Course Lectures Abstracts of Participants Fellows Project Reports Summer Study Program in Geophysical Fluid Dynamics Order and Disorder in Planetary Dynamos
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WHOI-88-16 1987 Course Lectures Abstracts of Participants Fellows Project Reports Summer Study Program in Geophysical Fluid Dynamics Order and Disorder in Planetary Dynamos Willem V.R.Malkus Edited by Mary Evans Berry Woods Hole Oceanographic Institution Woods Hole, Massachusetts 02543 May 1988 Technical Report Funding was provided by the Office of Naval Research under contract Number N00014-82-G-0079, and the National Science Foundation under grant Number DMS-85-04166. Reproduction in whole or in part is permitted for any purpose of the United States Government. This report should be cited as: Woods Hole Oceanog. Inst. Tech. Rept., WHOI-88-16. Approved for publication; distribution unlimited. Approved for Distribution: Charles D. Hollister Dean of Graduate Studies 1987 Summer Study Program in Geophysical Fluid Dynamics ORDER AND DISORDER IN PLANETARY DYNAMOS Woods Hole Oceanographic Institution Woods Hole, Massachusetts GEOPHYSICAL FLUID DYNAMICS PROGR.AM Summer 1987 Staff and Visitors NAME AFFILIATION Bayly, Bruce J. Courant Institute of Matheinatical Sciences Berger, Me1 S. University of Ma.ssachusetts at Amherst Bloxhain, Jeremy Harvard University Bolton, Edward W. Yale University Cattaneo, Fa,usto JILA, University of Colorado at Boulder Childress, Stephen Courant Institute of Mathelrlatical Sciences Fields, George B. Harvard University Flierl, Glenn R. Massachusetts Institute of Technology Griffa, Annalisa Scripps Institution of Oceanography Hide, Raymond Meteorological Office, Brachnell, U.I<. Hughes, David DAMTP, University of Cambridge Ierley, Glenn R. Michigan Technological University I-hasny, Robert Courant Institute of Mathematical Sciences Malkus, Willem V.R. Massachusetts Institute of Technology Proctor, Michael R.E. Cambridge University Salmon, Richard L. University of California., San Diego Shepherd, Theodore G. DAMTP, University of Cambridge Sommeria, Joel University of Grenoble Soward, Andrew M. University of Newcastle-upon-Tyne Spiegel, Edward A. Columbia University Stern, Melvin E. Florida State University Strauss, Henry R. Courant Institute of Mathematical Sciences Veronis, George Yale University Weiss, Nigel 0. Cambridge University Young, William R. Massachusetts Institute of Technologj~ Fellows Brazell, Lorna C. Emmanuel College, University of Ca,mbridge Gilbert, Andrew D. Gonville & Caius College, University of Cambridge Hollerbach, Rainer Scripps Institution of Oceanography Hu, Bin Bin Columbia University Jennings, Richard L. University of Newcastle upon Tyne Klapper, Isaac Courant Institute of Mathematical Sciences McMillan, Jack F. University of Hawaii Yano, Jun-Ichi Kyoto University Row 1: R. Jennings, A. Gilbert, M. Stern, B. Bayly, R. Salmon, 6. Veronis, A. Soward, A. Griffa, R, Hide Row 2: J. McMillan, R. Hozlerbach, J. Somrneria, B, Young, G, Ierley, 3. Yano, S. Childress, I. Klapper, B. Hu, L. Brazell, W. Malkus, N. Weiss, K. Rooth J* Whitehead, G, Field, E, Spiegel, M. Proctor, H. Strauss, T. Shepherd, J. Gollub iii - PREFACE Our principal lecturer, Stephen Childress, can be seen on a preceding page emerg- ing from the "magnetic cottage" he constructed to edify those of us who attended G.F.D. '87. His central theme was the kinematic properties of the "fast" dynamo, one whose growth rate is insensitive to electrical conductivity. These novel studies, and the seminars given by Andrew Soward and others, offer assurance of more mech- anistic understanding of evolving magnetic fields in stars and planets. The timely juxtaposition of Childress' lectures on kinematic fast dynamos and the seminars by Bruce Bayly on inertial three-dimensional instabilities of shear flow, may lead soon to a dynamic fast-fast dynamo. Faster than convecting continents, slower than Antarctic bottom water, waves in the Earth's magnetic field move to the west. Geophysicists' knowledge of the underlying process appears to advance at a similar pace. Yet the dynamos of the summer season already have suggested hydromagnetic flows which offer hope both of realizeable laboratory dynamos and more realistic planetary models. Perhaps the pace of testable predictions will quicken? The Fellowship lectures were more confined to the summer topic thaln is usual. Several of the Fellows reported on exciting discoveries of the season, several reported on sound extensions of work discussed in the principal lectures, and several struggled with problems which resisted swift resolution. The record of these lectures found in this volume may appear quite polished for a first paper, yet they have not been formally edited and must be treated as unpublished manuscripts. A reader who wishes to quote material found here should seek the permission of the author. We are indebted to the Office of Naval Research and to three branches of the National Science Foundation for financial support for the summer program. The polished execution of initial editing and IATS format of the principal lectures is due to Mary Berry. We are grateful to Mary and A. L. Peirson for their thoughtful administration of our program. Willem V.R. Malkus Director Contents 1 A Brief Tour of Dynamo Theory 1919 . 1978 7 1.1 Introduction ..................................... 7 1.2 Derivation of the Induction Equation ....................... 8 1.3 The Disc Dynamo ................................. 9 1.4 Toroidal-Poloidal Decomposition of Fields .................... 10 1.5 A Simple Induction Problem ........................... 12 1.6 Summary ...................................... 13 1.7 References ...................................... 13 2 The Alpha-Effect (How to Smooth a Rough Problem) 15 2.1 Introduction ..................................... 15 2.2 Parker's Cyclonic Events .............................. 17 2.3 Roberts' 2-D Cellular Flow (1972) ........................ 17 2.4 Braginskii's Models ................................. 21 2.5 References ...................................... 26 3 Induction at Large and Infinite R 2 7 3.1 Cauchy's Solution for R = CQ ........................... 27 3.2 The Effect of Diffusion ............................... 32 3.3 References ...................................... 36 4 Asymptotics for Large R in Steady Flow 37 4.1 Flux Expulsion ................................... 37 4.2 Alpha Effect at large R in Cellular Flow ..................... 40 4.3 Cat's Eyes ...................................... 44 4.4 Another Modification ................................ 49 4.5 Third Example of a-Effect in the Limit R + CQ ................. 50 4.6 The Density of Dynamos in One Model ..................... 50 4.7 References ...................................... 51 5 The Genesis of the Fast Dynamos 53 5.1 Introduction ..................................... 53 5.2 The Role of Diffusion ................................ 53 5.3 Liapunov Exponents ................................ 54 5.4 Some Models of Fast Dynamos .......................... 56 5.5 The Rope Dynamo ................................. 57 5.6 Problems Facing Fast Dynamos .......................... 59 5.7 Conclusions ..................................... 61 5.8 References ...................................... 61 2 CONTENTS 6 Fast Dynamos I: Steady Flow 6 3 6.1 Introduction ..................................... 63 6.2 Numerical studies of Beltrami's Flow ....................... 66 6.3 Some Aspects of the Geometry of the ABC Flows ................ 68 6.4 Soward's Analysis of Fast Dynamo Activity ................... 72 6.5 References ...................................... 73 7 Fast Dynamos 11: Analysis of Complex Flows 75 7.1 Soward's Analysis of Cellular Flow in the Limit R ca ............ 75 7.2 Heuristic Model ................................... 75 7.3 Boundary Layer Structure ............................. 77 7.4 Ponomarenko's Helical Dynamo .......................... 78 7.5 Dynamos from Non-Magnetic Instabilities .................... 79 7.6 Possible Approaches to Fast Dynamos ...................... 81 7.7 Overlapping Cat's Eyes .............................. 83 7.8 Blinking Eyes .................................... 83 7.9 Other Systems Worth Considering ........................ 83 7.10 References ......................................85 8 Fast Dynamos 111: Methods Based on Unsteady Maps and Flows 87 8.1 Introduction ..................................... 87 8.2 The Stretch-Fold-Shear (SFS) Map ........................ 87 8.3 Mapping the Magnetic Field ............................ 88 8.4 Digression on the "Tent Map" ........................... 90 8.5 Some Graphical Results .............................. 91 8.6 Some Analysis of the Model ............................96 8.7 References ......................................96 9 Dynamo Concepts Applied to Vortex Stretching 9 7 9.1 Preamble ...................................... 97 9.2 General Introduction ................................ 97 9.3 Possible Singularities of Euler and Navier-Stokes Flows ............. 99 9.4 The Taylor-Green Problem ............................ 101 9.5 Discussion ......................................101 9.6 Vortex Tube Models of the Energy Cascade ...................102 9.7 Vortical Automata .................................104 9.8 References ......................................107 ABSTRACTS OF PARTICIPANTS Fast Dynamos in Chaotic Flow Bruce Bayly Observational Constraints on Theories of the Geodynamo Jeremy Bloxham Nonlinear Convection in a Spherical Shell Edward W. Bolton Low Order Model of the Solar Dynamo Fausto Cattaneo Current Sheets in Force-Free Magnetic Fields George Field Wind-Driven Circulation and Free Equilibrium States