Vol.UMz 22, NUMsaa ]. JANUARY) 1950

. . .ie .& art. x's . .nterior anc Geoiuagnetisri-iWALTER M. ELSASSER

Randal Morgan Laboratory of Physics, University of Pennsylvania, Philadelphia, Pennsylvania

INTRODUCTION

HIS review represents an eGort to summarize thestate of our knowledge of the earth's far interior,

omitting the very extensive information that pertains tothe crust proper, i.e. , the topmost 30—50 kilometers ofthe earth. For the latter problems which are, of course,closely related to dynamical geology the reader is re-ferred to existing summaries (edited by Gutenberg,1939).

There are, in the main, two direct methods of ex-ploring the earth's far interior. These are, first, the studyof seismic waves that have penetrated to some depthinside the earth, and second the analysis of thegeomagnetic field and its secular variation. We shallgive a general survey of the state of quantitativeknowledge achieved in these subjects. Passing beyondthe organization of the data, one must proceed mainlyby indirect methods, and in practice this means that onehas to rely to a large extent on inductive reasoning. Thepower of logical induction is a notoriously controversialmatter. One can, of course, point to some of its familiarsuccesses in the history of research. Clearly, one of themain shortcomings of the inductive method lies in thecircumstance that the discovery of a hitherto unsus-pected fact may at any time overthrow past generaliza-tions. A classical example of this is to be found ingeophysics, in the discovery of radioactivity whichobliterated the sweeping cosmological consequences thatKelvin and others had drawn from the computed rate ofcooling of the earth on the assumption of thermal con-duction without internal sources.

%ith this proviso, it might be said that our knowledgeof the earth's interior seems to converge, as time goes on,toward a reasonably weH-defined picture. In this con-nection the chemistry of the earth plays a fundamentalrole. It would clearly make little sense to explore themechanical, thermal and electrical conditions prevailinginside the earth unless one obtained at the same time at

* Published as a contribution of a research project sponsored byThe OQR at t'he Institute for A,dvan&ed Study.

least some rough idea of its chemical constitution.Astrophysicists find that the mechanics of stellar modelsdepends on the "mixture" of elements assumed, pri-marily on the ratio of hydrogen to heavier nuclei;similarly it is almost trite to say that the physics of theearth's interior is dependent on its chemistry. At theextreme temperatures and pressures prevailing in thestars the equation of state of the matter involved be-comes fairly independent of finer details of the constitu-tion. This condition is not yet reached inside the earth,although a tendency in that direction might exist(Bullen, 1946, 1947).

In view of the situation just sketched a rather ex-tensive chapter of this review is devoted to geochemistry.A surprising amount of meticulous and fundamentalwork has been done on this subject, and we can onlyindicate some highlights. The writer is not acquaintedwith a good, comprehensive modern review of geo-chemistry, or with any review for that matter. Theseproblems have of late become of considerable interest inconnection with the cosmological importance of nuclearabundance ratios and a review written from the view-point of the geochemist would be most valuable. Thethird chapter deals with whatever little knowledge we

possess, or such inferences as we can draw, about themechanical and thermal state of the earth's interior.From the viewpoint of the geologist the most importantquestion by far is what mechanical mass motions couldtake place or have taken place during the earth' sgeological history. So far as the solid part of the earth isconcerned, we must confess that the question of possibleplastic Row under thermal or other influences is stillentirely open. We confine ourselves, therefore, to adescription of the static situation. Matters are somewhatmore favorable with respect to motions in the liquidcore where one ca,n perhaps gradually evolve a morespecific model. While this is of importance for thephenomena of terrestrial magnetism, it seems at presentunlikely that such motions have a significant influenceupon geological events of the earth's crust,

WA LTE R M. ELSASSER

l4 Q III

Focvs

Fro. 1. Seismic trajectories and travel times.

Success in a subject of this kind depends largely onthe confrontation of results obtained by various special-ists using widely di6ering tools. This imposes certainlimitations upon the author of a review who must needstake a large part of his information second-hand. Insome matters of a mildly controversial nature the writerhas tried to defer to the opinions of his better instructedcolleagues. The last chapter is of a somewhat di6'erent

character. It gives an account of the mechanics ofterrestrial magnetism developed in recent years by thiswriter and by E. C. Bullard. Not enough time has as yetelapsed to permit exhaustive criticism by others, but we

believe that this work is suKciently an integral part ofthe model of the earth's interior to warrant inclusion.

SEISMIC DATA

According to our present knowledge the earth consistsof three principal layers, the crust, the mantle, and thecore. The terminology used here is at present acceptedby practically all geophysicists. We shaH be concernedprimarily with the latter two strata which are separatedfrom each other by an extremely sharp discontinuity ata depth below the surface of 2900 km. This discontinuityforms the boundary of the core (the radius of the corebeing about 3500 km). The seismological work of thepast fifteen years has indicated that there exists near thecenter of the earth another distinct layer forming asphere of about 1300 km radius. No generally acceptedtermino1ogy for this has as yet been established; we

shall designate this layer as the "central body. " In theoMer literature, based on less complete seismic data,there are found more or less speculative subdivisionsinto a larger number of strata, but most of these havebeen gradually abandoned in favor of the simpler schemeindicated, with such minor subdivisions as will bediscussed later on.

The crust as revealed by seismology consists of atleast two comparatively thin layers composed of ma-terials of low density. For the purpose of investigatingthe structure of the earth's far interior, it is sufhcient toschematize the structure of the crust; it is found that theseismic data pertaining to the deeper parts of the earthare quite insensitive to assumed changes in the charac-teristics of the crustal layers. Jeffreys (1939) assumes asuperficial layer composed of granite, of mean thicknessfi km, density 2.6S, and below this a layer, termedintermediate, of thickness 24 km, density 2.87, Itg

chemical composition is not entirely established; itprobably consists in large part of basaltic rocks. Thelower limit of this layer is a surface of discontinuitywhich, in seismology, is usually taken as the lowerboundary of the crust (Mohorovicic discontinuity). Theterm "crust" is sometimes understood in a diGerentsense. Especially in the older literature it is customaryto designate as crust that part of the earth's body wherepermanent elastic stresses are of importance becausehydrostatic equilibrium does not obtain. Conceptions ofthis kind were first developed at a time when it was as-sumed that the phenomena of vulcanism indicate thatthe interior of the earth consists of a hot and viscousmagma. This interpretation of vulcanism is now aban-doned; it is conceded that the appearance of volcanicmagma is a secondary and rather superficial phenom-enon, the melting being caused by local processes whichare not yet fully identified. The earth's mantle, althoughsolid as the seismological evidence shows, yields tostresses by permanent deformation in plastic Row. Thisis shown by gravimetric measurements which provethat below a certain depth there prevails hydrostaticequilibrium to a high degree of approximation. Again,the hydrostatic equilibrium is not entirely complete asindicated by the phenomena of deep-focus earthquakes,reaching on occasion depths down to 700 km or evenmore. A large literature exists on deep-focus earth-quakes, but so far their appearance has not yet beensatisfactorily related to any dynamical concepts. Forthis reason, and because these quakes can obviously notbe dissociated from the mechanics of the crust, deep-focus earthquakes will not be further considered in thisreview. The transition layer, below which one can formany purposes assume hydrostatic equilibrium, is oftenschematically replaced by a single level whose depth hasbeen variously located below 30 and above 100 km.There is ample evidence that with respect to the trans-mission of seismic waves the earth below this level be-

haves as a body in hydrostatic equilibrium, that is, themechanical properties are constant on surfaces of con-

stant pressure which in equilibrium are also equi-potentials.

We shaH now give a brief review of the methods bywhich seismological information about the interior ofthe earth is obtained. One can either compare the times

at which signals from the same seismic shock arrive atdiferent seismographs, that is, determine the travel time

of the disturbance as function of the distance, or else,

compare the amplitudes produced at diGerent stations.The first method has been developed to a point where

very accurate data are obtained; the second often givescorroborative qualitative evidence. It may be assumedfor simplicity that the focus of the quake is at the sur-

face, the actual depth of the focus being accounted for

by a correction term which we shall disregard in thisbrief survey. Seismological tables were first constructedjn the early years of the century; they contmin the travel

EARTH'S INTERIOR AN D GEOMAGNETISM

time T as function of the distance 6 in angular measure

along a great circle connecting the focus with the pointof observation. Tables constructed in more recent years

by various authors show excellent agreement with eachother. Perhaps the most complete data, compiled justbefore the war, are those of Gutenberg and Richter andof Jeffreys and Bullen (Bibliography in Jeffreys, 1945).Jeffreys subjected the data to a thorough statisticalanalysis in order to achieve maximum consistency of thefinal travel times with the primary observations. Thedata of Tables I and III, below, which are simpleanalytical functions of the travel times are claimed byhim to have an intrinsic accuracy of 0.5 percent. %Pith afew exceptions to be considered later the data of otherseismologists agree with these results to within about 1

percent or better. XVith this accuracy it becomes neces-

sary to take into account the ellipticity of the earth; thiscan be done adequately by applying to the observedtravel times a small numerical correction term whose

magnitude depends on the location of the terminals ofthe path with respect to the earth's axis.

In the seismological literature the earthquake wavesare classified as follows: The longitudinal waves in themantle are designated as P ("Primary" ) waves, thetransverse waves as 5 ("Secondary" ) waves. Since theI' waves have the higher velocity they reach the seismo-

graph at a time when it is almost free from disturbance;the time of arrival of the I' wave is the most accuratedatum used in the construction of travel time tables.Composite waves in the mantle are designated by asuccession of symbols; PP, I'S, SS,etc. , are waves which

have been reflected once from the earth's surface; I'I'I'has been reflected twice; PcI', I'cS, ScS, etc. , are wavesthat have been reflected from the boundary of the core.The core itself being liquid, only longitudinal waves arepropagated in it; these are designated as E waves. Pathsthrough the core give rise to composite waves of thetypes PKP, SKS, PKS, etc. (The symbol PKP isusually abbreviated by P'.) There are certain othersymbols that refer to wave phenomena at or near thesurface of the earth which do not concern us here. In thepresent review we need not make much use of seismo-

logical terminology.Typical seismic waves have periods of the order of

several seconds and wave-lengths of the order of a fewto some hundred kilometers. Theory shows (Bullen,1947; Birch, 1939) that in an elastic medium dispersionarises only through the radial variation in the earth ofdensity and elastic constants. This e6ect being verysmall, there is no appreciable dispersion in the approxi-mation of ordinary seismology. The velocity increaseswith increasing depth in an otherwise homogeneouslayer; hence the trajectories of waves are convextowards the center of the earth. Some typical rays com-puted by methods to be described presently are shown in

Fig. 1.Some travel times of longitudinal waves are indi-cated in the figure at the point of emergence. The rays

TABLE I. Velocities in km/sec. within the mantle at diferentdepths. Ro 6338 km.

1.000.990.980.970.960.950.940.930.920.910.900.890.880.860.840.820.800.780.760.740.720.700.680.660.640.620.600.580.560.55

Long. (1)

7.757.948.138.338.548.758.979.509.91

10.2610.5510.7710.99)1.2911.5011.6711.8512.0312.2012.3712.5412.7112.8713.0213.1613.3213.4613.6013.6413.64

Long. (2)

7.767.948.128.328.548.809.129.479.81

10.1710.5010.8110.9711.2811.4911.6711.8612.0312.2112.3812.5612.7112.8713.0013.1813.3113.4213.5813.6513.62

Trans.

4.354.444.544.644.744.854.965.235.465.675.856.006.126.296.396.486.566.646.716.776.836.896.957.017.077.147.207.267.317.30

that pass through the core are shown only very sche-matically; some finer detail about their behavior will begiven presently.

'|A'e now describe briefly the procedure whereby thevelocity as function of the depth is obtained from thetravel time tables; the theory is due to Bateman andHerglotz (Jeffreys, 1929;Bullen, 1947).Let r, 6' be polarcoordinates about the earth's center and let V be thevelocity of the seismic waves where V is a function of ronly (the ellipticity being neglected). By Fermat'sprinciple the travel time along the actual path is aminimum; this time is

T=JI ds/V=) I/V/(dr/d6)2+r~j&d6.

Fro. 2.

If we write r'=dr/d6, we have the Euleria, n equation ofthis variational principle

d/d8(BL/Br') BL/Br =0— (2)

where L stands for the integrand in (1).A 6rst integralof (2), corresponding to the energy integral of ordinarymechanics, is

L r'(BL/Br') =p—

KVA LT E R M. ELSASSE R

p=dT/dA.

Eliminating p from (5) and (6) we get

V(r;„)= r;„(dT/dA)

(6)

(7)

where p is a constant. This gives

r2/V= p(r"+r')t

The parameter p has a simple physical meaning; from(1) and (3) we have

Vp/r = rd8/ds.

Simple geometry shows that

rd8/ds= cosa.,

where o; is the angle that the ray makes with the spherical.surfaces, r=const. Hence p=r cosn/V is a constantalong the ray; this is the generalized Snell's law. For thedeepest point of the trajectory we have in particular

V( min) min/p.

Now let P and P' in Fig. 2 be the points of emergence atthe surface of two neighboring rays and let Q be thepoint where the perpendicular from P upon the secondray intersects the latter. Then (I'I")A„=Rodhwhere 6 is

the angular distance of the point of emergence from thefocus, and (QP)A, = VdT where T is the travel time.Then cosn= UdT/R, dh and hence

On introducing (6) and (7) into (2) one can carry out thefinal integration of the differential equation; the resultmay be expressed in the form

Ro 1 p~ dT/dDlog =—

~ cosh —' da,r;„ir &0 (dT/dh)0

where the index 0 indicates that the derivative is to betaken at the upper limit of D. The relations (8) and (7)permit one to obtain the velocity as function of depth ifdT/dA is known from the travel time tables. If a truesurface of discontinuity intervenes, the analytical proc-ess becomes more complicated as well as indeterminateto some extent; we shall omit the details of this case.

The method has been used by a number of workers toobtain the variation of seismic velocities within theearth. We reproduce here the values given by Jeffreys(1939), not only because they are very detailed andaccurate, but also because K. E.Bullen has used them asbasis for additional computations of density, pressure,etc. , which will be reviewed later on. Table I (see alsoFig. 3) gives the velocities as function of depth in themantle. So=6338 km is the radius of the discontinuitymarking the lower boundary of the crust, taken byJeffreys to be at a depth of 33 km. The two sets of figuresgiven for the longitudinal (E) waves correspond todifferent assumptions regarding the existence of asecond-order discontinuity near 0.94RO (at a depth of

Velbcitgin kfnl sec.

I

JEFFREYSGUTENBERG

D E

FIG. 3. Seismic velocities.

II

II

II

I

I

I

I

I

I

II

I I

I

F'~II

l

Depth in km

5000 6000

EARTH'S INTERIOR AND GEOMAGNETISM

TABLE II. Velocities in the upper mantle in km/sec. $(G)Gutenberg, 1948; {J)Jeffreys, 1939;(GR) Gutenberg and Richter,1939; (MD) Macelwane and Dahm, 1939j.

Deptli, km

200250300350400450500550600650

G(a) G(d)

8.1 8.18.3 8.48.6 8.78,8 9.09.1 9.39.3 9.69.6 9.89.8 10.1

10.1 10.310.4 10.5

GR MD

8.3 8.1 8.38.5 8.4 8.58.6 9.Q 8.78.8 9.3 8.99.1 9.6 9.19.3 9.8 9.49.6 10.0 9.69.9 10.2 10.1

10.2 10.4 10.110.4 10.6 10.3

G J

4.50 4.604.58 4.684.68 4.764.80 4.854.95 4.945.11 5.125.28 5.325.45 5.555.62 5.655.80 5.80

about 400 km). This so called 20' discontinuity (it oc-curs near 6=20') has been much discussed by seis-

mologists; its weakness makes it difFicult to assure itsprecise character or even its existence. Column (1) ofTable I corresponds to the assumption of a second-orderdiscontinuity (discontinuity in slope) at 0.94Re, thecolumn (2) to a continuous curve showing a slight"waviness" in this region. The third column, fortransverse (5) waves, is computed under the same as-

sumptions as the first longitudinal column. It will beseen that the two sets of values for the I' waves arepractically identical except near 0.94EO, and even therethe discrepancy does not exceed 1.5 percent.

Recently Gutenberg (1948) has given an alternativeevaluation of the velocities in the upper part of themantle which he has developed over a number of years.According to these results there is a slight decrease ofthe seismic velocity in the top layers of the mantle downto about 150 km. From there on down the velocity in-

creases at a steady rate which is nearly linear and doesnot exhibit the discontinuity or distinct wave ofJeffreys' curves. Gutenberg thinks that the phenomenathat led to the assumption of irregularities near 400 kmcan be explained alternatively by means of the indicateddecrease of the velocity above 100 km coupled withcertain local efkcts, and by the difIiculties arising in theinterpretation of the travel time data.

In Table II, essentially taken from Gutenberg's paper,are reproduced comparative velocity values for theupper part of the mantle. Column (a) corresponds to theassumption of a linear decrease of the longitudinalvelocity from a value of 8.0 km/sec. at 40 km depth to7.76 km/sec. at 100 km depth. Column (d) correspondsto the assumption, instead, of a slight discontinuousdecrease of the velocity by 0.20 km/sec. at 80 km. Thedata do not as yet permit distinction between assump-tions of this type. The next three columns show oldervalues for purposes of comparison. The last two columnsgive Gutenberg's new values for the transverse ve-locities, and the corresponding values of Jeffreys.

We next proceed to the velocities in the core. Whereaswaves that have only traversed the mantle show asimple monotonical increase of the distance 6 of thepoint of emergence with increasing travel time, things

TABLE III. Velocity in km/sec. , within the core. 81=3473 km.

r/Rt

1.000.980.960.940.920.900.880.860.840.820.800.780.760.740.720.700.68

Vel.

8.108.188.268.358.448.538.638.?48.838.939.Q39.119.209.289.379.449.52

r!Rt

0.660.640.620.600.580.560.540.520.500.480.460.440.420.400.360.360.200.00

Vel.

9.589.659.729,789.849.909.97

10.0310.1010.1710.2310.3010.3710.44(9.7)11.1611.2611.31

are much more complicated in the core. Let us illustratethis behavior with reference to Fig. 1 for entirely longi-tudinal waves (P'), taking as variable the angle whichthe ray makes at departure with a plane tangent to theearth's surface. Since the refractive index is lower in thecore than in the mantle, the general behavior is as shown

by the group on the right-hand side of Fig. 1.A ray thatapproaches close to the core but does not penetrate to itemerges at 6=103', but a ray that actually grazes thesurface of the core emerges almost diametrally oppositethe focus, at 5=187'. As the angle of departure in-

creases further, 6 decreases until it has reached 143',then it increases again. These eGects, giving rise to theshadow zone shown in Fig. 1, may be described in anoptical analogue by saying that the core acts like aspherical lens with high refractive index that concen-trates the rays in the forward direction.

A further complication arises with rays that havelarge angles of departure, i.e., nearly normal incidence

upon the core. One would expect these rays to emerge atpoints nearly opposite the focus, but a group of them is

strongly deQected, showing lower values of 6 with aminimum at 6= 110'. (For the sake of clarity this grouphas been omitted from Fig. 1.) The phenomenon isinterpreted as caused by a sudden steep rise of thevelocity not far from the center of the core, at a depthof about 5000 km. (The inside of this region is the"central body. ")

Jeffreys' (1939) values of longitudinal velocities in thecore are given in Table III. (See also Fig. 3.) The radiusof the core has been determined as

Rg ——(3473&4) km, Rg/Re ——0.5480.

The velocity increases monotonically and withoutdiscontinuity from the boundary of the core down to0.40R&. Jeffreys then assumes a decrease of the velocityfrom there to 0.368~, a strong discontinuous rise at thispoint followed by an almost constant value of thevelocity from there to the center of the earth. Gutenbergand Richter (1938) in their earlier and very complete

WALTER M. ELSASSE R

analysis have simply a steep increase of the velocitybetween 0.428& and 0.368& again followed by an almostconstant velocity inside the central body (thin line inI'ig. 3).They do not at the present time believe that thedata require the "dip" introduced by Jeffreys (personalcommunication from Dr. Gutenberg).

The passage of transverse waves through the core hasnot been observed. (A few reported instances of this kindare generally regarded as spurious. ) This has early givenrise to the assumption that the core is liquid. Ke shallnow consider in some detail the evidence to this eGect.First we may well inquire into the nature of the dis-continuity at the boundary of the core. All the seis-mological evidence, derived from travel times as well asfrom intensities, indicates that this is a genuine first-order discontinuity. The accuracy, as we have noted, israther high. No contrary evidence has so far becomeknown. The nature of the discontinuity has beenstudied in some detail by Jeifreys (1945), who advancesan additional and very strong argument for the liquidcharacter of the core. He notes that while many cases ofrefraction of elastic waves at the interface of two solidshave been worked out, one thing is common to all ofthem, so long as the velocities of the longitudinal andtransverse waves are in about the same ratio: An inci-dent transverse wave gives a very small transmittedlongitudinal wave, and conversely. The wave of the typeSES has undergone two such transformations; never-theless the observations show that its intensity iscomparable to that of a transverse wave that hastravelled through the mantle only. This observed in-

tensity can easily be interpreted only if the matter belowthe boundary is a true liquid. Sometimes it has beenthought that the core might be a solid, viscous mediumof appreciably lower shear stability than the mantle sothat in it transverse waves can propagate but are veryrapidly attenuated. Je8reys has pointed out that theexistence of strongly damped transverse waves at thesame time with virtually undamped longitudinal wavesis not to be reconciled with the theory of elasticity ex-cept under altogether extravagant assumptions aboutthe elastic constants. The same conclusion has beenreached by Eucken (1944).

Additional evidence for the Quidity of the core is ob-tained from the analysis of the bodily tide of the earth.The lunar and solar tides of the earth's solid body havebeen measured with great accuracy by Michelson andGale (1919). These authors determined by an inter-ferometric method the change in water level at the endsof a long pipe. (The maximum tilt of the earth's surfaceproduced bz the lunar tide is about 0.02", the maxi-mum amplitude about 30 cm. ) The theory is relativelysimple only in the case of a homogeneous earth, thedeformation is clearly in inverse proportion to the shearmodulus. The theory of the non-homogeneous earthbeing extremely complicated, Jeffreys (1926) has treateda two-layer model. The shear modulus of the mantle can

be determined from the seismic velocity data, as will beshown below. Now it is possible to account for theobserved amplitude of the lunar bodily tide only if oneassumes that the shear modulus of the matter in thecore is negligibly small. This shows that the core behavesas a Quid in comparison with the shear stability of themantle; although it does not prove that the core has ashear stability as near to zero as do actual liquids, itconstitutes a valuable piece of independent evidence.On summarizing all the evidence brought forth we mayquote Jeff'reys (1945) "that any escape from the con-clusion that the core is liquid would have to be wildlyartificial. "

Nothing is known so far about the nature of thecentral body inside the core or about the character of thetransition layer surrounding it. Bullen (1946) has indi-

cated that the rise in velocity can be satisfactorily ex-

plained by assuming that the central body is solid, of thesame composition as the remainder of the core. So far noother evidence on the subject is known.

%e shall now review once more the concrete results ofseismological analysis, referring to the curves of Fig. 3.The heavy curves reproduce the data of JeBreys corre-sponding to the assumption of a second-order discon-

tinuity at a depth of 400 km. The thin curves give thedata of Gutenberg and Gutenberg and Richter at theplaces where they differ appreciably from Jeffreys'curves. (The most recent values have kindly been

supplied by Dr. Gutenberg. ) There is also shown aschematical division into "layers" designated by capitalletters, as used by Bullen (1947). This is related toJeffreys' analysis rather than to that of Gutenberg.According to Bullen's interpretation of Jeffreys' datathere are three principal layers in the mantle designatedas 8, C, D (A being the crust). Of these, Bullen considersB as most likely homogeneous physically, its lower

boundary being the 20 discontinuity discussed above.Layer C, between about 400 km and about 1000 km,might be considered as a transition layer between themore homogeneous layers B and D. Layer D, extendingfrom about 1000 km to 2900 km, has a fairly constantslope and might possibly be nearly homogeneous in

constitution.In Gutenberg's more recent analysis of the seismic

data for the upper part of the mantle there remain onlytwo principal layers in the mantle, the upper one ex-

tending down to a depth of about 800 km, the lower onefrom there on to the boundary of the core. In both layersthe slope of the velocity curves is fairly constant with

only a slight curvature. In Gutenberg's curve the transi-tion in the neighborhood of 800 km depth is ratherrapid; it represents perhaps a second-order discontinuity.This scheme according to Gutenberg might possibly beinterpreted as corresponding to a gradual change in theconstitution of the mantle in the layer below 800 km.

In the core Sullen distinguishes three consecutivelayers, E, F, and G. Layer E is almost certainly a

EARTH'S INTERIOR AND GEOMAGNETISM

homogeneous Quid (the reasons for this will appear lateron). Layer F marks the transition to the central bodyand layer G is the central body itself. The velocity valuesof Gutenberg and Richter (1938) for layer E differ verylittle from those of Jeffreys. Their values near and insidethe central body are shown in Fig. 3. As pointed outabove, layer G might possibly be solid.

CHEMICAL COMPOSITION

Since the earth's interior is not directly accessible tous, we must in many instances rely on indirect evidenceabout its physical state. This is not nearly as much of anultimate handicap as one might be inclined to think atfirst sight. Many inferences can be drawn from theseismic data; but on going beyond the results justpresented the conclusions will no longer be formallydeterminate in the same sense in which the seismicvelocities are solutions of a diR'erential equation intowhich the empirical data enter under the form of initialconditions. The methods to be applied from here on aremore in the nature of inductive inferences wherenarrower and narrower limits are imposed upon thepossible models of the earth's interior; variants of amodel that do not agree with the whole of the observedfacts are progressively eliminated. This method leavesus with a considerable body of quantitative information,as we shall see. From the viewpoint of such a method itis particularly desirable to gain first an insight into thechemical composition of the earth. It will appear thatthe inferences that can be drawn by reasoning alongchemical lines are rather stringent.

There are three sources of information on which weshall draw. These are the chemical composition of thesun, the chemical composition of meteorites, and thechemical composition of the igneous rocks of the earth' scrust. It is a fortuitous circumstance that definiteinferences can be had from such a heterogeneous collec-tion of data. The main results can also be stated withalmost no qualifications attached to them. We shall firstsummarily indicate these results and then discuss insome detail the findings upon which they are based. Thethree sources mentioned agree in showing that there is atremendous variation in the relative abundance ofelements. Even elements that are close neighbors in theperiodic system may differ by a factor of a hundred orthousand in abundance. In particular the elementsheavier than the iron group are so rare on a cosmic scalethat they can be neglected altogether for the purposes ofthis summary. A number of other, lighter elements arerare in the earth for diBerent reasons. There remain onlynine elements of appreciable abundance; these areoxygen, silicon, magnesium, iron, nickel, sulfur, alumi-num, calcium, and sodium. Of these, again, the first fourare much more abundant than the other five. Among allother elements only a few have an abundance in theearth of about one atom per thousand atoms of the fourmost abundant elements; the vast majority of elementsare even more rare. Only a small number of suKciently

TAsLE IV. Atomic abundances per 200 atoms Si.

689

111223141516171920222425262728

Element

C0FNaMgAlSiPSClKCaTlCrMnFeCoNi

Sun

10002760

9.816611.0

100

42.6

0.818.70.0111.951.48

2700.554.6

Meteorites

0.33347

0.024.42

87.248.79

1000.58

11.40.4-0.6?

0.695.710.471.130.66

89.10.354.60

Igneous rocks

0.27296

0.1512.48.76

30.5100

0.260.160.144.429.270.920.040.189.130.0070.017

stable compounds can be constructed from the abundantelements; specifically, oxides, sulfides, and silicates. It isseen, therefore, that the chemistry of the earth's interiorreduces to a rather simple scheme; and it appears likelythat the solid mantle contains the silicates, and that thecore consists of metallic iron and nickel.

The sources of the data of Table IV, below, are asfollows. The abundances for the sun refer to the solaratmosphere and represent the precision spectroscopicdeterminations of Unsold recently published (1947; seealso Wildt, 1947a). The question may be raised whetherthe solar atmosphere is representative of the averagecomposition of the sun. We need not grapple, however,with this problem since we are not concerned with theproportion of hydrogen relative to the other elements,nor with the abundance of the heavy elements. Theelements that interest us are comprised between oxygenand iron where the atomic mass changes only by about afactor of three. In view of the vast amount of turbulencein the sun one might think that separation by gravitycannot take place to any appreciable extent within thisgroup. Astronomers generally believe that (apart fromhydrogen) the chemical composition of the universe, orat least of the stellar system near the sun, is fairly uni-form, so that the data should be representative of thechemical composition of the aboriginal matter out ofwhich the earth and other planets were formed. Unsold'sfigures supersede the earlier solar abundance data byRussell, and Unsold has remarked that wherever hisresults deviate appreciably from the older data, the newdeterminations tend to be closer to the abundance ratiosfound in meteorites.

The chemical composition of meteoritic matter hasbeen a subject of much study. The older analyses(Farrington, 1915)already showed some definite charac-teristics. The most extensive set of data was given by I.and W. Noddack (1930); the results reproduced inTable IV are the revised data of Goldschmidt (1937)which do not diBer appreciably from the Noddacks'values. In forming mean values for the composition of

WALTE R M. E LSASSE R

meteorites an assumption must be made about the ratioof stone to iron meteorites. Owing to their diferentchemical behavior, diAerent resistance to erosion, andthe di6'erent likelihood of their being identified asmeteorites by persons finding them, this ratio is stilluncertain within wide limits. (Ni and Co are fixed rela-tive to iron. ) Borgstrom (1937) states that amongmeteorites that were witnessed to fall about 8 percent bymass are irons, 5 percent stony irons, the remainderstones; Watson (1939) has reason to believe that theratio of metal to stone masses should lie between 2:1and 1:4. The figure for the number of iron atoms inTable IV must be considered as rather arbitrary and, infact, the number given originally by the Noddacks isnearly twice the figure adopted by Goldschmidt andreproduced here.

iVote added December, $949.—In a paper just pub-lished, Harrison Brown (1949) uses all avs. ilable ma-terial to compile a table of nuclear abundances in theuniverse. The deviations from previous data do notseem so large as to require revision of our Table IV forthe limited purposes of the present review. The maindiscrepancy is again in the uncertain ratio of iron tosilicon which Brown determines from the mass ratio ofthe earth's core to mantle (admitting 10 percentmetallic iron in the mantle), getting 1.7 as comparedwith Goldschmidt's assumption of 0.89 for meteoritesand Unsold's figure of 2.7 for the sun.

The question arises naturally as to why the composi-tion of meteorites can give us clues to the chemicalconstitution of the earth. It must be realized that thecommon "shooting stars" whose orbits have beenphotographically studied in recent years have very smallmasses; the meteorites actually discovered on theground appear as "fireballs" on traversing the atmos-phere. Catalogs have been prepared of such eventsand the estimates of witnesses point to velocities so largethat the orbits are hyperbolic, i.e., these bodies wouldcome from outside the solar system. This is a view held

by a school especially of German astronomers. The mostrecent, mainly American students of the subject, believethat the eye witnesses tend to overestimate the velocitiessystematically and that the meteorites constitute mem-

bers of the solar system. The old suggestion thatmeteorites are fragments of a former planet is based on anumber of characteristics of the minerals found in them;particularly on the occurrence of large crystallites thatrequire slow cooling for their formation; more detailedchemical reasons have also been indicated (Brown andPatterson, 1947/48).

The values of Table IV represent the average compo-sition of meteorites. These bodies show a characteristicuniformity of chemical makeup. The main constituentsof the stones are the magnesium-iron silicates, olivineand pyroxenes which we shall encounter again, on somediBerent grounds, as the likely constituents of theearth's mantle. Enclosed in these is often a certain

amount of iron, either metallic or oxydized. The calcium-rich minerals, so characteristic of the earth's crust, arerarely found in meteorites; they make up only a fewpercent of the aggregated mass of the stones. A featuretypical of meteorites is, not only the occurrence ofmetallic iron practically never found in terrestrial de-posits, but also the joint occurrence of large masses ofstone and of metal in almost any proportion, in one andthe same meteorite (stony irons, pallasites). It has beenpointed out by the Noddacks and by Goldschmidt thatthis indicates clearly that the formation of meteoriticmatter must have taken place in a much smallergravitational Geld than that of the earth, since theseismic data make it practically certain (see the pre-ceding and the following section) that in the earth' sinterior the separation of silicates and metal is aboutcomplete. This need not necessarily be interpreted, asGoldschmidt wants, as an indication that the meteoriteshave condensed in small lumps in interstellar space, butit does mean that if there was a parent planet, this bodymust have been so small that it solidified before thegravitational separation of the silicate and metal phaseswas completed.

In the present context we need not take recourse tocosmological hypotheses. It must be admitted, however,that the assumption of a condensation of the earth fromoriginally hot and Quid matter is so natural and con-sistent with all data that it is hard to circumvene. Weneed not specify the mode and place of condensation ofthe meteoritic matter. The chemical composition ofthese bodies is in a characteristic way intermediate be-tween that of the sun and that of the crustal rocks;moreover the main chemical compounds are the same asthose that we believe, on other evidence, to constitutethe bulk of the earth. The meteorites diAer character-istically from the solar matter by the absence of such"volatile" elements as hydrogen, rare gases, carbon,nitrogen, and a large fraction of the oxygen; in this theyagree with the igneous rocks. On the other hand, thestudy of igneous rocks shows that if one goes from theupper strata to rocks which, by geological evidence,originated in regions near the bottom of the crust, thesilicates of calcium, aluminum, and the alkalis, charac-teristic of the ordinary rocks, are more and more re-placed by the silicates of iron and, primarily, of mag-nesium, so that the composition of these deepest rockstends to be much closer to that of the meteorites. On thewhole it is likely that the meteorites represent a con-densation of cosmic matter that is somewhat closer tothe aboriginal proportion of elements than the constitu-ents of the earth's crust that have been subjected to anextensive differentiation in the earth s strong gravita-tional held. In the present context the evidence drawnfrom the constitution of meteorites is mainly corrobo-rative. Since it fits so well into the over-all picture, itadds a good deal of strength to the conclusions drawnlater on about the composition of the earth's interior.

EARTH'S IN TE RIOR AX D GEOMAGXETISiXI

Finally we give, in Table IV, the mean chemicalcomposition of igneous rocks. The results of variousworkers as to these mean values dier very little, andthe earlier determinations of Clarke and Washington(1924) are in substantial agreement with those of laterworkers (e.g. , von Hevesy, 1932). By far the most ex-tensive analytical work in this field in recent times hasbeen done by V. Goldschmidt, who has with especial careinvestigated the relative abundances of all the rarerelements (not needed here). The values reproduced arefrom a summary of Goldschmidt's results (1937). Inview of the great consistency of the data from diGerentsources these figures may be considered as reliable meanvalues of the constitution of crustal igneous rocks.

Following Goldschmidt, the abundance data are givenas numbers of atoms per 100 atoms of silicon. Theelements omitted have abundances of less than 0.1 inboth meteorites and igneous rocks. (In the sun theabundances of the heavier elements near and beyond theiron group are roughly parallel to those in the meteorites,although sometimes slightly higher, for instance, Zn,0.31.) The following elements are absent from Table IV:

Noble gases, He, Ne, A.Hydrogen and nitrogen.The light elements, Li, Be, B.Two metals, Sc, V, just below iron, and all elements

heavier than the iron group.Very little need be said about the absence of the noble

gases. The very low abundance of the elements H and Nwill be discussed below. The comparative scarcity of thethree elements I.i, Be, B, not only on the earth and inmeteorites, but also in stellar atmospheres has anestablished explanation. It is attributed to the fact thatthese elements are subject to slow nuclear disintegrationat the temperatures prevailing in the interior of the sunand of ordinary stars. We may disregard these elementsas constituents of the earth's interior. The remainingelements not contained in Table IV are the heavy ele-ments beginning in the neighborhood of the iron group.According to Noddack, the plot of log abundancez.ersls Z shows that the points of the heavier elements inmeteorites fall into a rather narrow strip whose meanslope corresponds to a decrease of abundance as Z ' toZ ' in the middle of the periodic table, and flattens outtoward the very heavy elements; in the earth's crustthe abundance of the heavier elements is, in the mean,even smaller. Hence the heavier elements should notbe important for the constitution of the earth.

The last two columns of Table IV show that certainelements are concentrated in the earth's crust as com-pared to their abundance in the meteorites. Goldschmidtdesignates such elements as "lithophil, " the most com-mon ones being Ca, Al, Na, K. Four others are muchmore abundant in meteorites than in igneous rocks;these are Mg, Fe, Ni, S. This is what one should expectif the principal components of these elements have beensubjected to a process of sedimentation (rather, Rotation)

0FeSiMg

32.328.816.312.3

SNiAlCa

2.121.571.381.33.

Owing to the very large uncertainty in the amount ofiron present the precise figures have of course very littlesignificance, but the trend is obvious: These eight ele-ments make up 96 percent of the mass of the meteorites,the four most abundant ones make up 90 percent. It

in the earth's gravitational field so that the relativelylighter compounds of the "lithophil" elements, especiallyof the alkalis and earth-alkalis, are concentrated in thecrust. This view is strongly corroborated by the recentfinding that those igneous rocks that originated in thelowest layers of the crust (and are also rich in mag-nesium) show a much weaker radioactivity than averagecrustal rocks (Davis, 1947). This latter condition hadlong been postulated on the basis of theoretical con-siderations regarding the heat balance of the earth.

Finally there are a few light elements that are muchmore abundant in the sun than either in meteorites or inthe earth's crust. Unsold gives the following abundances,again relative to Si=100; hydrogen 1,380,000; carbon1000; nitrogen 2100; oxygen 2760. Clearly, almost allof the first three elements and the largest part of theoxygen must have escaped during the formation of theterrestrial and meteoritic matter in the form of theirmore volatile compounds. In this connection, sulfurshould also be mentioned as being rather an abundantelement in the sun. Even in meteorites the mean ratio ofsulfur to silicon is only about one-fourth of the ratio inthe sun. For chemical reasons sulfur can be assumed toexist inside the earth almost exclusively in form ofseveral types of iron-sulfides; this is found to be true inmeteorites. The older authors on the constitution of theearth often assumed the existence of a sulfide-oxidelayer, these compounds being intermediate in densitybetween the mantle and the core, and under laboratoryconditions sulfides and silicates are completely im-miscible. Jeffrey has made a careful search for such anintermediate layer just above the boundary of the coreand reports that it does definitely not exist. The presenceof sulfides in macroscopic dispersion in the mantleremains one of the problems of the chemistry of theearth's interior; however, the reduction in sulfur contentof the meteorites relative to the sun makes it possible tothink that a large part of the sulfur present in theaboriginal matter was able to escape from the earth inform of one of its more volatile compounds. The amountof sulfur in the earth's crust is negligible.

On reviewing the contents of Table IV it appears thatthere is a definite hierarchy of chemical elements in theearth; the picture is almost completely dominated by avery few abundant elements. Taking once more theproportions in the meteorites as an example, we findthat, expressed in mass as percentage of the total mass,the eight most abundant elements are as follows:

10 %ALTER. M. E LSASSE R

might be added that in terms of the conventional atomicand ionic radii the oxygen ions Ql up over 90 percent ofthe volume in the stony component of the meteorites,and a similar statement should no doubt apply to thesilicates in the earth's mantle.

%e shall now proceed from the elements to theircompounds. For this we require no more than someordinary mineral chemistry as contained in any refer-ence work (Dana-Hurlbut, 1941). It will suKce if weconan e ourselves to the more dense and stable com-pounds of the nine most abundant elements, and amongthese again the compounds of Fe, Si, Mg, S will be mostsignificant. Ke may readily disregard nickel altogetherin this context, considering it as a chemical homologof iron. Furthermore, the main constituents of theearth's interior are restricted by the fact that, accordingto results given later, the density of the material at thetop of the mantle cannot be very far from 3.3 and thecompressibility must be low enough to produce the ob-served, very large seismic velocities. There are only threesigni6cant types of compounds of the most abundantelements: oxides, sulMes, and silicates. (The sulfates arerare among mineral-forming compounds; they are notsu%ciently dense and stable to be of interest as con-stituents of the earth's interior. )

In the following enumeration the 6gures in paren-theses are densities.

Ondes. Apart from quartz or tridymite, Si02, themain representatives are the iron oxides, Fe20' (5.3) andFe304 (5.8). MgO occurs in minerals only in its hydratedfo™Mg(OH)2. Other compact oxides are A1~03 (4.O)

and MgA1204 (3.5—4.1).Sulfides. Only iron sulfides are of importance here.

Pyrrhotile is Fe~,S (4.6) with x between 0 and O.2.Troilite, found in meteorites, is close to FeS. There aretwo modifications of FeS, (4.9—5.0).

Sihcates. Of these there are several types. The physico-chemistry of the silicates enumerated here has beenstudied in great detail (see many papers of the CarnegieGeophysical Laboratory). The complex phase equilibriaof the magnesium-iron silicates at high temperatures in

particular are well investigated (Bowen and Schairer,1935).

Orthosi licates. The most important mineral in thisconnection is olivine, (Mg, Fe)2Si04. In these compounds

Oensity

grm /cm

,CalCur.'for

Z&26

Exl'er.

! Logo p(dyn/cms)

8 9 Io II IR I3 I4

FIG. 4. Dashed line: Probable density of iron.

magnesium and iron are freely substitutable; the wholeseries from forsterite Mg2SiOq (3.19) to fayalite FemSi04(4.14) exists and representatives are occasionally found.The common variety of olivine (near 3.3) contains ap-proximately 1 Fe to 9 Mg; in meteorites it is oftenslightly richer in iron. Crystals are orthorhombic. Therock, duni te, consists almost wholly of olivine and isoften quoted as representative of the constitution of theearth's mantle.

The term garmets designates silicates of the generalcomposition R3"R2"'(Si04)3, the most important combi-nations of positive ions being MgaA12 (3.5); Fe&1& (4.2);CaaA12 (3.5); Ca3Fe~ (3.7). Most of these are octahedral.

cVetasi locates. The common mineralogical term ispyroxeees. The magnesium-iron silicates range fromenstatite, MgSi03 (3.1—3.4), to minerals where part of the

Mg is replaced by iron. The minerals richer in iron arecalled hypersthene (3.3—3.5); if they contain iron oxidesthey are called bronsite. They cease to be stable if Feexceeds 90 percent. These minerals are common inmeteorites. Orthorhombic.

Diopside. CaMgSi206 (3.2—3.3). Here again Fe mayreplace Mg in any proportion up to CaFeSi20& (3.6).

Jadeite. NaAISi206 (3.3—3.5). Triclinic.The rock, eclogite, consists mainly of pyroxenes and

garnets.There are some other silicates of the abundant ele-

ments, but they are of an extremely complex structureand their existence in the earth s interior is not com-patible with the high seismic velocities observed there.

The simplest types of silicates enumerated are dis-

tinguished by their tendency to avoid the vitreous statein favor of straight crystallization. Olivine in particularhas only a very small vitreous range and normallycrystallizes directly from the liquid state. Also, thevolume contraction on devitrification is very large (near15 percent) for these simple silicates (Birch, 1942). All

this refers of course to laboratory pressures.

We can now discuss the approximate chemical consti-tution of the core, the mantle and the crust. We expectthe core to consist of iron with some slight admixture ofnickel. A.strong corroboration of this view is obtainedfrom an extrapolation of the density values of iron to therange of pressures prevailing in the core. Theoreticalcalculations about the behavior of metals under extremecompression have been carried out by Slater andKrutter (1935), Jensen (1938), and Feynman, Me-tropolis, and Teller (1949). These are based on aThomas-Fermi model for the electrons which becomeapplicable when the compression begins to obliterate the

specific structural effects of the valence electrons. Itappears from Jensen's results, and from general theoret-ical arguments, that this is the case at about 10' atmos-pheres. In Fig. 4, following Jensen, the density of iron is

plotted against logp, the values for layer E of the coreare bracketed between Bridgman's laboratory valuesand the Thomas-Fermi curve.

EARTH'8 INTERIOR AN D GEOMAGNETISM

If one tries to ascertain the composition of the centralbody, he finds at once that there are apparently notenough heavy elements to fill a volume of the requiredsize. In Goldschmidt's tables the abundance decreaseswith extreme rapidity between the iron group and thepalladium group. From there on the average abundancesremain fairly constant (on a logarithmic scale at least)up to the heaviest elements. The abundance of the heavyelements is about 19 +' of that of iron. Counting about50 such elements the aggregated number of atoms wouldbe 5.10 ' of that of iron, not enough to fill the centralbody whose volume is 5—6 percent of that of the core.The elements in the neighborhood of iron, on the otherhand, are mostly metals; since the iron group representsa minimum of the atomic volume it is very unlikely thatsuch elements will form a separate phase of greaterdensity. We need not discard, however, the idea thatwhatever fraction of heavy elements or compounds thereis has become concentrated to some degree in the centralbody,

Bullen (1946) suggests tha, t the central body consistsmainly of solid iron. This will almost account for therise in wave velocity, assuming a negligible change indensity and compressibility. By the formulas of the nextchapter a rise in longitudinal velocity from V= 10.44 toV= 11.16 (Table III) would correspond to an increase ofPoisson's ratio from zero to 0.37, somewhat large sinceother eGects have been neglected. It seems also possibleto assume a liquid-liquid phase transition correspondingto a rearrangement of electronic wave functions (El-sasser and Isenberg, 1949); assuming no change indensity this would correspond to a decrease in com-pressibility by 13 percent. It seems not quite impossiblethat this problem can ultimately be investigated bytheory, in view of the remarkable success achieved bythe theoretical analysis of the wave functions of solidiron by Manning and Greene (1943).

Turning to the mantle, it is characterized by extraor-dinarily high values of the seismic velocities. The pres-sures near the top of the mantle, of the order of some10,000 atmospheres, are well within the experimentallyaccessible range. The temperatures are unknown, butsince it is possible to make laboratory experiments onrocks under a few thousand atmospheres and at moder-ately high temperatures, no serious extrapolation isinvolved with regard to the physical conditions near thetop of the mantle. It can be shown that among theknown minerals only some few silicates, introduced byname above, yield the observed high velocities. Theseare olivine and the pyroxenes, some garnets and jadeite.It is characteristic that in the measurements of Bridg-man (1945/48) the minerals of this composition arefound to have a lower compressibility than any other ofthe many substances investigated, elements or com-pounds. As we go down below the earth's crust we mayexpect the magnesium-rich silicates, olivine and pyrox-enes to preponderate. Many geophysicists believe thatthe mantle consists largely of olivine. This view is held

for instance by Jeffreys and by Adams (1947). Birch(1939)points out that the experimental data do not per-mit one to decide between olivine and the pyroxenes.Both are the most common constituents of meteorites.As Jeffreys (1939)notes, the density of the moon is 3.33which on extrapolation to zero pressure becomes 3.29,practically the density of pure ordinary olivine. Thedensity at the top of the mantle is assumed by Jeffreysas 3.32, the density of olivine at the pressure prevailing.He indicates that for a variety of geophysical reasonsthe actual density there should not deviate from thisvalue by more than 0.1 or 0.2 at the most. As Birch re-marks, only the members of the olivine series with low ormoderate iron content have the requisite mechanicalstrength; this is also rejected in the fact that forsterite,the magnesium end of the series, has a melting point of1890' while fayalite, the iron end, melts at 1205'. Ensta-tite, the magnesium end of the pyroxene series, trans-forms to a monoclinic variant at about 1100', and at1560' it melts incongruently to forsterite and liquid Si02.This complex behavior at ordinary pressure and manyother traits of those silicates make it difFicult to be morespecific about the physical condition of the mantle be-yond the statement that there is no reason to doubt itscomposition almost wholly of silicates among which themagnesium silicates should greatly preponderate. Theseismic velocities are very large throughout the entiremantle, and this would seem to make it unlikely thatmore than rather modest proportions of other com-pounds, such as oxides and sulfides, could exist in themantle. The writer has been unable to find any specificindications in the literature: a quantitative upper limitfor the amount of iron oxides, iron sulfides and ironsilicates in the mantle would seem a highly desirabledatum.

In connection with the somewhat controversial ques-tion of second-order discontinuities in the mantle, asdiscussed in the preceding chapter, an interesting sug-gestion of Bernal (1936) should be mentioned. He notesthat most of the space in the silicate lattices is taken upby oxygen; the silicon ions, being comparatively small,are placed in the holes betmeen them. Compression ofthe lattice would have qualitatively the same eGect assubstitution of a larger ion for silicon. Goldschmidt hasstudied magnesium germanate, Mg2Ge04, and finds thatit has two stable modifications, one isomorphic witholivine, and another one which is cubic. The density ofthe latter is larger by 9 percent than that of the ortho-rhombic variety. This makes it possible to assume theexistence of a cubic modification of olivine at extremelyhigh pressures. Adams (1947) points out that if thereexists an upper boundary of cubic olivine, then this sur-face might have shifted in the course of the earth's lifeowing to a small change in temperature. It seems not,however, necessary to assume that such polymorphoustransitions occur almost instantaneously when the tem-perature changes. Experience shows that very high pres-sure often has an eGect similar to a lowering of the

KVALTE R M. ELSASSE R

TAsLz V. Mantle.

Depth

X10' km

0.0330.100.200.300.400.4130.500.600.700.800.901.001.201.401.601.802.002.202.402.602.802.90

r/Ro

10.9890.9740.9580.9420.9400.9260.9110.8950.8790.8630.8470.8160.7840.7530.7210.6900.6580.6260.5950.5630.548

Density

g/cm'

3.323.383.473.553.633.643.894, 134.334.494.604.684.804.915.035.135.245.345.445.545.635.68

Gravity

crn/sec. '

985989992995997998

100010011000999997995991988986985986990998

100910261037

Pressure

X 10"c.g.s.

0,0090.0310.0650.1000.1360.1410.1730.2130.2560.3000.3460.3920.490.580.680.780.880.991.091.201.321.37

temperature; it increases viscosity and reduces ion

mobilities and reaction rates. Thus the time required forthe formation of cubic olivine in the mantle might be

long even by geological scales. Hence Bernal's idea

might open up the possibility of a gradual contraction of

the earth during its life, a phenomenon that is strongly

suggested. by geological evidence.Turning now to the earth's crust we find it again

largely composed of silicates. Chemical analysis ofigneous rocks shows a Si02 content of about 60 percent.These silicates contain of course larger amounts of Ca,Al, Na, K than can be expected of the silicates of themantle. One of the great diGerences between the mantle

and the crust lies in their diferent degrees of homo-

geneity. It is of course more dificult to detect in-

homogeneities in the mantle than in the crust, but no

evidence for inhomogeneity in the mantle other than thesystematic vertical variation of the seismic velocities has

so far been given. The crust is very strongly inhomo-

geneous, both physically and chemically, so much so

that in contradistinction to the mantle great difficulties

are encountered in defining its average seismic character.To understand this behavior we best adopt a dynamicalview.

As will be shown below it is almost certain that thesolidification of the mantle took place from the bottom

up while the liquid overlying the solid stratum at anystage was in convective agitation. This explains why the

specifically lighter compounds can have become concen-

trated in the crust by a process of flotation quiteanalogous to the one encountered in a smelting furnace.We might expect that the substances that are congre-

gated at any one level in the mantle have densitiesreasonab1y close to each other. Ke might perhaps as-

sume that the original cosmic matter from which the

planet was formed was physically homogeneous. Thepronounced chemical dift'erentiation that we find in thecrust and that leads to concentration of differentminerals at diGerent places on the earth, must haveoriginated through the action of strong temperature orpressure gradients. Such gradients have been actingupon the crust since its inception and also during itsformation, but it is dificult to see how they can have acounterpart inside the mantle. This shows that it isdangerous to draw inferences about the composition ofthe mantle on starting from purely empirical observa-tions of the crust.

The question of the earth's internal constitution isclosely related to that of the constitution of the planets,a subject reviewed in detail by Wildt (1947).The pla, netsfall clearly into two groups, the inner small ones and theouter large ones. The outer planets have mean densitiesof the order of unity (e.g. , Jupiter 1.30, Saturn 0.69)whereas the inner planets are more like the earth(Mercury 2.86, Venus 4.86, Earth 5.52, Mars 3.86).Clearly only Venus (having about the same mass as theearth) can have an appreciable iron core; Mercury andMars have smaller mass and they are similar in densityto the moon (3.33); they may be suspected to consist ofsilicates. We do not know how this condition cameabout.

Kothari (1936) first derived a relation between themass and the radius of a "cold" celestial body in whoseinterior the temperature eBects are negligible comparedto the pressure eBects. He could show that the whitedwarf stars and the planets follow a common law of thistype. His assumptions about the compressibility werevery crude and were refined by Sommerfeld (1938),whoused a Thomas-Fermi model for the atoms. He derives aradius-mass curve for a body made of iron and anotherfor hydrogen and finds that the points for the innerplanets fall very close to the iron curve whereas thosefor the outer planets fall between the iron and hydrogencurves. Scholte (1947) has, apparently independently,derived the same relation in more detail and has com-puted such a curve for silicon. The points for the inner

I

Density

grm /cm

Depth jn km

KKO 8XO Xra aaa 51m

Fro. 5. Density variation after Bullen. Curve stops at boundaryof central body.

EARTH'S I NTE RIOR AND GEOMAGNETISM

planets fall rather closely on this curve. From this, andfrom the fact that the main constituent of these planetsshould really be oxygen, we may conclude that the re-lationship is rather insensitive to the choice of thematerial so long as the latter is appreciably heavier thanhydrogen.

In this connection we must mention an eGort at re-interpreting the data on the earth's interior by Kuhnand Rittmann (1941; Kuhn, 1942) which has thrownsome confusion among those inclined to geophysicalspeculations. These authors suggest that the earth's coreconsists mainly of hydrogen, or rather of undifferentiatedsolar matter of which hydrogen is the main constituent.In their opinion the viscosity in the central parts of theearth is so large that the escape of hydrogen from therecan never have taken place. They interpret the bound-ary of the core as a continuous but rapid (exponential)change in viscosity over a short distance such thattransverse waves that penetrate this region are rapidlydamped out by viscosity. These views have beenseverely criticized by Eucken (1944; also by othersquoted by Kildt, 1947). Eucken recapitulates and inpart redevelops the analysis of Jeffreys and otherseismologists given in the preceding chapter whichshows that the boundary of the core is a genuine andsharp discontinuity and that the data on the reflectioncoefFicients do not seem to permit any other interpreta-tion than that of a boundary between a solid and a trueliquid. Detailed calculations (Kronig, de Boer andKorringa, 1946) show that the density of hydrogenunder the pressures prevailing in the core is near unity,whereas the actual density in the core is about 9—13.On the other hand, the large planets, Jupiter, Saturn,etc. , have mean densities near unity, indicating thatthey do consist largely of hydrogen. There seems, then,scant hope of finding much hydrogen inside the earth.

Another suggestion has recently been made byRamsey (1948) on astronomical grounds. He indicatesthat he can satisfactorily account for the density of theinner planets on assuming them to consist of silicatesthroughout. Arguing from the fact that at sufFicient

compression any substance ultimately becomes a con-ductor, he interprets the boundary of the core as a phasetransition to a denser, liquid, and electrically conductingform of the silicates that constitute the mantle. A furtherpursuit of this line of thought leads into great difFiculties.

I

Pressurein million oem,

5(lO 4fnlOn }

TABLE VI. Core.

Depth

X10' km

Density

g/cm'

Gravity Pressure

cm/sec. ' X 10"c.g.s.

2.903.003.203.403.603.804.004.204.404.604.804.985.125.125.405.706.006.37

10.9710.9130.8550.7980.7400.6830.6250.5680.5100.4520.4000.360

0.2800.1930.1070

9.439.579.85

10.1110.3510.5610.7610.9411.1111.2711.4111.54

(14.2)(16.8)

(17.2)

10371019979936892848803758716677646626585

460320177

0

1.371.471.671.852.042.222.402.572.732.883.033.173.27

3.413.533.603,64

The compressibility can hardly increase on going to thedenser phase, and hence the change of wave velocityat the boundary of the core must be caused by acorresponding change in density. From the data of thenext chapter one finds readily that the change inthermodynamic potential, PhV, would be 14 electron-volts per molecule of olivine which seems rather ex-cessive for a substance so stable to begin with. (By thesame argument one can show that any density change ofiron at the boundary of the central body must be verysmall. )

emote added December, 1949.—There exist measure-ments of Bridgman (1945/48) on numerous elementsand compounds going up to 1 10' atmos. Using theseand using the Thomas-Fermi values from 1 10'atmos. on up one can successfully interpolate acrossthe remaining gap and can estimate densities withquite good accuracy. In this way one obtains not onlyisolated points but data for groups which have ahigher degree of consistency. The conclusions drawnfrom this material corroborate the picture of a silicatemantle surrounding an iron core; they do not lendsupport to Ramsey's assumption.

MECHANICAL AND THERMAL PROPERTIES

We have yet to determine the major mechanical andthermodynamical features of the earth's interior. Thefirst step in this direction consists in a further evaluationof the seismic data. Following the usage of seismologists,we introduce two elastic constants, the bulk modulusand the rigidity. The bulk modulus, k, is the reciprocalof the volume compressibility,

1/k = (1/p) (~~/~P). (9)

FIG. 6, Pressure variation.

The rigidity, p, , is identical with the conventional shearmodulus of elasticity. In terms of the conventionalelastic constants, X and p, we have k=X+2p/3. Thevelocities of longitudinal and transverse waves, t/'~ and

HALTE R M. E L$ASS E R

TABLE VII. Limiting densities.

Layer

X 10' km Assumpt. (1}

Density

Assumpt. (2}

0.4130.500.600.801.001.401.802.202.602.902.903.003.504.004.504.98

5.12

6.37

3.884.114.464.654.885.105.315.515.669.79.9

10.511.111.611.9

12.0

12.3

3.904.144.524.714.955.175.375.575.729.19.29.8

10.310.811.1

22.3

V&, can be expressed as follows:

V i'= (llp)(&+4p/3)Vt =p/p~

and hencek/p= Vi2 —4Vi2/3.

(10)

In the core where there are no transverse waves, (10)reduces to the familiar relation V = (RIP)~. FromTables I and II we at once get k/p and p/p as function ofdepth.

These elastic constants refer clearly to changes of themedium that are adiabatic in the sense of thermo-dynamics. Now assume that we have a honsogerleols

layer of the earth in which the temperature varies alongthe adiabat as the pressure increases with depth, i.e.,

(d T/dr) (dr/dp) = (BTIBp),q;,b

In this case we have by (9)

dpi«= (dpldP)(dPIdr) = gp(d p/dP) = -gp'I& (12)-Now let M(r) designate the total mass contained withina sphere of radius r about the center of the earth,

r

M(r) =M(EO) —4x )I pr2dr

Ro

Since g = y3f (r)/r where y is the gravitational constant,we finally have from (12)

(1Ip) (dpl«) = (p/&)L77rI(r)/r''j (13)

Since k/p is known from (11) it is possible to get p asfunction of r by numerical integration. This method,originally due to Williamson and Adams (1923)has beenused extensively by Sullen to obtain the data which arereproduced below. The meaning of the thermodynamics, l

limitation will be discussed later.

8= zmr', (14)

where for a homogeneous sphere z=0.40. For the earthone finds z=0.3341, indicating a strong increase ofdensity towards the center. Now if we determine thedensity distribution in the earth by a process of inte-gration from the outside inward, we know at any depth rthe residual mass M(r) and we also must get a plausiblevalue, slightly below 0.4, for the parameter z at thatdepth. These conditions are found to put extremelimitations on the admissible density distributions.

In Tables V and VI are given the results of densitycomputations of Bullen (1940/42) based on these princi-ples. These data as well as all the other numerical andanalytical work by Bullen and Birch reported in theremainder of this chapter use the velocity values ofJeffreys' analysis, as given above. The more recentvalues of Gutenberg will of course in all these cases leadto modifications for the upper part of the mantle. It isalmost certain, however, that this will have no seriouse6ect upon the results for the deeper layers or the core.

Table V refers to the mantle, Table VI to the core.The tables also contain numerical values for the ac-celeration of gravity, g, and the hydrostatic pressurewhich are readily computed when the density distribu-tion is known. The density is shown in Fig. 5, thepressure in Fig. 6. The pressure at the boundary of thecore is 1.37 million atmospheres, that at the center of theearth 3.64 million atmospheres.

In order to obtain these values Sullen proceeds asfollows: First, on deducting the mass and moment ofinertia of the crustal layers he finds a mass of 593.0)& 10"g and a moment of inertia of 79.80X10" gcm' for theremainder. Next he assumes that (13) is applicable tothe top layer, 8 of the mantle. The density at the top ofthe layer enters as a constant of integration, and this istaken as 3.32, being the density of olivine at that pres-sure, for reasons detailed above. Birch (1939) has voicedsome criticism of this particular value, but Bullen pointsout that there are many geophysical arguments for sucha value, and even if the initial density was in error by asmuch as 0.1 g/cm', this would not affect the densitiesbelow layer 8 by more than about 0.5 percent. If theapplication of (13) was not justified, considerable errorsin the slope of the density curve in layer 8 might result,but no serious errors in the density values below. Thereare some reasons to believe that (13) is applicable; oneof these has been advanced by Birch and will be indi-cated later on.

Any determination of the density distribution insidethe earth must comply with two extremely stringentconditions. The integrated density must give the correcttotal mass of the earth, 597.7X10"g (Olczak, 1938) andalso the correct moment of inertia as determined fromthe precession of the equinoxes. Its value (about theearth's axis) is 81.04X104' gcm'. Now let 8 be themoment of inertia of a sphere of mass m and radius r; weshall write

EARTH'5 INTERIOR AN D GEOMAGNF TISM

If the density distribution in the entire mantle isdetermined by the integration of (13), an impossiblylarge value, s=0.57, results for the core. This constitutesindependent proof that the mantle is not homogeneous.In order to compute densities in layers C and D, Bullenuses simple polynomials with undetermined coeKcients.He then makes some simple physical assumptions: thatthere is no discontinuity of p in the mantle, and that theupper part of layer D is fairly homogeneous so that (13)can be applied to it; finally, he assumes a suitable valueof 2, for the core. Using various consistency tests he findsx=0.380 to be a rather likely value, justified by theulterior density distribution in the core. These condi-tions determine the density values as given in Table V.The mass of the core under these assumptions is187.6)&10"g.

In the core things are somewhat simpler. Layer E isalmost certainly a physically homogeneous fiuid; in thiscase the temperature gradient cannot exceed the adiabat.As we shall show later, if it is less than the adiabat, thedifference is negligible; hence (13) may be applied inthis layer. Bullen finds that on grounds of consistencythe mean density in the central parts, layers F and G'

must be no less than 12.3 and no more than 22.3. Thevalues of Table VI are determined by an arbitrary as-sumption regarding the variation of density between thebottom of layer E and the top of layer 6, giving a meandensity in F and G about intermediate between the twolimits. Later on Bullen (1947) has also computed densitycurves for the two limiting cases of low or high density inthe central body. These are reproduced in Table VII. Itappeared to the present writer that there is someadvantage in having one internally consistent solutionof the density problem, carried through in all numericaldetail. This is the reason for giving Tables V and VI, notany superior accuracy. Bullen is emphatic about theprovisional character of his solution, and from theviewpoint of our present knowledge the values ofTable VII define an interval of equally good solutions ofthe density problem. On the other hand, the margin ofvariability is limited; Sullen believes that the densityvalues outside the central layers F and G are nowdetermined to within +3 percent. It might, moreover,prove diKcult to assume enough heavy elements tojustify a density as high as 22 or even 17 at the earth' s

center, and this would further restrict the admissibledistributions.

It is found that the pressure is rather insensitive tosmall changes in the density distribution; its values inTables V and VI are not likely to undergo much furtherchange. Gravity is found to be constant to within 1

percent at a mean value of 990 throughout almost theentire mantle; this circumstance has been used as anapproximation in dealing with the mechanics of themantle. The values of g in the central part of the earthare sensitive to changes in density; they change ratherconsiderably within the range de6ned in Table VII.

The clastic constants, k, p, , can now readily be com-

ALE VIII. Elastic constants.

Depth

X 10' kin

0.0330.100.200.300.4130.500.600.801.001.401.802.202.602.902.903.003.504.004.504.98

X 10"c.g.s.

1.161.241.381.541.732.152.573.193.594.204.875.576.236.516.26.58.19.7

11.112.6

X 10&2 c,g.s,

0.630.670.740.810.901.101.321.691.892.152.392.632.883.03

0.2690.2720.2750.2770.2800.2830.2820.2750.2760.2810.2880.2950.3000.3000.50.50.50.50.50.5

puted. The values corresponding to the density distri-bution of Tables V and VI are given in Table VIII(Bullen, 1947).For k and p the unit is 10"dyne/cm'. Thevalue of Poisson's constant, 0, is independent of p andis given by

[2(1+o)]/[1—2o]= VP/V, '.

The elastic constants in the central body are not in-cluded; they are subject to large uncertainties.

Another quantity that can be computed when thedensity is known is the ellipticity as function of thedepth, by Clairaut's theory (see Gutenberg, editor,1939, Chapter 13). Bullen (1947) gives numericalvalues; he finds that it decreases from 0.00337 at theearth's surface to 0.00257 at the boundary of the core.

Birch (1938/39) has shown that important advancesin the study of the earth's interior can arise from theapplication of Murnaghan's (1937) theory of finitestrain. Birch developed in particular the theory of thecase where the infinitesimal deformation by a seismicwave is superposed upon the finite deformation due tohydrostatic pressure. On applying this theory to theupper layer, 8, of the mantle, he assumes as given thetwo velocities at the top of the layer and the density atthe top. This permits him to determine the two elasticconstants of the theory and by means of those he cancompute the variation of the velocities and of densitythrough the layer. These agree with the seismic valuesof Jeffrey and with Bullen's values of p, respectively,within the errors of measurement. This result bears outBullen's assumption that Eq. (13) can be applied tolayer 8, since it shows that the same elastic constantsthat determine the seismic parameters also give thecompression with 6nite strain. Birch has carried outsimilar calculations for the layer D of the mantle, findinga somewhat less complete agreement with the observa-tions. It would seem that Murnaghan's theory is a

KVALTER M. ELSASSE R

valuable tool for a study of the elastic properties insidethe earth.

Up to now we have dealt with the mechanical prop-erties only as related either to the equilibrium case or towave motion. The phenomena of plastic ffow are ofgreat importance for the physics of the earth's interior.The theory of such phenomena is generally non-linear;it is notoriously complicated and not yet very wellunderstood. In order to survey the situation with re-

spect to the plastic ffow of solids, the rather simple linearrelation between stress and strain proposed by Maxwellhas often been used:

e=S/Ii, +(1/v)) SCh, (15)

where ~ is a shear strain, 5 the corresponding stress, p isthe shear modulus, v the viscosity. We may define atime

r= i/p, (16)

known as the relaxation time. For periods small com-pared to v., or for large values of v, the second term of(15) becomes small and we are left with the stress-strainrelation of classical elasticity. For periods large com-pared to r the first term of (15) becomes small and therelation between strain and stress reduces to that of theconventional Stokes' theory of the ffow of viscous ffuids.The thorough study of Birch and Bancroft (1942) hasshown that Maxwell's relation is not supported byobservational fact; moreover an eGort at founding it on atheoretical molecular model leads to contradictions. Par-ticularly the concept of viscosity loses most of itsordinary meaning for values above about 10" poises(roughly the viscosity of pitch).

Notwithstanding the inadequacy, Maxwell's relationhas had a certain heuristic value. It leads to the ideathat for one and the same body rapid motion can betreated by ordinary elastic theory and very slow motion

by viscous hydrodynamics. The former seems mell

enough justified and is employed consistently in seis-

mology; the latter is quite possibly erroneous. A fewresults have been obtained by a straightformard applica-tion of the Stokes-Navier equations of viscous ffow tothe mantle. Haskel! (1935/36) used the known rate ofrise of the Fennoscandian peninsula after the disappear-ance of the ice load to derive a coefTicient of viscosity forthe upper part of the mantle: v= 1.10".Meinesz (1937)on correcting an error in Haskell's treatment obtainsi =3.10".In this case (16) would give a relaxation timeof 1000 years, well above the periods of seismic wavesand well below the periods of geological changes thatrequire millions of years. Gutenberg (1941) has given avery detailed survey of all such uplift phenomena; heshows that those in North America and in other parts ofthe world are similar and lead to similar relaxationtimes. He also gives a review of the attempts at theo-rctlcal analysis, Using Haskell's value of v = 10~;

Pekeris (1935) calculated a model of viscous thermalconvection in the mantle introducing plausible devia-tions of the temperature distribution from sphericalsymmetry, of about 100'. He finds velocities of about1 cm/year. This seems to be typical of the general orderof velocity in quasi-viscous models. Meinesz (1948) hasrecently reviewed the question of cellular convection inthe mantle and has propounded some stimulating ideas.

Viscosity is very strongly dependent on temperatureand also on pressure. Most experiments on the pressureeffect of viscosity (Bridgman, 1931, 1946) have beenmade on organic substances such as oils; there is onestudy, by Dane and Birch (1938) undertaken especiallyin view of geophysical applications. The viscosity of820& glass was found to increase exponentially withincreasing pressure betmeen 1 and 1000 atmospheres,more rapidly at lower than at higher temperatures. Ifthe dependence of viscosity on temperature is alsoexponential, as is likely, there would still be a rather welldefined and relatively narrow temperature interval ofthe "softening" transition between the vitreous solidand liquid states.

It is, however, extremely questionable whether themodel of a quasi-liquid viscosity gives even a quali-tatively satisfactory description of the phenomena ofplastic ffow of solids. We cannot enter into a discussionof this subject, as its application to geophysical prob-lems is almost non-existent. We hope, however, thatstudies of this interesting question might be stimulatedby mentioning a number of experiments on the plasticffow of stressed solids which were at the same time sub-jected to extreme hydrostatic pressures (Bridgman,1935, 1945). A preliminary discussion of the bearing ofthe results on geology was given by Bridgman (1936).Under the applied very large hydrostatic pressures (upto 50,000 atmos. ) rupture of the solid is delayed so thatextremely large strains can occur before rupture takesplace. It is known that the stress-strain curve of manysolids consists, very roughly speaking, of two nearlylinear parts joined together by a transition region. Thelower part is the region of Hooke's law, the upper part,very much extended by hydrostatic pressure, is that ofplastic flow before rupture. The slope, dS/de, of thestress-strain curve in the region of plastic ffow is knownto be very small. Hence one can expect (as has beenpointed out to us by Professor Bridgman) that underlarge pressures and with large stresses the material as-sumes its strained configuration without appreciabletime-lag. The condition of the strained solid is then atany moment determined by the boundary conditions,with only such stresses remaining as cannot be released

by the plastic deformation.These observations apply not only to longitudinal,

but also to shearing stresses. Plastic ffow is extended tohigher strains and rupture is delayed, but, for manysubstances at least, rupture does occur under corre-spondingly higher shears. Hence the material does not

EARTH'5 INTERIOR AN D GEOMAGNETISM

approach the behavior of a liquid as the stress increases,it seems to retain some of its elasticity of form. Theexperimental evidence agrees with the geophysical factthat deep-focus earthquakes occur in the mantle todepths of 6—700 km; they generate very strong trans-verse waves and it is generally concluded from this thatthey represent in the main release of shearing stresses.

The above mentioned behavior of solids under pres-sure with respect to stresses, especially shearing stresses,applies to both crystalline substances and glasses. Noevidence has as yet appeared that would permit one totell mhether the matter of the mantle is vitreous orcrystalline. A number of arguments can be brought forthin favor of a crystalline state. Such are the extremelyhigh wave velocities and high densities derived fromthem, and the tendency of olivine to crystallize readilyfrom the melt without vitrification. On the other hand,the time interval during which the mantle solidifiedmight have been comparatively short, of the order of afew thousand years (see below). This would be enough topermit crystallization of an olivine mantle at atmos-pheric pressure, but whether this stil1. holds at thetremendous pressures actually prevailing has not beeninvestigated.

The viscosity of liquid metals behaves quite diRer-ently from that of silicates. The viscosity of mercuryincreases by about 30 percent on compression to 10,000atmospheres (Bridgman, 1931).On linear extrapolationto the 200 times larger pressure in layer E the viscositywould still be so low as to be characteristic of a liquid.Since at the same time the viscosity decreases veryrapidly with increasing temperature, there seems to bereasonable evidence that the matter of layer E of thecore is truly liquid in the conventional sense.

The final problem to be treated in this section is thatof the earth's internal temperature. There seems to existonly one eRort at estimating the temperature that willstand close criticism, and even if it should not be correctit represents at least a model with closely specified con-ditions. This is Jeffreys' hypothesis (1929) that thetemperature in the mantle is everywhere about that ofthe point of solidification, thus determined by thehydrostatic pressure alone. This is based on the fact thatheat conduction in the solid mantle is an extremely slowand inefhcient process, even over times as long as the ageof the earth (see below). The dependence of the meltingpoint upon depth in the earth is described by theClausius-Clapeyron equation:

dT-/«= g~(d T. /dP) = (g—T-/I )~I /I (—17)

where T is the melting point, L the latent heat of fusionand hp the density decrease at fusion. On substitutingnumerical values for olivine the gradient is found to be4.7'/km (Bowen and Schairer, 1935). This should thenapply to the upper part of the mantle. Jeffreys (1932)considered that a linear extrapolation from the labora-tory data for olivine to the pressure at the lowerboundary of the mantle should give an upper limit for

the temperature there; he finds this to be near 10,000'.One might be safe in expecting that hp decreases pro-nouncedly with increasing pressure whereas I should beless affected by the pressure. Daly (1943) on makingwhat amounts to an instructed guess at the meltingpoint curve arrives at temperatures of the order of4—5000' for the bottom of the mantle. A number ofauthors that have given thought to the problem arriveat comparable figures. The present writer is inclined toattribute also some weight to the fact, to be reportedlater on, that the electrical conductivity of the mantle israther low throughout, probably less than that of seawater. Since silicates are semiconductors it is hard to seehow they could fail to develop good electronic con-ductivity at sufFiciently high temperatures. Again, thisargument is in favor of a fairly low temperature.

Jeffreys' model presumes that the mantle has solidifiedfrom an originally liquid state. A fiuid earth is ratherreadily accessible to theoretical treatment (Jeffreys,1929). Heat transport is by convection, heat loss at thesurface by radiation. Jeffreys estimates the lifetime ofthe gaseous and liquid stages of the earth as about15,000 years. This is very short compared to the totallength of life of the earth, known from radioactivedeterminations to be 2—3X10' years (Jeffreys, 1948;Houtermans, 1947).

It was pointed out by Adams (1924) that the solidifi-cation of the mantle must have taken place from thebottom up. If this had not been the case the lower part ofthe mantle could hardly have solidified at all, since theheat conductivity of the solid part is inadequate tocarry oR the heat of solidification. Adams compares themelting point gradient with the adiabatic temperaturegradient that prevails in a convectively agitated liquid.For the latter we have by simple thermodynamics

dT/dr= gp(dT/dp) =— guT/c„, —(18)

where o. is the coefFicient of thermal expansion, c„specific heat at constant pressure. Kith the best valuesfor solid olivine Birch finds this gradient to be 0.4'/km,thus very much smaller than the melting point gradient.For liquid olivine the gradient should be somewhatlarger, but since the thermal expansion of liquids de-creases rapidly with increasing pressure, it is likely thatthis gradient also decreases at some depth. So long as themelting point gradient exceeds the adiabatic gradient itis readily seen that solidification proceeds from thebottom up.

Ke can apply similar arguments to the thermal stateof the core. Layer E of the core, being liquid, has a tem-perature gradient equal to the adiabat if thermal con-vection is occurring, and this is most probably the caseas will be shown later. Hence the application of formula(13) to compute the density variation in layer E isfully justified. This formula, as me have seen, holdsexactly for adiabatic conditions.

The temperature gradient in the mantle, provided itis the melting point gradient, is seen to be several times

M/ALTER M. ELSASSER

TAsr, E IX. Spherical harmonic analysis. Unit 10 ' gauss. ~

Source

GaussErman-PetersenAdamsAdamsFritscheSchmidtDyson-FurnerAfanasievaVestine-I. ange

Epoch

183518291845188018851885192219451945

-3235—3201—3219—3168—3164-3168—3095—3032—3057

—311-284—278—243—241—222—226-229—211

+625+601+578+603+591+595+592+590+581

+51—8+9—49—35—50—89—125—127

+292+257+284+297+286+278+299+288+296

+12—4—10—75—75—71—124—146—166

gs2

—2—14+4+6

+68+65

+144+150+164

+157+146+135+149+142+149+84+48+54

+ For the purposes of this review the number of gauss is equal to the number of oersted. This will also be considered to hold inside the core, assuming thatferromagnetic eEects are negligible. Non-rationalized electromagnetic units are used in conformity with the tradition of geomagnetic measurements.

the adiabatic gradient. Hence the stratification isunstable, much energy is available in principle forviscous convection, although actually the stratificationmight have a high degree of conditional stability owingto the large "viscosity. "We shall not enter here into thehighly controversial question as to the slow convectionmotions that might have taken place during the geo-logical history of the earth and their connection with thefolding of the large mountain ranges.

The old problem of the thermal history of the earthhas recently been resumed by Slichter (1941).He 6ndsthat with reasonable assumptions about the heat con-ductivity of the silicates the cooling efkct produced byconduction to the surface makes itself felt, at the presentage of the earth, to a depth of about 200 km. In a layerof this depth a stationary state of heat Qow has beenreached while the temperature lower down would beessentially unaBected by the presence of the surface.Lowan (1933—5) has applied rigorous mathematicalmethods to this problem.

Observations in deep shafts, etc., show a temperaturegradient of the surface layers that is extremely variabledepending upon the location, and is between 10'/kmand 50'/km. Most of the Row corresponding to thisgradient removes the radioactive heat output of thecrustal rocks. It is a well-known fact that the matter ofthe mantle must contain a much lower concentration ofradioactive substances than the crust does contain ac-cording to direct measurement; otherwise an impossiblylarge amount of heating of the whole earth would result.The recent work of Davis (1947) shows that eruptiverocks which originated near the bottom of the crust havea radium content so low, about 10 "

g/g, that thiscatastrophe is avoided. Moreover, as Slichter pointsout, an extreme rarefication of radioactive matter in theinterior need not be assumed, provided it be admittedthat the earth as a whole heats up very slightly in thecourse of time, not enough to melt; an eGect that couldhardly be perceived at the outside.

MAGNETIC PHENOMENA

A magnetic field observed at the surface of the earthcould be produced by sources in the interior of theearth, by sources outside the earth's surface, or by

electric currents crossing the surface. The first twocomponents of the field have a scalar potential. Here weare only concerned with the first component, the fieldwhose sources are inside the earth; it is known that thisis by far the largest component. Some older authorsestimated the external field and the non-potential fieldas about 3 percent each of the total, but these estimatesare now considered as extremely doubtful or spurious.

The separation of the field into its components can beachieved by means of spherical harmonic analysis; thismethod has been used since Gauss. I.et U designate thepotential of the combined internal and external field atthe earth's surface; we develop U into a series ofspherical harmonicsf

U=ROP (A "cosmic

+B„sinmrp)X "P (cos8), (19)

where 8 and q are the geographical co-latitude andlongitude and where X„ is a numerical factor, f

()„")'=2(n —m)!/(e+m)! (20)

Since (19) does not involve r, it is a potential for thehorizontal component of the held alone. Now let asimilar development of the vertical component, of theforce itself, be

Z =P (Z&y& „cosmy

+Z~2&„™sinmp)X„P„(cos8). (21)

A three-dimensional potential, @(r, 6, p), is related to(19) and (21) by

v=(q), =R., Z=(ay/ar)„„. (22)

If we write @ in the form

0 =Ro 2 I:&-"(r/Ro) "+(1—~-")(Ro/r) "+'jnm

XA„cosmq4„P„(cos8)+Rog[d„(r/Ro)"

+ (1—d„)(R0/r) "+'j8 "sinm q4„P (cos6), (23)

f +p is the radius of the earth, larger by 33 km than the value ofEp used previously.

f It may be noted that (2e+1)&) is the normalization factorof &» . The seminormalized functions P( )& )=X P whichapparently go back to Gauss are generally used by workers ingeomagnetism

EARTH'S INTERIOR AN D GEOMAGNETISM

the 6rst Eq. (22) is identically fulf&lied and the secondyields

(2n+ 1)c =Z&i& /A +n+ 1

(2'+ 1)d~ =Z&g&& /B~ +I+1,

so that the components of the internal and the externalfie1.d are determined separately if A, 8, and Z~&), Z~» areknown. The last two are directly obtained from theobserved vertical component by (21). The A's and 8'sare obtained from corresponding series for the horizontalforce components which are the derivatives of theseries (19).

The coefficients of the field of internal origin are

Epoch Mean Dipole Mean

183518291845

188018851885

192219451945

1.0611836 1.047

1.052

1.0361883 1.033

1.035

1.0111937 0.989

0.998

12.2' —63.5'1.033 11.'/ —64 7 ~

11.3 —64.3 I

11.6 —68.01.035 11.4 —67.8

11.3 —69.5

11.6 —69.11.000 11.8 —68.8

11.4 —70.0

ALE X. Dipole components.

Mean

—68.4

—69.3

g "=(1—c ")A„", h "=(1—d ")8 ". (24)

The symbols g„and h„are commonly used in thegeomagnetic literature. If the external field is negligible(24) becomes

m h m g m

In this approximation the potential is determined by thehorizontal components alone and its coefIicients mustagree with those found independently from the verticalcomponent of the 6eld. In practice the random errorsare very large. For this reason it is hardly worth while toeliminate the non-potential 6eld separately. There hasbeen much discussion about the reality of the externalfield and of the non-potential 6eld. Chapman andBartels (1940) indicate that the external field could besimulated by very small systematic errors in the mag-netic maps, and there seems at present no de6nite wayof ascertaining the reality of these components. Thesearguments do not apply to the rapidly variable com-ponents of the external field with periods of the order ofa day or several days. Such fields are produced by solarand lunar perturbations of the ionosphere and theirproperties are well known (Fleming, 1939; Chapmanand Bartels, 1940). Their spherical harmonic compo-nents have been determined, but they have no apparentrelation to the stationary components of the 6eld or tothe very slow secular variation of the latter. In investi-gations of stationary components of the non-potentialfield their magnitude is determined directly by com-puting the line integral of the magnetic force alongclosed contours on the surface of the earth.

In order to carry out a spherical harmonic analysisone prepares a table of the field components at a grid ofpoints with regular intervals in latitude and longitude.The values of the magnetic force components at thesepoints are scaled oG magnetic maps. The values of theLegendre functions at these points being tabulated, onecan then proceed to solve a system of normal equationsfor the spherical harmonic coefficients. Since some partsof the globe are not too closely surveyed magnetically,considerable errors are introduced into the final nu-merical data. In view of this and of the great irregularityof the field which results in relatively slow convergenceof the spherical harmonic series, students of the subject

prefer whenever possible to use the maps directly. Weshall give a brief survey of the results of the sphericalharmonic analysis, as some of the conclusions drawnfrom them are significant. The most complete analysesare those for the year 1885 by Schmidt (1895), for 1922by Dyson and Furner (1923), for 1945 by Vestine (1947)and an additional one for the same year by Afanasieva(1946). Schmidt is the only one who separated theinternal and external field, but in view of what has beensaid above no particular significance can be ascribed tothese results.

The numerical values of the dipole and quadrupolecoeKcients as collected by Dyson and Furner andsupplemented by Uestine are given in Table IX. Themost outstanding feature of this table is the secularvariation of the individual coeS.cients, which is clearlyapparent in spite of the individual scatter. The data forthe dipole terms are given in Table X in another form.The relative magnitude of the total dipole moment/is given in arbitrary units, putting the last value in the"mean" column equal to unity. Table X contains inaddition the polar angles of the dipole axis. The obser-vations fall conveniently into three groups about 50years apart, for which means have been taken. Theover-all decrease of the dipole moment amounting toabout 5 percent in a century can hardly be ascribed toobservational errors alone; its absolute magnitude mightnot be given correctly by Table X. Vestine (1947)carried through a spherical harmonic analysis for thesecular variation of the Geld; his values for the pre-ponderant dipole term, dg&'/dt, are as follows:

Epoch 1912.5 1922.5 1932.5 1942.5Percent per year —0.08 —0.09 —0.08 —0.03.

It is hard to say how accurate these values are, but therecan again be little doubt as to the over-all decrease ofthe dipole. It is likely that the reduced rate of change in1942.5 is real, since Uestine s maps indicate a slightgeneral change of the secular pattern at this epoch. Onewould of course not expect that the dipole moment hasdecreased or will continue to decrease over a long periodof time at the rate expressed in Table X; this change canbe taken as just one component of the general, highlyirregular variation of the GeM, other examples of which

$ The earth's magnetic moment in 1945 was 8.06&(10~ e.m.u.

KVALTER M. ELSASSE R

~Ion itude '. ~

~ II0-—-a— —--0

t

-30t500 l700

-30IQQQ Year

Fro. 7. Movement of thewesterly point of zero declina-tion along the equator.

will be encountered later. The inclination of the dipoleaxis has remained sensibly constant at 11.6' withvariations of only a few tenths of a degree. Bauer (1895)has investigated the inclination of the magnetic axisduring the period of I780—1885 by taking averages of theinclination over circles of geographical latitude. He findsthat the angle of inclination has remained constant with-in errors of the same order as those of Table X. On theother hand, the last columns of Table X show a slow ifsomewhat irregular precession of this axis in the direc-tion from east to west.

%e next inquire into the non-dipole part of the field.Some information about the secular variation of thequadrupole terms is contained in Table IX, but on thewhole it is better to rely directly on the magnetic maps.Anticipating the main result we may say that the non-

dipole components of the field are subject to strong and

comparatively rapid secular variations. There are ap-parently no constant components of the 6eld; there isevery reason to believe that the non-dipole field aver-aged over a sufliciently long interval of time wouldvanish. Consequently, the non-dipole part of the 6eldcannot be properly treated and understood separatefrom its secular variation.

In dealing with magnetic maps it might first be saidthat at many places there are small localized distortionsof the 6eld of limited extent caused by iron-bearingminerals in the crust. Such ore deposits are as a rule veryclose to the surface and permanent magnetic distortionsof this nature can rather readily be identified as such.As a matter of experience it is usually possible to recon-struct the 6eld that would be there in their absence. Noserious reasons have appeared as yet to indicate thatany appreciable part of the remaining non-dipole 6eldoriginates in the mantle. In the work of the U. S. Coastand Geodetic Survey it has been found that anomaliesof a scale intermediate between the distinctly local onesand those of very large extension are absent (Decl,1945). On considering large areas of the earth's surfaceand smoothing over the localized crustal. distortions oneis left with the field that is the subject of this review. Aconvenient tool for the study of the non-dipole field is

FIG. 8. Geomagnetic secular change in gammas per year, vertical intensity, epoch 1912.$.

EARTH�'5 INTER I OR AN 0 GEOMAGNETISM

maps obtained by subtracting the field of the dipoleterms from the total field. Unfortunately, very few suchmaps have been constructed, and so one has to adoptother, less satisfactory methods for the evaluation of theolder maps. The constant components of the fieM areautomatically eliminated in maps of the secular varia-tion. The most important set of these is due toVestine and his group (1947).

Magnetic maps go back to the second-half of theseventeenth century and some systematic records col-lected by voyagers cover an additional century. Owingto the irregular character of the field and its secularvariation many of the oMer investigations have virtuallyno systematic value. Gradually, however, a few moredistinct traits became apparent. Thus, Bauer (1895)states quite clearly that there is a trend of the irregular6eld configuration toward rotation about the earthfrom east to west, a westerly magnetic "wave" as hecalls it. Carlheim-Gyllenskold (1907) also finds a strongpreponderance of westerly displacements in his har-monic analysis of the secular variation. Given the over-all. irregularity of the field the value of such findings liesmore in cumulative evidence than in specific quanti-tative data. As one of the rare examples of the quan-titative type v e mention the westerly motion of the lines

of zero magnetic declination studied by Bauer by meansof data going back to the middle of the sixteenthcentury. The older references are based on a verycareful survey of records of navigators from 1540 to j680compiled by van Bemmelen (1893). Figure 7 shows thechange in longitude of the point at which the westerlyline of zero declination intersects the geographicalequator. Since the sixteenth century this point hasmoved from the center of Africa across the Atlantic andis now close to the head-waters of the Amazon. The totaldisplacement is just about 90' in 400 years, or 0.22' peryear. The corresponding point for the easterly line ofzero declination has also moved westward, but much lessregularly, with an occasional standstill, and altogetheronly by about 30—40'. This seems not due to anyinadequacy of the data, as this point has moved throughthe Malayan archipelago, a region well enough exploredfor a long time. A glance upon the maps of the magneticfield and the secular variation shows that the westerlyline of zero declination is fairly straight whereas theeasterly line has some pronounced distortions and passesthrough regions where the local deviations from thedipole field are large. We are unable at present to ex-plain the large rates of westerly motion as compared tothe slow westerly progress of the dipole axis shown in

'eI

(ll

MPSjfilfP ~~~I'.. ., : t

l, ', h$Xy~ ~&N$

I'lG, 9, Geomagnetic secular change in lammas per year, vertical intensity, epoch 1.94$.5,

KVALTE R M. ELSASSE R

Table X. This, together with the difference in speed ofthe two lines, indicates that the whole phenomenon is

highly complex, the dipole component being only re-sponsible for part of the motion; the rest must be due tohigher harmonic components which apparently move tothe west at a more rapid rate.

An extensive investigation of the secular variation forfour epochs at distances of ten years has been carried outby Vestine and collaborators (1947).This work is basedon an exhaustive evaluation of magnetic records for theentire globe between about 1905and 1945.Figures 8 and 9show two maps for the secular variation of the verticalintensity in units of 10 ' gauss per year at epochs 30years apart. Figures 10 and 11 are two similar maps forthe rate of change of the declination in minutes per year.The maps show that there are a number of centers ofsecular disturbance distributed rather irregularly overthe surface of the globe; the rate of change of the fie1.dmay have either sign. The lifetime of these disturbancesis apparently quite short, less than a century in theaverage to judge from the maps; some of them havedisappeared and others have newly appeared during theperiod of observation. In addition to their increase or

decrease in intensity the centers show irregular dis-placements, but there appears a quite distinct pre-ponderance of drift motion towards the west. This isvisible in the motion of individual centers as well as inthe displacement of the lines of zero secular change. Afew centers show a drift in north-southerly direction, butthis seems more in the nature of a random motion.Bullard!! has made a quantitative study of the westerlydrift, from Vestine's maps, using a least square method,and finds a value of (0.180&0.013) degree per year.The magnitude of this drift motion is found to be inde-pendent of the geographical latitude.

Many eGorts have been made to reconstruct the mag-netic 6eld of the past, previous to the advent of magneticmeasurements. All the methods have in common thatthey determine the direction or intensity, or both, of theremanent magnetization in a material that has beenformed or has solidi6ed at a known epoch of the his-torical or geological past. Such methods must be usedwith great caution and must be carefully checked forconsistency, and as in particular Haalck (1942) haspointed out, in many cases the results are unreliable.

Chevallier (1925) has studied the magnetization of

'«F', «ll' ' sF:' ~ . . NF. , 'sF "'sF ' W' " M " ~ W W': . sr .':W:: '%'. : sF ' 'sF ' 'M. ' W .F . :.W' sF::.: . Af':" sF. . "N' . .sF.

ti

I' 8!",.1

j . ! s

i I

FIG. 10. Geomagnetic secular change in minutes per year, declination, epoch 1912.5.

)~ Personal communication. Will be published soon.

EA 8 TH 'S I NTE RIOR AN 0 GEO MAGN ETISM

lavas of Mt. Etna, from dated eruptions, back to theyear 1280. The data for the declination give a curvethat, from about 1550 on, coincides with the directlyobserved values. Thellier (1938) determined the curve ofthe inclination at Paris back to the year 1400 by meansof bricks in historical buildings of known age, it beingassumed that a brick is conventionally baked in ahorizontal position and that its remanent field has thedirection of the earth's field at the time of baking. Theresults seem suKciently consistent to warrant the use ofthe method. An investigation of sediments was made byMcNish and Johnson (1938). Their main sample con-sisted of varved Pleistocene cl.ays from New England.Its time-scale (200 years) is determined, apart from anarbitrary origin, by comparison with an established tree-ring chronology. The curve of the declination obtained issmooth and shows considerable variation. Similar meas-urements with comparable results were made by Ising(1943) on varved clays in Sweden. A recent investigation

by Johnson, Murphy, and Torreson (1948) extends theprevious results. The values, for the declination, ob-tained again from New England varved clays, lie on areasonably smooth curve over an interval of 5000 years.The relative chronology of this period has been given byE. Antevs; the absolute time is probably from 15,000 to

10,000 B.C. The maxima and minima of the declinationare 20' east and 38' west, respectively. Sedimentationtests in the laboratory lead to the conclusion that thedeclination of the deposits follows quite closely that ofthe earth's field, while their inclination might dier veryappreciably from that of the external field. The authorsalso indicate evidence that the mean magnitude of theearth's field was of the same order as at present. On theother hand, Thellier and Thellier (1942/6) have tried todetermine the intensity of the field by the brick methodmentioned above and find that the present observeddecline in the field has apparently continued for severalcenturies in the past. The two results need not becontradictory, as there is no reason to presume that thelast-mentioned trend is more than a temporary Quctua-tion on the secular scale.

Systematic investigations of the remanent magnetiza-tion of sedimentary rocks have only been made quiterecently. The work of Graham (1949) is concerned withthe stability of the magnetization of sediments. If theway in which a bed of sedimentary rock has been foldedis known, it is possible by means of a graphical con-struction to restore the vector of the remanent mag-netization to its original undisturbed position, and thereconstructed bed should then appear uniformly mag-

FIG. 11.Geomagnetic secular change in minutes per year, dechnation, epoch 1942.5.

ELSASSE R

netized. This was done successfully for some pleistocenevarved clays in New England; it was also carried outwith great success for a silurian rock bed in Marylandwhose age is estimated at 3.5X10' years and in whichintense folding occurred some 2&(10' years ago. It maybe concluded that the magnetization has remained stablefor periods of this order. Another attempt, with anAppalachian limestone, showed, however, a large scatterof the points and further studies must be undertaken toascertain the cases in which stability can safely beassumed.

Torreson, Murphy, and Graham (1949) undertook astatistical sampling of the magnetization of sedimentaryrocks of various geological ages in the NorthwesternUnited States, the great majority being of tertiaryorigin. It may be assumed that the appearance of de6-nite regularities in such material indicates that at leastpart of it is magnetically stable. A frequency-distribu-tion curve of the declination showed a very sharpmaximum centered about true north (the present localdeclination is 17—21' east). Over half of the pointscontribute to this narrow peak, the remainder aredistributed at random. This would indicate that in themean the magnetic and the geographical poles coincide,but the authors indicate that the material is as yet toolimited to permit of final conclusions. The curve for theinclination shows an entirely similar maximum near anangle that is slightly smaller than the present inclination.

The literature on the magnetization of volcanic lavasis extensive, and mostly of little interest in the presentcontext, as hardly any dehnite conclusions have beendrawn from them. The question of the stability of themagnetization acquired on cooling below the Curie pointof the ferromagnetic component is still largely open.There seems to have been little systematic study of therelation of the magnetic behavior to the structure of theminerals involved.

An exception to this generally unruly behavior hasbeen found in studies of the magnetization of dikescarried out in recent years. ("A dike is a tabular body ofigneous rock that has been injected while molten into a6ssure" —Webster's dictionary. ) Two of these caseshave been very fully investigated. There is a system ofsuch dikes in Transvaal extending over an area about300 km long and up to 150km wide. The entire volcanicformation is magnetized in direction opposite to thepresent magnetic 6eld at the locality. The age of thesedikes is about 2X10' years. Other geological bodiesformed in the vicinity before and after this system aremagnetized in the same direction as the present 6eld(Gelletitch, 1937).This might conceivably be explainedby a shift of the magnetic equator sufficiently far (say30') to the south, but such an explanation fails foranother large system of dikes that extends across thenorthern part of England. The detailed investigation byBruckshaw and Robertson (1949) shows that nearly theentire system of dikes is magnetized in a directionopposite to the present field. Deviations from uniform

magnetization can apparently be explained by slightQow in the later stages of solidihcation, below the Curiepoint of the material (about 550'). Such Qow does notoccur in ordinary rocks at these temperatures, but it hasbeen observed to take place in material cooled downfrom higher temperatures and containing volatilesolutes. The regions near the walls of the dikes whichcool most rapidly show the most distinct reversepolarization. The authors themselves do not believe thatthere is another explanation to their endings than theexistence of a reversed geomagnetic field at the time ofthe formation of the dikes. If, however, smaller devia-tions can be explained by viscous Row below the Curiepoint, then it might be possible to explain the reversal asbeing caused by cellular convection that turns the massupside down before Anal solidification. Such "half-cycle"convection, though on a vastly larger scale, has beensuggested by Meinesz (1948) for the explanation ofcertain orogenic processes.

Other instances of reverse magnetization of volcanicintrusions have been reported; for instance two fromGermany (Schulze, 1930; Reich, 1935) and one fromBrazil (Malamphy, 1940). Not all igneous intrusions,however, have reverse magnetizations; in a number ofthem the magnetization is in the general direction of thepresent 6eld. These phenomena are quite obscure at thepresent time, especially if confronted with the resultsobtained from sedimentary rocks. The assumption ofnumerous reversals of the earth's field during geologicalages does not seem particularly attractive. Much workremains to be done, especially on the petrographicaspects of the problem.

Having now completed our survey of the direct mag-netic data, we shall next try to relate them more closelyto the other physical facts concerning the earth's in-terior. It is generally believed that ferromagnetism is notimportant in the strata below the crust. It is extremelyunlikely that the mantle contains large amounts offerromagnetic material. Moreover, the temperature atgreater depth should soon exceed the Curie point. Theefkct of pressure upon the Curie point can be studied inthe laboratory only to a very limited extent, and suchdata as exist indicate that the Curie point does not risewith rising pressure. It is possible to draw some infer-ences from the theory of ferromagnetism. In the latter,the permanent magnetization is attributed to exchangeeffects between electrons in the d shells of neighboringatoms. Calculations show that these exchange terms de-crease in absolute magnitude and eventually reversetheir sign (prohibiting ferromagnetism) as the distanceof neighboring atoms decreases. In the iron core wherethe linear distances of atoms are reduced by some 20percent, this eGect should be rather marked, andferromagnetism should not exist, quite apart from thetemperature eGect.

At various times it has been proposed that largecelestial bodies become magnetized by rotation owing to

EARTH'S I NTE RIOR AN D GEOMAGNETISM

fundamental properties of matter not adequately de-scribed by conventional theory. The latest suggestion ofthis type was put forth (Blackett, 1947—9) in conse-quence of Babcock's discovery (1947—8) of large mag-netic fields in rotating stars. Ke cannot analyzeBlackett's suggestion in the present review, but weshould remark that Runcorn and Chapman (1948) haveproposed a method whereby one might be able to decideexperimentally between a theory of this type and thoseother theories that seek the origin of the field in theearth's core. In the latter case, clearly, the field in-creases with depth in the earth, as r '. In a hypothesissuch as Blackett's on the other hand, one may assumethat any part of the rotating body contributes to themagnetization according to some law, e.g. , proportionalto its density. The calculations then show that the hori-zontal component of the field decreases in magnitude asone goes down in the earth. This conclusion can bechecked by measurements in deep mines, and a fewpreliminary measurements (quoted by Runcorn) indi-cate that an eGect of this type might exist. The devia-tions in question are, however, small and can be maskedby local variations of crustal origin. Lengthy series ofmeasurements must be made before the question can besettled by this means.

Within the framework of ordinary physics there re-mains the possibility that electric currents Row in theinterior of the earth. In this connection one wouldexpect the core to have a high electric conductivity andthe mantle to have a lorn one, and this is borne out bysuch limited evidence as is available at present. Ke mustthen expect that the largest part, if not all, of such cur-rents How inside the core. %e shall disregard for thepresent the permanent dipole part of the field and shalltentatively consider variable currents in the core assources of the variable part of the field. The questionarises whether sources in the core alone are sufhcient toproduce the field observed at the surface. An approxi-mate mathematical criterion for this can be found asfollows: Let the potential at the surface be written

U= Rp Q a„"y„"(i7,s), (25)

where the y„area set of spherical surface harmonics.The n„"might differ from the coeflicients (24) bynumerical factors depending on the normalization of the

. If it be assumed that there are no sources of thefield in the mantle, the potential at the boundary of thecore is

U, =Rp g (Rp/Ri) "+'n„"y„"(0,q). (26)

This potential can be constructed if the sphericalharmonic analysis at the surface has been carried out.Since the observational data yield only a finite numberof spherical harmonic coefficients, it is not possible toestablish a rigorous mathematical criterion for thevalidity of the extrapolation (26) as applied to a real

TABLE XI. Mean (r.m. s.) harmonic components of order m.

71 ~ 1

m~2ratio

m&4ratio

Model

1433

1433

2 3 4

370 196 920.26 0.53 0.47 0.47

370 177 730.26 0.48 0.41 0.440.46 0.51 0.53 0.54

43 8.50.20

32 110.340.54

field. Instead, one can use criteria of an approximatenature with a physical basis, and these will be found tobe sufficiently stringent.

McNish (1940), on surveying the non-dipole part ofthe field, found that it lent itself to an approximate re-production by means of 14 small dipoles of radialdirection situated at a depth of half the earth's radius,i.e., somewhat below the boundary of the core. Theymay have either sign and have an average magnitude ofabout 1 percent of the earth's moment. Elsasser (1941)compared the observed field with a model of a statisticalnature. He assumed that a number of small magneticdipoles are thrown at random into the volume of aspherical shell centered in the earth, and computed themean (r.m. s.) values of the spherical harmonic coeK-cients of the potential produced at the surface of theearth. If, for instance, all the dipoles have radialdirection and are located at the surface of a sphere ofradius R1 the result obtained is as follows: Let the y„in(25) and (26) be normalized. For the r.m.s. averageof the corresponding coefficients n„with fixed e andvariable m one finds

Lo j4" k(4K) /Rp Rll(Rp/R, )"(ri/2N+1)[MjA„(27)

where M is the moment of an individual dipole. Theratio of successive terms of this sequence for Ri/Rp=0.55 is given in the last line of Table XI. It isinteresting to note that the expression (27) is inde-pendent of the number of small dipoles involved. This isfound to be true so long as the number of dipoles remainsmoderately large (excluding, e.g., the case of a continu-ous, regular distribution). As the figures at the bottomof Table XI show, the ratios of successive terms ap-proach rather rapidly to the value 0.55=Ri/Rp as nincreases. A similar behavior is characteristic of anumber of statistical models investigated, provided theouter boundary of the volume available to the dipoles isthe sphere of radius Ri. This shows that for a randomdistribution of dipoles the series at the boundary of thecore converges extremely slowly. Now if one starts fromthe observed field at the surface and determines thepotential at the boundary of the core by means of (26),he would expect on physical grounds that this latterseries should not diverge. The statistical models withtheir extremely slow convergence thus appear as thelimiting case of surface potential distributions whichpermit location of all their sources in the core.

In order to compare these results with the observa-tions the rather complete analysis by Schmidt (1895)was used. All coeKcients were converted to normalized

HALTE R M. ELSASSER

harmonics and relative values were used, putting thecoefficient of the main dipole Geld parallel to the earth' saxis equal to 3.0,000. The values of Table XI are ob-tained by averaging for Gxed n over a limited range of m

(this is done because coeflicients with large m are likelyto be less accurate). The first and last columns of theTable have little significance; the former contains onlythe dipole terms perpendicular to the earth's axis, thelatter is rather unreliable on account of observationalerrors. The table shows that the ratio of successiveterms is somewhat below 0.55, thus securing conver-gence of the series at the boundary of the core. Thedistribution of sources is slightly more regular thanwould correspond to the statistical model, or else thesources are appreciably below the boundary of the core.The last alternative can be ruled out on physicalgrounds, as we shall see presently.

A somewhat diAerent approach was chosen by Bullard(1948). He investigated the secular cha, nge in theneighborhood of South Africa, as shown in Vestine's

maps. He then tries to represent the Geld of the secularvariation for 1922.5 by a time-dependent dipole locatedslightly below the surface of the core. The dipole ishorizontal and has roughly the direction from SE toNW. This dipole represents well the secular field at thesurface in the region vertically above it. At larger

distances the Geld of this dipole decreases somewhatmore rapidly than the actual Geld; this indicates thatthe agreement would not be improved by putting thedipole at a level above the core. By means of a lineararray of dipoles at the boundary of the core, in thedirection from SE to NW, one can represent both thehorizontal and the vertical component of the surfaceGeld over a rather large area. Bullard concludes that thegeometrical construction exhibits no contradiction tothe concept of sources concentrated in the core.

Using his spherical harmonic analysis Vestine (1947)has computed diagrams of spherical current sheets thatwould produce the observed Geld. He places these atvarious depths, the lowest at a depth of 3,000 km. Forgreater depth the extrapolated series would soon becomedivergent as we have seen. In Fig. 12 are shown thecurrents at 3,000 km depth that would produce the non-

dipole part of the field, as of 1945, there being 10' amp.between adjacent lines. It must be assumed that inreality the pattern is vastly more complex, as theharmonic analysis stops at n=6 and the higher har-monics omitted will be relatively large at the boundaryof the core. Such patterns therefore cannot be expectedto have much Gdelity in the details.

Next, we require some knowledge of the electricalconductivity in the earth's interior, especialLy in the

Pro. 12, Current-function in 106 amperes for thin spherical shell at depth 3000 km ~vithin earth to reproduce residual (non-dipole part)of main 6eld, epoch 1945.

EARTH'S I NTERIOR AN 0 GEOMAGNETISM

core. The conductivity of ordinary iron is 1.10' ohm-'cm ' (= 1.10-' e.m.u.) There will be pressure and tem-perature eBects which may be summarized roughly bywriting o =C 8'/T where 0 is the Debye temperature, Tthe absolute temperature. The factor C incorporates thechanges in conductivity owing to the changed shape ofthe electronic wave functions whereas the second factorexpresses the changes in the statistical behavior of theelectron gas. The Debye temperature is proportional tothe velocity of sound which in the core is about twicethat of ordinary iron. Taking T=9000' and assuming inthe absence of other knowledge that C remains constantwe find o = 1.3&(10'. Bullard (1948) has made an alter-nate estimate on using empirically determined pressureand temperature variations and on making an appreci-able allowance for alloying and melting. He arrives at0 =1.10'. This seems too small to the present writer; inhis latest paper (1949) Bullard uses an intermediate6gure, o.=3.10'. This value will also be adopted here.

We can at once draw an important conclusion fromthese estimates of the conductivity, namely, that thecurrents which produce the time-dependent part of thefield must Qow in a thin layer just below the boundaryof the core. A layer of metal acts as a shield for variablecurrents. Given a Fourier component of angular fre-quency 0, the depth of penetration, d, into a planesheet is expressed by the formula, used in the theory ofthe skin efI'ect,

1/d= (2+poQ)t. (28)

The situation here is opposite from that of the skineGect, the 6eld here originates inside the conductor andpenetrates from there to the outside. The detailedanalysis (1946) shows that variable currents Rowing at adepth much larger than the critical depth (28) do notproduce a field outside the core. Putting 2n./0=100years and o =3.10' we 6nd d=50 km. This thickness isso small that the result in turn justi6es the use of theplane-sheet approximation (28). Furthermore, owing tothe square root, the thickness changes only slowly when0 and 0 change. We may say with much generality thatthe currents producing the time-dependent part of thefield originate in a layer that is certainly very thincompared to the radius of the core.

It is possible to estimate the conductivity of themantle at least within certain broad limits. A glance atthe records of individual stations collected by Vestineshows that Fourier components of 50 years in thesecular variation are by no means uncommon. Hencethese frequencies are not damped out by the mantle.Using this value in (28) together with d= 2900 km weobtain 0.=0.5. This, of course, is a crude estimate andgives no more than an order of magnitude. The averageconductivity of the mantle should then be lower, say 0.1or less. The conductivity of surface rocks on the otherhand is 10-'—10 ' (all in ohm ' cm ').

There is another method of approaching the problemgf conductivity in the mantle, developed by Chapman

and several of his pupils. %e follow here the latest paper,by Lahiri and Price (1938).The method is based on thefact that there are currents in the ionosphere of shortperiod (e.g. , diurnal). These in turn induce currents inthe solid earth whose magnitude and phase depend onthe conductivity. The theory was developed for aconductivity that is a function of depth. The sphericalharmonic analysis of the observed 6eld has been carriedout for a number of leading terms and the external andinternal 6elds separated. Lahiri and Price find that theobservations can be fitted by the theory only if theconductivity remains very low for some depth and thenincreases rather rapidly to not much less than 10 ' at adepth of about 6—700 km. Not too much emphasisshould perhaps be placed on the quantitative side of theresults since they are based on a rather limited observa-tional material.

Coster (1948) has investigated the conductivity of anumber of rocks, olivine among them, and has foundthat they exhibit the typical temperature dependenceof semiconductors.

DYNAMICS OF THE CORE

We may assume that in the core there exist simul-taneously Quid motions and electric currents. The twointeract with each other, producing a number of rathercomplex dynamical phenomena which will be outlinedin this chapter. Ke shall first inquire into the causes ofthe motions so as to have as far as possible a concretephysical picture. Since the evidence for such motions asgathered from the geomagnetic variation is quite strong,the correct approach would be to postulate the existenceof motions in the core irrespective of their presumptiveorigin, leaving the discussion of their causes for later.The reader might, however, prefer to have a somewhatmore concrete picture in his mind; for this reason wehave inverted the logical order and shall first deal withassumptions about the causes of the motions. It is clearthat the theory of the interaction between the Geld andthe moving Quid, to be outlined thereafter, is essentiallya chapter of general dynamics and does not depend onany specific assumptions as to the nature of the primarypower supply.

Two causes of motions in the core suggest themselvesreadily; a rather diligent search has so far failed tounearth others. One of these is found in the variation ofthe earth's angular velocity of rotation as astronomicallyobserved, and attributed to tidal friction. The other isbased on the assumption of a sufhcient temperaturegradient in the core so as to produce thermal convection.These two mechanisms have been studied in detail byBullard (1949) whom we follow with some modifications.As will be indicated below, the magnetic 6eld and theQuid motions in the core are very closely coupled so thata transfer of energy from the Quid to the field and viceversa does constantly occur. It so becomes necessary toconsider two dissipative agencies, namely, mechanicalfriction in the Quid and Joule s heat in the electric cur-

%VALTE R M. EL SASSE R

&m~m + &c&c) (29)

where A: is the heat conductivity, v temperature gradient(m and c designating mantle and core, respectively). For7, we must use the adiabatic gradient, by (18).Bullardestimates this as 1.1'/km in the core. Again, if Jeffrey'smodel of the earth holds, v is the melting point gradientwhich will be several times larger than r,. It is com-paratively safe and easy to estimate x, by the Wiede-mann-Franz law, «/~T= const. , which gives a tempera-ture-independent « The value «. =0.18 cal./cm sec.

rent system. The postulated sources of power mustmaintain the motion against this dissipation. Numericalestimates given later make it likely that mechanicalfriction is negligible compared to electromagnetic dissi-pation. The latter can be estimated from the formulasgiven below to be about 5X10 " cal./1 cm' sec. Ourhypothetical model must generate this amount of power.

Bullard's study leads to the result that of the twopotential sources of motive power only thermal con-vection can produce the observed motions. The de-celeration of the earth's rotation which is attributable tofriction of the lunar tide (Jeffreys, 1929) is found to havea negligible effect upon the dynamics of the core. Thewriter had believed for some time past that lunarperturbations play a signi6cant role in the core's mo-tions, but it was more recently found that the con-clusions rested on an error. Recent studies of a varietyof lunar eGects (1950) have verified that no lunar per-turbation, tidal or precessional, has an effect on thecore large enough to maintain the motions whose mag-nitude is indicated by the geomagnetic secular variation.The astronomical data regarding the non-uniformity ofthe earth's rotation have been reviewed and re-evalu-ated by Spencer Jones (1939). The effects upon theliquid core have been studied from the purely hydro-dynamical viewpoint (Bondi and I yttleton, 1948) andturn out to be exceedingly small. This writer has nowcome to believe that the theory of geomagnetism canbe developed without reference to these astronomicalphenomena which seem to be of separate causation.

We therefore consider thermal convection in the coreas a means of generating Quid motions. An earlierattempt (Frenkel, 1945) led to satisfactory qualitativeresults; later Bullard gave a very clearcut thermo-dynamical analysis that we shall follow here with someadditions. The convective stratum may be presumed toconsist of aH of layer E (see Fig. 3) about 2000 km deep.Since a uniform generation of heat in the layer wouldnot produce convective conditions, there must either bean inQow of heat through the lower boundary or anoutQow through the upper boundary. The outQow,again, produces convection only if it exceeds the purelyconductive Qow of heat inside the core itself. We might,perhaps, expect that the conductive heat flow in themantle is less than that in the core, a condition that we

may designate as thermal insulation of the core, ex-pressed by

degree corresponds to the adopted value of 0. by virtueof the universal Wiedemann-Franz relation (it also is thelaboratory value for iron). It is well nigh impossible toobtain a good estimate of a as the subject of thermalconduction in insulators is very little explored. Omittingpressure efI'ects that are simply unknown, theory indi-cates that «~T ' (Peierls, 1928). Under laboratoryconditions the conductivity of olivine is about 1/14 ofthat of iron. Combining all these data, there results apresumption to the effect that (29) is fulfilled, and we

shall therefore assume for the time being that the coreis thermally insulated.

The simplest model under these conditions is one inwhich radioactive sources of heat are concentrated inthe central body. This seems a very likely assumptionsince the simple oxides, UO2 and Th02, have (underlaboratory pressure) densities around ten. To estimatethe amount of heat required, we note that the limitingefficiency, e, of the convective process considered as athermodynamical engine is, by the second law,

&=AT/T=Dr, /T,

where D is the mean depth over which convectivetransport is active. We take D= 10' cm, half the depthof layer E. Even with the extreme value, T=10,000',this gives an efficiency, &=11 percent. The assumedvalue of ~, might be too large, but in any event e is ofthe order of a few percent which will turn out to beample for the purposes required. As Bullard points out,there is good reason to believe that the convectivemachinery is rather efficient so that the actual efficiencyis not lower by orders of magnitude than the theoreticallimit.

In order that convection may actually occur, the heatavailable at the lower boundary of layer E must exceed~~, the heat carried away by conduction alone; we

might assume the total heat to be a small multiple ofKt say 2«r As (18) shows, .r is likely to be smaller at thebottom of the layer than at the top; furthermore thetemperature of 10,000' on which the previous estimateof 'T was based is rather extreme; we therefore proposeto adopt a somewhat lower estimate, r 0 3 =/km. 'valid,at the bottom of layer E. The convective heat suppliedto 1 cm' of the layer is then 7X10 "cal./sec. which is140 times the heat output due to electromagneticdissipation as quoted above. The total rise in tempera-ture of layer E during the lifetime of the earth is of theorder of 100', not enough to cause any appreciableeffects. Again, the heat supplied to the surface of thecentral body is 1.1 10 ' cal./cm' sec., and this might becompared to the mean outQow of heat from the earth' ssurface estimated (edited by Birch, 1942) at 1.3 10-'cal./cm'sec. Since the latter flow is principally due tothe radioactive heat output, we might say that theconvective mechanism must be operative if the radio-active material of the earth is preferential1y concen-trated not only in the crust, but also in the centralbody, and provided the radioactive content of the

k:ARI'H'S I ATE RIOR AN D GEOMAGNETISM

central body is to that of the crust at least in the ratioof their surface areas, that is 1:20.

In a forthcoming paper Bullard (1950) specifies hisprevious thermodynamical analysis by proposing amodel of convection in which it is assumed that the coreis not thermally insulated. This condition can arise intwo ways. Either (29) is not fulfilled which, in view ofthe large errors attendant especially on the extrapolationof the properties of the silicates to the lower mantle, isadmittedly possible; somewhat more likely is the pres-ence of slow viscous convection of the type brieflydiscussed in the third chapter. If such convection occursit can be shown to provide an adequate means ofremoval of the heat. In this model the radioactivematerial may be assumed to be uniformly distributedthrough the core; also the amount of radioactivity re-quired is somewhat less than in the previous model, ofthe order of that found in iron meteorites. There areadvantages and disadvantages to either of these modelsand the decision between them will have to await futureprogress.

XVe now proceed to the main subject of this chapter,the electromagnetic phenomena of the core and theirrelation to Quid motions. First, consider the core as alarge metallic sphere without internal motions. We areinterested in re]atively slow changes of the magneticfield and hence we can neglect the displacement currentin Maxwell's equations. The latter then read

yxE+aB/at=0, v B=0, (30)

(31)

where the susceptibility p, will be assumed constantthroughout, neglecting ferromagnetic eGects. In theahsence of mechanical motions we have J=OE, and o.

couM be assumed a function of r, but will for simplicitybe taken as constant throughout the core, having thevalue 0.=3)&10' ohm ' cm ' given previously. By afamiliar procedure we 6nd.

V2B = 47rpoBB/Bt. (32)

One obtains exponentially decaying solutions, "freemodes, " by putting

B=B'(r, 8, y)e ""

where the vector 6elds B' are expressible in terms ofBessel functions and spherical harmonics. They form anorthogonal system of vector functions over the sphere. 'It is important to note that there are two types of freemodes, schematically illustrated by Figs. 13 and 14. Weshall explain their difference for the special case ofrotationally symmetrical modes (zonal harmonics). Inthe 6rst type of free modes, the "magnetic modes, " thearrows of Fig. 13 designate electric currents (shownprojected upon the surface of the sphere). The corre-sponding magnetic 6elds will in general have the formshown by the solid arrows of Fig. 14 (in meridionalcross section). The field will extend to the outside of themetallic sphere, but current configurations are possiblewhere it is confined to the inside, as indicated by thedashed arrows. The second type of free modes, the"electric modes, " have currents given by the dashedarrows of Fig. 14. The corresponding magnetic fields arethose shown in Fig. 13, that is, the magnetic lines offorce are now circles about the axis. It can be shownthat the magnetic field of these modes vanishes rigor-ously outside the metallic sphere, so that if such a fieldexists inside the core it cannot be detected by measure-ments at the earth's surface. There are, however, reliableindirect methods of inferring its existence as we shall seepresently. This type of internal magnetic 6eld willbe designated as "toroidal" field, as distinct fromthe more conventional field of the magnetic modes(which might be called the "poloidal" field).

The mean life of the free, exponentially decayingmodes is

fp —0 E. '4' p, (T

where E is the radius of the core and u is a numericalconstant which becomes rapidly small for the higherorder modes. For the lowest magnetic dipole modea= ~, giving t&~15,000 years. The lifetime of the lowestelectric mode is of comparable magnitude. For the sun,by the way, Cowling (1945) has computed the variationof a with depth, arriving at lifetimes of the lowest freemodes of the order of 10"years.

We now consider the electromagnetic eGects ofmotions in the core. If a metallic conductor moves in amagnetic field a current is induced in the conductor, thetotal current being,

FIG. 13.

Fro. 14.

Dipole Quodru pole

Quadrupole

J=oE+ov x B,

where the second term represents the "motional" induc-tion. v is the local velocity of the fluid. It can be shownthat (34) accounts fully for the electromagnetic effectsof motions in a fluid conductor, apart from entirelynegligible relativistic terms. If (34) is substituted into(31) we have in (30) and (31) the full description of theelectromagnetic 6eld in the moving conductor. It shouldbe remarked that these equations do not change if we gofrom a system at rest to a uniformly rotating system of

' J. A. Stratton, E/ectromagnetic Theory {McGraw-Hi11 BookCompany, Inc. , New York, 1941), Chapter 7.

O'AL TF R. E LSASSE R

BS/N = V & (v ic 8)+(4n tier) 'v28. (35)

reference; hence it is for most purposes adequate to usecoordinates that are fixed relative to the rotating mantle.

The phenomena described by these equations werefirst noticed in connection with solar and sunspotmagnetism (Cowling, 1934; Alfven, 1942). They can bereproduced in the laboratory only under very favorableconditions, the reason being that in the cosmic andgeophysical phenomena the free decay periods of theelectric currents are large compared to the periods of themechanical motions, whereas the converse would be thecase in bodies of conventional size. Only quite recentlythe coupling between a magnetic field and sound wavesin mercury has been observed in the laboratory (Lund-quist, 1949).

In a special case these equations can at once beintegrated by conventional methods. If the Quid Movesin a large constant field, we may insert the correspondingvalue of 8 into (34) and interpret the current in (31) asdue to a given impressed e.m. f. Along these lines it ispossible to account for the secular variation of theearth's magnetic field as a perturbation of the main fieldcaused by the motion of the Quid in the top layers of thecore (Klsasser, 1946; Bullard, 1948).As the fluid moves,secondary electric currents are induced in it; these inturn give rise to the secondary non-dipole field and itssecular variation. Without going into details it might besaid that such an analysis verifies the interpretation ofthe centers of the secular variation as hydrodynamicentities, specifically, as regions of convergence anddivergence of the Quid in the top layers. It may well beassumed that they are at the same time vortices, but thecirculating motion of a vortex parallel to the surfacegives rise to a field of toroidal character on the insidethat does not have a counterpart outside of the core.What we perceive are the effects of convergence anddivergence which are connected with such vortices ongeneral dynamical grounds.

One can estimate the magnitude of the Quid velocitiesrequired to produce the observed secular variation. Thiscan be done in two independent ways. First we can usedirect inspection. The westerly drift motion of 0.18' peryear is typical of the rate of change of the geometry ofthe secular pattern; on the equator at the boundary ofthe core this represents a linear velocity of 0.034cm/sec. Secondly, one obtains a velocity from the induc-tion mechanism. Figure 12 gives values for the currentdensities pertaining to the secular variation in a two-dimensional sheet near the boundary of the core. As atypical average we may take j= 2 amp. /cm. By (34) wemay write for the order of magnitude of the velocities,v 47jr/doB, where d is the skin-depth; from (28) wehave d=50 km. Taking 8=4 gauss at the boundary ofthe core we find v=0.04 cm/sec.

Ke shall now obtain some more general resultsregarding the interaction of the moving Quid with themagnetic field. In place of (32) we have in the movingQuid

E. =4x'IJ, o I.v, (36)

where as usual I. is a representative length, and v is arepresentative velocity. Setting in the customary way1.=28 and taking v=0.04, we find E =1060. If v isactually larger inside the core, as we shall find reason tobelieve later on, E will be quite large. Since Rmeasures the ratio of the first to the second term on theright-hand side of (35), the dissipative term is small andmay be omitted in a first approximation. A knownvector identity then gives

88/Bt+(v v)8=(8 v)v —8(v v).

We may safely drop the last term, considering the Quidas incompressible. The left-hand side is nothing but the"substantial" derivative in the usual hydrodynamicalsense, thus

dS/Ct=(8 v)v. (37)

These fundamental equations can be given a simplephysical meaning. They are analogous to the well-known Helmholtz equations for the conservation ofvorticity in an incompressible Quid; they become indeedidentical with the Helmholtz formulas if B designatesthe vorticity. The proof of Helmholtz's main statementsis independent of the functional relation betweenvorticity and velocity (the former being the curt of thelatter); hence these theorems are true for an arbitraryvector fulfilling (37). We can enunciate them by re-placing the usual vortex lines and vortex tubes ofhydrodynamics by the lines or tubes of the magneticQux. The theorems then say in the first instance thatlines of force can neither be created nor destroyed in theQuid; as the Quid moves, the lines of force move so as toremain attached to the individual particles of the Quid.(Alfven used these results but did not provide a formalproof. ) In the second place one can show (Cowling,1934) that

JI S„dS=const. ,

where this is true in the sense that the substantialderivative, d/dt, of this integral vanishes, and also in thesense that the magnetic Qux remains constant along atube whose boundary is formed by the same lines offorce. All this is true, of course, only in the approxima-tion in which free decay is neglected. If free decay is

Fro. 15.

I

l

I

The first term on the right-hand side describes themotional induction, the second the eGects of free deray.In order to estimate their relative proportion in bodiesof diferent size we might, following Bullard (1949), usethe principles of dimensional similarity. Ke may definea "magnetic" Reynolds number as

EARTH'S INTERIOR AN D GEOMAGNETI SM

taken into account, its eBect upon the magnetic lines offorce is equivalent to the eGect of viscous friction uponthe vortex lines of an incompressible fluid (with 1/4' paas "magnetic" viscosity).

Before proceeding to applications of (37) we may notethat our equations imply an exchange of energy betweenthe field and the fluid motion. From (35) one finds aftersome calculations, on omitting the free decay terms,

—', (88'/Bt) = v (B x y x 8), (38)

where the term on the left is clearly 4' p, times the rateof change of the magnetic energy density; this quantitycan assume either sign, because the sign of the right-hand side depends on that of v. Since the Quid motionacts on the field, there must clearly be also an action ofthe 6eld upon the motion; this is the ponderomotiveforce,

F= Jxm= —(4'p)

—'8 x V x 8The work done on the Quid by this force per unit time,v F, is the negative of the change in field energy (38) asit must be.

%hen energy is transferred from the moving Quid tothe 6eld we may use the term "amplification. " Thesimplest such occurrence is shown schematically inFig. 15.Assume that there is originally a magnetic 6eldin the y direction (dashed lines) and that at time t =0 aQuid motion is started that is in the x direction but hasa gradient in the y direction as indicated by the velocityprofile at the left of the 6gure. In the absence of freedecay the lines of force become slanted as shown; an xcomponent of 8 is generated whose magnitude increasesproportional to t. Eventually the Quid motion will beslowed down by the ponderomotive forces exerted bythis field. Variations of this simple amplifier mechanismmust play a fundamental role in the dynamics of thecore, as may be seen as follows. The existence of thewesterly drift motion of the secular variation indicatesin all probability that the core does not rotate uni-

formly, as a rigid body„but that the angular velocityinside the core is variable with the depth. Such a motionwill deform the magnetic lines of force of the dipole fieldin a manner schematically shown in Fig. 16. Assumethat we 6rst have a dipole line of force in a meridionalplane (dashed hne in Fig. 16).Assume then for simplicitythat the inner part of the core rotates relative to theouter, the latter remaining fixed. The lines of force nowget dragged around the circles of latitude as shown, andif this non-uniform rotation continues long enough, the

FIG. 16.

lines of force are "wrapped around" the axis of rotationin nearly a system of circles. A more detailed analysis(1947) shows that if the initial field has the symmetry ofa magnetic dipole mode, the toroidal field generated isan electric quadrupole mode whose lines of force arecircles centered on the axis of rotation, the directions ofthe toroidal field vector being opposite in the twohemispheres. The most significant feature of this par-ticular mechanism is its lack of reciprocity: there is nosimilar interaction whereby, starting from a toroidalfield, one can produce a poloidal field of the type repre-sented by the original dipole. To prove this we note thatthe lines of force of the toroidal 6eld are circles centeredon the axis; it is clear that any rotationally symmetricalmotion transforms a family of circles into another suchfamily; hence it transforms any toroidal field again intoa toroidal field. Since the amplifier mechanism describedis very powerful, one can surmise the existence of atoroidal field that is larger than the dipole field. Bullardhas calculated the equilibrium between induction anddecay of a toroidal held produced by the mechanism ofFig. 16, the primary dipole 6eld being kept fixed. Forsimplicity he assumes a discontinuity rather than agradual change of angular velocity. He finds that therate of amplification is remarkably insensitive to thedetailed structure of the dipole field as well as to the sizeof the smaller volume that rotates relative to the outerpart of the core.

The preceding results suggest the idea that the whole

magnetic field of the earth is ultimately produced andmaintained by a mechanism of induction, the magneticenergy being drawn from the kinetic energy of the Quid

motion. This could be achieved if one could construct a"feed-back" mechanism whereby a part of the magneticenergy of the toroida1. field is returned to the originaldipole field in such a way that it reinforces the latter. Inview of the theorem stated in the preceding paragraphsuch a mechanism, if it exists, must lack rotationalsymmetry about the earth's axis. Hence one might ex-pect that the direct process of increasing the toroidalfield as shown in Fig. 16 is the main step of ampli6cationwhere most of the magnetic field energy is generated.From this viewpoint it becomes probable that thetoroidal 6eld is at least several times larger in theaverage than the observable dipole field. The figure forthe thermal dissipation of the magnetic field given in thebeginning of this chapter was estimated under theassumption of an average fieM in the core of 35 gauss.

If the non-symmetrical interactions are investigated(1946—47) they turn out to be very numerous but also inthe main very complicated. Bullard has found a simplemodel that involves only the superposition of two typesof motion. One of the two components of motion is thenon-uniform rotation about the axis, the other consistsof two ascending streams centered on diametricallyopposite points on the equator together with twodescending streams centered on points on the equator

%ALTER M. ELSASSE R

midway between the former. One finds by a geometricalanalysis that the combined motion will deform thetoroidal field in such a way that part of it is returned tot.he dipole mode with the correct sign so as to re-inforceit; this e6'ect takes place whatever the sign of thesubsidiary component of the motion.

If such a self-arnplificatory model of the earth's fieldis to be truly satisfactory it must exhibit a high degreeof stability so as to be able to account for the mainte-nance of the field over long periods. This requiresconsideration not only of the induction process but alsoof the ponderomotive forces. It would be very diS.cultindeed to prove stability for a specific model. Stabilitycan, however, be of a different type: it can consist in theaverage recurrence of certain patterns, that is, it can bestatistical. We shall very brieQy outline some statisticalaspects of the problem of amplification. The magneticcharts of the preceding chapter give evidence of a greatdeal of irregular motion in the core which, on empha-sizing its random character, we may designate as large-scale turbulence, a term commonly used in this sensein dynamic meteorology to designate the ensemble ofmajor atmospheric perturbations.

First, let us ask whether there exists a correlation be-tween the local velocity and the local magnetic field. By(34) the magnetic induction vanishes when v and 8 areeither parallel or antiparallel. By (39) the same is thecase for that part of the ponderomotive forces that isgenerated by the induction process. Considering thesame problem for sunspots, Gurevitqh and Lebedinsky(1946) have concluded that there must exist a verystrong correlation between field and velocity wheneverthe velocity is so large that free decay eAects becomenegligible. The streamlines and the lirM;s of magneticforce will tend to be parallel (or antiparallel) to eachother. This simple principle might be said to give us azero-order approximation to the relationship of motionand field. If the two vectors are not parallel or anti-parallel, energy transfer will in general take place andthis, as we have seen, can readily occur in eitherdirection.

It might be expected that the "spectrum" of theturbulent motion must have a fairly well-defined upper"cut-off." By virtue of (36) we can assign to eachvelocity a length, (4vrpov) ', such that for eddies smallerthan this critical size the magnetic quasi-viscositypredominates and tends to damp out the smaller eddies.This is in complete analogy to the action of mechanicalviscosity in ordinary turbulence. For the velocity givenabove, of 0.04 cm/sec. , the corresponding size is about10 km, much smaller than the observed elements of thesecular variation. Much of the fine detail of the secularvariation, however, if not almost all of it, is no doubtwiped out by the shielding action of the weakly con-ducting mantle.

The details of the turbulent pattern will depend onthe particular model of convective transport. If the heatis supplied by the central body, there must be a pro-

nounced increase of convective activity with depth inlayer E.The velocities should be larger at greater depthsand the eddies smaller. It might be noted that such amodel would also permit us to account for the pro-nounced preponderance of the earth's dipole terms (seeTable IX). The screening action of the outer layers ofthe core tends to suppress all but the lowest harmonicof the field originating at great depth. The observedhigher harmonics, from the quadrupoles on, can nodoubt be attributed to the motions in the top layers ofthe core.

VVe have seen that thermal dissipation of the currentsis small owing to the large numerical value of (36).Frictional dissipation in the Quid may be compared tothe electromagnetic dissipation by forming the non-dimensional quantity

R„/RI,= 47rpn(v/p), (40)

[pv'/2]A, ——[B'/Sm p],„. (43)

While this does not of course constitute a formal proof,it makes the existence of equal amounts of energy in theQuid and field rather plausible. It should be noted that

where E& is the usual "hydrodynarnical" Reynoldsnumber, and v/p the specific viscosity. If v is comparableto unity as it is in the liquids of simple molecularconstitution at laboratories pressures, then (40) isnumerically small, indicating that the heat of mechanicalfriction is small compared to Joule s heat. As all dissi-pative eSects are small compared to the mutualinteraction of Quid and field, it follows that the genera-tion of motion by thermal inhornogeneities of the Quid,which in the stationary state equals the dissipation,must also be small. Hence the core forms a nearly closedelectro-mechanical system. The turbulent motion maythen be expected to distribute the available energyin some definite statistical fashion over the accessibledegrees of freedom, mechanical and electromagnetic. Inorder to obtain an estimate of this distribution we con-sider the hydrodynamic equations of motion in whichthe ponderomotive forces (39) appear on the right-handside,

p8v/at+p(v v)v= —Vp —(4sp)-'Bx(~xS). (41)

On the left-hand side we use the vector identity

(v W)v=-,'V(v') —v x (~ x v).

We have neglected the frictional forces, and in thisapproximation we may put p= const. We now considerstationary isotropic turbulence and form a suitableaverage over time and space. The term Bv/Bt and all thegradient terms may be assumed to average to zero a,nd(41) becomes

p[v x (~ x v)]A„=(4mp)—'[8 x (~ x B)]A„. (42)

The simplest way of fulfilling (42) is clearly to assume"equipartition, "

EARTH'S INTERIOR AN D GEOMAGN F TISiQ 33

from the viewpoint of the present theory the fraction ofmagnetic 6eld energy outside the core itself is verysmall.

Taking 8=35 gauss in (43) we obtain an averagevelocity of 3 cm/sec. Since this is about 100 times theresult of the preceding estimates, the discrepancy isserious. It is unlikely that the mean value of 35 gauss isoverestimated by more than a factor of two; hence theprevious estimates must be too small. In the case of thewesterly drift motion it is indeed plausible that thisvelocity component is much smaller than the averageturbulent velocity. The drift motion must needs be of atransient and temporary nature. This follows from thefact that a differential rotation of core and mantleengenders a frictional and an electromagnetic torquewhich tend to equalize the angular velocities. Such adifferential rotation could only be maintained in thelong run by some perturbation effect of the moon, but itcan be established that all lunar sects are too small toproduce diGerential rotation of this magnitude. Hencethe westerly drift must be in the nature of a transient(perhaps comparable in its general character to thepresent-day decrease of the earth's magnetic dipolemoment). Hence we might well expect other velocitycomponents to be larger than those given by the drift.

As far as the second method of estimate, from thecurrents of Fig. 12, is concerned there are several reasonswhy the result might be considerably too small. In the6rst place this method gives us only the velocity com-ponent perpendicular to the magnetic lines of force; itsays nothing about the parallel component which shouldbe much larger. Secondly it is quite possible thatvelocities in the layer immediately adjacent to theboundary, which are the ones to be seen in the secularvariation, are smaller than the velocities in the freeliquid at greater depth, especially if a convective modelof the core heated from the inside holds. Finally, thepl'ojection from the surface of the earth to the boundaryof the core as carried out by spherical harmonic analysismight give currents that are too small since much detailis lost in the process.

Vote added Decenzber, le%.—Recent quantitativestudies carried out by C. Swift in this laboratory (to bepublished later) lead to the result that a series ofspherical harmonics is indeed inadequate for such pro-jection owing to its slow convergence. A satisfactorymethod of projecting the relatively localized features, asshown in Vestine's maps, has been developed. One findsthat the centers of the secular variation are stronger atthe boundary of the core than at the earth's surface by afactor of the order of ten. This should remove so muchof the above-mentioned discrepancy that the remainderis no longer very serious.

A model of isotropic turbulence would leave out ofaccount the earth's rotation. One might expect arealistic dynamical model of the core to be somewhereintermediate between isotropic turbulence and simpleamplification of the toroidal 6eld by non-uniform rota-

tion. Hence one should make allowance for the effect ofthe earth's rotation upon the turbulence motion. Let sbe the distance from the earth's axis. If s changes for agiven fluid particle, the latter will tend to conserve itsangular momentum, ps'co. For infinitely rapid turbulentmixing the fluid would approach a distribution ofangular velocity, ps're= const. , in which the outer partsrotate much slower than the inner parts of the core.Owing to the large electromagnetic and turbulentstresses the actual angular velocity gradient must bevery much smaller than this limiting value. It is seenthat any convective type of motion engenders ofnecessity a non-uniform rotation of the kind required toamplify the toroidal 6eld. The non-uniform rotation is ofthe same sign as the westerly drift but, for the reasonsindicated, the latter is likely to be a transient Quctua-tion, appearing near the surface, of a larger relative in-

crease of angular velocity at greater depth.Figure 17 shows the direction of the Coriolis forces

acting on a typical eddy which is here drawn inclinedrelative to the earth's axis. The solid line is a streamline(and also, in s. crude approximation, a magnetic line offorce). The Coriolis forces reverse their sign when thedirection of the motion is reversed. It is seen that there

Fzo. 17.

are now forces in the meridional plane which will giverise to motions and magnetic lines of force in theseplanes. In a theory of the maintenance of the 6eld it mill

have to be shown that the meridional components of theeddy 6eld do not average out, but have an excess in sucha direction as to reinforce the original dipole 6eld. Fromthe very simple notions given here to such a statistic-dynamical theory there is no doubt a long route v hichhas not yet been traveled. The general view which we

try to express is that the true state of the core is

probably intermediate between the large-scale modelscontaining the toroidal field and small-scale convectiveturbulence. Bullard's success with a large-scale model offeedback qualitatively worked out, is encouraging. Ifthe general theory of the state of the core as given hereis accepted on the basis of its agreement with a largenumber of geophysical data, then there must necessarilybe magnetic fields in the moving metallic Quid, but thesecould a Priori be in random directions. The existence ofthe main dipole 6eld might be taken as evidence for aline-up of eddy 6elds along the earth's axis, and there ishardly any other way this can be achieved than by theeGect of the Coriolis force. It might of course be that themagnetic field itself acts upon the Quid in such a way as

%VALTER M. E LSA SSE R

to be self-stabilizing once it has come into existence, butthis does not seem very likely. In a model with turbulenteddies there is evidently much leeway for changes inmagnitude and direction of the earth's field during itspast history. It seems therefore to be in good over-allagreement with the observational evidence as sum-marized previously.

The theory of magnetic fields in moving, electricallyconducting media forms a broad chapter of dynamics; itassumes hardly more than that the eGects of free decaybe not large compared to the motional induction.Cowling's studies of the sun indicate that a celestialbody of appreciably larger size than the earth willpractically never lose a magnetic field by the action ofdecay. There is, on the other hand, no limit to the rateof inductive amplification other than the velocity of theQuid. Thus magnetic fields should be a common phe-nomenon in the universe, and this is in agreement withrecent astrophysical findings (Babcock 1947—8). Large-scale motion of Quids is always turbulent, and practicallyall Quid matter in the universe is hot enough to be agood electrical conductor. Hence turbulent motionsshould be unstable until magnetic fields have beengenerated of the general order of magnitude given by(43). It would appear, therefore, that the earth's mag-netic field is a sample, close at hand, of a widespreadphenomenon of great cosmological interest.

ACKNOWLEDGMENTS

I am greatly indebted to a number of colleagues ofwide experience in geophysics with whom I was able toexchange views during the writing of this article. Drs.K. C. Bullard and Francis Birch have kindly read thedraft of the manuscript and have supplied numerousvaluable comments. Dr. B. Gutenberg has done thesame for the chapter on seismology. Several staff mern-bers of the Carnegie Geophysical Laboratory have dis-cussed with me pertinent chemical questions. Dr. P. %'.Bridgman and Dr. J. von Neumann have contributedvaluable suggestions. Dr. E.H. Vestine of the CarnegieDepartment of Terrestrial Magnetism has kindly sup-plied me with the originals of Figs. g—12. Dr. John vonNeumann, as supervisor of the Navy project mentionedon the title page, has been unfailingly cooperative inmaking the facilities of the project available.

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