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JOURNAL OF GEOPHYSICAL RESEARCH, VG ,. 96, NO. B$, PAGES 8407-8415, MAY 10, 1991 Differential Rotation BetweenLithosphereand Mantle' A Consequenceof Lateral Mantle Viscosity Variations YANICK RICARD 1 Dipartimento di Fisica, Settore Geofisica, Universitd di Bologna, Italy CARLO DOGLIONI Dipartimento di Scienze Geologiche, Universitd di Ferrara, Italy ROBERTO SABADINI2 Dipartimento di Fisica, Settore Geofisica, Universitd di Bologna, Italy The descriptionof plate motions in the so-calledhotspot referenceframe introducesa global rotation of the lithospherewith respectto the mantle. This rotation, called toroidal field of degree 1, is roughly westward. It reachesan amplitude of about 2 cm/yr and has been consistently found in the different generation of plate tectonic models. Various authors have tried to relate this observation to the deceleration of the Earth's rotation, to polar wander, or to tidal drag. However, these different physical mechanismscannot explain the requested amplitude. In this paper, we comparethe valuesof this rotation vector using different relative plate motion models expressedin the hotspot referenceframe. In a model Earth with lateral viscosity variations, a differential rotation is predicted. The observednet lithospheric rotation is consistent with the dynamics of a model Earth where the asthenosphericviscosity below the oceans is at least one order of magnitude lower than underneaththe continents. This relative westwarddrift of the lithospheremay account for the significantstructural differencesbetween east or west dipping subduction THE zones. DIFFERENTIAL LITHOSPHERE ROTATION BETWEEN equation: AND MANTLE f•L=3/VLx,.dS, (1) 2S0 The motions between plates are computed from three independent data sets. The first consists in the spreading where ur is the unit radial vector and So is the Earth's surrates on ridges deduced from the Earth's magnetic inver- face. This net rotation can also be deduced from the expansions recorded in the frozen mid-oceanic basalt. The secsion of surface velocities in vector spherical harmonics. The ond set includes the strikes of the transform faults which rotation is simply proportional to the toroidal coefficientsof are assumedto be parallel to the relative motions. Finally, degree 1. the earthquakes slip motions help to constraint the motions Minster et al. [1974]havecomputedthe lithosphericrotaparticularly at shear or converging boundaries. From these tion of their AM1 model. It amountsto 0.11ø/m.y. around data, only relative motions can be deduced. The choice of a pole situated at 129øE and 74øS. This correspondsto a an origin in the angular velocity space specifiesan absolute maximum velocity of 1.2 cm/yr. We compute this average reference frame. Of course, the most natural frame would rotation for the model AM1-2 [Minster and Jordan, 1978] be the one in which the deep mantle has no rotation. Differexpressedin the hotspot reference frame. In that case, the ent criteria have been proposed to define practically such a lithospheric rotationreaches 0.26ø/m.y.(2.8 cm/yr) around frame. Le Pichon[1968]implicitly suggests that the Antarc- 66øE and 54øS. We also used the HS2-NUVEL1 model tic plate remainsfixed. FollowingBurke and Wilson[1972], [DeMets et al., 1990; Gripp and Gordon, 1990]. Follow- it is the African plate that could be stationary with respect ing their results, the net lithospheric rotation should be of to the underlying mantle. The most widely accepted ref0.32ø/m.y.(3.6 cm/yr) around64øEand 52øS. erence frame is the one in which the hotspots remain fixed The last two models(AM1-2 and HS2-NUVEL1) have [Wilson,1965; Morgan,1971]. Thesethree hypotheses,al- been computed using different relative motion models but though conceptually different, lead to the same conclusion: fitting the same propagation rates of five volcanoesand the the lithosphere has a west trending average rota•ion. same trends of nine chains. The difference in their lithoQuantitatively, this lithospheric rotation f•œ c n be com- spheric rotations is thus only related to the difference in putedfromthe surface platevelocityVœby the followingthe relative velocity models. The selected hotspots were only located on, or very close to, the Pacific plate. On the 1Nowat Ddpartement de Gdologie, EcoleNormaleSup•- contrary, the AM1 model was constructed using 20 hotspot tracks sited on eight different plates. However, for the slow rieure, Paris, France. :•Alsoat Istitutodi Mineralogia, Universit& di Ferrara, moving plates, the actual azimuth of the hotspot chains cannot be precisely observed. Ferrara, Italy. To be sure that our estimation of the net lithospheric rotaCopyright1991by the AmericanGeophysical Union. tion is not strongly biased by the weight of the Pacific area nor unrealistically influenced by the selection of imprecise Paper number 91JB00204. traces, we perform another inversion. We only select traces 0148-0227/91/91JB-00204505.00 8407 8408 RICARD ET AL.: DIFFERENTIAL ROTATION OF THE LITHOSPHERE samphngdifferent plates where the absolutevelocity is a pri- relative plate motion modelsare only valid for the present ori knownto be of order or larger than 1.5 cm/yr. The traces time and the very last million years, whereasthe hotspots are all taken from the data that entered in the construction fixity applied for a much longertime. of the AM1 model. There are also consistent with the obserFigure2 showsthe pole,labelleda, of the net lithospheric vationsof Morgan [1972]. The inversionis performedwith rotation deduced from our inversion. This rotation is clearly the 14 hotspots listed in Table 1. As Minster and Jordan related to the total motion of the Pacific plate. The three [19781,we useboth observedvelocities(five data), and ob- poles labelled b, c, and d correspondingto modelsAM1, servedazimuths(14 data). The first nine traceswereused AM1-2, and HS2-NUVEL1 are also plotted with a circle for the determination of AM1-2 and HS2-NUVEL1; we add five more to ensurea better geographicalcoverage.We chose NUVEL-1 for relative plate motion model and we only readjust the global rotation to match our selecteddata. Our inversion indicates that the lithosphere has a net rotation of whosesize is proportional to the amphtude of the motion. The solid circle will correspond to the result of a model and will be discussedlater. Due to map distortion at high latitudes, the different poles look rather distant, but their angulardistanceis at most300. On the basis of geologicalobservations,various authors 56øS. In columns6 and 7 of Table 1, we show the velocity havealsoadvocatedfor this differentialrotation[Nelsonand and azimuth of the chosenhotspot traces according to our Temple,1972; Uyedaand Kanamori, 1979; Do91ioni1990]. model. All the computedvaluesbut two (Marquesasand Two types of thrust belts have to be distinguished,whether Galapagoson the Nazcaplate) lie within the prescribedun- they are related to westor eastdippingsubductions,that is, certainties. Figure 1 shows the localization of our hotspots whether they contrast or follow the relative eastwardrotaon top of a map depicting the tectonic plates taken into ac- tion of the mantle. The main differences are summarized in count by NUVEL-1. The hotspot velocities deduced from Figure 3. Of course,thesetwo typesof thrust belts should our inversion are also plotted with the observeddirections be consideredtwo end-members;obhqueand lateral subductions must be further distinguishedin between. and a priori uncertainties. West dipping subductionshave a steepinclinationof the Gar[unkelet al. [1986]have suggested that the discrepto backarcbasins(e.g.,WestPacific, ancy between the hotspot reference frame and the no-net slabandare associated rotation frame is a bias due to an overestimate of the miBarbados, Sandwich, Apennines, Carpathians). Thrust gration rates for the Pacific volcanoes.However,the AM1 belts related to this kind of subduction(contrastingthe reliefs, model was constructedby only fitting the trends of hotspots mantle flow) showlow structuraland morphologic without taking into account their absolute velocities. Simi- shallow upper crust rocks, very consistentforedeepgenerlarly, we alsoperformed an inversiononly usingthe azimuths ated by the roll-back of the subductionhinge, and coeval of our selected hotspots. A global rotation was still found back arc extensionwhich is eastwardpropagatingand eating is with a rotation pole consistantly located in the southern the accretiondrywedge. The area of active compression part of the Indian Ocean but with the smaller amplitude of very narrow, usually a few tens of kilometers. In thesewest dippingsubductions,the baseplate detachmentis nevercon1.0 cm/yr. Although different, the estimationsof the net lithospheric nected to the surface but rather folded and subducted. West rotation agreewith a roughly westwardrotation with a pole dippingsubductionsare alsocharacterizedby arcswith their located in the southern part of the Indian Ocean and an major convexity oriented toward the east, suggestingto be observedvelocity of a few centimeters per year. The Pacific obstacles to the westward flow of the lithosphere. hotspots seem to have a relative velocity with respect to East dipping slabs have a shallow dip and are not as- 0.15ø/m.y. (1.7 cm/yr) arounda polesituatedat 84øEand the other hotspots which explains the discrepancy between the models. However, it should be clear that this exercise cannot be a test of the hypothesis of hotspots fixity. The sociatedto back arc extensionalbasins (e.g., American Cordillera,WesternAlps, Dinarides,Zagros).Thrust belts relatedto thiskindof subduction (followingthemantleflow) TABLE 1. Hotspot data used to compute the Absolute Motion Model Name Plate Observed Observed Computed Computed Longitude Latitude Velocity, Azimuth, Velocity, Azimuth, øE ON cm/yr NøE cm/yr NøE Hawai Pacific -155 20 10.04-2. -644-10 8.4 -59 Marquesas Pacific -138 -11 9.8 4-2. -454-15 9.2 -65 Tahiti MacDonald Pitcairn Juan de Fuca Pacific Pacific Pacific Pacific -148 -140 -130 -130 -18 -29 -25 46 11.04-2. 10.54-2. 11.04-2. -654-15 -554-15 -65 4-15 -544-15 9.1 8.9 9.1 5.0 -63 -64 -67 -46 Galapagos Galapagos Coco Nazca -92 -92 -1 -1 454-10 954-10 8.8 5.1 43 81 - 110 - 17 -11 -11 56 - 14 45 65 -37 -38 -21 -8 - 120 4- 20 -43 4- 30 -734-30 474-20 454-10 55 4-10 2.0 1.7 1.9 1.7 1.4 1.5 - 101 -45 -87 60 40 58 Yellowstone Iceland Tristan Da Cunha Tristan Da Cunha Reunion Ascension N.America N.A merica S.America Africa Africa Africa The observed azimuths and rates with their uncertainties are shown in comparison with the predictions deduced from our inversion. RICARD 90ø 0 ET AL.: DIFFERENTIAL 90ø 8409 270ø 360 ø 90 ø • 45ø /' 00 45 ø 90 ø 0ø OF THE LITHOSPHERE 180 ø 45ø 0 ROTATION • I 90 ø 45 ø I 180ø I 270 ø 90 ø 360 ø Fig. 1. Selectedhotspotsused in our computationof the global lithosphericrotation. The azimuth of the observedtrends with their estimated uncertainties are also plotted. The arrows correspond to the velocities deduced from the NUVEL-1 amplitudes are listed in Table 1. model. Their showhugeexposures of basementrocks,highstructurMand the overthrusting lithosphere can account for this observamorphologicreliefsin contrastto limited and usuMlyshal- tion (Figure 3). In the east dippingsubductionsthe basal detachment can bring to the surfacedeeply buried materials. On the contrary, the metamorphosedrockssink into the mantle under West-dipping subductions. Oceanic ridges, continental rift zones, and subduction The two kinds of subductions provide different metamorphic paths for the relative thrust belts. Only in the east trenchesare generally perpendicular to the differential velocdippingthrust belt, coesite-pyrope bearingassemblages and ity field depictedin Figure 2. The main deviationsfrom this eclogites havebeenfound[Chopin,1984; Wanget al., 1989], rule are only found in the North Atlantic ridge, the western indicatingthat confiningpressures between20 and 30 kbar portion of the southwest Indian ridge, and the Philippine are reached. The position with respectto the eastwardman- subduction arc. On the contrary, the major shear zones tle flow of the decollementplane betweenthe subductedand appear to be parallel to the observeddifferential velocity. low foredeep. The Himalayan chain belongsgeologicallyto the samegroup, Mthoughit would appear as parallel to the global rotation depictedin Figure 2. 90o0 ø 90 ø 180ø 270 ø 360 ø 90 ø , 45 ø 0ø 45ø 45 ø 90ø 0ø , 90 ø 180ø ••/ / / 270 ø ] 90 ø 360 ø Fig. 2. Net rotation of the lithosphere with respect to the deep mantle. The rotation pole has been deduced from the observations of 14 hotspot tracesusing the relative motion model NUVEL-1 (a). The maximum velocity reaches1.7 cm/yr. We also showthe lithosphericrotation polesof the differentmodelsAM1 (b), AM1-2 (c), and HS2-NUVEL1 (d) with a circlewith sizeproportionalto the predicted rotation amplitude. The solid circle shows the prediction of a model with will be discuss later. 8410 RICARDET AL.: DIFFERENTIALROTATIONOF THE LITHOSPHERE w •:'-:'•..:!;i•ii tothelithosphere Fig. 3. Comparative sketch between thewestandeastdipping subductions connected to therelative eastward mantle motion. In thewest dipping subduction case, theslabactsasanobstacle totheeastward mantle flowandback arcextension develops dueto thelithospheric loss.Thiswillproduce aneastward migration ofthetectonic setting anda pronounced foredeep. Thebase plate detachment isinthiscase folded andsubducted. East dipping subduction has ashallow dip.Thebasal detachment oftheeastern plate isreaching thesurface; thisprovides a mechanism tobringdeepcrustal levels ofthethrusting plateat thesurface. Thesubduction hinge isinthiscase westward retreating. Themorphological relief ofthethrust beltrelated totheeastdipping subduction ismuch moredeveloped with respectto the eastdippingcase. FAILURE OF RADIALLY STRATIFIED EARTH MODELS torque of 1. x 1025Pas. Sucha torque willchange the The observation of thisdifferentialvelocityleadsto anold rotation period of the Earth whose moment of inertia is valueof2.5min andpuzzhngproblem.Mechanically, in a radiallystratified 8 x 1037kgm-2 bythetotallyunrealistic every day! Therefore, to explain the observation, we must Earth, the canonicalreferenceframe shouldbe the one in whichthelithosphere hasno-netrotation[Lliboutry, 1974]. understand how a net rotation can be induced without as- In effect,the Navier-Stokes equationsappliedto a mantle sociatednet torque. The motionsof the platesare inducedby the balanceof wherethe viscosityvariationsare only radial indicatethat driving andresistive torques.The drivingtorqueis related thetoroidalvelocity fieldof degree 1 (globalrotation) is uniform through themantle[Ha9erandO'Connell, 1981]. to the lateral densityvariationsin the mantle. It is induced buoyancy of the downgoing slabsor to the Thismeansthat thetoroidalstress fieldofdegree 1 iszeroas by the negative underthe ridges.This torquecanbe rea consequence of freeslipboundaryconditions whichprevail positivebuoyancy at the Earth surface. latedto the mantlecirculation [Ricardand Vigny,1989], boundaryforcessuchas Thepossibility ofinducing a westward driftbytidaldrag or alternativelyto the well-known slab pull and ridge push [Solomon and Sleep, 1974;Forsyth hasbeensuggested [Bostrom, 1971;Knopoffand Leeds, 1972; torqueis the consequence Moore,1973].However, Jordan [1974]hasclearlyshown that and Uyeda,1975].The resistive to the vistidal dragis far frombeingsignificant andthat thismech- of the dragimposedby the movinglithosphere cous asthenosphere. It is generally assumed that thedrag anismshouldbe abandoned.Furthermore,we sawthat the a simple viscous law,sothatthedrag force islinobserved globalrotationhasa polediffering significantly to •.Lobeys tothelithospheric velocity VL. Thedrag the Earth'spoleof rotation.A changein the Earth'srota- earlyproportional K maybe regionally variable.The relationship tion pole(true polarwander)wouldinducea poloidalfield coefficient betweenthesequantitiesreads relatedto thereadjustment oftheequatorial bulge[Sabadini et al., 1990].It is alsoresponsible for a globalrotationof thegeography with respect to theinertialreference frame, (2) but it doesnot produceanysignificant differential rotation betweenthe lithosphereand the mantle. Thisequationimpliestwoassumptions. The firstis rather Any other possibleexplanationof lithospheric rotation obvious andassumes that the velocityat the surfaceof the invokingthe applicationof a net torquewill fail because it mantleis equalto zero. This hypothesis couldbe validif wouldchange,or evenreverse,the Earth's rotation in a few thereisa strong decoupling between thelithosphere andthe monthsfor any realisticmantleviscosities. A simplenumericalestimationcanillustratethis point. To producea differential motionof 1.7cm/yroveranasthenospheric channelwitha thickness of 100km anda viscosity of 1019Pas, underlyingmantle.The second is moredelicatebut cannot bephysically sustained. It supposes thatevena nettorque applied to themantledoesnotinduce a globalrotation of theEarth.Thisimplies thatthemantle is notonlyrigid a equatorial stress of 5. x 104Pa s mustbe applied.This butalso ismaintained fixedinspace. Wemustthuschange lowlevelstress will produce onthewholelithosphere, a net equation (2)into equation (3) RICAR, D ET AL.: DIFFERENTIAL r = ROTATION OF THE LITHOSPHERE 8411 where Pi are simple matrices formally identical to inertial matrices for a body of equivalent surface density Ki = _ whereV M is the mantlevelocitybeneaththe lithosphere. If we make again the hypothesisthat the mantle is rigid, wemustimpose thatthenetaverage of r œis zeroandthat VM is a rigidrotation.Theseconditions werenotrealized q-z2) yz h=/ Ki a:y )ds. +z --(x•zy yz -(a: zz + zz by equation(2). By inspectionof equation(3), we seethat if the coupling (s) coefficient K isconstant, when thenetaverage ofrLisequalEquation (7)isclearly independent from thereference frame tozero, thenetdifferential rotation, which istheaverage of astheaddition ofa given rotation vector Qtoalltheplate Vœ- VM isalso equal tozero. Only thecoupling between rotation vectors Qi,induces thesame amount ofrotation Q to the mantle. Therefore we can deduce from equation the lateral variations of K and the surface velocity can lead to a zeroaverage of r œanda nonzero average ofVœ- VM. (7) the net rotation of the lithosphere-Q0 when we chose We can also see that even with variations in the coupling betweenlithosphere and asthenosphere,a pure rigid rotation at the surface induces a pure rotation of the whole planet without associated for Qi the rotation poles of the plates in a no-net rotation frame. We could have tried to find the function K which should be used in equation (6) in order to exactly fit the ob- stresses. served rotation pole. Unfortunately, the inversion of the Quantitatively, we can estimatethe value of K by considtwo-dimensionalcontinuousfunction K using only the obering that betweenthe surfaceand the depth H, the Earth served three componentsof the differential velocity is highly has a variableviscosityr/(r, 8, c•). In a thin shell approxinonunique. However, we saw that the rotation is clearly remation, the shear stresscan be consideredconstantand the lated to the Pacific plate motion, i.e., to the main oceanic vertically averagedrheologicallaw leads to plate. From seismic tomography, it also appears that con1 tinents and oceansare strongly differentiatedin the top of the upper mantle. Therefore, to test our approach,we use a very simple model for the couplingfunction K. We assume where(l/r/) is theverticalaverage otherthedepthH of l/r/ that the couplingcoefficientK is equal to 1 under oceanic and is a function of latitude and longitude. In this equation, areas and K½ under continental ones. VL- VM=H(•)r L, H is the thickness of an outer (4) shell which contains all the Figure 4 depictsthe misfit deducedfrom equation(7), lateralviscosity variations. Thisthickness is supposed to v/(f•at _ •bs)2 expressed incentimeter peryearasafuncbe smallerthan the characteristic lengthof the plates. By tion of Ke. When Kc = 1 the differentialvelocityis equal comparing equations (3) and(4) weseethat wecanwrite to 0 andthe misfitis equalto the net rotationthat wedeI ducedfrom NUVEL-1, 1.7 cm/yr. The misfitfunctionshows a clear minimum for a couplingcoefficient7 times larger un- -1 K= (H(•)) . (5) der continents than under oceans. In this case the observed misfit is around0.3 cm/yr, a valuewhichis smallerthan the DIFFERENTIAL VELOCITY FOR A RIGID MANTLE Our aim is to verify that using a realistic coupling func- differencesbetween the estimations of the observed global rotations deduced from AM1, AM1-2, and HS2-NUVEL1. tion, equation(3) can explainthe observedlithosphericro- 1 tation. Of course, this equation does not describe all the kinematics of the plates. On the real Earth, the driving forces,which are not consideredhere, induce a torque which cancels the one produced by the resistant stressesof equa- tion (3). 2 1 oo lO i i i i i i i i • i i i i 1.5 A simple calculation can be done in a model where the rigid plates are moving on top of a rigid mantle. We have seen that the mantle velocity is described by a pure rotation ft0 whereas the surface plate is described by i rotation vectors Qi, where i is the number of plates. The requirement I i i i 2 _ 1.5 _ 1 _ 0.5 that the net torquemust vanishleads,usingequation(3), to 0.5 i xx d,=f c(n0 x x d,, wheret•(i, 0•b)is 0 or 1 dependingwhether0•bpointsto the 01 i ith plateor not. Afterexpressing thedoublevectorial product, equation(6) reads COUPLING i i i i i O lOO RATIO Fig. 4. Misfit between the observed and computed net rotation of the lithosphere in centimeters per year as a function or the ratio between the continental and oceanic coupling coefficients. The coupling ratio is plotted on a logarithmic scale. 8412 RICARD ET AL.: DIFFERENTIAL ROTATION OF THE LITHOSPHERE At the minimum the computedrotation is of 1.7 cm/yr where arounda polelocatedat 93øEand 47øS.The corresponding pole is plotted on Figure 2 (solid circle). Our modelindicates a preferred value of around 7 for the viscosityincrease between oceanic and continental regions. This number representsthe ratio of the vertically averagedinverse viscosity of the two domains(equation(5)). At a givendepth, the lateral viscosityvariations can, of course,be larger. The good fit realized by our simple model suggeststhat the real lateral viscosity variations are indeed, closelyrelated to the ocean-continent distribution. Of course, the remaining misfit must not be interpreted as associatedto a nonzero net torque applied to the mantle. It rather indicates that the couplingfunction is not strictly proportional to the --H)2,/o*/1(1/*/) */0 --*/1' (12) C= •I + L(L H2 This equationexpresses the horizontalaveragevelocityof a Couettetiowembedded betweentwo boundaries of velocity V œ and V M. The mathematicalproblemrequiresthe solutionof a setof locMequations (equations (3) and(11)) andspectrMequations (10). The two local equationscan be written in the spectral domain' ocean fiifi'Ction. (vL)+ -- (vM)+ = M1(TL)4', (13) ((V))+ = M2(vL) + + (Id- M2)(vM) +. (14) The two matricesM1 and M2 are computedfrom the expansion of ((1/*/) andC in spherical harmonics. Theyread DIFFERENTIAL VELOCITY FOP, A VISCOUS MANTLE In 'Orderto solvethe equation(3) for a morerealistic M•(l•ra•,lsms)=•.12ra21ama(1/*/)12m •, (15) viscous mantle, we use the same mathematical formulation as describedby Ricard et al. [1988]. This formulationas- sumes that below the lithosphere, the vertical gradient of and the horizontal velocity is much larger than the ratio of the -12rn•larnaCl•rn (16) M•li mi,lsms)=•• •. surfacevelocity over the Earth's radius. Our Earth geom- l:•rn:•lsrna etry comprises one central sphere with radial mechanical The coefficientsUllml are computedusingWigner-3j properties, and an outer shell of thickness H in which the viscosityalsovarieswith latitude 0 and longitude•b. In this shell the lithosphericviscosityis */0 betweenthe surfaceand the depth œ(0,•b),then */1in the asthenospheric channelextendingfrom L(0,•b) to H. The radially averagedinverse symbols[Edmonds, 1960] Ixra• --(--1) 4•' ol•m2larna /x+rnx •/(2/1 -[1)(2/2 -{1)(2/3 -{1) viscositywhichentersequation(4) reads = - Z x --1 0 1 --mlm2m3 (17) (111213)(111213). (s) Thet•-12rn21arna coefficients vanish unless m2 = m1+ m3, l • rnx In the innersphere,the dynamicsis suitaMysolvedby the expansionof the differentquantities,suchas velocities and stresses, on the basisof generalizedsphericalharmonics [rn2]_<12,and [11--131_<12_<11+ 13. Usingequations (10), (13), and(14), the problemcanbe solved; whenVL is theplatevelocity in a no-netrotation frame,thetoroidal coefficients ofdegree 1ofVM aredirectly [PhinneyandBurridge,1973].Simplerelationships between proportionalto the lithosphericrotation. thespectral components ofrL andVM canbefound [Hager To avoidedgeeffectsin the descriptionof the inversevis- and O'Connell,1981;Ricardet al., 1984,1988].Theserela- cosityvariationsin sphericalharmonics,we havechosento take the smoothfunctionL(O,c•) whosenormalizedlaterM tionships can be summarized as follows: = + + variationsare depictedon Figure 5. This functionis a filtered versionof the oceanfunctionand only containshar- monicsof degrees smalleror equalto 10. The greyareas over continentscorrespondto a thick lithosphere,whereas wherethe superscript plusmeansthat this equationstands the oceansare underlaidby a thinnerlithosphere.'Wechose forthecomponents onthebasisof generalized spherical har- the amphtudeand averagevalueof the functionL in orderto monics. This equation indicates that when a horizontal ve- havea lithosphere with a thickness goingfrom0 to H = 100 locityof poloidal andtoroidalcomponents (vM)+ is im- km. Fromequation(9), the lateral variationsof the inverse posedat the surfaceof a viscoussphereof radiusR, the viscosity aregoingexactlyfromthelowestvMue1/*/0to the associated stressfieldis obtainedby applyingthe operator highestvalue F, whichmultipliesthe poloidalandtoroidalvelocitycomIn our formulation, onlythe average inverseof viscosity ponents of degree I by k•/ andk}/. Thevariations of the enters. The lithospheric thickness function that we have averaged horizontal velocityin the outerlayer(V} drivea verticalflowat the converging and divergingzones,which inducesanotherpoloidalstressfielddescribed by the coefficientk•/. Thisensures themassconservation of thelithospherethroughthe zoneof subduction andridges. In the outerlayer,the flowis described by the equation usedis only an easywayto scaleour viscosity variations; of course,the real lithospherecannot reach a zero thickness. Other heuristic models,such as a model with uniform viscosityunderlainby an asthenosphere with lateral viscos- ity variations, couldhaveleadto the sameinverse viscosity function. (4). Other relationships must be introducedto definethe Beneaththe outer shellwith its lateral viscosityvariaaveragehorizontalvelocity(V}. It reads tions,weconsider a mantlemadeup of twolayers,anupper part with viscosity*/u and a lowerpart with viscosity (v) = cv + (, - c)v (11) whichextendsfrom the depth D to the core-mantlebound- RICARD ET AL.: DIFFERENTIAL 90ø0ø 90ø 45ø ........................ ROTATION 180 ø ........................ OF THE LITHOSPHERE 270ø 8413 360 ø 90 ø 45 ø 0ø 0ø 45ø 0ø 90 ø 180ø 270ø 360 ø Fig. 5. Lateral variations of the averageinverseviscosityin our outer shell. The grey continentscorrespondto a high viscosity, whereasthe nonshadedoceansare underlaidby a lower viscosity.This normalizedfunctionis a filtered versionof the ocean-continent distribution and only containssphericalharmonicsof degreessmaller than 10. The local maxima in the oceansare due to Gibbs effectsrelatedto the spectraltruncation.The inverseaveraged viscosity is exactly1/•/0 at its minimumand1/•/1 at its maximum. ary. We ran the computation taking into accountall the cou- mantle. For a given lower mantle viscosity, a decreasein pling coefficientsup to a degree lmax going from lmax -- 10 the upper mantle viscosityleads to an increaseof the lateral to lmax -- 15. We verified that for lmax -- 15 an asymptotic viscosityvariations: an inviscid asthenospherewould lead to value was attained; the coupling of lateral viscosity varia- a zero differential velocity. tions of degreeslarger than 15 with velocity described by CONCLUSIONS vector harmonics of degree larger than 15 will not significantly change our estimation of the lithospheric differential The existenceof a global westwardrotation of the lithovelocity. spherewith respectto the hotspotsis stronglysuggested Figure 6 depicts by means of isolinesthe misfit between by data. This rotation is rather independantof the chosen the observed and modelled differential rotation as a function hotspot traces and of the absolute velocities chosenfor the of the ratios •0/•1 and •o/•u for different valuesof D and Pacific volcanoes. This rotation is a real one and is not an •11/•1o.Figures6a and 6b are for •/1= •/0; in Figures6c and artifact of the choiceof the referenceframein whichplate 6d, the viscosity in the lower part of the mantle has been motionsare defined. As a consequence, an anchoringefincreasedby a factor 10. In Figures 6a and 6c, the radiM fect of the subducted slabs in the mantle which is eastward viscosity transition within the mantle is at the upper-lower migratingis expected. This mechanism couldexplainthe mantleboundary(D = 650 km); in Figures6b and 6d the viscosityincreaseliesbelowa thin low-viscosity channel(D = 250 km). observedlarger dips of the westdippingsubductionswhich contrastthe mantle flow and the openingof the back arc basins. In all graphs,the right lowerpart with the darkestshading This net rotation is forbiddenby the modelswhere the representsa zone where the misfit is larger than 1.7 cm/yr Earth's propertiesare radially stratified. We showthat a net and therefore larger than the signal itself. The smaller mis- rotation naturally appearsin the modelswhere lateral visfits are attained in the zone without shading. The best solu- cosityvariationsare allowed,the amplitudeof this toroidal tionscanexplainmorethan 70% of the observation(a misfit field being directly related to the amount of lateral varia- of 0.5 cm/yr). This satisfactoryfit advocatesfor a strong tions. A similar conclusionhasbeendrawn by O'Connell et correlation between the lateral viscosityvariations of the real al. [1991].The observed rotationwith a polein the southern Earth and the ocean-continent distribution. The lateral vis- cosity variations required are Mways larger than what has been found in the simpler model with a rigid mantle. However, if a viscositycontrast by a factor of 70 is required for a part of the IndianOceanandan amplitudeof 1.7cm/yr can be simply explained by a viscositycontrastbetweensuboceanicand subcontinental mantle. Sucha viscositycontrast agreeswell with seismictomographyof the upper mantle, uniformmantle(Figures6a and 6b), an increaseof the lower that systematically indicates the existenceof fast roots bemantleviscosity(Figure6c) reducesthis contrastto a factor neath continental areas, in distinct contrast with the slow 30. These lateral variations are further reduced to a factor oceanicregions.If theseseismicvelocityanomaliesare theraround 20 if we impose a viscosityincreasejust below an as- mal or compositionM in origin,we mustexpectlateral visthenospheric channel(Figure6d). For a givenuppermantle cosityvariationsof 1 or 2 ordersof magnitude.Furtherviscosity,an increasein the lower mantle viscosity leads to a decreasein the necessarylateral viscosity variations' our more, these lateral variations are also consistent with heat flowdatabasedontheinterpretation of seismic tomography model tends to the asymptoticlimit we found for a rigid in termsof thermaland compositional anomalies [Yah et RICARDET AL.: DIFFERENTIALROTATIONOF THE LITHOSPHERE 8414 lOOO MISFIT CM/YR looo lO00• • oo MISFIT CM/YR looo 1oo 1 1 I 1o 1oo 1ooo •TERAL VISCOSITYVARIATIONS 1 •TERA • ISCOSI• ;AO•IATiONS 1000 MISFIT CM/YR •oo0 lOOO MISFIT CM/YR ,.,.'.v't..' .'.'.'"' 1000 . .,..,.,.,,,•,•...,.., . ß..:.:::: :•: ............ • 0 % lOO lO 1 ",",' 10 1o0 lOOO •1o lO 1000 •TERAL VISCOSITY VARIATIONS LATERAL VISCOSITY VARIATIONS Fig.6.Misfits depicted bymea•ofisolin•, between theobserved and computed netrotation ofthelithosphere incentimeters per ye•.Thehori•ntg a•sdepicts theratio between subcontJnent• •d suboce•ic viscosity •0/•1.Thevertic• axis depicts the ratio between su•continent• viscosity •d theviscosity ofthetopp•t oftheupper mantle •0/•. Below theouter shell, themantle viscosity v•esr•iily •d presents aviscosity impatthedepth D. This depth is650 kmforFi• 6aand 6c,•d 2S0 kmfor Fi• 6•and 6d.The viscosity o[thelower p•to[themantle is1inFibres 6aand 6b,and 10inFi• 6cand 6d. from the locationof the al., 1989].A substantial decrease of the uppermantlevis- the wholemantle,independently cositybeneaththe oceaniclithosphere is alsostronglysup- hotspots sources. portedby the analyses of the Holocene sealevelchanges in Acknowledgments. Wearegratefulto G. V. Dal Piazforuseful oceanic island sites located in the far field with respect to discussions. Oneof us (Y. Ricard)wasa fellowof ESA (EuroAgency) at theUniversity of Bologna (Italy). This the PleistocenicArctic and Antarcticice sheets[Nakadaand peanSpace Larnbeck, 1989]. A trade-off exists between lateral and radial viscosityvariations. 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