Earth and Planetary Science Letters 298 (2010) 229–243
Contents lists available at ScienceDirect
Earth and Planetary Science Letters
j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e p s l
Dynamical consequences in the lower mantle with the post-perovskite phase change
and strongly depth-dependent thermodynamic and transport properties
Nicola Tosi a,b,⁎, David A. Yuen c,d, Ondřej Čadek a
a
Department of Geophysics, Faculty of Mathematics and Physics, Charles University, Prague, Czech Republic
now at Institute of Planetary Research, Department of Planetary Physics, German Aerospace Center (DLR), Berlin, Germany
Department of Geology and Geophysics, University of Minnesota, Minneapolis, MN 55455, USA
d
Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, MN 55455, USA
b
c
a r t i c l e
i n f o
Article history:
Received 3 March 2010
Received in revised form 29 June 2010
Accepted 1 August 2010
Available online 30 August 2010
Editor: Y. Ricard
Keywords:
lower mantle dynamics
post-perovskite
phase transitions
thermal expansivity
lattice thermal conductivity
a b s t r a c t
We have carried out numerical simulations of large aspect-ratio 2-D mantle convection with the deep phase
change from perovskite (pv) to post-perovskite (ppv). Using the extended Boussinesq approximation for a
fluid with temperature- and pressure-dependent viscosity, we have investigated the effects of various ppv
phase parameters on the convective planform, heat transport and mean temperature and viscosity profiles.
Since ppv is expected to have a relatively weak rheology with respect to pv and a large thermal conductivity,
we have assumed that the transition from pv to ppv is accompanied by both a reduction in viscosity by 1 to 2
orders of magnitude and by an increase in thermal conductivity by a factor of 2. Furthermore, we have
analyzed the combined effects of a strongly decreasing thermal expansivity in pv and steeply increasing
thermal conductivity according to recent evidence from high-pressure experiments and first-principle
calculations. As long as the thermal expansivity and conductivity are constant, ppv exerts a small but
noticeable effect on mantle convection: it destabilizes the D″ layer, causes focusing of the heat flux peaks and
an increase of the average mantle temperature and of the temporal and spatial frequency of upwellings.
When the latest depth-dependent thermal expansivity and conductivity models are introduced, the effects of
ppv are dramatic. On the one hand, without ppv, we obtain a very sluggish convective regime characterized
by a relatively cool mantle dominated by large downwellings that tend to stagnate beneath the transition
zone. With ppv, on the other hand, we observe an extremely significant increase of the average mantle
temperature due to the formation of large sized and vigorous upwellings that in some cases tend to cluster,
thus forming superplumes. If a very large thermal conductivity at the core-mantle boundary is assumed
(k ~ 20 WK–1 m–1) we obtain a quasi steady-state regime characterized by large and stable plumes with long
lifetimes. The combination of strongly depth-dependent expansivity and conductivity is a viable mechanism
for the formation of long-wavelength, long-lived thermal anomalies in the deep mantle, even if a lowviscosity ppv atop the core-mantle boundary is included.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
The discovery of the exothermic phase transition from perovskite
(pv) to post-perovskite (ppv) from high-pressure experiments and
first-principle simulations (Murakami et al., 2004; Oganov and Ono,
2004; Tsuchiya et al., 2004) is having a great impact on our understanding of the structure and composition of the lowermost mantle.
While chemical heterogeneities are often invoked to explain the
seismic complexity of large low shear velocity provinces that are
likely associated with thermo-chemical piles or superplumes in the
lower mantle (Trampert et al., 2004; McNamara and Zhong, 2005;
Lay et al., 2006; Garnero and McNamara, 2008), growing seismolog⁎ Corresponding author. Department of Geophysics, Faculty of Mathematics and
Physics, Charles University, Prague, Czech Republic.
E-mail address: [email protected] (N. Tosi).
0012-821X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2010.08.001
ical evidence also supports the existence of the pv–ppv phase change
at least in regions, such as beneath Central America, where old
subducted lithosphere has reached into the D″ (Hutko et al., 2006; Lay
et al., 2006; Garnero et al., 2007; Ren et al., 2007; van der Hilst et al.,
2007; Hutko et al., 2008; van der Meer et al., 2010). The investigation
of the dynamical consequences on mantle convection due to the
presence of ppv has started shortly after its discovery in 2004.
Nakagawa and Tackley (2004) first incorporated the pv–ppv phase
change in numerical simulations of compressible mantle convection
in a half-cylindrical geometry. They found that this transition exerts
relatively small effects on mantle convection which, nevertheless, can
be easily identified. With respect to simulations where ppv is not
included, they encountered a generally hotter mantle characterized
by an increased number of small-scale upwellings that contribute to
enhance the heat flux through the core-mantle boundary (CMB). This
study was next extended to include compositional effects due to a
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N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
chemically distinct layer at the base of the mantle using a halfcylindrical (Nakagawa and Tackley, 2005) and a 3-D Cartesian
geometry (Nakagawa and Tackley, 2006). In both cases it was
observed a systematic anti-correlation between ppv regions and
sites where large piles of dense material accumulate, whose stability,
however, tends to be reduced by the presence of ppv. Matyska and
Yuen (2006) presented 2-D convection models whose focus was the
identification of the conditions that allow thermal superplumes to
form in the presence of ppv. They concluded that, for an exothermic
transition such as that from pv to ppv, it is necessary to take into
account a radiative thermal conductivity in the D″ region for the
formation of stable superplumes (Goncharov et al., 2006). Without
considering this mechanism, the destabilizing effect of ppv prevails
and reduces the spatial and temporal scale of upwellings and thus
prevents plume clustering and the formation of large-scale structures.
Using 3-D spherical convection models with a depth-dependent
viscosity, Monnereau and Yuen (2007) discussed the topology of the
pv–ppv phase boundary in relation to the CMB temperature and to
the so called temperature intercept (Tint), i.e. the temperature of the
phase transition at the CMB pressure. In fact, while the Clapeyron
slope and the density increase associated with ppv are relatively well
constrained (e.g. Hernlund and Labrosse, 2007; Catalli et al., 2009;
Tateno et al., 2009), there is still a large uncertainty in the estimate
of Tint (e.g. Hirose, 2006; Monnereau and Yuen, 2007). In particular,
Monnereau and Yuen (2007) found that seismological constraints can
be best satisfied by choosing a temperature intercept about 200 K
lower than the CMB temperature. Under this condition, the transition
to ppv does not occur everywhere above the CMB but only in
relatively cold regions where a double crossing of the Clapeyron curve
by the local geotherm takes place (Hernlund et al., 2005). In this case,
the lowermost mantle is characterized by ppv lenses, associated with
cold subducted material, underlain by a thin layer of pv, while holes,
where there is no ppv, are associated with regions of hot upwellings.
Monnereau and Yuen (2010) extended further the previous study
showing how information on the topology of ppv lenses can be used
to place constraints on the core heat flux.
In all the aforementioned modeling studies, no rheological change
is considered for ppv, consistently with the assumption that ppv
should deform via diffusion creep in the same way as pv, i.e. with the
same activation parameters. Although the current knowledge of ppv
rheology is far from being conclusive and complete (e.g. Yamazaki
and Karato, 2007), recent results suggest that the viscosity of ppv
may differ considerably from that of pv. Experimental measurements
conducted on calcium-iridate (CaIrO3), a low-pressure analogue of
pv, show that its ppv phase is significantly weaker than the pv one,
with minimum estimates ranging from a factor of 5 to 10 (Hunt et al.,
2009). CaIrO3 has also been employed for experiments of grain
growth by Yoshino and Yamazaki (2007). They showed that the
growth rate of ppv is much lower than that of pv. Therefore, if
the principal deformation mechanism for ppv is diffusion creep,
small grains should characterize this phase and induce a softening in
the D″ due to their slow growth rate. Diffusion rates for pv and ppv
have also been computed with first-principle simulations by Ammann
et al. (2009, 2010) who showed that diffusion in ppv occurs much
faster than in pv. Moreover, from results of experiments (Ohta et al.,
2008) and first-principle calculations (Oganov and Ono, 2005), the
electrical conductivity of ppv is expected to be several orders of
magnitude greater than that of pv. All these arguments indicate that
the viscosity of ppv is likely to be lower than the viscosity of pv. This
scenario is also consistent with the inversion of the long-wavelength
geoid conducted by Čadek and Fleitout (2006) who found a significant
correlation between the distribution of low viscosity areas in the D″
and the location of paleosubduction sites. Tosi et al. (2009a,b) further
explored the possible effects of ppv on the geoid and dynamic topography and argued that, over regions where slabs reach the deepest
mantle, the amplitude of the geoid can be significantly reduced
because of the presence of ppv if this has a lower viscosity than the
surroundings. Čížková et al. (2009) modeled the dynamics of lower
mantle slabs with a composite rheology and the pv–ppv transition.
Since it is well accepted that ppv is a highly anisotropic material
(Walte et al., 2009; Ammann et al., 2010), they investigated the effects
of modeling ppv deformation via dislocation creep. On the one hand,
they found that as long as no viscosity contrast for ppv is considered,
the effect of the phase transition is moderate no matter what deformation mechanism is used for ppv. On the other hand, as soon as ppv
is weaker than pv, the viscosity reduction induces dramatic changes
on the formation of upwellings and on the amount of heat transported
across the CMB. Matyska et al. (2010) analyzed the effects on mantle
dynamics due to a ppv phase occurring everywhere above the
CMB (i.e. with a single crossing of the Clapeyron curve) and having a
viscosity by one order of magnitude lower than the viscosity of pv.
They observed that the effects induced by a viscosity reduction in ppv
are similar to those generated by considering a radiative thermal
conductivity in the D″ layer, namely a general increase of the mantle
temperature and the tendency for upwellings to aggregate and form
plume clusters.
Besides ppv and its rheological variations, recent advances both in
experimental and computational measurements of thermodynamic
properties of lower mantle minerals can contribute substantially to
broaden our current view on the dynamics of the deep Earth. Mantle
convection simulations that incorporate depth-dependent thermal
expansivity (α) and conductivity (k) generally assume either that
these two quantities are constant or that they vary weakly with the
depth (e.g. Zhao and Yuen, 1987; Leitch et al., 1991; Hansen et al.,
1993; Tackley, 1996; van den Berg et al., 2002; Matyska and Yuen,
2005; Naliboff and Kellogg, 2006; Monnereau and Yuen, 2007;
Komabayashi et al., 2008). However, the latest high-pressure measurements of pv volume by X-ray diffraction (Katsura et al., 2009)
indicate that the decrease of the thermal expansivity of pv with
increasing pressure can be very large, with α decreasing by about one
order of magnitude from the surface to the CMB. Furthermore, recent
measurements (Xu et al., 2004; Beck et al., 2007; Hofmeister, 2008;
Goncharov et al., 2009, 2010) and first-principle calculations (de
Koker, 2009, 2010; Tang and Dong, 2010) of the lattice thermal
conductivity of pv and MgO periclase show that k increases strongly
across the whole mantle with values that can reach up to 20–30 Wm−1
K−1 near the CMB. Such strong depth variations of the thermodynamic
properties of the mantle can have important consequences on the style
and evolution of mantle convection (e.g. Hansen et al., 1993).
As the number of previous studies featuring variable ppv viscosity
is very restricted and, in particular, as the analysis of the combined
effects of ppv and strongly depth-dependent thermodynamic properties has received no attention so far, in this work we aim at
incorporating the most recent findings related to ppv rheology and
depth-dependent thermal expansivity and conductivity into 2-D
numerical simulations of mantle convection, which allows us to
sweep a wide parameter space. Starting from models in which α and k
are constant and no rheological variation is used to characterize ppv,
we add progressive complexity to our models and discuss the effects,
due to the reduction of ppv viscosity and variations with the depth of
thermal expansivity and conductivity, on the convective planform,
heat transport and long-term stability of large-scale structures.
2. Model description
2.1. Governing equations
We have performed our numerical simulations using the extended
Boussinesq approximation (EBA) for a fluid with infinite Prandtl
number (e.g. Ita and King, 1994). Although using the fully compressible anelastic liquid approximation (ALA) (e.g. Leng and Zhong, 2008)
is generally more correct, the EBA still represents a valid alternative
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
for several reasons. It is routinely employed in studies focused on
convection in the lower mantle (e.g. Matyska and Yuen, 2006; van den
Berg et al., 2010a) or even on convection in exoplanets with radii
larger than that of the Earth for which the effects of compressibility
should be even more pronounced (van den Berg et al., 2010b). The
EBA also accounts for two non-Boussinesq components, namely the
adiabatic and shear heating. In particular, thanks to the presence of
adiabatic heating in the equation of energy conservation (see Eq. (3)),
EBA captures an effect typical of compressible convection in that, with
constant thermal expansivity, it allows for the natural emergence of
an adiabatic temperature gradient, while the standard Boussinesq
approximation would yield an essentially isothermal mantle (Ita
and King, 1994). Furthermore, with the exception of Tan and Gurnis
(2005, 2007) who attribute to compositional dependent compressibility a fundamental role in regulating the stability of lower mantle
thermo-chemical structures, studies in which EBA and ALA have been
systematically compared have not evidenced any principal difference
in the qualitative behavior of the flow that can be ascribed to the lack
of a depth-dependent reference density profile in the EBA (Jarvis and
McKenzie, 1980; Lee and King, 2009; King et al., 2010; Phipps Morgan
and Rüpke, 2010).
The model domain is an aspect-ratio 10 box. This large aspect-ratio
is chosen to allow greater degrees of freedom to the flow in the
presence of strongly depth-dependent properties (Hansen et al.,
1993). With the variables and parameters listed in Table 1, the
equations of continuity, linear momentum and thermal energy in the
presence of multiple phase transitions take respectively the following
non-dimensional form:
∇⋅v = 0;
ð1Þ
t
−∇p + ∇⋅ η ∇v + ð∇vÞ
=
3
!
Ras αT + ∑ Rbi Γi ẑ;
i=1
DT
T
= ∇⋅ðk∇T Þ + Dis α T + s vz
Dt
ΔT
ð3Þ
3
Dis
Rbi
T
DΓ
t
+
η ∇v + ð∇vÞ : ∇v + ∑
Dis T + s γi i ;
Ras
ΔT
Dt
i = 1 Ras
where D/Dt represents the material time derivative, superscript t the
operation of transposition and : the double scalar product of tensors.
Eqs. (1)–(3) have been non-dimensionalized using surface values for
the thermodynamic and other physical parameters, the mantle depth
for the length, the time scale of thermal diffusion for the time and the
temperature drop across the mantle for the temperature. In the
momentum Eq. (2), the first and second terms on the right-hand side
account for the buoyancy forces due to temperature differences and
phase transitions, respectively. Buoyancy effects due to compositional
variations (e.g. van den Berg et al., 2010a) are here neglected. We
consider three phase transitions: at 410 km depth from olivine to
spinel (i = 1), at 660 km depth from spinel to pv (i = 2) and, according
to local pressure and temperature conditions, from pv to ppv (i = 3),
with the depth of this transition falling in general between 2700 and
2850 km in depth. Following Christensen and Yuen (1985), the effect
of the ith transition is taken into account using a phase function:
1
z−zi ðT Þ
Γi =
1 + tanh
;
2
w
0
γi
ðT−T0 Þ;
ρ0 g0
Symbol
Description
Scaling/numerical value
x
ẑ
t
v
vz
p
T
η
α
k
κ
Ras
Rbi
Dis
g0
ρ0
H
cp
ηs
αs
ks
κs
TCMB
Ts
ΔT
Tint
w
z 01
z 02
γ1
Position vector
Unit vertical vector
Time
Velocity vector
Vertical velocity
Dynamic pressure
Temperature
Viscosity
Thermal expansivity
Thermal conductivity
Thermal diffusivity
Surface Rayleigh number
Phase Rayleigh number
Surface dissipation number
Gravity acceleration
Reference density
Mantle depth
Heat capacity
Surface viscosity
Surface thermal expansivity
Surface thermal conductivity
Surface thermal diffusivity
Temperature at the CMB
Surface temperature
Temperature drop across the mantle
Temperature intercept
Phase transition width
Depth of the olivine–spinel transition
Depth of the spinel–pv transition
Clapeyron slope for olivine–spinel
transition (Bina and Helffrich, 1994)
Clapeyron slope for spinel–pv transition
(Bina and Helffrich, 1994)
Clapeyron slope for pv–ppv transition
(Hirose, 2006; Tateno et al., 2009)
Density jump for olivine–spinel transition
(Steinbach and Yuen, 1995)
Density jump for spinel–pv transition
(Steinbach and Yuen, 1995)
Density jump for pv–ppv transition
(Oganov and Ono, 2004)
H
H
H 2κ−1
s
κsH−1
−1
κsH
ηsκsH−2
ΔT
ηs
αs
ks
−1
ks = ksρ−1
0 cp
7
−1
ρ0 αsΔTH 3g0η−1
s κs =1.9∙10
−1
δρi H3g0η−1
κ
s
s
αsHg0c−1
p = 0.68
−2
9.8 m s
4500 kg m−3
2890 km
1250 J kg−1
1022 Pa s
3∙10−5 K−1
3.3 W m−1 K−1
5.9∙10−7 m2 s−1
3800 K
300 K
3500 K
3600, 4000 K
20 km
410 km
660 km
3 MPa K−1
γ3
δρ1
δρ2
δρ3
ð4Þ
ð5Þ
−2.5 MPa K−1
0, 9, 13 MPa K−1
273 kg m−3
342 kg m−3
67.5 kg m−3
where z 0i is the reference depth of the transition boundary (i.e. either
410 or 660 km), γi is the Clapeyron slope and T0 the reference
temperature. Post-perovskite is probably not ubiquitous above the
CMB. It likely occurs in isolated patches where pressure and temperature conditions suitable for its formation are met. In the pressure–
temperature space, the boundary between pv and ppv can be
described by a line:
p = pCMB + γppv ðT−Tint Þ;
ð6Þ
where pCMB is the hydrostatic pressure at the CMB, γppv ≡ γ3 is the
Clapeyron slope of the phase change and Tint is the intercept of the
pv–ppv phase boundary at the CMB. Therefore, if i = 3, zi(T) in Eq. (4)
takes the following form:
z 3 ðT Þ = H +
where z is the dimensional depth and w is the width of the phase
transition. If i = 1 or 2, the temperature-dependent function zi(T) is
defined as
zi ðT Þ = z i +
Table 1
Non-dimensional variables and numbers with the corresponding scaling and
dimensional parameters employed in this study. Multiple values are indicated for
parameters that are varied in the models tested.
γ2
ð2Þ
231
γ3
ðT−Tint Þ;
ρ0 g0
ð7Þ
where H is the mantle depth.
In the temperature Eq. (3), the advection of temperature on the
left hand side is balanced on the right-hand side by heat diffusion,
adiabatic heating/cooling, viscous dissipation and latent heat release
due to phase changes, respectively. The effects due to internal sources
of heat are not considered, consistently with the assumption that
Cartesian models generally yield mean temperatures that tend to
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N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
overestimate those found in more realistic spherical shell convection
models (O'Farrell and Lowman, in press).
2.2. Material parameters
2.2.1. Viscosity
We assume a Newtonian rheology with viscosity depending on
depth and temperature as follows (Matyska and Yuen, 2007):
n
n
oo
−1
ηðz; T Þ = η1 ðzÞ min f ; max f ; η 2 ðT Þ ;
ð8Þ
where the depth-dependent part η1(z) is defined as in Hanyk et al.
(1995):
2
η1 ðzÞ = 1 + 214:3 z exp −16:7ð0:7−zÞ
:
0:6
η 2 ðT Þ = exp 10
−1 :
0:2 + T
ð10Þ
The factor f in Eq. (8) determines the sensitivity of viscosity to
temperature differences and is taken equal to 10, with the
consequence that lateral viscosity contrasts due to temperature can
0.0
0.2
z
0.4
0.6
log10(η)
η ppv = cηpv ;
k1
k2
0.8
ð11Þ
where the prefactor c can take the values 1, 0.1 or 0.01 and ηpv denotes
the pv viscosity obtained from Eq. (8).
2.2.2. Thermal expansivity
Besides models where the thermal expansivity is held constant, we
consider models in which it decreases with depth according to the
following non-dimensional profile (Matyska and Yuen, 2007; Matyska
et al., 2010) (see Fig. 1):
ð9Þ
Eq. (9) implies a viscosity maximum in the mid lower mantle (see
Fig. 1) (Matyska et al., 2010; Morra et al., 2010). This is consistent with
modeling studies of dynamic geoid (Ricard and Wuming, 1991; Čadek
and van den Berg, 1998), postglacial rebound (Mitrovica and Forte,
2004; Tosi et al., 2005) and on predictions of the mantle viscosity
profile obtained using a realistic equation of state (Walzer et al.,
2004). Furthermore, the existence of such a maximum finds a
plausible explanation in the non-monotonicity of the activation
parameters of lower mantle minerals (Wentzcovitch et al., 2009)
caused by the high- to low-spin transition of Fe2+ in ferropericlase
(e.g. Badro et al., 2004; Speziale et al., 2005). Nevertheless, this
transition, which can be treated as an exothermic phase change
(Bower et al., 2009), is neglected in our calculations.
The temperature-dependent part is included as an Arrhenius law
(e.g. Schubert et al., 2001):
α
vary up to a factor of 100 which guarantees the mobility of the surface
thermal boundary layer (e.g. Moresi and Solomatov, 1995).
To account for the effects of a lower viscosity in ppv (Ammann
et al., 2009; Hunt et al., 2009), we assume that its viscosity is given by:
(
α=
ð1 + 0:78zÞ−5
0:44ð1 + 0:35ðz−0:23ÞÞ−7
if 0 ≤ z ≤ 0:23
if 0:23 b z ≤ 1
ð12Þ
Eq. (12) implies an overall decrease of α of about one order of
magnitude throughout the mantle and that the Anderson–Grüneisen
parameter is approximately 5 for the upper mantle and 7 for the lower
mantle, consistently with experimental estimates of olivine (Chopelas
and Boehler, 1992) and perovskite (Katsura et al., 2009). Although
potentially important, especially for the heat transport and the
distribution of flow velocities in the upper mantle, the temperature
dependence of the thermal expansivity (Ghias and Jarvis, 2008) is
here neglected.
2.2.3. Thermal conductivity
As the thermal conductivity plays a central role in regulating the
exchange of heat across the CMB and ultimately the thermal evolution
of both the mantle as well as the core, a great deal of attention has
recently been devoted to estimate the values of this parameter at
conditions of the lower mantle. Several experiments and theoretical
first-principles simulations have been conducted on the two principal
minerals that are thought to form the bulk of the lower mantle,
namely MgSiO3 perovskite and MgO periclase. Broad consensus has
grown that favors a large increase of the thermal conductivity with
pressure across both the upper (e.g. Xu et al., 2004) and lower mantle
(e.g. Goncharov et al., 2010) with a less important dependence on
temperature (e.g. Hofmeister, 2008). However, no conclusive agreement has been reached on the exact values that this parameter takes
on at lower mantle pressures. In fact recent experimental and
theoretical estimates of the total lattice thermal conductivity near
the CMB range from 6 Wm−1 K−1 (de Koker, 2010) to about 10 Wm−1
K−1 (Goncharov et al., 2009; Ohta, 2010; Tang and Dong, 2010) to
even 20–30 Wm−1 K−1 (Hofmeister, 2008).
In our simulations we consider then three different models in which
the thermal conductivity is either constant or, for simplicity, linearly
dependent on depth as shown in Fig. 1, with an increase over the mantle
of a factor of 3, corresponding to kCMB ~ 10 Wm−1 K−1 (hereafter
denoted as profile k1), or 6, corresponding to kCMB ~ 20 Wm−1 K−1
(hereafter denoted as profile k2). Furthermore, we also assume that
ppv has a conductivity larger than that of its surroundings by a factor of
2 (Hofmeister, 2007).
2.3. Numerical method
1.0
0
1
2
3
4
5
6
Fig. 1. Non-dimensional vertical profiles of the logarithm of viscosity (blue), thermal
expansivity (red) and thermal conductivity according to the models k1 (solid black) and
k2 (dashed black).
To solve the set of Eqs. (1)–(3), we use a newly developed code
named YACC (Yet Another Convection Code) based on a primitivevariable, 2nd order accurate finite-volume formulation (Patankar,
1980; Gerya and Yuen, 2003). The momentum Eq. (2) is integrated for
horizontal and vertical velocities located at staggered nodes, while
the continuity Eq. (1) is integrated for the pressure located at cell
centers. For the energy Eq. (3), we use an operator-splitting method
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
233
and the reduction of its viscosity. The models shown feature
temperature- and depth-dependent viscosity according to Eq. (8)
but constant thermal expansivity and conductivity. In Fig. 2a, no ppv is
included, while in Fig. 2b–d ppv is taken into account considering
Tint = 3600 K, γppv = 13 MPa K−1 and ηppv = ηpv (Fig. 2b), ηppv = 0.1ηpv
(Fig. 2c) and ηppv = 0.01ηpv (Fig. 2d). In Fig. 2b–d, below the snapshots
of the temperature field, also the occurrence of the corresponding
ppv phase is presented. The region shown comprises the bottom
780 km of the domain. Since in these calculations Tint b TCMB, the
boundary between pv and ppv is crossed twice by the local geotherm
(Hernlund et al., 2005). Therefore, all ppv regions are actually
“lenses” underlain by a thin layer of pv. Without ppv (Fig. 2a), the
temperature distribution is that typical of chaotic convection at
relatively high Rayleigh numbers (e.g. Matyska and Yuen, 2007) with
highly unstable TBLs and not clearly recognizable cells. Strong
downwellings develop from the top TBL and reach the bottom of the
domain. The bottom TBL is characterized by hot upwellings that tend
to lose their thermal signature while rising through the mantle
because of adiabatic cooling. Moreover large-scale horizontal flow
promotes the shearing and clustering of new thermal instabilities
forming at the bottom. It is convenient to describe the effects due to
ppv and its viscosity changes referring also to Figs. 3 and 4. For the
same models presented in Fig. 2, Fig. 3 shows the time evolution of
the temperature field as a function of the horizontal coordinate at
three non-dimensional depths: z = 0.1, 0.5 and 0.9, while Fig. 4
illustrates the vertical profiles of temperature and viscosity averaged
over the horizontal coordinate and over the second half of the time
evolution, the Nusselt number (top line), the time series of surface
and CMB heat fluxes and root mean square (RMS) velocity (central
line) and histograms along with the corresponding normalized
statistical distribution of the Nusselt number (Hansen et al., 1992)
(bottom line). As expected, ppv is present in those regions of the
bottom of the mantle where cold slabs reach the lowermost part of
the domain. Post-perovskite areas are disconnected from each other
where hot plumes are present (Fig. 2b–d). As already reported by
Nakagawa and Tackley (2004), introducing ppv with no change in
viscosity with respect to pv (Fig. 2b) causes a slightly hotter mantle
(Spiegelman and Katz, 2006) that combines a semi-Lagrangian
technique with bicubic interpolation to treat the temperature advection (Staniforth and Côté, 1991; Rosatti et al., 2005) and a semiimplicit Crank-Nicholson scheme for the diffusion. Because of its
unconditional stability (e.g. Bates and McDonald, 1982), the semiLagrangian method allows the use of relatively large time steps which
makes it particularly convenient when the conservation equations
have to be integrated over long time intervals. The systems of linear
equations arising from the discretization of the combined momentum
and continuity equations and of the thermal energy equation are
solved using the parallel direct sparse solver PARDISO (Schenk et al.,
2000). YACC has been thoroughly validated and has proven to be
accurate in the treatment of both Boussinesq and non-Boussinesq
convection (King et al., 2010).
All boundaries of the computational domain are impermeable
and free-slip. Boundary conditions for Eq. (3) consist of reflective
sidewalls and isothermal top and bottom boundaries with a temperature of 300 K and 3800 K respectively, the latter being consistent
with recent estimates of CMB absolute temperature obtained by
combining seismological and mineral physics models (van der Hilst
et al., 2007; Kawai and Tsuchiya, 2009). As initial condition, we use an
adiabatic temperature profile with a potential temperature of 1600 K
and thin boundary layers at the top, bottom and sidewalls. We run all
calculations up to the non-dimensional time t = 0.03, corresponding
to a time interval of about 13.4 billion years, i.e. roughly three times
the age of the Earth. To discretize the model domain, we use 1000
equally spaced nodal points in the horizontal direction and up to 200
points in the vertical direction with refinement by a factor of 4 in the
top and bottom thermal boundary layers (TBL), reaching a vertical
resolution of 5 km at the surface and CMB.
3. Results
3.1. Effect of low-viscosity post-perovskite
We start illustrating in Fig. 2 the long-term evolution of the
temperature field focusing on the effects due to the presence of ppv
α=const , k= const, no ppv
a
α=const , k= const, T int =3600 K, γ ppv =13 MPa/K, η ppv =η pv
b
α=const, k= const, T int =3600 K , γ ppv=13 MPa/K, η ppv=0.1 η pv
c
α=const , k= const , T int=3600 K , γ ppv=13 MPa/K, η ppv=0.01 η pv
d
0
T
1
Fig. 2. Long-term snapshots of the temperature distribution for four models featuring constant thermal expansivity and conductivity and no ppv (a) or ppv with Tint = 3600 K,
γppv = 13 MPa K−1 and ηppv = ηpv (b), ηppv = 0.1ηpv (c), ηppv = 0.01ηpv (d). In panels b–d the occurrence of the ppv phase is also shown below the temperature field in a box
comprising the bottom 780 km of the domain.
234
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
a
c
b
d
0.030
0.025
t
0.020
=0.1
0.015
0.010
0.005
0.000
0.030
0.025
t
0.020
=0.5
0.015
0.010
0.005
0.000
0.030
0.025
t
0.020
=0.9
0.015
0.010
0.005
0.000
0
2
4
6
8
10
0
2
4
x
6
8
10
0
x
0
2
4
6
x
T
8
10
0
2
4
6
8
10
x
1
Fig. 3. Time evolution of the temperature field for the same models shown in Fig. 2 as a function of the horizontal coordinate at three different depths: z = 0.1 (289 km, 1st line),
z = 0.5 (1445 km, 2nd line) and z = 0.9 (2601 km, 3rd line).
(compare black and red lines in Fig. 4) and contributes to destabilize
the bottom TBL by increasing the spatial and temporal frequency of
hot upwellings (Fig. 3b). The hotter mantle promotes in turn a
reduction of the viscosity profile over the whole mantle depth. In
particular, the viscosity drop from the depth z = 0.7 (2000 km) to the
CMB reduces from a factor of 80 to a factor of 60. The surface and
CMB heat fluxes undergo a slight increase and present a higher time
variability with respect to the simulation without ppv. Since the
RMS velocity also becomes larger, the convective heat transport is
enhanced, with the consequence that the Nusselt number increases
from 11.9 to 14.6.
From a quantitative point of view, as the viscosity of ppv is reduced
by one or two orders of magnitude (Fig. 2c and d, respectively), the
variations described above are further enhanced. We observe in fact
that the mean temperature, velocity, heat fluxes and Nusselt number
all tend to increase in a systematic way (Fig. 4, blue and green lines),
while the viscosity drop from the maximum in the mid-lower mantle
to the minimum above the CMB reaches up a factor of approximately
200. In particular, the time dependence of the CMB heat flux and RMS
velocity becomes more pronounced, being characterized by sharp and
sudden peaks due to the emergence of a large number of bottom
boundary layer instabilities. Indeed, the time evolution of the
temperature close to the bottom boundary (Fig. 3c and d, z = 0.9)
shows that low-viscosity ppv promotes the formation of plumes.
These, however, generally travel only short horizontal distances as
they tend to be sheared towards major upwellings. The increase of the
mantle temperature and number of plumes that accompanies the
reduction of ppv viscosity has a significant impact on the ppv volume
associated with each model. With higher mantle temperatures, ppv
forms progressively closer to the CMB with the consequence that the
average thickness of the ppv lenses is strongly reduced (compare ppv
panels in Fig. 2a, b and c). With the possible exception of the model
where ηppv = 0.01ηpv (Fig. 4, green lines), the other three models
seem to have reached statistical equilibrium because of the close
approximation of the Nusselt number distribution to a Gaussian
distribution function.
The case in which the viscosity of ppv is lowest (Figs. 2d and 3d) is
both characterized by a high convective vigor and by a relatively
stable evolution pattern. Three major downwellings and upwellings
are in fact observed that preserve their position over a nondimensional time span of about 0.01 (i.e. ~ 4.4 billion years) after
which the flow undergoes a major reorganization that disrupts the
stability of the convection planform. Nevertheless, as mentioned
above, care has to be taken in describing this case since a small trend is
still visible in the time series of heat fluxes and velocity (Fig. 4, green
lines) which suggests that the system is still approaching statistical
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
235
α = const, k=const, no ppv
α = const, k=const, T int =3600 K , γ ppv =13 MPa/K, η ppv = η pv
α = const, k=const, T int =3600 K , γ ppv =13 MPa/K, η ppv =0.1η pv
α = const, k=const, T int =3600 K , γ ppv =13 MPa/K, η ppv =0.01η pv
0.0
0.2
0.2
0.4
0.4
0.6
0.6
0.8
0.8
20
15
Nu
0.0
1.0
5
0
1.0
0.0
0.2
0.4
0.6
0.8
0.1
1.0
1
10
log10 (η)
T
30
1800
25
25
1500
20
20
15
vRMS
30
qCMB
qtop
10
15
1200
900
10
10
5
5
600
0.00
0.01
0.02
0.03
0.00
0.01
t
0.02
0.03
300
0.00
0.01
t
25 %
30 %
30 %
20 %
25 %
25 %
20 %
20 %
15 %
15 %
10 %
10 %
5%
5%
0.03
20 %
15 %
f
15 %
0.02
t
10 %
10 %
5%
0%
0%
10
12
14
5%
0%
0%
12
Nu
14
16
Nu
14
16
Nu
18
12
14
16
18
20
Nu
Fig. 4. Time-averaged vertical profiles of temperature and viscosity, Nusselt number (top panels), time series of the surface heat flux, CMB heat flux and root mean square velocity
(central panels), histograms and normalized Gaussian distributions of the frequency f of the occurrence of the Nusselt number evaluated for t ≥ 0.015 (bottom panels). The models
shown are the same of Fig. 2, i.e. a in black, b in red, c in blue and d in green. The dashed red line on the top left panel denotes the Clapeyron curve.
equilibrium, as also the distribution of the Nusselt number at the
bottom of Fig. 4 indicates.
3.2. Effect of post-perovskite phase parameters
For the calculations shown in the previous section, in order to
facilitate the formation of ppv and highlight its effects on mantle
dynamics, the Clapeyron slope of the pv–ppv transition was held
constant at 13 MPa K−1, in agreement with the X-ray diffraction
measurements performed by Tateno et al. (2009). Furthermore, only
the case in which the temperature intercept of the pv–ppv phase
boundary at the CMB is lower than the CMB temperature was
considered. In Fig. 5, we compare the average properties and time
series resulting from calculations in which we also employ a smaller
Clapeyron slope of 9 MPa K−1 (Tsuchiya et al., 2004) and a temperature intercept of 4000 K. This temperature, being larger than the
CMB temperature, causes the geotherm to intersect the pv–ppv
boundary only one time. As a result, ppv forms a layer of variable
236
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
α = const, k=
α = const, k=
α = const, k=
α = const, k=
α = const, k=
const , no ppv
const , T int =3600 K ,
const , T int =3600 K ,
const , T int =4000 K ,
const , T int =4000 K ,
γ ppv =9 MPa/K, η ppv = η pv
γ ppv =13 MPa/K, η ppv = η pv
γ ppv =9 MPa/K, η ppv = η pv
γ ppv =13 MPa/K, η ppv = η pv
0.0
0.0
0.2
0.2
0.4
0.4
16
14
12
Nu
10
0.6
0.6
0.8
0.8
8
6
4
2
1.0
0
1.0
0.0
0.2
0.4
0.6
0.8
0.1
1.0
1
T
log10(η)
20
1200
15
15
900
vRMS
qtop
qCMB
20
10
10
5
0.00
600
300
5
0.01
0.02
0.03
0.00
0.01
t
f
10
30 %
30 %
20 %
25 %
20 %
15 %
10 %
25 %
20 %
15 %
10 %
5%
0%
5%
0%
10 %
5%
0%
10
12
Nu
14
0.03
0.00
0.01
t
25 %
15 %
0.02
12
14
16
35 %
30 %
25 %
20 %
15 %
10 %
5%
0%
12
Nu
14
Nu
0.03
0.02
t
16
12
14
35 %
30 %
25 %
20 %
15 %
10 %
5%
0%
16
Nu
12
14
16
Nu
Fig. 5. As in Fig. 4 for models characterized by different parameters of the pv to ppv transition (see text for details).
height that covers the whole CMB. For simplicity, we show only the
results of calculations in which no viscosity changes are associated
with ppv, i.e. ηppv = ηpv, and both the thermal expansivity and
conductivity are constant. As the figure clearly shows, from a
quantitative point of view, the use of different Clapeyron slopes and
temperature intercepts affects the results very little. All the
temperature profiles are approximately adiabatic because of the use
of the EBA and of constant thermal expansivity, and are characterized
by no prominent difference in the size of the thermal boundary layers.
The presence of ppv tends to increase the average mantle temperature
and, as a consequence, to reduce the viscosity, no matter whether ppv
occurs in isolated lenses (Tint = 3600 K) or forms a ubiquitous layer
(Tint = 4000 K) and almost independently of the Clapeyron slope. As
shown in the panels illustrating the Nusselt number and the heat
fluxes, a slightly higher convective vigor is observed in models with
Tint = 3600 K with respect to models with Tint = 4000 K. With
Tint = 3600 K, in fact, we have Nu = 13.9 for γppv = 9 MPa K−1 (red
bar) and 14.6 for γppv = 9 MPa K−1 (blue bar), while, with Tint = 4000,
we have Nu = 13.6 for γppv = 9 MPa K−1 (magenta bar) and 13.4 for
γppv = 13 MPa K−1 (green bar), although the time series for the latter
model still exhibit a slight increasing trend. This is in fact not
surprising since the large Clapeyron slope and the high temperature
intercept favor the formation of pvv and thus of a higher degree of
instability which may prevent quasi steady-state to be reached even
on a long time scale. It must be also noted that Nakagawa and Tackley
(2004) reported larger differences in the average temperature and
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
α=α( ), k= const, no ppv
α=α( ), k=k 1( ), no ppv
a
c
α=α( ), k=k 1( ), T int =3600 K, γppv =13 MPa/K, ηppv =0.1η pv
T
-1
0
g
h
d
0
e
f
b
α=α( ), k= const, Tint =3600 K, γppv =13 MPa/K, ηppv =0.1η pv
237
1
1
log10(η)
2
3
Fig. 6. Long-term snapshots of the distribution of temperature (a, c, e, g) and logarithm of viscosity (b, d, f, h) for four models featuring depth-dependent thermal expansivity,
constant thermal conductivity (a–d) or depth-dependent thermal conductivity according to the profile k1 (e–h) and no ppv (a, b, e, f) or ppv with Tint = 3600 K, γppv = 13 MPa K−1
and ηppv = 0.1ηpv (c, d, g, h). Beneath panels d and h, the occurrence of the ppv phase, which reflects the viscosity distribution, is also shown.
heat transport than those observed here between models featuring
a standard Clapeyron slope of 8 MPa K−1 and an exaggerated one
of 16 MPa K−1.
3.3. Effect of depth-dependent thermal expansivity
A decrease with depth of the thermal expansivity (see Fig. 1 and
Section 2.2.2) lengthens the convective planform (Hansen et al.,
1993). In Fig. 6, on panels a–d, we show snapshots of the temperature
and viscosity distributions for models with depth-dependent thermal
expansivity and constant conductivity, either without considering the
pv–ppv transition (Fig. 6a, b) or including the pv–ppv transition
(Fig. 6c, d) with the same phase parameters used for the models
shown in Fig. 2 and with a viscosity reduction associated with ppv of
one order of magnitude. Without ppv, the strong reduction of the
expansivity causes a very sluggish regime dominated by large-scale
structures. Because of the decrease of α, cold slabs loose their
buoyancy while sinking deep into the mantle. As a consequence, they
tend to stagnate in the lower mantle causing an overall cooling of the
system as shown in Fig. 7 (black lines). The approximately adiabatic
temperature gradient obtained using constant thermal expansivity
(see Fig. 5, black lines) is now replaced by a nearly isothermal profile
in the bulk of the mantle caused by the diminished contribution of the
adiabatic heating. The widespread presence of cold slabs beneath the
transition zone combined with depth- and temperature-dependent
viscosity causes the lower mantle to have a generally high viscosity
(Fig. 7b) which only drops to smaller values in regions where
upwellings are present. In contrast to the downwellings, hot plumes
pick up buoyancy during their ascent and can maintain their
distinctive thermal signature throughout the whole mantle. While
the size of the upper TBL is only moderately influenced by the new
thermal expansivity, the bottom TBL thickens significantly because of
the presence of strong and large-sized upwellings. The average
viscosity profiles obtained without and with depth-dependent
α (black lines in Figs. 5 and 7, respectively) nearly overlap in the
top part of the upper mantle, exhibiting the same low-value in the
asthenosphere. Starting from a depth of about z = 0.7 (~340 km), they
progressively diverge from each other in the lower mantle. They reach
the largest difference at a depth of about z = 0.7 (~ 2000 km) while
they nearly overlap again over the bottom TBL.
Very differently from the models studied in Section 3.1, the
introduction of ppv in models with depth-dependent α and constant k
modifies dramatically the style of convection (compare Fig. 6a, b with
c, d). The convective regime is no longer sluggish but characterized by
sheet-like down- and upwellings of comparable importance that form
well-defined convection cells. This translates into a hotter mantle
with an approximately isothermal core and a much thinner bottom
TBL (see Fig. 7, red lines). As in the previous models, the presence of
ppv acts to make the mantle more unstable, favoring the formation of
plumes (see time series in Fig. 7). However, because of the depth
dependence of the thermal expansivity, plumes are intrinsically
hotter and stronger than those obtained with constant expansivity,
with the result that the increase of the average mantle temperature
becomes more pronounced. The most important lateral variations of
viscosity are associated with cold slabs. These tend to exhibit a large
viscosity contrast with respect to the surroundings, especially in the
lower mantle where the vertical viscosity profile has its maximum.
Broad ppv regions are associated with the spreading of subducted
material along the CMB and are separated from each other only at
locations where plumes rise from the bottom boundary layer.
3.4. Combined effects of depth-dependent thermal expansivity and
conductivity k1
In panels e to h of Fig. 6, we show snapshots of the temperature
and viscosity fields for models without (Fig. 6e, f) and with ppv
(Fig. 6g, h), in which, besides the thermal expansivity, we also assume
the thermal conductivity depending with depth according to the
profile k1 (see Fig. 1 and Section 2.2.3) and the thermal conductivity of
ppv twice as large as that of the surrounding pv. As long as ppv is not
taken into account, the presence of depth-dependent k does not
change significantly the convection planform with respect to the case
in which only α is a function of the depth (compare Fig. 6e, f and a, b).
Downwellings exhibit a sluggish behavior in the lower mantle. They
tend to thicken significantly because of viscous compression and
folding and to stagnate beneath the transition zone, causing the
mantle to be cool and highly viscous (Fig. 6f). The average temperature and viscosity profiles are shown in Fig. 7 and represented by
blue lines. The increase of k with depth facilitates heat conduction
from the core with respect to conduction to the surface. Compared to
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N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
α = α ( ), k= const ,no ppv
α = α ( ), k=const , T int =3600 K, γ ppv =13 MPa/K, η ppv =0.1η pv
α = α ( ), k=k1 ( ), no ppv
α = α ( ), k=k1 ( ), T int =3600 K, γ ppv =13 MPa/K, η ppv =0.1 η pv
0.0
0.0
30
0.2
0.2
25
0.4
0.4
0.6
0.6
Nu
20
15
10
0.8
0.8
1.0
0.0
0.2
0.4
0.6
0.8
5
1.0
0.01
1.00
0
0.1
1
log10(η)
T
40
40
30
30
10
100
1500
20
vRMS
qtop
qCMB
1200
20
900
600
10
300
10
0.00
0.01
0.02
0.00
0.03
0.01
t
0.02
0.03
0.00
0.01
0.02
t
0.03
t
15 %
40 %
20 %
30 %
15 %
20 %
10 %
10 %
10 %
5%
5%
20 %
10 %
f
15 %
5%
0%
8
10
12
Nu
0%
14
16
18
20
22
Nu
0%
24
0%
12
14
16
18
22
24
Nu
26
28
30
Nu
Fig. 7. As in Fig. 4 for models characterized also by depth-dependent thermal expansivity and conductivity (see text for details).
the results obtained with constant k (Fig. 7, black line), we observe
thus a slightly hotter geotherm, causing a lower viscosity profile, and a
thicker bottom TBL (compare black and blue lines in Fig. 7).
Again, introducing ppv with a viscosity by one order of magnitude
lower than that of the surroundings, induces first order changes in the
convection pattern (Fig. 6g, h) and in the temperature and viscosity
profiles and heat transport properties of the system (Fig. 7, green
lines). The increase of the mantle temperature due to ppv is now more
pronounced although the thickness of the bottom TBL is not reduced as
dramatically as in the case in which only α varies with depth.
Upwellings have no longer the form of relatively thin and isolated
conduits that connect the bottom of the mantle with the surface as in
Fig. 6c. Instead, they are now prominent features, being very broad and
hot and exhibiting a clear tendency to gather towards two regions,
located approximately at one- and three-fourths of the horizontal
length of the domain, where they form a morphology resembling two
superplumes. The dominance of hot lower mantle plumes also implies
a secondary role for downgoing slabs which appear now as isolated
cold features with still a relatively high viscosity. As a consequence,
there is a strong reduction in the amount of ppv which can only form in
isolated patches underlying the few downgoing plumes that are able to
penetrate into the deep mantle.
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
239
become essentially steady state with plumes (thin red trajectory in
Fig. 9a) and slabs (thick blue trajectory in Fig. 9a) keeping their
position throughout the rest of the computational time with
horizontal flow essentially confined in the two TBLs and with the
exception of the model shown in Fig. 9b for which the formation of a
new plume at t ~ 0.01 is observed. As in the previous models, the
mantle becomes hotter with the introduction of ppv. Downgoing slabs
conserve in this case a prominent role both if no viscosity contrast for
ppv is used (Fig. 8b) and if ppv viscosity is reduced by one order of
magnitude (Fig. 8c). The importance of cold slabs is also reflected by
the size of ppv lenses which appear now very broad and present the
greatest thickness observed among our models. Even with a strongly
depth-dependent conductivity, the presence of ppv still destabilizes
the dynamics. Yet it is remarkable that convection still remains in this
quasi steady state for several billion years with plumes anchored
at nearly fixed positions that exhibit very little lateral movements
(Fig. 9b,c).
In terms of complexity, the model shown in Fig. 6g and h is likely
the richest, among those considered in this study, as it features
multiple phase transitions and depth-dependent thermodynamic and
transport properties that give rise to a large time variability of the
flow. It is thus interesting to test for this special case the accuracy of
YACC in solving the equation of energy conservation (Eq. (3)). For a
given time step selected near the end of the calculation time, we have
integrated Eq. (3) over the domain and compared the volumetric heat
flow due to temperature advection (〈qDtT〉, l.h.s. of Eq. (3)) with the
sum of the following terms forming the r.h.s. of Eq. (3): surface heat
flow across the top and CMB (〈qtop〉 and 〈qCMB〉, respectively),
volumetric heat flow due to adiabatic compression (〈qadi〉), viscous
dissipation (〈qdiss〉) and latent heat due to phase transitions (〈qlat〉).
The values obtained are as follows: 〈qDtT〉 = 43.26, 〈qtop〉 = 239.14,
〈q CMB 〉 = −193.52, 〈q adi 〉 = −31.56, 〈q diss 〉 = 44.76 and 〈q lat 〉 =
−14.19. The sum of the last five terms yields 44.63 which implies a
difference of about 3% with respect to the first term. Imbalances of
comparable magnitude in the solution of the energy equation are
reported from much simpler convection calculations at lower
Rayleigh numbers and can also be related to the grid resolution
(King et al., 2010). Hence we can consider the accuracy of our solution
satisfactory.
4. Discussion and conclusions
In the past few years it has been generally recognized that mineral
physics along with geodynamical modeling and seismology forms the
third pillar for studying the geodynamics of the deep Earth. We have
therefore embarked on a systematic two-dimensional study of the
implications from the latest developments in mineral physics on
the dynamics of the lower mantle. This work follows the spirit of the
earlier work by Matyska and Yuen (2007). However, we have now
elected to employ a newly written numerical code because of its
rapidity in the 2-D computations and relative simplicity in analyzing
the voluminous amount of data in order to extract out the main
physical effects of each additional ingredient. The large impact on
mantle dynamics exerted by the combination of previously unappreciated components calls into question the standard use of models
featuring simple rheological, thermodynamical and transport properties. The main findings of this study can be summarized as follows:
3.5. Combined effects of depth-dependent thermal expansivity and
conductivity k2
In Fig. 8, we show snapshots of the temperature field for three
models in which the thermal conductivity increases strongly with
depth according to the profile k2 and ppv is either not included
(Fig. 8a) or it is included by considering either no viscosity contrast
with respect to the surroundings or a contrast involving one order of
magnitude (Fig. 10b and c, respectively). With k increasing by a factor
of 6 throughout the mantle, the effective Rayleigh number of the
system clearly drops in a dramatic way and convection becomes
much less time dependent. With no ppv (Fig. 8a), the temperature
distribution is now characterized by four major up- and downwellings. The downwellings are very broad cold structures that tend
to preserve a nearly constant temperature anomaly throughout most
of the mantle, while the upwellings are more localized and rise from
the bottom TBL in an essentially vertical fashion. As the plots of the
temporal evolution of the thermal field clearly show (Fig. 9a), as soon
as all these structure are formed at about t = 0.004, convection cells
1. Influence of the exothermic nature of the pv–ppv transition.
Nakagawa and Tackley (2004) and Matyska and Yuen (2005)
found that the exothermic character of the bottom phase transition
acts to destabilize the flow in the lower mantle but did not
elaborate this phenomenon in great detail nor dwelt on the
geophysical implications.
α=α( ), k=k2 ( ), no ppv
a
α=α( ), k=k 2 ( ), T int=3600 K, γ ppv=13 MPa/K, η ppv=η pv
b
α=α( ), k=k 2( ), T
c
int=3600
0
K , γ ppv=13 MPa/K, η ppv=0.1 η pv
T
1
Fig. 8. Long-term snapshots of the temperature distribution for three models featuring depth-dependent thermal expansivity and conductivity according to the profile k2 and no ppv
(a) or ppv with Tint = 3600 K, γppv = 13 MPa K−1 and ηppv = ηpv (b), ηppv = 0.1ηpv (c). In panels b and c the occurrence of the ppv phase is also shown.
240
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
a
c
b
0.030
0.025
t
0.020
z=0.1
0.015
0.010
0.005
0.000
0.030
0.025
t
0.020
z=0.5
0.015
0.010
0.005
0.000
0.030
0.025
t
0.020
z=0.9
0.015
0.010
0.005
0.000
0
2
4
x
6
8
10
0
2
4
0
x
T
6
8
10
0
2
4
x
6
8
10
1
Fig. 9. Time evolution of the temperature field for the same models shown in Fig. 8 as a function of the horizontal coordinate at three different depths: z = 0.1 (289 km, 1st line),
z = 0.5 (1445 km, 2nd line) and z = 0.9 (2601 km, 3rd line).
Differently from ours, the numerical models of Nakagawa and
Tackley (2004, 2005) and Matyska and Yuen (2005) did account for
internal heat sources. Nevertheless, even if it is well known that
internal heating acts to enhance the role of downwellings and to
produce more diffused and less vigorous upwellings (e.g. Hansen
et al., 1993; van den Berg et al., 2002), all the above mentioned
studies still reported a prominent role of ppv in the production
of small-scale plumes arising from a more unstable and chaotic
bottom TBL. Therefore, even though the bottom TBL in purely
bottom heated systems like those considered here is more easily
destabilized than in internally heated or mixed-heated systems,
the peculiar behavior of the pv–ppv transition in promoting the
formation of bottom TBL instabilities does not seem to be a direct
consequence of the heating mode. Furthermore, recent work
conducted in 2-D axisymmetric geometry by Shahnas and Peltier
(in press) demonstrates more quantitatively the destabilizing
character of the pv–ppv transition by looking at the influence on
layered convection between the upper and lower mantle circulation. These results concur well with our present findings.
2. Influence of a low viscosity ppv at the base of the mantle.
Our viscosity parameterization with depth (Hanyk et al., 1995)
features a maximum in the mid-lower mantle and is based on an
earlier result by Ricard and Wuming (1991), later echoed by Forte
and Mitrovica (2001) and Mitrovica and Forte (2004). Despite a
relatively high viscosity in the lower mantle that tends to stabilize
the flow, secondary instabilities induced by a reduction of ppv
viscosity by a factor of ten develop from the bottom TBL. Such a
viscosity reduction represents probably an upper limit. Otherwise,
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
taking into account the additional thermal contribution due to the
bottom boundary layer we would encounter an unreasonably large
viscosity hill between the D″ and the mid-lower mantle.
3. Influence of depth-dependent thermal expansivity in the presence
of ppv.
Earlier results preceding the discovery of ppv showed that the mantle
circulation was stabilized by depth-dependent thermal expansion
coefficient (e.g. Hansen et al., 1993). Our results show that even with a
strongly decreasing thermal expansivity the destabilizing character of
the pv–ppv transition is strong enough to override the earlier findings
of Hansen et al. (1993). This result thus has strong implications on
the stability of lower mantle plumes in the presence of ppv.
4. Impact of depth-dependent thermal conductivity in the lower
mantle.
This factor may be the most important new finding coming out of
this study, since we show that a strongly depth-dependent thermal
conductivity, in concert with a depth-dependent thermal expansivity, stabilizes lower-mantle plumes for a geologically long
time span in excess of billions of years, even in the presence of
the destabilizing influence of the pv–ppv transition. This reconciles
well with recent work by Dziewonski et al. (2010) who have
argued for the persistent existence of a longstanding longwavelength structure in the lower mantle that can exert a strong
influence on polar wander and by Torsvik et al. (2008) who have
proposed that the location of the large low shear velocity province
beneath Africa which is likely associated with a superplume has
not changed significantly over at least the past 300 million years.
Although our models are characterized by several simplifications,
such as the use of a 2-D Cartesian geometry, the lack of compressibility, internal heating, chemical heterogeneities (Nakagawa and
Tackley, 2005) and of the temperature dependence of thermal
expansivity (Ghias and Jarvis, 2008) and conductivity (Matyska and
Yuen, 2006), they clearly show the importance to consider all
components of a highly non-linear system such as the Earth's mantle.
We need to fill in all of the missing pieces, such as the interaction of
the pv–ppv transition with depth-dependent thermodynamic and
transport properties, as they can yield important information and
constraints. Our future task would be to include chemical heterogeneities in all of these interactions verifying the enduring influence
of chemical piles on lower mantle circulation.
Acknowledgments
We thank editor Yanick Ricard and two anonymous reviewers
whose comments helped to improve an earlier version of this work.
We also thank Arie van den Berg, Renata Wentzcovitch, Radek
Matyska, Hosein Shahnas, Julian Lowman, Anne Hofmeister and Dick
Peltier for stimulating discussions. Part of this work was conducted
while NT was hosted at the Department of Geophysics of the Freie
Universität Berlin which is gratefully acknowledged. This work has
been supported by the European Commission through the Marie Curie
Research Training Network C2C (contract MRTN-CT-2006-035957),
by the National Science Foundation through the CMG grant and by the
Czech Ministry of Education through the research project
MSM0021620860.
References
Ammann, M.W., Brodholt, J.P., Dobson, D.P., 2009. DFT study of migration enthalpies
in MgSiO3 perovskite. Phys. Chem. Miner. 36, 151–158. doi:10.1007/s00269-0080265-z.
Ammann, M.W., Brodholt, J.P., Wookey, J., Dobson, D.P., 2010. First-principles
constraints on diffusion in lower-mantle minerals and a weak D″ layer. Nature
465, 462–465. doi:10.1038/nature09052.
Badro, J., Rueff, J.P., Vanko, G., Monaco, G., Fiquet, G., Guyot, F., 2004. Electronic
transitions in perovskite: possible non-convecting layers in the lower mantle.
Science 305, 383–386. doi:10.1126/science.1098840.
241
Bates, J.R., McDonald, A., 1982. Multiply-upstream, semi-Lagrangian advective
schemes: analysis and application to a multi-level primitive equation model.
Mon. Weather Rev. 110, 1831–1842.
Beck, P., Goncharov, A.F., Struzhkin, V.V., Militzer, B., Mao, H., Hemley, R.J., 2007.
Measurement of thermal diffusivity at high pressure using a transient heating
technique. App. Phys. Lett. 91. doi:10.1063/1.2799243.
Bina, C., Helffrich, G., 1994. Phase transitions Clapeyron slopes and transition zone
seismic discontinuity topography. J. Geophys. Res. 99, 15853–15860.
Bower, D.J., Gurnis, M., Jackson, J.M., Sturhahn, W., 2009. Enhanced convection and fast
plumes in the lower mantle induced by the spin transition in ferropericlase.
Geophys. Res. Lett. 36. doi:10.1029/2009GL037706.
Čadek, O., Fleitout, L., 2006. Effect of lateral viscosity variations in the core-mantle
boundary region on predictions of the long-wavelength geoid. Stud. Geophys.
Geod. 50, 217–232.
Čadek, O., van den Berg, A.P., 1998. Radial profiles of temperature and viscosity in the
Earth's mantle inferred from the geoid and lateral seismic structure. Earth. Planet.
Sci. Lett. 164, 607–615. doi:10.1016/S0012-821X(98)00244-1.
Catalli, K., Shim, S.H., Prakapenka, V., 2009. Thickness and Clapeyron slope of the postperovskite boundary. Nature 462, 782–785. doi:10.1038/nature08598.
Chopelas, A., Boehler, R., 1992. Thermal expansivity in the lower mantle. Geophys. Res.
Lett. 19, 1983–1986.
Christensen, U.R., Yuen, D.A., 1985. Layered convection induced by phase transitions.
J. Geophys. Res. 90, 10291–10300.
Čížková, H., Čadek, O., Matyska, C., Yuen, D.A., 2009. Implications of post-perovskite
transport properties for core-mantle dynamics. Phys. Earth Planet. Inter. 180,
235–243. doi:10.1016/j.pepi.2009.08.008.
de Koker, N., 2009. Thermal conductivity of MgO periclase from equilibrium first principles
molecular dynamics. Phys. Rev. Lett. 103. doi:10.1103/PhysRevLett.103.125902.
de Koker, N., 2010. Thermal conductivity of MgO periclase at high pressure:
implications for the D″ region. Earth Planet. Sci. Lett. 292, 392–398. doi:10.1016/
j.epsl.2010.02.011.
Dziewonski, A.M., Lekic, V., Romanowicz, B.A., 2010. Mantle anchor structure: an
argument for bottom up tectonics. Eart Planet. Sci. Lett., submitted for publication.
Forte, A.M., Mitrovica, J.X., 2001. Deep-mantle high-viscosity flow and thermochemical
structure inferred from seismic and geodynamic data. Nature 410, 1049–1056.
doi:10.1038/35074000.
Garnero, E.J., McNamara, A.K., 2008. Structure and dynamics of Earth's lower mantle.
Science 320, 626–628. doi:10.1126/science.1148028.
Garnero, E.J., Lay, T., McNamara, A.K., 2007. Implications of lower mantle structural
heterogeneity for existence and nature of whole mantle plumes. In: Foulger, G.T.,
Jurdy, D.M., Foulger, G.T., Jurdy, D.M. (Eds.), Plates, plumes and planetary processes.
The Geological Society of America, pp. 79–101.
Gerya, T., Yuen, D.A., 2003. Characteristic-based marker-in-cell method with conservative
finite-differences schemes for modelling geological flows with strongly variable
transport properties. Phys. Earth Planet. Inter. 140, 293–318.
Ghias, S.R., Jarvis, G.T., 2008. Mantle convection models with temperature- and depthdependent thermal expansivity. J. Geophys. Res. 113. doi:10.1029/2007JB005355.
Goncharov, A.F., Struzhkin, V.V., Jacobsen, S.D., 2006. Reduced radiative conductivity of
low-spin (Mg, Fe)O in the lower mantle. Science 312, 1205–1208. doi:10.1126/
science.1125622.
Goncharov, A.F., Beck, P., Struzhkin, V.V., Haugen, B.D., Jacobsen, S.D., 2009. Thermal
conductivity of lower-mantle minerals. Phys. Earth Planet. Inter. 174, 24–32.
doi:10.1016/j.pepi.2008.07.033.
Goncharov, A.F., Struzhkina, V.V., Montoya, J.A., Kharlamova, S., Kundargi, R., Siebert, J.,
Badro, J., Antonangeli, D., Ryerson, F.J., Mao, W., 2010. Effect of composition,
structure, and spin state on the thermal conductivity of the Earth's lower mantle.
Phys. Earth Planet. Inter. 180, 148–153. doi:10.1016/j.pepi.2010.02.002.
Hansen, U., Yuen, D.A., Malevsky, A.V., 1992. Comparison of steady-state and strongly
chaotic thermal convection at high Rayleigh number. Phys. Rev. A 46, 4742–4754.
doi:10.1103/PhysRevA.46.4742.
Hansen, U., Yuen, D.A., Kroening, S.E., Larsen, T.B., 1993. Dynamical consequences of
depth-dependent thermal expansivity and viscosity on mantle circulations and
thermal structure. Phys. Earth Planet. Inter. 77, 205–223.
Hanyk, L., Moser, J., Yuen, D.A., Matyska, C., 1995. Time-domain approach for the transient responses in stratified viscoelastic Earth. Geophys. Res. Lett. 22, 1285–1288.
doi:10.1029/95GL01087.
Hernlund, J.W., Labrosse, S., 2007. Geophysically consistent values of the perovskite
to post-perovskite transition clapeyron slope. Geophys. Res. Lett. 34. doi:10.1029/
2006GL028961.
Hernlund, J.W., Thomas, C., Tackley, P.J., 2005. A doubling of the post-perovskite phase
boundary and structure of the Earth's lowermost mantle. Nature 434, 882–886.
Hirose, K., 2006. Postperovskite phase transition and its geophysical implications. Rev.
Geophys. 44. doi:10.1029/2005RG000186.
Hofmeister, A.M., 2007. Pressure dependence of thermal transport properties. Proc.
Natl. Acad. Sci. 104. doi:10.1073/pnas.0610734104.
Hofmeister, A.M., 2008. Inference of high thermal transport in the lower mantle from
laser-flash experiments and the damped harmonic oscillator model. Phys. Earth
Planet. Inter. 170, 201–206. doi:10.1016/j.pepi.2008.06.034.
Hunt, S.A., Weidner, D.J., Li, L., Wang, L., Walte, N.P., Brodholt, J.P., Dobson, D.P., 2009.
Weakening of calcium iridate during its transformation from perovskite to postperovskite. Nat. Geosci. 2, 794–797. doi:10.1038/ngeo663.
Hutko, A.R., Lay, T., Garnero, E.J., Revenaugh, J., 2006. Seismic detection of folded, subducted
lithosphere at the core-mantle boundary. Nature 441. doi:10.1038/nature04757.
Hutko, A.R., Lay, T., Revenaugh, J., Garnero, E.J., 2008. Anticorrelated seismic velocity
anomalies from post-perovskite in the lowermost mantle. Science 320, 1070–1074.
doi:10.1126/science.1155822.
242
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
Ita, J., King, S.D., 1994. Sensitivity of convection with an endothermic phase change to
the form of governing equations, initial conditions, boundary conditions and
equation of state. J. Geophys. Res. 99, 15919–15938.
Jarvis, G.T., McKenzie, D.P., 1980. Convection in a compressible fluid with infinite
Prandtl number. J. Fluid Mech. 96, 515–583.
Katsura, T., Yokoshi, S., Kawabe, K., Shatskiy, A., Manthilake, M.A.G.M., Zhai, S., Fukui,
H., Hegoda, H.A.C.I., Yoshino, T., Yamazaki, D., Matsuzaki, T., Yoneda, A., Ito, E.,
Sugita, M., Tomioka, N., Hagiya, K., Nozawa, A., Funakoshi, K., 2009. P-V-T relations
of MgSiO3 perovskite determined by in situ X-ray diffraction using a large-volume
high-pressure apparatus. Geophys. Res. Lett. 36. doi:10.1029/2008GL035658.
Kawai, K., Tsuchiya, T., 2009. Temperature profile in the lowermost mantle from
seismological and mineral physics joint modeling. Proc. Natl. Acad. Sci. 106,
22119–22123. doi:10.1073/pnas.0905920106.
King, S.D., Lee, C., van Keken, P., Leng, W., Zhong, S., Tan, E., Tosi, N., Kameyama, M.,
2010. A community benchmark for 2D Cartesian compressible convection in the
Earth's mantle. Geophys. J. Int. 180, 73–87. doi:10.1111/j.1365-246X.2009.04413.x.
Komabayashi, T., Hirose, K., Sugimura, E., Sata, N., Ohishi, Y., Dubrovinsky, L.S., 2008.
Simultaneous volume measurements of post-perovskite and perovskite in MgSiO3
and their thermal equations of state. Earth Planet. Sci. Lett. 265, 515–524.
doi:10.1016/j.epsl.2007.10.036.
Lay, T., Hernlund, J., Garnero, E., Thorne, M.S., 2006. A post-perovskite lens and D″ heat
flux beneath the central pacific. Science 314, 1272–1276. doi:10.1126/
science.1133280.
Lee, C., King, S.D., 2009. Effect of mantle compressibility on the thermal and flow
structures of the subduction zones. Geochem. Geophys. Geosys. 10, Q01006.
doi:10.1029/2008GC002151.
Leitch, A.M., Yuen, D.A., Sewell, G., 1991. Mantle convection with internal heating and
pressure-dependent thermal expansivity. Earth Planet. Sci. Lett. 102, 213–232.
Leng, W., Zhong, S., 2008. Viscous heating, adiabatic heating and energetic consistency in
compressible mantle convection. Geophys. J. Int. 173, 693–702. doi:10.1111/j.1365246X.2008.03745.x.
Matyska, C., Yuen, D.A., 2005. The importance of radiative heat transfer on superplumes
in the lower mantle with the new post-perovskite phase change. Earth Planet. Sci.
Lett. 234, 71–81.
Matyska, C., Yuen, D.A., 2006. Lower mantle dynamics with the post-perovskite phase
change, radiative thermal conductivity, temperature- and depth-dependent
viscosity. Phys. Earth Planet. Inter. 154, 196–207. doi:10.1016/j.pepi.2005.10.001.
Matyska, C., Yuen, D.A., 2007. Lower mantle material properties and convection models
of multiscale plumes. In: Foulger, G.T., Jurdy, D.M. (Eds.), Plates, plumes and
planetary processes. The Geological Society of America, pp. 137–163. doi:10.1130/
2007.2430(08).
Matyska, C., Yuen, D.A., Čížková, H., 2010. Thermomechanical influences from the nonmonotonicity of the rheological activation parameters in the lower mantle. Phys.
Earth Planet. Inter., submitted for publication.
McNamara, A.K., Zhong, S., 2005. Thermochemical structures beneath Africa and the
Pacific Ocean. Nature 437, 1136–1139. doi:10.1038/nature04066.
Mitrovica, J.X., Forte, A., 2004. A new inference of mantle viscosity based upon joint
inversion of convection and glacial isostatic adjustment data. Earth Planet. Sci. Lett.
225, 177–189.
Monnereau, M., Yuen, D.A., 2007. Topology of the post-perovskite phase transition and
mantle dynamics. Proc. Natl. Acad. Sci. 104, 9156–9161.
Monnereau, M., Yuen, D.A., 2010. Seismic imaging of the D″ and constraints on
the core heat flux. Phys. Earth Planet. Inter. 180 (3–4), 258–270. doi:10.1016/j.
pepi.2009.12.005.
Moresi, L., Solomatov, V.S., 1995. Numerical investigations of 2D convection with
extremely large viscosity contrasts. Phys. Fluids 7, 2154–2162.
Morra, G., Yuen, D.A., Boschi, L., Chatelain, P., Koumoutsakos, P., Tackley, P.J., 2010. The
fate of the slabs interacting with a viscosity hill in the mid-mantle. Phys. Eart
Planet. Int. 180, 271–282. doi:10.1016/j.pepi.2010.04.001.
Murakami, M., Hirose, K., Kawamura, K., Sata, N., Ohishi, Y., 2004. Post-perovskite phase
transition in MgSiO3. Science 304, 855–858.
Nakagawa, T., Tackley, P.J., 2004. Effects of a perovskite-post perovskite phase change
near core-mantle boundary in compressible mantle convection. Geophys. Res. Lett.
31. doi:10.1029/2004GL020648.
Nakagawa, T., Tackley, P.J., 2005. The interaction between the post-perovskite phase
change and a thermo-chemical boundary layer near the core-mantle boundary.
Earth Planet. Sci. Lett. 238, 204–216.
Nakagawa, T., Tackley, P.J., 2006. Three-dimensional structures and dynamics in the
deep mantle: Effects of post-perovskite phase change and deep mantle layering.
Geophys. Res. Lett. 33. doi:10.1029/2006GL025719.
Naliboff, J.B., Kellogg, L.H., 2006. Dynamic effects of a step-wise increase in thermal
conductivity and viscosity in the lowermost mantle. Geophys. Res. Lett. 33.
doi:10.1029/2006GL025717.
O'Farrell, K.A., Lowman, J.P., in press. Emulating the thermal structure of spherical shell
convection in plane-layer geometry mantle convection models. Phys. Earth Planet.
Inter. doi:10.1016/j.pepi.2010.06.010.
Oganov, A.R., Ono, S., 2004. Theoretical and experimental evidence for a postperovskite phase of MgSiO3 in Earth's D″. Nature 430, 445–448.
Oganov, A.R., Ono, S., 2005. The high pressure phase of alumina and implications for
Earth's D″ layer. Proc. Natl. Acad. Sci. 102, 10828–10831.
Ohta, K., 2010. Electrical and thermal conductivity of the Earth’s lower mantle. Tokyo
Institute of Technology.
Ohta, K., Onoda, S., Hirose, K., Sinmyo, R., Shimizu, K., Sata, N., Ohishi, Y., Yasuhara, A.,
2008. The electrical conductivity of post-perovskite in Earth's D″ layer. Science 320,
89–91. doi:10.1126/science.1155148.
Patankar, S.V., 1980. Numerical heat transfer and fluid flow. Hemisphere series on
computational methods in mechanics and thermal science. Taylor & Francis.
Phipps Morgan, J., Rüpke, L., 2010. Current mantle energetics. Geophys. J. Int., submitted
for publication.
Ren, Y., Stutzmann, E., van der Hilst, R.D., Besse, J., 2007. Understanding seismic
heterogeneities in the lower mantle beneath the Americas from seismic tomography
and plate tectonic history. J. Geophys. Res. 112. doi:10.1029/2005JB004154.
Ricard, Y., Wuming, B., 1991. Inferring viscosity and the 3-D density structure of the
mantle from geoid, topography and plate velocities. Geophys. J. Int. 105,
561–572.
Rosatti, G., Cesari, D., Bonaventura, L., 2005. Semi-implicit, semi-Lagrangian modelling
for environmental problems on staggered Cartesian grids with cut cells. J. Comput.
Phys. 204, 353–377. doi:10.1016/j.jcp. 2004.10.013.
Schenk, O., Gartner, K., Fichtner, W., 2000. Efficient sparse LU factorization with leftright looking strategy on shared memory multiprocessors. BIT 40, 158–176.
Schubert, G., Turcotte, D.L., Olson, P., 2001. Mantle convection in the Earth and planets.
Cambridge University Press, Cambridge.
Shahnas, H., Peltier, W.R., in press. Layered convection and the impact of the perovskitepost perovskite phase transition on mantle dynamics under isochemical conditions. J.
Geophys. Res. doi:10.1029/2009JB007199.
Speziale, S., Milner, A., Lee, V.E., Clark, S.M., Pasternak, M.P., Jeanloz, R., 2005. Iron spin
transition in Earth's mantle. Proc. Natl. Acad. Sci. 102, 17918–17922. doi:10.1073/
pnas.0508919102.
Spiegelman, M., Katz, R., 2006. A semi-Lagrangian Crank–Nicolson algorithm for the
numerical solution of advection–diffusion problems. Geochem. Geophys. Geosyst.
7, Q04014. doi:10.1029/2005GC001073.
Staniforth, A., Côté, J., 1991. Semi-Lagrangian integration schemes for athmospheric
models—a review. Mon. Weather Rev. 119, 2206–2223.
Steinbach, V., Yuen, D.A., 1995. The effects of temperature-dependent viscosity on
mantle convection with the two major phase transitions. Phys. Earth Planet. Inter.
90, 13–36. doi:10.1016/0031-9201(95)03018-R.
Tackley, P.J., 1996. Effects of strongly variable viscosity on three-dimensional
compressible convection in planetary mantles. J. Geophys. Res. 101, 3311–3331.
Tan, E., Gurnis, M., 2005. Metastable superplumes and mantle compressibility. Geophys.
Res. Lett. 32. doi:10.1029/2005GL024190.
Tan, E., Gurnis, M., 2007. Compressible thermochemical convection and application to
lower mantle structures. J. Geophys. Res. 112. doi:10.1029/2006JB004505.
Tang, X., Dong, J., 2010. Lattice thermal conductivity of MgO at conditions of Earth's
interior. Proc. Natl. Acad. Sci. 107, 4539–4543. doi:10.1073/pnas.0907194107.
Tateno, S., Hirose, K., Sata, N., Ohishi, Y., 2009. Determination of post-perovskite phase
transition boundary up to 4400 K and implications for thermal structure in D″ layer.
Earth Planet. Sci. Lett. 277, 130–136. doi:10.1016/j.epsl.2008.10.004.
Torsvik, T.H., Smethurst, M.A., Burke, K., Steinberger, B., 2008. Long term stability in
deep mantle structure: evidence from the ~ 300 Ma Skagerrak-Centered Large
Igneous Province (the SCLIP). Eart Planet. Sci. Lett. 267, 444–452. doi:10.1016/j.
epsl.2007.12.004.
Tosi, N., Sabadini, R., Marotta, A.M., Veermersen, L.A., 2005. Simultaneous inversion for
the Earth's mantle viscosity and ice mass imbalance in Antarctica and Greenland.
J. Geophys. Res. 110. doi:10.1029/2004JB003236.
Tosi, N., Čadek, O., Martinec, Z., 2009a. Subducted slabs and lateral viscosity variations:
effects on the long-wavelength geoid. Geophys. J. Int. 179, 813–826. doi:10.1111/j.1365246X.2009.04335.x.
Tosi, N., Čadek, O., Martinec, Z., Yuen, D.A., Kaufmann, G., 2009b. Is the long-wavelength
geoid sensitive to the presence of postperovskite above the core-mantle boundary?
Geophys. Res. Lett. 36. doi:10.1029/2008GL036902.
Trampert, J., Deschamps, F., Resovsky, J., Yuen, D.A., 2004. Probabilistic tomography
maps chemical heterogeneities throughout the lower mantle. Science 306,
853–856. doi:10.1126/science.1101996.
Tsuchiya, T., Tsuchiya, J., Umemoto, K., Wentzcovitch, R.M., 2004. Phase transition in
MgSiO3 perovskite in the Earth's lower mantle. Earth Planet. Sci. Lett. 224, 241–248.
van den Berg, A.P., Yuen, D.A., Allwardt, J.R., 2002. Non-linear effects from variable
thermal conductivity and mantle internal heating: implications for massive
melting and secular cooling of the mantle. Phys. Earth Planet. Inter. 129, 359–375.
van den Berg, A.P., De Hoop, M.V., Yuen, D.A., Duchkov, A., van der Hilst, R.D., Jacobs, M.,
2010a. Geodynamical modeling and multiscale seismic expression of thermochemical heterogeneity and phase transitions in the lowermost mantle. Phys. Earth
Planet. Inter. 244–257. doi:10.1016/j.pepi.2010.02.008.
van den Berg, A.P., Yuen, D.A., Beebe, G.L., Christiansen, M.D., 2010b. The dynamical
impact of electronic thermal conductivity on deep mantle convection of exosolar
planets. Phys. Earth Planet. Inter. 136–154. doi:10.1016/j.pepi.2009.11.001.
van der Hilst, R., De Hoop, M.V., Wang, S.H., Shim, L., Tenorio, P., 2007. Seismostratigraphy and thermal structure of Earth's core-mantle boundary region. Science
315, 1813–1817.
van der Meer, D.G., Spakman, W., van Hinsbergen, D.J.J., Amaru, M.L., Torsvik, T.H., 2010.
Towards absolute plate motions constrained by lower-mantle slab remnants. Nat.
Geosci. 3, 36–40. doi:10.1038/ngeo708.
Walte, N.P., Heidelbach, F., Miyajima, N., Frost, D.J., Rubie, D.C., Dobson, D.P., 2009.
Transformation textures in post-perovskite: understanding mantle flow in the D″
layer of the Earth. Geophys. Res. Lett. 36. doi:10.1029/2008GL036840.
Walzer, U., Hendel, R., Baumgardner, J., 2004. The effects of a variation of the radial
viscosity profile on mantle evolution. Tectonophysics 384, 55–90. doi:10.1016/j.
tecto.2004.02.012.
Wentzcovitch, R.M., Justo, J.F., Wu, Z., da Silva, C.R.S., Yuen, D.A., Kohlstedt, D., 2009.
Anomalous compressibility of ferropericlase throughout the iron spin cross-over.
Proc. Natl. Acad. Sci. 106, 8447–8452. doi:10.1073/pnas.0812150106.
N. Tosi et al. / Earth and Planetary Science Letters 298 (2010) 229–243
Xu, Y., Shankland, T.J., Linhardt, S., Rubie, D.C., Langenhorst, F., Klasinski, K., 2004. Thermal
diffusivity and conductivity of olivine, wadsleyite and ringwoodite to 20 GPa and
1373 K. Phys. Earth. Planet. Int. 143–144, 321–336. doi:10.1016/j.pepi.2004.03.005.
Yamazaki, D., Karato, S., 2007. Lattice preferred orientation of lower mantle materials
and seismic anisotropy in the D″ layer. In: Hirose, K., Brodholt, J., Lay, T., Yuen, D.A.
(Eds.), Post-perovskite: the last mantle phase transition: AGU. Geophysical
Monograph Series, 174, pp. 69–78.
243
Yoshino, T., Yamazaki, D., 2007. Grain growth kinetics of CaIrO3 perovskite and postperovskite, with implications for rheology of D″ layer. Earth Planet. Sci. Lett. 255,
458–493. doi:10.1016/j.epsl.2007.01.010.
Zhao, W., Yuen, D.A., 1987. The effects of adiabatic and viscous heatings on plumes.
Geophys. Res. Lett. 14, 1223–1226.