naar rechts alseen uitwijking naar links hebben. De gemiddelde uitwijking is dus eerst +1 maar neemt dan aftot nul doordat positieve en negatieve uitwijkingen steeds meer tegen elkaar wegvallen. Zo ookhet gemiddelde magneetveld van de zon. Dit betekent echter niet dat het echte magneetveld vande zon nul is, evenmin als de uitwijking van een willekeurige slinger uit het ensemble! Hieruitblijkt dat de interpretatie van het ensemblegemiddelde een subtiele aangelegenheid is.

Behalve op het magneetveld van de zon, kunnen we de ensemblemiddeling ook toepassenop de magnetische energie. Deze is evenredig met het kwadraat van het magneetveld en is dusaltijd positief (of gelijk nul). Hetzelfde geldt voor het kwadraat van de slingeruitwijking, omhet voorbeeld van de slingers weer aan te halen. Het effect van fasemenging op de magnetischeenergie is daarom anders dan op het magneetveld zelf, want er zijn geen negatieve bijdragen: degemiddelde magnetische energie neemt niet af met de tijd maar blijft constant.

De twee bovengenoemde conclusies, namelijk dat toepassing van het ensemblegemiddeldeleidt tot een uitdovend gemiddeld magneetveld en een constante magnetische energie, vormenhet uitgangspunt van de hoofdstukken 4 t/m 6. De wiskundige afleiding van de vergelijkingvoor de magnetische energie gaat gepaard met een moeilijkheid die niet aanwezig is in hetgeval van het gemiddelde magneetveld. Deze moeilijkheid heeft te maken met wat er in dezonnedynamo op zeer kleine lengteschalen (d.w.z. ongeveer 1 km) gebeurt. Op zulke afstandengedraagt een magneetveld zich namelijk zeer anders dan op grotere lengteschalen waarover totnu toe is gesproken. Het komt erop neer dat op kleine lengteschalen een magneetveld niet meeringevroren is maar kan uitdoven, waarbij magnetische energie wordt omgezet in warmte. Inhoofdstukken 4 en 5 wordt aangetoond dat dit omzettingsproces een belangrijke rol kan spelenin de energiebalans van de zonnedynamo.

Hoofdstuk 6 tenslotte gaat niet over de energiebalans, maar over het gemiddelde magneetveldvan andere sterren dan de zon. Hier wordt geprobeerd om de waargenomen lengtes van demagnetische cycli te verklaren uit de interne bouw en de draaisnelheid van deze sterren.

Referenties

Dijksterhuis, E.J. 1975, De mechanisering van het wereldbeeld, Meulenhoff (Amsterdam)Schrijver, C.J. 1990, Magnetische aktiviteit van zon en koele sterren, Zenit, september 1990, p.

317

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Schrijver, C.J. 1991, Helioseismologie, waarnemingen van het inwendige van de zon, Zenit,juli/augustus 1991, p. 269

Schrijver, C.J. en Zwaan, C. 1993, Het wisselend aangezicht van zon en sterren, Zenit,juli/augustus 1993, p. 293

Van den Oord, B. en Kuijpers, J. 1993, Magnetische explosies in het heelal, Zenit, juli/augustus1993, p. 300

Van Oss, R. 1995, Vuurwerk op de Zon, Zenit, oktober 1995, p. 438

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Dankwoord

Mijn dank gaat uit naar de personen die dit proefschrift mogelijk hebben gemaakt of die hetwerk eraan hebben veraangenaamd:

Mijn ouders, voor de steun en het vertrouwen die ze schonken, ondanks de ongrijpbaarheidvan mijn bezigheden.

Peter Hoyng, die mij begeleidde tijdens het promotieonderzoek en van wie ik niet alleen veelover dynamotheorie te weten ben gekomen maar ook onmisbare vaardigheden heb geleerd zoalshet gebruik van gezond verstand, het onderscheiden van hoofd- en bijzaken, het plaatsen van heteigen onderzoek in een breder kader en het schrijven van leesbare artikelen. Een proefschrift,zelfs een met een theoretisch onderwerp, was zonder deze scholing niet te schrijven.

Mijn promotoren Max Kuperus, voor zijn steun en aanmoediging, en Henk van Beijeren,voor zijn verhelderende suggesties op het statistische en mathematische vlak.

Kees Zwaan, voor zijn betrokkenheid bij de inhoud van dit proefschrift.

Gerard Stevens, mijn kamergenoot, voor boeiende gesprekken over het katholicisme en an-dere zaken en voor zijn gezonde cynisme over goedbedoelende vooruitgangsdenkers.

Hein Stadhouders, mijn docent Akkadisch, Egyptisch en Hebreeuws, voor zijn grote ken-nis van en enthousiasme voor de talen en culturen van het Nabije Oosten, waar ik door benaangestoken.

John Heise en Gerda Bergsma, al vele jaren mijn medestudenten Akkadisch en even enthou-siast over de Mesopotamische cultuur, voor de vele gesprekken over dit en over alles wat er nietsmee te maken heeft.

Ronald Machielse en Lucky Achmad voor het wegtoveren van computer- en programmeer-problemen.

Nico Sijm voor zijn ondersteuning op het gebied van IDL, en voor zijn gezagondermijnendehumor.

Van de (ex-)promovendi van SRON en het Sterrenkundig Instituut met name Erik Bakker,

Harry Blom, Jeroen van Gent, Diah Setia Gunawan, Wouter Hartmann, Rene

Noordhoek, Simon Portegies Zwart, Nick Schutgens, Kostas Tziotziou, Jacco

Vink en Robert Voors voor het goede A/OIO-klimaat.

Verder Henrik Spoon voor zijn audiovisuele begaafdheid en voor zijn vermogen de DDR weertot leven te brengen.

Dieter Schmitt, fur die Zusammenarbeit und fur die Gastfreundschaft wahrend meiner Be-suche in Gottingen.

Gunther Rudiger, Manfred Schussler und Michael Stix fur die Gastfreundschaftwahrend meines Aufenthalts in Potsdam und Freiburg.

Nigel Weiss, for the hospitality and the fruitful discussions on dynamo theory during my visitsto Cambridge.

Steve Saar, for the hospitality I enjoyed at the Center for Astrophysics at Harvard and forletting me stay in your house. I will never forget the evening tour of tacky Christmas decora-tions.

Wilfried Boland van de Stichting ASTRON voor alle financiele ondersteuning.

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Curriculum vitae

Ik zag het levenslicht op 3 augustus 1967 te Amersfoort, alwaar ik op het (niet meer bestaande)Eemlandcollege Noord op 29 mei 1985 mijn VWO-diploma heb behaald. Aansluitend ben ik aande Rijksuniversiteit Utrecht sterrenkunde gaan studeren. Tijdens de studie werd mijn interessegewekt in de theoretische natuurkunde en uiteindelijk kon ik geen keuze maken tussen beidedisciplines. Voor het eerstgenoemde vak deed ik een afstudeeronderzoek bij Max Kuperus metals onderwerp de magnetische viscositeit van accretieschijven. Dit onderzoek gaf mij een kleinvoorproefje op de rol van magneetvelden in de sterrenkunde, maar de afronding ervan heeft zekergeleden onder mijn inspanningen voor de theoretische natuurkunde. Het afstudeeronderzoekvoor dit vak deed ik bij Matthieu Ernst, met een onderwerp uit de statistische mechanica:diffusie in Lorentz-roostergassen. De hierbij benodigde computersimulaties hebben geholpen ommijn aversie tegen computers enigszins te overwinnen. Uiteindelijk behaalde ik op 31 oktober1991 mijn doctoraal theoretische sterrenkunde en theoretische natuurkunde. Matthieu Ernststelde me toen in de gelegenheid om nog een maand op een korte FOM-positie door te gaan methet onderzoek aan roostergassen.

Sterrenkunde en natuurkunde hebben altijd met het verschijnsel taal moeten wedijveren ommijn aandacht. Aanvankelijk waaierde mijn interesse voor talen uit in diverse richtingen zondersamenhang, maar in het vierde studiejaar vond ik mijn refugium in de colleges Akkadisch, Egyp-tisch en later ook Hebreeuws, allen gegeven door Hein Stadhouders aan de Faculteit Theologie.Deze colleges hebben mijn liefde gevestigd voor de filologie - ook een exacte wetenschap - envoor het oude Nabije Oosten.

Na mijn afstuderen stond ik opnieuw voor de keuze tussen sterrenkunde en natuurkunde,maar deze werd eenvoudig doordat zich de mogelijkheid voordeed een promotieonderzoek te doenwaarin aspecten uit beide disciplines worden verenigd. Aan dit onderzoek, oorspronkelijk getiteld”Stochastische aanslag van grootschalige magneetvelden”, ben ik in januari 1992 begonnen alsonderzoeker in opleiding (OIO), betaald door de stichting ASTRON, onder begeleiding vanPeter Hoyng, met als promoteres Max Kuperus en Henk van Beijeren. Mijn werkplek werdde Stichting Ruimteonderzoek Nederland (SRON) te Utrecht. Het onderzoek begon moeizaam,maar na een bijstelling van het onderzoeksthema is uiteindelijk een resultaat bereikt in de vormvan dit proefschrift.

Tijdens mijn promotieonderzoek heb ik conferenties bezocht in Cambridge (Engeland), Gott-ingen (Duitsland) en Boulder (Verenigde Staten). Later heb ik nog eens met Peter een bezoekgebracht aan Cambridge (Engeland) en tenslotte heb ik een rondreis gemaakt langs institutenin Freiburg, Gottingen en Potsdam (Duitsland), wederom Cambridge (Engeland) en Harvard(Verenigde Staten).

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