10.1098/rsta.2002.1079
Geodynamic and seismic constraints on the
thermochemical structure and dynamics of
convection in the deep mantle
By Alessandro M. Forte1 , Jerry X. Mitrovica2
a n d A u d e Espesset1 †
1
Department of Earth Sciences, University of Western Ontario,
London, Ontario N6A 5B7, Canada
2
Department of Physics, University of Toronto, Toronto,
Ontario M5S 1A7, Canada
Published online 27 September 2002
We revisit a recent study by Forte & Mitrovica in which global geophysical observables associated with mantle convection were inverted and the existence of a strong
increase in viscosity near a depth of 2000 km was inferred. Employing mineral-physics
data and theory we also showed that, although there are chemical anomalies in
the lowermost mantle, they are unable to inhibit the dominant thermal buoyancy
of the deep-mantle mega-plumes below Africa and the Pacific Ocean. New Monte
Carlo simulations are employed to explore the impact of uncertainties in current
mineral-physics constraints on inferences of deep-mantle thermochemical structure.
To explore the impact of the high-viscosity peak at a depth of 2000 km on the evolution of lower-mantle structure, we carried out time-dependent convection simulations.
The latter show that the stability and longevity of the dominant long-wavelength
heterogeneity in the lowermost mantle are controlled by this viscosity peak.
Keywords: geodynamics; seismic tomography; mantle viscosity; three-dimensional
(3D) structure; thermochemical anomalies; mantle convection
1. Introduction
Recent progress in developing an integrated understanding of the three-dimensional
(3D) structure, composition and dynamics of the Earth’s lower mantle (Gurnis et
al . 1998; van Keken & Ballentine 1998; Kellogg et al . 1999; van der Hilst & Kárason
1999; Sidorin et al . 1999; Forte 2000; Tackley 2000; Forte & Mitrovica 2001; Helffrich
& Wood 2001) has been achieved through joint studies involving a variety of Earth
Science disciplines, especially global seismic tomography (e.g. van der Hilst et al .
1997; Grand et al . 1997; Bijwaard et al . 1998; Ekström & Dziewonski 1998; Ritsema
et al . 1999; Mégnin & Romanowicz 2000; Masters et al . 2000), mineral physics (e.g.
Karato 1993; Boehler 1996; Wang & Weidner 1996; Stacey 1998; Jackson 1998; Brodholt 2000), geochemistry (e.g. Allègre & Lewin 1995; Allègre et al . 1996; O’Nions &
† Present address: Atlantis Scientific Inc., Nepean, Ontario K2E 7M6, Canada.
One contribution of 14 to a Discussion Meeting ‘Chemical reservoirs and convection in the Earth’s
mantle’.
Phil. Trans. R. Soc. Lond. A (2002) 360, 2521–2543
2521
c 2002 The Royal Society
2522
A. M. Forte, J. X. Mitrovica and A. Espesset
Tolstikhin 1996; Hofmann 1997; Albarède 1998; Albarède & van der Hilst 1999) and
geodynamics. Recent global geodynamic analyses, with references to other studies,
are provided by Forte (2000) and Panasyuk & Hager (2000).
In this paper we revisit and further elaborate on the recent analysis of lowermantle dynamics by Forte & Mitrovica (2001) (henceforth referred to as ‘FM2001’).
FM2001 focused on the dynamical properties of the seismically imaged lower-mantle
mega-plumes below the central Pacific Ocean and below Africa. The issue they considered is whether these mega-plumes are stagnant structures, sitting passively at the
base of the mantle (e.g. Kellogg et al . 1999), or whether they are thermally buoyant
and actively ascending. In the latter case, geodynamical and geological arguments
for an active, hot mega-plume rising below Africa have been presented by LithgowBertelloni & Silver (1998) and by Gurnis et al . (2000). A global-scale geodynamic
analysis by FM2001 suggested that both the Pacific and African mega-plumes are
thermally buoyant and that they contribute to driving the mantle convective circulation.
2. Geodynamically constrained mantle-flow models based on
high-resolution seismic tomography
Models of the 3D mantle flow field may be derived to satisfy a wide range of geodynamic surface observables associated with the thermal-convection process (e.g. Hager
& Clayton 1989; Forte et al . 1993). The mantle is assumed to behave as an extremely
viscous fluid on geological time-scales and its effective viscosity is usually considered
to vary with depth only. The latter assumption significantly simplifies the spectral
solution of the viscous-flow problem using spherical harmonic basis functions (e.g.
Hager & O’Connell 1981; Richards & Hager 1984).
(a) Geodynamic observables and kernel functions
We employ here the spectral Green function solution of the dynamic-flow equations
recently developed by Forte (2000), which is based on a reformulated description of
the internal-loading problem. This spectral method provides a gravitationally consistent solution for viscous flow in a compressible spherical mantle. A distinguishing
feature of this new method is that the dependence of the flow solution on the radial
viscosity profile is described by a non-dimensional parameter, Λη , defined as
η(r)
d
,
Λη = r ln
dr
η0
in which η(r) is the radial viscosity profile of the mantle and η0 (= 1021 Pa s) is a
reference scaling value for the viscosity. With this logarithmic dependence on viscosity, it is possible to resolve extremely large radial variations in mantle viscosity
numerically.
The principal flow-related surface observables that have been studied on the basis
of the global tomography models are the geoid or corresponding free-air gravity
anomalies, the dynamic surface topography, the topography of the core–mantle
boundary (CMB), and the tectonic-plate motions (see Hager & Clayton 1989; Forte
et al . 1993 for reviews). The theoretical dependence of these observables on internal
density anomalies is expressed in terms of kernel functions, which are calculated in
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
2523
the spherical harmonic domain using the viscous-flow Green functions. These spectral kernel functions relate the spherical harmonic coefficients of the internal density
perturbations δρm
(r) to the corresponding harmonic coefficients of the surface data
dm
as
a
=
c
K (Λη ; r)δρm
(2.1)
dm
(r) dr,
b
where l and m are spherical harmonic degree and order, a is the mean radius
of the solid surface, b is the mean radius of the CMB and K is the (nondimensional) kernel function which depends on radius and on the mantle-viscosity
profile, the latter expressed by the non-dimensional parameter Λη defined above.
The precise expression for the multiplicative coefficient c depends on the corresponding surface data: for the non-hydrostatic geoid field, c = 3/[(2 + 1)ρ̄],
where ρ̄ is the mean density of the Earth; for the surface or CMB topography,
c = 1/∆ρ, where ∆ρ is the density jump across the mantle–ocean or mantle–core
boundary.
Fully self-consistent modelling of surface-plate motions requires a special treatment that goes beyond classical fluid mechanics (e.g. Tackley 2000). It is possible, nonetheless, to model the impact of plates on the mantle’s convective flow
by specifying appropriate surface boundary conditions (e.g. Ricard & Vigny 1989;
Gable et al . 1991; Forte & Peltier 1991, 1994). The modelling of plate motions
employed here (and in FM2001) uses the method of Forte & Peltier (1994), which
is based on the explicit consideration of the limited class of surface flows that are
in accord with the requirement that plate motions may only occur by rigid-body
rotations.
As shown by Forte & Peltier (1994), the presence of rigid surface plates enforces
a partitioning of internal density anomalies into two separate families: δρ = δ ρ̂ + δ ρ̄,
where the flow produced by δ ρ̂ is consistent with the geometry of possible rigidplate motions at the surface and the flow driven by δ ρ̄ is orthogonal to any possible plate motions. In other words, the plates completely resist the flow produced by δ ρ̄. This implies that free-slip and no-slip surface boundary conditions
must be applied in order to solve for the mantle flow produced by δ ρ̂ and δ ρ̄,
respectively.
Surface-plate velocities, v, may be characterized by their horizontal divergence
field ∇H · v (Forte & Peltier 1987). To predict the harmonic coefficients of the plate
divergence driven by the density anomalies δ ρ̂, we employ expression (2.1), in which
c = g0 /η0 , where g0 is the mean surface acceleration due to gravity. The presence
of the reference value of viscosity η0 implies that plate motions, unlike gravity or
topography, will be dependent on the absolute value of mantle viscosity.
The geodynamic kernels calculated on the basis of the mantle’s viscosity profile
obtained by FM2001 (see § 2 b) are illustrated in figure 1.
(b) Radial profile of mantle viscosity
FM2001 inferred viscosity profiles (figure 2) using two mantle-flow models based
on density anomalies derived from the tomography models of Grand et al . (1997) and
Ekström & Dziewonski (1998). The viscosity profile (dotted line, figure 2), inferred
on the basis of the most recent version of Grand’s (2001) tomography model, was
Phil. Trans. R. Soc. Lond. A (2002)
2524
A. M. Forte, J. X. Mitrovica and A. Espesset
depth (km)
0
1000
2000
(a)
3000
–0.4 –0.2
(b)
0
0.2 0.4
geoid: free-slip
0.6 –0.4 –0.2
0
0.2 0.4
geoid: no-slip
0.6
depth (km)
0
1000
2000
(c)
(d)
3000
–1.0
–0.5
0.0
topography: free-slip
–1.0
–0.5
0
topography: no-slip
depth (km)
0
1000
2000
(e)
3000
–0.20 0.15 –0.10 0.05
divergence
(f)
0
0
0.5
1
CMB: free-slip & no-slip
Figure 1. Geodynamic kernel functions. In (a)–(e) the kernels corresponding to harmonic degree
= 2 are represented by a bold line; = 4 by a solid line; = 8 by a dotted line; = 16 by a
dot-dashed line and = 32 by a dashed line. The geoid kernels for free-slip and no-slip surface
conditions are shown in (a) and (b), respectively. The surface topography kernels for free-slip and
no-slip conditions are shown in (c) and (d), respectively. The horizontal divergence kernels are
shown in (e). The kernels for the dynamic core–mantle boundary (CMB) ellipticity, defined in
(2.1) for = 2 and m = 0, are shown in (f ). The bold line represents the CMB ellipticity kernel
for a no-slip surface, while the solid line represents the kernel for a free-slip surface. All kernels
were calculated for the mantle-viscosity profile obtained by FM2001 on the basis of Grand’s
tomography model (solid line in figure 2).
obtained using the same nonlinear-inversion procedure described by FM2001 and
Forte (2000). All profiles are characterized by an upper-mantle asthenospheric channel with strongly reduced viscosity, a viscosity maximum at the top of the lower
mantle and a strong increase in viscosity at a depth of 2000 km. As we will show
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
2525
0
depth (km)
1000
2000
3000
10–2
10–1
100
101
21
viscosity (10 Pa s)
102
103
Figure 2. Geodynamically inferred mantle viscosity. The solid and dashed lines represent the
viscosity profiles inferred by FM2001 using mantle-flow models based on the tomography models
of Grand et al . (1997) and Ekström & Dziewonski (1998), respectively. The dotted curve is the
profile obtained on the basis of Grand’s (2001) tomography model.
below, this deep-mantle viscosity maximum has a major impact on mantle-convection
dynamics in the lower mantle.
(c) Geodynamically inferred viscosity and seismic Q
Seismic attenuation and long-term mantle creep may occur through the same
microphysical processes and can thus be dependent on the same physical variables,
especially temperature (e.g. Karato 1998). When relating mantle-viscosity profiles to
seismic Q profiles, we must contend with the different resolving power of the seismic
and geodynamic data.
The shear-attenuation Qµ model of Durek & Ekström (1996) is parametrized in
terms of five constant Qµ layers (four in the upper mantle and one for the lower
mantle), consistent with the resolving power of their seismic data. In each of these
five layers, we calculated the average viscosity using the geodynamically inferred viscosity profiles (figure 2). Assuming the frequency dependence Qµ ∼ ω α (e.g. Karato
1998), the five-layer viscosity model is translated into an equivalent Qµ model and
vice versa. Both viscosity, η, and Qµ depend on temperature through an associated
activation energy as Qµ = Q0 exp(αgQ Tm /T ), η = η0 exp(gη Tm /T ), in which gQ and
Phil. Trans. R. Soc. Lond. A (2002)
2526
A. M. Forte, J. X. Mitrovica and A. Espesset
gη relate the activation energies for attenuation and creep, respectively, to mantle
melting temperature Tm (e.g. Weertman & Weertman 1975). Combining these two
expressions, we obtain
Q0
αgQ
αgQ
η0
η
Qµ
= ln
−
+
,
(2.2)
ln
ln
ln
Qsurf
Qsurf
gη
ηsurf
gη
ηsurf
in which Qsurf and ηsurf represent the value of Qµ and viscosity, respectively, at the
top of the mantle.
Assuming that the activation energies for attenuation and creep are identical
(implying gQ = gη ), we may then determine the α value that minimizes the leastsquares misfit between the left- and right-hand sides of equation (2.2) in all five
mantle layers employed in the radial Qµ parametrization. Employing the viscosity
profile obtained on the basis of Grand’s tomography model (solid line, figure 2),
we infer a value of α = 0.275, with misfits to seismic Q−1
µ not exceeding 20%. The
inferred α falls within the range of acceptable values: 0.1 < α < 0.3 (e.g. Karato &
Spetzler 1990; Jackson et al . 2000).
There thus appears to be a good indication that seismic inferences of Q and our
current geodynamic inferences of mantle viscosity are mutually consistent. This consistency also reinforces our view (FM2001) that the Occam-inferred radial viscosity profiles have not been substantially biased by the neglect of lateral viscosity
variations, particularly in the upper mantle, where inferences of Q and η are both
characterized by strong depth variations.
(d ) Mantle-density anomalies
Seismic anomalies in tomographic models may be translated into equivalent density
and/or temperature anomalies on the basis of experimental and theoretical mineral
physics (e.g. Karato 1993; Čadek et al . 1994; Boehler 1996; Wang & Weidner 1996;
Sobolev et al . 1997; Stacey 1998; Zhang & Weidner 1999; Trampert et al . 2001;
Stixrude & Lithgow-Bertelloni 2001; Oganov et al . 2001). Another method for constraining mantle-density anomalies involves the direct inversion of convection-related
geophysical observables (e.g. Forte et al . 1993; Corrieu et al . 1994; Forte & Perry
2000; Panasyuk & Hager 2000; FM2001; Pari 2001; Deschamps et al . 2001).
Some of the difficulties faced in estimating mantle-density anomalies from seismic
tomography models may be appreciated by considering a comparison of various ‘highresolution’ models in figure 3. (A more extensive comparison of recent tomography
models may be found in Becker & Boschi (2002).)
With the exception of the top 400 km of the mantle, the average amplitudes of the
relative perturbations of seismic shear velocity δVS /VS in the tomography models of
Grand et al . (1997) and Ekström & Dziewonski (1998) differ significantly (figure 3a).
There are also differences in the spatial pattern of the seismic anomalies, especially
in the depth interval 300–1000 km (figure 3b).
These comparisons show that direct application of mineral physics to translate
seismic anomalies into density anomalies gives rise to large variations in the estimated density, depending on the choice of tomography model. To avoid this difficulty, FM2001 determined optimal d ln ρ/d ln VS scaling coefficients by inverting
global convection-related data, which included free-air gravity anomalies, horizontal
divergence of present-day plate velocities, dynamic surface topography and excess
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
0
2527
(a)
(b)
depth (km)
1000
2000
3000
0.0
1.0
2.0
RMS amplitude (%)
3.0 0.0
0.2
0.4
0.6
cross correlation
0.8
1.0
Figure 3. Comparison of seismic tomography models. (a) Root-mean-square (RMS) amplitude
of relative perturbations of seismic shear velocity δVS /VS , as a function of depth. The solid,
dotted and dashed lines represent the RMS amplitudes obtained from the tomography models
of Grand et al . (1997), Grand (2001) and Ekström & Dziewonski (1998), respectively. (b) Global
cross-correlation, as a function of depth, between the tomography models of Ekström & Dziewonski (1998) and Grand et al . (1997), shown by the solid line, and between the models of Ekström
& Dziewonski (1998) and Grand (2001), shown by the dotted line.
CMB ellipticity. Figure 4 shows the average amplitudes of mantle-density anomalies
obtained by translating the seismic anomalies in the tomography models using the
geodynamically inferred d ln ρ/d ln VS scaling.
Global fits to the convection-related data that are obtained on the basis of geodynamically constrained density anomalies are summarized in table 1. Estimated
global surface heat flux at the top of the mantle, ca. 36 TW, provides an independent, a posteriori check on mantle-flow models (e.g. Hager & Clayton 1989). The
tomography-based flow models in table 1 predict a vertical heat advection that never
exceeds 30 TW across the mantle.
(e) Implications of deep-mantle high viscosity on convective flow
Figure 5 shows the lower-mantle flow field predicted using the density anomalies
and viscosity profile inferred from the Ekström & Dziewonski (1998) tomography
model. We note that the predicted convective flow at mid-mantle depths (figure 5a)
is characterized by numerous individual columns of buoyant upwelling flow, below
the Pacific and below Africa. The Pacific upwellings lie below well-known hotspot
Phil. Trans. R. Soc. Lond. A (2002)
2528
A. M. Forte, J. X. Mitrovica and A. Espesset
0
depth (km)
1000
2000
3000
0
0.1
0.2
RMS amplitude (%)
0.3
Figure 4. Mantle-density anomalies. The solid, dotted and dashed lines represent the RMS
amplitudes of the density anomalies derived from the tomography models of Grand et al . (1997),
Grand (2001) and Ekström & Dziewonski (1998), respectively. The seismic shear-velocity perturbations in these tomography models were converted to density perturbations using d ln ρ/d ln VS
scaling factors derived in joint inversions of convection-related datasets (FM2001).
groups (e.g. Hawaii, French Polynesia) and the main upwelling below Africa appears
to lie directly below the Tanzania craton. Recent studies of the shallow mantle
in this region indicate that this deep upwelling is not currently impacting on the
base of the craton (Brazier et al . 2000). We also note a large, semicircular ring of
mantle upwelling, apparently encircling the southern half of the African continent,
which extends from below the southern Indian Ocean to the mantle below the midAtlantic Ocean. This immense arc of upwelling flow appears to directly coincide
with location of the ‘DUPAL’ geochemical province (Dupré & Allègre 1983; Hart
1988).
The convective flow changes character at depths approaching the location of the
high-viscosity peak at a depth of 2000 km (figure 5b). At these depths, the shorter
horizontal wavelengths of the flow field are suppressed and the flow is strongly dominated by very long horizontal wavelengths. This is similar to the ‘doming’ regime
that Davaille (1999) identified in chemically stratified convection experiments. The
flow at these depths is dominated by a quadrupolar (harmonic degree = 2) pattern. Below southern Africa it is characterized by a significant broadening of the arc
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
2529
90ºN
60ºN
30ºN
0º
30ºS
60ºS
90ºS
0
60ºE
2cm/yr (horizontal)
120ºE
180º
120ºW
+2.4
60ºW
0
60ºW
0
–2.4
cm/yr (vertical)
90ºN
60ºN
30ºN
0º
30ºS
60ºS
90ºS
0
60ºE
2cm/yr (horizontal)
120ºE
180º
+1.6
120ºW
–1.6
cm/yr (vertical)
Figure 5. Present-day convective flow in the lower mantle. Mantle flow shown here is predicted
on the basis of geodynamically constrained density anomalies (figure 4, dashed line) derived
from the Ekström & Dziewonski (1998) tomography model and using the corresponding inferred
viscosity profile (figure 2, dashed line). (a) Predicted flow field at a depth of 1300 km. Colours
represent the vertical component of the flow velocity (red, upwards; blue, downwards), while
black arrows represent the horizontal components. (b) Predicted flow field at a depth of 1800 km.
(Figure adapted from FM2001.)
Phil. Trans. R. Soc. Lond. A (2002)
2530
A. M. Forte, J. X. Mitrovica and A. Espesset
depth (km)
0
l= 4
l= 8
1000
2000
3000
0
depth (km)
l= 2
0
0.1 0.2 0.3 0.4 0.5
0
l= 16
0.1 0.2 0.3 0.4 0.5
0
0.1
l= 20
0.2
0.3
l= 120
1000
2000
3000
0
0.05 0.10 0.15 0.20
0
0.05 0.10 0.15 0.20
0
0.5
1.0
1.5
RMS vertical flow velocity (cm/yr)
Figure 6. Spectral analysis of predicted vertical flow in the mantle. Each frame shows the depth
variation of the RMS amplitude of vertical flow corresponding to the indicated harmonic degree
. The flow is predicted on the basis of the density anomalies (dashed line, figure 4) derived
from the Ekström & Dziewonski (1998) tomography model. The solid lines represent the flow
predicted on the basis of the geodynamically inferred viscosity profile (dashed line, figure 2). The
dashed lines correspond to the flow predicted by a simple two-layer viscosity profile obtained by
averaging the viscosity in the upper mantle and in the lower mantle. This profile is characterized
by a factor of 50 jump in viscosity at a depth of 670 km.
of upwelling flow extending from the Indian to Atlantic Oceans. The geographical
coincidence of this predicted deep-mantle flow with the DUPAL anomaly supports
Hart’s (1988) hypothesis for a quadrupolar pattern of convection, near the CMB,
which transports the DUPAL chemical components.
The deep-mantle high-viscosity peak evidently behaves as a low-pass filter and
this is more clearly illustrated in the spectral analysis of flow shown in figure 6. The
shorter-wavelength components of vertical flow are strongly damped below a depth
of ca. 1800 km, contrary to what is predicted with a constant-viscosity lower mantle
(dashed lines, figure 6).
A more complete illustration of the geometry and dynamics of the buoyant lowermantle ‘mega-plumes’ below Africa and the central Pacific is provided by the wholemantle cross-section shown in figure 7. The intensity and scale of the upwelling flow
in these two mega-plumes is similar in the lower mantle; however, in the upper mantle
the character of the flow below Africa is quite different from that below the central
Pacific.
We note that the buoyancy-driven flow associated with the African mega-plume is
sharply deflected northwards as it enters the upper mantle, flowing sub-horizontally
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
2531
Table 1. Fits to global convection-related datasets
(All fits are expressed in terms of variance reductions. The free-air gravitational anomalies are
calculated on the basis of the non-hydrostatic geopotential derived from satellite data (Marsh
et al . 1990). The horizontal divergence of the plate velocities are calculated using plate motions
specified by the NUVEL-1 model (De Mets et al . 1990). The dynamic surface topography is
obtained by correcting for the effects of crustal isostasy (Forte & Perry 2000; Pari 2001). The
excess or dynamic CMB ellipticity is constrained by space geodetic analyses of free-core nutation
periods (Mathews et al . 1999). The numbers in parentheses are the variance reductions to the
equivalent non-hydrostatic geoid anomalies derived from satellite data (Marsh et al . 1990).)
tomography-based
density anomalies
Grand et al . (1997)a
Ekström & Dziewonski (1998)b
Grand (2001)c
free-air gravity
[ = 2–20]
divergence
[ = 1–32]
topography
[ = 1–20]
CMB
ellipticity
32% (64%)
43% (71%)
42% (69%)
59%
69%
62%
38%
22%
38%
100%
100%
100%
a
Predictions obtained using the geodynamically inferred viscosity profile (solid line, figure 2)
and density anomalies (solid line, figure 4).
b
Predictions obtained on the basis of geodynamically inferred viscosity (dashed line, figure 2)
and density (dashed line, figure 4).
c
Predictions obtained on the basis of geodynamically inferred viscosity (dotted line, figure 2)
and density (dotted line, figure 4).
below the African craton before emerging at the surface along the Red Sea plate
boundary. This dominant horizontal, rather than vertical, transport of plume
material is in accord with regional tomography studies (e.g. Brazier et al . 2000;
Nyblade et al . 2000) of the East African Rift system. Global seismic studies of
transition-zone thickness also show no significant variation below East Africa, supporting the absence of deep-seated plume anomalies directly below this region
(Chevrot et al . 1999).
The convective flow field associated with the Pacific mega-plume is more symmetrical as it enters the upper mantle, directly below the location of the South
Pacific Superswell (McNutt & Judge 1990). This symmetry gives way to a more
complicated pattern of flow in the shallow upper mantle, especially below the
entire length of the (eastern and southern) Pacific ridge system, where it performs a striking ‘demi-tour ’† (figure 7). The relationship between upper-mantle seismic anisotropy in the Pacific hemisphere and the underlying flow driven by the
ascent of the deep-mantle mega-plume has been recently analysed by Gaboret et
al . (2001).
3. Compositional heterogeneity in the deep lower mantle
The dominant thermal buoyancy of the two lower-mantle mega-plumes, shown clearly
in figure 7, may at first seem to imply that opposing chemical or compositional heterogeneity is absent in the lower mantle. FM2001 showed, however, that this dominant thermal buoyancy does not rule out the presence of large-scale compositional
anomalies in the lower mantle.
† French for ‘U-turn’.
Phil. Trans. R. Soc. Lond. A (2002)
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A. M. Forte, J. X. Mitrovica and A. Espesset
90º
0º
60
12
º
30
15
0º
º
60
90
0
30
12
0
15
0
180
0º
180º
0
21
24
33
0
0
0
270
0º
33
21
0º
30
24
0º
0º
30
5 cm/yr
270º
–1.5 %
+1.5%
0
dVS/VS
Figure 7. Dynamic upwelling mega-plumes in the lower mantle. A great-circle slice through
the tomographic model of Ekström & Dziewonski (1988). The trajectory of the great circle is
indicated by a yellow line in the inset map. The red triangles in this map identify the Hawaii,
Tahiti and Marquesas hotspots. The colour contours represent the relative perturbations in
seismic shear velocity identified by the coloured scale bar. The superimposed black arrows in
this cross-section represent the mantle-flow velocities (scale shown at bottom right) predicted
on the basis of geodynamically inferred mantle-density anomalies (dashed line, figure 4) and
viscosity profile (dashed line, figure 2). The location of the 670 km seismic discontinuity is
illustrated by the thin black circle superimposed on the velocity anomalies. (Adapted from
Gaboret et al . (2001).)
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
2533
Table 2. Thermal and compositional derivatives
(All values not enclosed by parentheses have been obtained using the approximations and data
described in detail by FM2001. Values enclosed in square brackets have been obtained on the
basis of the finite-strain and thermoelastic calculations described in this work. Round brackets
enclose values estimated by Stacey (1998), while the curly braces enclose the recent determinations obtained by Oganov et al . (2001).)
elastic
property
thermal (−105 × ∂/∂T (K−1 ))
anharmonic
anelastic
compositional
10 × ∂/∂XFe 100 × ∂/∂XP v
depth = 1970 km
ln ρ
1.3 [1.2] (1.3) {1.4}
ln VS
5.0 [4.4] (4.7) {3.9}
ln Vφ
0.8 [1.5] (0.6) {0.9}
—
2.5
0
+3.2 [+3.0]
−2.2 [−2.5]
−1.6 [−1.5]
+1.1 [+2.8]
+4.9 [+3.5]
+5.5 [+3.4]
depth = 2740 km
ln ρ
1.0 [0.8] (1.1)
ln VS
4.7 [4.0] (4.1)
ln Vφ
0.7 [1.2] (0.4)
—
2.4
0
+3.2 [+3.0]
−2.2 [−2.5]
−1.6 [−1.5]
+0.4 [+2.2]
+4.5 [+1.9]
+4.8 [+2.0]
(a) Constraints from seismic tomography and mineral physics
Joint tomographic inversions for lateral variations in seismic shear velocity δVS /VS
and bulk-sound or acoustic velocity δVφ /Vφ clearly show that these two types of
seismic heterogeneity are anti-correlated with each other in the lowermost mantle
(Su & Dziewonski 1997). More recent joint inversions for δVS /VS and δVφ /Vφ confirm
the anti-correlation of these two fields (Kennett et al . 1998; Masters et al . 2000) and,
as argued by Stacey (1998), it is difficult to reconcile this anti-correlation with pure
thermal heterogeneity.
To determine the relative importance of thermal and compositional anomalies,
FM2001 estimated temperature and compositional derivatives of density and seismic velocity, assuming a lower mantle whose composition and state involves three
components: MgO, FeO and SiO2 . We parametrized compositional heterogeneity in
terms of variations of the molar fraction of perovskite (XPv ) and of iron (XFe ).
These derivatives were obtained on the basis of theoretical approximations and data
in the mineral-physics literature (e.g. Jackson 1998; Stacey 1998). The derivatives
estimated for a pressure of 127 GPa (a depth of 2740 km) are presented in table 2.
In this table we also present derivatives, estimated as in FM2001, for a pressure of
85 GPa (a depth of 1970 km).
As an alternative to the approximations employed in FM2001, we have also calculated the thermal and compositional derivatives throughout the mantle using published ambient-pressure (P = 0) thermal equation-of-state results, which were subsequently projected to deep-mantle pressures using adiabatic finite-strain models. The
approach followed here is very similar to that employed by Trampert et al . (2001),
who have modelled finite-strain compression using third-order Birch–Murnaghan
equations, whereas we employed a finite-strain equation based on the Rydberg potential. We found the latter to be especially simple to use from a numerical perspective.
A detailed presentation and evaluation of the different finite-strain equations are proPhil. Trans. R. Soc. Lond. A (2002)
2534
A. M. Forte, J. X. Mitrovica and A. Espesset
vided by Stacey (1998), who concludes that acceptable fits to compression parameters
(e.g. zero-pressure bulk modulus K0 , and K0 K0 ) derived from the preliminary reference Earth model (PREM) (Dziewonski & Anderson 1981) require a fourth-order
Birch equation and that the Rydberg potential provides an equally good fit. These
finite-strain equations provide a means of extrapolating zero-pressure values of density ρ and bulk modulus K to lower-mantle pressures. Similar extrapolations of shear
modulus µ are instead performed using the linear relationship between µ, K and P ,
derived by Stacey (1992).
The calculation of density and elastic moduli for a particular lower-mantle composition is performed using standard Voigt–Reuss–Hill averages of the end-member
components, modelled here as (Mg,Fe)SiO3 perovskite and (Mg,Fe)O magnesiowüstite. High-temperature values, at ambient pressure, for the elastic moduli and
density are determined on the basis of the thermal equation-of-state models and
thermoelastic parameters presented in Jackson (1998, tables 1 and 3). All calculations assumed that the partition coefficient k for iron between perovskite and magnesiowüstite is 0.45, as in Jackson (1998). The procedure followed in this stage of the
modelling is again very similar to that of Trampert et al . (2001), and we explored a
range of reference compositions and lower-mantle potential temperatures T0 , choosing those values which yielded a close match (within 2%) to the seismic velocity and
density in the PREM. The temperature and compositional derivatives were then
calculated numerically by perturbing the reference composition and temperature. In
table 2 we present average values, enclosed in square brackets, of the numerically
determined derivatives.
For comparison, we also include in this table the temperature derivatives, enclosed
in round brackets, estimated by Stacey (1998). (No compositional derivatives were
determined in Stacey’s analysis.) Substantial progress has recently been made in ab
initio molecular-dynamics simulations of the thermoelastic properties of candidate
lower-mantle phases (e.g. Brodholt 2000). Oganov et al . (2001) recently determined
new estimates of the temperature derivatives of pure magnesium perovskite (MgSiO3 )
and these are also included in table 2, enclosed in curly braces. Considering the 20%
uncertainties cited by Oganov et al . (2001), their estimated derivatives agree well
with the other values appearing in table 2.
FM2001 assembled the theoretical expressions for the perturbations in seismic
velocity and density due to thermochemical anomalies into the linear system



 
δ ln VS
δT
c11 c12 c13
δ ln Vφ  = c21 c22 c23  δXPv  ,
(3.1)
δ ln ρ
c31 c32 c33
δXFe
in which the matrix elements cij are the thermal or compositional derivatives in
table 2.
On the basis of this matrix equation, FM2001 showed that specific linear combinations of δ ln VS and δ ln Vφ can be used to extract effective representations of
compositional heterogeneity δXeff (containing contributions from both δXP v and
δXFe ) and thermal heterogeneity δTeff (containing contributions from both δT and
δXFe ). FM2001 found that δXeff is strongly correlated to the bulk-sound velocity
anomalies at the same depth and that δTeff is strongly correlated to shear-velocity
anomalies. Shear-velocity anomalies appear to provide an excellent probe for deepmantle temperature anomalies, despite the presence of compositional heterogeneity.
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
2535
800
700
number of trials
600
500
400
300
200
100
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
buoyancy ratio (Rρ)
Figure 8. Monte Carlo exploration of buoyancy ratios at a depth of 2740 km. The large
(unshaded) histogram represents the distribution of buoyancy ratios that are consistent with
current uncertainties in global geodynamic constraints on mantle-density structure and uncertainties in joint tomography/mineral-physics constraints on thermochemical structure. (See text
for more details.) The smaller (shaded) histogram represents the distribution of buoyancy ratios
that are consistent with the present-day dynamic topography and estimated uplift rates of Africa
(Gurnis et al . 2000).
(b) Geodynamic constraints on thermochemical structure
Figure 4 shows that geodynamically constrained density anomalies are relatively
independent of the choice of shear tomography model. Accordingly, the inferred
density field, together with the joint bulk-sound/shear tomography models, may be
used in equation (3.1) to separately resolve δT , δXP v and δXFe anomalies in the
lower mantle. FM2001 found that the regions of hotter mantle below the central
Pacific and below southern Africa are also enriched in iron (the global correlation
coefficient between δT and δXFe is +0.7). This implies that, at these depths, the
thermal buoyancy of these mega-plumes is partly offset by the intrinsic negative
buoyancy due to iron enrichment.
The relative importance of thermal and chemical buoyancy may be quantified by
a buoyancy ratio defined as Rρ = (δρ/ρ)chem /(αδT ). This ratio is a key parameter
in thermochemical convection models and its value determines the dynamics and
evolution of the convective flow (e.g. Tackley 1998; Kellogg et al . 1999; Davaille
1999). The buoyancy ratios estimated by FM2001 are in the range ∼ 0.1 to ∼ 0.3,
which is in accord with the dominant thermal buoyancy of the deep-mantle megaplumes (figures 5 and 7). Davaille (1999) finds that this range of buoyancy ratios is
associated with the distinct ‘doming’ regime for thermochemical convection.
In view of the uncertainties in the estimated thermal and compositional derivatives
in table 2, we wish to test whether our estimates of the buoyancy ratio are robust. We
explored the impact of these uncertainties by allowing all elements cij of the matrix
in equation (3.1) to vary by 50% in each direction (which is of the order of the range
of values in table 2). The independent geodynamic constraints on density anomalies,
Phil. Trans. R. Soc. Lond. A (2002)
2536
A. M. Forte, J. X. Mitrovica and A. Espesset
derived from Grand’s tomography model (solid line, figure 4), are allowed to vary
by 50% in both directions. The shear and bulk-sound-velocity anomalies were taken
from the tomography model of Masters et al . (2000) and remained fixed. We then
carried out a Monte Carlo search for values of the matrix elements cij and density
anomalies which were consistent with the above ranges. For each trial we determined
the corresponding buoyancy ratio Rρ , and the results of all trials are summarized in
the histogram shown in figure 8.
The histogram shows that the most probable values for Rρ span the range 0.1–0.4
and values above Rρ = 0.5 appear to be unlikely. The range of probable values for
Rρ may be further refined by considering other datasets that are independent of
the global convection-related observables we have so far employed. Inferences of the
present-day dynamic topography of Africa and geologically based estimates of Cenozoic uplift rates of Africa may be used to place direct constraints on the buoyancy
of the deep-mantle mega-plume below Africa (Gurnis et al . 2000). The analysis by
Gurnis et al . (2000) determined an African dynamic topography in the range 300–
600 m and average uplift rates in the range 5–30 m Myr−1 . We applied each of the
trial buoyancy values generated in the previous Monte Carlo search to the African
mega-plume and then ran the same axisymmetric convection simulations as described
in Gurnis et al . (2000). Those simulations that predicted dynamic topography and
uplift rates consistent with the observational constraints were employed to construct
the smaller, shaded, histogram in figure 8. From this shaded histogram we note that
the probable buoyancy ratios, applicable to the African mega-plume, range from 0.1
to 0.3.
4. Time-dependent convection dynamics and the impact
of a high-viscosity peak in the deep mantle
The question we wish to address here is whether the viscosity-induced, low-pass filtering of flow in the deep lower mantle (§ 2 e) can produce a quadrupolar pattern of
heterogeneity that persists over long geological time-intervals. If so, does the convective flow give rise to an amplitude spectrum of thermal heterogeneity over time that
is similar to that of seismic tomography models in the deep lower mantle. Efforts
to explain the spectrum of heterogeneity in seismic tomography models with timedependent mantle-convection simulations are not new (e.g. Jarvis & Peltier 1986),
but significant progress has recently been made on the basis of a new generation
of convection models in 3D spherical geometry (e.g. Bunge & Richards 1996). To
explore the stability of deep-mantle quadrupolar convection we developed a thermalconvection model in 3D spherical geometry, which is based on the recent geodynamically inferred viscosity profiles. A complete discussion of all numerical aspects (and
tests) of this convection model may be found in Espesset (2001).
To illustrate the impact of radial viscosity variations, we carried out thermalconvection simulations using the following two viscosity models.
Model 1: employs a constant viscosity, η = 95 × 1021 Pa s, which is 10 times greater
than the mean (depth-averaged) viscosity inferred on the basis of Grand’s tomography model (solid line, figure 2).
Phil. Trans. R. Soc. Lond. A (2002)
depth (km)
Deep-mantle structure and convection dynamics
0
0
–600
–600
–1200
–1200
–1800
–1800
–2400
–2400
4
8
12
spherical harmonic degree
4
16
0.0
2537
8
12
0.5
16
1.0
Figure 9. Spectral heterogeneity map of lateral temperature variations. The lateral temperature
variations in the convection simulations have been expanded in terms of spherical harmonics, and
the RMS amplitude at each harmonic degree (horizontal axis) is plotted as a function of depth.
The spectral amplitudes in these maps have been normalized by the greatest overall amplitude.
(a) Spectral amplitudes of the temperature anomalies in model 1. (b) Spectral amplitudes of the
temperature anomalies in model 2.
Model 2: employs the viscosity profile inferred on the basis of Grand’s model (solid
line, figure 2), but with overall amplitude multiplied by 10, yielding a mean viscosity which is identical to that in model 1.
The viscosity values in both models 1 and 2 have been increased by a factor of
10 because the effective (mantle-averaged) Rayleigh number in both models would
otherwise be too high (ca. 108 ) to be resolved numerically with our current computational resources.
Both models include internal heating which is set to the value estimated by Stacey
(1992), that is 5.1 × 10−12 W kg−1 . Employing Boehler’s (1996) estimate of the temperature at the CMB, we assume a temperature increase ∆T = 3700 K across the
depth of the mantle. Both convection simulations are started at time t = 0, with
lateral temperature anomalies derived from Grand’s tomography model (solid line,
figure 4). For both models, the convection simulations are run for ca. 12 000 timesteps, corresponding to 20 Gyr for model 1, and 4 Gyr for model 2. In both cases, the
heat fluxes have not attained completely steady-state values. After 12 000 time-steps,
the surface heat flux is ca. 23 × 1012 W for model 1, and ca. 40 × 1012 W for model 2.
A useful summary of the spectral content of temperature heterogeneity in thermalconvection simulations is provided by spectral heterogeneity maps (SHMs) (e.g.
Bunge & Richards 1996). The SHM obtained by averaging over the last 5 Gyr in
the model-1 simulation, and by averaging over the last billion years in the model-2
simulation, is shown in figure 9. Both models are characterized by maximum spectral amplitude in the uppermost and lowermost mantle, signalling the presence of
horizontal thermal boundary layers (e.g. Jarvis & Peltier 1986). The isoviscous conPhil. Trans. R. Soc. Lond. A (2002)
2538
A. M. Forte, J. X. Mitrovica and A. Espesset
vection simulation (model 1) is characterized by spectral amplitudes distributed over
a broad range of wavelengths in the deep mantle.
In contrast, the spectrum for model 2 is much redder and is characterized by the
predominance of harmonic degree = 2 heterogeneity in the lower mantle. In the
uppermost mantle, both harmonic degrees 2 and 3 prevail. Of particular interest is
the strong dominance of degree 2 in the lowermost part of the lower mantle. All global
seismic tomography models show a clear dominance of degree-2 heterogeneity in the
lowermost mantle. Our convection simulations show that this quadrupolar structure,
characterized by antipodal hot mega-plumes below the Pacific and African plates,
is stable and controlled by the strong increase in viscosity at a depth of 2000 km
(cf. figure 2). In addition, these simulations suggest that this stable pattern can
persist over billion-year (model) time-scales.
5. Concluding remarks
Geochemical data present significant challenges in efforts to develop consistent, comprehensive models of time-dependent mantle convection. More specifically, the constraints provided by observational studies of mantle abundances and distributions of
rare-earth elements (U and Th) and noble gases (Ar, Xe and He) are a major concern.
The nature, interpretation and robustness of these constraints have been reviewed by
Albarède & van der Hilst (1999) and Helffrich & Wood (2001). These studies suggest
that as much as 55% of the mantle might have a primordial or undepleted chemical
composition. This conclusion is the basis of the long-held view that the mantle must
be chemically layered and it has motivated the development of some of the most
recent stratified-convection models (e.g. Kellogg et al . 1999; Davaille 1999).
Although the proposed fraction of mantle that may be undifferentiated seems fairly
consistent from one geochemical study to another, what are much less clear are the
topology, shape or geographical demarcation of the putative mantle reservoirs (e.g.
Manga 1996; Becker et al . 1999; Coltice & Ricard 1999; Helffrich & Wood 2001).
Indeed, there are viable geochemical models which are based on the possibility of
‘blobs’ of undifferentiated mantle rather than on continuous layers (e.g. Manga 1996;
Becker et al . 1999). The importance of topology, or size distribution, of the reservoirs
is emphasized by geodynamically constrained mantle-flow models, such as the one
presented in this study, which show significant transport of material across the bulk
of the mantle (figure 7), contrary to the notion of chemically stratified layers in the
mantle.
Some progress towards addressing this issue may be achieved by performing mixing
simulations with time-dependent thermal-convection models. Such calculations have
already been performed with simplified two-layer viscosity models in two-dimensional
cylindrical geometry (van Keken & Ballentine 1998). To date, mixing calculations
carried out in 3D spherical geometry have assumed a constant, time-independent,
mantle-flow pattern and have again assumed that the viscosity is constant throughout
the lower mantle (van Keken & Zhong 1999). Time-dependent mixing calculations in
a 3D spherical mantle with a high-viscosity peak near a depth of 2000 km (figure 2)
have not yet been performed.
The radial viscosity profile also plays a key role in geodynamic inversions for the
mantle-density structure (cf. § 2 d) and hence in determining the thermochemical
Phil. Trans. R. Soc. Lond. A (2002)
Deep-mantle structure and convection dynamics
2539
structure of the mantle. Moreover, from the perspective of time-dependent thermalconvection dynamics, the radial viscosity profile in the deep lower mantle appears to
exert fundamental control on the planform and stability of lateral heterogeneity of
temperature (cf. § 4). It is therefore important to establish whether our inference of
a high-viscosity peak at a depth of 2000 km, based on a suite of convection-related
observables, is robust.
To this end, Mitrovica & Forte (2002) recently carried out a new series of nonlinear
inversions of global convection data and glacial isostatic adjustment (GIA) data.
These simultaneous inversions of convection and GIA data differ from our previous
joint inversions (Forte & Mitrovica 1996; Mitrovica & Forte 1997) in two major
respects. First, Mitrovica & Forte (2002) included new GIA data associated with the
relaxation spectrum of Fennoscandian post-glacial uplift and also a set of site-specific
decay times for Hudson Bay and Scandinavia. Second, in the earlier inversions, the
only constraints on absolute values of viscosity were provided by the GIA data. In
contrast, Mitrovica & Forte (2002) included tectonic-plate velocities in the viscosity
inversions.
The Mitrovica & Forte (2002) inversions indicate that the absolute viscosity values
preferred by the GIA data are compatible with those required to explain present-day
plate-motion data. Most important, from the perspective of the present study, is that
Mitrovica & Forte (2002) obtained viscosity profiles that are again characterized by
a strong increase in viscosity near a depth of 2000 km. This feature of the mantleviscosity profile thus appears to be robust and it strengthens our conclusion that
this viscosity peak plays a major role in deep-mantle dynamics.
We acknowledge an anonymous reviewer for detailed comments. We thank C. Gaboret for preparing figure 7. This work is supported by grants from the Canadian National Science and Engineering Research Council. A.M.F. acknowledges the Canada Foundation for Innovation and
the Ontario Innovation Trust for supporting the computational infrastructure employed in this
study. J.X.M. and A.M.F. acknowledge support provided by the Canadian Institute for Advanced
Research—Earth Systems Evolution Program.
Note added in proof
The importance of geochemical constraints on the rheology, and hence the mixing
efficiency, of mantle convection is clearly discussed in Allègre (2002). Through consideration of effective stirring times for the sources of xenon- and lead-isotopic ratios,
Allègre suggests that the mantle viscosity above a depth of 400 km (i.e. the asthenosphere) should be much less than the effective viscosity of the transition-zone region
between depths of 400 and 670 km. This conclusion is well supported by the geodynamically inferred viscosity profiles (figure 2) and the more recent profiles inferred
by inverting convection and GIA datasets (Mitrovica & Forte 2002). The analysis of
instantaneous mixing efficiency presented by FM2001 shows that, for these viscosity
profiles, the convective mixing in the asthenosphere is at least an order of magnitude
greater than in the transition zone.
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