Alignment of the Earth's Magnetic Field with the Axis of Rotation and Reversals of Polarity:Laboratory Experiments on a MechanismAuthor(s): H. R. CraneSource: Proceedings of the National Academy of Sciences of the United States of America,Vol. 71, No. 11 (Nov., 1974), pp. 4400-4403Published by: National Academy of SciencesStable URL: http://www.jstor.org/stable/64204 .

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

Alignment of the Earth's Magnetic Field i

Reversals of Polarity: Laboratory Experin (magnetic screening/rotating screen/anisotropic screenin

H. R. CRANE

Department of Physics, University of Michigan, Ann Arbor, Mich. 481

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

ABSTRACT A mechanism that can cause the earth's external magnetic field to be aligned with the axis of rotation and to reverse at random times is described, It depends upon two arbitrary assumptions: (a) A dipole magnetic source, of unspecified nature, deep within the core, wanders randomly in direction. (b) The conducting fluid at the outer boundary of the core circulates in a pattern that is symmetrical with respect to the earth's axis of rotation. It is shown that such a circulating layer will act as an anisotropic screen, which will suppress the field of the transverse component of the source dipole. 'The field observed outside the core will be mainly that of thle axial component of the source, and it will reverse abruptly whenever the direction of the source crosses the equatorial plane. Quantitative experimental studies, made on small-scale models, of the effects and their properties are described. The only datum that even suggests a value that may be used for the angular velocity of the circulating outer layer withi respect to thle source is the angular velocity of the westward drift of the earth's nondipolar field. If that value is used, the anisotropic screening effect comes out to be strong enough to give alignment and reversal characteristics that are similar to those found from paleomagnetic studies.

The idea that the earth's magnetic field is maintained by some kind of self-excited dynamo process, involving motion of the conducting fluid of the core in its own magnetic field, has been accepted for a long time. The problem as to what the

pattern of flow of the fluid must be in order to be self-exciting and to generate the observed external field has not been solved. Characteristics of the external field, which include its

approximate alignment with the earth's axis of rotation and its rather sudden reversals at random intervals, are puzzles whose solutions will, it has been assumed, have to follow the 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 the features observed at the surface. The literature is far too extensive to be summarized here. There are excellent review articles 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 part of the core to another through magnetic forces is strong. If that is true, it directs one to seek a solution for the system as a whole. Such a solution, ideally, would contain not only the dynamo mechanism but all the secondary features such as the reversals and the alignment. Unfortunately, the solution of the total problem in magnetohydrodynamics is extremely difficult, perhaps insurmountable. It has been attacked as a total system; also the expedient of dividing off parts of the

44

vith the Axis of Rotation and ients on a Mechanism g)

)4

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 to the principle of strong coupling throughout the core, in that the process is treated as separable into parts in considerable degree. There is no apparent way to justify such a treatment a priori, and no attempt is made to find a way. The end result alone makes the model interesting and worthy of description: it predicts a behavior for the alignment and reversals that is similar in character to that known from paleomagnetic rec-

ords, and it does so on values of the physical parameters that are thought to be within reasonable bounds. Therefore, the model may, even if totally on the wrong track, contain some useful suggestion for a problem that is otherwise baffling. The

starting assumptions are: (a) The source of magnetism is a dipole, residing (or generated) in a region deep within the

core, and its direction can wander at random with respect to the direction of the earth's axis of rotation. (b) The outermost

layer of the liquid core circulates, relative to the deeper part, in some pattern that is symmetrical about the earth's axis of rotation (as might be expected as the result of the Coriolis force acting upon liquid that is undergoing upward or down- ward convection). With the help of experiments I show how the circulating outer layer will have the effect of an aniso- tropic screen which will suppress the field of the transverse

component of the dipole only, and how, consequently, the field observed outside the earth will be that of an axial dipole which will reverse abruptly whenever the direction of the

wandering 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 simplest form. A hollow copper "can" of diameter and height about 7 cm and wall thickness 0.48 cm was mounted so that it could be spun on its axis of symmetry, at speeds up to 60 revolu- tions/sec. Inside, at the center, a rod-shaped permanent dipole magnet, 2 cm long, was mounted so that it could be fixed 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, of course, stationary in laboratory coordinates, while the can rotated. The field intensity outside was measured with a Hall effect magnetometer. With the dipole set transverse to the axis of rotation, the plots of the external field shown in Fig.

)0

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

180

120 240

90 ? ?)?_270

60 30 00 330 300

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 fieldl 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 is parallel 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 90? 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

Earth's Field Alignment and Reversals 4401

not a rigid body, but is a fluid in which there is shear, as it would be in the earth's core. The possibilities, both theoreti- cal and experimental, for answering this question are quite restricted. First, only cases in which the shear (or flow) pat- tern is sym]netrical about an axis can be handled; all others are too complicated. But this happens to correspond to the conditions of the model we are exploring. The case in which the shear pattern is symmetrical about an axis and is in the presence of a poloidal magnetic field that is symmetrical about the same axis is well understood. Toroidal fields are generated, which are totally inside the conductor. Further, as proven by Cowling (5) and generalized by Backus and Chandrasekhar (6), the toroidal fields, being everywhere orthogonal to the poloidal field, cannot reinforce (or modify) the latter. It follows from this that axially symmetric shear in the screen cannot affect the transparency of the screen to the field of an axial dipole. There is no corresponding theory from which to infer what the effect of axially symmetric shear would be in the case in which the dipole is aligned transverse to the axis, which of course is the other part of the anisotropic screening effect. I have therefore done an experiment on such a case. In addition, I have checked the case of the axially aligned poloidal field. These two experiments are described next.

The ideal experiment would be done with a screen con- sisting of a vessel of conducting fluid, stationary at the center and rotating at the outer boundary. But, due mainly to the very poor conductivity of mercury, which is the best avail- able conducting fluid, a laboratory table-size experiment is not feasible. Instead, differentially rotating copper cylinders, made electrically 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 a thin (0.012 cm) mercury film. The maximum speed of revolu- tion, 20 revolutions/sec was only 1/3 that used with the single can; nevertheless, measurements of the attenuation could be made to about ?2% and of the azimuthal displacement to about i5%. The internal, toroidal, field was not measured directly, but was calculated. It was approximately equal to the poloidal field. Measurements of the attenuation and azi- muthal displacement with and without the mercury were compared. The results, for the magnet both axially and transversely oriented, were the same with and without the mercury, 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. The top cylinder rotated at 20 revolutions/sec. A dipole magnet above the top cylinder, parallel-to and on the axis, supplied the poloidal field. The toroidal field was measured in a small hole in the copper near the mercury film. It was three times the poloidal field at the same place. The field was measured all around the cylinders when (a) both were stationary, (b) one was rotating and supported on an air film so that there was no toroidal field, and (c) one was rotating, with the mer- cury film (and the toroidal field) present. The external field was the same in all three cases to within the accuracy of the measurement, which was about ?0.5%. The absence of an effect is in accord with the theorems of Cowling and of Backus and Chandrasekhar.

The anisotropic screening effect will be expected to hold valid in the presence of axisymmetric shear provided (a) the

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4402 Geophysics: Crane

shear does not destroy the screen's transparency to the field of an axial dipole, and (b) it does not destroy the screen's

opacity to the field of a transverse dipole. The first of these is shown quite well by both theory and experiment. The second is shown by the double can experiment, but only for a particu- lar configuration. However, there is little basis for thinking that in any case shear will render the screen transparent to the 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 a metal 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 the thickness of the screen and 8 is the skin depth.* The formulas will be recognized as those for the attenuation and phase change 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 practical applications if the radii of curvature in the enclosure are large compared to 8 and if Aa is larger than 8. There are additional

factors, which are sometimes lumped under the heading of reflection from the inner wall of the enclosure (9). In the

present experiments they are best thought of as being associ- ated with several coupling coefficients: source to sensor before insertion of the screen; source to screen and screen to sensor after insertion. We may write the reduction in intensity and

phase shift as R exp(-Aa/8) and S + Aa/8. Needless to say, R and S are dependent upon the particular geometry and are not readily calculated. Values for some situations of engi- neering interest are available, but for sets of parameters that

might apply (with change of scale) to the earth, little guidance is to be found. I, therefore, had to try to get the necessary information experimentally, and, for reasons already given, from rigid rather than liquid screens. Of especial interest are the behavior of R under a change of scale and the sensitivity of R to the size of the source (coil) relative to the size of the

cavity. The latter point is of concern because in the model we are considering, the source is a "black box" whose size must remain unspecified. Partial answers, at least, are provided by the experimental data that follow.

Measurements over a sufficient range of angular velocities were not feasible using the rotating can. Therefore, the frame of reference was changed to one in which the enclosure was

stationary and the dipole was rotating. A transverse rotating dipole was made by a pair of small coils at right angles to one another, and carrying alternating currents at 90? phase difference. The enclosures were long (effectively infinite) cylinders. The radial magnetic intensity and the phase of the

alternating field outside were measured. (The latter was determined by locating the azimuth at which the phase of the field outside matched that of the current in one of the coils.) The ranges of the parameters were: frequency 25 Hz to 20 kHz, wall thickness 0.17 to 2.2 cm, and La/8 up to 7. All the cylinders were copper except one, which was lead.

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

* a = (109p/47r2uf)/2), where p is the specific resistivity (ohm cm), u the magnetic permeability, andf the frequency (Hz). For copper at 20?C and 100 Hz, 8 = 0.662 cm.

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

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

radius of the cylinder

R Inner radius Aa f Coil Coil

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

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

Cylinder C is lead; all others copper. The radii of coils 1 and 2 were 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 same diameters and different wall thicknesses. On the incremental basis, the attenuation agreed with exp(-Aa/6) to within 12% and the phase shift with Aa/8 to within 3%, over the full range measured, the increment in Aa/6 running from 0.5 to 4.5. The deviations of the incremental values from those predicted by the formulas were systematic, and could reason- ably be attributed to the curvature of the screen, the radius of curvature of which was, typically, 10 to 20 6.

The total phase shift was about 0.7 radian greater than

Aa/6, throughout the range of parameters. Since the phase shift will not be used in the application to the earth, it will not 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 scaling properties. A general test is not possible, but we can at least ask if R will remain constant when the radius of the source

(coil), the radius of the screen (r), and 8 are all multiplied by the same constant. Aa need not be multiplied by the same constant because it enters only into the internal part of the

attenuation, exp(- Aa/), which is known. Anticipating that the ratio of r to 8, which will be of interest in the application to the earth, will be about 15, I made a set of determinations of R, all with that ratio (Table 1).

Several points are of interest in Table 1. (a) R is not very sensitive to the ratio of the coil radius to the cavity radius.

(Compare cylinder E, coil 1, with cylinder D, coil 2, where the ratio changes from 0.85 to 0.12.) (b) A large change in

specific resistivity, p (Cu to Pb, a factor of 13), makes little

change in R. (c) Since R is less than unity, it enhances the

anisotropic screening effect. The results test the scaling over

ranges in p and in ratio of coil and cavity diameters which seem to be satisfactorily large, for the application to the earth. The

range 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 necessary to adopt effective values for the angular velocity and the thickness of the circulating fluid layer that does the screening. These will be effective values, because the actual angular velocity certainly will have a dependence upon both latitude and radius. The only available observation that even suggests a magnitude for the angular velocity is that of the westward drift of the nondipolar component of the earth's fieldl (10). That value, 0.18 degree/year, is used for the calculation. This,

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

w -

8c o

0 90

LI-

0 'l

0

0 30 60 90 120 150 180 DIRECTION INSIDE, DEGREE

FIG. 2. Upper curve: Ratio of the magnetic moment of the dipole source to the apparent magnetic mnoment outside the rotating screen, plotted against the direction of the source. Lower curve: Direction of the dipole with respect to the axis of rotation of the screen as it will appear outside, plotted against the direction of the source. Both curves are calculated for Aa/8 -3.

and the generally accepted value of about 3.3 X 10-4 ohm cm for the specific resistivity, give 8 = 230 km, or about /15 the radius of the liquid core. The effective thickness of the screen- ing layer, Aa, is arbitrary except that it should be a small fraction 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 those of the source, inside, can now be calculated. The results, for Aa/8 = 3, are shown in Fig. 2. The values were calculated without taking advantage of the "reflection" screening factor, R. If R is taken as 0.25 (Table 1) and is included, the depth of the circulating layer necessary to give the curves in Fig. 2 is reduced from 690 km to 370 km. It hardly needs to be re-

emphasized that these figures come from highly speculative input data.

The assumption that the magnetic source wanders in direction at random, combined with the relationships of Fig. 2, gives the result that the external field stays nearly aligned with the axis of rotation most of the time, and thait when it reverses it does so rapidly, accompanied by a decrease in

apparent dipole moment. However, the sharpness of both of these changes leaves something to be desired. cThis will be discussed 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

Earth's Field Alignment and Reversals 4403

characteristics are fairly well exhibited in paleomagnetic records. If we term the field "aligned" when its direction is within 15? of the axis of rotation, then according to the records of the past megayear or so (11) the field has been aligned for 97-99% of the total time (my estimate). The same quantity, when computed from the data of Fig. 2 on the assumption 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/i. Values of 4.5 for 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 to speculate on such a modification, there is one piece of factual information that came out of the experiments with the rotat- ing can and that may be worth simply noting. When the magnet was inclined to the axis of rotation of the screen, there was a mutual torque tending to make the magnet and the axis of rotatioln of the can parallel. This introduces a bistable property to the system. When competing with random disturbing effects, it would have the result of increasing the fraction of time spent in the aligned conditioi.

The lackl of sharpness of the intensity dip (Fig. 2) is more of a problem than is that of the reversal of direction, although the paleonlagnetic data on intensity during reversals are less definitive than those on direction (12). This would not be made sharper simply by increasing Aa/8. An intensity satura- tion effect would flatten the curve and make the dip sharper, but at the present stage of knowledge, there would be nothing to justify introducing such a remedy.

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

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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

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(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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