Download Notes on Continental Lithosphere

Notes on Continental Lithosphere
Definitions:
First, crust. This is easier than most, as in most places it is the same. Seismologists define Moho
from steep P wavespeed gradient near 7.5 km/s (this produces a refracted arrival, termed Pn, that
was basis for Moho in the first place). Petrologists usually like crust to be material derived from
the mantle by melting, where the residuum is in the mantle. Occasionally you can have such
material be eclogitic (due to intracrustal differentiation), which is seismologically mantle but
petrologically crust. Otherwise both definitions work well together.
In contrast, lithosphere is really vague.
Petrologically: Upper mantle isolated from convective processes (so having distinct trace
element and isotope signatures).
Seismologically: Material above low-velocity zone (depth where velocities become lower with
depth). LVZ is well defined in oceans, but difficult (often) in continents. Usually best
developed as an S-wave LVZ. This is often best measured with surface waves.
Thermally: Thermal boundary layer is sometimes called “lithosphere”. Because the thermal
boundary layer is not convecting, this often comes close to the petrologic definition if maintained
over a long time. It is defined as where you go from a conductive to a convective (adiabatic)
geotherm.
Mechanically. This is actually where the definition started, but even it is somewhat vague.
Lithosphere has higher stresses than asthenosphere below; it is in this manner a viscosity
contrast. One definition is that the lithosphere can transmit (shear) stresses over geologic time.
Thus the depth of compensation for isostasy is the asthenosphere through this definition. An
extreme version of this is the thickness of an elastic plate, which by definition retains stresses
over an infinitely long time. Such thicknesses are typically tens to maybe just over 100 km.
Most workers do not take such thicknesses literally.
Basic observations: Lang will talk about petrologic constraints. Seismologically, we rely
mostly on surface waves. This is largely because surface waves have a strong dependence on
depth, as was illustrated in class. It is also because boday waves generated at the surface will not
yield a refraction from a low-velocity zone; they might yield a reflection (depending on how
abrupt the velocity reversal is) and they will show a “shadow zone” suh as we discussed from the
outer core. Neither of these observations yields much information on the velocities within the
low velocity zone, particularly for the continents, where lateral variations in structure can
complicate the interpretation of these phenomena.
Generally, the low-velocity zone is found about 80-100 km down under oceanic crust and 200
km or more under cratons, with continental margins having a variety of possible thicknesses.
Thermal lithosphere
A key here is, what is the thermal structure of the lithosphere? We’d like to know where the
geotherm switches from the more conductive shallow geotherm to the mantle adiabatic
geotherm. As we have seen, the changes in the thermal structure of the lithosphere are
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.2
responsible for the topography of most of the ocean basins. Is there something equivalent in
continents? To address these, we consider heat in continental lithosphere
Our observations are directly made only at the surface. The conductive heat flow q is
dT
, where T is the temperature with depth and k is the coefficient of thermal conductivity.
q=k
dz
For typical crustal rocks, k is 2-3 W m-1 °C-1. In most areas dT/dz = 20 – 30 °C/km, yielding a
heat flow of 40-90 mW m-2. Another unit currently falling out of favor is the heat flow unit
(HFU): 1 HFU = 41.84 mW m-2. The continental average heat flow is about 65±1.6 mW m–2; for
oceans it is 101±2.2 mW m-2 (from Turcotte and Schubert’s 2nd edition).
Actual measurements of the heat flow are usually made in boreholes. Difficulties that have
to be overcome include thermal transients associated with drilling the hole, hydrological effects
(hot water bringing heat in or cold water taking heat out), and topographic effects. Most of the
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
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practitioners of heat flow are well aware of these problems and usually will have avoided holes
that will not yield reasonably representative values, but it is well to be aware of them as
occasionally maps of thermal gradients are presented that include a lot of bad holes.
In the ocean basins, where the concentration of radioactive elements is not so great and not
very variable, we could ignore it in plotting heat flow vs. age. On continents, which, as we have
seen, are global repositories for things like potassium, thorium , and uranium, we cannot.
Heat in the Earth comes both from heat being transferred out of the mantle (the end result of
both gravitational energy from formation and differentiation of the core and radioactive decay in
the Earth's interior) and from radioactive decay in the crust. Locally there can be heat sinks or
sources from chemical reactions and shear heating; in a global sense, these are unimportant (see
sections 3.2.2 and 3.2.3 of Stüwe for details) but can be important in the vicinity of magmas and
large shear zones. Because most radioisotopes are concentrated in the crust, this term needs to be
considered before we can infer the thermal structure of the deeper levels of the lithosphere.
It turns out, somewhat surprisingly, that in many areas the surface heat flow is linearly
related to the heat production of the surface rocks (measured as a volumetric pruduction in µW
m-3 or by mass as µW kg-1). The Sierra Nevada turns out to be a classic example:
The linear fit allows us to remove the effect of heat production in the crust by seeing where the
line hits the axis of 0 heat production; this value is called the reduced heat flow qr. This is
frequently interpreted to be the heat coming from the mantle, but this is in fact the very
uppermost bound on what
can be coming from the
mantle. First, a constant
contribution from
radioactivity in the lower
crust is likely present.
Second, the lateral variations
in radioactivity will cause
heat to flow laterally
(Jaupart, C., Horizontal heattransfer due to radioactivity
contrasts - causes and
consequences of the linear
heat-flow relation,
Geophysical Journal of the
Royal Astronomical Society
75 (2): 411-435 1983). We
shall return to this point after
considering the onedimensional case.
The very simplest
explanation for the linear
relationship of heat flow to
heat production is that the
radiogenic isotopes are in a
layer of some thickness D. In
this case, qo = qr + DA, where
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
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A is the volumetric heat production. The slope of the line above then determines the thickness
of the slab. An alternative that will produce the same result is if the heat production decreases
exponentially with depth (discussed in Saltus and Lachenbruch's Sierra Nevada paper and in
Turcotte and Schubert, pp. 145-147). While the Sierra and many parts of the eastern U.S. have
the linear behavior with heat production, other areas do not, including the Rocky Mtns. and the
Basin and Range.
The geotherm in this case requires a new term. If we are at steady state, so there is no
change in temperature, then the heat flow out of a volume of rock has to equal that in plus the
heat produced in the rock. If heat is flowing upward along the z axis and our volume is a slab of
thickness dz with a heat production A (expressed as W m-3), then we find
q( z − dz) = q( z) + A dz
(1)
dq
= −A
dz
dT
We have made a mild change in definitions so that q = k
for z positive down but heat flow
dz
€
positive up, so
dq d  dT 
d 2T
=  k €  = k 2 = −A
dz dz  dz 
dz
(2)
Over a slab where A and k are constant, we can simply integrate this to get
€
k
dT
= −Az + c1
dz
(3)
This is the heat flow, and if z= 0 is the surface, then c1 = qo, the observed surface heat flow. We
integrate again to get (within the radioactive body)
€
A
q
(4)
T = − z 2 + o z + Ts
2k
k
where the second constant of integration falls out easily as the temperature at the surface. This is
a very helpful equation, though, as we shall see, it has had some misleading implications. We
can rewrite this in terms€of the heat flow entering from below the slab, as we noted that q0 = qr +
DA:
A
q
z(2D − z) + r z + Ts
2k
k
A 2 qr
=
D + z + Ts
2k
k
T=
z<D
(4a)
z>D
Note that the two righthand terms are the linear conductive geotherm in the crust without
radioactivity. Thus this would require the temperatures to be hotter beneath the radioactive
material than to€the sides.
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
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Although useful, the derivation above has a problem encountered in many places, such as
the Sierra, is that the depth of erosion of plutons is variable, suggesting that the thickness D
should vary, but the heat flows still plot on the nice regression line. Although there are a number
of possible solutions, an elegant one suggested by Art Lachenbruch some years ago is an
exponential decay of heat, A = A0e−z / hr , where hr is a length scale of decay of radioactive heat
production with depth. We can redo the integral from (2) to (3):
€
q=k
dT
= hr A + c1
dz
(5)
In this case, we might specify the heat flow at great depth be a constraint, which we will
designate qm, and so c1 = qm. Thus we see that q will depend linearly on A independent of the
€
depth z, satisfying our observations
in some areas. We also get (once again) that the heat flow at
the surface when A equals zero is the heat flow at depth. Another integration yields
hr 2
q
T =−
A0e−z / hr + m z + c 2
k
k
2
h
q
= r A0 (1− e−z / hr ) + m z + Ts
k
k
q
(q − q )h
= Ts + m z + 0 m r (1− e−z / hr )
k
k
(6)
This alternative geotherm isn’t very different from what we would get with a slab, but the
geotherms will diverge with greater depth such that an exponential decay will have higher
€ the slab below the slab thickness.
temperatures than
Neither a slab of constant heat production nor an exponential decay of heat production is
consistent with what we observe in the field (usually it seems to be somewhere in between).
These two models provide us some bounds for the shallow subsurface and for greater depths
under certain circumstances.
Real heat flow from the mantle
We return to what the real deep thermal structure looks like. This can most easily be seen by
calculating one-dimensional geotherms from the surface down assuming a constant deep heat
flow (qr) but differing heat productions in a slab using equation (4):
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
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Sierran geotherms
-2
A = 0.3
A=4
A=2
0
A = 0.3, q =2q (today)
r
r
A = 4, qr=2qr(today)
A = 2, qr=2qr(today)
Depth (km)
2
4
6
8
10
12
0
20
40
60
80
100
120
Temperature Above Surface Temp (°C)
As you go down from a fixed surface temperature, the geotherms diverge, and while they
bend back to being parallel as you emerge from the bottom of the radioactive part of the crust
(see eqn 4a), the temperatures would seem to be higher under the more radioactive crust. At first
blush, this would seem to require the lithosphere to vary in thickness with variations in heat
production, which is really rather ridiculous unless the length scales of radioactive heat
production are on the order of the thickness of the lithosphere. For much shorter variations
(which is true in the Sierra to a large degree) we suspect that the heat from the mantle under high
heat producing regions will instead flow into the colder regions under lower heat producing
regions. This means that the actual heat being produced under a given measurement is actually
lower for low heat producing areas and higher for high heat producing areas:
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
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Thus the reduced heat flow is no better than the maximum heat flow from the mantle. Mike
Sandiford has made some nice plots of this effect
(http://jaeger.earthsci.unimelb.edu.au/msandifo/Essays/Hansen/Hansen.html). The summary is
illustrated here:
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.8
Here the lateral length scale of variations in heat production is hx and the actual heat flow from
depth is 20 mW m-3 (the dashed line is the relationship you would expect if there was no lateral
conduction of heat).
From this analysis, the thickness of the lithosphere calculated from a conductive
geotherm assuming a heat flow equal to the reduced heat flow will be too thin. What you really
need is the properly averaged heat production over the region (notice that the dashed lines cross
the “observed” curves above at some point likely to represent the average heat production) with a
proper (lower) mantle heat flow.
Assumptions, assumptions, assumptions
The discussion above includes a number of assumptions we should be very careful about.
One is that the thermal conductivity k is constant with depth. This is not true in the mantle (there
is a depth dependence) and it is not true for crustal rocks. Most notably, this is not true for
shales, which tend to have a lower k than most other crustal rocks. This means that a pile of
shale will act as a blanket, and temperatures will tend to be higher under them than other rocks.
It turns out this is a big issue in interpreting fission track ages in the Rocky Mountains.
Another huge assumption is that things are in equilibrium. You can go out of equilibrium
in two easy ways: change the temperature or heat flow at the bottom of the lithosphere, or add or
remove material from the top by sedimentation or erosion, from the middle through tectonism, or
from the bottom through convective processes.
The final big assumption is that heat is conducted. In fact, within the crust there are
processes like hydrothermal systems, regional aquifers, and magmatic systems that all are
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.9
capable of moving heat through advection. We will not explicitly consider this issue here, but
you should be aware that these can be largescale effects. For instance, low heat flow over much
of the Colorado Plateau has been attributed to water flow transporting heat sideways in aquifers
such as the Coconino sandstone; a similar anomaly is along the Snake River Plain. Also, we
previously mentioned that heat flow measurements in oceans were highly scattered towards the
ridges because of geothermal systems.
Thermal disequilibrium
The plot below from Saltus and Lachenbruch (1991) helps us understand this:
The idea is that if a new fixed temperature is emplaced at some depth H, then the heat flow
with time will change as shown by the "∆T at boundary" curve above. If instead of a fixed
temperature we change the heat flux, then the surface heat flow will follow the "∆q at boundary"
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.10
curve, where the original heat flow is at the 0 position on the y-axis and the new steady-state is at
100. The two extra axes below are for emplacement of a body at 5 km depth and for the depth of
a body emplaced 20 m.y. ago and producing the changes. Using the bottom curve, for instance,
if we suspect that the surface heat flow is exactly halfway from old values to new, and we
believe this was caused by a change in heat flow, the depth of that change would be about 40 km.
Thus we can see that changes to the thermal structure at the base of the lithosphere take tens of
millions of years to show up at the surface. Heat flow is in fact the only geophysical observable
that is more a measure of past conditions than present ones.
Mechanical lithosphere
As we shall see, the definition of mechanical lithosphere is fraught with ambiguity, but
let us start with a simple conceptualization and move forward from there. A useful
approximation to the lithosphere is an elastic plate over a fluid substrate. There is a well
developed theory from engineering for the behavior of plates, and Earth scientists have only had
to modify it a little to produce something of use. The full derivation can be found in Chapter 3
of Turcotte and Schubert; we will begin with the general equation in two dimensions (x and z)
that applies to plates
D
d 4w
d 2w
+
P
+ ρ a − ρ f gw = qa ( x )
dx 4
dx 2
(
)
(7)
where D is the flexural rigidity, P is the force parallel to the plate, ρa is the density of the
underlying fluid, ρw is the density of material filling in the top of the plate (water if in the oceans,
air or sediments on land, depending on the situation), w is the deflection downwards of the plate,
and qa is the load applied to the plate. Basically, when you plop a load on a plate, it doesn’t go
into local isostatic equilibrium, it distributed the load by bending. The broader it distributed the
load, the stronger the plate. If the plate is of a constant Young’s modulus E, Poisson’s ratio ν,
and a thickness Te, then
D≡
ETe 3
12(1− ν 2 )
(8)
Periodic topography (Turcotte and Schubert, sec. 3-14)
The simplest place to start is with a solution that requires little effort and yet gives us our
€ where isostasy takes over from rigidity. Consider some region
first real insights into the scales
where topography is sinusoidal with x such that the elevation e(x) = e0sin (2πx/λ) and does not
vary with y. The load on the lithosphere, qa, is then the variation of weight that accompanies this
deflection:
x
(9)
qa (x) = ρ c ge0 sin2π
λ
where ρc is the density of the crust associated with the height variation. If we now assume there
is no end load pressure P, we find in eqn. (7) that
D
d 4w
x
4 + ( ρ a − ρ c )gw = ρ cge0 sin2π
dx
λ
(10)
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.11
Note that we use a fill density of ρc, which means that our applied load is added on a level upper
surface and so is the final (observed) topography of the system. Given the periodic nature of the
load, we would expect that the deflection of the lithosphere would also be periodic. Let us guess
a solution of the form
x
(11)
w(x) = w0 sin2π
λ
Putting this into equation 10 produces
 2π  4
x
x
x
Dw 0   sin2π + ( ρ a − ρ c )gw 0 sin2π = ρ c ge0 sin2π
 λ
λ
λ
λ
  2π  4

w 0 D  + ( ρ a − ρ c )g = ρ c ge0
  λ

e0
w0 =
4
D  2π  ρ a
−1
  +
gρ c  λ 
ρc
(12)
We notice two endmembers: one where the wavelength is very small, one where it is very large.
 D 1 / 4
 ), the first term in the denominator dominates
When€λ is small (much smaller than 2π 
 gρ 
c
and w0 approaches zero. This is the familiar case of the rigid earth not yielding beneath a
landfill; the wavelength of the landfill turns out to be much smaller than the flexural strength of
the lithosphere. In the other extreme, as λ goes to infinity (at least much greater than
 D 1 / 4
 ) then the first term drops out and we are left with w0 = e0 ρc/(ρa - ρc), which is the
2π 
 gρc 
equation for local, Airy isostatic support of mountains. (For references, you might note that the
factor ρc/(ρa - ρc) is how much more the Moho is perturbed for a given topographic signal; it
varies from about 4 to 8, so 1 km high mountains should have 4-8 km deeper Mohos than
adjacent lowlands). For values of the wavelength in between, the topography is partially
compensated; this is illustrated in Figure 3-26 of Turcotte and Schubert.
The sinusoidal solution can be the basis for a Fourier expansion of the response to a load
and is one potential strategy for solving more complex loads. The wavelength where support for
topography transitions from rigid to isostatic depends on the flexural rigidity D of the plate, and
determining this transition in the basis for so-called admittance analysis of gravity (a proxy for
the deflection w of the Moho) against topography that has been very successful in revealing the
rigidity of oceanic lithosphere. For the cases of greatest interest here, there are simpler solutions.
Linear load on unbroken lithosphere (sec. 3-16)
Things are somewhat more complicated when we make our load finite. Let us start with a simple
load isolated along a line; let the magnitude of this force be V0 at x = 0. Elsewhere the load is
zero. We once again let the end load be negligible and wish to solve (7) everywhere but x = 0
and thus face
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
D
d 4w
+ ( ρ a − ρ w )gw = 0
dx 4
p.12
(13)
There is a general solution to this equation:
x
x
x
x


w = e x /α  c1 cos + c2 sin  + e − x / α  c3 cos + c 4 sin 
α
α
α
α
(14)
where the flexural parameter α is defined as
1/4

4D


α=
 g(ρa − ρ w ) 
(15)
There is a natural symmetry about x = 0, so we can limit our efforts for x > 0. For distances far
from the load, we expect the deflection to go to 0; this causes the positive exponent terms to be
dropped (c1 and c2 = 0). Because our plate is continuous and symmetric, the slope = dw/dx must
be zero at x = 0. This in turn forces c3 = c4.
The last constant is related to the magnitude of our load V0. We can show that V = dM/dx, which
is
V=
− Eh 3 d 3w
d 3w
=
−
D
dx 3
12 1 − ν 2 dx 3
(
)
(16)
Since our load is pointing down, and since we are solving the half of this for x > 0, we find that
V0 4Dc3
(17)
=
2
α3
We can now return to (14) and find our solution to be
x
x

w = w0 e − x /α  cos + sin 
α
α
(18)
where our maximum deflection w0 is at x = 0 and is
V0α 3
w0 =
8D
(19)
There are several characteristics of this solution worth examination. First is to note that
α does not depend on V0. Thus an increase in the load does not alter the shape of the solution,
only its amplitude. This is particularly noteworthy because the solution varies about w = 0:
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.13
Here the vertical axis is w/w0 and the horizontal is x/α. The crossing of w = 0 at x = 0.75πα
separates the depression from the forebulge. The position of this crossing is unaffected by
changing w0 (or V0); thus the position of the forebulge is only affected by the position of the load
and α, which depends on the physical characteristics of the lithosphere. Note that the forebulge
is in general much smaller than the depression itself, only reaching a maximum amplitude just
under 5% of the main depression. (Also note that α depends on the density contrast of the
infilling material and the asthenosphere--what happens when a sediment starved basin is filled?).
Flexure of a broken plate (sec. 3-17)
The final instance we are interested in is if the plate is broken. Strictly speaking this
simply doesn't happen--there is never an open edge to a lithospheric plate. But once again this
kind of approximation gives us insight to situations where the strength of the lithosphere might
vary dramatically where the load is applied.
In this instance we no longer require that the slope of the plate be flat at x = 0; we replace
this with the requirement that the moment be zero on the end of the plate (there is no source of
moment in our initial analysis). This means that we require that the second derivative of w with
x is 0 at x = 0. Returning to eqn (23) we now find that c1, c2, and c4 must be zero. Repeating the
exercise that went into equations (25) and (26), we find that
V0 2Dc3
(29)
=
2
α3
Plugging back in, we find
w = w0 e − x /α cos
x
α
(30)
where
w0 =
V0α 3
4D
(31)
Plotting on the same horizontal scale as above (and with w/w0 with the new w0), we see several
things.
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.14
As before, the zero crossings and forebulge positions rely on α exclusively. The zero
crossing has moved in considerably closer to the load for a given α, as has the forebulge. The
amplitude of the forebulge is now nearly 7% of the maximum depression (which itself is twice
the amplitude as for the unbroken plate). When trying to fit a given deflection profile using a
broken plate will require about a 50% thicker effective elastic plate thickness than fitting the
same profile with a continuous plate. The largest difference in the shapes of the plots is near the
load, which is where the assumption of a load only at x = 0 breaks down.
Additional variations are possible where a moment is placed on the end of the elastic
plate; this is covered in detail in Turcotte and Schubert; it is most applicable when a large
subsurface load (usually a subducted plate) is present. Use of this formulation predicts the
observed topography at ocean trenches quite well, as well as the deflection of (unfaulted)
continental sedimentary basins, as we shall discuss further.
Application to lithospheric thickness
Ref: Forsyth, D. W., Subsurface loading and estimates of the flexural rigidity of
continental lithosphere, J. Geophys. Res., 90, 12,623-12,632, 1985.
There are two basic classes of approaches to measuring the flexural rigidity of the
lithosphere, one a spectral approach and the other a forward modeling approach. In the first, we
use the sinusoidal solution from (11) and (12). The first attempt at this used what is generally
termed an admittance calculation. This assumed that the topography at the surface was the load
on the plate, and that the deflection of the plate could be estimated from the Bouguer gravity
anomaly. In this case, if you spectrally divide the Bouguer anomaly by the topography, you will
get an admittance Q(ω). As the wavelength gets longer, there should be perfect correlation of
the two signals and Q goes to 1. When the topography is supported by the strength of the
lithosphere with essentially no deflection, it drops to zero. How the shift occurs determines the
elastic plate thickness. This approach worked well in the oceans, where most of the shorter
wavelength topography is due to volcanic edifices. However, when applied to continents the
result seemed silly: continental lithosphere had values of Te of about 5 to 10 km, much less than
the thickness over which earthquakes are found to occur. The problem is that there are loads
applied at depth that may not produce much topography.
A second spectral approach developed by Forsyth allows loads to be both in the
subsurface and the surface. In this case it is the coherence between the topography and the
gravity that is exploited; the more coherent these two signals (which is a value of 1), the more
the plate acts as though isostasy was local. The less coherent (a value of 0), the closer to a rigid
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.15
plate. Again, the transition determines the elastic plate thickness, but the problem of determining
the load location is mitigated. This approach yields numbers more in line with expectations.
Forward modeling requires that a load be applied in some known manner and then
examining the resulting flexure. These often are of the form of the line load solutions (above). It
has the advantage of avoiding some assumptions in all the spectral techniques, but can only be
applied in some areas. Foredeeps, subduction zones, and moats around island chains are all good
examples where it has been applied.
Strength of the lithosphere
References:
Brace and Kohlstedt, Limits on lithospheric stress imposed by laboratory experiments, J.
Geophys. Res., 85, 6248-6252, 1980.
Turcotte and Schubert, Geodynamics, chapter 7 (and 8 to some degree).
Rheological layering of the lithosphere.
There appear to be two fundamental modes of deformation of rocks: brittle failure and ductile
flow. The former is related to earthquakes, which represent such failure; the latter should not
generate earthquakes. The boundary between the two regimes is termed the brittle-ductile
transition. Although it is often represented as a major discontinuity within the Earth, it is in fact
most probably a broad zone within which a number of different mechanisms of rock deformation
occur. Somewhere within this zone should be the base of that part of the lithosphere that
generates earthquakes (the seismogenic layer).
Our understanding of deformation within the Earth is largely based upon laboratory
experiments and, to a lesser degree, observations of naturally deformed rocks. Lab experiments
are necessarily over a much shorter time than actual rock deformation and are of a smaller scale;
these substantial differences suggest that some caution must be exercised when employing these
observations. (One possible example of the lab being too small has been suggested for the
frictional behavior of faults during an earthquake. A theoretical mode of motion on a fault
would cause the effective frictional resistance of the fault to go to zero during rupture. This
mode is observed in experiments with foam rubber blocks but not with rocks. It turns out that to
look for this kind of deformation in rocks would require an apparatus with blocks of rock several
meters on a side, an experiment not yet conducted).
Brittle upper crust
An examination of rock properties reveals a large variation in rock strength with lithology.
This would suggest that lithology would be the most important factor in determining the strength
of rocks. However, experiments have revealed that this variation is almost entirely restricted to
the effort to break a rock; once broken, the frictional behavior of a fault remains pretty uniform
from lithology to lithology. What is more, this behavior depends almost entirely on the effective
pressure on the fault, with little dependence on temperature or strain rate. This observation is
termed Byerlee's Law and is expressed as a relationship of the shear stress and effective normal
stress on a fault. This can be reformed as a relationship between the principal stresses:
σ1 ≅ 5 σ 3
σ 3 < 110 MPa
σ1 ≅ 3.1σ 3 + 210
σ 3 > 110 MPa
(1)
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.16
These stresses are effective stresses, meaning that the pore pressure has been subtracted from
them ( σ n = σ n − Pp , where Pp is the pore pressure and n is between 1 and 3).
Eqn. 1 can be rewritten for compression (where σ3 is the vertical stress, ρgz) as
σ1 − σ 3 ≅ 4 (ρ − λρ w )gz
(ρ − λρw )gz < 110 MPa
σ1 − σ 3 ≅ 2.1( ρ − λρ w ) gz + 210 (ρ − λρw )gz > 110 MPa
(2)
where λ is the fraction of hydrostatic pore pressure and ρw is the density of the pore fluid. A
similar manipulation can be done for the extensional case by setting σ1 to the weight of the rock
above.
This makes pore pressure one large uncertainty in determining the strength of the brittle part
of the lithosphere. Observations down to a couple of kilometers depth in continents suggest that
a pore pressure between 0 (dry) and hydrostatic (Pp = ρw gz) are probably about correct for much
continental crust. If you recall, there is some evidence in accretionary wedges and possibly foldand-thrust belts for higher pore pressures in these environments.
Ductile lower lithosphere
At higher pressures, rocks begin to flow ductilely. In this environment, there is no longer a
maximum strength per se. Instead, there is a certain stress that will cause flow at a certain rate,
usually expressed as a strain rate ε˙ . Drop below that stress, and deformation is at a slower rate.
Increase the stress, and deformation will go faster. Unlike frictional slip, the relations here
depend on the rock type involved, the temperature, and the stress difference σ1 - σ3. The general
form of this is a power-law flow:
n −E
ε˙ = C(σ 1 − σ 3 ) e a
RT
(4)
For olivine (from Goetze, 1978), C is about 7 x 104, n is 3, Ea (the activation energy) is 0.52
MJ/mol, and R (the ideal (or universal) gas constant) is 8.317 J K-1 mole-1. T is in Kelvin. (For
dry quartzite, Ea is 0.19 MJ/mol, n is 3, and C is about 5 x 10-6 (not 106 as shown in Brace and
Kohlstedt).
For purposes of strength of the lithosphere, we may rewrite (4) in terms of a stress difference
as a function of temperature:
1/n
 ε˙e E a RT 
σ1 − σ 3 = 

 C 
(5)
If we assume a linear change of temperature with depth, T = A + bz, then we find a fairly
rapid decrease in strength with depth for a given strain rate. (Goetze's work indicates a lower
differential stress than this equation at high differential stresses; we will use this form for
illustrative purposes, but you'd want to use proper values for any research). We can combine (2)
and (5), taking the minimum of the two, to get a feel for the strength of the lithosphere:
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
Compressional strength
Compressional strength, varying strain rate
0
0
20
20
40
40
depth, km
depth, km
p.17
60
15°/km, total strength 6.6 10
20°/km, total strength 4.0 10
25°/km, total strength 2.7 10
80
10-16/s
10-15/
10-14/
60
13
N/m
N/m
13
N/m
13
80
100
100
120
120
0
500
1000
1500
2000
2500
3000
0
500
1000
σ1 - σ3, MPa
1500
2000
2500
3000
σ1 - σ3, MPa
depth, km
Note that these curves are for a pore pressure of zero (dry rock) in compression.
The upper part of the lithosphere fails as a brittle/elastic medium, the lower part by creep,
which fails aseismically. Thus the boundary between the two is often termed the brittle-ductile
transition and is often considered the base of the seismogenic part of the crust. Note that this
transition moves upward in response to higher temperatures. This variation in the brittle-ductile
transition has been observed, at least crudely, around the globe in continental areas.
The figure on the left shows the variation in strength for a strain rate of 10-15 s-1; that on the
right shows a variation for several strain rates with a fixed temperature increase of 20°C/km.
Note that the change in strain rate over two orders of magnitude is fairly small compared to the
change caused by temperature variations. The area under each curve is the total strength of the
lithosphere, meaning that deviatoric horizontal forces equal to that area would have to be applied
to get the entire lithosphere to strain at the rate assumed. The strengths for the three curves on the
left are listed in the caption.
The curves above are for a
Compression, strain rate 10-15 s-1 , quartzite over olivene
monolithologic lithosphere made
0
entirely of olivine, perhaps a good
first approximation for the oceanic
20
lithosphere. We know that the
continental lithosphere is stratified,
40
with more quartz-rich rocks above the
Moho and ultramafic rocks below.
60
Thus we would like to see the effect
of this stratification, if any. Using the
25°/km, total strength 0.55 10 N/m
80
20°/km, total strength 0.89 10 N/m
flow laws for quartzite above a depth
15°/km, total strength 3.6 10 N/m
of 35 km and that for olivine below,
100
we can redo the left figure above for a
stratified lithosphere (at right).
120
Because quartzite is quite weak at
0
500
1000
1500
2000
2500
3000
much lower temperatures than olivine,
σ - σ , MPa
the net effect is that lithosphere
13
13
13
1
3
Physics and Chemistry of the Solid Earth-Continental Lithosphere Geophysics
p.18
should be considerably weaker with quartz present in the crust than when it is not. Once again
we can integrate the curves to get the gross strength of the lithosphere and we find it has
decreased by 50-80% for the case above.
It is important to remember that these are extrapolations of laboratory measurements; the
constants used, not to mention the stress exponent n, are subject to considerable uncertainty.
Additionally, the brittle-ductile transition itself probably deforms by neither of the mechanisms
assumed above but instead by other modes of failure, making the lithosphere somewhat weaker
at these depths. The general effects of temperature in weakening the lithosphere, in strain rate
mildly strengthening the lithosphere, and in quartz weakening the lithosphere are probably solid
results. Understanding these variations of strength allows us to consider the role of strength and
force in continental deformation.