Spherical shell models of mantle convection with tectonic plates
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Earth and Planetary Science Letters 184 (2001) 575^587
www.elsevier.com/locate/epsl
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 not account for the ¢rst order tectonic features of
the Earth, subduction zones and mid-oceanic
For the last decade, three-dimensional spherical ridges. As plates are integral parts of the mantle,
models have improved our understanding of man- plate tectonics is an integral part of whole mantle
tle dynamics. Thermal structure [1^3], rheological convection, not just a convective mode which can
e¡ects [4^6], or the spectacular pattern of layering be superimposed on models. A description of
induced by an endothermic phase change [6^8] mantle convection consistent with plate tectonics
have been extensively investigated. However, in is essential for our insight of the Earth mantle
spite of progresses in numerical accuracy and de- dynamics.
scription of mantle mineralogy, these models do 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 rheo-
* Corresponding author. Tel.: +33-561-332-968; logical law enable to mimic ocean ridges, trans-
Fax: +33-561-332-900; E-mail: marc.monnereau@cnes.fr form faults and trenches. A strong temperature
0012-821X / 01 / $ ^ see front matter ß 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 0 1 2 - 8 2 1 X ( 0 0 ) 0 0 3 3 4 - 4
dekorasjon
EPSL 5687 4-1-01 dagbok Geel Zwart
Cyaan Magenta
576 M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587
dependence is clearly necessary, but not su¤cient. a mantle with viscosity strati¢cation. In essence,
Christensen and Harder [9] show that a plate-like the plate motion balances the torque of stresses
pattern emerges only with a non-Newtonian induced by mantle convection and the torque of
rheology. However, using the expected parameters stresses resisting the plate motion. The rotation of
for mantle rocks creep fails to yield signi¢cant each plate determines a surface velocity ¢eld that
toroidal kinetic energy. A most successful plate- will be used as top condition in the equation of
like behavior is obtained with an ad hoc `self-lu- motion. This is quite di¡erent superimposing a
bricating' rheology, especially in the description given velocity ¢eld as do Hager and O'Connell
of strike^slip motion [10^14]. Although this is [22] or Bunge and Richards [23]. In the latter
the only rheologically self-consistent approach, case, the prescribed velocities may be not consis-
the great viscosity contrasts it involves raise seri- tent with the density ¢eld. It leads to a driven
ous numerical di¤culties which still restrict its convective £ow where movement can take place
application to low Rayleigh numbers and small in the absence of buoyancy forces and where en-
cartesian domains [9,13^16] or to the two-dimen- ergy may not balance. Conversely, this method
sional shallow £uid layer formulation developed ensures that the interactions between plates and
by Bercovicci [10^12]. £ow in the interior are not altered by external
In fact, the plate tectonics problem addresses applied forces. The coupling between plates and
two complementary kinds of questions: the self- convection is self-consistent : plates are driven by
generation of plates and the long-term coupling the mantle £ow and, in return, modify its mass
between plates and mantle convection. If a rheo- distribution. This approach prescribes the plate
logical approach is essential for the former, it ap- geometry. As a ¢rst step, we choose to ¢x the
pears less central in the later case. For that pur- boundaries preventing any changes there. This re-
pose, a simpler way may be to assume pre-existing striction, the strongest, is not inevitable still, mo-
plates, and so to specify the location of weak bile margins require rules, more obvious for
zones or faults [17,18] or, in the extreme, to in- ridges than for trenches. Inside the boundaries
clude plates in boundary conditions of numerical and over a thickness of 90 km, only a pure solid
models. It has the advantage of avoiding the nu- rotation is authorized, equivalent assuming that
merical problems related to horizontal viscosity the strength of the lithosphere can support the
contrasts. Such an approach was fruitful in the local variations in stress. There is no direct inter-
reconstruction of plate velocities from tomogra- action between plates, but only through the
phy models converted in density ¢eld [19] and underlying mantle. When a plate moves, it gener-
has been extended to convection in cartesian ge- ates poloidal and toroidal ¢elds inside the whole
ometry [20] and recently in spherical geometry mantle which transmit stresses to other plates.
[21]. In this paper, we also propose to reach this The poloidal and toroidal equations are expanded
goal in spherical geometry. This approach clearly in spherical harmonics (up to degree and order
sacri¢ces the ability to study the conditions of 90) and solved by ¢nite di¡erences in the radial
plate emergence, but in return, its simplicity al- direction (100 grid points). The temperature equa-
lows investigation on the long-term coupling be- tion is solved by ¢nite volumes [24] and Alternate
tween whole mantle convection and plate tecto- Direction Inversion method over 100 by 180 by 90
nics, in the range of realistic parameters. grid points, so that the mesh area is 2 by 2³ and
around 30 km thick. Lastly, the computation re-
mains in the Boussinesq approximation, without
2. Model set-up phase change.
It is important to notice that the velocity ¢eld,
The present model is substantially the same as resulting from the torque balance, imposed at the
the one developed by Gable et al. [20] and is ex- top surface is discontinuous at plate boundaries.
tensively described in spherical geometry by Ri- It leads to a non-integrable logarithmic stress sin-
card and Vigny [19]. It consists of plates overlying gularity when the mesh size tends toward zero.
EPSL 5687 4-1-01 Cyaan Magenta Geel Zwart
M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587 577
The resulting e¡ect is that plate velocity depends mantle convection. The most signi¢cant part of
on the model resolution. Actually, stresses vanish this budget, 22 TW, is imputable to the contents
at plate boundaries because of the presence of in radiogenic elements of the mantle (i.e.
partially molten mantle below oceanic ridges or 5.5U10312 W/kg), the remaining part, 12 TW,
faults elsewhere. In our model, the mesh size (2 resulting from the cooling of the mantle (75^
by 2³) is large enough to reduce this e¡ect. 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
3. Main experiments heating of the mantle has two origins: its contents
in radiogenic sources and its cooling rate. With
In order to highlight the e¡ects of tectonic these values, the basal heating for the Earth's
plates on mantle convection, we present three ex- mantle ranges from 5% to 15%. So in experi-
periments where only the top condition of the ments, we set the total internal heating to 30
equation of motion varies, all the other parame- TW, which appears as an intermediate value.
ters remaining unchanged: ¢rst a free-slip case as This ¢xes the non-dimension intensity of internal
a reference case, then a case with four schematic heating to 41.8 and the associated Rayleigh num-
plates and lastly a case with an Earth's-like plate ber to 9.2U107 .
geometry. Hereafter, these three cases will be re-
ferred to as cases 1, 2 and 3, respectively. These 3.1. Free-slip case: case 1
experiments are performed with a range of pa-
rameters as consistent as possible with the Earth's As reference, a top free-slip case, case 1, is run.
context. The top and bottom temperatures are set It depicts (Fig. 1a,b) a classic pattern found in
to 0³C and 2000³C, respectively. Clearly, the val- spherical geometry: the £ow is driven by large
ue chosen for the CMB temperature lies below the cylindrical hot plumes surrounded by a network
lowermost estimate for the core surface temper- of downwelling sheets. The unusually low number
ature that ranges from 3800 K to 5000 K [25,26]. of upwellings results from the stepwise increase in
Removing 900 K related to the adiabatic gradient, viscosity at 670 km depth. Such an increase in
it leads to a temperature step through the whole lower mantle viscosity reddens the thermal hetero-
mantle between 2900 K and 4100 K. Since the geneity spectrum [7]. It also favors the develop-
Rayleigh number is more dependent on the great ment of sheet-like and elongated downwellings.
uncertainty on the bulk viscosity, the choice of a This pattern is often highlighted because of its
2000 K step, whose reasons will be discussed in resemblance to the Earth's mantle slabs. How-
Section 6, does not a¡ect the contents of the ex- ever, the analogy remains questionable. The sim-
periments in term of pattern and behavior of the ilarity of both objects lies in their linear aspect,
mantle convection. The viscosity above 670 km but subduction zones do not form a closed curve
depth is 1021 Pa s and 3U1022 Pa s below, what separating di¡erent cells of convection. This re-
realizes a viscosity increase by a factor 30 needed £ects an essential di¡erence in their relations to
to explain the relation between low degrees of convection. Subduction, consisting of convergent
geoid an internal mass distribution [19]. Other zones, is associated with diverging zones, ocean
parameters are classical and lead to a Rayleigh ridges, comparable in size and shape. Actually,
number based on the higher viscosity of slabs drive plate tectonics [19] and ridges appear
2.2U106 . The choice of internal heating intensity more or less passive. Conversely, the counterpart
is more crucial because it has a major impact on of downwellings, in our free-slip case, is cylindri-
the convection planform. An estimate may be de- cal upwellings whose surface expression is a di-
duced from global geophysical data, e.g. Stacey verging point. Besides, as the maximum velocities
and Loper [27]. If the total geothermal £ux is are realized inside the plumes, the downwellings
42 TW [28] and the radiogenic heat of the behave like a return £ow.
crust is 8 TW, 34 TW are released by the Earth's
EPSL 5687 4-1-01 Cyaan Magenta Geel Zwart
578 M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587
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 sin-
gle 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 tions are run over several billion years until the
di¡erence between the top and the basal heat £ow
Next, we investigate the in£uence of the plate equals the internal heating and the averaged ther-
condition on mantle convection in a simple case mal pro¢le reaches a steady state). The plate drift
with only four plates; two polar and two equato- also reveals the strong in£uence of the downwel-
rial (case 2). The substitution of the top condition lings: the two plates bounding the converging
completely reorganizes the mantle dynamics. All zone move fastest. Clearly, the downwellings drive
the downwellings focus under a plate boundary the dynamics, but the situation is not the opposite
(Fig. 1c,d), setting a single vigorous sinking £ow of the free-slip case. In the presence of plates, the
whose surface area is much smaller than in the converging zone connects with diverging zones
previous case. As a consequence, the maximum that are not related to plumes. Naturally, they
speed now occurs in this current and reaches twice are located at plate boundaries, but do not corre-
the speed inside plumes. The plumes themselves, spond to a thermal anomaly ((B) in Fig. 1d). They
less vigorous than with free-slip, are swept away remain completely passive.
by the cold £ow spreading out at the core surface.
This dynamical structure then remains stable, un- 3.3. Earth's geometry plate case: case 3
til thermal equilibrium is reached (in order to
compare cases with the Earth in terms of heat The peculiar geometry and symmetry of plates
£ow, plate velocities or geotherm, the computa- used in the previous case can mask a part of the
EPSL 5687 4-1-01 Cyaan Magenta Geel ZwartM. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587 579
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 down-
wellings (A), passive diverging lines (B) and independent plumes (C). Purely strike^slip zones (D) also appear. Note that the sur-
face extent of downwellings decreases with time.
EPSL 5687 4-1-01 Cyaan Magenta Geel Zwart580 M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587
complexity inherent in the coupling between 4. Additional experiments
plates and convection. Accordingly, we now in-
vestigate the e¡ect of the current plate distribu- If cases 1, 2 and 3 shed some light on the in-
tion (case 3). Of course, we do not expect to ¢nd £uence of the tectonic plates on mantle convec-
the present day plate motion, also related to the tion, they remain restricted to a single set of pa-
mass distribution in the mantle and so to the past rameters. We now extend the investigation,
evolution of plate tectonics. Surprisingly, the ma- varying the viscosity pro¢le and the amount of
jor features of case 2 are preserved in this last internal heating. Following our ¢rst approach,
experiment (Fig. 2). Here again, downwellings for each new di¡erent set of parameters we
form a continuous and open shape. Plumes stand present a free-slip case and a case with plates.
far from these and the diverging zones remain We also compare constant viscosity cases with
passive. However, small di¡erences appear. The layered viscosity cases where the viscosity in-
converging line is now longer and may comprise creases by a factor of 30 beneath a depth of 650
a branch (Fig. 2d). Sometimes, in calculations km as in the previous set of experiments. In order
started from di¡erent initial conditions, downwel- to isolate the dynamics of downwellings, the ¢rst
lings establish in two or three separate segments. four experiments are run with 100% of internal
In fact, this dynamical structure remains in all the heating. The temperature ¢eld at 700 km depth
experiments we run, despite varying the size, the of these experiments is shown in Fig. 3a^d. The
geometry, the number of plates, and even the vis- free-slip layered viscosity case (Fig. 3c) features
cosity pro¢le, the amount of internal heating or the ability of viscosity increase with depth to pro-
the Rayleigh number. It appears to be a charac- mote elongated cold structures instead of point-
teristic of the model with plates whatever the oth- like downwellings characterizing the pure inter-
er conditions. nally heated isoviscous case (Fig. 3a) [7,23]. This
The stability of the dynamical structure found contrasts with the similarity in the convection
in the four plate case becomes less marked as the planform depicted by plate cases (isoviscous in
number of plates increases. The time evolution Fig. 3b and layered viscosity in Fig. 3d). Never-
depicted in Fig. 2 reveals an intense reorganiza- theless the in£uence of the depth increase in vis-
tion of downwellings with a strong reduction in cosity remains perceptible. In the layered viscosity
area during the ¢rst billion years. Then, the evo- case (Fig. 3d), downwellings form a continuous
lution slows down and downwellings go on mov- line located at plate boundaries as observed in
ing slowly along the plate boundaries, toward a cases 2 and 3, whereas, in the isoviscous case
triple junction structure (Fig. 2d), which may be (Fig. 3b), they are divided in several segments.
more stable than the previous ones. Note that this Also, a few point-like structures persist beneath
sequence cannot mimic Earth's plate tectonic evo- plate interiors. In addition, the whole extension
lution whose essential factor is the mobility of of linear downwellings is reduced by a factor of
plate margins. The high degree of freedom of 2 from the isoviscous case to the layered viscosity
plate geometry in this case also allows a pure case. This denotes that the viscosity increase with
strike^slip zone to develop (Fig. 2c (D)). Its extent depth has a focusing e¡ect on the convective
represents only a small fraction of plate bound- structures, the consequence of which being a red-
aries. Nor are purely diverging or converging dening of the temperature spectrum.
zones the dominant feature. Most boundaries The four following cases (Fig. 3^h) are per-
combine roughly equal toroidal and poloidal formed with 85% of internal heating, correspond-
movements. This is similar to the Earth, if we ing to the lower estimate. Here again, the in£u-
consider that small transform faults along ridges ence of viscosity strati¢cation is more conspicuous
belong to a single diverging system oblique to the with the free-slip condition where the focusing
spreading direction. mostly concerns the upwellings. Strong hot cylin-
EPSL 5687 4-1-01 Cyaan Magenta Geel ZwartM. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587 581
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.
EPSL 5687 4-1-01 Cyaan Magenta Geel Zwart582 M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587
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 ampli-
tude is contoured as a function of mantle depth (vertical axis, surface at the top and CMB at the bottom) and spherical harmon-
ic degree. Each panel has been normalized to the maximum amplitude. There are 10 contour intervals.
drical plumes develop when the viscosity step is the same features: the dynamics are driven by
introduced. The same feature may be noted on strong downwellings, preserving a continuous lin-
case with plates, where hot plumes are only ob- ear shape from one case to the other.
served in the case with layered viscosity. Note that Fig. 4 displays the spectral heterogeneity map
no in£uence on the downwellings appears clearly. (SHM) of these eight cases, drawing contours of
The focusing e¡ect on hot plumes has to be re- spectral root mean square amplitude as function
lated to the strong cooling of the mantle induced of depth and spherical harmonic degrees. The
by the viscosity increase with depth [29], which free-slip isoviscous cases (Fig. 4a,e) are dominated
enhances the buoyancy of hot thermal structures. by short wavelength. The reddening induced by
This is a strong argument in favor of the presence the depth-dependent viscosity [23] appears clearly
of a viscosity increase with depth in the Earth's for the SHM of the layered viscosity free-slip case
mantle. As discussed before, the internal heating with pure internal heating (Fig. 4c) and seems
in the Earth's context would amount between enhanced by a small amount of basal heating
85% and 95%, inhibiting the development of hot (Fig. 4g). Cases with plates present a similar
plumes in case of constant viscosity. SHM feature for the upper mantle, exhibiting a
Besides this corollary concerning the role of strong component at low degrees up to 10 (Fig.
depth-dependent viscosity, there is a simple re- 4b,d,f,h). For the very low degrees up to 5, the
mark which raises from the juxtaposition of the signal extends throughout the mantle except in
eight cases of Fig. 3. While the di¡erent condi- the pure internal heated isoviscous case (Fig.
tions, basal heating or not, layered viscosity or 4b), where there is no signi¢cant amplitude in
not, strongly a¡ect the convection planform in the lower mantle. Bunge and Richards [23] have
the free-slip cases, the cases with plates exhibit shown that viscosity increase with depth is a way
EPSL 5687 4-1-01 Cyaan Magenta Geel ZwartM. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587 583
to transport in the deep mantle a signi¢cant part remains furtive during the reorganization of plate
of thermal heterogeneity produced at low degrees velocities.
by the plate tectonics in the upper mantle, and so Another di¡erence lies in the puzzling weak
appears to be an important factor in producing number of plumes. This feature is not a character-
SHM ¢tting that is determined by seismic tomog- istic of the only plate model, but still holds in the
raphy. Here, we show that basal heating may also free-slip case or in various studies previously pub-
produce signi¢cant thermal heterogeneities at low lished [1,30]. Actually, if hotspots identify by their
degrees and great depth, even with a constant geochemical signature, most of them di¡er from
viscosity. Although SHM is useful to characterize the Hawaii paradigm in size, activity or track.
the convection in terms of wavelength, it remains Broad hotspots, as Hawaii or Iceland, are obvi-
ambiguous. For instance, while the convection ously rare. Our model may account for the largest
planform of the free-slip and plate layered viscos- ones, but clearly not for the tens observed on the
ity cases with 85% of internal heating strongly Earth.
di¡er (Fig. 3g,h), their associated SHMs (Fig. On the other hand, other plume features re-
4g,h) display similar features. As a matter of main. Fig. 5 reveals that the interaction between
fact, SHMs reveal only an aspect of mantle con- a plume and a moving plate remains localized at
vection and have to be associated to other observ- the base of the lithosphere. The plume ascends
ables featuring the Earth's mantle dynamics as the vertically. Only its head is swept downstream, as
dynamical structures, poloidal^toroidal partition- expected for the Hawaiian hotspot [31,32]. Note
ing, heat £ow or geotherm. 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 ap-
pear clearly. Earth's subduction zones are notably
asymmetric with one plate plunging beneath an-
other, while at each converging zone in cases 2 or
3, two plates dive and collapse in a same down-
welling. 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 sepa-
rating plates as in Zhong and Gurnis [18]. Con-
sequently, the nearly equal partitioning in plate Fig. 5. Radial section of the thermal and velocity ¢eld of
margin of ridges and subduction observable on case 3 at 5.6 Gyr (Fig. 2d), (A), (B) and (C) corresponding
the Earth is not present in cases 2 and 3 where to (A), (B) and (C) in Fig. 2d. Note the absence of thermal
diverging zones prevail, except at the beginning of anomaly at the diverging zone (B), the thickening of the
case 3 (Fig. 2a) where some downwellings between thermal boundary layer from (B) to (C), the lithospheric ero-
sion by the upwelling impinging the surface (C) and then the
plates converging at di¡erent rates are tilted by re-thickening of the thermal boundary layer. Note also the
the sublithospheric mantle £ow. This feature dis- large scale £ow from (B) to (A) in the upper mantle and in
appears as the thermal equilibrium establishes and the reverse way in the lower mantle.
EPSL 5687 4-1-01 Cyaan Magenta Geel Zwart584 M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587
layer (Fig. 5C) which may account for the topo- years in case 2, where the small number of plates
graphic swell developing around a hotspot [33]. In and the symmetry of shape strongly restrict the
fact, plumes are less a¡ected by plate motion than possible solutions. Actually, varying several pa-
by the large scale £ow standing in the high-viscos- rameters, we ¢nd only two. Further, in case 3,
ity layer (below 670 km depth). The velocity ¢eld where the plate distribution is more complex,
in the low-viscosity layer is strongly correlated the dynamical structure evolves slowly, with a tor-
with plate motion, but the underlying mantle oidal/poloidal ratio £uctuating around a mean
£ows in a di¡erent way, setting up large cells be- value (0.66 at 0.4 Gyr, 0.79 at 1.1 Gyr, 0.75 at
tween downwellings and diverging zones (Fig. 5). 2.1 Gyr, 0.78 at 5.6 Gyr, and 0.80 at 6.5 Gyr). A
These deep currents drag the plumes below the most important feature related to the time evolu-
diverging zones. Their surface drift can reach tion of the thermal structure is that the dynamics
2000 km in a billion years, so that, when the ex- tend to a strong degree one. This is observable on
periment reaches thermal equilibrium, after ¢ve or the case with four plates but also on all cases with
six billion years, most of plumes have been cap- 15 plates except on the one performed with 100%
tured by the diverging zones. This situation is not of internal heating and constant viscosity. Cold
relevant for the Earth, except perhaps for the Ice- downwellings converge toward a single thermal
land or Azores hotspots where the ridge migration structure. This feature may be seen as the reason
speed is small in the hotspot reference frame. This of the continent gathering that periodically oc-
re£ects the shortcoming of the model in which the curs. Of course, because of the lack of a complete
plate boundaries remain ¢xed. In reality, they modelling of plate tectonics in the model, this
change continuously, particularly the ridges which only remains as a possible inference of the phys-
may migrate as fast as the plates. In case 3, the ical process that drives the long-term evolution of
plume motion and the average plate velocity fall plate tectonics.
in the range of the values for the Earth, i.e.
2 mm/yr and 4 cm/yr, respectively, so that the 5.3. Surface velocity, heat £ow and geotherm
plumes in our model can be regarded as ¢xed
with respect to plate drift. This is a feature well The impact of tectonic plates on dynamics is
highlighted in Zhong et al. [21]. not restricted to the convective structures, but
also a¡ects integral quantities as the mean surface
5.2. Poloidal versus toroidal components velocity, the heat £ow, and the mean temperature
pro¢le. Besides the developing of a strong rota-
An other element of comparison is the parti- tional component, plates also alter the mean sur-
tioning of kinetic energy between the poloidal face speed. In case 1, it reaches 5.6 cm/yr, slows to
and the toroidal components. Conversely in 4.1 cm/yr in case 3 and to 3.5 cm/yr in the four
most models with variable viscosity, the plate con- plate case, all parameters being identical in the
dition naturally leads to a strong toroidal compo- three cases. The smaller the number of plates,
nent. The ratio of toroidal to poloidal component the less vigorously the convection. The plate con-
is 0.67 in the four plate case. In case 3, this ratio dition appears intermediate between two asymp-
reaches 0.8, close to the value observed for the totic conditions, no-slip and free-slip, equivalent
Earth. This agreement is not surprising. O'Con- to a single plate and to an in¢nite number of
nell et al. [34] presume that, since the toroidal plates, respectively. On Fig. 6, the Nusselt number
velocities are not involved in the heat transfer, is plotted as a function of the Rayleigh number
the convective process develops a dynamical for four di¡erent surface conditions : free-slip, 15
structure reducing the loss of kinetic energy, so plates, four plates and one plate. The plate design
that the partitioning essentially depends on the for four and 15 plate cases are the same as for
plate distribution. Besides the partitioning, this cases 2 and 3, respectively. The viscosity pro¢le
assumption also explains the notable stability of used is the same as in the cases 1, 2 and 3, and
the thermal structure during over several billion there is no internal heating. The plot depicts a
EPSL 5687 4-1-01 Cyaan Magenta Geel ZwartM. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587 585
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 re-
mains insulated below the thermal boundary
layer.
The experimental parameters for case 3 are de-
signed to ¢t Earth's observables. If the heat £ow
and the surface velocities derived from the free-
slip 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
Fig. 6. Nusselt number as a function of the Rayleigh number 1600³C [38].
for four di¡erent plate conditions. The experiment has been
performed with no internal heating. There is a stepwise in-
crease in viscosity with depth by a factor of 30 at 650 km
6. Concluding remarks
depth. The Rayleigh number is based on the lower mantle
viscosity.
It is important to notice that the sublithospher-
ic temperature is known with a small uncertainty.
classic power law relationship between the Nusselt It is deduced from two independent observables:
number and the Rayleigh number. As expected, the heat £ow, topography and age relationship in
the exponent increases from the single plate con- oceanic domains, and the geochemical composi-
dition up to the free-slip condition, the four plates tion of mid-ocean ridge basalts. It is one of the
and the 15 plates conditions corresponding to in- strongest constraints on the Earth's geotherm.
termediate values. Even if the exponents for four Thus a one layer convection established in the
and 15 plates conditions di¡er, they remain very
close indicating only a small in£uence of the num-
ber of plate, and so of the plate geometry on the
Nusselt^Rayleigh relationship. This behavior is
also observed in cases with internal heating, con-
vection 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%, respec-
tively).
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 con-
dition (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 inver- Fig. 7. Averaged temperature pro¢le in cases 1, 2 and 3. In
sion observed in cases with plate. It results from cases with plates, the mantle is hotter and the pro¢les are
the combination of two e¡ects: (1) the concentra- clearly marked by a temperature inversion.
EPSL 5687 4-1-01 Cyaan Magenta Geel Zwart586 M. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587
Earth's mantle is only consistent with a very low Acknowledgements
temperature at the CMB, close to the value set in
case 3, i.e. below 3200 K corresponding to a 2000 We thank G. Ceuleneer, K. Feigl and M. Ra-
K super-adiabatic temperature step throughout binowicz for helpful discussion. Useful sugges-
the whole mantle. This is 500 K colder than the tions and comments by Yanick Ricard and by
lower estimate [25,26]. A greater viscosity step an anonymous reviewer are gratefully acknowl-
should reduce the mean mantle temperature al- edged. This work was supported by a grant
lowing higher CMB temperatures, but also should from the Institut des Sciences de l'Univers. Com-
increase the heat £ux from the core up to unreal- puting resources were provided by the Centre Na-
istic values. On the other hand, the endothermic tional d'Etudes Spatiales.[AC]
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 References
with a signi¢cant thermal boundary layer at the
phase change [7,39]. A more promising way to [1] G.A. Glatzmaier, Numerical simulation of mantle convec-
conciliate the paradox existing between the subli- tion: time-dependent, three-dimensional, compressible,
spherical shell, Geophys. Astrophys. Fluid Dyn. 43
thospheric temperature, the CMB temperature (1988) 223^264.
and the small heat £ux at the core surface, per- [2] D. Bercovici, G. Schubert, G.A. Glatzmaier, Three-di-
haps lies in the existence of geochemical strati¢- mensional spherical models of convection in the Earth's
cation of the mantle as proposed by Davaille [40] mantle, Science 244 (1989) 893^955.
[3] G.A. Glatzmaier, G. Schubert, D. Bercovici, Chaotic,
and Kellogg et al. [41]. As a matter of fact, a
subduction-like down£ows in a spherical model of con-
stable deep layer with a high content in radiogenic vection in the Earth's mantle, Nature 347 (1990) 274^277.
elements would insulate the core, preventing a [4] J.-T. Ratcli¡, G. Schubert, Three-dimensional variable
great heat £ux and would favor the creation of viscosity convection of an in¢nite Prandtl number Bous-
more hot plume than in our models. sinesq £uid in a spherical shell, Geophys. Res. Lett. 22
(1995) 2227^2230.
We have shown that the simple introduction of
[5] S. Zhang, D.A. Yuen, Various in£uences on plumes and
a piecewise continuous surface generates features dynamics in time-dependent, compressible mantle convec-
that look like Earth's plate tectonics. Beyond this tion in 3-D spherical shell, Phys. Earth Planet. Int. 94
spectacular reorganization, the in£uence of plates (1996) 241^267.
on the inner structure of convection and on global [6] H.P. Bunge, M. Richards, J.R. Baumgardner, E¡ect of
quantities as heat £ow or geotherm highlights the depth-dependent viscosity on the planform of mantle con-
vection, Nature 379 (1996) 436^438.
inseparable character of plate tectonics and Earth [7] H.P. Bunge, M. Richards, J.R. Baumgardner, A sensitiv-
mantle convection. Clearly plate tectonics is not a ity study of 3-D spherical mantle convection at 108 Ray-
simple consequence of the mantle convection but leigh number: e¡ects of depth-dependent viscosity, heat-
the mantle dynamics mode acting on the Earth. ing mode and an endothermic phase change, J. Geophys.
This casts doubts on the applicability of free- Res. 102 (1997) 11991^12007.
[8] P.J. Tackley, D.J. Stevenson, G.A. Glatzmaier, G. Schu-
slip or no-slip models of convection to the Earth. bert, E¡ects of an endothermic phase transition at 670 km
These classical approaches are perhaps more rel- depth in a spherical model of convection in the Earth's
evant for the dynamics of Venus where the sur- mantle, Nature 361 (1993) 699^704.
face temperature, 500³C, limits the strength of the [9] U. Christensen, H. Harder, 3-D convection with variable
viscosity, Geophys. J. Int. 104 (1991) 213^226.
lithosphere, and distributes the surface deforma-
[10] D. Bercovici, A simple model of plate generation from
tion. The topography of this planet is character- mantle £ow, Geophys. J. Int. 114 (1993) 635^650.
ized by more or less circular highlands sur- [11] D. Bercovici, A source-sink model of the generation of
rounded by plains, similar to the pattern of plate tectonics from non-Newtonian mantle £ow, J. Geo-
dynamical topography generated by free-slip phys. Res. 100 (1995) 2013^2030.
spherical models of convection. [12] D. Bercovici, Generation of plate tectonics from litho-
sphere-mantle £ow and void-volatile self lubrication,
Earth Planet. Sci. Lett. 154 (1998) 139^151.
EPSL 5687 4-1-01 Cyaan Magenta Geel ZwartM. Monnereau, S. Quërë / Earth and Planetary Science Letters 184 (2001) 575^587 587
[13] P.J. Tackley, Self-consistent generation of tectonic plates leigh number convection in a medium with depth-depen-
in three-dimensional mantle convection, Earth Planet. Sci. dent viscosity, Geophys. J. R. Astron. Soc. 85 (1986) 523^
Lett. 157 (1998) 9^22. 541.
[14] R. Trompert, U. Hansen, Mantle convection simulations [30] H.P. Bunge, M.A. Richards, C. Lithgow-Bertelloni, J.R.
with rheologies that generate plate-like behaviour, Nature Baumgardner, S.P. Grand, B.A. Romanowicz, Time
395 (1998) 686^689. scales and heterogeneous structure in geodynamic Earth
[15] P.J. Tackley, Self-consistent generation of tectonic plates models, Science 280 (1998) 91^95.
in time-dependent mantle convection simulations 1. Pseu- [31] N.M. Ribe, U.R. Christensen, Three-dimensional model-
doplastic yielding, G3 1, 2000GC000036, 2000. ing of plume^lithosphere interaction, J. Geophys. Res. 99
[16] P.J. Tackley, Self-consistent generation of tectonic plates (1994) 669^682.
in time-dependent mantle convection simulations 2. strain [32] W.B. Moore, G. Schubert, P. Tackley, Three-dimensional
weakening and astenosphere, G3 1, 2000GC000043, 2000. simulations of plume^lithosphere interaction at the Ha-
[17] S.A. Weistein, P.L. Olson, Thermal convection with non- waiian swell, Science 279 (1998) 1008.
Newtonian plates, Geophys. J. Int. 111 (1992) 515^530. [33] M. Monnereau, M. Rabinowicz, E. Arquis, Mechanical
[18] S. Zhong, M. Gurnis, Mantle convection with plates and erosion and reheating of the lithosphere: A numerical
mobile faulted plate margins, Science 267 (1995) 838^843. model for hotspot swells, J. Geophys. Res. 98 (1993)
[19] Y. Ricard, C. Vigny, Mantle dynamics with induced plate 809^823.
tectonics, J. Geophys. Res. 94 (1989) 17543^17559. [34] R.J. O'Connell, C.W. Gable, B.H. Hager, in: R. Sabadini
[20] C.W. Gable, R.J. O'Connell, B.J. Travis, Convection in et al. (Eds.), Glacial Isostasy, Sea-Level and Mantle
three dimensions with surface plates: generation of toroi- Rheology, Kluwer Academic Publishers, 1991, pp. 535^
dal £ow, J. Geophys. Res. 96 (1991) 8391^8405. 551.
[21] S.J. Zhong, M.T. Zuber, L. Moresi, M. Gurnis, Role of [35] S.T. Kirby, W.B. Durham, L.A. Stern, Mantle phase
temperature-dependent viscosity and surface plates in changes and deep-earthquake faulting in subducting litho-
spherical shell models of mantle convection, J. Geophys. sphere, Science 252 (1991) 216^225.
Res. 105 (2000) 11063^11082. [36] S.T. Kirby, S. Stein, E.A. Okal, D.C. Rubie, Metastable
[22] B.H. Hager, R.J. O'Connell, A simple global model of mantle phase transformations and deep earthquakes in
plate dynamics and mantle convection, J. Geophys. Res. subducting oceanic lithosphere, Rev. Geophys. 34 (1996)
86 (1981) 4843^4867. 261^306.
[23] H.P. Bunge, M. Richards, The origin of long-wavelength [37] B. Parsons, J.G. Sclater, An analysis of the variation of
structure in mantle convection, Geophys. Res. Lett. 23 ocean £oor bathymetry and heat £ow with age, J. Geo-
(1996) 2987^2990. phys. Res. 82 (1977) 803^827.
[24] S.V. Patankar, Numerical Heat Transfer and Fluid Flow, [38] G. Ceuleneer, M. Monnereau, M. Rabinowicz, C. Rosem-
Hemisphere, New York, 1980. berg, Thermal and petrological consequence of melt mi-
[25] R. Jeanloz, The nature of the Earth's core, Annu. Rev. gration within mantle plumes, Phil. Trans. R. Soc. Lond.
Earth Planet. Sci. 18 (1990) 357^386. 342 (1993) 53^64.
[26] R. Boehler, Melting temperature of the Earth's mantle [39] M. Monnereau, M. Rabinowicz, Is the 670 km phase
and core: Earth's thermal structure, Annu. Rev. Earth transition able to layer the Earth's convection in a mantle
Planet. Sci. 24 (1996) 15^40. with depth-dependent viscosity?, Geophys. Res. Lett. 23
[27] F.D. Stacey, D.E. Loper, Thermal history of the Earth: a (1996) 1001^1004.
corollary concerning non-linear mantle rheology, Phys. [40] A. Davaille, Simultaneous generation of hotspots and
Earth Planet. Int. 53 (1988) 167^174. superswells by convection in a heterogeneous planetary
[28] J.G. Sclater, C. Jaupart, D. Galson, The heat £ow trough mantle, Nature 402 (1999) 756^760.
oceanic and continental crust and the heat loss of the [41] L.H. Kellogg, B.H. Hager, R.D. vanderHilst, Composi-
Earth, Rev. Geophys. Space Phys. 18 (1980) 269^311. tional strati¢cation in the deep mantle, Science 283 (1999)
[29] M. Gurnis, G.F. Davies, Numerical study of high Ray- 1881^1884.
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