Chapter 1
Stretching continents
The most important isostatic consequence of crustal deformation relates to the changes
in the density structure of the lithosphere which dictates the
isostaticallysupported elevation of the lithosphere. For convergent deformations,
involving crustal thickening, the isostatic response is (generally) uplift (or mountain
building). For extensional deformations the isostatic effect is in part subsidence (or
basin formation) and in part uplift, depending on:
 how the
extensional strain is distributed within the lithosphere; and
 whether mantle melts
have been added to the lithosphere during extension.
Basins formed as the
isostatic consequence of extensional deformations are termed stretch basins and include
most continental passive margins formed during continental breakup events.
1 Isostatic calculations
1.0.1 Rift Phase Subsidence
The treatment of the isostatic consequences of an
extensional deformation is similar to te treatment of the uplift caused by convergent
deformation, only now we have also to consider the density distribution of the medium
filling the basin. We first consider the effects of a homogeneous extensional deformation
on the scale of the lithosphere with no magmatic additions (Figure ):
Figure 1: Lithosphericscale extensional geometry used to quantify
subsidence.
Using this
condition we can parameterize the deformation at any point within the basin in terms of
the one dimensional vertical strain:
Which for plane strain ([(e)\dot]_{yy} = 0) gives b = [(e)\dot]_{xx} or, equivalently, the horizontal stretch
across the basin. Note that b is the reciprocal of f, the thickening factor used
in Chapter 9.
We can formulate the isostatic effects of active rifting in the following
manner. Assuming an intial linear lithospheric geotherm, which amounts to ignoring the
effects of any internal heat production, gives the temperature T_{z} at depth z (Figure
) :

T_{z} = T_{s} + (T_{l}  T_{s}) 
z z_{l}




 (3) 
 

where T_{l} is the temperature at the base of the
lithosphere, T_{s} is the temperature at the surface of the lithsophere and z_{l} is the
thickness of the lithosphere prior to stretching (Figure ). Letting
T_{m} equal the temperature difference across the lithosphere, and setting T_{s} = 0, gives:
After
homogeneous stretching by a factor b the thermal structure is given by:
Figure 2: Temperature (a) and density (bd) structure assumed in
determining subsidence in a stretched basin. (b) shows denstiy structure prior to
stretching, (c) shows density structure immediately after instantaneous stretching, (d)
shows density structure after thermal sag.

T_{z} = T_{m} b Ê 
z z_{l}




 

 (5) 
 

The initial vertical density
structure is given by (Figure 2):

r_{z} = r_{c} 
æ ç
è

1Ê+Êa ÊT_{m}Ê 
æ ç
è

1ÊÊ 
z z_{l}

ö ÷
ø

ö ÷
ø





r_{z} = r_{m} 
æ ç
è

1Ê+Êa ÊT_{m}Ê 
æ ç
è

1ÊÊ 
z z_{l}

ö ÷
ø

ö ÷
ø




 (6) 
 

where r_{c} and r_{m} are
respectively, the density of the crust and mantle at the temperature T_{m} and a
is the volumetric coefficient of thermal expansion. Following stretching by b the
density structure is:



r_{z} = r_{c} 
æ ç
è

1Ê+Êa ÊT_{m}Ê 
æ ç
è

1ÊÊ 
z b z_{l}

ö ÷
ø

ö ÷
ø




r_{z} = r_{m} 
æ ç
è

1Ê+Êa ÊT_{m}Ê 
æ ç
è

1ÊÊ 
z b z_{l}

ö ÷
ø

ö ÷
ø



S_{i} + 
z_{c} b

< z < S_{i} + 
z_{l} b


 
S_{i} + 
z_{l} b

< z < z_{l} 
 (7) 
 

where r_{f} is the density of the medium filling the rift
basin of thickness S_{i}. The condition of isostasy amounts to equating the vertical
stress, s_{zz}, at a common depth, z_{l}, beneath the unstretched and stretched
basin. Beneath the unstretched lithosphere s_{zz} at z_{l} is given by (allowing
that r_{c} a_{c} = r_{m} a_{m}):



 

g z_{c} r_{c} + g (z_{l}  z_{c}) r_{m} + 
z_{l} 2

gÊÊ aÊ r_{m} ÊT_{m} 
 (8) 
 

Beneath the stretched basin s_{zz} at z_{l} is given by:



 

g r_{f} S_{i} + g Êz_{c} Êr_{c} b+ g Ê(z_{l}ÊÊz_{c})Êr_{m}Êb 
 

+ 
g Êz_{l}Ê aÊ r_{m}ÊT_{m} 2Êb

+ g r_{m} 
æ ç
è

z_{l}ÊÊS_{i}ÊÊ 
z_{l} b

ö ÷
ø


 (9) 
 

equating Eqn 8 with Eqn 9 and solving for S_{i}
gives (Figure ):
S_{i} = (bÊÊ1)Ê(Ê2 Êz_{c}Ê(r_{c}ÊÊr_{m})Ê+ÊT_{m}Ê a Êz_{l} Êr_{m} )Ê2 Êb Ê(r_{f}ÊÊr_{m})Ê 
 (10) 
Figure 3: Rift phase subsidence as a function of
stretching for a basin completely filled by sediment (r_{f} = 2500 kg m^{3}), and
water (r_{f} = 1000 kg m^{3}).
1.0.2 Sag Phase Subsidence
We assume that in the steady state the thickness of
the lithosphere is dictated by balancing the rate of heat flow through the lithosphere
with the heat supplied to its base by the convective motion in the interior. Any
deformation involving changes in the thickness of the lithosphere therefore induces a
departure from this equilibrium condition which in turn influences the ensuing thermal
evolution of the lithosphere. Since heat loss is proportional to the temperature
gradient in the lithosphere extensional deformation brings the asthenosphere closer to the
surface and consequently increases the heat loss through the lithosphere (Chapter 4).
This imbalance causes heat to be lost through the lithosphere more quickly than it is
supplied. The consequent cooling and thickening of the lithosphere causes it to become
more dense, inducing a kind of thermal subsidence or sag. There are two questions of
interest here:
 what is the relationship between the stretching
factor b and the induced thermal subsidence; and
 over what time scale does
the thermal subsidence take place (this question will be addressed in Section 10.4).
We can formulate the isostatic effects of the thermal or sag phase
subsidence, S_{t}, following a stretching by a factor b in the following manner.
Following reestablishment of the equilibrium lithospheric thickness, z_{l}, by conductive
heat loss the density distribution will be:




r_{z} = r_{c} 
æ ç
è

1Ê+Êa ÊT_{m}Ê 
æ ç
è

1ÊÊ 
z z_{l}

ö ÷
ø

ö ÷
ø




S_{t} + S_{i} < z < S_{t} + S_{i} + 
z_{c} b



r_{z} = r_{m} 
æ ç
è

1Ê+Êa ÊT_{m}ÊÊ 
æ ç
è

1ÊÊ 
z z_{l}

ö ÷
ø

ö ÷
ø




S_{t} + S_{i} + 
z_{c} b

< z < z_{l} 
 (11) 
 

where S_{t} is the subsidence due to thermal sag. Letting the total
subsidence, S = S_{t} + S_{i}, then s_{zz} at z_{l} beneath the saged basin is given
by:



 

g Ê r_{f} S + gÊ z_{c}ÊÊ r_{c} Êb+ 
 

g Ê r_{m} 
æ ç
è

z_{l}ÊÊSÊÊ 
z_{c} b

+ 
Ê z_{l}ÊÊ a ÊT_{m}Ê 2

Ê 
ö ÷
ø


 (12) 
 

Solving for S gives:
S = Ê 
z_{c}Ê(bÊÊ1)ÊÊ( r_{c}ÊÊr_{m})) bÊ(r_{f}Êr_{m})


 (13) 
and S_{t} (Figure ):
S_{t} = 
T_{m}Ê a Êr_{m}Ê z_{l}Ê (bÊÊ1) 2 ÊbÊ(r_{f}ÊÊr_{m})


 (14) 
1.0.3 Heterogeneous stretching of the lithosphere
The f_{c}f_{l} diagrams (note
that f is the reciprocal of b) shown in Figure allows the rift phase
subsidence to be calculated for any inhomogeneous stretching deformation, in which the
deformation is not equally partitioned between the crust and mantle (although in these
diagrams there has been no allowance made of the density of the medium filling the
depressions).
Figure 4: Sag
phase subsidence as a function of stretching for a basin completely filled by sediment
(r_{f} = 2500 kg m^{3}), and water (r_{f} = 1000 kg m^{3})
The f_{c}f_{l}Ê
parameterisation allows us to consider the isostatic effects of magma additions to the
crust during extension, as for instance has occurred during stretching of the Basin and
Range Province in the western USA. Magma additions add mass to the crust, with the result
that f_{c} > Êf_{l}Ê. In the Basin and Range Province magma addition appears to have
been sufficent to maintain a crustal thickness in the vicinity of 30 km, while the
underlying mantle part of the lithosphere has been severely attenuated (Gans, 1988). The
resulting path shown in Figure explains the anomalously high elevation of the
Basin and Range Province. Finally, the isostatic effects of stretching are dependent to
a large extent on the vertical density structure of the lithosphere prior to stretching.
For typical continental lithosphere we have shown that homogeneous (or pure shear)
thinning results in subsidence. In contrast, Figure shows that homogeneous
thinning of old oceanic lithosphere results in uplift or Ridge formation. Indeed, this is
exactly as we showed in Chapter 7, and this is exactly why the ocean ridges are
important engines for lithospheric motion and deformation.
Figure 5: f_{c}f_{l}Ê diagrams for stretching of
typical continental lithosphere (z_{c} = 35 km, z_{l} = 125 km, elevation contours in 1
km intervals) and typical oceanic crust (z_{c} = 6 km, z_{l} = 125 km, elevation contours
in 0.5 km intervals). Note that homogeneous stretching of continental lithosphere leads to
subsidence (elevation contours for subsidence assume no sedimentation or water filling,
to change to a filled basin multiply subsidence by r_{m}/(r_{m}  r_{f})) while
homogeneous stretching of oceans leads to uplift (i.e., ridge formations).
2 Mechanical consequences of extension
2.0.4 Inttroduction
So far we
have treated basin formation in terms of a kinematic model only, that is without
consideration of the forces that drive the deformation. In order to develop a mechanical
model we must consider the thermal evolution of stretch basins explicitly, because the
mechanical strength of the lithosphere is closely allied to its thermal state. We use the
mechanical model described in Chapter 2.
2.0.5 The time scale of thermal sag
As discussed in Chapter 4.3 a convenient measure of the response time
to thermal perturbations in media in which heat is transported only by conduction is
given by the thermal time constant, t:
where l is the length scale over which the perturbation occurs, k
is the thermal diffusivity (units m^{2} s^{1}). The thermal time constant provides a
measure of time for the decay of approximately 50% of the thermal perturbation, with
measurable deacy occuring up until about 3 t. The thermal time constant of the
continental lithosphere is of the order of 60 Ma (k = 10^{6} m^{2}s^{1}, l = 10^{5} m), and thus we may expect that subsidence associated with sag of a rifted basin
for a period of up to 150200 Ma (Figure ).
Since the decay of a thermal perturbation is proportional to the second derivative of the
temperature gradient with respect to depth (Chapter 4) and is therefore an exponentially
decreasing function of time. Consequently, the thermal sag subsidence should also decline
exponentially with time.
Figure 6: Evolution of the geotherm during sag following
instantaneous stretching at time t = 0.
2.0.6 Thermal evolution
at finite extensional strain rates
The instantaneous thinning of lithosphere will result in isothermal decompression of all
material points in the lithosphere. The resultant increase in the geotherm for an
instantaneous stretch is b. However, lithosphere stretched at finite rates will
evolve geotherms lower than the theoretical ïnstantaneous" geotherm for two reasons.
with important consequence for the evolution of strength within the lithosphere
.
 Firstly any stretching event will necessarily cause the
vertically integrated volumetric heat production of the lithosphere to diminish (by a
factor of b for homogenous stretching events). Assuming a steady state geotherm
prior to stretching then the declining heat production caused by stretching will lead to
cooling of individual material points within the lithosphere (Figure ), with
the amount of cooling for a given stretch inversely proportional to the strain rate.
Figure 7: Stretching leads to an upward migration of isotherms, and
thus an increase in the geotherm. However, for finite stretching rates individual
material points cool (as shown by arrowed lines), because the stretching reduces the heat
production in any vertical column through the lithosphere by the factor b (the
cooling is most pronounced in the crust where most of te heat producing elements reside).
This model shows the homogeneous stretching of a 100 km thick lithosphere by b =
2.0 over 30 Ma ([(e)\dot]_{z} = 2 x 10^{15} s^{1}).
 Secondly, lithosphere stretched at finite rates will evolve geotherms lower
than the theoretical ïnstantaneous" geotherm because the processes of rifting and sag
will overlap in time and space. The extent of overlap will depend on the both the strain
rate and the finite strain.
2.0.7 Mechanical Evolution
We have shown that
convergent deformation of the continental lithosphere induces weakening due largely to
the increase of the Moho temperature associated with increasing heat production within
the lithosphere. This provides a self localising mechanism for convergent orogens, with
the length scales dictated by the rheology, strain rate relationship. Similarly the
length scales for divergent deformation will be also controlled by rheology. The
lithosphere is, however, weaker in tension than in compression because brittle failure in
tension occurs at lower stress difference than in compression (Figure ).
Figure 8: See
text for discussion Cconsequenlty, for an equivalent geotherm and strain rate the across
strike length scale of an extensional orogen will be significantly less than
a compressional orogen.
For small initial finite stretches the lithosphere will undergo essentially isothermal
decompression (Figure 7). The consequent decrease in the depth of the power
law failure envelopes for quartz and olivine will induce a reduction in the strength of
the lithosphere; a form of thermal weakening (Figure a). However, with
increasingly large finite strains, the cooling of material points as they are
decompressed (Figure ) ultimately leads to an increase in whole
lithospheric strength (Figure ). This analysis suggests that early in
their evolution rift zones should be selflocalising but gradually becoming more
distributed. The point at which the switch from weakening to strengthening
occurs depends critically on the strain rate with the critical strain decreasing with
decreasing strain rate.
2.0.8 The role of magmas
Of course the thermal and mechanical evolution of
extensional orogens is depenedent not only on the rate of deformation but also on the
involvement of asthenospheric magmatic additions. Decompression of the asthenosphere
beneath stretched lithosphere may induce melting in an analogous fashion to melting
beneath ridges. For the normal potential temperature of the asthenosphere (1280^{o}C)
this requires decompression of asthenosphere beneath depths of approximately 45 km.
Assuming initial lithospheric thicknesses of the order of 100 km or more the minimum
b value required for melting of the mantle beneath the stretched lithsophere is about
2. Once melting occurs, the rapid advection of heat into the overlying lithosphere will
cause dramatic strength reductions, potentially overriding the strain hardening
associated with large strains mentioned in the previous section. Moreover, the isostatic
consequences of magma addition are to increase the elevation of the lithosphere, as is
the case in the Basin and Range Province, potentially giving rise to extensional buoyancy
forces which augment the original driving forces for stretching. This magmatically
augmented weakening may be an essential requirement for the complete rupturing of
stretched continental lithosphere to form new spreading oceans.
2.0.9 The role of mantle plumes
In the discussion above we have not explicitly
considered the origin of the forces that drive extension of the continental lithosphere.
While it is clear that compressional forces within the continents can be generated by the
tractions exerted by subducting slabs it is not at all clear that substantial extensional
force may be exerted on normal thickness continetal lithosphere. This is because normal
thickness lithosphere (35 km thick crust and 125 km thick lithosphere) is in approximate
potential energy balance with the midocean ridges; that is the ridges seem to neither
exert compression nor tension on normal continental lithosphere. Of course regions of
elevated topography within the continents, such as associated with thickened crust
formed in zones of compression, will naturally experience tension when the forces driving
convergence are relaxed. Another important way of increasing the potential energy of the
lithosphere may be through the impingement of a mantle plume at the base of the
lithosphere, which may have the effect of jacking up the lithosphere by as much as 1 km.
The associated potential energy gain (for a column of unit area) is given to a first
approximation by:
U_{plume} = 
g r_{c} h^{2} 2


 (16) 
where h is
the additional elevation caused by the jacking effect of the mantle plume, and for crustal
density r_{c} = 2800 kg m^{3} is equivalent to a tensional force per unit length of
topography of about 1.4 x 10^{12} N m^{1} for every km of additional elevation.
An important additional effect of extension initiated by a plume relates to the higher
mantle temperatures which may be up to about 300^{o}C higher than the mean convective
mantle temperatures. The implication is that decompression melting in the asthenosphere
will begin at a much earlier stage in the rifting process. For example for a potential
mantle temperature of 1580^{o}C melting begins at about 120 km depth, implying that plumes
may begin generating melts without any appreciable deformation of the overlying
lithosphere. Moreover as mentioned in the previous section the transport of such melts
into the lithosphere must further enhance the potential energy of the lithosphere and
weakens it in such a way to augment any extensional deformation that has already begun.
Figure 9: Schematic representation of how lithospheric
strength changes as a function of stretching. For low finite strains and/or high strain
rates decompression is essentially isothermal and thus the plastic failure envelopes move
to shallower depths (since they are dependent on temperature and thus only indirectly on
depth through the geotherm  Eqn 5.2) inducing a reduction in lithospheric strength by an
amount equal to Area 1 + Area 3. A decrease in the depth of the Moho results in a strength
increase by an amount equivalent to Area 2. The net change in the lithospheric strength
is given by Area 1 + Area 3  Area 2. At low strain rates and/or high finite strains
cooling of material points occurs such that there is no rise in the plastic failure
envelopes with further deformation (i.e., Area 1 and Area 3 tend to 0).The consequent
strain hardening is equivalent to Area 2. White and McKenzie (1989) point out that in
mantle upwelling zones the temperatures may be in excess of 100^{o}C above the
typical potential temperature of the mantle. Stretching above such abnormally hot
asthenosphere will induce melting at rather lower b values than for normal mantle, and
for a given b value contribute a much greater amount of melt to the overlying
lithsophere. The variable magmatic history of extensioanl provibnces and rifted margins
may therefore relate to the thermal state of the subjacent convective interior during
rifting.
3 Sedimentation in stretched basins
We have seen
that the evolution of rift basins can be viewed in terms of two distinct phases: the rift
phase, and the sag phase. In the the rift phase sedimentation is associated with the
isostatic subsidence induced by the active deformation. Rift phase sediments show signs of
active tectonism, such as growth faults with rapid facies and thickness changes. Sag
phase sedimentation follows rift phase and represents the isostatic effects of freezing
asthenosphere onto the base of the thinned lithosphere as the geotherm returns to steady
state. Sagphase sediments are typically laterally extensive without abrupt facies
changes. Subsidence rates decrease exponentially through the sag phase with a
characteristic time scale indicative of the thermal time constant of the lithosphere.
3.0.10 The Steer's Head
Stretched basin stratigraphy commonly exhibits a Steer's
Head or Texas Longhorn shape with the sag phase sediments extending beyond the zone of
observable crustal stretching (Figure 8.9). This characteristic stratigraphy may be a
function of either the finite flexural rigidity of the lithosphere or by differential
stretching of the crust and mantle parts of the lithosphere (White & Mckenzie, 1988).
In the calculations above we made the simplyfying assumption that the lithosphere is
everywhere in local isostatic equilibrium. This amounts to treating the lithosphere as
infinitely weak (to vertical shearing) and therefore incapable of supporting bending or
flexural stresses. The lithosphere is not infinitely weak but rather has a finite
flexural rigidity. The consequences of the finite flexural rigidity are that loads on the
lithosphere, such as sedimentary or tectonic loads, are generally compensated on a
regional scale rather than a local scale, with significant local departures from
isostatic equilibrium. In the case of the deposition of sediment in rift basins the
finite flexural rigidity of the lithosphere allows compensation of the sedimentary load
beyond the zone of stretching allowing downward flexing of the lithosphere beyond the
stretched zone. Since the active rift phase is usually associated with a severe thermal
perturbation the flexural rigidity of the lithosphere is expected to be very low during
active rifting. However cooling attendent with sag pahse subsidence will increase the
flexural rigidity causing the loading due to emplacement of sediment to be compensated
on length scales which increase with time (Figure ).
Figure 10: Schematic cross section showing characteristic steer's head
formation of rift basins. White and McKenzie (1988) argue that the
flexural rigidity of continental lithosphere is insufficient to explain the Steer's Head
geometry and regard the characteristic geometry as a function of heterogeneous stretching
of the lithosphere, with the characteristic length scale for mantle stretching greater
than the length scale for crustal stretching. Indeed there is no good reason that
stretching should be homogeneous on the scale of the lithsophere, and the resulting
values of b_{c} and b_{l} (the reciprocals of f_{c} and f_{l}  f_{c},
respectively) are shown in Figure . This allows subdivision of the basin
into three regions which show characteristic elevation changes. Region a where the rift
phase subsidence is greater than for homogeneous stretching; region b where the rift
phase elevation change is either subsidence or uplift, but which show substantial sag
phase sedimentation with sedimetation onlapping the basin margins progressively through
the sag phase, and region c where there is only uplift during rifting and sag restores the
initial elevation of the lithosphere (i.e., no net subsidence). The respective f_{l}f_{c}
paths are shown in Figure .
Figure 11:
Heterogeneous stretching model for Steer's Head formation (after White and McKenzie,
1988). See text for discussion.