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

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of the subducted material eventually rea-

ches the bottom of the mantle.

(x) Mantle-derived rocks indicate that isotopi-

cally distinct source regions have survived

convective mixing.

A convection model satisfying all of these

conditions is demanding of ingenuity. With the

very limited direct evidence of just how the

lower mantle is convecting, it is not surprising

that there are divergent ideas. We can make the

general observation that, since the heat flux per

unit area at radius r decreases with depth

(decreasing r), then so does the speed of convec-

tion. This is qualitatively consistent with increas-

ing viscosity, but an increase by a factor of order

ten, as suggested by post-glacial rebound, can be

accommodated only by assuming that convec-

tive motion is much less concentrated than in

the upper mantle. We need to note also that the

assumption that the mantle heat loss is derived

from throughout the mantle is equivalent to

the assumption that the subducted coolness is

also distributed throughout the lower mantle

(although not necessarily steadily). Rather little

reaches the base of the mantle but two thirds

passes through 660 km depth. This makes it eas-

ier to understand that lower mantle convection

is slower, but it is not as easy to see just how the

coolness is distributed through a medium with

the lower mantle viscosity.

We can reasonably ask: is it necessary to

assume that convection is the only significant

mechanism for mantle cooling? The alternative

is conduction, but the conductivity would need to

be about 80 W m

1

and the only possibility

for this would be radiative conduction, as has

sometimes been suggested. Applying Eq. (19.62)

we see that this would mean an opacity

" 150 m

1

, corresponding to an optical depth

of 7 mm. It appears implausible that iron-bearing

lower mantle minerals could be so transparent. In

any case, if the lower mantle conductivity were as

high as this then the thermal structure would be

quite different from our present understanding.

Core heat would be conducted up through the

lower mantle with no D

thermal boundary

layer, or supply of hot material for deep mantle

plumes, eliminating the explanation of the

mismatch of core and mantle temperatures.

Convection would be weak, if it occurred at all,

with negligible power for tectonics. So, we

require bodily convection of the whole mantle,

and lack of a thermal boundary layer at 660 km

depth means that the lower mantle cannot be

considered separately. Observation (vi) above

means that, since the upward return flow in the

upper mantle is very slow and broad in scale, it is

inevitable that this motion extends into the lower

mantle. These conclusions alert us to the problem

that the lower mantle is cooled by subducted

material rapidly enough to ensure that there are

no persistent large-scale temperature variations

that would cause implausibly large buoyancy

forces. In the final paragraph of Section 12.6 we

suggest that the logical explanation is that the

convective pattern repeatedly changes, on a

time scale much shorter than 10

years, so that

different regions of the lower mantle are cooled

sequentially and not simultaneously.

13.3 Buoyancy forces in deep

mantle plumes

The contrast in temperature between the lower

mantle, extrapolated to the core–mantle boun-

dary from the 660 km deep phase transition, and

the outer core, extrapolated from the solidifica-

tion point at the inner core boundary, is at least

several hundred degrees and probably nearer to

1000 K. This is the temperature increment in the

thermal boundary layer at the base of the man-

tle. It is identified as the D

layer, although there

are also compositional and phase heterogene-

ities that complicate the interpretation of D

Since viscosity is a strong function of temper-

ature, the mantle material adjacent to the core,

being at core temperature, is much less visc-

ous than the material higher up. We can use

Eq. (10.27) to show how big the effect is. Taking

n ¼1 for linear (Newtonian) viscosity,  ¼ =

",the

ratio of viscosities at temperatures T

and T



=

¼ exp½gT

ð1=T

 1=T

Þ: (13:8)

We cannot choose plausible values of T

, T

that give a value of this ratio smaller than 10

:1.

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The ratio is smallest if the higher temperature,

, is near to the solidus temperature, T

. Taking

¼2785 K, T

¼3739 K from Appendix G with

¼4000 K, 

/

¼6 10

. Variations of 200 K

in the assumed value of T

change this ratio by

a factor less than two. This is the factor by which

the material immediately adjacent to the core

is softened relative to the bulk of the lower

mantle.

A thin layer of the most softened material at

the bottom of the mantle (the base of D

), being

thermally dilated and strongly buoyant, feeds

the deep mantle plumes. Its reduced viscosity

explains why the plumes can be narrow and

fast flowing. Not all of the plume material can

be at the boundary temperature and its average

viscosity is a compromise with the viscosity of

material drawn from slightly higher up in D

But the softest material flows fastest and the

flow in D

is restricted to a very thin layer of

the lowest viscosity at the bottom and involves

only a very small fraction of the boundary layer.

It is convenient to refer to an average viscosity,

which cannot plausibly exceed 10

4

of the gen-

eral lower mantle viscosity. Taking this to be

Pa s, we have an average plume viscosity

of 10

Pa s. The dynamics of plumes transport-

ing this softened material upwards through the

mantle are presented by Loper and Stacey (1983)

for Newtonian viscosity (n ¼1 in Eq. 10.27) and

generalized to non-Newtonian viscosity by Loper

(1984). The material is softest at the centre,

where the flow is concentrated, and the buoy-

ancy gives it a reduced pressure, causing a slow

inward collapse of the mantle, which restricts

the diameter and throttles the flow in a self-

stabilizing way. Plumes have thermal halos

with diameters of order 10 times their effective

fluid diameters. Here we consider a simplified

version, a vertical pipe of fixed radius, a, convey-

ing ‘fluid’ of uniform viscosity, 10

Pa s.

If there are 20 mantle plumes conveying alto-

gether 4 10

W of core heat to the surface, or

to the asthenosphere, that is 2 10

W each,

and, by Fig. 22.5, 39% of this is converted to

mechanical energy, then the mechanical power

of the buoyancy of each plume is E ¼0.8 10

This is E/l ¼2.4 10

W per metre of plume,

assuming a vertical rise, so that if the volume

rate of flow is

V ¼ pa

v for mean speed v the

buoyancy force per metre of plume is

F=l ¼ð

E=lÞ=v ¼ð

E=lÞpa

V: (13:9)

But

V is given by the heat transported,

Q ¼

VC

T ¼ 2  10

W: (13:10)

With  ¼4500 kg m

3

averaged over the mantle

depth, C

¼1200 J K

1

and taking DT ¼800 K

(slightly less than (T

T

) by the plume averag-

ing calculation of Stacey and Loper (1983)), we

have

V ¼ 46 m

1

. Then by Eq. (13.9),

F=lðNm

1

Þ¼1900 a

ða in metresÞ: (13:11)

This force is balanced by the viscous drag, for

which we adopt Poiseuille’s formula for the vol-

ume rate of flow driven by a pressure difference

DP over length l,

V ¼ pPa

=8l ¼ðF=lÞa

=8; (13:12)

in which we have substituted pa

DP ¼F. With (F/l)

given by Eq. (13.11) and taking  ¼10

Pa s, we

have

a ¼ð8

V =1900Þ

1=4

¼ 21 km: (13:13)

There are obvious uncertainties and approx-

imations in this calculation but the

power

in Eq. (13.13) ensures that the calculated

plume radius is not very sensitive to them.

The p lumes are narrow, although the esti-

mated radius refers o nly to the very hot flow-

ing channels which are surrounded by much

wider thermal halos that may be seen by seis-

mological observatio ns. The average speed of

plume material,

v ¼

V=pa

¼ 3:3  10

8

1

¼ 1:0m=year: (13:14)

is much greater than the plate speed. These cal-

culations ignore the effect of partial melting

which further reduces viscosity and plume

radius and increases the speed.

13.4 Topographic stress

The lithosphere is thought to be strong enough

to sustain some level of deviatoric stress for

at least 10

years. Deviatoric stresses relax in

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the weaker asthenosphere, where the dominant

stress is pressure. The pressure of elevated asthe-

nosphere at mid-ocean ridges exerts a horizontal

force on the oceanic lithosphere. ‘Ridge push’ is

the horizontal component of the distributed

force from the asthenospheric pressure across

the thickening lithosphere. It compresses the

plates. Plate acceleration is insignificant, so

ridge push is balanced by resistive forces, oppo-

sitely directed ridge push from distant ridges,

viscous drag by the asthenosphere, friction on

transform faults and forces from adjacent plates

or subduction zones. Of the forces that act on

the lithosphere, calculation of ridge push is

most straightforward. Figure 13.2 shows the

situation of the Africa plate, which is geometri-

cally symmetrical, with ridges both east and

west, and is believed to be nearly stationary

with respect to the underlying mantle. This

allows us to neglect any viscous drag of the

asthenosphere, or the effect of subduction

zones, simplifying the calculation of internal

stresses. The ridges are assumed to be spreading

symmetrically, so that they are retreating from

the continent and leaving fresh oceanic litho-

sphere essentially stationary with respect to

the underlying mantle.

The initially uplifted lithosphere thickens as

it subsides, and ridge push is the force per unit

length of ridge, given by the line integral of the

horizontal component of the asthenospheric

pressure on the underside of the lithosphere

between points A and C in Fig. 13.2,

RP ¼

dl cosðÞ¼

dz; (13:15)

where P

is the asthenospheric pressure, dl is the

differential line length along AC and  is the

angle dl makes with the z axis, taken as down-

wards (note that in this chapter we adopt the

rock mechanics sign convention that compres-

sive stress is positive). This force is balanced by

the stress within the lithosphere, so that the

total integrated force on the lithosphere can

be calculated as the line integral of P

over AB,

avoiding the need to integrate over the curved

surface. In the Africa case, ridge push simply

imposes a static compressive stress on the litho-

sphere, but in other situations, with the litho-

sphere moving across the asthenosphere, the

ridge push is distributed across the lithosphere–

asthenosphere boundary and overcomes the

viscous drag. Noting that the dimension AB in

Fig. 13.2 is about 0.02 of the horizontal extent of

a lithospheric plate (5000 km), the average

shear stress between the lithosphere and astheno-

sphere is 0.02 of the ridge push calculated for

the Africa case. If this stress is assumed to be

sufficient to overcome the viscous drag, then the

asthenospheric viscosity does not exceed

4 10

Pa s. We use the ridge as a reference in

calculating stresses elsewhere in the lithosphere.

To examine the stresses that develop in

uplifted regions in isostatic equilibrium, con-

sider a block of floating material with its upper

surface a distance e above the fluid surface, as in

Fig. 13.3. For the analysis that follows, its lateral

extent is assumed to be much greater than its

thickness. Isostasy requires that at the depth of

compensation, at the bottom of the block,

the vertical stress in the block and pressure

in the fluid are the same. However, elsewhere

Ridge

push

Lithosphere

Ocean

Continent

Water

Crust

Water

z = z

z = 0

FIGURE 13.2 A schematic representation of the Africa plate with ridges on either side. The symmetry simplifies

the calculation of stresses because we assume it to be stationary, with no asthenospheric drag. At a mid-ocean

ridge the asthenosphere exerts pressure on the lithosphere and this is resolved to give the horizontal stress. The

integral of the resolved stress over the boundary is called ridge push. Vertical stresses are attributed to gravity.

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the vertical and horizontal stresses within the

block are different. We shall show that the ver-

tical stress is larger, on average, by (1/2)

ge,

where 

is the density of the solid, and this

difference is proportional to the difference in

the geoid height in the block and fluid areas.

Let z be measured downwards from the solid

surface (Fig 13.3). The horizontal pressure

exerted by the fluid is



¼

gðz  eÞ; z  e

¼0 z

(13:16)

(recalling the convention that compression is

taken as positive). The vertical stress in the

solid is



¼ 

gz: (13:17)

The average vertical stress is found by integrat-

ing Eq. (13.17) from 0 to the compensation depth,

, and dividing by z





: (13:18)

The average horizontal stress is found by inte-

grating Eq. (13.16) from e to z

and dividing by z





ðz

 eÞ

: (13:19)

Combining Eq. (13.19) with the isostatic relation

(for uniform gravity), z



¼(z

e)

, the average

stress difference is



 



ge: (13:20)

This stress difference acts in a manner that

would cause the block to spread out. Such a

deviatoric stress, with the maximum principal

stress vertical, is the Anderson condition for nor-

mal faulting (Section 11.5).

We have developed this result for uniform

densities. However, it is true for arbitrary depth

variation of density in regions of isostatic equi-

librium. Consider two regions in isostatic equi-

librium with density variation 

(z) and 

(z),

joined at a common vertical boundary, with

depth of compensation D. Let region 2 be fluid-

like, with no deviatoric stress. We can calculate

the average vertical stresses in region 1 due to

the vertical gravity body force and the average

and horizontal stresses from the horizontal trac-

tion applied by region 2,





ðz

Þgdz

dz;





ðz

Þgdz

dz:

(13:21)

Integrating (13.21) by parts,





ðzÞgdz 

z

ðzÞgdz





ðzÞgdz 

z

ðzÞgdz:

(13:22)

The average stress difference is

ð

 

Þ¼



ðzÞgdz 

z

ðzÞgdz





ðzÞgdz þ

z

ðzÞgdz:

(13:23)

Isostatic balance requires



ðzÞgdz ¼



ðzÞgdz; (13:24)

so that Eq. (13.23) becomes

ð

 

Þ¼ ¼

zðzÞgdz; (13:25)

z = 0

= z

FIGURE 13.3 Geometry for calculation of stresses

in an isostatically balanced block.

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where D(z) ¼

(z) 

(z).

As in Eq. (9.20), the difference in geoid height

between the regions is given by

N ¼

2pG

½

ðzÞ

ðzÞzdz (13:26)

where G is the gravitational constant. Comparing

Eqs. (13.25) and (13.26) we see that, in general,

 ¼

2pGD

N; (13:27)

with g assumed to be the same in both regions.

In areas where the geoid is high compared with

surrounding areas, the vertical stress is greater,

leading to normal faulting. The discussion here

considers a first-order treatment that suffices for

an understanding of tectonic stresses. For some

purposes higher orders may need to be consid-

ered (Chambat and Valette, 2005).

Figure 13.1(a) is a plot of the geoid filtered to

select long wavelengths (5000 km < l < 15 000 km)

and Fig. 13.1(b) shows the higher spatial fre-

quencies (100 km < l < 5000 km), with the long

wavelengths filtered off. The long wavelength

geoid (harmonic degrees 6) is attributed to fea-

tures of the deep mantle (Hager and Richards,

1989), especially the lower mantle where the

viscosity is high (Section 9.5), and changes

caused by convection are necessarily slow.

Richards and Engbretson (1992) found a strong

correlation of the long-wavelength features

with the pattern of deep mantle densities

expected by extrapolating back for 100 million

years the present pattern of slab subduction.

This is evidence that slabs penetrated, in at

least some cases, to the base of the mantle and

this is confirmed by the correlation with tomo-

graphic models. The fact that the correlation is

limited to the hundred million year extrapola-

tion is consistent with the convection model

discussed in Section 12.6, which requires the

pattern of convective motion to change at inter-

vals of this order, so that surviving pieces of ear-

lier remnant slabs would not be correlated with

the present pattern, although they would take

much longer than this to decay by thermal dif-

fusion. (The thermal halo of an old slab would

not be distinguishable in observations of long

wavelengths from a young, thin slab with the

same integrated coolness).

The shorter- wavelen gth features of the geoid

cannot arise from de ep density variations, but

are caused mainly by the topography of litho-

spheric blocks that, for features larger than

a few hundred kilometers, are isostatically

compensated by flow in the ast henosphere.

Therefore Fig. 13.1(b) can be used to infer the

stress state in the lithosphere. Comparison w ith

the stress map (Fig. 11.11) sho ws that there is a

correlation with the stress state. The correlatio n

between long-wavelength features of the

geoid and lithospheric stress al lows the long-

wavelength geoid features to b e identified with

the lower mantle. The asthenosphere is not

completely inviscid, but for features hundreds

of kilometres in extent it adjusts to isostat ic

equilibrium in thousands or tens of thousands

of years. On this time scale it decouples the

lithosphere from the deeper mantle, where den-

sity anomalies are presum ed to persist for tens

or hundreds of millions of y ears.

13.5 Stress regimes of continents

and ocean floors

We now apply the arguments of Section 13.4 to

oceanic and continental lithosphere. Figure 13.2

is a schematic cross-section of the lithosphere for

the Africa plate from the Indian Ocean ridge

to the mid-Atlantic ridge. Applying Eq. (13.25)

to this situation,

 ¼



zdz þ

Rþe

ð

 

Þzdz þ

Rþeþt

Rþe

ð

 

Þzdz

Rþeþt

ð

 

Þzdz þ

ð

 

Þzdz

;

(13:28)

where depths and densities are as shown in

Fig. 13.4. By using average densities for each of

the layers the above integrals can be written

 ¼

ðz

iþ1

 z

Þ

; (13:29)

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with z

and D

being the depths and density

differences in the terms in Eq. (13.28).

For the comparison of the continent with the

ridge the isostatic constraint on the densities and

layer thicknesses is



þ 

ðz

 z

Þ¼

R þ 

t þ 

ðz

 R  e  tÞ:

(13:30)

Similar equations apply to the comparison of old

ocean and ridge. Table 13.1 gives numerical val-

ues for the application of this equation. Using

these values in Eq. (13.28) and the corresponding

equation for the old ocean-ridge comparison, we

obtain the stresses and crustal thicknesses listed

in Table 13.2 for various surface elevations, rela-

tive to sea level (including deep ocean). This is

a simplified model and the lithosphere is very

variable, so these results must be taken as a

guide and not a definitive statement on crustal

stresses, but they suffice to show what to expect.

We have assumed a flat Earth, with Cartesian

force balance, a perfect fluid at the centres of

ridges and a perfectly fluid asthenosphere. The

development as summarized in Table 13.2 refers

to ocean floors and continents adjacent to

spreading ridges with no intervening subduction

zones. Tensions are observed in areas of high

elevation, above about 2 km. Regions of lower

elevation (with negative D in Table 13.2), are

in compression. In this situation reverse faulting

is expected. This is consistent with the mecha-

nisms of earthquakes in old ocean areas (Wiens

and Stein, 1985). More elevated continental areas

are subjected to extensional stresses that cause

normal faulting.

The model results in Table 13.2 assume that

the depth of compensation, below which low

asthenospheric viscosity ensures effectively

hydrostatic conditions, is 100 km. If we assume

60 km instead, then the transition from com-

pressional stress to extensional stress occurs at

an elevation of 1 km, instead of 2 km, and con-

versely, compensation at 140 km depth raises

this transition to 3 km elevation. We can now

z = 0

+ e

+ e + t

= L+e = z

z = z

Ridge Old Ocean Continent

FIGURE 13.4 Layer thicknesses and densities for ocean

ridges, deep ocean and continents used in calculating

lithospheric stresses.

Table 13.1 Numerical values of layer

thicknesses and densities in Fig. 13.4, used

in calculating lithospheric stresses. R, D and

L are depths below sea level of the ridge,

deep ocean floor and base of the

lithosphere, respectively

R 2500 m 

1020 kg m

3

D 6000 m 

2800 kg m

3

L 100 km 

3300 kg m

3



3392 kg m

3

Table 13.2 Stress difference, stress ratio and

crustal thickness for different lithospheric

elevations. Crustal thickness, z

,is

estimated from isostatic balance for various

assumed values of surface elevation, e

Elevation

e (m) D ¼



 



/

(km)

6000 39.6 MPa 0.974 7.00

0 19.75 MPa 0.987 31.04

1000 10.33 MPa 0.993 36.78

1963 0 1.0 42.30

2000 0.43 MPa 1.0003 42.51

3000  12.51 MPa 1.0084 48.24

4000  25.88 MPa 1.0176 53.97

5000  40.49 MPa 1.0278 59.71

6000  56.3 MPa 1.0390 65.44

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consider what this means for major mountain

ranges (Himalaya and Andes) and high plateaux

(East Africa). In the East African rift zone we

have a large area of 2 km elevation and this is

consistent with the idea that higher crust falls

apart under the tensional stress calculated

in Table 13.2. The high Himalaya and Andes

require recognition of the tectonic forces that

are actively pushing them up, imposing addi-

tional compressive stresses, south–north in the

Himalaya (Kong and Bird, 1996) and west–east in

the Andes, causing reverse faulting in the foot-

hills. On the crests of these ranges normal faults,

that mark the collapse process, are transverse

to the ranges (Yin, 2000; Yuan et al., 2000; Zho

et al., 2001). The additional stress produces

higher mountains than are allowed in the quasi-

static situation assumed in Table 13.2. The high

Himalaya (Coblentz et al., 1998) and Andes (Lamb

and Davis, 2003) appear to arise from special

cases of plate convergence in which subduction

is inhibited over sections of these ranges by

increased friction from an absence of the normal

lubrication by wet marine sediment.

Since the compressive stress apparent in old

ocean, even as it approaches a subduction zone,

is presumed to be driven by push from a distant

ridge, the implication is that the asthenospheric

drag on oceanic plates is not significant, and

therefore that the asthenospheric viscosity is

very low. This is deduced also in Sections 13.2

and 13.4. It is probably less true under conti-

nents, which tend to ‘drag their feet’ in the

plate motions, but it is still possible to relate

the features of the world stress map (Fig. 11.11)

to the same forces (e.g. Richardson, 1992).

The emphasis on ridge push appears to rele-

gate subduction zones to a secondary role in the

generation of tectonic stresses, but this empha-

sis must be tempered by recognition that the

energy of thermal convection, which drives the

whole tectonic process, is derived from vertical

motion, and that this is most obvious in the

subduction zones. Tectonic stresses may appear

to be derived from the elevations of ocean ridges,

but this is due to buoyancy and cannot be iso-

lated from the upward displacement of hot man-

tle material by subduction of cooled lithosphere.

The ridge elevation above old ocean floor is a

measure of the lithospheric contraction that

gives the negative buoyancy driving subduction.

Loads on the lithosphere with dimensions

up to a few hundred kilometers can be supported

by elastic flexure and depart from local isostasy.

Such support has been called regional isostasy

in that the balancing force is spread over a

larger region than that occupied by the load.

For example, the Hawaiian island chain has

depressed the surrounding sea floor by several

kilometers. By modelling this as a load on an

elastic plate floating on an inviscid astheno-

sphere (e.g. Watts, 2001) and fitting either top-

ography or gravity to the model, an effective

thickness of the elastic plate can be determined.

Elastic thicknesses of about 30 km, no more

than 30% of the seismologically estimated litho-

spheric thickness, are inferred, as discussed in

Section 20.4.

13.6 Coulombic thrust wedges

Earthquakes in subduction zones occur at all

depths to about 700 km, but most of the major

ones are shallow, occurring in the upper few tens

of kilometres on shallow dipping (208) slabs.

Because of overburden pressure the mechanism

of frictional sliding of deep earthquakes requires

that either deviatoric stresses are extremely

large or, the more likely explanation, that the

effective friction is low because of increased pore

fluid pressures (Eq. 11.42). The second of these is

consistent with global models of stresses within

plates (Bird, 1998), which favour effective fric-

tion coefficients that have low values of about

0.1 to 0.2 compared with values determined

in the laboratory of 0.6–0.85 (Section 11.5). The

material in the seismogenic zone above a slab

undergoes brittle deformation that involves pil-

ing up of wedges of scraped-off sediments and in

some cases crustal deformation and thickening

in the overriding plate. In this section we exam-

ine the associated failure and stresses.

We consider the pile of sediments or crustal

rock that builds up above a subducted plate

using a simplified treatment of the analysis by

D. Davis et al. (1983). The sedimentary rocks form

a wedge of material, the so-called accretionary

13.6 COULOMBIC THRUST WEDGES 193

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wedge, which deforms continuously as it grows.

The form of the deformation includes folding of

the rock or thrusting where slabs of rock push

over other slabs along low-angle planes called

decollement surfaces. The time averaged behav-

iour of the fold and thrust belts, so formed, can

be treated as if the material is in a continuous

state of failure, according to Coulomb’s equation

(Eq. 11.41). Such a material is called a Coulombic

material. As the wedge develops, the slope of the

wedge topography adjusts to balance the stresses

within it. A bulldozer pushing a pile of sand up a

hill represents an analogous process. Initially the

sand deforms internally and the slope steepens,

until an equilibrium taper is reached. The wedge

then moves uphill and grows self-similarly as

more material is added at the toe. The critical

taper is the condition for which the wedge is on

the verge of failure everywhere, including slip

along its base. For the case of a wedge accreting

on a subducting plate, the dipping plate forms

the sloping interface, while the arc or over-riding

plate, which provides the compressive stress, 

in Fig. 13.5, corresponds to the bulldozer in the

sand-pile analogy.

We examine the process for a sub-aerial

wedge in order to relate frictional properties to

topographic slope, , and dip of the base, .The

effective friction depends on (1 l

)

where 

the coefficient of internal friction (Eqs. 11.41 and

11.42) and l

is the ratio of water to lithostatic

pressures. As an example, we apply the theory to

calculate the friction on the base of the wedge

formed by subduction of the Asia plate beneath

Taiwan, where  and  are known and l

in the

wedge is determined from pressure measure-

ments in boreholes.

Let x be measured along the base and z at

right angles (Fig. 13.5). Consider the forces on

an element dx of the wedge per unit length

in the y-direction. The gravitational force/length

resolved in the x-direction is gHdx sin . The

frictional force/length is 

dx.If

is the

normal stress acting across any face perpendi-

cular to the x-axis (as elsewhere in this chapter,

we assume the rock mechanics convention

with compression positive), the net force in

the x-direction is

d

ðxÞ

dz,whereH is the

thickness of the wedge at x. Balance of these

forces in the x-direction gives

gH sin  þ 

d

dz ¼ 0: (13:31)

We assume that the vertical stress arises from

the weight of the material. In the small angle

approximation for ,



¼ gðH  zÞ: (13:32)

Allowing for fluid pressure by introducing the

Hubbert–Rubey coefficient, as in Eqs. (11.41) and

(11.42), the effective normal traction becomes

+ β

σ'

sin

FIGURE 13.5 Geometry of an accretionary wedge. The gradient of an x -directed compressional stress in the wedge

balances the x-component of gravity and friction on the base.

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



¼ð1  l

ÞgðH  zÞ: (13:33)

The traction at the base is



¼ 





¼ 

ð1  l

ÞgH; (13:34)

and failure in the wedge gives



¼ 





¼ 

ð1  l

ÞgðH  zÞ; (13:35)

where 

is the coefficient of friction along the

base and l

are the Hubbert–Rubey ratios at

an arbitrary level in the wedge and at the base.

We treat the cohesionless case, that is, S

¼0in

Eq.(11.42).Forthebaseofthewedgetobeanon-

slip surface, the frictional stress must be less than

the stress required to cause internal deformation,

which means that ð1  l

Þ

ð1  l

Þ

Regional Coulombic failure means that any

depth within the wedge 

adjusts to be greater

than 

by an amount that generates sufficient

shear to cause failure. For a critical wedge H is

linear in x and so 

/ 

. Let



¼ A

: (13:36)

Using Eq. (13.32) and dH=dx ¼tanð þ Þ,

Eq. (13.31) becomes

gH sin  þ 

 AgH tanð þ Þ¼0; (13:37)

where the proportionality constant, A, is deter-

mined from the failure criterion as follows.

The assumption that the whole wedge is at

the point of failure means that, if we know A,we

can use Eq. (13.36) to obtain (



) ¼(A 1)

and substitute for 

, using Eq. (13.32), to obtain



. Consider the Mohr circle of Fig. 13.6(a). The

maximum and minimum effective compressive

stresses are 



and 



. Let the a ngle between

the axis of maximum compression 



and the x-

axis b e w. At the top of the wedge 

must be

parallel to the topography, so w ¼ þ and,

for shallow dips, we make the approximation

w(z) ¼constant, that is, w is independent of

depth. (The more general case applicable to

steeper dips is treated by D. Davis et al., 1983).

Since the wedge material is taken to be at fail-

ure at every point, the failure criterion defines

two planes that are at the point of failure and

are orie nted at angles (p/4 /2) with respect

to the 

axis (Fig. 13.6(b)). From triangle BCA in

Fig 13.6(a),

ð



 



Þ¼

ð



 



Þcos 2w; (13:38)

and from triangle OAC,

ð



 



Þ¼

ð



þ 



Þsin : (13:39)

Combining Eqs. (13.32), (13.33), (13.38) and (13.39),

ð



 



Þ¼

ð

 

Þ¼

ð1  l

Þ

csc  sec 2w  1

(13:40)

(

−

)/2

∗

π/4 − φ/2

(a)

(b)

(

∗

−

∗

)/2

xx zz

(

∗

−

∗

)/2

xx zz

∗

FIGURE 13.6 Mohr circles (a) within the wedge. The

geometry for the maximum principal stress, and failure

planes (dotted) in the wedge are shown in (b).

Philippine Sea plate

Eurasia plate

FIGURE 13.7 Application of

Coulombic failure theory to the

Taiwan accretionary wedge,

modified after D. Davis et al. (1983).

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Solving for A in Eq. (13.36),

A ¼ 1 þ

1  l

csc  sec 2w  1



: (13:41)

then, with substitution for 

by Eq. (13.34),

Eq. (13.37) becomes

 þ  ¼

ð1  l

Þ

þ 

1 þð1  l

Þ= csc  secð2 þ 2Þ1ðÞ

(13:42)

Then

¼1 fð þ Þ½1 þð1  l

Þ=

csc  secð2 þ 2Þ1ðÞg=

: ð13:43Þ

Thus, if we assume a laboratory value of



¼

¼0.85, the friction on the base can

be calculated from measurements of ,  and l

An exam ple of a pplication of this formula is

showninFig.13.7fortheTaiwanwedge.The

Asian plate is subducting beneath west Taiwan

forming a mountain belt that is described as a

Coulombic wedge. The su rface slope, ,hasa

mean value of about 3.08. Offshore seismic

reflection profiling has determined the slope 

of the decollement surface at the toe of t he

wedgetobe68, and it is presumed to remain

constant landwards. Fluid pressures measured

in deep boreholes indicate l

¼0.7. If fluid

pressure were hydrostatic, the value would be

about 0.4. Higher values mean that the fluid

is over-pressured; it has been trapped in t he

sediments and compressed as the formation

has been buried. We assume the laboratory val-

ues for friction coefficients (the Byerlee l aw,

Section 11.5), 

¼

¼0.85. This leaves one

parameter to be determined in Eq. (13.42), l

Substituting these values in Eq. (13.43) gives

¼0.85 for the base, which satisfies the

condition ð1  l

Þ

ð1  l

Þ

, that is the

wedge material has a larger effective friction

coefficient (Eq. 11.38), 



¼ð1  l

Þ

¼ 0:25,

than the base 



¼ð1  l

Þ

¼ 0:13. This is

consistent w ith the global value of 0.1 to 0.2,

mentioned in the opening p aragraph of this

section.

In another example, the importance of var-

iable friction is seen in the greater topographic

slope of the central high Andes compared with

the southern and northern low Andes. Because

of the dry climate, the trench in front of the

high Andes is almost devoid of sediment,

whereas that to the north and south has abun-

dant sediments. The main source of water at the

wedge base and in the wedge itself is thought to

come from dehydration of subducted sediments.

Greater effective friction on the dry slab can

account for the larger wedge and elevation of

the central Andes and its cumulative effect over

time accounts for the indentation of South

America (Problem 13.7).

The critical wedge theory (using a more

advanced treatmen t than described here) has

been applied to fold and thrust belts and acc-

retionary wedges worldwide, including the

Himalaya, the fold and thrust belts on the land-

ward side of the Andes, western North America

(Horton, 19 99) and the Zagros mountains of

Iran (Bird, 1978). It is used to examine the

interrelated effects of pore pressure, failure,

sedimentation, erosion and thrusting. Near

the toe of the w edge, the base slips steadily,

but at greater depths, in the seismogenic zone,

slip occurs quasi-periodically in large earth-

quakes. Stress that causes sliding along the

base is given by Eq. (13.34), which, for the

Taiwan case, becomes 

¼0.13 gH. At a depth

of 20 km this is 76 MPa, that is an average

deviatoric stress of 38 MPa over this depth

range. This value is comparable to the internal

deviatoric stress in old ocean derived from ridge

push, 37 MPa (Table 13.2), but is more than two

orders of magnitude higher than the astheno-

spheric shear stresses (0.1 MPa) calculated in

Section 13.2.

The magnitude of deviatoric stress, , in slabs

is limited by the convective energy allowed by

the thermodynamic arguments in Section 22.4.

The analysis in this section demonstrates the

importance of pore pressure for frictional slid-

ing at depth. For Taiwan we find l

¼l

¼0.85

compared with 0.7 in the wedge itself. At greater

depths l

must increase to near 1.0 if frictional

siding is the mechanism of deep earthquakes,

so that  

(1 l

)

. Otherwise overburden

pressure would preclude earthquakes at depths

greater than a few tens of kilometres.

196 CONVECTIVE AND TECTONIC STRESSES