Stacey F.D., Davis P.M. Physics of the Earth

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a thermal history with this conflict in relative

cooling rates.

When we introduce radiogenic heat to the

argument, we have more early heat, but if

Q is constrained by the mantle control, the

added heat merely slows the early cooling

rate by substituting a heat source of even lower

thermodynamic efficiency. Precession introdu-

ces some early energy of high efficiency but, in

Section 24.7, we conclude that it is not important

at the present time and would have been only a

minor contribution, even early in the life of the

Earth.

Now we take a closer look at Eqs. (23.24) and

(23.26) to see how drastically we must modify the

constant

Q assumption. Suppose that 50% of the

heat (Q

þQ

) is used, as we might assume to be

the case two billion years ago. The values of V

and T are obtained by writing

 T



þ Q



¼ Q

 T



þQ

 T



3=2

þ Q

ðÞ:

(23:27)

With the values of Q

and Q

above, we have a

numerical solution (T

T)/(T

T

) ¼0.5576.

The corresponding fractional volume is V/

¼0.4164, that is the inner core is less than

half formed although the required temperature

drop has passed the half-way point. Now we

can use (T

T)/(T

T

) ¼0.7467 in Eq. (23.26)

to find the fraction of

Q contributed by the Q

term at that stage of inner core development.

The result is 0.435, compared with the present

value, 0.508.

A conclusion from Eq. (23.27) is that the Q

term was dominant early in the development of

the inner core but that for the second half of its

development, perhaps the last two billion years,

the ratio of the Q

and Q

contributions varied

too little to question the constant

Q assumption.

In Section 25.4 we mention reports of paleomag-

netic observations suggesting that the geomag-

netic field was systematically weaker 2 to 2.5

billion years ago than in more recent times.

This is consistent with a slightly smaller contri-

bution to core energy by the inner core at that

time. However, when the inner core was much

smaller, there is no alternative to a stronger core-

to-mantle heat flux, whether attributed to more

rapid cooling or to radioactivity. The constant

assumption is not compatible with the observa-

tion that the geomagnetic field existed 3.5 bil-

lion years ago. But the inference that it appears

satisfactory for the last two billion years is con-

sistent with our argument that radiogenic heat

has a minor role.

We have little observational control on theo-

ries of the early state of the core, but conclude

that it was never more than 200 K hotter than at

present. There is even a suggestion, reviewed by

Sumita and Yoshida (2003), that the core was

initially stably stratified. The idea is that, as the

Earth accreted, the core formed by liquid iron

sinking through the proto-mantle, with its sol-

utes in local chemical equilibrium with mantle

minerals. But the solubility of oxygen in liquid

iron increases with pressure, so that the early

accumulating inner part of the core would have

had less oxygen than the later part because it

percolated through the mantle at lower pres-

sure. The resulting stable stratification would

not have broken down until development of

the inner core began to release excess oxygen

into the liquid. It is important not to contem-

plate any theory that allows doubt about the

viability of the dynamo. If this stable stratifica-

tion occurred it could not have survived for a

billion years, and requires a very early start to

inner core development, but that is compatible

with the cooling model presented here.

We see that, with the core conductivity that

we assume, it is marginally possible to avoid the

need to assume radiogenic heat in the core. In

view of the uncertainty in the conductivity, we

must consider the case for radioactivity in the

core to remain open for discussion. This is not

the only doubt. The calculations all assume that

the rejected light solute from inner core solidifi-

cation mixes uniformly into the outer core. If we

assume instead that it all finds its way to the top

of the core then the gravitational energy is 1.84

times as much as for uniform mixing. Such a

dramatic enhancement of compositional power

would greatly strengthen the case against core

radioactivity, but there is no serious evidence

for it. A consequence would be a gravitationally

23.5 COOLING HISTORY OF THE CORE 387

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stable light layer at the top of the core, with a

mass deficit Dm (Eq. (22.30)), or perhaps a large

fraction of it, relative to uniform composition. It

would appear as a surface layer of thickness t (m)

with a density deficit D ¼(2.8 10

/t)kgm

3

rel-

ative to the deeper core. If the volume of the layer

is comparable to the inner core volume then its

thickness is t  50 km and D  560 kg m

3

The existence of a stably stratified layer with a

thickness of this order has been suggested on the

basis of geomagnetic studies (Whaler, 1980;

Braginsky, 1993, 1999) but doubted (Fearn and

Loper, 1981). If even a small fraction of the light

concentrate reached the core–mantle boundary

it would establish a stable layer, so the possibility

must be allowed, but the fraction would have

to be a large one if it is to have a significant

influence on core energetics. There is no seismo-

logical evidence requiring such a layer, but it

would not be easily obtained. No P-waves have

their deepest penetration in the outermost part

of the core and observations would need to use a

weak S-to-P conversion at the core–mantle

boundary.

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The geomagnetic field

24.1 Preamble

The study of geomagnetism has a longer history

than other branches of geophysics, partly

because of its use as an aid to navigation. In the

earliest times, the spontaneous alignment of

magnetized iron oxide (lodestone) was believed

to be due to an extra-terrestrial influence.

Properties of spherical lodestones were described

in a letter, written in 1269 by Petrius Peregrinus

(Pierre de Maricourt of Picardy in France).

He introduced the word ‘poles’ in connection

with magnets because their influence was

believed to be derived from the celestial poles.

Recognition of the similarity of the magnetic

fields of lodestones to the field of the Earth

appears to have awaited the work of Robert

Norman and William Gilbert of England in the

sixteenth century. A review of their evidence

that the Earth is a great magnet, with its axis

approximately north–south, was presented, in

Latin, in Gilbert’s book De Magnete,publishedin

1600. An English translation by P. F. Mottelay

appeared in 1893 and has been reprinted (Dover

Publications, 1958) as an early milestone in scien-

tific literature.

Magnetic declination, the difference between

the direction of the field, as indicated by mari-

ners’ compasses, and true (geographic) north,

was indicated on navigational maps by the mid-

sixteenth century, but was attributed to a mis-

alignment of magnetic and geographic axes and

not to a flaw in Gilbert’s concept of a dipole field.

There were also observations of a progressive

change in declination in London that were not

mentioned by Gilbert, although it would be sur-

prising if he were unaware of them. This is a

manifestation of what we now know as the geo-

magnetic secular variation, which was first

reported in 1634 by H. Gellibrand. Edmund

Halley, of comet fame, made a detailed study of

magnetic declination and its secular variation,

initially over the Atlantic Ocean and then more

widely. By 1700 he recognized not only that

there are significant departures from a dipole

field, but that features of the field showed a

generally westward drift. He attributed this to

inner concentric shells, rotating slightly more

slowly than the outer shell of the Earth and

carrying with them embedded magnets, so

anticipating by more than two centuries modern

ideas about differential rotation within the core.

The westward drift has featured prominently in

geomagnetic studies and is discussed in

Section 24.6.

Following Halley’s work, there was only one

notable development before the twentieth cen-

tury. In 1838, C. F. Gauss applied a development

of potential theory (the basis of spherical har-

monic analysis – Appendix C) to confirm beyond

doubt the inference of Gilbert and Halley that the

field is of internal origin. By that time the con-

nection between rapid disturbances of the field

and aurorae was well documented and the only

observed natural events that appeared to be

related to geomagnetism occurred outside the

Earth. Thus, Gauss’s analysis may have stalled a

drift back to Peregrinus’s notion of an externally

generated field. Gauss must have been aware of

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experiments on electromagnetic induction by

M. Faraday, published six years earlier, but he

did not make the connection and there is no evi-

dence of any suggestion that the geomagnetic

field was driven by deep electric currents until

about 1900. Even then it was not readily accep-

ted, and studies of geomagnetism were com-

pletely dominated by relationships between

magnetic disturbances and extra-terrestrial

effects, solar activity and aurorae.

The self-exciting dynamo mechanism that we

now accept as the cause of the geomagnetic field

faced conceptual difficulties that were gradually

removed by four developments.

(i) In 1908, G. Hale, director of the Mount

Wilson astronomical observatory, reported

the splitting of spectral lines in the radia-

tion from sunspots that could be caused

only by very strong magnetic fields (the

Zeeman effect).

(ii) Hale’s observation aroused speculation on

the mechanism for spontaneous generation

of sunspot fields by motion of a fluid con-

ductor, prompting Larmor (1919) to suggest

that this could explain not only a solar field

but also the terrestrial field.

(iii) H. Alfven developed a theory of cosmic-scale

magnetic fields controlled by and control-

ling the motion of tenuous plasma. He origi-

nated the frozen flux concept of a magnetic

field carried around and deformed by a fluid

conductor, showing that amplification was

not only possible but inevitable with suit-

able turbulent motion.

(iv) Seismology established that the Earth has a

dense fluid core. The cosmic abundance of

iron, apparent in meteorites and in the

solar atmosphere, pointed to a liquid

iron core.

In the 1940s, W. M. Elsasser began putting

these ideas together in the first serious study of

the magnetohydrodynamic dynamo mecha-

nism. He was soon followed by E. C. Bullard,

who was quick to recognize the significance of

what Elsasser had started. But rival hypotheses

were slow to die. Theoretical physicists, led by

Einstein, were groping for evidence of a connec-

tion between gravity and electromagnetism,

postulating a fundamental relationship between

the rotations and magnetic fields of large bodies.

This particular hypothesis received its fatal

blow only with very sensitive observations on

rotating spheres, which induced no magnetic

fields (Blackett, 1952), and observations that

the strength of the geomagnetic field generally

increased, rather than decreased, with depth

in mines.

In spite of early statements, such as one

in 1902 by L. A. Bauer (cited by Parkinson,

1983, p. 108) that the field was due ‘doubtless to

a system of electric currents embedded deep

within the interior of the Earth and connected

in some way with the earth’s rotation’, dynamo

theory faced strong and influential opposition.

Especially negative was S. Chapman who, as

late as 1940 (Chapman and Bartels, 1940, p 704),

reasserted an earlier statement by A. Schuster:

‘the difficulties which stand in the way of basing

terrestrial magnetism on electric currents inside

the earth are insurmountable’. Ideas about geo-

magnetism have evolved over many years.

Complementary perspectives on its history are

presented by Elsasser (1978, pp. 225–230),

Parkinson (1983) and Merrill et al. (1996).

Our observations of the geomagnetic field are

limited by the fact that it is generated in the core,

which is little more than half the radius of the

Earth. Small-scale features are invisible, not just

because they are diminished by distance, but

because they are obscured by magnetization in

the Earth’s crust. Core-generated features of the

field smaller than about 1500 km are concealed.

Our view of the field is restricted also by the

electrical conductivity of the mantle, which is

low compared with that of the core but high

enough to attenuate magnetic fluctuations with

periods shorter than a year or so. Estimates of the

conductivity profile of the mantle have been

based on the assumption that it varies with

radius, but not laterally. Although this gives a

global view it can be no more than an approx-

imation and, with respect to the lowermost man-

tle, may be seriously wrong. A long-standing

observation that some features of the field

appear to be stationary while others drift

(Yukutake and Tachinaka, 1969) is consistent

with the idea that the ‘standing’ features are

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held in place by high conductivity patches in the

mantle. In view of the heterogeneity of D

which is presumed to be both compositional

and thermal, this is the most plausible location

for such patches. Although this is conjectural, it

introduces a complication to the inference of

core motions from extrapolations of the surface

field to the core–mantle boundary. For a recent

discussion of inferred core motion see Eymin

and Hulot (2005).

Unmaintained electric currents in the core

would decay by ohmic dissipation with a time

constant of order 10

years. The general features

of a regeneration mechanism, involving a

combination of convection and rotation, are

understood (Section 24.5). Both thermal and

compositional convection are presumed to

occur, with the compositional effect dominant

at the present time. Core energetics are dis-

cussed in Section 22.7. The possibility of a con-

tribution by precessional torques is also

canvassed (Malkus, 1963, 1989; Vanyo, 1991),

but, by our estimate of the angular difference

between core and mantle angular momentum

axes (Section 7.5), is only a minor effect. There

is no precise estimate of the energy demand

by the dynamo, but we can appeal to the princi-

ple that whatever mechanism occurs most

easily is the one that dominates. 10

may well suffice, but a consideration of nuta-

tional coupling suggests three times as much

and, in calculating core energy, we assume a

dynamo power requirement of 3 10

W, as in

Fig. 21.4.

Although the secular variation of the geo-

magnetic field is precisely observed, the rate

of change is slow and the historical record of

direct observations gives only limited insight.

To see how the field behaves on a longer time

scale we turn to paleomagnetism, the record

of the field in ancient times, preserved in the

magnetizations of rocks and archeological

specimens. When we look at the field on a geo-

logical time scale, we see a new range of phe-

nomena, that could not even be guessed at from

the historical record. This is the subject of

Chapter 25, but the discoveries of paleomagnet-

ism, especially field reversals and the axial

dipole principle, have had a profound effect

on our understanding of the field, as discussed

in this chapter.

24.2 The pattern of the field

As William Gilbert noticed, the magnetic field of

the Earth is similar to that of a magnetized

sphere, or to a small but powerful bar magnet

at the centre. Such a field is termed a dipole field

because it could, in principle, be produced by a

pair of magnetic poles of equal strengths but

opposite signs a small distance apart. The mag-

netic moment or dipole moment, m, is then

envisaged as the product of pole strength and

separation. Since magnetic moments are pro-

duced by circulating currents, the equivalent

definition in terms of a current loop is the prod-

uct of current i and loop area A,

m ¼ iA: (24:1)

Although the Earth’s field is predominantly dipo-

lar, non-dipole components contribute about 20%

of its strength at the surface. We refer to a best-

fitting dipole and need to be specific about what

this means. The dipole most closely fitting the

observed field is slightly off centre, but if we

refer to the best-fitting geocentric dipole then its

moment (in 2005) is

Earth

¼ 7:768  10

: (24:2)

The axis of this dipole is inclined to the geo-

graphic or rotational axis by 108. If we discount

the equatorial component of this moment and

consider just the geocentric axial dipole then its

moment is 7.644 10

The field of a dipole is conveniently repre-

sented in terms of the scalar magnetic potential,

, which may be differentiated to obtain any

component of the field,

 r

4pr

m cos 

4pr

; (24:3)

where  is the angle between the dipole axis and

the radius vector r from the dipole to the point

considered. These equations assume the dimen-

sions of the current loop to be negligible com-

pared with r. The field is

24.2 THE PATTERN OF THE FIELD 391

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B ¼

grad V

: (24:4)

The horizontal (circumferential) and vertical

(radial) components are



¼



@





sin ; (24:5)

¼





cos : (24:6)

Applying these equations to the Earth’s surface,

taken to be a sphere of radius a, and noting that

they refer to the magnetic axis, not the geo-

graphic axis, the horizontal and vertical compo-

nents of the field, in conventional notation, are

H ¼B



ðr ¼ aÞ¼B

sin ; (24:7)

Z ¼B

ðr ¼ aÞ¼2B

cos ; (24:8)

where



Earth

¼ 3:004  10

5

T ¼ 0:3004 Gauss

(24:9)

is the strength on the magnetic equator of the

best-fitting geocentric dipole field. By choosing

the coordinate axis to be the axis of the dipole

we avoid consideration of a latitudinal compo-

nent of the field, since @V

=@l ¼ 0. In the SI

system of units, used here, 

¼4p 10

7

1

is the permeability of free space. In the cgs sys-

tem B

¼0.3004 Gauss and is the same as

¼0.3004 Oersted, with 

¼1 Gauss/Oersted.

The total field strength at the Earth’s surface

(r ¼a)is

B ¼ B



þ B



¼ B

1 þ 3 cos





(24:10)

Its inclination to the horizontal is I, where

tan I ¼ B



¼ 2 cot  ¼ 2 tan 

(24:11)

and  ¼(90 8 ) is the magnetic latitude.

Equation (24.11) is used to calculate paleopole

positions from the dip angles of magnetic rema-

nence in rocks. It is also the differential equation

for a magnetic line of force because

tan I ¼

r d

¼ 2 cot ; (24:12)

which integrates to

sin



sin



; (24:13)

where 

is the magnetic co-latitude at which the

field line crosses radius a, generally taken to be

on the magnetic equator.

By subtracting the dipole field from the

observed field, we are left with the non-dipole

field. Its features, as they were in 1945, are clearly

displayed in Fig. 24.1, which is one of the maps

produced by Bullard et al. (1950), who sought clues

to the origin of the field by studying the behav-

iour of the non-dipole field. For the International

Geomagnetic Reference Field (IGRF2005,

Table 24.1), the rms strength of the non-dipole

field over the Earth’s surface, 1.11 10

5

T, is a

quarter of the rms strength of the dipole field,

ﬃﬃﬃ

¼4.248 10

5

T, but, as discussed below,

the ratio is quite different at core level.

The separation of the dipole and non-dipole

fields is accomplished by spherical harmonic

analysis. Equation (24.3) is the first term of an

infinite sum with the form of Eq. (C.11). As noted

in Appendix C, by convention in geomagnetism,

tesseral and sectoral harmonics are normalized

to have the same rms values over a spherical

surface as the zonal harmonics of the same

degree, but they are not fully normalized in the

sense of the p

harmonics in Appendix C. The

system used in geomagnetism is referred to as

Schmidt normalization after A. Schmidt, who

introduced it. Harmonic coefficients of the

field, g

and h

, obtained with this convention

are the Gauss coefficients of the field. For the

complete internal field, that is omitting terms

that represent a field of external origin,



l¼1



lþ1

m¼0

cos ml þ h

sin ml



 P

cos ðÞ: ð24:14Þ

This is so written that the coefficients, g and h,

have the dimensions of field; they are normally

given in nanoTesla (nT), that is, 10

9

Tor10

5

Gauss, often referred to as gammas in older liter-

ature. Coefficients up to (l,m) ¼(8,8) are given

in Table 24.1a. The use of satellite data for global

magnetic field modelling, discussed by Olsen

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FI G U R E 24.1 The non-dipole field for 1945. Contours give the vertical component in intervals of 2 T and

arrows represent the horizontal component. Reproduced, by permission, from Bullard et al. (1950).

Table 24.1a Spherical harmonic coefficients of the International Geomagnetic Reference Field

(IGRF) 2005 to degree and order 8, 8. For each (l,m) the coefficients are g

followed by h

nanoTesla, as defined by Eq. (24.14), referred to a surface of radius 6371.2 km

l 0 1 2345678

1 29556.8 1671.8

5080.0

2 2340.5 3047.0 1656.9

2594.9 516.7

3 1335.7 2305.3 1246.8 674.4

200.4 269.3 524.5

4 919.8 798.2 211.5 379.5 100.2

281.4 255.8 145.7 304.7

5 227.6 354.4 208.8 136.6 168.3 14.1

42.7 179.8 123.0 19.5 103.6

6 72.9 69.6 76.6 151.1 15.0 14.7 86.4

20.2 54.7 63.2 63.4 0.0 50.3

7 79.8 74.4 1.4 38.6 12.3 9.4 5.5 2.0

61.4 22.5 6.9 25.4 10.9 26.4 4.8

8 24.8 7.7 11.4 6.8 18.0 10.0 9.4 11.4 5.0

11.2 21.0 9.7 19.8 16.1 7.7 12.8 0.1

24.2 THE PATTERN OF THE FIELD 393

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(2002), leads particularly directly to the spherical

harmonic representation in Eq. (24.14).

There is no l ¼0 term in Eq. (24.14) because

that would correspond to a magnetic monopole.

If monopoles exist they certainly do not contrib-

ute to geomagnetism. The l ¼1 terms represent

the dipole. For this general representation of the

field, the axis is the geographic axis but the

co-latitude, , is the geocentric value, which differs

from the geographic co-latitude by a small angle

due to the ellipticity of the Earth (Fig. 6.2). The

coefficient g

is the axial component of the

dipole field and the equatorial components are

given by g

(through the Greenwich meridian)

and h

. The strength of the dipole field on the

magnetic equator is

¼ g



þ g



þ h



1=2

; (24:15)

as given by Eq. (24.9). The angle between the

magnetic and geographic axes is , where

tan  ¼ g



þ h



= g



¼ 0:192;

(24:16)

giving  ¼10.268. Higher degree terms in

Eq. (24.14) represent quadrupole, octupole and

higher multipole components of the field.

Field components in the northward (X ), east-

ward (Y ) and downward (Z ) directions are deriv-

atives of V

X ¼



@

; (24:17)

Y ¼



r sin 

; (24:18)

Z ¼ 

; (24:19)

at geocentric colatitude  and longitude l. All

of these components are directly observable,

whereas V

is not. If all three components

are measured, then there is some redundancy

in calculating the g and h coefficients, but

the additional information is needed if the field

is not assumed to be entirely internal in origin.

Then another set of coefficients, corresponding

to C

and S

in Eq. (C.11) and representing

Table 24.1b Secular variation of the IGRF 2005. These are the rates of change (nT/year) of the

coefficients in Table 24.1a

0 1 2 345678

1 8.8 10.8

21.3

2 15.0 6.9 1.0

23.3 14.0

3 0.3 3.1 0.9 6.8

5.4 6.5 2.0

4 2.5 2.8 7.1 5.9 3.2

2.0 1.8 5.6 0.0

5 2.6 0.4 3.0 1.2 0.2 0.6

0.1 1.8 2.0 4.5 1.0

6 0.8 0.2 0.2 2.1 2.1 0.4 1.3

0.4 1.9 0.4 0.4 0.2 0.9

7 0.4 0.0 0.2 1.1 0.6 0.4 0.5 0.9

0.8 0.4 0.1 0.2 0.9 0.3 0.3

8 0.2 0.2 0.2 0.2 0.2 0.2 0.5 0.7 0.5

0.2 0.2 0.2 0.4 0.2  0.3 0.5 0.4

394 THE GEOMAGNETIC FIELD

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external sources, must be separated from g and h.

This is discussed in Section 24.3 in connection

with the separation of fields induced within the

Earth by external, disturbing fields.

The normalization of spherical harmonics is

discussed in Appendix C. When this is applied to

the vector magnetic field, to obtain the strength

or intensity, and averaged over the surface of a

sphere, the rms field strength at the Earth’s sur-

face, due to any harmonic term represented by

or h

, is (Lowes, 1966)



rms

¼ðl þ 1Þ

1=2

; h



: (24:20)

Spherical harmonics are orthogonal functions,

so that in summing terms we add their squares,

and the mean square surface field due to all

harmonic terms of degree l is

¼ðl þ 1Þ

m¼0



þ h



: (24:21)

We can regard this as the sum of all terms

representing wavelengths (2pa/l ), so that a plot

of R

vs l (Fig. 24.2) is a power spectrum of the

spatial variation in field strength. This plot

shows two distinct ranges. At low harmonic

degrees the core field is dominant but falls

off rapidly with l, whereas for l > 14 it is masked

by the field due to crustal magnetization, which

has an almost white spectrum (R

nearly inde-

pendent of l ). Cain et al. (1989) fitted the data to

an equation with two terms, one representing

each source,

¼ 9:66  10

ð0:286Þ

þ 19:1ð0:996Þ

(24:22)

Extrapolation to core–mantle boundary

Surface field

10 20

Harmonic de

ree, l

(nT

)

FI G U RE 24.2 Mean square field strength,

, (Eq. 24.23), for each harmonic degree, l,

plotted as a function of l. Solid circles are

data points from Cain et al. (1989). The open

circle and the upper line are downward

continuations to the core–mantle boundary

of the dipole and the trend of harmonics l ¼2

to 14. Harmonics up to l ¼63 were included

in the original analysis but are increasingly

affected by noise.

24.2 THE PATTERN OF THE FIELD 395

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The form of Eq. (24.14) allows the surface

field to be extrapolated downwards to its source,

a mathematical process known as downward con-

tinuation in exploration geophysics, in which

it is used to estimate the depths and shapes of

sources of magnetic and gravitational anomal-

ies. Since the potential of an lth degree har-

monic falls off with radius as r

(l þ1)

,thefield

components (Eqs. (24.17) to (24.19)) fall off as

(l þ2)

and, therefore, the squ are of t he field

and the sum of squares of all degr ee l co mp o-

nents, that is R

,fallsoffasr

(2l þ4)

.Thus,the

mean square field at radius r, for all terms of

degree l,is

ðrÞ¼R

ðaÞða=rÞ

2lþ4

; (24:23)

where the value at the surface (radius a)is

obtained from the coefficients in Table 24.1a.

The upper line in Fig. 24.2 is the downward con-

tinuation to the core–mantle boundary by appli-

cation of Eq. (24.23) to the first term in

Eq. (24.22). At that depth the core-generated

field has a spectrum that is pink (almost white,

but still slightly red). Similarly, the crustal field,

observed at the surface, is spectrally slightly

pink, although that is not obvious in Fig. 24.2.

The field within a source region cannot be

estimated by downward continuation without fur-

ther assumptions because it can be represented

by a potential, such as Eq. (24.14), only outside

the source volume. However, the spectrum at the

surface of a source can be a useful indicator of its

dimensions. If we consider a source of infinites-

imal thickness at a single depth, then from the

spectrum at any higher level we can determine

the depth if the source spectrum is known, or

determine the source spectrum if the depth is

known. If spectral contributions from different

levels in a distributed source are independent,

their effect is simply additive at the surface. For a

spectrally uniform source occupying a depth

range that is small compared with the radius r,

the spectrum at the surface is very similar to that

for a spectrally similar but very thin source

located in the middle of the depth range. So,

although downward continuation into a source

region is formally disallowed, by downward

continuing to the mid-depth of a thin source

we can obtain a satisfactory measure of the

source spectrum.

The near whiteness of both crustal and core

fields at source level invites the inference that

the sources are in fact white and thin. This is

readily justified in the case of the crustal field.

By Eq. (24.22), the mid-depth of a spectrally white

crustal field would be at a radius r=aðÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

0:996

13 km below the surface. This coincides with the

median depth of the magnetized crust, because

material below 20 to 25 km is at temperatures

above the Curie points of magnetic minerals.

Hence, the crustal magnetization has a white

spatial spectrum. The significance of a similar

extrapolation for the core field (the first term of

Eq. (24.22)) is less obvious. The spectral

calculations of Cain et al. (1989), used in this

equation, give a median depth for a spectrally

white source 80 47 km below the core surface.

Several authors (e.g. Langel and Estes, 1982) have

given similar estimates, in spite of the long

extrapolation from the Earth’s surface.

We take the view that the nearly white spatial

spectrum of the field at core level is a physically

significant feature with a fundamental cause,

although there is not, to our knowledge, an

explanation in dynamo theory. Also, it is not

clear that the principle applies to other planets

and, in any case, it is necessary to exclude the

dipole and probably the quadrupole terms,

which do not fit the spectrum. But we suppose

that the white spectrum describes the smaller-

scale features that indicate core turbulence. We

consider a simple model that is consistent with

the observations and is useful to the discussion of

the secular variation of the field in Section 24.3.

Adopting the 80 km half-depth estimate, we

assume that a characteristic depth of about

160 km in the surface of the core is a physically

real feature. The current pattern is spectrally

white but the field at greater depths is unobserv-

able. By the frozen flux principle (Sections 24.3

and 24.5), the observable field moves with the

outer layer of the core, which screens the deeper

parts from view. The observed spectrum is char-

acteristic of the field in the surface layer and gives

no direct clue to the deeper pattern. The 160 km

396 THE GEOMAGNETIC FIELD