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

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India plate about 40 million years ago, so this

problem requires further thought.

12.5 The pattern of mantle

convection

We summarize by a few key points the observa-

tions on which our ideas about mantle convec-

tion are based.

(i) Unless the thermal conductivity of the

mantle is much higher than we have reason

to believe (30 to 100 times that of familiar

mantle rocks), diffusive cooling of the deep

mantle is insignificant. Without convec-

tion, radioactivity would cause the temper-

ature to rise by about 120 K/10

years and

much more in the distant past.

(ii) The electrical conductivity of the lower

mantle is low enough to transmit compo-

nents of the geomagnetic secular variation

with periods as short as a year, disallowing a

temperature much higher than that of the

upper mantle (Dobson and Brodholt, 2000).

This is consistent also with the viscosity var-

iation discussed in the following paragraph.

There can be no substantial mid-mantle

thermal boundary layer; the mantle must

be convecting as a whole and not as separate

upper and lower mantle circulations.

(iii) Post-glacial rebound (Section 9.5) indicates

a general increase in viscosity with depth

below the asthenosphere. Although viscos-

ity is very temperature-sensitive and must

be presumed to be locally quite variable,

the general trend is consistent with a

decrease in convective speed with depth.

The extreme viscosity variations occur in

the boundary layers, the cool lithosphere,

which is quasi-rigid, and the base of the

layer, which is at core temperature,

1000 K hotter than the adjacent mantle,

with a viscosity lower by at least 10

. It may

even include pockets of partial melt.

(iv) The almost rigid surface plates are moving

about with speeds of several centimetres

per year. Oceanic crust generated at ridges

and disappearing at subduction zones

amounts to about 3.4 km

/year; the oceanic

surface area is 3 10

, so the average

age of the lithosphere at subduction is 90

million years.

(v) Continents move about with their plates,

although they appear less mobile than

purely oceanic plates, but continental

crust is not subducted, being too buoyant

(although erosion deposits continental

material in the oceans, from which it is

subducted). The continental crust has pro-

gressively accumulated around ancient

cores (shields) that are more than 20

times as old as the oldest ocean floor.

(vi) Isolated volcanic hot spots, of which

Hawaii is the most studied example, are

not moving with the plates, but more

slowly than all but the slowest plates and

apparently independently of them. The

chemistry and isotopic abundances in the

lavas that they produce (ocean island

basalt – OIB) are distinct from the mid-

ocean ridge basalt (MORB).

(vii) The total surface heat flux from the Earth

is 44.2 10

W, of which 8 10

Wis

attributed to crustal radioactivity, almost

all of it in the continents. Disallowing also

a core component of the heat loss, which

we account for separately, thermal convec-

tion of the mantle must be explained by a

heat flux of 32 10

(viii) Most of the subduction zones are marked by

inclined planes of earthquake foci, in some

cases extending to 700 km depth. Seismic

tomography indicates that cool subducting

slabs penetrate much further, but aseismi-

cally. However, they are evidently impeded

by the phase transition at 660 km depth.

(ix) Subduction zones are marked also by lines

of volcanos, concentrated where the sub-

ducting slabs have reached about 100 km

depth. Their chemistry includes material

derived from sea water.

(x) Vertical motion over most of the mantle

at more than a small fraction of the plate

speed is disallowed by the sharpness of

mantle phase boundaries, especially that

at 660 km depth, as pointed out in

Section 22.5.

12.5 THE PATTERN OF MANTLE CONVECTION 177

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Loper (1985) pointed out that convection is

driven by sources of buoyancy (positive or neg-

ative) generated at boundaries and we empha-

size this in Section 13.1. Both the upper and

lower boundaries of the mantle are sources of

buoyancy and they drive two quite different

modes of convection that are superimposed

and at least semi-independent. Plate tectonics is

driven by cooling of the lithosphere and the

plumes that are responsible for the hot spots

carry up heat conducted into the D

layer from

the core. The essential reason for the two differ-

ent convective styles is that they are controlled

by strong but opposite viscosity contrasts. The

negatively buoyant lithosphere forms a layer so

much more viscous than the underlying mantle

that it is virtually rigid and is subducted in broad,

coherent slabs, with shearing motion taken up

by the asthenosphere and upper mantle.

Conversely, the hot material from the base of

is readily deformed and self-adjusts to pass

up through the mantle with minimum deforma-

tion of the surrounding mantle. This means that

it forms narrow, axisymmetric plumes, to which

the flow is effectively confined. In each case the

style of convection is governed by the principle

that what happens is what occurs most easily.

Deformation is concentrated in the least viscous

materials and this situation is reinforced by the

heat which is generated by it.

A graphic and convincing illustration of plate

tectonic motions was recognized by Duffield

(1972) in a natural analogue, the solidifying

crust on a convecting lake of lava. Plate spread-

ing centres, subduction zones and transform

faults, where plate motions mismatch, were all

recorded on a film that shows the motion in ‘real

time’. The effectiveness of this natural model can

be attributed to the fact that the viscosity con-

trast between the crust and the underlying lava

realistically mimics the contrast between the

lithosphere and the asthenosphere. Attempts to

model mantle convection without this contrast do

not include necessary physics. The return flow

of asthenospheric material into the ocean ridge

spreading centres can be regarded as a passive

consequence of subduction-driven convection.

Satisfactory modelling of plumes also requires

a high viscosity contrast. Laboratory experiments

on liquids heated from below go some way to

achieving this if the viscosity is sufficiently tem-

perature dependent and the temperature incre-

ment high enough. Viscosity scaling can be

brought closer to the terrestrial situation if two

different liquids are used, but a model with

immiscible fluids does not properly represent

the terrestrial situation. But all such models

give the same convective pattern, axisymmetric

plumes with narrow, rapidly flowing cores. This

is inevitable if convection is driven by buoyancy

of material with much lower viscosity than the

rest of the medium within which it moves.

A feature of plume models is that they

are initiated by large, approximately spherical

plume heads that melt or soften their way up

through the overlying material and are supplied

with fresh hot material via narrow conduits.

Established mantle plumes, such as those respon-

sible for the hot spots considered in Section 12.4,

are simply the narrow conduits of continuing

flow, as in Fig. 12.3. When a new plume head

reaches the lithosphere it includes partial melt

that may appear as massive basalt flows (flood

basalts) that completely dwarf any recent volcan-

ism. Courtillot (1999) identified continental areas

of such flood basalts, finding seven with dates

coinciding with mass extinctions of species

(Section 5.5). The best documented is in the

Deccan area of India, extending into the ocean

and centred near Bombay, from which the hot

spot trace leads, via the Maldive Islands, to the

centre of current volcanism on Reunion Island,

800 km east of Madagascar. The process by which

a new plume head works its way up through the

mantle is obviously very slow and on average they

have appeared at about 50 million year intervals.

So, we suppose that one or several are currently

on the way up. Can they be identified with exten-

sive areas of uplift, presumed to be due to heat

(superplumes), under Africa and the South Pacific

(McNutt, 1998)?

To see why new plumes start and established

ones are extinguished, we need to consider the

structure and mobility of the D

layer, where

they originate. Core heat is conducted into it,

producing a highly mobile layer that is skimmed

off into the plumes, but D

is not uniform.

Seismologically, it gives the appearance of

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having areas where its boundary layer action is

fully effective and others where dense material

has accumulated, imposing blankets on the

core heat. By analogy with the crust we refer to

these areas as crypto-oceans and crypto-continents

(see Fig. 12.3). The mobile layer moves towards the

plume bases and the crypto-continents are carried

along with it. When one of them reaches a plume

base it cuts off much of the core heat in the vicinity

of the plume, starving it and encouraging a new

plume to start in a neighbouring crypto-ocean area.

The thin, mobile, layer at the base of D

is less

viscous than the mantle well above it by a factor

of at least 10

, but it could be much more than this.

Thin, ultra-low velocity zones (ULVZs) at the base of

the mantle are recognized seismologically and

have been explained in terms of partial melting.

However, an alternative explanation is that they

are patches of the lowermost mantle where

the post-perovskite phase, which is probably the

dominant mineral at that level, has absorbed a

high proportion of iron from the core (see

Sections 17.7 and 19.5).

Plume activity is controlled by the structure

and behaviour of D

and not by the mantle tec-

tonics. The reverse appears not to be true.

Courtillot (1999, Chapter 5) traces the evidence

that several of the areas of extensive flood

basalts mark the initial openings of new ocean

basins, implying that continental break-up was

triggered by the appearance of massive plume

heads. In at least some cases, changes in the plate

tectonic pattern, with rearrangement of conti-

nental blocks, appear to have been stimulated

by the plume activity. But if new plumes arrive at

the surface more or less randomly, then they

arrive in existing ocean basins at least as often.

In this case also they may prompt development

of new ridges, but probably rather more easily

‘steal’ nearby ridges, causing them to shift. The

largest single flood basalt province is the sub-

marine Ontong–Java plateau, north-east of the

Solomon Islands, with a volume that could be as

large as 10

. But, unlike continental flood

basalts, submarine events are not accompanied

by extensive faunal extinctions, presumably

because they have little effect on the atmosphere.

The flow of low-viscosity material in the

plumes is rapid, at least 1 m/year. The total flow

of plume material, estimated to be 40 km

/year,

is much greater than the average global rate of

eruption of hot spot basalt, even including occa-

sional new flood basalt provinces. Most of the

plume material either underplates the crust or

lithosphere, or is injected into the astheno-

sphere. Sleep (1990) calculated that the eleva-

tions of hot spot ‘swells’, such as that along the

Hawaiian chain, are consistent with the buoy-

ancy introduced by the total plume heat flux

(4 10

W). It can be regarded as localized

contributions to the asthenosphere, but globally

it is not a major effect because the rate of accre-

tion of asthenospheric material on to the litho-

sphere is about 10 times the plume flux.

12.6 Tectonic history and mantle

heterogeneity

A current example of the opening of a new ocean

basin is the Red Sea, which is the early stage of

the separation of Africa from Arabia. There are

also areas where convergence of plates appears

to have started, but without developed subduc-

tion zones; one, marked with a query in Fig. 12.2,

occurs in the Indian Ocean, between the Indian

and Australian plates. Some new or potential

plate boundaries may fail to develop, but others

will certainly do so, with the plate tectonic

pattern changing on a time scale of tens to hun-

dreds of millions of years. Other boundaries have

become inactive. We expect extinct ridges, which

are relatively shallow features, to disappear from

geological view more rapidly than subduction

zones and former plates now deep in the mantle.

The durability of remnants of subducted slabs

depends on how they are assimilated by the man-

tle and is a key to a significant tectonic question.

It is especially important to consider the lower

mantle because convection there is slowest but

the heat source is greatest. A mechanism for the

distribution of subducted coolness is suggested

in the final paragraph of this section.

As mentioned in Section 12.5, without cool-

ing, radioactivity would raise the temperature of

the mantle by about 120 K/10

years at the

present time and much more when radioactivity

12.6 TECTONIC HISTORY AND MANTLE HETEROGENEITY 179

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was stronger. This is superimposed on any tem-

perature changes resulting from adiabatic com-

pression or decompression and occurs whether

or not the material is moving. In Section 23.4 it

is concluded that long-term cooling of the man-

tle reduces its potential temperature by about

55 K/10

years and in the lower mantle this

means a cooling rate of 62 to 88 K/10

years,

depending on depth, so the total heat loss (at

the present rate) is equivalent to cooling by an

average rate of 195 K/10

years. By this we mean

that if a selected volume of the mantle is isolated

from convective cooling for 10

years while its

surroundings are cooled at the average rate, then

a temperature contrast of 195 K develops. We

consider what this means in terms of tomogra-

phically observed heterogeneity (Section 19.7)

and convective instability. With the temperature

dependences of seismic velocities for the lower

mantle in Table 19.1, a 195 K temperature cont-

rast corresponds to a P velocity variation of 0.2%

to 0.6% and an S velocity variation of 0.5% to 1.1%.

These are at the upper bound of observed varia-

tions, so, even though temperature is not a com-

plete explanation of observed heterogeneity,

we can use these values to argue that no part of

the mantle can be thermally isolated for 10

years. It is all convectively cooled.

Convective cooling of the lower mantle

presents an interesting problem, bearing in

mind that it is more viscous than the upper

mantle and its convective motion is slower. The

problem is that it must be cooled throughout,

although cool lithospheric material reaches it

only via thin subduction zones that are separated

from one another by thousands of kilometres.

Fragmentation of the subducted slabs and wide

distribution through the viscous lower mantle

material must be discounted as is also thermal

diffusion, which is much too slow. The physical

scale of thermal diffusion with a relaxation

time  ¼10

years is indicated by the value of

thermal diffusivity,  ¼1to210

6

1

, giv-

ing ()

1/2

¼180 to 250 km.

We can emphasize the nature of the problem

by considering the buoyancy of a large volume

with a temperature contrast D T 200 K, as

would occur in an isolated volume in 10

years,

and corresponding density contrast D/ ¼DT

3 10

3

, relative to its surroundings. It is con-

venient to consider a spherical volume, radius

r ¼1000 km, because that allows us to use

Stokes’s law for the viscous drag caused by its

motion at speed v through a medium of viscosity

 (not to be confused with thermal diffusivity,

above),

F ¼ 6prv: (12:4)

This is equated to the buoyancy force arising

from the density contrast,

F ¼ð4=3Þpr

Tg  6:3  10

N; (12:5)

with g ¼10 m s

2

and  ¼5000 kg m

3

so that,

with  ¼10

Pa s and other numerical values

from Appendices F and G,

v ¼ð2=9Þr

Tg=  3:3  10

9

1

 10 cm=year: ð12:6Þ

This is so much faster than any plausible lower

mantle convective speed that the existence of

such a large volume with a 200 K density con-

trast must be discounted. Volumes with this tem-

perature contrast can be no more than a few

hundred kilometres in size.

The requirement for subducted coolness to

be distributed through the lower mantle on a

scale not greater than a few hundred kilometres

may appear to present difficulty in devising a

plausible convective pattern, but it compels us

to recognize that the mantle is not all cooled

simultaneously. From time to time the convec-

tive pattern changes, so that different parts of

the mantle are cooled in turn. New subduction

zones (and spreading centres) appear and estab-

lished ones die out. The intervals between these

changes cannot be more than about 100 million

years, that is the conventional ‘overturn time’ of

mantle convection. To the extent that tomo-

graphically observed irregularities in the lower

mantle can be attributed to temperature varia-

tions they reflect not just the present convective

pattern but relics of earlier patterns that slowly

fade. We must presume that the lower mantle is

not cooled steadily, but that different regions

are cooled in turn, as the convection pattern

changes.

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Convective and tectonic stresses

13.1 Preamble

Decades of general disbelief in continental drift

preceded recognition of plate tectonics in the

1960s. In the 1800s, Kelvin noted that the

Earth’s response to tidal forces indicated an

average rigidity modulus exceeding that of steel

and, when seismology provided details of inter-

nal structure, it showed that the Earth is solid

to nearly 2900 km depth. Early proponents of

continental drift faced the difficulty that this

evidence of solidity appeared incompatible

with yielding to any stresses then envisaged.

Convection had been contemplated in the

1800s as a means of conveying heat from the

deep interior and resolving the age-of-the–Earth

problem (Section 4.2). But the idea was stifled,

first by difficulty with solar energy and then,

when radioactivity was discovered, by recogni-

tion that its concentration in continental rocks

suggested that the Earth’s heat sources were

shallow. But by the 1960s, paleomagnetic evi-

dence of continental drift had become over-

whelming, mobility of the ocean floors had

been recognized and an explanation in terms of

convection had become unavoidable. As we now

understand, all tectonic processes are ultimately

driven by convection. What we see is the surface

expression of motion that is necessarily deep.

Thermodynamic arguments (Chapter 22) are

central to understanding convective energy and

tectonic stresses. The mechanical energy can be

derived only by upward transport of heat from

very deep sources. This energy is the product of

heat flux and a thermodynamic efficiency, giving

an estimated power of 7.7 10

W (Section 22.4).

In this chapter we consider the implications for

tectonic stresses, the point being to demonstrate

that the convective power suffices to explain

tectonics.

The power generated by mantle convection is

dissipated in the mantle and crust. Thermo-

dynamics does not specify how the dissipation

is distributed but observations of plate motion

give a good measure of the speed of convection

and, using this, the calculated power gives an

estimate of the stresses involved. The fact that

the convective stress, so calculated, is compara-

ble to the stresses apparent in subduction zone

earthquakes (Chapter 15) confirms both that

convective stresses are responsible for earth-

quakes and that earthquakes are localized irreg-

ularities in the grand pattern of convection

(Chapter 12). Thus, the earthquake distribution

is a direct indication of that pattern. Moreover,

comparison with the gravitational energy

released by the negative buoyancy of subducting

slabs shows that they provide the principal driv-

ing force of convection.

We can make a more general case for this

conclusion. As Loper (1985) pointed out in a dis-

cussion of the principles of mantle convection, it

must be driven by sources of buoyancy (positive

or negative) generated at boundaries and our

attention must focus on them. We can emphasize

this point by imagining a mantle approximating

the real one, with an adiabatic temperature gra-

dient and a uniform distribution of heat sources

(radioactivity), but supposing it to be thermally

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isolated from all surroundings so that its temper-

ature rises uniformly. It is large enough for ther-

mal conduction to be ineffective in equalizing

the temperature. As the temperature rises, the

temperature gradient becomes sub-adiabatic

(preventing convection), because the adiabatic

gradient is proportional to absolute temperature

and to remain adiabatic all temperatures would

have to rise proportionately and not uniformly.

Internal heat alone does not cause convection, it

inhibits convection, and it is only by cooling at

the surface that the necessary buoyancy is gener-

ated. Of course, heating at the lower boundary

(by the core) has the same effect, but if we appeal

to heterogeneity of internal sources, then the

radioactively well-endowed materials would

tend to rise and, if unmixed, stay at the top.

Thus, any convection driven by internal sources

without reference to surface cooling would have

finished early in the life of the Earth.

The conclusion that subduction of cooled

lithospheric plates is the essential driving mech-

anism of convection carries the inference that

subduction zones are also areas where dissipa-

tion is concentrated. Although this is true, there

is no 1:1 correspondence between sources and

sinks of convective energy. Subduction does not

occur in isolation but is part of a convective cycle

with motion throughout the mantle and crust.

Stress and energy dissipation occur wherever

there is material deformation and consequential

stresses extend to areas where there is no notice-

able deformation. Observations of crustal stress

(Sections 11.5 and 11.6), combined with topogra-

phy, isostatic balance or unbalance and geolog-

ical features indicative of ongoing deformation,

all contribute to the global picture of a dynamic

system.

Our discussion of the mechanism of plate

tectonics assumes that it is driven entirely by

the loss of heat to the surface from the body of

the mantle. The convective plumes driven by

core heat are geometrically very different from

plate-driven convection and operate apparently

independently of it. However, plumes influence

plate motion. They contribute to ‘ridge push’, as

in Iceland, and probably aid the initiation of new

spreading centres, as in East Africa. To reach the

surface, core heat must traverse the entire depth

of the mantle, giving the plumes a high thermo-

dynamic efficiency (39% by Fig. 22.5).

Stresses within the lithosphere are required

to support topography. In Chapter 9 we examine

large-scale loading of the Earth’s surface by

ice sheets that cause relaxation in the mantle

towards isostatic equilibrium. The relaxation

is rapid compared with geological processes

because the material deformation is quite small.

Similarly, topography with scale lengths greater

than a few hundred kilometres is in isostatic equi-

librium. However, at shorter wavelengths, loads

are supported by elastic flexure of the lithosphere,

which indicates that the lithosphere can retain

elastic strains over geologic time and so must

have an effective viscosity several orders of mag-

nitude higher than that of the underlying mantle.

A high-floating lithospheric block, such as a pla-

teau, has internal stresses that would spread it out

to sea level were it not held together elastically.

On the other hand low-lying blocks are in com-

pression. The stress state can be calculated by

integrating over boundary forces, and superim-

posing the body force of gravity. For an internally

homogeneous fluid planet with blocks floating on

its surface the difference between vertical and

horizontal stresses in the blocks is linearly related

to variations in geoid height. Regions of positive

geoid correspond to a state of extension and neg-

ative to compression. While the Earth is more

complicated because of internal heterogeneity,

an overall correspondence between geoid height

and stress state can be recognized.

The very long wavelength features of the

geoid are attributed to density variations in the

lower mantle and are believed to indicate resi-

dua of past subduction. These effects must be

subtracted before the geoid features related to

lithospheric structure can be recognized. Only

the long wavelength density variations in the

lower mantle are gravitationally apparent at

the surface as the higher harmonic terms are

geometrically attenuated. The geoid features of

intermediate wavelengths can be interpreted in

terms of topography and density variations in an

isostatically balanced lithosphere–asthenosphere

system. A plot of the geoid, with harmonic

degrees of six and less subtracted (Fig. 13.1(b)),

relates to details in the stress map (Fig. 11.11).

182 CONVECTIVE AND TECTONIC STRESSES

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–50 –40 –30 –20 –10 0 10 20 30 40 50

Geoid (m)

120° E

180°

120° W

60° W

0°

60° E

60° S 60° S

30° S 30° S

0° 0°

30° N 30° N

60° N 60° N

(a)

–20 –15 –10 –5 0 5 10 15 20

Geoid (m)

120° E

(b)

180°

120° W

60° W

0°

60° E

60°S 60° S

30°S 30° S

0° 0°

30°N 30° N

60°N 60° N

FIGURE 13.1 Filtered versions of Geoid EGM96 by Lemoine et al. (1998) selecting (a) wavelengths from 5000 km to

15 000 km and (b) wavelengths from 100 km to 5000 km. Figures by King (2002). In (b) East Africa is an area of positive

geoid anomaly, consistent with extensional stress. Other elevated regions in extension include the Tibetan plateau,

the Basin and Range of west North America, and the Andes.

13 . 1 P R EA M B LE 183

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Regions of high elevation, plateaux and moun-

tain ranges, are in tension and correspond to

positive anomalies in the filtered geoid. The

most dramatic effect of filtering is in seen in

the plateau of East Africa, where the world’s

largest continental rift is found. Without har-

monic filtering East Africa is in a geoid low, but

when low harmonics are subtracted it appears in

a high.

Topography is built up through inelastic pro-

cesses, such as folding and earthquakes. A parti-

cular case occurs above subduction zones in the

material that accumulates above subducting

plates, as the so-called accretionary wedges. The

subduction acts as a conveyor belt, piling sedi-

mentary material onto the over-riding plates that

act as backstops. The material fails in a brittle

fashion and adjusts to a wedge shape that is con-

trolled by friction between broken surfaces

with internal fluid pressure. The wedge angle

is referred to as a critical taper. This is modelled

by a theory that balances the gravitational and

frictional forces that are calculated in terms of

the topographic slope and dip of the sliding base-

ment. It applies equally well to large-scale fea-

tures, such as the Himalaya and Taiwan, and to

small-scale models such as sand box experiments.

13.2 Convective energy, stress

and mantle viscosity

The mechanical energy of mantle convection is

estimated in Chapter 22 from a fundamental

thermodynamic argument. For the whole man-

tle the power is 7.7 10

W. It may seem sur-

prising that, of this total, only 2.4 10

Wis

produced in the lower mantle and 5.3 10

in the upper mantle. The reason is that the

lower mantle heat is distributed and much of it

is transported over a limited temperature range,

whereas all of the lower mantle heat traverses

the entire upper mantle. The significance of

these numbers is that they are values of power

that must be used in deforming mantle (and

crustal) material and not merely power that is

available in principle. They must be equated to

integrals of the product of stress and strain rate

through the whole volume of the mantle. For the

upper mantle we have direct evidence of stress

and strain rate and can demonstrate that the

thermodynamically calculated energy suffices

to explain them. It is not so clear what is happen-

ing in the lower mantle but, as these numbers

show, the power generation per unit volume is

more than five times smaller than in the upper

mantle. While we cannot require that dissipa-

tion be distributed in the same way, it is inevi-

table that convection is more vigorous in the

upper mantle.

In making a quantitative assessment of the

relationship between convective energy and

stress, we use details of plate geometries and

speeds from a comprehensive survey by Bird

(2003) and especially his Table 3. The rate of pro-

duction of new crust is 3.36 km

/year, more than

90% of it at the ocean ridges, with a small contri-

bution by their extensions on to the continents.

Concentrating attention on the 67 000 km of oce-

anic spreading ridges, the average spreading rate

is 5.1 cm/year or 2.5 cm/year (7.4 10

10

1

)

each way from the ridge axes. By assuming this

to be the average speed, v, of the surface plates

across the underlying mantle, with total area

equal to the surface area of the Earth (A ¼5.1 

), we can write the viscous dissipation by

this motion,

, as a fraction f

of the upper man-

tle convective energy,

E ¼ 5:3  10

¼ f

E ¼ Av; (13:1)

where  is the shear stress between the plate and

the mantle below. The deformation is concen-

trated in the layer of lowest viscosity (the asthe-

nosphere). Let this have an effective thickness H

and viscosity . Then the strain rate is

¼ v=H (13:2)

and the viscosity is

 ¼ = "

¼ H=v: (13:3)

We have independent evidence of the relation-

ship between  and H from post- glacial rebound

(Section 9.5). Writing H ¼n 100 km, Eq. (9.35)

gives

 ðPa sÞ¼6:5  10

: (13:4)

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Identifying this with Eq. (13.3) and substituting

the value of v (7.4 10

10

1

), we have

 ðPaÞ¼4:8  10

: (13:5)

If we assume n ¼2, that is a 200 km thick astheno-

sphere, which we regard as the upper limit of

the plausible range, then  ¼1.9 10

Pa. This is

much less than the stress release typical of large

earthquakes, 10

Pa or so, and indicates that

the plates slide relatively freely over the astheno-

sphere. Taking n ¼1.5, that is a 150 km average

asthenospheric thickness,  ¼1.1 10

Pa and, by

Eq. (13.1), the viscous dissipation is 4.1 10

with f

¼0.0077. The corresponding average asthe-

nospheric viscosity is 2.2 10

Pa s. Although

there are obvious uncertainties in this calculation

it suffices to indicate that the surface motion

of the plates is little more than a passive conse-

quence of mantle convection. An explanation of

the lithospheric stress in terms of the gravitational

potential energy of ridge elevation is given in

Section 13.4.

Now consider the energy of subduction. The

process becomes somewhat confused at 660 km

depth so we concentrate initially on the upper

mantle, that is we calculate the gravitational

energy of subduction to this depth of 3.36 km

year of lithosphere (assuming that it all reaches

that depth). Its negative buoyancy is due to

the thermal shrinkage apparent from ocean

depth observations, as discussed in Section 20.2.

With the allowance for the isostatic component of

ocean floor deepening we take the average shrin-

kage to be 2.1 km, of which Dz

¼200 m occurs

in the igneous crust, density 

¼2900 kg m

3

leaving Dz

¼1900 m to be accounted for by

shrinkage of mantle material, with density



¼3370 kg m

3

. The thermal expansion coeffi-

cient, , decreases with depth, being only 0.7 of

the zero pressure value at 660 km, and the density

increases by the factor 1.18 over the same range

(including effects of the phase transitions above

660 km). Thus the product () decreases by

the factor 0.83 over this depth range and we take

the average of this product in the upper mantle

to be smaller than the surface value by h i/

()

¼0.915. The negative buoyancy arising from

shrinkage is multiplied by this factor. It is partly

offset by the positive buoyancy of the intrinsic

density difference, (



) ¼470 kg m

3

,between

the crustal component, thickness z

¼ 7 km,

and the mantle into which it sinks. With these

numbers the net downward buoyancy force

per square metre of oceanic lithosphere is

F=A ¼g½ð

z

þ 

z

Þhi=ðÞ

ð

 

Þz

¼5:66  10

2

;

(13:6)

where g ¼9.92 m s

2

is the average gravity

over the 660 km depth of subduction. With

3.36 km

/year ¼0.106 m

1

of lithosphere des-

cending 6.6 10

m, the rate of gravitational

energy release is 4.0 10

W, more than 70% of

the upper mantle energy estimated thermo-

dynamically in Section 22.4. If we assume that a

significant fraction of the crustal component

escapes full subduction, then the energy release

by subduction is even nearer to the thermody-

namic result, but there is still a difference to be

accounted for by convective forces elsewhere in

the upper mantle. The allowance for crustal

buoyancy introduces the problem that basaltic

oceanic crust is believed to convert to a denser

form, termed eclogite, at depth, removing its

buoyancy, with an effect similar to the assump-

tion that the crust does not subduct. However,

this is a phase transition and needs special con-

sideration in the calculation of convective

energy (Section 22.3). If we remove the crustal

term in Eq. (13.6), then we must allow for

a compensating energy term elsewhere in the

convective cycle.

We estimate the mantle viscosity by equating

the gravitational energy release by subduction

to the viscous dissipation by subducting slabs.

Their effective surface area is the 51 000 km

length of the subduction zones multiplied by

the 660 km/sin  down-slab distance, where  is

the dip angle, which we take as 45 8, all multi-

plied by 2 because the slabs experience drag

on both top and bottom surfaces, giving a total

area A

¼9.5 10

. Since the total surface

area is conserved and the 3.36 km

/year of cru-

stal generation is matched by the same rate of

subduction, the average subduction speed is

6.6 cm/year, that is v ¼2.1 10

9

1

. We iden-

tify the energy dissipation with the gravitational

energy release estimated above, so that

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E ¼ A

v  ¼ 4:0  10

W; (13:7)

where  is the consequent mantle stress, for

which this argument gives 2.0 10

Pa. This is

at the high end of the range of subduction zone

stresses inferred from earthquakes and confirms

the adequacy of convection to maintain tectonic

activity, including earthquakes.

In applying Eq. (13.3) to estimate the viscosity

of the mantle as a whole from its resistance to

slab subduction, there is no obvious value of H

to assume. Although the shearing action may be

spread quite widely, the self-softening effect of

the dissipation will tend to concentrate it. If we

assume H ¼150 km, with v ¼2.1 10

9

1

,as

above, then  ¼1.4 10

Pa s. Acknowledging

the wide variations in viscosity, this must be

regarded as agreement with the post-glacial

rebound estimate of upper mantle viscosity

(Section 9.5).

So far we have considered only surface motion

of the plates and subduction in the upper mantle,

essentially because these are the observed fea-

tures of mantle convection. Discussion of the

pattern of return flow and the involvement of

the lower mantle is necessarily more speculative,

but there are several considerations that provide

constraints on hypotheses.

(i) The adiabatic temperature gradient is pro-

portional to the absolute temperature, T,

as in Eqs. (19.55) and (19.56), so, assuming

the mantle to remain approximately adia-

batic as it cools, the temperature drop, and

therefore the heat lost by any material

element is proportional to T. The high tem-

perature of the lower mantle means that

the heat lost by the lower mantle is a

greater proportion of the total heat loss

(77%) than its mass is of the total mass

(73%). The 73% factor applies to radiogenic

heat, which we assume to be distributed

according to density. Core heat is treated

separately.

(ii) The mechanical power generated in the

lower mantle is about 30% of that gener-

ated in the mantle as a whole, although it

comprises 2/3 of the volume. Convective

energy is calculated thermodynamically,

but in relating it to buoyancy forces, it is

necessary to recognize that the thermal

expansion coefficient decreases by a factor

of three over the depth range of the man-

tle. Also the temperature ratios are lower

in the lower mantle than in the upper

mantle, reducing the Carnot efficiency of

convection.

(iii) Viscosity of the lower mantle is higher

than that of the upper mantle, perhaps

by a factor of about ten.

(iv) At the present rate of subduction a volume

equal to the upper mantle is cycled

through the lithosphere in one billion

years or less. For a volume equal to the

whole mantle the corresponding time is

three billion years.

(v) Two thirds of the mantle heat lost to the

surface is accounted for by radioactivity,

which is distributed throughout the man-

tle. Unless we allow the implausible

hypothesis that parts of the mantle are

consistently heating up over billions of

years, subducted coolness must, by some

mechanism, be distributed throughout the

volume.

(vi) Except in subduction zones, mantle phase

boundaries are observed to be seismically

sharp and essentially uniform in depth.

As shown in Section 22.4, this restricts

the speed of upward return flow in the

upper mantle to a small fraction of the

speed of subduction (<10%), requiring it

to be very broad in scale and probably

mantle-wide.

(vii) The temperature of the lower mantle

inferred from its electrical conductivity is

not dramatically higher than that of the

upper mantle (Dobson and Brodholt,

2000), disallowing the hypothesis of a

mid-mantle thermal boundary between

separate upper and lower mantle

circulations.

(viii) At least some of the slab material main-

tains its coherence in penetrating the

lower mantle, but the phase transition at

660 km is an impediment to convection.

(ix) The thermodynamically calculated con-

vective energy disallows the idea that all

186 CONVECTIVE AND TECTONIC STRESSES