IUGG Union Lectures Reprinted from the IUGG Chronicle Published by International Union of Geodesy and Geophysics TABLE OF CONTENTS Preface Helmut Moritz iii TheE arth'sM agnetismP:a stA chievemenatsn dF utureC hallenges H.K. Moffatt 1 Predictability of Chaotic Phenomena A.S. Monin 21 IonosphereT:h e Edgeo f Space SiegfriedJ . Bauer 37 Tidal Interactionsi n the Earth Moon System Paul Melchior 47 ¸ 1992 International Union of Geodesya nd Geophysics Distributed by the American Geophysical Union 1992 Printed in the United States of America PREFACE The various disciplines of geodesya nd geophysicsg row so rapidly that it is difficult to follow recent developments outside one's particular field of research. For this reason the Union Lectures were established, in which prominent speakers review various topics in geodesy and geophysicsi n a way that should be understandable to their colleagues from other geophysical and related disciplines. Union Lectures, usually five at each General Assembly, were first introduced at the XVII General Assembly of the Interna- tional Union of Geodesy and Geophysics( IUGG) held at Canberra in 1979 and since then have enjoyed considerable interest. To underline their importance, no other scientific events of IUGG are permitted during the time of a Union Lecture. The four Union Lectures collected in this reprint volume cover a broad spectrum of topics in geophysics, ranging from the Earth's core (H.K. Moffatt) and mantle (P. Melchior) to hydrosphere and atmosphere (A. Monin) and ionosphere (S.]. Bauer), if such a grossly simplified characterization is permitted. The dynamo theory which is to explain the origin of the Earth's magnetic field, is exerting a constant fascination to theoreticians. The same holds for the theory of nonlinear dynamical systems( popularly known as chaos theory), which is a mathematical theory founded by the great mathematician Henri Poincar(cid:127) a hundred years ago in a remarkable attempt to solve a difficult problem of planetary motion, and revived around 1960 by a group of mathema- ticians among whom the meteorologist E.N. Lorenz has played a fundamental role, as popular expressions such as the butterfly effect in weather prediction and strange attractors in the mathematics of chaos show. Geophysicala pplicationso f this theory range from earthquake prediction (remember the Union Lecture of V.I. Keilis-Borok at the Vancouver General Assembly 1987, published in Reviews of Geophysics 1990) to turbulent phenomena in geophysical fluid dynamics and magnetohydrodynamics.T he theory of Earth tides goes back to Laplace, even to Newton, but only recently, thanks to highly precise measurements and reœmedt heories, it has permitted to obtain valuable information on the structure of the Earth's interior. It forms a link between geodesy, geophysicsa nd astronomy (tides are due to the attraction of the Sun and the Moon), in a similar way as the study of the ionosphere links geophysicst o space sciences. Thus the lectures collected in this volume provide a comprehensivep icture, not only of various disciplines of geophysics,b ut also of their relation to other sciences. These Union Lectures were published first in various issues of the IUGG Chronicle 1991-1992, from camera-readym anuscriptsf urnished by the authors,w ithout undergoing a reviewing process. Since this journal is essentially only an information bulletin on IUGG events, it is not widely accessible.T herefore, IUGG is very grateful to the American GeophysicalU nion for publishingt his reprint volume for IUGG, and to Prof. P. Melchior, Honorary Secretary General of IUGG and editor of the Chronicle, for his preparatory work. It is hoped that the present monograph will be of great interest to geophysicistsa s well as to other scientists who are engaged in the study of our planet. Helmut Moritz, President IUGG iii Special Publications IUGG Union Lectures Vol. 35 IUGG General Assembly,V ienna, August 1991 Union Lecture THE EARTH'S MAGNETISM PAST ACHIEVEMENTS AND FUTURE CHALLENGES H.K. Moffatt Department of Applied Mathematics and Theoretical Physics University Cambridge Silver Street, Cambridge CB3 9EW, UK 1. Historical Perspectives h,ly subject is the magnetic field of the Earth and the xnannero f its variation in space and time, a subject that has challengedt he imagination of mankind since the da,wno f civilisation. My particular concerni s the root causeo f terrestrial magnetism,t he mech- anism that I shall describe as the 'dynamo process'b y xvhicht he magnetic field of the Earth has been generateda nd maintained sincet he earliest geologicatl imes. The history of the magnetic field is revealed by the study of ancient rocks which were susceptiblet o and magnetisedb y the Earth's field as they solidifieda nd cooledt hrough the Curie point - anythingf rom 100 t( to 700 K dependingo n the mineral. From palaeomagnetirce cords, we know that the magneticf ield has existed at roughly its presents trength for at least 3.5 x 109y ears(cid:127)a nd that reversalso f its polarityh aveo ccurreda t randomi ntervalst,y p- ically betweeno ne and 5 times 105 years,t hroughoutt hesea eonso f time. But only in the last 50 yearsh ave we come anywheren ear to an understandingo f this extraordinary global phenomenon.I want to describes omeo f the developmentso f these last 50 years, to indicate where we stand now, and to considert he nature of the critical problemst hat still confrontu s. But first, allow me to place this problemi n proper historicalp erspective. Every child who has played with a magneticc ompassk nowst hat the compassn eedle points North; but he learnsa s he growso lder that magneticN orth is not quite the samea (cid:127)s 'true' North definedb y the Pole star; or to put it differently,t hat the magnetic dipole axis is slightly inclined to the rotation axis of the Earth. This worrying mismatch was already known to the Chineseo f the Sung dynasty. In his great work 'Sciencea nd Civilisation in Copyright 1992 International Union of Geodesy and Geophysics Special Publications IUGG Union Lectures Vol. 35 China'J, osepNhe edhamqu otetsh eM &ngC hhiP i Thano f ShenK ua( c1088w) hichh e translateass f ollows": Magicianrsu b the pointo f a needlew ith the lodestoneth; eni t is ablet o pointt o thes outh.B uti t alwa!Iisn clinessl iyMl!tlo tl(cid:127)ee asta, ndd oesn otp oint directla!lt tl(cid:127)es ourlys"o;t he' declinatioonf t' hef ieldw ask nowna,t leastto theC hinese, morteh an90 0y earasg ogIt. w arse discovaenrdce hda rtoeudbt yt hee arlnya vigatoofr s the1 5tha nd1 6thc enturieasn di np articulabry C hristophCeor lumbu(1s4 92w) hosger eat voyageo f discoverwy ill be celebratende xt year. We recognizteh is declinationno wa s a manifestationo f a cruciald eparturefr om axisymmetryw hichi s essentiMfo r the Earth's internal dynamo to operate. The distinctionb etweenl ocal magneticn orth and 'true' north is often indicatedo n large-scamlea ps.T he smallp rintu suallyw arnst hat thea ngleb etweenth e twod irections changeirsr egularblyy u p to 1ø in 6 yearsT. hisi s the' seculavra riationo'f them agnetic fieldw hichw ask nownt o navigatorosf the 17th centurya, ndw asn o doubta considerable nuisanceto them. EdmundH alley consideredth e possiblec auseso f this secularv ariation in 1693,a ndc oncludetdh at "thee xternapl artso f the globem ayw ellb e reckoneads the she1a1n, dt hei nternMa sa nucleuosr innerg lobein cludewd ithino ursw, itha fluidm edium between.. . onlyt hiso uterS phereh avingit s turbinatingm otions omes mM1m attere ither swiftero r slowerth an the innerB all." - This wasa propheticv isiona s far as the inner structureo f the earthi s concernebdu, t alsor emarkablien its perceptioonf the needf or differentiraol tationn,o wr ecognizeadsa furtherk eye lemenint thed ynampor ocess. Am.o dernu nderstandionfg th isd ynampor ocesiss b asedo nM ichaell: 'aradayla'sw of inductionT. heB anko f Englanhda st hisy earm arkedth eb icentenaoryf F araday's birthb yp rintinag n ewœ 20n oteb earinhgi si mageB. yp ainstakienxgp erimenFtasr, aday discovereind 1 832t hat if a conductomro vesa crosas magnetifcie ld,a ndi f a pathi s availablefo r the completioonf a currentc ircuit,t hen,i n generalc, urrentw ill flowi n that circuitF. ort hisa chievemeFnatr, adawy asa wardetdh eC oplaMy edaol ft heR oyaSl ociety of LondonT. heR oyaSl ocietcyi tationre cordtsh at "... heg ivesin disputabelev idencoef (cid:127) FootnoAtleth: ouqguho taelds ion C hapm&aB na rr(e1l9 4V0o, IlI , p.90t2h)e q uo- tation is thereg ivena lessp ositivein terpretation. Copyright 1992 International Union of Geodesy and Geophysics Special Publications IUGG Union Lectures Vol. 35 electric action due to terrestrial magnetisma lone. An important addition is thus made to the facts which have long been accumulatingf or the solutiono f that most interesting problem, the magnetismo f the earth." It was in fact more than an important addition; it was the key ingrediento f the dynamop rocessa, lthought his was not recognisedti ll much later. Faraday is not alone this year among eminent magneticians to be commemorated on a banknote. Carl FriedrichG auss( 1777-1855) alsoe njoyst his distinctiono n the new 10 mark notei ssuedth is yearb y the DeutscheB undesbankI.n two greatp apers,G auss( 1832, 1838) establishedth e sphericalh armonicd ecompositiono f the Earth's magneticf ield and the techniqueb y which secular variation of the field could be quantified. The traditional unit of field intensityi n Geomagnetismis of courset he gauss( G), and it is regrettable that the Syst(cid:127)meI nternationaol f unitsn owf avourst he tesla( T) (IT = 104(7). Gauss's sphericahl armonicd ecompositioanl lowsu s to extrapolatet he field (assumedp otential) downt o the core-mantleb oundary( CMB), to map the contourso f constantr adial field at the CMB, and to do so at different epochsu sing all available data; Bloxham, Gubbins &: Jackson( 1989) presentt heser esultsi n brilliant colourc ontours.I n thesem aps,t he dipole ingrediento f the field is still quite evidenta t the CMB, but there is alsoa strongp resence of quadrupole,o ctupolea nd higher order ingredients,a s is to be expectedf rom the nature of downwarde xtrapolationt owardst he region where the 'source'c urrents are confined. The slowe volutiono f the pattern( i.e. its secularv ariation)i s alsoe vident. The high point and climax of electromagneticth eory in the 19th century was the publicationi n 1873 of Maxwell'sT reatiseo n Electricitya nd Magnetism. Maxwell built on Faraday'sd iscoveriesa, nd completedt he systemo f equationst hat bear his name. It is interesting to note howevert hat in the chapter of the treatise devoted to "Terrestrial Magnetism", Maxwell comesn owheren ear to any explanationo f the real nature of the phenomenon.H e confinesh imselft o a descriptiono f Gauss'st echniquesfo r the determi- nation of the Earth's field and its time variation,a nd for demonstratingth at the dominant sourcesfo r the field are of internal rather than externalo rigin;b ut as to the root causeo f the phenomenonh,e writes: "The fieldo f investigationin to whichw e are introducedb y the Copyright 1992 International Union of Geodesy and Geophysics Special Publications IUGG Union Lectures Vol. 35 study of terrestrialm agnetismi s as profounda s it is extensiv.e.. . What cause,w hether exterior to the earth or in its inner depths, producess uche normousc hangesi n the earth's magnetism,t hat its magneticp olesm ove slowlyf rom one part of the globet o another? ... These immensec hangesi n so large a body force us to concludet hat we are not yet acquainted with one of the most powerful agents in nature, the sceneo f whose activity lies in those inner depths of the earth, to the knowledgeo f which we have so few means of access".I t was the scienceo f seismologyth at was to providet he vital meanso f access, establishingth e existencefi rst of a liquid outer core( Jeffreys( 1926): "the centralc orei s probablyf luid, but its viscosityis uncertain"),a nd secondlyo f an inner core (Lehmann 1936)t1h ati ss olid(B ulle1n9 46)b, othn owb elievetod b ee ssentfioarlt heo peratioonf the geodynamo. One of the earliest discussionso f possible causeso f terrestrial magnetism was given by Arthur Schusteri n his Presidential addresst o the Physical Societyo f London in 1912. Schusterd iscussedth e argumentsf or and againsta current systemi n the earth's interior, and concludedt hat "the difficultiesw hich stand in the way of basingt errestrial magnetism on electkicc urrentsi nside the earth are insurmountable"- strongw ords,w hich have since been invalidated with the passageo f time, and the birth and advanceo f magnetohydrody- namics. Nevertheless,e ven as late as 1940, Chapman &: Barrel in their great treatise on Geomagnetism, came to the same defeatist conclusiona s Schuster. They discussedL ar- mor's( 1921) suggestiocno ncerningth e possibilityo f self-excitingd ynamoa ctionb ut stated that "Cowling, however,h as shownt hat suchs elf-excitationi s not possible.C onsequently, Schuster'sv iew still holds,t hat 'the difficulties. .. are insuperable[s ic]!" . Cowling( 1934) had not in fact shownt hat suchs elf-excitationi s not possible:h e had merely shownt hat it was not possiblef or axisymmetric systems,a nd the tilt of the magnetic dipole which had beenk nownf or centuriess howst hat we are dealingw ith an emphaticallyn on-axisymmetric system. Neverthelesst he fact that Chapman &: Barrel could be so easily persuadedt hat I Ani lluminatidnigs cussoioft nh ed evelopmelenatsd intgo thesdei scoveraiensd,t he reasonsf or attributing them to Jeffreys and Bullen respectively, is given in the paper "Discoveryo f the Earth's Core"b y Brush( 1980). Copyright 1992 International Union of Geodesy and Geophysics Special Publications IUGG Union Lectures Vol. 35 Cowling'sa nti-dynamot heorem( see,f or example,M offatt 1978) closedt he matter is an indication of the powerful influence that this theorem then had- this no doubt becausei t was one of the few exact resultso f the subject. The year 1940 marks a high point in the collectiona nd systematisationo f geomagneticd ata, but also marks the nadir as regards real understandingo f the origins of terrestrial magnetism. The post-wary earsh aves eena profoundt ransformationin the situation,t o the point at which a dynamo theory of the origin of the earth's magneticf ield is now universally accepteda mongg eophysicistsT. he progressin dynamot heory has been dramatic, and the theory applies with equal force to planets other than the Earth. Statements in textbooks of the 1980'sa re as vigorouslyp ositivea s Schuster'(s1 912) statementw asn egative.T hus, for example,C ook (1980) writes "There is no theory other than a dynamo theory that showsa ny signso f accountinfgo r the magneticfi eldso f the planets";a nd Jacobs(1 984) writes "There has been much speculationo n the origin of the Earth's magneticf ield ... The only possiblem eans seemst o be some form of electromagneticin duction, electric currentsf lowingi n the Earth's core". It is a dynamo theory basedo n the principles of magnetohydrodynamicasn, d ultimately on a suitablee xploitationo f Faraday'sl aw of induction,t hat has led to this remarkablere volutionin our understandinogf Nature. We now turn to the essentiali ngredientso f this dynamo theory. 2. Essential Ingredients of the Dynamo Process The most plausibles cenariof or the geodynamo,n ow widely accepted,i s that first advancedb y Braginski(i 1964a). In this scenariot,h e ultimates ourceo f energyf or the dynamois gravitationael nergyr eleasedb y the slowc oolingo f the Earth, and consequential slowg rowtho f the solidi nner core. As this inner cores olidifiesa, mushyz onei s created at the inner coreb oundary( iCB), within which buoyancyis generatedd ue to the slow releaseo f the lighteri ngredients(s ulphur,o xygen,s ilicon. ..) in the liquid iron alloy of whicht he outerc orei s composedth, is buoyancyfo rceb eingc ompensatepda rtly by the Coriolisf orce due to the Earth's rotation, and partly by the Lorentz force due to the magnetic field, whose presencew e seek to explain. The Coriolis force inducest he vital propertyo f helicityi n the convectivem otion, while an associatedte ndencyt o conserve Copyright 1992 International Union of Geodesy and Geophysics Special Publications IUGG Union Lectures Vol. 35 angular momentum leads to a state of differential rotation in the core about the axis of rotation(d efinewd ithr espectot them antle)T. hesep ropertieasre n owk nownto beh ighly conducivteo self-excitindgy namoa ction,r epresentebdy the 'bootstrapc' ycle Electromagnetic .(cid:127)'"'(cid:127) InductixoXnX (cid:127) Magnetic Electric Fie(cid:127).l(cid:127)d(cid:127) AmpLearwe/'-s- (cid:127)Current The magneticfi eldg rowsa sa n instabilitya ssociatewdi th thisc yclea, nde quilibrates whent he Lorentzf orcei s stronge noughto exerta brakinga ndc ontrollinagc tiono n the convectionW. ithin this scenariore, versalosf polarityc ana ppears implya s a flipping betweentw ou nstables tateso f a complexn onlineadr ynamicasl ystem. Thesee ssentiianl gredienctsa nb em odellebdy a combinatioonf t woi dealisesdy stems (figure la,b); the first is the self-excitingd iscd ynamo,i n whicha discr otatesi n the presencoef a magnetifcie lda ssociate(vdi a Ampere'lsa w)w itht hec urrenIt inducebdy Faxaday'sla w. The simpleste quationg overningth is systemi s of the form dI L(cid:127) + nI = M(cid:127)2I (1) whereL andR axet hes elf-inductanacned r esistancoef thet otalc urrenct ircuitM, is the mutuailn ductancbee tweetnh ew irel oopa ndt her im of thed isca, ndF /is thea ngular velocitoy ft hed isc.T her ight-hansdi deo fe qn.( 1) representthse F axadaiyn ductioenf fect. Actually,e qn. (1) oversimplifiethse situationf;o r it impliesth at whenF /is constanat nd greatert han R/M, the currentg rowsl ike ep (cid:127) where p = L-(cid:127)(M(cid:127)2- R) , (2) a dynamoin stabilityA. s R -, 0, thisg ivesp ,,, M(cid:127)2/L, a finiteg rowthr ate,t heh all-maxk of whati s nowd escribeads a 'fast'd ynamoI.n thisr espectth, em odeel quatio(n1 ) is Copyright 1992 International Union of Geodesy and Geophysics Special Publications IUGG Union Lectures Vol. 35 lnisleadingb, ut the deficiencyis not lethal and may be corrected(M offatt 1979) in a.w a.y that showst hat the discd ynamoi s not in fact fast, but,s low( p--} 0 a.sR --} 0). (cid:127)(t) Cooling (cid:127)(t) T(O,t)----A (t) cos 0 +B(t) sinO Heating (a) (b) Figure 1. Marriageo f (a) the Bullard homopolard iscd ynamoa ,nd( b) the 5Velandor fluid loop, coupled through gravity, to provide a model thermally driven dynamo. The secondm odeli ngredient( figure lb) is a fluid loopo f radius a in a vertical plane, heated from below and cooledf rom above, a potentially unsta.bles ituation (SVclandcr 1967). If the fluid flowsi n the O-directionw ith angularv elocityf /(t) then the temperant,r e perturbationT (O, t) satisfiesa n equationo f the form OT OT 02T 0-7+ n(cid:127) = D37r + S( O ) (3) where D is a measureo f thermal diffusivity in the O-directiona nd S(O) representst he differentialh eating,w hichf or simplicityw e may take to be s(o) = -(cid:127)si(cid:127)0 (4) where a is a positivec onstant. The solutiono f (3) then has the form T(O,t ) = A(t) cos0 + B(t) sin0 (.(cid:127)) Copyright 1992 International Union of Geodesy and Geophysics