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

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For surface material of ‘standard reference den-

sity’ 2670 kg m

3

, the Bouguer gradient becomes

0.1967 mGal m

1

. Neither of these methods is

satisfactory where there is more than slight topo-

graphy and in general detailed topographic cor-

rections are needed.

A comparative glance at continent-scale free-

air and Bouguer anomaly maps shows that the

free-air anomalies are generally much smaller.

This is evidence for continental scale isostatic

balance, by which surface elevation is compen-

sated by reduced density at depth. Isostasy is

considered further in Section 9.3. On a scale of

100 km or less, free-air anomalies may be much

greater, because they can be supported by the

strength of the lithosphere (the cool uppermost

100 km or so of the Earth).

6.4 The approach to equilibrium

ellipticity

As indicated in Section 6.1, there are four causes of

an excess ellipticity of the Earth, relative to hydro-

static equilibrium. It is not a simple matter to iso-

late them for independent study. But, although

calculation of the equilibrium ellipticity itself is

mathematically tricky, it is not subject to question.

The complexity of the problem arises from the fact

that, although the surfaces of equal density are

equipotentials, their ellipticities decrease with

depth. The basic reason for this can be seen by

considering the equilibrium flattening of a rotating

body of uniform density  as in Problem 6.4, which

shows that flattening is proportional to 

1

.This

means that in a body such as the Earth, in which

density increases with depth, the ellipticities of the

equipotential surfaces are affected by the material

above as well as below the surfaces and so cannot

be written in terms of a straightforward integral.

An approximate first order theory (e.g.

Jeffreys, 1959) yields the equilibrium (hydro-

static) flattening

5=2ðÞm

1 þ 5=2ðÞ1  3=2ðÞC=Ma

ðÞ½

; (6:39)

where m is given by Eq. (6.32). A higher-order

treatment, with resort to numerical methods, is

necessary to obtain a value that allows a satisfac-

tory comparison with the observed flattening.

Nakiboglu (1982) gives

¼ 1=299:627 ¼ 3:337 48  10

3

(6:40)

with corresponding hydrostatic geoid coefficients

¼ 1:072 70

 10

3

; (6:41)

¼2:992  10

6

: (6:42)

Comparison of Eqs. (6.23) and (6.40) gives an

excess flattening, relative to hydrostatic equili-

brium, of 0.5%, which corresponds to a differ-

ence in equatorial and polar radii 100 m greater

than equilibrium. We can note that if equili-

brium is assumed then Eq. (6.39) allows an esti-

mate of the moment of inertia from the surface

flattening. On this basis the observed flattening

of the Earth gives C/Ma

¼0.3309, which is quite

close to the measured value, 0.3307. Thus, it is

reasonable to apply Eq. (6.39) to Mars, for which

observations of precession give a value of the

moment of inertia, to show that, within the

accuracy of observations, it has an equilibrium

ellipticity.

The dynamic oblateness, J

, (Eq. 6.14), is

slowly decreasing as the Earth adjusts to dimin-

ished polar ice caps. Cox and Chao (2002) have

documented this effect, pointing out that for two

decades of precise measurement, 1979–1998,

there was a more or less steady rate of change,

/dt 2.8 10

11

per year, but that for four

years, 1998–2002, the trend was temporarily

reversed. Chao et al. (2003) suggested that this

could be explained by a climatically driven

decade-scale oscillation in the Pacific Ocean.

The earlier trend has resumed and is interpreted

as the long-term effect attributable to post-glacial

rebound, a topic of Section 9.5.

The total excess J

, DJ

¼9.9 10

6

, the differ-

ence between Eqs. (6.14) and (6.41), suggests a

very long relaxation time, DJ

/(dJ

/dt) ¼350 000

years, but this is meaningless. We need to con-

sider more carefully what DJ

means. As men-

tioned in Section 6.1, there are four causes of

the excess ellipticity and not all of them can be

considered as departures from equilibrium. The

tidal effect, that is the ellipticity resulting from

6.4 THE APPROACH TO EQUILIBRIUM ELLIPTICITY 87

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the difference between the tidal potential at the

poles and the average on the equator (Chapter 8)

must obviously be subtracted, but only reduces

the effective DJ

by 1.0 10

8

. The contribution

to dJ

/dt by slowing rotation, caused by tidal

friction (Chapter 8) is seen to be negligible by

noting that d ln !/dt ¼2.8 10

10

year

1

and

that the equilibrium J

varies with rotational

speed ! as !

,sothat(dlnJ

/dt)

tidal

¼5.6 10

10

year

1

and therefore (dJ

/dt)

tidal

¼6 10

13

year

1

. For any plausible lag of the flattening

this corresponds to a negligible contribution to

. In fact it is smaller than the human contribu-

tion to dJ

/dt caused by impoundment of water in

reservoirs, for which Chao (1995) gives a mini-

mum estimate of 1 10

12

year

1

The major problem is heterogeneity of the

mantle and the fact that the Earth self-adjusts its

orientation to minimize rotational energy by

aligning the heterogeneities to maximize the

moment of inertia about the rotational axis.

We have no reliable way of estimating the result-

ing contribution to DJ

, but an idea of how

big it is likely to be is obtained from the ellipti-

city in the equatorial plane. From the spherical

harmonic coefficients of the gravitational

potential in Table 9.1 we see that the coefficien ts

representing equatorial ellipticity give

[(C

)

þ(S

)

] ¼2.8 10

6

,comparedwiththe

excess axial ellipticity, 4.4 10

6

,obtainedfrom

the difference between observed and equilibrium

values of C

. Thus, subtracting the equilibrium

ellipticity, we see the Earth as a triaxial ellipsoid

with moments of inertia C > B > A such that (C B)

and (B A) are comparable. This is what would

be expected statistically, so we have no basis for

supposing that more than a small fraction of

the excess ellipticity is attributable to glacial

depression and we have no direct way of estimat-

ing the magnitude of this fraction. Note that

Eqs. (6.11)to (6.18) assume Eq. (6.10) and therefore

that the value of A is really (A þB)/2.

The observed progressive decrease in J

gives

a tantalizing glimpse of deep mantle rheology. If

the component of DJ

representing residual gla-

cial depression of the poles were precisely

known, then its decay, together with the other

low order geoid harmonics (Chapter 9), would

give a measure of the depth variation of mantle

viscosity. Mitrovica and Peltier (1993) and Han

and Wahr (1995) discussed this problem, con-

cluding that available data did not suffice for a

clear conclusion, and it remains true that there is

no satisfactory independent evidence of DJ

However, post-glacial rebound does provide esti-

mates of mantle rheology and most recent stud-

ies agree that there is an increase in viscosity

with depth in the mantle by a factor of order

100 (e.g. Mitrovica and Forte, 1997; Kaufman

and Lambeck, 2000).

Variations in J

and in the rotation rate are

linked. In considering the effect of the tidal slow-

ing, we can apply Eq. (6.20) to the hydrostatic

state of the Earth and see that both f and J

vary

with rotation in the manner of m, which depends

on !

. Thus the equilibrium value of J

, that is J

is proportional to !

and so the variation caused

by the slowing rotation is

d J



: (6:43)

With the total slowing by lunar and solar tides

given by Eq. (8.31), ! ¼6.5 10

22

rad s

2

, this

yields

d J

=dt ¼1:9  10

20

1

¼6:1  10

13

year

1

(6:44)

This is below the level of detectability. If we can

estimate the time constant, , for relaxation of

the excess ellipticity, then the excess attribut-

able to the slowing rotation is

J

¼ dJ

dtðÞ: (6:45)

For   10

years, J

 6:1  10

9

, which is

close to the resolution limit for measurements

of J

and is not a significant effect.

Now consider the change in rotation rate

due to the variation of J

caused by the post-

glacial rebound (Section 9.6). The angular

momentum of the Earth is conserved in this

process, so that

dðC!Þ=dt ¼ 0; (6:46)

where C is the axial moment of inertia, and

therefore

_!=! ¼

C=C: (6:47)

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In relating this to

, the rebound component

of dJ

/dt, it is convenient to introduce the para-

meter C=Ma

ðÞ¼0:330 695, because this is deter-

mined by the Earth’s density profile and is not a

function of time (i.e. C /a

). Then, rewriting the

definition of J

(Eq. 6.14),

C  A

¼ 1 



; (6:48)

and noting that

A ¼

C=2, we have



¼ 0:4948

: (6:49)

Thus, by Eq. (6.47), there is a rebound component

of rotational acceleration

! ¼!

C=C ¼2:02!

: (6:50)

Identifying

¼2:8  10

11

year

1

with

we have a rebound component of rotational

acceleration

=! ¼ 5:7  10

11

year

1

¼1:8  10

18

1

; (6:51)

and therefore

¼ 1:3  10

22

rad s

2

: (6:52)

This opposes the frictional slowing by Eq. (8.31),

tidal

¼6:5  10

22

rad s

2

, giving a net slowing

! ¼5:2  10

22

rad s

2

In addition to the tidal and rebound contribu-

tions to

! there are irregular effects of angular

momentum exchange with the atmosphere,

oceans and core, but if we consider an average

over 1000 years or more we have some prospect

that these short-term effects are averaged out.

This is the approach of Stephenson and

Morrison (1995), who concluded, from 2000

years of eclipse data, that allowance for the

rebound effect removed what would otherwise

be a discrepancy in tidal friction observations.

However, the rotational effects of glaciation

and rebound occur over thousands to tens of

thousands of years and are brief transients on

the time scale of tidal friction which acts over

the entire life of the Earth (Sections 8.3, 8.4).

6.4 THE APPROACH TO EQUILIBRIUM ELLIPTICITY 89

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Precession, wobble and rotational

irregularities

7.1 Preamble

The axis of the Earth’s rotation is inclined to the

pole of the ecliptic (normal to the orbital plane) by

.45. This gives us the seasons. But the axis does

not maintain a constant orientation in space.

Gravitational interactions between the equatorial

bulge and the Moon and Sun cause a slow preces-

sion of the axis about the ecliptic pole. The axis

describes a cone with a 478 angle in a period of

25 730 years, fast enough for precise astronomical

measurement, but not so fast that navigation by

the stars is seriously inconvenienced. This is the

most obvious of the complications to simple rota-

tion. In an illuminating survey of the history of

the subject, Ekman (1993) notes that the preces-

sion was observed by Hipparchus, who, in about

125 BC, reported a measure of its rate in remark-

able agreement with modern estimates.

To see how the precession arises, consider a

small satellite in a circular orbit that is inclined

to the equatorial plane. Since the Earth is not

spherical, its gravitational force on the satellite

is directed precisely towards the centre of the

Earth only when the satellite is above the equa-

tor (or, of course, one of the poles if it is in a polar

orbit). At intermediate latitudes the latitude var-

iation of gravity imparts a torque tending to pull

the satellite orbit towards the equatorial plane.

The torque acts in a sense perpendicular to the

angular momentum vector, causing a precession

of the orbit. The points at which the orbit crosses

the equatorial plane are referred to as nodes. As

seen in space, not relative to the rotating Earth,

they drift progressively in a prograde sense, east-

wards for a satellite with a west-to-east compo-

nent of orbital motion. The rate of this drift gives

a precise measure of the Earth’s ellipticity, as

represented by the coefficient (Eq. 6.14)

¼ðC  AÞ=Ma

¼ 1:082 626  10

3

; (7:1)

where C, A are the moments of inertia about

polar and equatorial axes and M, a are the mass

and equatorial radius. There is a corresponding

opposite torque on the Earth, but for a man-

made satellite this is not significant.

The same principle applies to interactions

with the Sun and Moon, but in these cases

there is a noticeable effect on the Earth. The

torque, proportional to (C A), acts on the

Earth’s rotational angular momentum, C!,so

that the rate of the resulting precession provides

a measure of the quantity known as the dynam-

ical ellipticity:

H ¼ðC  AÞ=C ¼ 3:273 79  10

3

¼ 1=305:4567:

(7:2)

Combining Eqs. (7.1) and (7.2) we obtain the

moment of inertia coefficient

C=Ma

¼ J

=H ¼ 0:330 698: (7:3)

This is a measure of the density profile of the

Earth. For a uniform sphere the value would be 0.4

and the smaller value in Eq. (7.3) is a measure of

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the concentration of mass towards the centre. The

moments of inertia of other bodies in the Solar

System are mentioned in Eq. (1.17) and Table 1.2.

The result in Eq. (7.3) was first used in modelling

the Earth by K. E. Bullen in the 1930s and remains

an important parameter that any model must

match, in the same way as the total mass must

agree with the observed mass (Section 17.6).

The dynamical ellipticity, H, appears also in

the explanation of another phenomenon, the

Chandler wobble (Section 7.3). This is a motion

of the Earth that involves no interaction with

any other body. It arises because the rotational

axis departs by a small angle, , normally about

0.15 arcsec (0.7 microradian) from the symmetry

axis, which is the axis of maximum moment of

inertia. This means that the rotational energy

exceeds the energy of symmetrical rotation

with the same angular momentum by an amount

proportional to 

. The resulting gyroscopic tor-

que tends to turn the Earth so that its symmetry

axis coincides with the axis of angular momen-

tum (which remains fixed for the purpose of this

discussion because we are considering an effect

that is entirely internal to the Earth). The torque,

acting on the angular momentum, causes a pro-

grade precession of the rotational axis about the

angular momentum axis that is apparent as a

cyclic variation of latitude.

For a rigid Ea rth the wobble period would be

1/H days ¼305 days (see Eq. (7.2)). This was the

period sought for many years before the discov-

ery, by S. C. Chandler in 1891, of the observed

period, about 432 days. The difference is explained

by the elastic deformation of the Earth, with

accompanying responses of the oceans and core

to the gyroscopic torque. The deformation par-

tially adjusts the equatorial bulge towards sym-

metrical rotation, reducing the torque and so

lengthening the period of the motion. Thus, the

wobble provides a measure of the global average

rigidity of the Earth, a check on the elasticity

inferred from seismology (Chapter 17).

The wobble is damped with a time constant of

about 30 years (Eq. (7.25)), which means that it

must be continuously maintained. The mecha-

nism for its maintenance has been a contentious

issue for more than a century. Coupling to irreg-

ular core motions, atmospheric motions and

earthquakes have all been repeatedly examined

and found inadequate. We now have better evi-

dence of a cause, involving interaction of the

atmosphere and oceans, expressed primarily by

ocean floor pressure variations (Gross, 2000). The

14-month Chandler wobble is superimposed on a

slightly smaller 12-month variation, driven by

seasonal mass re-distribution, and is apparent

as a cyclic variation in the latitudes of observa-

tories, with an amplitude that is a bea t of the

14- and 12-month periods.

The astronomical observations that are the

subject of this chapter have a long history.

The relevance to various solid-Earth problems

emerged more recently. As already mentioned,

the determination of the dynamic ellipticity is

particularly important, but other observations of

interest include the elasticity of the Earth at the

wobble frequency and evidence of coupling of

the core to rotation of the mantle. Lambeck

(1980) gave a comprehensive review of the sev-

eral causes of rotational irregularity discussed in

this chapter and the following one.

7.2 Precession of the equinoxes

The two principal terms in the external gravita-

tional potential of the Earth are given by Eq. (6.13).

The central, r

1

, term is dominant, but the second

term is non-central, that is, it has a latitude

dependence, due to the equatorial bulge. In addi-

tion to the central gravitational force m @V=@rðÞ,

exerted on a mass m at (r,), there is a torque

m @V=@ðÞ, with a corresponding equal and

opposite torque exerted by the mass on the

Earth.Themagnitudeofthetorqueisproportional

to (m/r

). For small bodies, such as man-made satel-

lites, the only consequence is a regression of the

nodes of the satellite orbits, that is, a precessional

motion of the orbits about the equatorial plane

(Section 9.2). But the torques exerted on the

Moon and Sun are balanced by the torques exerted

by these bodies on the Earth and cause the preces-

sion of the Earth’s rotational axis. The process is

almost non-dissipative, although the possibility

that there is some precessional dissipation in the

core is considered in Section 24.7, in connection

with the power of the geomagnetic dynamo.

7.2 PRECESSION OF THE EQUINOXES 91

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The cause of the precessional torque is illus-

trated in Fig. 7.1. The Moon and Sun pull the

equatorial bulge towards alignment with them-

selves. The solar torque is a maximum at the

solstices, when the Sun is 23.58 degrees from

the equatorial plane, and it vanishes at the equi-

noxes, when the Sun is directly above the equa-

tor. But the sense of the torque is the same at

both solstices, so that, although it occurs in semi-

annual pulses, it has a cumulative effect.

Similarly, the lunar torque occurs in semi-

monthly pulses and its average effect is added

to that of the Sun. The lunar contribution is

slightly more than twice the solar one, because

(m/r

) is larger for the Moon.

Coupled with the precession are oscillatory

motions of the axis towards and away from the

pole of the ecliptic, that is, in a direction perpen-

dicular to the precessional motion. They are

referred to as nutations (‘nodding’). Semi-annual

and semi-monthly nutations arise from compo-

nents of the solar and lunar torques perpen-

dicular to the precessional component. These

nutational torque components are zero at the

solar and lunar solstices, when the torques are

purely precessional, and at the equinoxes, when

the torques vanish because the Sun or Moon are

in the equatorial plane. A larger amplitude nuta-

tion (9.21 arcsec) of 18.6 year period arises from

the coupling of the Earth to the precession of the

pole of the lunar orbit about the pole of the

ecliptic (Earth’s orbital plane). The consequent

variation in the inclination of the lunar orbit to

the equator causes the Earth’s nutation with the

same period.

To examine quantitatively the precessional

torque on the Earth, consider the geometry of

Fig. 7.2, with the Sun, mass M

, instantaneously

at geocentric latitude  and distance R from the

Earth’s centre. The gravitational potential at the

centre of the Sun due to the Earth is given by

Eq. (6.13), so that the torque exerted on the Sun,

and therefore by the Sun on the Earth, is

L ¼ M

@

3GM

C  AðÞsin  cos : (7:4)

This torque acts about an axis in the equatorial

plane normal to the Earth–Sun line, that is, it

tends to pull the bulge into line with the instan-

taneous Earth–Sun axis. We can resolve it into

components L

about O x and L

about Oy. Then L

is the torque component causing the semi-

annual nutation and L

is the solar precessional

torque.

Pole of

Ecliptic

Axis

Force

Ecliptic

Plane

Sun

(or Moon)

Equator

FIGURE 7.1 Origin of the precessional torque. The gravitational action of the Sun (and Moon) on the alignment

of the Earth’s equatorial bulge exerts a torque that tends to pull the bulge into alignment with the instantaneous

Earth–Sun (or Earth–Moon) axis. The torque vanishes when the Sun (or Moon) crosses the equatorial plane, but

appears with the same sign for both halves of the orbit, causing a net average precessional torque.

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3GM

C  AðÞsin  cos  sin ; (7:5)

where  is the longitude of the Sun, relative to its

longitude at the equinox. Two trigonometric

identities relate ,  to , ,where is the azimu-

thal angle of the Sun in its orbital plane and  is

here the maximum value of  (23.58):

sin  ¼ sin  sin ; (7:6)

tan  ¼ tan  sin ; (7:7)

so that, in terms of  and ,

3GM

C  AðÞsin  cos  sin

: (7:8)

Since  gives the Sun’s position in the orbital

plane,

sin

 ¼ 12

; (7:9)

and therefore the mean precessional torque,

averaged over a year, is

C  AðÞsin  cos : (7:10)

This annual average torque acts, in the sense of

the arrows in Fig. 7.1, about the axis Oy (Fig. 7.2).

Since Oy is perpendicular to Oz, which is

the axis of the Earth’s angular momentum, C!,

the solar precessional torque causes an angular

rate of change in the orientation of the rotational

axis

¼ L

=C! ¼

C  AðÞ

sin  cos 

C  A

sin  cos ;

(7:11)

where !

¼ GM

is Kepler’s third law

(Eq. (B.23), Appendix B), !

being the angular

speed of the Earth–Sun orbital motion.

Still supposing the solar torque to act alone,

as the Earth’s rotational axis moves, the equato-

rial plane moves with it, causing the positions of

the equinoxes (Oy in Fig. 7.2) to move around the

orbit. The axis of the precessional torque moves

with Oy, so that the torque causes the cyclic

precession of the rotational axis about the pole

of the ecliptic, describing a cone of semi-angle .

If the solar torque acted alone, one cycle

of the precession would take a time 

2p=!

¼ 2p sin =O

, where

¼ O

= sin  ¼

C  A

cos : (7:12)

FIGURE 7.2 Geometry of the precessional torque. Oxy represents the equatorial plane and z is the rotational axis.

The Sun is at an instantaneous geocentric latitude , on an orbit of radius R in the plane Ox

y, inclined at  ¼23.58 to

the equatorial plane. The projection of the orbit onto the equatorial plane is shown as the dashed ellipse. For this

purpose we can consider the orbit of the Sun about the Earth as equivalent to the orbit of the Earth about the Sun. A

similar figure applies to the Moon.

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A similar expression applies to the lunar contri-

bution, !

(except that in substituting Kepler’s

third law it is the mass of the Earth that is dom-

inant for the Earth–Moon system), and the

observed rate of precession, caused by both

solar and lunar torques, is

¼ !

þ !

¼ 50:3846ð13Þ arcsec=year: (7:13)

Since !

is well observed, as are the other quan-

tities in Eq. (7.12) except the moments of inertia,

the precession gives a measure of the dynamical

ellipticity, H ¼(CA)/C (Eq. (7.2)). The fact that this

is less than the geoid ellipticity demonstrates that

the deeper, denser layers of the Earth, notably the

core–mantle boundary, are less elliptical than the

surface (see problem 6.4, Appendix J). Combining

Eqs. (7.2) and (6.14) we obtain the coefficient of the

Earth’s moment of inertia, C/Ma

(Eq. (7.3)). This is

an important parameter which Earth models are

constrained to fit (Section 17.6).

Since the precession progressively changes the

orientation of the rotation axis relative to the peri-

gee and apogee of the orbit, there is an associated

climatic cycle. There are also cyclic variations in

the inclination of the ecliptic and the orbital eccen-

tricity. These periodicities in orbital characteristics

have climatic implications first studied in detail by

M. Milankovitch in the early 1940s. Climatic cycles

are observable in the sedimentary record at these

periods (Berger, 1988), and Williams (1991) has

found evidence of Milankovitch cycles as early as

440 million years ago. These observations provide

ausefulcheckontheevolutionofthelunarorbit

(Section 8.4). Implications for climate are consid-

ered in Section 26.4.

7.3 The Chandler wobble

Independently of its gravitational interactions

with external bodies, the Earth undergoes a

free, Eulerian precession, sometimes called the

free nutation in geophysical literature, although

it is not strictly a nutation. To distinguish it from

the forced motions due to interactions with

other bodies, the term wobble, or, in the name

of its discoverer, Chandler wobble, is preferred.

The wobble results from the rotation of the Earth

about an axis that departs slightly from its axis of

symmetry (or greatest moment of inertia). The

total angular momentum remains constant, in

magnitude and direction, but the symmetry axis

follows a circular path about the spin axis, which

remains nearly fixed in absolute orientation.

The wobble is apparent as a cyclic variation of

latitude with a period of about 432 days (1.2

years) and variable amplitude averaging about

0.15 arcsec (Vondr

ak, 1999). The Chandler wob-

ble is superimposed on a 12-month (seasonal)

latitude variation of 0.1 arcsec amplitude, so

that the two periods are observed to beat, result-

ing in a six-year cycle in the amplitude of the

latitude variation (Fig. 7.3).

The theory of the wobble period is usually

presented in terms of Euler’s equations, but

greatest geophysical interest arises from the

mechanisms of excitation and damping and, for

this purpose, it is convenient to use a derivation

that makes use of wobble energy.

Consider a hypothetical, rigid Earth with

principal moments of inertia C, A, A, where

C > A, rotating with angular velocity ! about an

axis at a very small angle, , to the C axis. The

total rotational energy is the sum of the energies

of the components of rotation about the three

principal axes,

þ Am



; (7:14)

where m

, m

, and m

are the direction cosines of

the rotational axis to the A, A and C axes. Thus

þ m

¼ 

; (7:15)

where m

þ m

¼ 1:

The energy of rotation with the same angular

momentum about the C axis, that is, with  ¼0

and no wobble, is

: (7:16)

Thus, the kinetic energy of the wobble is the

difference between Eqs. (7.14) and (7.16).

Making use of Eq. (7.15), this is

¼ E

 E

1 

C  A









(7:17)

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The angular momentum is the vector sum of its

components about the principal axes and is con-

served, so that it is equal to C!

(with no wobble),

that is

¼ C

þ A



1=2

¼ 1 

 A







C!:

(7:18)

Substituting for !

by Eq. (7.18) in Eq. (7.17), we

have

C  A





AH!



; (7:19)

where H is the dynamical ellipticity (Eq. (7.2))

obtained from the precession.

Since the excess rotational energy is a func-

tion of the angle , it follows that there is a

gyroscopic torque, L

, that tends to turn the

Earth to its lowest energy state ( ¼0):

¼

d

¼AH!

: (7:20)

This torque acts on the angular momentum com-

ponent in the equatorial plane, A!, that is, the

component associated with the axial misalign-

ment, and, for the hypothetical rigid Earth,

causes free precessional motion at an angular

frequency

¼

A!

¼ H!: (7:21)

This is a prograde motion with a period C/(C A)

days ¼305 days.

A 305-day latitude variation was sought for

many years before the discovery of the 432 day

period by S. C. Chandler in 1891. The difference is

primarily due to the elastic deformation of the

Earth, which partially accommodates the equato-

rial bulge to the instantaneous rotation axis,

reducing the gyroscopic torque. There are also

smaller effects of the oceans and core. Neglecting

the lesser contributions, the solid deformation

reduces the gyroscopic torque, L

,bythefactor

305/432 ¼0.7, which means that the bulge is

deflected by an angle 0.3, requiring a shear strain

" ¼ 0:3f  1  10

3

; (7:22)

where f 1/300 is the equatorial flattening.

As mentioned in Section 7.1, for more than a

century a satisfactory explanation for the excita-

tion of the wobble was elusive and the same is

true of its damping. If the excitation is random

then the observed spectrum of the wobble reso-

nance gives a measure of the damping and this is

expressed in terms of the Q of the wobble, for

which there are two alternative (but formally

equivalent) definitions:

–100

–200

100

200

300

600 500 400 300 200 100 0

–100

Jan 83

Jan 82

Sept 80

Jan 81

Sept 1985

Jan 85

Y, milliarc sec

X, milliarc sec

Jan 84

FIGURE 7.3 Path of the pole

from late 1980 to late 1985,

approximately one beat period

of the combined 432- (Chandler)

and 365-day (seasonal) motions.

X and Y represent displacements

in the direction of the Greenwich

meridian and 90

East. Accuracy

of the observations improved

during this period. By 1985 it was

clear that the irregularities in the

pole path are real. From a record

presented by Carter (1989).

7.3 THE CHANDLER WOBBLE 95

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¼2pE

=E

; (7:23)

¼ !

=!

: (7:24)

Here DE

is the diminution in one wobble

cycle of the energy, E

, of an unmaintained

wobble, not a directly observed quantity. !

the spectral line width (at half power) of the

wobble, so that Q

is defined as the sharpness

of the wobble frequency. These definitions have

equivalents in electrical circuit theory, where

the term Q – quality factor – originates. Jeffreys

(1959) presented a mechanical analogue that is

easy to visualize – a circular pendulum and a

group of boys with pea-shooters around it, ran-

domly firing peas at it. The pendulum swings

in response to the impulses and if it is only

lightly damped it may develop a large amplitude,

so that individual impulses have only slight

effects. This is the high Q situation with little

dissipation of energy per cycle and a sharply

defined period of oscillation. If the Q is lowered

by additional damping, the amplitude is

reduced, individual impulses become propor-

tionately more important and the period

becomes more blurred. Application of Eq. (7.24)

to observations gives

 80; (7:25)

corresponding to decay of an unmaintained

wobble with a time constant 2Q

30

years, but with a wide range of uncertainty

(30 < Q

< 180). An analysis by Vondr

ak (1999)

suggests that the wobble is not a simple linear

phenomenon, which the concept of Q assumes,

but that the amplitude and period are related.

However, this may be an artifact of the excitation

mechanism.

The role of mantle anelasticity in damping

the wobble can be examined by considering the

anelastic Q required to give the observed wobble

Q (Eq. (7.25)) with a strain amplitude given by

Eq. (7.22). The volume-averaged rigidity modulus

of the Earth is  124 GPa, so that the elastic

strain energy for a wobble of amplitude  is

ð1=2Þ"

V ¼ 6:7  10



Joules; (7:26)

where V is the volume of the Earth. This com-

pares with

¼ 7:0  10



Joules (7:27)

by Eq. (7.19). Thus, for mantle anelasticity to

damp the wobble would require an anelastic Q



/10 8 and this is so much smaller than the

seismologically observed Q

as to be implausible.

Themantlecanmakeonlyaminorcontribution

to Q

It is important to note that an excitation event

becomes ineffective if it extends over more than

half a wobble period (7 months). Internal torques or

mass displacements that move the symmetry axis

away from the rotation pole at one instant would

move it closer half a period later, when the sym-

metry axis is on the opposite side of the pole. This

leaves coupling of the Earth to the atmosphere and

oceans as the only viable mechanism, as concluded

by Gross (2000). A suggestion that the atmosphere

alone suffices is kept alive by Aoyama and Naito

(2001), although their argument is based on evi-

denceforasharp14-monthpeakinthespectrum

of atmospheric (wind plus pressure) excitation,

which is hard to understand. A reasonably sharp

‘line’ spectrum of excitation for the annual wobble

can be understood as a seasonal effect, so that in

this case the line width is unrelated to damping,

unlike the case of random excitation.

The existence of the forced annual wobble

demonstrates that seasonal movements of the

atmosphere and hydrosphere cause an annual

polar shift of 0.05 arcsec. The 14-month wob-

ble, of average amplitude about 0.15 arcsec, with

dissipation at a rate corresponding to a free

decay time constant of 25 periods, requires exci-

tation by a random source of amplitude 0.15/

ﬃp

25 ¼0.03 arcsec, a surprisingly large fraction

of the annual excitation. But the wobble Q is not

well observed and these numbers are insecure.

Superimposed on the wobble there is a steady

drift of the pole towards 798 W, at a rate which

Vondr

ak estimated to be 0.351†/century (11 cm/

year). It is attributed to asymmetry of the mass

redistribution caused by post-glacial rebound.

The total displacement of the pole in this process

is only of order 1 km. It is not an observation of

polar wander, but it is possible in principle that

true polar wander, driven by mass displacements

in the mantle accompanying changes in convec-

tion pattern, is also occurring.

96 PRECESSION, WOBBLE AND ROTATIONAL IRREGULARITIES