Earth and Planetary Science Letters 413 (2015) 59–69
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
www.elsevier.com/locate/epsl
Generation of tectonic over-pressure inside subducting
oceanic lithosphere involving phase-loop of olivine–wadsleyite
transition
Byung-Dal So a,b,∗ , David A. Yuen c,d,e
a
School of Earth, Atmosphere and Environment, Monash University, Clayton, 3800 Victoria, Australia
Research Institute of Natural Sciences, Chungnam National University, Daejeon 305-764, South Korea
c
Minnesota Supercomputing Institute, University of Minnesota, Twin Cities, Minneapolis, MN 55455, USA
d
Department of Earth Sciences, University of Minnesota, Twin Cities, Minneapolis, MN 55455, USA
e
School of Environment Studies, China University of Geosciences, 430074 Wuhan, China
b
a r t i c l e
i n f o
Article history:
Received 5 August 2014
Received in revised form 26 December 2014
Accepted 26 December 2014
Available online 15 January 2015
Editor: Y. Ricard
Keywords:
tectonic over-pressure
phase transition
phase loop
olivine–spinel coexistence zone
shear heating
subducting slab
a b s t r a c t
We conducted a two-dimensional numerical model to analyze the generation of tectonic over-pressure,
which is a positive deviation from lithostatic pressure, for deep slabs which are anchored at the
660 km phase boundary. The formation of the ductile shear zone under a compressional tectonic
setting induces tectonic over-pressure. We first propose that an apparent shear zone originated from an
elastic heterogeneity in the phase loop, which is the two-phase (i.e., olivine and wadsleyite) coexistence
interval around the 410 km boundary within subducting oceanic lithospheres, can cause tectonic overpressure with a range from 0.3 to 1.5 GPa. This over-pressure significantly impacts the formation of the
olivine–wadsleyite phase transition. The flattening of the olivine–wadsleyite interface by over-pressure is
well-resolved. Therefore, we argue that the over-pressure should be applied when analyzing the phase
boundary within the subducting lithosphere. Our results provide a new insight on the interplay among
the phase transition, shear zone formation and tectonic over-pressure.
© 2015 Elsevier B.V. All rights reserved.
1. Introduction
Pressure at depth in terrestrial planets, which is one of the
most important environmental variables controlling geodynamical
phenomena such as solid–solid phase transitions and brittle/ductile deformation, has been simply interpreted to be the hydrostatic
pressure. The pressure distribution inside the Earth is more monotonic than that of temperature and deviatoric stress, which can
largely vary with short-wavelength perturbations such as mechanical damage (Bercovici and Ricard, 2003; Karrech et al., 2011),
viscous dissipation (John et al., 2009; Ogawa, 1987) and frictional
heating (Lachenbruch, 1980; So and Yuen, 2013). Thus, one might
have supposed that the pressure can directly indicate the depth at
which high-pressure metamorphism generally occurs. For this reason, the influence of dynamical pressure in geodynamics has been
relatively underestimated compared with that of temperature and
deviatoric stress. However, the lithostatic pressure assumption is
appropriate only for the case of a static or quasi-static lithosphere
*
Corresponding author.
E-mail address: [email protected] (B.-D. So).
http://dx.doi.org/10.1016/j.epsl.2014.12.048
0012-821X/© 2015 Elsevier B.V. All rights reserved.
without shear stress (Mancktelow, 2008). As a result, the depthestimation based on exhumed high-pressure metamorphic rocks
may have an inaccuracy (Schmalholz and Podladchikov, 2014). For
more realistic situations where dynamic deformation regimes, including compression, extension, bending and buckling, are taken
into account, we should consider tectonic pressure which is an additional pressure generated by the deformation process itself (e.g.,
Moulas et al., 2013; Schmalholz and Podladchikov, 2013). Henceforth we can deduce that the total pressure is the sum of lithostatic and tectonic pressures.
The tectonic pressure has positive (i.e., tectonic over-pressure)
or negative (i.e., tectonic under-pressure) values depending on tectonic situations. For the case of tectonic over-pressure, the total pressure is higher than the lithostatic pressure (e.g., Rutland,
1965). Conversely, tectonic under-pressure means that the total
pressure is lower than the lithostatic pressure in corner flow situations (e.g., Jischke, 1975). For instance, the compressional deformation (Mancktelow, 2008) and the formation of the ductile
shear zone (e.g., Li et al., 2010; Schmalholz and Podladchikov,
2013) can lead to tectonic over-pressure. On the other hand, tectonic under-pressure appears in lithospheres undergoing extension
60
B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
(Soesoo et al., 1997) and brittle fracturing (Mancktelow, 2006).
There is still debate whether tectonic pressure has the capability
to influence the geodynamical process. While some authors argue
that the effect of tectonic pressure is too weak to significantly influence the whole geological process (Burov and Yamato, 2008),
many suggest that tectonic pressure high enough to show a geological impact (from several hundred MPa to several GPa) can
be created under various tectonic settings (e.g., Burg and Gerya,
2005). In addition, the pressure gradient from tectonic pressure
to lithostatic pressure can cause an additional suction force on
fluid in the lithosphere (Tovish et al., 1978). Using the concept
of tectonic pressure, previous studies proposed plausible explanations for deep slab hydration (i.e., deeper than 40 km; Faccenda
et al., 2009), the mismatch between subduction/exhumation rate
and petrological pressure record in exhumed (ultra) high-pressure
metamorphic rocks (e.g., eclogites and blueschists; Kamb, 1961;
Smith, 1988) and flow directions around the brittle and ductile
shear zone. Despite these efforts, the influence of tectonic pressure
on solid–solid phase transitions in subducting lithospheres has not
been addressed. Inside slabs, there are notable pressure-sensitive
phase transitions (Ringwood, 1968) such as olivine → wadsleyite
and wadsleyite → perovskite + magnesiowüstite.
In subducting lithospheres, compression is exerted on the lithosphere with a resistance force from the 660 km phase boundary
(e.g., Fukao et al., 2009). Moreover, the drop in deviatoric stress in
a localized shear zone can induce tectonic pressure (Schmalholz
and Podladchikov, 2013). The shear zones within the slab can
easily form in accordance with transformational faulting by the
olivine–wadsleyite phase transition (Green and Houston, 1995;
Kirby, 1987) and spatial heterogeneities in density (Gerya et al.,
2004) and elastic modulus (So et al., 2012). Thus, it is highly probable that tectonic over-pressure operates in the slab, and it is natural that the phase transition inside the slab should be impacted by
this tectonic pressure. Since high-pressure laboratory experiments
focusing on phase transition within deeply subducted lithospheres
have been performed under the assumption of lithostatic pressure
(e.g., Schmidt and Ziemann, 2000), the effect of compression/extension and shear zone formation on the pressure distribution has
not been considered. Therefore, to obtain a better understanding of the phase stability zone within the subducting lithosphere,
we should include the generation and effect of tectonic pressure
into numerical simulations containing dynamic deformation and
shear zone formation.
In this study, we have focused on the olivine–wadsleyite phase
transition at the depth of 410 km. This is because the zone around
the 410 km phase boundary has considerable complexity that requires tectonic pressure with the phase transition and shear zone
formation. Especially around the 410 km phase boundary, the reaction rate of olivine–wadsleyite phase transition is kinetically decelerated under the low temperature conditions in the cold core
of subducting lithospheres (Sung and Burns, 1976). As a result, the
phase loop, which refers to the olivine–wadsleyite coexistence zone,
forms around the 410 km phase boundary (Akaogi et al., 1989;
Rubie and Ross, 1994). In recent mineral physical studies, the
elastic modulus in the phase loop is lower by a factor of 3 to
10 than that in the surrounding region (Li and Weidner, 2008;
Ricard et al., 2009). The phase loop with lower elastic modulus can
accumulate a larger elastic energy per unit area than the surrounding region (Regenauer-Lieb et al., 2012; So et al., 2012). Due to a
large elastic energy storage in the phase loop, the energy is much
efficiently released around the phase loop. Consequently, the apparent shear zone can be achieved, and it is a preferable condition
for tectonic over-pressure. The existence of tectonic over-pressure
in the ductile shear zone is related with the force balance along a
parallel-direction with far-field force (e.g., ridge push or slab pull)
(Schmalholz and Podladchikov, 2013). Previous studies estimated
that the magnitude of tectonic pressure within ductile shear zones
is several GPa (e.g., Burg and Gerya, 2005), which may be larger
than deviatoric stress (Schmalholz et al., 2014). Therefore, we cannot ignore this large tectonic over-pressure in the investigation of
the phase transition around 410 km depth. To understand how the
phase loop and shear zone formation affect tectonic pressure and
phase stability, we perform a two-dimensional finite element modeling with different amounts of elastic modulus drop within the
phase loop and the thickness of phase loop.
2. Numerical techniques used in modelling
We have performed fully coupled thermal-mechanical calculations in a two-dimensional domain. A commercial finite element
code, ABAQUS (Hibbitt, Karlsson & Sorensen, Inc., 2009), was employed to solve governing equations with an elasto-plastic constitutive relation under a plane-strain assumption. We have focused on only elasto-plastic deformation, disregarding a viscous
deformation. In previous studies (e.g., Zhong et al., 1998) dealing with plastic deformation within the subducting lithosphere, the
authors have widely employed a pseudo-plastic rheology, in which
the normal plasticity can be described in a viscous manner with
temperature/stress-dependent viscosity. Even though people have
been conscious of the need for Maxwell visco-elasto-plastic model,
which refers to a serial combination of elastic, viscous and plastic
elements, they used the pseudo-plasticity instead of normal plasticity. This is because that the pseudo-plasticity shows a reasonable numerical convergence (Zhong et al., 1998). Few studies have
tried to investigate the method for handling Maxwell visco-elastoplastic rheology without pseudo-plasticity. In our study, the most
important aspect is calculating a plastic deformation with plastic
rheology leading to shear heating and tectonic over-pressure. Thus,
we employed elasto-plasticity as a rheology of subducting lithosphere. The viscous behavior is described by dash-pot elements
being attached on the boundary of subducting lithosphere (see
Fig. 1).
Our time-stepping method was full Newtonian implicit scheme
(e.g., Choquet et al., 1995). We solve three important governing
equations (e.g., Kaus and Podladchikov, 2006). The first is the law
of mass conservation written as
∂ vi
=0
∂ xi
(1)
where v i and xi are velocity and spatial coordinate in ith direction.
The second is the law of momentum conservation with Boussinesq
approximation (e.g., Christensen, 1996),
∂ σi j
+ ρ 1 − α ( T − T ref ) g δi2 = 0
∂xj
where σi j = − p δi j + τi j
(δi j refers to Kronecker delta).
(2)
σ , τ , ρ , α , T , T ref , g and p are, respectively, Cauchy stress, deviatoric stress tensors, density, thermal expansivity, temperature,
reference temperature (i.e., the lowest temperature in the domain,
100 ◦ C) and gravitational acceleration and mean pressure (i.e., average value of principle components of σ ). The third is the law of
energy conservation,
∂T
∂T
∂2T
1
1
+ vi
=κ 2 +
g α ( T − T ref ) w +
τi j : ε̇iplj
∂t
∂ xi
c
ρ
cP
∂xj
P
(3)
where t, κ , c P and w are time, thermal diffusivity, specific heat
and vertical velocity aligned with gravity, respectively. The second
term in the right-hand side of Eq. (3) refers to adiabatic compression term (e.g., Yoshioka et al., 2013). The plastic strain-rate tensor
B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
61
Fig. 1. (a) The domain for numerical simulation. The black and red springs correspond to behaviors of upper and lower mantle, respectively. A wedge-shaped zone (see
shaded zone) indicates the phase loop having a lower shear modulus compared with the surrounding region. (b) The pre-calculated thermal structure of numerical domain.
This temperature distribution is calculated during the subduction of a 50 Myr old oceanic lithosphere with a constant rate (i.e., 5 cm/yr). We define the olivine–wadsleyite
phase boundary using a typical phase diagram with Clapeyron slope of 2 MPa/K (Bina and Helffrich, 1994). The maximum thickness (i.e., 20 km) of phase loop must be
shown in the coldest part of subducting lithosphere. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
ε̇iplj is one of the components of the total strain-rate tensor ε̇itotal
j .
total
We can simply obtain ε̇i j using the relation
∂v j
1 ∂ vi
pl
total
el
total
.
ε̇i j = ε̇i j + ε̇i j where ε̇i j =
+
(4)
2 ∂xj
∂ xi
ε̇ielj means the elastic strain-rate defined by
ε̇ielj =
1 D τi j
2G Dt
(G is shear modulus)
(5)
where D / Dt is the objective Jaumann derivative (Kaus and Podladchikov, 2006). We observe J 2 (i.e., the second invariant of τi j ) in
each element for every time step. When the value of J 2 exceeds
pl
the predefined plastic yield strength σY , the ε̇i j is taken into account following the power-law rheology for dislocation creep (e.g.,
Chopra and Paterson, 1981)
ε̇iplj = A J n2−1 τi j exp −
Ea + p V ∗
RT
(6)
.
n, A, E a , V ∗ and R are, respectively, the power-law exponent, the
pre-factor, activation energy, activation volume and universal gas
constant. In the last term of Eq. (3), the tensor contraction between
pl
deviatoric stress and plastic strain-rate tensors (i.e., τi j : ε̇i j ) mathematically describes the time rate of shear heat generation per unit
volume. In this way, all of Eqs. (1)–(6) are fully coupled, and both
elastic and plastic behaviors are simultaneously contained in our
calculations. We list detailed values and description of parameters
for the calculation in Table 1.
The slab in this study is assumed to have already subducted
to the 660 km phase boundary, and not to touch the boundary
yet. As soon as the calculation starts, a compression due to resistance force against the further penetration at the depth of 660 km
(Frohlich, 1989) is activated with touching the 600 km phase
boundary. In Fig. 1a, we describe the inclined rectangular domain
that represents the slab and has a width of 100 km (e.g., Schellart
et al., 2007) and length of 950 km with 45◦ dip. The phase loop,
the olivine–wadsleyite coexistence zone, is defined around the
410 km phase boundary. In the phase loop, the elastic modulus
is greatly reduced by a factor of 3 to 10 (Li and Weidner, 2008).
By the definition of the phase loop, the mole fraction of olivine in
the loop varies from 1 to 0 at the top and bottom of the loop, respectively (Rubie and Ross, 1994). Therefore, it is very likely that
we may define a quadratic distribution of the shear modulus with
the minimum shear modulus at the center of the loop. We perform a series of calculations with a wide range of shear modulus
contrast (i.e., R s ) from 1 to 10. The R s is the shear modulus ratio of the surrounding to phase loop. The thickness of phase loop
is identical to the depth-interval of olivine–wadsleyite phase transition around the 410 km depth. Some high-pressure experimental
studies reported that the depth-interval of two-phase coexistence
is ∼15 km (e.g., Zhang and Herzberg, 1994). It has been known
that water content in the slab significantly alters the thickness of
two-phase coexistence. For instance, Smyth and Frost (2002) reported that the thickness can be enlarged with increasing water
content, from ∼12 km to ∼40 km under nearly anhydrous and
saturated environments, respectively. On the other hand, Hosoya
et al. (2005) showed that the thickness of two-phase coexistence
zone becomes narrower when the larger amount of water exists,
even though their studies assumed that the lithosphere is quickly
moving downward without any resistance force from the 660 km
phase boundary. Seismic reflectivity studies on the transition zone
around the depth of 410 km proposed that the depth-interval is
4 km or less (Yamazaki and Hirahara, 1994). We adopt here 20 km
(see shaded zone in Fig. 1) as the thickness of the phase loop,
which is consistent with general consensus.
We assign finer rectangular elements (i.e., 0.1 km × 0.1 km)
around the loop where shear heating and tectonic pressure are expect to be localized. The size of mesh progressively increases up to
0.5 km × 0.5 km as the distance from the phase loop increases.
The total number of elements is ∼2 × 106 . Viscous behavior of asthenospheric mantle has not been included in our numerical setup.
Instead of including a dynamical mantle, we choose a dash-pot element, which can mimic the velocity-dependent restoring force of
a Newtonian viscous mantle (e.g., Capitanio et al., 2010). The bottom boundary is composed of the dash-pot element with larger
viscosity (i.e., 4 × 1022 Pa s; Steinberger, 2000), which implies the
slab is partially fixed at the 660 km boundary. The top boundary
is compressed with a constant velocity of 5 cm/yr corresponding
to a far-field ridge push. The oceanic lithosphere, with an age of
50 Myrs, which has a thermal structure based on half-space cool-
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B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
Table 1
Notations and detailed values of input parameters.
Symbol
Definition
Unit
Value
κ
ρ
thermal diffusivity
density
specific heat
shear modulus
plastic yield strength
thermal expansivity
power-law exponent
prefactor
universal gas constant
activation energy
activation volume
surface temperature
mantle temperature at 100 km depth
age of oceanic lithosphere
vertical coordinate in oceanic lithosphere
m2 s−1
kg m−3
J kg−1 K−1
Pa
Pa
K−1
10−6 a
3300a
800a
8 × 1010 a
2 × 108 a
2 × 10−5 b
4.48b
4.3 × 10−16 b
8.314
4 × 105 b
1.4 × 10−5 c
0
1300d
1.577 × 1015
0–105
cP
G
σY
α
n
A
R
Ea
V∗
T0
Tm
t age
y
a
b
c
d
Pa−n s−1
J mol−1 K−1
J mol−1
m3 mol−1
◦C
◦C
s
m
Chopra and Paterson, 1981.
Regenauer-Lieb and Yuen, 1998.
Karato and Ogawa, 1982.
Turcotte and Schubert, 2014.
ing model (see Eq. (7); Turcotte and Schubert, 2014), is subducting
at the trench with 5 cm/yr. The temperature at depth y in the
oceanic lithosphere with a certain age t age is written as a form of
error function,
T ( y , t age ) = ( T m − T 0 ) erf
(κ is thermal diffusivity)
2
√
y
κ t age
+ T0
(7)
where T 0 and T m are temperature at the surface and mantle
temperature at 100 km depth, respectively. We pre-calculated the
temperature distribution of the whole slab for a highly resolved
thermal-mechanical calculation including shear heating and tectonic over-pressure with resistance force by the 660 km phase
boundary. In order to pre-calculate a temperature distribution of
the subducting lithosphere right before the lithosphere touches
the 660 km phase boundary, we employed a constant subducting
velocity (i.e., 5 cm/yr) and widely-accepted geotherm of mantle
(Turcotte and Schubert, 2014). At that moment (i.e., right before
touching the 660 km boundary), we defined a phase loop following the Clapeyron slope as we explained earlier (see Fig. 1b). Then,
the calculated distribution of temperature and shear modulus is directly transferred to the highly resolved domain for calculation of
shear heating and tectonic over-pressure as soon as the lithosphere
touches the 660 km boundary.
3. Results
We will provide an insight into the effect of tectonic pressure on solid–solid phase transitions within subducting lithosphere. Hence, we investigate systematic trends in (1) shear heating, (2) deviatoric stress and (3) the generation of tectonic pressure with and without the olivine–wadsleyite phase loop based on
high-resolution two-dimensional finite element simulations.
3.1. Case without phase loop
We first observe an overall pattern of tectonic pressure in the
case without the phase loop (i.e., R s = 1) as a reference model
for numerical experiments dealing with the loop. The model without the loop does not show significant variations in temperature
and deviatoric stress as well as pressure under constant rate compression (i.e., 5 cm/yr) during a time span of one million years.
Because the stress level of domain does not reach the predefined
plastic yield strength, plastic strain-rate and its subsequent shear
Fig. 2. The distribution of tectonic pressure (i.e., total pressure–lithostatic pressure)
for the case without the phase loop. Although both tectonic under- and overpressure are generated, the amplitude is smaller than 40 MPa, which is not strongly
influential in the olivine–wadsleyite phase transition. (For interpretation of the references to color in this figure, the reader is referred to the web version of this
article.)
heating cannot initiate. If the slab was completely anchored (i.e.,
fixed) at the 660 km phase boundary, the stress level would easily
rise up to the yield strength. However, we use the dash-pot element because we intend to mimic viscous behavior of lower mantle with a finite viscosity of 4 × 1022 Pa s. Thus, the stress buildup
is not efficient enough to allow fast plastic yielding and shear instability. Although the domain is not plastically yielded in this
homogeneous medium, a long-wavelength buckling occurs through
the whole domain. We may speculate that tectonic pressure may
be generated around regions with extension and compression due
to the compression-induced buckling. We plot the distribution of
tectonic pressure (i.e., total pressure–lithostatic pressure) in Fig. 2.
As we predicted based on previous theoretical and numerical studies, the tectonic over-pressure and under-pressure can be recognized in parts of compression (see red arrows in Fig. 2) and extension (see blue arrows in Fig. 2). However, the level of tectonic
pressure is less than 40 MPa, which is not sufficiently large to influence the olivine–wadsleyite phase transition. The pressure of 40
MPa corresponds to the overburden weight of typical subducting
material with ∼1.2 km thickness.
B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
63
To determine whether an elastic heterogeneity assigned to the loop
can induce an apparent shear zone, we compute the temperature
increase with varying values of R s . The tensor contraction between
ε̇iplj and τi j (i.e., ε̇iplj : τi j ; see Eq. (3)) is directly converted into the
shear heating (per unit volume and time). Fig. 4 shows the distribution of temperature elevation in the slab with 1 Myr compression. A well-localized thermal structure around the loop indicates
that the dominant physics is a positive feedback where the elevation of plastic strain-rate and temperature promote each other.
We may define this localized structure of thermal anomaly as the
ductile shear zone.
The formation of over-pressure within the ductile shear zone
has been intensively investigated (Schmalholz and Podladchikov,
2013). However, those studies focused only on over-pressure in
the shallow portion of lithosphere (Angiboust et al., 2012). To test
the possibility of over-pressure generation under
the condition of
deep slab, we map the distributions of
Fig. 3. The horizontal distribution of pressure gradient (MPa/km) along the red lines
in Fig. 2 for the case without the phase loop. The pressure gradient around 410 km
depth is almost zero. It directly means that the fluid transport in the phase transition zone is weak when the loop (i.e., elastic heterogeneity) is not imposed. The
positive and negative pressure gradients indicate that the fluid flow directions are
right and left, respectively. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
We also display pressure gradients (from the left to right) along
the lines for each hundred km depth (see red lines in Fig. 2) to
determine fluid transport capability by pressure gradient. In the
region far from the center of slab (see black and orange lines in
Fig. 3), the magnitude of pressure gradient at the slab–mantle interface is 10–15 MPa/km. This value is similar with that from
previous study (Faccenda et al., 2009) and suffices for promoting
fluid transport. On the other hand, the pressure gradient around
the transition zone (see yellow line in Fig. 3) is almost zero, which
cannot drive the transport. In addition, the amplitude of pressure
gradients inside the slab are smaller than 2 MPa. This means that
the fluid flow in the slab is static when the phase loop and its elastic heterogeneity are absent. If we adopted real mantle, instead of
viscous dash-pot elements, the zone of shear heating and/or buckling can be propagated from the slab inside to the mantle. Thus,
we may expect that the pressure gradient can be smoother in the
case with viscous mantle, not the dash-pot element.
3.2. Cases with phase loop
We calculate the state of stress, temperature and pressure when
the phase loop with a lower shear modulus is imposed within
the subducting lithosphere undergoing constant rate compression.
τv =
3
2 ij
τ : τi j (i.e., von
Mises deviatoric stress) and tectonic pressure with different R s values (see Figs. 5a and 5b, respectively). For all values of R s , the
τ v in the loop is smaller than that in surrounding region. Since
the localized heating and strain-rate cause a thermal- and strainsoftening, the loop is not able to support a large deviatoric stress
for a given strain or strain-rate. For instance, in the case of R s = 4,
the areal average of τ v over the loop and the outside medium
are relatively ∼0.5 and 1.3 GPa. Although there is no strong consensus regarding the level of deviatoric stress within subducting
lithospheres, some numerical studies dealing with subduction slabs
(without the formation of shear zone) estimated ∼1 GPa of deviatoric stress (e.g., Čížková et al., 2007). On the other hand, when a
seismic slip and/or shear zone is involved, the stress drops down
to 0.1–0.5 GPa (John et al., 2009). Our results on the stress evolution including a stress buildup and the stress release as a form of
shear heating seems to be consistent with those previous works.
For the cases of R s smaller than a certain critical value (i.e., 2),
weak tectonic over- and under-pressure (i.e., ∼40 MPa) with a
large spatial-scale is shown around compression and extension induced by slab buckling. It is a similar trend for the model without
phase loop. Otherwise, when the R s is larger than 2, a localized
tectonic over-pressure appears around the phase loop, which corresponds to the shear zone. In comparison with the models of
1 ≤ R s < 2 (see Fig. 2), the over-pressure is likely to form in the
inner cold part of the slab, which exhibits the elevated interface of
olivine–wadsleyite phase transition. Therefore, we argue that the
over-pressure with the phase loop and shear heating has a definite
impact on the pattern of phase transition.
The average ratio between the increase of pressure and temperature (i.e., P tec /
T ) is calculated over the phase loop (see
Fig. 4. The temperature distribution after 1 Myr compression with different R s values. The case of a larger R s (i.e., ≥2) develops a more localized shear zone with larger
temperature elevation.
64
B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
Fig. 5. Distribution of (a) von Mises deviatoric stress and (b) the tectonic pressure with different R s values from 2 to 10. The opposite movement of deviatoric stress and the
over-pressure is observed. (For interpretation of the references to color in this figure, the reader is referred to the web version of this article.)
Fig. 6a). The peak of P tec /
T is ∼4 MPa/K when R s = 2.5. For
cases of R s from 1.1 to 2.5, the P tec /
T increases abruptly from
∼1.18 MPa/K to ∼4 MPa/K. On the other hand, after R s = 2.5,
the P tec /
T converges to ∼2.8. Most cases show the P tec /
T
obviously larger than a typical Clapeyron slope of the olivine–
wadsleyite phase transition (i.e., 1.5–3 MPa/K; Bina and Helffrich,
1994). This excessive P tec /
T value over the Clapeyron slope obviously means that the feature of phase transition within the phase
loop is significantly perturbed. We plot, in Figs. 6b and 6c, the
total pressure (i.e., lithostatic pressure + tectonic pressure) along
the central line of subducting lithosphere (see the red line AA in
Fig. 5b). The dashed line refers to static lithostatic pressure without tectonic pressure and before the start of compression. Around
the phase loop, the over-pressure clearly appears for all cases of
the shear zone. Although the magnitude of over-pressure is different depending on R s , the slope of total pressure is almost twice
as steep as that of the lithostatic one. The maximum magnitude
of over-pressure increases from 0.1 GPa (at R s = 2) to 1.5 GPa
(at R s = 10).
We compute the temporal evolution of deviatoric stress and
tectonic pressure to analyze the mechanism of over-pressure formation around the phase loop. The opposite movement of deviatoric stress and the over-pressure are expected for force balancing in a parallel direction with overall compression (Schmalholz
and Podladchikov, 2013). The solid lines in Fig. 7a display the
areal average of τ v over the shear zone with passing time. All
of the models catastrophically release the stored elastic energy
as a form of shear heating as soon as strain-softening initiates
following the cessation of strain-hardening. The strongly coupled
interplay between thermal- and strain-softening drops the stress
level sharply. Because the phase loop with a lower elastic modulus
(i.e., the case of larger R s ) can accumulate a larger elastic strain
energy (Regenauer-Lieb et al., 2012), a more vigorous dissipation
of the stored elastic energy and its subsequent greater strainsoftening should appear. Right after the stress drop, the tectonic
over-pressure will be generated to compensate the stress drop for
a force balance. Hence, it is a natural deduction that the model
with a larger stress drop induces a larger over-pressure. Fig. 7b
shows time evolution of over-pressure over the phase loop. As we
expected above, we can find that the over-pressure is abruptly
increased (see black line in Fig. 7b) at the moment of strainsoftening and stress drop. In addition, a smaller R s causes a lower
over-pressure. We plot the time derivative of the over-pressure
with different R s in Fig. 7c. When the R s is smaller than 2, the
tectonic pressure is almost steady. Otherwise, we recognize that a
pulse-like elevation of the over-pressure when the R s larger than 2
is adopted.
To see explicitly how the olivine–wadsleyite interface is changed by the shear zone formation and its subsequent over-pressure,
we visualize, in Fig. 8, the interface on the subducting lithosphere
with varying R s values. In the case of R s = 1 (see dashed black
line), the topography from the 410 km depth is ∼30 km, which lies
within the range of earlier estimates (e.g., Chambers et al., 2005).
In contrast to the model without the phase loop, the topography
after deformation in the models with the phase loop is enormously
affected by the over-pressure due to shear zone formation. When
a R s larger than 2 is imposed, the topography is broader than that
of a slab with no phase loop because the zone of shear localization
is not limited to the phase loop. Rather, it diffuses toward the outside depending on initial temperature and rheological properties.
This indicates that the topography of olivine–wadsleyite boundary
in subducting lithospheres can rise with tectonic over-pressure following shear heating around the phase loop. For instance, in the
B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
65
Fig. 7. (a) The time evolution of areal average of τ v over the shear zone with varying
R s . We can clearly recognize the strain-hardening and strain-softening in all cases.
During the strain-softening, the case with a larger R s (see the black line) shows
a large stress-drop. (b) The areal average of tectonic over-pressure over the shear
zone with different R s . The case of a smaller R s generates a weaker over-pressure.
The timing of strain-softening onset and the abrupt pressure elevation are similar. (c) The time derivative of tectonic over-pressure (GPa/yr) with varying R s . The
larger R s induces a quicker and larger elevation of tectonic pressure.
Fig. 6. (a) The average ratio between the increase of pressure and temperature (i.e.,
P tec /
T ) with different R s . There is a local maximum P tec /
T (i.e., ∼4 MPa/K)
when R s = 2.5. Most cases show the P tec /
T larger than a typical Clapeyron
slope of the olivine–wadsleyite phase transition (see the shaded area). (b) The tectonic pressure profile along the red line AA in Fig. 5b. The larger R s induces the
larger tectonic over-pressure. The magnitude of over-pressure around the 410 km
depth can be high up to 1.5 GPa (at R s = 3). Otherwise, the over-pressure in cases
of a smaller R s is very small (i.e., 0.3 GPa at R s = 2).
case of R s = 10, the topography elevation is ∼15 km (see black
solid line in Fig. 8).
Lastly, we depict, in Fig. 9, the pressure gradient around the
phase transition zone (i.e., 410 km depth), which has revealed as
a larger water reservoir (Huang et al., 2005). Compared with the
case without the phase loop (see red dotted line in Fig. 9), the
magnitude of pressure gradient is significantly larger by a factor
of 2 to 5. Thus, we may certainly infer that the exchange of water
and melt across the slab–mantle interface can be huge. Moreover,
a strong fluctuation of pressure gradient occurs inside the slab,
which argues for the existence of some sort of complicated fluid
flow within the slab for large enough R s .
4. Discussion
We have argued here that the shear zone formation around the
phase loop (i.e., olivine–wadsleyite two-phase coexistence zone),
Fig. 8. The olivine–wadsleyite phase interface. The dashed black line corresponds to
the interface in the case without phase loop. There is a trend that a broader and
higher topography of the olivine–wadsleyite phase boundary when the larger R s is
assigned.
which has a lower elastic modulus compared with the surrounding
region, can significantly influence the feature of olivine–wadsleyite
phase transition and fluid transport at the slab–mantle interface.
Our results support that R s values representing the magnitude
66
B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
Fig. 9. The horizontal distribution of pressure gradient (MPa/km) along the red
line BB in Fig. 5b. The strongly fluctuating pressure gradient inside the slab is
generated. It indicates that the fluid flow within the slab is also vigorous. (For interpretation of the references to color in this figure, the reader is referred to the web
version of this article.)
of elastic heterogeneity can control the shape of the olivine–
wadsleyite phase boundary. For the cases where the R s values
are assigned below the critical value (i.e., ∼2), the shear zones
do not appear even though there is a phase loop. Similarly inside a slab without the phase loop (i.e., R s = 1), models with
the R s smaller than the critical value (i.e., 1 < R s < 2) generate a
weak (i.e., ∼40 MPa) and long wavelength tectonic pressure due
to compression-induced buckling. Hence, the phase loop with a
smaller R s hardly influences the phase transition. Moreover, the
pressure gradient around the phase transition zone is too weak
to drive fluid transport. On the other hand, when the R s value
is large enough (i.e., ≥2), the positive feedback between powerlaw rheology and plastic deformation facilitates apparent ductile
shear zones. We note here that the critical values of R s for the onset of shear instability may vary with differences in rheology and
interfacial shape of elastic heterogeneity. Other numerical studies
concentrating on the instability by the contrast in elastic modulus (e.g., Ben-Zion and Shi, 2005; So et al., 2012) suggested slightly
different critical points of R s from 1.5 to 1.7.
Previous numerical studies employed a wide range of plastic
yield strength of subducting oceanic lithosphere from 200 MPa
(relatively weak, Christensen, 1996) to 1 GPa (relatively strong,
Billen, 2010). The amount of shear heating and amplitude of overpressure complicatedly interacts with the plastic yielding. For instance, a recent numerical study reported (So and Yuen, 2014) that
the interplay among rheology (e.g., activation energy) and deformation history, plastic yield strength can cause a persistence of
the yielding over the yield strength. Thus, the model with a wide
range of plastic yield strength should be investigated.
The positive feedback causes a strong thermal weakening,
which leads to the relaxation of deviatoric stress within the
shear zone. The pressure increases to compensate for a loss
of stress to achieve a force balance in the parallel direction
of far-field compression (Schmalholz and Podladchikov, 2013;
Schmalholz et al., 2014). The range of tectonic pressure induced
around the shear zone is between 0.3 GPa (at R s = 2) and 1.5 GPa
(at R s = 10), which are, respectively compatible with ∼9 km and
∼45 km vertical thickness of typical oceanic lithosphere with a
density of 3300 km/m3 (e.g., Turcotte and Morgan, 1992). When
we consider that a widely accepted topography of the olivine–
wadsleyite interface in subducting lithospheres ranges between 30
to 60 km (e.g., Chambers et al., 2005), the tectonic pressure should
not be ignored when analyzing the phase transition in slabs. Since
the temperature of shear zone increases, a higher pressure is required for the olivine–wadsleyite phase transition. However, the
ratio of pressure to temperature changes (i.e., P tec /
T ) are be-
tween 4 MPa/K (at R s = 2) to 2.8 MPa/K (at R s = 10), which
are larger than a typical Clapeyron slopes of olivine–wadsleyite
transition (i.e., 1.5–3 MPa/K; Bina and Helffrich, 1994). This large
P tec /
T means that the tectonic over-pressure still can alter
the phase transition despite the temperature elevation. Moreover,
in contrast to the case without the loop, the pressure gradient
along the horizon of 410 km depth becomes larger and more complicated.
The pressure fluctuations from lithostatic conditions may create
the pressure gradient necessary to act as a driving force for fluid
flow (Faccenda et al., 2012). Because recent studies have continually revealed a large water reservoir in the mantle transition zone
(e.g., Huang et al., 2005; Schmandt et al., 2014), the tectonic overpressure around the phase loop will be highlighted as a plausible
mechanism for water transport across the slab–mantle interface.
If the large scale tectonic over-pressure from the phase loop is
employed for fluid exchange, this mechanism may contribute to
understand how the slab can provide a huge amount of water to a
certain depth range down to the phase transition zone. We found
that the pressure gradient for the case with the phase loop (i.e.,
R s ≥ 2) is sufficiently large to cause fluid flow at the slab–mantle
boundary. In addition, the pressure gradient is complicated even
inside the slab, which argues for complex fluid flow inside the
slab.
The important point to emphasize here is the thickness of
phase loops within subducting lithospheres in a realistic geological setting. Although many thermodynamic (e.g., Akaogi et al.,
1989) and seismological (e.g., Shearer, 2000) studies have reported
the two-phase coexistence zone (i.e., the phase loop) around the
410 km boundary, the width of phase loop remains uncertain.
If the phase transition occurs very quickly, then the loop is very
thin (i.e., at most 5 km; e.g., Yamazaki and Hirahara, 1994) or even
cannot be detected. In this case, a seismic reflection wave at the
olivine–wadsleyite phase boundary would not be able to identify
the phase loop. On the other hand, even if the thickness of phase
loop is ∼35 km (Shearer, 2000), this thickness is still too small to
be seismically resolved (Ricard et al., 2009). In addition, the content of water can vary the loop thickness (e.g., Smyth and Frost,
2002). In spite of the uncertainty in phase loop, its elastic heterogeneity compared with the surroundings (Li and Weidner, 2008)
can cause the formation of a shear zone and the subsequent tectonic over-pressure, which has a strong impact on the feature of
olivine–wadsleyite phase transition. From our results, we may surmise that the olivine–wadsleyite interface becomes broader than
the case without the loop because the shear zone is broadly distributed around the loop.
The temperature condition has been known as the most important factor for controlling the thickness of olivine–wadsleyite
phase loop. For instance, the thickness of phase loop varies from
∼30 km to ∼9 km with temperature condition from 700 K to
1250 K (Akaogi et al., 1989), which is consistent with this study.
However, other studies showed that the thickness of phase loop
can be affected by Mg–Fe composition (e.g., Katsura et al., 2004)
and water content (e.g., Smyth and Frost, 2002). Thus, the temperature dependence of phase loop thickness can be different with
our study, which means the phase loop thickness can be uniform. Even though So and Yuen (2015) showed that shear heating
also appears within the phase loop with a uniform thickness (i.e.,
weak temperature dependence), more detailed investigation of tectonic over-pressure with a dynamic changing thickness of the loop
should be performed.
Our model also has an implication for the distribution of deep
earthquakes in the phase transition zone. Many seismological studies have found that deep earthquake activity ceases in the depth
range of 300 to 450 km in subducting lithospheres (Frohlich, 1989
and references therein). However, it is natural that the faulting
B.-D. So, D.A. Yuen / Earth and Planetary Science Letters 413 (2015) 59–69
associated with olivine–wadsleyite phase transition (i.e., transformational faulting; Green and Houston, 1995) should occur and
trigger deep earthquakes in this seismically quiet zone. Based on
our study, shear zone formation around phase loop, which accompanies a deviatoric stress drop and tectonic over-pressure, can
restrict the transformational faulting. This is due to the presence
of a large deviatoric stress (i.e., ∼2 GPa; Schubnel et al., 2013)
and low confined pressure which promote necessary conditions
for faulting. We emphasize here that the phase loop may be a
new way to explain deep earthquake distribution along subducting
lithospheres. Even if we do not accept the existence of the phase
loop and its consequence on tectonic over-pressure, ductile shear
zones (at least small scale) can be originated from preexisting
faults such as oceanic transforms (Jiao et al., 2000). This shear zone
will also cause tectonic over-pressure. Thus, preexisting faults and
the associated over-pressure should also be considered when we
estimate the metamorphic depth of exhumed rocks in subduction
zones. We need to discuss about metastable olivine wedge where
olivine polymorph persists down to ∼600 km depth in a cold core
of a slab. This is because the low temperature in the core retards
the reaction rate of olivine–wadsleyite phase transition (Kawakatsu
and Yoshioka, 2011). We focused on the over-pressure generated
by shear zone formation due to the phase loop, which arises not
from metastability (Yoshioka et al., 1997). However, the metastable
olivine wedge might also contain a portion of wadsleyite, which
implies that there may be a two phase zone that causes an elastic heterogeneity as the phase loop does. If the elastic modulus in
the metastable olivine wedge partially containing the wadsleyite
is constrained, our study can contribute in investigation of the
tectonic over-pressure within the metastable olivine wedge. Thus,
the relationship between the over-pressure and metastable olivine
wedge should be understood to determine the areal extent of the
wedge. This will help us to adopt more realistic density structure of subducting lithosphere into numerical simulations dealing
with the metastable olivine wedge as a density heterogeneity (e.g.,
Tetzlaff and Schmeling, 2009).
5. Conclusion
We performed a series two-dimensional thermal-mechanical
simulations to assess the existence of the shear zone on the phase
loop and the associated tectonic over-pressure and their effects on
the olivine–wadsleyite phase transition. We applied a wide range
of R s values, which represents the shear modulus ratio between
the surrounding region and the phase loop, to observe the effect
of the magnitude of elastic heterogeneity on the tectonic overpressure and its related feature of phase transition. Our calculations show that the cases of a larger R s induces a larger tectonic
over-pressure during the deformation. When the R s values are 2
and 10, the magnitude of over-pressure are 0.3 and 1.5 GPa, respectively. Due to the diffused shear zone around the phase loop,
the sharp topography before the generation of over-pressure becomes broader after the over-pressure forms. When there is no
phase loop, the pressure gradient around the 410 km transition
zone is almost zero. Otherwise, the gradient for the case of phase
loop is large even inside the slab. It supports that the fluid flow
within the slab can be vigorous. We propose that over-pressure
should be considered in studying complicated phenomena in slabs,
such as the interface variation of phase transition, fluid transport
from the slab to mantle and the deep earthquake. If the slab and
mantle around 410 km olivine–wadsleyite phase transition zone
is almost anhydrous (i.e., less than several tens to hundreds wt.
ppm) (Kawakatsu and Yoshioka, 2011), we may expect that the
coupling between low pore-pressure and high over-pressure can
affect the distribution of deep earthquake. Moreover, the rheology
of olivine and wadsleyite is strongly depending of water content
67
(Karato, 1986), which can control the amount of shear heating.
Thus, we should investigate an interplay among water content,
shear heating and tectonic over-pressure.
Acknowledgements
We thank to two anonymous reviewers for their careful reviews, which significantly improved our manuscript. This research
was supported by the National Research Foundation of Korea
(NRF-2014R1A6A3A04055841) for B.-D. So and U.S. National Science Foundation grants in the Collaboration of Mathematics and
Geosciences (CMG) program and Geochemistry for D.A. Yuen.
We also thank discussions with Yuri Podladchikhov.
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