On Modeling Magnetic Fields 

on a Sphere with 

Dipoles and Quadrupoles A <*Nfc/ JL 

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1118
On Modeling Magnetic Fields 

on a Sphere with 

Dipoles and Quadrupoles

By DAVID G. KNAPP

GEOLOGICAL SURVEY PROFESSIONAL PAPER 1118

A study of the global geomagnetic field with 

special emphasis on quadrupoles

UNITED STATES GOVERNMENT PRINTING OFFICE, WASHINGTON: 1980
UNITED STATES DEPARTMENT OF THE INTERIOR 

CECIL D. ANDRUS, Secretary

GEOLOGICAL SURVEY 

H. William Menard, Director

Library of Congress Cataloging in Publication Data

Knapp, David Goodwin, 1907-

On modeling magnetic fields on a sphere with dipoles and quadrupoles.

(Geological Survey Professional Paper 1118)

Bibliography: p. 35

1. Magnetism, Terrestrial Mathematical models. 2. Quadrupoles. I. Title. II. Series: United States

Geological Survey Professional Paper 1118. 

QC827.K58 538'.?'ol84 79-4046

For sale by the Superintendent of Documents, U.S. Government Printing Office

Washington, D.C. 20402 

Stock Number 024-001-03262-7
CONTENTS

Page

/VOStrHCt TMTM-TMTMTMTM"TM-TMTMTM"TM«TMTM-«—"—————————————— j_

Introduction——————————-—--——-———........—..——....... i

Characteristics of the field of a dipole —————————————— 1

Multipoles ——————————-—--——--—-——--——-——...._. 2

The general quadrupole ——————-———._....................—.— 2

Some characteristics of quadrupole fields -————————- 4

Character of lines of force ———————————————— 6

Interconversion of quadrupoles —————————————— $

Relations with spherical harmonic analysis —————————— 9

Quadrupole secular change and its conservative aspect — 12

The eccentric dipole ——————————————————— 13

Fertile and sterile parts of the quadrupole ———————— 13

Relations with spherical harmonic analysis—Continued

Other remarks concerning eccentric dipoles ———— 

Application to extant models -—————————-——-

E ccentric-dipole behavior ———————————————• 

Results of calculations --—--—------——-——-—-

Additional results ————————————————

Hypothesis on the character of the quadrupole —————-

Some further implications of the quadrupole constraint — 

Some qualitative remarks on octupoles ——————.—...

Summary and conclusions -----———————————-—~

Selected references ——————————————————————

Page

14

14

20

21

26

27

29

34

35

35

ILLUSTRATIONS

Page

FIGURES 1-4. Sketches of:

1. Linear quadrupoles -—--————————-———---——————--———————————...——————— 2

2. Three conceptions of the rhombic quadrupole ————-——--———————---—————————-----——— 3

3. Linear quadrupoles on a sphere directed toward P0 and Pj ——————--—-—-——-——-—---———--— 3

4. Attitude of quadrupole shown in figures 5-7 -----——————————————————-———-——-————-— 6

5-7. Graphs of normal-quadrupole field on the sphere showing:

5. Potential, vertical intensity, and inclination ————————————————————————......————— 7

6. Magnetic meridians, declination, and horizontal and total intensity ——————————————————-—-— 7

7. North and east components ———--—--------———--—--—-_-———————————-———————————————— 7

8-11. Sketches of:

8. Dipole field line -.-----.-------------------------------------.---------.-----.----------------------------------------------------------------------------———- 8

9. Linear-quadrupole field line ---------—————-———————-—-——————————--——--—-----——- 8

10. Normal-quadrupole field line-----------------------------—--—--—-———-—-—-----————————————— 8

11. Relations of cardinal axes to primitive lines and identification of angular parameters ————---————— 11 

12-17. Graphs of drift of the geomagnetic quadrupole:

12. Primary axis since 1550 —————————————————————————————————————————————— 15

13. Primary axis since 1930 ---------------------------------———-——-——---————————-———...——— jg

1 J-^T. A. J.1/CU T?pH \^CLL (*ai*Hinfi U111CIJ. 1 CUVlO avic OlllVsC GITI/"»O J.tltlV/ 1 ^^O -.—..—_.._-.___— ————————— _ _..______ _ _....._..—————————_—.._———..—————.——— j_ 1 7i

15. Red cardinal axis since 1930 ————————————————————————————————————————————— 17

J.O. XJlUc CHiQlrlHI HX1S 31HC6 J.O«5vJ ———————---------.————-—---——----————-———————————--—--—————————————————---»«-•••-•--•-- J.o

1 j. 7 i • Rind JJAUC i^suvlifisil \~>CLL miicu. d.A.i.0 avio onicc cini^o 1 J. Q^fl zftj\j ____. _________ __ .——————— _ 

.... ——__ __ __ ..._———_———————.—————._.———..———————————.———— 1Qj. ^

18-20. Graphs of change of U:

18 Since 1550 _________________________________________________________ 20

19 Since 1925 ______________________________________—_________________ 21

£A\J 9fl f O111CC7 C!inr»f> 1 J. Q1 17-1 fl \J _ ———————"—-«———"-———---———--————————-———TM...TM......TM....TM.....TM....———..................................... — .... . ————— 99£t£t

21. Direct plot of change of the quadrupole moment ——————————————————————————————————————— 23

22. Sheared plot of change of the quadrupole moment.----.--.-——-————-————————-—-—--—————---—--— 24

23. Graph of coordinates of displacement vector for eccentric dipole —————-—-——--————--——--————— 27

24. Graph of change of dc for selected analyses ———-----------------———————————-——------———-———————-—————- 28

25. Graph of coordinates of the centered-dipole axis ————--——-—————----——-——————————————-—————— 30

26. Graph of quadrupole rotation rate using Class II analyses —————-—--——-—-——————----—-——--—----- 31

27. Graph of center of clockwise quadrupole rotation using chiefly Class I analyses ———————————————————— 32

28. Graph of center of clockwise quadrupole rotation using chiefly Class II analyses —-——-——-—-——--—--——— 33

29. Schematic diagram of current loops in normal quadrupole —--———-—-----————---—-—------------—— 33

30. Sketch of arrangement of four dipoles forming a cubic octupole, and equivalent quatrefoil array of coplanar current

lOOyS ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------^ o4 in
IV CONTENTS

TABLES

Page

TABLE 1. Expressions for the fields of dipoles and quadrupoles ——————————————————————— — ————————————— — . 5

«. '9fU3.Q.rupO16 |j£Lr£lIll6u6rS OI SpGCiriC IlGlQ IT1OCL61S --------------------------------------------------------------------------- —-—-——"—--— TMTM-—TMTM—-TMTMTM- J_Q

Q fjroiirjiriff of pxttirit snh.6riC3.1 hflrmoriic flmilvsps -•-------•-•-»•--•-------------------"----—-----•"--—-----—--«----—"------—----------—--—--——•—»-»• 2^S

4 Ecc6ntriC"diriol6 nt

5. Changes of coefficients caused by imposing the constraint for U = 1 — ----- — - — - — — — - — - — — —— — - — — — — —— -- — - 34
ON MODELING MAGNETIC FIELDS ON A SPHERE 

WITH DIPOLES AND QUADRUPOLES

By DAVID G. KNAPP

ABSTRACT

This paper assists in the understanding of the global geomagnetic 

field as it is manifested in models slightly more complex than the 

centered dipole, with special emphasis on the quadrupole, which 

among the harmonic components is the chief determinant of the 

nondipole field configuration. To this end it examines the geometric 

properties of the three kinds of quadrupoles, as well as their interconvertibility, and the ways in which they can be combined or resolved into constituent parts. Improved methods are developed for 

establishing the quadrupole parameters (especially the cardinal 

axes) from spherical harmonic coefficients and for using the various 

extant analyses to examine the secular change of the quadrupole. 

Earlier reports of the quadrupole's westward drift are confirmed, its 

rotation being found to be about 15 minutes per year, but the center 

of clockwise rotation is markedly displaced from the geographic 

pole and is situated in the region of the Aleutian Islands. A graphic 

display clarifies the disparities among different models, promotes 

the study of relative benefits of refinements in spherical harmonic 

analysis, and points the way toward a definitive assessment of the 

quadrupole and especially of its secular change. The way in which 

one part of the quadrupole combines with the centered dipole to produce the eccentric dipole is also examined (with some possible bearing on radial-dipole models). Support is presented for the hypothesis 

that the geomagnetic quadrupole tends to hold rather closely to the 

aspect of a "normal" quadrupole—one with identical configurations 

for its positive and negative regions. Some properties of octupoles 

are discussed qualitatively.

INTRODUCTION

Ever since the time of William Gilbert nearly four 

centuries ago, there has been perennial interest in the 

devising of models to simulate the patterns of the geomagnetic field. The simpler models, though easy to understand and to describe, are not very faithful to nature; as the models are made more complex in order to 

improve the fit, they depart more and more from the 

physically comprehensible. This report attempts to 

dispel some of the obscurity and stresses the geometric 

aspects of the models, particularly those that are 

slightly more detailed than the centered dipole. The 

quadrupole has never been investigated with sufficient 

geometric insight to afford a satisfying conception of 

its characteristics. The eccentric dipole is likewise of 

great interest, not only for its unitary characteristics 

(falling somewhat short of the dipole-plus-quadrupole

model in fidelity to nature but giving a somewhat truer 

picture than the centered dipole alone), but also for its 

bearing upon the characteristics of models comprising 

an array of two or more radial or other eccentric dipoles.

The centered dipole fails to embody two striking 

features of the world charts, namely (1) the oblique or 

"corkscrew" aspect of the agonic lines, and (2) the 

elongation of the intensity loops enclosing the Arctic 

dip pole. The eccentric-dipole model does depict feature 

(1), which, however, is not intrinsic to the field but 

rather an effect of its relation to the coordinate system 

(reflecting a longitude difference between the dipole's 

displacement vector and the meridian plane of the centered dipole). To account for (2), a genuine field characteristic, requires still greater complexity in .the model.

CHARACTERISTICS OF THE FIELD OF A DIPOLE

The simplest useful model is the one developed by 

Biot (Humboldt and Biot, 1804), namely, a magnetic 

dipole at the center of the globe. The basic notion of a 

dipole grew out of the then rather novel concept of 

magnetic point poles (or as Biot preferred to say, "centers of action") and represents the limiting case of a 

pair of point poles of opposite kind, as they are caused 

to approach one another to within an infinitesimal separation, while their pole strength is increased to preserve a constant magnetic moment. That is, they are 

conceived as separated by so small a distance that further approach (short of coalescence) has no observable 

effect on the field patterns. Physicists today accord to 

the dipole a reality usually denied to the point pole 

itself; the dipole moment is a fundamental parameter 

of elementary particles, conceived as the effect of spin 

and orbital motion of electric charge, whereas little 

progress has been made (despite strenuous efforts) in 

identifying an isolated magnetic monopole. This gradual shift in point of view away from the monopole notion reflects the concept that magnetism is but a manifestation of electric charge in motion.

The configuration of the dipole field is specified by 

the equation for its scalar potential, Vd; namely,

1
ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

Vd = (Mlr2) (COS 0), (1)

where M is the magnetic moment of the dipole, r is the 

radial distance of the point of observation from it, and 

0 is the angle between the position vector of the point 

and the reversed vector moment of the dipole. If the dipole is axial, centered, and southward-directed, 6 is the 

colatitude. From this equation it is readily possible to 

develop others for the field strength and its various 

components and angular elements; thus, the total intensity, F, is given by

F = (Mlr3) (1 + 3 cos2 0) °' 5 .

MULTIPOLES

(2)

To improve upon the fit of a dipole model requires 

some increased complexity, such as that obtained from 

multipoles. Two equal but opposite dipoles at an infinitesimal separation constitute a quadrupole, two 

quadrupoles of this configuration constitute an octupole, and so on. The quadrupole field is the harmonic 

component ranking next below the dipole in magnitude, and may be the dominant constituent during intervals in the remote past when the dipole field was undergoing reversals for other bodies as well as the 

Earth.

Currently, the quadrupole may be seen as the chief 

determinant of the character of the nondipole field, and 

(unlike the centered dipole) as the vehicle of a substantial part of the directional secular change of geomagnetism. The rationale of the latter characteristic will 

become clear with the development of the field geometry.

The quadrupole can be characterized by the direction 

in which the reversed dipole is displaced from the forward one. This direction, taken in conjunction with a 

dipole axis, will usually define a characteristic plane; 

the angle 2co, which that direction makes with the forward dipole axis, may take on any value. If 2co is 180°, 

or zero, we have a linear quadrupole (Howe, 1939) with 

the north poles outward (called positive or "red," to 

revive an old sign convention) or with the south poles 

outward (called negative or "blue"). These cases are exceptional in that no characteristic plane is defined in 

the sense just set forth. If 2co is 90°, or 270°, we have 

Howe's "planar" quadrupole, which I prefer to designate as a normal quadrupole. If 2co is neither zero nor 

any multiple of 90 °, the result is a rhombic quadrupole.

Consider now the field of a linear quadrupole. Like a 

dipole field, it has circular symmetry about an axis, so 

that when alined with the polar axis of coordinates it is 

zonal (Chapman and Bartels, 1940, p. 615). Unlike the

dipole field, it also has mirror symmetry (as to both 

magnitude and sign) about the plane through the quadrupole and normal to its axis. On the sphere, both axial 

points are dip poles of the same sign—for a positive or 

red quadrupole, N; for a negative or blue one, S—and 

no other singular points occur, but the equator is a continuous locus of radially directed field, with polarity 

contrary to that of the dip poles, and field strength 

equal to one-half.

The scalar potential can be obtained by replacing cos 

0 with ylr in equation (1), and differencing in the denominator to allow for the effect of the second dipole 

through the altered r. In this way we obtain for the 

potential of the linear quadrupole field,

Vt = (MJr3) (3 cos2 0 - 1), (3)

and for total intensity,

Fl = (3M,/2r4 ) [(5 cos 20 + 1) (cos 20 + 1) + 4f5 . (4)

The procedure given by Chapman and Bartels (1940, 

p. 11) for deriving the line-of-force equation for the dipole is likewise applicable to the linear quadrupole. The 

equations for the dipole and linear quadrupole are, 

respectively,

and

r = k sin2 0

r2 = k 2 | tan 0 sin3 20 |,

(5)

(6)

where k is the value of r at the remotest point on the 

line of force. One way to think of a linear quadrupole is 

to suppose the separation between the centers of the 

two constituent dipoles to be just equal to the pole 

separation of each dipole, forming a row of three poles, 

the two exterior ones being of a given pole strength 

and sign, and the interior one of twice the pole strength 

and the opposite sign (fig. 1).

FIGURE 1.—Linear quadrupoles; N 

is positive (red) and S is negative 

(blue).

N) ©

THE GENERAL QUADRUPOLE

Conventionally, the general quadrupole may be conceived as an array of four point poles, in either of the 

two equivalent configurations shown in figure 2 (A and
THE GENERAL QUADRUPOLE

^->

,N

Primitive lines

Primitive lines

(N)

Primary axis

Cardinal axes

FIGURE 2.—Three conceptions of the rhombic quadrupole. For a normal quadrupole the rhombus reduces to a

square.

B), and forming a parallelogram whose sides mark the 

two directions in space separated by angle 2co. Maxwell (1873, v. 1, art. 219, p. 160) designated these lines 

as "axes hi and h2 ." Either axis may be regarded as the 

direction of a constituent dipole with its antiparallel 

counterpart; the other axis is then the line along which 

the dipoles are infinitesimally separated from one 

another to generate the quadrupole. But in the general 

case, neither of the lines is an axis of the resulting field. 

I find it preferable to denote them not as axes but 

rather as the primitive lines of the quadrupole.

The same array of poles may also be regarded as a 

pair of orthogonal linear quadrupoles of contrary 

kinds, with the double-strength interior poles superposed and neutralizing one another (fig. 2C). This interpretation has the merit of showing unambiguously, 

and giving physical meaning to, two authentic axes of 

the field—the bisectors (as noted by Umov, 1904) of 

the acute and obtuse angles between the primitive 

lines. The bisectors may be appropriately designated 

the cardinal axes. The linear quadrupoles associated 

with them are red for the acute angle and blue for the 

obtuse angle, thereby imparting contrasting field 

characteristics to the corresponding regions. We shall 

see later that the cardinal axes can be used in a unique 

way to examine secular change. An orthogonal system 

(Howe's principal axes; Howe, 1939) is formed by the 

cardinal axes along with a third axis (the normal to the 

characteristic plane) that is here called the primary 

axis. When the constituent linear quadrupoles are of 

equal magnitude, the resulting quadrupole is a normal 

one, and in that case the primitive lines are likewise 

mutually perpendicular, forming angles of 45 ° with the 

cardinal axes.

The concept of orthogonal linear quadrupoles with 

neutralized central poles is fully valid for the rhombic 

case; the poles have equal strength with separation adjusted to achiev.e the differing strengths of the red and 

blue linear quadrupoles. That is, they mark the vertices of a rhombus.

Equation (3) applies to a linear quadrupole alined 

with the axis of coordinates. It can be adapted to any 

attitude by replacing 8 with an expression denoting 

the angle between the point of observation and the 

desired quadrupole axis. If we let 00 and AQ denote the 

colatitude and longitude of the axis at P0 of a red linear 

quadrupole (fig. 3), we can readily write an equation for 

its potential distribution on the sphere. Then if we let 

0t and A! similarly refer to the axis at P^ of a blue linear 

quadrupole, making a right angle with the red one, we 

can form an expression for the potential of the blue 

quadrupole; and by combining these two expressions 

we have an equation for their combined potential; it is

V = (MISr3) { (1 + 3 cos 20) (1 + 3 cos 2 00

- 4U cos2 0t)

+ 8U sin2 0t + 12 [sin 200 cos (Ao - A)

- U sin 20t cos (At - A)] sin 20 

+ 3 [(1 - cos 200 ) cos 2 (Ao - A)

- 2U (1 - cos 20t cos2 (At - A)] [1 - cos 20]}. (7)

Equation (7) describes a general quadrupole in terms 

of two linear quadrupoles, one red and one blue, at 

right angles to each other. No other restrictions are imposed on the attitudes of the cardinal axes. Here, U is 

the ratio of the blue to the red linear quadrupole moment, M is the moment of the red linear quadrupole,

PI

FIGURE 3.—Linear quadrupoles directed toward P0 and Pi. 00, AO, 

and 0!, A, are the colatitudes and 

east longitudes of the red and 

blue quadrupoles, respectively.

P PO
ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

and —MU that of the blue one. In this equation the 

configuration is governed by the five parameters M, U, 

00 , 61, and AQ. Parameter Xl is not independent but is retained to simplify the equation; it is related to the others by

sin (A! — AO) = sin £ /sin 0lt (8)

where £ is the true azimuth of the line extending from 

P0 to Pl given by

cos f = cos 0!/sin Q0 . (9)

If we set 00 and 0: at 90 ° in equation (7), we describe a 

general quadrupole that is in the equatorial plane; and 

it may be further particularized by setting AO at zero 

and A! at 90 °, thus placing the red linear quadrupole in 

longitude zero and the blue one in longitude 90 °. Equation (7) can then be reduced to

V = (M/r3) {3[1 - (1 + C7)sin2 A]sin2 0 + U - 1} (10) 

and this may be further reduced to

V = (d2/4r 3) [3(cos 2A + cos 2co)sin2 0

- 2 cos 2co], (11)

where

and

62 = 2 M(l + £7), (12)

cos 2co = (1 - £/)/(! + U). (13)

For a normal quadrupole in the same attitude we need 

only set U = 1, reducing equation (10) to the form

Vn = (3Mn/r3) (cos 2A sin2 0). (14)

For the quadrupole represented by equation (14), the 

primitive lines have longitudes of 45° and 135°.

The total intensity of the fields stipulated by equations (10) and (14) may be written, respectively, as

F = (3d2 sin 0/2T-4) {sin2 2A + qf cos2 0

+ [—cos 2o> esc 0 (15) 

+(3/2)gnsin0]2}0 - 5,

and

F = (3M/r4) (4 + 5 sin2 0 cos2 2A)°- 5 sin 0, (16)

where

il = cos 2A + cos 2o>.

Table 1 compares the expressions for the various elements of the field of a dipole, of a linear quadrupole 

along the polar axis, and of a normal quadrupole in the 

equatorial plane, as stipulated for equations (1), (3), 

and (11).

SOME CHARACTERISTICS OF QUADRUPOLE FIELDS

The field of every quadrupole differs fundamentally 

from a dipole field in that the dipole field is of odd 

degree and shows asymmetry of sense (inward versus 

outward) along any straight line going through the 

center, whereas the quadrupole is of even degree and 

does not show this asymmetry. A linear quadrupole 

has a characteristic sign, and a rhombic one may be 

dominated by the sign of its stronger linear-quadrupole constituent. The field of a rhombic quadrupole, 

or of a normal one, has positive and negative aspects, 

distributed in relation to its cardinal axes.

A general quadrupole cannot be fully described 

simply by specifying its primitive lines and its (unsigned) moment. It is necessary somehow to distinguish between the two sorts of quadrupoles that can 

exist with identical directions of the primitive lines. 

The two are related in that the positive aspects of one 

are exactly matched by the negative of the other. The 

situation may be examined as follows: If we look at the 

parallelogram that represents the four point poles of a 

rhombic quadrupole, the signs of the point poles alternating as we go round the figure, they may (as has been 

noted) be coupled in either of two ways to form the two 

dipoles as shown in figure 2. The angle 2co represents 

the interior angle of the rhombus at one of the positive 

or north-seeking poles. Thus, 2cu is unambiguously 

either obtuse or acute. (The definition given earlier 

made 2o> the angle from the forward direction of either 

dipole to the direction in which the other one is displaced from it.)

If 2o> is obtuse (fig. 2B), the dominant sign of the 

quadrupole is negative. In this case the points bearing 

the <S label fall in the obtuse sectors of the characteristic plane, and the neighboring lines of force are outward-directed. Each of these statements must be altered appropriately if 2o> is acute (fig. 2A). To avoid 

ambiguity, therefore, we need to state whether 2o> is 

obtuse or acute, indicating which of the angles between 

the primitive lines is 2o> and which is its supplement.

If instead of the primitive lines one is dealing with 

their angular bisectors, that is, the cardinal axes (fig. 

2C), the distinction between the axes is clear, in that 

one of them marks the positive (red) and the other the 

negative (blue) linear quadrupole of an orthogonal pair. 

The ambiguity now takes a different form. It is now 

necessary to stipulate not only which is which but also 

the strength of each, or the strength of one and their
THE GENERAL QUADRUPOLE 

TABLE 1.—Expressions for the fields ofdipoles and quadrupoles

[M is the magnetic moment, r is the radial distance, 8 is the angle of the position vector, A is the longitude. See text for complete definitions.)

Function

Dipole,

southward

directed

Negative (blue)

linear 

quadrupole

Normal 

quadrupole1

TanD

Tan/-

Common factor in component exCoefficients:

For north field —

For east field —

Resultant horizontal field 

For downward vertical field

For component parallel to polar 

axis (positive northward)

For component normal to (positive toward) polar axis and copxQ.iiQ.r witn. it ---------------——«~

0

2ctn 6

M/r3

sin 6

sin 6

2 cos 6

1-3 cos2 6

1.5 sin 26

0

3 cos 20

2 sin 26

2 sin 26

0

|2 sin 20| 

3 cos 26 + 1

(1-5 cos 20) cos 6 

(5 cos20 + 3) sin 0

sec0 tan 2A 

-1.5(csc2 0 sec2 2A- I)'06

3 M«/r<

sin 20 cos 2A 

2 sin 0 sin 2A

2(1 - sin2 0 cos2 2A)° 6 sin 0 

-3 sin2 0 cos 2A

2.5 sin 20 sin 0 cos 2A 

0.5(5 cos 20 - 1) sin 0 cos 2A

1 In equatorial plane, with red linear constituent in the zero meridian.

ratio, U, in order to round out the rhombic quadrupole 

specification.

For a normal quadrupole (primitive lines orthogonal 

and U = 1), the quadrupole moment is not predominantly of either sign, and the two possible varieties of 

the quadrupole that have the same primitive lines are 

alike except for a 90° rotation. Thus, equation (14) is 

affected only by a sign change when A is altered by 90 °. 

In order to complete the description of a normal quadrupole that is framed by its primitive lines, we must 

stipulate which of the quadrants of the characteristic 

plane are positive by designating which of the cardinal 

axes is the red one.

For the general quadrupole, even though the primitive lines taken separately are devoid of any sense 

characteristic, such a characteristic is implicit in their 

interrelationship. Thus, notwithstanding their inherent ambiguity, they may be treated in such a way as to 

bring out the distinct and unambiguous character of 

the cardinal axes—one red and one blue. Theoretically 

it would be possible for the cardinal axes of the geomagnetic quadrupole to be unaffected by secular 

change, even while the primitive lines substantially 

changed their angular separation, provided the motions of the primitive lines were at the same rate in opposite senses. Such a change would affect £7, the ratio 

of blue to red linear quadrupole moments.

The field developed by the normal quadrupole of 

equation (14) may be examined by constructing a set of 

schematic diagrams to map its several characteristics. 

In view of the symmetries discussed later, it is sufficient to deal with a semilune comprising 45 ° in longitude and 90° in latitude, as indicated by the shaded 

area in figure 4. The characteristics of the normal 

quadrupole field may be visualized or deduced with the 

aid of figures 5-7, showing the surface distribution of 

the conventional magnetic elements which would be 

found on the sphere.

In both the normal and rhombic quadrupole, the field 

has genuine mirror symmetry about each of the three 

planes whose intersections are the primary and the cardinal axes; the zero —180° and the ±90° meridians are 

both agonic lines (reminiscent of a naive medieval view 

of the real isogonic pattern). There are two N and two S 

dip poles, all on the equator where it crosses the aforesaid meridians. With four ordinary dip poles, it follows 

that there must be two "false" poles where the horizontal component vanishes but where the magnetic 

meridians do not converge; these are found at the geographic poles. The meridian circles defined by the longitudes of the primitive lines are loci of ±90° declination. The equator of the coordinate system is likewise a 

line of ±90° declination, with eight segments of alternating sign.
ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

FIGURE 4.—Attitude of quadrupole shown in figures 5-7.

Other properties of the normal quadrupole (as 

distinct from the rhombic one) are of interest: The 

"false" poles (fig. 6) are null points, where all components of the field vanish. The magnetic meridians 

containing the primitive lines are planes of quasisymmetry, in that although the configurations are mirrored in them, the signs are continuous (not inverted) 

in passing across the reflecting surface, just as they 

are for the field pattern of a single dipole in respect to 

its equatorial plane. These meridians are aclinic lines, 

and four of the eight junctions along the equator 

(where it is crossed by the aclinic meridians) are maxima and minima of the east component, Y (fig. 7), and 

also maxima of the horizontal component, H (fig. 6), 

the value there being two-thirds of the total-intensity 

maximum attained at the other four junctions (the dip 

poles). The north component, X (fig. 7), has four maxima and four minima, all falling on the agonic meridians at latitudes ±45°. These are saddle points of the 

H isolines (fig. 6), with values equal to one-third of the 

maximum total intensity, F (fig. 6).

The quadrupole part of the Earth's present magnetic 

field is very nearly that of a normal quadrupole. The 

foregoing description, however, pertains to a hypothetical one in the equatorial plane, whereas the 

Earth's quadrupole has a quite different attitude.

CHARACTER OF LINES OF FORCE

For a dipole or linear quadrupole alined with the polar axis, the lines of force are the traces of meridional 

planes in surfaces of revolution described by equations 

(5) and (6), just as the magnetic meridians are traces of

the same planes in the sphere. To illustrate such lines 

of force, it suffices to depict only one for the dipole and 

one for the quadrupole, figures 8 and 9, respectively. 

Others are replicas of the one curve, expanded or contracted appropriately.

For a normal quadrupole in the equatorial plane as 

stipulated by equation (14), the character of the lines of 

force, which are now nonplanar curves, may be deduced from the expressions for the angular elements D 

and /. A line of force emanating from a point on the 

sphere with small positive latitude and longitude will 

trace a path reaching almost to the geographic pole 

before it crosses the 45 ° meridian and returns to the 

sphere at a corresponding point near latitude zero and 

longitude 90 °, as shown in figure 10.

Both rising and falling portions will conform to the 

same surface of revolution. The slope of any such surface where it traces a given parallel of latitude on the 

sphere is the ratio of the vertical magnetic component 

to the (horizontal) north magnetic component, or —1.5 

tan 9. This function is independent of longitude, so 

that any line of force emanating from or returning to 

the sphere in that latitude will conform to this same 

surface; the shape is a useful aid to the perception of 

the character of the lines of force. By an integration 

similar to that in developing equations (5) and (6), we 

can develop the equation of the surface of revolution, r:

r = k(cos 0)15. (17)

Any given one of these surfaces of revolution will contain a family of lines of force, forming a nest of halfloops that are symmetrical about the meridian marked 

by the primitive line.

Since the expressions for D and / are functions of 9 

and A only, not of r, each line of force has its direction in 

space at a given point, the same as for any higher and 

lower lines of force of the same latitude and longitude. 

Hence, all lines of force intersecting a given radius vector fall on the surface generated when that radius vector moves so as to trace on the sphere a magnetic meridian curve.

The equation of a magnetic meridian curve can be derived as follows: First we note than tan D is the reciprocal slope of such a curve at any point. To allow for 

the convergence of the meridians we divide tan D by 

cos 9 and write

dA

d0 —tan D esc 9

—2 tan 2A . 

sin 29

(18)

the last step is accomplished by substituting for tan D 

the expression in the last column of table 1. Equation
THE GENERAL QUADBUPOLE

FIGURE 5.—Normal-quadrupole field on the sphere: V, potential; Z, vertical intensity; and

I, inclination.

'False" pole

Magnetic meridians

D

(5 cos2 2X + 4)0.5

F

2.0

FIGURE 6.—Normal-quadrupole field on the sphere: magnetic meridians; D, declination; and horizontal (H) and total (F) intensities.

X

FIGURE 7.—Normal-quadrupole field on the sphere: 

X, north component; and Y, east component.
ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

Axis of symmetry

Plane of quasisymmetry

FIGURE 8.—Dipole field line.

FIGURE 9.—Linear-quadrupole field line.

(18) states the basic condition that a magnetic meridian curve must satisfy. It is readily integrated to give

sin 2A tan2 0 = k, (19)

where the constant k is the value of tan2 9 for A = 

45°—that is, for the minimum 0. Thus (19) is the equation of the magnetic meridian curves on the sphere; it 

also describes in space the singly curved surfaces 

whose intersections with the surfaces of revolution described by equation (17) are the lines of force of a normal quadrupole having its attitude defined by equation (14).

INTERCONVERSION OF QUADRUPOLES

With U = 0, equation (7) gives the potential of a red 

linear quadrupole:

V = (M/8r3) {(1 + 3 cos 20)(1 + 3 cos 200 ) 

+ 12[sin 200 sin 20 cos(Ao - A)] 

+ 3[(1 - cos 200 )cos 2(Ao - A)] (1 - cos 20)}. (20)

Iat0° 

,long 90°

< Equatorial plane

-Iat0° 

long 0°

FIGURE 10.—Normal-quadrupole field line.

Now if we set 00 and A,, first at 90 ° and zero, respectively, then both at 90 °, then both at zero, we have expressions for the potentials of three orthogonal linear 

quadrupoles, and their sum is found to be

V, = (M/8r3) {(4 - 2 - 2)(1 + 3 cos 20) 

+ 3(2 cos 2A)(1 - cos 20) 

+ 3[2(-cos 2A)) (1- cos 20)}=0 (21)

That is, the combination of three equal, orthogonal linear quadrupoles of the same sign is a nullity. Consequently, two equal mutually perpendicular linear 

quadrupoles are rigorously equivalent to a third one of 

the contrary sign (and of the same magnitude), orthogonal with both given ones. It follows that if two 

normal quadrupoles of equal strength are combined in 

such a way that their constituent linear quadrupoles of 

one sign coincide and those of the other sign are
RELATIONS WITH SPHERICAL HARMONIC ANALYSIS

mutually perpendicular, the result is a new linear quadrupole of triple strength. Hence, any linear quadrupole 

can be resolved into two orthogonal normal quadrupoles.

It can be shown that any set of three normal quadrupoles having their constituent linear quadrupoles 

along orthogonal axes can be reduced to two such normal quadrupoles.

The randomly oriented quadrupole, whether it is linear, normal, or rhombic, cannot be resolved into an arbitrarily directed orthogonal set of three linear or of 

three normal quadrupoles. However, any quadrupole 

or combination of superposed quadrupoles may be resolved into a set of five elements—namely three normal quadrupoles in the three coordinate planes, each 

with its cardinal axes directed at 45 ° from the coordinate axes, plus two more normal quadrupoles in two of 

the coordinate planes, directed along the axes. Furthermore, the general quadrupole can always be reduced to a pair of orthogonal linear quadrupoles (not 

necessarily equal and of unlike sign) if the axes are 

suitably chosen.

A rhombic quadrupole may be resolved in several 

ways that seem quite different but are in fact equivalent. The simplest way is perhaps the above-mentioned 

combination of two orthogonal linear quadrupoles of 

contrary sign. Suppose the red one is the stronger so 

that U < 1. We may now add a set of three orthogonal 

blue linear quadrupoles of equal magnitude, two of 

them alined with the given ones. Let their strength be 

such that the given (unequal) blue and red components 

are now replaced with equal ones. Since the added set 

is a nullity, the result is merely a different way of expressing the original quadrupole. The new model has 

three linear quadrupoles, two blue and one red. The 

red one matches the strength of one of the blue ones, 

forming a crossed pair and comprising a normal quadrupole, whereas the other blue one has a magnitude 

that is (1 — £/)/(! + U) times that of either member of 

the crossed pair.

If we now add another set of three blue ones, this 

time choosing the magnitude to match the members of 

the crossed pair, the red one will be cancelled and the 

result will be an orthogonal combination of but two linear blue ones with a magnitude ratio again given by £7, 

and still rigorously equivalent to the original quadrupole.

Flux emanating from a centered normal quadrupole 

exits from the enclosing sphere in the red sectors and 

reenters it symmetrically in the blue ones. For a rhombic quadrupole, this kind of quasisymmetry is lost, 

though true symmetry about each of the three principal planes is retained. If U is reduced from unity, there 

is increasing departure from the regularity of the initial pattern until for U = 0 the quadrupole becomes a 

red linear one, alined along what was the red axis of the 

original normal quadrupole, which axis now becomes 

an axis of circular symmetry. Conversely, if U is increased to infinity, the quadrupole becomes a linear 

blue one alined on the former blue axis. Thus, U is a 

direct index of the character of the geomagnetic quadrupole field, and as such is grossly descriptive of the 

overall nondipole field, since, as will be seen, the latter 

is dominated by the quadrupole constituent.

RELATIONS WITH SPHERICAL HARMONIC

ANALYSIS

As usually undertaken, a mathematical description 

of the internally generated magnetic field on the surface of a sphere or spheroid involves potential analysis 

by means of spherical harmonic functions. The result 

of such analysis is a set of coefficients of which the 

three first-degree terms represent a dipole, the five second-degree terms a quadrupole, the seven third-degree 

terms an octupole, and so on. The quadrupole terms 

dominate the nondipole field, one or more of them being usually larger than any of those of higher degree.

Equations (3) and (14) gave the potential distribution 

of a linear and of a normal quadrupole with specified 

attitudes. It is easy to write corresponding expressions 

for other axis-related quadrupoles. When these expressions are compared with the expansions of the associated Legendre functions for the second-degree 

terms of the spherical harmonic analysis (see Chapman 

and Bartels, 1940, p. 639, for notation used herein), the 

following conclusions are established:

1. The gl term depicts a linear quadrupole alined with 

the polar axis of the coordinate system. As noted 

on page 2, this constituent of the quadrupole part 

of the field being analyzed is zonal.

2. The gl term denotes a normal quadrupole with attitude as specified by equation (14)—that is, one in 

the equatorial plane with its red and blue cardinal 

axes in the 0° and 90° meridian planes, respectively.

3. The h\ term denotes a normal quadrupole in the 

equatorial plane but rotated 45° from the 

preceding one. The g\ and h\ terms together depict 

a single normal quadrupole in the equatorial 

plane, having its cardinal axes at an intermediate 

orientation, the red one in longitude Vi arctan 

(h|/g|). This is the sectorial part of the quadrupole 

field. As long as g\ and h\ are excluded, incorporating the g\ term will have no effect on the 

characteristic plane or on the cardinal axes, but it 

will cause the resultant to be a rhombic 

quadrupole, with U governed by gl.
10 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

4. The g\ and h\ terms represent normal quadrupoles 

whose characteristic planes are the zero meridian 

and the 90° meridian, respectively, each with its 

cardinal axes tilted 45 ° from the equatorial plane 

of the coordinate system. Together, the two terms 

represent a normal quadrupole (the tesseral part) 

with similar attitude in a meridional plane whose 

longitude is given by arctan (h2lg\). The inclusion 

of nonzero values of g\ and h\, along with the other 

three terms, will cause the primary axis of the resultant quadrupole to be inclined from the polar 

axis in the aforementioned meridional plane, and 

(unless g\ = h\] will contribute to U, so that the 

resultant quadrupole may be rhombic even if g\ is 

zero. With g\ ? h\ and #2^0, the effects on U 

might balance out, so the quadrupole could still be 

a normal one.

It will be seen that with trivial substitutions regarding (1) and (2), these five elements comprise the set 

earlier mentioned as sufficing to portray the field of 

any quadrupole. Note further that by rotating the coordinate system to bring the polar axis into coincidence with the primary axis of the unresolved 

quadrupole, the tesseral part (the g\ and h\ terms) 

could be extinguished, leaving only zonal and sectorial 

parts.

When a general quadrupole is dissected by spherical 

harmonic analysis, the first effect is to segregate the 

zonal part—the linear quadrupole corresponding to the 

g\ term. What then remains is a combination of two 

normal quadrupoles, each having a potential distribution that comprises four segments with alternating 

signs delimited by a pair of orthogonal great circles. 

For the sectorial quadrupole both circles are meridians; for the tesseral quadrupole one is the equator 

and one is a meridian.

The sectorial-tesseral distinction is clearly an artifact of the way the original unresolved quadrupole 

relates to the coordinate system. In this paper the 

designations "sectorial" and "tesseral" are used in the 

sense in which they were originally introduced by Maxwell (1873) rather than the variant usage of Jory (1956) 

who related them to the centered-dipole axis. (See 

p. 6).

The dipole and quadrupole, as specified respectively 

by the three first-degree and five second-degree spherical harmonic coefficients, are both artifacts in that all 

the parameters depend upon the choice of coordinate 

axes. The three dipole vectors are readily compounded 

to form the single vector that represents the aggregate 

dipole moment, the magnitude of which is invariant 

with respect to choice of axes. The quadrupole 

moments cannot be compounded in the same way as 

simple vectors. However there are means of treating

the second-degree coefficients to express the resultant 

quadrupole in a unitary form that is less dependent on 

the choice of coordinate axes. Thus, we might end up 

with a specification of the two equal constituent 

dipoles of the quadrupole, along with the value of 2cu 

and the directions of the primitive lines. This is the 

method most commonly used (for example, Winch and 

Slaucitajs, 1966). Another scheme would specify the 

moments of the essential two orthogonal linear quadrupoles and the directions of their axes (the cardinal 

axes), as set out in equation (7).

An expression describing the distribution on the 

sphere of the aggregate potential of all five axis-related 

quadrupoles can be obtained by simply combining the 

five separate expressions. Now, if we expand the terms 

of equation (7), insofar as they involve functions of 

(Ao — A) or (A! — A), and sort out the terms in cos A and 

sin A, we obtain an equation which can be compared 

directly with the one that depicts the composite field of 

the five axis-related quadrupoles; this comparison 

leads to the following equations:

(22)

(23)

(24)

(25)

= 1 - U + 3(cos 200 - t/cos 20,), 

3-° ' 5 g\ rVM = sin 200 cos AO - U sin 20, cos Alt 

3-° 5 h\ rVM = sin 200 sin AO - U sin 20, sin A,,

2 .3-0 5 gi ,3/M _ cos 2Ao (i _ Cos 200 ) 

- t/cos2A, (1 -cos 20,),

and

2 • 3-°' 6 Af rVM = sin 2A,, (1 - cos 200) 

- t/sin2A,(l-cos20i). (26)

From these equations we can obtain directly the 

spherical harmonic coefficients corresponding to any 

general quadrupole that is specified by its angular 

parameters in terms of an orthogonal pair of linear 

quadrupoles of unlike sign, with moments in a 

specified ratio.

Consider next how to find the coordinates of the 

principal axes (the two cardinal axes and the primary 

axis) when those of the primitive lines are known. 

Referring to figure 11, routine manipulation of 

triangles will yield the equations

cos 2o> = cos HI cos u2 + sin u, sin u2 cos (A2 — A,,), (27) 

cos 00 = (cos u2 + cos u,)/2 cos cu, (28) 

cos 0, = (cos u2 — cos u,)/2 sin co, (29)
RELATIONS WITH SPHERICAL HARMONIC ANALYSIS 11

FIGURE 11.—Relations of the cardinal axes to the primitive lines, and identification of angular parameters. See p. 3 for explanations of

labels.

COS (An — A,,) =

_ COS CO — COS I/! COS 00 ——————;——————;——-—————'

sin M! sin 60

(30)

COS (A, —

_ — ————:———:—o———' sin co + cos ul cos 0t 

sin 1/1 sin 60

cos Y] = cos

cos 6P = sin 60 sin rj,

cos (Ap — AO) = — cot 00 cot 6P.

(32)

(33)

(34)

These are supplemented by equation (12) and a variant 

of equation (13); namely

U= 1 — cos 2co 

1 + cos 2co

(35)

A more troublesome requirement is that of finding 

the principal axes when the second-degree spherical 

harmonic coefficients are known. This might be done 

by an inverse, iterative application of equations (22) to 

(26), but a closed solution, if attainable, would be less 

cumbersome. Umov (1904) developed a closed solution 

for the analogous problem of finding the primitive 

lines from the spherical harmonic coefficients, and we 

can apply his procedure explicitly and then use equations (27) to (31) to get the desired parameters. However, for this purpose it is helpful to transform two of
12 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

his equations (the last two in the group he labeled "I") 

into the form

cot MI = x =

h21 cos AI —

and

cot uz= y =

., .- ,, 1.5M2 sm(A2 —

g21 sinA2 — h21 cos A2 

1.5M2 sin(A2 — Aj)

(36)

(37)

Umov's procedure involves the numerical solution of a 

cubic equation having three real roots, the well-known 

irreducible case. Although algebraic solution of this 

cubic is impossible, a trigonometric procedure may be 

used (Dickson, 1922, §47). Only one of the resulting 

roots will yield a meaningful value of cos 2co.

QUADRUPOLE SECULAR CHANGE 

AND ITS CONSERVATIVE ASPECT

A distinction that is rather technical, yet still fundamental, may be recognized between the dipole on the 

one hand and the quadrupole and higher multipoles on 

the other. Under secular change, the dipole axis could 

move on the sphere along a track which might or might 

not trace a great-circle path. Since there is but one 

axis, its instantaneous direction of motion contains no 

clue to the curvature of the path it traces out. Only by 

comparing the motion at two successive epochs could 

we observe such curvature and locate the center of gyration. A multipole, on the other hand, has at least 

three axes (unless it is a linear one). By assessing the 

motion of any two of its axes we can fix the center of 

gyration describing the drift from the secular-change 

data of a single epoch.

It is well known that the secular change shows predominantly regional characteristics, although it is now 

also accepted that an important part must have a global character. To evaluate and isolate the global features may be a vital step in gaining a fuller understanding of the remaining constituents.

What about the dipole field? The centered dipole 

with its wandering and reversals is important in paleomagnetic studies. However, recognition is increasing 

that dwelling exclusively on the dipole presents the 

hazard of a simplistic approach that can ignore significant phenomena (Harrison, 1975). Here, our concern 

with the secular change pertains to structure observed 

in historic times; and in this context the secular drift of 

the dipole is difficult to pin down. The centered dipole, 

though varying somewhat in strength, is almost fixed

in direction. That is, its very slow drift in position contributes a nearly negligible part of the overall patterns 

of the secular-change field. Thus, the multipoles (terms 

of degree 2 or higher) must be the bearers of the dominant features of the secular-change configurations.

The study of the character of secular change thus 

seems to link up with the study of the nondipole 

field—a task that has more than one approach. A currently favored and promising technique is to model the 

nondipole field (or even the entire field) by postulating 

an array of current loops just within the core boundary—or as more expediently approximated, by assuming a distribution of satellite dipoles. However, if we 

seek to focus on the global aspect of secular change, it 

may be also instructive to examine the nondipole field 

by the alternative approach of studying the behavior 

of the centered multipoles of higher complexity. In this 

approach, the quadrupole, as the dominant constituent 

of the nondipole field, is most likely to epitomize any 

global features that may be present in the secular 

change. Hence, the quadrupole is clearly the first thing 

to study. The techniques developed may be adaptable 

to some of the higher multipoles as well. The secularchange problem, then, offers a distinct and cogent incentive for probing the character and behavior of the 

geomagnetic quadrupole.

The quadrupole may exhibit the effects of secular 

change in one or more of the following ways: (1) the 

combined quadrupole moment may change; (2) the 

three principal axes may undergo systematic rotation 

about a fourth axis; or (3) Umay change. If the effect is 

(2), the primitive lines would necessarily reflect it; but 

these lines would be affected in a different way by any 

change in U— they would undergo opposing drifts in 

the characteristic plane, which might obscure the situation. To study the global constituent we should distinguish between effects (2) and (3).

With the lapse of time, rotation of a field having axial symmetry, such as that of a centered dipole or of a 

linear quadrupole, will be scarcely perceptible in the 

pattern of the change parameters unless the axis of 

such rotation makes a considerable angle with the axis 

of symmetry; but rotation of the field of a normal 

quadrupole, or of a rhombic one that is not much different from a normal one, will be apparent no matter 

where the axis of rotation lies. That is, the surface field 

on the sphere has a two-dimensional configuration, so 

that any kind of rotation is bound to displace some of 

the zero lines of the pattern. Consequently, if the 

secular change has any global constituent we may expect it to be prominently manifested as an angular 

drift of the principal axes of the quadrupole, and 

perhaps of the higher multipoles as well.
RELATIONS WITH SPHERICAL HARMONIC ANALYSIS 13

THE ECCENTRIC DIPOLE

The resultant of the vectors represented by the three 

first-degree terms of the spherical-harmonic expansion 

is the well-known inclined, centered dipole of best fit. 

It was shown by Schmidt (1918) that an eccentric 

dipole is uniquely defined by these same three terms in 

conjunction with the five second-degree terms (those of 

the quadrupole), and that this eccentric dipole has the 

same moment and the same attitude in space as the 

centered dipole based on the first-degree terms alone.

Six parameters suffice to describe Schmidt's eccentric dipole. These could be, for example, the three orthogonal components of the centered dipole and the 

three orthogonal components of the vector stipulating 

the displacement of the eccentric dipole away from the 

Earth's center. The field of the eccentric dipole is not 

identical with that of the combined quadrupole and 

centered dipole with which it corresponds (governed by 

eight coefficients). Nevertheless, it is of some interest 

to consider just how the displacement of the eccentric 

dipole from the center is affected by manipulating the 

quadrupole.

When a linear quadrupole is superimposed on a dipole (both centered in the sphere), the eccentric dipole 

that best approximates the composite field is one that 

is displaced, (1) along the dipole axis if the quadrupole 

is either alined with or perpendicular to that axis, or (2) 

perpendicular to the dipole axis if the quadrupole is 

tilted 45 ° from it. The effect of a normal quadrupole (or 

of a rhombic one) may be examined by considering the 

separate effects of its constituent linear quadrupoles.

FERTILE AND STERILE PARTS 

OF THE QUADRUPOLE

We have seen how the general quadrupole is resolved 

(by spherical harmonic analysis) into five constituents 

governed by the coordinate axes. A similar resolution 

could be conducted relative to any set of orthogonal 

axes. Thus, if we define the polar axis to coincide with 

the centered dipole, and define the other axes so that 

one of the resulting planes would contain the displacement vector of the eccentric dipole, the resolution 

would break up the quadrupole into "fertile" and 

"sterile" parts, in the sense that the fertile parts would 

be responsible for the displacement of the dipole and 

the sterile parts would not, though of course they 

would still be a necessary part of the description of the 

original quadrupole field. The sterile parts ((2) and (3), 

p. 9) make up a normal quadrupole in the equatorial 

plane of the dipole, whereas the fertile parts ((1), (4), 

p. 9) constitute a rhombic quadrupole in the plane defined by the dipole axis and the displacement vector. 

Separately, the fertile constituents comprise a linear

part whose axis coincides with the dipole axis and a 

normal part in the plane just mentioned, with its cardinal axes 45 ° from the dipole axis. The displacement 

of the dipole along its axis is proportional to the 

strength of the linear part, but the displacement of the 

dipole normal to its axis is proportional to the strength 

of the normal part.

If the field subjected to spherical harmonic analysis 

happens to be that of an eccentric dipole, the sterile 

part of the quadrupole will be zero. Further, if the 

dipole is displaced only along its axis (if it is a radial 

dipole), the quadrupole will be only a linear one along 

the same axis. If the dipole is displaced only in its 

equatorial plane, the quadrupole will be a normal one 

coplanar with the dipole, with its cardinal axes at 45 ° 

with the dipole axis. (This is approximately true of the 

Earth's field.)

It is now clear why Chargoy (1950) found that if one 

deducts from the geomagnetic field its eccentric-dipole 

component, the residuum has for its quadrupole part a 

normal quadrupole in the plane perpendicular to the 

dipole. As noted by Macht (1950), this result is not a 

fortuitous circumstance of the geomagnetic field but is 

a mathematical necessity; for the dipole by its eccentricity allows for a certain quadrupole constituent, and 

the only remaining quadrupole constituent is a normal 

one in the equatorial plane defined by the dipole.

When the sterile part of the quadrupole is zero, the 

six parameters of the eccentric dipole describe a field 

corrresponding as nearly as possible to that of the 

eight parameters of the centered-dipole-plus-quadrupole model, in the sense that no quadrupole component 

is neglected. The eccentric-dipole field, of course, embodies not only the centered dipole plus the fertile 

quadrupole, but also an infinite series of higher 

multipole terms, and there is no reason to expect these 

components to be at all similar, or even related, to the 

terms of corresponding degree in the original spherical 

harmonic analysis, since the eccentric-dipole displacement is governed strictly by the parameters of 

the fertile quadrupole alone. Only if that displacement 

is large, however, would a large number of multipole 

terms be required to approximate well the eccentricdipole field.

The eccentric dipole specified by Schmidt's (1918) 

procedure, based on the first eight terms of the spherical harmonic expansion, is the dipole that best simulates the field represented by those terms. It is usual 

to consider that this dipole is likewise the one of best 

fit for the field of higher approximation represented by 

the more detailed analysis. Bartels (1936, p. 230) 

wrote, "If the field of a magnet like the Earth is to be 

approximated by that of a dipole not necessarily
14 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

situated at the Earth's center, it can be shown that 

there is one point C in the magnet, called the magnetic 

center, which gives the most suitable location." 

Schmidt's eccentric dipole is defined so as to fall at the 

magnetic center, defined originally by Kelvin (1872, 

p. 374).

As a matter of fact, C is usually specified to be determined solely by the first- and second-degree terms; 

thus it is not necessarily the site of the best fitting 

dipole if the higher terms are to be considered. This 

may be seen most readily if we simplify the circumstances by supposing that the field being modeled is 

zonal, that is, symmetrical about the axis of coordinates, hence capable of being completely depicted by 

the terms of various degrees of order 0. In this case all 

terms are zero except #?, gl, gl, — gm and Schmidt's eccentric dipole will be displaced along the axis through 

a distance determined by gl. But if higher terms are 

considered, it is clear that each term of even degree 

will represent an incremental distribution of potential 

having two N or two S dip poles at the ends of the axis 

(like the N or S dip poles of the quadrupole term) and 

will result in a separate and independent displacement 

of the dipole along its axis; so the dipole that best approximates the composite field of the whole series will 

be displaced from the center by a greater or smaller 

distance than the one that depends only on the terms 

of degrees 1 and 2, unless it happens that the 

displacements due to the higher terms add up to zero.

The fallacy of regarding the eccentric dipole determined from the first- and second-degree terms as the 

dipole that best fits the actual field has also been 

pointed out by Bochev (1965). However, the discrepancy may well be so small as to have theoretical interest only.

OTHER REMARKS CONCERNING 

ECCENTRIC DIPOLES

Given the six parameters of any eccentric dipole, one 

can evaluate, by means of an equation given by Hurwitz (1960), those of its nearest equivalent centereddipole-plus-quadrupole combination, thus finding the 

spherical harmonic coefficients of degrees 1 and 2 (of 

the eccentric-dipole field) without the necessity of conducting a spherical harmonic analysis. (The Hurwitz 

equation is not restricted to the first two degrees but 

these are of special interest here.) And if more than one 

eccentric dipole were assumed, the resulting coefficients could be determined for each one, and they could 

be summed to get those of the synthetic field generated by all the dipoles, insofar as it was reproducible 

in terms of eight coefficients. However, the individual 

quadrupoles so derived would evidently be of a special 

character in that each of them would be a rhombic one

in the plane defined by the particular dipole and its 

displacement vector.

This constraint on the second-degree terms (limiting 

the quadrupole to a plane established by the eccentric 

dipole) would be operative no matter how the coefficients were derived and no matter how many higher 

degree coefficients were also determined. It implies interrelations among the five second-degree coefficients, 

such that g\ and h\ would vanish if the coordinate axis 

were chosen to conform with the dipole. Nevertheless, 

such a capability is a useful tool for studying satellitedipole field models.

The quadrupole so constrained is coplanar with the 

dipole. The longitude of its characteristic plane (again 

assuming the coordinate system chosen to conform 

with the dipole) is equal to arctan (h\lg$; and if in addition gl is zero, the quadrupole is a normal one. Alternatively, if all the second-degree terms except gl are 

zero, the dipole and quadrupole (now linear) are on the 

coordinate axis. More generally, for any radial dipole 

the quadrupole will be a linear one coaxial with it.

Although the five second-degree coefficients so derived constitute a unique and correct reflection of any 

given eccentric dipole, they do not fully depict the 

quadrupole aspect of any more generalized field of 

which the eccentric dipole may be only an approximation, inasmuch as they fail to incorporate the former's 

sterile quadrupole constituent. Assuming a given set 

of real data subjected to spherical harmonic analysis, 

its quadrupole will contain not only the fertile constituent disclosed by the eccentric-dipole approach but 

also the sterile part, which is lost through that approach. For a field free of artificial constraints, knowledge of the six parameters of the eccentric dipole 

could not suffice to recover the eight parameters of the 

slightly more detailed dipole-plus-quadrupole description.

APPLICATION TO EXTANT MODELS

By means of a computer program invoking the procedures explained earlier (equations (27)-(37)), the parameters given in table 2 have been calculated from the 

second-degree terms of a number of extant analyses, as 

indicated. 1 Figures 12-22 show similar information 

graphically for most of the models depicted in extant 

analyses, which are documented in table 3. There is of 

course redundancy in giving eight parameters in table 

2 when five would suffice to define the quadrupole, but 

the redundancy is useful in the search for the most inJ The program is arranged to accept coefficients in any of the three alternate modes of 

normalization that have been used in the past, as well as those in the present standard 

Schmidt mode. It also includes a routine based on equations (22)-(26) for using the 

calculated quadrupole-axis parameters to reconstitute second-degree spherical-harmonic 

coefficients, and for comparing these with the input values as a safeguard against calculation errors.
APPLICATION TO EXTANT MODELS 15

TABLE 2—Quadrupole parameters of specific field models

[60 , Ol , AO and Aj are the coordinates of the cardinal axes; 6P, Ap are the coordinates of the primitive axis; U is the ratio of the quadrupole moments;

d2 = 2M (1 + U); parameters in degrees and minutes]

Model — - —

Epoch ———

Barraclough Gauss

1650 1835

DysonFurner- AWC-70 AWC-70 

Fanselau

1922 1937.5 1970

IGRF

1965

Parameter:

U

l/2(J2

52 00

89 36

60 30

205 50

52 00

322 00

6.403

2417.9

46 31

24 32

56 24

153 36

62 01

264 16

0.848

1938.9

57 59

4 22

45 30

132 17

61 30

254 32

0.966

2143.5

59 11

355 41

39 57

131 05

67 13

251 10

0.950

2290.5

61 58

347 26

34 24

128 28

71 44

247 19

1.13

2516.2

61 14

348 15

35 21

128 57

71 15

247 31

1.134

2461.2

structive way to depict the secular change of the geomagnetic quadrupole. The primary axis is of interest 

because it is the normal to the quadrupole's characteristic plane and hence defines that plane with only 

two parameters. Both cardinal axes lie in that plane 

and mark those points on the surface of the sphere 

where the quadrupole field is normal to the surface. As

we have seen, these points are the authentic surface 

dip poles of the quadrupole field (N and S).

Note that for the recent models, the colatitude and 

longitude of the red axis (00 and AJ place it off the Atlantic coast of Morocco; those of the blue axis (61 and 

AJ place it in Amur, Siberia; and those of the primary 

axis (6P and Ap) place it in the Pacific Ocean about 120

120° 105° 90° 75'

FIGURE 12.—Drift of the primary axis of the geomagnetic quadrupole since 1550. Letters refer to the groups listed in table 3.
16

114'

ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

112° 110° 108° 106°W.

18° N

FIGURE 13.—Drift of the primary axis of the geomagnetic quadrupole since 1927. Letters refer to groups listed in table 3.

km from the extremity of Baja California. The three 

axes are, of course, orthogonal. Comparison of the 

Gauss model with recent ones shows clearly that all 

three axes have undergone a pronounced westward 

change of longitude, along with latitude changes signifying that the axis of rotation of the quadrupole is by 

no means coincident with the geographic axis. The 

1650 model likewise supports this trend, at least 

qualitatively.

The Dyson-Furner-Fanselau parameters in table 2 

agree well with the treatment by Howe (1939), which 

gives for that model the coordinates of the primary 

axis and of the cardinal axes. Most other published 

discussions of the quadrupole field lay chief stress on 

the primitive lines, and some go so far as to impute to 

their locations on the sphere the character of surface 

poles; but this concept is denied by the geometry as 

shown here.
30° 15°W.

APPLICATION TO EXTANT MODELS

15°E. 30° 45° 60° 75°

17

90° 105° 120°

0°

FIGURE 14.—Drift of the red cardinal axis of the geomagnetic quadrupole since 1550. Letters refer to groups listed in table 3.

14° 12° 10° 8° 6° 4°W.

FIGURE 15.—Drift of the red cardinal axis of the geomagnetic quadrupole since 1930. Letters refer to groups listed in table 3.
18

90°

ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

105° E. 120° 135° 150° 165° 180° 195° 210° 225° 240° E.

0°

FIGURE 16.—Drift of the blue cardinal axis of the geomagnetic quadrupole since 1550. Letters refer to groups listed in table 3.

The antipodal ends of the primary axis have no 

polarity; that is, they lack any magnetic characteristic 

that would distinguish one end from the other. The 

same is true of each of the cardinal axes, and of the 

primitive lines as well. The end of each axis chosen for 

display in table 2 and figures 12-17 is the end lying 

north of the equator. The pivot of clockwise rotation 

may fall close to the 90 ° arc that connects the blue and 

primary axis points, although it is usually somewhat 

farther south.

Since many epochs are involved, the computer work 

is arranged so that models may be grouped in sequences of two or more for comparison to bring out the 

secular change. This phase of the program incorporates a routine for locating the axis about which the 

quadrupole has turned in the interval linking the 

members of each coupled pair, and for determining the 

angle of rotation during that interval.

The analyses documented in table 3 can be thought 

of as consisting of two classes: Class I covers those 

analyses taken as models for single epochs (whether or 

not accompanying secular-change parameters were 

provided); and Class II comprises sequences of closely

related models, worked up for two or more epochs to 

determine secular change. The latter are identified in 

table 3 by listing the number of epochs for each sequence, and in figures 12-22 by using full lines to connect the plotted points.

Although not shown in the figures, the coordinates 

of the primitive lines are also calculated by the computer program, and where the models used are identical with those selected by other authors, the coordinates are in good agreement. An exception is the 

1829 model by Erman and Petersen (1874) (Sequence 

F), which as calculated by Umov (1904) gave a conflicting result. As already noted by Winch and 

Slaucitajs (1966), Umov's result in this instance 

reflected some error in calculation and must be 

disregarded.

The Class I models are assembled in convenient sequences, and in some instances are linked with Class II 

sequences. No systematic principle governs the grouping, but an effort has been made (with only moderate 

success) to associate models not differing too greatly 

in the value of n (degree and order), and at the same 

time to ameliorate the random criss-crossing of tracks
52

128°E

APPLICATION TO EXTANT MODELS

130° 132° 134°

19

50°N

136°

FIGURE 17.—Drift of the blue cardinal axis of the geomagnetic quadrupole since 1930. Letters refer to groups listed

in table 3.
ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

1500 1600 1700 1800 1900 2000

YEARS

FIGURE 18.—Value of U (ordinate) since 1550. Letters refer to groups listed in table 3.

manifested in preliminary versions of figures 12-17. 

The models designated as Ttt T2 , and Nlt N2 ,...N9 are 

constructed synthetically by applying specific secularchange models to an arbitrarily chosen base model, 

and for this reason are omitted from figures 12-17, but 

are included in figures 19 and 20. As regards older 

models (say before 1850), it is not to be supposed that 

their clustering along seemingly definite pathways is 

necessarily indicative of their precision, for all such 

models may be similarly biased by the absence of data 

over vast regions of the globe.

ECCENTRIC-DIPOLE BEHAVIOR

Table 4 shows the parameters of the eccentric dipole 

for several models reckoned according to Schmidt's 

procedure, and figures 23-24 show some of these parameters for many of the models referenced in table 3. 

The parameter 6C shown in table 4 and figure 24 is the

angle (measured at the Earth's center) between the 

displacement vector and the north leg of the centereddipole axis. As was noted by Bartels (1936), the eccentric dipole that Schmidt's procedure yielded on the 

basis of his adaptation of the Dyson-Furner-Fanselau 

analysis for epoch 1922 was notable in that its situation was almost exactly in the equatorial plane of the 

centered dipole. That is, for this model 6C is very nearly 

90°, and the fertile quadrupole is almost exactly represented by its normal constituent in the meridional 

plane in which the dipole is displaced. Other models 

(earlier and later) do not exhibit this characteristic so 

markedly; the value of 6C seems to vacillate in much 

the same way as U, and of course there should be a relation between these parameters, although no attempt 

is made here to formulate it rigorously.

The eccentric dipole (as defined by the first eight 

spherical harmonic coefficients) is characterized by its
APPLICATION TO EXTANT MODELS 21

1.20

1.15

1.10

1.05

1.00

0.95

0.90

1930 1940 1950 

YEARS

1960 1970

FIGURE 19.—Value of U (ordinate) since 1925. Letters refer to groups listed in table 3.

strength, its attitude, and its position. The first two 

features (and likewise their secular changes) are identical to those for the centered dipole. The third—that 

is, the vector stipulating how the dipole is displaced 

from the center of the sphere—is all that distinguishes 

the eccentric dipole from the centered one. This displacement vector does exhibit significant secular 

changes or drifts, not only a westward drift but also a 

northward shifting; this was pointed out by Nagata 

(1965) and is confirmed in figure 23 for the past 8 or 10 

decades. Those drifts in fact reflect changes in the 

strength and attitude of the quadrupole relative to the 

centered dipole. Hence it may be preferable to examine 

directly the quadrupole's changes so that its influence 

can be seen separately, uncontaminated by any 

"noise" or spurious constituents that might stem from 

uncertainties of the centered dipole.

For most of the models listed in table 3, the centereddipole axis (south-seeking end) falls within the shaded 

area of figure 25; the exceptions are shown individually 

in the figure, and all relate to older models based on 

sparse data.

For an interesting treatment of multipole characteristics relative to centered-dipole parameters, see Jory

(1956); his "sectorial quadrupole" appears to be what 

has been designated here as the sterile quadrupole.

RESULTS OF CALCULATIONS

The behavior brought out in figures 12-17 confirms 

the conclusions of Bullard and others (1950) and of 

other studies that the quadrupole is currently undergoing a pronounced drift. It may be called "westward" 

in the loose sense that the pivot or pole of clockwise rotation is certainly in the Northern Hemisphere; but 

since that pole lies far from the geographic pole, the 

drift is not "westward" in the strictest sense. (In the 

Chukchi Sea, for example, the drift is roughly eastward.)

Figures 12-24 are offered to illustrate the capabilities of the technique rather than to portray the exact 

quadrupole parameters. Any significance attached to 

the diagrams resides in the plotted points. The lines 

linking them in sequences should be regarded primarily as identification aids, not as attempts to depict 

precise drift paths, particularly when the time span is 

more than 5 or 10 years. Some of the major discrepancies manifested in figures 12-17 involve models that 

are derived wholly or chiefly from observatory data.
22

1.30

ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

1.20

R,,

1.10

\

^*N,

1.00

0.90

1910 1920 1930 1940 1950 1960 1970

YEARS 

FIGURE 20.—Value of U (ordinate) since 1910. Letters refer to groups listed in table 3.

Models so constructed include the L sequence, Z1958, 

Z1959, and A1960. This outcome may be due in part to 

a severe global asymmetry in the distribution of 

observatories. The wide disparity between Z1958 and 

Z1959 (of nearly the same epoch though differing in 

their selection of observatories) may indicate how 

radically a change in data distribution can affect a 

model. However, disparities in the situation of the 

quadrupole axes are evident for some other models 

that are not observatory based (for example, models 

D1955, H1945, H1955, and Z1955). Perhaps some of 

the disparities arose from variance in the way in which 

the analysts dealt with the problem of latitude 

weighting of the input data. In any event, the portrayal of quadrupole parameters as here exemplified 

affords one means of assessing the effects of various 

refinements in technique.

Notwithstanding their limitations, the figures do depict (at least for recent decades and discounting the 

main-field aspects of the observatory-based models) 

systematic trends that are believed valid. The rate of 

rotation seems to have been fairly consistent for 

several decades (fig. 26); the mean of 37 Class II determinations for intervals after 1900 is 15.3 ± 2.5

minutes per year. The corresponding mean coordinates 

of the pivot of clockwise rotation are: colatitude 47.4° 

± 16.1°, longitude 204° ±15°. There is no clear 

evidence of systematic secondary drift in the center of 

rotation itself (figs. 27, 28), but this is only a tentative 

conclusion because the center of rotation seems to skip 

about rather erratically. Its random motion is no doubt 

due at least in part to uncertainty in the models, arising from inadequacy of data. Some of it can be attributed to differences in analytic techniques and to the 

coupling of Class I analyses that differ from each other 

in their maximum degree and order (such as the aliasing affect on the second-degree terms entailed in a 

fluctuating limitation on higher degree terms).

It seems quite possible that the center of rotation is 

in fact rather stable at its site in the North Pacific 

Ocean, that the rate of rotation of the quadrupole 

about that center may be nearly constant, and that U 

is subject to only moderate change, perhaps none at 

all. (See next section, "Additional results.") Should 

these stabilities become well established, they would 

make possible a composite determination of the 

angular parameters of the quadrupole more credible 

than that of any individual analysis; for it would be
APPLICATION TO EXTANT MODELS 23

'E

J*"/

/

16

1800, 1820 1840 1860 1880 1900 

YEARS

1920 1940 1960 1980

FIGURE 21.—Value of the quadrupole moment. Ordinate is 1A62 in units of 100 nanotesla. Letters refer to groups listed in table 3.

founded on a selection of various analyses ranging over 

an interval of several decades. The parameters so 

determined as a function of time would be a definitive

representation of the geomagnetic quadrupole, which 

constitutes the most important global constituent of 

the nondipole field and of the secular change. I have
ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

-op

1.0

\K \

1800 1820 1840 1860 1880 1900 

YEARS

1920 1940 1960 1980

FIGURE 22.—Change of the quadrupole moment over time (sheared plot). In rectangular coordinates, the graph depicts the ordinate as

( l/262 + 94.2 — 0.06 X year). Letters refer to groups listed in table 3.

refrained from attempting such a determination; in 

view of the diverse provenance of the constituent 

models and the possible impact of the results upon 

future modeling efforts, it is felt that a joint undertaking by various agencies and individuals concerned 

would be more authoritative than any unilateral effort.

If future analyses with more evenly distributed data 

show that the pivot of rotation holds its position better 

than do the quadrupole axes, this finding will enhance 

the significance of the quadrupole and afford further 

testimony as to its conservative tendency.

The fact that the pivot of rotation is clearly far from 

the geographic pole lends support to a similar finding 

by Malin and Saunders (1973) with respect to the rotation of the composite of multipole fields up to degree 

six. The further fact that the latter's pole of rotation as 

reported by those authors differs radically from that 

for the quadrupole alone as found here tends likewise 

to support their suggestion that the higher degree 

multipoles are important in establishing the secular 

drift of the field as a whole.

Figure 21 shows, for some of the Class II models, the 

changes in the combined quadrupole moment, which is 

the numerical sum of the red and blue linear components marking the cardinal axes, and is denoted in 

equation (12) and in table 2 as 1A62 . These changes are

fairly steady at about 0.3 percent per year, as previously reported by Winch and Slaucitajs (1966). This 

graph is drawn to exclude results earlier than 1800, because such early values are necessarily conjectural for 

lack of pertinent intensity data.

The quantity actually given in table 2 and figure 21 

is yz62 , expressed in nanoteslas (nT) for convenient 

comparison. To convert into quadrupole moment (unit, 

weber-km2 ), multiply by 1.6477 X 1015 . The data shown 

in figure 21 are replotted in figure 22 on a sheared 

basis—that is, with a fixed linear slope of +0.06 nT per 

year subtracted from each value before plotting, thus 

permitting an expanded ordinate scale and clarifying 

the relation of the different sequences. The absolute 

value for any plotted point may be recovered by 

scaling vertically from the inclined grid lines.

The several parameters of the quadrupole have been 

worked up as a demonstration of technique and a step 

in advancing the understanding of the nondipole field 

and its secular change. It remains to be seen to what 

extent analogous procedures and spatial concepts may 

apply to the octupole and higher multipoles. Meanwhile, the situation of the quadrupole is well established, with its primary axis emerging in the Revilla 

Gigodo Islands off the Gulf of California, its red and 

blue axes respectively between southern Morocco and
APPLICATION TO EXTANT MODELS 25 

TABLE 3.—Grouping of extant spherical harmonic analyses

Group

A

B

C

D

E

F

G

H

J

K

Epoch

1550-1700

1750-1800

1885

1958

1960

1965

1600-1910

1960

1965

1600-1800

1964.8

1600-1770

1830

1880

1945

1955

1965

1968-1970

1780

1800

1835

1885

1922

1945

1955

1960

1965-1970

1975-1980

1600-1780

1829

1845-1880

1922

1942.5

1955

1960

1965

1835

1860

1885

1945

1960

1965

1945

1955

1907-1945

1957.5

1960

1965

1907-1955

1964.8

1975

"rnaxt

4

5

4

6

8

10

4

7

9

4

9

4

5

4

15

6

12

22

4

5

4

7

6

4

6

6

8

8

4

4

6

6

6

6

9

10

6

4

4

15

10

12

4

6

6

6

9

10

12

9

12

Model designation and reference

Braginskii (1972a) (4 epochs)

Braginskii (1972b) (3 epochs)

Neumayer and Petersen (Schmidt, 1917)

Ben'kova and Tyurmina (1961)

Fougere (1965)

AFCRL 3-15-68 (Fougere, 1969)

Barraclough (1974) (8 epochs)

GSFC (4/64) (Cain and others, 1965)

Pogo (3/68) (Cain and Cain, 1971)

Braginskii (1972b) (5 epochs)

Cosmos (9/68) (Cain and Cain, 1971)

Yukutake (1971) (4 epochs)

Kamtz (1851)

Quintus-Icilius (1881)

Fanselau and Kautzleben (1958)

Kautzleben (1965 a,b)

Malin nominee (Cain and Cain, 1971)

Pogo n-22 (Cain and others, 1974) (2 epochs)

Braginskii (1969)

Braginskii (1972b)

Gauss (1839)

Fritsche (1897)

Dyson-Furner-Fanselau (Schmidt, 1934)

Chakrabarty (1954) (ramax=2)

Finch and Leaton (1957)

Jensen and Cain (1962)

IGRF (IAGA Comm. 2, 1969) (2 epochs)

IGRF (Leaton, 1976) (2 epochs)

Fritsche (Yukutake, 1971) (4 epochs)

Erman and Petersen (1874)

Adams (1900) (2 epochs)

Dyson and Furner (1923)

Jones and Melotte (1953)

Adam and others (1963a,b) (Izmir charts)

GSFC (9/65) (Hendricks and Cain, 1966)

IZMIRAN nominee (Cain and Cain, 1971)

McDonald and Gunst (1967)

Carlheim-Gyllenskb'ld (1896); Schmidt (1917)

Schmidt (1917)

Fanselau and others (1964)

GSFC (12/66) (Cain and others, 1967)

Hurwitz and others (1966)

Afanasieva (1946) (nmax=2)

Adam and others (1962) (B. A. charts)

Vestine and others (1947) (5 epochs)

Malin and Pocock (1969)

GSFC (7/65) (Cain, 1966)

AFCRL 11-1-67 (Fougere, 1969)

Vestine and others (1963) (6 epochs)

Tyurmina (1968)

Barraclough and others (1975)
26 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

TABLE 3.—Grouping of extant spherical harmonic analyses—Continued

Group Epoch Model designation and reference

L 1932-1958 4 Adam and others (1963a,b) (7 epochs)

M 1937-1970 12 AWC-70 (Hurwitz and others, 1974) with secular change applied from Hurwitz and Fabiano

	(1969) (7 epochs)

N 1942-1962 6 Malin (1969) (5 epochs)

1965 6 N1962.5 adjusted to epoch using RGO-2 isoporic model (Cain and Cain, 1971)

*N, 1970 8 N1965 plus AFCRL (3/68) isopors

N2 1970 8 N1965 plus AFCRL (11/67) isopors

N3 1970 8 N1965 plus GSFC isopors

N4 1970 8 N1965 plus Pogo (10/68) isopors

N5 1970 8 N1965 plus Izmir isopors

N6 1970 8 N1965 plus RGO-1 isopors from P1965

N7 1970 8 N1965 plus RGO-2 isopors

N8 1970 8 N1965 plus Pogo (3/68) isopors

N9 1970 8 N1965 plus CGS isopors

P 1958.5 6 Nagata and Oguti (1962)

1965 8 Leaton, Malin, and Evans (1965)

Q 1955 12 Vestine and others (USSR charts) (1963)

1960 11 Pogo (10/68) (Cain and Cain, 1971)

1965 12 AWC-70 (Hurwitz and others, 1974) (adjusted to epoch using Hurwitz and Fabiano, 1969)

R 1960 10 Pogo (8/69) (Cain and Sweeney, 1970)

1965 8 RGO-2 nominee (Cain and Cain, 1971)

S 1960 6 Adam and others (1963) (Izmir charts)

1964.8 9 Tyurmina and Cherevko (1967)

T 1965-1970 12 CGS nominee (Hurwitz and others, 1966) (2 epochs)

U 1975 8 AWC-75 (Fabiano and Peddie, 1975)

U, 1970 8 AWC-75 less AWC-75 isopors

U2 1970 8 AWC-75 less IGS (UK) isopors

Z 1955 24 Jensen and Whitaker (1963); Heppner (1963); ramax = 17

1958 6 Adam and others (1962) (99 observatories)

_________1959_______4 Zmuda and Neuman (1961) ______________________________

t "max refers to the maximum degree and order of the analysis.

*Formed by using each of the nine secular-change adjustments described in Cain and Cain (1971) to reduce N1965 to epoch 1970. 

Similarly, Group U is formed by applying two adjustments to reduce U1975 to epoch 1970.

the Canary Islands and in a remote part of Amur, 

Siberia, and the pivot of its clockwise drift in the 

region of the Aleutian Islands.

ADDITIONAL RESULTS

The past several decades have witnessed substantial 

growth in the Earth's quadrupole moment along with 

an unmistakable drift in the principal axes of the 

quadrupole, but these trends have not been accompanied by comparable changes in the relative 

strengths of the red and blue components. Their ratio, 

C/,has shown indecisive small changes since 1900 (figs.

18-20), with a near-unity long-term mean; and this parameter seems to attain greater stability as the data 

coverage improves. The older, classical analyses show 

values of U that scatter more widely but still manifest 

no clear secular trend, suggesting that the scatter is 

very possibly due to such poor data coverage that no 

genuine change of U can be discerned. It may be pertinent that when the original Gaussian data were subjected to a new analysis by D. Watson (sequence G of 

table 3, this report) for epoch 1835, using a modern approach for filling in some of the data gaps (McDonald 

and Gunst, 1967), the value of U was thereby altered 

from 0.845 to 1.041. I do not imply that the new treat-
140° E.

HYPOTHESIS ON THE CHARACTER OF THE QUADRUPOLE

150° 160° 170° 180° 190° 200° 210°

27

220° E.

FIGURE 23.—Coordinates of the displacement vector for the eccentric dipole. Letters refer to groups listed in table 3.

ment is automatically a major improvement on the 

original analysis, but this result does demonstrate that 

improved data distribution can substantially affect the 

value of U. In any event, recent data suggest that U 

has a pronounced conservative tendency.

HYPOTHESIS ON THE CHARACTER 

OF THE QUADRUPOLE

The efforts of recent investigators to extend spherical harmonic analyses backward in time are of course 

attended by great uncertainty, as pointed out by Barraclough (1974) with respect to his sequence of models 

beginning with 1600. The U values corresponding to 

the first four of his models are 2.1, 6.4, 3.8, and 1.4 (fig. 

18, sequence B). If such values were taken seriously, 

we would conclude that the quadrupole had undergone 

some profound transformations, with a strongly rhombic model first approaching and then receding from the 

status of a linear quadrupole before finally settling 

down in recent decades to approximate that of a normal quadrupole. Such vicissitudes would necessarily 

denote marked transformations of the whole character 

and aspect of the nondipole field. Can such radical and 

erratic changes be reconciled with the conservative 

tendency otherwise characterizing the low-degree constituents of the field? Would the needed energy 

transformations comport with reasonable assumptions as to the parameters of core dynamics? These 

questions deserve further study, but it seems intuitively unlikely that the field has really undergone such 

mutations, in view of the manifest stability of U in the 

recent models.

It is suggested rather that the erratic variability of 

U in the classical models and its excessive magnitude
28

110°

ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

B B

60° E.

1600 1700 1800 1900 2000

YEARS

FIGURE 24.—Change of Bc for selected analyses. 0C is the angle measured at the Earth's center between the displacement vector and the

north leg of the centered-dipole axis. Letters refer to groups listed in table 3.

for the early Barraclough models are artifacts of the inadequacy of the data used, and it would seem to be a 

plausible hypothesis that in fact U has a strong 

tendency to remain at or near unity—that is, for 2co to 

have only insignificant departures from 90 °, as noted 

by Zolotov (1966). The indicated departures of U from 

unity are rather unsystematic, especially for recent 

years with gradually improved data distribution. Even 

now, the vector data are blighted by large gaps, and it 

is conceivable that filling in the gaps would appreciably modify the coefficients and possibly bring U 

still closer to unity.

Zolotov's suggestion was questioned by Winch and 

Malin (1969), who found a gradual progression of 2o>, 

which passed through 90° in 1952. The sequence N 

plot of figure 20 does indeed show U ascending 

through unity in early 1952, but persistence of this 

drift as a long-term phenomenon is not strongly supported by other data depicted in figures 19-20. The 

secular-change data are even more seriously deficient 

(in global coverage) than those for the main field. If improved secular-change data should in the future be 

found to reduce the apparent rate of drift of U, this 

would further support the hypothesis. And if the drift
SOME FURTHER IMPLICATIONS OF THE QUADRUPOLE CONSTRAINT 29

TABLE 4—Eccentric-dipole parameters of specific field models

[Of the two versions of the displacement-vector parameters, those on the lower h'ne of each pair are modified values obtained by imposing the constraint for U = 1, holding the 

cardinal axes unchanged. Rc, magnitude of the displacement vector in terms of Earth radii. 6C, angle measured at the Earth's center between the displacement vector and the 

north leg of the centered-dipole axis. Angles in degrees and minutes]

Model - —— —

Epoch ——— -

Dipole vector:

e

A

Displacement 

vector:

ec 

AC

6C

Rc

Barraclough

1650

7 16 

319 09

100 31 

97 33

210 01 

239 51

102 51 

96 09

0.05164 

0.03267

Gauss

1835

12 10 

296 28

103 06 

103 01

185 24 

185 24

106 36 

107 06

0.04540 

0.04452

DysonFurnerFanselau

1922

11 30 

291 10

83 28 

83 25

161 44 

161 08

90 50 

90 52

0.05397 

0.05395

AWC-70

1937.5

11 30 

291 57

78 36 

78 25

156 22 

155 37

86 53 

86 49

0.06240 

0.06242

AWC-70

1970

11 26

289 47

71 27 

72 01

148 02 

149 39

80 32 

80 54

0.07270 

0.07224

IGRF

1965

11 26 

290 14

72 50 

73 24

148 46 

150 25

81 53 

82 15

0.07067 

0.07025

is real, it may change sign from time to time so that U 

never gets far from unity.

Different attempts to derive isoporic models yield 

discordant rates of change of U, despite the use of recent data and updated techniques. Nine proposals were 

formally entered in 1968 as candidates for the time 

derivative terms of the International Geomagnetic 

Reference Field. Their second-degree terms are 

reflected in the cluster of nine short lines in figure 20 

for the interval 1965-1970. No doubt the slope disparity stems partly from differences in the weighting of 

data; if the well-known global imbalance of observatory distribution could be remedied, such a spread 

might well be minimized. (Note that the attachment of 

this grouping to a particular main-field model is arbitrary and holds no significance as to the absolute 

value of U.)

If the hypothesis that U is constrained to remain at 

or close to unity is tenable, then many of the older 

models, if not the current ones, could be improved by 

imposing the empirical constraint that U = 1 . 

Geometrically, this constraint implies that the zonal 

element delimited by any two small circles parallel to 

the characteristic plane of the quadrupole and by 

"meridians" of the primary axis that are 180° apart 

will encompass zero net quadrupole flux (the total 

amounts that enter and leave that part of the sphere 

being equal). The hypothesis, then, in its boldest form 

asserts that the configuration of the several electriccurrent vortices comprising the nondipole part of the 

geomagnetic core-dynamo is in some way physically 

regularized so that it cannot give rise to a quadrupole 

that differs significantly from a normal one.

SOME FURTHER IMPLICATIONS 

OF THE QUADRUPOLE CONSTRAINT

It is pertinent to inquire whether the hypothesis has 

any physical basis. None of a rigorous character appears, but a limited facet of the matter is to be noted. 

Let us regard a quadrupole as the result of two nearby 

current loops (of contrary sense) in a plasma, depicting 

two dipoles. Their detailed interplay, rather than their 

gross equivalence to simple dipoles, will govern the 

outcome.

For a normal quadrupole the loops would be 

coplanar, and their affinity would lead to an 

equilibrium configuration (fig. 29A), wherein the attraction between uncrossed remnants of the inner 

limbs is balanced by that between the overlapping segments (now urging the loops apart) plus the net repulsion involving the outer limbs. If the loops are considered to merge in their shared region (fig. 29B), the 

forces change but the equilibrium spacing is much the 

same. For a linear quadrupole the loops must be coaxial, one over the other, rather than coplanar. In this 

case (irrespective of the separation) there is no net 

attraction but only repulsion, precluding any
30 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

85° N.

60° 50° 40° 30° W.

FIGURE 25.—Coordinates of the centered-dipole axis. Letters refer to groups listed in table 3. For most models listed in 

table 3 the south-seeking end of centered-dipole axis falls in the shaded area.
SOME FURTHER IMPLICATIONS OF THE QUADRUPOLE CONSTRAINT 31

50

40

DC 

LLJ 

Q.

O

DC

<30

u.

O

I

\

20

D""' V

10

1600 1700 1800 

YEARS

1900 2000

FIGURE 26.—Quadrupole rotation rate. Class II analyses for groups from table 3.

equilibrium mode. And for a rhombic quadrupole the 

interacting forces would cause the loops either to 

separate entirely or to become coplanar. The only 

viable sort of conjunction is that yielding the normal 

quadrupole.

These concepts apply also in a space that is 

populated by a multiple distribution of loops in various 

attitudes, free to undergo random encounters with one 

another. Normal quadrupoles may well be the only sort 

producible by such activity.

Thus it seems physically plausible to postulate that 

the normal quadrupole is the only sort that can endure 

in a simple electromagnetic situation. Although this 

analog does not preclude the existence of rhombic or 

linear quadrupoles in an artificially controlled environment, it does afford a rationale for hypothesizing

that in nature their occurrence, if encountered at all, 

might be regarded as anomalous and transient.

The bifoliate current of figure 295 could occur in 

either of two varieties, one the inverse of the other and 

rotated 90° about the line of conjunction of the paired 

loops. The fields of the two species would be indistinguishable at a distance. In fact, it would be (at 

least mathematically) possible for both forms of loops 

to coexist, each contributing its portion of the aggregate quadrupole field; and if they happened to be of 

equal strength, the resultant current along the line of 

conjunction (the primary axis) would be zero. Such 

coexistence and equality are not necessary to account 

for a quadrupole field, but the symmetrical configuration they afford is helpful in visualizing the kinds of 

symmetry appearing in the field.
32 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES
120°E. 140°

SOME FURTHER IMPLICATIONS OF THE QUADRUPOLE CONSTRAINT

160° 180° 200° 220° 240° 260° 80° N. 280° 300° 320° 340°

33

360°

20° N.

0°

FIGURE 28.—Center of clockwise quadrupole rotation. Chiefly Class II analyses for groups from table 3.

These considerations will perhaps lend support to 

the suggestion that any substantial deviations from 

this rule, such as those that appear to occur with the 

older analyses, are but artifacts of well-known inadequacies of the data. At least we have a common-sense approach for inquiring whether the imposition of such a 

constraint might afford an improvement on existing 

and future models. Furthermore, if the constraint is a 

valid one for the Earth it may apply to the other 

planets as well.

The earlier history of geomagnetic sphericalharmonic analysis left undetermined whether the 

monopole, nonpotential, and external terms were 

significant; but as the data and techniques have improved it has come to be recognized that the monopole 

and nonpotential terms are almost certainly negligible

FIGURE 29.—Current loops in a schematic normal quadrupole. A, 

Attraction in uncrossed remnants balanced by that region between overlapping segments. B, Loops considered to merge in 

their shared region.

if not zero, and the upper limit on the exterior terms is 

assuredly much smaller than could have been deduced 

from the earlier work. The constraint hypothesized 

here may be of a 9haracter such that it can never be 

rigorously established, but if it should be found 

nominally valid to a sufficiently high accuracy, it may 

well afford another tool for refining our knowledge of 

the geomagnetic field and its secular change.

If constrained to be a normal one, the quadrupole 

would need for its specification a minimum of four parameters, that is, the moment, the coordinates of the 

primary axis, and the angle between, say, the blue axis 

and the meridian plane containing the primary axis.

Ideally, if the constraint is valid it should be imposed as one of the initial conditions of a spherical harmonic analysis. This is not easily done when the objective is to modify an existing classical analysis, since 

the original technique and input data may not be 

recoverable. Alternatively, the constraint may be applied as an empirical adjustment of the red and blue 

moments to a mean value, with the effect of rotating 

the two primitive lines in contrary senses about the 

primary axis until they are 90° apart. As so applied, 

the constraint has no effect on any of the principal 

axes, hence this is not a satisfactory way to test the actual effect that the constraint would have if applied 

during an initial analysis. However, it can give some
34 ON MODELING MAGNETIC FIELDS ON A SPHERE WITH DIPOLES AND QUADRUPOLES

TABLE 5—Changes of coefficients caused by imposing the constraint for U = 1

[For ease of comparison the changes (in nanotesla) are expressed in terms of Schmidt-normalized coefficients irrespective of the mode in which the original coefficients were stated]

Model Barraclough Gauss

DysonFurnerFanselau

AWC-70 AWC-70 IGRF

Epoch 1650 1835 1922 1937.5 1970 1965

Ag-£ -275.8

Agi -1684.6

Lh\ +21.6

Ag| +491.6

Lh\ +1309.2

-28.7

-21.2

-113.7

+8.1

-557.5

-5.4

-7.7

-25.9

-21.6

+ 12.7

-15.8

-12.8

-33.9

-34.8

+26.3

+58.4

+25.2

+78.2

+82.4

-86.8

+56.5

+25.2

+ 79.1

+81.7

-84.9

idea of the order of magnitude of the effects on the 

spherical harmonic coefficients and on the displacement of the eccentric dipole. Accordingly, my program 

for calculating the quadrupole parameters has been extended to include an adjustment of the red and blue 

quadrupole moments to their arithmetic mean and the 

derivation of modified spherical harmonic coefficients 

and modified eccentric-dipole parameters corresponding with the adjustment. Table 5 shows how 

much the second-degree coefficients for several models 

would be affected by imposing the constraint in this 

limited way.

It is clear that the constraint so applied causes the 

coefficients to be changed substantially even for the 

latest models. If the constraint hypothesis is valid to 

within close limits, these changes give a rough indication of the order of magnitude of the uncertainties in 

the extant coefficients, arising from the circumstance 

that none of the existing analyses has hitherto incorporated the constraint. It is certainly not argued 

that any analysis could be most effectively improved 

upon by this makeshift manner of imposing the constraint.

SOME QUALITATIVE REMARKS ON OCTUPOLES

Just as the quadrupole may be regarded as the combination of two dipoles, so the octupole may be taken 

as the combination of two quadrupoles. The constituent quadrupoles are inversely related—that is, the 

positive elements of each one correspond in strength 

and direction to the negative elements of the other. 

Taking one sense of the red cardinal axis of one of the 

quadrupoles as a reference direction, the other 

quadrupole is displaced from the first one in a definite 

direction, which may be specified by two angles. One of 

these may be the angle measured in a plane normal to 

the characteristic plane of the quadrupoles, from the 

latter plane to the line of displacement. If this angle is 

zero the two quadrupoles are coplanar. The other angle 

may be reckoned in the characteristic plane, to the 

above normal plane, or if the first angle is zero to the 

displacement direction itself. (If the first angle is 90°,

the second one is indeterminate; and if in addition the 

component quadrupoles are normal ones, the octupole 

has the configuration of a cube.)

Thus the octupole may be specified by seven parameters—the five parameters that define its constituent quadrupoles (these being identical except for the 

sign change) and the two angles mentioned above. 

Note that the constituent quadrupoles may be rhombic, normal, or linear. The resultant octupole may be 

linear (but only if the quadrupoles are linear and both 

of the angles discussed above are zero). The octupole 

may be planar (if the two quadrupoles are coplanar). In 

all other cases, it need be neither linear nor planar.

Of the several possible forms of octupole, it appears 

that the simplest nonlinear one would be cubic, formed 

of two normal quadrupoles separated along their common primary axis—that is, in a direction normal to 

their characteristic planes. The eight corners of the 

cube would correspond with the eight point poles making up the four constituent dipoles (fig. 30). A cubic 

octupole may be regarded as the result of a cloverleaf 

or quatrefoil array of coplanar current loops, and may 

well be the only stable kind of octupole. There are three 

attitudes of the quatrefoil current array that could proFIGURE 30.—Arrangement of four dipoles to form a cubic octupole, and equivalent quatrefoil array of coplanar current loops.
SELECTED REFERENCES 35

duce the identical octupole field (that is, the fields 

generated would be indistinguishable at a distance) 

corresponding to three ways of peripherally coupling 

the eight point poles of the cubic array. All three current patterns might coexist, each contributing perhaps 

one-third of the aggregate octupole field. However, if 

two similar quatrefoil current patterns were combined 

so that their planes intersected normally but they both 

showed the same sense of current flow across their 

plane of symmetry, the result would be a multipole 

with n = 4—what is called a sedecimupole by Winch 

(1967).

The primitive lines of a quadrupole are established 

by its constituent dipoles under the two possible ways 

of assigning their directions in space. Similarly, the 

alternate ways of assigning the octupole field to two 

quadrupoles lead to certain directions in space, now 

three in number, which may likewise be called 

primitive lines. For a cubic octupole, they are an orthogonal set of directions corresponding to the face 

centers of the cube, and they mark the necessary six 

false poles or saddle points of the potential configuration on the sphere.

A quadrupole has two cardinal axes, each being a 

bisector of an angle between the two primitive lines. 

The cardinal axes mark the true dip poles of the surface field on the sphere. The octupole has four 

analogous axes, each of which is separated by equal 

angles from the three primitive lines. In the cubic octupole these are the diagonals of the cube, so even for 

this simple case they are not orthogonal with each 

other. However, they still mark the dip poles (eight in 

number) of the octupole field on the sphere. These 

poles, like those of a dipole field but unlike those of a 

quadrupole, will occur in pairs with unlike signs, each 

pair marking opposite ends of an axis.

It remains for further investigation to ascertain 

whether or not the geomagnetic octupole and higher 

multipoles are subject to any constraint that would 

restrict them to specific categories, analogous to the 

hypothesized restriction of the quadrupole to a normal 

one. To anticipate somewhat, if it were stipulated that 

each of the constituent quadrupoles of every higher 

multipole must be a normal one, and that they are invariably coupled in orthogonal relationships, with 

coplanar current loops, it would follow that the octupole (a cubic one) could be described by but four independent parameters, the fourth-degree multipole by 

another set of four, and so on.

SUMMARY AND CONCLUSIONS

In this report I have sought to develop improved 

understanding of quadrupole behavior by emphasizing 

the concept of the cardinal axes as distinguished from

the less fruitful primitive lines, and to bring out the 

quadrupole's unique significance in relation to the 

secular change, especially in relation to global constituents. New insights are presented regarding the 

geometry of quadrupole fields, the way in which they 

govern the displacement of Schmidt's eccentric dipole, 

and the isolation of those quadrupole elements that 

have no such influence. The secular change of various 

parameters is graphically depicted on the basis of more 

than a hundred extant spherical harmonic analyses 

from many sources. Earlier work reporting the growth 

and westward drift of the quadrupole is confirmed and 

refined. The approaches here developed facilitate isolation of the asymmetric components of the quadrupole 

and thereby promote understanding of the nondipole 

field as it may be manifested in extant and future 

spherical harmonic analyses. And we see that when an 

analysis is proposed on the basis of radial dipoles, 

every such dipole is associated with a series of 

multipoles having especially restricted characteristics 

that are of interest in relation to spherical harmonic 

analyses. Finally, there is offered for further consideration and testing the hypothesis that the composite 

geomagnetic quadrupole is naturally constrained to be 

approximately normal.

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nach den von der Deutschen Seewarte herausgegebenen magnetischen Karten fiir 1880.0: Hamburg, Archiv der Deutschen 

Seewarte, 4 (2). Pages unknown.

Schmidt, A., 1917, Erdmagnetismus, in Enzyklopaie der mathematischen Wissenschaften, mit Einschluss ihrer Anwendungen: 

Leipzig, B. G. Teubner, v. 6 (1), p. 266-396.

___1918, Geomagnetische Koordinaten, in Archiv des Erdmagnetismus: Potsdam, Heft 3, p. 14.

___1934, Der magnetische Mittelpunkt der Erde und seine Bedeutung: Gerlands Beitrage zur Geophysik, v. 41, p. 346-358.

Tyurmina, L. O., 1968, Analytical model of the geomagnetic field 

according to Cosmos 49 data: Geomagnetism and Aeronomy, v. 

8, p. 786-788.

Tyurmina, L. O., and Cherevko, T. N., 1967, Some results of a spherical harmonic analysis of the geomagnetic field from Cosmos 49 

data: Geomagnetism and Aeronomy, v. 7, p. 917. [For theory 

see v. 7, p. 207-210.]

Umov, N. A., 1904, Postroenie geometricheskogo obraza potentsiala Gaussa, kak priem izyskaniya zakonov zemnogo magnetizma: Obschchestvo Liubitelei Estestvoznaniia Antropologii i Etnografii, 107 (1). [Condensed German translation in Journal of 

Geophysical Research, v. 9, no. 3, p. 105-112.]

Vestine, E. H., 1953, On variations of the geomagnetic field, fluid 

motions, and the rate of the Earth's rotation: Journal of Geophysical Research, v. 58, no. 2, p. 127-145.

Vestine, E. H., Lange, I., LaPorte, L., and Scott, W. E., 1947, The 

geomagnetic field, its description and analysis: Carnegie Institution of Washington, Publication 580.

Vestine, E. H., Sibley, W. L., Kern, J. W., and Carlstedt, J. L., 1963, 

Integral and spherical-harmonic analysis of the geomagnetic 

field for 1955.0, part 1: Journal of Geomagnetism and Geoelectricity, v. 15, no. 2, p. 47-89.

Winch, D. E., 1967, The fourth-order geomagnetic multipole—the 

sedecimupole: Pure and Applied Geophysics, v. 67, p. 112-127.

Winch, D. E., and Malin, S. R. C., 1969, Changes in geomagnetic 

multipole parameters 1942.5-1962.5: Pure and Applied Geophysics, v. 74, p. 16-22.

Winch, D. E., and Slaucitajs, L., 1966, Geomagnetic multipoles, 

1829 to 1960: Pure and Applied Geophysics, v. 63, p. 121-132. 

[Gives coefficients for several models including F 1932 and six 

averaged ones.]

Yukutake, T., 1971, Spherical harmonic analysis of the Earth's 

magnetic field for the 17th and 18th centuries: Journal of Geomagnetism and Geoelectricity, v. 23, no. 1, p. 11-31. [The 

Fritsche models are discussed also in Braginskii (1969).]

Zmuda, A. J., and Neuman, M., 1961, The correction and mutual 

dependence of harmonic coefficients, in Space Research: International Space Science Symposium, 2nd, Florence, Proceedings, v. 2, p. 701-707. [Coefficients also given in McDonald 

and Gunst (1967).]

Zolotov, I. G., 1966, Representation of the geomagnetic field by 

means of multipoles: Geomagnetism and Aeronomy, v. 6, 

p. 413-421.

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