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Introduction to
Tectonophysics
Patrice F. Rey
CHAPTER 1
The Earth’s Geotherm
The geotherm, i.e. the distribution of temperature with depth, is an important characteristics of the Earth's lithosphere
because temperature impacts on all physical properties of rocks (e.g. density, viscosity, conductivity, elasticity,
magnetism etc). In particular, temperature controls the rheology of rocks and therefore how they deform in response to
applied deviatoric stresses, and how the Earth's lithosphere reacts to tectonic forces. In this chapter we derive, from first
principles, a simple expression for the geotherm.
1
SECTION 1
Heat Transfer in the Earth’s lithosphere
In this section
1. Heat transfer in the Earth’s
Lithosphere
2. Heat energy and
temperature
3. Heat conduction
4. Heat advection
5. Radiogenic heat production
The continental geotherm is a function of the i/ rate at which
heat is produced or consumed within the lithosphere, ii/ the
rate at which the lithosphere looses heat to the atmosphere/
ocean system, and iii/ the rate at which the lithosphere gains
heat from the hot convective mantle. When the heat lost by the
lithosphere balances the heat gained by the lithosphere, an equilibrium is reached and the geotherm is said to be steady state
2
(i.e. the temperature at any given depth does not change
through time). In contrast, when the lithosphere has a net gain
or a net loss of heat, the geotherm is said to be transient (i.e the
temperature changes through time) until a new equilibrium is
reached. On a billion year time scale, the geotherm is always
transient because the primordial accretionary heat and the
Earth’s supply in radiogenic isotopes progressively decrease,
•Heat conduction (transfer of kinetic energy between molecules
or atoms from a hot to a less hot region)
•Heat advection (replacement of a volume of rock at temperature T1 with an equivalent volume at temperature T2)
•Heat production (heat produced by radioactive isotopes, viscous heating, exothermic metamorphic reactions)
• Heat consumption (heat consumed by endothermic metamorphic reactions, in particular partial melting)
however, on a scale of 100 myr, and in the absence of tectonic activity, the geotherm can approach an equilibrium which expresses the
balance between heat gained and heat lost by the lithosphere.
Here, we first review the processes involved in heat generation and
heat transfer, and we derive from the rate of these processes a general equation which describes the change in temperature with
depth and through time. From this general equation we derive a
particular solution for the so called "steady state" continental geotherm (temperature changes with depth but not with time, i.e. zero
net heat gain or loss). In a second part, we discuss how the steady
state continental geotherm is affected by a number of geological
processes including, lithospheric thinning and thickening, burial
via sedimentary or volcanic processes, and basal heating via the
spreading of mantle plumes at the base of the Earth's lithosphere.
The variation of temperature dT over an increment of time dt depends on the sum of heat variations dE due to each process. In
what follows, we derive three expressions for i/ the rate of heat
conduction, ii/ for the rate of heat advection, and iii/ for the rate
of radiogenic heating. From these, we derive the 1D conductionadvection heat transfer equation from which an expression for the
steady state geotherm can be derived. Sounds more complicated
than it really is. So bear with me ...
Temperature and Heat
The temperature (degree of hotness or coldest) of a small volume
of rock somewhere in the lithosphere varies if heat energy (a form
of kinetic energy) is gained or lost. The relationship that gives the
variation of temperature dT as a function of a variation of heat dE
is:
Heat conduction
Conduction transports heat from hot to cold regions. The flow of
heat (Q) is proportional to the negative temperature gradient (dT/
dz) between the cold and the hot region, with the coefficient of proportionality being the conductivity (k). Mathematically this translates into the Fourier's law where Q is in W.m-2 and k is in
W.m-1.K-1. In our reference frame, z increases downward
(T(z+dz)>T(z)). Conduction occurs in the direction of decreasing
temperature (i.e. dEc is a gain for upward conduction) hence the
dT=dE/(Cp.m)
with Cp the heat capacity, and m the mass.
The main processes able to change the amount of heat energy in
the lithosphere are:
3
sign "-" insures that Q is positive upward (dEc is positive when T
increases downward).
Q =−K
E1=Q(z).a.dt
z
Q(z)
dEc = E2 - E1
dEc = (dQ/dz).dz.a.dt
a
z+dz
E2=Q(z+dz).a.dt
Radiogenic heat production
Radiogenic desintegration of radioactive isotopes (238U, 235U, 232Th, and
40K) releases heat. The increment of
radiogenic heat (dEr) produced in a
small cylinder of rock of section a and
length dz over an increment of time dt
is:
dT
dz
Let's consider a small cylinder of rock
of section a (area in m2). If the incoming and outgoing heat at both ends of
the cylinder are the same, there is no
net heat gain or loss, and the temperature remains unchanged. Temperature
changes when the total heat
E1=Q(z) ⋅ a ⋅ dt leaving the volume over
a time interval dt is different to the total
heat E2=Q(z + dz)⋅a ⋅dt entering it.
A⋅a⋅dz⋅dt = A⋅dV⋅dt
z
A
dEr = A.a.dz.dt
z+dz
a
with A the rate of radiogenic heat production. Radioactive heat is
the main internal heat source for the earth as a whole (it is measured in W.m-3).
The entering heat flow Q(z + dz) can be
Q(z+dz)
approximated with a Taylor serie in
which only the two first terms are of significance. (nb: mathematically f (xn + d x) can be approximated from
f (xn) and the derivatives at location xn : f ′(xn) , f ′′(xn) , etc:
Heat Advection
Advection of heat implies that
an amount of material at temE1 =Cp.ρ.a.uz .dt.T
z
perature T (in yellow on the
sketch) is being pushed out of
dEu = E2 - E1
our cylinder and replaced by
dEu = C .ρ.a.u .dt.dT
an equivalent amount of matez
p
rial at temperature T+dT. The
increment of heat gained or
a
lost (dEu) over an increment of
E
)
time dt is proportional to the
z+dz 2 =Cp.ρ.a.uz .dt.( T+dT
mass of material displaced (
ρ ⋅a⋅uz ⋅dt ), the heat capacity
dQ dz 2 d 2Q
Q(z + dz) = Q(z) + dz⋅
+
⋅
+ ⋅⋅⋅
dz
2 dz 2
Therefore the increment of heat (dEc) gained or lost in an increment
of time dt is dEc = E1 − E2 :
dQ
d 2T
a⋅Q(z)⋅dt − a⋅Q(z + dz)⋅dt = − a⋅dz⋅
⋅dt = K⋅ 2 ⋅dV⋅dt
dz
dz
4
a.uz . dt
a.uz . dt
to right, the conduction term, the term of radiogenic heat production, and the advective term. In 3D, this equation becomes:
of the material (Cp, J kg-1 K-1: the amount of energy required by
1kg of the material to increase its temperature by 1 K), and the temperature contrast of the two exchanged volumes. (nb: because z increasing with depth, uz is negative for upward convection, however, dEu is a gain for upward convection hence the sign - insures
that when uz is negative (i.e. upward motion) the advective heat is
positive). We get that the rate of heat advection is:
∂T
K
∂2T ∂2T ∂2T
A
∂T
=
⋅[ 2 +
+
]
+
−
u
⋅
z
∂t
ρ⋅Cp ∂x
∂y 2
∂z 2
ρ⋅Cp
∂z
−Cp ⋅ρ⋅a⋅uz ⋅dt ⋅dT
The geotherm in the continental crust
Total heat gained or lost
We derive here the equation for the steady state continental geotherm. For a steady state geotherm we get that dT/dt=0 and uz = 0
(no convection in the lithosphere and no erosion or sedimentation);
therefore the heat transfer equation simplifies to:
Adding up the rate of conductive heat (dEc), the rate of convective
heat (dEu), and the rate of radiogenic heat (dEr) gives the total rate
of heat gained or lost. This variation of heat triggers a change in
temperature dT = dE/(Cp.m) therefore :
d 2T
A
=−
dz 2
K
Cp.m. dT= dEr + dEu + dEc and:
This is a second order differential equation. This kind of equation
is solved by integrating twice and by using two boundary conditions. For example we may know the temperature at the surface
let's say: T=To at z=0; and we may know the surface heat flow for
instance at z=0 the heat flow is -Qo (remember this is positive). Assuming that A is constant with depth, the first integration led to
the temperature gradient dT/dz:
d 2T
Cp ⋅ρ⋅dV⋅dT = A⋅dV⋅dt − Cp ⋅ρ⋅a⋅uz ⋅dt ⋅dT + K⋅ 2 ⋅dV⋅dt
dz
This leads to the 1D conduction-advection heat transfer equation:
dT
K d 2T
A
dT
=
⋅ 2 +
− uz ⋅
dt
ρ⋅Cp dz
ρ⋅Cp
dz
dT
A
= − ⋅z + C1
dz
K
This equation describes the variation of temperature with depth
and through time due to heat conduction, radiogenic heat and heat
advection. This equation assumes no lateral heat flow (hence 1D).
On the right end side of the equation the three terms are, from left
This gives the slope of the geotherm as a function of depth. From
this function we get at the surface: dt/dz=C1
5
relationships, one for the crust (equation in the previous slide), one
for the lithospheric mantle.
The second boundary condition demands, via the Fourier's law,
that at z=0:
dT Q0
=
dz
K
If the production of radiogenic heat is zero in the mantle then we
get:
d 2T
=0
dz 2
by combining the last two equations for z=0 we get C1 = Qo/K, we
can therefore write that...
Q
dT
A
= − ⋅z + 0
dz
K
K
Integrating twice we get:
T(z) = C2 ⋅z + C3
Integrating a second time led to:
With the two following boundary conditions: T=Tc at z=zc=Moho,
and Q=-Qm (the basal heat flow), we get that:
A 2 Q0
z + C2
T(z) = −
⋅z +
K
2K
C3 = Tc −
The first boundary condition says that T=To at z=0, therefore we
get that:
T(z) = −
and that
A 2 Q0
z + T0
⋅z +
K
2K
C2 =
Qm
⋅z
K c
Qm
K
The geotherm in the lithospheric mantle is therefore:
This relationship is the steady state geotherm. It gives the distribution through depth of temperature in a layer with homogeneous
radiogenic production A, conductivity K, with a surface temperature of To and a surface heat flow of Qo.
T(z) =
Qm
⋅ (z − zc) + Tc
K
Hence, the geotherm in the lithosphere is defined by a two steps
function:
The geotherm in the continental lithosphere and beyond
A 2 Q0
T(z) = −
z + T0, for 0 < z < zc
⋅z +
K
2K
Qm
T(z) =
⋅ (z − zc) + Tc, for z > zc
K
The continental lithosphere consists in two layers with contrasted
thermal properties. In particular the radiogenic heat production in
the mantle is negligible compared to that of the crust. The geotherm in continental lithosphere is therefore best described by two
6
alternatively, if Qm instead of Qo is known the crustal geotherm
(i.e. when 0<z<zc):
T(z) = −
This is called a differential equation because this equation links todT
dT dT d 2T
gether the derivatives ( , , 2 ) of a function T(z, t). Here,
dt dz dz
dt
dT
is the rate of temperature change at a given depth;
is the gradidz
d 2T
ent of temperature change at a given time; and 2 is the gradient
dz
of the gradient of temperature change at a given time.
Q
A 2
A
⋅z + [ m + ⋅zc]⋅z + T0, for 0 < z < zc
2K
K K
In the asthenospheric mantle, convective motion is such that the
temperature shows relatively little variation as convection acts as
an efficient mixing process. For z>zl (zl being the base of the lithosphere) the geotherm follows the adiabatic gradient ~0.3 K per kilometer. For instance, from the top of the convective mantle (i.e. the
base of the lithosphere) to the core/mantle boundary (i.e. a distance of ca. 2700-2800km) temperature increases by only ~3000ºC.
This contrast with the evolution of the temperature within the lithosphere where temperature increases from ca. 0ºC at the surface to
1300ºC at depth of 100-200 km.
From this differential equation, one can derive a solution for the
particular case when thermal equilibrium is reached (i.e. there is no
dT
change of temperature through time:
=0 and uz=0). This is the
dt
case of steady geotherm for which the differential equation simplifies to:
d 2T
A
T’’(z) = 2 = −
dz
K
This tells us that the change of the temperature gradient with
d dT
) is equal to a constant (-A/K). This says that the
depth ( )(
dz dz
geotherm in the continental crust is not a straight line.
Appendix 1: Solving differential equation
Very often the rate of natural processes (i.e. rate of heat conduction,
rate of heat advection, the rate of radiogenic heating etc) can easily
be calculated or measured. In the context of the Earth’s geotherm,
we have seen that from the rate of individual heat transfer mechanisms, we can express the overall rate of temperature change
through time (t) at any depth (z) by:
Lets plot T’’(z):
T''(z)
dT
K d 2T
A
dT
=
⋅ 2 +
− uz ⋅
dt
ρ ⋅Cp dz
ρ ⋅Cp
dz
-A/K
7
T''(z)
z
d 2T
A
The differential equation 2 = − can be solved to find out the
dz
K
geotherm T(z). For this the differential equation is integrated
twice:
The curve in blue is the geotherm, however bear in mind that this
curve is only relevant for 0 < z < zmoho.
T(z)
The first integration leads to the temperature gradient :
T’(z) =
dT
A
= − ⋅z + C1
dz
K
T(0)
zmoho
Graphically:
z
T'(z)
c1
-A/K
One may recognizes that the integration constants C1 and C2 have
the following meanings: C1 is the temperature gradient (the slope
of the geotherm) at the surface (i.e. z=0); and C2 is the temperature
at the surface. The temperature at the surface is typically in the
range of 0 to 30ºC, and the temperature gradient at the surface can
be determined by measuring the temperature at the surface and at
the bottom of a well.
z
-A/K
T''(z)
The second integration leads to the geotherm:
T(z) = −
Appendix 2: Steady state geotherm with decreasing RHP
A 2
⋅z + C1 ⋅z + C2
2K
The steady state crustal geotherm we derived earlier assumed that
the radiogenic heat production is constant with depth. This is usually not the case as cycles of partial melting and crustal differentiation, through upward flow of granitic melt, tend to deplet the
lower crust and enrich the upper crust in radiogenic elements.
Hence, the decrease with depth of the radiogenic heat production
is often described by an exponential law:
Graphically:
T'(z)
c1
-A/K
T(z)
c2
−z
A(z) = A0 . Exp( )
hc
z
-A/K
T''(z)
8
This means that the radiogenic heat production measured at the
surface ( A0 ) in divided by e (e=2.71) every hc meters (typically
5000m< hc <25000). Therefore the 1D conduction-advection heat
transfer equation is:
After substituting C1 and C2 into T(z), expanding and simplifying,
one gets:
dT
K d 2T
A(z)
dT
=
⋅ 2 +
− uz ⋅
dt
ρ⋅Cp dz
ρ⋅Cp
dz
T(z) =
If one knows the surface heat flow instead of the mantle heat flow
then:
In the case of a steady state geotherm, this equation becomes:
d 2T −A(z)
=
2
dz
K
replacing A(z) we get that
T(z) =
A0
d 2T
−z
⋅Exp(
=
−
)
2
dz
K
hc
Following two successive integrations one gets:
A
dT
−z
= 0 ⋅hc ⋅Exp( ) + C1 and
dz
K
hc
T(z) = −
A0 2
−z
⋅hc ⋅Exp( ) + C1z + C2
K
hc
To find out the two integration constants C1 and C2 we need to call
upon two boundary conditions. Lets say that we know the temperature at the Earth' s surface T0 and the heat flow Qm entering the
base of the lithosphere.
The first boundary condition leads to: C2
... and second to: C1
=
A0 2
A
−z
Q
−z
⋅hc ⋅(1 − Exp( )) + ( m − 0 ⋅hc ⋅Exp( c ))⋅z + T0
K
hc
K
K
hc
= T0 +
A0 2
⋅hc
K
−z
Qm A0
−
⋅hc ⋅Exp( c )
K
K
hc
9
A0 2
Q
−z
⋅hc ⋅(1 − Exp( )) + 0 ⋅z + T0
K
hc
K
SECTION 2
Earth’s Geotherm
In this section
0
1. Steady state geotherm
2. Transient geotherm
400
800
1200
TºC
1600
20
40
0
400
800
TºC
1200 1600
20
Moho
40
0
400
800
1200
TºC
1600
20
40
Moho
to
Moho
t1
60
60
to
Mechanical boundary layer
80
80
60
t5
t100
t10
80
t
t
8
100
8
t20
100
t10
t20
t100
t1
Thermal boundary layer
t5
100
to
8
t
Plume spreads under the lithosphere
Steady state geotherm
In the previous section we have derive the 1D conductionadvection heat transfer equation and we have derive an equation desbribing the equilibrium geotherm, also called “steady
state geotherm” because the temperature at every depth doesn’t
change through time. Here we explore how sensitive is the geotherm to the radiogenic heat production when it is not constant
10
120
140
140
Plume spreads at the top of the
thermal boundary layer
Depth (km)
Depth (km)
140
120
Depth (km)
120
Plume spreads at the Moho
with depth (as we have assumed in the first section). We then
look at how sensitive is the geotherm to chance in mantle heat
flow, conducvity and thickness of the radiogenic crust.
Distribution of radiogenic heat production
We have assumed so far that the volumetric radiogenic heat production A was depth-independent ( A remains constant with
* nb: These are elements that tend to concentrate into the melt phase during partial melting. Due to the buoyancy of the melt phase, incompatible
elements concentrate over time into the upper part of the crust. Incompatible elements such as U, Th and K have a large radius and therefore
do not fit easily into crystals lattice, hence their tendency to move into
the melt phase when partial melting occurs.
depth). However, because the upper crust is enriched in incompatible elements* the radiogenic heat production decreases with depth.
A common model assumes that A is divided by e (e=2.71) every h
metres, h being the length scale of the exponential law. This model of
distribution is given by:
A(z) = A0 ⋅Exp( − z /h)
Basal Heat Flow (Qm)
The graph illustrates the sensitivity of the geotherm to the mantle
heat flow Qm, also called basal heat flow. It shows three geotherms
calculated assuming same
TºC
rate of radiogenic heat pro0
500
1000
1500
0
duction, same conductivity
and same crustal thickness.
20
Only the mantle heat flow
varies.
Moho
40
Depth (km)
100
=24
80
Depth (km)
=36
Qm
11
60
Qm=12
A basal heat flow of 12 x 10-3
W.m-2 (yellow geotherm) is
characteristic of cold and
thick cratonic lithospheres,
whereas a basal heat of 36 103 W.m-2 is representative of
thinned lithosphere. Increasing the mantle heat flow
from 12 to 36 10-3 W.m-2 increases the temperature at
the Moho from ~320ºC to
720ºC.
Qm
Depth (km)
The graphs shows the geotherm for A decreasing exponentially
with various length scale h (10, 25, 50km). Ao is adjusted so that the
total radiogenic heat production (R.H.P which is given by the integration of the radiogenic heat
Temperature (TºC)
Radiogenic Heat (mW.m-3)
profile with
500 1000
6
2
4
8
depth) is the
same in all models. The larger is
25
25
h (i.e. the deeper
are the radioMoho
50
50
genic elements)
TMoho:
the hotter is the
724,
geotherm. This
935,
10 km
1284
75
75
suggests that at
25 km
50 km
an early stage of
R.H.P.: 0.078 W.m-2
the Earth evolu100
100
Archaean (3.5-3 Ga)
tion (before the
extraction of the
crust) the Earth
had a warmer geotherm.
the geotherm increases as warmer temperatures are reached at
lesser depths (the slope of the geotherm increases). During thickening the geotherm decreases as cooler conditions are met deeper in
the crust (the slope of the geotherm decreases). Second, as the thickness of the crust changes, the steady state geotherm is affected.
This is because a thicker crust will produce more radiogenic heat,
therefore crustal thickening leads to warmer geotherm whereas
crustal thinning leads to cooler geotherm.
Thermal conductivity (K)
The graph below illustrates the sensitivity of the geotherm to the
thermal conductivity K (in W.m-1.K-1). It shows three geotherms calculated assuming same rate of
radiogenic heat production,
TºC
same same crustal thickness,
0
500
1000
1500
0
and same mantle heat flow.
The conductivity is known to
20
vary with temperature. Here
however, we assume that K is
Moho
40
constant through depth. K is
proportional to the ability of
60
material to conduct heat away.
Therefore the larger the ther80
mal conductivity the lesser
100
heat they can stored and the
cooler is the geotherm. Decreasing the thermal conductivity from 2.75 to 1.75 W.m-1.K-1
increases the temperature at
the Moho from 500 to 750ºC.
Lithospheric thickening
5
.25
k=2
1.7
k=
Thickening produces heat advection as a volume of rock, and the
heat attached to it, is displaced vertically resulting in a rapid cooling of the geotherm. Thickening also increases the thickness of the
radiogenic layer therefore increasing the production of radiogenic
heat in the lithosphere. Isostasy (cf. section of Isostasy and Gravitational Forces) produces uplift of the lithosphere leading to erosion
which in turn affects the amount and distribution of the radiogenic
heat elements in the crust which also impacts on the geotherm. The
interplay between the mode of the thickening (heterogeneous via
thrusting vs homogeneous), the thickening rate and the erosion
rate leads to contrasted thermal histories.
The graphs show transient geotherms (at 0, 0.5, 2, 10...Myr) following heterogeneous thickening (thickening is achieved by doubling
the thickness of the crust via a single thrust, which explains the
temperature discontinuity at t=0), and homogeneous thickening
(the thicknesses of the crust and the lithospheric mantle are doubled by pure shear deformation). Erosion is discarded here. The discontinuity in the case of heterogeneous thickening is smoothed out
in a few Ma. Transient geotherms in both cases are similar for time
> 10 Myr.
Depth (km)
.75
k=2
Thickness of the radiogenic crust
Crustal thickening and thinning, via tectonics, modify the geotherm in two different ways. First, during deformation, heat is advected mainly upward (in the case of thinning) and mainly downward (in the case of thickening) as rocks carry their temperature
with them during fast thinning or fast thickening. During thinning
12
0
200 400
600
800 1000 1200
TºC
0
50
600
100
150
150
Heterogeneous thickening via thrusting
and doubling the thickness of the crust
Depth (km)
100
Homogeneous thickening via pure shear
doubling the thickness of the lithosphere
Sedimentation of burial of the crust
Lithospheric thinning
0
0
200 400
600
TºC
800 1000 1200
Sedimentation and burial of the continental crust under a few kilometre of volcano-sedimentary rocks have a significant long-term
effect on the geotherm. The newly deposited layer effectively insulates the heat producing layer. If the conductivity of the upper
layer is lower or equal to that of the buried layer, then heat accumulates increasing the geotherm.
This effect could have played a major role in the differentiation of
the continental crust in the Archaean. At that time, radiogenic heat
production in the crust was 2 to 6 time larger that of present day
rate of radiogenic heat production in the continental crust. Furthermore it was a time when 5 to 15 km thick continental flood basalts
(the so called greenstones) where deposited at the surface of the
Moho
50
yr
m
Depth (km)
0
yr
m
100
10
50 my
m r
yr
10
0
25
Thinning drives heat advection as rock masses, and the
heat attached to them, are displaced vertically mainly upward resulting in an instantaneous warming of the geotherm (the geothermal gradient increases). However, it
also decreases the thickness of
the radiogenic crust therefore
reduces the production of radiogenic heat in the litho-
r
my
250
yr
Moho
yr
0m
25
yr
m
50
m
Moho
r
my
10
yr
2
yr
sphere. Isostasy leads the subsidence of the surface of the lithosphere leading to sedimentation which in turn affects the amount
and distribution of the radiogenic heat elements in the crust which
also impacts on the geotherm. The interplay between the geometry
of the thinning, the thinning rate, and the sedimentation rate lead
to contrasted thermal histories.
The graph on the right shows transient geotherms (0, 10, 50, 100,
250...Myr) following homogeneous thinning (the thicknesses of the
crust and the lithospheric mantle are halfed by pure shear deformation). Sedimentation is discarded here. Following the increase of
the geothermal gradient due to extensional deformation, thermal
relaxation leads to cooling and therefore the thickening of the lithosphere. Slowy, the transient geotherm approaches the steady state
geotherm.
TºC
800 1000 1200
m
50
yr
0.
5m
Depth (km)
200 400
m
10
50
0
0
60% homogeneous thinning of
the lithosphere via pure shear
13
Archaean continental lithosphere
Crust
to+50Ma
50
Geotherm t
Depth (km)
100
125
8
to+100Ma
75
Mantle
25
Geotherm t
Geotherm to
to+10Ma
to+20Ma
0
to+30Ma
to+200Ma
75
Mantle
to+400Ma
to+1Ga
100
to+40Ma
400
800
1200
TºC
1600
20
to+50Ma
40
0
400
800
TºC
1200 1600
20
Moho
40
0
400
800
1200
TºC
1600
20
40
Moho
to
Moho
to+100Ma
125
t1
to+200Ma
60
60
to
Mechanical boundary layer
150
150
175
175
80
100
100
t5
t100
t10
t20
t
t10
t100
t1
Thermal boundary layer
80
t5
60
t20
80
t
8
Geotherm to
8
Crust
100
to
120
Depth (km)
Earth, insulating the heat producing Earth's crust.
The graphs show transient geotherms following the emplacement
of a 6 km (graph on the left) and 12 km thick greenstone covers,
with no radiogenic heat production in them.
The temperature increase is large enough to lead to profound partial melting in the crust.
t
8
140
Plume spreads under the lithosphere
120
120
140
140
Plume spreads at the top of the
thermal boundary layer
Depth (km)
50
Greenstone 200 400 600 800 1000 1200 1400
Depth (km)
Crust
25
mantle plume assuming that the plume's head spreads into a 50km
thick layer with a temperature of 1700ºC. The three graphs show
the results for various depth of emplacement.
TºC
8
TºC
Greenstone 200 400 600 800 1000 1200 1400
Depth (km)
Archaean continental lithosphere
Plume spreads at the Moho
Transient geotherms
Calculation of the steady state geotherm is relatively straightforward as it is not time dependent. Calculation of transient geotherms, such as those displayed above, requires computational
trickery as the temperature changes in both space and time. Many
analytical solutions have been proposed for a range of problems.
The well-known book from Carslaw and Jagger, first published in
1946 "Conduction of Heat in Solids", provides hundreds of analytical solutions to many problems. Here, we give a couple of them.
Emplacement of a mantle plume
Mantle plumes initiate at the core-mantle boundary and rise
through the convective mantle. Upon approaching the more rigid
lithosphere, they spreads laterally under the lithosphere. They may
also thermally erode to base of the lithosphere and spread higher
up. This is equivalent to put a hot layer, a few tens of kilometre
thick, underneath or within the colder lithosphere. In the Archaean, the Earth was warmer and plumes were most likely more
numerous. The graphs below document the thermal impact of a
14
Progressive cooling of the oceanic lithosphere
In the 19th century, Lord Kelvin used
this last relationship to find out the age
of the Earth. Kelvin made the assumption that the Earth had formed as a molten body at the temperature at which
basalt melts (he assumed Tm=2000ºC)
and that it had cooled by conduction to
its present surface heat flow (he assumed Q0= -30 10-3 W.m-2). He considered the Earth to be spherically symmetric, and assumed that all heat was
lost at the surface by conduction (with
κ =10-6.m-2.s-1). Thus the problem is reduced to that of finding the
temperature within a cooling half-space of infinite extent as a function of time after the half space is set at a specific temperature, a
problem for which the equations above apply. Kelvin found that
the Earth was 50 Myr old...
The formation of oceanic lithosphere at mid-oceanic-ridge is a thermal problem involving the progressive cooling of the asthenosphere. The temperature at the seafloor Ts is maintained constant,
so is the temperature in the asthenosphere Tm. The lithosphere becomes cooler and therefore thickner as cooling proceeds. Assuming
no heat production and no sedimentation the 1D advectionconduction heat transfer equation becomes:
dT
d 2T
= κ⋅ 2
dt
dz
with κ the thermal diffuvity:
κ = K⋅ρ⋅Cp
The analytical solution of this differential equation is:
T(z, t) = Ts + (Tm − Ts)⋅erf(
z
2 κ⋅t
)
By differentiating with respect to z one get the temperature gradient ...
Cooling of a dike
Tm − Ts
dT
z2
=
)
⋅Exp( −
dz
4⋅κ⋅t
π⋅κ⋅t
The cooling of a dike with half width w is another problem which
has an analytical solution in the following form:
Ts
w−x
w+x
T(x, t) = ⋅(erf(
) + erf(
))
2
2 κ⋅t
2 κ⋅t
...from which the variation of the surface heat flow through time
can get extracted as:
Tm − Ts
dT
=
dz
π⋅κ⋅t
If the dike has a width of 2m, i.e. w=1m, and its initial temperature
was Ts=1000°C, and κ = 10-6m2s-1, then the temperature at the center of the dike would be about 640°C after one week, 340°C after
one month, and only 100°C after one year.
15
solid-liquid (L<0 for partial melting, L>0 for crystallization) X the
melt fraction is a function of T and therefore z.
Some swarms, such as
the giant McKenzie
swarm in Canada, appear to radiate from a
point, commonly interpreted as a plume source
for the magmas. The
mid-Proterozoic Coppermine River flood basalts
were erupted at the
same time near the
plume head. Individual dykes range from 10-50 m in width, with
some up to 200 m wide. Some dykes can be traced for up to 2000
km.
∂T
∂ 2T
A
∂T ∂X L
= κ⋅ 2 +
−U
+
⋅
∂t
∂z
ρ⋅Cp
∂z
∂t Cp
re-arranging we get:
∂T
1
∂T
∂ 2T
A
∂T
−(
⋅ ) = κ⋅ 2 +
−U
∂t
Tliq − Tsol ∂t
∂z
ρ⋅Cp
∂z
and therefore:
(1)
∂T
L
∂ 2T
A
∂T
− (1 −
) = κ⋅ 2 +
−U
∂t
Cp ⋅(Tliq − Tsol)
∂z
ρ⋅Cp
∂z
Boundary conditions: T(z, 0)=0; T(0, t)=T0; dT/dz(zl, t)=-Qm/K
Numerical solutions
An approximation of this differential equation can be obtained by
replacing ∂t and ∂z with finite differences ∆t and h. The central
tenet of this computational technique is that for a given time t=n,
the temperature at a depth z=i (Tin) can be calculated from knowledge of the temperature at depth z=i-h and z=i+h. In a similar fashion, for a given depth z=i, temperature at time t=n can be caculated
from knowledge or the temperature at time t=n-∆t and temperature at time t=n+ ∆t. The level a accuracy depend of the size of the
time and space finite differences. There are many ways to express
T(z, t) as a function of T(z-h, t), T(z+h, t), T(z, t- ∆), T(z, t+ ∆). Here
we present the Crank-Nicholson scheme where, equation (1) is rewriten as:
Since the arrival of computers, which allow for large numbers of
operation to be performed in routine, new techniques based on numerical algorithms have been designed to solve differential equations. These numerical recipes are based on "discretization" of the
differential equation...
•Consider the 1D heat conduction-advection equation in a slab zl
thick. The upper surface of the slab has at temperature To. The transient temperature is then described by the conductive-advective
equation of heat balance. Here we consider a situation where internal radiogenic heat is creation at a rate A, heat is lost or gain by advection at speed U (since z increases downward, if erosion U<0, if
sedimentation U>0), and heat is lost through phase transition
(2)
16
Tin+1 − Tin
L
− 1−
Δt
Cp ⋅(Tliq − Tsol) )
(
= κ⋅
n
n
n+1
n+1
Ti+1
Ti+1
− 2Tin + Ti−1
− 2Tin+1 + Ti−1
1
⋅
+
2 (
h2
h2
)
where: Tin
=
T(zi, tn), T n+1
i
and ai = Rd, ci = Rd, bi = (1 − 2Rd + V − Lt) Rearranging terms, we obtain the following set of implicit equations for the temperature at the n+1 time step in terms of the temperature at the n-th time step (4):
Tin+1 − Tin
A
+
−U
h
ρ⋅Cp
A ⋅Δt
= din
k
This formulation results in N equations and N+4 unknowns, the
four additional unknowns being u0n and uN+1n, u0n+1, uN+1n+1. However, these four additional unknowns lie outside the computational
grid (time, space). Also, from the boundary conditions the following conditions must be satisfied at the n-th and n+1 time steps:
1+n
1+n
n
n
a1i ⋅T−1+i
+ b1i ⋅Ti1+n + c1i ⋅T1+i
= ai ⋅T−1+i
+ bi ⋅Tin + ci ⋅T1+i
+
= T(zi, tn+1), etc The finite difference equation (2) is known as the Crank-Nicholson
scheme. It is based on central differences for the spatial derivatives averaged forward in time over time steps n and n+1.
Expanding (2) and re-arranging to express temperature at time n+1
as a function of the temperatura at time n we get equation (3):
T1n = 1, TNn = 0, T1n+1 = 1, TNn+1 = 0 L
−κ ⋅ Δt 1+n
−κ ⋅ Δt 1+n
κ ⋅ Δt UΔt
1+n
⋅
T
+
1
−
+
+
⋅ T1+i
+
⋅T
i
−1+i
)
( 2h 2
2h 2
Cp ⋅(Tliq − Tsol)
h2
h )
(
=
This gives the following set of N equations (5) for the temperature
at the n+1 time step :
κ ⋅Δt UΔt
A ⋅Δt
κ ⋅Δt n
L
κ ⋅ Δt n
n
−
+
⋅T
+
1
−
+
+
⋅T
⋅T
i
−1+i
1+i
) ρ ⋅ Cp
( 2h 2
2h 2
Cp ⋅ (Tliq − Tsol)
h2
h )
(
a11 ⋅ T01+n + b11 ⋅ Ti1+n + c11 ⋅ T21+n = a1 ⋅ T0n + b1 ⋅ T1n + c1 ⋅ T2n +
remplacing and substituing:
Rd =
A ⋅ Δt
, i = 1
k
A ⋅ Δt
, i = 2 to N − 1
k
A ⋅ Δt
1+n
1+n
n
n
a1N ⋅ TN−1
+ b1N ⋅ TN1+n + c1N ⋅ TN+1
= aN ⋅ TN−1
+ bN ⋅ TNn + cN ⋅ TN+1
+
, i = N
k
1+n
1+n
n
n
a1i ⋅ Ti−1
+ b1i ⋅ Ti1+n + c1i ⋅ Ti+1
= a1 ⋅ Ti−1
+ bi ⋅ Tin + c1 ⋅ Ti+1
+
κ ⋅Δt
U⋅Δt
L
,
V
=
,
Lt
=
2h 2
h
Cp(Tliq − Tsol)
we get...
1+n
1+n
−Rd ⋅T−1+i
+ (1 + 2Rd + V − Lt) ⋅ Ti1+n − Rd ⋅T1+i
n
n
= Rd ⋅T−1+i
+ (1 − 2Rd − Lt + V ) ⋅Tin + (Rd ⋅T1+i
)+
with (6):
A⋅Δt
ρ ⋅Cp
b11 = 1, c11 = 0, a1N = 0, b1N = 1 b1 = 1, c1 = 0, aN = 0, bN = 0
further replacement...
These latter conditions ensure that the boundary conditions are always satisfied.
a1i = − Rd, c1i = −Rd, b1i = (1 + 2Rd + V − Lt)
17
Numerical solutions an example
The N equations above can be rearranged into the following matrix
equations:
200 400
0.
5m
600
800 1000 1200
TºC
yr
yr
0m
25
m
50
yr
Moho
100
Depth (km)
The figure on the right shows the instantantaneous and steady state geotherm
150
(potential geotherm reached after infinite
Heterogeneous thickening via thrusting
and doubling the thickness of the crust
time). The aims is to display transient
geotherms down to a depth z=zl+zt at
various time intervals, where zl is the thickness of the lithosphere
before thickening zl=120km.
with:
b1
d1n
a2
d2n
0
d3n
= 0
.
.
0
n
dN−1
0
dNn
0
50
0
r
my
10
yr
m
..
..
b11 c11 0 0
0
T1n+1
d1n
..
..
a12 b12 c12 0
0
T2n+1
d2n
..
0
0
0 a13 b13 c13
d3n
T3n+1
..
.. ..
..
..
=
(7)
.
0
0 ⋅ .
.
.
..
.. ..
..
..
0
0
n
n+1
dN−1
T
N−1
..
..
..
a1N−1 b1N−1 c1N−1
0
dNn
n+1
T
N
..
.. ..
..
a1N
b1N
0
0
2
• We illustrate here a probleme where a
continental crust with initial thickness zc
is thickened by a factor 2 via the emplacement of one single zt thick thrust.
zc=40km, zt=40km.
..
..
0 0
0
1
..
..
b2 c2 0
0
T2n
..
a3 b3 c3 0
0
T3n
A⋅Δt
.. .. ..
..
..
⋅
⋅
.
0
.
k
.. .. ..
..
..
0
n
TN−1
.. .. .. aN−1 bN−1 cN−1
TNn
.. .. ..
..
aN
bN
c1
• We choose a spatial finite diffence h=4000m (spatial grid). Therefore the number of column in the tridiagonal matrixes will be zl/h.
The Crank Nicholson scheme imposes the maximum time step of
h2/(2.κ) = 253,000 years.
• We get that Rd=(κ.∆t)/(2h2)=0.25. We assume no erosion: v=0; and
we allow for partial melting:
Lt=Latent_heat/(Cp.(Tliquidus-Tsolidus)). With Rd, v and Lt we can
determine ai1, bi1, ci1, the coefficient of the tridiagonal matrix at
time n+1, and ai, bi, and ci the coefficient of the tridiagonal matrix
at time n.
Thus the RHS of (7) is determined explicitly from the solution at
the n-th time step. Note that the boundary conditions are satisfied
by ensuring that the coefficients given in (6) are satisfied. Hence in
order to find the solution at the time step, one must solve N linear
equations. Since the coefficient matrixes in (7) and (8) are tridiagonal, one can make use of efficient algorithms (e.g. Thomas algorithm) to find the solution.
We choose to get the transient geotherm at 20 Myr interval up to
400 Myr. The depth of the tridiagonal matrixes (number of row) is
therefore 400/20=20.
18
• With this, we construct the tridiagonal coefficient matrixes (7)
and (8) , the equation system (7) is solved via matrix inversion
(here Numpy, R, MatLab or Mathematica are well equiped to do
the dirty work...). Graph on the right show transient geotherms
from 20 to 400 Ma.
(7)
(8)
19
CHAPTER 2
Isostasy & Gravitational Forces
Isostasy is the physical process that explains
the surface elevation of the Earth’s lithosphere
at rest. It relates to the Archimedes’ principle
showing that the elevation of a floating body
above the surrounding fluid depends on the
density contrast and vertical length of the
floating body.
Isostatic equilibrium must not be confused
with gravitational equilibrium. A gravitational
force (a volume force) appears when density
interfaces are no longer parallel to gravity
equipotential surfaces. Hence, the action of
isostasy, which forces density interfaces away
from equipotentials, introduces gravitational
forces driving the flow of rocks from regions of
high pressure to regions of low pressure.
The Tibetan plateau stands 5000 m above sea
level. It is the surface expression of a 75 km
thick continental crust, which imparts a
significant gravitational push on its
surrounding.
20
ISOSTATIC EQUILIBRIUM AND MECHANICAL EQUILIBRIUM
The sketch below represents a schematic cross-section through a lithospheric plate. Although the thicknesses of both the crust and the lithospheric mantle vary, isostatic equilibrium is maintained.
• Isostatic equilibrium implies that at - and below - a parA
B
C
ticular depth, called the compensation depth, the pressure
becomes hydrostatic (ie pressure shows no lateral variation). In other terms, at or below the compensation level,
the weight of vertical columns with the same crosssectional areas standing on the same gravitational equipotential surface are the same.
• Isostasy controls the elevation of the Earth's surface.
Lithospheres with thin/thick continental crusts have
lower/higher surface elevation compared to the average
lithosphere. However, isostatic equilibrium does not mean
Compensation level
mechanical equilibrium. Lateral variations of density
Isostatic equilibrium=>Lithostatic pressure@z = constant for z > Compensation level
along a given equipotential surface produce gravitational
forces (body forces) acting inside the lithospheric plate.
B
σzz = ρ.g.z(MPa)
C
GRAVITATIONAL POTENTIAL ENERGY AND GRAVITATIONAL FORCE
(B)
C)
z(
ρg
21
z
ρg
The sketch bellow illustrates the variation with depth of the lithostatic pressure along two
lithospheric columns, B and C, in isostatic equilibrium. It is assumed here that the density of
the crust (pink) and that of the mantle (green) are laterally constant (i.e. they are the same in
both columns) and that there is no density difference between the lithospheric mantle and the
asthenosphere. Because it has a thicker crust, elevation of column C is higher than that of column B. Consequently the lithostatic pressure at any depth within the crust of column C is a
higher than the lithostatic pressure at the same depth in column B. However, at or below the
compensation depth (here the base of crust in column C) the lithostatic pressure is the same in
both column.
GPE (C)
GPE (B)
GPE (B) < GPE (C) => Horizontal force acting from C to B
C)
z(
∫
ρg
GPEC =
σzz = ρ.g.z(MPa)
C
B)
z(
Top
B
ρg
The magnitude of the Gravitational Potential Energy (GPE, in Pa) for a given lithospheric column is given by the surface area of the region delimited (i.e. bounded) by i) its ρ.g.z function
(lithostatic pressure), ii) the vertical axis (z), and iii) the compensation level. For column B this
is the orange region, and for column C the yellow region which extends underneath the orange region. The gravitational force (Fg in N.m-1) that C and B applies on each other is equal
to the difference of their respective GPE (difference between the two colored surface areas, ie
the thin yellow wedge on the graph below). An excess in GPE in column C (Fg>0) will drive
extension, whereas a deficit (Fg<0) will drive contraction.
GPE (C)
GPE (B)
Top
ρC(z)⋅g⋅z dz
and
Bottom
GPEB =
∫
ρB(z)⋅g⋅z dz
Bottom
Fg = GPEC − GPEB
Movie 2.1 Collapse of orogenic plateaux
This movie shows a 65 km thick
crust adjacent to a 40 km thick crust
(brown), in green the upper mantle.
It starts at a stage when thermal relaxation has produced 20 to 25% of
melt (dark pink) in the lower crust.
The only force acting is the gravitational force. It is the same physics
that is responsible of the spreading
of camembert, or that of warm butter.
22
GPE (B) < GPE (C) => Horizontal force acting from C to B
THE CASE OF LITHOSPHERIC THICKENING AND DEBLOBBING
Δσzz (MPa)
Compression
50
The figures on the right show the gravitational force (Fg) due to instantaneous homogeneous thickening (at time to), and following convective thinning (to+30 Myr). Convective thinning is the gravitational process upon which the cold and dense lithospheric keel is dragged into the convective
mantle. Following homogeneous thickening, the excess in gravitational potential energy in the upper part of the thickened lithosphere is more or less balanced by a deficit in gravitational potential
energy in the deeper part of the lithosphere. Indeed the integration of σzz with depth gives a very
small graviational force (Fg = 0.77 1012 N.m-1). 100 Tension
Depth (km)
50
100
150
Following convective thinning (the cold and heavy lithospheric keel has been removed), the deformed lithosphere has a larger excess in gravitational potential energy. This excess in GPE gives a
gravitational force Fg = 6.7 1012 N.m-1. Such an Fg could easily balance or overcome tectonic forces,
and in some circumstances could be sufficient to drive extensional collapse of the mountain belt.
-1
Fg=0.77xTNm
to
Δσzz (MPa)
50
-50
100
150
20
40
60
THE CASE OF LITHOSPHERIC THICKENING AND DEBLOBBING
80
-1
100
Gravitational force (Fg) due to instantaneous homogeneous stretching (at time to) and following
thermal relaxation (at time to+200Myr). The yellow shaded areas represent the gravitational force
acting on the lithosphere. Fg is given by the integratation with depth of the difference in lithostatic
pressure (σzz) between the thinned and the undeformed lithosphere.
As thermal relaxation and cooling proceeds, the thickness of the lithospheric mantle increases. This
leads to a decrease in the extensional gravitational force acting on the lower part of the lithosphere
and an increase in the compressional gravitational force acting in its upper part. The integration of
Fg with depth gives a value > 1012 N.m-1. For contraction structures to develop Fg must be larger
that the integrated strength of the thinned lithosphere.
23
Δσzz (MPa)
50
Compression -50
Tension
Depth (km)
50
100
to
-150 -100
-1
Fg=0.17xTNm
Δσzz (MPa)
-50
50
150
100
20
40
Depth (km)
Following homogeneous thinning by a factor of 2 (50% reduction of the thickness of the crust and
that the whole lithosphere), Fg is rather small (< 1012 N.m-1).
to+30Ma
Fg=6.7xTNm
to+200Ma
60
80
100
-1
Fg=1.2xTNm
GLOBAL VS LOCAL GRAVITATIONAL STRESS FIELDS
The Earth’s plate's mean gravitational potential energy (GPE) defines a global reference level. A lithospheric column with an excess of
GPE with respect to this reference will be under extensional stresses, or under compressional stresses if it has a deficit. Continental and
oceanic columns with an elevation of ~70 m and -4.3 km respectively are in mechanical equilibrium with the global reference GPE. Midoceanic ridges (MOR) have an excess of GPE which explains why they are in extension. Let's focus on the column B on the graph below.
Depending on its elevation, B has either an excess (B2) or a deficit (B1) of GPE with respect to the GPE of MOR. What would be evolution
of the gravity-related stress acting on B during thinning, which will shift the geometry of B toward that of C?
Potential Energy
We consider first the case where the potential energy of column B is B1< GPE of MOR. As B becomes thinner, its GPE must increases to
reach that of MOR (path B1->C1). Therefore, column C is in extension with respect to its surrounding (column B). It is also in extension
with respect to the global Earth's lithoB
A
C
sphere since C has a GPE C1 greater
Thick
Normal
Thin
than that of the global mean potential.
Hence, the regional gravity-related
Extension Pe>0
B2 Elevation B2>~70m
state of stress enhances the global
M.O.R.
C2
gravity-related state of stress.
-4.3km
C1
Plate's mean potential energy
In contrast, if the column B has a GPE
B1 Elevation B1~70m
Compression Pe<0
B2 then thinning of B will lead to a deM.O.R.
M.O.R.
A
crease in GPE (path B2->C2) . With a
B
C
GPE C2, column C is in compression
Crust
Oceanic
lithosphere
since it has a deficit of GPE with its imLithospheric
mediate surroundings (B2). This reMantle
gional compression opposes the global
Not to scale
extensional state of stress related to the
∆GPE between the thinned lithosphere (C2) and the global mean potential.
From this analysis we can define two gravitational stress field components: The Global Gravitational Stress Field (GGSF), and the Regional Gravitational Stress Field (RGSF). The GGSF results from ∆GPE between a lithospheric column and the global mean potential energy, whereas the RGSF results from the contrast in ∆GPE between a deformed lithospheric column and its immediate surrounding. The
Effective Gravitational Stress Field (EGSF) is the superimposition of both gravitational stress fields.
24
GRAVITATIONAL FORCE AND THE FORMATION OF CONTINENTAL PLATEAUX Depending on its sign, GPE may oppose or
enhance thickening.
In this example (Vanderhaeghe et al., 2003) a
mountain belt develops due to the subduction of the continental lithospheric mantle.
a) A localised mountain belt has formed.
Crustal thickening produces an extensional
gravitational force (Fg) that opposes the tectonic driving force (Fd).
b) and c) When Fg balances Fd, thickening migrates into adjacent areas of the foreland region: the mountain belt grows laterally and a
plateau develops.
The mountain belt is supported by the basal
traction exerced by the subduction at the base
of the overriding plate. Should the subduction stop, there would be no force to oppose
Fg, which will then drive extensional collapse.
25
GRAVITATIONAL FORCE AND CONVECTIVE THINNING
Convective thinning describes the drag, into the convective mantle, of the lower part of the lithospheric mantle. This mechanism corresponds to the development of a Rayleigh-Taylor gravitational instability (driven by a density inversion): The heavy yet weak lower part of
the lithospheric mantle is gravitationaly instable with respect to its surrounding.
Depending on the rheology of the lithospheric mantle Houseman and Molnar (1997) estimated that the part comprised in between the isotherm 900±100º and the isotherm 1300º is gravitationally unstable. The detachment of this heavy keel from the rest of the lithosphere dramatically changes the balance of forces. Convective thinning results in a sudden increase in GPE stored in the thickened crust, and therefore promotes extensional collapse.
ioo
26
CHAPTER 3
Tectonic Forces
M.O.R.
Fg
Frp
Oceanic
dFr
Fo
dFf
Continental lithosphere
Fox
100 KM
dFr
dFaz
Mantle flow
Mantle flow
Fss
dFa
Fss
Mantle flow
dFr
e
ntl
Ma
dFp
w
flo
z
w
flo
e
ntl
Ma
x
Fsp
Ever since plate tectonic theory took hold in the late 1960s, geoscientists have argued over what
drives plates: mantle upwelling at ridges that pushes plates apart, mantle circulation that drags plates
along, or slab pull have been, independently or in combination, proposed as driving forces for plate
motion. In this chapter we explore the notion of tectonic forces.
27
FORCES DRIVING PLATES MOTION
Mantle flow
Ma
w
flo
w
flo
e
ntl
e
ntl
Ma
FORCES DRIVING PLATES DEFORMATION
A fourth force is involved in the dynamic of continental margins. This force is
called the gravitational force (Fg). It is a volume force that acts between the
high-standing continental plate and the low-standing adjacent abyssal plain.
The interplay between Frp , Fg, Fss and Fsp can result in contrasting tectonic regimes.
FG
ental
n
Conti
phere
Asthenos
here
lithosp
rce
nal Fo
atio
Gravit
FSS
FSP
FSS
28
Ocea
ati
Gravit
phere
thos
nic li
Force
onal
FRP
100 KM
Based on a simple model of lithosphere and mantle M.O.R.
Fg
interactions, geophysicists have found that three
Frp
Oceanic
Fo
dFf
major tectonic forces — slab pull, slab suction and
Continental lithosphere
dFr
Fox
ridge push — can interact to explain most observed
dFr
plates motion at the Earth surface. A subducting
dFa
Fss
Fss
dFaz
slab “pulls” the rest of the oceanic plate behind it.
Mantle flow
Mantle flow
This is refers to as the slab pull force (Fsp). According
to Don Anderson, “... slabs drive tectonics. There is no
dFr
need for other driving mechanisms such as plumes or
dFp
mantle convection that are independent of plate tectonx
ics”. Others consider that the viscous drag imFsp
posed by the convective mantle, and/or the asz
thenospheric flow forced by the nearby subducting
slab, can also contribute to drive the subduction of
the oceanic lithosphere. This is the slab suction force
(Fss). When slabs detach and sink into the asthenosphere the suction force does disappear. The mantle flow associated to the sinking slab
can drive the dynamic subsidence of the lithosphere above. Finally, the gravitational force acting between the high-standing mid-oceanicridge and the distant, low-standing oceanic abyssal plain is called the ridge push force (Frp). It is of second order importance when compared to the slab pull. Hence, plates motion results from the interplay between mantle flow and forces acting at
plates boundary.
In the context of an active continental margin, the tectonic regime experi-
M.O.R.
Trench
enced by the upper plate above the subduction zone can vary from no tec-
Frp
tonics, extensional tectonics, contractional tectonics, or transcurrent tecton-
Plate
Oceanic lith
osph
ere
Fg
Continental lithosphere
ics. The tectonic regime expresses the balance of forces acting on the conti-
Extensional tectonics is the tectonic regime when the subduction trench
moves away from the upper continental plate. The trench retreats from the
continent in response to slab rollback driven by the slab pull force, or under
the push from the gravitational force. This tectonic regime leads to the detachment of continental fragments from the main continental plate.
MOR-Trench
nental margin.
Fsp
Trench-Plate
M.O.R.
Trench
Plate
Fg
Frp
Oceanic lith
osph
ere
When the distance between the trench and a reference location inside the
Continental lithosphere
continental plate decreases, the continental margin is shortened. The tecorogen. Through times, the interplay between, ridge-push, slab-pull, slabsuction and gravitational force results in a changing tectonic regime with
alternating periods of contraction, gravitational collapse and extension.
MOR-Trench
tonic regime is contractional and leads to the development of a cordillera
Fsp
Trench-Plate
These changes can be triggered by small variations in the velocity oo the direction of plates motion.
M.O.R.
Trench
Fg
Frp
An active yet tectonically stable continental margin indicates that the result-
Plate
Oceanic lith
osph
ere
ing effective force acting on the margin is very small and unable to deform
Continental lithosphere
MOR-Trench
the margin. The margin is in mechanical equilibrium.
Fsp
Trench-Plate
29
MANTLE FLOW: THE CASE OF STRONG COUPLING
Underneath the lithosphere, the flow of the asthenosphere most likely
exerts an influence on the overriding plate motion. The extent of this influence is strongly debated as it depends on the level of mechanical coupling between the lithosphere and the convecting asthenosphere.
v
The sketches on the right illustrate two cases where the mantle exerts a
strong but opposite influence on the plates. The red arrows show the velocity profile across the subducting lithosphere and into the adjacent asthenospheric mantle.
A/ The subducting slab drags the surrounding asthenosphere into the
subduction zone. The asthenosphere resists and slow down the subducting slab. In this case, the sub-lithospheric mantle flow is the result of
plate tectonic processes, as the slab sucks the nearby asthenospheric
mantle into the subduction zone.
A
B/ The flow in the asthenosphere drives plate motion by dragging the
oceanic lithosphere into a downwelling zone (suction force through
shear traction). Here, it is the flow in the sub-lithospheric mantle that
powers plate-tectonic, as mantle convection sucks the slab into the subduction zone.
If the strength of the asthenosphere/lithosphere coupling is not that significant, the convective asthenosphere can still interfer with plate motion. In particular, when the flow in the asthenosphere is at an angle to
the subducting slab it can forces the steepening or shallowing of the
slab. See next slide.
30
v
B
MANTLE FLOW: THE CASE OF WEAK COUPLING
When the flow lines in the asthenosphere are at an angle to the subducting slab, the “mantle wind” can force the steepening or shallowing of the
slab, hence promoting trench retreat or advance.
C/ The asthenospheric wind under the subducting slab forces the slab to
bend downward, enhancing the slab-pull. This effect is here enhanced
by the downward asthenospheric flow above the slab forcing the steepening of the slab.
D/ In this case, the asthenospheric wind supports the subducting slab,
opposing the slab-pull. The slab is pushed upward forcing the shallowing of the slab, and perhaps squeezing the mantle wedge.
C
Overall, the forces Frp , Fg, Fss and Fsp acting on the plates, the sublithospheric flow and the flow of the mantle in the vicinity of the slab all
influence plate motion, and the tectonic regime at continental plate margins.
D
31
RIDGE PUSH AND OTHER HORIZONTAL GRAVITATIONAL FORCES
Gravitational force at mid-ocean ridge
Mid-oceanic-ridges (MOR) form a ~55,000 km long mountain belt in the middle of the ocean. This mountain belt
stands ~2500 m above the average depth of oceanic abyssal
plain (-5000 m), and the crest of the MOR stands in average
2500 m below sea level. MOR, including the region of hot
mantle underneath, applies an horizontal force directed from
the MOR to the adjacent abyssal plains and adjacent continents. MORs are in isostatic equilibrium, but not in mechanical equilibrium since they are the locus of important and sustained extensional deformation, which accommodates the
continuous production of new oceanic crust through decompression partial melting of the asthenopheric mantle.
The map on the right shows the bathymetry of the north Atlantic ocean beween North Africa and Newfoundland. The
bathymetric profile (along the red line) across the midoceanic ridge reveals the bathymetry across the MOR, with its crest at around 2000 m depth, and the abyssal plains between -4500 and
-5000 m.
Assuming temperature independent densities, an estimate of the total ridge-push per unit length parallel to the ridge axis is: &
L e
Frp = g⋅e⋅ (ρm − ρw)⋅
+
(3 2)
where e is the elevation of the mid-oceanic ridge above the cooling lithosphere, ρm is the density of asthenosphere (3320 kg m-3), ρw is the
density of sea water (1030 kg m-3), L is the thickness of the oceanic plate (85 km), and g is the acceleration of the gravity field (10 m s-2).
With these values we get Frp = 2×1012 Nm-1. This estimate, we will see, is an order of magnitude less than the slab-pull force. However, the
value of the ridge-push force may increase up to 6.2×1012 Nm-1 when the ridge is underlain by a hot spot. A more thorough formulation
must integrate the temperature dependence of the densities. Using the concept of gravitational force, the total ridge push can be expressed
32
as the integration with depth of the difference in lithostatic pressure between the isostatic column at the mid-oceanic ridge and an
oceanic column far away from the mid-oceanic ridge. In other
terms, the ridge push corresponds to the difference between the
gravitational potential energy of the MOR and than of the adjacent
abyssal plain.
Fg = GPEMOR − GPEPLAIN
Continental margins are also regions with large contrasts in elevation, in particular when these margins support mountain belts. For
instance, the South American Andes stand in average ~8000 m
above the adjacent abyssal plain of the east Pacific. The large contrast in gravitational potential energy produces a gravitational
force directed from the continent towards the oceanic basin. The Andes are also in dynamic equilibrium supported by the convergence
of the South American and Nazca plates.
Gravitational force at continent-ocean transition
Even in the absence of cordillera, the transition between continents, which stand in average a
few hundred meters above sea level, and adjacent abyssal plains are regions of significant gravitational stresses. Not surprisingly, earthquakes tend to concentrate at plate boundaries even
when they involve passive margins. For instance, earthquakes tend to concentrate in a broad
zone at the edge of the Australian continent. This zone shows a significant gradient in gravitational potential energy and therefore a strong gravitational force that is at the origin of these
earthquakes.
33
SLAB PULL FORCE
M.O.R.
A simple formulation for the slab-pull per unit length parallel to the
trench is given by : zm. (zl. ∆ρ).g. This formulation assumes that the density of the plate and that of the asthenosphere are temperatureindependent. Assuming an average density contrast of about 60 kg.m-3, a
depth of the subducting slab zm = 660 km, a thickness of the slab zl = 100
km and g = 10 m s-2, we get an absolute maximum for the slab pull Fsp ~
4×1013 N m-1.
Trench
Frp
zl
Oceanic lith
osph
ere
Plate
Fg
Continental lithosphere
zm
Fsp
Note: the value would be an order of magnitude greater (Fsp ~
N
m-1.) for a slab going all the way down to the core-mantle boundary.
2×1014
Asthenosphere
A more accurate formulation takes into account the temperature dependence of density, the diffusion of heat, and the velocity of the subducting slab. An estimate of the slab-pull force per unit length of subduction zone, Fsp(z), acting at depth Fsp and caused by the density contrast between the cold oceanic plate and the mantle is given by:
8⋅g⋅α⋅ρm ⋅Ta ⋅L 2 ⋅Re
π 2 ⋅z
π 2 ⋅d
Fsp =
⋅ Exp( −
) − Exp( −
)
π4
2⋅Re⋅L
2⋅Re⋅L )
(
with:
Re =
ρm ⋅Cp ⋅ν⋅L
2⋅k
where Re is the thermal Reynolds number (the ratio of heat convection to heat conduction), z is the depth beneath the base of the oceanic
plate, α is the coefficient of thermal expansion (3×10-5 K-1), Ta is the temperature of the asthenosphere (1350ºC), k is the thermal conductivity (2 W m-1 K-1), L is the thickness of the plate (85 km) and d is the depth of the upper mantle (d+L = 660 km), Cp is the specific heat
(1.17×103 J kg-1 K-1), and v is the rate at which the oceanic slab sinks into the mantle (~ 10 cm yr-1). Using the above values we get Fsp =
2.5×1013 N m-1.
However, it is worth noting that Fsp linearly depends on the ill-defined coefficient of thermal expansion. With a coefficient of thermal expansion varying in the range of 2×10-5 to 4×10-5 K-1 the slab-pull force varies between 1.7×1013 and 3.4×1013 N m-1. In addition, the olivinespinel phase change which occurs at around 350 to 420 km increases the density in the subducting slab, providing an extra pull. Note: This
34
extra pull depends on the thermodynamical characteristics of the phase transition, notably the slope of its Clapeyron curve (dP/dT). The
value of the slope of the Clapeyron curve ranges between 3 and 4 MPa K-1. The resulting extra pull ranges between 1.2×1013 and 1.6×1013 N
m-1.
The slab-pull force is opposed by the friction force between the slab and the lithosphere viscosity of the asthenospheric mantle. This force is
proportional to the velocity of the subducting plate and to the viscosity of the asthenosphere, that is also poorly constrained. Numerical calculations based on the differential equations for the flow of a viscous fluid suggest that the resistive force is of the order of 1013 N m-1.
MAXIMUM EFFECTIVE TECTONIC FORCE
Overall, the maximum effective tectonic force available for lithospheric deformation is of the order of a few 1013 Nm-1. This force must be
able to double the thickness of the continental crust, to account for Tibet, the highest and largest plateau on Earth, and to sustain the gravitational force associated with this thickening and with mountain belts at the edge of continents (e.g. the Andes).
Averaged over a lithospheric thickness of 100 to 200 km, the maximum differential stress available to deform the Earth lithospheres is in
the range 100-500 MPa. This places some constraints on the rheology of Earth’s material.
a/
re
sphe
ic litho
Ocean
FRP
here
hosp
nic lit
e
l Forc
Ocea
tationa
Gravi
ental
Contin
FG
sphere
Astheno
here
hosp
Cont
l lit
inenta
rce
nal Fo
tatio
Gravi
b2/
b1/
FSS
FSP
here
FRP
hosp
nic lit
FSS
Ocea
FG
FG
sphere
here
en
Contin
l lit
inenta
e
l Forc
tationa
Gravi
sphere
Astheno
FSS
FSS
FSS
FSS
35
here
hosp
nic lit
Ocea
Cont
e
l Forc
tationa
Gravi
here
hosp
Astheno
hosp
tal lit
FRP
here
lithosp
here
Asthenosp
APPENDIX: This R script grabs bathymetry and topography data from the US National Oceanographic and Atmospheric Administration
(NOAA), creates a topographic map, and plot a topographic profile (see p.31).
# Load the marmap library
require(marmap)
#setwd stands for set working directory
setwd('/Users/Patrice/Documents/GIS/')
# Grab data from National Oceanographic and Atmospheric Administration (National Geophysical Data Center)
# More data available here: https://www.bodc.ac.uk/data/online_delivery/gebco/
map_Altiplano<-getNOAA.bathy(lon1=-60, lon2=-80, lat1=-30.00, lat2=-10, resolution=2)
# Hypsometry of the region
# zbreaks = quantile(map_Altiplano, seq(0, 1, length.out=256))
zbreaks = zbreaks <- seq(-8000, 8000, by=100)
plot(zbreaks)
# Make colour palette
ocean.pal <- colorRampPalette(c("#000000", "#000413", "#000728", "#002650", "#005E8C", "#0096C8", "#45BCBB", "#8AE2AE",
"#BCF8B9", "#DBFBDC"))
land.pal <- colorRampPalette(c("#336600", "#F3CA89", "#D9A627", "#A49019", "#9F7B0D", "#996600", "#B27676", "#C2B0B0",
"#E5E5E5", "#FFFFFF"))
#land.pal <- colorRampPalette(c("#467832", "#887438", "#B19D48", "#DBC758", "#FAE769", "#FAEB7E", "#FCED93", "#FCF1A7",
"#FCF6C1", "#FDFAE0"))
mypalette <-c(ocean.pal(sum(zbreaks<=0)-1), land.pal(sum(zbreaks>0)))
# Plot the map
plot(map_Altiplano, image=T, bpal=mypalette, land=T, deep=c(-6000, -3000, 0), shallow=c(-3000, 500, 0), step=c(1000, 500, 0),
col=c("grey10", "grey40", "black"), lty=c(1, 1, 1), drawlabel=c(T, T, T), asp=0, ylim=c(-30,-10), xlim=c(-80,-60))
# Topographic profile
points(c(-78, -61),c(-20, -20), type="o", col=2)
profile <- get.transect(map_Altiplano, x1 = -78, y1 = -20, x2 = -61, y2 = -20, locator=F, distance = TRUE)
plotProfile(profile, ylim=c(-9000,5000), xlim=c(0,1800))
# The following organize map, legend and profile into one single image
def.par <- par(no.readonly = TRUE)
nf<-layout(matrix(1:4,nc=2), height=c(3,1), width=c(5,1))
par(mar=c(4,4,1,1))
layout.show(nf)
plot(map_Altiplano, image=T, bpal=mypalette, land=T, deep=c(-6000, -3000, 0), shallow=c(-3000, 500, 0), step=c(1000, 500, 0),
col=c("grey10", "grey40", "black"), lty=c(1, 1, 1), drawlabel=c(T, T, T), asp=1, ylim=c(-30,-10), xlim=c(-80,-60))
points(c(-78, -61),c(-20, -20), type="o", col=2)
plotProfile(profile, ylim=c(-9000,5000), xlim=c(0,1800))
image(x=0, y=zbreaks, z=matrix(zbreaks, 1, length(zbreaks)), col=mypalette, breaks=zbreaks, useRaster=FALSE, xlab="", ylab="",
axes=FALSE)
36
CHAPTER 4
Introduction to Rheology
Rheology is the study of flow, the mechanical response of material to applied deviatoric stresses. “Flow” is
used here in its broader meaning, which includes both viscous flow and frictional flow. The relationships
between, on one hand applied deviatoric stress and resulting strain, and on the other hand deviatoric stress
and strain rate characterise the macroscopic mechanical behavior of rocks. These relationships lead to the
constitutive equations linking deviatoric stress and strain rate; "constitutive" as they depend on the
constitution of the material.
37
SECTION 1
Elastic, Plastic and Viscous Flows
In this section
1. Brittle vs ductile
deformation
2. Elastic, plastic, and
viscous flow curves
Elastic
Viscoelastic
σ ε = σ (1 - e-t/τ)
ε=
E
E
Viscoplastic
3. Rheology of
polycrystalline rocks
4. Steady-state flow laws
5. Sensitiviy of flow laws
Primary Creep
Secondary Creep
Rocks display a very large range of response
when submitted to deviatoric stresses. Over millions of years, hot rocks can flow under small deviatoric stress like a very low viscosity fluid.
Rocks can break under a sudden deviatoric stress
load (e.g. hammer). Rocks can resist moderate
deviatoric stress, until it reaches a threshold from
38
which permanent strain accumulates. Rocks can
deform elastically since they transmit mechanical
waves (sound wave, seismic wave). The rheology of rocks is therefore complex and covers a
broad range of mechanical behavior from elastic,
plastic and viscous.
BRITTLE VS DUCTILE DEFORMATION: The role of temperature, composition and strain rate...
Objects deform as a continuum (continuous deformation via elastic deformation or via viscous flow) when the flow unit are atoms moving
in the lattice of crystal. In contrast, when the flow unit are fragments of crystals or fragments of rocks moving relatively to each other because of fractures or faults, deformationis said to be brittle (discontinuous deformation via frictional flow).
Discontinuous deformation:
• Case 1: The media is pre-fractured, its strength depends on the frictional strength of pre-existing fractures and faults.
• Case 2: The media has no pre-existing fractures, its strength depends on its constitution in particular on the cohesion of its grains. This
cohesion must be overcome before fractures and faults can accommodate frictional sliding.
Continuous deformation:
• Elastic deformation
• Plastic deformation
• Viscous deformation
Ductile deformation. Photograph P. Rey
Brittle deformation. Photograph Pui-Leng, flickr
39
CONTINUUM MODELS OF ROCK’S MECHANICAL BEHAVIOR
Each type of flow has a characteristic flow curve in the deviatoric stress vs strain space. The graph on the right shows the flow curves for
three elementary deformation mechanisms:
σ
σ
Ideal Plastic
Elastic
σ*
Deviatoric stress
• Linear elastic flow (in blue): For small stresses, most material are elastic. The characteristics of elastic flow are: 1/ Elastic flow occurs as soon as stress is applied. 2/ Strain increases
as long as stress keeps increasing. The elastic flow curve is linear, its slope is 1/E with E the Young modulus, a physical
properties characteristics of elastic flow. 3/ Strain does not
accumulate if the stress is maintained constant. 4/The material recovers its original shape when stress is relaxed.
The mechanical analogue for elastic deformation is a spring.
ε=
σ
E
σ*
ε=
. f(t)
E
A
Plastic
ε
B
Viscous
• Plastic flow (in green): In most material elastic flow is limε= σ . t
η
ited to a certain level of stress beyond which the flow
switches from elastic to plastic. This limit is called the yield
stress (σ ∗). The characteristics of plastic flow are: 1/ Plastic
D
flow occurs as soon as the yield stress is reached (i.e. from
point A onwards). 2/ Plastic strain accumulates at a level of
C
1/E
stress which is constant and equal to the yield stress. Hence,
Plastic
Elastic
Cumulative strain (ε )
the amount of strain depends on the duration t over which
stress is applied. 3/ The plastic component of strain is permanent. When the imposed deviatoric stress is removed (B>C),
the material recovers the component of elastic deformation only. 4/ For and ideal plastic material there is no elastic component.
The mechanical analogue of plastic deformation is a rigid block sliding on a rough planar surface.
• Linear viscous behavior (in purple): Linear viscous material (newtonian viscosity) have no yield stress. The flow curve is characterized
by a linear relationship between stress and strain. Strain accumulates at varying and also constant stress (dashed purple line from D) and
when the stress is removed the flow stops but the material does not return to its undeformed state. The mechanical analogue of viscous deformation is a dashpot.
40
In a graph strain vs time, in experiments where the deviatoric stress is applied at t = 0 and maintained constant for t > 0, elastic, plastic and
viscous flows are characterised by three contrasting relationships.
Elastic strain: When the deviatoric stress is imposed, the material records a finite amount of elastic strain. This amount of elastic strain remains constant
σ
through time (and equal to ) as long as the deviatoric stress is maintained.
E
Ideal Plastic
ε = σ*E-1 + α . t
ε=
Cumulative strain (ε )
Plastic strain: The deviatoric stress is constant and equal to the yield stress. At
σ∗
t = 0, a component of elastic strain ( ) is instantaneously recorded before the
E
onset of accumulation of plastic strain. Plastic strain accumulates as long as
the deviatoric stress is maintained. For ideal plastic flow, the rate of accumulation (α) is constant.
σ = Constant
Viscous
Elastic
ε=
σ
E
Viscous strain: Viscous strain accumumates even for a very small deviatoric
σ
stress. This strain accumulate at a constant rate ( ), which depends of the
η
shear viscosity (η) of the material. Viscous strain accumulates as long as the deviatoric stress is maintained.
Time
σ
RHEOLOGY OF POLYCRYSTALLINE ROCKS
Elastic
Plastic
Hardening
C
41
C''
σ*'
Creep
B
Failure
σ*
Deviatoric stress
Crystalline rocks display a mechanical behavior that incorporates the three elementary flows; rocks are elasto-visco-plastic material. This graph shows a characteristic flow curve for a polycrystalline material. At stresses below the yield
stress (σ ∗), polycrystalline material behave elastically (blue curve). Above the
yield stress, the material behave plastically. At low level of strain (green curve
from A to B) the material becomes stronger (hardening) as the applied stress
must increases in order to keep the material deforming (i.e. the yield stress
must be exceeded for strain to accumulate). At high level of strain (red line beyond B) the material flows under a constant stress. Under the hardening plastic
σ .t
η
A
E
C'
Cumulative strain (ε )
regime, the removal of the driving stress leads to the removal of the elastic component of the deformation (curve CC'). If the sample is restressed, elastic deformation occurs under an extended domain as the yield stress has increased (curve C'C''). This means that the material
has become stronger.
To understand how hardening occurs one must understand how strain is achieved at the microscopic scale. Plastic strain affects the arrangement of atoms in the lattice of minerals. During strain, this arrangement is perturbated by the introduction of defaults or gaps in the
lattice called dislocations. Each dislocation introduces a local elastic strain. Under stress, these defaults move around leading to permanent
deformation. In the hardening plastic regime, the density of defaults increases but low stress levels impede their displacement, hence the
elastic energy in the crystal lattice increases making the material stronger. The animations show how dislocation moves about:
Dislocation wall
Motion of dislocations is driven by internal elastic
energy. Dislocations organize themselves into arrays such as dislocation walls to minimize internal
elastic energy. This results in the multiplication of
sub-grains and therefore permanent plastic strain.
sub-grain
Movie 4.1 Dislocation glide
Movie 4.2 Dislocation climb
42
Movie 4.3 Dislocation glide and climb
σ = Constant
Secondary creep
A
stic
Plastic
covered whereas the viscoelastic deformation is recovered over a period of time (t'1 - t'2). The sample, however, records a permanent plas-
la
oe
Elastic
V isc
oe
c
strain rate. Upon unloading elastic deformation is instantaneously re-
Failure
ta
Cons
Vis
strain and time implying that the material is deformating at constant
A'
Rate
n
i
a
r
t
nt S
Elastic
Primary
creep
σ*
Secondary creep is characterised by a linear relationship between
Tertiary creep
Elastic
Primary creep corresponds to a reversible flow for which elastic deformation is instantaneously removed following unloading of the sample (at time t1), whereas another component of strain called viscoelastic deformation is also recovered but over of a time window t1 - t2.
Cumulative strain (ε )
The viscous flow of polycrystalline material can be illustrated on a
space strain vs time for experiment performed at constant stress. In
this graph, the plastic flow curve can be divided into three regimes
called: Primary, Secondary and Tertiary creep.
t1
la s
tic
t2
t'1
t'2
Time
tic deformation. It is within the secondary creep regime that the creep
parameters that govern the rheology of rocks in the ductile regime are determined. The tertiary creep corresponds to the development of a
mechanical instability in which an increase in strain rate leads to the mechanical failure of the stressed sample. Real rocks display a complex elasto-visco-plastic behavior. This mechanical behavior can be represented by a
combination of springs (elastic component), blocks on a rough surface (plastic component), and dashpots (viscous component) connected in series or in parallel to fit a
real flow curve.
The rock analogue on the right has the same flow curve than than presented above.
At depth > 10 to 20 km, the deformation of rocks is characterised by slow steadystate creep at constant strain rate, that can accommodates large amounts of ductile
deformation. It is generally assumed that steady-state constitutive equations of
rocks, or flow laws, can be used to characterise the large-strain high-temperature
ductile deformation that occurs in the Earth.
43
Elastic
Viscoelastic
σ ε = σ (1 - e-t/τ)
ε=
E
E
Viscoplastic
Primary Creep
Secondary Creep
SECTION 2
Strength envelopes
In this section
1. Low-Moderate
stress
Yield
stress (MPa)
2. High-stress3000
regime
2000 1000
Yield stress (MPa)
1000
0
2000 1000
Yield stress (MPa)
1000
0
2000 1000
Yield stress (MPa)
1000
0
2000 1000
1000
0
3. Brittle regime
4. Sensitivity of flow curves
20
20
20
20
5. Rheological profiles
Moho
6. Sensitivity of profiles
40
40
60
80
789MPa 255MPa
[12] 100
40
60
40
60
80
80
598MPa 193MPa
[11] 100
584MPa 213MPa
[16] 100
The notion of strength envelope is relative to the
evolution with depth of the deviatoric stress necessary either to exceed the yield stress, or to
achieved a pre-defined strain rate through viscous
flow. At low temperatures, where plastic strain
dominates, the strength envelope is mainly a function of pore fluid pressure and the tectonic re44
60
80
582MPa 212MPa
[1] 100
gime. At higher temperature where viscous creep
dominates, the stength envelope is sensitive to
rock types, strain rate, and the magnitude of the
deviatoric stress. Here we start by giving the relationship deviatoric stress vs temperature (i.e.
depth).
FLOW LAWS FOR STEADY-STATE CREEP: LOW TO MODERATE-STRESS REGIME
The constitutive equation that accounts for most low to moderate-stress, steady-state strain, is the power-law creep equation, so called because the absolute value of the steady-state strain rate is proportional to the differential stress raised to a power n. The following equations
•
give the strain-rate ( ϵ ) as a funtion of the differential stress, and the differential stress as a function of the strain rate:
•
1/n
ϵ
E
−E
⋅Exp
ϵ = A⋅σ ⋅Exp
re-aranging σ =
( n⋅R⋅T )
(R.T)
(A)
•
n
where A is a constant (Pa-n s-1), n is the stress exponent that characterises the sensitivity of strain rate on the differential stress (n is dimensionless), E is the activation energy per mole for the creep process (J mol-1), it is the energy barrier that inhibits the creep mechanism, R is
the Boltzmann constant (8.3144 J mol-1 K-1), and T is the temperature (K). The constants A, E and n are characteristic of material. The power
law creep shows that both temperature and differential stress have a large effect on the strain rate. Thus an increase in temperature increases the strain rate for a constant stress, or lowers the stress required to produce a given strain rate.
n=1
n=3
n=
8
σ1−σ3
This effect is accounted for by the rapid increase, with increasing temperature, of the exponential term in the equations above. The graph shows that as n increases from 1 to large values,
power-law material evolves from viscous material (for n=1 the power law becomes a linear relationship between stress and strain rate) to near ideal plastic material. For most rocks at moderate stress level 2<n <5, whereas at low level stress, 1<n<2.
ε
FLOW LAWS FOR STEADY-STATE CREEP: HIGH-STRESS REGIME
Power-law creep implies that at 500ºC olivine would only deform at unrealistically high stresses.
A better description of the behavior of olivine at high-stress regime (> 200MPa) is given by the
Dorn's law, a relationship that is not as temperature dependent as the power law, in which Qd is
an activiation energy, σd a critical stress that must be exceeded, and εd is the critical strain rate.
45
σ = σd ⋅ 1 −
•
ϵ
R⋅T
⋅ln •d
Ed
(ϵ)
FLOW CURVES IN THE BRITTLE REGIME
With pre-existing fractures: At low temperature or at high strain rate or under high pore-pressure, but mainly in the upper crust and the upper mantle, the failure mechanism is modelled as frictional sliding:
σ = β⋅(1 − λ)⋅(ρ⋅g⋅z)
where g is the gravitational acceleration, λ is the ratio of fluid pore pressure to the normal stress, ρ the density. β is a parameter dependent
on the type of faulting given by:
β=
R−1
1 + ( σzz − σ 3 ) ⋅(R − 1)
σ −σ
1
with:
R=
(
)
1 + μ2 − μ
−2
3
and μ the coefficient of internal friction that characterises the roughness of the fracture plane.
Because the coefficient of internal friction varies little around 0.75 (pretty much independent on the rock composition), we get that R=4,
and therefore β varies continuously from 0.75 for normal dip slip faults, to 1.2 for strike-slip fault, to 3 for reverse dip slip faults depending
of the exact value of the principal stresses ratio. The frictional sliding equation shows that it takes less differential stress to achieve brittle
failure under extensional stress regime (maximum principal stress vertical) than under a compressional stress regime (maximum principal
stress horizontal). This makes sense since in a compressional stress regime gravity acts against reverse faulting but enhances the principal
stress in extension.
Without pre-existing fractures: In this case, the strength of rocks includes the cohesion (C) holding the grains together at atmospheric pressure. The failure mechanism is modelled by the Coulomb's criteria linking the shear stress ( τ ) acting on a fault plane and the normal stress
( σn ) acting perpendicular to it: τ
= C + μ⋅σn
The normal stress and the shear stress are related to the orientation of the fault plane with respect to the principal stress axes ( σ1, σ2 and σ3):
σn =
σ1 + σ3 σ1 − σ3
−
⋅cos(2α)
2
2
and
τ=
σ1 − σ3
⋅sin(2α)
2
with α the angle between the fault plane and σ1.
46
SENSITIVITY OF FLOW CURVES TO THERMODYNAMIC PARAMETERS AND CHEMICAL ENVIRONMENT
Confining Pressure and Temperature
250
Wombeyan Marble
σ3=100 MPa
200
150
σ3=35 MPa
100
σ1−σ3 (MPa)
Confining pressure prevents rocks from falling apart. It is therefore not surprising that increasing confining pressure increases the amount of deformation a sample can accumulate before failure. For instance, in the example of the Wombeyan
marble brittle failure is either reached for a larger amount of accumulated strain
or even impeded when a higher confining pressure is applied. This higher confining pressure allows the sample to sustain a larger differential stress. Temperature
enhances the ductility of materia. For instance, in the example of the Solenhofen
limestone, raising temperature decreases the magnitude of the deviatoric stress
that can be supported, and increases the amount of accumulated strain before
brittle failure . Under high temperature material can therefore accumulate more
strain but they can do so for a smaller amount of differential stress.
σ3=10 MPa
50
Failure
Failure
1
400
2
Cumulative strain (ε )
ε
Solenhofen Limestone
300ºC
Failure
300
500ºC
Failure
600ºC
200
σ1−σ3 (MPa)
3
Failure
100
Pc~40MPa
2
Paterson and Wong, 2006. Experimental Rock Deformation – The Brittle Field. Surveys
in Geophysics, 27, 4, pp 487-488.
47
4
6
8
Cumulative strain (ε )
10 ε
H2O and Pore Fluid Pressure
Water in rocks acts as a softening agent. Indeed, compared to a dry sample, a wet sample deforms at lower deviatoric stress (bottom left figure). The presence of fluids enhance mechanism of deformation controlled by coupled dissolution (at high stress region) and precipitation
(in low stress regions). Pore fluid pressure acts against the confining pressure and therefore promotes failure at lower differential stress and
lower strain (figure on the right). This process is called “hydraulic fracturing”, this is the process explaining micro-seismicity following
dam water loading.
300
Yule Marble
Dry
600
Solenhofen Limestone
0.5% H20
60 MPa
200
400
100
T=300ºC
Pc~500MPa
4
σ
σ
ε
200
Failure
93 MPa
3
Failure
65 MPa
Failure
78 MPa
6
9
Cumulative strain (ε )
1
σ
3
σ
8
12
18
Cumulative strain (ε )
σ1−σ3 (MPa)
σ1−σ3 (MPa)
2.9% H20
1
3
Fluid assisted diffusive mass-transfer through the
lattice (Nabarro-Herring creep) or along the grain
boundaries (Cobble creep)
@ Mervin Paterson, ANU
48
12
ε
RHEOLOGICAL PROFILE OF THE CONTINENTAL LITHOSPHERE
It is time now to synthesise what we have learn about the rheology of crystalline rocks to determine the strength profile (i.e. envelop) of the
continental lithosphere.
The strength of the lithosphere can be defined as the vertical integration, from the top to the bottom of the lithosphere, of the differential
stress required to trigger either brittle failure or the flow failure of rocks. Failure can occur by power-law creep or Dorn law creep at high
temperature and low strain-rate, or by frictional sliding at low temperature and high strain rate, in which case the differential stress depends of the tectonic regime. However at any given depth, the failure mechanism is the one that requires the minimum differential stress to
operate.
The strength of rocks at any depth in the crust is the lowest differential stress to reach either brittle or flow failure:
σ1 − σ3 = β⋅(1 − λ)⋅(ρc ⋅g⋅z)
•
or
1/nc
ϵ
σ1 − σ3 =
( Ac )
Ec
⋅Exp
( nc ⋅R⋅T(z) )
The strength of rocks at any depth in the lithospheric mantle is the lowest differential stress to reach either brittle, or power law, or Dorn
law failure:
•
or
σ1 − σ3 = β⋅(1 − λ)⋅(ρm ⋅g⋅z)
1/n
Em
ϵ
for (σ1 − σ3) < 200MPa
σ1 − σ3 =
⋅Exp
)
(
A
n
⋅R⋅T(z)
( m)
m
or σ1 − σ3
= σd ⋅ 1 −
•
ϵd
R⋅T(z)
for (σ1 − σ3) > 200MPa
⋅ln •
Ed
(ϵ)
49
The figure below shows the rheological profiles for a compressional and an extensional tectonic regime. The rheological parameters for the
crust and the mantle are that of a granite (quartz >40%) and a dunite (peridotite with >80% olivine) respectively. The power law and Dorn
creep law are dependent on both the temperature T and the strain rate. Therefore, these parameters must be defined first. The geotherm
here is such that the temperature at the Moho is 460ºC and the choosen strain rate is 10-15 s-1.
The straight parts of the profiles (in the upper crust and the upper mantle) represent brittle failures. The red lines are the Dorn law creep
curves, whereas the black curved lines are the power law flow curves. One can see that the differential stress for both the power law and
Dorn law creep strongly decrease with T.
β=3
Strength in extension
β=0.75
Crust
Ac = 5x10-6 MPa-ns-1
nc = 3.0
Ec = 190 kJ.mol-1
Mantle
Am = 7x104 MPa-ns-1
Ed=540 kJ.mol-1
nm = 3.0
Stress Treshold=8500 MPa
ed=3.05x1011s-1
Em = 520 kJ.mol-1
σ1-σ3 (MPa) 1000
0 1000
Compression
Extension
20
Moho
40
60
Dorn Law +
Power law
TMoho=460ºC
Depth (km)
Strength in compression
80
100
A few things to note:
• With the Dorn law, brittle failure does not occur in compression. Dorn law creep significantly reduces the strength of the upper mantle.
• The brittle part of the crust is thicker in extension that it is in compression.
• For a normal geotherm (TMoho <650ºC), the upper crust and the upper mantle are the strongest layers of the lithosphere, the lower crust
and the lower lithospheric mantle are comparatively much weaker.
• Because quartz deforms by ductile flow at lower temperature (~300ºC) than olivine (ductile at ~600ºC) the upper mantle is much stronger
than the lower crust for a normal geotherm (TMoho <650ºC).
50
SENSITIVITY OF STRENGTH ENVELOPS TO TEMPERATURE
The figure illustrates the dependence of the strength envelops on temperature. From left to right the temperature at the Moho increases
from 400ºC up to 700ºC. The strength of the lithosphere, in both compression and extension (the surface area of the dark and pale blue regions respectively) has been averaged over the lithospheric thickness assuming for the mantle either a power law creep or a combination of
power law and Dorn law. As temperature increases, the averaged strength of the lithosphere decreases significantly.
0 1000
2000 1000
0 1000
80
80
Dorn law+ 438 MPa 221 MPa
Power law -40%
100 -7%
TMoho=400ºC
410 MPa 151 MPa
-41%
239 MPa 142 MPa
100 -6%
-41%
TMoho=460ºC
100 -6%
TMoho=540ºC
40
60
60
236 MPa 108 MPa
106 MPa
158 MPa
99 MPa
100 -8%
-33%
61 MPa
80
80
101 MPa
-5%
TMoho=620ºC
The integrated strength of the continental lithosphere (in Nm-1) is defined by the vertical integration (i.e. over depth) of the rheological profile. Since the frictional sliding depends on the tectonics regime, three "strengths" can be defined for compressional (Fedt), transcurrent (Feds), and extensional (Fedc) tectonic regime:
The figure on the right shows the integrated strength as a function of temperature at the Moho.
At TMoho about 500ºC the upper mantle is the stronger layer of the lithosphere (in blue in the inset). Because the power law creep is exponentially dependent on temperature, a small increase in
temperature significantly reduces the integrated strength. Indeed, when TMoho is close to 700ºC
the strength of the upper mantle drastically decreases. Past 700ºC, the stronger layer of the lithosphere is the brittle upper crust the rheology of which has no dependence on temperature. The
red and the blue curves are the integrated strength in extension and compression respectively.
51
20
40
80
333 MPa 183 MPa
0 1000
20
60
570 MPa 195 MPa
1000
0 1000
40
60
60
2000 1000
20
40
40
Power law 737 MPa 237 MPa
0 1000
20
20
Moho
2000 1000
57 MPa
100
-7%
TMoho=700ºC
500ºC
40
700ºC
30
20
10
TMoho (ºC)
0
-10
400
600
800
1000
Strength
-20
-30
-40
-50
Depth
1000
Integrated Strength x 1012 N.m-1
σ1-σ3 (MPa) 2000
700ºC
500ºC
SENSITIVITY OF STRENGTH ENVELOPS TO LITHOLOGIE
The continental crust displays a wide range
of compositional variation from mafic
rocks (gabbros, amphibolites) to quartz
dominated composition (granites). This
contrast with the relative homogeneity the
mantle (peridotite with 60%<olivine<95%).
The figure on the right illustrates the dependence of the rheology of the lithosphere
on its lithological composition. All rheological profiles involve the same geotherm
with a temperature at the Moho of 540ºC
(thin dashed line TMoho = 600ºC).
Depending of the composition and the
rheological parameters the integrated
strength in contraction of the continental
lithosphere varies over one order of magnitude from 790 MPa down to 78 MPa. In extension the integrated strength ranges from
255 MPa down to 50 MPa.
The large variability of the rheological properties of common crustal and mantellic
rocks, prevents the definition of a “standard” rheology for the continental lithosphere.
Yield stress (MPa)
3000 2000 1000
Yield stress (MPa)
1000
0
2000 1000
20
Moho
Yield stress (MPa)
1000
0
[12] 100
[11] 100
1000
2000 1000
20
584MPa 213MPa
2000 1000
582MPa 212MPa
20
[1] 100
1000
0
2000 1000
1000
0
20
40
40
60
60
60
80
80
80
460MPa 178MPa
433MPa 166 MPa
[6] 100
[4] 100
3000 2000 1000
1000
0
2000 1000
20
1000
0
20
2000 1000
60
20
Moho
40
40
60
60
60
80
80
80
406MPa 163MPa
[13] 100
1000
40
TMoho=540ºC
80
100
0
Strength in
extension
40
425MPa 155 MPa
[14-15]
2000 1000
TMoho=600ºC
1000
0
20
20
40
40
60
80
404MPa 160 MPa
379MPa 149MPa
365MPa 153MPa
355MPa 147MPa
[5] 100
[7] 100
[9] 100
[10] 100
52
Strength in
compression
20
Moho
40
Moho
Mantle
80
[16] 100
1000
0
Lower crust
60
80
598MPa 193MPa
Upper crust
40
60
80
789MPa 255MPa
1000
0
20
40
60
80
2000 1000
20
40
60
0
2000 1000
20
40
3000 2000 1000
Yield stress (MPa)
1000
0
60
80
78MPa 49MPa
[2wet]
100