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

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Appendix J

Problems

1.1 Calculate the moments of inertia of (a) a

spherical shell and (b) a uniform sphere of

mass M and radius R. (c) Consider a sphere

within which density is inversely propor-

tional to radius (giving gravity independent

of radius, as in Problem 17.1, and a reason-

able approximation for the Earth’s mantle).

What is the moment of inertia coefficient,

I/MR

1.2

(a) Consider a planet with a uniform mantle

and a uniform core of half the total radius

R, with density f times the mantle den-

sity. What values of f would be required

to give moments of inertia 0.3307MR

0.365MR

and 0.391MR

, corresponding

to the Earth, Mars and Moon

respectively?

(b) Consider instead that the ratio of core to

mantle densities is 3.0, both being uni-

form. What fractions of the total radius

must the core have in the three cases?

1.3 Allen (1973) gives the following model

for the density of the Sun as a function of

radius r. The outer radius is r

¼6.9 10

and the total mass is M

¼1.989 10

kg.

Use the values to obtain a rough estimate

of the moment of inertia of the Sun.

(Allen’s estimate from finer details is

5.7 10

kg m

.) What fraction is this of

the moment of inertia of a uniform sphere

of the same mass and radius?

1.4 By virtue of the rotation of the Sun, radiation

from opposite (approaching and receding)

limbs is seen to be Doppler shifted (to blue

and red respectively). Thus the radiation

field carries away some of the solar angular

momentum. What is the fractional rate of

slowing of the Sun’s rotation by this effect?

(Use the solar moment of inertia from

Problem 1.3 and other parameters from

Table A.5, Appendix A.)

1.5 Consider a spherical black body at 1 AU from

the Sun to be rotating and tumbling in such a

way as to equalize the temperature over its

whole surface. What is that temperature?

(The solar constant is 1370 W m

2

1.6 What would be the equilibrium temperature

of the subsolar point on the body in

Problem 1.5 if it presented a constant face

to the Sun? (Neglect conduction within the

body.)

1.7 What is the equivalent black-body temper-

ature for the surface of the Sun (radius

6.96 10

m)?

1.8 How does the equilibrium temperature of a

planet depend on the radius of its orbit?

1.9 Consider a space-craft approaching the Solar

System in a search for planets. As it gets

closer it begins to observe the planets in

an order according to the total amount of

r/r

(kg m

3

) r/r

(kg m

3

)

0 160 000 0.6 350

0.04 141 000 0.7 80

0.1 89 000 0.8 18

0.2 41 000 0.9 2

0.3 13 300 0.95 0.4

0.4 3600 1.0 0

0.5 1000

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sunlight reflected. Assuming that, in the

wavelength range used, all the albedos

(reflection coefficients) are equal, what is

this order? (Planetary parameters are listed

in Table 1.1.)

1.10 Suppose that collisions in the asteroidal

belt are continuously producing interplan-

etary dust with a distribution of sizes such

that the number of particles dN within any

mass range m to (m þdm) is given by

dN /m

n

dm, where n > 1. Once produced,

the dust spirals towards the Sun by the

Poynting–Robertson effect (Eq. (1.21)).

What is the distribution of particle masses

intercepted by the Earth? Assume the mass

intercepted to be only a small fraction of

the total and to be determined by residence

time near to the Earth’s orbit and that

all particles have equal densities. (Note the

evidence, discussed in Sections 1.8 and 1.9,

that the flux of dust originating in the aster-

oidal belt is not steady.)

1.11 What is the average speed of arrival on the

Earth of meteoroids falling freely from out-

side the Solar System? How does it compare

with the escape speed from the Earth?

2.1 Assuming meteorites to be of two kinds,

irons that are 100% metal and chondrites

that average 10% metal, and that the total

metal content of the meteorite bodies is the

same fraction of their total mass as the core

is for the total of the Earth, what fraction of

the total mass of meteorites is in the irons?

(Masses of the Earth and the core are given

in Appendix A, Table A.4.)

2.2 The uncompressed density of the planet

Mercury is estimated to be 5280kg m

3

.If

it consists of a core and mantle with compo-

sitions similar to those of the Earth, what is

the ratio of core radius to total radius?

2.3 The elements in Table 2.1 are listed in the

order of their abundances in the solar

atmosphere. Below are the nuclear mass

excesses for their most abundant and next

most abundant isotopes. The reference

standard (zero) is the isotope

C and the

other values are departures of the mass per

nucleon from the

C mass, in parts per

million. The atomic weights are given in

parentheses. Is there a direct correlation

between abundances and nuclear mass

excesses? What other factors need to be

taken into account? Note: see the proton

and neutron masses in Table A.1

(Appendix A).

3.1

(a) Calculate the gravitational energy released

bythecollapseofamassM of material,

initially dispersed to infinite separation,

to a uniform sphere of radius R.

(b) Repeat the calculation for the Sun,

assuming the density distribution in

Problem 1.3.

3.2 Derive Eqs. 1.8 and 1.9.

3.3 Show that for atoms of a radioactive species

with decay constant l, the mean life is l

1

3.4 Consider a rock that yields the following iso-

chron data:

Ar/d

K ¼0.098 0.005,

Sr/

Sr)/d(

Rb/

Sr) ¼0.0215 0.0007,

207

Pb/

204

Pb)/d(

206

Pb/

204

Pb) ¼0.090 0.005.

(a) Estimate the age and uncertainty, from

each of these isochrons.

(b) Suggest reasons for the discrepancies

between the alternative estimates and a

probable history.

3.5 The decay constants for the uranium

isotopes are known with greater accuracy

than the decay constant for

Rb. Use the

H (1)7825 (2)3913

He (4)651 (3)5343

O (16) 318 (18) 47

C (12)0 (13)258

Ne (20) 378 (22) 392

Fe (56) 1162 (54)  1118

N (14)220 (15)7

Si (28) 824 (29) 811

Mg (24) 623 (26) 670

S (32) 873 (34)  945

Ar (36) 902 (40) 940

Ni (58) 1115 (60) 1154

Ca (40) 935 (44) 1012

Al (27) 684 

Na (23) 445 

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Rb–Sr isochron for meteorites (Eq. (4.4)) with

the meteorite age, 4.54 10

years, from the

Pb–Pb isochron (Eq. (4.3)) to calculate the

decay constant, l, for

Rb. Assuming that

the uncertainty in this calculation is due

entirely to scatter of values in the Pb–Pb

isochron, what is the uncertainty in l

for

Rb?

4.1 If the processes of nuclear synthesis pro-

duced

127

I and

129

I in equal atomic abundan-

ces and proceeded at a uniform rate for a

time , and did not operate either before or

after this, and if accumulation of

129

Xe in a

particular meteorite commenced at time t

after the cessation of nuclear synthesis,

obtain the expression relating the ratio

(

129

Xe/

127

I) in the meteorite to , t, and the

decay constant l for

129

I. Show that Eq. (4.6)

is the special case for  l

1

4.2 If the heavy elements were formed 4.67 10

years ago (i.e. about 10

years before solidifica-

tion of the meteorites ), what was the ratio of

235

238

U at that time? (Note: the estimate

includes all heavier species that decayed to

these isotopes. The original

235

Uabundance

itself was probably very small because, in a

neutron-rich environment,

235

U undergoes

rapid neutron-fission.)

4.3 Two chondrules, A and B, from the same

meteorite have ratios of excess

129

Xe to

127

differing by a factor two, the value for A

being larger. If this is interpreted as a differ-

ence in formation dates, which chondrule

formed first and by how long? (See

Table H.3 for

129

I decay.)

4.4 Measurements on a freshly fallen meteorite

give an isotopic ratio

He/

H ¼4.52 10

Noting that

H decays to

He and assuming

that the two isotopes were produced in equal

abundances by cosmic ray bombardment,

what was the duration of the cosmic ray

exposure of the Meteorite?

5.1 Use the constants in Table H.1 to derive

Eqs. (5.2) and (5.3).

5.2 Consider a model for the growth of

Ar in

the Earth and the atmosphere, according to

which the Earth starts with no argon t years

ago, but accumulates it by decay of

K, of

which there is now a mass K remaining.

Argon is lost to the atmosphere at a rate

proportional to its concentration in the

Earth, that is the rate is L times the total

argon in the Earth.

(a) How much argon is there in the Earth

now, in terms of K, L, and the decay con-

stant, l, for

(b) What is the argon content of the

atmosphere?

(d) If t ¼4.5 10

years and the argon con-

tents of the Earth and atmosphere are

equal, what is the value of L?

(To solve a differential equation of the form

dy/dx ¼Ae

þBy, put y ¼ue

5.3 What is the gravitational energy release

resulting from settling of a core of density

 and radius 0.55R from an initially uni-

form Earth of density

 and radius R, assum-

ing as an approximation that the density of

each element of material is unaffected? Use

values for mass and radius of the Earth. What

average temperature rise would this energy

release cause? Assume an effective heat

capacity of 9.93 10

1

. The result of a

more rigorous calculation is given as an

entry in Table 21.1.

6.1 If the whole of the angular momentum of the

Solar System, 3.15 10

kg m

1

, were put

into the Sun, without changing its size or

density profile, would it be rotationally sta-

ble? This question can be answered at two

levels. The simpler one is to assume that the

Sun remains spherical and compare the grav-

itational attraction with the centrifugal

‘force’ at the equator. A better approxima-

tion is to allow the Sun to deform to an

equilibrium oblate ellipsoid and use the

theory of Sections 6.3 or 8.5 to calculate the

flattening. A value of flattening, f, less than

unity (c/a > 0) indicates stability. Assume the

solar moment of inertia given in

Problem 1.3.

6.2 Consider the Moon to be momentarily

directly between the Sun and Earth (during

a solar eclipse) and calculate the ratio of its

gravitational attractions to the Sun and

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Earth. Since the solar gravity is stronger,

explain how the Moon remains in orbit

about the Earth instead of going into an

independent orbit about the Sun.

6.3 Consider a f luid plane t, of uniform densi ty

, rotating with angular speed !,givinga

slight consequent flattening, f. Show that

f /

1

. (Note: this is the reason why t he

core is less flattened than the Earth as a

whole.) Show that your result is consistent

with Eq. (6.39).

6.4 Show that, for a planet of uniform density, or

one in which the surfaces of equal density

are homologous ellipsoids (all having the

same ellipticity), the dynamical ellipticity,

H (Eq. (7.2)), is related to the surface flatten-

ing, f,by

H ¼ f 1 



6.5 (a) For Mars ! ¼7.0882 10

5

rad s

1

and

¼1.825 10

3

. Assuming hydrostatic equi-

librium, calculate f and C/Ma

. (For mass and

radius see Table 1.1, with values for the Earth

in Table A.4).

6.6 Show that for a spherical planet of mean

density

, equatorial gravity vanishes at rota-

tional speed given by

crit





1=2

;

and that if the density is uniform the corre-

sponding angular momentum is

crit

¼ 0:32G

1=2

5=3



1=6

;

where M is planetary mass. It appears that

many planetary and asteroidal angular

momenta roughly follow a law of this form,

but with a numerical coefficient of about

0.07, that is 20% of the critical value. What

values of this coefficient apply to the Earth

and to Jupiter (for which you will need to

estimate the moment of inertia). What reason

is there for the Earth to have a low value?

6.7

(a) If the following gravity measurements

were made at noon each day on a ship

sailing due north at constant speed, at

what time on which day did it cross the

equator and what was its speed?

(b) If at latitude ’ the ship turned due East,

what was the resulting change in gravity

measured on the moving ship?

7.1 The equator of Mars is inclined at 248 to the

orbital plane. Using parameters from

Problem 6.5, what is the rate of precession

due to the solar torque?

7.2 By making guesses for the total mass and

mean speed of traffic and the average separa-

tion of opposing traffic lanes, estimate the

effect on the Earth’s rotation of the diurnal

development of traffic in North America.

7.3

(a) Consider two concentric spheres that are

constrained to rotate about axes that are

inclined to one another by an angle .Ifa

thin layer of viscous fluid between them

exerts a linear frictional drag on the dif-

ferential motion and the outer sphere is

maintained at angular speed !

, what is

the rotational speed of the inner one in

equilibrium with it?

(b) If the fluid layer has radius r, thickness

d r and viscosity , what is the power

dissipation?

between the core and mantle given by

Eq. (7.33) and precession maintains an

angular difference of 6 10

6

rad

between the core and mantle axes, what

is the consequent dissipation?

(d) What effect does the dissipation in

plausible? If plausible and a layer with

core viscosity 10

2

Pa s is responsible,

what is its thickness?

Day g Day g

1 9.802 22 9 9.780 50

2 9.798 05 10 9.780 33

3 9.794 15 11 9.780 83

4 9.790 59 12 9.781 97

5 9.787 47 13 9.783 74

6 9.784 84 14 9.786 09

7 9.782 78 15 9.788 97

8 9.781 32 16 9.792 32

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7.4 Using Eq.(7.22), estimate the elastic strain

energy associated with the Chandler wobble,

as a fraction of the total wobble energy.

7.5 Assuming equilibrium flattening of the

Earth, proportional to !

, how does the

period of the Chandler wobble depend on

!? What rate of rotation is required to

make the wobble period 1 year? When

might this have occurred?

7.6 Consider a mass m of water to be impounded

in a reservoir at latitude , effectively with-

drawing it from the ocean.

(a) With the simple assumption that the axis

of rotation is unchanged, what value of 

would give no change in LOD?

(b) When considered more closely, the

centre of mass of the Earth and the axis

of maximum moment of inertia are

changed. By how much?

wobble period (7 months), then its filling

would contribute to excitation of the

Chandler wobble. What mass of water

and latitude would be required to make

this significant?

8.1 Substituting Eq. (8.2) in Eq. (8.1) we find a

cos w term in W. Why does an asymmetrical

term of this form not appear in Eq. (8.9)?

8.2 Show that for a fluid Earth of uniform den-

sity, k

¼3/2.

8.3 What is the approximate amplitude of the

tide raised in the planet Mercury by the Sun?

How does this compare with the probable

elastic limit?

8.4 Suppose that the primeval Earth had a rota-

tion period half of the present value and that

subsequent slowing has been due entirely to

angular momentum exchange with the

Moon, all motions being coplanar.

(a) If the Moon was formed in a circular

orbit, what was the orbital radius?

(b) If the Moon was captured from a para-

bolic orbit (as has sometimes been sug-

gested), what was the distance of its

closest approach?

8.5 Assuming conservation of angular momen-

tum in the Earth–Moon system, what will be

the angular velocities of the Earth’s rotation

and the Moon’s orbital motion and the dis-

tance of the Moon when the tide in a 5 km

deep equatorial channel is resonant? (In

water of depth h, the speed of a wave of

wavelength l h is

ﬃﬃﬃﬃﬃ

8.6 If the Moon were rotating at one revolution

per day at its present distance from the Earth,

with a tidal phase lag  ¼0.28 in Eq. (8.20) and

following equations, and k

¼0.3, how long

would it take for tidal friction to stop the

rotation (relative to the Earth)?

8.7 The tidal potential Love number for the Earth

as a whole, as observed by satellites, is

¼0.245, with a phase lag,  ¼2.98,ofthe

tidal bulge. As noted in Section 8.2, for the

solid Earth alone the Love number would be

¼0.298. The difference represents an

inverted tide of the oceans, with negative k

Marine tides are complicated, but we can

regard k

as a vector representing their total

effect. Tidal dissipation is dominated by the

oceans, so we can consider k

as a vector

aligned with the Moon (or Sun). What are the

magnitudeandphaseofk

,asthevectordif-

ference between k

and k

? What amplitude of

themarinetidedoesthiscorrespondto?

9.1

(a) If the Greenland ice sheet (at 758N)

decreases by the melting of 1000 km

ice annually, distributing melt water uni-

formly over the Earth with no other

adjustment, estimate

(i) the angular migration of the pole,

(ii) the associated slowing of the Earth’s

rotation.

(b) What influence would isostatic rebound

have on the conclusion to part (a)?

9.2

(a) Calculate the gravity due to a thin disc

(thickness Dz) of density  and radius R at

a point on its axis at distance h from it.

(b) What is the simple approximation for

R h, but R < 1?

gravity due to an infinite sheet. Note that,

since the result is independent of h,there

is no restriction on Dz in this case.

(d) Why cannot the result in (c) be obtained

by considering a spherical shell and

allowing its radius to become infinite?

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9.3 In the 1850s G. B. Airy, whose explanation of

isostasy is illustrated by Fig. 9.6(b), reported

an experiment to determine the value of

the Newtonian gravitational constant, G,

from the variation of gravity, g, with depth

in a mine and crustal density, . The results

were reported as a measurement of the

mean density of the Earth,

.

(a) Why is the determination of

 equiva-

lent to the determination of G?

(b) What is the ratio /

 that would give

zero gravity gradient in a mine?

mean Earth density. What is the typical

crustal gravity gradient?

(d) What is the gravity gradient in the

ocean?

(e) By how much does the free air gradient

differ from the hypothetical ‘free vac-

uum gradient’, with no atmosphere?

10.1 Obtain the expression for Poisson’s ratio in

terms of the ratio of P- and S-wave speeds.

(See Appendix D for relationships between

moduli.)

10.2 Consider a laminated medium composed of

alternate layers with equal thicknesses of

material with P-wave speeds V

and V

for

bulk material. The values of density and

Poisson’s ratio are the same for both materi-

als. Show that for P-waves with wavelengths

that are large compared with the layer thick-

nesses (but small compared with the overall

dimensions of the medium) the velocities

perpendicular and parallel to the layers are

perpendicular

ﬃﬃﬃ

ðV

þ V

1=2

;

parallel

þ V



1 



ð1  Þ

ðV

 V

ðV

þ V

"#)

1=2

What values of V

are required for

velocity anisotropies of 1%, 5%?

(Hint: calculate the elastic moduli for both

cases, using Eqs. (10.4) for each layer. The

expression for Young’s modulus, E,in

terms of  and  is given in Appendix D.)

10.3 Consider two seismic stations, one close

to an earthquake and the other on the oppo-

site side of the Earth. In a particular fre-

quency range the first station sees a white

P-wave spectrum and the spectrum from

the remote one gives d ln (amplitude)/

d(frequency) ¼6. Assuming Q to be inde-

pendent of frequency,

(a) what is the effective Q

for the trans-

Earth path?

(b) what is the average Q

for the mantle if

the core has infinite Q?

for the mantle would be

expected?

(d) what value of d ln(amplitude)/d(fre-

quency) would the S-wave spectrum

at the remote station give, if the S-wave

spectrum at the near station is white?

10.4 Assuming elliptical anelastic hysteresis

loops (Fig. 10.3), what resolution in strain

recording is required to measure to 10% a

rock Q of 200, using a strain cycle of ampli-

tude 10

6

11.1 Solve for the stresses and displacements for

a1010 10 km elastic block in the grav-

itational field of the Earth, with zero fric-

tion at its base. What is the maximum

displacement if the elastic moduli are

given by  ¼l ¼3 10

Pa?

11.2 Consider a granular material which has a

cohesion S

and coefficient of friction 

subject to a body force that gives rise to a

vertical stress 

¼gz. Find, from the

Coulomb shear stress, the angle of repose

above which sliding occurs.

11.3 An approximate solution to the stresses

from a pressurized spherical magma cham-

ber of radius R and internal pressure P has

radial stress given by 

and a tangen-

tial stress given by 



¼ 



¼

(a) Find the coefficient A, and give an

expression for the displacement field.

(b) Now consider the case of a spherical

cavity with radius R and zero internal

pressure in an infinite medium. Let the

pressure at infinity be a uniform

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pressure. Show that radial and tangen-

tial stresses are



¼ P 1 



;





¼ P 1 þ



;

11.4 Suppose that a hydrofracture experiment is

carried out at various depths in a bore hole,

and that above a depth of 1 km the fractures

are horizontal, but below 1 km they are all

vertical. Let the maximum stress, 

,be

horizontal and equal to 100 MPa. Estimate

the value of all the principal stresses at a

depth of 2 km.

11.5 Consider a hydrofracture experiment at

1 km depth in a medium with density

 ¼3000 kg m

3

. A crack extends north and

south of the borehole. The break-down

pressure P

¼40 MPa. The shut-in or closure

pressure is P

¼25 MPa. The vertical princi-

pal stress is determined by gravity (assume

g ¼10 m s

2

(a) Calculate the principal stresses and

their directions.

(b) What type of earthquake would be

expected in this region? Explain.

region and the coefficient of friction is

0.6, what would be the expected geom-

etry of the fault planes?.

11.6 An approximate solution to the stresses

from a pressurized cylindrical magma

chamber has radial stress given by



and a tangential stress given by





¼

;



¼ 0:

(a) Show that these equations satisfy the 2D

equilibrium equations.

(b) If P

is the pressure at r ¼ R, find the

coefficient A.

ment field.

11.7 The stresses caused by a line load (between

x ¼1and x ¼1) of forces applied in the

horizontal direction at the surface of an

elastic half space, z > 0, are



2Xz

; 

2Xy

; 

2Xzy

: (J:1)

(a) Show that these equations satisfy the

equilibrium equations.

(b) Confirm that they satisfy the free sur-

face boundary conditions (zero normal

and tangential stresses) except along

the line of application of the force.

on the x axis, by integrating horizontal

traction about a cylindrical region sur-

rounding the line of application, find

the value of X intermsofthisforce.

11.8 Stresses generated by a normal line load of

strength X per unit length at the surface of

an elastic half space are given by



2Xz

; 

2Xzy

; 

2Xz

Show that this is equivalent to a horizontal

line (1

1) of vertically directed

forces (z  directionÞ, strength F per unit

length, by integrating stress about a cylin-

drical region surrounding the line of appli-

cation and find the value of X in terms of F.

11.9 The stresses in an elastic half space caused

by a horizontal line load at the surface are

given in Problem 11.7. Use these to consider

the stresses arising from a long strip of

material of thickness h and width 2W h

that is welded to a half space, z > 0, with the

same elasticity, and is heated, causing it to

expand thermally. Changes of the dimen-

sion in the direction of its length x are

prevented, and the surface is free. The

change in the width of the strip (in the y

direction) is constrained by the half space.

Show that along the mid-line of the strip

the y strain is (4/p)(h/W)e,wheree is the

strain that would o ccur without the con-

straint of the half space..

(a) First use Eqs. (10.24) to show that the

elastic modulus describing the plane

strain situation (with zero strain main-

tained in the x direction) is E/(1 n

(b) The problem can be solved by first con-

sidering the unconstrained strip to be

compressed by line forces along its

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edges of magnitude to reduce the strain

to the constrained value, that is by caus-

ing strain e[1 4p(h/W)]. Then apply

opposite line forces of the same magni-

tude a distance 2W apart in the surface of

the half space to produce strain e(4p)(h/

W) in the half space at the midpoint line,

so that when the strip and the half space

are welded the forces cancel and the

strain e(4p)(h/W)remains.

12.1 DeMets et al. (1990) give the motion of the

Indian plate relative to the Pacific plate as

1.1539 10

6

degrees per year about a pole

at (60.494N 30.403W). Show that, with the

motion of the Pacific plate relative to the

Hawaii hot spot given as the first entry in

Table 12.1, the motion of the Indian plate

given in the table follows.

12.2 If the ocean floor heat flux, discounting hot

spot heat attributable to core cooling,

27 tW, is explained by cooling, to a depth

of 100 km, of lithosphere that is replaced at

a rate of 3.4 km

/year, what is its average

temperature change? Is this reasonable, or

should the estimated depth of cooling be

adjusted?

12.3 Bird (2003) estimated the total linear length

of spreading centres to be 67 000 km and

zones of convergence (subduction) to be

51 000 km. if we take the sum of them to

be the total length of plate boundaries,

what is the mean plate size, that is the

‘reach’ or distance between source and

sink? Is this consistent with the estimates

of mean plate speed in Section 13.2 and

average plate lifetime from Section 20.2

(90 million years)? Such average numbers

can be misleading. There is a range of plate

sizes and speeds and the relationship

between averages depends on how size

and speed, or lifetime and speed, are corre-

lated. Suppose the speeds to be uniformly

distributed over the range 0 to n

max

with an

average



 ¼ 

max

=2, and approximate this

with a distribution in which half of the

ridge length produces crust spreading at



=2 and the other half produces crust

spreading at 3



=2. Suggest how the

lifetimes of the plates may be correlated

with their speeds and show how the corre-

lation influences the relationship between

mean values.

12.4 Consider a continental block of thickness

40 km and diameter 4000 km, with a sur-

face elevation 840 m above sea level, iso-

statically balanced with ocean floor at

an elevation of 4500 m, to drift from

Antarctica to the equator. Estimate the

change in the Earth’s rotation rate and the

shift of the pole of rotation with respect to

all other features, which remain fixed in

relative positions, under two alternative

conditions, (a) there is no change in elliptic-

ity, (b) the equatorial bulge readjusts to

equilibrium.

12.5 Some geological reports indicate that, 100

million years ago, sea level was 200 m

higher than a present. We assume this to

be a global effect and not a local effect of

land movement. It cannot be accounted for

by complete ice cap melting (which would

cause a rise of about 80 m) combined with

any plausible warming and thermal expan-

sion of the oceans. Consider the changes in

ocean floor topography that could have

been responsible. Isostatic balance of con-

tinents and ocean basins ensures that there

was no net transfer of mass between such

large areas and therefore that sea level var-

iations would have been caused by varia-

tions in the thermal structure of the

lithosphere. The diffusive cooling model

for the oceanic lithosphere (Section 20.2)

gives the total cooling, and therefore

shrinkage, at age t as proportional to t

1/2

This causes ocean deepening, relative to the

ridge crests, z ¼z*(t/)

1/2

, where z* ¼3000 m

is the value at age * ¼90 million years, the

present average duration (lifetime) of the

oceanic lithosphere. Flooding of the conti-

nents would be caused by a decrease in the

average value of z, and therefore t

1/2

, requir-

ing the ocean floors to be hotter and

younger, on average, than at present. This

could have arisen in several ways. One pos-

sibility is a plate redistribution such that

average plate ages are now more nearly

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equal than 100 million years ago, but with

no change to the overall average, effectively

replacing with middle-aged lithosphere

areas that were previously both younger

and older. Consider two simpler alterna-

tives, in the approximation that all plates

have the same lifetime: (i) faster plate

motion with no change in geometry, and

(ii) more (smaller) plates, with correspond-

ingly more ridges and subduction zones,

and no change in plate speed. By how

much must the lifetime decrease to give a

200 m rise in sea level? What is the implied

change in heat flux? In Section 23.2, we

allow for hydrothermal cooling by replac-

ing t

1/2

in the above expressions with t

1/3

How much difference would this make?

13.1 Using Eq. (13.6) and Fig. 20.2, with the iso-

static correction factor, 1.437, as applied to

Eq. (20.11), estimate the absolute minimum

age of oceanic lithosphere that may

subduct.

13.2 Assuming that the rate of subduction of

lithosphere is 3.36 km

/year and that its

thermal shrinkage averages 2.1 km, calcu-

late the rate at which energy would be

released by subduction to the base of the

mantle of all the coolness that this repre-

sents. Approximations that suffice for this

purpose are g ¼10 m s

2

(constant) and the

variations in density and thermal expan-

sion coefficient are such that the product,

a, decreases linearly with depth, by a fac-

tor 2 over the whole mantle depth.

Compare your result with the 7.7 10

of convective energy derived thermody-

namically in Chapter 22. Your answer will

be about twice this allowed maximum.

What is the explanation?

13.3 Calculate the gravitational energy released by

the separation of the crust from the mantle.

This is energy of compositional convection in

thesamesenseastheenergyreleasedby

redistribution of light solute by the solidify-

ing inner core (Section 22.6) and is 100% effi-

cient in contributing to the mechanical

energy of convection. How significant a

contribution is it? Compare your answer

with the relevant entry in Table 21.1.

13.4 Ignoring core heat, and assuming mantle

convection to be driven only by its own

internal radiogenic heat, over what depth

at the base of the mantle would this heat be

inadequate to drive convection? Assume

the total mantle radiogenic heat to be

20 10

W and uniform conductivity

5Wm

1

. To calculate the temperature

gradient, use Eq. (19.55) with thermal pro-

perties in Appendix G.

13.5 If the viscosity of plume material is

(a) underestimated or (b) overestimated by

a factor 100 in Section 13.3, what would

be the corrected plume diameter for each

case?

13.6 Derive Eq. (13.20), using Eq. (13.25).

13.7 The slope of the accretionary wedge that

forms the high Andes (roughly latitudes

138Sto278S) is  ¼ 48, twice that of the

low Andes, further north and south, for

which  ¼ 28. It has been proposed that

this difference arises from the cumulative

effect of greater friction on the top of the

down-going plate (Lamb and Davis, 2003)

beneath the high Andes. Because of climate

conditions the trench in front of the central

Andes is almost devoid of sediment,

whereas to the north and south the trench

has thick sediments. Assume that the

amount of sediment determines the fric-

tional properties on the slab, mainly by

water pressure both in the wedge and at

its base, and that the effective friction

beneath the high Andes is double that

beneath the low Andes. Use Eq. (13.41) to

analyse these differences.

(a) Assuming that for the low Andes

¼ 0:7 and 

¼ 

¼ 0:85 (as used

for Taiwan in Section 13.6) and that

 ¼ 68, what is the value of l

for the

low Andes that gives rise to  ¼ 28?

(b) What values of l

and l

for the high

Andes give  ¼ 48, along with the

doubledfrictiononthebase?Comment

on whether the solution is reasonable,

given the arguments above.

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14.1 Equation (14.1) gives strain as a function

of distance from the axis of a screw disloca-

tion (of infinite length) in a uniform (infin-

ite) medium. Using the fact that strain

energy per unit volume is

"

, where 

is rigidity, calculate the strain energy per

unit length of a screw dislocation. What is

wrong with the assumptions in the prob-

lem that cause the answer to tend logarith-

mically to infinity? How can the question

be posed more realistically?

14.2 Equation (14.34) defines surface-wave mag-

nitude, M

, and Eq. (14.36) gives an empirical

relationship to earthquake energy. Using

the fact that (2pa/T) is the peak ground strain

in a seismic wave and the square of this gives

the wave energy density, relate the total

energy to the wave energy density. What

does the result imply about the variation

with magnitude of the duration of the seis-

mic wave train? Is this plausible? Note that

the same conclusion does not follow if body-

wave magnitudes m are used with the equa-

tion equivalent to (14.36):

log

E ¼ 2:3 m  0:5:

14.3 Consider a train of P-waves of period T,

amplitude a and total duration , observed

at distance R from an earthquake. The local

rock has density  and P-wave speed V

.If

these observations are representative of all

directions from the earthquake, what is the

total energy in the P-waves?

14.4 Use the general expression for the phase

speed of an ocean wave, v (Eq. (14.53)),

with the expression for group velocity, u

(Eq. (16.49)), to obtain the relationship

between u and v for water of arbitrary

depth, h. Show that for long wavelengths

(l h) u ¼v and that for short wavelengths

(l h) u ¼v/2.

14.5 Using the equation for displacements by a

unit point force

4p





4ð1  Þ



;

(a) Show that the displacements are of the

form

¼B

ðl þ 3Þ

r ð l þ Þ



; u

Byx

; u

Bxz

;

and show that B ¼

4p

4ð1  Þ

(for unit

point force B has units of length

(b) Obtain the static displacement field for

a double couple source with slip in the

x-direction across a plane oriented in

the y-direction.

displacement b on area S in a medium of

modulus  and show that the displace-

ment field at any point in the medium is

given by

bS

12p

ðy

þ c

3=2

12p

ðy

þ c

3=2

(d) Consider a magnitude 6 strike-slip earth-

quake on a square fault plane oriented

east–west, with its top boundary at a

depth of 4 km. Seismic records indicate

a stress drop of 3 MPa. With the assump-

tions that the slip is uniform across

the fault plane and that the rigidity of

the medium is  ¼ 3  10

Pa, use the

moment–magnitude relationship and

the definition of moment to estimate

the dimension of the fault plane and

magnitude of slip.

(e) Calculate and plot the horizontal dis-

placements on the surface at x ¼0, y

¼5, 4, 3, 2, 0, 1, 2, 3, 4, 5 km.

14.6 Integrate Eq. (14.27) for different time inter-

vals to obtain the trapezoid shapes in

Figure 14.11(c).

14.7 The Brune (1970) spectrum (Eq. (14.32)) of

a far field seismic pulse corresponds to a

seismic pulse of shape given by PðtÞ¼kt

expðatÞ. Show that

(a) k ¼ uð0Þ!

;

(b) a ¼ !

;

ðtÞ¼M

½1 ðat þ 1Þ

expðatÞ:

15.1 If a transcurrent (transform) fault, across

which there is a shear stress of 10

Pa,

creeps aseismically at an average rate of

5 cm/year, estimate the magnitude of the

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