JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 89, NO. B7, PAGES 5987-6002, JULY
10, 1984
Geoid Anomalies in a Dynamic Earth
MARK
A. RICHARDS AND ]]RADFORD H. HAGER
SeismologicalLaboratory, California Institute of Technology,Pasadena
In order to obtain a dynamicallyconsistentrelationshipbetweenthe geoidand the earth'sresponseto
internal buoyancyforces,we have calculatedpotential and surfacedeformationLove numbersfor internal loading. These quantitiesdepend on the depth and harmonic degreeof loading. They can be
integratedas Green functionsto obtain the dynamic responsedue to an arbitrary distribution of internal
density contrasts.Sphericallysymmetric,self-gravitatingflow models are constructedfor a variety of
radial Newtonian viscosityvariationsand flow configurationsincludingboth whole mantle and layered
convection.We demonstratethat boundarydeformationdue to internal loading reachesits equilibrium
value on the sametime scaleas postglacialrebound,much lessthan the time scalefor significantchange
in the convectiveflow pattern, by calculating relaxation times for a seriesof sphericallysymmetric
viscousearth models.For uniform mantle viscositythe geoid signaturedue to boundarydeformationsis
larger than that due to internal loads, resultingin net negativegeoid anomaliesfor positivedensity
contrasts.
Geoidanomalies
fromintermediate-wavelength
density
contrasts
are amplified
by up to an
order of magnitude. Geoid anomalies are primarily the result of density contrasts in the interior of
convectinglayers; density contrastsnear layer boundariesare almost completelycompensated.Layered
mantleconvectionresultsin smallergeoidanomaliesthan mantle-wideflow for a givendensitycontrast.
Viscositystratificationleadsto more complicatedspectralsignatures.
Becauseof the sensitivityof the
dynamicresponsefunctionsto model parameters,forward modelsfor the geoid can be usedto combine
severalsourcesof geophysicaldata (e.g., subductedslab locations,seismicvelocity anomalies,surface
topography)to constrainbetter the structureand viscosityof the mantle.
modelsand theirimplications
for geoidinterpretation.
The
INTRODUCTION
aim is to providequa•ntitative
relationshipsbetweendensity
The relationship between large-scalegeoid anomalies and
contrasts
within
the
earth
and
other
geophysical
observables,
thermally driven flow in the earth's mantle was discussed
including boundary topography,as well as the geoid. At the
almost 50 years ago by Pekeris [1935]. He showed that the
gravitatio•nal
effectof the surfacedeformation
causedby the
flow is oppositein sign and comparablein magnitudeto that
of the driving density contrast. Consequently,in a viscous
earth the net gravity or geoid anomaly is also dependentin
both sign and magnitudeupon the dynamicsof the mantle.
This representsa complete departure from the result for a
rigid or elastic earth in which positive internal density contrastsare alwaysassociatedwith positivegravitationalanomalies.
Studiesof postglacialrebound [e.g., Haskell, 1935; Cathles,
1975] as well as the very existenceof plate motions show that
presenttime we cannotsolvethe full problem of thermalcon-
vectionfor a givenmodelto determine
thesedynamical
relationships
for the Wholesyste•(seeMcKenzie[1977]for
tWo-dimensional
numerical
exampleS).
Sinceboththe temperature structureof the mantle and the temperaturedependence
of the densityand viscosityof mantle mineralsare unknown,
and since even the geometry of the convectivecirculation is
not known(i.e.,wholemantleversuslayeredconvection),
a
simpler and more direct approach is desirable.If the thermal
density anomaly is treated simply as a "load," the resulting
surface
deformati
øn andgeoidanomalycanbe determined
by
the mantle respondsto stressesapplied over geologictime
scalesby slow creepingflow. Therefore any interpretation of
solvingonlytheequilibrium
equations
for a viscous
earth.
corn [1964, 1967] addressed
the relationshipbetweenlongwavelengthgravity anomaliesand the flow field in a selfgravitating, uniform viscous sphere. Each of these studies
showedthat the deformationof boundaries,especiallythe
upper surface,has a major effectupon the net gravity or geoid
anomalyarisingfrom a densitycontrastat depth. Moreover,
the effectsof viscositystratificationand layeredconvectionin
functions
of thedepthandharmonic
degree
of theload,thus
The standard characterizationof the earth's responseto
tidal
loading in terms of Love numbers [Love, 1911' Munk
long-wavelength
geoidanomalies
shouldincludethedynamicaleffects
firstdescribed
byPekeris
[i935].These
effects,
par- and MacDonald,1960] suggestsa usefulway to characterize
ticularlyboundarydeformationcausedby flow,havebeenin- dynamic responsefunctions.Love numbersfor internal loadby normalizing
residual
geoid
vestigatedby Morgan [1965], McKenzie [1977], and Parsons ing of the earthareobtained
and Daly [1983] for intermediate-wavelengthfeatures using anomalies and boundary deformationsby the gravitational
two-dimensional
modelswith uniformmantleviscosity.Run- potential of the driving load. We obtain these quantitiesas
yieldingLove numbersthat are equivalentto Green functions.
A major question currently is whether chemical stratification of the mantle, associated with the 670-km seismic dis-
continuity, presentsa barrier to vertical flow and divides the
mantle into separatelyconvectinglayers. In order to address
this issueour flow modelsinclude not only radial viscosity
the mantleCansignificantly
alterthe calculated
relationship variations but also the possibilityof either mantle-wideor
chemicallystratifiedflow in the mantleas illustratedin Figure
betweengeoid elevationsand driving density contrasts[Rich1. Both the geoid and boundary deformation responsefUncardsandHager,I98I; Ricardetal.,1983].
in this PaPer,we developand discussseveraldynamical tions (Love numbers)show a strong model dependence.For
example, for mantle-wideflow, positive driving density conCopYright1984by the AmericanGeophysicalUnion.
trasts cause net negative geoid anomalies for uniform mantle
viscosity,sincethe negativeanomaly causedby upper surface
Papernumber
4B0594.
deformationoverwhelms
the geoidanomalydue to the density
0148-0227/84/004B-0594505.00
5987
5988
RICHARDS AND HAGER: GEOID ANOMALIES IN A DYNAMIC EARTH
Reference Boundary
DeformedBoundary
powerful constraint on geoid interpretation, and a large
amountof informationon crustalthickness,
topography,and
densityhave yet to be consideredin relation to the geoid. It is
thereforeresonableto expectincreasinglyaccurateand useful
observationsof the earth's densityanomaliesand effective
boundary deformations.Cast in the form of dynamic response
functionsas discussedin this paper, thesedata provide means
Density
Contrast
for discriminatingamong variousdynamicmodelsfor the
mantle.
MODELING
ReferenceBoundary
DeformedBoundary
/
œ=5
Upper
Mantle
Density
Contrast
CONSIDERATIONS
Quantitative models for the geoid derive from constitutive
laws, equations of motion and material continuity, and
boundary conditions.It is impossibleat the present time to
specifyfully the earth'stheologyor to solveall theseequations
exactly.We must make various approximationsand assumptions in developingmathematicalmodels;in doing so we try
to includethe important physicaleffectswhile avoidingunnecessarycomplicationin the method of solution.In this section
we discussour assumptionsconcerningmantle rheology and
flow, boundary conditions,and the thermal driving forcesinvolved.Boundary deformationis affordeda detailedtreatment
in a separatesection.
Rheologyand Flow
(b)
Fig. 1. Illustrations of flow models for spherical earth calculations (l = 3). (a) Whole mantle flow. (b) Flow with a chemicalbarrier
at the 670-km discontinuity.Plus and minus signsindicate positive
and negativedensitycontrasts.The dashedlines are referenceboundaries,and the solid lines representthe displacedboundaries.Streamlines indicate
the sense of flow.
The selectionof appropriate modelsfor the mechanicalbehavior of the lithosphere,mantle, and core dependsupon both
the time and length scalesinvolved. Here we are interestedin
length scales for which lithospheric strength is negligible,
roughly defining what is meant by "long-wavelength"geoid
anomalies,and time scalesof the order of those required for
substantial changes in the convective flow pattern in the
mantle.As we show below, this impliesharmonicdegreesI less
than 40 (wavelengthsgreater than 1000 km). If mantle flow is
reflectedin plate motions,the mantle flow pattern is stablefor
times far in excessof 1 m.y., which we take as a characteristic
contrast itselfi However, net positive geoid anomalies are obtained for a channelof sufficientlylow viscosityin the upper
mantle. This occurs becauselow upper mantle viscosityre- time scale. The core is inviscid for the time scales of interest
ducesthe deformation of the upper surface.The core-mantle here; it may also be assumedto be in a state of hydrostatic
boundary deformation increasesbut has less effect upon the equilibrium.
geoid becauseof its great depth. As shown in Figure lb, the
The lithospherepresentsseveralproblems,including those
layered flow case introducesmuch more complicatedbehav- of finite elasticstrengthand of lateral variationsin theological
ior. It is preciselythis strong model dependencethat makes properties, density, and thickness.For loads of wavelength
these models useful in geodynamics.The observedspectral greater than about 1000 km the elastic strength of the lithoand loading-depthdependenceof theseresponsefunctionscan sphereis negligible[McKenzie and Bowin,1976; Watts, 1978]
be used to discriminateamong various proposedmodels for so that surfaceloads are supportedby buoyancyand the remantle structureand rheology.
sultingflow in the mantle. The lithosphereis essentiallytransAlthough observations of satellite orbits provided the parent to long-wavelengthnormal tractionsfrom flow in the
meansfor determiningthe lower-orderharmonicsof the geo- mantle.
potential over two decadesago [Kaula, 1963a; Guier, 1963],
Lateral variations in theological properties of the lithosubsequent
efforts
tdibterpret
thelong-wavelength
geoid
have sphereare responsiblefor the plate tectonicstyle of convection
beenlargelyunsucc6ssful.
Somecorrelations
withtectonicfea- in the earth's mantle. The plates move as distinct units with
tureshave beensuggested[e.g.,Kaula, 1972], notably a gener- respectto each other and effectivelyform a rigid lid for any
al correspondence
betweensubductionzonesand geoid highs. sublithosphericsmall-scaleconvectionwhich may exist. Plate
Chase [1979] and Crough and Jurdy [1980] demonstrateda boundaries,on the other hand, are relatively weak, allowing
remarkable correlation betweenthe spatial distribution of hot the plates themselvesto participate in mantle convection
spots and the nonhydrostaticsecondharmonic geoid. Hager [Hager and O'Connell,1981]. This lateral heterogeneityof the
[this issue] has shown that the fourth through ninth geoid effectiveviscosityof the lithosphereallowsdensitycontrastsin
harmonicsare strongly correlatedwith the seismicity-inferred the interior to excite significant toroidal flow [Hager and
presenceof subductingslabs,thus yielding quantitative esti- O'Connell,1978] not just the poloidal tT0•vwhich would result
mates over a definite spectralrange for the dynamic response from a mantle with sphericallysymmetricviscositystructure.
functions which are the subject of this paper. Additionally,
The choice of boundary conditionsat the surface is not
recent seismologicaldeterminations of lateral variations in obvious,and the analytical techniquewe use here does not
seismicvelocities [e.g., Nakanishi and Anderson,1982; Dzie- accountfor lateral viscosityvariations.We argue that the mewonski,this issue;Clayton and Comer, 1983] provide another chanicaleffect of the lithosphereon small-scaleflow beneath
RICHARDS AND HAGER: OEOID ANOMALIES IN A DYNAMIC EARTH
plate interiors can be representedby a no-slip boundary condition at the earth's surface.Flow involving the plates themselvesis probably best approximated by a free-slipboundary
condition. We presentcalculationsfor both casesand find that
the resultsare similar. This suggeststhat a more complicated
boundary condition that would better representthe effectsof
lithosphericplateswould also be similar.
The effect on the geoid of lateral variations in lithospheric
thickness and density has been discussedby Chase and
McNutt [1982] and Hager [1983]. These variations are primarily the result of variations in crustal thicknessand in the
age of the lithosphere.Sincethey are closeto the surface,they
are generallywell compensated,
and their effecton the geoidis
small (lessthan 20 m out of a total geoid variation of greater
than 200 m). However, their effect on topography is large. If
surfacedeformationand the geoid are to be usedconcurrently
to obtain sublithosphericdensitycontrastsas discussedbelow,
correctionsmust be made to compensatefor the topographic
effectsof large densitycontrastswithin the lithosphere.
The appropriateconstitutivelaw (or laws)for modelingflow
in the mantle cannot be determinedwith certainty at the present time. Possible creep mechanisms for deformation of
mantle minerals include dislocation climb [Weertman, 1968],
which impliesa nonlinearrheology,and grain boundarydiffusion or superplasticity[Twiss, 1976; Ashbyand Verrall, 1977;
Berckhemeret al., 1979], which at low stresslevelsmight result
in a linear relationshipbetween shear stressand strain rate.
Mathematical tractability has led most researchersto employ
linear rheology,either Maxwellian or Newtonian, in modeling
flow in the mantle. Furthermore, for some surface loading
problemsin which the magnitudeof shear stressdecayswith
depth, nonlinear rheologymight not be distinguishablefrom
layeredlinear rheology;the lower stresslevelsfound at depth
would correspondto higher apparent viscosity.Estimatesfor
effectivemantle viscosityhave been obtained for a variety of
loading problems.Values given for averagemantle viscosity
5989
Chemical stratificationand multilayer convectionhave been
suggested[e.g., Anderson,1979] to explain the major seismic
discontinuityat 670 km. Geochemicalbudget modelsas well
as the lack of seismicitybelow 670 km are thought by someto
suggestthat upper mantle flow does not penetrate this level
[Jacobsen and Wasserburg, 1980; Richter and McKenzie,
1981]. We include the effectof sucha boundary in our investigation in order to understandhow geoid and geodeticdata
might be usedto test the chemicallayer hypothesis.A chemical discontinuityis modeledby settingthe (steadystate) vertical velocityto zero at the boundary; horizontal velocitiesand
normal
and shear tractions are continuous.
This results in a
two-layer, shear-coupled,antisymmetricflow system,as illustrated in Figure lb. Another possibilityassociatedwith both
the 400- and 670-km discontinuitiesis that of an abrupt phase
changewithin the mantle, which in the simplestcasemight be
modeled as a spike in the compressibilitycurve for the mantle
assumingthat the transitionis adiabatic and ignoring thermal
effects.We have not treated this case since compressibility
introduces nonlinearity into the field equations and makes
solutions much more difficult to obtain.
Driving Forces and Loads
The relationshipsamong loading, gravity, and deformation
can be obtainedwithout solvingfor the thermodynamics.This
is accomplishedby calculating the flow driven by arbitrary
densitycontrastsat any given depth. Kernels (Love numbers)
representingthe viscousresponsefunctions so obtained can
then be integrated over depth in accordancewith any prescribeddistributionof thermal densityanomalies;the linearity
of the problem (with the caveat of linear, spherically symmetric viscosity)allows for superpositionof solutions. Our
method is to solve for loading due to a surfacedensity contrast at a given depth and sphericalharmonic degree,thereby
characterizingthe responseas a function of spatial wavelength
and depth in the mantle. In this way we can isolate the rehavegenerally
beenof theorderof 1021Pa s (1022P) [O'Con- lationshipsdesiredfor geophysicalobservablesfrom the thernell, 1971; Cathies, 1975; Peltier, 1976; Yuen et al., 1982], mal part of the convectionproblem.
althoughestimates
as smallas 10•8 Pa s havebeenobtained
for the upper mantle or asthenospherefor loads of smaller
scale[Passey,1981]. Although viscoelasticmodelshave found
application to shorter-termproblemssuch as glacial loading
and unloading [Clark et al., 1978; Wu and Peltier, 1982], the
time scalesof 1 m.y. or greaterof interesthere are in excessof
Maxwell times for the mantle so we ignore elasticeffects.For
the purposeof exploringthe basic physicsof internal loading
The Field Equations
With the above qualifications and simplificationswe can
specifytractable field equationsto investigatethe loading
problem for a variety of rheologicaland structural configurations in the mantle. The mantle will be assumed to behave as a
self-gravitating,spherically symmetric, incompressible,Newtonian viscousfluid. Since the Reynold's number is very large
problems
andfor mathematical
simplicity,
weemployNew-' owingto the mantle'shigh viscosity,
inertialor timetonian models in which viscosity is dependent upon depth
only, althoughwhen this theory is applied to actual data, the
resultssuggestthat lateral variationsin effectiveviscositymay
be important.
dependentterms are omitted from the equations of motion.
The only time dependenceis introduced by changesin position with time of the driving density contrastsand relaxation
of the boundariesto a steady state condition of deformation.
We
addressthe relaxation problem in detail in Appendix 2,
Boundary Conditions
the result being that boundary deformations decay rapidly
Three possibleboundariesare consideredin our spherically comparedto the time scaleof flow in the interior.
symmetric,layered earth models: (1) the core-mantleboundThe equationsof motion can be written
ary, (2) the upper surface,and (3) a change in composition
and/or viscosityacrossthe 670-km seismicdiscontinuity.
V. '• + pg = 0
(1)
We model the core-mantle boundary as one at which there
is no shear traction and no steadystate vertical transport.As where x is the stresstensor, p the density, and g the gravidiscussedabove, the mantle-lithosphereboundary is more tational acceleration. The mantle will be assumed to be incomplicated.We have investigatedboth no-slip and free-slip compressiblethroughout;although radial densitylayering can
conditionsand have includedboth typesin the resultspresent- be arbitrarily imposed in our method of solution, allowance
ed here, although, as we noted above, the differencebetween for finite fluid compressibilityis mathematicallydifficult and is
the two is not profound.
generally ignored by most authors [Cathies, 1975; Peltier,
5990
RICHARDSAND HAGER' GEOID ANOMALIESIN A DYNAMIC EARTH
/--
Deformed
,•..•n dory
_•__
.zU_
I -z
.Reference
b_0u_ndory
•z(x)
Fig. 2. Illustration of the geometryfor the analyticaltreatmentof
deformation of the boundary betweentwo fluid half spaceswith densities p• and P2- The actual boundary (solid line) is displacedan
amount &(x) from the referenceboundary (dashedline).
Since,flow-induced stressesare always much smaller (first
order) than the lithostaticstresslevel in the mantle (zeroth
order), flow and stressvariables are first order also; their derivatives behave like the product of first-order terms and the
approximatespatialwave number.The only first-ordercorrec-
tion due to deformationis the hydrostaticcorrectionto the
normal
stress'
i
i
'•....f = Zzzde,
- Pig cSz
(6)
In passingfrom the referenceboundary as seenin medium 2
to that seenin medium 1, we get an apparentjump in normal
1981] since the dynamic effect is probably small [Jarvis and
McKenzie, 1980]. Ricard et al. [1983] have shown that the
stress'
ATzz
1• '•ZZdef
2__Ap12
Qc•z
(7)
ref12: '•ZZdef
effectof compression
from lateral gravity variationsis negligible. The incompressible
continuityequationis
where Apl 2 = pl - Pa. By continuity of stressat the deformed
v. v = 0
(2) boundary,
Azzzf12= _Apa2gcSz
where v is the velocity vector. The Newtonian constitutive
relation
re
(8)
is
(A similar argumentwill imply an effectivejump in the gravix = - pl + 2r/e
(3) tational accelerationat the referenceboundary in the fully
self-gravitating
sphericalcase.)
wherep is the pressure,I the identity matrix, r/the viscosity,
This result can be readily applied to a simple half-space
and e the strain rate tensor.
problem. Figure 3 illustratesa surfacedensitycontrast(i.e., a
For global scale-loadingproblems,self-gravitationeffects thermal density anomaly), ad(k)cos (kx), at depth d, exciting
cannotbe ignored[Love, 1911;Clark et al., 1978]. The gravi- flow in a viscoushalf spaceof viscosityr/and densityp, with a
tational effects of deformed boundaries must be included in
traction free surfaceat the top. For simplicity we will first
any self-consistent
model. The gravitationalpotential V must assumethat the density contrast is not advectedwith the resatisfy
sulting flow so that it remainsfixed in space(this could be
V2V = 4r•Gp
(4) done experimentallyusinga heat pump, for example).We will
then show that the density contrast would not be advecteda
where we have chosenthe sign convention such that g =
significantdistancein the time it takesfor the boundarydefor- V V. These equationsare linear in all the variablesand can mation to reach equilibrium. Solving (1)-(3) using the twobe straightforwardlysolved by either propagator matrix dimensionalpropagator [ttager and O'Connell,1981], we find
[Hager and O'Connell,1981] or numerical techniques.Before that the boundary displacement3z evolvesfrom its initial unproceedingto a fully three-dimensional
(spherical)solution,we deformedpositionas
present some useful results from the simple two-dimensional
ad(k) COS
half-spaceproblem.Resultsfrom the viscousrelaxationprobcSz
= (1 + kd)exp( - kd)(1- e-'/•)
(9)
P
lem that justify the hypothesisof steadystateflow are givenin
Appendix 2.
The boundaryrelaxeswith timeexponentiallytowarda steady
state of deformation
with time constant
ANALYTICAL TREATMENT OF BOUNDARY DEFORMATION
Loading of the earth by gravitationalpotential (e.g.,tidal
loading), by external loads (e.g., glacial loading), or internal
densitycontrasts(e.g.,thermalconvection)will producedeformations of both the surfaceand any internal boundaries.In
this section we analyticallytreat boundary deformationto
first-order accuracyand derive someusefulresultsfor the two-
dimensionalproblem. Figure 2 illustratesthe warping of a
z = 2rlk/pg
(1O)
This is the sametime as that derivedfor the surfaceloadingor
unloading problem (e.g.,postglacialrebound [Haskell, 1935;
Cathles, 1975]). For example, with r/= 10:• Pa s,
2 = 2r•/k_>1000km, p = 3.5 Mg/m3, g = 9.8 m/s:, we obtain
z < 11,000 years. Assuming velocities in the mantle of the
order of 100 mm/yr or less,we seethat mantle transport of at
materialboundary
relativeto its deformed
or reference
state,
with densitiesPl and p•_above and below the boundary,respectively.The velocityand stressfieldsmust be continuousat
the deformedboundary.However,our solutiontechniquerequiresthat we propagatesolutionvectorsfrom one horizontal
/--Deformed
z
boundaryto the next,so we requireexpressions
for the velocity and stressfieldsat the reference
(undeformed)
boundaries.
If
the magnitude of deformation,gz, is sufficientlysmall compared to the thicknessof either of the adjacentlayersand the
spatial wavelength,•t, of interest(as in this study),any vari-
able,ui in mediumi, maybecontinued,
to first-order
accuracy,
from the deformedboundaryto the referencelevel by
-•x
z--0....
free •-•},:2
......
%(K)• ,• '• ,••
Fig. 3. Illustration of the Fourier flow analysis in a twodimensionalhalf space.The surfacedensitycontrastad(k) cos(kx) at
depth d excitesflow, resultingin deformationof the free surface.We
assumehere that the advectionof the densitycontrast by the flow is
negligible on the time scale for establishingthe boundary deformation.
RICHARDSAND HAGER' GEOID ANOMALIESIN A DYNAMIC EARTH
most a few kilometers (much less than the depth scale of
mantle convection)occursbeforethe free surfaceis completely
For sphericalmodelsin generalthe normalizedpotential
relaxed.
Kt(r) -
Alternatively,we can assumethat boundarydeformationis
rapid and calculateflow velocitiesunder the assumptionthat
vertical flow at the deformed surfacevanishes(i.e., boundary
deformation is complete).In this case,
v=(d)
= - ad(k)
cos
(kx)
gdexp
(-2kd)
4r/
(11)
5:• - 2(1+ kd)exp(kd)e
vx(d)
kd
(12)
Once again, we see that long-wavelengthboundary deformation is rapid comparedto changesin the convectiveflow
pattern independentof aa(k).Note that this result holds even
for "thin" layers which are normally associatedwith long
relaxation times. In Appendix 2 we show that viscousrelaxation occurs on a time scale much shorter than that for mantle
=
2r•G cos (kx)
k
[-(1
+ kd) + 1] exp (-kd)ad(k)
or
5 vres(0)
-
kd
(13)
+
i•v/geoid
__4toGa
Kt(r)(r/a)
•+25pt(r
) dr (16)
r=a
2/+1 or=o
cal earth models.
•vres(z= 0) = •V eff+ (•Va'•
(14)
This responsefunction is measurableif the driving density
contrasts within the earth are known a priori. Hager [this
issue]has usedthis in his discussionof geoid anomaliesfrom
subductedslabs, where density contrasts can be estimated.
Another applicationis in comparingseismicvelocityheterogeneitiesto the geoid (B. H. Hager et al., unpublishedmanuscript, 1984).By assuminga relation betweenseismicvelocity
and density the long-wavelengthgeoid coefficientsare obtained from the integral
convectionby calculatingrelaxation times for severalspheriThe long time limit of (9) shows that the effective mass
deficit associatedwith the surfacedeformation, •eff = pSZ, is
of oppositesignand of the sameorder of magnitudeas •a. It is
now evident for at least two reasonsthat the assumptionthat
gz is sufficientlysmall for the application of a linear continuation of the boundary condition is probablyjustified' (1) Thermal density contrastsin the earth, with the possibleexception
of subductedslabs,are probably not large enough to cause
gross deformation of either internal or external boundaries,
and (2) the earth's topographya priori precludeslithospheric
deformationsgreater than 10 km, while seismicdata do not
suggestlarge deformationsof the core-mantleboundary[Dziewonski and Haddon, 1974] or the 670-km discontinuity, although coverage is limited, especially in subduction zones
where deformation is expectedto be the largest [Hager, this
issue].
From (9) we can obtain the relationship between the observedgravitational potential and the load as well as the relationship betweentopography and geoid due to %(k). The
residual potential calculatedat the referencesurfacecontains
contributionsfrom both % and
•l res
is the Green function for the earth's surfacepotential per unit
loading at radius r and sphericalharmoinic degree I. This
quantity is a function of the earth model in general and is
related to Kaula's [1963b] elastic internal loading potential
Love number, kt", by
g,(r) =
Comparing this to the characteristicsurfacevelocityobtained
by differentiating(9), we find that
5991
where a is the radius of the earth, cSpt(r
) is the /th harmonic
density contrast at radius r, and G is the gravitational constant.
The other
observable
we can calculate
is a dimensionless
"impedance,"definedas the ratio of geoid elevationto boundary deformation:
Zl(r) = 5•/geoid/•]
CJrl
(17)
where•r• is the lth harmonicdeformationof the surface,and •
is the gravitational acceleration.(Note that this is not a true
impedance
sinceit involvesthe observedpotential•geoid instead of the driving potential •[,'•'.) Defining a surfacedeformation Love number h•" [e.g., Munk and MacDonald, 1960],
we have
kl"
Zl(r ) = 1 + •
hi"
This quantity could be estimatedfor the surfaceby taking the
ratio of harmonic geoid coefficientsto topographiccoefficients
with the effects of crustal thickness variations
removed.
To
estimateZl(r) for a givendensitydistributionand earth model,
the numerator and denominator of (17) must be integrated
separately.
We have now defined two observablesrelating the geoid
directly to internal loading and earth structure for a density
contrast at a given depth. Also, (16) shows how to interpret
thesequantities for models with distributed density contrasts.
We have not yet introduced the gravitational interaction between the load and the massanomaliesdue to boundary deformation.
This means that for a uniform half spacea positive density
SPHERICAL EARTH MODELS
contrast at depth results in a negative geoid anomaly. The
geoid anomaly goesto zero as the densitycontrastapproaches
the surface. Furthermore, for depths greater than the wave- Formal Solution
length the geoid can be much larger in magnitude than that
Analytical solutions to field equations (1)-(4) with radial
obtained for a rigid half spacefor which there would be no
variations in viscosityand density and for arbitrary, laterally
boundary deformation. This occurs because the stress that
varying internal loading are given by Hager and O'Connell
causesboundary deformation falls off less rapidly with the
[1981]. Internal densitycontrastsdrive poloidal flow fieldsfor
which the relevant stress,flow, and gravitational potential
depth of loadingthan the potentialfrom the load itself.
5992
RICHARDS AND HAGER' GEOID ANOMALIES IN A DYNAMIC
Now, as was indicated previously in (14)-(17), we can
- CrdTspl/or
b
characterize
E[• '•\ 1:2
_•
!:7 S/..•
'"/I1:20
IJ \
EARTH
6
m
all solutions in terms of harmonic
order I and
radial level or depth of the driving density contrastssince,
owing to the linearity of the field equations,thesesolutionsor
kernelscan be superposedto representany arbitrary density
contrast in the mantle.
,
•
,Surface
c[• T *",-,,-,,-1670km
'
o,, \ •7 =u'uu ..,:
tI.&W
½:o..
'
above
J
J k above
/ •
½
•
\,, 670km
J j
.[/,
o',.x%.
(d),
:5500
64000
n.'"
I
20
•670km
I •
I
20
A familiar and useful property of the propagator matrix
formulation is that solution vectors can be propagated
through a seriesof differentmaterial layers by simply forming
the product of the individual layer matrices:
P(r, ro)= P(r, r•)P(r•, r,o)
I
(24)
20•
Surface
)--'! ,{:
Therefore changesin viscosity(and density) with depth are
16?Om'i
m
easilyincorporatedinto this formalism.
' ,,,-r
,, ,,' f
Boundary Conditions
Fig. 4. Mass displacementat the boundariesas a function of
loading depth for harmonicdegrees2, 7, and 20 (whole mantle flow).
Plots are normalized to a unit densitycontrastload. The "S" curve is
for the upper surface,"C" is for the core-mantleboundary,and "T" is
the total mass displaced.Figures 4d and 4e show the effect of low
viscosityin the upper mantle for harmonic degree7. Figures 4a-4e
are for FF boundary conditionsand Figures 4f-4h are for NF conditions.
variables can be propagated from one radial layer to another
accordingto
We have discussedtwo typesof boundary conditions:(1) A
free-slip (denoted "F") boundary condition requires zero
radial velocity(v•)and zero shearstress(z•0),a conditionwhich
appliesat the core-mantleboundary, and (2) a no-slip (denoted "N") boundary condition requireszero radial and tangential velocities(v• and Vo).Good argumentscan be made for
applying either of these boundary conditions at the surface.
For completenessand to gain insight into the physicsof the
problem we have modeled both combinations. For example,
for no slip at the deformedsurface(r = a + t•ra)and free slip
at the core-mantle boundary (r = c + 5rc), we have, to first
order,
u(r) = P(r, ro)u(ro)+
P(r, •)b{•) d•
(18)
po
a2•_•_V.]
r
rio It3rJ
uN(a
+t•ra):
[0,
0,0,a•rrOa/rio,
poaSV•/rio,-
o
where u is the six vector given by
(25)
u(r)
= vr,
Vo,
rrr•/rio,
rZ•o/rio,
porSV/qo,
pør2
riot3r
j
(19) ur(c
+5r•)=O,•o•,
c•:m/rio,
O,poSV•/rio,
rior3r
poC_
]
with radial and tangential velocitiesv, and Vo,normal radial
and shear deviatoric stressesz, and Z,o,perturbed potential
5V, and referencedensity and viscosityPo and rio. In these
expressionsand for the remainder of this paper all dynamical
variables contain an implicit sphericalharmonic dependence
which has been suppressedfor simplicity. The 6 x 6 matrix
P(r, ro) can be expressedanalytically [Gantmacher,1960] as a
function of r/r o, normalizedlayer densityp* = P/Po, normalized layer viscosityri* = ri/rio,and harmonic order I. The driving term for this systemis the integral on the right in (18) in
which the densitycontrastsare introduced by
b(r)= [0, O,rg(r)Sp(r)/rio,
O,O, -4•r2GpoSp(r)/rio]
t
(20)
where g(r) is the unperturbed(hydrostatic)gravitational acceleration and 5p(r) is the densitycontrastat radius r.
The problem is greatly simplifiedmathematicallyby casting
the driving densitystructurenot only as a sum over spherical
harmonicsbut also as a sum over radial surfacedensity con-
where we have also set the normal stress to zero at the surface.
These boundary conditions apply at the actual deformed
boundaries(see Figure 1); however, (22) shows only how to
propagatefrom one sphericalreferenceboundary to another.
Therefore(25) must be analyticallycontinuedto the reference
boundaries via (5)-(8) cast in sphericalcoordinates.This is a
tediousoperationwhichinvolvesfindingexpressions
for • ....
l•rrc,
5Va,5Vc in termsof theresulting
harmonicsurface
deformations(t•ra,5rc)and the detailsas well as the resultingsystem
of equations are included in Appendix 1. This procedureinTABLE 1. Model Parametersfor Basic Models for SphericalEarth
Calculations
Model*
Pt
A
B
C
4.43
4.43
4.43
0.0
0.0
0.0
4.43
4.43
4.43
D
E
4.92
4.92
0.5
0.5
3.50
3.50
(21)
u(r)= P(r,ro)u(ro)
+ • P(r,bi)bi
•/., Pas
d, km
1021
1019 or 1020
1019 or 1020
'"
200
670
1021
1019 or 1020
670
670
Flow
1021
1021
l021
1021
l021
Parametersinclude earth and core radii, c and a; core, upper, and
Pc,P,, and Pt (in Mg/m3);upperand lower
(22) lowermantledensities,
b,
where
bi = [0, 0, big(bi)ci/rio,
O,0, -4•bi2Gpoo'i/rio]
r
tit, Pas
Layered Flow
b•
where 5(r) is the Dirac delta function and the ai are the surface
densitycontrasts.Equation (18) becomes
p.
Whole Mantle
trasts, that is,
ap(r)= •, a(r- bi)oi
6Pt..
(23)
mantleviscosities,
•/, and •/t' effectivedensityjumps at the core-mantle
boundaryand the 670-km discontinuity,6pcmand t•ptu' and the depth
of the upper flow or viscositylayer, d.
*For all modelsc = 3480 km, a - 6371 km, p• - 11.0, and 6Pcm=
4.5.
RICHARDS AND HAGER: GEOID ANOMALIES IN A DYNAMIC EARTH
Normalizedpotentialcontribution
valvestwo physicaleffects:(1) When solution vectorsare referencedto the underformedboundaries,there is an apparent
jump in normal stressat each boundary given by an expressionsimilar to (8):
•'r,.,.= --•3pg(r)•3r
O0
n.'
,
,/../..•
.
,
: .....
race
0c•---'..."Z..•
05...........ISu
r.
/ .... ' -'•'"•..."'"•"'"•'-"
.....
•.....
6?•km
II, , .......
"'....... ,
, Iol F , , , lu /,
/ ...... t =2
/."
6400)
and (2) there is a similarjump in gravity at eachboundary(see
Appendix 1). Accordingly,each boundary deformation makes
a first-order contribution to the perturbed potential. This
occursbecause,as demonstratedabove, the massdisplacedis
of the sameorder of magnitudeas the driving densitycontrast.
The important thing to note is that we can castthe problemin
a form whosesolutiongivesthe deformationof boundariesas
well as the gravitationalpotential at thoseboundariesas functions of the harmonicorder and depth of loading.From these
solutionswe can generatethe desiredquantities (Love numbersand impedances)definedby (14)-(17).
Equation (24) showshow to treat layeringeffectsin material
properties,but a layered flow system(Figure lb) requires a
separate boundary condition at the flow barrier. We model
this boundary as a compositionalchange accompaniedby a
densityjump which resultsin a simpleflow barrier with shear
coupling betweenthe two layers. Mathematically, this can be
expressedas
0.5
V.
I 0
t=7
0.5
,, .....
to,
o;-'........
.......
0
½--;t
/c/'.........
:.........
ø '"."'
rr
i.¾./.::::.j..
, , (d)[........
, (e
o ,,,
......
::
........
, , ,=z
3500
05
t:20
, (C)/core
Io
c,.__...•,
...•....•..
.......
.-'•."•--'t
øurTø
, ' ' (f)Cor
?
Fig. 6. Relative values of gravitational potential contributions
from the density contrast at the indicated depth ("a"), the deformation of the core-mantle boundary ("c"), and the upper surface
deformation ("a") for harmonic degrees2, 7, and 20. (The %" and "c"
curvesactually have oppositesign from the "a" curve).Values plotted
are normalized by the maximum value of curve "a" for conveniencein
comparison.Figure 6d showsthe upper surfacedeformation not corrected for self-gravitation (curve "a'"). Figures 6a-6c are for FF
boundary conditionsand Figures 6d-6f are for NF conditions,all for
uniform mantle viscosity.
u(r
+•r)=[0.
vo.
rz./rlo.
rz•o/rlo.
poT•V/rlo.
•r/o •Orj ß
in which the radial velocityis setto zero at the boundary.This
also representsanother boundary which will deform under
loading and an additionalapparentjump in normal stressand
gravity will occur when (27) is analytically continued to its
referencesurface(e.g., 670 km depth). For this two-layer flow
problem we have two systemsof equations(22) that are coupled at an internal boundary whosefield variablesare given
by (27). We have not included the details here, but solution of
these propagator equationsproceedsstraightforwardly,as in
Appendix 1, where details are given for the case of whole
,
4
(26)
(27)
5993
well as the gravitational potential and its radial derivative at
sphericalreferenceboundariescorrespondingto the unperturbed layer boundaries.These referenceboundary vectors
can be propagated(seeequation(18)) to any radial level in the
earth, so each solution implicitly containsthe stress-flowfield
and gravitationalfield throughoutthe mantle.For any specific
model,solutionsvary with the depth of loadingand harmonic
degreeso that even for the limited variety of modelswe have
consideredhere a very large amount of information is generated. The sphericalearth results presentedin this section are
restrictedto thoseinvolving either the geoid or boundary deformations.The resultsthat follow involve only the approximations discussedabove and are analytic, although the resultingalgebraicexpressions
are evaluatedon a computer.
Whole Mantle
Flow
The simplestmodel is that for mantle-wideflow and most of
the physical ideas from sphericalmodeling can be demonmantle flow.
stratedwith this model. Figures4a-4c show for model A (see
Table 1) the amount of massper unit area, normalizedby the
RESPONSE FUNCTIONS
amplitudeof the load, that is displacedby deformationof the
The mathematicalformalismwe have developedfor solving core-mantleboundary and the upper surfaceas a function of
the internal loading problem yields solutionsin the form of the depth of loading for representativeharmonicdegrees2, 7,
boundary vectorsthat give the fluid velocitiesand stressesas and 20. The displacedmassis oppositein sign to that of the
driving mass anomaly so its negative is plotted for ease in
comparison.Figures4a-4c are for free-slipat both the coreMode excitation
mantle interfaceand the upper surface("FF" case)while Figures4f-4h are for no slip at the uppersurface("NF" case).The
O01;.0-0.5
0 0.5I.O-1;0-q.5
? O;,.Sj
I.9 [I.
9-q.5
? O;5/•O]$urfoce
closerthe load is to a boundarythe larger are the resulting
mass displacementand deformation at that boundary. Also
plotted is the total amount of massdisplacedat both bound/•
i
I
F'"• /Core
aries. By analogy to the Airy or Pratt principlesof isostatic
6400-1.0-0.5
0 0.5
1.0 -I.0 -0.5 0 0.5 1.0Surfoce
,
,
,
•,
compensationin the lithosphere,these curvesrepresentdyE L•*:0.'j.•.//•/•0
J •'•*:b.O'
i -"•'•(•'670
km
namic isostasyfor mantle loads in a sphericalearth. The total
mass displacedis oppositein sign and comparable,but not
•670
km
identical,in magnitudeto that of the load (dashedline) for
long wavelengths. For the uniform viscosity model,
/
,oov
,,,,,,.ov1
.•t |670kin
•
/
f=7(e
Fig. 5. Viscousnormal mode excitation as a function of loading
depth for whole mantle models.The "C"(core) and "M0" (mantle)
modesare unit normalized,and their excitationamplitudesare plotted for harmonicdegrees2, 7, and 20. Figures5d and 5e illustratethe
effectof low viscosityin the uppermantle.
--O'disp
(total)/abis of orderunity for 1= 2 and 1= 7, but for
higher values of I this ratio becomesmuch smaller if the load
is not closeto a boundary: this meansthat the load is almost
entirelydynamicallysupportedby flow in the interior.Figures
4d and 4e showthe effectsof one and two ordersof magnitude
5994
RICHARDS AND HAGER: GEOID ANOMALIES IN A DYNAMIC EARTH
K
7
14
-14
-7
0
NF
/
!
//I
7
•
14
Surface
NN/ 6 km
vector.This resultsin a decoupledfour-vector systemwhere
u(6 x 6)-• u'(4 x 4) = Evr,vo,rr,/rlo + pr•iV/rlo,rr,o/rlo]r
Physically,the normal stressterm has been augmentedby a
"gravitational pressure"term, prSV/rlo, to form a systemof
/79"
equations that is otherwisesimilar to the 4 x 4 propagator
, Io
systemusedin two-dimensionalproblems [e.g., Cathles,1975].
This
formulation then shows explicitly how self-gravitation
7 14 -14 -7 0
7
14Surfac
e
entersinto the dynamicsof the loading problem. Upon examFF
670km
ination of the excitation vector (23) we notice that there are
two driving terms: (1) The third term of the vector correspondsto the stressdue to the density contrast being acted
upon by the zeroth-orderfield, and (2) the sixth term repre'
Core
sentsthe driving force due to the first-order field perturbation
from the density contrast, that is, a gravitational pressure
Fig. 7. Surface potential response K as a function of loading
term. These extra pressureterms do not drive flow in steady
depth for harmonic degrees 2, 7, and 20. Illustrated are the four
state,but they do affectboundary deformations.
possiblecombinations of free-slip (F) and no-slip (N) boundary conditions, all calculated for a uniform visocisty mantle. The light,
In Appendix 2 we discussthe problem of viscousrelaxation
straight lines in Figures 7a and 7b show the two-dimensional halfto steady state in terms of the largest decay time associated
spaceresult, K = -kd. The dashedline at K = 1 givesthe rigid earth
with a given earth model. However, this approach constitutes
result for comparison.
a worst caseanalysissinceall of the relaxation modesare, in
general,excitedby loading. Although we were able to justify
the steady state assumptionfor our models even for these
worst cases,it is possiblewith the analytical tools here to
viscositycontrast above 670 km depth (model C) for 1 = 7 for solvefor mode excitationas a functionof the depth of loading
the "FF" case.The lower viscosityin the upper mantle lessens and harmonic degree.An eigenmodefor the simple two-layer
the couplingbetweenthe flow and the upper surface,thereby case can be representedby a unit normalized two vector
decreasingthe deformation of the upper surfaceand increas- giving the relative amount of mass displacedat the upper
ing that of the core-mantleboundary. In Figure 4e the cou- surfaceand the core-mantleboundary. For modelsA and C
pling is so weak that self-gravitationactuallycausesthe defor- there is a mantle model (MO) and a core mode (C). For MO,
mation to reverse,resultingin the slightly negativeexcursion both boundaries flex in the same sense; for the C mode their
of its mass displacementcurve. This effect will be addressed flexure is oppositely directed. The relative amounts of mode
more fully below.
excitation are determined by finding the appropriate linear
Comparison of Figures4a and 4f showsthat the main effect combination of MO and C required to give the boundary
of the no-slip condition,as opposedto the free-slipcondition, massdisplacementsin Figure 4. Note that this matchingalso
at the upper surfaceis to restrictthe flow near that boundary, solvesthe unloadingproblem,that is, excitation of modesdue
resultingin more deformation and mass displacementat the to the suddenreleaseof an internal load of long duration; the
upper boundary. This effect diminisheswith increasing1 as loading and unloading problems are equivalent in terms of
seenby comparisonof Figures4b and 4g as well as Figures4c relative mode excitation.Figures 5a-5c show the resultsof the
I
I
,'
,"!1
II
,(c).
co!
7=.o•
i
and 4h. Notice
that for NF
conditions
the maximum
mass
displacementat the upper surface occurs with the load at
depth rather than when it is nearest to the surface.A similar
effectcan be derived analytically for the two-dimensionalcase
and is the result of flow restrictionin a channeldue to longwavelengthloading.In the three-dimensional(spherical)case
this subsurface
maximum
in deformation
is also enhanced
considerablyby the self-gravitationof the boundary. In addition to the stressesgeneratedby the load "sinking"in the
ambient(zeroth order) potential field there is a first-orderperturbation
in the ambient
calculationsfor the FF modelsof Figure 4.
Figure 4 showsthat at least for long-wavelengthloads, the
amount of massdisplacedat the boundariesis comparableto
the mass of the load itself, so the total geoid anomaly at the
surface involves significantcontributions from these sources.
K
-2
6400,
,
• /
field due to both the load and the
mass displacementsat the boundaries.Although this idea is
no more complicatedthan that of a self-consistentgravity
field, the effect is physically subtle and warrants some discussion.
-I
,
0
2
20','.'.œx7'
-2
-I
o
I
2
v*:o.
L'
' ""• ' /,Surface
•.•)V
m
a7"?
i ]670
, km
.-',: I above / ./
/
3500 ' ' .... •'--•"•-•
-2
-I
0
2
-2
-I
0
iSurface
6400l'
' -"•' I
.....
The basic propagator equations (18) are written for field
variable six vectors,the last two terms of which are the perturbed geopotential field and its radial derivative (gravitational acceleration).These two variablesmust satisfy Poisson'sequation independently,and it has recentlybeen shown
;5oo-'---'.....
i (c,)
(d Car.
by R. J. O'Connell et al. (unpublishedmanuscript,1984) that
the 6 x 6 set of equationscan be reducedto coupled4 x 4
Fig. 8. Surfacepotential responseK as a function of loading
and 2 x 2 systemsin which the 2 x 2 systeminvolvesonly the depth
for viscositycontrastsof 0.1 and 0.01 in the upper mantle.
potential variablesand Poisson'sequation.The 4 x 4 system Figures8a and 8b are for FF boundaryconditionsand Figures8c
, •')"?
'r/'k'=
O.I]--'r/*
=0.0,
7-'----•,,
16•
km
-•..
is obtained by substitutionof u3 + p'u5 for u3 in the six
/ /. 670
km /67O
km
/ / l,
and 8d are for NF conditions.
"-
RICHARDS AND HAGER: GEOID ANOMALIES IN A DYNAMIC
EARTH
5995
- O-dJsp/ob
Figure 6 shows the relative contribution from each of the
6400? -I
threesources6Va,6 V,•,and 6V,,as functionsof the depth of the
0
I
2-2
-I
0
I
2 -? -!
,.? J
2•Surfoce
load. The 6V,,("c") curvehas a simple(a/r)t+2 dependence
derived solely from potential theory (see equation (A7)) and
the 6Va ("a") and 6• ("c") curves are proportional to the
product of the massdisplacementcurvesof Figure 4 and the
(air)t+2 factor.Potentials
6Vaand6V•areof opposite
signthan
6V,,.Their absolutevaluesare plottednormalizedby the maximum value of 6Va to facilitate direct comparison. Figures
6a-6c are for FF conditions and Figures 6d-6f are for NF
conditions.In most of the figuresto follow we refer to potential anomaliessincethey are relatedto geoidanomaliessimply
through 6N = 6V/g, where 6N is the geoid height due to 6V
and g is the gravitational accelerationat the surface.As was
the casefor the two-dimensionalhalf space,the geoid contribution due to the deformationof the upper surfaceis generally
larger than that due to the load. The contribution from the
core-mantle boundary is generally small except for loads at
great depth. Again, comparing Figures 6a-6c with Figures
6d-6f, the effect of strongerupper surfacecouplingdue to the
no-slip condition is evident. Notice that for l= 2 and l-7
with NF conditions,the maximum 6Vacontribution occursat
depth. In Figure 6d we have plotted (seecurve "a'") the result
obtained ignoring self-gravitationin order to demonstrateits
importance for lower-degree harmonics. This was accomplishedby ignoring the self-gravitationterms describedabove
(at the expenseof a self-consistentfield). Since the difference
betweencurve 6V,, and the sum of 6Vaand 6 V• determinesthe
surfacepotential anomaly, this effectcannot be ignored for the
lowest harmonic degrees.
ß
"
/ ,
clco,,
Fig. 10. Mass displacement at the boundaries as a function of
loading depth for two-layer flow modelswith uniform viscosity(same
as Figure 4 with additional displacementcurve "M" for the 670-km
discontinuity).
compensationof the geoid is completeto first order sinceall
the loading stressis absorbedby deflectionof the boundary.
The geoid is much more sensitiveto densitycontrastsin the
middle regionsof the mantle than to comparabledensitycontrasts near boundaries.
The dominatinginfluenceof the upper surfacedeformation
is diminishedby the effectof low viscosityin the uppermantle
resultingin lessnegativeor even positivevaluesof K. This is
shown in Figure 8 for both FF and NF conditions. In this
case the different boundary conditions result in more markedly different geoid signatures.The effect of the low-viscosity
channel in the upper mantle is strongestfor shorter wavelengths(larger/), whereasthe channelis almost transparentto
l-2
loading. We have not presented many of the other
modelsof viscositystratificationwhich are also plausible,but
their effectcan be roughlyextrapolatedfrom thesefigures.For
example, a thinner channel, say 200 km thick, remains transparent to much shorter wavelengthsthan for the 670-km case.
The ratio of geoid anomaly to surfacedeformation, the imThe total surfacepotential •Vtot normalized by the load
potential 6V,, resultsin the responsefunction K, the modified pedancefunction Z (equation (17)), is shown in Figure 9 for
Love number definedin (14). Figure 7 showsK as a function models A and C for both FF and NF conditions. For uniform
of loading depth and harmonic degreefor the four possible viscosity,Z is positivesincethe signof the geoidis determined
combinationsof boundaryconditions.The differences
among by the upper surfacedeformation.With a viscositycontrast
theseresultsare not great although the relative couplingef- the functionsbecomemore complicatedand larger due to a
fects due to N or F conditionscan be seen, especiallyfor
low-order harmonics.The caseswith no slip at the coremantle boundary are includedbecausethey simulatehigh viscosityin the lowermostmantle.In the more pertinentFF and
NF cases,K is invariably negativefor model A (no viscosity
contrast). As predicted by (13), the magnitude of K can be
much greaterthan unity; consequently,
the geoid signatureof
a densitycontrastat depthis amplified.The straightlight lines
in Figures 7a and 7b show the two-dimensionalhalf-space
values for K. Note that with the load at either boundary,
reducedsurfacetopographic
signature.
Note that singularities
in Z can occur sincethe surfacedeformationcan changesign
(go through a zero) as seenfor l = 7 in Figure 9f In practical
applicationsthese singularitieswill be smoothedby integration over a depth distributiono.fdensitycontrasts.
Layered Flow
The flow model representinga chemicaldiscontinuityat 670
km depthis illustratedin Figure lb corresponding
to models
D and E. Mass displacementsat the boundariesare shown in
Figure 10, comparableto Figure 4 for a uniform composition
mantle. In these cases there is deformation
z
640(•0.5
0
0.5 -0.5
0
0.5
-I0
0
I0
½ d
/ 670km'!i'.\,7
• / / ---',il obove/,
\1
3500
o
o.5
-0.5
0
0.5
J
-2
0
2•
.
and effective mass
displacementat the layer boundary as shown by the "M"
curves.For loads near the 670-km discontinuity the stressis
taken up principallyby the deformationof that boundary.The
curvesfor total massdisplacementin Figure 10, computedfor
NF boundary conditions,again representdynamiccompensation, as discussedfor the case of whole mantle flow. The sense
of flow in both the upper and lower mantle is reversedfor
I--.,-•..•-•.•-•,
"7:o.o1_167o
•m loadingin the upper mantle from that resultingfrom loading
/ obove
• \ / •/ •0"-•,,,7
i\ obove
/ •'% i\670km/
in the lower mantle.Consequently,
the signof both the coremantle
boundary
and
upper
surface
deformation depends
20-•, •
upon whether the load is above or below the 670-km discontinuity; singularities occur in the correspondingimpedance
Fig. 9. SurfacedeformationimpedanceZ as a functionof loading functions,and the behavior of K becomesmore complicated.
/
depth illustratingthe effectof low viscosityin the upper mantle for
harmonicdegrees2, 7, and 20. Figures9a-9c are for FF boundary Figure 11 shows the relative excitation of viscous relaxation
conditions and Figures 9d-9f are for NF conditions. The dashed line
normal modesfor theselayeredmodelswhere we now have an
at Z = 0 givesthe "perfect"compensation
resultfor comparison.
additional mode, M1, associated with deformation of the
5996
RICHARDS AND HAGER: GEOID ANOMALIES IN A DYNAMIC EARTH
Mode Excitation
-•
0
64001
r...
•
a
3-•
0
•
a
3-e
-4
0
4
e•f
......
,... j / [,......,... r%MO('•
";'
.........
•oo/
Fig. 11. Viscousnormal mode excitation as a fu•ctio• of loading
depth for two-layer flow models with unifo• viscosity (same as
Figure 5 with the additional M1 mode due to deformation of the
6?O-kindiscontinuity).
670-km discontinuity.The depth dependencefor the MO and
C mode is strikingly similar to that for the whole mantle case.
The M1 mode is, as expected,dominant near the 670-km
discontinuity.
The chemicallayer responsefunctionsK and Z for models
D and E are shownin Figures12 and 13. The potentialfunction K exhibits a more complicateddepth dependencethan
for the whole mantle case.In particular,for no viscositycontrast the sign of the geoid anomaly reversesas we crossthe
670-km discontinuitydue to the dominanceand reversal of
the upper surfacedeformation; geoid anomaliesdue to correlated upper and lower mantle density contrasts are anticorrelated. Also, all functions K have an additional zero at
670 km depth.For a viscositycontrastat 670 km the coupling
at the upper surfaceis reducedsufficientlyso that the deformation of the 670-km discontinuitydominatesthe surfacepotentiM resulting again in negative values for K. Therefore a
wide variety of behavioris possiblefor a relativelysmall range
of viscositycontrasts(lessthan one order of magnitude).
Note that the maximum valuesof K for densitycontrastsin
the upper mantle are small compared to those for the whole
mantle flow model (Figure 7) and thosefor densitycontrasts
in the lower mantle for the stratified models.The physical
interpretationof this behavioris usefulin developingintuition
about dynamic geoid anomalies.As a first approximation,dynamic isostasyresults in the conservationof mass in any
column, at least at long wavelengths(see Figures 4 and 10).
The total geoid anomaly resultsfrom a massquadrupoleconsistingof a driving massanomaly at depth and compensating
mass anomaliesat the deformed boundaries.The magnitude
of the anomaly dependsupon the separation of the boundaries: the "arm length" of the quadrupole.For a given mass
anomaly the deeper the convectinglayer, the larger the arm
length and the greaterthe geoid anomaly.In the limit of zero
thicknessthe geoid anomaly in a convectinglayer goes to
among internal loading, surfacedeformation, and the geoid is
a strong function not only of the depth and harmonic degree
of loading but also the mechanical structure of the mantle.
The dashed referencelines in Figures 14a-14c representthe
value of K that would be observedfor a rigid earth, that is, if
we ignore the dynamicresponse.The impedanceplots,Figures
14d-14e,have a Z = 0 referenceline ("perfect"compensation)
since Z becomesinfinite for a rigid earth. Even the limited
range of models we have explored exhibit a wide range of
valuesfor K and Z that indicatethe sensitivityof the observables to structure.Interpretation of the earth's geoid in terms
of internal processes
demandscareful considerationof a variety of physicaleffects,but much of the nonuniqueness
inherent
in surfacegravity problemsis removed becauseof the distinct
signatureof differentmodels.
Figures 7-9 and 12-14 consititute"maps" that show how to
relate geoid anomaliesand surfacedeformationsto the depth
and harmonic degreeof driving density contrasts.In most of
the models there is roughly an order of magnitudeamplification of the higher harmonic geoid anomaliesfor loads at
great depth. For example,K attains its largestvalue of - 12 in
model
A for l = 20 with the load several hundred
kilometers
above the core-mantleboundary. This requiresmodification
of simple state-of-stress
type argumentsconcerningthe maximum geoid anomalieswhich can be generatedby loads supported at great depth [Kaula, 1963b]. Required deviatoric
stresses
up to an order of magnitudesmallercan supportdensity contrastsgeneratinga given geoid anomaly in dynamic
earth modelsas opposedto an elasticmodel. Of course,these
modifiedLovenumbersmuststill be multipliedby (r/a)•+2 to
give the total potential (see Hager, this issue,Figure 4). Also
from the figuresshowingK as a function of depth we seethat
to first order, K is zero at the boundaries,which impliesthat
compensationof loadsnear boundariesis essentiallycomplete.
This means that bumps due to a variable thermal boundary
layer in a convectingmantle are essentiallymaskedout of the
geoid signature.Sincethesedensitycontrastsare likely to be
among the largest associatedwith convection,this becomesa
seriousconstrainton the resolvabilityof thesefeaturesin the
geoid. A good example of this is the observationthat midoceanic ridges have very little long-wavelengthgeoid signature. Also in referenceto the upper boundary layer, crustal
and lithosphericthicknessand densityare not in generalvery
well known for the earth. Application of the impedanceresponsefunctionsrequiresa more complete synthesisof information on lithosphericthicknessand surfacetopographythan
zero.
is currently available, and this problem is currently under
Upper surfaceimpedancevalues,plotted in Figure 13, exhi- study.In addition to thesecomplicationsit shouldbe remembit a singularityat 670 km and becomevery large for loads bered that for a given densityanomaly "map" for the mantle,
below this boundary with two ordersof magnitudeviscosity say from seismic heterogeneity data or from a threecontrast.The sign and magnitudeof Z for the chemicallayer dimensionalconvectionmodel, one must integrateK(r, l) and
caseis consequently
a stronglyvaryingfunctionof depth,har- Z(r, l) over depth, as in (16) (the numerator •V and the de-
monic degree,and viscositystratification;like K, it exhibits
distinct (althoughnot unique) characteristics
that are highly
model dependent.
INTERPRETATION
The rangeof solutionsfor K and Z obtainedfor the simple
modelswe have describedare illustratedin Figure 14. Instead
of plotting K and Z as functionsof depth and harmonic
degree,we have now plotted them as functionsof harmonic
degreeand earth model for representativedepthsof 300, 1400,
and 2600 km in order to emphasizethe most importantconclusionresultingfrom this study: The relationshipthat exists
Fig. 12. Surfacepotential responseK as a function of loading
depth for two-layermodelsillustratingthe effectof low viscosityin
the upper mantle. Boundaryconditionsare no slip (N) at the surface
and freeslip (F) at the core-mantleboundary.
RICHARDSAND HAGER' GEOID ANOMALIESIN A DYNAMIC EARTH
K
nominator g •ir of Z must be integratedseparately).This will
tend to smooth the respectivemodels summarized in Figure
14. Another important point illustrated by thesefiguresis that
long-wavelengthgeoid anomaliesare influencedmore by density contrastsin the middle mantle than in the uppermost or
lowermost mantle. Also, for a given range of density anomalies,whole mantle convectionresultsin larger geoid anomalies
than layered convection.
An example of the processof interpretation using dynamic
responsefunctionsis found in Hager's [this issue] analysisof
the correlation betweenthe geoid and subductedslabs as evidenced by deep focus earthquakes.Seismicallyactive slabs
representknown positive density contraststhat correlate spatially at better than the 99% confidencelevel in a positive
sensewith the degree4-9 geoid. The positive correlation in
this wavelengthband requires an increasein viscositywith
depth of two orders of magnitude between the upper and
lower mantle in regions of active subduction.The amplitude
of the observed geoid anomalies in the context of dynamic
earth models requires much more excessmass than can be
provided by subducted slabs alone in the upper mantle. A
straightforward explanation is that the positive density contrasts
associated
with
subduction
extend
into
the
5997
20•2 -I
O
I
2 -I0 -5
/
0
".•'1
5
/
I0 -;E)
-I0
/
\•
• i
o
'i
io
2o
/
I- o',.',f
i /d J
I-
o IO
,,
1
I-
I!
\i ,I
-!
4OOk•,
z
-0.12 -0.06
20
0
o,,o•o.12-
-0.8 -0.4
0
0.4
0,,8
18
16
8 •o
E
6
i
I
t ',',,,:,::
Fig. 14. Harmonic dependenceof responsefunctionsK and Z for
representativeloading depths of 300, 1400, and 2600 km for a variety
of models.Curves "a," "b," and "c" are for whole mantle flow with FF
boundary conditionsand upper mantle viscositycontrastsof 1.0, 0.1,
and 0.01, respectively.Curves "d," "e," and "f" are for two-layer flow
with NF boundary conditionsand upper mantle viscositycontrastsof
lower
mantle.
Seismologyis now reachingthe point where it is possibleto
map lateral velocity variations in the mantle. Examplesinclude the determinationof degreetwo lateral heterogeneityin
the upper mantle by Masters et al. [1982], more detailed surface wave studiesincluding odd and higher-orderharmonics
1.0, 0.1, and 0.01.
The other geophysicalobservablewe have discussedis surface
deformation,which is expectedto showa correlationwith
thisissue],and body wavestudiesof lateralvelocityvariations
in the lower mantle [Dziewonskiet al., 1977; Dziewonski,this the long-wavelengthgeoid.Beforethis signatureof mantle dyissue;Clayton and Comer, 1983]. When the velocity anomalies namicscan be measured,however,the large effectsof crustal
determined by these studies are compared to the observed thickness variations on topography must be removed. A
geoidby assigningreasonabledensitycontraststo the velocity simple, preliminary result is obtainable if we limit our comanomalies(neglectingthe dynamicaleffectswe have discussed), parison to old shield areas. For these areas, erosion can be
the geoids predicted are several times larger than those ob- assumed to have established a constant continental freeboard
over geologictime. Also limiting our comparisonto regions
servedand are of oppositesign.
As we have shownin this paper, the dynamicsof flow in the removed from collision zones, we find that the African shield,
mantle can reducethe long-wavelengthgeoid anomaliesfrom in a major geoid high, is high standing,while the Siberianand
those resulting from the driving density contrasts alone and the Canadianshields,in major geoidlows, are relativelylow
can even lead to a reversal in sign. Thus the seismological standing [NOAA, 1980]. Similar conclusionscan be reached
resultsare not qualitatively surprising.They are useful,when from the hypsographiccurvesof Harrison et al. [1981]. From
combinedwith the theory describedhere,in placingmeaning- theseobservationswe estimatethat the impedanceZ at long
ful constraintson mantle dynamics.By combining observa- wavelengthsis of order +0.1. This is consistentwith the retional seismologyand the quantitative theory of dynamic sults for the long-wavelengthcorrelation betweenseismicvegeoid anomalies we can learn far more than we could by locity heterogeneityin the lower mantle and the geoid. More
either techniquealone. For example, B. H. Hager et al. (un- detailed analysis using crustal thicknessand density data
publishedmanuscript,1984)have shownthat 70% of the vari- should yield more accuratequantitative resultsover a broader
ance of the degree2-3 geoid can be accountedfor by seismi- spectralrange.
Within the framework of a sphericallysymmetricmodel we
cally inferreddensitycontrastsin the lower mantle, usingthe
are
unable to reconcile the evidencefrom geoid anomalies
kernelsof Figure 8c, for a model with uniform composition
and an increasein viscosityof a factorof 10 acrossthe 670-km over subductionzonesthat the effectiveviscosityincreasesby
two orders of magnitudewith the preliminaryevidencefrom
discontinuity.
seismicstudiesand elevationof shieldareasthat the viscosity
z
increasesby only one order of magnitude. Perhaps not sur,
! {•,-..,'..
,
....
:aurmce prisingly,lateral variationsin effectiveviscosityare suggested.
Further theoretical improvementsin our understandingof
geoid anomalies in a dynamic earth are clearly desirable.
Modeling of the effectsof lateral viscosityvariationsand nonlinear rheologywould be particularlyusefulin understanding
Fig. 13. Surfacedeformation impedanceZ as a function of load- the geoid signatureof subductedslabssincethey existin zones
ing depthfor two-layermodelsillustratingthe effectof low viscosity characterizedby large deviatoric stressesand temperature
in the uppermantle.(Dashedcurvesindicatesingularitiesin Z.)
gradients.We would also like to model the effectsof adiabatic
[Nakanishi and Anderson, 1982; Woodhouseand Dziewonski,
I 0 0.1
-0.15
0 (115
-'200 20,-,
,,, ri?/lr,
,, ,, ! I/ZI I<</ i ,Ill , , ;,,,
__
5998
RICHARDS AND HAGER' GEOID ANOMALIES IN A DYNAMIC EARTH
compressibilityand adiabatic phase changesin the mantle.
These improvementswill require numerical modeling and
would therefore imply a major departure from the analytical
methods we have described.
or
ur(a)
=I0,Vo,•,
--pmg(a)cSr/rlo,
O,
poacSV,•/rlo,
pøa2
c3c5V'•
T
(Alb)
SUMMARY
We have used sphericalNewtonian earth modelsto investigate the relationship between driving loads and their geoid
and surface topographic signatures.Normalized surface potential K and deformation impedanceZ have been calculated
for representativecasesof viscousand chemical stratification
in the mantle. The following dynamical effectsare found to be
important for geoid interpretation:
1. The responseof the upper surfaceto loadinghas a large
effect upon the behavior of the geoid signature,with negative
geoid anomaliescorrelated with positive driving density contrastsfor the simplestmodelswithout viscositycontrasts.
2. Considerable amplification of deep, higher harmonic
loads is reflectedin the geoid due to the manner in which flow
stressesdrive boundary deformation.
3. The choice of the upper surface boundary condition
(free slip versus no slip) does not strongly effect the basic
behavior of the responsefunctions.
4. Lower viscosityin the upper mantle tends to drive K
positiveand Z toward larger values.For a very large viscosity
contrast the upper surfacedeformation may reversesign due
to gravitational pressureresultingin a singularityin Z.
5. The introduction of a flow barrier correspondingto a
chemical boundary has a pronounced effect on the magnitude
of the responsefunctionsK and Z. In particular, densitycontrasts in the upper mantle have a much smaller geoid signa-
At the core, using equation (6) and the perturbed hydrostatic
stress,-p•g•, at the referencelevel as seenin the mantle we
obtain
ur(c)
=[0,
Voc,
gPcmg(C)Cgrc/qO
--pcgVcc/qo,
O,
PoC•
/qO,
pøc2
•0 7[J
Note that the normal
8.
Viscous
relaxation
of boundaries
stress term in this vector contains not
only the effectivestressdiscontinuity(proportionalto •rc) but
also a gravitational pressureterm (proportional to •)'
this
respresents
the pressurefield within the inviscidcore.
The deformations•r• and •rc causefirst-orderperturbations
in the potential •V in addition to that due to the driving
density contrast a at depth b. These perturbationsare appropriately treated as effective surface massesat the reference
boundary levels, as discussedabove in our analytical treatment of boundary deformation. We now calculatethe potential at r = a (lithosphere)due to a surfacemass distribution
a = Zl al at r = b. The perturbed potentialsjust above and
below the surface are
al
+=
I
r
ov- =
l
a
Fl
ture; this might help distinguishthe two basicflow models
when loadswithin the upper mantle can be estimated.
6. Near-boundary density contrastsare masked by the deformation of the boundary.
7. Self-gravitation is important for low harmonic degree
loading.
(A2)
sinceV2•V = 0 awayfromthe surfacedensitycontrast.
At the surfaceV2•V = 4•Gp so,integratingover a volume
• enclosedby a surfaceS,
occurs on a much
shorter time scalethan convectiveflow so that boundary deformation due to internal loading can be consideredsteady
by Green'stheorem.Integratingover a "pill box" containinga
9. Applicationsof this theory to global data from geodesy piece of the surfacedensity contrast and shrinking the radial
and seismologyshow that the dynamicaleffectswe have pre- thicknessof the box to zero, p• a dr, n• F, V(•V)• a(•V)/ar
dicted can be observed for the earth. Improved analysis and (A4) becomes
shouldyield a better understandingof mantle dynamics.
state.
APPENDIX 1' ANALYTICAL DETAILS
FOR THE WHOLE MANTLE PROBLEM
For our perturbed boundary conditions we impose an efIn order to solve equations(22) the boundary conditions
fective surface mass ffeff soch that the apparent jump in
(25) must be analyticallycontinuedto their respectivespherical reference surfaces which are the mean earth radius a and
the mean
core radius
c. The
field variables
are continued
normalstressat the reference
boundaryis givenby &• =
g(a)%ff.Expanding
within the medium through which they are propagated in
equations(22), in this casethe mantle.To first order, the only
terms in vectors(25) affectedare the radial normal deviatoric
stres,
s and the gravitational acceleration.Sincethe stressabove
where we have made the /th harmonic dependenceexplicit,
and using the above expansions for •V to calculate
the earth's surface is zero, the normal deviatoric stress at the
[0(• V)/Or]_+ weobtainthedesiredresult'
referenceboundary is just the apparent jump describedby
equation (8), so
•V•-=-4•Ga21
•r•l
+ (r/a)l•
uS(a)
po
a2&SV]
r
Similarly, at r = c (core-mantleboundary)we obtain
=[0,
O,
--pmg(a)aeSr/rlo,
arrow/rio,
poaeSV•/rlo,
(Ala)
(A6)
• + 4•Gc
• •r•t(c/r)
t+
••
a
g(c)
21 + 1
(A7)
RICHARDS AND HAGER: GEOID ANOMALIES IN A DYNAMIC EARTH
5999
ized by setting a to unity. Exact values of the propagator
Again, for the densitycontrastat b,
,:5
V•+= 4r•Gb t•21
+ 1(b/r)
at
t+'Yt
elementsPcau, Pba
q can be calculatedaccordingto the pro(A8)
at1(r/b)tyt
,5
V•_ =4r•Gb•t21+
cedures of Hager and O'Connell [1981] and (All) can be
solvedin a straightforwardmanner. With these solutionsfor
the boundary deformationsand the potentials via (A10) we
can calculate the kernels defined by (14)-(17).
APPENDIX 2: VISCOUS RELAXATION TIMES
The total potentialsat a and at c are givenby
FOR SPHERICAL EARTH MODELS
{•Va
tøt-- ({•Va--it-{•Vc
+ -it-{•Vb
+)r=a
•Vcto'= (•v•- + •v• + + •v•-),=c
constants
These expressionscontain explicitly the perturbed potentials
due to boundary deformation that are required for a self-
gravitatingmodel.Writing the &rrt in termsof the 6r, we
obtain the following expressionsfor the potentials in terms of
the stress variables and at:
5V•- 21+ 1
06Va
4r•G
Or
21 + 1
much
smaller
that the boundaries
than
relax with time
the time scales for convective
flow. These time constants can be obtained
from consideraton
of the normal mode problem for relaxation in a radially stratified viscousearth model. Viscoelasticsolutionshave been presented by Wu and Peltier [1982-1,but since convectivetime
scalesare far in excessof Maxwell times in the earth, the effect
of elasticitycan be ignored. The purely viscousnormal mode
problem is much less complicatedand was first investigated
by Parsons[1972]. Using propagator matrix methods which
are describedin detail by R. J. O'Connell et al. (unpublished
manuscript,1984),we have solvedfor the relaxation spectraof
the self-gravitatingsphericalmodels usedin the geoid calculations that follow. By setting all the stress-flowvariables and
boundary deformationsproportional to exp (-t/z•) and solving the resultingsystemof homogeneousequations[seeHa#er
and O'Connell, 1979] the eigenvalueszi as well as the eigen-
pmg(a)5%
l
We need to demonstrate
(A9)
Pmg(a)
6ra
5Pcmg(C)Sr
c -- (1+ 1)
o't
modes for flow, stress, and deformation are obtained. Each
flow boundary introducesan additional relaxation time constant and eigenmode.For example,for a simplemodel with an
(A10) inviscidcore overlain by a uniform mantle we obtain a mantle
and a core mode. These modes are symmetric and anti4r•G
pmg(a)6r
a +- 6pcr•g(c)6r
c+ b
symmetric
respectively,with respectto the senseof boundary
6Vc-21+
1
g½
relaxation, and the time constant for the antisymmetricmode
growsrapidly as the thicknessof the flow layer decreases[Solomonet al., 1982].
Or- 21+ 1 • Pmg(a)•ra
In Table 1 we have listed the parameters used for the
models presented in this paper. Density values have been
chosento match the total mass of each layer (as well as the
gc
earth) and the gravity at each boundary [Dziewonskiet al.,
Combining the first four of equations(21) with reference 1975], although with sucha layering schemeit is impossibleto
boundary expressions(Ala) and (A2) and potentials(A10), we match simultaneously other earth parameters, such as
obtain the following equationsfor 6ra and 6rc as well as Voc moment of inertia, which are not important to this study.
gc
(l+ 1)6Pcmg(C)grc
+I
at
and Z,oa:
Pcai2t;Oc
+ {Pcai3(1
-- Pc*CC/rlo)
+[Pca
i5--(1+ 1)Pcai6]c}C•Pcmg(C)C•rc
•
+ [(Pca
i5 + lPcai6)
A --•i3-
k?
•
Pcai3pc*aA/•o]
e
pmg(a)•ra--6maz,o•/•o
2
=
12 • I0 14 I• I
'::......MO
W'
=0'0•
....
above
2•km
•
lb Ih I• I
•
where
Pca= P(a, c)
e
' .¾.0
above
670
km
•
I0 14
)/
eß
+o'o4aOo/o ...
+l'o]o
.. ß
.. '"
o•
ee©e
'•" ' :
-C.....
C ''''''
-c
.........
'"
,
Pt,a-- P(a, b)
4r•Gpø
(g-•a)a))(•)t+l
2/+
1
B •
2/+
1
C I
2/+
1
and wherethe driving termson the right are usuallynormal-
Harmonicdegree
Fig. A1. Relaxation times and relative boundary discplacement
amplitudesfor viscousnormal modesas functionsof harmonicdegree
for whole mantle flow. M0 refersto the symmetricmantle mode and
C refersto the asymmetriccore mode.The notation "-C" in Figures
Alb, Ald, and All emphasizesthat the boundary deformationsare of
opposite sign for the core mode. The models representedinclude
uniform mantle viscosityand low viscositychannelsabove 200 and
670 km depth. Figure Ala also showsthe relatively minor effect of a
high-viscosity"lithosphere"layer.
6000
RICHARDS
ANDHAGER'GEOIDANOMALIES
IN A DYNAMIC
EARTH
I
5 -
•,
I
I
eeeeeeeeeeeee
MI
i
_
e
i i i i i-
upper 100 km of mantle ("lithosphere")with two orders of
magnitudehigherviscositythan the mantle(r/- 1023Pa s, or
r/* = r//r/o= 100with% - 102•Pa s denotingthereference
or
lower mantle viscosity).For the wavelengthsof interesthere
sucha layeris essentially
transparent.In FiguresAlc and Ald
•J;veO'60170
km
and Ale and A lf we illustratethe effect of low viscosity
(r/* = 0.01)in the uppermantleabove200 and670km, respectively.There are threeprincipaleffectsto be noted:(1) From
4 øø©øø
Ceeeeeeeeeeee
4
• øø'ø
12
•
I10 14 18
21ß..M.O
.............
2
6
I0
14
18
6,
i
I
'•/
I
I
I
I
(a)
4
i
i
I
I
•
-C
-C
•"+/'
--•,/
-
(3) the strength of theseeffectsincreaseswith the thicknessof
obove670km
) •MO•
_
2
•
i
I
I
6
I0
the amplitude plots we see that the two modes tend to be
decoupledby the low viscositychannel,(2) the C moderelaxation times are essentiallyunaffectedwhile the mantle mode
timesare decreased
by one to two ordersof magnitude,and
14
18
I
the low-viscosity
channel.From thesesimplecasesillustrating
the effectsof viscositylayeringwe concludethat for a broad
classof wholemantleflowmodels,no relaxationtimesgreater
than 10½yearsare obtainedfor a lowermantleviscosity
of
102• Pa s.
We now considermodelsfor two-layer,shearcoupledflow
4-
-
2
in the mantle(modelsD and E) in whichthedepthof the top
layer corresponds
to the 670-km discontinuity.
Sinceupper
-
and lower mantle material do not mix across the 670-km
-MI
0"
-2,
2
MO....
i
6
i
I0
0
i
14
i
18
2
obove
6YO
km
I
6
I0
14
18
Harmonicdegree
discontinuityin thesemodelswe have introducedan internal
boundary whose deformation contributes another mantle
mode(M1). This boundarycouldbe a chemicaldiscontinuity
or a phaseboundarywith sluggishkinetics.FiguresA2a-A2c
Fig. A2. Relaxationtimesand relativeboundaryamplitudesfor
viscous
normalmodesasfunctions
of harmonic
degreefor two-layer
flow.The M 1 modeis generatedby deformationof the flow barrierat
670 km depth,and the right-handcolumnof figuresare for a lowviscosity
uppermantle.Plusandminussignson theamplitudecurves
show the relaxation times, the relative deformation of the
core-mantle boundary, and the relative deformation of the
670-km discontinuityfor each of the modesC, M0, and M1,
respectively.The relaxation times for the C and M0 modesare
essentiallythe same as those obtained for the whole mantle
indicate sign reversalsin the senseof deformation relative to that of
case.However, the M1 mode has a much longer relaxation
thesurface.
FigureA2a alsoshowstheeffectof usinga density
jump
of0.3instead
of 0.5Mg/m3at the670-kmdiscontinuity.
Also,for the viscousrelaxationproblemwe usesmaller,more
time(about105years).
Theboundary
deformation
amplitudes
exhibita muchmorecomplicateddependence
uponl than for
previousmodels,and the meaningof theseeigenmodes
will
becomemoreapparentwhenwe address
the problemof mode
realisticdensitycontrastsacrossinternalboundaries(adia- excitation; for now we will concentrate on the relaxation
batic compression
in the mantlebeingignored)sincethese times. In particular,when the densitycontrastacrossthe
valuesstronglyaffectthe time constantsobtained.
670-kmdiscontinuity
is decreased
from 0.5 to 0.3 Mg/m3 a
The two basic models used are those of whole mantle and
significant
increase
in the M1 time occurs(FigureA2a).This
layeredmantleflow,bothwithan inviscidcore.Arbitraryvis- canbe easilyunderstood
physically
sincethe buoyancy
force
cosityanddensitylayeringcanbe treated,sothemodelspre- that tendsto restorea boundaryto its reference
configuration
sented here are chosen to be illustrative rather than exhaus- is proportional
to the densityjump, 6p, at that boundary.
tive. In FiguresA1 and A2 we plot the relaxationtime con- Thereforeas 6p is madesmaller,the associatedrelaxationtime
stants(eigenvalues)
andthe associated
boundarydeformations increases
accordingly.
Sincetheactualdensitycontrasts
within
associated
with each eigenmode.
An individualeigenmode the earth are not exactlyknown,it is importantto remember
consistsof a flow field throughoutthe mantle and could be this effectwhenmodelingrelaxationtimes.In FiguresA2drepresented
assuch.However,for our purposes
the boundary A2f we illustratethe effectsof low viscosityin the upper
deformations
serveto identifyboth the appropriatebranch mantle for two-layer flow. Both the M0 and M1 modesac(mode) and the relative excitation of each mode as we demonstrate later. Figures A la and A lb show the resultswhich are
cordingly exhibit smallerrelaxation times, the effect on M1
beingone half to one order of magnitude.We concludethat
obtainedfor the uniformmantlemodel(modelA). For each for the two-layerflow model the longestrelaxationtime exharmonic number l, following the nomenclatureof Peltier pectedis about105years.An uppermantleflowlayerinvolv[1976],thereis a coremode(C) and a mantlemode(MO). ing chemicalboundariesat say 400 or 220 km [Anderson,
For I < 20 the largestrelaxation time obtainedis lessthan 10½
1979] would resultin longerrelaxationtimes,but we have not
yearsand is associated
with the C mode.Note that for high includedthesemorecomplicated
casesin our geoidmodels.
harmonicorderthe relaxationtimeincreases
with increasing None of the relaxation times calculated so far have been in
wavenumberas in equation(10). In Figure Alb we plot the excessof 105years,whichfor reasonablemantleflow velocicoredeformationamplitudenormalizedby the surfacedefor- tieswouldallowfor about10km transportin themantle.This
mationamplitudefor eachmode.As we wouldexpect,the is indeedsmallcomparedto the flow dimensions,
so for lower
modesare stronglycoupledat low harmonicorder and rela- mantleviscosities
of 1021Pa s,theassumption
of steadystate
tivelydecoupled
at highervaluesof l, thusdistinguishing
the flow is verified.Recentstudiesby Peltier [1981] and Yuenet
C and MO modebranches.
Alsoshownin FigureAla is the al. [1982] indicatethat the viscosityof the lowermantleis less
veryslightchangein relaxationtimescaused
by modeling
the than1022Pa s. Since
therelaxation
problem
scales
linearly
RICHARDS AND HAGER: GEOID ANOMALIESIN A DYNAMIC EARTH
with referenceviscosity(which we always take to be that of
the lower mantle),it is not likely that the steadystate hypothesis for boundary deformationis seriouslyviolated for the
overall convective circulation in the mantle. Alternatively,
computingthe ratio of flow velocityto the velocityof relaxation of the boundaryas in equation(12), the viscositycancels,
indicatingthat boundariesrelax rapidly relativeto changesin
the flow regimewhateverthe mantleviscosity.
6001
Hager, B. H., and R. J. O'Connell,A simpleglobal model of plate
dynamicsand mantle convection,J. Geophys.Res.,86, 4843-4867,
1981.
Harrison, C. G. A., G. W. Brass,E. Saltzman,J. Sloan II, J. Southam,
and J. M. Whitman, Sea level variations, global sedimentation
rates,and the hypsographiccurve,Earth Planet.Sci.Lett., 54, 1-16,
1981.
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Acknowledgments.We thank Don L. Anderson for stimulating
discussionand R. J. O'Connell, Roger Phillips, and an anonymous
reviewerfor very usefulcommentson the manuscript.This work was
supported by NASA grants NSG-7610, NAG-5315 and NAS5-27226
and an Alfred P. Sloan Foundation Fellowship awarded to B. H.
Hager. Contribution 3981, Division of Geological and Planetary
Sciences, California Institute of Technology, Pasadena, California
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(ReceivedSeptember28, 1983;
revisedApril 18, 1984;
acceptedApril 18, 1984.)