Lecture Notes in Physics

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1 Lecture Notes in Physics Editorial Board R. Beig, Vienna, Austria J. Ehlers, Potsdam, Germany U. Frisch, Nice, France K. Hepp, Zilrich, Switzerland R. L. Jaffe, Cambridge, MA, USA R. Kippenhahn, Uttingen, Germany I. Ojima, Kyoto, Japan H. A. Weidenmfiller, Heidelberg, Germany J. Wess, Mfinchen, Germany J. Zittartz, K61n, Germany Managing Editor W. BeigIb8ck Assisted by Mrs. Edwina Pfendbach c/o Springer-Verlag, Physics Editorial Department II Tiergartenstrasse 17, D-6912i Heidelberg, Germany Springer Berlin I-1eidelberg New York Barcelona HongKong London Milan Paris Singapore Tokyo

2 quickly, The Editorial Policy for Monographs The series Lecture Notes in Physics reports new developments in physical research and - teaching informally, and at a high level. The type of material considered for publication in the monograph Series includes monographs presenting original research or new angles in a classical field. The timeliness of a manuscript is more important than its form, which may be preliminary or tentative. Manuscripts should be reasonably selfcontained. They will often present not only results of the author(s) but also related work by other people and will provide sufficient motivation, examples, and applications. The manuscripts or a detailed description thereof should be submitted either to one of the series editors or to the managing editor. The proposal is then carefully refereed. A final decision concerning publication can often only be made on the basis ofthe complete manuscript, but otherwise the editors will try to make a preliminary decision as definite as they can on the basis of the available information. Manuscripts should be no less than ioo and preferably no more than 400 pages in length. Final manuscripts should preferably be in English, or possibly in French or German. They should include a table of contents and an informative introduction accessible also to treated. Authors are free to use the material readers not particularly familiar with the topic in other publications. However, if extensive use is made elsewhere, the publisher should be informed. Authors receive jointly 50 complimentary copies of their book. They are entitled to purchase further copies of their book at a reduced rate. As a rule no reprints of individual contributions can be supplied. No royalty is paid on Lecture Notes in Physics volumes. Commitment to publish is made by letter of interest rather than by signing a formal contract. Springer-Verlag secures the copyright for each volume. The Production Process The books are hardbound, and quality paper appropriate to the needs of the author(s) is used. Publication time is about ten weeks. More than twenty years of experience guarantee authors the best possible service. To reach the goal of rapid publication at a low price the technique of photographic reproduction from a camera-ready manuscript was chosen. This process shifts the main responsibility for the technical quality considerably from the publisher to the author. We therefore urge all authors to observe very carefully our guidefines for the preparation of camera-ready manuscripts, which we will supply on request. This applies especially to the quality of figures and halftones submitted for publication. Figures shouldbe submitted as originals or glossyprints, as very often Xerox copies are not suitable for reproduction. For the same reason, any writing within figures should not be smaller than 2.5 mm. It might be useful to look at some ofthe volumes alreadypublished or, especially if some atypical text is planned, to write to the Physics Editorial Department of Springer-Verlag direct. This avoids mistakes and time-consuming correspondence during the production period. As a special service, we offer free of charge B-TEX and TEX macro packages to format the text according to Springer-Verlag's quality requirements. We strongly recommend authors to make use of this offer, as the result will be a book of considerably improved technical quality. Manuscripts not meeting the technical standard of the series will have to be returned for improvement. For further information please contact Springer-Verlag, Physics Editorial Department II, Tiergartenstrasse 17, D-6912i Heidelberg, Germany.

3 Friedrich H. Busse Stefan C. Müller (Eds.) 4 '. Springer

4 Berlin Editors Friedrich H. Busse Physikalisches Institut Universitift Bayreuth D-9544o Bayreuth, Germany Stefan C. Müller Fakultät für Naturwissenschaften Institut für Experimentelle Physik Otto-von-Guericke-Universitdt Magdeburg UniversitiftsplatZ 2 D Magdeburg, Germany Library of Congress Cataloging-in-Publication Data. Die Deutsche Bibliothek - CIP-Einheitsaufnahme Evolution of spontaneous structures in dissipative continuous systems / Friedrich Busse ; Stefan C - Mliller (ed.). ; Heidelbergr ; New York ; Barcelona ; Hong Kong ; London ; Milan Paris ; Singapore ; Tokyo : Springer, 1998 (Lecture notes in physics. N.s. M, Monographs; 55) ISBN ISSN (Lecture Notes in Physics. Monographs) ISBN Springer-Verlag Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Springer-Verlag Berlin Heidelberg 1998 Printed in Germany The use ofgeneral descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting- Camera-ready by the authors/editors Cover design. design &production, Heidelberg SPIN: / o - Printed on acid-free paper

5 Preface In the past decades the subject of the spontaneous formation of structures in systems far from equilibrium has grown into a major branch of physics research with strong ties to neighboring disciplines. It has become evident that a diverse range of phenomena can be understood within a common mathematical framework which has been called nonlinear dynamics of continuous systems. This name emphasizes the close relationship to the field of nonlinear dynamics of systems with few degrees of freedom which has evolved into a mathematically mature subject in the recent past. Many dynamical features of continuous systems can actually be described through a reduction to a few degrees of freedom and properties of the latter type of systems continue to inspire the study of continuous systems. The goal of this book is to demonstrate through numerous examples the opportunities that exist for the study of nonlinear phenomena through the use of common tools of mathematical analyses and dynamical interpretations. Instead of providing a comprehensive overview of the rapidly evolving field, the contributors to this book are trying to communicate to a wide scientific audience the essence of what they have learnt about the formation of spontaneous structures in dissipative continuous systems and about the competition between order and chaos that characterizes these systems. It is hoped that the book will be helpful even to those scientists represented by the contributing authors. whose disciplines are not The first chapter introduces the mathematical foundations of the subject and some of the mathematical methods used in its study. This chapter clearly plays a central role even though it is not a prerequisite for understanding the following chapters. All chapters of the book are self-contained. But the first one challenges most strongly the mathematical abilities of the reader. Fluid dynamics traditionally has played a leading role in the development of the nonlinear dynamics of continuous systems. The basic equations of motion are well known in this case and the mathematical analysis has proceeded further into regimes of spatio-temporal chaos than has been possible in other continuous systems. Even the mathematical analysis of the dynamics of complex fluids such as liquid crystals has advanced sufficiently such that quantitative comparisons with observations can be achieved. The seventh and eighth chapters focus on recent progress in this area.

6 VI The examples of structure formation in simple Newtonian fluids are far too numerous to be surveyed in the present volume. Several books have been devoted to the classical cases of spontaneous formation of patterns such as Rayleigh-B6nard convection in a fluid layer heated from below and the Taylor-Couette system. Recent progress in the understanding of unusual dynamical features of the latter system is described in the second chapter. Convection driven by centrifugal buoyancy and surface tension induced instabilities are other topics of this chapter. In the third chapter a modified Rayleigh-B6naxd problem is discussed where the pure fluid is replaced by a mixture of two fluids. Also addressed in this chapter is the influence of a mean shear flow on problems of pattern formation. Examples of purely hydrodynamic instabilities in shear layers and the subsequent transition to turbulence axe considered in the fourth chapter. In comparison with the closed systems treated in most of the other chapters, the formation of patterns cannot be exhibited as easily in open systems and sophisticated methods of data analysis are often required to identify "coherent structures". In contrast to the usually assumed time independence of the external conditions, the Faraday experiment depends on time periodic vaxiations of the effective gravity acting on a fluid layer with a free surface. A fascinating variety of standing surface waves is observed in this case. Recent theoretical developments together with experimental observations are reported in the fifth chapter. Periodic conditions can also be imposed in the spatial domain. The resulting dynamics gains complexity through the interaction of modes with forced and spontaneous structures. These and other heterogeneity effects axe discussed in the sixth chapter. But examples can also be found in other chapters as, for example, in the eighth chapter where the influence of noise on the onset of electrohydrodynamic instability in liquid crystals has been analyzed. After hydrodynamical systems, chemical oscillations are the next best understood pattern forming systems. Although the details of the chemical reactions are often too complex to be modelled accurately, quantitative comparisons with measurements are still possible on the basis of models with reaction-diffusion equations for just the major chemical species. Particularly interesting examples of pattern dynamics have been found in recent years in chemical reactions on catalytic surfaces, a topic which is presented in the ninth chapter. Because of buoyancy driven flows caused by concentration gradients, chemical reactions in the fluid phase are intimately connected with convection phenomena as is discussed in the tenth chapter. The wide area of pattern formation in flames could be added here if sufficient space were available. It is remarkable to what extent the reaction-diffusion type equations can be applied for the description of structure forming processes in semiconductors. EspeciaRy useful are coupled activator and inhibitor equations. The activator part corresponds to ionisation or to electron-hole pair cre-

7 VII ation, while the buildup of space charges acts as an inhibitor. In the eleventh chapter a wide variety of experimentally observed phenomena are described within this framework. The basic coherent structure is the current filament. It appears as a single entity or in regular patterns and its dynamics can be influenced by magnetic fields or with electron or laser beams. The numerous electronic techniques available will undoubtedly ensure increasing applications of the formations of structures in semiconductors. The last three chapters are devoted to subjects that are further removed from the central theme of the book. Granular media such as ordinary sand exhibit a variety of different dynamical patterns when shaken or rotated. But a quantitative continuum description is not yet possible in spite of the similarity between sandripples and wave phenomena. The theoretical interpretations in the chapter on granular materials thus rely primarily on computational simulations of the molecular dynamics type which have become a highly successful tool in- studying the dynamics of collections of little solid spheres. The dynamo problem outlined in the next to last chapter, on the other hand, represents a typical bifurcation problem in the dynamics of electrically conducting fluids. There exists little opportunity for pattern formation, however, since the structure of the generated magnetic field will be governed by the outer boundary conditions. This property is caused by the huge magnetic diflusivity of the order of Im2/S of typical liquid metals. The formation of spontaneous magnetic structures thus becomes a problem of cosmic dimensions and only with special efforts it may be realized in the laboratory. Finally, the last chapter presents an example of the formation of spatiotemporal structures in biology and demonstrates the extent to which the development of multicellular organisms can be understood on the basis of reaction-diff-usion type equations together with equations of motion. It is fascinating to see how the morphogenetic mechanisms in the evolution of a slime mould can be explained on the basis of rather simple physical and chemical principles. Acknowledgement: The authors of the chapters of this volume are grateful to the Deutsche Forschungsgemeinschaft for its support of the research focus on Evolution of Spontaneous Structures in Dissipative Continuous Systems. They also wish to thank the evaluation committee and especially its chairman, Prof. S. GroBmann, for the guidance it has provided throughout the past seven years. Bayreuth and Magdeburg, October 1998 F.H. Busse S. C. Miiller

8 Contents Mathematical Tools for Pattern Formation G. Dangelmayr... and L. Kramer.. 1 Formation of Dynamical Structures in Axisymmetric Fluid Systems F.H. Busse, G. Pfister and D. Schwabe Pattern Formation in Binary Fluid Convection and in Systems with Throughfiow M. Licke, W. Barten, P. Bichel, C. Fitterer, St. Hollinger and Ch. Jung Dynamical Structures in Open Fluid Systems D. Rempfer Theoretical and Experimental Investigations of the Faraday Instability H. W. Miller, R. Friedrich and D. Papathanassiou Pattern Formation in an Inhomogeneous Environment W. Zimmermann, B. Painter and R. Behringer Electrically Driven Instabilities in Smectic Liquid Crystal Films H. Pleiner, R. Stannarius and W. Zimmermann Electrohydro dynamic Convection in Nematics W. Pesch and U. Behn....J'J V Dynamics of Patterns of Chemical Reactions on Surfaces R. Imbihl, H. Engel and M. Eiswirth Convection and Pattern Formation Induced by Autocatalytic Chemical Reactions H.R. Brand and S.C. Miller

9 x Formation of Spatio-Temporal Structures in Semiconductors E. Schill, F.J. Niedernostheide, J. Parisi, W. Prettl and H. G. Purwins Formation of Patterns in Granular Materials A. Betat, CAVI. Dury, L Rehberg, G.H. Ristow, M.A. Scherer, M. Schr5ter and G. Straflburqer Spontaneous Generation of Magnetic Fields in the Laboratory F.H. Busse, U. MiWer, R. Stieglitz and A. Tilgner From Single Cells to a Multicellular Organism: The Development of the Social Amoebae Dictyostelium Discoideum B.N. Vasiev and CJ Weijer