Chapter 12
Dynamics of the Lithosphere
In this chapter we study how the lithosphere deforms in response to external forces, and
how factors such as the thermal regime and lithology affect the deformation. First, we
study several factors controlling the strength of the lithosphere. Then, the strength is
related to the instability of the lithosphere that determines the fate of continental rifts
and the creation of spatially periodic deformation zones.
12.1
Strength profile
We have studied several deformation mechanisms, including elasticity, brittle faulting, and ductile
flow, that are relevant to describe the deformation of rocks. They exhibit different stresses for a
given tectonic condition with a specific depth, temperature, strain, strain rate, etc. When a rock mass
deforms, it has options for the mechanisms. Then, the deformation mechanism that minimizes the
resistance to the deformation is chosen. If the stress accompanied by brittle deformation is less than
that of ductile deformation, the latter deformation mechanism is chosen. As the product of force and
distance is equal to energy, tectonic deformation proceeds with the minimum energy dissipation.
Consider large-scale deformations such as a continental breakup resulting from extensional deformation of the lithosphere. In those cases we do not take elasticity into account because those
deformations surpass the elastic limit. Brittle and ductile deformations are important. If the deformation of a rift can be treated as plane strain on the vertical section across the rift zone, the resisting
stress is described by Eq. (6.21) as follows. No brittle faulting occurs if the differential stress,
σh − σv , is less than the critical one1 :
!
2μf (1 − λf )
(12.1)
σv ,
Δσbrittle = −
(1 + μ2f )1/2 + μf
where σv is the vertical stress that is usually equal to the overburden stress (Fig. 12.1). In the case
1 Note
that rifting occurs when σh < σv , so that σh − σv < 0.
303
304
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
of compressional tectonics, the critical stress is given by the equation
!
2μf (1 − λf )
σv ,
Δσbrittle = +
(1 + μ2f )1/2 − μf
(12.2)
and faulting occurs when the differential stress exceeds this. A rock mass breaks when it is subject
to stresses over this limit. Therefore, Eqs. (12.1) and (12.2) represent the brittle strength of the
rock mass under extensional and compressional tectonic regimes, respectively. The brittle strength
is insensitive to lithology.
The vertical stress, σv , is determined by overburden so that the critical differential stress describes the magnitude of horizontal stress. If tectonic stress is gradually built up and the horizontal
stress reaches the critical stress, faulting ‘immediately’ occurs in the geological sense. The brittle
deformation relieves the state of stress under the critical state. Consequently, the state of stress is
at or just below the strength under tectonically active regions: Eqs. (12.1) and (12.2) describe the
approximate magnitude of horizontal stress.
On the other hand, the stress due to ductile deformation depends on the lithology, strain rate, and
temperature. The deviatoric stress for the ductile deformation is given by the equation (§10.8.3),
1 1
Q −
−1
s = A n ε̇En exp
ė,
(12.3)
nRT
where ε̇E is the equivalent strain rate
(
1
ė : ė.
(12.4)
2
Δσductile = τ1 − τ3 decreases rapidly with increasing temperature. If the lithology and strain rate do
not change within a rock mass, temperature rises with depth and, consequently, Δσductile falls rapidly.
Because of the temperature increase with depth, the ductile strength decreases with it. Hence, the
abscissa in Fig. 10.18 that displays temperatures can be translated into depths. Taking the horizontal
x- and vertically downward z-axes, and assuming a pure shear
⎛
⎞
ε̇xx 0 0
ė = ⎝ 0 0 0 ⎠
(ε̇xx = −ε̇zz ),
0 0 ε̇zz
√
we have ε̇E = 2 |ε̇xx | from Eq. (12.4). If rocks are isotropic, we have Δσ = |τxx − τzz | and the
ductile strength
Q − 1 1 −1
ε̇xx − ε̇zz .
Δσ = 2A n ε̇xx n exp
(12.5)
nRT
Temperature T increases generally with depth, therefore Δσ decreases exponentially with it. The
ductile strength strongly depends on the lithology, as Eq. (12.5) includes material constants A and
Q. When differential stress applied to the lithosphere is smaller than the ductile strength or brittle
strength of rocks over the entire profile of the lithosphere, the lithosphere supports the stress by
elasticity.
ε̇E =
12.1. STRENGTH PROFILE
305
Figure 12.1: Schematic strength profile of the lithosphere with a uniform lithology. Brittle and ductile strengths are shown by solid lines. Rocks deform by brittle failure above the brittle-ductile
transition (BDT) because brittle deformation is easier than ductile deformation in that the former
needs less differential stresses than the latter, whereas ductile deformation prevails under BDT. The
strength of quasi-plastic deformation is shown by dotted lines. Since the brittle strength in extensional tectonic regimes is smaller than that in compressional tectonic regimes (§6.7), the lithosphere
is more vulnerable to extensional than compressional tectonics.
The brittle strength increases and the ductile one decreases with increasing depth. Deformations
are accommodated either by a brittle or ductile mechanism that needs lower differential stress for the
deformation. Hence, there must be a transition region at which brittle deformation gives way to ductile deformation. This is known as the brittle-ductile transition2 . It is often found that earthquakes
occur in the crust no deeper than 15–20 km. This is probably because the transition occurs at that
depth.
The solid lines in Fig. 12.1 shows the yield strength of constituent rocks at each depth, so that the
graph exhibiting such strengths is called the strength profile of the lithosphere. There is a difference
in the brittle strengths in normal and reverse faulting stress regimes (Fig. 6.14). The former strength
is smaller than the latter one so that the brittle-ductile transition in the normal faulting regime is
deeper than the other.
The oceanic lithosphere has abundant olivine minerals which control the mechanical strength of
both the crust and the lithospheric mantle. Therefore, the oceanic lithosphere has strength profiles
like those in Fig. 12.1. By contrast, the continental crust has a much more complex structure that
has been shaped in the long history of the crust. It has significant heterogeneity in lithology. Ductile
strength strongly depends on the lithology. In addition, impermeable rocks make overpressured rock
2 The fact is that there is a regime where the deformation of rocks is accommodated by ductile and brittle mechanisms at
the same time, known as quasi-plasticity , around the brittle-ductile transition (Fig. 12.1) [213]. However, the quasi-plastic
regime is not well understood, so that we do not refer to that regime any more.
306
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
Figure 12.2: Schematic picture showing the strength heterogeneity of the crust associated with the
heterogeneity in lithology and other factors including pore fluid pressure and temperature. Strength
is designated by the gray scale.
Figure 12.3: Diagram showing the strength profile of continental lithosphere with homogeneous
crust. Dashed lines indicate brittle strengths for two stress regimes. In this figure a brittle regime
exists in the upper mantle for the case of extensional tectonics. However, the existence depends on
the temperature of the depths.
masses. Rocks with abundant radioactive elements affect the thermal regime in the crust. Therefore,
the continental crust must have an intricate strength distribution, which is schematically shown in
Fig. 12.2. However, we assume simple structures for the following considerations. Namely, we
assume that the entire crust is composed of one type of rock, or that the crust has two layers with
different lithology. If the crust has a single layer, the continental lithosphere has a strength profile
like those in Fig. 12.3.
12.2. STRETCHING INSTABILITY OF THE LITHOSPHERE
12.2
307
Stretching instability of the lithosphere
Infinitesimal strain until whole lithosphere failure
Consider that Δσ(z) denotes the strength profile, then the integral
tL
Fc =
Δσ dz
(12.6)
0
is the maximum force for the lithosphere to support without finite pure shear deformation, where
the depth tL denotes the base of the lithosphere. The force is designated by the area bounded by the
strength profile.
Unfortunately, the lithosphere-asthenosphere boundary is fuzzy, i.e., the strength decreases con
tinuously as exp Q/nRT (z) in the boundary zone. Accordingly, the base of the lithosphere is
sometimes defined for convenience as the depth where Δσ σzz is satisfied. More simply, the
depth is sometimes defined as the depth where the differential stress reaches a small specific value
in the range 10–20 MPa [191].
What if a tectonic force greater than Fc in Eq. (12.6) is applied to a continental lithosphere
that was initially at rest? Consider the initial distribution of Δσ being equal to that of the Earth
pressure at rest down to a certain depth that defines the base of the lithosphere (Fig. 12.4). Under
that depth, the initial state of stress is assumed to be lithostatic. Once the force is applied, rocks near
the surface yield because of the near-surface low brittle strengths. The differential stress in the rocks
near the base of the lithosphere exceeds the ductile strength so that the rocks yield immediately
after the force is applied. Namely, the elastic core begins to be eroded at the top and base. The
reduced thickness of the elastic core leads to an increase of force that the core have to support. The
increase further reduces the elastic core to form a positive feedback cycle [109] and, the entire depth
of the lithosphere eventually yields. This is called whole lithosphere failure. Until this phenomenon
occurs, the existence of the elastic regime prohibits finite deformation of the lithosphere. After the
failure of the whole lithosphere, finite deformation of the lithosphere begins.
Finite strain
The history of finite deformation of a rift can be revealed from syn- and post-rift sedimentary records.
Continental rifting can lead to continental breakup. However, some rifting is sometimes aborted. The
North Sea rift is an example of failed rifts. Intra-arc rifts have the two ends, also. Intra-arc rifting
leads to back-arc spreading as the rifting in the Izu-Bonin-Mariana arc resulted in the spreading
in the Shikoku and the Mariana basins of the Philippine Sea plate. On the other hand, there are
the aborted rift basins of the Paleogene under the continental shelf behind the Ryukyu arc to the
northwest of the plate.
The duration of intra-arc rifting is one order of magnitude smaller than that of intra-continental
rifting [263]. If rifting has several phases, each phase lasts tens of million years in continents. The
activity of the Triassic rift in Fig. 3.24 is an example. By contrast, intra-arc rifting lasts a few million
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
308
z
Figure 12.4: Schematic picture showing the evolution of differential stress in the continental lithosphere leading to whole lithosphere failure.
years. The extensional tectonics that led to the opening of the Japan Sea backarc basin lasted 5 m.y.
[263].
The difference is explained by the initial thermal structure of the lithosphere. Intra-arc rifting
occurs in volcanic arcs because magmatism has heated up and reduced the ductile strength of the
lithosphere before rifting. For example, temperature is estimated by a petrological method in the
Northeast Japan volcanic arc at 850 and 1400◦ at 25 and 80 km, respectively [107, 238]. The crust
has a steep geothermal gradient there. On the other hand, temperatures at depths of 25 and 40 km are
estimated to be 525 and 850◦ in the Basin and Range Province [179], indicating a lower geothermal
gradient.
Based on a plausible structure of the lithosphere, the duration of rifting can be estimated by
numerical simulations [55, 236]. For this purpose, we take the present thermal structure under
Northeast Japan and the Basin and Range Province as the references for volcanic arcs and continents.
In addition, we assume pure-shear rifting by a constant force. The evolution of a pure-shear rift is
calculated by the following one-dimensional model. The upper and lower crust is considered to
be composed of granite and gabbro, respectively, so that their ductile strengths are determined by
those of quartz and plagioclase. That of the mantle lithosphere is determined by olivine. Thermal
evolution is calculated by the one-dimensional equation
∂T
∂T
∂2 T
+v
=κ 2,
∂t
∂z
∂z
(12.7)
where T is the temperature, z is depth, t is the time since the beginning of rifting, v is the downward
velocity and κ is thermal diffusivity. Because of the pure-shear, v = zĖzz = −zĖxx(Eq. (3.81)).
Based on the petrological estimates, the boundary conditions T z=0 = 0◦ C and T z=a = 1300◦
12.2. STRETCHING INSTABILITY OF THE LITHOSPHERE
309
Figure 12.5: Strength profiles for model rifts [236].
C, where a is a depth of 50 km for the intra-arc rift and 74 km for the intra-continental rift. The
temperature distributions designated by the dotted lines in Fig. 12.5 are used for the initial conditions
for the rifts. The strength profiles for the strain rates of 1.432 × 10−15 s−1 and 7.954 × 10−17 s−1 ,
respectively, are shown in the same figure. The rate of the intra-continental rift is two orders of
magnitude smaller than that of the intra-arc rift, merely because the former has a smaller geothermal
gradient. Although the strength profile for the former has the maximum in the upper mantle, the
strength of the mantle part for the intra-arc rift is significantly smaller than the strength of the crust.
This is the most conspicuous difference in the strength profiles, and is due to the sensitivity of the
ductile strength of olivine to temperature compared to those of quartz and plagioclase. It is assumed
that the asthenosphere is passively uplifted under the rift by the same amount as the thinning of the
lithosphere. That is, mantle rocks are supplied into the region between the surface (z = 0) and the
depth z = a from below to compensate the thinning of the crust. We neglect the vertical movement
of the surface by rifting because the amount of movement is smaller than those of the upper-lower
crustal interface and of the Moho interface.
The thinning of the lithosphere raises isotherms to steepen the geothermal gradient. The temperature changes are calculated with Eq. (12.7) for every time step in the numerical simulation.
Consequently, the transient temperature structure is designated by a convex upward curve in Fig.
12.6. If rifting proceeds very slowly, the change in the temperature structure is small. In this case, a
largely isothermal thinning of the crust increases the occupation of the olivine-rich layer in the range
between z = 0 and a. Because the layer is stronger than quartz and plagioclase if the temperature
is the same, the increased occupation augments the strength of the lithosphere that is defined by
&a
Δσ(z) dz. The applied force is assumed to be constant, so that the raised strength decelerates the
0
rifting. The slowdown of the rifting decreases the rate of temperature change to complete a negative
310
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
Figure 12.6: Schematic picture showing the temporal variation of temperature structure.
feedback cycle to stop rifting.
On the other hand, if the lithosphere thins much more rapidly than thermal conduction through
the lithosphere, the thinning is substantially adiabatic. The result is a increased geothermal gradient
which further leads to a weakening of the lithosphere. If the tectonic force is the constant through
time, the reduced strength accelerates the rifting. Accordingly, we have a positive feedback cycle in
this case. Then, the rate of lithospheric attenuation goes to infinity. We interpret this as the breakup
of the lithosphere.
Consequently, the fate of rifting, either leading to breakup or cessation, depends on the rate
of thinning relative to thermal conduction through the lithosphere [55]. The lithosphere exhibits
stretching instability that determines the fate of rifting. The rate depends on several factors including
the tectonic force, the composition of the lithosphere, geothermal gradient, and pore fluid pressure.
Figure 12.7(a) shows the strain rate versus time from the beginning of the intra-arc rift. The rate
shows divergence if the force applied to the lithosphere is greater than a critical value of about 0.396
TN−1 [236]. In this case, the duration of rifting is a few m.y. If the force is just under this value, the
rifting is aborted after several m.y. from the initiation of rifting. The duration is consistent with the
ones observed for intra-arc rifts in the world.
The critical force for intra-continental rifts is one order of magnitude greater than that for intraarc rifts3 . Namely, the island arc lithosphere is much weaker than the continental lithosphere, so that
inter-plate deformations are concentrated in the arcs (Fig. 12.7(b)). The critical force depends not
only on the strength of the lithosphere but also on the thermal diffusivity κ, because the transition
is determined by the speed of lithospheric thinning relative to thermal conduction through the lithosphere. Rising magmas carry heat toward the surface to increase the effective thermal diffusivity of
3 The critical force shown in Fig. 12.7(b) is valid specifically for the conditions assumed in the numerical simulation.
It is suggested that diffusion creep of olivine instead of dislocation creep greatly reduces the critical force by one order of
magnitude for intra-continental rifts [82].
12.2. STRETCHING INSTABILITY OF THE LITHOSPHERE
311
Figure 12.7: Unstable behavior of the model rifts [236]. Thermal diffusivity κ is assumed to be
3.5 × 10−6 m2 s−1 . (a) Temporal variation of the strain rate of the intra-arc rift for different tectonic
forces indicated in the dimensions of TN−1 . (b) The relationship between thermal diffusivity and the
critical force to separate the fate of intra-continental and intra-arc rifts.
the lithosphere4 . The numerical simulation showed that intra-arc rifts tend to be formed in volcanic
arcs. However, the critical stress is insensitive to κ for intra-arc rifts compared to intra-continental
rifts (Fig. 12.7(b)).
When the Japan Sea back-arc basin opened, the lithosphere under NE Japan was extended to form
an intra-arc rift. The rapid subsidence of the rift is designated by the increase of paleobathymetry
from ∼16 Ma to ∼15 Ma shown in Fig. 3.14. Figure 12.8 shows the temporal change in the rate
of subsidence estimated from the stratigraphic records of the rift. The rifting began at around 20
Ma. Since then, the subsidence accelerated to reach ∼3 km/m.y. The rapid subsidence was abruptly
stopped at 15 Ma when the rifting was aborted. The duration of rifting was 3–5 m.y. Assuming
the local isostasy and initial thickness of the crust to be 35 km, which is estimated from the present
crustal thickness of the Shikhote Alin (Fig. 2.12), the amount of subsidence can be transformed into
the reduced thickness of the crust. The total subsidence at around 2 km corresponds to the crustal
thickness at 25–30 km at the end of rifting5 . The total subsidence corresponds to an extensional
strain of 20%. Therefore, the average strain rate was 2 × 10−15 s−1 . This value agrees roughly with
the strain rate for the failed intra-arc rift that is estimated by the numerical simulation (Fig. 12.7).
4 Numerical simulation by Honda to estimate the thermal structure under the volcanic arc uses the effective thermal diffusivity at 10−5 ms−2 , which is one order of magnitude larger than that of rocks [81]. The intra-arc rifting in Northeast
Japan was associated with massive volcanism. The representative thickness of the volcanics is about 2 km. Assuming
a heat capacity of 1.0 × 103 J kg−1 and a density of 2.5 × 103 kg m−3 , we have the heat capacity per unit volume at
(1.0 × 103 ) × (2.5 × 103 ) = 2.5 × 106 Jm−3 . In addition, assuming the temperature of magma to be 1000◦ C, the heat
transported by the volcanism was 5.0 × 1012 Jm−2 . Dividing this by the duration of volcanism (ca. 3 m.y.), we obtain the heat
flux of 50 mWm−2 . This is as great as the average heat flow from the solid Earth. A significant amount of intrusion may have
accompanied the volcanism so that the increased effective thermal diffusivity by one order of magnitude seems reasonable
for the Early Miocene rifting in Northeast Japan.
5 The present crustal thickness of Northeast Japan is greater than this thickness by 5–10 km, probably due to igneous
underplating since 15 Ma.
312
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
Figure 12.8: Temporal change in the subsidence rate of the Northeast Japan volcanic arc when the
Japan Sea back-arc basin opened [263].
In addition, the duration is consistent with the calculated one.
12.3
Factors controlling te of the continental lithosphere
It was seen in Section 8.3 that the effective elastic thickness of the oceanic lithosphere is approximated by the depth of the 400–600◦ isotherm, and that of the continental lithosphere has a positive
correlation with tectonic age and heat flow (§8.7). Figure 12.9 indicates that the latter has a much
better correlation. Those observations suggest that the effective elastic thickness of the lithosphere
is controlled by its thermal regime. Stable continents are stable because they generally have a low
average heat flow. However, the relationship between heat flow and te is complicated by the strength
profile of the continental lithosphere which is more intricate than the oceanic lithosphere. In this section, we consider the effective elastic thickness of the continental lithosphere in terms of its strength
profile.
We assume that the continental crust is composed of a single rock type for brevity, and that the
lithosphere behaves as a elastic-perfectly-plastic body, i.e., we deal with the lithosphere that has
strength profiles as shown in Fig. 12.10. The figure shows the vertical distribution of differential
stress in the flexed lithosphere for three cases. While the curvature is small, yielding occurs only
near the top and base of the lithosphere to leave a thick elastic core in the lithosphere. However, as
the curvature increases, the differential stress generated by the bending exceeds the strength of the
lower crust so that the elastic core is divided into two layers (Fig. 12.10(b)). If the interface between
the elastic layers with thicknesses of t1 and t2 is lubricated, the whole system has the effective elastic
thickness te ≈ max(t1 , t2 ) (§8.4.2). The effective elastic thickness before the separation was t1 + t2 ,
which is larger than max(t1 , t2 ). Therefore, te may decrease abruptly at a critical curvature [28].
12.3. FACTORS CONTROLLING T E OF THE CONTINENTAL LITHOSPHERE
313
Figure 12.9: Effective elastic thickness of the continental lithosphere te . (a) te versus tectonic age
[28]. (b) te versus heat flow in Africa [72]. The error bars in this plot that are designated in the figure
in [72] are omitted here for simplicity.
Figure 12.10: Strength profiles of a model continental lithosphere and the vertical distribution of
the differential stress (gray) generated by the concave upward bending of the lithosphere, where
Δσ = σH − σv . (a) Bending with a small curvature K of the lithosphere gives rise to yielding near
the top and base of the lithosphere. There is a thick elastic core indicated by the linear part of
the stress distribution in the lithosphere. (b) As the curvature increases, the linear part reaches the
curve that designates the ductile strength of the lower crust (dashed line). Yielding in the lower crust
divides the elastic core into two parts, one in the crust and the other in the lithospheric mantle (gray).
Namely, the crust and mantle is mechanically decoupled. (c) Vertical distribution of the differential
stress caused by the bending and simultaneous horizontal compressive force F .
This phenomenon is observed in collisional orogens [140]. Figure 12.10(c) shows a case where a
horizontal tectonic force F is applied to the bent lithosphere.
Given the flexure w(x) of the lithosphere, it is straightforward to calculate the curvature K = w .
Then, the horizontal strain εxx is obtained via Eq. (8.12). Accordingly, if ẇ(x) is also given, ε̇ xx
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
314
is obtained. The rate of strain is transformed to ductile strength. Therefore, we can calculate stress
from w(x) and ẇ(x). The bending moment is obtained from the stress as
tL
M=
(12.8)
σxx (z − zn ) dz.
0
Then, using the formula M = −DK, the flexural rigidity D and the effective elastic thickness te of
the flexed lithosphere can be estimated.
There may be depth ranges of yielding for given w(x) and ẇ(x) (Fig. 12.10(b)). In that case,
Eq. (12.8) is replaced by the piecewise integrals
M=
N i=1
(i)
σxx z − zn dz,
(12.9)
ith layer
(i)
where zn is the depth of the ith layer in which deformation is accommodated by a brittle ductile or
elastic mechanism. Using the bending moment in Eq. (12.9), the effective elastic thickness of this
rheologically multi-layered lithosphere is estimated. It should be noted that te is different from the
thickness of the elastic core.
If the horizontal force
F =
tL
σxx dz
(12.10)
0
does not vanish, the horizontal stress σxx in Eq. (12.9) should be replaced by (σxx − Fmean ), where
Fmean = F/tL is the mean horizontal stress in the lithosphere. Figure 12.11 shows (σxx − Fmean )
for regions of compressive and extensional tectonics. Since the brittle strength for the former region
is larger than that for the latter if pore fluid pressure is the same, te of the horizontally stretched
lithosphere is smaller than that of the horizontally constricted lithosphere [141]. Whole lithosphere
failure occurs more easily in extensional than in compressive tectonics.
Figure 12.12 shows the effect of the curvature, geothermal gradient and tectonic force Fmean on
te for a model continental lithosphere. In addition, te is also affected by curvature, even if other
factors are the same. Increased curvature increases differential stress to widen the yielded zones,
which reduces te . Figure 12.12(a) shows that sensitivity on the geothermal gradient decreases with
increasing curvature. The radius of curvature at the right side of this graph is R = 1/K = 100 km,
comparable with the thickness of the continental lithosphere. Thus, the model of a thin elastic plate
has a large error at this end. Figure 12.12(b) shows te versus Fmean . The effective elastic thickness
declines more rapidly in a normal fault regime than in a reverse fault regime with increasing tectonic
force |Fmean |.
The continental lithosphere has complex structures, leading to an intricate strength profile. Thus,
the effective elastic thickness of the lithosphere depends on various factors including lithology, curvature, thermal structure, horizontal tectonic force, radioactive heat generation in the crust, etc.
[140]. The Canadian shield has a considerable spatial variation in te , although those factors have not
so large differences. The variation suggests that geological structures such as old faults and suture
zones play important roles in determining te [27, 254].
12.4. PERIODIC DEFORMATION OF THE LITHOSPHERE
315
Figure 12.11: Schematic strength profiles of the oceanic lithosphere and vertical distribution of differential stresses in regions of (a) compressive and (b) extensional tectonics. Due to the asymmetric
brittle strength for those cases, the effective elastic thickness for the former case is larger than for
the latter. The width of the gray area indicates Δσ − Fmean .
Figure 12.12: Effective elastic thickness te of a model continental lithosphere that has a dry granitic
crust [126]. The strain rate is assumed to be 10−15 s−1 . (a) The effect of curvature K and geothermal
gradient. (b) Effect of horizontal stress Fmean with an assumed curvature K = 10−5 km−1 .
12.4
Periodic deformation of the lithosphere
Faults and folds are sometimes aligned parallel and uniform intervals in wide deformation zones.
Horst and grabens in the Basin and Range Province are examples. The wide rift zone called sulcus
on Jovian satellite Ganymede have grooves that are surface manifestations of normal faults (Fig.
3.6). Many icy satellites have such zones. Figure 12.13 shows a rift zone separating the heavily
cratered, therefore older, terrains on Enceradus.
The Fletcher-Hallet model explains those spatially periodic structures using the linear stability
analysis of a multi-layered system with a free boundary [63]. The model is also applied to the
periodic surface undulations of ropy lavas [62].
316
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
Figure 12.13: Saturnian satellite Enceradus (Voyager image 1715S2-001). The diameter of this
moon is about 500 km. Note the difference in the number density of craters. Smooth planes with a
smaller crater density have grooves. The heavily cratered terrain is divided by a rift zone (arrow).
Consider the crust of an icy satellite instead of the Earth’s crust, because the latter has complex
structures. For example, Ganymede has a very thick icy layer (§9.3), so that the shallow part of the
layer may be brittle as grooves and other geological structures indicate, but deep-seated ice may be
ductile. Accordingly, we assume a surface brittle layer floating on a viscous fluid (Fig. 12.14(a)).
The brittle behavior of the layer (1) is simulated by a power-law fluid with a very large power-law
exponent n(1) 1. The layer (2) is also a power-law fluid but has the exponent n(2) ≈ 3. Plane
strains on the xz-plane across the wide deformation zone are assumed, where the z-axis is defined
upward with the origin O at the base of the initial brittle layer (Fig. 12.14). H is the initial thickness
of the layer (1). All the layers are assumed to be incompressible.
Following Section 9.4, horizontally extending pure shear is regarded as the mean flow for this
12.4. PERIODIC DEFORMATION OF THE LITHOSPHERE
317
Figure 12.14: Schematic illustration showing the Fletcher-Hallet model. The layer (1) is a brittle
layer with a free boundary at its top and is underlain by the viscous layer (2). The layer (0) is
composed of inviscid fluid or space. The interface between the layers (1) and (2) is the brittleductile transition. Dashed lines indicate the initial levels of the surface and brittle-ductile transition.
multi-layered system, and we have assumed incompressibility. Therefore, the results in Section 10.8
are applicable to this problem. Namely, the differential equation in Eq. (10.89) holds for each of
layers (1) and (2), so that the growth rates of the coefficients of their separable solutions gives the
stability of this system.
In this model, infinitesimal strains are assumed so that we can ignore temperature perturbations,
i.e., temperature is a function of z in layer (2):
T = T0 − Γz
(z < 0),
(12.11)
where Γ is the geothermal gradient. Since the perturbation is very small, the effective viscosity in
layer (2) may be approximated by
η (2) = η0 exp(−γz).
(12.12)
The effective viscosity of layer (1) is assumed to be constant.
In the layers (1) and (2), we obtain the stress ratio Φ = 1/2 using the same argument as we did
to derive the slip line theory in Section 10.5. Substituting this stress ratio into Eq. (4.15), we have
TII = (ΔS/2)2 and the Mohr circles in Fig. 12.15. Therefore, we have
1
2
(Sxx − Szz )2 + Sxz
.
(12.13)
4
The perturbations of the pure shear parallel to the coordinate axes are very small, therefore, (Sxx −
2
Szz )2 Sxz
(Fig. 12.15). The effective viscosity of the fluid η satisfies
(n−1)/2 −1
(12.14)
2η = BTII
TII =
B = B ∗ exp(−Q/RT ) .
(12.15)
Layer (2) has the constitutive equation
Sxx = −p + 2ηDxx ,
Szz = −p + 2ηDzz ,
Sxz = 2ηDxz .
(12.16)
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
318
Combining the incompressibility Dxx + Dzz = 0 and Eq. (12.16), we obtain
S xx − S zz = 4ηDxx .
(12.17)
The principal axes of the mean flow are parallel to the coordinate axes. Hence, we have S xz = 0.
Combining Eq. (12.15), we obtain
2
2
1
1
(12.18)
T II =
S xx − S zz = S ,
S≡
S xx − S zz > 0.
4
2
Thus, the effective viscosity satisfies
−1
2η = B S
(1−n)
=B
− 1n
( 1 −1)
Dxxn .
(12.19)
Substituting Eq. (12.15) into (12.19), and using the temperature in Eq. (12.11), we have
∗ 1n (1− 1n ) −1
Q
.
η= 2 B
Dxx
exp
nR(T0 − Γz)
The operand of the exponential function is expanded about z = 0 as
Q
ΓQz
Q
+
.
≈
nR(T0 − Γz)
nRT0 nRT 2
0
Therefore, using Eq. (12.12), we obtain
Q ∗ 1n (1− 1n ) −1
η0 = 2 B
,
exp
Dxx
nRT0
ΓQ
γ=
.
nRT02
(12.20)
(12.21)
These equations give the effective viscosity of layer (2). That of layer (1) is assumed to be constant at
η0 , and the state of stress in this layer is determined by Eq. (12.17) and on the Mises yield criterion.
For a linear analysis of this system, we have to solve Eq. (10.89) under appropriate boundary
conditions. Unlike single layer folding, the boundary between layers (0) and (1) is free. The local
rectangular coordinates are defined at the boundary to which the s- and n-axes are taken to be parallel
and perpendicular (Fig. 12.16). The function h(x) denotes the height of the boundary. Then, the
components of S satisfy the conditions at the boundary
Ssn z=h = 0,
Snn z=H = ρgh,
(12.22)
where the normal stress is linearized as Eq. (D.31). These equations are transformed into the
expressions with the coordinates x and z by considering the force balance at the boundary. Let N
be the unit normal to the boundary, then the force balance is expressed as
S(0) · N = S(1) · (−N ).
(12.23)
12.4. PERIODIC DEFORMATION OF THE LITHOSPHERE
319
Figure 12.15: Mohr circles for the stress state with Φ = 1/2.
Let θ be the inclination of the s-axis with respect to the x-axis (Fig. 12.16). The stress components
are transformed as
(i)
(i)
(i)
(i) cos θ sin θ
Sxx Sxz
cos θ − sin θ
Sss Ssn
=
.
(i)
(i)
(i)
(i)
− sin θ cos θ
sin θ cos θ
Sns Snn
Szx Szz
Therefore,
(i)
(i)
(i)
Snn = Sxx sin2 θ − 2Sxz sin θ cos θ + Szz cos2 θ,
(i)
(i)
(i) Sns = − Sxx − Szz sin θ cos θ + Sxz cos2 θ − sin2 θ .
(12.24)
(12.25)
These trigonometric functions are rewritten as sin θ = ∂$
h/∂s and cos θ = ∂x/∂s. Ignoring higherorder terms, we have
∂$
h
sin θ ≈
,
cos θ ≈ 1.
∂x
Substituting these equations into Eqs. (12.24) and (12.25), and again ignoring the higher-order
terms, we obtain
$
(i) ∂h
Snn = −2Sxz
∂x
$
(i)
(i) ∂h
(i)
Sns = − Sxx − Szz
+ Sxz .
∂x
(i)
+ Szz ,
(12.26)
$
$
The stress tensor is divided
into
the mean and perturbation, S = S + S. Because of S ≈ O, the
higher-order terms of ∂$
h/∂x S can be neglected so that Eq. (12.26) is transformed into
(i) ∂$
h
Snn = −2S xz
∂x
(i)
(i)
+ S zz + S$zz ,
h
(i)
(i) ∂$
(i)
(i)
Sns = − S xx − S zz
+ S xz + S$xz .
∂x
(12.27)
(12.28)
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
320
Figure 12.16: Schematic picture showing the boundary condition between layers (1) and (2).
On the other hand, a non-slip condition and force balance are appropriate bondary conditions at
the boundary between layers (0) and (1):
(1) (2) (1) (2) $
$
vx = $
vx ,
vz = $
vz z=0
z=0
z=0
z=0
(1)
(2)
(1)
(2)
S$zz = S$zz ,
S$xz = S$xz .
z=0
z=0
z=0
z=0
Note that the layers are of the same density. If they are not, buoyancy force should be taken into
account.
Under these boundary conditions, the differential equation in Eq. (10.89) is solved. We use a
solution of the form
$
vx = −
1 dW
sin kx,
λ dz
$
vz = W cos kx,
h − H = A(t) cos kx,
where k is the horizontal wavenumber. Consequently, we obtain
dA
vz = (q − 1)Dxx A,
= −Dxx A + $
dt
z=H
(12.29)
where q(k) denotes the growth rate6 . The waves with q(k) > 1 are amplified, but those with q(k) < 1
are erased. The waves with the maximum growth rate may emerge as macroscale boudins. According to Fletcher and Hallet [63], q(k) is determined by the four dimensionless parameters, n(1) , n(2) ,
γH and ρgH/2cY . The third one is the characteristic depth over which the effective viscosity of
the layer (2) decreases by 1/e. The fourth one designates the strength of layer (1). Figure 12.17(a)
shows a graph of q(k), where kd indicates the wavenumber that maximizes the growth rate. It
is demonstrated that kd is insensitive to n(1) if this power-law exponent is greater than 102 (Fig.
12.17(b)). This is convenient to simulate the brittle behavior of layer (1). Rocks and ice have a
value of n(2) about 3. Therefore, the remaining parameters γH and ρgH/2cY determine the kd . The
6 See
[62] for the concrete expression of q(k).
12.4. PERIODIC DEFORMATION OF THE LITHOSPHERE
321
Figure 12.17: Growth rate of crustal boudins predicted by the Fletcher-Hallet model [63]. (a) Growth
rate versus dimensionless wavenumber. The values n(1) =104 , n(2) =3, γH=10 and ρgH/2cY =3 are
assumed. (b) Relationship between the dominant wavenumber kd and the power-law exponent n(1) .
The parameter values n(2) =5, γH=10 and ρgH/2cY =3 are assumed here.
brittle-ductile boundary goes down with an increasing strain rate. Therefore, H depends on Dxx
(Eq. (12.20)). Likewise, the critical stress cY depends on the strain rate. On the other hand, γ is
determined by Γ via Eq. (12.21). Consequently, Dxx and Γ control the dominant wavenumber kd .
Many icy satellites have grooves with roughly uniform intervals. Thus, the dominant wavelength
is observable there. However, we need one more line of evidence to constrain Dxx and Γ in the
Fletcher-Hallet model. Let qd be the growth rate at the dominant wavelength kd . Then, from Eq.
(12.29), we obtain
ln(A/A0 ) = (qd − 1)Dxx t,
(12.30)
where A0 and A are the initial amplitude and that of the time t. The left-hand side of Eq. (12.30)
represents the logarithmic strain of the crust.
This equation enables an order-of-magnitude estimation of the controlling factors. The initial
amplitude is unknown, but A0 ∼ 100 m may be acceptable. A is known from the present topography.
For example, grooves on icy satellites have amplitudes of the order of A ∼ 102 m. The global strain
of the moons is in the order of 1%. Thus, we estimate Dxx t ∼ 10−2 . Using these estimates, we
obtain qd ∼ 500 from Eq. (12.30). The Fletcher-Hallet model gives appropriate values of Dxx and
Γ from this growth rate at the observed kd . If the values are unrealistic, the model is found to be
inapplicable to the object. It was found that the model is applicable to Ganymede, Enceradus, and
Miranda [44, 76]. As for Ganymede, the values Γ ≈ 20 K km−1 and Dxx ≈ 10−14 s−1 were obtained.
Similar values were determined for Enceradus and Miranda. Thus, the extensional tectonics lasted
1%/10−14 s−1 ∼ 104 –105 years, surprisingly short for the global tectonics compared to the present
tectonics of the Earth.
The lithosphere of the Earth has a much more complex structure than that assumed in the
CHAPTER 12. DYNAMICS OF THE LITHOSPHERE
322
Fletcher-Hallet model. The density difference of the crust and mantle gives rise to buoyancy. The
strength of the brittle layer depends on depth, unlike the strength in the model. In some active
regions, the hot and fluent lower crust mechanically separates the upper crust and the mantle lithosphere, so that there are two strong layers there. If the upper and lower crusts have significantly
different rheology, such as granite and gabbro, the lithosphere may exhibit complex behavior. Models have been developed to take account of those factors [16, 17, 196, 282]. Such a model was
successfully applied to the folded oceanic lithosphere in the northeastern Indian Ocean [281], where
the simple elastic model failed to explain the dominant wavelength (§8.4.3). Linear stability analysis assumes very small perturbations. Numerical simulations are used to link the analysis and finite
amplitude deformations as the rifts discussed in Section 12.2 [25].
Exercises
12.1 In the model introduced in §12.2, the fate of rifts is determined by the competition between
the rate of lithospheric thinning and vertical heat transfer through it. Consider how to define the
Péclet number, which is a dimensionless number indicating the representative time of motion of a
mass relative to the time constant of heat conduction.
12.2 Consider that pure shear rifting begins in the lithosphere with a negligible strength of the
crust compared to that of the mantle lithosphere, and that the geothermal gradient Γ is constant in
the lithosphere. Let
1
Q
−1
− 1n
n
Ė
(12.31)
exp
T = A Ė
nRT
be the constitutive equation of the mantle, where
⎛
Ė 0
Ė = ⎝ 0 0
0 0
⎞
0
0 ⎠
−Ė
(12.32)
is the rate of strain tensor. Show that the approximate equation
a
0
A− n Ė 1/n TM2 nR
exp
ΓQ
1
Txx dz ≈
Q
nRTM
indicates the tectonic force needed for the rifting [55], where Txx is the horizontal component of the
deviatoric stress, TM is the Moho temperature.