Gubbins D., Herrero-Bervera E. Encyclopedia of Geomagnetism and Paleomagnetism

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in th e sa me di rectio n as th e a ppl ied f ield an d t he field nea r th e in ner walls of

the cylinder will oppose the applied field. In the design of a magnetic

shield for the SRM the field change measured along the shield axis is nor-

mally used to determine the required shield length. The on-axis attenuation

of axially applied fields is given by 31

(x/r)

wh ere x is th e on-a xis d istan ce

into the cylinder from an open end and r is the radius of the superconduct-

in g s hi eld. The on -axis att enu atio n of f ield s a ppl ied tran sve rse t o t he shi eld

axis is given by 36

(x/2 r)

(Deaver and Goree, 1967). A typical design attenua-

tion for transverse fields is 10

and this gives a minimum shield length of

7.7 times shield radii. For this shield length and transverse attenuation of

the ax ial at tenu atio n w i ll be 3 1

ð7: 7r Þ =r

¼ 3 :0  10

.Thus,theaxial

attenuation of a cylindrical shield is orders of magnitude better than its

transverse attenuation, and it is important to place the SRM where the

amb ient m agn etic tran sve rse fi eld no ise is a m in im um.

Cryog enics

Another important performance variable in an SRM is the cryo-

genic environment. The SRM normally uses liquid helium as the

thermal environment. A few SRMs have used liquid nitrogen as a sec-

ondary coolant, but most have used superi nsulated dewar construction

(Kropschot et al ., 1968). In a superinsulated dewar the liquid helium

reservoir and SQUID volume are enclosed in a vacuum region sur-

rounded by many layers of aluminized mylar placed within this vacuum.

Thermally conducting shields are also placed within this region and

are cooled by utilizing the heat capacity of the evolving helium gas.

The first SRMs used about 3 l of liquid helium per day. Since the initial

development of the SRM cryocoolers have been incorporated into the

thermal design. The cryocooler is used to provide external cooling to

the thermal shields placed within the superinsulation. A typical two

stage cryocooler as used with the 2G Enterprises SRM will absorb

about 0.5 W of heat at 15 K and 15 W at 80 K (CTI model 350,

2004). The extra cooling provided by the cryocooler will reduce the

helium loss rate to below 0.1 l day

 1

in a properly designed dewar.

This greatly reduced loss rate has made it practical to use SRMs in

laboratories where liquid helium is difficult to obtain and/or is very

expensive.

It is also possible to use cryocoolers that will either reduce the

helium loss rate to essentially zero, or produce the required low tem-

perature without a reservoir of liquid helium. Cryocoolers that will

produce 4 K temperatures and/or liquid helium have been available

for many years but the cost and poor long-term reliability have made

them unacceptable for use with SRMs. Recent developments with

pulse tube cryocoolers appear to change this situation. The author, in

October 2004, began the development of a new magnetometer that

would not use any liquid helium. The first SRM using this new pulse

tube cryocooler technology (Cryomech , Inc., 2005) was completed

and delive red to Dr. J. Hus at Dourbes, Belgium in May 2005 (Institut

Royal Metrologiq ue, Belgium). SQUID and superconducting shield

temperatures of approximately 4.0 K were obtained from room tem-

perature to operation in about 3 days and the magnetic measurement

performance is identical to that achieved with the liquid helium cooled

systems. The pulse tube cryocooler has no moving mechanical parts

at low temperatures and is predicted to have a very long mean

time between failure, probably in excess of 15 years for the cold head,

10 years for the easily replaced drive motor, and about 10 years for

the compressor. The projected excellent reliability of the pulse tube

cryocooler has made it practical to use with this new SRM.

High tem peratur e supercondu ctors

The discovery of materials that become superconducting at “high ”

temperatures in 1986 (Bednorz and Muller 1986) caused much excite-

ment in all fields of superconductivity. If instruments could be built

with these new materials and operate at 70 –90 K the fabrication

and operating cost could be dramatically reduced. Unfortunately, the

high temperature materials that have been discovered to date (2004)

are very brittle and it has been difficult to make high temperature

superconduc ting wires and SQUIDs. In particular, wires have not been

produced where superconducting pickup coils and transformers can be

constructed (Kleiner et al., 2004).

William S. Goree

Bibl iogra phy

2G Ent erpri se s, G or ee, W.S ., at Wil li am S. G or ee, I nc ., 2 040 S uns et D ri ve ,

Pacific Grove, CA, 93950 (wgoree@wsgi.us), and Goodman, W.L.,

at Applie d Physics Systems, Inc., 1245 Space Park Way, Mountain

View, CA 94043 (goodman@appliedphysics.com).

Bednorz, J.G., and Muller, K.A., 1986. Z. Phys. B. 64 : p. 189.

Collinson, D.W., 1983. Methods in Rock and Paleomagnetism: Techni-

ques and Instrumentation. London: Chapman and Hall, 503 pp.

Cox, A., Doell, R.R., and Dalrymple, G.B., 1964. Reversals of the

earth ’ s magnetic field. Science , 144: 1537 – 1543.

Cryomech, Inc., 2005. 113 Falso Drive, Syracuse, NY 13211 (http://

www.cryomech.com).

CTI, 2004. Cryogenic Technology, Inc., a division of Helix Technology

(http://www.helixtechnology.com).

Deaver, B., and Goree, W.S., 1967. Some techniques for sensitive

magnetic measurements using superconducting circuits and mag-

netic shields. Review of Scientific Instruments, 38:311–318.

Fuller, M., 1987. Experimental methods in rock magnetism and paleo-

magnetism. Methods of Experimental Physics , 24: 303– 471.

Galbrun, B., At CEA, Grenoble, France.

Good, J., Cryogenic Limited (http://www.cryogenic.co.uk).

Goree, W.S., and Fuller, M.D., 1976. Magnetometers using RF-driven

SQUIDs and their applications in rock magnetism and paleomag-

netism. Reviews of Geophysics and Space Physics , 14 : 591 –608.

Helmholtz, H.V., 1849. Vortrag in der Sitzung , Volume 16, III. Berlin:

Physikalische Gesellschaft; Loeb, L.B., 1947. Fundamentals

of Electricity and Magnetism , 3rd edition. New York: John Wiley

& Sons.

Kleiner, R., Koelle, D., Ludwig, F., and Clarke, J., 2004. Supercon-

ducting Quantum Interference Devices: State-of-the-Art and Appli-

cations. Proceedings of the IEEE, Special Issue on Applications of

Superconductivity, Volume 92, October 2004, pp. 1534 – 1548.

Kropschot, R.H., et al., 1968. Technology of liquid helium. National

Bureau of Standards Monograph , 111 : 170.

Nagy, E.A., and Valet, J.P., 1993. New advances in paleomagnetic

studies of sediment cores using U-channels. Geophysics Research

Letters , 20: 671– 674.

ODP, 1985. Please see the ODP website: http://www.odp.edu/.

Tauxe, L., Labrecque, J.L., Dodson, R., and Fuller, M., 1983. “U”

channels—a new technique for paleomagnetic analysis of hydrau-

lic piston cores. Eos, Transactions, American Geophysical Union,

64: 219.

Vrba, J., CTF Systems, Inc., 9 Burbidge Street, Coquitlam, BC,

Canada Y3K7B2 (corp@vsmmedtech.com).

Weeks, R., Laj, C., Edignoux, L., Fuller, M., Roberts, A., Manganne, R.,

Blanchard, E., and Goree, W.S., 1993. Improvements in long-core

measurement techniques: applications in paleomagnetism and paleo-

ceanography. Geophysical Journal International, 114: 651–662.

Zimmerman, J.E., and Campbell, W., 1975. Test of cryogenic Squid

for geomagnetic field measurements. Geophysics, 40: 269–284.

RUNCORN, S. KEITH (1922–1995)

I first met S.K. Runcorn in June of 1955. I had just graduated from

Columbia College with a degree in Geology. He reached me at my

grandparent’s home on a Saturday morning and asked if I would like

to be his field assistant working in Arizona for the summer and by

the way I had to leave with him on Monday morning. I thought it

886 RUNCORN, S. KEITH (1922–1995)

was a joke but the longer we talked the more it seemed to be a real

offer. So after a lot of discussion in the family I decided to take the risk

on Runcorn so off we went on Monday to the wild west. Runcorn was

supremely self-confident and somehow always expected things to turn

out the way he wanted them to. By the end of the summer I was on

my way to Cambridge England instead of Wyoming where I had

expected to do my graduate studies. We first met when Keith was

33-years old having been born in 1922 in Southport Lancashire. He

loved to swim and played rugby until he was past 40. He had a great

interest in religion and during a trip to France, Italy, and Switzerland,

which we took in late 1955, we made a tour of the great Cathedrals of

France and northern Italy. In his later years he was to serve on the

Vatican Council even though he was a lifelong member of the Church

of England.

Without a doubt however his great love was science. His undergrad-

uate education was obtained at Cambridge University where he was a

member of Gonville and Caius College. He studied engineering,

receiving his degree in 1943 and spent the rest of the war researching

at the Royal Radar Establishment. In 1946 he was appointed to a lec-

tureship in the Physics Department at Manchester University were

P.M.S. Blackett was Department Chairman. Research in the Physics

Department was dominated by cosmic ray research. However, Blackett

was becoming interested in the generation of Earth’s magnetic field

and proposed a theory that rotating matter would generate a magnetic

field and suggested that Runcorn get involved in this research. The com-

peting theory was proposed by Elsasser and expanded by Bullard that the

Earth’s magnetic field arises from electric currents in the Earth’s core.

The simple test to differentiate between the two is to measure the field

at the surface and at depth in the earth, if the field increases then the field

is generated within the Earth. This experiment was carried out by

Runcorn and others in the coalmines of Lancashire and Yorkshire. The

field was shown to increase with depth showing that the field originated

within the earth. This study served as Runcorn’s PhD thesis.

Blackett had produced a very sensitive astatic magnetometer in his

attempt to measure the magnetic field of a rotating gold ball, this failed

but the magnetometer was put to work trying to find a record of secu-

lar variation in sediments and Runcorn became interested in this pro-

blem. Runcorn accepted a position at the department of Geodesy and

Geophysics at Cambridge at about the same time (1950) that Blackett

moved to Imperial College London, paleomagnetic research began

at both institutions, which of course became a competition. When

Runcorn arrived at Cambridge he found that paleomagnetic research

had already begun on lavas from Iceland by Jan Hospers who found

superimposed lavas of normal and reverse polarity, which were aligned

along the axis of Earth’s rotation however there was no laboratory.

Runcorn set about to construct a new Paleomagnetic laboratory with

the help of his students, K.M. Creer, E. Irving, D.W. Collinson, and

J.H. Parry. The emphasis was still on secular variation but it soon

became apparent that directions obtained by Irving in the pre-Cambrian

and by Creer in Triassic red sediments were widely divergent from the

present field and it became apparent that red sediments and igneous

rocks gave internally consistent paleomagnetic results. The group used

statistics developed by Fisher and the first apparent polar wander curve

was produced by 1954. These results were interpreted as true polar

wander an idea that was proposed by T. Gold. The early results from

Runcorn’s group were mostly from British rocks. However by 1954

Runcorn was traveling to the United States to collect rock samples

and the first results had been obtained. By 1954 the original group began

to disperse; Irving to Australia and Creer to the British Geological

Survey.

Runcorn did not limit his research to paleomagnetism and he built

a research group at Cambridge that studied various aspects of mag-

netic field behavior, e.g., R. Hide on convection in rotating fluids,

P.H. Roberts on magnetohydrodynamics, F. Lowes on detecting earth

currents which he believed might be leaking from the core. After the

departure of Irving and Creer the paleomagnetic group at Cambridge

consisted of J.C. Belshe, A.E.M. Nairn, D.W. Collinson, and J. Perry.

This group was joined in the autumn of 1955 by N.D. Opdyke.

Runcorn was offered the position of Professor of Physics in the

Department of Physics at the College of Newcastle on Tyne, which

at the time was King’s College of Durham University. He transferred

the paleomagnetics laboratory to Newcastle and it was up and running

in 1956. At the time of the move to Newcastle he was still maintaining

that the data from North America and Europe could still be explained

by Polar Wander alone. However the data was becoming better and a

separation of the polar wander paths of Europe and North America

persisted with the North American curve falling systematically to the

West of the European curve. In the spring of 1956 he decided that

the data could not be explained solely by Polar Wandering and he

embraced Continental Drift. It is also true that new data was coming

in from Australia that certainly could not be explained by Polar Wan-

der alone. He became a fervent supporter of Continental Drift and lec-

tured widely on the subject in Europe and North America where the

Earth Science community was hostile to the idea. He and all paleo-

magnetists would see these ideas vindicated in the late 1960s by the

success of the Plate Tectonic Theory. Runcorn adopted a strategy to

convince the scientific community that crustal mobility was real.

Ken Creer had rejoined him at Newcastle and was given the task of

producing a polar wander curve for South America and Alan Nairn

was sent to Africa were Anton Hales and Ian Gough were already

working. Ted Irving had gone to Australia where his results were

beginning to appear in print. The last of the easily accessible Gondwana

continents was India where Blackett’s group from Imperial College was

working hard. It is interesting that Blackett was a supporter of Continen-

tal Drift in the early 1950s but did not say so in print until 1960. North

American paleomagnetists were very skeptical of Continental Drift, in

particular John Graham was a strong opponent of Continental Drift

and he and Runcorn had some rather heated public debates, which on

Graham’s part became rather personal. Cox and Doell were not convinced

that paleomagnetic data supported Continental Drift as late as 1960.

Runcorn was always cool under verbal attack and I cannot ever recall

him responding in kind. He was always interested by the ideaof convection

in the mantle and proposed that solid-state creep under stress from thermal

and gravitational effects was responsible for the motion of continents.

He stopped doing active research in paleomagnetism in the mid-

1960s and turned his attention to other scientific problems. He became

fascinated by the origin and evolution of the Moon and the possibility

of convection during the early history of the moon. He took advantage

of the fact that the USA was going to the moon, which meant that

funds would become available for lunar research. Moon rocks were

returned from the moon in 1969 and in subsequent years till the end

of the Apollo program. The magnetic properties of these rocks were

studied at Newcastle and in other laboratories and all were found to

be magnetic. Runcorn concluded that the Moon must have had a mag-

netic field early in its history and that this field must have been gener-

ated by magnetohydrodynamic motions in the Moon’s core. Satellites

orbiting the moon detected small magnetic anomalies, which Runcorn

attributed to magnetizations being acquired at different times during

the early history of the moon prior to 3.9-Ga years ago and resulting

from the moon having a dipole field. The different pole positions he

attributed to polar wander on the moon, which he suggested was

caused by impacts on the Moon’s surface. His last paper published

in 1996 was titled “

The Formation of the Lunar core.”

Runcorn had a inquiring mind and during his career he published on

many different subjects for example wind directions determined from

eolian sandstone, the changing length of the day through time as

recorded in coral growth, earth currents using abandoned cable in the

Pacific, earthquakes and polar motion, excitation of the Chandler wob-

ble, and convection in the Planets.

Runcorn was regarded highly by the scientific Community and this

high regard resulted in many honors. He was elected a fellow of the

Royal Society in 1965, the Indian National Academy of Science

(1980), and he was an honorary member of the Royal Netherlands

Academy of Arts and Sciences (1985), The Royal Norwegian Acad-

emy of Science and letters (1985), the Bavarian Academy of Science

(1990), and the Royal Society of New South Wales (1993). He won

RUNCORN, S. KEITH (1922–1995) 887

a series of awards and medals beginning with the Napier Shaw prize of

the Royal Meteorological Society (1959), the Charles Chree prize from

the Institute of Physics (1969), the Vetlesen prize of Columbia Univer-

sity (1971), the John Adams Fleming Medal from the AGU (1983), the

Gold Medal From the RAS (1984), and the Wegener Medal from the

EGU (1987). He was appointed to Papal Academy of Sciences and

helped to get Galileo reinstated by the Catholic Church.

Runcorn retired as the Professor of Physics at Newcastle in 1988 but

he never stopped working. He was appoint ed to the Chapman Chair of

Physical Sciences at the University of Alaska at Fairbanks, spending

three months a year in Alaska teaching and doing research. He was

found dead in his hotel room in San Diego on 5th December 1995

being the victim of a violent attack. He is mourned by many to whom

he was a friend and mentor. He was a unique individual who was mar-

ried to science and whose home was the Universe.

Neil Opdyke

RUNCORN’S THEOREM

This remarkable theorem is concerned with the exterior magnetic field

generated by magnetized materials at the surface of a planet. Usually

magnetized materials above their Curie temperature lead to patterns

of magnetic anomalies (q.v.); however, under idealized circumstances

it transpires that some arrangements of magnetization lead to no exter-

nal magnetic field, because of extremely subtle and fortuitous cancel-

lation of the magnetic fields generated by one area of magnetization by

the magnetic fields generated by another area of magnetization. The

proof of this fact is quite complex, and the reader uninterested in the

details should skip to the last two paragraphs to discover the applica-

tion of this theorem.

We fix attention on the exterior magnetic field B(r

) (at observa-

tion point r

) generated by a magnetization distribution M(r)

enclosed in a spherical shell whose volume we denote by V and sur-

faces by ]V. The field B(r

) can be derived from a potential f

Bðr

Þ¼rf

ðr

Þ;

where f

is given by

ðr

Þ¼

Gðr; r

ÞMðrÞd

r; (Eq. 1)

where the vector Green’s function Gðr; r

Þ is given by

Gðr; r

Þ¼r

 rj

: (Eq. 2)

The potential and thus the magnetic field is in general nonzero for

arbitrary M(r). There are, however, a few distributions of M(r)

(termed annihilators) that generate no exterior, a simple example is

the arrangement MðrÞ¼AðrÞ

r for arbitrary function A(r), a purely

radial magnetization. This spherical arrangement is the equivalent of

the infinite plane layer with constant magnetization in a Cartesian geo-

metry, which also generates no external field. In the spherical case it

generates no magnetic field because the radial part of the vector

Green’s function contains no spherical harmonic of degree zero (see

(7) below), whereas the magnetization MðrÞ¼AðrÞ

r contains only

degree zero; consequently the orthogonality of spherical harmonics

(q.v.) leads to a zero result.

In 1975 Keith Runcorn proved that a more complex arrangement

of magnetization was also an annihilator, namely that arising when

a homogeneous material is magnetized by an arbitrary internal source.

The theorem applies equally well to the case of remanent or induced

magnetization, but we treat the case of induced magnetization here.

Assuming that a linear relation exists between magnetization and

applied magnetic field via a constant a, we have an equation of

the form

MðrÞ¼aBðrÞ

(in SI the constant a ¼ w=m

; see Crustal magnetic field). It is the case

that the inducing field is itself a potential field satisfying Laplace’s

equation, so we can write B ¼rV

where V

is the internal potential

(see Main field modeling); then the integral (1) reads

ðr

Þ¼a

 rj

rV

r; (Eq. 3)

where the “self-magnetization” due to terms of order a

have been

dropped.

We expand V

in terms of fully normalized spherical harmonics with

coefficients b

, analogous to the conventional Gauss coefficients

; h

¼ c

l¼1

m¼l



lþ1

ðy; fÞ; (Eq. 4)

where c is the core radius; by assuming that the mantle is an insulator,

is harmonic everywhere outside of c. The spherical harmonics

satisfy an orthogonality relation when integrated over the unit sphere

O on which jrj¼1 as follows:

dO ¼ d

: (Eq. 5)

For constant susceptibility, (3) can be written as

ðr

Þ¼a

n r

 rj

ðrÞd

r (Eq. 6)

by an application of Gauss’ Theorem, where

n is a unit normal and ]V

represents both the inner and outer surfaces of the shell. We use the

expansion of the reciprocal distance in spherical harmonics for

j > jrj

 rj¼

l¼0

m¼l

2l þ 1



ðy; fÞY

ðy

; f

Þ (Eq. 7)

to obtain

ðr

Þ¼a

l;m

2l þ 1



lþ1

ðy; fÞY

ðy

; f

c

p; q



pþ1

ðy; fÞdO (Eq: 8)

We invoke equation(s) and find that the orthogonality of the Y

cou-

ples l and p in the sums, leading to

ðr

Þ¼am

l; m

2l þ 1



lþ1

ðy

; f

ÞdO:

(Eq. 9)

Since the integrand consists of two spherical surfaces with opposing out-

ward normals, then each surface integral gives a contribution of the same

absolute value but of opposite sign, and therefore the external potential

generated is zero. This is Runcorn’s (1975a,b) theorem. Note that

the theorem also allows a to be a function of r simply by noting that the

theorem applies equally well to infinitesimally thin shells; each shell

generates no field even when a is different in each one.

888 RUNCORN’S THEOREM

The importance of this theorem was in explaining the absence of

any appreciable exterior field outside Earth’s moon, despite the pre-

sence of apparently significantly magnetized rocks on the Moon’s sur-

face. Runcorn (1975a,b) hypothesized that the mantle rocks had been

magnetized by an internally generated dynamo field (now switched

off) as they cooled through their Curie temperature; provided the Curie

isotherm is everywhere spherical and the remanent susceptibility is

everywhere only a function of radius, this even allows a time-dependent

internal source field.

For a higher order analysis in a and the treatment of an ellipsoidal

Earth, see Jackson et al. (1999) and Lesur and Jackson (2000). Further

description of the general form of annihilators can be found in Arkani-

Hamed and Dyment (1996) and Maus and Haak (2003).

A. Jackson

Bibliography

Arkani-Hamed, J., and Dyment, J., 1996. Magnetic potential and mag-

netization contrasts of Earth’s lithosphere. Journal of Geophysical

Research, 101(B5): 11401–11426 (doi: 10.1029/95JB03537).

Jackson, A., Winch, D.E., and Lesur, V., 1999. Geomagnetic effects of

the Earth’s ellipticity. Geophysical Journal International, 138:

285–289.

Lesur, V., and Jackson, A., 2000. Exact nonlinear solutions for intern-

ally induced magnetisation in a shell. Geophysical Journal Interna-

tional, 140: 453–459.

Maus, S., and Haak, V., 2003. Magnetic field annihilators: invisible

magnetisation at the magnetic equator. Geophysical Journal Inter-

national, 155: 509–513.

Runcorn, S.K., 1975a. An ancient lunar magnetic dipole field. Nature,

253: 701–703.

Runcorn, S.K., 1975b. On the interpretation of lunar magnetism. Phy-

sics of the Earth and Planetary Interiors, 10: 327–335.

Cross-references

Crustal Magnetic Field

Harmonics, Spherical

Magnetic Anomalies, Long Wavelength

Main Field Modeling

RUNCORN’S THEOREM 889

SABINE, EDWARD (1788–1883)

General Sir Edward Sabine (Figure S1), soldier-scientist, made many

of the first magnetic surveys to include magnetic force using Gauss’

newly devised method for determining absolute intensity, was instru-

mental in setting up magnetic observatories and Ross’ magnetic survey

of the Antarctic regions, and completed the definitive survey of the

geomagnetic field for epoch 1818–1876.

Born in Dublin to a distinguished military family, he was educated

at Marlow and Woolwich where he excelled in mathematics. He

obtained his first commission in 1803 at the age of 15, rising to

General in 1870. After service in Canada, which involved capture by

the American privateer the Yorktown after a running fight of 24 h

and a close engagement of an hour with a greatly superior force, he

began his work on terrestrial magnetism and gravity in 1816. In

1827 he obtained from the Duke of Wellington general leave of

absence from military duties as long as he could usefully be employ ed

in scientific pursuits. He was knighted in 1869 and served as President

of the Royal Society of London from 1861–1871.

His work on gravity, influenced by Henry Kater, dominated over his

work on magnetism early in his career (Anon, 1892) but we shall not

dwell on it here. His first expedition was as astronomer to John Ross’

expedition to find a northwest passage in 1818; this was followed by

Parry’s expedition in the following year. Gravity occupied most of

his time on these early expeditions, but he also observed magnetic

inclination, for the first time in many places, thus forming the basis

for later measurements of secular variation.

In 1834, while posted in Ireland, he began the first systematic mag-

netic survey of the British Isles, starting in Ireland before moving on to

Scotland and finally completing England and Wales in 1837, working

with Rev. Humphrey Lloyd (q.v.) and John Ross. They repeated the

survey twenty three years later, from 1858–1861. A meeting with

Baron Alexander von Humboldt in Berlin in 1836 led to von

Humboldt’s urging the British Government to set up magnetic obser-

vatories throughout the Empire. Sabine played a major role in recom-

mending the establishment of observatories in both hemispheres

(Toronto, St Helena, Cape of Good Hope, and Hobart Tasmania) and

in recommending a magnetic survey of the Antarctic regions. The

magnetic observatories were run by Lieutenants of the Royal Artillery

trained by Lloyd in Dublin, and the magnetic survey was carried out

by John Ross’s expedition of 1840.

Sabine’s campaign for global magnetic surveys is regarded by his-

torians of science as the first scientific endeavor requiring substantial

government support, both financial and in kind in the form of navy

ships and army logistics. It is known as the Magnetic Crusade and is

described in detail by Cawood (1979).

Sabine was almost fanatical in his collection and archiving of observa-

tions, which he believed to be the essential foundation of advancement of

knowledge. His main scientific legacy is his survey of all magnetic obser-

vations made from about 1818 until 1876, when his scientific activity

ceased. The compilation is thorough and accurate; it is published in fifteen

contributions to the Philosophical Transactions of the Royal Society, a

gigantic task. His compilation forms the major part of modern models of

the geomagnetic field for the first half of the 19th century (Bloxham,

1986; Bloxham et al., 1989; Jackson et al., 2000); so carefully was it con-

structed that it could be used virtually unedited in a modern inversion.

Sabine’s work was certainly not restricted to data compilation: by careful

statistical analysis he demonstrated a relationship between sunspots

and magnetic storms (Sabine, 1851, 1852), thus heralding the start of a

completely new discipline of solar-terrestrial physics.

David Gubbins

Figure S1 Bust of Edward Sabine by Joseph Durham. It currently

stands in the cafeteria of the Royal Society of London (reproduced

with permission of The Royal Society, ã The Royal Society).

Bibliography

Anon, 1892. Obituary notices of fellows deceased. Proceedings of the

Royal Society of London, 51: xliii– lii.

Bloxham, J., 1986. Models of the magnetic field at the core-mantle

boundary. Journal of Geophysical Research, 91: 13954–13966.

Bloxham, J., Gubbins, D., and Jackson, A., 1989. Geomagnetic secu-

lar variation. Philosophical Transactions of the Royal Society of

London, 329, 415–502.

Cawood, J., 1979. The Magnetic Crusade: science and politics in early

Victorian Britain. Isis, 70 : 493–519.

Jackson, A., Jonkers, A.R.T., and Walker, M.R., 2000. Four centuries

of geomagnetic secular variation from historical records. Philo-

sophical Transactions of the Royal Society of London , 358:

957–990.

Sabine, E., 1851. On periodical laws discoverable in the mean effects

of the larger magnetic disturbances. Philosophical Transactions of

the Royal Society of London, 141: 123–139.

Sabine, E., 1852. On periodical laws discoverable in the mean effects

of the larger magnetic disturbances. II. Philosophical Transactions

of the Royal Society of London, 142: 103–129.

Cross-references

Gauss, Carl Friedrich (1777–1855)

Gauss’ Determination of Absolute Intensity

Humboldt, Alexander von (1759–1859)

Humboldt, Alexander von and Magnetic Storms

Lloyd, Humphrey

Time-Dependent Models of the Geomagnetic Field

Voyages Making Geomagnetic Measurements

SEAMOUNT MAGNETISM

It has been estimated that there are more than a million seamounts in

the oceans of the world (Smith and Jordan, 1988). These rise from

all depths in the oceans and may reach the ocean surface or even rise

above the surface. If they rise above the surface they are called islands.

Although many oceanic islands have the same origin as seamounts,

they will not be discussed here. Shallow water corals are frequently

found on the top of seamounts. Because the seamount is denser than

the surrounding seafloor they may sink with time so the coral tops

are now deeply underwater.

Seamount shape and location

Most seamounts are approximately conical in shape, many with a base

diameter of 100 km or more and sides with slopes of about 5



.In

recent decades, many shipboard surveys and even submarine investi-

gations have been made of seamounts. Many seamounts are isolated

features; however, they frequently lie in a chain, with the direction

of the chain extending away from a present or past mid-ocean spread-

ing center. Some, however, exist in clusters, do not display a conical

shape, and may appear more like a small underwater plateau. Most

are believed to have been formed as a rapidly cooled volcanic eruption

near the axis of mid-ocean spreading centers. While this would make a

linear progression in age along a chain of seamounts, there are some

chains that do not have a simple increase in age with distance along

the chain. Why seamounts form in some areas and not in others is

unknown. When the magma forming the seamount cools it is magne-

tized in the direction of the Earth’s magnetic field at that time. Because

it is on a moving plate, both the location of the seamount and the

direction of the Earth’s magnetic field change with time. Studies of

seamount magnetism attempt to learn about each or both of these

two variables. Such studies are aided by knowing the age of the

seamount from radiometric or other absolute dating means; however,

this absolute age is frequently not known.

If one takes the direction of the past geomagnetic field as known,

then finding the direction of magnetization of the seamount will give

its age. That can be compared to the age of the adjacent seafloor, as

determined from magnetic anomalies, and provide information about

tectonic plate motions.

Why to study seamount magnetization?

To a first approximation the two-dimensional magnetic anomaly pat-

tern over a seamount is like that of a magnetic dipole; however, for

many seamounts, especially those of an irregular shape, this is a poor

approximation. When a reference field, such as the IGRF, is subtracted

from the observed magnetic field reading, the anomaly pattern generally

shows a high area and a low area. If the anomaly high is on the equator

side of the seamount, then it is approximately normally magnetized.

If the high is on the polar side it is largely reversely magnetized. Model

calculations can be made to better determine the strength and direction of

magnetization of different parts of the seamount.

Alternatively, the magnetization of a seamount may be studied from

the magnetization of rocks recovered by dredging or deep-sea drilling

cores. Dredging allows the intensity of magnetization to be deter-

mined, but not its direction, since the in situ orientation of the dredged

rock is unknown. Drilled rocks allow the intensity and inclination

(or dip) of magnetization to be determined, but the azimuth is less well

known because the azimuth of the core is not accurately determined.

In principle both the magnetic anomaly pattern and the rocks could be

studied for the same seamount, but that has rarely been done. Opinion is

divided as to which method gives the more accurate information

about the direction of magnetization of the seamount. Furthermore, in

seamount magnetic studies it is usually assumed that the seamount was

formed quickly, i.e., all in the same place and with little change, or secu-

lar variation of the geomagnetic field during formation. Studies have not

clearly shown that the body of the seamount has different magnetizations

in its different parts, except some models that indicate a nonmagnetic cap

on the top.

Some seamount magnetization studies

In the 1960s, when marine magnetometers came into use, the first

magnetic surveys were made over simple, conical seamounts. A

review of early geomagnetic studies, especially for Pacific Ocean

seamounts, has been given by Grossling (1970). One early study was

made of Vema Seamount, which lies in the South Atlantic Ocean about

1000 km west of Cape Town. It was discovered in 1959 that Vema

Seamount rises from a depth of 4600 m, coming to within 40 m of

the sea surface. This seamount was found to be approximately rever-

sely magnetized. Model studies showed it has a magnetization of about

0.0015 emu, which is characteristic of basalt (Heirtzler and Hadley,

1966). This study had no ages for this seamount.

Several magnetic field studies have been made over the New

England seamounts in the North Atlantic. Mayhew (1986) treated the

magnetic field over several of these seamounts as due to a dipole.

From Bear Seamount, nearer to shore, to Nashville Seamount, further

off shore, the radiometric ages vary from 103 to 83 my and are of the

same age as the adjacent seafloor as determined by magnetic anoma-

lies and the basal age of sediments. The direction of the equivalent

geomagnetic dipole was about what is expected for this Cretaceous

normal epoch and showed that this simple means of determining the

magnetization direction may be useful.

In the eastern central Pacific Ocean, studies have been made of the

northern Cocos Seamounts (McNutt and Batiza, 1981), of the magnet-

ism of the Line Island Seamounts (Sager and Keating, 1984), and a

rather different type of study of the Emperor Seamounts (Tarduno

et al., 2003).

SEAMOUNT MAGNETISM 891

McNutt and Batiza (1981) studied the magnetic survey data from

six scattered seamounts, ranging in page from 1 to 7 my in age in the

Cocos plate in the eastern Central Pacific. Over this time the magnetiza-

tion of the seamounts showed a counterclockwise rotation of



agreeing with models of plate motion. Model studies showed a mag-

netization of up to 10 Am

1

is required, with better fits to observations

if the upper layer of the seamount is not included in the body shape.

The Line Islands form a 4200 km chain in the central equatorial

Pacific, south of the Hawaiian Islands. Several investigators have

studied these features. Sager and Keating (1984) have studied eight

of these seamounts. Paleomagnetic directions of four, with ages of

72–85 my, agree with other geomagnetic data. Some central sea-

mounts are much younger (Eocene) in age. Model studies show that

they have magnetizations of 3–5Am

1

After the Ocean Drilling Program drilled several seamounts of the

Emperor Seamount chain in the northwest Pacific Ocean, Tarduno

et al. (2003) studied the magnetic properties of recovered rocks with

radiometric ages of 81–47 my after reasoning that secular variation

of the geomagnetic field did not influence their measurements. Paleo-

latitude determinations indicated a progressive change and that the sea-

mounts were formed by a rapid motion over the Hawaiian hotspot.

Model calculations

The calculations of magnetic anomalies due to magnetic bodies of irre-

gular shape have advanced considerably since the 1960s, largely due

to the use of digital computers. Special computer codes have been

written for seamounts. One of the earlier codes calculated the anomaly

due to a thin “pancake” with its individual magnetization and approxi-

mated the seamount structure as a stack of pancakes, and summing the

anomaly components of the entire stack (Talwani, 1965). In contrast to

this forward modeling technique, it is possible to start with the

observed anomalies and calculate the approximate magnetization con-

strained by the shape of the body (Parker, 1991). This can yield infor-

mation about how magnetization is distributed over the body structure

although the solutions are not unique.

J.R. Heirtzler and K.A. Nazarova

Bibliography

Grossling, B.F., 1970. Seamount magnetization. In Maxwell, A.E.

(ed.), The Sea, Volume 4, Part 1. New York: Wiley-Interscience,

Chap. 4, pp. 129–156.

Heirtzler, J.R., and Hadley, M.L., 1966. Magnetic anomaly over Vema

Seamount. Nature, 212: 912–913.

Mayhew, M.A., 1986. Approximate paleomagnetic poles for some of

the New England Seamounts. Earth and Planetary Science Letters,

79: 183–194.

McNutt, M., and Batiza, R., 1981. Paleomagnetism of northern Cocos

seamounts: constraints on absolute plate motion. Geology, 9:148–154.

Parker, R., 1991. A theory of ideal bodies for seamount magnetism.

Journal of Geophysical Research, 96: 16101–16112.

Sager, W.W., 1992. Seamount age estimates from paleomagnetism and

their implications for the history of volcanism on the Pacific plate.

In Keating, B.H., and Bolton, B.R. (eds.), Geology and Offshore

Mineral Resources of the Central Pacific Basin. Circum-Pacific

Council for Energy and Mineral Resources Earth Science Series.

New York: Springer.

Sager, W.W., and Keating, B.H., 1984. Paleomagnetism of Line Island

Seamounts: evidence for late Cretaceous and early Tertiary volcan-

ism. Journal of Geophysical Research, 89: 11135–11151.

Smith, D.K., and Jordan, T.H., 1988. Seamount statistics in the Pacific

Ocean. Journal of Geophysical Research, 93: 2899–2918.

Talwani, M., 1965. Computation with the help of a digital computer of

magnetic anomalies caused by bodies of arbitrary shape. Geophy-

sics, 30: 797–817.

Tarduno, J.A., Duncan, R.A., Schell, D.W., Cottrell, R.D., Steinberger, B.,

Thordarson, T., Kerr, B.C., Neal, C.R., Frey, F.A., Torii, M., and

Carvallo, C., 2003. The Emperor Seamounts: southward motion of

the Hawaiian hotspot plume in Earth’s mantle. Science, 301:

1064–1069.

Cross-references

Magnetization

Oceanic Crust

SECULAR VARIATION MODEL

The secular variation of Earth’s magnetic field has been studied for

almost as long as paleomagnetic measurements have been made. Many

models have been proposed to explain the latitudinal variation of the

secular variation. It was found simpler to study the scatter of the vir-

tual geomagnetic poles (VGPs) rather than the directions because

VGPs usually have a uniform scatter around a mean pole whereas

directions can have an elongated distribution for observations made

close to the equator. A short history of secular variation models will

be presented, where it will be shown that the present day field has been

an important factor in determining many model parameters. It will be

argued that one of the drawbacks of most models is the practice of

ignoring data that give low-latitude VGPs falling within a certain band,

centered on the equator, whose width varied from model to model. No

rigorous explanation is produced for the specific choice of the width

of the band. The model proposed here uses information from VGPs fall-

ing in all latitude bands following the Camps and Prévot (1996) model

which used VGPs from all latitudes. The model has a central dipole and

a number of off-centered dipoles arranged within the core, to represent

the nondipole field. The strength of all the dipoles is allowed to vary

statistically in a predetermined manner, and the scatter of VGPs is cal-

culated for various observational latitudes. The model fits the data as

well as the Camps and Prévot model. It has a larger number of off-

centered dipoles close to the geographic poles than to the equator, this

being necessary to give the higher scatter of VGPs at high observation

latitudes compared to low observation latitudes.

Introduction

The study of Earth’s magnetic field has been materially influenced by

the measurements of Earth orbiting satellites, starting by MAGSAT,

which was launched in 1979 and measured the field for about

6 months before its orbit decayed. Satellites launched more recently

are Ørsted, Champ, and SAC-C. A good source of information about

recent geopotential field satellites is given at http://denali.gsfc.nasa.

gov/terr_ma g/e_missions.html. Although these satellites allow a very

accurate picture of Earth’s magnetic field to be seen, they do not give

the full picture of the field because of the long time scale of field

changes. For this we have to turn to paleomagnetic methods.

The advantage of paleomagnetic methods is that they cover a much

wider range of time than do direct observations. The disadvantage is

that the coverage of measurements over Earth’s surface is usually quite

poor, and so it is difficult to obtain viable spherical harmonic represen-

tations of the field. We are, therefore, left with spot readings of the

geomagnetic field at certain locations on Earth’s surface, from which

we need to infer something about the geomagnetic field. The quantity

that paleomagnetists have chosen to look at is the scatter of directions

or VGPs as a function of observation latitude. It turns out that this is

frequently a useful parameter to work with because it does tell us

something interesting about the generation of the geomagnetic field.

Edmund Gunter (the inventor of a precursor to the slide rule and

also of Gunter’s chain, still used in surveying today) was the first

892 SECULAR VARIATION MODEL

person to notice that the variation or declination of a compass needle

from true north changed with time. Gellibrand, who was Gunter’s

colleague at Gresham’s College, published the first paper attempting

to describe the secular variation measured using compass needle

directions (Gellibrand, 1635).

A parade of secular variation models

Model A (Irving and Ward, 1964) supposed that the field was due to

an axial dipole field plus randomly directed fields at the surface. These

surface fields are expected to be spherically symmetric with respect to

the axial dipole field, and in the first use of this model the surface per-

turbing fields were uniform as a function of latitude and constant. This

resulted in a total field dispersion (as measured by the angular standard

deviation) that decreased with absolute latitude. Irving and Ward

(1964) showed that this dispersion was very close to the dispersion

obtained when the present day field was rotated around the geographic

pole. These results are shown in Figure S2. The data used for demon-

stration of this model are not those used by Irving and Ward (1964),

who gave no list of the data. They come from Creer (1962) who pro-

posed model B and who generated the results from the rotation of the

1945 field around the spin axis of Earth that is shown in Figure S2 too.

Model A proposes that the scatter in directions becomes lower by a

factor of 2 on going from equator to pole, and as can be seen this is

roughly what the 1945 field does and what the paleomagnetic data

suggest. This is a direct result of the strength of the dipole field, which

increases by a factor of 2 on going from equator to poles, and of the

fact that the perturbing field was of constant size.

Model B (Creer et al., 1959) simply uses a dipole wobble to explain

the angular scatter. The data studied in this paper were from the Paleo-

zoic Great Whin Sill which has roughly horizontal directions of mag-

netization. These directions form an elongated pattern but when used

to calculate VGPs, give circularly symmetric patterns, thus suggesting

that they are caused by dipole wobble with a Fisherian (1953) distribu-

tion of dipole directions. This observation has meant that many of the

SV models proposed have been based on an analysis of VGP scatter

rather than on direction scatter. One other notable thing is that this

model gives no latitudinal variation of VGP scatter. Creer et al.

(1959) also showed that angular dispersion of field data was lower

in the northern hemisphere than in the southern hemisphere.

Creer (1962) used a number of different geomagnetic field models

and rotated them around the spin axis to determine angular standard

deviation of field directions. He compiled four data sets from 1882

to 1955 and suggested that the hemispheric difference was likely to

be temporary. So he combined the northern and southern hemispheres

and showed that the angular dispersion of the field was similar for each

of the data sets. A curve drawn through the data showed a drop from

about 21



at the pole to 10



at a latitude of 60



. This was compared with

scatter data from igneous rocks (shown here in Figure S2) and

there was considerable agreement.

When going from Fisherian poles to directions or Fisherian direc-

tions to poles, there are equations that determine the ratio of scatter

(Merrill et al., 1996). Two different equations have been suggested

for the transformation of Fisherian VGPs to directions (Creer, 1962;

Cox, 1970). The solution to this dilemma is shown in Figure S3.

The Monte Carlo distributions were done using 4000 randomly gener-

ated VGPs with a precision parameter of 60. The upper of these two

curves was calculated using the input precision parameter of 60 and

the lower curve was calculated using the calculated VGP precision

parameter for the 4000 generated VGPs. It can be seen that the Cox

(1970) version of the ratio is closer to the experimental results than

is the equation of Creer (1962).

Model C was proposed by Cox (1962). He allowed for both dipole

wobble and variations in the nondipole field. He also advocated the

use of VGPs rather than directions because dipole wobble is simple

to deal with when using VGPs as there is then no latitudinal variation.

Figure S2 This shows the angular standard deviation of the field

as a function of latitude. Model A is from Irving and Ward (1964).

The line marked by diamonds is the scatter observed at a specific

point when the 1945 field is rotated through 360



. The black dots

are results obtained from igneous rocks collected at different

latitudes.

Figure S3 This shows the ratio of field scatter to VGP scatter

(as calculated by the ratios of the precision parameters) as a

function of latitude. The curve marked by triangles uses the

equation for Fisherian directions to poles from Cox (1970), for

comparison purposes. The two curves marked by diamonds and

plusses are from Cox (1970) and Creer (1962). The Monte Carlo

curves were calculated as described in the text.

SECULAR VARIATION MODEL 893

He showed that the VGPs calculated from the field as it rotates around

the geomagnetic pole, thus showing the variation of the nondipole

field only, has a low angular standard deviation (ASD) at the equator

of about 8



which rises as the pole is approached, so that close to

the pole the ASD is about 16



. He also showed that there was a hemi-

spheric difference, the high-latitude reading for the South Pole being



greater than that close to the North Pole. Cox used an equation pro-

posed by Runcorn (1957) to express the precision parameter of VGP

scatter from both dipole and nondipole sources, which is

; (Eq. 1)

where k is Fisher’s precision parameter for polar scatter, T shows total

contributions, N shows nondipole contributions and D shows dipole con-

tributions. The corresponding equation for angular standard deviation is

¼ d

þ d

: (Eq. 2)

Cox did not compare his model of the present day VGP dispersion

with paleomagnetic data. The results of his calculations are shown in

Figure S4. The bottom curve shows the result of adding a dipole field

scatter equivalent to half of the poles lying within 11.5



of the

geographic pole, giving a k

of 34.5. The scatter, therefore, rises from



at the equator to 20



at the poles.

Model D was also proposed by Cox (1970). Cox allowed for changes

in intensity and direction of the nondipole field, and also changes in the

direction and intensity of the dipole field. The nondipole field intensity

was described by Rayleigh (1919) function that was calculated to

explain the directions of magnetization in small rock samples when there

was a random component added to some stable component. It is not clear

whether this is a reasonable distribution for the nondipole field portion of

Earth’s magnetic field. This distribution is given by:

pðf Þ¼

ﬃﬃﬃ



exp

f



; (Eq. 3)

where f

completely specifies the distribution and is the mode. This is

not the same as the more usual Rayleigh function that describes things

like the distribution of wave heights given by:

pðf Þ¼



exp

f



: (Eq. 4)

This distribution function produces a value for the angular variance of

the population of nondipole ( f ) and dipole (F ) fields:

lðÞ



1  r

ðÞ

3=2

1 þ 3 sin



: (Eq. 5)

Assuming that the VGPs are distributed according to Fisherian distri-

bution and not the field directions, Cox determined that the variance

in pole positions should follow the equation below.

¼ S

ðlÞ



1  r

ðÞ

3=2

: (Eq. 6)

is the contribution from dipole wobble and so has no latitudinal

variance. In the variable W

NP,

N stands for the nondipole part of the

field and P stands for the poles that are Fisherian in distribution.

is given by:

¼ 5

1 þ 3 sin

5 þ 3 sin



1=2

: (Eq. 7)

This function increases by a factor of 1.6 on going from equator to

pole. Since the rotated 1945 field produces poles whose scatter rises

by a factor of more than 2 on going from equator to pole, it is natural

to suppose that the function f

(l) must produce a scatter that rises

modestly by a factor of 1.3 on going from equator to pole. No attempt

was made by Cox to compare his models with paleomagnetic results.

Figure S5 shows some of the results from Cox’s model recalculated

for this paper. The red line marked by diamonds is the “observed” scat-

ter, obtained by rotating the present day field around the geomagnetic

pole axis and adding an 11



dipole wobble. The green curve marked

by circles is the model result using a dipole wobble of 11



plus a non-

dipole field that is constant, i.e., it does not depend on the dipole field.

The blue curve marked with plusses uses the same dipole wobble but

assumes that the nondipole field is directly related to the dipole field

(i.e., it increases by a factor of 2 on going from equator to pole).

Model E was proposed by Baag and Helsley (1974). They wished to

explain why the green dotted curve in Figure S5 does not increase

enough with latitude to match the red diamond curve. They suggested

that the nondipole and dipole fields were correlated by some amount,

as expressed by the following:

¼ S

þ S

þ 2r

; (Eq. 8)

where r

is variable correlation coefficient between the dipole field

and the nondipole field. This means that the following holds:

þ S

 S

þ S

ðÞ

; (Eq. 9)

thus allowing us to have lower and upper bounds to the angular stan-

dard deviation. Figure S6 shows all five models discussed so far using

the angular standard deviation of VGP distribution as the common

ordinate, so that comparisons may be made (data from Baag and

Helsley, 1974). The data shown by Baag and Helsley (1974) show a

general increase of scatter with latitude, with several results falling

above most of the curves except E

Many of the models discussed above have not considered the

growing number of paleomagnetic results that might contribute to

Figure S4 Cox’s (1962) model C. Nondipole field scatter shown

by the top three curves, and dipole plus nondipole scatter shown

on the bottom curve.

894 SECULAR VARIATION MODEL

understanding the variation of angular standard deviation as a function

of latitude. The paper introducing model M (McElhinny and Merrill,

1975) made the first serious attempt to analyze paleomagnetic data.

They used information for the past 5 Ma and rejected data whose aver-

age VGP was further than 40



away from the poles to remove results

showing transitional directions that might be recording a reversal. This

is a more extreme position that other workers who have often used

directions more than 45



away from the poles to signify transitional

behavior. Their data showed that the VGP scatter rose from about



at the equator to almost 20



at high latitudes. Data from the

Brunhes epoch showed similar results, with slightly less scatter at high

latitudes but with significantly greater uncertainty, rendering the two

results indistinguishable. A minor error (Harrison, 1980) rendered

model M incorrect in that it predicted a much greater latitudinal varia-

tion of VGP scatter than actually existed in the data that they pre-

sented. Model M is basically similar to the blue curve (marked by

plusses) in Figure S5, in which the latitudinal variation of the nondi-

pole field was equivalent to the latitudinal variation of an axial dipole

field. Harrison (1980) suggested a modification to model M in which

the nondipole field had two components, one of which increased with

latitude like a dipole field and another component that depended on the

rate of change of angular momentum in a rising convection cell, a

function that is responsible for creating core surface current rings that

are probably responsible for the nondipole field. Figure S7 shows an

excellent agreement between model and data. With the low latitude

cutoff proposed by McElhinny and Merrill (1975) VGP scatter rose

from about 13



at the equator to 20



at high latitudes.

Model F was proposed by McFadden and McElhinny (1984). It

assumes that the VGPs are symmetrically distributed about the mean,

as was done for some earlier models. This agrees with observation

and also is convenient because dipole wobble in this system is

latitude-independent. The VGP scatter is usually written as:

¼ S

þ S

ðlÞ: (Eq. 10)

They point out that using paleosecular variation data cannot solve for

the unknowns in this equation because they cannot separate out dipole

from nondipole effects. The authors, therefore, suggest that the model

predict a value of W(l) that is consistent with the latitudinal variation

of the present day nondipole field. They then have an extended discus-

sion concerning the distribution of field directions that will result in a

Figure S6 VGP angular dispersion as a function of latitude plotted

for models A-E. Model A is in red (plusses), model B is in blue

(squares), model C is in yellow (rhombus) (same as model E

model D is in green (circles), and model E

is in gray (triangles).

Figure S7 Model proposed by Harrison (1980) compared with

data compiled by McElhinny and Merrill (1975).

Figure S5 Model D (Cox, 1970). Angular standard deviation of

VGPs as a function of latitude. The red line marked by rhombi is

the present day nondipole field variation to which has been added

an 11



dipole wobble. The green line (dots) is model D using a

nondipole field that does not vary with latitude. The blue line

(plusses) is model D using a nondipole field that varies in intensity

in the same way that a dipole field does. This is equivalent to

model M.

SECULAR VARIATION MODEL 895