MAGNETOHYDRODYNAMICS AND THE EARTH’S CORE The fluid mechanics of astrophysics A series edited by Andrew Soward University of Exeter, UK and Michael Ghil University of California, Los Angeles, USA Founding Editor: Paul Roberts University of California, Los Angeles, USA Volume 1 Volume 7 Solar Flare Magnetohydrodynamics Earth’s Deep Interior Edited by E R Priest The Doornbos memorial volume Edited by D J Crossley Volume 2 Stellar and Planetary Magnetism Volume 8 Edited by A M Soward Geophysical and Astrophysical Convection Edited by P A Fox and R M Kerr Volume 3 Magnetic Fields in Astrophysics Volume 9 Ya B Zeldovich, A A Ruzmaikin and Advances in Nonlinear Dynamos D D Sokoloff Edited by Antonio Ferriz-Mas and Manuel Nuñez Volume 4 Mantle Convection Volume 10 Plate tectonics and global dynamics Magnetohydrodynamics and the Earth’s Core Edited by W R Peltier Selected works of Paul Roberts Edited by A M Soward Volume 5 Differential Rotation and Stellar Convection Sun and solar-type stars G. Rüdiger Volume 6 Turbulence, Current Sheets and Shocks in Cosmic Plasma S I Vaintshtein, A M Bykov and I N Toptygin This book is part of a series. The publisher will accept continuation orders which may be cancelled at any time and which provide for automatic billing and shipping of each title in the series upon publication. Please write for details. MAGNETOHYDRODYNAMICS AND THE EARTH’S CORE Selected works of Paul Roberts Edited by Andrew Soward University of Exeter, UK First published 2003 by Taylor & Francis 11 New Fetter Lane, London EC4P 4EE Simultaneously published in the USA and Canada by Taylor & Francis Inc, 29 West 35th Street, New York, NY 10001 Taylor & Francis is an imprint of the Taylor & Francis Group This edition published in the Taylor & Francis e-Library, 2005. “To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.” © 2003 Taylor & Francis Publisher’s Note This book has been prepared from camera-ready-copy. Printed and bound in Great Britain by T J International Ltd, Padstow, Cornwall All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Every effort has been made to ensure that the advice and information in this book is true and accurate at the time of going to press. However, neither the publisher nor the authors can accept any legal responsibility or liability for any errors or omissions that may be made. In the case of drug administration, any medical procedure or the use of technical equipment mentioned within this book, you are strongly advised to consult the manufacturer’s guidelines. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging in Publication Data A catalog record for this title has been requested. ISBN 0-203-27222-X Master e-book ISBN To Maureen CONTENTS Foreword xi Commentary on publications xii The 1999 John Adam Fleming Medal xxi Acknowledgements xxv PART 1 Geomagnetism and dynamos 1 Propagation of induced fields through the core 3 P.H. ROBERTS On analysis of the secular variation – A hydromagnetic constraint: theory 15 P.H. ROBERTS AND S. SCOTT Some comments on the theory of homogeneous dynamos 30 R.D. GIBSON AND P.H. ROBERTS Nearly symmetric hydromagnetic dynamos 43 J.G. TOUGH AND P.H. ROBERTS Magnetic core–mantle coupling by inertial waves 52 P.H. ROBERTS A three-dimensional kinematic dynamo 54 S. KUMAR AND P.H. ROBERTS Are planetary dynamos driven by gravitational settling? 78 DAVID E. LOPER AND PAUL H. ROBERTS A study of conditions at the inner core boundary of the Earth 80 DAVID E. LOPER AND PAUL H. ROBERTS Structure of the Earth’s inner core 86 DAVID R. FEARN, DAVID E. LOPER AND PAUL H. ROBERTS Time-dependent electromagnetic core–mantle coupling 88 M. STIX AND P.H. ROBERTS vii CONTENTS On topographic core–mantle coupling 100 PAUL H. ROBERTS A three-dimensional convective dynamo solution with rotating and finitely conducting inner core and mantle 107 GARY A. GLATZMAIER AND PAUL H. ROBERTS A three-dimensional self-consistent computer simulation of a geomagnetic field reversal 120 GARY A. GLATZMAIER AND PAUL H. ROBERTS An anelastic evolutionary geodynamo simulation driven by compositional and thermal convection 127 GARY A. GLATZMAIER AND PAUL H. ROBERTS Rotation and magnetism of Earth’s inner core 141 GARY A. GLATZMAIER AND PAUL H. ROBERTS PART 2 Fluid mechanics and MHD 147 Characteristic value problems posed by differential equations arising in hydrodynamics and hydromagnetics 149 P.H. ROBERTS On the motion of a liquid in a spheroidal cavity of a precessing rigid body 169 K. STEWARTSON AND P.H. ROBERTS The ellipticity of a slowly rotating configuration 189 S. CHANDRASEKHAR AND P.H. ROBERTS Singularities of Hartmann layers 197 P.H. ROBERTS The critical layer in stratified shear flow 211 P. BALDWIN AND P.H. ROBERTS On the hydromagnetics of rotating fluids 229 IBRAHIM A. ELTAYEB AND PAUL H. ROBERTS A transformation of the stellar wind equations 232 P.H. ROBERTS viii CONTENTS A hamiltonian theory for weakly interacting vortices 234 P.H. ROBERTS A unified approach to mean field electrodynamics 245 P.H. ROBERTS AND A.M. SOWARD MAC waves 261 P.H. ROBERTS On the motion of a fluid that is incompressible in a generalized sense and its relationship to the Boussinesq approximation 264 R.N. HILLS AND P.H. ROBERTS Nonlinear theory of localized standing waves 272 BRUCE DENARDO, ANDRÉS LARRAZA, SETH PUTTERMAN AND PAUL ROBERTS Ideal instabilities in rapidly rotating MHD systems that have critical layers 276 SIKUN LAN, WEIJIA KUANG AND PAUL H. ROBERTS Structure and stability of a spherical implosion 304 P.H. ROBERTS AND C.C. WU Thermal inertial waves in a rotating fluid layer: Exact and asymptotic solutions 310 KEKE ZHANG AND PAUL H. ROBERTS The decay of bubble oscillations 318 P.H. ROBERTS AND C.C. WU PART 3 Superfluidity 321 Roton–phonon relaxation times in helium II 323 R.J. DONNELLY AND P.H. ROBERTS Nucleation of quantized vortex rings by ions in helium II 325 RUSSELL J. DONNELLY AND PAUL H. ROBERTS Theory of the onset of superflow 330 PAUL H. ROBERTS AND RUSSELL J. DONNELLY Motion of ions at finite velocities in helium II 335 DONALD M. STRAYER, RUSSEL J. DONNELLY AND PAUL H. ROBERTS ix CONTENTS Dynamics of rotons 340 P.H. ROBERTS AND R.J. DONNELLY A simple theory of temperature-dependent energy levels: application to He II 342 P.H. ROBERTS AND R.J. DONNELLY The electron bubble in liquid helium 345 P.H. ROBERTS Calculation of the static healing length in helium II 363 P.H. ROBERTS, R.N. HILLS AND R.J. DONNELLY Superflow in restricted geometries 367 R.J. DONNELLY, R.N. HILLS AND P.H. ROBERTS Non-linear hydrodynamics and a one fluid theory of superfluid He4 371 S.J. PUTTERMAN AND P.H. ROBERTS What happens to vortex rings that shrink? 375 C.A. JONES AND P.H. ROBERTS PART 4 Mixed phase regions 377 On the life cycle of a mush 379 R.N. HILLS, P.H. ROBERTS AND A.M. SOWARD Relaxation effects and the evolution of a mush 405 R.N. HILLS AND P.H. ROBERTS Publication list 422 x FOREWORD Between Thursday 15th and Saturday 17th July 1999 the Paul Roberts’ Seventieth Anniversary Meeting was held at the School of Mathematical Sciences at the University of Exeter. During that meeting a number of Paul’s close colleagues gave substantial talks on topics which focused on their past research interactions, examples of which will be found amongst the selected works that appear here. As part of those celebrations, we encouraged Paul to collect together those papers, that he saw fit, into this single volume. His research contributions are remarkable both in their multitude, diversity, depth of content and interna- tional importance. A particular merit of this volume is Paul’s interesting choice of papers, which includes some very fine articles that appeared in diverse and possibly remote jour- nals. It is therefore extremely valuable to have them all readily accessible in this one volume, even though it can only catch the flavour of his remarkable achievements. Paul has, however, kindly provided a Commentary of the papers included to help guide us through; he has put his work into historical context focusing for us on the issues of the time. The result is a wonderful compendium giving a clear overview of many of Paul’s seminal contributions, which have served as the foundation for much subsequent research. It was extremely timely that in 1999 the American Geophysical Union honoured Paul with the award of the John Fleming Medal. The complete Citation, made by Masaru Kono, is included here as well as Paul’s Response. Any further Editorial words seem superfluous; the Citation, the Response and Paul’s Commentary say it all. Andrew Soward, 2000 xi COMMENTARY ON PUBLICATIONS Anyone who troubles to look at my publication list, is sure to wonder how someone can possibly have moved so randomly amongst such a diverse set of topics, and this is meant to explain how it happened. It also talks about some interesting scientists I have met. My career in research started in 1951 when Herman Bondi, whose lectures to us as undergraduates had been so inspirational, agreed to advise me for the next three years.