Local numerical modelling of magnetoconvection and turbulence - implications for mean-field theories Petri J. K¨apyl¨a Department of Astronomy, Faculty of Science University of Helsinki Academic dissertation To be presented, with the permission of the Faculty of Science of the University of Helsinki, for public criticism in Auditorium XII on 13th October 2006, at 12 o’clock noon. Helsinki 2006 ISBN 952-10-3396-7 (paperback) ISBN 952-10-3397-5 (pdf) Helsinki 2006 Yliopistopaino Acknowledgements First, I would like to acknowledge the financial support from the Magnus Ehrnrooth foundation and the Finnish Graduate School for Astronomy and Space Physics during the thesis work. Furthermore, the Kiepenheuer-Institut f¨ur Sonnenphysik, DFG gradu- ate school “Nonlinear Differential Equations: Modelling, Theory, Numerics, Visualisa- tion”, and the Academy of Finland grant No. 1112020 are acknowledged for providing travel support. The hospitality of Observatoire Midi-Pyr´en´ees in Toulouse and Nordita in Copenhagen during my numerous visits is also acknowledged. Secondly, I wish to thank my supervisors, Dr. Maarit Korpi and Prof. Ilkka Tuominen for their invaluable help, ideas, and support during the thesis work. Furthermore, Prof. Michael Stix, whose knowledge, experience and integrity I greatly admire, deserves spe- cial thanks. I also wish to thank my collaborators Dr. Mathieu Ossendrijver for his help on various problems on physics and numerics, and Prof. Axel Brandenburg for discussions about life, universe and everything (among other things). Majority of the research for this thesis was done while I was staying at the Kiepenheuer- Institut f¨ur Sonnenphysik (KIS) in Freiburg. I wish to thank the staff of KIS for provid- ing a relaxed atmosphere for research and for the occasional movie- and football-related activities during my stay. Special thanks go to Tayeb Aiouaz, Christian Beck, Christian Bethge, Sven Bingert, Peter Caligari, Wolfgang Dobler, Christian Hupfer, Lars Krieger, Daniel M¨uller, Reza Rezaei, Markus Roth, Rolf Schlichenmaier, and Oskar Steiner, some of whom have already moved on from the KIS. I also wish to thank friends I made in Oulu (many of whom are no longer there) Altti, Antti I., Antti S. & Susanna, Arto & Silva, Heidi & Michael, Jouni, Kerttu, Klaus, Marko, Pertti, Petri, Raine, and Satu. Furthermore, I thank Aku and Mikko, my friends since childhood. Very special thanks to my parents Veijo and Raija and to my sisters Tiina and Katri for their patience and support during my studies. iii Included publications The publications included in this dissertation are: Paper I: Brandenburg, A., K¨apyl¨a, P. J. & Mohammed, A. (2004), ‘Non-Fickian dif- fusion and tau approximation from numerical turbulence’, Physics of Fluids, 16, 1020 Paper II: K¨apyl¨a, P. J., Korpi, M. J. & Tuominen, I. (2004), ‘Local models of stellar convection: Reynolds stresses and turbulent heat transport’, Astronomy & Astrophysics, 422, 793 Paper III: K¨apyl¨a, P. J., Korpi, M. J., Stix, M. & Tuominen, I. (2005), ‘Local models of stellar convection II: Rotation dependence of the mixing length relations’, Astronomy & Astrophysics, 438, 403 Paper IV: K¨apyl¨a, P. J., Korpi, M. J., Ossendrijver, M. & Tuominen, I. (2006), ‘Lo- cal models of stellar convection III: The Strouhal Number’, Astronomy & Astrophysics, 448, 433 Paper V: K¨apyl¨a, P. J., Korpi, M. J., Ossendrijver, M. & Stix, M. (2006), ‘Magne- toconvection and dynamo coefficients III: α-effect and magnetic pumping in the rapid rotation regime’, Astronomy & Astrophysics, 455, 401 Paper VI: K¨apyl¨a, P. J., Korpi, M. J., & Tuominen, I. (2006), ‘Solar dynamo models with α-effect and turbulent pumping from local 3D convection calculations’, Astronomis- che Nachrichten (in press) v Contents Acknowledgements iii Included publications v 1 Introduction 1 2 Convection and turbulence in late-type stars 3 2.1Principlesofstellarconvection....................... 3 2.1.1 Stability criterion . ..................... 3 2.1.2 Convectioninmainsequencestars................. 4 2.2Viscosityandresistivityinstellarconvectionzones............ 5 2.3Emergenteffectsofturbulentconvection.................. 6 2.3.1 Differentialrotationandmeridionalcirculation.......... 6 2.3.2 Heattransport............................ 9 2.3.3 Penetrationandovershooting.................... 10 2.3.4 Magnetism.............................. 11 3 Mean-field magnetohydrodynamics 15 3.1Basicsofthemean-fieldtheory....................... 15 3.1.1 Firstordersmoothingapproximation(FOSA)........... 16 3.1.2 Minimaltauapproximation(MTA)................. 17 3.2Angularmomentumtransport........................ 18 3.2.1 ReynoldsstressesandtheΛ-effect................. 19 3.2.2 Meridionalcirculation........................ 21 3.3Convectiveheattransport.......................... 22 3.3.1 Themixinglengthconcept..................... 23 3.4Magneticfieldgeneration.......................... 25 4 Modelling 29 4.1Turbulencemodel............................... 29 4.2Convectionmodel............................... 30 4.2.1 Geometryandgoverningequations................. 30 4.2.2 Boundary conditions and dimensionless quantities . .... 32 4.2.3 Dimensionlessparameters...................... 33 4.2.4 Stratification............................. 34 4.2.5 Thenumericalmethod........................ 35 4.3Limitationsofthenumericalapproach................... 35 vii 4.3.1 Comparisonofnumericalmodelstorealstars........... 37 4.4Axisymmetricmean-fielddynamomodel.................. 39 5Results 41 5.1Passivescalartransport:FOSAorMTA?................. 41 5.2Strouhalnumberforconvection....................... 44 5.3Convectiveangularmomentumtransport................. 47 5.3.1 Effectsofshear............................ 51 5.3.2 Effectsofanimposedmagneticfield................ 52 5.4Convectiveheattransport.......................... 53 5.5Effectsofrotationonthemixinglengthrelations............. 55 5.6 Convective α-effect and turbulent pumping in the rapid rotation regime 58 5.6.1 The α-effectandkinetichelicity.................. 58 5.6.2 Turbulentpumpingofmeanmagneticfields............ 61 5.6.3 ComparisontoFOSA........................ 63 5.6.4 Representative results from axisymmetric mean-field models . 65 6 Conclusions and further work 71 Bibliography 74 Paper I 91 Paper II 101 Paper II 127 Paper IV 137 Paper V 145 Paper VI 159 Publications not included in the thesis 172 viii Chapter 1 Introduction Solar and stellar observations reveal a multitude of complex phenomena that are related to the outer convection zones of those stars. The most prominent emergent effects of convection are the differential rotation and large-scale magnetic fields, strikingly visible from the solar surface. In the longer term, the 11 year sunspot cycle and the grand minima seen in the sunspot number are the most well-known examples of solar activity. During the past decade or so, the surface features of a variety of other active late-type stars, discernible with the Doppler imaging technique, have also been obtained. These stars are usually more rapid rotators and exhibit starspot distributions and cycles that are significantly different from those of the Sun: the large scale magnetic fields are highly non-axisymmetric and concentrated near the poles. During the last decades mean-field dynamo models, in which large scale magnetic fields are thought to arise due to the combined inductive action of differential rotation and small scale turbulence, have been enormously successful in reproducing many of the observed features of the solar magnetic activity on a global scale. In the meantime, new observational techniques, most prominently helioseismology, have yielded invaluable information about the interior of the Sun. Asteroseismology promises to reveal the inner structure of stars other than the Sun in the near future. This new information, however, imposes strict conditions on dynamo models. Moreover, basically all of the present dynamo models depend on knowledge of the small-scale turbulent effects that produce the large-scale phenomena such as differential rotation and global magnetic fields. In many dynamo models these effects have to be prescribed in a rather ad hoc fashion with little or no support from observations or theoretical considerations. With powerful enough computers it would be possible, at least in principle, to numer- ically solve the equations of magnetohydrodynamics under stellar conditions, distinguish the dominant physical processes and study their behaviour as functions of different pa- rameters. However, scales of several orders of magnitude need to be resolved in the same model in order to accurately capture all the relevant dynamics, which renders the full problem unsolvable with the present day, or for that matter, any forseeable computers. In our view, a combination of mean-field modelling and local 3D calculations is a more fruitful approach. The large-scale structures of convection and magnetic fields are well described by global mean-field models, provided that the small-scale turbulent effects are adequately parameterized. The latter can be achieved by performing local calcula- tions because they allow a much higher spatial resolution than what can be achieved in direct global calculations. 1 In the present dissertation three differing aspects of developing mean-field theories and models of stars are studied. Firstly, the basic assumptions of different mean-field theories are tested in the contexts of passive scalar diffusion due to isotropic turbulence (Paper I) and the turbulent electromotive force due to turbulent convection