mixing and deformations in mantle plumestheir storage in the earth’s mantle for at least 1 billion...

Mixing and deformations in mantle plumes

Cinzia G. Farnetani a;*, Bernard Legras b, Paul J. Tackley c

a Institut de Physique du Globe de Paris, Laboratoire de Dynamique des Systemes Geologiques, Boite 89, 4, place Jussieu,75252 Paris Cedex 05, France

b Ecole Normale Superieure, Paris, Francec University of California, Los Angeles, CA, USA

Received 30 July 2001; received in revised form 26 November 2001; accepted 3 December 2001

Abstract

The long standing idea that the source of oceanic island basalts includes ancient subducted material is strengthenedby recent geochemical observations for Hawaii [Lassiter and Hauri, Earth Planet. Sci. Lett. 164 (1998) 4833496] andIceland [Kempton et al., Earth Planet. Sci. Lett. 177 (2000) 2553271]. In particular, the isotopic variations in Hawaiianshield lavas indicate the presence of two distinct recycled components: ancient oceanic crust+sediments, and alteredultramafic lower crust or lithospheric mantle. Lassiter and Hauri [Earth Planet. Sci. Lett. 164 (1998) 4833496] suggestthat both components are from the same packet of recycled oceanic lithosphere, thus implying that chemicalheterogeneities a few km thick can be preserved in the convecting mantle. In this paper we investigate the role of mantleplumes in stirring mantle heterogeneities and we address the following questions: (1) Is the heterogeneous nature ofplumes inherited at the source or does it develop through entrainment? (2) Is stirring more efficient in the plume head orin the long-lived plume tail? (3) Are the geochemical implications consistent with fluid dynamical models? We use athree-dimensional numerical model in Cartesian geometry to simulate the dynamics of an isolated plume. Transportcalculations, conducted on a vertical plane of symmetry, allow us to advect passive tracers forward or backward in timeto investigate mixing. We also calculate the finite-time Lyapunov exponents in order to quantify the deformationsassociated to the plume rise. Our results show that: (1) the thermal boundary layer, where the plume forms, is the regionmost efficiently sampled by a mantle plume. Since the overlying mantle is not entrained in the plume head, we speculatethat the geochemically heterogeneous nature of plumes is inherited from the source. Our results also predict the absenceof present-day upper mantle, source of MORB, in plume lavas. (2) Heterogeneities initially located in the source regionundergo a series of stretching and folding events while rising in the plume head and may be reduced to narrowfilaments. We find that stirring is more important in the plume head than in the long-lived plume tail. Therefore, ourresults predict that distinct geochemical heterogeneities are more likely to be found in hotspot lavas rather than in floodbasalt lavas, associated to partial melting of a plume tail and a plume head [Richards et al., Science 246 (1989) 103^107],respectively. (3) High Lyapunov exponents, indicating high deformations, are found at the frontier between the plumehead and the sublithospheric mantle surrounding the plume head. We speculate that the arrival of a large plume headcould induce seismic anisotropy in the shallow upper mantle. ß 2002 Elsevier Science B.V. All rights reserved.

Keywords: mantle plumes; mixing; heterogeneity; deformation

0012-821X / 02 / $ ^ see front matter ß 2002 Elsevier Science B.V. All rights reserved.PII: S 0 0 1 2 - 8 2 1 X ( 0 1 ) 0 0 5 9 7 - 0

* Corresponding author. Tel: +33-1-4427-4796; Fax: +33-1-4427-2481. E-mail address: [email protected] (C.G. Farnetani).

EPSL 6093 7-3-02 Cyaan Magenta Geel Zwart

Earth and Planetary Science Letters 196 (2002) 1^15

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1. Introduction

Geochemical analysis on lavas from Hawaiiand Iceland, the most vigorous present-day hot-spots, provide compelling evidences on the heter-ogeneous nature of mantle plumes. A plume prob-ably consists of a mixture of enriched and morefusible streaks in a depleted and refractory matrix.The depleted component in plumes seems to bedi¡erent from the depleted upper mantle sourceof MORBs. According to Kempton et al. [2] thedepleted component present in the Iceland plumedi¡ers from the depleted upper mantle sampled bythe present-day North Atlantic mid-ocean ridge.The high hafnium isotope ratio of the depletedcomponent is a long-lived and intrinsic featureof the Iceland plume and is probably generatedby ancient melting events. Moreover a long termisolation of the di¡erent components is necessaryto generate the isotope ratios observed in Icelan-dic basalts. Similarly, for Hawaii, the depletedupper mantle component is very minor or non-existent [4]. This is in agreement with oxygenôs-mium^lead isotope correlations by Lassiter andHauri [1] precluding the generation of the Keaisotopic signature from astenospheric upper man-tle. It is also in agreement with the lead^hafniumisotopic relations [5] indicating that upper mantlematerial is not involved in the production of theHawaiian basalts.

The enriched component in plumes is mostlikely associated to the presence of recycled oce-anic crust, as ¢rst suggested by Hofmann andWhite [6]. Recently Lassiter and Hauri [1] haveused oxygenôsmium^lead isotopes to constrainthe nature of the plume material feeding the Koo-lau and the Kea trends in Hawaii. They ¢nd thatthe radiogenic Os isotopes and anomalouslyheavy oxygen isotopes for the Koolau lavas re£ectmelt generation from recycled oceanic crust pluspelagic sediment, while for the Kea lavas the un-radiogenic Os isotopes and anomalously lightoxygen isotopes re£ect the presence of recycled,hydrothermally altered, ultrama¢c lower crust orlithospheric mantle. The presence of old pelagicsediments in the source of Koolau componenthas also been identi¢ed by Blichert-Toft et al.[5], using hafnium isotopic data.

Therefore, the most recent geochemical studieson lavas from Iceland and Hawaii seem to indi-cate that the most important geochemical compo-nents in a mantle plume are both related to thesame dynamical process: the subduction of theoceanic lithosphere with crust and sediments,their storage in the Earth's mantle for at least1 billion years, their incorporation in mantleplumes. For vigorous mantle plumes, such as Ice-land and Hawaii, one more component may bepresent: a small fraction of relatively undegassedmaterial that could explain the helium and neonisotopic ratios. The high 3He/4He ratio for Hawaiiand for Iceland has been generally attributed tothe presence of an ùndegassed' reservoir, rich inprimitive 3He. Also the identi¢cation of near-solarneon isotopic ratios in some glasses from Loihiand Kilauea [7] and from Iceland [8,9] suggeststhe presence of an ùndegassed' mantle compo-nent with high [Nesolar]/[U+Th].

In mantle plumes there are therefore at leastthree distinct components and the understandingof their origin and evolution may provide majorconstraints on the style of mantle convection, onthe e¤ciency of mixing in the Earth's interior, andon the £uid dynamics of plumes.

A ¢rst important geodynamic implication con-cerns the fate of subducted material. Seismic to-mography (e.g., [10] and references therein) showsthat slab penetrate in the lower mantle in spite ofthe inhibiting e¡ect of the phase transition at 660km depth. However, one can speculate that thestyle of mantle convection changed over timeand that convection in the past was predomi-nantly layered [11]. The age of the recycled mate-rial in mantle plumes could represent a key pa-rameter to test this hypothesis : if the subductedlithosphere remained isolated for 1 Ga, or may beup to 3 Ga (the age of the sediments estimated by[5]) then the subducted material was more likelystored in the lower mantle, rather than in theupper mantle continuously sampled by spreadingridges. This would support the idea that wholemantle convection was active also in the past.

A second important geodynamic implicationconcerns the e¤ciency of mixing of the subductedmaterial. According to Lassiter and Hauri [1] thepreservation of both upper- and lower-crustal


C.G. Farnetani et al. / Earth and Planetary Science Letters 196 (2002) 1^152

oxygen isotope signatures in Hawaiian lavas, in-dicates that chemical heterogeneities with lengthscales of only a few kilometers can be preserved inthe convecting mantle for long periods of time.Similarly, Sobolev at al. [12] working on `ghostplagioclase' trace-element signature in inclusionsin Mauna Loa olivines, infer that subducted andrecycled crustal gabbro can retain much of itsoriginal chemical identity through the convectivecycle. These observations support the idea thatthe subducted crust does not mix completelywith the peridotitic matrix, but remains distinctforming a `marble cake' mantle, as ¢rst suggestedby Alle©gre and Turcotte [13]. We also note thatdue to the negligible rate of chemical di¡usionany compositional di¡erence in the mantle isonly stirred, not mixed.

The third important geodynamic implicationconcerns the preservation of more undegassedmaterial in a convecting mantle [14] and the exis-tence of two distinct OIB and MORB sources.

In order to understand the geochemical mes-sage brought to the surface by plumes we needto address some fundamental questions such as`Which regions of the mantle are more e¤cientlysampled by mantle plumes?' and `Is the heteroge-neous nature of mantle plumes inherited at thesource or does it develop through entrainmentof various components during the ascent to thesurface?' In this paper we use a three-dimensionalnumerical model in Cartesian geometry to solvefor the solid-state dynamics of an isolated mantleplume from the lower mantle. Passive heterogene-ities are modeled by tracers that can be advectedforward in time as well as backward in time. Wealso calculate the ¢nite-time deformations inducedby the rise of a plume, thus enabling us to addressanother question: `Where does most of the defor-mation take place when a plume rises : within theplume or in the surrounding mantle?'

Our conceptual model is designed to study thedynamics of a strong plume, such as Iceland orHawaii for which there is ample geochemical andgeophysical evidence that they originate in thelower mantle. In the models we vary the mantleviscosity structure and we can include the dynam-ic e¡ect of the endothermic phase transition at660 km depth.

We ¢nd that the region most e¤ciently sampledis the thermal boundary layer where the plumeoriginates, while the surrounding mantle, throughwhich the plume rises, is very poorly sampled.Our results, showing negligible entrainment inmantle plumes are consistent with previous nu-merical models [15], while they disagree with earlylaboratory experiments [16]. The most importantreason is the di¡erent initial condition for plumeformation: instability at a thermal boundary, ver-sus injection of hot £uid from a point source. We¢nd that the upper mantle material is not en-trained in the plume head, rather it is displacedby the plume and forms a sheath surrounding thehot plume. Therefore, our results predict the ab-sence of any geochemical signature of present-dayupper mantle in plume lavas. By calculating theLyapunov exponents we quantify the deforma-tions and we ¢nd that mantle material formingthe plume head is more deformed (by a seriesof stretching and folding events) than the materialin the plume tail, that undergoes mainly stretch-ing.

2. Numerical model

We use the Cartesian three-dimensional numer-ical model STAG3D by Paul Tackley. The modelsimulates the dynamics of solid-state convectionby solving the equations for the conservation ofmass, conservation of momentum, and conserva-tion of energy for a viscous £uid at in¢nitePrandtl number. It also solves the advection equa-tion of an active chemical ¢eld which allows us toinvestigate the role of chemically denser materialat the base of the mantle. The slight numericaldi¡usion that arises when advecting sharp inter-faces is corrected by applying the ¢lter of [17]. Itis out of the purpose of this paper to describe indetail the model which has been thoroughly pre-sented elsewhere, (e.g., [18] and references there-in).

The size of our domain in the x, y, and z direc-tions is 2900U1450U2900 km, respectively,meshed by 128U64U128 elements. Hence, the el-ement size is 22.6 km/element in the horizontaldirections, while it is 15 km/element in the vertical


C.G. Farnetani et al. / Earth and Planetary Science Letters 196 (2002) 1^15 3

direction for the bottom 300 km, and 24 km/ele-ment above.

The physical parameters used to normalize theequations are: the mantle depth D = 2900 km, thepotential temperature contrast between the sur-face and the bottom va= 1800³C, the mantle den-sity b= 4000 kg m33, which is constant by theBoussinesq approximation, the thermal expansioncoe¤cient K= 2U1035 K31, the thermal di¡usiv-ity U= 1036 m2s31, the reference mantle viscosityRm = 1022 Pa s for am = 1300³C. Using the abovevalues the Rayleigh number is 3.5U106.

The viscosity is temperature-dependent accord-ing to the exponential law:

R �a � � Rm exp�23�a m3a �� 1�

which gives a min Rplume = 0.01 Rm and maxRlithosphere = 30 Rm, where Rm is the viscosity ofthe surrounding mantle. We also explore strati¢edmodels with viscosity ratio between upper andlower mantle RUM/RLM = 1/30.

The velocity boundary conditions are free slipfor the top and the bottom and no incoming oroutgoing £ux is allowed through the walls. Forthe vertical wall xz (at y = 0) we use re£ectingboundary conditions since we only model half ofthe plume. In fact the plume is centered in themiddle of the x-axis (at x = 1450 km) and aty = 0. We remark that although our calculationsare conducted in three dimensions, all the resultspresented in this paper are relative to an axisym-metric case. The boundary conditions for the po-tential temperature ¢eld are constant at the sur-face (a= 0³C) and at the bottom (a= 1800³C),while the vertical walls are insulated. At the initialtime, the potential temperature is constantthroughout the mantle (a= 1300³C), while thereis a thermal boundary layer 120 km thick at thebottom, where the temperature pro¢le increasesacross as an error function. In order to triggerthe plume formation we apply a small perturba-tion (max is 1% va) which decreases linearly withradial distance. Because of this initial setting, themotion stays essentially axisymmetric during thewhole simulation period.

All the boundaries are impermeable to thechemical ¢eld. The initial value of the chemical

¢eld is zero everywhere, except for a layer 20km thick at the bottom of the mantle. This thinlayer has a chemical excess density of 2% withrespect to the surrounding mantle (vbch/vbth = 0.55). We use the active chemical ¢eldonly for one model since it is out of the purposeof this paper to vary systematically the thicknessand the excess density of the chemically denserlayer. Our choice of a 20-km thick layer corre-sponds to an average thickness of the ultra lowvelocity layer detected at the base of the Earth'smantle [19], which has a variable thickness be-tween 0 and 40 km. For a more extensive inves-tigation of the role of chemical strati¢cationacross DQ on mantle plume dynamics see [20].

2.1. Transport calculations

The advection of passive tracers is performedo¡-line from the archived velocity ¢eld of themodel. Because of the axisymmetric symmetrywe only use velocity components in the verticaly^z plane at x = 1450 km saved at each timestep. The advection is performed using a two-di-mensional bicubic spline interpolation within themesh of the model [21] and using a second-orderRunge^Kutta method for the time scheme. Morethan half a million trajectories are calculated for-ward and backward in time.

The forward integration starts at the initial timet0. It is particularly interesting to put the tracerswithin a few hundred km above the core mantleboundary. The backward integration starts at the¢nal time t1 when the plume head has spreadbeneath the lithosphere. Tracers are distributedregularly in each grid element and by advectingthem backward to the initial time we can under-stand where each tracer comes from.

We calculate the deformation associated withthe rise of the plume, represented by the ¢nite-time Lyapunov exponents. The Lyapunov expo-nents are a convenient indicator of the sensitivityto small perturbations of the initial position andprovide the exponential rate of divergence of in-¢nitesimally nearby initial trajectories. By inte-grating the deformation along each tracer trajec-tory we can understand (i) where is initiallylocated the material which deforms the most dur-



ing the formation and the ascent of the plume and(ii) where is located this highly deformed materialwhen the plume head arrives beneath the litho-sphere.

To calculate the ¢nite-time Lyapunov expo-nents we consider a trajectory x, starting at theposition x0 at time t0 and ending at the positionx1 at time t1. The velocity vector at each positionx along the trajectory and time t is given by:

dxdt� u�x; t� �2�

The in¢nitesimal displacement h0 at time t0 istransformed into h1 at time t1 by:

h 1 �M�x0; t0; x1; t1�h 0; �3�

where M is a linear operator given by the integra-tion of:

dMdt� Du

DxM �4�

along the trajectory, where Du/Dx is the materialderivative of the velocity. At the initial time t0 thematrix M corresponds to the identity matrix:

M�x0; t0; x0; t0� � Identity �5�

The variation of the displacement over the in-terval [t0,t1] is given by the ratio:

Mh MMh 0M

� MMh 0M

Mh 0M� h T

0 MT Mh 0

h T0 h 0

� �1=2

This ratio is governed by the real eigenvaluesc� and c3 of MT M, the right Cauchy^Greenstrain tensor [22], and the projection of h0 ontoits eigenvectors. The ¢nite-time Lyapunov expo-nents or ¢nite-time deformations are de¢ned as:

V � � 1t13t0

lnc� �6�

V 3 � 1t13t0

lnc3

They are calculated by numerical integration ofEq. 4 for the discretized version of Du/Dx along

the particle trajectory. See [23] for a detailed dis-cussion.

A circular blob of tracers with unit radius sur-rounding x0 at time t0 is turned into an ellipsoidsurrounding x1 at time t1 with semi-axis of lengthc� and c3. Conversely a circular blob of tracerswith unit radius surrounding x1 at time t1 was anellipsoid with axis 1/c� and 1/c3 surrounding x0

at time t0. Because the motion preserves volumein three dimensions:

V � � V 3 � 1t13t0

lnr0

r1�7�

where r0 and r1 are the initial and the ¢nal radialdistances.

Because of the exponential separation of c�and c3, the eigenvalue problem is badly condi-tioned for M and, in practice, only V� can beobtained with some accuracy in forward integra-tion. Since M is transformed into M31 by revers-ing time, V3 can be obtained by backward inte-gration.

3. Results

3.1. Plume dynamics and tracers advection

The simplest model, thereafter called Model 1,has a uniform background mantle viscosityRm = 1022 Pa s and varying Rplume which can beas small as 0.01Rm. Fig. 1a shows the initial con-dition for the temperature ¢eld on the y^z planeat x = 1450 km, colors indicate the excess temper-ature with respect to the mantle. Fig. 1b showsthe initial position of the passive tracers, each`pack' of tracers is 100 km high and 180 kmlong. The plume head radius is 250 km wide(Fig. 1c) at the elapsed time te = 105 Myr (noticethat a long time of ca. 80^90 Myr, elapses fromthe initial condition to the stage when a verysmall plume forms). Fig. 1d shows that tracersinitially close to the axis form a stretched bound-ary of the plume head, while tracers that are asfar as 1000 km are stretched and travel rapidly atthe bottom of the thermal boundary layer wherethe horizontal velocities are important. At te = 135



Myr the plume is at 800 km depth, its radius is380 km and there is no evidence of e¤cient en-trainment within the plume head (Fig. 1e). Thetracers initially up to 700 km far from the axis(Fig. 1f) are in a torus within the plume headthat has been stretched and folded. This torus

rises more slowly than material closer to theaxis. Tracers in the axial part of the plumecome from a great radial distance and rise veryrapidly in the plume conduit. Fig. 1g shows alarge plume head, 650 km radius, spread beneaththe lithosphere at te = 175 Myr. The correspond-

Fig. 1. Plume evolution for Model 1. Top: excess temperature, (a) initial condition, (c,e) later stages, (g) ¢nal stage. Middle: trac-ers advected forward, (b) initial condition, (d,f,h) later stages. (i) Detail of the plume head shown in 1h. (l) Tracers advectedbackward. Color scale indicates the height above the CMB (see text).



ing tracers position (Fig. 1h) indicates that all thematerial in the plume head comes from the ther-mal boundary layer. Material in the tail has beenstretched, while material in the head has beenstretched and folded. A detail of the plume headfor this ¢nal step is shown in Fig. 1i. The plumehead is heterogeneous since tracers with di¡erentinitial position get close by in the plume head.The results of the backward calculation are shownin Fig. 1l. Colors now correspond to the calcu-lated depth of origin of each tracer. Tracers withcalculated initial position in the upper mantle arein light gray, tracers that come from the lowermantle are in dark gray, while tracers comingfrom the bottom of the mantle are shown withrainbow colors. The good agreement betweenthe the shape of the head Fig. 1i,l is an indicatorof the accuracy of the numerical integration. Sincethe tracers are put on a regular grid at t1, Fig. 1lis plotted with a uniform spatial resolution whilethe dispersion of tracers in Fig. 1i reduces theresolution in the plume core. A clear result isthat upper mantle material is pushed away bythe rising plume head and does not enter in theplume head. By comparing Fig. 1i,l, it is apparentthat the tail is formed by concentric cylindersoriginating from sloping surfaces across the DQlayer and that the motion is more e¤cient at stir-ring lateral inhomogeneities than vertical inhomo-geneities.

In Model 2 the mantle is strati¢ed with a vis-cosity jump at 660 km. The viscosity ratio be-tween the upper mantle and the lower mantleRUM/RLM = 1/30 and, as above, minRplumeW0.01Rm. The excess temperature (Fig.2a) and the tracers position (Fig. 2b) when theplume head is at 1700 km depth (at te = 115Myr) are similar to the previous model since theconditions in the lower mantle are unchanged.Fig. 2c shows the plume at 660 km depth atte = 130 Myr. The part of the plume closer tothe axis ascends rapidly in the upper mantle, whilethe part of the head still in the lower mantle risesmore slowly. The e¡ect of the viscosity jump at660 km depth profoundly a¡ects the shape of thehead and the degree of stretching of the risingmaterial. At te = 132 Myr a relatively small plumehead (radius = 300 km) reaches the base of the

lithosphere, while a long stretch of hot materialremains in the lower mantle (Fig. 2e). The head iscomposed of tracers that were originally at morethan 1000 km from the axis (Fig. 2f), while ma-terial initially closer to the axis constitutes thelong stretch still in the lower mantle. After 139Myr the plume head has spread beneath the litho-sphere (Fig. 2g), its excess temperature is com-plex: a hot part (va= 350^400³C) surrounds acolder central part (va= 100³C). The hot part cor-responds to the location of the tracers from thelowermost 100 km (Fig. 2h). It is also evident thatsome tracers `packs' are reduced to narrow ¢la-ments and the deformation is considerable. A de-tail of the plume head is shown in Fig. 2i, thedi¡erent colors indicate that the head is heteroge-neous, mainly in the distal part of the plume head.The results of the backward calculation indicatethat the colder central part of the plume head ismade of material coming from 100^150 km abovethe CMB (Fig. 2l). Compared to the previousmodel, the viscosity jump at 660 km depth hasthe e¡ect of reducing the volume of the plumehead, increasing stirring of the deep mantle mate-rial that constitutes the head, while the entrain-ment of upper mantle in the head is absent inboth models.

For Model 3 a thin and chemically denser layeris present at the base of the mantle, while themantle viscosity is identical to Model 2. The pres-ence of chemical heterogeneities signi¢cantly af-fects the dynamics and the ¢nal structure andcomposition of a plume. Since the hottest partof the thermal boundary layer remains at the bot-tom, the excess temperature of the plume is re-duced to va= 200^300³C and it is more realisticthan in the previous models. The colder temper-ature induces a weaker viscosity contrast betweenthe plume and the surrounding mantle, thereforethe shape of the plume is now more similar to a`spout' than to a `mushroom' (Fig. 3a). Fig. 3bshows that part of the tracers pile up at the bot-tom and form a large `foot', while the head iscomposed of stretched deep material. The pres-ence of a `foot' facilitates the rise of tracers ini-tially at more than 1000 km from the axis (Fig.3d,f). The plume head narrows while entering inthe low viscosity upper mantle (Fig. 3e) further



stretching the ascending material (Fig. 3f). Theplume head spreading beneath the lithosphere isshown in Fig. 3g at the elapsed time te = 290 Myr.This long elapsed time is due to the presence ofdenser material in the thermal boundary layerwhich hinders plume formation: nearly 200 Myrare necessary to form a small initial plume which

then ascends in ca. 90 Myr. The tracers position(Fig. 3h) shows that the plume head is stronglystretched and heterogeneous, and that a con-siderable fraction of tracers is trapped at thebottom. A detail of the forward calculation isshown in Fig. 3i. Material originally at distanceless than 300 km (black and red tracers) can be

Fig. 2. Plume evolution for Model 2. Top: excess temperature. Middle: tracers advected forward. (i) Detail of the plume headshown in 2h. (l) Tracers advected backward.



found in the plume head both close to the axis,than stretched in a rim all around the head,also in this case the plume head is very heteroge-neous. The results of the backward calculation(Fig. 3l) indicate that tracers from the lowermost25 km are absent from the plume head, whilematerial originally 150 km above the CMB occu-

pies the zone in the head with lowest excess tem-peratures.

3.2. Vertical displacements and deformations

The vertical displacement is shown in Fig. 4 forthe three models. This ¢gure is obtained by ad-

Fig. 3. Plume evolution for Model 3. Top: excess temperature. Middle: tracers advected forward. (i) Detail of the plume headshown in 3h. (l) Tracers advected backward.



vecting backward and forward in time 100 tracersin each grid element. For each model, the coloredregions contain the same material at both the ini-tial and the ¢nal time. The descending motion (indark gray) occurs on the periphery of the domainto replace the material pumped into the plume.

Backward tracers advection (top panels) showthat (i) the material with the highest vertical dis-placement is concentrated in the head, (ii) theinterface with the surrounding material with lowdisplacement is sharp on the top side of the head.Forward tracers advection (bottom panels) shows

Fig. 4. Vertical displacements over the interval [t0,t1]. Top: Plotted as a function of the ¢nal position, using backward advection.Bottom: Plotted as a function of the initial position, using forward advection. The color scale is the same for all panels.



that material at the base rises the most, except fora thin chemically denser layer in Model 3 (Fig.4f). In all models there is a sharp transition inthe thermal boundary layer between tracers withhigh vertical displacement and tracers with low(or negative) vertical displacement. Highest valuesalong the axis are due to the e¤cient vertical dis-placement in the plume tail. The reverse verticalordering of colors at the initial and ¢nal time isdue to the penetration of the plume across themantle with very little entrainment. The yellowand green regions consist of lower mantle materialwhich rise signi¢cantly but it remains screenedfrom the plume head by deeper material.

Fig. 5 shows the ¢nite-time Lyapunov expo-nents, or ¢nite-time deformation, plotted at the¢nal and initial time (top and bottom panels, re-spectively). It is clear that the strongest deforma-tion is highly localized. We recall that the defor-mation is the exponential of the Lyapunovexponent. For Model 1 (Fig. 5a) the most de-formed region corresponds to the stretched rimof the plume head where tracers of di¡erent radialdistance come in close contact. It is interesting tonote that high values extend also in the portion ofthe upper mantle that has been displaced by therising plume. In other words, the most deformedpart in the upper mantle is the frontier betweenthe head and the surrounding mantle in contactwith it. The fact that maximum stretched regionsare also dynamical barriers is often observed inchaotic geophysical £ows [24^26]. At the initialtime (Fig. 5b), the maximum stretched region islocated in a sloping annular layer which is alsomarking the boundary between the ascending andthe descending material in Fig. 4b. For Model 2(Fig. 5c) the viscosity contrast between upper andlower mantle signi¢cantly increases the deforma-tions. Once again the highest values correspond tothe boundary of the plume head. Since a tongue isleft behind in the lower mantle, and material isentrained between this tongue and the tail, theboundary is much more convoluted than forModel 1 and the whole head exhibits higher de-formation. It is apparent that the maximumstretched layer divides the plume head in severalsubdomains originating from di¡erent reservoirs.The location of the maximum stretched region at

initial time (Fig. 5d) is also more convoluted andsplitted in several parts corresponding again tothe lines of sharp variation of the vertical dis-placement in Fig. 4d. The same features areshown in Fig. 5e,f for Model 3 with smaller valuesof the Lyapunov exponent since the elapsed timeis much longer than in the other two cases, whilethe accumulated deformation, measured by c�,remains of the same order.

We ¢nd that for all models the deformations inthe plume tail are weaker than in the leadingplume head. We also ¢nd that the viscosity con-trast between the upper and lower mantle has themost dramatic e¡ect on the dynamics and on stir-ring, while the presence of the chemically denserlayer in Model 3 reduces the plume ascent andenhances the entrainment of lower mantle bytrapping the denser material in the foot of theplume.

3.3. Heterogeneities in the long-lived plume tail

The results presented above indicate that defor-mations in the plume tail may be lower than inthe plume head. Here we investigate in more de-tail the dynamics of a long-lived plume tail, bycarrying on the calculations for 80 Myr after theonset of partial melting.

The new model includes the mechanical e¡ectof the endothermic phase transition at 660 kmdepth, with Clapeyron slope Q=33 MPa/K. Thedensity contrast induced by the phase transition is8%, while the viscosity contrast between upperand lower mantle is one order of magnitude. Inorder to obtain plume excess temperatures around250³C, comparable with values inferred from pet-rology, the initial excess temperature across thethermal boundary is now 350³C. The calculatedbuoyancy £ux across the plume tail, at 250 kmdepth, is 7.3 Mg/s (after 20 Myr from the onsetof melting), 4.9 Mg/s (after 40 Myr) and 4.4 Mg/s(after 80 Myr). For comparison, the estimatedbuoyancy £ux for Hawaii is 8.7 Mg/s and for Ice-land is 1.4 Mg/s [28], thus indicating that we aremodeling a fairly strong and long-lived plume.

We advect packs of tracers randomly distrib-uted in the lowermost mantle with an arbitraryinitial size of 25U25 km, the total volume of



this heterogeneous material represents 10% of thethermal boundary layer volume. Fig. 6 shows theplume head 40 Myr after the onset of melting.The heterogeneities are reduced to narrow ¢la-

ments in the plume head, while they are less de-formed in the long-lived plume tail. Material froma 50-km thick zone just above the phase transition(gray color in Fig. 6) does not enter in the melting

Fig. 5. Lyapunov exponents over the interval [t0,t1]. Top panels: V3(x1) for backward integration. Bottom panels: V�(x0) for for-ward integration.



region but closely surrounds the plume head.Therefore, the phase transition does not representa barrier to the ascending £ow and entrainment ofupper mantle material is not enhanced. However,the joint e¡ect of a phase transition and of theviscosity contrast between upper and lower man-tle induces the formation of a `reservoir' of hotplume material ponding beneath the phase tran-sition. Similar to Model 2 the plume head is fedby a secondary conduit parallel to the axial con-duit. This complex structure may reduce the vol-ume of material satisfying the pressure^tempera-ture conditions required for partial melting. Toinvestigate which fraction of plume materialundergoes partial melting we advect thousand oftracers and at each time step we compare the realtemperature of each tracer with the correspondingsolidus temperature, given by the empirical for-mulation of Zhang and Herzberg [27] which best¢ts the melting data on anhydrous peridotite atpressures between 5 and 15 GPa: Ts = 100230.96P23465 ln(1/P), where P is the lithostatic pres-sure. We ¢nd that only the axial part of the plumehead partially melts, while the lower part of theplume head, fed by the tongue trapped in the low-er mantle, never melts. (For sake of clearness the

position of all the tracers crossing the zone ofpartial melting is not shown in Fig. 6).

Although the melting model is very crude andthe extent of the melting region depends on sev-eral parameters, including the thickness of thelithosphere, the pressure^temperature conditionsand the £uid content, this result suggests thatplumes e¤ciently inject deep heterogeneous mate-rial into the upper mantle and that just a fractionof it partially melts. Therefore, unmelted plumehead material may `pollute' the upper mantleand we speculate that it could be responsible forsome of the geochemical diversity of MORBs.

4. Conclusions

We have used a numerical model to investigatemixing and deformations for plumes that origi-nate from the thermal boundary layer at thebase of the Earth's mantle, while we did not con-sider plumes that may ascend from the top ofdome-like structures associated to large scale ther-mo-chemical convection [18,29]. Therefore weconsider that our models are more appropriatefor hotspots like Hawaii and Iceland, rather

Fig. 6. Plume head 40 Myr after the onset of melting. Black tracers: stretched heterogeneities initially in the thermal boundarylayer. Gray tracers: come from 660^620 km depth. The white contour surrounds the region of partial melting.



than for the numerous hotspots of the South Pa-ci¢c.

Our results indicate that

1. the thermal boundary layer, from whichplumes originate, is the region most e¤cientlysampled. Therefore, we may speculate that thegeochemically heterogeneous nature of mantleplumes is inherited at the source, rather thanbeing entrained during the ascent. Since we donot model magma ascent we cannot investigatethe role of lithospheric and crustal contamina-tion on the geochemical signature of hotspotlavas.

2. Upper mantle material is not entrained, but itis displaced by the plume, thus we predict thatpresent-day upper mantle, source of MORB, isnot involved in the production of plume lavas.Therefore, we may speculate that the geo-chemically depleted component in vigorousplumes, such as Hawaii and Iceland, is dueto the presence of subducted and recycled an-cient depleted material. This is in good agree-ment with geochemical observations for Ha-waii [4] and Iceland [2].

3. Material in the plume head is highly deformedand stirred by a series of stretching and foldingevents occurring during the plume rise. Heter-ogeneities initially present in the source regionmay be reduced to narrow ¢laments in theplume head. We also ¢nd that stirring and de-formation in the plume tail are lower than inthe plume head. Therefore, we may speculatethat a same pack of subducted/recycled crust+lithosphere may still show a distinct geochem-ical signature in hotspot lavas, consistent withgeochemical data [1].

4. By including a very crude melting model we¢nd that only part of a plume head partiallymelts. This allows us to speculate that hetero-geneous deep mantle material may be injectedin the upper mantle and may give rise to geo-chemical heterogeneities when sampled byspreading ridges, even at great distance fromthe plume axis.

5. The phase transition at 660 km depth does notrepresent a barrier to vigorous ascendingplume, in agreement with previous works

(e.g., [3]) and it does not have a profound ef-fect on stirring and entrainment of upper man-tle in the plume head. We ¢nd that materialjust above the transition zone (i.e., from 660to 620 km depth) closely surrounds the plumehead. Although in our simple model this`sheath' of material does not melt, we cannotexclude that it may melt in case of decompres-sional melting induced by rifting, such as forIceland.

6. The ¢nite-time Lyapounv exponents, indicatorsof deformations, show that the boundary be-tween the plume head and the upper mantleundergoes signi¢cant deformation. This couldinduce seismic anisotropy in the shallow uppermantle all around the large initial plume head.To our knowledge this testable hypothesis hasnot been investigated by seismologists.

Acknowledgements

We thank both anonymous reviewers for theirconstructive comments, and Patrick Stoclet for hisuseful technical advice.[AC]

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mixing and deformations in mantle plumestheir storage in the earth’s mantle for at least 1 billion...

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