GEOPHYSICAL RESEARCH LETTERS, VOL. 22, NO. 11, PAGES 1317-1320, JUNE 1, 1995
Cenozoicplate driving forces
C. Lithgow-Bertelloni
Institutfor Geophysik,UniversiflitG6ttingen,G6ttingen,Germany
Mark
A. Richards
Dept. of GeologyandGeophysics,
Universityof California,Berkeley
Abstract. Past studiesof plate driving forces have concluded
that the forces due to subductedslabsin the upper mantle and
those due to the thickening of the oceaniclithosphereare the
principal driving forces.We reexaminethe balanceof driving
forcesfor the present-dayand extendour analysisthroughthe
Cenozoic, using an analytical torque balance method which
accountsfor interactionsbetween plates via viscouscoupling
to the inducedmantleflow. We use an evolvingmantledensity
heterogeneityfield based on the last 200 Myr. of subduction
to drive plate motions,an approachwhich has proven successful in predicting the present-daymantle heterogeneityfield.
We find that for plausibleupper mantle viscositiesthe forces
due to subducted slabs in the Cenozoic
and Mesozoic
account
for in excessof 90% of plate driving forces and those due to
lithosphericthickening for less than 10%.
Introduction
Understandingglobal plate motions is one of the central
problems of geodynamics, and determining the balance of
forces acting on plates remains one of the ultimate goals of
mantle convectionmodeling. Previousstudieshave concluded
that the plates are mainly driven by a combination of the
"pull" of slabs on subductingplates and the "push"from the
ridges, opposed by basal drag and collisional resistanceat
plate boundaries.What has been called the "ridge push"force
is actuallydistributedover the entire area of oceanicplatesand
resultsfrom the thermal thickening of the denseoceaniclithosphere[Lister, 1975; Hager & O'Connell, 1981]. "Slab pull"
is also a buoyancy force due to the dense downgoing slab,
which is coupled to the lithosphereby some combinationof
elasticand viscousforcesin the slab and surroundingmantle.
Most previousstudieshave useda processof trial and error
or inversion of plate velocities and the intraplate stressfield
to estimate the relative magnitude of driving and resisting
torques[Solomonet al., 1975; Forsyth& Uyeda,1975; Chapple
seismictomographyand, when convolvedwith the appropriate dynamicalresponsefunctions,can be usedto predictnot
only plate velocities [Ricard & Vigny, 1989] but also the
geoid[Ricardet al., 1993], dynamictopography
andco.ntinental flooding [Gurnis, 1993]. Our model differs from other forward modelsbasedon density heterogeneityfields in two respects.First, we assumethat densityheterogeneitybelow the
lithosphereis subductedmaterial,so that our modelsare based
on geologically constrainedpast subductionhistories and
lithospheric heterogeneityrather than being inferred from
seismictomography[e.g. Ricard & Vigny,1989; Woodwardet
al., 1991]. The seconddifference is that our approachallows
us to predict past plate velocities.
We
solve
for
the
instantaneous
3-D
flow
and
viscous
stressesinduced by the internal density contrastsarising from
the subductedlithosphere,and thosearisingfrom a thickening
oceaniclithosphere.Solutionsare obtainedusing an analytical torquebalancemethodsimilarto that of Hager & O'Connell
[1981]. We present calculations at the end of each of six
stagesof the Cenozoicand assessthe successof our modelby
comparingpredictedand observedplate velocities,along with
a measure of estimated
uncertainties.
We examine
the relative
force balance between subductedslabs and lithosphericthickening for the entire Cenozoic. Finally, we investigatethe effect of mantle viscositystructureon the relative magnitudeof
plate driving forces.
& Tullis, 1977; Gordon et al., 1978; Richardson et al., 1979;
Richardson, 1992]. In this work we use a forward modeling
approach to predict plate motions. Our starting point is a
physical model of plate driving forces, with plates viscously
coupledto a continuouslyevolving model of density heterogeneity in the Earth's mantle and lithosphere.This approach,
first used by Hager & O'Connell [1981], has distinct advantages over empirical models:The density heterogeneityfield
can be comparedwith the structureof the mantleinferredfrom
Copyright 1995 by the AmericanGeophysicalUnion.
Paper number 95GL01325
0094-8534/95 / 95GL-01325 $03.00
1317
Model and Methodology
The forces that drive plate motions arise from buoyancy
forcesdue to internal density heterogeneities.Near the surface,
such forces include the thermal thickening of the oceanic
lithosphere. In the deep mantle, it is likely that subducted
slabscontributethe largestbuoyancyforces [e.g. Richards &
Engebretson, 1992]. In our model these density heterogeneities act on the plates through the viscous mantle flow
which they induce,and we ignore elasticforces.The resisting
forcesarise from viscousdrag beneaththe lithosphere,and we
ignore local interplate forces such as collisional resistanceat
plate boundaries.Our goal is to constructthe most straightforward possiblemodel of plate motionswhich is consistent
with a broad range of geological and geophysicalconstraints
on global plate motionsand mantle densitystructure.
Our model for the density heterogeneityfield of the Earth's
mantle
is constructed
from
subduction
histories
for the 200
Myr. precedingthe end of eachplate stageof the Cenozoicreconstructionsof Gordon & Jurdy [1986]. We use their plate
boundariesand velocitiesfor the Cenozoicand our own compilations for the Mesozoic [Lithgow-Bertelloni et al., 1993].
Subductedslabs are assigneddensity contrastsaccordingto an
oceanic lithosphere cooling model, and these density con-
1318
LITHGOW-BERTELLONI
& RICHARDS PLATE DRIVING
FORCES
trastsare prescribedto sink verticallythroughthe upperman- Table 1: Viscosity (Pa-s) (rl) StructureModels: (a) Fit to
tle at the initial rate of convergence.
We assumethat the slabs the geoid;(b) LVC-300 km thick; (c) LVC~70 km thick;
freely penetratethroughthe 670 km discontinuity
into the (d) Hager & O'Connell[ 1981]
.
lower mantle,sinkingat 1/4 the uppermantlerate.This slowDepth
(a)
(b)
(c)
Depth
0-130 lx1022 lx1022 lx1022
0 - 64 lx1021
ing factoris equalto theinverseof thenaturallogarithm
of the
130- 200 lx1021 lx1019 lx1019 64- 128 lx10 •9
viscositycontrastbetweenupperandlowermantle[Gurnis&
200-410 lx102• lx10 •9 lx102• 128-410 lx1021
Davies, 1986; Richards, 1991] for our preferred viscosity
410 - 670 1x 102• 1x 102• 1x 102• 410 - 670 1x 102•
model(Table 1). With theseassumptions,
our modelgivesexcellent agreementwith the observedgeoid and plate velocities
670-2890 5x 1022 5x 1022 5x 1022 670-2890 lx1021
[Ricard et el., 1993].
el., 1993]. The lithosphericviscosityis lower than that of the
lower mantle, becauseit representsan averageover relatively
Royeret el. [1993].The densitycontrast
in kg/m3 is givenby high viscosity regions beneath continentsand old oceans,
(80h-4)/(h+8), where the lithosphericplate thicknessin km is low viscosityregionsbeneathyoungoceans,and plate boundh=l 0• m, andtheageof theplatex is in Myr. Thiscorresponds arieswhich appearto be very weak. Viscositystructuresb and
to an 8 km thick oceaniccrustof density2900 kg/m3 anda c are usedto investigatethe effectsof a low viscositychannel.
The only free parameterin our model is the referenceabsosuboceanic
lithospheric
mantleof density3380 kg/m•.
We predict plate motionsby computingthe driving torques lute viscosity, r/r. Plate velocitiesscalelinearly with the aband then finding plate rotation poles such that the resisting solute viscosity so that this parameter affects only plate
torquesexactly balance them. What allows us to predict the speeds,not relative motiondirections.The value of •r is deThe effects of thickening of the oceanic lithosphereas a
function of age are based on the reconstructedisochronsof
plate rotationvectorsis the fact that the resistingtorquesare
linearly dependenton them, while the driving torquesare in-
terminedby minimizingthe Z2 misfit betweenthe observed
dependent.
Th_.e
torque
balance
is expressed
as Qint= M•,
vii= II(j x )idA/Aj, where
i= x,y,z,theindex
j runs
over
and predicted
components of
the linear
momenta
vector,wjthe rotationvector
where• and Qintare3N dimensionalvectors(N is the number the 12 plates,rj is theposition
andAjtheareaof platej.
of plates) containing,respectively,the 3-componentEuler rotation vectorof eachplate and the driving vectortorquesact-
in.gon eachplate.For eachplate,P, we obtainQint,from Analysis of Driving Forces
Qint,
=IA•x•'intdA'
where
•'int
isthebasal
shear
stress
on
eachpla{eobtainedwith the propagator
matrixsolutionof
Hager& O'Connell
[1981],F is theposition
vector
andApis
the area of the plate. The 3N x 3N matrix M relatesthe resisting torquesto the Euler rotationvectors.It dependson the viscosity structureand containsinformationon the plate geometry. Its off-diagonalelementsare non-zerobecausemovement
of any one plate will generateviscousstressesat the base of
all the othersthroughthe inducedmantle flow.
The problem thus consistsof two parts whosesolutionscan
be superposed,becausethey are both subjectto the sametype
Our model reproducespresent day plate motions well, as
shownby the comparisonof the observedand predictedvelocity fields in Figure 1. We quantify the agreementby computing
the variance reduction of the horizontal divergenceand radial
vorticity fields as defined in Lithgow-Bertelloni et el. [1993]
and global correlation coefficients between the predictedand
observedcartesiancomponentsof the linear momenta.
We compute both weighted and unweightedlinear correlation coefficients between observed and predicted velocity
fields.Fortheformerwe weightthe VO.'s
by thefraction
of the
of bound_.ary
condition:1) To calculatethe driving shear total surface area of the Earth that each plate occupies.This
stresses•'intinducedby the densityheterogeneityfield the surface velocities are set to zero. 2) The resisting shear stresses
are calculated by imposing the plate geometry and applying
unit rotationsin each of the three cartesiandirectionsto every
plate, in the absenceof internal loads. An exact solutionfor
plate motions(with zero net torqueon eachplate) is obtained
by superpositionof these two solutions.This is equivalentto
the methodof Ricard & Vigny [ 1989].
When modeling piecewisecontinuousplate motionsa technical problem arisesdue to the viscousstresssingularitythat
forms at a "fluid" plate boundary[Hager & O'Connell, 1981].
Viscous resistanceat plate edges increases(unphysically)as
the logarithm of the highest harmonic degree retained in the
flow field. We truncateour calculationsat degree20 as Ricard
& Vigny [ 1989] and Ricard & Wurning[ 1991]. We find that
truncationsat higher degrees(up to 50) do not changethe results significantly, i.e. the correlation coefficients between
predictedand observedplate motionschangeby lessthan 0.01
and the best fitting absoluteviscosityby less than 15%.
We have calculated solutions for the viscosity structures
given in Table 1. Our preferredviscositystructure(a) has a
lithosphere which is 10 times more viscous than the upper
mantle
and a lower
mantle
which
is 50 times more viscous.
This viscosity structure gives the best fit to the observed
geoidusingour mantledensityheterogeneity
model[Ricard et
approachgives the global correlation between observedand
predictedsurfacevelocity fields, which we feel is the bestmeasure of model success.For 36 degreesof freedom (3 cartesian
componentsfor each of 12 plates) the correlationsare significant above the 95% confidencelevel for any correlationcoefficient greater than 0.33. The overall weighted correlationcoefficient is -.90 (.62 unweighted). Our variance reductionfor
the horizontaldivergence(96%) and the radial vorticity (66%)
fields is substantiallybetter than previous studiesbased on
seismictomography[Ricard & Vigny, 1989; Woodwardet el.,
1993] which achieved instead 66% and 20% variance reduction, respectively. The good agreementsuggeststhat flow induced by mass anomalies in the lithosphereand mantle accountsfor most of the forces driving the plates, and that our
density heterogeneitymodel maybe a good approximationto
the structureof the mantle and lithosphere.The model is also
successfulat reproducing past plate motions with weighted
correlation coefficients ranging from 0.7-0.9 (.6-.8 unweighted) (Fig. 2). The decreasein the correlationcoefficients
for the stagespostdating43 Ma reflects an inability to reproduce the changein motion of the Pacific plate for thesestages
[Richards & Lithgow-Bertelloni, 1995].
One deficiency of our model is that the velocitiesof plates
with large continentalareasare, in general,overpredicted.The
magnitudeand directionof the velocity of the Nazca and South
LITHGOw-BER'TELLONI
& RICHARDS
?LAT• DRIVINGFoRcEs
Observed
PlateVelocities
f0i'the0-10MaStage
1319
1
HA'WAIl-EMPEROR
.'"';•
<'"
'"'""
'"•'<'
0.9
(41•EMN/•>
•'•''"• '•
0.8
0.7
.
ß
0.5
i
8
0
Predicted
PlateVelocities
forthe0-10
MaStage
16
i'
i
24 .. 32
i
40
"1
I
48
56
64
TIME (Ma)
Figure2. Weighted
correlation
coefficients
between
theob2
served
andpredicted
components
of platevelocities
andassociatedlc• confidence
contours
(hatched
areas).Circlesmarkthe
beginning
andendof eachstage.Predicted
velocities,
in the
no-net
torque
reference
frame,arecompared
toobserved
veloc-
itiesin theno-netrotation
reference
frame,obtained
by subtracting
thenetrotation
of thelithosphere.
Witlirespect
to ttie
mantlefromthepolesof rotation
of Gordon& Jurdy[1986].
(This viscosity structureproduces•,ery poor fits to the ob-
Figure 1. (Top) Present-day
plate velocitiesand plate
boundai'iesin the no-net rotationreferenc6frame. The arrows
servedgeoid.)
A contribution
toplate
forces
might
arise
fromtheinherent
defisity difference between continentsand oceans.
are pr6porti0nai
tO the magnitude
.of the platevelocity
(--Y=5cm/yr).
Plates:
AF=Africa;
AN=Antarctica
AR=Arabia;
CA=Caribbean;
CO=Cocos;
EU=Eurasia;
IN=india;NA=North
America;NZ=Nazca;PA=Pacific;PL=Philippine;
SA=South
America;(Bottom)Predicted
plateVelocities
in the no-net
100%:
'aI( )
torquereference
frame.
'......
Lower
Mantle
. ......
*'......* ...........*
.....
60%[ Thickening
oi:
/ Oceanic
LithOsphere
American
p•"•esarepooi'lypredicted.
Thesource
of thediscrepancies
h...ybeduetO:1) apProximae
{reatment
of subduc-
40%1-
I
/
tiondynamics;
2) othersources
of densityheterogeneities
not
.. I
.
l
/
ß
Upper.Mantle
/
[
included in the model; 3) unmodeled f6rces such as collisional
resistance
or the elasticcomponent
of slabpull.
,•,,•- ..... •-•--- •-...... l ............ ,-•---• .....
VVOO 8 16 24 32 40 48 56
Wenowanalyze
in detailtherelative
magnitudes
oi:the
platedrivingforces(Fig. 3a).We separate
theeffects.
of each
-.
load:Lithospheric
thickening,
andtheupper
andlowermantle
64
TIME (Ma)
•(b)
[LT+AL
L)
slabsareeachallowedto drivetheplatesin theabsence
of the
others.
Foreachloadwe calculate
themagnitude
of thetorques
actingOneachplate,usingthe sameupper.
mantleviscosity
structure,
andcalculate
its percentage
contribution.
The totalcontribution
of uppermantlepluslowermantle
e,•,
e•eeaee'
1027-"
_.
slabsis in excessof 95% for all times,while that from litho-
sphericeffectS;
is of order5% or less.The large•,alueof our
slabcontribution
is in largepartd[ieto thelowermantleslabs
in ourdensity
heterogeneity
model.
Theslightdecrease
in the
cy
o
102(
contribution
of theforcesdueto lithos•3h•ric
thickening
prior
10•7
10•8
10•9
102ø
1021
to thepresent
stageis 'partially
an artifactof thepaucityof
VISCOSITY (Pa s)
isochron
dataaswe go backin time.The.lackof datatendsto
lowertheestimaied
agecontrast,
andtherefore
thethickness Figure
3(a)i•ercent
coniribution
ofdriving
loads
asa funcand density contrast, acrossan oceanic plate. If we include
tion of time; (b) Effects of an increasinglyless viscouslow
onlytheuppermantleslabsthetotalcbntribution
fromlitho- viscositychannel(LVC) on the magnitudeof the torquesexSphericthickening
is -12-15%.for our preferredviscosity ertedby: all the Slabs(ALL) + lithospheric
thickening(LT)
structure.
Using•he viscositystructure
of Hager & O'Connell
[198!] (Table1) chfinges
therelativecontribution
to -70% for
the slabsand -30% foi' lithospheric
thickening,in agreement
with theirresults.Includingthe lowermantleslabsyieldsa to-
(solidand opentriatigles),uppermantleslabs(UM) + LT
(open
andsolidsquares)
andUM alone
(open
andsolidcircles).
Themagnitude
Ofthetorqueexerted
by LT aloneis shown
for
reference.
SolidlinesandopenSymbols
forviscosity
structure
tal contribution
in excessof 95% for the slabcomponent.(b) in Table 1; dashedlinesand solid symbolsfor (c).
1320
LITHGOW-BERTELLONI
& RICHARDS PLATE DRIVING
Unfortunately, its effect is difficult to assess.If continentsare
rigid then there is no contribution to the driving torques. If,
on the other hand, we consider the tendency towards gravitational spreading, then continents contribute a large negative
mass anomaly, which will tend to opposethe effect of thickening of the oceanic lithosphere. [Franck, 1972; Hager &
O'Connell, 1981]. In the latter case, the difference between
oceanic and continental density structurecontributesa somewhat larger torque on the plates than lithosphericthickening.
Here we have, for simplicity, ignored the ocean-continent
function.
To examine the sensitivity of our conclusionsto the as-
sumedviscosityStructure,
we investigatethe effectsof a low
viscosity asthenospheric channel. This type of viscosity
structure maximizes the relative contribution of lithospheric
thickening to plate driving forces by partially decoupling the
overlying lithosphere from the deep mantle flow induced by
subduction-related density heterogeneity; in the inviscid
limit, flow inducedby massanomalieslying below the lithospherewill exert no torque on the plates. Fig. 3b showsthe
various plate driving torque contributionsas functions of the
viscosity of a low viscosity channel. To obtain a lithospheric
thickening contribution to plate driving forces equal in magnitude to that of upper mantle slabsrequiresmore than 3 orders
of magnitude viscosity contrast between the upper mantle
transition zone and the low viscosity channel. In other words,
the viscosity of the low viscosity channel must be lower than
1017Pa-s and the uppermantleviscositymustbe lower than
1020Pa-s.This resultholdsfor a low viscositychanneleither
-100 or -300 km in thickness.Given the estimatesfrom postglacial reboundon the viscosityof the upper mantle [Nakada
& Lambeck, 1987; Mitrovica & ?eltier, 1993] and a possibly
"weak asthenosphere",
these values are unreasonablylow.
Summary
We have assumedthat plate driving forcesarise from the
flow induced by density heterogeneitiesin the mantle due to
slabs and horizontal differences in lithosphericdensity structure. We have constructeda simple model for mantle density
heterogeneitybased on the last 200 Myr. of subduction.With
this model we predict presentand past plate motionsin excellent agreement with observations.
Examining the relative contribution of plate driving forces
we have shown that the plate driving torquesdue to subducted
slabs are in excess of 95%, and lithospheric contributions
only of order 5%. To achievea 1:1 ratio betweenlithospheric
thickening forces and the forces due to subductedslabs, requires a low viscosity channel many ordersof magnitudeless
viscous than the deep upper mantle. We conclude that plates
are mostly driven by buoyancy forces due to subductedlithosphere and that these forces can be modeled with reasonable
accuracythrough the Cenozoic. Improved plate motion models
require a better characterizationof resistingforces along plate
boundary faults.
Acknowledgments. We thank R. Richardsonand two anonymousreferees for thoughtfulreviews, N. Ribe and U. Christensenfor helpful discussions.This work was partly supportedby NSF grantEAR-9117538
and CH77-4-3 to U. C. from the DeutscheForschungsGemeinschaft.
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Institutftir Geophysik,U. G6ttingen,
HerzbergerLandstrasse
180, 37075 G6ttingen,Germany.
Mark A. Richards,Dept. of Geology and Geophysics,301 McCone
Hall, Universityof California,Berkeley,CA 94720-4767.
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(ReceivedJuly 12, 1994;revisedNovember21, 1994;accepted
February5, 1995)