Download Differential Rotation Between Lithosphere and Mantle: A

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. An increase in the lower mantle viscosity confines
workhasbeenpartly supported
by the INSU-DBT (Dynamique
et Bilandela Terre)program
(contribution
270),andbytheASI
(AgenziaSpazialeItaliana).
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acceptedJanuary14, 1990.)