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

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localized heat flow anomaly along it.

Assume that in the uppermost layer the

shear stress is reduced to the same value as

the normal stress. (The reason for lack of a

heat flow anomaly of the estimated magni-

tude is considered in Section 15.6. Measure-

ments along the San Andreas fault in

California drew attention to this problem.)

15.2 Can the absence of a heat flow anomaly,

referred to in Problem 15.1, be explained

by retention of heat (or energy in any other

form) in the fault zone? Calculate the total

heat generated by friction if the slip accumu-

lates to 200 km and the heat does not

escape? If it is applied to the melting of

granite, latent heat 4.5 10

Jkg

1

,estimate

the thickness of rock on either side of the

fault that would melt.

15.3 Let the time to failure for aftershocks be

failure

¼A

n

, where  is stress and n is a

large positive integer. Show that this gives

rise to Omori’s law (Eq. (15.26) or (15.27)) for

aftershocks, and give the restrictions on the

distribution of stress in the aftershock

zone.

15.4 Calculate the radiation efficiency for the

Brune spectrum (Eq. (14.32)) using the follow-

ing steps. The Fourier transform

of the displacement field of a seismogram

(Eq. (14.32)) can be written

ðf Þ¼

1þðf =f

Given Parseval’s relation

1

uðtÞ

dt ¼

1

uðf Þ

df and the radiated energy

(Eq. (15.47)),

(a) perform the energy integral in the fre-

quency domain and show that

5V

;

(b) use Eq. (15.41) to show that

¼ðV

=lÞ=2p ¼ 3:5  0:9  V

=ð2plÞ,

and that

¼ 

3:15



;

49.7%.

15.5

(a) Give an expression for corner frequency

as a function of magnitude.

(b) Ass uming that earthquakes have

an average stress drop of 3 MPa in

rocks of rigidity  ¼ 3  10

Pa, use

formulae from Chapters 14 and 15

to complete the following table.

Assume that faults are equi-dimen-

sional and that rupture velocity is

3kms

1

15.6 Equation (15.49) treats the sliding frictional

block with displacement y ¼ y

ð1  cosð!tÞÞ

and a moment acceleration that goes as

€

¼ 2S

€

y ¼ 2S!

cos !t. That is, the

block acceleration is

€

y ¼ !

cos !t ¼

cos !t; for 0

T=2:

Note that the duration of sliding is T/2,

where T is the period of the cosine. Now

consider an alternative step-model in

which

€

y ¼ a

; 0

t  T=4;

€

y ¼a

; T=4

t  T=2;

max

¼ 2y

;

where values are adjusted so that the same

moment is developed during the same

time interval, but the acceleration is a

Magnitude Moment l (Size) T, rupture time b (slip) f

(corner frequency)

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positive step followed by negative step

function that brings the block to rest over

the same time interval.

(a) Recalling that the distance travelled,

y ¼ 1=2

€

, find the acceleration a

ensuring that the block travels 2y

(b) Compare the maximum accelerations in

the two cases (cosine and step functions).

€

dt in the two cases

(noting that

cos

xdx ¼

sinð2xÞ).

(d) The seismic and radiation efficiencies

for the frictional blocks are 0.5% and

2.5%, respectively. By obtaining the

ratios of the integrals in (c), calculate

the efficiencies for the step-function

case analysed in this problem.

(e) Are the efficiencies larger or smaller

than for the sliding block model? Why?

Is this model realistic?

16.1 Show that Eq. (16.22) can be derived by

considering wave energies instead of ampli-

tudes. What is the meaning of the possible

alternative signs of R?

16.2 Consider a refracted ray that crosses a plane

interface separating layers of velocities V

and V

. Show by considering travel times of

adjacent rays that Snell’s law (Eq. (16.6)) is

equivalent to Fermat’s principle that the

travel time for a seismic wave between

source and receiver along a ray path is sta-

tionary (normally a minimum) with respect

to adjacent paths, that is, the wave takes the

fastest path.

16.3 Derive Eq. (16.75) by substitution of

Eq. (16.71) for u in Eq. (16.74) and solving

for the coefficients a

in that equation by

taking a Fourier transform and using

Eq. (16.72) to resolve individual coefficients.

16.4 Given Eq. (16.35) for an S-wave incident on a

boundary:

inc

¼½u

; u

¼½cos j; sin jexp i!ðpx  z  tÞ½;

show that

(a) the reflected P-wave is

refl

¼½u

; u



¼ R

½sin i; cos iexp i!ðpx þ z  tÞ½

(b) the reflected S-wave is

refl

¼½u

; u



¼ R

½cos j; sin jexp i!ðpx þ z  t

½

2pV

R

þð1  2V

Þð1  R

Þ¼0;

ð1  2V

ÞR

þ 2V

p=V

ð1 þ R

Þ¼0;

follows from setting tractions on the

surface equal to zero.

16.5 The Sl ichter mode of free oscillation is

an oscillation, or rotation, of the inner

core about the centre of the Earth. In prin-

ciple, it provides the most precise observa-

tion of the density contrast between

the inner and outer cores. Observations

(by sensitive gravity meters) have been elu-

sive, b ut it may now have been seen

(Pagiatakis et al., 2007). C onsider a simple

model of a uniform inner core, density 

in a non-rotating earth (with no splitting of

themodefrequency),inauniformouter

core of density 

.Calculatetherestoring

force for an off-centre displacement and

allow for the fact that outer core material

must mov e to accommod ate inner core

motion by assuming that the inertial

mass of the motion is that of the inner

core plus an equal volume of outer co re

material. Show that the pe riod of the

motion is

T ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

3p=G

ðÞ

þ 

ðÞ=

 

ðÞðÞ

Assuming an inner–outer core density

contrast of 820 kg m

3

(Maste rs and

Gubbins, 2003), estimate the period (in

hours). What are the significant approxi-

mations?

17.1 Consider a planet within which gravity is

independent of depth (or radius). How does

density vary with radius? Express the result

in terms of mean density and total radius.

(Note: gravity is approximately independent

of depth in the Earth’s mantle – Fig. 17.11(b).

See also Problems 1.1(c) and 21.1.)

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17.2 Deduce the velocities and thicknesses of

plane horizontal layers that would give

the following (hypothetical) first P-wave

arrivals:

17.3

(a) A seismic refraction method is to be

used to measure the thickness of sedi-

ment overlying igneous rock. If the

P-wave velocities are estimated to

be 2.5 km s

1

and 4.5 km s

1

and the

postulated depth of sediment is 0.2 km,

over what distance range should seis-

mometers be placed to distinguish

clearly between refractions from the

igneous rock and direct P-waves from a

surface explosion?

(b) A seismometer is buried at depth z in a

uniform layer of P-wave velocity V

and

thickness z

> z. There is another seis-

mometer on the surface directly above

it. Below z

is a thick layer of velocity

> V

. Both seismometers observe sig-

nals from a surface source that is suffi-

ciently far away for the first arriving

waves to have penetrated the deeper

layer. What is the difference between

first arrival times?

17.4 Derive Eq. (17.8). What is the maximum

value of  for which head waves could, in

principle, be observed down-slope?

17.5 Consider a highly simplified model of the

Earth, with three uniform layers: a core

of radius 3500 km with P-wave speed

V ¼9kms

1

, a lower mantle of thickness

2300 km (outer radius 5800 km) and

V ¼12 km s

1

, an upper mantle 600 km

thick with V ¼9kms

1

(total radius

6400 km). What are the epicentral ranges

of triplication of arrivals due to the mantle

transition and the core shadow zone?

17.6 Show that in a flat Earth model in which

seismic velocity increases linearly with

depth, the seismic rays are circular arcs.

17.7 The P-wave speed at the base of the mantle

(at 0.55 Earth radius) is 13.7 km s

1

and

the velocity at the base of the crust (at negli-

gible depth) is 6.4 km s

1

.Whatisthesemi-

angle of the cone in the crust to which seis-

mic rays from the core are confined?

18.1

(a) Obtain an expression for the pressure at

arbitrary radius r in a planet of uniform

density, in terms of its mass, M, and

radius, R.

(b) Show that the central pressure can be

expressed in terms of the surface grav-

ity, independently of radius or mass.

18.2 To draw attention to the increase in central

pressure caused by an inward concentra-

tion of mass, consider a simple model, in

which density varies with radius, r,as

 ¼ a  br=R;

where R ¼6.371 10

m is the outer

radius, a ¼13 000 kg m

3

is the central

density and b ¼10 000 kg m

3

, the surface

density, being (a b).

(a) Calculate the total mass of the model, to

verify that it is very close to the mass of

the Earth.

(b) Calculate the internal pressure as a

function of radius and compare the cen-

tral pressure with that of the Earth and

of a uniform sphere of the same mass

and radius.

cient, I/MR

, for the model and com-

pare it with a uniform sphere and

the Earth. (See also the models in

Problems 1.1 and 1.2.)

18.3

(a) Show that Eq. (18.21) with m ¼2, n ¼4

and all higher terms zero gives the sec-

ond-order Birch finite strain equation,

P ¼





5=3





2=3

1

;

where K

is the bulk modulus at P ¼0.

S (km) T (s) S (km) T (s)

1 0.33 80 15.45

3 1.00 100 18.53

5 1.53 120 21.61

10 2.53 140 24.62

20 4.53 160 27.05

30 6.53 200 31.93

50 10.53 250 38.03

60 12.38 300 44.13

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(b) Show that, for this equation, K

¼ (dP/

dK)

P ¼0

¼4.

18.4 A harmonic solid would be one in which the

potential energy of an atomic displacement

from equilibrium separation from a neigh-

bour, r

, is proportional to the square of

displacement,

ðrÞ¼Aðr  r

;

so that atomic oscillations are sinusoidal

or harmonic. Using this, with Eqs. (18.15)

to (18.20), obtain expressions for

(a) P and K in terms of K

and =



;

(b) K

¼d K/dP, in terms of =



, showing

that K

¼1,

free volume-type theories (Eq. (19.39))

in terms of f and =



. (Note that 

and therefore also the thermal expan-

sion coefficient are negative for a har-

monic solid.)

19.1 For an ideal gas the familiar equation of

state (n moles) is PV ¼ nRT and the principal

specific heats are related by C

 C

¼ R=m

for atomic mass m.

(a) Show that for an ideal gas the Gr

uneisen

parameter is  ¼ C

 1 :

(Note: we are not using the ideal gas

notation, in which  represents C

(b) Use this result to show that the temper-

ature variation under adiabatic com-

pression is

¼ðV



compression is represented by

þ1

¼ constant:

19.2

(a) Ignoring the differences between K

and

and their pressure derivatives, use

Eqs. (19.33) and (19.39), with elasticity

data in Tables F.2 and F.3, to obtain

rough estimates of the Gr ¨uneisen param-

eter at the top and bottom of the lower

mantle and outer core.

(b) With C

for the mantle by Eq. (19.3) and

a multiplying factor 1.5 for the core, to

allow for the electronic heat capacity,

use the results in (a) with Eq. (19.1) to

estimate the ranges of the expansion

coefficient, .

the temperature ranges in Tables G.1

and G.2, estimate the ratio K

ð1 þ TÞ. This is a measure of the

error in ignoring differences between

and K

and their pressure derivatives

(Eqs. (E.1) to (E.3) in Appendix E).

19.3

(a) Use Eq. (19.55) or (19.56) to derive the

second term in Eq. (17.32), that is Birch’s

modification of the Williamson–Adams

equation for a non-adiabatic gradient.

(b) Show that the factor (1 g

1

d/dr),

used by K. E. Bullen as a test for homo-

geneity in the Earth, is equivalent to dK/

dP for a homogeneous, adiabatic layer.

(Note that  ¼K/ and r is radius in this

problem.)

19.4

(a) Consider a model for a series of planets

of different radii, R , with similar heat

concentrations per unit volume,

, and

the same surface temperature, T

Assuming diffusive equilibrium, obtain

expressions for the variation of central

temperature with radius for each of two

assumptions about conductivity:

(i) constant conductivity, ,

(ii)  ¼AT

, where A is a constant,

(Case (ii) refers to radiative conduc-

tivity that can become important at

high temperatures in minerals that

are reasonably transparent. See Eq.

(19.60).

(b) Consider constant k,but

varying with

radial distance r fromthecentreas

ðr=RÞ

,wherel is a constant, inde-

pendent of R,and

is the surface value

,whichvarieswithR in such a way

that the average heat generation per unit

volume is independent of R. Find the cen-

tral temperature as a function of R.

19.5 In spherical symmetry it is convenient to

write the thermal diffusion equation with

no internal heat sources (Eq. (20.1)) in terms

of radius, r,

@T=@t ¼ð=r

Þ@=@rðr

@T=@rÞ:

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Consider a sphere of radius R and constant

diffusivity, , with an initial temperature

profile T

(r) which decays exponentially,

TðrÞ¼T

ðrÞexpðt=Þ:

Calculate the thermal relaxation time, ,

in terms of R and , and the functional

form of T(r) that does not change with

time.

19.6 Using the Lindemann relationship, T

2/3



, with the Debye approximation for

 (Eq. (19.29)), derive Eq. (19.43).

19.7 What lithospheric temperature gradient

gives compensation of temperature and

pressure effects on density, so that d/

dz ¼0? Assume uniform composition and

apply Eq. (17.32).

19.8 Derive Eq. (19.52) from Eqs. (19.50) and

(19.51).

19.9 Using Table F.2 for numerical integration

and assuming T ¼T

¼5000 K at the inner

core boundary (r ¼1221.5 km), estimate T

and T

at the core–mantle boundary

(r ¼3480 km), using Eqs. (19.19) and (19.52)

respectively. The difference is a measure of

the cooling required for complete core sol-

idification. (Note: use Eq. (19.39) for  with

f ¼ 1:44.)

20.1 The oceanic lithosphere shrinks with age,

due to cooling, resulting in a progressive

increase in ocean depth with age of the

ocean floor (Fig. 20.2). Isostatic balance

(Section 9.3) is maintained and the added

load of sea water causes a subsidence of the

lithosphere, additional to its shrinkage. If

sea water density is 

¼1025 kg m

3

and

the lithospheric density is 

¼3350 kg m

3

show that the depth increase exceeds the

shrinkage by the factor (1 

/

)

¼1.437, as used in Eq. (20.11).

20.2

(a) Assuming a thermal oscillation at the

surface of the Earth, T ¼T

sin !t, and

that at depth z the temperature varia-

tion is given by Eq. (20.23), show by

differentiation and substitution in

Eq. (20.1) that a and b satisfy Eq. (20.24).

(b) If the annual surface temperature oscil-

lation has a peak-to-peak amplitude of

30 K and the diffusivity of surface rocks

is 1.3 10

6

1

, at what depth in the

crust must an instrument be buried to be

subjected to a peak-to-peak annual tem-

perature oscillation of 10

3

20.3 Show that, at depth z, the temperature

wave in Problem 20.2 has an oscillatory

gradient of amplitude



max

ﬃﬃﬃ



exp 

ﬃﬃﬃﬃﬃﬃ

2



20.4 Show that the temperature difference over

the lower half of a borehole of depth z due

to penetration of the thermal wave consid-

ered in Problems 20.2 and 20.3 oscillates

with amplitude fT

, where

f ¼ e

x

þ 1  2e

x=2

cos

hi

1=2

;

x ¼



ﬃﬃﬃﬃﬃﬃ

2

z* being the effective penetration depth

(skin depth) of the wave. Show further

that f has a maximum value of 0.347

at x ¼1.335. If z ¼1000 m and  ¼ 1:26

6

1

, what is the corresponding

period of the thermal wave?

20.5 Consider a region in which the crust has a

total thickness z

, with radioactive heat

sources decreasing linearly with depth to

zero at depth z

. The heat flux from the

mantle into the crust is a quarter of the

surface heat flux and the temperature at

the mantle–crust boundary is T

. Heat

through the crust is by conduction only

and the temperature profile is in equili-

brium. Assuming that surface tempera-

ture is zero , obtain an expression for

temperature as a function of depth z,

down to z

,intermsofT

,zandz

.By

what factor does the gradient near the

surface exceed the av erage crustal temper-

ature gradient?

21.1 As in Problems 1.1(c) and 17.1, consider a

self-gravitating body with density inversely

proportional to radius. Compare the

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gravitational energy released by its forma-

tion with Eq. (21.2). What value of f is

required?

21.2 If the heat flux from the Moon is in equili-

brium with internal radioactivity, which

has the same average concentration as the

(mantle plus crust) of the Earth, what is the

heat flux per unit area? (Relevant numerical

details are given in Table 21.3.)

21.3 What is the total classical heat capacity of

the Earth if the mantle (4 10

kg) has a

mean atomic weight of 21.1 and the core

(2 10

kg) has a mean atomic weight of

44.8? What average cooling would be

required to produce the present heat flux

for 4.5 10

years? Is this plausible? If so,

why did Kelvin’s age-of-the-Earth problem

arise? (See Section 4.2.)

22.1 Consider a series of planets, as in

Problem 19.4(a), except that they are con-

vecting and maintain adiabatic gradients

below their thin lithospheres, which have

a common basal temperature, T

. Assume

that material compression follows

Murnaghan’s equation (see Eq. (18.37))

and that the Gr ¨uneisen parameter, ,is

a constant, so that Eq. (19.22) applies.

Central pressure may be adopted from

Problem 18.1(a) (which ignores the

increase in density with depth or planet

size). How does central temperature vary

with radius?

22.2 Suppose that convected heat is derived

uniformly from the whole volume of the

mantle and the thermodynamic efficiency

for each element of heat is propor tional to

the depth from which it is derived. The

inner and outer radii are R

and R

and

for heat from R

the efficiency is 

max

What is the average efficiency of whole

mantle convection in terms of 

max

, R

and R

22.3 If the Gr ¨uneisen parameter of the mantle is

assumed to be constant,  ¼1.2, and the

relationship between incompressibility K

and pressure P is

K ¼ 2  10

Pa þ 4P;

what is the thermodynamic efficiency of

the convective transport of heat to the

surface from the base of the mantle,

where P ¼1.3 10

Pa?

22.4 Using the model of the core of Mercury in

Section 24.8, with an inner core that has

grown to 0.8 of the core radius, and assum-

ing that a compositional density contrast

of 5% was maintained between the inner

and outer cores during inner core growth,

estimate the total gravitational energy

released during inner core growth. How

significant would this be to the Mercury

dynamo?

23.1 Is the decrease in moment of inertia of

the Earth due to thermal contraction suffi-

cient to cause an observable effect on

rotation?

23.2 If the convected heat flux,

, from the man-

tle is proportional to the square root of the

speed of convection, but the thermody-

namic efficiency with which mechanical

power is generated is independent of the

speed, show that the effective viscosity is

proportional to

3

23.3 Use the heat balance equation (Eq. (23.14))

to set up a simplified numerical calculation

for mantle cooling, by which radiogenic

heat production in the mantle can be esti-

mated. The targets of the calculation are the

present value,

, and the present cooling

rate. Assume:

(i) radiogenic heat decays as a single expo-

nential with a half life of 2  10

years,

(ii) convected heat,

Q ðTÞ varies with tem-

perature by a simplified form of Eq.

(23.21):

Q ðTÞ¼

exp

2gT

3ðn þ 1Þ





;

with

¼1700 K, the present value of T,

¼2500 K, the value of T 4.5 10

years

ago,

¼32 10

W, the present convected

heat loss,

g ¼25, n ¼1, 

¼7.4 10

1

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The calculation is iterative. Start from

the present conditions and by repeated

trials integrate backwards in time to find

the value of

that leads back to the

required starting condition, T ¼ T

t ¼4:5  10

years. The value of

that

you obtain will be less than

and the

difference gives the present cooling rate.

23.4 Taking the melting point of core alloy at the

inner/outer core interface as 5000 K, and

the temperature at the 660 km phase tran-

sition, 1950 K, from mineral physics, with

elasticities from Appendix F and the

Gr ¨uneisen parameter from Appendix G,

(a) extrapolate the temperature to the core–

mantle boundary from both directions,

(b) if the difference between the two esti-

mates is identified as the temperature

increment across a thermal boundary

layer 200 km thick, what is the average

temperature gradient in it? If the ther-

mal conductivity of the layer is 5

1

, what is the conducted

heat? Is this consistent with heat con-

duction at the top of the core?

23.5 The mantle must have cooled much more

than the core to leave a 1000 K thermal

boundary layer at its base, and continuing

faster mantle cooling is a feature of the

thermal history, as presented in Chapter 23.

Assuming this to be correct, calculate the

maximum core cooling rate permitted by

each of the three mantle cooling models,

with different radiogenic heat, summar-

ized by Figs. 23.2 and 23.3. What are the

corresponding values of core-to-mantle

heat flux if core radioactivity gives

(a) 2tW, (b) zero? Note: take the mantle

and core heat capacities to be given by

Eqs. (21.7) and (21.20). The core value refers

to cooling at the core–mantle boundary,

but the mantle value refers to changes in

the potential temperature and requires

adjustment to an adiabatic extrapolation

to the core–mantle boundary by the factor

CMB

¼1/1.64.

24.1 Consider a spherical conductor with a pat-

tern of currents that gives a uniform axial

magnetic field throughout its volume. The

field outside the sphere is equivalent to that

of a dipole. Show that the field energy out-

side the sphere is half of the field energy

within the sphere.

24.2 Show that, if the core is a sphere of radius R

and uniform electrical resistivity, 

, carry-

ing a current that is a simple circulation

about the axis, then

(a) if the current density is uniform it gives

a dipole moment by Eq. (24.49) and

therefore a power dissipation by

Eq. (24.50),

(b) if the current density is proportional to

radial distance, r, from the axis,

i ¼ i

r =R;

then the magnetic moment is

m ¼ 4p=15ðÞi

and the ohmic dissipation is





This is the minimum possible dissipa-

tion for specified m.

24.3 Assuming the field strength in the core to be

uniform with strength 2B

,whereB

is the

field on the equator just above the core sur-

face, and to be related to dipole moment by

Eq. (24.9), estimate the total field energy in

terms of dipole moment (using the result of

Problem (24.1)). By combining the resulting

expression with the dissipation obtained in

Problem 24.2(a), estimate the free decay

time of the core currents and compare

with the result from Eq. (24.43). (Note that

this is only a rough calculation because the

field is not uniform for the assumed current

patterns.) Assume a uniform core resistivity

of 4mO m.

24.4 What is the electromagnetic skin depth of

the core for a magnetic fluctuation with

(a) a 1000-year period? (b) a 10-year period?

(Static layers at the top of the core have

been considered. Their thickness is limited

by the fact that secular variation gets

through. These periods are extremes for

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the secular variation of harmonic terms in

Fig. 24.4.)

24.5 Consider a model of the core rotating coher-

ently about an axis at a small angle  to the

mantle axis, as in Problem 7.3. What is the

value of  that would explain the westward

drift?

24.6 The gravitational energy released by adding

to the inner core a mass dm of material that

is more dense than the outer core, from

which it is taken, and allowing the excess

of light material to mix into the outer core,

is given by Eq. (22.35). Obtain the corre-

sponding expression for the energy

released by mixing into the outer core a

similar excess mass of material that is

more dense than the outer core and is

deposited from the mantle. By what factor

does this energy differ from that given by

Eq. (22.35)?

24.7 For a geocentric dipole field of strength

given by Eq. (24.10), show that

(a) the rms surface field strength is

ﬃﬃﬃ

(b) the mean (unsigned) field strength is

1.38B

25.1 If the fraction of intrusive igneous rocks that

have self-reversing remanent magnetism is

and the fraction of the intruded rocks that

self-reverse is f

, show that the fraction of

the intrusive rocks that disagree in polarity

with their baked contacts is

F ¼ f

þ f

 2f

ðÞ:

Observations suggest F  0:02. If f

¼ f

¼ f ,

what are the possible values of f ?

25.2 As mentioned in Section 25.5, paleointen-

sity measurements use natural thermore-

manence of igneous rocks, with the

assumption that blocking temperatures

are independent of cooling rate. This can-

not be quite true and natural cooling rates

may be 10

times slower than laboratory

cooling. To estimate the magnitude of the

resulting error, calculate first the differ-

ence in E/ kT

for the two cooling rates by

Eq. (25.2) and then use Fig. 25.2 to estimate

the variation in spontaneous magnetiza-

tion of magnetite with temperature at a

blocking temperature T

¼0.9

. Given

that E/ m

, calculate the variations in m

and T

for the different cooling rates. The

values so calculated correspond to  and T

in Eq. (25.4). Assuming B/kT  1, this equa-

tion gives the magnetization at the block-

ing temperature, which is stronger at the

lower blocking temperature. But cooling to

low temperature causes a further increase

in m

to m

and this increase is less for the

slower cooling because of the higher initial

value of m

. Therefore, only the T

factor

remains to cause an error in paleointensity

estimates. How big is this error for a 10

difference in cooling rates?

25.3 If polar wander occurs at about 0.38

per million years, and the rms scatter

of individual measurements of apparent

instantaneous pole positions, caused by

geomagnetic secular variation, is 158, how

many suitably averaged measurements

are required to establish with 95% confi-

dence that a pole position differs from one

30 million years earlier?

25.4 Viscous remanent magnetization (VRM) is

the slow development of magnetic rema-

nence at a fixed temperature. The process

is not fundamentally different from ther-

moremanence, but involves blocking tem-

peratures that are at ambient temperature

on the relevant time scale. VRM induced by

a field (or its decay after removal from a

field) has often been reported to vary as

the logarithm of time. This is not character-

istic of grains all with the same activation

energy, E, which would show an exponen-

tial relaxation towards equilibrium. Noting

the sharpness of the blocking condition for

individual grains, that is a small variation

in T has a big effect on relaxation time, ,

use Eq. (25.1) or (25.2) to show that an

assembly of grains with a uniform distribu-

tion of values of E can give a log t variation

in VRM. (For a detailed discussion of

VRM see Dunlop and

Ozdemir, 1997,

Chapter 10.)

25.5 For reasons discussed in Section 25.4, we

now discount the possibility that, immedi-

ately after a geomagnetic reversal, further

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reversals are inhibited. We may, not unrea-

sonably, suggest that the opposite is the

case, and that further reversals are more

likely. For what may appear to be a rather

subtle reason, this could be mistaken for

inhibition in the reversal record. Why?

26.1 Variations in insolation (energy received

from the Sun) are caused by regular

variations in orbital characteristics

(the Milankovitch cycles, discussed in

Section 26.4). Assuming orbital angular

momentum to be conserved, calculate the

variation in annual average insolation due

to the variation in orbital eccentricity e.

(Equations in Appendix B are helpful to

solution of this problem. A simple method

is to calculate the total energy received on

one complete orbit and then relate the orbi-

tal period to e.)

26.2 What is the effect on the Earth’s orbit of

friction of the solar tide? Could this have

a significant influence on the global climate?

26.3 Calculate the effect on the length of the

day of a 2 mm per year rise in sea level,

assuming that it is due to (a) melting of

the polar caps, and (b) thermal expansion

of the surface layers of the oceans.

26.4 What is the required net heat input for each

of cases (a) and (b) in Problem 26.3?

26.5 If the total energy use in Table 26.1 were

derived entirely from tidal power stations,

by what factor would the rate of change in

the length of day increase?

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