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

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Wilson, D., 1993. Confirmation of the astronomical calibration of the

magnetic polarity timescale from seafloor spreading rates. Nature,

364: 788–790.

Cross-references

Core motions

Geomagnetic dipole field

Geomagnetic excursion

Geomagnetic hazards

Magnetization, chemical remanent (CRM)

Magnetization, depositional remanent (DRM)

Magnetization, natural remanent (NRM)

Non-dipole field

Paleomagnetic secular variation

Paleomagnetism, deep-sea sediments

Polarity transitions: Radioisotopic Dating

GEOMAGNETIC SECULAR VARIATION

Introduction

The term secular comes from the Latin Seculum which means the

duration of the influence of a powerful family becoming steadily less.

The Romans accounted 100 years for that, why it also was synon-

ymous for century. In context of geomagnetism it implies a long-term

variation of the Earth’s magnetic field.

The observed temporal variations of the Earth’s magnetic field

cover timescales from milliseconds to a few million years and origi-

nate in two distinct (external and internal) source regions with respect

to the Earth’s surface. With this respect the fluctuations of the external

field ranges from milliseconds to a few decades, where the longer per-

iods are related to variations of the solar magnetic field, e.g., the turn-

over of solar magnetic field (about 22 years). Changes of the internal

field are of the order of a few years to millions of years. This variation

results from the effect of magnetic induction in the fluid outer core and

from effects of magnetic diffusion in the core and the mantle. Here, we

distinguish geomagnetic secular variation and paleomagnetic secular

variation, where the latter includes temporal changes longer than sev-

eral hundreds of years, such as reversals (see Reversals, theory). The

overlap of periods of internal and external sources in the range of a

few years to decades can be separated in internal and external contri-

butions applying spherical harmonic analysis (see Internal external

field separation).

The geomagne tic secular variation was first noticed by Gunter and

“Gellibrand” in 1635, who collected measurements of magnetic decli-

nation made at Limehouse near London between 1580 and 1634. This

component gradually changed over the period of 350 years from 11



to 24



W in 1820, before turning eastward again. Figure G47 shows the

declination measured in London and Danzig (Gdansk) for about 350

years. Whereas the main field is dominated by its dipolar nature, the

secular variation is clearly nondipolar, which is reflected in regions

of different magnitudes of secular variation. For example, in the paci-

fic region the secular variation appears to be crestless. However, the

(geomagnetic) secular variation has been observed to be the feature

of the main field and not of the local field.

One other prominent feature of the secular variation is the tendency

of isoporic foci (areas of maximum secular variation) to drift west-

ward. Analyzes by Vestine et al. (1947) and Bloxham et al. (1989)

found an averaged drift rate of about 0.3



1

In addition to the slowly varying secular variation event-like fea-

tures appear, the so-called geomagnetic jerks. Such jerks show up as

a change of sign in the slope of the secular variation, a discontinuity

in the second time derivative of the field, most clearly seen in the east

(Y ) component of the geomagnetic field. For the last 100 years at least

seven jerks have been reported (1912, 1925, 1969, 1978, 1983, 1991,

and 1999), some of them of global extent. The 1969 event (first

described by Courtillot et al., 1978) was widely investigated; on the

basis of observatory records. Courtillot et al. (1978) and Malin and

Hodder (1982) showed its global extent, although it was not evident

in all field components. This fact and the coincidental occurrence

of jerks and sunspot maxima set off a lively discussion between

Alldredge and McLeod in the 1980s (Alldredge, 1984; McLeod, 1985;

Backus et al., 1987) about the causative processes of jerks. The gen-

eral understanding is that jerks are of internal origin, mainly because

of two reasons: First, the potential of the solar cycle has approximately

the form of zonal spherical harmonics, therefore any contribution

to the East component would be small. Jerks are most clearly visible

in the East component of European observatories. The second argu-

ment is based on a comparison of the strength of the solar maxima

adjacent to the 1969 jerk, 1958 and 1980. At both epochs the solar

maxima were more pronounced than in 1969, so we would expect

two jerks at those epochs, but nothing obvious happened in 1958

and the jerks around 1978 and 1983 do not fit well to the solar max-

imum in 1980 (see also entry on Geomagnetic jerks).

Determination of secular variation models

Before considering the generation of secular variation and its link to

other observables of processes in the Earth’s core, we shall first

describe a modeling approach to map the secular variation at the

source region, the core-mantle boundary (CMB).

In principle, the magnetic field and its variation can be separated in

parts due to external and internal sources by spherical harmonic analysis

(Gauss, 1839). The geomagnetic field is then represented by the so called

Gauss coefficients (see entries on Internal external field separation and

Main field modeling). The variations of the internal field can be modeled

by expanding the internal Gauss coefficients in a Taylor series in time

about some epoch, t

, e.g.,

ðtÞ¼g

ðt

Þþ

ðt

Þðt  t

Þþ

€

ðt  t

þ (Eq. 1)

where the first time derivative

is the secular variation and the sec-

ond time derivative

€

the secular acceleration. The determinations

usually have been truncated after some derivatives. The GSFC(12/

66) model (Cain et al., 1967) included second time derivatives and

the GSFC(9/80) model of the magnetic field between 1960 and 1980

by Langel et al. (1982) included third time derivatives. These

Figure G47 The declination at London and Danzig (Gdansk).

346 GEOMAGNETIC SECULAR VARIATION

representations are only adequate to represent the temporal variation

over short periods, e.g., the GSFC(9/80) is only sufficient for the per-

iod 1955–1980. Outside this period the misfit increases drastically.

An alternative attempt to model the secular variation was put for-

ward by Langel et al. (1986). The methodology is to model the secular

variation directly from first time derivatives of observatory annual

means. For example, using the differences between annual means for

different years, i.e.,

X ¼ DX =Dt. Then performing with those

X ;

Z spherical harmonic analysis. Langel et al. (1986) achieved a

continuous representation of the secular variation model for 1903 to

1982 by fitting each coefficient with cubic B-splines.

In a series of publications Bloxham (1987); Bloxham and Jackson

(1989); Bloxham and Jackson (1992) developed a method which gives

the most favorable description of the secular variation. This method

bases on the simultaneous construction of a time-dependent model of

the secular variation and main field. Their description of the time-

dependent geomagnetic potential at the CMB follows:

Vðr; y; fÞ¼

l¼1

m¼0

n¼1



lþ2

ðl þ 1Þðg

cosðmfÞ

þ h

sinðmfÞÞP

ðcos yÞM

ðtÞ; (Eq. 2)

where a is the Earth’ radius, c is the radius of the Earth’s core, M

ðtÞ

are the temporal basis functions, i.e. cubic B-splines. The expansion

(2) involves coefficients fg

; h

g which are related to the standard

Gauss coefficients fg

; h

g by

n¼1

ðtÞ (Eq. 3)

and for the h

likewise. The model is derived to meet some con-

straints, first it should be spatially smooth. Spatial smoothness, for

instance could be controlled by the minimum ohmic dissipation based

on the ohmic heating bound of Gubbins (1975). The second constraint

to meet is the temporal smoothness of the model, which can be con-

trolled by minimizing the second time derivative of the radial compo-

nent of the field at the CMB. Further constraints, invoking satellite

field models for certain epochs are conceivable as shown by Wardinski

(2005) (see entry on Time-dependent models of the geomagnetic field).

Periodicities in the geomagnetic secular variation and

related phenomena

An analysis of the geomagnetic activity and the solar activity reveals

periods which can be attributed to the effect of the sunspot cycle.

These periodicities are linked to external field variations, such as fluc-

tuation of the strength and position of the ring current and variations of

the strengths of current systems in the magnetosphere. Therefore, geo-

magnetic field variations exhibit nearly the same periods. If we assume

that the effect of sunspot cycle and its harmonics is independent of

observatory longitudes, then the effect can be approximated by a series

of zonal harmonics. We would expect that these contributions domi-

nantly map into the first degree Gauss coefficients of the internal field.

Langel et al. (1986) found evidence that some of the periods also exist

in higher degree secular variation coefficients ð

;

Þ. It is most

likely, that this is due to induction in the mantle.

Beside the periods related to solar variations there exists a bundle of

periods which are supposed to be inherent features of the geomagnetic

secular variation. One prominent long period is the 23 (22.9) years

period. It is found in the data of geomagnetic observatories (Alldredge,

1977b) as well as in secular variation coefficients (Langel et al., 1986).

Although the closeness of this period with the double solar cycle per-

iod may refer to a common cause, the origin of the 23 years period seems

to be internal. Alldredge (1977b) showed that it does not appear in all

analyzed observatories and is not in phase at those observatories at

which it is present. Further, Langel et al. (1986) argued for an internal

origin, because it is only present in a few secular variation coefficients,

namely

;

The picture is less clear when considering longer periods such as the

60 years period. Slaucitajis and Winch (1965) provided evidence of

this period by an analysis of five observatory data sets. They found

an averaged period of 61  6 years. These results were confirmed by

Jin and Jin (1989) with a slightly different period of 60  12 years,

where the uncertainty is due to an average of the results from an ana-

lysis of inclination and declination data. But its true nature is not clear

since the match with a period of about 60 years in geomagnetic activ-

ity and sunspot numbers suggests an external origin which may cause

an inductive 60 years period. On the other hand, there also may be an

internal origin of this signal related to the variation of the westward

drift and torsional oscillations of the fluid outer core. A comparison

of the variation of the “westward drift” of the eccentric dipole (which

is part of the secular variation) with the variation of the “length of day”

reveals a significant correlation between both variations at about

60 years (Vestine, 1953; Vestine and Kahle, 1968). Braginsky (1970)

developed a theory which is capable of characterizing both observa-

bles as two aspects of one phenomenon related to torsional magneto-

hydrodynamic oscillations in the Earth’s core. These oscillations can

have periods of about 60, 30, and 20 years, and via coupling processes

between mantle and core they should produce fluctuation of the

Earth’s rotation. (see Core-mantle coupling, Length of day variations

and Torsional oscillations (q.v.)). Bloxham et al. (2002) suggested that

the occurrence of geomagnetic jerks should be linked to “torsional

oscillation,” therefore also to the length of day variations. Indeed,

recent analyzes by Holme and de Viron (2005) and Shirai et al.

(2005) reveal that “geomagnetic jerks” are directly linked to the pro-

cesses responsible for changes in the core angular momentum.

Further long periods may exist, such as 18.6 and 9.3 years resulting

from the periodic gravitational action of celestial bodies on the equa-

torial bulge of the Earth. However, their spectral peaks are fairly close

to those of the solar activity. Another was predicted by Yukutake

(1972) to be the free modes of the electromagnetically coupled core-

mantle system, which has a length of about 6.7 years. Currie (1973)

may have found this period. For other periods, e.g., 70, 55, 32, 29,

and 13 years (Langel et al., 1986) a theory is only vaguely outlined

and needs further investigation.

Generation of secular variation

Secular variation results from the effect of magnetic induction in the

fluid outer core and from effects of magnetic diffusion in the core

and the mantle. These processes are constituted by the induction equa-

tion, which follows from Faraday’s induction law

¼rE; (Eq. 4)

where E is the electric field, showing that a spatially varying electric

field can induce a magnetic field. And further, Ampere’s law

j ¼ E þ u  B ¼

rB; (Eq. 5)

where s is the electric conductivity, j the current density, and m the

magnetic permeability. Combining both and after some algebra we get

¼rðu  BÞþr

B: (Eq. 6)

This is the induction equation, where the first term on the right-hand

side displays the advection of the magnetic field due to the fluid

motion in the liquid outer core, the second represents the action of

magnetic diffusion.

GEOMAGNETIC SECULAR VARIATION 347

It is generally assumed that the diffusive timescale is factor 300 to

500 to the advective timescale. This implies that on short timescales,

i.e., less than 100 years, the secular variation is entirely caused by

the rigidly coupled movement of the magnetic field lines with the fluid

motion in the liquid outer core and therefore diffusion can be

neglected.

¼rðu  BÞ (Eq. 7)

That is the so-called frozen flux hypothesis (see Alfvén’s Theorem and

the frozen flux approximation (q.v.)).

The “frozen flux assumption” simplifies the inversion for the fluid

motion u of the liquid outer core (see Core motions), but it imparts

an incomplete description of the mechanism of the secular variation

generation, for two reasons. First, it reduces the algebraic order of

the induction equation by one degree, therefore the solution for the

fluid flow may not be expected to be complete or correct (Love,

1999). Second, there is evidence that the frozen flux assumption is vio-

lated in the last 40 or 50 years. Bloxham and Gubbins (1985); Blox-

ham (1986) evaluated the magnetic flux through individual flux

patches at the CMB. A necessary condition for the “frozen flux

approximation” to apply is that the flux F through a patch S on the

core surface bounded by a contour of zero radial field must be constant

with respect to time

F ¼

dS ¼ const: (Eq. 8)

Patches located in the southern hemisphere show a significant change

of the flux during the period 1960–1980. On a global scale the

unsigned flux integral

jFj¼

CMB

jdS ¼ const:; (Eq. 9)

must also be constant with respect to time. Recent analyzes by Holme

and Olsen (2005) and Wardinski (2005) suggest that this is not the

case for the period 1980–2000. Figure G48 shows the differences of

this integral for successive years averaged over the period 1980 to

2000. The given error bars are the sample variance of the integral

for a specific truncation degree. For truncation degrees less than 9 a

conservation of unsigned flux is not achieved, for higher degree this

seems to be achieved within error margins, but it should be mentioned

that the higher degrees of the model more and more reflect the a priori

beliefs, i.e., spatial and temporal damping applied in the computation

of the secular variation model.

In a related discussion about the origin of the “westward drift” it has

been suggested that waves rather than a bulk advective motion account

for the “westward drift” and most of the secular variation. The possible

set of wave types are: inner core oscillation, which can have very short

periods of about a year (Gubbins, 1981; Glatzmaier and Roberts,

1996). And, “torsional oscillations” of coaxial cylindrical shells oscil-

lating in a solid-body rotation, for which the force balance is between

Lorentz force and inertia. The periods are of the order of decades

(Braginsky, 1970; Zatman and Bloxham, 1997, 1999). On longer time-

scales (300 years) magnetohydrodynamic waves which are depen-

dent on Magnetic, Archimedean (buoyancy), and Coriolis force

could play a role. The difference between torsional oscillation and

MAC-waves is the additional acting of buoyancy in MAC-waves.

An interaction of these waves, core-surface flows and the morphology

of the magnetic field on a timescale shorter than 300 years is likely. How-

ever, it should be mentioned that our distance to core surface and the

mantle low conductivity attenuate any details in time and space.

The influen ce of the mantle on the secular variation

The secular variation as observed at the Earth’s surface undergoes two

principal processes when it permeates from the core through the man-

tle. First a geometrical attenuation determined by the factor



lþ2

;

where a=c  1:8. The higher the harmonic number l, the stronger is

the geometrical attenuation. The second effect concerns the magnetic

diffusion. Here the temporal change of the field is entirely given by

the diffusion of the magnetic field through the mantle implicitly

assuming that there is no magnetic field generation in the mantle.

The shortest temporal variation of the core’s magnetic field that can

be seen at the Earth’s surface is determined by the conductivity of the

mantle and in particular boundary between core and mantle. This

region of compositional variation, partial melting and heterogeneous

high conductivity, known as D

, may have a thickness of about 200

km and should have a significant impact to the permeability of the

magnetic field (see D

However, our understanding of the electrical conductivity of the

mantle and its lower part is very poor. All method, which have been

employed to deduce the mantle conductivity from the observed secular

variation are based on the assumption that the mantle conductivity s

varies as a power law of the radius r

s ¼ s



: (Eq. 10)

This model permits high conductivity in the boundary layer as well

as low values of conductivity in the upper mantle, well in agreement

with “magnetotellurics.” The values of s

ranges from 223 S m

1

(McDonald, 1957) and 100 000 S m

1

(Alldredge, 1977a) depending

on the method of analysis (see Electrical conductivity of the mantle).

A value between 600 and 3000 S m

1

(the later preferred by Backus,

1983) would be in good agreement with length of day variation and

electromagnetic core-mantle coupling (Stix and Roberts, 1984; Paulus

and Stix, 1989; Stewart et al., 1995).

Our knowledge of the mantle conductivity is far from being com-

plete and a rigorous analysis, along theoretical lines as set by Backus

(1983) remains to be done. This requires a consolidated knowledge

Figure G48 The mean change of the unsigned flux integral for

1980–2000.

348 GEOMAGNETIC SECULAR VARIATION

about the secular variation. In order to achieve this the observation of

the Earth’s magnetic field have to be improved, either in terms of more

geomagnetic observatories or satellite missions such as MAGSAT,

ØRSTED, and CHAMP.

Ingo Wardinski

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GEOMAGNETIC SECULAR VARIATION 349

Cross-references

Alfvén’s Theorem and the Frozen Flux Approximation

CHAMP

Core-Mantle Coupling, Electromagnetic

Core Motions

Core-Mantle Coupling, Electromagnetic

Core-Mantle Coupling, Thermal

Core-Mantle Coupling, Topographic

and F Layers of the Earth

as a Boundary Layer

, Anisotropy

, Composition

, Seismic Properties

Gauss, Carl Friedrich (1777–1855)

Gellibrand, Henry (1559–1636)

Geomagnetic Jerks

Internal External Field Separation

Length of Day Variations, Decadal

Length of Day Variations, Long Term

Magnetotellurics

Main Field Modeling

Mantle, Electrical Conductivity, Mineralogy

Oscillations, Torsional

Reversals, Theory

Time-dependent Models of the Geomagnetic Field

Westward Drift

GEOMAGNETIC SPECTRUM, SPATIAL

Introduction

Just as for a function defined round a circle it is useful to separate the

contributions of different wavelengths/spatial frequencies, so for a

function defined on the surface of a sphere it is useful to separate

the contributions of different (two-dimensional) wavelengths/spatial

frequencies. In the context of geomagnetism it is useful to apply this

concept of a spatial spectrum to the main magnetic field (coming from

the core) and its time variation (the secular variation), and to the field

from the magnetization of the crust.

In the one-dimensional case we call this process Fourier analysis; if

we have a variable F, which is known round a circle (and which is there-

fore a periodic function round the circle), we can express it in the form

FðlÞ¼S

ðlÞ¼

ða

cos ml þ b

sin mlÞ; (Eq. 1)

where l is the angle round the circle, measured from some origin.

and b

are numerical coefficients; the integers m, which start from

zero, can be thought of as spatial frequencies, or wave numbers. Each

term cos ml or sin ml is called a harmonic, though this name is some-

times also applied to the numerical values of a

cos m l and b

sin m l.

The concepts of frequency and harmonic come from the analogous

situation of a variable which is a periodic function in time; this func-

tion of time can be separated into harmonics specified by frequencies

(in time) which are multiples of the fundamental frequency.

These harmonics are orthogonal; in the sense that the average value

round the circle of the product of any two different harmonics is zero:



¼ 0 (Eq. 2)

unless m ¼ m (and both harmonics use only either cos ml or sin ml),

where <x> represents the average of x round the circle. Therefore, if we

expand an arbitrary F in terms of its harmonic components F

, and then

square it, all the cross-terms vanish when we take the average round the

circle. So the mean-square value of F round the circle, <F

>, becomes



¼ S



cos

ml þ b

sin



ða

þ b

Þ=2; (Eq. 3)

the factor of (1/2) is the mean-square (ms) value of cos ml or sin ml

round the circle. (For m ¼ 0 the mean-square value is in fact 1, but I

will ignore this minor complication.) We see that the two harmonics

of a given spatial frequency m contribute to the overall mean-square

value independently of all the other harmonics.

We can write F

(l) ¼ (a

cos ml þ b

sin ml) alternatively as

cos(mle); it is a sinusoidal variation of amplitude

¼ða

þ b

1=2

, and phase e. Our choice of the origin of the angle

l is arbitrary; while the numerical values of a

and b

, and the phase

e, will depend on this choice of origin, the variation of F

round the real

circle, and its amplitude c

, is independent of this choice.

A plot of c

against m is called the (power) spectrum of F. (The

notation arises becaus e for a wave-motion periodic in time, c

is pro-

portional to the average power (over the cycle) carried by the wave at

that frequency.)

Application to the geomagnetic field

The case of the geomagnetic field is more complicated for two rea-

sons. The first complication is that the field is a vector (at each point

in space it has direction as well as magnitude) rather than a scalar

(which has only magnitude). However, in regions where there is no

significant electric current (in practice this means throughout the upper

mantle and the lower atmosphere) we can represent the vector mag-

netic field B as the gradient of a scalar potential, V,

B ¼grad V; (Eq. 4)

and can then work in terms of this scalar variable V.

The second complication is that the relevant geometry is three-

dimensional space rather than a one-dimensional circle. As we are

concerned with a roughly spherical Earth, it is convenient to use a

spherical polar coordinate system (r, y, l) based at the center of the

Earth, with y ¼ 0 being the direction of the north geographic pole,

and l ¼ 0 the direction of the Greenwich meridian. (The angle l is

the conventional longitude, but because the Earth is a slightly oblate

ellipsoid the (geocentric) y is not quite the same as the (geodetic)

colatitude.)

Provided that there is also no significant magnetization in the region

of interest, it is convenient to approximate the scalar potential V as a

finite sum of “spherical harmonics” (see Harmonics, spherical). (In prac-

tice, although the rocks of the crust are magnetized, even in the crust they

produce a field which is small compared with that of the electric currents

in the core, and we can often ignore this minor complication. See the last

section below for the relative magnitude of the fields.)

We then have

Vðr; y; lÞ¼

n;m

ðr; y; lÞ

¼ a

n;m

ðcos yÞðg

cos ml þ h

sin mlÞða=rÞ

nþ1

(Eq. 5)

for that part of the field having sources internal to the region of inter-

est. The P

ðcos yÞ are (scaled versions of ) the mathematical functions

of y called Legendre polynomials. The different harmonics are labeled

by two integers, the degree n and the order m, with m  n. (Note that

some authors use the symbol l rather than n for the degree.) The factor

a, the mean radius of the Earth, is incorporated so that the numerical

coefficients g

and h

have the dimensions of magnetic field, conven-

tionally expressed in units of nanotesla.

The particular variation with radius in Eq. (5) is valid only for

internal sources, and only in a region free from sources of magnetic

field. But on any given spherical surface, whatever the source

350 GEOMAGNETIC SPECTRUM, SPATIAL

distribution, a scalar such as V can be represented uniquely as a func-

tion of (y, l) given by the sum of a set of surface harmonics, where

any radial variation is incorporated numerically into the coefficients.

For simplicity, considering a field of internal origin on the surface

r ¼ a (approximately the surface of the Earth), we have

Vðy; lÞ¼

n;m

ðy; lÞ¼

ðy; lÞ (Eq. 6)

where

ðy; lÞ¼aP

ðcos yÞðg

cos ml þ h

sin mlÞ: (Eq. 7)

We can think of these “surface harmonics,” P

ðcos yÞcos ml and

ðcos yÞsin ml, as being the two-dimensional analogs of the one-

dimensional Fourier harmonics on a circle.

For a given physical field, the numerical values of the coefficients,

and h

will depend on the origin of our coordinate system. If, for

example, we used the meridian of Paris, rather than that of Greenwich,

to define l ¼ 0, we would find that the coefficients would be different.

However, for a given pair of g

and h

we find that the sum of their

squares, ðg

þðh

is a constant, independent of our choice of the

origin l ¼ 0 (exactly as in one-dimensional Fourier analysis). More

generally, even if we also move the y ¼ 0 axis to another point on

the sphere, it turns out that we have

½ðg

þðh

¼constant; (Eq. 8)

where the sum is over m ¼ 0,1,2,...,n. (This particularly simple result

is true provided that we use the Schmidt seminormalized definitions

for the Legendre polynomials P

, as is conventional in geomagnetism.

It is also true if we use fully normalized P

(as do workers in gravity),

but not if we use the basic mathematical definitions (sometimes

denoted by P

) which give unnormalized functions, the mean-square

of which varies with m as well as with n.) In fact any one of the V

Eq. (6) is physically the same, whatever our choice of coordinate

system; the processes producing the fields do not “know” of our

arbitrary choice of coordinate system.

We can think of all the contributions of a given degree n as having

essentially the same minimum (surface) wavelength on the sphere, of

value (approximately) the circumference divided by degree n . Mathe-

matically, the three terms for n ¼ 1 correspond to the potentials given

by three dipoles at the center, the n ¼ 2 terms to five quadrupoles at

the center, and so on, but of course physically the sources are distrib-

uted current systems.

Just as in Fourier analysis the various harmonics are orthogonal

round the circle, these surface harmonics are orthogonal over the sphe-

rical surface; the average value over the sphere of the product of two

different surface harmonics is zero. Now using <...> for the average

over the sphere r ¼ a, the mean-square value of V over the surface can

be expressed as the sum



where V



¼ a

½ðg

þðh

=ð2n þ 1Þ; (Eq: 9Þ

the factor 1/(2nþ1) is the mean-square value of each harmonic over

the surface for our Schmidt seminormalized P

However we are using the scalar potential V only as a convenient

mathematical simplification. What we are really interested in is the

magnetic field B itself. It turns out that if we represent each harmonic

ðcos yÞcos ml or P

ðcos yÞsin ml by (say) W

, and put

¼gradW

; (Eq. 10)

then the vector fields B

are themselves also orthogonal over the

sphere: using the scalar product for the multiplication, we have

 B



¼ 0

unless n ¼ n, m ¼ m (and both harmonics use only either cos ml or

sin ml);

 B



¼ðn þ 1Þ: (Eq. 11)

The factor (n þ 1) is the mean-square value over the sphere of the vec-

tor fields corresponding to each of the individual harmonics of the

degree n part of the scalar potential V (Lowes, 1966). (This factor

(n þ 1) is valid only if the sources of the field are internal to the

sphere. For external sources, e.g., ionospheric or magnetospheric cur-

rents, the factor is n.)

So if B is the observed field (X, Y, Z), where X, Y, Z are Cartesian

components of the vector, the mean-square value of the field vector

over the sphere r ¼ a becomes

< B  B >¼ B

¼ X

þ Y

þ Z



(Eq. 12)

Where

¼ðn þ 1Þ



þ h



: (Eq. 13)

If we plot R

as a function of n, we have t he geomagnetic spatial

spectrum.

An alternative interpretation of Eq. (13) is that if it is multiplied by

1/2m

(and the coefficients are expressed in units of tesla) it gives the

stored magnetic energy density B·H/2 (the energy stored per unit

volume in the magnetic field, in J m

3

) averaged over the surface; so

this spatial spectrum (of the vector field) is sometimes called the geo-

magnetic power spectrum, or the geomagnetic energy spectrum. (R

roughly analogous to the “degree variance” used in gravity. But note that

while R

is zero for n ¼ 0 in geomagnetism, as there are no magnetic

monopoles, the n ¼ 0 term completely dominates in gravity.)

Results for the geomagnetic field

If we apply Eq. (13) to the results of a spherical harmonic analysis of

the geomagnetic field of internal origin, as observed globally by satel-

lites such as Ørsted and CHAMP, and plot R

on a logarithmic scale

against n, we obtain the spectrum of Figure G49. Formally, given

observation of the field only at and above the Earth’s surface, we can-

not tell whereabouts inside the surface are the currents producing the

field. But if we look at the plot, we see that to a first approximation

it can be represented as the sum of two straight lines (two power-law

functions on a linear scale). Although there is argument about the

exact shape of the best-fitting curves, it is now accepted that (almost

all of) the field from harmonics up to about degree n ¼ 14 comes from

electric currents in the Earth’s core, while (almost all of) the field of

higher degrees comes from the magnetization of crustal rocks.

As is well known, at the Earth’s surface the geomagnetic field is

dominated by its dipole component, and indeed the n ¼ 1, dipole,

point lies well above the first line. But the line is a good fit for values

of n from 2 to about 12. If we go back to the definition of Eq. (5),

and remember that taking the gradient in effect introduces another

power of (a/r), we see that to draw the equivalent plot for a radius r

different from a, we have to multiply each R

by the factor (a /r )

2nþ4

This has more effect on the points for larger n, so that for r < a the

line becomes less steep. In fact, if we go down to the radius of the

core-mantle boundary (CMB) the line becomes nearly horizontal,

and if we continued the extrapolation a little further we would find that

just below the CMB the line becomes horizontal; each degree would

make about the same contribution to the total ms field. Strictly,

because there will be electric currents in this region, this further extra-

polation is not allowable, but this result is a convincing argument that

these fields do have their origin in the core.

GEOMAGNETIC SPECTRUM, SPATIAL 351

The other line, for higher spatial frequencies/shorter wavelengths,

is already sloping upward, so we cannot use such a simple argument

to indicate the source depth. The results for even higher frequencies

(see below) show that the curve does in fact eventually turn

downward, as it must for the crust to contribute only a finite field,

and the overall shape is consistent with reasonable models of crustal

magnetization.

So there is no doubt that this separation into core and crustal origin

must be largely correct; the division is usually put at about n ¼ 14. But

note that the spectrum of the core field will continue above this value,

and that of the crustal field will continue below this value. We do not

know the detailed physics that lies behind each of the two spectra, so

we can only guess at how they are to be extrapolated. But if we simply

extrapolate the straight lines, then we find that at the Earth’s surface

the core field (of total root-mean-square (rms) magnitude about

45000 nT) would contribute about 10 nT rms from degrees beyond

13, while the crust (total rms about 300 nT) would contribute about

10 nT from degrees below n ¼ 14.

While the spectrum of the main field itself is now quite well

determined (by measurements from polar-orbiting satellites such as

Magsat, Ørsted, and CHAMP), the spectrum of the crustal field (see

Crustal magnetic field) is much more uncertain. All the points shown

on Figure G49 are obtained from satellite measurements. However at

satellite altitudes the crustal field is small, and for individual measure-

ments is often not much larger than the instrument noise. Probably

more importantly, there are also comparable or larger time-varying

fields of external origin, and the separation of the spatial-variation

and time-variation of the field seen by a moving satellite is very diffi-

cult. In fact, as techniques improve there has been a tendency for the

derived crustal spectrum to become lower on a graph such as that of

Figure G49! For degrees above about 30–40 it is at present difficult

to do a formal spherical harmonic analysis of the satellite data, though

it is sometimes possible to do a Fourier, one-dimensional, analysis

along individual tracks. Similarly we can use the long tracks of the

Project MAGNET airborne vector magnetometer (see Project

MAGNET), and a much larger number of tracks from oceanographic

vessels towing a scalar magnetometer; these approaches can extend

the data out to about n ¼ 1000. However these are only one-

dimensional cross sections of the field variation over the two-dimen-

sional sphere, so some assumptions have to be made (see, e.g., Korte

et al,, 2002); also, the near-surface tracks are almost all over the

oceans, where the crust is systematically thinner than under the conti-

nents. Such results however do suggest that the mean-square field per

degree does in fact start to decrease at higher degrees. In theory, for the

very high degrees, we could analyze data from the large number of

detailed aeromagnetic surveys which have been carried out (though

mostly over land), but there are many problems, and this has not yet

been attempted.

This concept of a spatial spectrum can obviously be extended to the

secular variation, the time variation of the main field. A complication

is that there are many external sources of field variations, of period of

a year or less, the effects of which are most easily removed by taking

averages over one year. This averaging is most easily done with the

long-term surface observations at magnetic observatories and repeat

stations; unfortunately these have only a very poor global coverage.

Satellites have much better global coverage, and are now also giving

continuous coverage in time, though unfortunately their measurements

are subject to more interfering effects. However, at present it looks as

though we can estimate the secular variation over the last year or so,

and hence its spectrum, up to about n ¼ 10 (see, e.g., Langlais

et al., 2003). The spectrum for 2000.0 is shown on Figure G49;itis

much flatter than that of the main field, so that (in this range) at the

CMB the shorter wavelengths are varying more rapidly than the longer

wavelengths.

History

Lowes (1966) was the first to introduce to the English-speaking com-

munity the expression for mean-square field, Eq. (11) of this article, in

the context of geomagnetism; he also produced the first power spec-

trum, analogous to Figure G49, in Lowes (1974). However, essentially

the same expression had been introduced earlier in Germany, by Lucke

in a colloquium in 1955, referred to by Fanselau and Lucke (l956).

Lucke worked in terms of the energy per unit volume stored in a

magnetic field, which (in modern notation) is

E ¼ð1=2ÞB  H ¼ð1=2m

ÞB  B: (Eq. 14)

So, when he averaged over the sphere r ¼ a, he obtained

¼ð1 =2m

Þ B:B

¼ð1=2m

ðn þ 1Þ g



þ h



¼ð1 =2pm

; (Eq. 15)

Ehi¼ð1=2m

; (Eq. 16)

in the notation of this article.

Lucke’s lecture was also referred to by Mauersberger (1956), who

produced the same expression for the mean energy density, but using

a different derivation. Hence graphs such as those of Figure G49 are

sometimes referred to as a “Lowes spectrum,”“Lowes-Mauersberger

spectrum,” or “Mauersberger-Lowes spectrum.”

Frank Lowes

Figure G49 Spatial spectrum of the internal part of the

geomagnetic field at 2000.0. The circles are the values of R

, the

mean-square vector field produced by the harmonics of degree n

over the surface r ¼ a, plotted on a logarithmic scale against n;

right-hand scale. The squares are the corresponding values for the

secular variation; left-hand scale. The coefficients used are those

of CM4, Sabaka et al. (2005).

352 GEOMAGNETIC SPECTRUM, SPATIAL

Bibliography

Fanselau, von G., and Lucke, O., 1956. Über die Veränderlichkeit des

erdmagnetischen Hauptfeldes und seine Theorien. Zeitshrift für

Geophysik, 22: 121–216.

Korte, K., Constable, C.G., and Parker, R.L., 2002. Revised magnetic

power spectrum of the oceanic crust. Journal of Geophysical

Research, 107(B9): doi:10.1029/2001JB1389.

Langlais, B., Mandea, M., and Ultrée-Guérard, P., 2003. High-resolution

magnetic field modeling: application to MAGSAT and Ørsted data.

Physics of the Earth and Planetary Interiors, 135:77–79.

Lowes, F.J., 1966. Mean-square values on sphere of spherical harmo-

nic vector fields. Journal of Geophysical Research, 71: 2179.

Lowes, F.J., 1974. Spatial power spectrum of the main geomagnetic

field, and extrapolation to the core. Geophysical Journal of the

Royal Astronomical Society, 36: 717–730.

Mauersberger, P., 1956. Das Mittel der Energiedichte des geomagne-

tischen Hauptfeldes an der Erdoberfläche und seine säkulare

Änderung, Gerlands Beitrage Geophysik, 65: 207–215.

Sabaka, T.J., Olsen, N., and Purucker, M., 2005. Extending compre-

hensive models of the Earth’s magnetic field with Ørsted and

CHAMP data. Geophysical Journal International, 159: 521–547.

Cross-references

Crustal Magnetic Field

Harmonics, Spherical

IGRF, International Geomagnetic Reference Field

Main Field Modeling

Nondipole Field

Project Magnet

GEOMAGNETIC SPECTRUM, TEMPORAL

The geomagnetic field varies on a huge range of timescales, and one

way to study these variations is by analyzing how changes in the geo-

magnetic field are distributed as a function of frequency. This can be

done by estimating the spectrum of geomagnetic variations. The power

spectral density S( f ) is a measure of the power in geomagnetic field

variations at frequency f. When integrated over all frequencies it

measures the total variance in the geomagnetic field.

Figure G50/Plate 2 shows a schematic of the various processes that

contribute to the geomagnetic field, and these can be roughly divided

according to the frequency range in which they operate. The bulk of

Earth’s magnetic field is generated in the liquid outer core, where fluid

flow is influenced by Earth rotation and the geometry of the inner

core. Core flow produces a secular variation in the magnetic field,

which propagates upward through the relatively electrically insulating

mantle and crust. Short-term changes in core field are attenuated by

their passage through the mantle so that at periods less than a few

months most of the changes are of external origin. The crust makes

a small static contribution to the overall field, which only changes

detectably on geological timescales making an insignificant contribu-

tion to the long-period spectrum. Above the insulating atmosphere

the electrically conductive ionosphere (q.v.) supports Sq currents with

a diurnal variation as a result of dayside solar heating. Lightning gen-

erates high-frequency Schumann resonances in the Earth/ionosphere

cavity. Outside the solid Earth the magnetosphere (q.v.), the manifesta-

tion of the core dynamo, is deformed and modulated by the solar

wind, compressed on the dayside and elongated on the nightside. At

a distance of about 3 earth radii, the magnetospheric ring current

acts to oppose the main field and is also modulated by solar activity.

Although changes in solar activity probably occur on all timescales the

associated magnetic variations are much smaller than the changes in

the core field at long periods, and only make a very minor contribution

to the power spectrum.

With an adequate physical theory to describe each of the above pro-

cesses one could predict the power in geomagnetic field variations as a

function of frequency. The inverse problem is to use direct observa-

tions or paleomagnetic measurements of the geomagnetic field to esti-

mate the power spectrum. Power spectral estimation is usually carried

out using variants of one of the following well-known techniques:

(1) direct spectral estimation based on the fast Fourier transform, and

using extensions and improvements to the time-honored periodogram

method introduced by Schuster in 1898 to search for hidden periodici-

ties in meteorological data; (2) the autocovariance method which

exploits the Fourier transform relationship between the time-domain

autocovariance and frequency-dependent power spectral density of a

process; (3) parametric modeling schemes like the maximum entropy

method based on a discrete autoregressive process. The relative merits

of these techniques have been widely discussed (see Constable and

Johnson, 2005: or Barton, 1983, for the geomagnetic context), while

all have been used in analyzes of geomagnetic intensity and directional

variations. It is generally acknowledged (e.g., Percival and Walden,

1993) that direct spectral estimation combined with tapering and

averaging of nearly independent spectral estimates can provide high-

resolution estimates and be used to optimize the unavoidable trade-offs

between variance and bias.

The parameter often chosen to represent the geomagnetic spectrum

is the field strength at midlatitudes, or a proxy form for times where

it is not possible to obtain a direct measurement. This is the case for

paleomagnetic time series derived from lacustrine or marine sediments

which provide only directional information and/or relative intensity

variations. Barton (1982) combined spectra from lake sediment direc-

tional paleomagnetic records with those from full vector data recorded

at magnetic observatories and used periodogram analysis in the first

attempt to provide an integrated power spectrum for periods ranging

from less than a year to 10

years. Courtillot and Le Mouël (1988)

merged Barton’s result with other spectral estimates at longer and

shorter periods extending the timescales from seconds to millions of

years. They debated whether the result was compatible with an overall

1/f

spectrum, and concluded that it was too early to make such an

inference.

A recent version of such a composite spectrum (Constable and

Constable, 2004) uses spectral estimates from relative paleointensity

variations (Constable et al., 1998) at long periods and is shown in

Figure G51 (note that this is an amplitude rather than a power

spectrum, that is the square root of power spectral density). Between

10

and 1 Hz, the spectrum is from Filloux (1987). Above 1 Hz,

the results are those of Nichols et al. (1988). Internal variations reflect-

ing motions of the fluid core dominate at periods longer than a few

months, and the spectrum generally rises toward longer period (low

frequency) with reversals (q.v.) of the dipolar part of the field the

dominant influence on 10

to 10

year timescales. The 11-year sunspot

cycle, solar rotation, and Earth ’s orbit modulate the distortions of the

field associated with geomagnetic storms, which themselves have

energy in the several hour to several second band. Energy at the daily

variation and harmonics comes from diurnal heating of the ionosphere.

Lightning creates high frequency energy in the Earth/ionosphere

cavity, which resonates at 7–8 s and associated harmonics. At the

highest frequencies there is presumed to be a continued fall-off

in the natural spectrum. The upturn seen in Figure G51 reflects the

dominant influence of human-made sources.

The spectrum in Figure G51 lacks information at very long periods

where the spectrum is dominated by the intensity variations associated

with changing geomagnetic reversal rate, and between about 10 ka and

10-year periods, where the timescales for processes of internal and

external origin overlap. Figure G52/Plate 4c provides a range of esti-

mates at centennial to 50 Ma periods for the spectrum of the geomag-

netic dipole moment (q.v.). The longest period power spectrum is

estimated from reversal times given by the magnetostratigraphic

GEOMAGNETIC SPECTRUM, TEMPORAL 353

timescale and shown in black. Further information comes from various

sedimentary relative paleointensity records with varying accumulation

rates and the dipole moment estimate of a time varying global

paleomagnetic field model for the past 7 ka.

In constructing a paleomagnetic power spectrum like that in

Figure G52/Plate 4c there are a number of challenges. A basic require-

ment for spectral analysis is a time series of observations, but there is

no single record that covers the time span of interest. The magnetostra-

tigraphic record appears to be nonstationary with long-term changes in

reversal rate, and provides no information about intensity variation on

long timescales. The relative paleointensity records from sediments not

only lack an absolute scale, but are usually unevenly sampled in time

so that some stable calibration and interpolation scheme is required

before using the standard analysis techniques. It is likely that some

sediments record a smoothed version of the geomagnetic signal

because of low sedimentation rates, while in others it may be neces-

sary to consider the possibility of aliasing. Nongeomagnetic signals

may be inadvertently interpreted as arising from geomagnetic varia-

tions with time.

The choice of dipole moment (as opposed to some other geomag-

netic field parameter) is motivated in large part by the dominance of

the geocentric axial dipole when the field is averaged over long time

intervals: the strength of the axial dipole is representative of the global

field, and may be related to the amount of energy required by the geo-

dynamo or to the geomagnetic reversal rate. Although it is possible

that other properties of the field (such as nondipole field contributions)

directly reflect particular physical processes controlling the secular

variation, resolving such variations in paleomagnetic time series

remains controversial. Overall, it is likely that the power in geomag-

netic field variations is underestimated.

There is substantial scope for improving the spectra in both

Figures G51 and G52/Plate 4c. Relative intensity records from sedi-

ments are steadily improving, and new modeling techniques may

extend time-varying geomagnetic field models providing better dipole

moment estimates on million year timescales. This may give new

insight into what controls very long-period secular variation. Many

newer observations could be used for direct spectral estimates to

replace the current schematic spectrum at periods from decades down

to 1 s. Although the general form of the spectrum is quite well under-

stood in this region, detailed analyzes could provide further insight

Figure G51 Composite amplitude spectrum of geomagnetic

variations as a function of frequency (Constable and Constable,

2004): annotations indicate the predominant physical processes at

the various timescales. Copyright American Geophysical Union,

2004, reproduced with their and the authors’ permission.

354 GEOMAGNETIC SPECTRUM, TEMPORAL

into the underlying physical processes. Techniques such as wavelet

analysis, that can take account of nonstationarity in the underlying

geomagnetic processes, have yet to be fully exploited for geomagnetic

data (Guyodo et al., 2000) and may prove useful. However, despite the

relatively crude nature of existing spectral estimates it is apparent that

the geomagnetic power spectrum does not follow a simple power law

fall-off with increasing frequency. The form is instead influenced by

the characteristic timescales that reflect the distinct physical processes

contributing to the geomagnetic field.

Catherine Constable

Bibliography

Barton, C.E., 1982. Spectral analysis of palaeomagnetic time series

and the geomagnetic spectrum. Philosophical Transactions of the

Royal Society of London A, 306: 203–209.

Barton, C.E., 1983. Analysis of paleomagnetic time series-techniques

and applications. Geophysical Survey, 5: 335–368.

Constable, C.G., and Constable, S.C., 2004. Satellite magnetic field

measurements: applications in studying the deep earth. In Sparks,

R.S.J., and Hawkesworth, C.J., (eds.), The State of the Planet:

Frontiers and Challenges in Geophysics. Washington, DC: American

Geophysical Union, doi: 10.1029/150GM13, pp. 147–160.

Constable, C.G., and Johnson, C.L., 2005. A paleomagnetic power

spectrum. Physics of the Earth and Planetary Interiors, 153:

61–63.

Constable, C.G., Tauxe, L., and Parker, R.L., 1998. Analysis of 11 Ma

of geomagnetic intensity variation. Journal of Geophysical

Research, 103: 17735–17748.

Courtillot, V., and Le Mouël, J.-L., 1988. Time variations of the

Earth’s magnetic field: from daily to secular. Annual Review of

Earth and Planetary Science, 16: 389–476.

Filloux, J.H., 1987. Instrumentation and experimental methods for

oceanic studies. In Jacobs, J.A. (ed.), Geomagnetism. London:

Academic Press, pp. 143–248.

Guyodo, Y., Gaillot, P., and Channell, J.E.T., 2000. Wavelet analysis

of relative geomagnetic paleointensity at ODP Site 983. Earth

and Planetary Science Letters, 184: 109–123.

Nichols, E.A., Morrison, H.F., and Clarke, J., 1988. Signals and noise

in measurements of low-frequency geomagnetic fields. Journal of

Geophysical Research, 93: 13743–13754.

Percival, D.B., and Walden, A.T., 1993. Spectral Analysis for Physical

Applications: Multitaper and Conventional Univariate Techniques.

Cambridge: Cambridge University Press.

Cross-references

Dipole Moment Variation

Harmonics, Spherical

Ionosphere

Magnetosphere of the Earth

Mantle

Nondipole Field

Reversals, Theory

Secular Variation Model

GEOMAGNETISM, HISTORY OF

In its present form, the geophysical discipline of geomagnetism is of

relatively recent origin (the term “geomagnetism” was coined in

1938 by Sydney Chapman,(q.v.)), yet interpretations of the Earth’s

magnetic field have been propounded by scholars from the Middle

Ages onward, in the broader context of cosmologies, natural philoso-

phies, and oceanic navigation. In this historical overview, geomagnet-

ism will be considered as the scientific study of the Earth’s internal

field and its secular change, encompassing both descriptive and causal

hypotheses; paleomagnetism and the external field are treated else-

where. Furthermore, since most of this volume is dedicated to current

issues, stress will here be placed on earlier times.

Main developmental stages

The history of geomagnetism can be subdivided into three main peri-

ods: firstly, a proto-scientific stage (up to the 16th century), during

which awareness slowly grew of the existence of a global magnetic

property worthy of investigation. At this time, causal hypotheses

almost exclusively identified the heavens as the seat of magnetic

attraction. Secondly, an early-modern stage took place (16th to early

19th century), during which directional data (magnetic declination

and inclination) were increasingly measured, compiled, and mapped.

These efforts led to the discovery of secular variation and causal mod-

els involving one to three crustal or nuclear dipoles (terrestrial polar

attraction). Thirdly, a modern stage emerged (from the 1830s), charac-

terized by measurement of the full magnetic vector (direction and

intensity) in dedicated scientific surveys, observatories, and satellites;

by the description of the field as a whole, both at the surface and at

the top of the source region; and by the mid-20th century introduction

of geodynamo theory (q.v.), leading to a multitude of numerical and

laboratory magnetohydrodynamic simulations.

These developments are summarized per century in Table G8, which

pays separate attention to empirical (data and mapping) and theoretical

aspects (hypotheses). As is apparent, the main watershed between the

early-modern and modern stage pervades both. The most influential

scientist in bringing about this change was Carl Friedrich Gauss

Figure G52/Plate 4c Composite spectrum for the geomagnetic

dipole moment constructed from the magnetostratigraphic

reversal record with and without cryptochrons (frequencies

2

–20 Ma

1

), various sedimentary records of relative

paleointensity (10

–10

1

), and from the dipole moment of a

0–7 ka paleomagnetic field model (10

–10

1

). Figure

redrawn from Constable and Johnson (2005).

GEOMAGNETISM, HISTORY OF 355