WHO 1-78-67 Hole VOLUME NOTES ON THE 1979 SUMMER STUDY PROGRAM ON DYNAMO MODELS OF GEOMAGNETISM IN GEOPHYSICAL FLUID DYNAMICS AT THE WOODS HOLE OCEANOGRAPHIC INSTITUTION Willem V. R. Ma1 kus, Director and Mary Thayer, Editor November 1978 TECHNICAL REPORT Prepured for the Office of Naval Research under Contract NOOOZ4-78-G-0072. App~oved for public release; distribution wz Zimi ted. WOODS HOLE, MASSACHUSETTS 02543 WHO 1-78-67 1978 VOLUME n LECTURES of the FELLOWS NOTES ON THE 1978 SLJMMER STUDY PROGRAM ON DYNAMO MODELS OF GEOMAGNETISM IN GEOPHYSICAL FLUID DYNAMICS AT THE WOODS HOLE OCEANOGRAPHIC INSTITUTION Willem V. R. Malkus, Director and Mary Thayer, Editor WOODS HOLE OCEANOGRAPHIC INSTITUTION Woods Hole, Massachusetts 02543 November 1978 TECHNICAL REPORT Prepared for the Office of IVuvaZ Research under Contract iVO0014-78-60072. Reproduction in whole or in part is permitted for any purpose of the UrrCted States Govsmunent. This report shouZd be cited as: Woods Hole Oceanographic Institution Technical Report WHO1-78-67. Approved for pubZic release; distribution unt-inrited. fi A Approved for Distribution: Robert W. Morse Dean of ~raduateStudies STAFF MEMBERS and PARTICIPANTS Benton, Edward R. University of Colorado, Boulder Busse, Frederick H. University of California, Los Angeles Chi ldress, Stephen . Courant Institute of Mathematical Sciences Gilman, Peter A. N.C.A.R., Boulder, Colorado Howard, Louis N. Massachusetts Institute of Technology Huppert, Herbert E. California Institute of Technology Keller, Joseph a. Courant Institute of Mathematical Sciences Kraichnan, Robert Dub 1in , New Hampsh ire tayzer, David Harvard Observatory, Cambr idge Loper, David Florida State University, Tallahassee Malkus, Willem V.R. Massachusetts Institute of Technology Melcher, James R. Massachusetts Institute of Technology Moffatt, Keith Bristol University, England Olsen, Peter The Johns Hopkins University Pedlosky, Joseph University of Chicago Proctor, Michael R. E. Cambridge University, England Robbins, Kay A. University of Texas Roberts, Paul H. University of Newcastle-on-Tyne, England Sowa rd, Andrew University of California, Los Angeles Spiegel , Edward A. Columbia University Stern, Melvin E. University of Rhode Island Weiss, Nigel 0. Cambridge University, England Whitehead, John A. \Joods Hole Oceanographic Institution Widnall , Sheila E. Massachusetts Institute of Technology Postdoctoral Fel low Knobloch, Edgar Harvard University Predoctoral Fellows Chapman, Christopher J. Bristol University, England Condi, Francis J. The Johns Hopkins University Cuong, Phan Glen University of Cal ifornia, Los Angeles Frenzen, Christopher L. California Institute of Technology, Pasadena Hart, David University of California at Berkeley Hoiyer, Judith University of Cambridge, England Hukauda, Hisashi University of Tohoku, Japan lerley, Glen Massachusetts Institute of Technology Mitsumoto, Shigek . University of Tokyo, Japan Oliver, Dean S. University of Washington, Seattle. EDITOR'S PREFACE VOLUME I I This volume contains the manuscripts of research lectures by the eleven fellows of the summer program. Five of the lectures overlap significantly with the central summer theme of geomagnetism. The other six lectures cover a broad range of current G.F.D. topics from col lective instability to strange attractors. Several of these research efforts are quite polished and probably wi 1 l appear in journals soon. The middle half represent reports of sound progress on studies of thesis calibre. But then, a few of the lectures report on only the very first conse- quences of a novel idea. These lecture reports have not been edited or reviewed in a manner appropriate for published papers. They, therefore, should be regarded as unpublished manuscripts. Readers who wish to reproduce any of the material recorded here should seek permission directly from the authors. These two volumes represent both what we brought with us to the program and the excited first product of our scientific interactions. More sedately worded professional results invariably emerge as the year progresses. For this opportunity, we wish to thank the \Joods Hole Oceano- graphic Institution, the Office of Naval Research, and N.A.S.A. for encouragement and financial support. Mary C. Thayer Willem V. R. Malkus _ HI _ " • "• •• ", , •" ·. 1,-• -0 -6, E L "3 ,mvl" , 34o ---72 m :0:1:ZE ,e, le •-.a, " .- L· .v• -u,, ~z a-" Em 5 U c ~ ~m ~ - ,K - ln *U ••cn m· 0" ••," •~, trad) L- 72 '" ~8 •· -(I .-- ,lL &! ,-c a u>, " '- C- "rg, ..-0-- , E CI a "i -m .-" L c ,-c, ,a,-. r:r * toc• " TI t'3 "C •" N OG$, ,"ln L Ib m mu." I .-"e, • .. ,2 -a r-0• - ••e, 3 "--.. " ,Y~ L. L 3 0" ax LI .- " - •, , ! - iv - Contents of Volume 1: Course Lectures, Seminars, and Mini-symposium Abstracts. m CONTENTS OF VOLUME 11 Lectures of the Fellows Page No. /' "Benard Convect ion wi th Constant Heat Flux Boundaries" 1 Christopher J. Chapman "Sidewall Boundary Layers in Rapidly Rotating Hydromagnetic Convect ion" 18 Francis J. Condi I "Long Wave Motions in an Adiabatic Atmosphere . 24 Pham Giem Cuong "Slurry Dynamics and the Geodynamo" . 37 1 Christopher L. Frenzen "Strange Attractors due to Feedback in Potential . Systemst1 . 4 5 - David C. W. Hart "Barotropic and Barocl inic Sol i tons" . 4 7 Hisashi Hukuda "The Collective lnstabil i ty of Salt Fingers" 57 Judith Y. Holyer "Mean Field Equations for Certain Moments of the Magnetic Field" 70 Glenn R. lerley "On Turbulent Ekman Layers: The Effect of Finite Rossby Numbers etc." 7 9 Sh lgeki Mitsumoto "Steady Buoyant Plumes in a Fluid of Large Prandtl Number and Temperature Dependent Viscosity" 9 1 - Dean S. Oliver d Two papers, "Statistical Dynamics of the Lorenz Model" on p.91 and "Turbulent Diffusion of Magnetic Fields", p.133, will be found as Edgar I Knobloch's Postdoctoral lectures in Volume 1. LECTURES OF THE FELLOWS BEINARDC ONVECTION w 1 TH CONSTANT HEAT FLUX BOUNDAR I ES I Christopher J. Chapman d 1 (1) Introduction The convection which occurs when fluid between two infinite horizontal planes is heated sufficiently strongly from below has been intensively studied. In most of the published analysis it is assumed that the horizontal boundaries of the fluid are perfect conductors, so that the temperature on each is con- stant. (The term 'constant' wi 11 invariably be used to mean ' independent of position'.) In this paper it is assumed instead that the heat flux across the boundaries is constant, so that their temperatures will depend on position once convection has begun. It would be possible in the laboratory to supply heat at a lower boundary at a rate independent of position and temperature, and one means of removing heat from the top at a constant rate would be to have cool- ing by evaporation. An approximation to constant heat flux is obtained by placing the fluid between two poor conductors, and a linear analysis of this situation has been given by Hurle et aZ. (1967). (2) Effect of Fluid Motions on Temperature Distribution Suppose that at some initial instant the fluid is motionless and the temperature varies linearly with height z, from To at the bottom of the layer to Ti at the top (To > TI). If we now impose a steady roll-type motion on the fluid, then in a region where the fluid is rising, the advection of the tem- perature profile will cause the temperature in the centre of the layer to rise; since our boundary conditions are such that the temperature gradient at the boundaries does not alter, the temperature at the boundaries must then rise and the resulting temperature profile will be approximately a linear function of z with the same gradient as before. Similarly, in regions of sinking fluid, the temperature wi 11 be its value before less an amount independent of z. The flow of heat is in the horizontal direction, and the appropriate length scale for estimating the magnitude of diffusion is the horizontal length scale of the motion, since diffusion does not alter the shape of a linear temperatureprofile. Thus we deduce the rather surprising fact - that a roll motion of given small velocity can produce arbitrarily large changes in the temperature if the width of the rolls is taken large enough. (This fact is important later.) The equilibrium isotherm pattern is: Low Temperature l sotherm I Sinking Fluid Rising ~luipHigh Temperature l sotherm The difference in temperature between top and bottom of the layer is approxi- mately independent of position. If the width of the rolls is very large, it will clearly take a long time for the equilibrium temperature distribution to be reached; there is thus a long initial period during which heat is slowly transferred from the regions of sinking fluid to the regions of rising fluid far away. Note how different this is from what happens when the boundaries are held at constant temperature. In this case diffusion in the vertical direction limits the alteration of the temperature, however large the horizontal length scale. (3) Definitions and Governing Equations Assume that fluid of kinematic viscosity ?) and thermal diffusivity X lies between the planes z = -d and z = +d: Fluid We take the equations describing the motion to be: 0 -3~ +y*vp=-pq rad p-- e.. +-tc'vRe, at s, 9- where -u is the velocity, T is the temperature, ,,a the density, p the pressure, the density at temperature To, d the coefficient of thermal expan:.ion, and -qgE the acceleration due to gravity. The Boussinesq approximation is 2 made, that the fluid can be taken to be incompressible except insofar as changes in density produce buoyancy forces. At the boundaries we assume that here is no stress in the fluid and that the temperature gradient is -/51(/3>@); so writing -u = (u, v, w) we have: The equations admit the steady conduction solution: .- Define. 8 , dp , and 4by the equations 7=Ts+ 0, P=,P,cbp Then from (1) and (2) we obtain We shall consider only motions in the (x, z) plane and independent of y.