Proc. Nat. Acad. Sci. USAVol. 71, No. 11, pp. 4400-4403, November 1974

Alignment of the Earth's Magnetic Field with the Axis of Rotation andReversals of Polarity: Laboratory Experiments on a Mechanism

(magnetic screening/rotating screen/anisotropic screening)

H. R. CRANE

Department of Physics, University of Michigan, Ann Arbor, Mich. 48104

Contributed by H. R. Crane, August 23, 1974

ABSTRACT A mechanism that can cause the earth'sexternal magnetic field to be aligned with the axis ofrotation and to reverse at random times is described Itdepends upon two arbitrary assumptions: (a) A dipolemagnetic source, of unspecified nature, deep within thecore, wanders randomly in direction. (b) The conductingfluid at the outer boundary of the core circulates in apattern that is symmetrical with respect to the earth'saxis of rotation. It is shown that such a circulating layerwill act as an anisotropic screen, which will suppress thefield of the transverse component of the source dipole.The field observed outside the core will be mainly that ofthe axial component of the source, and it will reverseabruptly whenever the direction of the source crosses theequatorial plane. Quantitative experimental studies, madeon small-scale models, of the effects and their propertiesare described. The only datum that even suggests a valuethat may be used for the angular velocity of the circulatingouter layer with respect to the source is the angularvelocity of the westward drift of the earth's nondipolarfield. If that value is used, the anisotropic screening effectcomes out to be strong enough to give alignment andreversal characteristics that are similar to those foundfrom paleomagnetic studies.

The idea that the earth's magnetic field is maintained bysome kind of self-excited dynamo process, involving motionof the conducting fluid of the core in its own magnetic field,has been accepted for a long time. The problem as to what thepattern of flow of the fluid must be in order to be self-excitingand to generate the observed external field has not beensolved. Characteristics of the external field, which include itsapproximate alignment with the earth's axis of rotation andits rather sudden reversals at random intervals, are puzzleswhose solutions will, it has been assumed, have to followthe understanding of the dynamo itself.Many efforts have been made to work out models for what

goes on in the earth's core, and thereby to account for thefeatures observed at the surface. The literature is far tooextensive to be summarized here. There are excellent reviewarticles and monographs, examples of which are cited (1-4).The problem is, first of all, one of magnetohydrodynamics.The available criteria indicate that the coupling of one partof the core to another through magnetic forces is strong. Ifthat is true, it directs one to seek a solution for the system asa whole. Such a solution, ideally, would contain not only thedynamo mechanism but all the secondary features such as thereversals and the alignment. Unfortunately, the solution ofthe t'otal problem in magnetohydrodynamics is extremelydifficult, perhaps insurmountable. It has been attacked as a

total system; also the expedient of dividing off parts of the

problem has been used. But results, in terms of the effort,have been meager.

In this paper I examine a model that does not conform tothe principle of strong coupling throughout the core, in thatthe process is treated as separable into parts in considerabledegree. There is no apparent way to justify such a treatmenta priori, and no attempt is made to find a way. The end resultalone makes the model interesting and worthy of description:it predicts a behavior for the alignment and reversals that issimilar in character to that known from paleomagnetic rec-ords, and it does so on values of the physical parameters thatare thought to be within reasonable bounds. Therefore, themodel may, even if totally on the wrong track, contain someuseful suggestion for a problem that is otherwise baffling. Thestarting assumptions are: (a) The source of magnetism is adipole, residing (or generated) in a region deep within thecore, and its direction can wander at random with respect tothe direction of the earth's axis of rotation. (b) The outermostlayer of the liquid core circulates, relative to the deeper part,in some pattern that is symmetrical about the earth's axis ofrotation (as might be expected as the result of the Coriolisforce acting upon liquid that is undergoing upward or down-ward convection). With the help of experiments I show howthe circulating outer layer will have the effect of an aniso-tropic screen which will suppress the field of the transversecomponent of the dipole only, and how, consequently, thefield observed outside the earth will be that of an axial dipolewhich will reverse abruptly whenever the direction of thewandering internal dipole crosses the equatorial plane.

EXPERIMENTS AND THEIR INTERPRETATION

A demonstration

Since the anisotropic screening effect is not generally familiar,I will describe an experiment that shows it in its simplestform. A hollow copper "can" of diameter and height about7 cm and wall thickness 0.48 cm was mounted so that it couldbe spun on its axis of symmetry, at speeds up to 60 revolu-tions/sec. Inside, at the center, a rod-shaped permanentdipole magnet, 2 cm long, was mounted so that it could befixed at any angle to the axis of rotation of the can. The can

was closed except for a hole at one end through which a

spindle projected, to hold the magnet. The magnet was, ofcourse, stationary in laboratory coordinates, while the can

rotated. The field intensity outside was measured with a Halleffect magnetometer. With the dipole set transverse to theaxis of rotation, the plots of the external field shown in Fig.

4400

Earth's Field Alignment and Reversals 4401

60 30 0° 330 300

FIG. 1. Polar plots of the radial magnetic intensity in the

plane normal to the axis of rotation of the can. The speeds of

rotation of the can are, reading clockwise, 0, 26, and 52 revolu-

tions/sec. The sense of rotation of the can was clockwise. Points

represent experimental data.

1 were obtained. It should be noted that besides the decrease

in intensity with increasing speed of rotation, the field pat-

tern was displaced in azimuth. When the dipole was oriented

parallel to the axis of rotation, the external field was un-

affected by the rotating can, to within the accuracy of mea-

surement, which was 0.5%.The behavior may be understood in several ways. The

absence of any effect on the magnetic field when the dipole isparallel to, and on, the axis of rotation of the can is under-stood by noting that the magnetic flux linking any closed

path in the can is invariant under the rotation. Therefore, no

eddy currents are generated. A more general way to view it is

from frame of reference of the rotating can. If the dipole is

parallel to the axis, the can "sees" only a static dipole. If the

dipole is transverse to the axis the can sees a rotating dipole,which is equivalent to the combination of two oscillating

(ac) dipoles at right angles to each other and the axis of rota-

tion, and 900 different in phase. The problem thus reduces

to the standard one of an oscillator screened by a metal en-

closure, for which formulas are available. In addition to the

attenuation of the field intensity, a phase lag is introduced. In

the frame of reference in which the screen is rotating and the

dipole is stationary, this shows up as an azimuthal displace-

ment of the field pattern, just as we observed in the demon-

stration. The quantitative side of the problem will be de-

ferred until later. We are especially interested in the case in

which the dipole is inclined to the axis, and is stationary while

the can rotates. It follows from the above that only the trans-

verse component will be screened; therefore, as the angular

velocity of the can increases, the apparent direction of the

dipole, as observed from outside, becomes more and more

nearly parallel to the axis of rotation. The rotating screen

may be thought of as a filter, letting out preferentially the

axial dipole component.The problem of the liquid screen

A crucial question that must be met before going further is

whether the effect just described is valid when the screen is

not a rigid body, but is a fluid in which there is shear, as itwould be in the earth's core. The possibilities, both theoreti-cal and experimental, for answering this question are quiterestricted. First, only cases in which the shear (or flow) pat-tern is symmetrical about an axis can be handled; all othersare too complicated. But this happens to correspond to theconditions of the model we are exploring. The case in whichthe shear pattern is symmetrical about an axis and is in thepresence of a poloidal magnetic field that is symmetricalabout the same axis is well understood. Toroidal fields aregenerated, which are totally inside the conductor. Further,as proven by Cowling (5) and generalized by Backus andChandrasekhar (6), the toroidal fields, being everywhereorthogonal to the poloidal field, cannot reinforce (or modify)the latter. It follows from this that axially symmetric shearin the screen cannot affect the transparency of the screento the field of an axial dipole. There is no corresponding theoryfrom which to infer what the effect of axially symmetric shearwould be in the case in which the dipole is aligned transverseto the axis, which of course is the other part of the anisotropicscreening effect. I have therefore done an experiment on sucha case. In addition, I have checked the case of the axiallyaligned poloidal field. These two experiments are describednext.The ideal experiment would be done with a screen con-

sisting of a vessel of conducting fluid, stationary at the centerand rotating at the outer boundary. But, due mainly to thevery poor conductivity of mercury, which is the best avail-able conducting fluid, a laboratory table-size experiment is notfeasible. Instead, differentially rotating copper cylinders, madeelectrically continuous by a thin mercury film, were used.The apparatus for the first experiment was a modification

of the rotating can. It was a can within a can, separated by athin (0.012 cm) mercury film. The maximum speed of revolu-tion, 20 revolutions/sec was only 1/3 that used with the singlecan; nevertheless, measurements of the attenuation could bemade to about ±2% and of the azimuthal displacement toabout ±5%. The internal, toroidal, field was not measureddirectly, but was calculated. It was approximately equal tothe poloidal field. Measurements of the attenuation and azi-muthal displacement with and without the mercury werecompared. The results, for the magnet both axially andtransversely oriented, were the same with and without themercury, which is to say, with and without the toroidal field.The apparatus for the second experiment was two solid

copper cylinders, each 15 cm in diameter and 7.5 cm high,one on top of the other, separated by, and lubricated by,mercury which was fed in at the center under pressure. Thetop cylinder rotated at 20 revolutions/sec. A dipole magnetabove the top cylinder, parallel-to and on the axis, suppliedthe poloidal field. The toroidal field was measured in a smallhole in the copper near the mercury film. It was three timesthe poloidal field at the same place. The field was measuredall around the cylinders when (a) both were stationary, (b)one was rotating and supported on an air film so that therewas no toroidal field, and (c) one was rotating, with the mer-cury film (and the toroidal field) present. The external fieldwas the same in all three cases to within the accuracy of themeasurement, which was about ±t0.5%. The absence of aneffect is in accord with the theorems of Cowling and of Backusand Chandrasekhar.The anisotropic screening effect will be expected to hold

valid in the presence of axisymmetric shear provided (a) the

Proc. Nat. Acad. Sci. USA 71 (1974)

Proc. Nat. Acad. Sci. USA 71 (1974)

shear does not destroy the screen's transparency to the fieldof an axial dipole, and (b) it does not destroy the screen'sopacity to the field of a transverse dipole. The first of these isshown quite well by both theory and experiment. The secondis shown by the double can experiment, but only for a particu-lar configuration. However, there is little basis for thinkingthat in any case shear will render the screen transparent tothe field of the transverse dipole.

Quantitative measurements of screening,especially boundary effects

Finally, the screening of the transverse dipole must be ex-amined quantitatively. The screening of an oscillator by ametal box is a standard engineering problem (7). The rule-of-thumb formulas are: an intensity attenuation factor, exp-(- Aa/6) and a phase shift Aa/6 radians, where Aa is thethickness of the screen and a is the skin depth.* The formulaswill be recognized as those for the attenuation and phasechange of a plane wave in an infinite conducting medium,where a is the coordinate along the direction of propagation(8). They are presumed to hold well enough for practicalapplications if the radii of curvature in the enclosure are largecompared to a and if Aa is larger than 6. There are additionalfactors, which are sometimes lumped under the heading ofreflection from the inner wall of the enclosure (9). In thepresent experiments they are best thought of as being associ-ated with several coupling coefficients: source to sensor beforeinsertion of the screen; source to screen and screen to sensorafter insertion. We may write the reduction in intensity andphase shift as R exp(- Aa/6) and S + Aa/b. Needless to say,R and S are dependent upon the particular geometry and arenot readily calculated. Values for some situations of engi-neering interest are available, but for sets of parameters thatmight apply (with change of scale) to the earth, little guidanceis to be found. I, therefore, had to try to get the necessaryinformation experimentally, and, for reasons already given,from rigid rather than liquid screens. Of especial interest arethe behavior of R under a change of scale and the sensitivityof R to the size of the source (coil) relative to the size of thecavity. The latter point is of concern because in the model weare considering, the source is a "black box" whose size mustremain unspecified. Partial answers, at least, are provided bythe experimental data that follow.Measurements over a sufficient range of angular velocities

were not feasible using the rotating can. Therefore, the frameof reference was changed to one in which the enclosure wasstationary and the dipole was rotating. A transverse rotatingdipole was made by a pair of small coils at right angles toone another, and carrying alternating currents at 900 phasedifference. The enclosures were long (effectively infinite)cylinders. The radial magnetic intensity and the phase of thealternating field outside were measured. (The latter wasdetermined by locating the azimuth at which the phase of thefield outside matched that of the current in one of the coils.)The ranges of the parameters were: frequency 25 Hz to 20kHz, wall thickness 0.17 to 2.2 cm, and Aa/6 up to 7. All thecylinders were copper except one, which was lead.

First, the attenuation and phase shift within the metal,without the boundary (reflection) effects, were checked. Thesewere separated from the boundary effects by the standard

TABLE 1. Test for the constancy of R under changes ofparameters, while holding the skin depth equal to 1/15 the inner

radius of the cylinder

RInnerradius Aa f Coil Coil

Cylinder (cm) (cm) (kHz) 1 2

A 3.17 0.478 0.975 0.281 0.225B 3.17 0.170 0.975 0.256 0.243C 3.13 0.303 12.8 0.235 0.235D 4.96 0.279 0.402 0.242 0.250E 1.59 0.318 3.90 0.263 0.216

Cylinder C is lead; all others copper. The radii of coils 1 and 2were 1.35 and 0.60 cm, respectively. R is the mea ured att- nu-ation divided by exp(- Aa/6).

substitution method, using pairs of cylinders having the samediameters and different wall thicknesses. On the incrementalbasis, the attenuation agreed with exp(- Aa/6) to within12% and the phase shift with Aa/b to within 3%, over thefull range measured, the increment in Aa/b running from 0.5to 4.5. The deviations of the incremental values from thosepredicted by the formulas were systematic, and could reason-ably be attributed to the curvature of the screen, the radiusof curvature of which was, typically, 10 to 205.The total phase shift was about 0.7 radian greater than

Aa/8, throughout the range of parameters. Since the phaseshift will not be used in the application to the earth, it willnot be pursued further here.The "reflection" part of the attenuation (R) was important

enough to warrant a test, such as could be made, of its scalingproperties. A general test is not possible, but we can at leastask if R will remain constant when the radius of the source(coil), the radius of the screen (r), and a are all multiplied bythe same constant. Aa need not be multiplied by the sameconstant because it enters only into the internal part of theattenuation, exp(- Aa/6), which is known. Anticipating thatthe ratio of r to 6, which will be of interest in the applicationto the earth, will be about 15, I made a set of determinationsof R, all with that ratio (Table 1).

Several points are of interest in Table 1. (a) R is not verysensitive to the ratio of the coil radius to the cavity radius.(Compare cylinder E, coil 1, with cylinder D, coil 2, wherethe ratio changes from 0.85 to 0.12.) (b) A large change inspecific resistivity, p (Cu to Pb, a factor of 13), makes littlechange in R. (c) Since R is less than unity, it enhances theanisotropic screening effect. The results test the scaling overranges in p and in ratio of coil and cavity diameters which seemto be satisfactorily large, for the application to the earth. Therange in r is of course miniscule, by the same comparison.Nevertheless, some confidence may be gained from the tests.

APPLICATION TO THE EARTH

Before applying the screening effect to the earth, it is necessaryto adopt effective values for the angular velocity and thethickness of the circulating fluid layer that does the screening.These will be effective values, because the actual angularvelocity certainly will have a dependence upon both latitudeand radius. The only available observation that even suggestsa magnitude for the angular velocity is that of the westwarddrift of the nondipolar component of the earth's field (10).That value, 0.18 degree/year, is used for the calculation. This,

* a = (l0p/47r2,f)'/2, where p is the specific resistivity (ohm cm),,u the magnetic permeability, andf the frequency (Hz). For copperat 20'C and 100 Hz, a = 0.662 cm.

AA.02 Geophysics -. Cmne

Earth's Field Alignment and Reversals 4403

0j

0 30 60 90 1 20 150 180

DIRECTION INSIDE, DEGREE

FIG. 2. Upper curve: Ratio of the magnetic moment of thedipole source to the apparent magnetic moment outside therotating screen, plotted against the direction of the source.

Lower curve: Direction of the dipole with respect to the axis ofrotation of the screen as it will appear outside, plotted againstthe direction of the source. Both curves are calculated for Aa/5= 3.

and the generally accepted value of about 3.3 X 10-4 ohm cm

for the specific resistivity, give a = 230 km, or about 1/15 theradius of the liquid core. The effective thickness of the screen-

ing layer, Aa, is arbitrary except that it should be a smallfraction of the radius of the core.The relations between the apparent magnetic moment of

the dipole and its direction, as seen outside the core, and thoseof the source, inside, can now be calculated. The results, forAa/6 = 3, are shown in Fig. 2. The values were calculatedwithout taking advantage of the "reflection" screening factor,R. If R is taken as 0.25 (Table 1) and is included, the depth ofthe circulating layer necessary to give the curves in Fig. 2 isreduced from 690 km to 370 km. It hardly needs to be re-

emphasized that these figures come from highly speculativeinput data.

The assumption that the magnetic source wanders indirection at random, combined with the relationships of Fig. 2,gives the result that the external field stays nearly alignedwith the axis of rotation most of the time, and that when itreverses it does so rapidly, accompanied by a decrease inapparent dipole moment. However, the sharpness of both ofthese changes leaves something to be desired. This will bediscussed in the following section.

DISCUSSION

In the sharpness, or lack of it, in the reversal and intensity dip,lies one of the few sensitive tests for any model, since these

characteristics are fairly well exhibited in paleomagneticrecords. If we term the field "aligned" when its direction iswithin 15° of the axis of rotation, then according to therecords of the past megayear or so (11) the field has beenaligned for 97-99% of the total time (my estimate). The samequantity, when computed from the data of Fig. 2 on theassumption of random wandering of the source direction,comes out to be less than 80%. This is a large discrepancy.However, the fraction is quite sensitive to Aa/5. Values of 4.5for Aa/6 and 0.25 for R would give 99%.The fraction of time spent aligned would also be sensitive

to any systematic departure from randomness in the wander-ing of the source direction. While it would be fruitless tospeculate on such a modification, there is one piece of factualinformation that came out of the experiments with the rotat-ing can and that may be worth simply noting. When themagnet was inclined to the axis of rotation of the screen, therewas a mutual torque tending to make the magnet and the axisof rotation of the can parallel. This introduces a bistableproperty to the system. When competing with randomdisturbing effects, it would have the result of increasing thefraction of time spent in the aligned condition.The lack of sharpness of the intensity dip (Fig. 2) is more of

a problem than is that of the reversal of direction, althoughthe paleoniagnetic data on intensity during reversals are lessdefinitive than those on direction (12). This would not bemade sharper simply by increasing Aa/6. An intensity satura-tion effect would flatten the curve and make the dip sharper,but at the present stage of knowledge, there would be nothingto justify introducing such a remedy.

I am indebted to Prof. G. W. Ford for many helpful discussions.

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3. Malkus, W. V. R. (1971) in Proc. Int'l School of Physics-Enrico Fermi (Academic Press, New York), Vol. 50, pp.38-50.

4. Rikitake, T. (1966) Development in Solid Earth Geophysics(Elsevier, Amsterdam, New York and London), Vol. 2,pp. 1-308.

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Acad. Sci. USA 42, 105-109.7. Terman, F. A. (1955) in Electronic and Radio Engineering

(McGraw Hill Co., New York), p. 36.8. Stratton, J. A. (1941) in Electromagnetic Theory (McGraw

Hill Co., New York), p. 504.9. Ficchi, R. F. (1964) in Electrical Interference (Hayden

Book Co., New York), pp. 30-33.10. Bullard, E. C., Freedman, C., Gelman, H. & Nixon, J.

(1950) Phil. Trans. Roy. Soc. London, Ser. A 124, 67-92.11. Ninkovich, D., Opdyke, N., Heezen, B. C. & Foster, J. H.

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Proc. Nat. Acad. Sci. USA 71 (1974)