JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 102,NO. Bll, PAGES 24,655-24,667,NOVEMBER 10, 1997
Partial advection of equidensity surfaces:
A solution for the dynamic topography problem?
Yves Le Stunff I and Yanick Richard
Laboratoire des Sciencesde la Terre, Ecole Normale Sup6rieure, Lyon, France
Abstract. Although one-layerdynamic modelsof the Earth's mantle have
successfully
explainedthe geoid,they generatea surfacedynamictopographythat
seemstoo large relative to geologicalobservations.In this study, we hypothesize
the possibility of partial advectionof mantle equidensitysurfacesby vertical
motion inducedby "driving"loads. Theselarge-scale"flow-dependent"
loadswould
greatly reducethe dynamictopographyamplitude,while preservinga goodfit to
the observedgeoid. Variousphysicalprocesses
related to nonequilibriumphase
changesor to the existenceof chemicalheterogeneityin the mantle couldjustify
a partial advection of the mean density. In this paper, we simply considerthe
flow-dependentloads as proportional to the vertical flow velocity. Two density
mantlemodelsa,re considered,
onefrom subductionreconstruction
[Ricardet al.,
1993]andonefromseismictomography
[Li andRomanowicz,
1995].We showthat
a very moderateentrainment(a few kilometers)of the equidensitysurfacesin the
transition zone is sufficientto reducedynamic topographyamplitude by a factor
of 2 or 3. The seismicvelocitysignal associatedwith this entrainmentwould be
hidden by the signal of thermal origin. Using this new hypothesis,we compute
sea level changesassociatedwith epeirogenyfor the Cretaceous,Paleocene,and
Oligoceneperiods. The amplitude and phaseof thesechangesare in fairly good
agreementwith geologicalhypsometriccurves. Our resultssuggestthat not only
the thermodynamics,but also the kineticsof mineralogicalphasechangesin the
transition zone are of crucial importance.
1. Introduction
Dynamicmodelsof the Earth'smantlehaveexplained
surfaceobservations,suchas geoid,plate motions,and
topography,in termsof largescaleheterogeneity
as revealed by global tomography. Two types of models
wherecirculationis either wholemantle (onelayer) or
stratifiedat 670 km depth (two layers)havebeenconsidered. Up until now, these models have considered
the Earth's mantle as being in thermodynamicequilib-
the amplitude of dynamic topographyhas been extensivelydebatedin the publishedliterature [seeForte et
al., 1993a;Gurnis,1993a;Le Stunffand Ricard, 1995],
we repeat its definitionhere and remind the readerof
the discrepancybetweentheoreticalpredictionsand observations.Most of the Earth's topographyis obviously
related to crustal thickness variations.
Another
ma-
jor componentis relatedto the thermal coolingof the
oceaniclithosphere.This componentis roughlyvarying
as the squareroot of the seafloorage,and, within a few
rium. This means that if a neutrally buoyant mantle
hundredmeters,the sameagerelationshipholdsfor all
parcel is moved upward or downward, it keepsits neuoceans[Steinand Stein, 1992]. The thick lithosphere
tral buoyancy, unlessit crossesa chemical barrier.
under continentsis generallyseen as chemicallydisWhile the •ne-layer modelshave been quite successtinct and lighterthan the oceaniclithosphere
[Doinet
ful in fitting the long-wavelength
geoid (degrees2 to al., 1996]. This view hasbeenchallenged,
however,by
10) [e.g., Ricard et al., 1993], the dynamictopogra- Forteet al. [1993b]who arguedthat cratonicrootscorphy has not yet been properly explained. Although respondto activedownwellings.
Whetherthesecrustal
and lithosphericcomponentsof the surfacetopography
are called static or dynamic is more a semanticissue
•Now at Centre Scientifique et Technique, Total, Saint- than a scientificone; they are simply related by the isoR•my-Les-Chevreuse, France.
static rule to the density anomaliesin the top -•200 km
of our planet. In this paper, we defineas "isostatic"
Copyright 1997 by the American GeophysicalUnion.
the topographydue to densityvariationsin the cold
thermalboundarylayerof mantleconvection(crustand
Paper number 97JB02346.
lithosphere).We defineas "dynamic"the topography
0148-0227/97/97JB-02346509.00
24,655
24,656
LE STUNFF AND RICARD: PARTIAL ADVECTION
that is related to the flow driven by deepmantle hetero-
geneities.Observingthe present-daydynamictopography consistsof stripping off the effectsof the isostatic
component(asdefinedabove)fromthe total surfacetopography.
The amplitude of dynamic topographyis predicted
to be •-q-2 km by models of whole mantle circulation
AND DYNAMIC TOPOGRAPHY
2. Mantle
Flow
With
Partial
Advection
The usual approach to mantle circulation is to solve
the continuity,the gravity and the Navier-Stokesequationsfor a fluid with infinite Prandtl number.Although
in recentyears,compressible
mode]s[Forteand Peltier,
1991;Thoravalet al., 1994;Corrieuet al., 1995]or models with laterally variableviscosity[Cadeket al., 1993;
Zhang and Christensen,1993]have beendeveloped,in
this paper, we limit ourselvesto a purely incompressible
fluid with depth-dependent
NewtonJanviscosity.
In previousmodels,mantleinternaldensityvariations
drivenby lowermantleheterogeneities
[e.g.,Hagerand
Clayton, 1989; Ricard et al., 1993]. This seemsmuch
too large, consideringthe two followinggeologicalobservations.First, the departureof seafloortopography
from a pure age relationshipis certainly smallerthan 1 were derived from the distribution of subducted slabs
km [Colinand Fleitout, 1990],indicatingthat the dy-
[Hager,1984;Ricardet al., 1993]or wereproportional
namic topography componentover oceansis of small
to seismicvelocityanomalies[e.g.,Hagerand Clayton,
amplitude. This has also been corroboratedby sev-
1989].In additionto theseloads,long-wavelength
intereral other authors [Cazenaveand Lago, 1991; Colin,
facedeflections
havebeenconsidered[Thoravalet al.,
1993;Kido andSeno,1994;Le StunffandRicard,1995].
1995; Defraigneet al., 1995]. In what follows,we see
Second,the departureof Cenozoicsealevel changes
that
there may be somedifficultiesin derivingdensity
recordedby different continentsfrom a uniformeustatic
anomaliesfromslabdistributionor seismictomography.
curveis smallerthan I km [Gurnis,1990]. Strictly
speaking,this is not a measureof the absolutedy- 2.1. Density Field Derived From a Distribution
namic topography but, rather, of its relative variations of Subducted
Slabs.
through time. However, becausecontinentsare carried
In the exampleof a modeldrivenby subductingslabs,
by plate tectonicsat a presumablyfaster velocitythan
the
load is describedby a densityfield 5p. This density
that at which the sluggishlower mantle evolves,the relis
mostly
controlledby the temperature of the slabsand
ative topographyvariationsshouldbe indicativeof the
andpetromaximumamplitudeof the absolutedynamictopogra- includes(or couldinclude)the compositional
logical variations occurringduring subduction. These
phy.
The circulationmodelslayeredat 670 km depth gen- loadsforcethe "normalmantle" (i.e., the mantlewitherate a much smaller dynamic topography but are not out thermal anomaly) to undergoa vertical motion,
as successfulin fitting the geoid. They predict a huge even at a great distance. Various processescan resist
topography on the 670 km discontinuity and require
large density variations in the transition zone and the
this motion.
2.1.1 Latent
heat release.
The dynamic conse-
lowermantle. Wen and Anderson[1997]havehypoth- quenceof a phasechange(assumingequilibrium)is
esizeda deeperlayeringof the mantle (•900 km) to twofold [Schubertand Turcotte, 1971]. The main ef-
reconcilegeoidmodelingwith low dynamictopography. fect, which is purely local and can be accounted for
All theselayeredmodelsseemto contradictvarioushigh in the slab model, is due to the sign of the Clapeyron
precisiontomographicmodels[e.g.,Grand, 1987,1994; slope. An exothermicphasechange(i.e., at 410 km
Van der Hilst et al., 1991,1997],wheresubductingslabs depth) tendsto enhancethe flow whilean endothermic
are observed in the lower mantle.
phasechange(i.e., at 670 km depth) impedesit [ChrisThe whole or layered mantle modelsoffer a simplified tensenand Yucn, 1985]. Another effect,the releaseof
view of mantle dynamics and neither is totally satisfac- latent heat, always counteractsthe flow regardlessof
tory. Models where a layered convectionis punctuated
the sign of the Clapeyron slope. It is named the Ver-
by events(avalanches)
of wholemantleflowshavealso hoogeneffect[Verhoogen,1965]. The phasetransition
beenproposed[Macheteland Weber, 1991;Peltier and is advected by the flow at a distance of order h:
Solheim,1992;Tackleyet al., 1994].However,from the
point of view of computing an instantaneousdynamic
topography, they are akin to whole mantle modelsduring avalanchesand to layered mantle modelsotherwise.
In this study, we propose an alternative to the existing models by including the possibility of a partial
advection of the equidensity surfacesof the mantle by
the convectiveflow. This approach allowsus to derive a
continuousrangeof modelsfrom purely adiabaticwhole
mantle convectionto total layering. This advectioncan
mimic different possibleeffectsthat are discussed
in section 2, such as an effect of the releaseof latent heat,
h- - ApT72
pa#
Cp'
(1)
where T is temperature, p is density, Ap is the density jump acrossthe phasechange,'/is the Clapeyron
slope,Cp is heat capacityand # is gravity. Equation
(1) showsthat the advectionis independent
of the sign
of the Clapeyronslope. A downgoingmotion(toward
a densityincrease)inducesa depression
of the phase
boundary,and an upgoingmotion(towarda densitydecrease),raisesthe phaseboundary.The exactamount
a kinetic delay for phasechanges,or the resistanceof of advectionis alsorelated to the vertical velocityof the
compositionalheterogeneitiesto crossthe mantle den- flow and to the thicknessof the transition for real, tnulsity discontinuities.
tivariantphasechanges
[Christensen,
1985,1997].How-
LE STUNFF AND RICARD: PARTIAL ADVECTION
AND DYNAMIC
TOPOGRAPHY
24,657
ever, (1) predictsthe correctscaleof advection,which in a deflectionof the equidensitysurfacesby the radial
amounts to 4.2, 3.3, and 4.4 km for the 400, 520, and
670 km interfaces,respectively(assuming,in the tran-
flOW.
All of thesestabilizingphenomena(latent heat, ki-
sitionzone,T- 1800K, g - 10m s2 andCp- 1250 netics,and chemicalheterogeneities)
tend to advectthe
J kg-• K-1 with p - 3.6 3.8 4.2 kg m-a 7 - 3 5 compositionor the petrologyby a quantity 5C, so that,
-3 GPa K-• and Ap - 150,50,250 kg m-a, for the c• assumingagain that the driving forcesare due to sinkolivine -+/• spinel,,• spinel-• 7 spineland 7 spinel-• ing slabs,
perovskite+magnesiowfistite
transitions,respectively).
The loadsarisingfrom this Verhoogeneffectcannotbe
includedin the slabmodelas nonlocal,i.e., not directly
5/:)total
--5/:)slab
q- •'• T
related to the slabs but to the flow itself. The deflection
due to the Verhoogeneffectis negligiblecomparedwith
the deflection of the phase transition induced by the
Clapeyron effect insidecold subductingslabs. However,
at a large distance from strong thermal anomalies,the
Verhoogeneffect will be important. The Verhoogeneffect is, in principle, includedin globalconvectionmodels
but not in circulation modelsdriven by a priori loads.
2.1.2
Kinetics
effects.
The nucleation
where the index T means that the derivative
is not due
to temperature variations. Clearly, 5U is related to the
radial flow velocity ur. Although there is no indication
that the relationship between these two quantities has
to be linear, we simply assumein our heuristic approach
that the change5C is proportional to ur. Therefore we
write
5Ptotal- 5Pslab
-- •(r)Ur,
barrier re-
(3)
quires an overshoot of the phase domain for a phase wherea(r) controlsthe amplitudeof thesenonlocaleftransformation to occur. Although poorly constrained, fects at radius r. Here a is likely to be positive, as a
this overshoot can be of the order of 1-10 km downpositive radial velocity advectsdense,negatively buoy-
stream[Solomatovand Stevenson,1994]. After nucle-
ant mantle.
ation, the phasechangesare controlledby chemicaldiffusion. With characteristic diffusion constants of 10-•
2.2. Density Field Derived From Seismic
m2 s-• appropriateto hot mantlein the transitionzone Tomography
and grain sizesor of the order of i cm, the phasechange
takes10a-104years,duringwhichan advectionaslarge
In
the case where
the
loads
are derived
from
seis-
mic tomography,the distinction between "local" drivas i km can occur. However, in a multiphase compoing loads and "resistive,"flow-dependentloadsis much
nent, only part of the material reacts and the chemiless obvious. Up until now, the velocity heterogeneity
cal diffusion processis therefore coupledwith a percomapped by seismologistshas been consideredby geoid
lation problem that might increasethe advectiondismodelers to be proportional to lateral density varia-
tances[Solomatovand Stevenson,1994]. Thesethe-
oretical considerationsseem to contradict experiments
where very fast reactionswith no hysteresisare reported
[Chopelas
et al., 1994;Martinezet al., 1997].However,
a possiblehysteresisin the transitions, corresponding
to only a few kilometers, would require a resolution
of order 0.04 GPa, well beyond the resolutionof highpressureexperiments. Moreover,the grain size of material in experiments does not correspond to that of
mantle conditions, and the experiments are performed
in isothermal rather than adiabatic conditions. Exper-
tions.
(•Ptomo
-- (Op)
•ssC5Vs.
(4)
The index C indicates that the partial derivative is
for a constant compositionand petrology. In the lower
mantle, this derivative has been estimated to be of order
0.2 kg s m-4 by geoidmodeling[Corrieuet al., 1994].
This correspondsto what is expected when p and V•
are dueto tempera,
ture variationsonly [Andersonet al.,
1968]. It haslongbeenrecognizedthat heterogeneity
in
imentally,the phasetransition (at least for a olivine the upper mantle could be either thermal or composi-• • olivine and • olivine-• 7 olivine) shouldbe --•40 tional. Proportionality between density and velocity is
km thick but the fact that they reflect high frequency still assumedalthoughthe density/velocityderivative
seismicwavesindicatesthat they are thinner(--•10km) used for geoid modeling does not correspondto that
[Vidaleand Benz, 1992].This is an indicationthat real expected from temperature variations only.
However, a distinction must be made between the
phase transitions may not be that simple. Of course,
in cold subducting slabs, the diffusion is much slower driving loads and the flow-dependentloads when transand metastable phasescan persist far from equilibrium lating velocityanomalyinto density.Equation(4) neglectsthe fact that the relationshipsbetweenp and V•
[Kirby et al., 1996].
are not the samewhether the temperature or the miner2.1.3. Chemical compositionalheterogeneities.
Even if the upper and lower mantle are, on average, alogy changes. Indeed, both density and velocity varichemically identical, they certainly contain chemical ations depend on temperature and mineralogy:
heterogeneities. For example, blobs of eclogitizedmaterial might be present. Their density is such that they
are buoyantjust belowthe transitionzone [Ringwood,
1990]. They will resista flow that crosses
the 670 km
discontinuity but will be partially advected, resulting
24,658
LE STUNFF AND RICARD' PARTIAL ADVECTION AND DYNAMIC TOPOGRAPHY
av,- 77'j aT+ acj
c
T
per mantle, which is itself 40 times lessviscousthan the
lower mantle. This viscosityprofile corresponds,more
(6) or less,to the radial viscosityneededto fit the geoidfor
Lateral density variations are thus equal to
slabs or tomography-deriveddensity models. Kernels
for a - 0 are depictedin Figure I by solidcurves,while
op
)C6¬
t•Ptotal
-- •s
dashedcurvesare for a - 3 101økg s m-4 anddotted
linesare for c•- 3 10TMkg s m-4. Thesetwo different
valuesof c• imply that an upwardflow of I cm yr-•
inducesa densityexcess
of 10 and 100kg m-a, respectively.The geoidkernelis in kgm-•, the topography
is
r
Therefore
(7)
c
-1 in the caseof perfectisostasy,and the radial velocity
we can write
5Ptotal-- 5Ptomo
--
(8)
is inm as-• kg-•.
The geoidkernelsfor c• - 0 (Figure 1, solidcurves)
correspond to the usual whole mantle circulation mod-
keepingin mind that 5Ptomo
computedfrom (4) is not els. Dottedcurvekernels(c• - 3 10TMkg s m-q/m)
correct when density variations other than purely thermal ones are present.
are closeto thoseobtainedwith pure chemicallayering.
The differencelies in the fact that in the previously
published stratified models,the barrier to the flow was
localizedat 670 km depth, whereashere it is diffuseand
are,in principle,different.An estimatefor (Op/OV•)T extends throughout the whole transition zone. Loads
directly computed from a radial seismic model like
locatedbetween410 and 670 km depth are locally comIASPEI91 [Kennettand Engdahl,1991]is aroundi kg
s m -4 in the transition zone. As this value is 5 times pensated by the advection of the equidensitysurfaces
and thereforeyield a small gravity signal. The dashed
The functionsc•(r) and/3(r) in equations(3) and (8)
largerthan (Op/OVs)c,oneexpectsc•(r)and /•(r) to
differ only moderately. In what follows,we identify the
two functionsand call them c•(r).
curvekernels(c•- 3 10•ø kg s m-4) areintermediary
results, but they keep the characteristicsof whole man-
tle models,namely a negativesignin the lowermantle
and a positive peak around the 670 km depth.
3. Geoid and Topography Kernels
In the whole mantle case (c• - 0), lower mantle
anomaly
contributessignificantly
to the dynamictopogAssumingthat internal loadsare definedby (3) or
raphy
amplitude
(Figure
lb,
solid
curve). The dotted
(8), we can computethe associated
mantleflow. As
curve
kernels
(c•
3
10
TM
kg
s
m
-4)
arecloseto those
is usually done, we first consider the responseof the
obtainedfor a chemicallystratifiedcasewhereonly upmodelto a surface
density
anomaly
5(r - re)Y•,•
(0,c)),
i.e., to a unit surfaceload of given sphericalharmon- per mantle mass anomaliescontribute to dynamic to-
curve),
ics degree I and order m, localized at a given depth pography.For c• - 3 10•ø kg s m-4 (dashed
the
lower
mantle
contribution
has
been
significantly
rer0. The response,
calledkernel(Gt(r) andTt(r) for the
duced compared with the casewithout advection.
geoidand topography,respectively),is independentof
The radial flow velocity at 670 km is displayedin
the order m and contains the effects of the load itself tothe
Figure lc. As c• becomeslarger, the flow velocity
gether with thoserelated to boundary deformationsand
is
reduced
and mantle circulationbecomestwo layered.
flow-inducedbuoyancy.By meansof radial convolution
and sphericalharmonicssummation,the responsefor a For c•: 3 10TMkg s m-4 the transitionzoneactsas
complexinternal massdistribution is readily obtained. a barrier for the flow. In the intermediate case, the
The detailsof numericalresolutionare givenin the ap- amplitude of the radial flow is reducedby •030%.
pendix.
By inspectionof the new expressions
of load (see(3)
and (8)), one seesthat when a = 0, the problemis
similarto wholemantleconvection.Whena(r) = 5(rre), the systemadjustsitself to a zeroverticalvelocity
at depth re and is therefore similar to previoustwolayer convectionmodels.Thereforethis study includes
and generalizesall modelsthat have been previously
discussed.
For simplicity,we assumethat a(r) is zero
4. Geoid Fitting
The effect of partial density advectionis tested on
two differentmodelsof mantledensityheterogeneity,
onederivedfromsubduction
reconstructions
[Ricardet
al., 1993]and onededucedfrom tomography
[Li and
Romanowicz,1995]. We consider,aspreviously,a man-
tle model in which c• is zero except in the transition
zone and where the lithosphericand upper mantle viszone.
cositiesare in a ratio of 10. We vary simultaneouslyc•
Variouskernels(degrees
2, 4, 8) are displayedin Fig- and the viscosityjump betweenupper and lowermanurel for different values of c•. Geoid kernels are distle in order to fit the geoidand decreasethe dynamic
played in Figure la, topographykernelsare shownin topography.
Figure lb. The kernels for the radial flow velocity at
Geoidand topographyhavebeencomputedsumming
670 km depth, induced by unit surfaceloads located at all degreesand ordersup to ! - 8. Only internal loads
variousdepthsin the mantle, are depictedin Figure lc. deeperthan 300 km depth have been consideredfor the
The lithosphereis 10 times more viscousthan the up- dynamic topography,as the shallowerheterogeneities
in the whole mantle and is constant in the transition
LE STUNFF AND RICARD' PAP•FIAL ADVECTION AND DYNAMIC TOPOGRAPHY
GEOID
24,659
KERNELS
6000
4
55OO
5000 !
4500
-
i
I
/\
\
4000
a
3500
-1.5
-1
-0.5
0
DYNAMIC
0.5
TOPOGRAPHY
1
2
xlo
KERNELS
i
i
•
•
•
•
-0.8
-0.6
-0.4
-0.2
6000
55OO
ß-• 5000
4500
4000
3500
-1
b
RADIALVELOCITY (670 KM) KERNELS
7000
i
i
i
i
•
•
'
'
-0.8
-0.6
-0.4
-0.2
4
6000
-
5000
-
4000
-
3000
-1.4
•
-1.2
•
-1
c
o
xlo
Figure 1.
Response
functions
(kernels)
of (1i) asa function
of Earthradius(verticalaxis)
for sphericalharmonicdegrees2, 4, and 8 showing(a) geoid,(b) snrfacedynamictopography,
and (c) radial flowvelocityat 670 kin. Solidcurvesare obtainedfor advectioncoefficienta - 0
(noadvection),
dashedcurvesarefor ct- 3 10lø kg s m-4 anddottedcurvesaretbr a- 3 10TM
kg s m-4. As seenin Figureslb and lc, increa.sing
a cancels
the contribution
of lowermantle
anomaliesto the topography and decreasesthe radial velocity.
give rise to the isostatic topography and must not be
squareamplitude of the predicteddynamictopography.
included[e.g., Le Stunff and Ricard, 1995]. The to- The massmodel is derivedfrom slabsin Figures2a and
pographyhasbeencomputedassuming
a uniformpres- 2c, and from tomography in Figures 2b and 2d. The
ence of ocean at the surfaceof the Earth, i.e., using a residualgeoidand the dynamictopographyamplitudes
surfacedensity contrast correspondingto the difference are depicted by means of isolinesthat are 5 and 20 m
apart, respectively. The x axis givesthe value of the
between
mantleandwaterdensities
(2.2kg m-3).
Figure 2 showsthe results of this search. Figures 2a viscosityjump at 670 km depth, the y axis givesthe
and 2b depict the amplitude of the residual geoid, i.e., value of a.
the root-mean-square
difference
betweenpredictedand
The best fit for the geoid in models without advecobservedgeoid. Figures2c and 2d showthe root-mean- tion (a = 0) is obtainedfor a viscosityjump around
24,660
LE STUNFF AND RICARD' PARTIAL ADVECTION
AND DYNAMIC
TOPOGRAPHY
ResidualGeoidAmplitude
0
?O3
5
• 15
20
40
60
80
100
20
viscosityincrease
40
60
80
100
viscosityincrease
Dynamic TopographyAmplitude
........... [ ..................................I ..................................I.....
I
E
lO
15
c
I
20
40
60
80
100
viscosityincrease
20
I
40
I
60
I
80
100
viscosityincrease
Figure 2. Residualgeoidamplitudederivedfrom (a), subauctionreconstruction,
and (b), seismic tomographyand averagedynamictopographyderivedfrom (c), subductionreconstruction,
and (d), seismic
tomography
for variousviscosity
jumpsbetweenupperandlowermantle(x axis)
and variousadvectioncoefficients
c• (y axis). Contourlinesare 5 rn apart for the residualgeoid
and 20 rn apart for the dynamic topography. White areasare consideredsatisfactoryresults. Notice the different patterns for geoidand topographyand the wide range of solutionsthat predict
a reasonablegeoidbut yield differentamplitudesfor averagedynamictopography(from ,•600 rn
to 0).
40, already found in previousstudies[e.g., Ricard et the velocity flow at 670 km depth by -030%, compared
al., 1993]. However,the white valleysin the residual with the ]nodel without density advection.
Figure3 showsmapsof dynamictopography(degrees
geoid show that models with larger viscositycontrasts
contrastof 40 (Figand largerresistances
in the transitionzoneare equally 1-15)for the casesc•= 0, viscosity
contrastof
suitable. It. has already been observedthat chemically ure 3a) and c•= 3 101økg s m-4, viscosity
60
(Figure
3b).
Contour
lines
are
250
m
apart.
The
stratifiedmodels(infinite c•) with very largeviscosity
slab model has been used. The peri-Pacific minimum
of topography related to the existenceof subductions
Residualgeoidand dynamictopographycontourlines since the Mesozoic decreasesfrom-1700 m in Figure
do not have the same pattern. It is thus possibleto find 3a to -900 m in Figure 3b. With a small amount of
a point of satisfactorygeoidresidualfor any root-mean- advection, the dynamic topography over oceansis negsquaredynamictopographyfrom around600 rn (a = 0 ligible with respect to the normal subsidencedue to
and viscositycontrast of 40) to basicallyzero. For ex- lithosphericcooling. A similar reductionin the dynamic
ample,a modelwitha=3 10løkgs m-4 anda viscositytopography is obtained with the tomographicloads. As
jump of 60 explains87% of the geoidvarianceand de- with the slab model, the pattern of dynamic topography
creasesthe dynamic topography by a factor of 2 and mostly consistsof a peri-Pacific low, which is reduced
contrasts,
up to 104, canexplainthe geoid[Colin,1993].
LE STUNFF AND RICARD' PARTIAL ADVECTION AND DYNAMIC TOPOGRAPHY
from-1600
to-600
m when advection
with
the same
is included.
5. Amount
of Advected
Mass
24,661
that the crustal propertieswould changeto compensate
and decreasethe total topography, continentsshould
move up and down with an amplitude comparable to
that of the dynamic topography. The comparisonbetween sedimentologicalrecordsof continentalplatforms
and modelsof dynamic topographyshouldthereforebe
a severeand stringent criterion for the quality of these
The question is whether the hypothetical advected
massthat might reducethe topographyis physicallyacceptableor not. It is easyto computethis massanomaly models[Gurnis,1990,1993b;Mitrovicaet al., 1989].
from the radial velocityin the transitionzone(see(3)
Assumingthat most mantle heterogeneities
are the
and (8)). It canthen be translatedinto seismicvelocity resultsof past subductions,we can easilycomputesea
anomaly
levelchanges
in the last -•70 Myr [Ricardet al., 1993].
Computingsealevel changeson a longertime-scaleappearsdifficult,consideringthe lack of knowledgeof subet al.,
and can be compared to the total velocity anomaly seen ductionsprior to 120Myr ago[Lithgow-Bertelloni
1993]
and
the
time
needed
by
slabs
to
cross
the
mantle,
by tomography. Both maps in Figure 4 have been computed at a depth of 410 km. Contour lines are 5 m -070 Myr. The sealevelvariationsare recordedgeologis-• apart. The anomalyinducedby advectionwould cally in sedimentsat the continentalmargins.Sealevel
be, in our interpretation, -•20% of the observedsig- is controlledby the dift•rencebetweentopographyand
nal. The signal due to advectionwould be included in geoid. Therefore we compute this quantity as a functhe observed anomaly, so that the difference between tion of time and then substractthe sea level predicted
au•
(9)
the maps in Figure 4a and 4b representsthe velocity
anomaly of pure thermal origin. The predicted seismic
anomaly derived from advected massesis somewhat anticorrelated with the observedseismictomography,as
fast and dense areas correspondto downwellingsthat
entrain the light and shallowdensitiesof the mantle.
We can also interpret the advectedmassanomaly in
terms of deflection of the equidensitysurfaces.The total massanomaly integrated over the transition zoneof
thickness L - 260 k•_•.is c• < u• • L, where • u• • is
the average vertical velocity. This mass can be translated into a constant deflectionh of the equidensitysurfacesbetween the densitiesPminand Pmax,so that
-
Pmax -- Pmin
A simplecalculationwith • u• •-
.
(10)
1 cm yr-•, whichis
the order of magnitude of the long-wavelengthvertical
velocity in the transition zone, gives a downwarddeflection of 12 km on the 410 km discontinuityalone or,
equivalently,8 km on the 670 km discontinuityaloneor
3 km for all the equidensitysurfacesthrough the transition zone.
The previous estimates show that, if it exists, the
amount of advectedheterogeneityis associatedwith little deflection of the equidensity surfacesand is small
comparedto the total seismicheterogeneityof thermal
origin.
6. Sea Level Change
When a dynamic topographyis predictedfor a given
continent, one might always argue that the uncertainties in the crustal density or thicknessare such that the
total predictedtopography,dynamic plus isostatic,can-
at time t and at t = 0.
Our results are characteristic of the differences in sea
levelrecordsobservedat differentcontinentalmargins.
The absolute sea level curve should take into account a
global eustatic signal, which is not,presentlyincluded.
This signalis due not only to the changesin average
seafloor
age[e.g.,TurcotteandSchubert,
1982]but also
to the changesin dynamictopography[Hager,1980].
This eustaticsignal is recordedby all continentalmarginsand constitutesan additionalsignalthat couldbe
added to each local sea level curve predictedby our
model.
We performedour computationfor three differentpe-
riods: late Oligocene(27.7 Ma), late Paleocene(59.2
Ma), and late Cretaceous(88 Ma), althoughthe accuracy of the slab distribution for the last period is poor.
The valueof a in the transition zonehas been kept to 3
10•ø kg s m-4. A lowviscosity
asthenosphere
of .1 viscosity and 300 km thicknesshas been added under the
lithosphere. The existenceof this asthenosphereis not
required by geoidmodeling. However,it further reduces
the dynamictopographywithout affectingthe geoidtoo
much. With this viscosity profile, the distribution of
slabs at presentexplains84.6% of the geoid variance.
For comparison,the best model without asthenosphere
or advectionexplainsup to 87% of the geoidvariance.
As sedimentary recordshave been made on continental
margin,a densityjump of 3.2kgma hasbeenconsidered
at the surface.
Figure 5 depicts the results for all three periods. We
movedthe continentsin time followingScotese[1992].
The white areas have undergonea subsidencelarger
than 200 m, while thoseshadedin dark gray had an uplift larger than 200 m. The maximum sea level change
since Cretaceous is moderate and correspondsto an
not be assessed within 4-1 km and therefore calmot be
uplift of --•800 m located over Alaska. Indonesia has
comparedwith observations.However,through geolog- subsidedby a few hundred meters, although details of
ical times, the continents move on top of the dynamic the paleogeographyare required to propose a precise
topographyandundergolarge-scale
verticalmotions(or sea level curve. South America has been fairly stable
epeirogenicmotions). As we haveno reasonto believe and has subsidedby less than 100 m. The relative sea
24,662
LE STUNFF AND RICARD: PARTIAL ADVECTION AND DYNAMIC TOPOGRAPHY
Dynamic Topography(degrees 1-15)
6O
4O
2O
0
-20
-40
-60
-80
a
0
20
40
60
80
100
120
140
160
180
200
220
240
260
"
'"
280
300
320
340
No advection
:
'
..::•½,i!iiiii'•-'"'•?-.:•:--'-":-•
............
!!i?"•ii•.
•
--:•!!
-•.'!i•!•',::::•'.
':'
....
....
•;i•:•;.-.'i:.•:•ii•i•iii:•i:•it..iiii•'.i•:"'
.?•i•ii½:.-'"'""---'-"':----"•:•?'--."---'-:•
•-'"-":•:-"'?-:•':•
.......
...:;:•::•
i•:'•
-.-..:J:
.... ::s:•:•s
:.•:•:::•
::--:.::•:..--•:::::.•
..........
::--
ß
.-:.•
.:................
;-:
ß
----
i
!ii:!:?:"
''
-40
-60
-80
I
0
I
20
I
40
I
60
I
80
I
100
I
120
I
140
I
160
I
180
I
200
I
220
I
240
I
260
I
280
I
300
I
320
I
340
Partial advection
Figure 3. Dynamictopographymapscomputedfrom a densitymodeldeducedfrom subduction
reconstruction
(degrees1 to 15) for (a) no advection(c• - 0). and a viscosity
jump at 670 km
equalto 40 and (b) c•- 3 10lø kg s m-4 anda viscosity
jump at 670km equalto 60. Contour
lines are 250 m apart. Depressedarea are shaded,and uplifted area are white. Notice the periPacificdepressionin both mapsand the reductionof topographyamplitudeaccountingfor partial
advection.
level of Australia at presentand Cretaceousis the same, 7. Discussion
althoughAustralia height was around 100 m above
Presentlevel in between. India has undergonea slight
downwardmotion (,-•100m ) beforeOligocenebut has
been fairly stable sincethen. These findingsare in excellentagreementwith the relative hypsometriccurves
compiledby Lithgow-Bertelloni
and Gurnis[1997]from
varioussources[Crossand Pilger, 1978; Bond, 1979;
It is obviousthat variousphysicalprocessesrelated to
phasetransitions, suchas releaseof latent heat, kinetic
effects,and motion of chemicallydistinct mantle bodies
can slow down the vertical
flow in the mantle.
We rec-
ognizethat the amplitudeof thesephenomenais, as of
yet, mostly unconstrainedand that the parameterizaHarrison et al., 1983; Veevers,1984; Fleitout and Singh, tion of theseeffects(takenasproportionalto u,.), their
amplitude(the choiceof c•), and localization(continu1988;Mitrovicaet al., 1989;Ronovet al., 1989].
LE STUNFF AND RICARD: PARTIAL ADVECTION
AND DYNAMIC
TOPOGRAPHY
24,663
from Tomography at 400 km
40
.......
2O
0
-20
-40
-60
-80
0
20
I
I
I
I
I
I
I
I
I
I
I
I
I
I
I
"•
:'•'
•"
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
from
Non-Adiabatic
Mass
at 400
km
80
6O
4O
20
0
-20
-40
-60
-80
b
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
Figure 4. (a) Map of seismicvelocityanomalyat 400 km depth as seenfrom tomography
[ri and Romanowicz,
1995],(b) Part of the velocityanomalythat couldbe due to advected
masses
according
to (8). Contourlinesare 5 rn s-z. Shadedareasare zonesof slowerthan
averagevelocity, and white areas are zonesof faster velocity. The advectioninduced anomaly
is only -•20% of the total observedanomaly.The differencebetweenFigures4a and 4b can be
interpreted as the velocity anomaly of pure thermal origin.
ousthroughoutthe transitionzone)is ad hoc. However,
even minor, theseeffectswould have a large impact on
the dynamic topography.
In models without advection, large-amplitude dynamic topography is implied by the resolution of the
Navier-Stokes equation. This surface load is also required in order to fit the geoid. The effect of partial ad-
controlledby the massdipoleconsisting
of surfacedeflection and lower mantle loadsin previousmodelsis
nowexplainedby the massdipoleconsisting
of advected
equidensitysurfacesand lower mantle loads.
It is important to emphasizethe largedifferences
between our model and models where the mantle is chem-
icallylayeredat 670km depthor deeper[Wenand Anderson,1997].The variousphenomena
that ]nightslow
anomaly at depth. The long-wavelengthgeoidthat was down the flow in our model do not requesta change
vection is to transfer
the surface load into a diffuse mass
24,664
LE STUNFFANDRICARD:PARTIALADVECTIONANDDYNAMICTOPOGRAPHY
ley of reasonablemodel goestoward very large viscos_
Cretaceous
itywhen
aincreases.
Chemically
stratified
models
must
alsocompensatethe mantle anomaliesat 670 km depth,
which leadsto very large deflectionsof this interface.In
contrast, our model needsa moderate viscosityincrease
30ø
0*
-30 ø
30ø andinduces
asmall,
equivalent
topography
atthephase
changes.
"•
'....................................................
!i•
................
'.......
...........
0* flow
Our
also
greatly
from
models
where
the
\........................
'i.•:•)•:iii!•i•ii•i•ii•.•..•ii!•i•:!•i?f•i!•?•::•{i•i?....•.
('.....
00 ":"?'•'•'
ismodel
reduced
bydiffers
theeffect
ofendothermic
phase
tran'........
• ??==
...........
-::½}j•':•:•;?
:•?'"•:•::;.?:.•
......
'....................
30 ø sitionat 670 km depth, althoughthe massanomaly
• •;•::'•::"
::::;?::::::::
..........
:•::•:•
............
-60 .....
- '•'-' ••••:••
...........
60
thatiscreated
in thetransition
zonebyadvection
is
comparable
inpattern
and
amplitude
tothat
inferred
by Thoravalet al. [1995].This anomalyis alsocomparable to that observedat long wavelengthby seismol-
ogists[Shearerand Masters, 1992]. However,whereas
this anomaly is interpreted as thermal by Thoraval et
Paleocene
al.
[1995], our interpretationis different. We con-
sider that the deepeningof the phasetransition at 670
km depth inside slabs, as documented at short wave-
length by Wicks and Richards[1993]and Vidale and
Benz [1992],is mostly a consequence
of the endothermic phase change. We suggest,on the contrary,that
the long-wavelengthpattern found by Shearerand Masters [1992]couldbe rather due to advection.Assuming
30
o
.313
ø
60ø
that slabs cross the interface, their cold temperature
"30
Oligocene
depressesthe interface by 10-30 km, assumingexperimentally determined Clapeyron slopes. However, this
depressionshouldbe very localizedover the slab thicknessand shouldnot be visible at long wavelength.It. is
easy in our slab model to take into accountthe depression of the 670 km depth phase boundary within the
slabs. The volume for which density has to be changed
is sosmall that the predictedgeoidis not.muchaffected.
Although, in section5, we discussa possibleadvection in terms of deflectionof the upper-lowermantle
boundary, we think this phenomenoncould occur in a
diffuseway all over the transition zone. Many more
phasetransitionsthan thoseclearlyseenin seismology
(a olivine-+/3 spinel,7 spinel-+ perovskite+ magnesiowiistite)take placein the uppermantle(orthopyroxene -+ clinopyroxene,clinopyroxene--+ garnet,/3 spinel
-+ 7 spinel, garnet --+ perovskit.
e...). Each of them,
when advected by the flow, could induce a new load of
stabilizing buoyancy. The advectionof phasechanges
Figure 5. Sea level changespredictedbetweenthe is alwaysdownstreamand doesnot dependon the sign
presentand threedifferentperiods:late Cretaceous
(88 of the Clapeyron slope. This means that the deflection
Ma), late Paleocene
(59.2Ma), andlateOligocene
(27.7 of the 410 km (exothermic)and 670 km (endothermic)
couldbe correlated(if advectiondominates)
Ma). Paleogeography
reconstructions
are fromScotese boundaries
(if thermaleffectsdominate).
[1992]. Contourlines are 100 m apart. Shadedareas ratherthan anticorrelated
have undergoneuplift and white areashaveundergone Advection and thermal effects act in an opposite way
subsidence.
at 410 km, and therefore one might expect.the topography on this discontinuityto be of lower amplitudethan
in the major element compositionsuchas iron or sili-
that of the 670 km discontinuity,where theseeffectsadd
up. This would explain why lesstopographyis actually
cium[e.g.,AndersonandBass,1986],throughthe tran- seenby seismologyat 410 km than at. 670 km and why
do not look anticorrelated[Shearer
sitionzone. The exchangerate betweenupperand lower thesetopographies
mantleis still large,and slabscouldcrossthe interface. and Masters, 1992; M.P. Flanagan and P.M. Shearer,
Chemicallystratified modelsrequirevery large viscos- Global mappingof topographyon transition zonedis-
ity contrast(•0104)at 670 km depthto fit the geoid continuitiesby stackingSS precursors,submitted to J.
[Colin,1993].This is clearin Figure2, wherethe val- Geophys.Res., 1997].
LE STUNFF AND RICARD: PARTIAL ADVECTION
Lithgowand Gurnis[1997]recentlyshowedthat past
sealevelvariationscanbe computedfrom a geodynamic
model,whereall mantle heterogeneityare deducedfrom
ancient subducted slabs as suggestedby Ricard et al.
[1993]. When comparedwith geologicalobservations,
AND DYNAMIC
TOPOGRAPHY
24,665
Bax
= 12
•----ln
r _ r a <c•ga
a-r•)
'
r]0
p > r]0 (•-(r
+r•ln
(__r)(PcPro)),
r0
the computed sea level curves are in phase but display
lnuch too large amplitudes. Partial advectionenables
us to reduce
the sea level variations
to a reasonable
amount, matching both the amplitude and the phaseof
OZgaPm
geologicalrecords.
Appendix:
(14)
ra <p>
•lo
_
0a).
Mathematics
where Pc and rc are the core densityand radius, respectively. When B is computed, the propagator matrix
The continuity,Navier-Stokes,and Poissonequations correspondingto exponential of B is readily obtained
are solvedby usinga sphericalexpansionof all variables. through seriesexpansion.
A six-dimensional
vector U is then built from the radial
variationsof radial velocity,poloidalhorizontal velocity,
Acknowledgments.
We are grateful to C. Thoraval for
vertical stress, poloidal shear stress,gravity potential, discussionsand to C. Lithgow-Bertelloni, M. Gurnis and U.
and gravity acceleration. When incl,.•dingthe new ex- Christensen for providing us with their latest unpublished
pressionof 5p in the equationsof mantle circulation, manuscripts.
one obtainsthe classicalpoloidalequation[e.g.,Hager
and Clayton, 1989]:
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AND DYNAMIC TOPOGRAPHY
24,667
Y. Le Stunif, Centre Scientifique et Technique, Total, Domaine de Beauplan, Route de Versailles, 78470, Saint-Rdmy-
Les-Chevreuse,France. (e-mail: [email protected])
Y. Ricard, Laboratoire des Sciences de la Terre, Ecole
Normale Supdrieure, 46 Allde d'Italie, 69364, Lyon, France.
(e-mail: [email protected])
(ReceivedDecember10, 1996; revisedJuly 8, 1997;
acceptedAugust 7, 1997.)