Slabs in the lower mantle — results of dynamic modelling compared ...

244( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257Fig. 1. Slab input model: The figure shows locations, times and masses of slab ‘‘particles’’ inserted into the mantle at a depth 380 km. Itcorresponds to the subduction history model shown in Steinberger and O’Connell Ž 1998 ., which in turn is based on the data of Gordon andJurdy Ž 1986. and Lithgow-Bertelloni et al. Ž 1993 .. Areas of dots correspond to masses of particles, however, overlap between dots occurs.See text for details on how slab input model was obtained.calculated on the grid, and subsequently expanded inspherical harmonics up to degree and order 31.Spherical harmonic expansion is facilitated by thespecific choice of gridpoints. After this, the methodof Hager and O’Connell Ž 1979, 1981.is used tocalculate the flow field. This method essentiallycomputes a spherical harmonic expansion of the flowfield, given spherical harmonic expansions of thedensity field and surface plate motions, and a radialviscosity structure. A more detailed description ofthe method has been given by Steinberger andO’Connell Ž 1998 .. Žn.b.: There should be a minussigninstead of a plus-sign in the third lines of Eqs.Ž A2. and Ž A3.of that paper. These equations weremisprinted, but used correctly..Here, the original routines were modified to includethe effects of compressibility and phase boundaries:The latter are included as mass sheets on theundeflected boundary that correspond to the massanomalies caused by the phase boundary deflection.Two phase boundaries Žat depth 400 km with aproduct of Clapeyron slope and density jump 0.5P10 3MParKkgrm 3 and at depth 670 km with y0.3P10 3MParK kgrm 3. are included. The value at 400 kmis based on Akaogi et al. Ž 1989.for a Pyrolite mantlewith 60% Olivine content. The value at 670 km isgiven by Akaogi and Ito Ž 1999 .. It is smaller thanvalues assumed in many convection calculations, asit also includes the effects of the Majorite–Perovskitetransition, which occurs at a similar depthwith a positive Clapeyron slope. With these parameters,the effects of phase boundaries are found to berather small, in agreement with the results by Bungeet al. Ž 1998 .: They compared two convection simulations,one with rather strong phase boundaries andone without phase boundaries and found a better fit

( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257 245to Grand’s model in the case without phase boundaries.Including compressibility does not have a largeeffect either.Only slabs that have been subducted below 380km are used to calculate flow. Since the motion ofslabs above 380 km is calculated kinematically, itwould be the most consistent approach to use theslabs as kinematic boundary conditions for the flow.However, this has not been done since the numericalimplementation would take considerable effort.Moreover, since plate motions are already used askinematic boundary conditions at the surface, mantleflow already has a downward component of magnitudesimilar to convergence rate below subductionzones. Thus, including subducted slabs as furtherboundary conditions would presumably not changeresults by much. Whether or not slabs in the uppermost380 km are included in the flow calculationdoes not have much effect on the flow field in thelower mantle, hence on the results.Anomalous masses of slab particles are assumedto decrease proportional to their radial coordinateŽ sEarth radius minus depth ., thus approximatelyaccounting for the decrease of thermal expansioncoefficient with depth Ž Chopelas and Boehler, 1989 ..Results do not strongly depend on the exact depthdependence of anomalous mass chosen.2.3. Viscosity structureŽ .The method of Hager and O’Connell 1979, 1981enables flow calculation for a viscosity varying withFig. 3. Calculated present depth of slab particles vs. time ofsubduction Ž below 380 km .. Each dot corresponds to one slabparticle. The black line shows the average depth of slab particlesfor a given time of subduction.radius only. The viscosity structure used here isshown in Fig. 2, and is the same as in SteinbergerŽ 2000 ., where it was discussed that it is in agreementwith evidence from postglacial rebound and thegeoid, and a review of this evidence was given. Therelative values of lithospheric, asthenospheric andFig. 2. Viscosity structure used, and corresponding geoid kernels Žcalculated for a free upper boundary with the method described byPanasyuk et al., 1996.for spherical harmonic degrees 2–12, graytone corresponding to degree.

246( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257lower mantle viscosity were especially chosen toallow an acceptable fit of the observed geoid; kernelsare very similar to the viscosity structure for whichRicard et al. Ž 1993.found an optimum fit of thegeoid, using a similar, but somewhat simpler modelof subduction. Specifically, geoid kernels that aresignificantly negative in the lower mantle are requiredto predict a geoid similar to the observed one.Obviously, the neglect of lateral viscosity variationsis one of the fundamental limitations of thisand other studies on this subject. Lateral viscosityvariations are, however, poorly known and difficultto implement numerically Ž e.g., Gurnis et al., 1998 ..Moreover, like the sinking speed of a viscous spherethat is mostly determined by the viscosity of thesurrounding fluid, it can be expected that the sinkingspeed of slabs depends mostly on ambient mantleviscosity and not so much on the internal slab rheology,in which case the use of a viscosity structurevarying with radius only would be reasonable.2.4. Motion of slab particles in mantle flowIn principle, the motion of slabs can be calculatedby simply interpolating the flow field from the gridto the locations of the slab particles. However, themodel of plate boundaries used starts at 120 Ma,whereas in the real Earth, there has been subductionfor much longer. Therefore at early times our slabmodel would significantly underpredict flow speedsand the calculated slab sinking speed would be tooslow. To compensate for this, and to obtain a morerealistic initial condition, we start the calculation 220Ma ago, with constant subduction locations for thefirst 110 Ma, similar to Bunge et al. Ž 1998 .. Whendiscussing the distribution of slabs, inferred densityFig. 4. Calculated vertical flow component at depth 0.72r . Downward flow only is shown. Thick black lines indicate locations of crossEŽ .sections shown in Figs. 7–9. Letters s,t,u indicate places mentioned in the text.

( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257 247distribution and comparison with tomography andthe geoid Ži.e., for the results displayed in Figs.5–11 ., we will however only use slabs inserted after120 Ma, which are based on an actual plate motionhistory model.3. Results of mantle flow and slab distribution3.1. Sinking speed of slabsThe previously found good correlation betweenlowermost mantle seismic heterogeneity and Cenozoicand Mesozoic subduction history Že.g., Richardsand Engebretson, 1992.suggests that slabs take ofthe order of 100 Ma to sink to the lowermost mantle.Our results are in accordance: Fig. 3 shows that slabsthat have been subducted 120 Ma ago or earlier havesunk to within a few hundred kilometers of the CMBon average. The average depth of slabs subductedlater than 120 Ma as a function of subduction timeindicates an average sinking speed of the order of2000 kmr120 Maf1.7 cmryear. This value isprobably slightly over-estimated, because a constantsubduction location prior to 110 Ma was assumed,and in this case, sinking speeds tend to be higherthan in the case of rapid trench migration, as will bediscussed below. Calculations for a variety of othermodels Žwith different viscosity structures, phaseboundary parameters, thermal expansivity, etc..yielded rather similar results. The largest uncertaintiesarise from the rather poor knowledge of lowermantle viscosity, which might be a few times higheror lower, but even so, comparison of various modelresults indicate uncertainties in average sinking speedof a few tens of percent at most.Lateral variations of sinking speed are, however,much larger, and thus need to be considered whencomparing predicted density anomalies with resultsfrom seismic tomography. Fig. 4 shows the verticalcomponent of the mantle flow field at a depth halfwayFig. 5. Calculated present distribution of slabs. Areas of dots correspond to masses of particles, however overlap between dots occurs.Ž .Letters n,o,p,q,r indicate places mentioned in the text. Only slabs that have been inserted into the mantle below 380 km depth are plotted.

248( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257between transition zone and CMB: The maximumsinking speed is about twice the average slab sinkingspeed inferred from Fig. 3. Correspondingly, thefastest sinking slabs reach the lowermost mantleafter only about 60 Ma, as can also be seen in Fig. 3.High sinking speeds tend to occur in regions withlarge amounts of subduction and little trench migration.Fig. 6. Predictions of density anomalies in the mantle from subduction history compared with tomographic models — depth averages offour different mantle depth intervals. Only positive anomalies are shown. Thick black lines indicate locations of cross-sections shown inFigs. 7–9. Letters Ž a,b,c,d,e,f,v,w.indicate places mentioned in the text.

( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257 249Ž .Fig. 6 continued .3.2. Magnitude and lateral distribution of densityanomalies due to slabsFig. 5 shows the calculated present distribution ofslabs: in order to compare results with tomographicmodels of wave speed, it is however more useful tocompute the corresponding density field: Because ofthe uncertainties in our model parameters, hence slabsinking speed, slab locations may be predicted somewhattoo deep or too shallow. It is therefore moreappropriate to compare not at given depth levels, butaveraged over certain depth intervals: In Fig. 6, the

250( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257mantle is divided in four intervals spanning 600 kmeach. The p-wave model of Karason ´ et al. Ž 1997 .,and the latest s-wave model by S. Grand Ž1998, pers.comm. ., which is similar to Grand et al. Ž 1997.areused for comparison. These models were chosenbecause of their high resolution, but the choice doesnot imply judgment regarding the quality of anymodel. These models were converted from the originalgrid to spherical harmonic coefficients and subsequentlyre-evaluated in space domain, using coefficientsup to degree and order 31, as for the slabmodel.The computed magnitude of the density anomaliesis up to about 0.5% in the upper three layers andsomewhat larger in the lowermost mantle. For comparison,below 500 km depth the anomalies of thes-wave model have similar magnitude Žexcept in thelowermost mantle, where they are larger ., theanomalies of the p-wave model are even smaller.Based on mineral physics arguments, however, relativedensity anomalies should be of the order of0.2–0.3 times relative s-wave speed anomalies, or0.4–0.5 times relative p-wave anomalies Že.g.,Karato, 1993 ., indicating that either the anomalies ofthe tomographic models are too small or the predicteddensity anomalies are too large. Compared toother tomographic models, Že.g., Su et al., 1994; Liand Romanowicz, 1996.the amplitude of the modelsby Grand Ž 1998. and Karason ´ et al. Ž 1997.is indeedsomewhat low.In the uppermost layer Ž 500–1100 km ., the predicteddensity anomaly in most places represents therecent subduction history, with positive anomalies inthe regions surrounding the Pacific, and in an areastretching from the Mediterranean to Indonesia, correspondingto the former Thetys subduction. Most ofthe positive velocity anomalies of the seismic modelsare also in these regions. The models also agree insome of the details: all three models show an elongatedanomaly stretching through the Americas, withFig. 7. Predicted density anomalies and flow due to subduction atthree different times, and seismic p- and s-wave anomalies underNorth America — vertical cross-section at 388N. Length ofarrows corresponds to total flow in 20 Ma; only positive anomaliesare shown. A thick black line indicates where flow is displayedin Fig. 4.

( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257 251a maximum beneath the northern part of SouthAmerica Ž. a . Also, all three models show a maximumbeneath Indonesia Ž. b . In many other places,the models show considerable differences, which canbe due to a variety of reasons: The interpretation ofseismic anomalies in terms of density anomalies isnot obvious, because p- and s- wave anomalies donot correlate well everywhere, as seen in Fig. 6; thesubduction history model used may be inaccurate inplaces; the mantle flow model used may be toosimplistic.In the next layer Ž 1100–1700 km ., the maximumbeneath the Americas is further north — roughlybeneath the Caribbean in all three models Ž. c . Themaximum of the anomaly corresponding to Thetyssubduction is further west — roughly beneath Indiain all three models Ž. d . Again, the models disagree inother areas for the reasons mentioned above.In the lower part of lower mantle, the predicteddensity anomalies remain in similar areas Žbelow theThetys region and around the Pacific.and an additionalanomaly corresponding to the Mesozoic subductionbetween Izanagi and Farallon plates in theNorthern Pacific Ž. e becomes prominent. Also, as ageneral feature, when going down in the mantle, thepredicted anomalies become more patchy and lesslinear features. Comparison with seismic anomaliesbecomes even more difficult in the lowermost twolayers: p- and s-wave anomalies look considerablydifferent, which may be due to chemical heterogeneitiesŽ Kennett et al., 1998 .. The interpretation ofseismic anomalies in terms of density anomaliestherefore becomes less obvious. Also, the lowermostmantle must contain the remains of slabs subductedearlier than 120 Ma ago, which are not included inthe model. Especially in the lowermost layer, wecannot expect that the model would predict all densityanomalies. For example, although the predicteddensity anomaly beneath Asia extends further westthan the locations of subduction during the past 120Fig. 8. Predicted density anomalies and flow due to subduction atthree different times, and seismic p- and s-wave anomalies underEast Asia — vertical cross-section at 388N. Length of arrowscorresponds to total flow in 20 Ma; only positive anomalies areshown. A thick black line indicates where flow is displayed inFig. 4.

252( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257Ma, the anomaly seen in both tomographic modelsbeneath Asia is considerably larger in extent andstretches even further west Ž. f . This is most likelydue to earlier subduction, as discussed by van derVoo et al. Ž 1999 ..3.3. Cross-sections through mantleIn the following, some cross-sections through theflow and density fields are shown: The regions displayedare on the surface of a cone, but they aremapped on to rectangles for convenience. Thecross-sections only show vertical and east–westcomponent of the flow, although flow calculationsare done in three dimensions in spherical geometry.A cross-section beneath North America is shownin Fig. 7, because the eastward dipping positiveanomaly in both p- and s-wave speed has been oneof the first strong evidences for slabs penetrating intothe lower mantle Ž Grand, 1994 .. Here, it is thereforetested whether such a feature can be reproduced. Thecalculated density anomaly is in a similar locationand has a similar shape: It also dips eastward, however,it is not as straight and dips less steeply. Thismight indicate that in this region the model somewhatunderpredicts the slab sinking speed. The panelat 40 Ma ago shows the effect of the phase boundaryin the calculations: Slabs pile up at the Spinel–Perovskitetransition before they eventually penetrateinto the lower mantle.Fig. 8 shows a cross-section beneath East Asia.Again, the prediction and the tomographic imagesshow an anomaly in a similar region and of similarshape: beginning in the upper mantle beneath JapanŽ ;1408E.it stretches further west across the transitionzone, and is more or less vertical in the lowermantle, going all the way to the CMB. Again, especiallyin the lowermost mantle, the predicted anomalyis not as extensive as observed, as the model doesnot include subduction earlier than 120 Ma. In thisFig. 9. Predicted density anomalies and flow due to subduction atthree different times, and seismic p- and s-wave anomalies underAustralia — vertical cross-section at 388S. Length of arrowscorresponds to total flow in 20 Ma; only positive anomalies areshown. A thick black line indicates where flow is displayed inFig. 4.

( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257 253region, the calculated effect of the phase transition isvery small.Finally, Fig. 9, showing a cross-section south ofAustralia, rather serves to indicate problems still tobe overcome: Here, the tomographic models do notagree well, and the predicted density anomaly doesnot resemble either of them.4. Slab distribution and the geoidThe theory relating mantle density heterogeneitiesand geoid has been developed by Richards and HagerŽ 1984. and Ricard et al. Ž 1984 .. Hager Ž 1984.showedthat a good fit to the geoid at spherical harmonicdegrees 4–9 can be achieved if these density anomaliesare inferred from the locations of subductedslabs. More recently, Ricard et al. Ž 1993.were ableto explain 73% of the first eight spherical harmonicdegrees of the geoid, using a subduction historymodel. Both studies indicated a significant viscosityincrease with depth Ž factor 30–100 .. Here, geoidkernels are computed with the method described byPanasyuk et al. Ž 1996 .; the geoid is computed byconvolving these kernels with the density anomalyinferred from slabs. There is a slight inconsistency inthis procedure, as the geoid kernels were computedfor a ‘‘free’’ upper boundary Žalthough with a lithosphere15 times more viscous than the mantle beneath., whereas the flow was calculated with thegiven plate motions as boundary condition. Theboundary condition for the geoid was chosen becausein this way the best fit can be achieved, as waspreviously noted Ž e.g., Thoraval and Richards, 1997 ..However, the inconsistency is not a serious one, asthe results do not depend much on the assumedboundary condition. Even for a fixed surface boundarycondition Ži.e., zero plate motions as boundarycondition for the flow, but the same slab inputmodel.results look rather similar. Because of thelow viscosity in the uppermost mantle, in the modelplate motions are not strongly coupled to flow in thelower mantle, and motion of the slabs in the lowermantle depends mostly on the density anomaly of theslabs themselves and not much on the boundarycondition. In Fig. 10, predicted and observed geoidare compared. There is good visual agreement inessential features: localized geoid highs over Indone-sia Ž. g and South America Ž. h , and a rather broadgeoid high over Africa Ž. i , geoid lows over northeasternNorth America Ž. j , Siberia Ž k ., the IndianOcean Ž. l , and Antarctica Ž m ..A more qualitative comparison can also be madedirectly between the slab distribution Ž Fig. 5.and thegeoid: Slabs primarily in the lower mantle Žsuch asbelow the south of India Ž. n , central Siberia Ž. o ,Arctic Canada Ž.. p tend to be associated with geoidlows, whereas slabs primarily in the upper mantleŽsuch as below New Guinea and surroundings Ž q .,South America Ž.. r tend to be associated with geoidhighs. Again, this points toward geoid kernels thatare negative in the lowermost part of the mantle andpositive in the upper part. The viscosity structureused here yields geoid kernels that show these characteristicsat least for low spherical harmonic degrees.Predicted and observed geoid have similarmagnitude, however, predicted geoid highs are higherthan observed, predicted lows are not as low asobserved. One reason for this discrepancy may bethat geoid kernels ought to be less positive in theupper mantle and more negative in the lower mantlethan the ones used here.No attempt was made to optimize the fit to thegeoid, with a more systematic search through parameterspace. This would be beyond the scope of thispaper, especially since this model does not includeall density anomalies expected in the mantle: subductedslabs originate at the upper thermal boundarylayer of the mantle, but there is also a lower thermalboundary layer that presumably gives rise to densityheterogeneities in the form of mantle plumes. Becauseof higher viscosities in the lowermost mantle,it can be expected that plume conduits in the lowermostmantle are significantly thicker than closer tothe surface. If, as in this model, geoid kernels arenegative in the lowermost mantle at low sphericalharmonic degrees, a high number of plumes per areawill correspond to geoid highs at low spherical harmonicdegrees. The observed distribution of hotspotsand calculations of the shape of plume conduitsŽ Steinberger, 2000.suggest that plume conduits inthe lowermost mantle are particularly frequent undersouthern Africa and the South Central Pacific, whichare in fact regions where the predicted geoid is, on alarge scale, lower than observed. A positive correlationof the residual geoid Žafter subtracting the ef-

254( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257Fig. 10. Geoid ‘‘predicted’’ by convolving the calculated density anomaly due to subducted slabs with the kernels shown in Fig. 2. TheŽ .observed geoid is shown for comparison. Letters g,h,i,j,k,l,m indicate places mentioned in text.

( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257 255fects of subducted slabs.with the hotspot distributionhas also been previously reported Že.g., Richardset al., 1988 ..5. ConclusionsThe results presented here do not differ dramaticallyfrom previous results. The average sinkingspeeds that were determined here are similar to whatpreviously has been assumed. In most cases, lateralmotion of slabs was found to be small, as shown inFig. 11, thus largely justifying the assumption ofvertical sinking that was used in some previousmodels. Lateral variations of sinking speed are, however,significant: The fastest sinking speeds result,where a lot of anomalous slab mass is subducted inthe same area Ž i.e., without much trench migration ..Fig. 4 shows the present maxima of sinking speed inthe mid-mantle beneath ChinarSouth East Asia Ž. s ,the Caribbean Ž. t , and the Scotia Sea Ž u ..In regions where the sinking speed has been fast,the model predicts that slabs subducted during thepast 120 Ma have already sunk close to the CMBand spread laterally: Examples can be seen in Fig. 6beneath Asia Ž. f , the northern Pacific Ž. e , and theAtlantic east of the Caribbean Ž. v , as well as theFig. 11. Horizontal displacement of subducted slab particles betweentheir initial location at depth 380 km and their presentlocation. Each gray dot corresponds to one slab particle. The blackline shows the average displacement of particles in sphericalshells of thickness 0.01 Earth radii.South Atlantic Ž w .. In these areas, strong positiveanomalies are predicted in the lowermost mantle,without subduction having occurred immediatelyabove in the slab input model Ž Fig. 1 .. In accordance,both p- and s-wave models show large positiveanomalies in the same region beneath Asia Ž. f ,and positive anomalies in a similar region beneaththe northern Pacific Ž. e ; the s-wave model also haspositive anomalies in the same two regions beneaththe Atlantic Ž. v and Ž w .. A rather large offset betweensubduction locations and slab location in thelowermost mantle is also predicted in the Indianocean Ž. n . Again, there are positive p- and s-waveanomalies close to where the slabs are predicted.Both tomographic models also have some localizedstrong negative velocity anomalies Ž not shown .,which are most likely unrelated to subduction. Additionally,the tomographic models have strong positiveanomalies in some regions of the lowermostmantle where none are predicted, and which arelikely due to earlier subduction. A significant shortcomingof this and other slab models is that themodels of plate motion history and hence subductionused start 120 or 180 Ma ago. Our knowledge of thehistory of plate motion before that is not very good,and even the model used after 120 Ma, is probablywrong in some places. Fortunately, it may not benecessary to know subduction history accuratelymuch earlier than about 150 Ma ago, as the results ofBunge et al. Ž 1998.reveal a 150-Ma time scale forgenerating thermal heterogeneity in the mantle. Thehigh-resolution tomographic images can now help tofurther constrain models of plate motion and subductionhistory; an example shown here concerns thesubduction zone in the Northern Pacific, betweenIzanagi and Farallon plates, implied by Mesozoicplate reconstructions. The location of this subductionzone is not well constrained by plate reconstructionsalone and the good agreement between predicted andobserved anomalies Ž. e in that region helps to confirmthe location. A more satisfactory approach thanusing a given subduction history model, as done hereand in most previous studies, would be to vary platereconstructions within their uncertainties and testwhich model best predicts tomography. Such anapproach was taken by Gurnis et al. Ž 1998.for aparticular region; to do this on a global scale will bea considerable interdisciplinary research effort and is

256( )B. SteinbergerrPhysics of the Earth and Planetary Interiors 118 2000 241–257beyond the scope of the present paper, which ismainly intended to demonstrate the feasibility ofsuch an effort.The slab model does not strongly depend ondetails of the viscosity structure chosen. It can onlybe inferred that the average viscosity of the lowermantle must not be much less than about 10 22 Pa.For the viscosity structure chosen, which reaches avalue of 4P10 22 Pa in the lower mantle, slabs haveon average sunk into the lower half of the lowermantle after about 100 Ma, but some slabs comeclose to the CMB already after about 60 Ma Žsee Fig.3 .. For a viscosity much lower than that, modelresults show slabs reaching the lowermost mantlefaster and subsequently spreading out close to theCMB, before they would presumably get thermallyequilibrated — a result not in agreement with theobservation that positive lowermost mantle seismicanomalies tend to be beneath areas of Mesozoicsubduction and not much displaced laterally. If, onthe other hand, a viscosity contrast much higher than100 between upper and lower mantle is chosen,predictions of the geoid become less satisfactory.Although there are still considerable differences, aconsensus between various current models of mantleviscosity, inferred from models of postglacial reboundand the geoid Že.g., Mitrovica and Forte,1997; Lambeck and Johnston, 1998 ., and fromhotspot motion ŽRichards, 1991; Corrieu and Ricard,1998; Steinberger and O’Connell, 1998.begins toemerge. All those models indicate a significant viscosityincrease with depth, with absolute viscosity inthe upper mantle typically a few times 10 20 Pa andlower mantle viscosity of the order of 10 22 Pa ormore. The viscosity model used here is in agreementwith these general features.AcknowledgementsSpecial thanks goes to Svetlana Panasyuk forproviding the MATLAB code to calculate geoidkernels, and to Steve Grand and Rob van der Hilstfor sharing their most recent tomographic models viaanonymous ftp. A substantial part of this work wasdone during a postdoctoral fellowship at AcademiaSinica Ž Taiwan .. Some preliminary work was donewith Rick O’Connell, who also provided variousFORTRAN routines and datasets for past plate motionsand geometries, and whom I wish to thank forcontinuing encouragement to pursue this work. Furtherinformation and suggestions were given by ArtCalderwood, Gabi Marquart, Harro Schmeling,Manuela Tetzlaff and Catherine Thoraval. Commentsby Yanick Ricard and an anonymous reviewer areappreciated. Figures were prepared using GMTgraphics Ž Wessel and Smith, 1995 ..ReferencesAkaogi, M., Ito, E., 1999. Calorimetric study on majorite–perovskitetransition in the system Mg 4Si 4O 12–Mg 3Al2Si 3O 12:transition boundaries with positive pressure–temperatureslopes. Phys. Earth Planet. Inter. 114, 129–140.Akaogi, M., Ito, E., Navrotsky, A., 1989. Olivine-modifiedspinel–spinel transitions in the system Mg 2SiO 4–Fe2SiO 4:calorimetric measurements, thermochemical calculations, andgeophysical application. J. Geophys. Res. 94, 15671–15685.Bunge, H.-P., Richards, M.A., Lithgow-Bertelloni, C., Baumgardner,J.R., Grand, S.P., Romanowicz, B.A., 1998. Time scalesand heterogeneous structure in geodynamic Earth models.Science 280, 91–95.Chopelas, A., Boehler, R., 1989. Thermal expansion measurementsat very high pressure, systematics, and a case for achemically homogeneous mantle. Geophys. Res. Lett. 16,1347–1350.Corrieu, V., Ricard, Y., 1998. Hotspots and mantle dynamics.Geophys. J. Int., submitted.Gordon, R.G., Jurdy, D., 1986. Cenozoic global plate motions. J.Geophys. Res. 91, 12389–12406.Grand, S.P., 1994. Mantle shear structure beneath the Americasand surrounding Oceans. J. Geophys. Res. 99, 11591–11621.Grand, S.P., van der Hilst, R.D., Widiyantoro, S., 1997. Globalseismic tomography: a snapshot of convection in the Earth.GSA Today 7, 1–7.Gurnis, M., 1993. Phanerozoic marine inundation of continentsdriven by dynamic topography above subducting slabs. Nature364, 589–593.Gurnis, M., Muller, ¨ R.D., Moresi, L., 1998. Dynamics of Cretaceousvertical motion of Australia and the Australian–Antarcticdiscordance. Science 279, 1499–1504.Hager, B.H., 1984. Subducted slabs and the geoid: constraints onmantle rheology and flow. J. Geophys. Res. 89, 6003–6015.Hager, B.H., O’Connell, R.J., 1979. Kinematic models of largescalemantle flow. J. Geophys. Res. 84, 1031–1048.Hager, B.H., O’Connell, R.J., 1981. A simple global model ofplate dynamics and mantle convection. J. Geophys. Res. 86,4843–4867.Karason, ´ H., van der Hilst, R.D., Creager, K., 1997. Improvingseismic models of global p-wave speed by the inclusion ofcore phases Ž abstract .. EOS Trans AGU 78 Ž 46 ., F17, FallMeet. Suppl.

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