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Earth and Planetary Science Letters 184 (2001) 575^587
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Spherical shell models of mantle convection with
tectonic plates
Marc Monnereau *, Sandrine Quërë
UMR5562, CNRS, Observatoire Midi-Pyrënëes, 14 avenue Edouard Belin, 31400 Toulouse, France
Received 2 May 2000; received in revised form 16 October 2000; accepted 30 October 2000
Abstract
A simple three-dimensional spherical model of mantle convection, where plates are taken into account in the top
boundary condition, allows to investigate the plate tectonics^mantle convection coupling in a self-consistent way.
Avoiding the strong difficulties inherent in the numerical treatment of rheology, the plate condition appears efficient in
reproducing the Earth-like features as subduction, mid-oceanic ridges and hotspots. Whereas the free-slip condition
leads to a classical polygonal cell pattern with cylindrical hot plumes surrounded by downwellings, the plate condition
favors the development of strong linear downwellings associated to passive diverging zones along plate boundaries.
These cold currents, very similar to subductions, act the main role in mantle convection: they drive the whole
circulation. In that context, hot plumes remain almost independent, except if on the long term, cold material spreading
at the core surface induces a slight migration, below a few mm/yr, of their surface impingement. The main result is that
plate tectonics appear to be more than a simple mode of organization of the surface movements, it is the essence of the
Earth mantle dynamics. ß 2001 Elsevier Science B.V. All rights reserved.
Keywords: mantle; convection; plate tectonics; three-dimensional models; £uid dynamics
1. Introduction
For the last decade, three-dimensional spherical
models have improved our understanding of mantle dynamics. Thermal structure [1^3], rheological
e¡ects [4^6], or the spectacular pattern of layering
induced by an endothermic phase change [6^8]
have been extensively investigated. However, in
spite of progresses in numerical accuracy and description of mantle mineralogy, these models do
* Corresponding author. Tel.: +33-561-332-968;
Fax: +33-561-332-900; E-mail: [email protected]
not account for the ¢rst order tectonic features of
the Earth, subduction zones and mid-oceanic
ridges. As plates are integral parts of the mantle,
plate tectonics is an integral part of whole mantle
convection, not just a convective mode which can
be superimposed on models. A description of
mantle convection consistent with plate tectonics
is essential for our insight of the Earth mantle
dynamics.
The best way to reach this end seems to include
the complexities of the lithosphere rheology in
models. Besides the numerical di¤culties inherent
in this approach, it requires a choice of the rheological law enable to mimic ocean ridges, transform faults and trenches. A strong temperature
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dependence is clearly necessary, but not su¤cient.
Christensen and Harder [9] show that a plate-like
pattern emerges only with a non-Newtonian
rheology. However, using the expected parameters
for mantle rocks creep fails to yield signi¢cant
toroidal kinetic energy. A most successful platelike behavior is obtained with an ad hoc `self-lubricating' rheology, especially in the description
of strike^slip motion [10^14]. Although this is
the only rheologically self-consistent approach,
the great viscosity contrasts it involves raise serious numerical di¤culties which still restrict its
application to low Rayleigh numbers and small
cartesian domains [9,13^16] or to the two-dimensional shallow £uid layer formulation developed
by Bercovicci [10^12].
In fact, the plate tectonics problem addresses
two complementary kinds of questions: the selfgeneration of plates and the long-term coupling
between plates and mantle convection. If a rheological approach is essential for the former, it appears less central in the later case. For that purpose, a simpler way may be to assume pre-existing
plates, and so to specify the location of weak
zones or faults [17,18] or, in the extreme, to include plates in boundary conditions of numerical
models. It has the advantage of avoiding the numerical problems related to horizontal viscosity
contrasts. Such an approach was fruitful in the
reconstruction of plate velocities from tomography models converted in density ¢eld [19] and
has been extended to convection in cartesian geometry [20] and recently in spherical geometry
[21]. In this paper, we also propose to reach this
goal in spherical geometry. This approach clearly
sacri¢ces the ability to study the conditions of
plate emergence, but in return, its simplicity allows investigation on the long-term coupling between whole mantle convection and plate tectonics, in the range of realistic parameters.
2. Model set-up
The present model is substantially the same as
the one developed by Gable et al. [20] and is extensively described in spherical geometry by Ricard and Vigny [19]. It consists of plates overlying
a mantle with viscosity strati¢cation. In essence,
the plate motion balances the torque of stresses
induced by mantle convection and the torque of
stresses resisting the plate motion. The rotation of
each plate determines a surface velocity ¢eld that
will be used as top condition in the equation of
motion. This is quite di¡erent superimposing a
given velocity ¢eld as do Hager and O'Connell
[22] or Bunge and Richards [23]. In the latter
case, the prescribed velocities may be not consistent with the density ¢eld. It leads to a driven
convective £ow where movement can take place
in the absence of buoyancy forces and where energy may not balance. Conversely, this method
ensures that the interactions between plates and
£ow in the interior are not altered by external
applied forces. The coupling between plates and
convection is self-consistent : plates are driven by
the mantle £ow and, in return, modify its mass
distribution. This approach prescribes the plate
geometry. As a ¢rst step, we choose to ¢x the
boundaries preventing any changes there. This restriction, the strongest, is not inevitable still, mobile margins require rules, more obvious for
ridges than for trenches. Inside the boundaries
and over a thickness of 90 km, only a pure solid
rotation is authorized, equivalent assuming that
the strength of the lithosphere can support the
local variations in stress. There is no direct interaction between plates, but only through the
underlying mantle. When a plate moves, it generates poloidal and toroidal ¢elds inside the whole
mantle which transmit stresses to other plates.
The poloidal and toroidal equations are expanded
in spherical harmonics (up to degree and order
90) and solved by ¢nite di¡erences in the radial
direction (100 grid points). The temperature equation is solved by ¢nite volumes [24] and Alternate
Direction Inversion method over 100 by 180 by 90
grid points, so that the mesh area is 2 by 2³ and
around 30 km thick. Lastly, the computation remains in the Boussinesq approximation, without
phase change.
It is important to notice that the velocity ¢eld,
resulting from the torque balance, imposed at the
top surface is discontinuous at plate boundaries.
It leads to a non-integrable logarithmic stress singularity when the mesh size tends toward zero.
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The resulting e¡ect is that plate velocity depends
on the model resolution. Actually, stresses vanish
at plate boundaries because of the presence of
partially molten mantle below oceanic ridges or
faults elsewhere. In our model, the mesh size (2
by 2³) is large enough to reduce this e¡ect.
3. Main experiments
In order to highlight the e¡ects of tectonic
plates on mantle convection, we present three experiments where only the top condition of the
equation of motion varies, all the other parameters remaining unchanged: ¢rst a free-slip case as
a reference case, then a case with four schematic
plates and lastly a case with an Earth's-like plate
geometry. Hereafter, these three cases will be referred to as cases 1, 2 and 3, respectively. These
experiments are performed with a range of parameters as consistent as possible with the Earth's
context. The top and bottom temperatures are set
to 0³C and 2000³C, respectively. Clearly, the value chosen for the CMB temperature lies below the
lowermost estimate for the core surface temperature that ranges from 3800 K to 5000 K [25,26].
Removing 900 K related to the adiabatic gradient,
it leads to a temperature step through the whole
mantle between 2900 K and 4100 K. Since the
Rayleigh number is more dependent on the great
uncertainty on the bulk viscosity, the choice of a
2000 K step, whose reasons will be discussed in
Section 6, does not a¡ect the contents of the experiments in term of pattern and behavior of the
mantle convection. The viscosity above 670 km
depth is 1021 Pa s and 3U1022 Pa s below, what
realizes a viscosity increase by a factor 30 needed
to explain the relation between low degrees of
geoid an internal mass distribution [19]. Other
parameters are classical and lead to a Rayleigh
number based on the higher viscosity of
2.2U106 . The choice of internal heating intensity
is more crucial because it has a major impact on
the convection planform. An estimate may be deduced from global geophysical data, e.g. Stacey
and Loper [27]. If the total geothermal £ux is
42 TW [28] and the radiogenic heat of the
crust is 8 TW, 34 TW are released by the Earth's
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mantle convection. The most signi¢cant part of
this budget, 22 TW, is imputable to the contents
in radiogenic elements of the mantle (i.e.
5.5U10312 W/kg), the remaining part, 12 TW,
resulting from the cooling of the mantle (75^
80%) and the cooling plus the solidi¢cation of
core (20^25%). This last part only is accountable
for the basal heating. Accordingly the internal
heating of the mantle has two origins: its contents
in radiogenic sources and its cooling rate. With
these values, the basal heating for the Earth's
mantle ranges from 5% to 15%. So in experiments, we set the total internal heating to 30
TW, which appears as an intermediate value.
This ¢xes the non-dimension intensity of internal
heating to 41.8 and the associated Rayleigh number to 9.2U107 .
3.1. Free-slip case: case 1
As reference, a top free-slip case, case 1, is run.
It depicts (Fig. 1a,b) a classic pattern found in
spherical geometry: the £ow is driven by large
cylindrical hot plumes surrounded by a network
of downwelling sheets. The unusually low number
of upwellings results from the stepwise increase in
viscosity at 670 km depth. Such an increase in
lower mantle viscosity reddens the thermal heterogeneity spectrum [7]. It also favors the development of sheet-like and elongated downwellings.
This pattern is often highlighted because of its
resemblance to the Earth's mantle slabs. However, the analogy remains questionable. The similarity of both objects lies in their linear aspect,
but subduction zones do not form a closed curve
separating di¡erent cells of convection. This re£ects an essential di¡erence in their relations to
convection. Subduction, consisting of convergent
zones, is associated with diverging zones, ocean
ridges, comparable in size and shape. Actually,
slabs drive plate tectonics [19] and ridges appear
more or less passive. Conversely, the counterpart
of downwellings, in our free-slip case, is cylindrical upwellings whose surface expression is a diverging point. Besides, as the maximum velocities
are realized inside the plumes, the downwellings
behave like a return £ow.
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Fig. 1. Comparison between models of mantle convection with di¡erent top conditions. (a, b) Case 1: free-slip condition. (c, d)
Case 2: plate condition with four plates. Plate boundaries are shown in (c) (lines). (a, c) The lateral temperature variations at
700 km depth with surface velocity ¢eld. (b, d) Radial section of temperature and velocity ¢eld corresponding to the dashed line
in (a) and (c). Cases 1 and 2 di¡er only in the top condition. In contrast to case 1, downwellings, in case 2, are focused in a single open line at a plate margin (A) and spread at the core surface. Note the appearance of passive diverging lines (B) unrelated
to thermal structure.
3.2. Four plate case : case 2
Next, we investigate the in£uence of the plate
condition on mantle convection in a simple case
with only four plates; two polar and two equatorial (case 2). The substitution of the top condition
completely reorganizes the mantle dynamics. All
the downwellings focus under a plate boundary
(Fig. 1c,d), setting a single vigorous sinking £ow
whose surface area is much smaller than in the
previous case. As a consequence, the maximum
speed now occurs in this current and reaches twice
the speed inside plumes. The plumes themselves,
less vigorous than with free-slip, are swept away
by the cold £ow spreading out at the core surface.
This dynamical structure then remains stable, until thermal equilibrium is reached (in order to
compare cases with the Earth in terms of heat
£ow, plate velocities or geotherm, the computa-
tions are run over several billion years until the
di¡erence between the top and the basal heat £ow
equals the internal heating and the averaged thermal pro¢le reaches a steady state). The plate drift
also reveals the strong in£uence of the downwellings: the two plates bounding the converging
zone move fastest. Clearly, the downwellings drive
the dynamics, but the situation is not the opposite
of the free-slip case. In the presence of plates, the
converging zone connects with diverging zones
that are not related to plumes. Naturally, they
are located at plate boundaries, but do not correspond to a thermal anomaly ((B) in Fig. 1d). They
remain completely passive.
3.3. Earth's geometry plate case: case 3
The peculiar geometry and symmetry of plates
used in the previous case can mask a part of the
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Fig. 2. (a^d) Time evolution of case 3 (plate model with 15 plates). Except for the number of plates, all parameters are identical
to case 2. Characteristics of (a^d) are the same as Fig. 1c. The main features of case 2 are preserved: concentered linear downwellings (A), passive diverging lines (B) and independent plumes (C). Purely strike^slip zones (D) also appear. Note that the surface extent of downwellings decreases with time.
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complexity inherent in the coupling between
plates and convection. Accordingly, we now investigate the e¡ect of the current plate distribution (case 3). Of course, we do not expect to ¢nd
the present day plate motion, also related to the
mass distribution in the mantle and so to the past
evolution of plate tectonics. Surprisingly, the major features of case 2 are preserved in this last
experiment (Fig. 2). Here again, downwellings
form a continuous and open shape. Plumes stand
far from these and the diverging zones remain
passive. However, small di¡erences appear. The
converging line is now longer and may comprise
a branch (Fig. 2d). Sometimes, in calculations
started from di¡erent initial conditions, downwellings establish in two or three separate segments.
In fact, this dynamical structure remains in all the
experiments we run, despite varying the size, the
geometry, the number of plates, and even the viscosity pro¢le, the amount of internal heating or
the Rayleigh number. It appears to be a characteristic of the model with plates whatever the other conditions.
The stability of the dynamical structure found
in the four plate case becomes less marked as the
number of plates increases. The time evolution
depicted in Fig. 2 reveals an intense reorganization of downwellings with a strong reduction in
area during the ¢rst billion years. Then, the evolution slows down and downwellings go on moving slowly along the plate boundaries, toward a
triple junction structure (Fig. 2d), which may be
more stable than the previous ones. Note that this
sequence cannot mimic Earth's plate tectonic evolution whose essential factor is the mobility of
plate margins. The high degree of freedom of
plate geometry in this case also allows a pure
strike^slip zone to develop (Fig. 2c (D)). Its extent
represents only a small fraction of plate boundaries. Nor are purely diverging or converging
zones the dominant feature. Most boundaries
combine roughly equal toroidal and poloidal
movements. This is similar to the Earth, if we
consider that small transform faults along ridges
belong to a single diverging system oblique to the
spreading direction.
4. Additional experiments
If cases 1, 2 and 3 shed some light on the in£uence of the tectonic plates on mantle convection, they remain restricted to a single set of parameters. We now extend the investigation,
varying the viscosity pro¢le and the amount of
internal heating. Following our ¢rst approach,
for each new di¡erent set of parameters we
present a free-slip case and a case with plates.
We also compare constant viscosity cases with
layered viscosity cases where the viscosity increases by a factor of 30 beneath a depth of 650
km as in the previous set of experiments. In order
to isolate the dynamics of downwellings, the ¢rst
four experiments are run with 100% of internal
heating. The temperature ¢eld at 700 km depth
of these experiments is shown in Fig. 3a^d. The
free-slip layered viscosity case (Fig. 3c) features
the ability of viscosity increase with depth to promote elongated cold structures instead of pointlike downwellings characterizing the pure internally heated isoviscous case (Fig. 3a) [7,23]. This
contrasts with the similarity in the convection
planform depicted by plate cases (isoviscous in
Fig. 3b and layered viscosity in Fig. 3d). Nevertheless the in£uence of the depth increase in viscosity remains perceptible. In the layered viscosity
case (Fig. 3d), downwellings form a continuous
line located at plate boundaries as observed in
cases 2 and 3, whereas, in the isoviscous case
(Fig. 3b), they are divided in several segments.
Also, a few point-like structures persist beneath
plate interiors. In addition, the whole extension
of linear downwellings is reduced by a factor of
2 from the isoviscous case to the layered viscosity
case. This denotes that the viscosity increase with
depth has a focusing e¡ect on the convective
structures, the consequence of which being a reddening of the temperature spectrum.
The four following cases (Fig. 3^h) are performed with 85% of internal heating, corresponding to the lower estimate. Here again, the in£uence of viscosity strati¢cation is more conspicuous
with the free-slip condition where the focusing
mostly concerns the upwellings. Strong hot cylin-
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Fig. 3. Temperature ¢eld at 700 km depth. Top boundary condition is free-slip in left panels and with plates in right panels.
(a), (b), (c) and (d) correspond to cases without basal heating. (e), (f), (g) and (h) correspond to cases with 15% of basal heating.
(a), (b), (e) and (f) are isoviscous cases. In (c), (d), (g) and (h) the upper mantle viscosity has been reduced by a factor of 30.
Note that cases with plate conditions are less sensitive to the variation of the heating mode and to the viscosity pro¢le than the
free-slip cases.
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Fig. 4. SHM for the eight cases shown in Fig. 3 (panels have the same label in Figs. 3 and 4. Root mean square spectral amplitude is contoured as a function of mantle depth (vertical axis, surface at the top and CMB at the bottom) and spherical harmonic degree. Each panel has been normalized to the maximum amplitude. There are 10 contour intervals.
drical plumes develop when the viscosity step is
introduced. The same feature may be noted on
case with plates, where hot plumes are only observed in the case with layered viscosity. Note that
no in£uence on the downwellings appears clearly.
The focusing e¡ect on hot plumes has to be related to the strong cooling of the mantle induced
by the viscosity increase with depth [29], which
enhances the buoyancy of hot thermal structures.
This is a strong argument in favor of the presence
of a viscosity increase with depth in the Earth's
mantle. As discussed before, the internal heating
in the Earth's context would amount between
85% and 95%, inhibiting the development of hot
plumes in case of constant viscosity.
Besides this corollary concerning the role of
depth-dependent viscosity, there is a simple remark which raises from the juxtaposition of the
eight cases of Fig. 3. While the di¡erent conditions, basal heating or not, layered viscosity or
not, strongly a¡ect the convection planform in
the free-slip cases, the cases with plates exhibit
the same features: the dynamics are driven by
strong downwellings, preserving a continuous linear shape from one case to the other.
Fig. 4 displays the spectral heterogeneity map
(SHM) of these eight cases, drawing contours of
spectral root mean square amplitude as function
of depth and spherical harmonic degrees. The
free-slip isoviscous cases (Fig. 4a,e) are dominated
by short wavelength. The reddening induced by
the depth-dependent viscosity [23] appears clearly
for the SHM of the layered viscosity free-slip case
with pure internal heating (Fig. 4c) and seems
enhanced by a small amount of basal heating
(Fig. 4g). Cases with plates present a similar
SHM feature for the upper mantle, exhibiting a
strong component at low degrees up to 10 (Fig.
4b,d,f,h). For the very low degrees up to 5, the
signal extends throughout the mantle except in
the pure internal heated isoviscous case (Fig.
4b), where there is no signi¢cant amplitude in
the lower mantle. Bunge and Richards [23] have
shown that viscosity increase with depth is a way
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to transport in the deep mantle a signi¢cant part
of thermal heterogeneity produced at low degrees
by the plate tectonics in the upper mantle, and so
appears to be an important factor in producing
SHM ¢tting that is determined by seismic tomography. Here, we show that basal heating may also
produce signi¢cant thermal heterogeneities at low
degrees and great depth, even with a constant
viscosity. Although SHM is useful to characterize
the convection in terms of wavelength, it remains
ambiguous. For instance, while the convection
planform of the free-slip and plate layered viscosity cases with 85% of internal heating strongly
di¡er (Fig. 3g,h), their associated SHMs (Fig.
4g,h) display similar features. As a matter of
fact, SHMs reveal only an aspect of mantle convection and have to be associated to other observables featuring the Earth's mantle dynamics as the
dynamical structures, poloidal^toroidal partitioning, heat £ow or geotherm.
583
remains furtive during the reorganization of plate
velocities.
Another di¡erence lies in the puzzling weak
number of plumes. This feature is not a characteristic of the only plate model, but still holds in the
free-slip case or in various studies previously published [1,30]. Actually, if hotspots identify by their
geochemical signature, most of them di¡er from
the Hawaii paradigm in size, activity or track.
Broad hotspots, as Hawaii or Iceland, are obviously rare. Our model may account for the largest
ones, but clearly not for the tens observed on the
Earth.
On the other hand, other plume features remain. Fig. 5 reveals that the interaction between
a plume and a moving plate remains localized at
the base of the lithosphere. The plume ascends
vertically. Only its head is swept downstream, as
expected for the Hawaiian hotspot [31,32]. Note
the dynamical erosion of the thermal boundary
5. Comparison with the Earth
5.1. Dynamical structures
Obviously, the dynamical structures naturally
developed with plate condition, intense linear
downwellings, passive diverging zones and
plumes, are strikingly reminiscent of slabs, ridges
and hotspots. However, some di¡erences do appear clearly. Earth's subduction zones are notably
asymmetric with one plate plunging beneath another, while at each converging zone in cases 2 or
3, two plates dive and collapse in a same downwelling. The same behavior was also obtained by
Zhong et al. [21] where plates are included in a
similar way: boundaries are assumed purely £uid
and so cannot capture the essence of faults separating plates as in Zhong and Gurnis [18]. Consequently, the nearly equal partitioning in plate
margin of ridges and subduction observable on
the Earth is not present in cases 2 and 3 where
diverging zones prevail, except at the beginning of
case 3 (Fig. 2a) where some downwellings between
plates converging at di¡erent rates are tilted by
the sublithospheric mantle £ow. This feature disappears as the thermal equilibrium establishes and
Fig. 5. Radial section of the thermal and velocity ¢eld of
case 3 at 5.6 Gyr (Fig. 2d), (A), (B) and (C) corresponding
to (A), (B) and (C) in Fig. 2d. Note the absence of thermal
anomaly at the diverging zone (B), the thickening of the
thermal boundary layer from (B) to (C), the lithospheric erosion by the upwelling impinging the surface (C) and then the
re-thickening of the thermal boundary layer. Note also the
large scale £ow from (B) to (A) in the upper mantle and in
the reverse way in the lower mantle.
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layer (Fig. 5C) which may account for the topographic swell developing around a hotspot [33]. In
fact, plumes are less a¡ected by plate motion than
by the large scale £ow standing in the high-viscosity layer (below 670 km depth). The velocity ¢eld
in the low-viscosity layer is strongly correlated
with plate motion, but the underlying mantle
£ows in a di¡erent way, setting up large cells between downwellings and diverging zones (Fig. 5).
These deep currents drag the plumes below the
diverging zones. Their surface drift can reach
2000 km in a billion years, so that, when the experiment reaches thermal equilibrium, after ¢ve or
six billion years, most of plumes have been captured by the diverging zones. This situation is not
relevant for the Earth, except perhaps for the Iceland or Azores hotspots where the ridge migration
speed is small in the hotspot reference frame. This
re£ects the shortcoming of the model in which the
plate boundaries remain ¢xed. In reality, they
change continuously, particularly the ridges which
may migrate as fast as the plates. In case 3, the
plume motion and the average plate velocity fall
in the range of the values for the Earth, i.e.
2 mm/yr and 4 cm/yr, respectively, so that the
plumes in our model can be regarded as ¢xed
with respect to plate drift. This is a feature well
highlighted in Zhong et al. [21].
5.2. Poloidal versus toroidal components
An other element of comparison is the partitioning of kinetic energy between the poloidal
and the toroidal components. Conversely in
most models with variable viscosity, the plate condition naturally leads to a strong toroidal component. The ratio of toroidal to poloidal component
is 0.67 in the four plate case. In case 3, this ratio
reaches 0.8, close to the value observed for the
Earth. This agreement is not surprising. O'Connell et al. [34] presume that, since the toroidal
velocities are not involved in the heat transfer,
the convective process develops a dynamical
structure reducing the loss of kinetic energy, so
that the partitioning essentially depends on the
plate distribution. Besides the partitioning, this
assumption also explains the notable stability of
the thermal structure during over several billion
years in case 2, where the small number of plates
and the symmetry of shape strongly restrict the
possible solutions. Actually, varying several parameters, we ¢nd only two. Further, in case 3,
where the plate distribution is more complex,
the dynamical structure evolves slowly, with a toroidal/poloidal ratio £uctuating around a mean
value (0.66 at 0.4 Gyr, 0.79 at 1.1 Gyr, 0.75 at
2.1 Gyr, 0.78 at 5.6 Gyr, and 0.80 at 6.5 Gyr). A
most important feature related to the time evolution of the thermal structure is that the dynamics
tend to a strong degree one. This is observable on
the case with four plates but also on all cases with
15 plates except on the one performed with 100%
of internal heating and constant viscosity. Cold
downwellings converge toward a single thermal
structure. This feature may be seen as the reason
of the continent gathering that periodically occurs. Of course, because of the lack of a complete
modelling of plate tectonics in the model, this
only remains as a possible inference of the physical process that drives the long-term evolution of
plate tectonics.
5.3. Surface velocity, heat £ow and geotherm
The impact of tectonic plates on dynamics is
not restricted to the convective structures, but
also a¡ects integral quantities as the mean surface
velocity, the heat £ow, and the mean temperature
pro¢le. Besides the developing of a strong rotational component, plates also alter the mean surface speed. In case 1, it reaches 5.6 cm/yr, slows to
4.1 cm/yr in case 3 and to 3.5 cm/yr in the four
plate case, all parameters being identical in the
three cases. The smaller the number of plates,
the less vigorously the convection. The plate condition appears intermediate between two asymptotic conditions, no-slip and free-slip, equivalent
to a single plate and to an in¢nite number of
plates, respectively. On Fig. 6, the Nusselt number
is plotted as a function of the Rayleigh number
for four di¡erent surface conditions : free-slip, 15
plates, four plates and one plate. The plate design
for four and 15 plate cases are the same as for
cases 2 and 3, respectively. The viscosity pro¢le
used is the same as in the cases 1, 2 and 3, and
there is no internal heating. The plot depicts a
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Fig. 6. Nusselt number as a function of the Rayleigh number
for four di¡erent plate conditions. The experiment has been
performed with no internal heating. There is a stepwise increase in viscosity with depth by a factor of 30 at 650 km
depth. The Rayleigh number is based on the lower mantle
viscosity.
classic power law relationship between the Nusselt
number and the Rayleigh number. As expected,
the exponent increases from the single plate condition up to the free-slip condition, the four plates
and the 15 plates conditions corresponding to intermediate values. Even if the exponents for four
and 15 plates conditions di¡er, they remain very
close indicating only a small in£uence of the number of plate, and so of the plate geometry on the
Nusselt^Rayleigh relationship. This behavior is
also observed in cases with internal heating, convection providing 40 TW in case 1, 35 TW in case
3, and 33 TW in case 2 (consequently the basal
heating represents 25%, 14%, and 9%, respectively).
The mean temperature pro¢le does not escape
the plate's in£uence. As expected, the mantle is
hotter in cases with plates than with free-slip condition (Fig. 7). The temperature drop through the
top boundary layer spans 47% of the temperature
range in case 1, 72% in case 3 and 82% in case 2.
More surprising is the strong temperature inversion observed in cases with plate. It results from
the combination of two e¡ects: (1) the concentra-
585
tion of downwellings makes it more di¤cult to
reheat cold material piling up at the core surface,
(2) most plume material, swept away by the plate
movement, is unable to reach the surface and remains insulated below the thermal boundary
layer.
The experimental parameters for case 3 are designed to ¢t Earth's observables. If the heat £ow
and the surface velocities derived from the freeslip case remain comparable to the Earth's values
(i.e. 34 TW and 4 cm/yr), the temperatures are
clearly unrealistic : 600³C in cold currents at 670
km depth, 940³C just below the thermal boundary
layer and only 1300³C in plumes at 200 km depth.
As a comparison, case 3 gives 850³C, 1400³C and
1700³C, respectively, the estimation for the Earth
ranges around 700³C [35,36], 1350³C [37] and
1600³C [38].
6. Concluding remarks
It is important to notice that the sublithospheric temperature is known with a small uncertainty.
It is deduced from two independent observables:
the heat £ow, topography and age relationship in
oceanic domains, and the geochemical composition of mid-ocean ridge basalts. It is one of the
strongest constraints on the Earth's geotherm.
Thus a one layer convection established in the
Fig. 7. Averaged temperature pro¢le in cases 1, 2 and 3. In
cases with plates, the mantle is hotter and the pro¢les are
clearly marked by a temperature inversion.
EPSL 5687 4-1-01 Cyaan Magenta Geel Zwart
586
M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587
Earth's mantle is only consistent with a very low
temperature at the CMB, close to the value set in
case 3, i.e. below 3200 K corresponding to a 2000
K super-adiabatic temperature step throughout
the whole mantle. This is 500 K colder than the
lower estimate [25,26]. A greater viscosity step
should reduce the mean mantle temperature allowing higher CMB temperatures, but also should
increase the heat £ux from the core up to unrealistic values. On the other hand, the endothermic
phase change responsible for the 650 km depth
seismic discontinuity does not appear to be able
to stratify the convection in a two layer mode
with a signi¢cant thermal boundary layer at the
phase change [7,39]. A more promising way to
conciliate the paradox existing between the sublithospheric temperature, the CMB temperature
and the small heat £ux at the core surface, perhaps lies in the existence of geochemical strati¢cation of the mantle as proposed by Davaille [40]
and Kellogg et al. [41]. As a matter of fact, a
stable deep layer with a high content in radiogenic
elements would insulate the core, preventing a
great heat £ux and would favor the creation of
more hot plume than in our models.
We have shown that the simple introduction of
a piecewise continuous surface generates features
that look like Earth's plate tectonics. Beyond this
spectacular reorganization, the in£uence of plates
on the inner structure of convection and on global
quantities as heat £ow or geotherm highlights the
inseparable character of plate tectonics and Earth
mantle convection. Clearly plate tectonics is not a
simple consequence of the mantle convection but
the mantle dynamics mode acting on the Earth.
This casts doubts on the applicability of freeslip or no-slip models of convection to the Earth.
These classical approaches are perhaps more relevant for the dynamics of Venus where the surface temperature, 500³C, limits the strength of the
lithosphere, and distributes the surface deformation. The topography of this planet is characterized by more or less circular highlands surrounded by plains, similar to the pattern of
dynamical topography generated by free-slip
spherical models of convection.
Acknowledgements
We thank G. Ceuleneer, K. Feigl and M. Rabinowicz for helpful discussion. Useful suggestions and comments by Yanick Ricard and by
an anonymous reviewer are gratefully acknowledged. This work was supported by a grant
from the Institut des Sciences de l'Univers. Computing resources were provided by the Centre National d'Etudes Spatiales.[AC]
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