Earth and Planetary Science Letters 312 (2011) 348–359
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Earth and Planetary Science Letters
journal homepage: www.elsevier.com/locate/epsl
Bent-shaped plumes and horizontal channel flow beneath the 660 km discontinuity
Nicola Tosi a, b,⁎, David A. Yuen c, d
a
Department of Planetary Physics, German Aerospace Center (DLR), Berlin, Germany
Department of Planetary Geodesy, Technical University Berlin, Germany
c
Department of Earth Sciences, University of Minnesota, Minneapolis, USA
d
Minnesota Supercomputing Institute, University of Minnesota, Minneapolis, USA
b
a r t i c l e
i n f o
Article history:
Received 1 August 2011
Received in revised form 23 September 2011
Accepted 9 October 2011
Available online 20 November 2011
Editor: Y. Ricard
Keywords:
mantle plumes
phase transitions
temperature dependent viscosity
thermal expansivity
lattice thermal conductivity
a b s t r a c t
Recent high-resolution seismic imaging of the transition zone topography beneath the Hawaiian archipelago
shows strong evidence for a 1000 to 2000 km wide hot thermal anomaly ponding beneath the 660 km boundary
west of Hawaii islands [Q. Cao et al. Seismic imaging of transition zone discontinuities suggests hot mantle west
of Hawaii. Science (2011), 332, 1068–1071]. This scenario suggests that Hawaiian volcanism may not be caused
by a stationary narrow plume rising from the core–mantle boundary but by hot plume material first held back
beneath the 660 km discontinuity and then entrained under the transition zone before coming up to the surface.
Using a cylindrical model of iso-chemical mantle convection with multiple phase transitions, we investigate the
dynamical conditions for obtaining this peculiar plume morphology. Focusing on the role exerted by pressuredependent thermodynamic and transport parameters, we show that a strong reduction of the coefficient of
thermal expansion in the lower mantle and a viscosity hill at a depth of around 1800 km allow plumes to have
enough focused buoyancy to reach and pass the 660 km depth interface. The lateral spreading of plumes near
the top of the lower mantle manifests itself as a channel flow whose length is controlled by the viscosity contrast
due to temperature variations ΔηT. For small values of ΔηT, broad and highly viscous plumes are generated that
tend to pass through the transition zone relatively unperturbed. For higher values (102 ≤ ΔηT ≤ 103), we obtain
horizontal channel flows beneath the 660 km boundary as long as 1500 km within a timescale that resembles
that of Hawaiian hotspot activity. This finding could help to explain the origin of the broad hot anomaly observed
west of Hawaii. For a normal thermal anomaly of 450 K associated with a lower mantle plume, we obtain activation energies of about 400 kJ/mol and 670 kJ/mol for ΔηT = 102 and 103, respectively, in good agreement with
values based on lower mantle mineral physics. If an increase of the thermal conductivity with depth is also
included, our model can predict both long channel flows beneath the 660 km discontinuity and also values of
the radial velocity of local blobs sinking in the lower mantle of around 1 cm/yr, in good agreement with those
inferred for the sinking rate of cold remnants of the subducted lithosphere.
© 2011 Elsevier B.V. All rights reserved.
1. Introduction
According to the traditional plume hypothesis (see e.g. Campbell,
2007, for a recent review), volcanism at Hawaii and Emperor chain is
attributed to a narrow plume that rises from the core–mantle boundary
(CMB) and remains essentially stationary for at least 100 Myr below the
Pacific plate. This model is supported by geochemical (e.g. Hofmann,
1997) and seismic observations according to which a distinct thermal
anomaly beneath Hawaii can be tracked deep into the lower mantle
(e.g. Montelli et al., 2006; Wolfe et al., 2009). However, the seismic
evidence of deep rooted plumes is controversial and considered as not
conclusive (e.g. Boschi et al., 2006). On the other hand, seismic studies
have been conducted that, underneath Hawaii, do not show any anomalous decrease in the wave speed nor in the thickness of the transition
⁎ Corresponding author. Department of Planetary Physics, German Aerospace Center
(DLR), Berlin, Germany.
E-mail address: [email protected] (N. Tosi).
0012-821X/$ – see front matter © 2011 Elsevier B.V. All rights reserved.
doi:10.1016/j.epsl.2011.10.015
zone nor are compatible with high temperatures or extensive amounts
of melt (e.g. Deuss, 2007; Ritsema et al., 2009; Tauzin et al., 2008). To
explain intra-plate volcanism then, different dynamical mechanisms
have been invoked that are controlled by upper mantle processes
(Anderson, 2010) such as small-scale convection (Ballmer et al., 2010,
2011) or shear-driven generation of upwellings (Conrad et al., 2011).
While still advocating the deep plume hypothesis, on paleomagnetic
grounds (Tarduno et al., 2003), Tarduno et al. (2009) propose the prominent bend in the Hawaiian hotspot track to be a consequence either of
large-scale deep mantle circulation advecting the plume conduit
(Steinberger et al., 2004), the so-called mantle-wind, or of the motion
of the base of the plume (Hansen et al., 1993) or of the capture, and
subsequent release, of the plume head by a migrating ridge.
Yet another interesting point of view is offered by recent highresolution seismic imaging of the transition zone topography that has
displayed a great complexity in the shape of phase transitions boundaries beneath Hawaii (Cao et al., 2011). In particular, Cao and
co-authors reported an unexpected correlation of the 410, 520 and
N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
660 km boundaries about 2000 km west of the Hawaiian hotspot, with
the three interfaces exhibiting marked downward deflections with
respect to their reference depths. While the phase transitions from olivine to wadslyite at 410 km and from wadslyite to ringwoodite at
520 km have positive Clapeyron slopes and are indeed expected to
occur anomalously deep in a region of high temperatures such as
beneath a hotspot, the transition from ringwoodite to perovskitepericlase at 660 km has negative Clapeyron slope and is expected
instead to take place at shallower depths. These authors interpreted
the downward deflection of the 660 km boundary as a plausible
evidence for an exothermic post-garnet transition (e.g. Hirose, 2002;
Yu et al., 2011) forming the main seismic contrast and being caused
by a temperature anomaly as high as 450 K with a broad lateral extent
that ponds beneath the transition zone. Moving closer to the hotspot,
updoming of the 660 km interface is observed, consistently with the
expected deformation of the post-spinel phase boundary. From a geodynamic point of view, this scenario suggests that, despite the presence
of an exothermic post-garnet transition, the top of the lower mantle can
act as a barrier to an upwelling. Its deep mantle source would then not
be aligned with the present-day location of the hotspot which in turn
could represent the surface expression of complex pathways taken by
the upwelling itself in the upper mantle.
In our previous works, we followed the latest theoretical and experimental results on constraining thermodynamic properties of minerals
at lower mantle conditions (e.g. de Koker, 2010; Goncharov et al.,
2010; Katsura et al., 2009; Tang and Dong, 2010). In particular, we
focused on the combined effects of strongly depth-dependent thermal
expansivity and conductivity in global plane-layer models of mantle
convection with multiple phase transitions (Tosi et al., 2010), including
the one from perovskite to post-perovskite (Murakami et al., 2004;
Oganov and Ono, 2004). We showed that despite the propensity of
post-perovskite to destabilize the bottom thermal boundary layer
(Matyska and Yuen, 2006; Nakagawa and Tackley, 2004), low thermal
expansivity (Katsura et al., 2009) and high thermal conductivity values
at depth (de Koker, 2010; Hofmeister, 2008; Ohta, 2010; Tang and
Dong, 2010) allow for the formation of long-lived, large-scale thermal
and thermo-chemical anomalies atop the CMB, with interesting consequences for the long-term stability of thermo-chemical piles
(Dziewonski et al., 2010; Torsvik et al., 2010). Furthermore, we showed
that strongly depth-dependent coefficients of thermal expansion and
conduction turn out to be important ingredients for the formation of a
regional plume-fed asthenosphere with low viscosity in the vicinity of
hot upwellings in oceanic mantle (Yamamoto et al., 2007; Yuen et al.,
2011).
A plume morphology resembling that hypothesized by Cao et al.
(2011) was observed, though not analyzed in detail, in numerical
models of axisymmetric mantle convection with a modest resolution
by Yuen et al. (1996, 2007) and by Yuen et al. (2011) in Cartesian
models. Farnetani and Samuel (2005) and Samuel and Bercovici
(2006) observed similar structures in the framework of regional,
thermo-chemical plume dynamics. In this work we aim at studying
the conditions under which this type of plumes can be obtained in
connection with the key-role played by depth-dependent thermodynamic parameters. Moreover, as according to Cao et al. (2011) the hot
material ponding near the top of the lower mantle can have a large lateral extent (about 1000 and possibly up to 2000 km), we will use this
new piece of seismic information to place constraints on the mantle
rheology. We will show that the magnitude of the lateral viscosity
contrast due to temperature plays an important role in regulating the
amount of lateral spreading of plume material beneath the 660 km
boundary. Finally, our findings will be reinforced by demonstrating
that, in our preferred models, the velocity of sinking slab remnants in
the lower mantle matches well the constraint of about 1 cm/yr recently
derived by van der Meer et al. (2010) by correlating cold remnants of
subducted lithosphere, as seen in a tomographic model, with the corresponding tectonically-induced orogenic belts.
349
2. Model description
2.1. Governing equations and numerical method
We have solved the conservation equations of mass, linear momentum and thermal energy with multiple phase transitions in a full cylindrical annulus using the extended Boussinesq approximation (e.g. King
et al., 2010):
∂j vj ¼ 0;
ð1Þ
!
3
X
Rbl
¼ Ra αT−
Γ δ ;
−∂j p þ ∂j η ∂j vi þ ∂i vj
Ra l jr
l¼1
ð2Þ
3
X
DT
Di
Rb DΓ
¼ ∂j κ∂j T þ Diαvr ðT þ T 0 Þ þ Ф þ
Di l l γl ðT þ T 0 Þ þ R:
Dt
Ra
Ra Dt
l¼1
ð3Þ
Symbols and parameters are explained in Table 1. The continuity
Eq. (1) describes the conservation of mass for an incompressible fluid.
In the momentum Eq. (2), where an infinite Prandtl number is assumed,
the first and second terms on the right hand side account for the buoyancy forces due to temperature differences and phase transitions,
respectively. Three phase transitions are considered: at 410 km depth
from olivine (ol) to spinel (sp) (l=1), at 660 km depth from spinel to perovskite (pv) (l=2) and from perovskite to post-perovskite (ppv) (l=3).
The effect of the l-th transition is taken into account through the traditional approach of Christensen and Yuen (1985) using a phase function Γl:
Γl ¼
1
z−zl ðT Þ
1 þ tanh
;
2
w
ð4Þ
where z is the depth, w is the width of the phase transition and zl(T) is defined as
0
0
zl ðT Þ ¼ zl þ γl T−T l ;
ð5Þ
where γl, zl0 and Tl0 are the Clapeyron slope, transition depth and transition temperature. In the thermal energy Eq. (3), heat advection is
balanced by heat diffusion, adiabatic heating/cooling, viscous dissipation,
latent heat release/absorption due to phase changes and internal heating
due to radioactive sources. Since the models have been ran to steadystate, secular cooling has been neglected.
Eqs. (1)–(3) have been solved with the 2D–3D spherical code Gaia
(Huettig and Breuer, 2011). Gaia is based on a finite volume formulation with second-order accuracy in space and time. It uses the SIMPLE
algorithm (Patankar, 1980) to enforce the incompressibility
constraint (1) and a fully implicit three-levels time scheme to solve
the advection–diffusion Eq. (3) (Harder and Hansen, 2005).
This study is different from our previous works that were conducted in Cartesian geometry with a large aspect-ratio of 10 (Tosi
et al., 2010; Yuen et al., 2011). Here we have adopted a cylindrical domain consisting of a full annulus. As discussed by Yuen et al. (2011),
the choice of this geometry is appropriate in models with depthdependent thermal expansivity in which thermal plumes play a
prominent role. In fact, in Cartesian models, the large size of the
CMB surface allows for the formation of a large number of plumes
that can cause an excessive increase of the mantle temperature. The
smaller ratio of the inner to outer surface along with the increased degrees of freedom characteristic of the cylindrical geometry help to deliver smaller and more realistic temperatures.
The use of full cylindrical or spherical domains in connection with
free-slip boundary conditions can induce in the solution a spurious
component of pure rotation of the mantle as a whole since the velocity
field is determined only up to a rigid body rotation. To remove this pure
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N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
Table 1
Non-dimensional variables and numbers with the corresponding scaling and dimensional
parameters employed in this study. Note that while Eqs. (4) and (5) are written in nondimensional form, here, for clarity, the dimensional values of the reference depth, reference temperature and Clapeyron slope of the phase transitions are reported.
Symbol/
definition
Description
Scaling/
numerical value
t
z
vj
vr
p
T
T0 = Ts/ΔT
Φ = η(∂jvi +
∂ivj)∂ivj
η
α
k
k
κ¼
ρ0 cp
Time
Depth
Velocity component
Radial velocity
Dynamic pressure
Temperature
Surface temperature (non-dimensional)
Viscous dissipation
d2κs− 1
d
κsd− 1
κsd− 1
ηsκsd− 2
ΔT
0.85
ηsκs2d− 4
Viscosity
Thermal expansivity
Thermal conductivity
ηs
αs
ks
Thermal diffusivity
κs
Thermal Rayleigh number
1.9 × 107
l-th phase Rayleigh number
Rayleigh number for ol–sp transition
Rayleigh number for sp–pv transition
Rayleigh number for pv–ppv transition
–
1.1 × 107
1.4 × 107
2.7 × 106
Dissipation number
0.68
v′ ¼ v−ω r:
ð7Þ
The surface and CMB are treated as free-slip and isothermal boundaries. At the CMB, the temperature is set to 3800 K consistently with
recent estimates obtained by combining seismological and mineral
physics models (Kawai and Tsuchiya, 2009; van der Hilst et al., 2007).
As initial condition we employ a randomly perturbed adiabatic profile
with a potential temperature of 1600 K along with 100 km thick thermal boundary layers at the top and bottom. To discretize the model
domain, we use 256 equally spaced radial shells corresponding to a
radial resolution of about 11 km. Each shell is then subdivided into
2749 portions in such a way that the lateral resolution of the grid equals
the radial resolution at the mid-mantle depth. A total of more than
7 × 105 grid points is then employed for all calculations. Resolution
tests conducted using 384 shells and about 1.2 × 106 grid points have
not shown any significant difference in the results.
3
ρ α s ΔTd g 0
Ra ¼ 0
ηs κ s
where V is the shell volume and r the position vector. Then we subtract
the rotational motion from the solution vector v to obtain a corrected
velocity v′ (Kameyama et al., 2008):
3
δρ d g 0
Rbl ¼ l
ηs κ s
Rb1
Rb2
Rb3
α s dg 0
Di ¼
cp
2.2. Rheology and thermodynamic parameters
We have assumed a Newtonian rheology with viscosity dependent
on depth and temperature as follows:
2
R¼
d
g0
ρ0
cp
ηs
αs
ks
κs
Ts
ΔT
w
T10
T20
T30
z10
z20
z30
γ1
γ2
γ3
δρ1
δρ2
δρ3
H0
H0 d
cp κ s ΔT
Internal heating number
20
Mantle depth
Gravity acceleration
Reference density
Heat capacity
Reference viscosity
Reference thermal expansivity
Reference thermal conductivity
Reference thermal diffusivity
Surface temperature
Temperature drop across the mantle
Phase transition width
Reference temperature for ol–sp
transition
Reference temperature for sp–pv
transition
Reference temperature for pv–ppv
transition
Reference depth of ol–sp transition
Reference depth of sp–pv transition
Reference depth of pv–ppv transition
Clapeyron slope for ol–sp transition
(Bina and Helffrich, 1994)
Clapeyron slope for sp–pv transition
(Bina and Helffrich, 1994)
Clapeyron slope for pv–ppv transition
(Hirose, 2006)
Density jump for ol–sp transition
(Steinbach and Yuen, 1995)
Density jump for sp–pv transition
(Steinbach and Yuen, 1995)
Density jump for pv–ppv transition
(Oganov and Ono, 2004)
Internal heating rate
2890 km
9.8 m s−2
4500 kg m−3
1250 J kg−1
1022 Pa s
3 × 10− 5 K− 1
3.3 W m−1 K−1
5.9 × 10− 7 m 2s−1
300 K
3500 K
25 km
1760 K
ηðz; T Þ ¼ η1 ðzÞη2 ðT Þ;
1870 K
3600 K
ð8Þ
where the depth-dependent part η1(z) is defined as in Hanyk et al.
(1995):
2
:
η1 ðzÞ ¼ 1 þ 214:3 z exp −16:7ð0:7−zÞ
ð9Þ
Eq. (9) implies a viscosity maximum in the mid lower mantle (solid
black line in Fig. 1) (Morra et al., 2010; Tosi et al., 2010) which is consistent with modeling studies of the dynamic geoid (Čadek and van den
Berg, 1998; Ricard and Wuming, 1991), postglacial rebound (Mitrovica
and Forte, 2004; Tosi et al., 2005) and with molecular dynamics simulations of self-diffusion of MgO-periclase (Ito and Toriumi, 2010).
410 km
660 km
2890 km
3 MPa K−1
− 2.5 MPa K−1
13 MPa K−1
273 kg m−3
342 kg m−3
67.5 kg m−3
5.6 × 10− 12 W
kg−1
rotation from the solution, we calculate at each time step the average
angular velocity ω of the cylindrical shell:
1 rv
ω¼ ∫
dV;
V V ‖r‖2
ð6Þ
Fig. 1. Non-dimensional vertical profiles of the thermal conductivity (blue), thermal
expansivity (red), decadic logarithm of the viscosity according to Eq. (9) (solid black)
and according to Eq. (8) with ΔηT = 10 (dashed black), ΔηT = 102 (dashed-dotted black)
and ΔηT = 103 (dotted black) calculated for a typical mantle geotherm.
N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
For the temperature-dependent part we have employed a standard
linearization of the Arrhenius law
351
3. Results
3.1. Global convection patterns
η2 ðT Þ ¼ expð−βT Þ;
ð10Þ
where β = log(ΔηT) regulates the magnitude of the viscosity contrast
due to temperature variations. In Fig. 1 we have plotted the viscosity
profiles obtained from Eq. (8) using an adiabat with potential temperature of 1600 K, 100 km thick thermal boundary layers and β = 2.3
(dashed black line), 4.6 (dash-dotted black line) and 6.9 (dotted black
line), corresponding to changes in viscosity by one, two or three orders
of magnitude respectively, i.e. ΔηT = 10, 102 and 103. Admittedly,
Eq. (10) does not allow us to account for the viscosity gradient in the
upper mantle and for the strong viscosity localizations that would be
present if a proper plastic rheology for plate-like behavior were considered (e.g. (Tackley, 2000)). Nevertheless, it represents a good approximation of the Arrhenius law in the lower mantle where the average
temperature is higher.
Beside the phase transitions at 410 and 660 km depth, all models
also include the transition from pv to ppv near the CMB. The current
knowledge of ppv rheology is extremely uncertain (Karato, 2010).
Nevertheless, as experimental (Hunt et al., 2009) and theoretical evidence (Ammann et al., 2010) suggest that ppv may be significantly
weaker than pv, we have assumed the viscosity of the ppv phase to be
one order of magnitude lower than that of the surrounding pv phase.
With this assumption, the ratio between the maximum lower mantle
viscosity and the average viscosity of the D″ layer can become quite
large. Such a viscosity structure is in accord with a recent study of
Nakada and Karato (in press). In fact, by analyzing the decay time of
the Chandler wobble and time-dependent tidal deformations in the
framework of viscoelastic deformation models, Nakada and Karato estimated the viscosity in the bottom 300 km of the mantle to be 2–3
orders of magnitude lower than the average lower mantle viscosity
and up to 4 orders of magnitude lower in the bottom 100 km. Note
that even though the viscosity of ppv can influence the stability of the
bottom thermal boundary layer (Tosi et al., 2010), our results are not
affected significantly by the assumption of a weaker ppv.
In those models where the thermal expansivity α depends on depth,
we have assumed the Anderson-Grüneisen parameter to be 5 in the
upper mantle and 6.5 in the lower mantle. These values, which are
consistent with experiments conducted on olivine (Chopelas and
Boehler, 1992) and perovskite (Katsura et al., 2009), imply an overall
decrease of α of about one order of magnitude throughout the mantle
(red line in Fig. 1). The temperature dependence of the thermal expansivity, which is known to be mostly relevant for the heat transport and
distribution of flow velocities in the upper mantle (Ghias and Jarvis,
2008; Schmeling et al., 2003), is here neglected.
Although no conclusive agreement has been reached on the values
that the lattice thermal conductivity k takes on at lower mantle pressures, it is generally well accepted that it increases steadily both across
the upper (e.g. (Xu et al., 2004)) and lower mantle (e.g. (Goncharov
et al., 2010; Ohta, 2010)). Ab-initio calculations of the thermal conductivity of MgO periclase yield values of the total lattice conductivity near
the CMB that range from 6 Wm− 1 K − 1 (de Koker, 2010) to
10 Wm − 1 K − 1 (Tang and Dong, 2010), the latter being also consistent
with experiments conducted on perovskite (Ohta, 2010). Nevertheless,
values as high as 20–30 Wm− 1 K− 1 have also been proposed (Hofmeister, 2008). In our simulations in which k depends on depth, we have
assumed for simplicity a linear increase across the mantle by a factor
of 3, corresponding to k ~ 10 Wm − 1 K − 1 at the CMB (blue line in
Fig. 1). Although under high pressure conditions of the deep mantle a
T − 1 dependence of the lattice conductivity on temperature is likely to
be present (de Koker, 2010; Ohta, 2010), for simplicity it is here
neglected.
We start showing in Fig. 2 long-term snapshots of the temperature
field. Fig. 2d shows a model where both the thermal expansivity α
and thermal conductivity k are constant and the temperature viscosity
contrast ΔηT is 102. The temperature distribution is that characteristic
of convection at relatively high Rayleigh numbers with thin and highly
unstable thermal boundary layers (TBL). Cold downwellings interact
with the 410 and 660 km discontinuities and are often deflected horizontally by the endothermic phase transition at 660 km depth that
tends to retard slabs from sinking into the lower mantle. Despite the
relatively high viscosity imposed in the mid-lower mantle (Fig. 1), hot
upwellings are short-lived and loose their thermal signature before
reaching the upper mantle. Models that feature a constant thermal
expansivity and conductivity but different values of ΔηT (not shown
here) exhibit a very similar behavior.
The effects induced by depth-dependent thermal expansivity can be
readily observed in the top line of Fig. 2 which shows global temperature snapshots obtained using three different temperature viscosity
contrasts: ΔηT = 10 (Fig. 2a), 102 (Fig. 2b) and 103 (Fig. 2c). A thermal
expansivity that decreases strongly with depth promotes the formation
of large-scale plumes (Hansen et al., 1993; Tosi et al., 2010). In fact, as
soon as a thermal instability forms at the bottom TBL and starts to
rise, it acquires buoyancy during its ascent because of the growing
thermal expansivity with height. The opposite argument applies to
downwellings that lose their buoyancy while sinking into the mantle
because of the reduction of α with the depth. As a consequence, and
also because of the presence of a viscosity hill, they stagnate in the
mid mantle causing an overall cooling of the lower mantle.
The effects of employing different values of ΔηT can be readily seen by
comparing panels a, b and c. For small viscosity contrasts (ΔηT =10,
panel a), plumes are thick and highly viscous. Thus, they tend to pass
the endothermic phase transition at 660 km depth essentially unperturbed, although occasionally short channeling of the flow beneath the
transition zone can take place as for the plume portrayed in the bottom
right part of panel a. For ΔηT =102 (panel b), the interaction of plumes
with the 660 km phase change is now much more pronounced. Plumes
have a reduced buoyancy, being less viscous and thinner. Locally layered
convection generally causes them to travel horizontally along the top part
of lower mantle cells giving rise to pronounced flow channeling beneath
the 660 km discontinuity. As an alternative mechanism, tilting of the
plume conduit by large-scale mantle flow is also observed (Kerr and
Mériaux, 2004; Richards and Griffiths, 1988; Skilbeck and Whitehead,
1978). A more significant layering is observed when ΔηT is raised to 103
(panel c). A few plumes still retain enough buoyancy to reach the
surface, normally after interacting and being horizontally deflected by
the spinel–perovskite boundary. However, since plumes are thin and
have a low viscosity, they often fail to develop completely, as large-scale
mantle flow tends to shear and diffuse them.
It is well known that the presence of internal heat sources influences
the planform of convection by increasing the importance of downwellings over that of upwellings (e.g. (van den Berg et al., 2002)). Nevertheless, with a strong decrease of the thermal expansivity, the role of
plumes is so prominent that the convection pattern is qualitatively
unaltered by the presence of internal heating. This can be readily seen
by comparing the models shown in Fig. 2b (without internal heating)
and 2e (with internal heating), which look very similar, apart from an
expected increase of the average temperature. From a quantitative
viewpoint, the volume-averaged mantle temperature is 1750 K in the
first case and 1840 K in the second one. Furthermore, plume buoyancy
fluxes are not much dissimilar in the two cases: 1200 kg/s and 850 kg/
s are typical values obtained in the two simulations shown in Fig. 2b
and e, respectively. In the following we will then consider for simplicity
only basally heated situations.
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N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
Fig. 2. Long-term snapshots of the temperature distribution for six models. All models are basally heated except for (e) in which also internal heating is considered. Top line: constant
thermal conductivity, depth-dependent thermal expansivity and ΔηT = 10 (a), 102 (b) and 103 (c). Bottom line: ΔηT = 102 and constant thermal conductivity and expansivity (d), constant
thermal conductivity and depth-dependent thermal expansivity (e), depth-dependent thermal conductivity and expansivity (f).
To highlight the effects due to variable thermal conductivity, in
Fig. 2f we show a snapshot from a model that features both depthdependent thermal expansivity and conductivity along with a temperature viscosity contrast of 102. This plot can be directly compared
with Fig. 2b in which k is constant. The morphology of plumes and the
way these are bent by the 660 km boundary and form long channel
flows is very similar. However, with an increasing thermal conductivity
that favors heat transport by conduction at depth, plumes become
thicker and stronger. They rise through the lower mantle in an essentially vertical fashion and tilting of the plume conduit due to largescale mantle flow is not observed.
3.2. Channel flows beneath the 660 km discontinuity: morphology
As described in the previous section, models that do not include a
depth-dependent thermal expansivity fail to deliver plumes that are
buoyant enough to pass through the transition zone and reach the
surface. Therefore, as our goal is to characterize mantle upwellings
that interact clearly with the transition zone, from now on we will
focus only on models that incorporate a depth-dependent thermal
expansivity, with the possible addition of depth-dependent thermal
conductivity.
In Fig. 3 we show contour plots of the temperature anomaly δT ¼
T−T ðrÞ (where T ðr Þ is the instantaneous, radially averaged temperature profile) for six different models, while in Fig. 4 we show the corresponding viscosity distribution. Since in all models the viscosity of ppv
is assumed to be lower than that of pv by one order of magnitude, in
Fig. 4 the occurrence of the ppv phase can be readily seen. Figs. 3 and
4 portray the typical behavior of upwellings interacting with the
660 km depth discontinuity, which depends on the three parameters
α, k and ΔηT. The thermal conductivity is constant in the top line (panels
a, b, c), while it increases with depth in the bottom line (panels d, e, f). In
the first column (panels a, d), the temperature viscosity contrast ΔηT is
set to 10, while in the second (b, e) and third columns (c, f) is set to 102
and 103, respectively. In order to emphasize the differences in the
amount of hot material that accumulates near the top of the lower
mantle, the snapshots shown in Figs. 3 and 4 have been selected for
the maximum offset between the upper and lower mantle branches of
the plume (see also Figs. 5 and 6). Quantitatively, the excess temperature associated with upwellings does not present large differences
among the models shown here. Plumes are typically characterized by
a compact hot core close to the CMB where δT = 1000–1250 K. Near
the top of the lower mantle, temperature anomalies of ~500 K are
observed that decrease to ~250 K in the upper mantle.
Plume size and dynamics are strongly influenced by the value
employed for the viscosity contrast due to temperature. For ΔηT = 10
(panels a and d in Figs. 3 and 4), plumes are only slightly deflected by
the 660 km discontinuity. As their viscosity is close to that of the ambient mantle, they do not deform easily and the negative buoyancy arising
from the upward deflection of the spinel–perovskite boundary is not
sufficient to make hot rising material accumulate at the top of the
lower mantle. For values of ΔηT of 102 and 103 (panels b, e and c, f in
both Figs. 3 and 4), plumes have a significantly lower viscosity with
respect to the surroundings, which facilitates their deformation. The increased effectiveness of this transition in hindering the motion of rising
material causes plumes to assume a bent-shaped form. They tend to
flow preferentially beneath the phase boundary, following a horizontal
track which generally extends over several hundred kilometers, in
N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
353
Fig. 3. Effect of the temperature viscosity contrast ΔηT on plume-channel flows beneath the 660 km discontinuity. The color scale indicates temperature variations with respect to the average
radial profile. Top line: constant thermal conductivity and depth-dependent thermal expansivity with ΔηT =10 (a), ΔηT =102 (b) and ΔηT =103 (c). Bottom line: depth-dependent thermal
conductivity and expansivity with ΔηT =10 (d), ΔηT =102 (e) and ΔηT =103 (f). Dashed lines indicate the actual shape of the boundaries of the olivine-spinel and spinel–perovskite phase
transitions.
some cases over 1000 km or more (Fig. 3b and e). Unlike the stagnant
slab flows, which are sluggish, these hot channel flows are rapid, with
horizontal velocities beneath the 660 km boundary as high as 15 cm/yr.
From a qualitative point of view, the effects due to the presence of
depth-dependent thermal conductivity can be easily recognized by
comparing the top and bottom panels of Fig. 3. Using a depthdependent k promotes a slightly hotter mantle with plumes that tend
to be thicker than their counterparts obtained with a constant conductivity (Tosi et al., 2010). Nevertheless, the manner in which plume motion is affected by the phase transitions remains essentially unchanged.
The temperature distribution portrayed in Fig. 3e is particularly
noteworthy (see also Fig. 6 for a plot of the time-evolution). Here a
cold downwelling splashes on a hot plume channel flowing beneath
the 660 km boundary and causes the formation and rise of a secondary
plume. This behavior reveals that the plume channels observed in our
models can also serve as local boundary layers and hence as a source
of secondary thermal instabilities.
3.3. Channel flows beneath the 660 km discontinuity: formation and
evolution
We analyze now more closely the formation of channel flows and
discuss quantitatively the effects due to variable conductivity. To this
end, we show, in Figs. 5 and 6 respectively, the temporal evolution of
plumes observed in a model with and without depth-dependent thermal conductivity obtained using ΔηT = 102. In both cases the way the
channel flow forms is similar. First, a large-sized plume head reaches
the transition zone. Its thinned tip flows through the upper mantle,
helped by the exothermic transition at 410 km depth. As more and
more hot material is supplied from the bottom, upward distortion of
the 660 km boundary becomes more and more significant and causes
the plume to bend. The convection becomes locally layered. Hence,
instead of flowing in a vertical fashion, the rising plume flows now preferentially along the top of the lower mantle while advecting its upper
mantle branch (Fig. 9). At the same time, the base of the plume near
the CMB moves in the opposite direction.
For the two plumes shown in Figs. 5 and 6, and generally for the
majority of those generated in models that feature the same parametrizations, the horizontal channel flows beneath the 660 km boundary
reach a considerable length, over 700 km. Hot plume material with an
excess temperature as high as 500 K can eventually become spread
laterally over long distances of the order of 1000 to 1500 km.
While the plumes shown in Figs. 5 and 6 do not exhibit principle qualitative differences both in their shape and in the way they form, they are
characterized instead by a remarkably different temporal evolution. With
constant thermal expansivity (Fig. 5), the time interval measured from
the moment the plume hits the 660 km discontinuity until the resulting
channel flow reaches its maximal length is about 250 Myr, while it is
much shorter (~100 Myr) for the case with variable thermal conductivity
(Fig. 6). In fact, large values of k near the CMB favor a diffusive regime for
the coldest portions of the mantle (Tosi et al., 2010). This amount of additional heat in turn powers the advection of plumes which tend to flow faster. Thus the lifetimes of the hot channel flows below the transition zone
are influenced by the thermal conductivity of the deep mantle, which is a
somewhat surprising finding.
3.4. Flow velocities in the lower mantle
As long as a variable thermal expansivity is used, long plume channel flows are obtained both using constant and variable conductivity
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N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
Fig. 4. Viscosity distribution for the same models plotted in Fig. 3.
Fig. 5. Time-evolution of plume-channel flow for a model with constant thermal conductivity, depth-dependent thermal expansivity and ΔηT = 102. The color scale indicates temperature
variations with respect to the averaged radial profile. Dashed lines indicate the actual shape of the boundaries of the olivine–spinel and spinel–perovskite phase transitions.
N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
355
Fig. 6. As Fig. 5 but for a model with depth-dependent thermal conductivity and expansivity and ΔηT = 102.
Fig. 7. Time-evolution of sinking slab remnants and a rising plume in the lower mantle for a model with constant thermal conductivity, depth-dependent thermal expansivity and
ΔηT = 102. Top line: temperature variations with respect to the averaged radial profile. Bottom line: distribution of the radial velocity. Dashed lines indicate the shape of the boundaries
of the olivine–spinel and spinel–perovskite phase transitions.
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N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
even though different time-scales characterize their evolution. In order
to establish which is the preferable model, we have looked at characteristic mantle flow velocities taken from our simulations. In Fig. 7 we
show three snapshots, taken from a simulation in which k is constant
and ΔηT = 102, that portray the motion of typical remnants of cold
downwellings that sink in the lower mantle (top line) along with the
corresponding distribution of radial flow velocity (bottom line). As a
clear consequence of mass conservation, the nearly vertical downgoing
motion of subducted material is accompanied by the simultaneous generation of a plume that rises with a very similar speed (~0.5 cm/yr). This
indicates that, in first approximation, values of the average radial flow
velocity can be considered as representative of both down- and upwellings. Under this assumption, we show in Fig. 8 a summary of vertical
profiles of the radial (red lines) and tangential velocities (blue lines)
for nine different models. In the top line both α and k are constant, in
the middle line only α is depth-dependent while in the bottom line
both α and k are depth-dependent. The three columns correspond to
values of ΔηT of 10, 102 and 103, respectively. As a reference, we have
also plotted the value of 1.2 cm/yr (dashed black line) recently
estimated by (van der Meer et al., 2010) for the sinking rate of ancient
slab remnants. This value was obtained by identifying in a global tomographic model a large number of oceanic plates subducted in the lower
mantle and linking them to the present-day mountain ranges to which
they are likely to have given rise. On the one hand, using constant thermodynamic parameters (panels a–c), a characteristic arc-shaped profile
with a peak in the mid lower mantle and no evidence of the presence of
phase transitions is obtained for the radial velocity. On the other hand,
panels d–i, in which either α (d–f) or both α and k (g–i) are depthdependent, clearly show partial layering of the mantle. In fact, local
minima and maxima near the 660 km boundary are observed for the
radial and horizontal velocity, respectively.
With ΔηT = 102 (Fig. 8b), a relatively good match with the value
proposed by van der Meer et al. (2010) is observed, although this
model is not able to generate ponds of hot material beneath the transition zone as hypothesized by Cao et al. (2011). While this can be easily
achieved using depth-dependent thermal expansivity only, panels d–f
of Fig. 8 also show that the radial velocity profiles obtained for these
models clearly underestimate the reference value of 1.2 cm/yr no
Fig. 8. Summary of profiles of the average absolute radial (red) and horizontal velocity (blue) for different models. Top line (a, b, c): constant thermal expansivity and conductivity.
Mid line (d, e, f): depth-dependent thermal expansivity and constant conductivity. Bottom line (g, h, i): depth-dependent thermal expansivity and conductivity. Left column (a, d,
f): ΔηT = 10. Mid column (b, e, h): ΔηT = 102. Right column (c, f, i): ΔηT = 103. The green box highlights the two models whose radial velocity profile best approximates the value of
1.2 cm/yr proposed by (van der Meer et al., 2010) for the rate of sinking velocity of slabs remnants in the lower mantle (black dashed lines).
N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
matter what value of temperature viscosity contrast is used. The addition of depth-dependent thermal conductivity plays a fundamentally
important role in regulating mantle flow velocities (see Section 3.3). Indeed if thermal expansivity and conductivity are considered together
(Fig. 8g–i) and, in addition, the parameter ΔηT is chosen between 10 2
and 103, also a good match to the value of radial velocity predicted by
van der Meer et al. (2010) can be attained (Fig. 8h and i). It is difficult
to obtain these values of the radial flow velocity on the basis of Stokes'
flow analysis using only a depth-dependent viscosity and depthdependent thermal expansivity.
4. Discussion and conclusions
Recent seismic imaging investigation with multiply reflected
S-waves has revealed a broad, hot, pond-like structure beneath the
transition zone west of Hawaii that may represent a plausible source
for the volcanism at Hawaii itself and perhaps other islands (Cao et al.,
2011). This finding, albeit controversial (e.g. Wolfe et al., 2009), has
allowed us to infer the existence of a channel flow under the 660 km
discontinuity by using a numerical model of mantle convection in a
2D cylindrical geometry. Such a flow, when applied to the context of
the central Pacific, would connect a lower mantle plume located west
of Hawaii and a conduit-like plume in the upper mantle that fuels
Hawaiian volcanism.
We have found that this behavior is largely influenced by the depthdependent properties of the lower mantle, such as the thermal expansivity and thermal conductivity. The importance of the depthdependent expansivity on plumes has been known for a long time
(Zhao and Yuen, 1987). In fact, a strong decrease of the expansivity
with depth allows plumes to maintain their buoyancy and speeds
them up on their upward passage. It is essential that plumes remain
highly buoyant when they reach 660 km in order that they interact
strongly with the negative Clapeyron slope phase transition so that
the accompanying upward distortion of the phase boundary is significant (Schubert et al., 1975). This helps to stop the plumes and bend
them at the same time, thus generating the channel flow we observe.
In the framework of our models, without any depth-dependence of α,
it is difficult to generate ponds of hot material beneath the phase
boundary. However, it is important to note that this condition on the
thermal expansivity can be not so strict in a wider context. The numerical models of Farnetani and Samuel (2005) and Samuel and Bercovici
(2006) show in fact that accumulation of plume material beneath the
transition zone can be easily achieved with a constant α if chemical
heterogeneities are also accounted for.
The importance of depth-dependent conductivity is qualitatively
smaller than that of thermal expansivity. The plumes and channel
flows look very similar both using constant and depth-dependent k.
However, the important differences are in the faster flow velocities
and shorter lifetimes of the channel flow in the cases in which k
increases with the depth. Our models show that the characteristic
time scale of a plume from its encounter with the phase transition at
the top of the lower mantle to its eventual rise to the surface and subsequent advection in the upper mantle can range between approximately
100 Myr to 250 Myr, depending on whether a depth-dependent k is
taken into account or not. From the viewpoint of global mantle dynamics, with large values of the thermal conductivity in the deep mantle,
our models predict radial flow velocities which are in good agreement
with those estimated recently by van der Meer et al. (2010).
The range of lateral viscosity contrast between the hot plume and
the adjacent mantle is also quite critical in maintaining this delicate
configuration of channel flow and plume in dynamical equilibrium.
On the one hand, for a small value of the temperature viscosity contrast
(ΔηT = 10), our models predict large-scale plumes that, because of their
large viscosity, tend to pass through the phase transition undergoing
little deformation and thus failing to generate ponds of hot material
near the top of the lower mantle. On the other hand, for larger values
357
of ΔηT, we obtain channel flows with a large lateral extent that could
explain the origin of the broad anomaly observed west of Hawaii. In
particular, when both α and k are depth-dependent, a typical channel
flow thermal anomaly of 450 K, as observed in our models and as proposed by (Cao et al., 2011), implies an activation energy of about
190 kJ/mol for ΔηT = 10, 400 kJ/mol for ΔηT = 102 and 670 kJ/mol for
ΔηT = 10. These values have been calculated from the Arrhenius law
assuming that a given value of ΔηT corresponds to the mantle/plume
viscosity contrast for a plume with an excess temperature of 450 K.
Note that the latter two values bound the estimate of 500 kJ/mol
proposed by Yamazaki and Karato (2001) for the lower mantle, which
makes our preferred models also consistent with mineral physics
estimates.
The goal of this study was not to provide a detailed quantitative
model for the Hawaiian hotspot. Nevertheless, when both α and k are
depth-dependent, typical buoyancy fluxes that we obtain are approximately 1900 kg/s if ΔηT = 102 (Fig. 3e) and 4700 kg/s if ΔηT = 103
(Fig. 3f). These values compare reasonably well with estimates based
on the magnitude of the swell that predict buoyancy fluxes in the
range of 4000–6000 kg/s (Vidal and Bonneville, 2004). Thus, despite
the simplified 2D geometry, our models do not imply unrealistically
strong plumes.
Fig. 9 schematically shows the channel flow scenario which leads to
the buildup of the large pond of hot material beneath the transition
zone. The plume advances with the growth of the pond. This mechanism can be thought of as a new form of plume advection being different from the large-scale mantle wind and plume capture by a ridge
proposed by Tarduno et al. (2009). The length d of the pond being
around 1000 to 1500 km is an important constraint of this model and
could be tested using other data sources besides seismic imaging such
as petrological, paleomagnetic and plate reconstruction data acting in
concert. In this connection, another site where the channel flowplume scenario may be present is the Hainan plume (Lei et al., 2009).
However, here the situation is more complex than for Hawaii because
of the influence of the neighboring Philippine subducting slab.
In a sense, there is a complementary relationship between channel
flow and stagnant slab in the transition zone, both having been detected
by seismology. The two phenomena represent opposite ends of the
spectrum but they both yield important information about the physical
properties of the mantle, such as the rheology. Research devoted to the
detection and characterization of plume channel flows beneath the
transition could represent an interesting counterpart of the research devoted to stagnant slabs which has been a major focus in Japan for over a
decade (Fukao et al., 2009; Suetsugu et al., 2010).
Fig. 9. Schematic diagram (not drawn to scale) illustrating the formation mechanism of a
bent-shaped plume with the accompanying channel flow below the transition zone. Time
t0: a highly buoyant plume reaches the top of the lower mantle. Time t1: after being retarded and slightly deformed by the 660 km phase transition, the plume reaches the surface at
a location x1. Time t2: hot material supplied from below is held back by the phase transition and flows preferentially horizontally while the upper mantle branch of the plume is
advected towards a new position x2. Time t3: horizontal motion beneath the transition
zone continues until the tip of the plume at the surface reaches the position x3 and the
channel flow attains its maximal horizontal length d.
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N. Tosi, D.A. Yuen / Earth and Planetary Science Letters 312 (2011) 348–359
Our models are characterized by a series of simplifications that should
be addressed in the future. We should include the phase changes in the
garnet system and test the effects of the majorite–perovskite transition.
If exothermic (Hirose, 2002), this transition would speed up the hottest
portion of the plumes, reducing the propensity to local layering and to
the formation of bent-shaped plumes. Nevertheless, the exact nature
of the Clapeyron slope is still fraught with some uncertainty from
both theoretical and experimental standpoints (Wentzcovitch et al.,
2010). The temperature dependence of α and the influence of the T− 1
term in the phonon part of the thermal conductivity should be ultimately included in global mantle convection models as much as a plate-like
rheology from which a more realistic convective behavior of the upper
mantle and lithosphere is to be expected. Eventually the plume
morphology that we have discussed in this work should be studied in
a regional, three-dimensional context.
Acknowledgments
We are grateful to Maxim Ballmer and an anonymous reviewer for
their thorough and constructive reviews and to the editor Yanick Ricard
for his comments. We thank discussions with Maarten DeHoop. This
work has been supported by the Czech Science Foundation (project
P210/11/1366), the Helmholtz Association through the Research
Alliance “Planetary Evolution and Life”, the Deutsche Forschungs
Gemeinschaft (grant number TO 704/1-1) and the NSF Geophysics
Program. D.A. Yuen was supported by the Senior Visiting Professorship
Program of the Chinese Academy of Sciences.
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