Download s Crust and Upper Mantle

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 85, NO. B I 1, PAGES 6377-6396, NOVEMBER 10, 1980
Implicationsof Regional Gravity for State Stressin
the Earth's Crust and Upper Mantle
MARCIA
MCNUTT
U.S. GeologicalSurvey,Menlo Park, California 94025
Topographyis maintainedby stressdifferenceswithin the earth. Dependingon the distributionof the
stresswe classifythe supportaseitherlocal or regionalcompensation.In general,the stresses
implied in a
regionalcompensationschemeare an order of magnitudelarger than thosecorresponding
to local isostasy.Gravity anomalies,a measureof the earth'sdeparturefrom hydrostaticequilibrium,can be usedto
distinguishbetweenthe two compensation
mechanismsand thusto estimatethe magnitudeof deviatoric
stressin the crustand uppermantle.Topographycreatedat an oceanicridgecrestor in a major continental orogeniczoneappearsto be locallycompensated.
Suchfeatureswere formedon weak crustincapable
of maintainingstressdifferencesmuch greaterthan the stressfrom the applied load. Oceanic volcanoes
formed on an already cooled,thickenedlithosphereare regionallysupportedwith elasticstresses.Similarly, the broad topographicriseseawardof subductionzonesis elasticallysupportedasthe lithosphereis
bent near the plate margin. Althoughthe implied stressis to somedegreedependenton the theological
model assumed,the gravity anomaliesand surfacedeformation producedby thesefeaturesdemonstrate
that the upper 30-40 km of the oceaniclithosphereis capableof regionallysupportingstressdifferences
in the 100-MPa range. Given certain conditionsof load emplacement,continentalcrustcan also support
loads regionallyover 100-m.y. time scales,but the effectsof erosiononly allow an estimateof a lower
bound on stress.Data from spaceprobesindicate that the upper layers of other terrestrialplanetsalso
supporttopographic-inducedstressdifferencesin excessof 100 MPa.
INTRODUCTION
permost mantle that supportssignificant deviatoric stress,a
Seismologists
tell us that abovethe core the mantle reactsas 'lithosphere.'Note that the thicknessof the lithospherecan be
an elasticsolid to the passageof seismicdisturbances.The ef- defined on the basisof seismic,thermal, compositional,or mefective seismicrigidity, or resistanceto deformation by shear- chanical properties. We will use the term in the restricted
ing stresses,
is comparableto the strengthof steel,but this esti- senseof the mechanicallithosphere,that portion of the crust
mate only applies to short-periodstressesof relatively small and upper mantle with long-term strength.
Within the contextof this discussionwe define 'isostasy'as
magnitude. The strength may be much smaller for large
stressesof long duration, and in general, the problem dis- the condition in which all stressesare hydrostaticbelow some
cussedhere is one of determiningthe earth's 'permanent' re- compensationdepth. This definition encompassesboth local
sistanceto shearingstressesas a function of depth, amplitude, and regional compensationmechanisms.A feature is said to
and wavelength of the disturbance.This question is funda- be locally compensatedif the total mass in any vertical colmental in that it bears on many other geophysicalproblems, umn above the compensation depth is constant. In this
suchas the developmentof mountain chains,long-term verti- scheme, elevated regions are pointwise compensated.A recal motionsof the earth'ssurface,the temperaturestructurein gional mechanismdistributescompensationlaterally around
the crust and uppermost mantle, the scale and rate of con- the feature as well as vertically beneath it, implying that shear
stressescan be transmitted horizontally.
vection, and other aspectsof earth dynamics.
We will begin with a brief discussionof Jeffreys' [1924,
From an historical viewpoint the first estimatesof stressin
1943,
1976] work, becausehis conceptshave in some way inthe earth were basedon gravity anomalies.The fact that stress
differencesexist cannot be denied. Topographicfeaturesalone fluenced the thoughtsof almost all subsequentinvestigators.
representa departure from hydrostaticequilibrium, and the The secondsectionconcentrateson strengthestimatesderived
earth's reaction to the surfaceload accordingto certain rheo- from measurementsof the longest-wavelengthcomponentsof
logical laws providesa meansof distributingthe stresses
over the earth's gravity field. The following two sectionsdeal with
depth within the earth. Gravity anomaliesare mostoften used oceanicand continental studiesof a more regional dimension.
to estimatethe earth's responseto surfacestress,and thus the Finally, the resultsfor the earth are comparedwith stressestimates for the moon and Mars based on recent elevation and
stressissuenaturally becomesinvolved with the questionof
isostatic compensation.The discussionof stresswithin the gravity measurementsby spaceprobes.
earth mustnot be limited to that which is topographica,Hy
inJEFFREYS'
duced;
gravityanomalies
alsotell us thatthereexis(mass
anomalies, and therefore deviatoric stresses,that are unrelated
to existingsurfaceelevations.
The purposeof this paper is to review estimatesof earth
strength based on gravity observations, although consideration will also be given to studiesof isostaticcompensation
based on surface deformation. The prevailing theme that
emergesis that there exists a region in the crust and upThis paper is not subjectto U.S. copyright. Publishedin 1980 by
the American GeophysicalUnion.
Paper number 80B0277.
WORK
Given only a surfacedistributionof topography,Jeffreys
[1976]describes
threeapproaches
to determiningstress:
1. Assumean earth rheologyand mechanismfor isostatic
compensation.
Apply the load,allowthe earthto respond,and
calculate
the stress.
2. Assumethe dynamic processesthat form the topography, and work out the stressconsequences.
3. Calculateall possiblestressdistributions
consistent
with
the surface load. The distribution that attains the least maxi-
6377
6378
MCNUTT:
IMPLICATIONS OF GRAVITY FOR STRESS
-500
0 0
ioo
,
,
.12
%
.....
+.....
50
i
,
Crmo
x = hgp/e
ø-mex
:.Kn$n
h
I
ot z: • Tr x w0velength
of10ud
Airy
Compensetion•
•
h0
T
-•---a0uguer
An•mol•
. _•(elestic
pluto)
'•'"•"•••
.... TI
TITI•
f • •
Pm
Buoyancy
Forces
Asthen0sphere
(fluid
O-max--up
t0I0 hg/o/e
O-ma
x; hgp
Fig. 1. Modelsusedby Jefftoys[1976]to calculatestressin the earthfrom surfaceloading.(a) Harmonicloadingon an
elasticearth.Maximum stress(amax)dependson the root-mean-square
of the topographicstress(s,) and a multiplicative
factor(k,), whichdependsonly on harmonicdegree.(6) Harmonicloadingon a flat earth. Orientationof the stressellipsoidis alsoshown.(c) Loadingon an elasticplateoverlyinga fluid. From Bank• et M. [1977],reprintedwith permissionof
Blackwell ScientificPublications.(g) Airy compensation.
mum stressdifferenceprovidesa lowerboundfor the strength
Jeffreysbeginswith the first approach,assumingan elastic
rheologyfor the earth. This is basicallya Bouguertheory corThe first two methodsare examplesof the 'forward'prob- rectedfor earth elasticityand makesno assumptionof isoslem in geophysics;the stressanswersare no more valid than tatic compensationby requiringthat the stressbe hydrostatic
the assumptions.The first method is often used to test the below somedepth. The value of this approachis twofold. An
plausibilityof variousrheologies,usingpredictedstresslevels, analyticsolutioncan be easilyobtained,thusavoidingtedious
gravity anomalies,and surfacedeformation as a measure of numericalcalculationof stressdistributions.Secondly,as Jefthe model'sacceptability.The secondapproachis rarely used freysargues,the elastictheory uncorrectedfor isostasygivesa
of the earth.
owingto a lack of informationon the fundamentals
of orog-
better estimate of the lower bound on the stress differences be-
eny. The third type of analysisrepresentsthe true 'inverse'
problem in that the resultsdependonly on the observations,
not on the assumptions.It should be remembered that the
bound itself is the only quantity of importancein the inverse
approach. The stressdistribution that attains the least maxi-
causethe entire earth contributesto the supportof the load. If
we assumethat the interior of the earth is hydrostaticand thus
supportsno stressdifferencesbelow a certaindepth, the maximum stressdifferencein the overlyingelasticlayer must increaseto compensatefor the lossof supportfrom below. Jefmum stressmay not resemble that of the earth, since it was freysalso findsthat the leastmaximum stressdifferencegiven
not requiredthat the solutionresultfrom any known or even by the elastictheory is not much more than the value deterplausible earth behavior.
mined by the inverseapproach.The reasonfor this perhaps
surprising result is that any nonelastic solution for stress
TABLE 1. StressFrom Loadingon an ElasticEarth
within the earth must increasethe total strainenergyper unit
n
k.
s•, MPa Ao,,,MPa
r/a
Depth,km volume (Castigliano'sprinciple) even if it does reduce the
maximum deviatoricstress.Thus while the inverseapproach
2
3.03
21.3
64.6
0
6400
3
10
30
1.70
1.13
1.025
oo
1.040
106 Pa = 10 bars.
23.4
9.6
3.4
39.8
10.8
3.4
0.591
0.895
0.966
2600
670
220
finds stress solutions that decrease the maximum
of a func-
tion, it does so at the expenseof increasingthe volume integral of that same function. The result is that stressvalues
nearly comparableto the maximum are spreadover a greater
volume.
MCNUTT:
IMPLICATIONS
OF GRAVITY
The first caseconsideredby Jeffreysis that of a surfaceharmonicload restingon an incompressible
elasticsphere(Figure
la). Let snequal the root-mean-squarevariation of the topographicstressof degreen as given in termsof fully normalized
spherical harmonics. The maximum stressdifference in the
sphere from the applied normal stressis given by kns, in
which k,is a multiplicativefactor dependentonly on n and is
located at a depth of approximately 1/n times the sphere's
radius.Using the valuesfor k,and depth from Jeffreys[1943]
and the topographycoefficientsfor the earth from Balmino et
al. [1973], we can constructTable 1 of stressdifferencesAo,
defined
as the difference
between
the maximum
FOR STRESS
6379
the plate leads to departuresfrom perfect local isostasyfor
short-wavelength loads, but longer-wavelength loads are
nearly locally compensated.For a 1-km-amplitude harmonic
load of density2500kg/m3 and wavelength450 km the maximum stressdifferencein the plate reaches300 MPa. In general, for a floating crustthe maximum stressmay be as much
as 10 times the amplitude of the surfacestress.Jeffreysconcludes that if mountainsare supportedby a floating elastic
plate, laboratory measurementson the strengthof rocks indicatethat long-wavelengthinequalities5 km in height should
cause fracture
of the crust.
and mini-
Taking again the case in which strengthis uniform with
mum principle stresses,and distancefrom the center of the depth, nonelasticsolutionsdo not reduce the maximum stress
earth in fraction of earth radius that the stress maximum ocsignificantly.The greatestimprovementis found for harmonic
curs.The numbersin this table reveal a theme that reappears loads of degree2 and 3, for which the reductionin necessary
in many later studies.For degreeshigher than n -- 10 the strength is from 5 to 30%. For two-dimensional harmonic
maximum stress difference is less than 10 MPa and occurs at
loading on a planar boundary,however,the reductionin the
decreasingdepths.Although estimatesof the strengthof the maximum stressdifference from the elastic solution is only
about 7%.
earth's uppermostregionsvary [Lainbeck, 1972],the 10 MPa
necessaryto support the topographicstressassumingan elasA familiar example of a nonelasticsolution for a floating
tic rheology is well within even the most conservativeesti- crust is the Airy isostaticmechanism,shown in Figure ld.
mates.The low-order harmonicsdo presenta problem, how- Compensationfor surfacefeaturesis achievedby thickening
ever. While it is highly probable that rocks withstand stress the•:rust
belowelevated
regions.
Forperfectisostatic
equilibdifferencesat leastastarge as 70 MPa in crustalenvironments, rium the vertical pressurefrom the load is balanced by the
it is unlikely that the requisitestrengthis presentin the mantle buoyancypressurefrom fluid displacedby a root of depth w:
betweenthe lithosphereand the core. If we require that the wgAp -----hgp, where Ap is the density differencebetween the
supportfor the low-orderharmonicslie in the lithosphere,the fluid and the crustand h the land elevationof densityp. The
maximum stress difference can increase to several hundred
maximum stressdifferencein the crust for this compensation
megapascals.
distributionis equal to the magnitudeof the load. For smallFor features less than a few hundred kilometers in horizonwavelength featuresthe elastic solution describedabove gives
tal scale,Jeffreysadoptsa flat earth model, shownin Figure a smaller stressmaximum, but for wavlengths longer than
lb. Given a normal stress distribution with harmonic form
2.6Te the Airy mechanismis optimal. Therefore the local isostasy statement that mass per unit area is constant does not
p• = t•ghcos(•x)
lead to the smallest maximum stress differences in all cases. If
where p is the load density,the maximum stressdifferenceis
of the order of 2pgh/e and occursat a depth of 1/2•r times the
wavelengthof the load. The stresssolutionsfor loads modeled
as raisedstripswith rectangularor triangular sectionare similar; in general, the greateststressdifferencein the elastic theory is betweenone half and two thirdsof the rangeof the load
stress,and the greateststrengthis neededat a depth about one
quarter of the width of the load. Applying theseapproximate
equationsto the Himalayas, for which Jeffreysassumesa
height differenceof 5 km betweenpeaks and troughsand a
chain width of 100 km, he estimates that stress differences
reach 450 MPa at depths of 20-30 km if the mountains are
elasticallysupported.
Jeffreys'flat earth treatment can be modified to include the
effectsof isostasyby assumingthat the elasticsupportresides
only in a thin elasticplate overlyinga fluid that can support
no shearstresses
(Figure lc). For loadswith small amplitude
and wavelengthlessthan Te, the thicknessof the plate, the
abovesolutionapproximatelyholdsbecausethe regionof appreciable stressesresidesabove the fluid. For loads whose
wavelengthis long in relation to the thicknessof the elastic
layer, the thin-plate equationsfrom elasticplate theorycan be
applied. Jeffreysconsidersa casein which the plate is 50 km
thick. The flexural rigidity
ETe3
D= 12(1-v2)
(1)
where E is Young's modulusand v is Poisson'sratio, correspondingto this Tevalue is 9 x 10•3 N m. The supportfrom
the definitionof isostasyis amendedto requirethat the maximum stress difference
be minimized
and in addition
that all
stressesbe hydrostaticbelow the compensationdepth, then
the Airy mechanismwould not be acceptable.However, free
air gravity anomaliesin the minimum stressstate would be
larger than what is actually observed. Although the Airy
mechanismbetter agreeswith observationsbecausethe compensation is total and extendsto all wavelength features, the
model is unrealistic.The requirementthat small changesin
the load producevertical motion of crustalblocksthat rigidly
opposeany horizontalmovementimpliesthat crustalmaterial
is infinitely anisotropicto deformation.
Artyushkov[1973, 1974]has elaboratedon this point. While
local isostasyrequiresonly that vertical forcesequilibrate,the
true stable position of the crust balancesvertical forces,horizontal forces, and all moments. Deviatoric stressescannot be
homogeneouslydistributed over depth if a layer is locally
compensated.Departures of a locally compensatedcrust from
true equilibrium are inverselyproportional to the characteristic horizontal scale of the topography or density inhomogeneity. Thus appreciable displacementsof the crust
from isostasywill occur only for very narrow features.
Given only the horizontal and vertical structure of lithosphericdensity inhomogeneities,Artyushkov[1973] devisesa
method of estimatingdeviatoric stressaveragedover the lithospheric thickness.These stressesarise from lateral density
and thicknessinhomogeneitiesin an isostaticallycompensated
crust.The buoyancyforce at any density discontinuityis
ented along the normal to the interface, which may not coincide with the vertical direction. For sloping interfaces, only
6380
MCNUTT:
IMPLICATIONS OF GRAVITY FOR STRESS
TABLE 2. Artyushkov'sStressEstimates
Feature
Stress,MPa
T•, km
MidoceanRidge
Crest
-
24
50
Margin (EastPacificRise)
Margin (Mid-Atlantic Ridge)
- 55
-140
50
50
10
80
60
80
110
80
200
80
Continents
Margin
3-km uplift, supportedby
crustal root
5-km uplift, supportedby
While it is not yet possibleto discountfinite strengthentirely
in the lower mantle, the attenuationof upward-continuedpotential fieldsmakes it unlikely that any but the longest-wavelength anomalieswould have sourcedepthsgreater than 900
km. Therefore if the gravity anomalies originate below the
lithosphere,they more likely pertain to the convectionproblem. Any gravity anomalies that do reside within the lithospheremay provide an estimateof its strength,but the possibility still remains that mass anomalieslocated within the
lithosphereare dynamically maintained by vertical forces at
its base.
crustal root
A method adoptedby severalinvestigators[Cruierand Newton, 1965; Allen, 1972; Khan, 1977] for determining source
depthsfor low degreesof the global gravity field is basedon
the assumptionthat the gravity anomalies are produced by
the vertical componentof the buoyancyforce is balancedby randomly distributeddensityvariations.The assumptionof a
the weightof the topography,leavingan unbalancedhorizon- white spectrum for the density variations is sufficiently retal component.For severalexamplesof areas with strong strictiveto permit a unique inversionfor either the depth from
crustalstructure,estimatesof the averagedeviatoricstressare which the density anomaliesshould extend uniformly downgiven in Table 2.
ward or the depth of a single-densityinterface concentrating
Note that only the product of averagestresstimes litho- the anomalies.The resultsare dependenton the particular set
sphericthicknessTe is determinedby Artyushkov'sanalysis. of potential coefficientsused [Higbie and Stacey, 1971],but a
The resultsin Table 2 may underestimatethe deviatoricstress recent analysisby Khan [1977] showsthat the latest satellite
for the assumedmodels,sinceit is unlikely that the earth sup- and combination solutions, WGS 72, Gem 7, Gem 8, and
2-km uplift, supportedby
low-densitymantle
portslarge stressdifferencesbelow 40-50 km. The assumed PGS 110, are consistentto degree 10 or 11.
densitystructureis also critical to Artyushkov'scalculation.
Assuminga single-densityinterface, the sourcedepths acHowever, with deep seismicsoundingand gravity data it is cording to Khan [1977] are 600-800 km for n -- 2 to n -- 11
possibleto resolvesomeof thesedetailsof crustalstructure. and 300-600 km for n -- 11 to n -- 30. Shallower depths are
Further data are needed to determine the actual stress disfound on the assumptionof a disorderedmantle below the
tribution within the lithosphere.In particular,the tectonicim- specifieddepth: 150-370 km for n -- 2 to n -- 11 and 150-450
plicationsof the stresses
in Table 2 dependprimarily on the km for n -- 11to n -- 30. Only the anomalous
part of C2ø relahorizontal and vertical viscositydistribution in the litho- tive to the best fitting satellitereferenceellipsoidwas used in
sphere,which we are only beginningto understand.Never- the analysis.The nonhydrostatic
part of C2ø is aboutan order
theless,Artyushkov[1973]proposesa schemeof globaltecton- of magnitude greater than the next largest coefficientand
ics in which horizontal motions are driven by lateral
tendsto bias the third- and fourth-degreecomponentstoward
spreadingof crustaldensityand thicknessinhomogeneities. greaterdepth.
Continued differentiation of the core releasesbuoyant mateWe could conclude from Khan's analysis that the major
rial that ascendsthrough the mantle, forming the roots for contributionto the global gravity field is from sublithospheric
new uplifts.Thuschemicaldifferentiationprovidesthe driving depth and thereforemust be maintainedby convection.Howmechanism.At presentthere is little supportfor Artyushkov's ever, Goodacre [1978] has questioned the assumption that
proposal;sinceit is not possibleto fix all ridgecrestswith re- densityvariationsare uncorrelatedwithin the earth. The fact
spectto the lowermantle,it becomesdifficultin Artyushkov's that the sourcedepth dependson n may reflect the existence
schemeto explain the continuedexistenceof individual ridge of severalwarped discontinuitysurfaces,but it alsomay result
systemsover long geologictime spans.Many other mechani- from a red-shiftedspectrumof densityvariations.To illustrate
cal, geochemical,
and observational
objectionscouldbe listed, this point, Goodacre[1978] considersthe earth's surface tobut they do not directly bear on the resultsof Table 2. The pography.The amplitude spectrumis not white; it varies as
fact remainsthat in regionsof relief on internal densityinter- (2n + 1)-•/: [Balminoet al., 1973].Usingthe potentialcoeffi-
faces as well as on the earth's surface, deviatoric stressesexist,
and they are of the order of a hundredmegapascals.
GLOBAL
STUDIES
The globalgravityfield revealsdeparturesfrom hydrostatic
equilibrium that extendover thousandsof kilometers.If these
featuresare supportedin an isostaticsenseby the lithosphere,
their continuedexistenceimplies significantfinite strength.
There is, however,a dynamicalternativeto staticsupport;the
anomaliescould be maintained by flow in the mantle. On the
assumptionthat the lithosphereis the only regionof the earth
with appreciablestrength,depth of the causativemasscan be
usedasthe criterionfor distinguishing
betweenthesetwo possibilities. Evidence from glacial rebound, isostatic compensation,and seismicstudiesconfirmsthe existenceof an extremely weak asthenosphereunderlying the lithosphere.
cientsfor the gravity field from the topography,the depth estimate for the topography on the assumptionthat the density
variations have a fiat spectrumis a few hundred kilometers,
rather than the expectedzero depth. There is no reasonto believe that internal density variations are any more random
than those at the surface,and therefore the above depth estimates may be meaningless.Overall, this approach to interpretingthe globalgravity field is not highlypromising,sinceit
involvesvirtually untestableassumptionsconcerningthe statistical behavior of inhomogeneitiesdeep within the mantle.
While the above procedure was designedto directly estimate source depth for gravity anomalies, McKenzie [1967]
posedthe problemin the reversemanner:If we supposethat
gravity anomaliesare supportedby the lithosphere,then what
stressesare implied? McKenzie then rejects a lithospheric
sourcefor anomaliesthat require stressesabove an assumed
MCNUTT:
TABLE
3.
Lambeck's
IMPLICATIONS
Stress Estimates
n'
Omax,
MPa
4
5
6
7
875
487
313
219
8
9
10
11
12
13
14
166
129
103
84
70
59
49
15
16
,
OF GRAVITY
39
31
ultimate strength.He restrictsdiscussionto gravity anomalies
that are produced by uncompensatedwarping of an elastic
lithosphere, which could reasonably apply to subducting
platesand broad lithosphericswells.Although it might be argued that these large-scale plate deformations are a consequenceof mantle flow, the only concernhere is whether the
lithospherecan maintain the configurationonce it has been
established.
From observedgravity and geoidanomalyamplitudesand
wavelengths, McKenzie calculates a minimum stressof 83
MPa to supportplate flexurenear the Tonga and PuertoRico
trenches,assuminga 50-km-thicklithosphere.For a 100-kmthick plate the stressestimateis reduced to 22 MPa. The decision as to whether the support for these features is derived
from the lithosphere hinges on the choice for its ultimate
FOR STRESS
6381
errorin McKenzie's
equations
andhisuseof a two-dimensional geometry.
For a three-dimensionalgeometry, Lambeck constructsa
table of averagemaximum stressarisingfrom gravity anomalies of degreen' and higher, assumingthey are supportedby a
100-km-thick lithosphere (Table 3). With Lambeck's preferred 100- to 150-MPa estimatefor the critical strengthof the
lithosphere,it is possiblestaticallyto supportanomaliesof degree 8 or 9 and greater.
Lambeck [1972] emphasizesthe point that the broad positive anomaliesdescribedby harmonicdegreesn -- 8 and n = 9
over spreadingcentersneed not be maintainedby convective
forces,but this conclusiondependson the assumedthickness
of the lithosphere.Isostaticstudiesindicate that the mechanical lithosphereis only 30 km or so thick. For a more realistic
lithosphericthickness,the stressesin Table 3 would increase
by a factor of 9.
Chase[1979], like McKenzie [1967] and Lambeck[1972], favors the forward approachto modeling the long wavelengths
of the gravity and geoid anomalies.Rather than considering
the effectsof a warped lithosphere,he attributesthe anomalies
to uncompensatedpoint massesat depth. Much of the geoid
character in harmonics 10 through 20 can be explained by
positive mass anomalies in the worldwide subductionzone
system. The stressesimplied by the uncompensatedmass
range from a minimum 22 MPa for the Zagros subduction
zone to a maximum 162 MPa for the Ryukuyu Trench, assuminga 100-km-thick plate. The required massexcessis, in
most cases,lessthan the amount predictedby thermal plate
models [e.g., McKenzie, 1969]. Thus stressestimatesbased on
a mathematicalformulation of densityexcessin a subducting
slabwill be larger than thoseconsistentwith the gravity field.
Although the models of McKenzie, Lambeck, and Chase
are useful for estimatingstressimplied by the global gravity
field, thesemodels are static and thus do not answerthe question of whether the lithospherealone supportsits anomalous
strength.McKenzie citeslaboratoryexperimentsby Griggset
al. [1960] that produce shear failure in dunite at 400-MPa
stress.He rejectsthis strengthestimatein favor of the 20 MPa
impliedby earthquakestressdrops[BruneandAllen, 1967].As
McKenzie admits, there is little justificationfor this choice.
The amountof stressdrop during a seismicrupture can only
be a lowerboundon the shearstrengthof the lithosphere.On mass and sustains its deformation. Nevertheless, the reoccurthe basis of his choice of 20 MPa a 100-km-thick elastic lithoring themefrom thesestudiesis that for a varietyof plausible
sphereis necessary
to maintainthe subductionzone gravity anomalysources,gravityanomalieswith wavelengthsbetween
anomalies against shear failure. The longer-wavelength 5000 and 1000 km require that the earth in someway (conanomalies representedin the satellite gravity field and the vectivelyor otherwise)maintain stressdifferencesfrom 20
geoid, however,require 100 MPa or more of stresseven for a MPa to near 200 MPa.
100-km-thick lithosphere. McKenzie concludes that these
anomaliesmustbe maintainedby flowin themantleand may
OCEANIC STUDIES
be a consequence
of small temperatureinhomogeneities.
Kaula [1969, 1972] continueswith McKenzie's theme that
the supportfor the satellite-derivedgravity field must lie below the lithosphereand entailsflow in the mantle. He proposesa tectonicclassification
of the long-wavelength
gravity
anomaliescorrespondingto degreesn -- 6 to n -- 16 basedon
an associationwith tectonicfeaturessuchas subductionzones,
ridge crests,orogenicbelts, sedimentarybasins,and areas of
Pleistocene
glaciation.Of the 11 classifications,
6 correspond
to currentlyactivetectonicfeatures.If indeedthe lithosphere
is incapable of supportingstatically the gravity anomalies
overthesefeatures,thenthe magnitudeandsignof the gravity
anomaliesprovideinformationconcerningmantleconvection.
Lambeck [1972] challengesthe assumptionthat the anomalies observedin the sateBRite
gravityfield cannotbe supported
by the lithosphere.His greatestobjectionis with McKenzie's
20-MPa stresslimit, which he considersto be too low by at
least a factor of 4. Lambeck also concludes that McKenzie's
stressestimatesare overestimatedby a factor of 2 due to an
Long- WavelengthAnomalies
The vastimprovementin satelliteand surface-shipgravimetry fosterednew attemptsin the mid-1970'sto determinethe
sourceof long-wavelengthgravity anomalies[Andersonet al.,
1973; Menard, 1973; Weisseland Hayes, 1974; $clater et al.,
1975; Watts, 1976]. These investigationsall center on oceanic
observationsin order to avoid the complexitiesof continental
crustal structureand tectonic history, and they share a common line of reasoning.Numerical simulationsof mantle convectionin a NewtonJanfluid [e.g.,McKenzie et al., 1974] predict a positive surface elevation and gravity anomaly over
risingconvectionlimbs for both high and low Rayleigh number flow. Thus positivelycorrelatedgravity and depth anomaliesprovidea strongcasefor convectionin the mantle,particularly when lithospheric sourcesfor the anomaliescan be
ruled out. Although the calculationof residualdepth anomalies is tedious in that it involves systematicallycorrecting
bathymetricdata for sedimentloading and the empirical age-
6382
MCNUTT:
IMPLICATIONS OF GRAVITY
Free o•r Grovdy
Anomaly
FOR STRESS
more convincingby comparingthe ratio of Fourier transforms
of the gravity and bathymetrydata with the theoreticaladmittance from convection
models:
Z(k) = C•)/H(k)
+ 32_.5
•=75km
Te:50
km
.•
--
+221
Observed
+16.6
Te=30km
[McKenzie, 1977], where uppercasevariables denote Fourier
transformsof gravity g and topographyh and k is the modulus of k. Most of the analysescomputeda regressionfrom 5ø
x 5o data averagesand thereforegive an admittanceestimate
at only one k value.
In addition, from what is known of lithosphericbehavior
throughisostaticcompensationand glacial reboundstudies,it
is uncharacteristic
of the lithosphereto sustainindefinitelyuncompensatedwarps thousandsof kilometersin extent without
some incentive from below. Even if we do supposethat upward flexuresare causedby the risingof low-densitymaterial
from the mantle, the gravity anomaliesare too ambiguousto
define the distribution of the load. The magnitude of the
deviatoric
Te
•O•m_.
F
stresses associated with the observed surface strain
could be several tens of MPa or several hundred MPa, de+198
pending on how the supportis applied.
Isostatic Compensation
+ 395
Te=lOk
Topography
I00
[Watts, 1978;Cochran,1979;McNutt, 1979].Thesegravity
MGAL
anomalies,in the wavelengthrange from 20 to 1000 km, are
distinctlyof lithosphericoriginand thuspertainmoredirectly
,
IOO0]
•.
Meters
01 .•!.'::,
. ?-'
_1000•
...
In the oceans,about 50% of the power in the gravity spectrum is related to the compensation of oceanic features
problemof lithosphericstrength.The choicebetween
IOOKM
0 :37906 tolocaltheand
regionalcompensation
for a particularfeaturecan
Fig. 2. Comparison of observedand computed free air gravity
anomaly profiles of the Hawaiian ridge near Oahu. The computed
profilesare basedon the elasticplate model and assumedvaluesof Te
of 10, 20, 30, 50, and 75 km. The best overall fit to the observeddata is
for Te -- 30 km. From Watts et al. [1980].
be decidedon the basisof gravity anomaliesalone.Figure 2,
from Wattset al. [ 1980],comparesthe observedfree air grav-
ity anomalyover Oahu with theoreticalgravityfrom elastic
plate models.An extremelythin platemodel(Te -- 10 km)
poorlypredicts
theobserved
gravity;theanomaly
froma local
compensation
mechanism
(Te -- O)wouldgivea worsefit yet.
Largepositivefreeair gravityanomalies
flankedby encircling
depthrelation[Sclateret aL, 1971],the remainderof the anal- troughsoverOahuandotherislandsandseamounts
demand
ysisconsistsof simply calculatingregressionlines of gravity regional compensation.Such featurestherefore are more
on bathymetry.
likely to placea lower boundon the strengthof the lithoAndersonet al. [1973],for example,claim that a 0.33 gravity sphere;they are very large,relativelyuneroded,and regionunit (gu)/m correlation between gravity and bathymetry
holdsfor the worldwide ocean-ridgesystem.On an ocean-byocean basis,however, the correlation is convincingonly for
the Atlantic and the SouthwestIndian Ridge [seeAndersonet
aL, 1973, Figure 3], and even the Indian Ocean correlation
disappearsif the data from the Madagascarplateau are removed. In addition, Watts [1976] questionsthe reliability of
the resultsof Andersonet al. [1973], Menard [1973], and Weissel and Hayes [1974],which are all basedon the SE 2 gravity
field [Gaposchkinand Lambeck, 1971]. A comparisonof the
ally supported.
Models proposedto explain the bathymetry,gravity anomalies, and deformation of the Moho boundary in the vicinity
of theselargeislandsand seamounts
assumethat the asthenospherebehaves,for loadingtime scalesgreaterthan 30,000
years,as an inviscidfluid and that the lithosphere
can be describedby oneof the followingtheologies:(1) perfectlyelastic
[Gunn,1943; Walcott,1970b],(2) elasticwith discontinuig.
y at
a free edgeunderthe load [Walcott,1970b;Wattsand Cochran,
1974], (3) viscoelastic,(4) layered viscoelasticwith viscosity
SE 2 field with sea surface data and the more recent Gem 6
decreasingwith depth [Suyenaga, 1977], and (5) elasticsolution[Lerchet aL, 1974]revealsthat anomalypeakscan be perfectlyplastic[Liu and Kosloff,1978].The simplestmodel,
offsetas much as 900 km, half the wavelengthof interest,in a continuouselasticplate, almostcertainly over-estimatesthe
the SE 2 field.
stressfor a given load. Any attempt to incorporatemore realIn any case it is unlikely that analysisof very long wave- istic theologicalbehavior into the lithospherecan only serve
lengthgravity and topographicanomalieswill provide much to reducethe implied stress.For the purposeof this discussion
information concerningthe strengthof the lithosphere. Re- we wish to determine how much the stress can be lowered
gardlessof whetheror not the lithospherecould supportthe without violating the observations.
The trend in recentyearstoward increasingcomplexityand
observedgravity anomalies,the agreementbetweentheoretical calculations and observed regressionslopes favors con- number of free parametersin the models has been dictated
with the implied stresses
than by revectivesupport.The casefor convectioncould be made even more by dissatisfaction
MCNUTT:
IMPLICATIONS OF GRAVITY FOR STRESS
6383
Fig. 3. Examples
of twoelasticplatemodels.Uppermodel:continuous
elasticplateof thickness
T•;Pcis thedensityof
the materialoverlyingthe lithosphere
andPmis thedensityof the asthenosphere.
Lowermodel:elasticplatefracturedunder the load axis. From Walcott [1976].
fmement in the observations.Consider, for example, the thinplate equation
3 km high and 30 km wide is 200 MPa. Even taking into account the three-dimensionalityand distributednature of real
loads, McNutt and Menard [1978] calculate 200-MPa stress
Dvnw(x)+ Apgw(x)= P
under Tahiti usinga best fitting 14-km elasticplate thickness.
where D is the flexural rigidity given by (1), w the plate deflec- Walcott [1976] considers200 MPa to be the 'crushingstrength
tion, Ap the density contrastfor materials overlying and un- of rock' and predictsthat failure should occur.
Model 2 in Figure 3 reducesthe maximum stressby removderlying the plate, and P the applied load. The solution for
plate deflectionof a continuouselasticsheet(Figure 3) under ing the point of maximum curvature.The equationfor the deflection of a plate with a free edge beneath the load is
a line load, such as a seamount chain, is
w = B exp (-x/a)
w = exp (-x/a)A[cos (x/a) + sin (x/a)]
in which the flexural parameter a is related to D by
where
B = P/Apga
Ol
4 '- 4D/Apg
The value of A is determinedby the isostaticcondition
cos (x/a)
The bending stressis
-2EzB
P= 2
Apgw(x)dx
and therefore
o•=
a2 exp(-x/a)sin(x/a)
and reachesits maximumat x/a = •r/2 (the firstnodalpoint):
A = P/2Apga
The bendingstressox is proportionalto d2w/dx2.
'
-2EzA
-2EzP
O.... -- Apa3
exp
(-•r/2)
A load P that would produce 100 MPa of stresson a continuousplate would produceonly about 40 MPa bendingstresson
a fractured plate. While the discontinuousplate reducesthe
where E is Young's modulusand z the vertical distancewithin
implied stress,Walcott[1970b]found that he wasunableto sithe plate from the neutral plane. The maximum for this funcmultaneouslyfit Moho depth, flexural wavelength,and bathtion occursat x -- 0 and is given by
ymetrywith one valuefor the flexuralrigidity of the fractured
plate. Moreover, while it might be reasonableto supposethat
-EzP
during magmatic activity the lithosphericplate is weakened
Oxmax
= Apga3
and possiblydecoupledbelow the load, it is difficult to justify
Walcott [1976] calculatesthat the maximum bendingstress a free edge boundary condition for regionsin which volcanic
at the baseof a 60-km-thick plate loaded by a seamountchain activity has ceased[Liu and Kosloff, 1978].
o,,=
a2 exp(-x/a)[cos
(x/a)- sin(x/a)]
6384
MCNUTT: IMPLICATIONSOF GRAVITY FOR STRESS
A viscoelasticplate can alsoreducestressfor a given flexure
profile. The behaviorof a plate with a viscousin addition to
an elasticresponseis to relax its elasticstresses
as loading time
increases.Given an initial viscoelasticrigidity Do for loading
times short in relation to the viscoelasticrelaxation time r, the
plate appearsto be perfectlyelastic.For loadingtimeslong in
relation to r the flexuralwavelengthdecreases,
and the amplitude increases.If we were to interpret a viscoelasticplate profile in termsof the perfectlyelasticequations,it would appear
that the flexural rigidity waslessthan Do. The importantpoint
for the implied stressesis that the observedcurvature of the
plate is no longer proportionalto the stressbecausesomeof
the strainis nonrecoverable,nonelasticdeformationcausedby
viscousflow. For any t > 0 then the stresswill alwaysbe overestimatedif the strain is assumedto be perfectlyelastic.Nadai
[1963]derivesthe followingexpression
for the remainingelastic deformation
in which
a04=
4Do
Apg Yo
=X/ao y=x/a
and
4t/r = (ao/a)4- 1 - In (ao/a)4
The integrationof (4) is not easily accomplishedwith the
systemof equationsin this form. For small t, however,we can
estimatean upper bound for the remaining elasticdeflection
w' by puttinga lower boundon $w exp (-t/r) dt:
w'(t) = w(t) - e-'/'w(O)
= w(t) - w(O){1- exp(-t/r)}
w' in terms of the viscoelastic deflection w at a
time t after the emplacementof the load:
w'(t)
--w(t)
- exp
(-t/r)
t w(t')exp(t'/r)dt'
Doexp[-t/r(1+ 14k4)]
1+ ink
4{1- exp
[-t/r(1+/4k4)]}
(5)
Substitutingfor w from (4) into (5), we obtain
(2)
The reductionin maximum stressfor a viscoelasticrheology
as comparedwith an elasticone dependson the ratio t/r, and
thereforewe need an estimateof r for the oceaniclithosphere.
Walcott [1970a] interpretsnumerousapparent flexural rigidities D' from continental and oceanic loading studies in
terms of a viscoelasticplate model. From a trend toward decreasingD' with increasingload age he estimatesr -- 105
years. There are severalproblemscomplicatingWalcott's interpretation. Known differencesin continental and oceanic
thermal structuresuggestthat apparentelasticthicknesses
will
vary. Firstly, some of the differencesin D' may result from
variations in initial rigidity Do quite apart from any relaxation. Secondly,the loadingtimesfor PleistocenelakesAlgonquin and Agassiz are less than 10,000 years. The asthenospherecannot be treated as a fluid on suchshort time scales,
and the loadsno doubt did not reachequilibrium. Finally, the
relation between D' and Do for linearly viscoelasticplate is
[McNutt and Parker, 1978]
D'--
er/' dt'
(3)
where
P {1(e_•o
cos
Yo
- e-•cos
y)
2-pg
+ SinYO)(e-t/'
I [e_yo
(cos
Yo
- In(ao/a))]
}
+ --
0lo
The bendingstressis proportionalto the secondderivative of
w', which has a singularityat the origin. In the physicalworld,
however, we would not encounter this point of infinite viscoelasticcurvature,sincepoint loads do not exist. For points
other than x -- 0 the curvature
d•w '
remains finite and is
p
•dx= Apgao
3{yo-3(e
-yo
cos
yo- e-ycos
y)
+ yo-2[e
-yø(cosYo+ sinYo)- (ao/a)e-• (cosy + siny)]
+ yo-'[e-yøsinYo- (ao/a)2e
-y siny]
+ [e-'/' - In (ao/a)le-yø(sinYo- cosYo)}
This expressionwas evaluated at the crest of the first flexure
arch for valuesof t/r = 0.02568and t/, -- 3.0568,with ao -- 50
km. In each casethe stresswas comparedwith the stressimplied by a purely elastic plate with the same flexural wavelength.For the shortloadingtime t/r -- 0.02568the reduction
elasticplate modelby using(2) and the equationfor viscoelastic
deflection
at timet froma constant
lineload[Nadai,
in stressfor the viscoelastic
rheologyis 5%.When the loading
time is of the orderof 3 timesthe viscoelastic
decaytime, the
stressreductioncalculatedis 15%, but this is only a lower
bound, becausefor large t the approximationmade in evaluatingthe integralin (2) is no longervalid. Thereforewe may
concludethat interpretingflexure profilesin terms of a viscoelasticrather than an elastic plate model leads to lower
stressestimates,but the differenceis significantonly for loads
much older than r. If r is of the order of 100,000years,we
couldpresumethat for loadsa million yearsor older, appre-
1963]:
ciable stress relaxation
P = Do/Apg
Apparentflexuralrigidities
canonlybe compared
if theyare
determinedfrom loadsof the samewavelength
• -- 2•r/k.
A valuefor r aslow as 10• yearswouldindeedgreatlydecreasethe elasticstresses
impliedby present-dayloads.We
can approximatethe reductionin stressfor a givenflexural
profile when interpretedin termsof a viscoelasticrather than
w(x)
--2-•pg'(e-yø
cos
Yo
- e-ycos
y)
1[e_•o
(cos
Yo Yo)l
[1In(ao/a)l}
(4)
+ --
+ sin
-
has occurred.
However, flexural evidencefrom the Hawaiian-Emperor
seamountchain is incompatiblewith a 10• year value for r.
Assumingan initial plate thicknessof 90 km, Watts [1978]
findsthat the present-day20- to 30-km apparentplate thicknessunder the islandof Hawaii requiresa r value of 105-106
years.Flexure beneaththe older Emperorseamountsnorth of
MCNUTT: IMPLICATIONS OF GRAVITY FOR STRESS
0 Km
20
0
60
I00
140
I
I
I
•
6385
strain rate;
•1 = 3.4 x 108kbar-n s--l;
n=3;
G-- 30;
I
80
o•
,ooo"•] r ø
b0•Om '
Tm
T
•ooo
o
melting temperatureat ambient pressure;
actual temperatureat depth z;
deviatoric
stress.
m
The numericalconstantsin this empiricalcreeplaw are taken
from the work of Carter [1975], Weertman and Weertman
[1975], and Kirby [1977]. The creep strain is estimatedfrom
the
elasticstresses
and homologoustemperature(T/Tm) curve
Fig. 4. Creep in a finite element viscous-elasticplate under constant load. (a) Cross section of an 80-km-thick plate showing the from Mercier and Carter [ 1975].
3. Assume the creep strain is an initial strain within the
spreadof fluid elementsas time progresses.
The numbersindicate
whichregionsof the plate have relaxedto the point of becomingfluid plate and return to step 1. When the creepstrainin a plate eleelementsat 1, 3, and 5 x 10n yearsaftertheloadis applied.Thesere- ment becomesgreater than the elastic strain, the element is
•iI
Deflection
gionssupportno shearstress.(b) Vertical displacementw at the plate
considered to be a fluid.
surface.The arrows indicate the location of maximum displacement
at 0 and at I and 3 x 10n years.After Suyenaga
[1977].
40øN impliesTe-- 10-20 km and r -- 106-107years.Watts interprets this discrepancy in viscoelasticrelaxation times to
mean that significantrelaxation does not occur in the Pacific
plate over 50-m.y. time scales.He prefers a model in which
flexural rigidity doesnot decreasein time but is a function of
the age and therefore of the thicknessof the oceanicplate
when the load is applied. The lower flexural rigiditiesfor the
Emperor seamountsrelative to the Hawaiian Archipelagoare
explained by a younger age, and therefore a lower elastic
thickness,for the regionof the Pacificplate beneaththe Emperor Seamountswhen the chain wascreated50-60 m.y. ago.
The very low flexural rigidity and thin elasticplate thickness
found by Cochran [1979] and McNutt [1979] for the topography in the vicinity of PacificspreadingcenterssupportsWatts'
interpretation.
Regardlessof whether or not viscoelasticrelaxation does
occur, the implications for stressare not affected for long
decay times. For r greater than 50 m.y., no appreciablestress
relaxation has occurred for Hawaii, Tahiti, and other relatively young oceanicloads.Both elasticand viscoelasticplate
modelspredict stresses
exceeding100 MPa.
Another variation on the elasticplate model is the layered
plate in which relaxationbeginsat the baseof the lithosphere
and migratesupward. This schemeis appealing from a theologicalviewpoint becausethe temperaturedependenceof viscosity predictsthat viscositydecreaseswith depth. Suyenaga
[1977] usesa finite elementscheme[Zienkiewicz,1971]to investigatethe time history of flexure for a plate with depthdependent viscosity. He terms this plate model 'viscouselastic' to distinguish it from the viscoelasticmodel whose
propertiesare constantwith depth. Suyenaga'sprocedurecan
be outlined
as follows:
Elastic elementsbegin to become fluid at the base of the
lithosphere first where T/Tm is a maximum, although, as
shownin Figure 4a, the pattern is alsoinfluencedby the stress
distributionfrom the flexure.The flexuralrigidity of the plate
thereforedecreaseswith time becausethe nonfluid portion of
the thermal (• 100 km) lithospherethat actually supportsthe
loaddecreases
with time. The upwardspreadof fluid elements
practicallyhalts after about 106yearsat depthsbetween15
and 30 km, where T = 0.3Tmto 0.5Tm[Murrell, 1976]. Below
this depth interval, stressrelaxation is complete,while above,
relaxation is only partial and does not change appreciably
over loading time scales.As a resultof the partial relaxation
the stressin the upper, long-termelasticportion of the plate is
lessthan what would be estimatedfrom elastic theory. The
evolutionof the surfacedisplacementasthe fluid elementsmigrate upward, shownin Figure 4b, is similar to that of the viscoelasticlithospherewith no depth dependencein the viscosity. The important difference in the two models is that
relaxation eventually ceasesin a viscous-elasticlithosphere
before the effective rigidity reacheszero.
The viscous-elasticmodel is intended to explain why the
apparentelasticthicknessof the oceaniclithosphereis only a
fraction of its seismicthickness[Hanks, 1971, 1977]. We can
also estimate its long-term stressimplicationsby considering
the strain rate at the baseof the 30-km-thick elasticlayer as-
sumingT = 0.5Tin.From (6), • -- 2.4 X 10-m7/S,
which equals
7.5 x 10-n/m.y., for a deviatoricstressof 200 MPa. In the
viscoelasticplate model it is assumedthat strain rate depends
linearly on stress:
• = (1/•t)o
•t = Er/3
•_•o
1. Apply a load at time zero to a plate 100 km thick, and
using the finite element method, numerically calculate the
elastic displacementand stressaccordingto the elastic plate
equations.
2. Calculate the amount of creep that will occur at any
point within the plate during a time incrementAt accordingto
a creep law basedon experimentaldata [Weertman, 1970]
• øo
• = •1on exp (-GTm/T)
(6)
5
STRAIN,•
I0
15
(0/0)
Fig. 5. Stress-strain
curvefor dunite at 500-MPa confiningpressure,
in which
800øC,and 5 x 10-4 s-• strainrate.After Griggset al. [1960].
6386
Predictionof the Rheologyof the Flexed Lithosphere
,
,
from OlivineDeformation Maps
Depth(km.)
for olderoceaniclithosphere
0
20
40
[
:;'
/
•
60
80
i00
I.......[..... I......•...... I .. [ ,._.1
,7'•
'%,
'•
•
-
.........
•
. •,
/N?,
. •-- .',
o i Moxirnurn
/ • ,•-•,•
X¾,
• xx•.Xl
• xxx-- .•
nd
10 •, bending
/ '.,'VI•%•i • '• ,I • x •- I0'1o, - io2
I/_.
?
/
/
X ,¾|lJ•'\x\ ,E
•, _'•J
..-, !/ / ';/ / /
/!.'
qj\
"• ,"P,
_..
J/ CREEP
dJ=lV
'._•\,,-r16i• Ira,-,-•_ -•
/ . J,".;,-r18
2"
I0
r-•Jvvr_.• L_/-WV
t,\ CREEP
/I -
j/,..
I
•
,
ø
---
,,
"Elastic '
!
Temperature (øC x l0 )
I
C'i!!!i/
' ' ' ' ' /•.
Base of the Lithosphere
..... ( Newtonian
)
C•?•.•.,•c
Diffusional
Law : •:•:•
.......................................
•;•Plastic
/,
Cree•'
• /
///
////.
[
RhealogicalZonation of the Lithosphere
Fig.6. Olivine
deformation
mapwithsuperimposed
upper
bound
stress
distribution
in thelithosphere
below
a point
load.Thedeformation
mapisforolivine
witha 0.1grainsizeandincludes
theeffect
ofincreased
confining
pressure
with
depth.
Thefieldsarelabeled
according
to thesteady
staterheology
thatdominates
at thatstress
andtemperature.
Also
includedarethepredicted
strainrates(dashedlines);thereforeeffectiveviscosities
canbecalculated.
The baseof the litho-
sphere
(thickness
100km)isdefined
bythetemperature
of 1300øC.
Fora thinner
lithosphere
thedepthscale
canbereadas
'percent
oflithospheric
thickness.'
Thelithospheric
stress
profile
plotsmagnitude
ofstress
versus
depth.Stress
in thiscontextmaybeinterpreted
asbending
stress,
shear
stress
intensity,
ordeviatoric
stress,
allof whichhaveverysimilarvalues.
The lowerpartof thefigureshows
a projection
of thefieldscrossed
by thestress
distribution
ontoa section
of the litho-
sphere
andislabeled
according
tothedominant
rheology
predicted
forthatzone.'Elastic'
implies
thatthestrainratefor
thatzonewillbesufficiently
slowthatit willappear
elastic
overlongtimescales.
FromAshby
andVerrall
[1977],
asmodi-
fiedbyBeaumont
[1979].
Reprinted
withpermission
ofBeaumont
[1979]
andElsevier
Scientific
Publishing
Company.
A strainratesimilarto theviscous-elastic
ratewouldoccurin lithospheresupportsthe load. The increasein stressas the
theviscoelastic
platefor •7-- 8 x 1024
N s/m2or •--• 10m.y.
Thusat leastinitially,therelaxation
rateat 30-kmdepthin a
viscous-elastic
plateis quantitatively
similarto therateof relaxationin a viscoelastic
platewith•-= 10m.y.,assuming
the
samestress
level.A majordifference
in thetwomodels
is that
whileviscoclastic
relaxation
proceeds
throughout
thedepthof
theplate,in theviscous-elastic
model,relaxation
at deeplevelsincreases
thestresses
in theupperlevels,
sincelessof the
platethinsalsoincreases
thestrainrate,butthefactorT/T,.
dominates
thebehavior.
For example,to obtaina strainrate
as highas7.5 x 10-4/m.y.at a depthof 15 km whereT -•
0.3T,,,,thestress
wouldhaveto increase
to 4,400,000
MPa.
In general,theultimate,long-termstress
on timescales
of
several
hundred
millionyearsfor a viscoclastic
platemodel
wouldbeoftheorderoftheweightoftheload(several
tensof
megapascals),
since
thecompensation
wouldapproach
thatof
MCNUTT:
IMPLICATIONS OF GRAVITY
Airy. For a similartime scalethe stressin the upper 15 km of
a viscous-elastic
plate would continueto increasein the range
of several hundred megapascals,since the low T/T,, ratio
would precluderelaxation at shallow depths.In this respect
then the stressestimateis quite similar to that of the purely
elastic plate. Consideration of laboratory creep data has
merely provideda mechanismto thin the seismiclithosphere
from 100 to the observed30 km on a million-year time scale.
It is perhapssignificantto point out that if we were to interpret the changein flexuralprofilein a viscous-elastic
plate in
termsof a viscoelasticplate equation,it would appear that ?
increaseswith time. This is exactly the behavior noted by
Watts[1978]alongthe Hawaiian-Emperorchain.
The elastic-plasticlithosphere[Liu and Kosloff, 1978]is a
further refinement of plate mechanismsincorporatingrock
deformation data. The stress-strainrelation for an elastic-perfectlyplasticmaterialis shownin Figure $. The elasticbehavior is representedby the slopedline along which stressand
strainare linearly proportional.At a critical stresslevel Ocdetermined by the rate at which the load deformsthe lithosphere,the materialbehavioris plastic.Strain increaseswithout changein stress.The strainrate dependenceof the yield
stressis given by a powerlaw equationextrapolatinglaboratory data to geologicloadingrates[Carter, 1976]:
FOR STRESS
6387
nessof the lithospherewill be reducedfrom above by brittle
fracture and from below by ductile deformation. Strengthdecreaseswith depth according to the temperature structure,
leaving an elasticcore severaltens of kilometersthick with a
yield strengthbetween300 and 800 MPa.
Beaurnont[1979] demonstratesthe theological complexity
in a flexedlithosphereby superimposing
the deviatoricstress
predictedby a 100-kin-thickelasticplate on Ashby and Verrall's [1977] deformation map for olivine (Figure 6). His
model only applies to the instant in time before creep and
plasticfailure redistributethe stress,but in a generalway it indicatesthe type of responseexpected.If preexistingzonesof
weaknessoccur in the upper 10 to 15 km of the plate, the region yieldsby brittle failure on faults.Even in the absenceof
faults the lithospherecan deform by cataclasticflow involving
stablemicrofracturingat depthsbetween2 and 20 km [Kirby,
1980]. The lower 20 km of the lithosphereyields plastically.
Rapid relaxation by diffusionand power law creep at 50- to
80-km depthsalsoreducesthe effectiveelasticthicknessof the
lithosphere.The central part of the plate between 15 and approximately50 km relaxesso slowlythat for million-year time
scalesit appearselastic.
To only a first approximation therefore can we model the
mechanicallithosphereas an elasticplate 30-40 km thick. The
more completepicture from the rock mechanicsliterature pre(7)
• = AOcn exp (-Q/R2)
dictsthat with detailed observationswe shouldalso detect(1)
in which Q is the activationenergyand R the gas constant. a time dependencein the flexural rigidity as diffusionalcreep
The ultimate strengthof the lithosphereis equal to Ocand de- gradually thins the elasticplate and (2) a stressdependencein
pendson temperature(and thusdepth) as well as strain rate. the flexural rigidity as the volume of plasticallyyielded lithoFor lower strain rates correspondingto geologicloading, the sphere increaseswith deviatoric stress.The time-dependent
aspectsof flexureare not evidentover the loading times(~200
yield stressalso is lower.
Liu and Kosloff [1978] model flexure under the Hawaiian m.y.) observedin the oceans.If relaxation in the elastic core
archipelagousing an elastic-plasticlithosphere.The rate de- doesoccur,the time scalefor the creepis greaterthan 50 m.y.
pendenceof the elasticresponseis incorporatedby using90% For oceanic islands and seamounts,Beaumont [1979] finds
that the limited zone of plasticfailure changesthe surfacedisof the elastic moduli, based on a 2% decrease for each 3-decade decreasein loadingrate. The plasticpart of the deforma- placement by 'only 10%. In the absenceof reliable data on
tion is given by (7) usingthe parametersfrom Carter and Ave Moho displacement,deviatoric stressesassociatedwith topoLallement [1970]. Liu and Kosloff successfullyfit a profile graphicloading may not be large enoughto resolvedetailsof
northeastof the island of Oahu, but there are many free pa- plasticdeformationusingsurfaceobservationsalone.
rametersin the model, and the fit is in no way unique. Since
The Outer Rise
stresscausedby any load cannot exceedOc,the stressimplications for the elastic-plasticplate are determined by the asThe casefor plasticyielding is far better documentedin the
sumedtemperaturestructureand deformationrate in (7). For lithospheric flexure profiles seaward of subduction zones.
the parametersselectedby Liu and Kosloff, strengthvaries Here the strain reachesabout 2%, compared with strains of
greatly with depth, and the nonyieldedportion of the litho- lessthan 1% for seamountloads [Watts et al., 1980]. In the resphereis about 30 km thick. This study is meant to illustrate gion of the outer rise, approximately100 km from the trench
how rock deformation data can be employed in flexure stud- axis, the lithosphericdeformationis adequatelydescribedby
ies. For the Hawaiian ridge, at least, there appears to be no the elastic plate model. The effective elastic thicknessin the
conflict between the extrapolatedresultsof deformation ex- range 30-40 km [Hanks, 1971; Watts and Talwani, 1974]
perimentsand observedlithosphericflexurefrom large long- agreeswell with the resultsfrom seamountloading studies.
Nearer to the subductionzone, on the outer trench slope,the
term geologicloads.
To summarizethis section,we find that the proposedrheo- extremely large curvature in the plate requiresplastic yieldlogicalmodelscan explain the regionalcompensationfor fea- ing. McAdooet al. [1978]estimate470-720 MPa for the yield
tures like Hawaii as long as they predict a 30-km-thick, pre- stress.Similar resultsare obtained in other studies[ Turcotteet
dominantly elasticlayer that sustains100 MPa or more stress al., 1978; Bodine and Watts, 1979].
on a time scalefrom 1 to 50 m.y. Model 1, the homogeneous
Not all investigatorsagree that the outer rise is supported
elastic plate, is merely the simplestmechanismthat satisfies by the strengthof the lithosphere.Melosh [1978] proposesan
this requirement.The fact that a one-parameterelasticmodel alternative schemein which the outer rise is produced by a
can explain the data testifiesmore to limitationsof the obser- momentum changein the flow of a viscouslayer beneath an
vations than to the rheologicalsimplicity of the earth. Rock elasticlithosphere(Figure 7a). Sinceit is not requiredthat the
strengthdependson a number of parameters,includingrock outer rise result from forces and moments applied at the
type, confiningpressure,temperature,and strainrate. Accord- trench axis, the wavelengthof the deformationdoesnot detering to a review by Kirby [1980]the effectivemechanicalthick- mine the thickness of the elastic lithosphere. Deviatoric
6388
MCNUTT:IMPLICATIONS
OF GRAVITYFORSTRESS
SPERE
Fig.7. (a)Dynamic
model
forouter
risctopography.
A thinelastic
platepassively
rides
overa moving
viscous
layer.(b)
Variationof dynamic
modelin whichtheelastic
plateisfaulted,formingindependent
blocks.
stresses
can be madearbitrarilysmallby thinningthe elastic
plate.For the old oceaniclithosphere
commonto the outer
riseregion,however,
it wouldbeunreasonable
to suppose
that
CONTINENTAL
STUDIES
There is little doubt that oceanictopographyformedon a
is regionallycompensated
with
its mechanical thicknessis lessthan the 30- to 40-km thickness cooled,thickenedlithosphere
high stresses
(hundredsof megapascals).
Can the
beneathseamounts.
Regardless
of the dynamicsof formation associated
and, if so,what is
for the outer rise,the observedstrainstill impliesseveralhun- samemodelbe appliedto the continents,
over billion-year
dredmegapascals
of deviatoric
stress
in a 30- to 40-km-thick the rheologyof the continentallithosphere
time
scales?
We
can
anticipate
several
problems
in the analyplate.
A variationof the dynamicsupportmodelassumes
that the sis of continental compensation:
Natureof theload. Implicitin mostloadingstudiesis the
upperlithosphere
consists
offaultedblocks
independently
ridthat the lithosphere
passively
responds
to a load
ing the deeperviscousflow, as shownin Figure7b. This assumption
canbe
model assumesthat the lithosphere is devoid of shear appliedfromabove.A goodcasefor thissimplification
strength.
Seismic
refraction
profiles(seeWattsetal. [1980]for made for sedimentarybasinswith distantsources.While for
is lessjustified,the plate is alan example)showthat normalfaultsare indeeda common volcanicloadsthe assumption
feature of outer trench walls. Brittle behavior in the upper 10 teredand weakenedby magmaticactivityin the area leastre-
kmoftheoceanic
plateunder
largetensile
stress
ispredictedsolved
bythedata.Onlywhen
complete
refraction
dataunder
bytherockdeformation
experiments
[Kirby,
1980].
In fact,if volcanic
loads
become
available
willit benecessary
todealreevidence
of surface
fracturing
werelacking,
therewouldbe alistically
withtheload-forming
process.
Theloading
assumpgoodreasonto doubtwhetherlaboratory
datado applyto
geologic
processes.
At thistime,however,
theredoesnot appearto beanybasisfor extending
thefaults40 km to thebase
of the elasticplate,and the modelcan be rejectedfor both
rheological
[Murrell, 1976;Ashbyand Verrall,1977;Kirby,
1980]and seismicreasons
[Hanks,1979].
tion is leastappropriatefor continentalmountainbelts.The
detailsof orogenyare sketchy,but in mostcases,topography
is createdas lithosphericplatescollide.The distinctionbetween the 'load' and the lithospherebecomesmeaninglessif
the entire thicknessof the lithosphereis deformed.
Complexityof geologichistory. Overprintingof several
MCNUTT:
IMPLICATIONS
OF GRAVITY
orogenic,intrusive,and extrusivevolcaniceventsis common
for older continentalregionsbut rare in the oceans.Ever present erosion in subaerial
environments
introduces
a time de-
pendenceto the loadinghistory.
6389
FOR STRESS
O(k)-- -2•rpGexp(-kz½)
(8)
wherep is the densityof the topography,zcthe compensation
depth, and G Newton's gravitationalconstant.For the elastic
plate compensationmodel,
SedimentaryBasins
{I +k4O•
-'exp(-kz½)
Q(k)
---2•rpG
apgJ
While the sedimentary basins involve loading conditions
which best fit the idealization in the theoretical model, the
theologicalimplicationsfrom various studiesshow no consensus.For example, Haxby et al. [1976] infer from the configurationof successive
sedimentaryfaciesthat the effectiveri-
gidity in the MichiganBasinhasincreasedto 4 x 1023N m
over a time span of approximately100 m.y. They invoke uplift of the gabbro-eclogitephasetransitioncausedby shoaling
isotherms as the driving mechanism for gravitational subsidenceof the basin.An increasein the flexural rigidity would
be an expected consequenceof cooling and thickening a
purely elasticlithosphere.Nunn and Sleep[1979] successfully
model the samebasinasloading on either an elasticplate with
D -- 2 x 102'N m or a viscoelastic
platewith Do -- 10•4 N m
and • = I m.y. Thermal contractionis used as the driving
mechanism for subsidence. For the North Sea, Beaumont
(9)
whereAp is the densitycontrastfor materialsoverlyingand
underlying the plate. Figure 8 showstheoreticalisostaticresponsefunctionsderivedfrom (9) for two z½values.The D = 0
curvescorrespondto local compensation.For identicalz, val-
uesthe elasticplate response
(D = 1022N m) has a sharper
curvatureand falls off more quicklyto zero at shortwavelengths.The explanation for this behavior is that narrow features, supportedby the strength of the plate, have smaller
Bougueranomaliesand thereforelower Q valuesthan similar
featuresthat are locally compensated.Broad featuresdo not
feel the effect of the elasticplate, and thereforelocal and regional responsesare similar at long wavelengths.Because
compensationdepth varies,curvature,rather than amplitude
of the observedQ, is the best estimateof elasticstresses.For
example,referringto Figure 8, the local compensation(D -- 0)
responseat medium to long wavelengthswith a 40-kin compensationdepth is actually lower in amplitude than the D --
[1978]prefersthe viscoelastic
crustalmodel to explain the apparentdecreasein flexuralrigidity with time. In a manner expectedfrom a viscoelastictheologythe subsidingpart of the
basin narrowswith time, and erosionof the peripheral uplift 1022N m response
with shallower(30 km) compensation.
regions results in an apparently regressivestratigraphicseA closerelative of the isostaticresponsefunction is the free
quence at the basin surface.This pattern of progressively air responseZ:
youngersedimentsoutcroppingtoward the centerof the basin
FA(k) -- Z(k). H(k)
is typical of many basins.Sclaterand Christie[1980] explain
the observed crustal thickness, heat flow, and subsidence of where FA is the Fourier transform of the free air gravity
the North Sea by invoking 50-100% stretchingof the Central anomaly. The Z responseis most commonlyusedfor oceanic
Graben. They deliberatelyignoreflexuralloadingof the litho- surveys.
The theoretical
response
2 is relatedto 0 via
sphereon the assumptionthat thermal subsidenceand fault• = {) + 2•rpGexp(-kz,)
ing are the dominant factorscontrollingthe evolutionin the
centerof the basin.Clearly, the interpretationof sedimentary wherez, is the averagedepth from the seasurfaceto the topofaciesin terms of a theologicalmodel is nonunique,with the graphic surface.
mechanismof subsidence
contributingthe most uncertainty.
There are two major limitations in usingthe responsefuncStudiesof the structureand evolutionof sedimentarybasins tion technique.The first problem is that all gravity signal is
and their one-sidedanalogues,passivecontinentalmargins, assumedto be related to compensationof topography with
will not allow unambiguousestimatesof rheologicalparame- constantdensity. Density variations unrelated to topography
tersuntil otherfactorsarebetterdeterm'med.
Thesefactorsindude the initiating mechanism,the sedimentarybudget,and
sea level history [Beaumont,1979].
are a sourceof noise,but their effectcan be minimized by statistically separatingout only the gravity field correlatedwith
the topography.In regionsof low topographicsignal, how-
DistributedTopography
Individual
volcanic features such as the island of Hawaii
are rare on the continents.Elevatedregionsoccur in broad
mountain belts where deformationfrom any one load is indistinguishablefrom that of another.A usefulmethod for analyzing the isostaticcompensationin regionsof complextopography is the response function technique. On the
assumptionthat isostaticcompensation
for a point load is linear and isotropic,the isostaticresponse
functionQ(k) can be
calculatedfrom Fourier transformsof topographicand Bouguer anomaly data [Dormanand Lewis, 1970]:
C•) -- Q(k). H(k)
whereG and H are gravityand topographytransformsand k
-12
-10
-O8
-0.6
-O4
- 2N
rn '•x•L --
-O2
o
5000
DD
!01,
, ,'
2000
I000
500
0
200
100
50
X,km
= Ikl - (k•• + ky•)-'/•. Theresponse
Q calculated
fromactual
Fig. 8. Theoretical isostatic responsefunctions for local comdatasetscanbe directlycompared
to theoretical
0 fromthe pensation
(solidcurves)andregionalcompensation
withD = 10•2 N
linearizedforms of regionaland local compensationmodels. m (dashedcurves).Eachresponseis calculatedassumingtwo different
For example,for linear Airy isostasy,
valuesof compensationdepth z½.
6390
MCNUTT: IMPLICATIONS OF GRAVITY FOR STRESS
-I.4
-I.2
--
-I.0
--
-0.8
--
-0.6
--
-0.4
--
-0.2
--
0
--
AUSTRALIA
McNutt & Perker , 1978
Lewis & Dotman,
U.S.A.
1970
Stephenson & Beaumont• 1979
CANADA
0.2
5000
2000
I000
500
200
I00
50
WAVELENGTH X , KM
Fig.9. Comparison
ofUnited
Statc•
andAustralian
isostatic
response
functions
withtheresponse
fromtheCanadian
Shield.Barsindicatethe standarderror in eachspectralestimate.
ever, the uncertaintyin Q is large. A more seriousproblem inv01vesthe necessarilylarge dimensionsof the regionunder investigation.For the continentsthe characteristicfalloff region
for Q lies between2000-km and 500-km wavelengths.The calculatedQ may thereforegiveonly an averagerigidity estimate
from severaltectonicprovinceswithin the surveyboundary.
From the curvature in Lewis and Dorman's [1970] response
function for the United States(Figure 9), Banks et al. [1977]
determine that the apparent flexural rigidity in the United
Statesis between102•and 1022N m. They concludethat Walcott[1970a]foundhigherrigidityvalues(1023-10
24N m) from
other North American features because he measured the deat the surface rather than at the Moro. Thus Banks
formation
et al. [1977]implythatsomesortof depth-dependent
relaxa-
MississippiValley (Figure 11). The responsefunctionsfrom
the two grids are comparedin Figure 12. The isostaticcompensationis obviouslydifferent in the two regions,and Lewis
and Dorman's [1970] responsefunction was a hybrid of the
two. The overallresponsereflectsthe response
signalfrom the
regioncontainingthe highestcorrelationbetweenthe gravity
and the topography.
The Q functionsin Figure 12 can be directly inverted for
parameters in compensationmodels. We will consider the
data in terms of the best fitting flexural rigidity D and compensationdepth zcin an elasticplate mechanism.This model
is appropriate for two reasons:the uncertainty in the data
doesnot permit inversionfor more than two parameters,and
the resultscan be interpretedquite generally.If the compensationis actually local rather than regional,the preferred
D value will be extremelylow. If viscoelasticrelaxationoccurs, the resultingD can be interpreted as a time-dependent
apparentrigidity.
The inversionresultsare presentedin Figure 13. The horizontal scaleplotsflexural rigidity, and the verticalscalerepresentsthe one-normmisfit betweenthe observedresponseand
the theoreticalmodel. Each curve correspondsto a differentzc
value. The dashedlines indicate the probability level at which
tion reconcilesthe two rigidity estimates.
McNutt and Parker [1978] find an even lower value for apparent flexural rigidity in Australia and invoke a viscoelastic
relaxation model to explain the difference in United States
and AustralianD values.Assuminga 200-m.y. age for the latest Australian orogeny and a 50-m.y. age for the Laramide
orogenyin the United States,the viscoelasticrelaxation time
•- = 45 m.y. If the viscoelastictheory is correct,the isostaticresponsefrom easternCanadashouldresemblethat of Australia
becausethe region consistsof Precambrian shield and oro- the model is consistent with the data. For the western section
geniczonesno youngerthan the Paleozoic.Figure 9 compares the best fitting compensationdepth is between30 and 35 kin,
the isostaticresponsefrom the Canadian Shield [Stephenson and the response
rulesout flexuralrigiditiesabove10•9N m.
and Beaumont, 1979] with the Q estimatesfrom the United Since this D value correspondsto a plate thicknessof only 1
States and Australia. The viscoelasticmodel fails this test; the km, we conclude that compensationin the western United
Canadian responsein no way resemblesthat of Australia.
States is local. For the eastern grid, regardlessof what comThe validity of McNutt and Parker's [1978] viscoelastic pensation
depthwe assume,
the bestfittingD is 5 X 10•2N m,
,model
hingeson theassumption
thattheresponse
functionis which implies a 20-kin-thick plate. The misfit valley is exdominatedby the signal from the most recentorogeny.This tremely sharp and stronglyrules out local compensation.
assumptionisjustified for Australia.When the continentis diResidual gravity maps give an indication of which features
vided into eastern,central, and westerngrids,the most coher- determine the isostaticresponse.The residualgravity map is
ent Q which dominatesthe averageQ is that of the geologi- producedby subtractingfrom the observedBouguergravity
cally youngesteasternsection.In a similar mannerthe United the anomaly predictedby filtering the topographywith the
Statesdata setcan be dividedinto two subsets:
a westerngrid observedresponsefunction. The residualsare 'isostaticanomcontainingthe Rocky Mountains, the Basin and Range, and alies,' although no particular compensationmechanismhas
the Sierra Nevada (Figure 10); and an easterngrid containing been assumed.The residual gravity map for the easterngrid
the AppalachianMountains, the Great Lakes region,and the (Figure 11) showsa prominentpositiveanomalyoverthe mid-
MCNUTT:
IMPLICATIONS OF GRAVITY FOR STRESS
6391
crusthas lost whateververticalmobility it possessed
during
orogeny.The stressimpliedby the gravityanomalyis 50 MPa,
assumingthat it originatesat about 50-km depth.
In the westerngrid thereare few surprises
(Figure 10). The
Rocky Mountains display anomaliesconsistentlypositivein
sign, indicating that the compensationdepth is generally
deeperthan the 30- to 35-km grid average.The Basin and
Range showsnegativeresiduals,indicatingthat the Moho is
slightly shallowerthan the regional average.Both observations are consistent with the notions that the Colorado Plateau
hasa root and the Basinand Rangehasan anomalouslythin
crust.
WESTERN UNITED
STATES
ISOSTATIC
ANOMALY
'"'•
...... >100
WITHIN+_IO0 g.u.
:.'".....'•
<-I00
•
200-g.u. CONTOURS
500
KM
I
These resultssuggestthat the tectonicprocessthat formed
the topographyin the westerngrid, by someperhapsthermal
or mechanical means, left the crust and upper mantle incapable of transmitting stresslaterally. In the easternUnited
States,at leastpart of the lithosphereis capableof responding
elastically.A very simple-mindedmodel which could explain
theseresultsis shownin Figure 14. Perhapsthe thin-skinned
tectonicsresponsiblefor the Appalachian Mountains left the
lower part of the elastic lithosphereessentiallyundisturbed.
We would expect to observean anomalouslyshallow compensation depth becausethe Bouguer gravity signal comes
from the flexureof two interfaceswith densitydiscontinuities:
one at the base of the deformed sedimentarylayer and the
other at the Moho locatedsomewherewithin the elasticlayer.
This type of compensationwould result from shallow, compressionaltectonics.In the westernUnited Statesthe topographic elevation is caused by tensional and/or vertical
stressesaffectingthe entire crust and upper mantle. Whether
differential movementoccurredalong faults, as shownin the
schematicdiagram, or high temperaturescausedthe rocks to
behave viscouslyrather than elasticallyis immaterial for the
purposeof explainingthe gravity and the compensation.Firm
conclusionsrequire that individual loads be modeled; at this
point we cannotrule out the possibilitythat severalisostatic
mechanisms,supporting features with different characteristic
Fig. 10. Residual gravity anomaliesin the westernUnited States. wavelengths,have combined to produce a meaninglessreAnomalies are the Fourier transform of G•k) - Q(k)H(k), where G
sponsefunction.
and H are the wave number domain representationsof Bouguer grav-
Other studiessupportthis conclusionthat continental crust
can sustainelastic stresses,given the right conditionsof load
emplacement.Both the Boothia Uplift [Walcott, 1970a] and
continentgravity high and anomalouslowsin the Mississippi the midcontinentgravity high [Cohenand Meyer, 1966] apValley. Thesegravity anomaliesarisefrom subsurfacedensity pear to be regionally compensatedwith a flexural rigidity of
ity andtopography
maps,respectively.
variationsuncorrelatedwith the topographyand therefore 2 x 1023N m. The associated
platethickness,
30 km, is 10 km
cannot be analyzed by the responsefunction approach.The
residuallows in the Hudson Bay area show the effect of still
incomplete isostatic rebound from the Pleistoceneglaciers.
For the purposesof this study,the mostimportant observation
is that the Bouguer gravity signal from the Appalachian
Mountains has been essentiallyremoved. The isostaticresponsein the easterngrid is determinedby the gravity signal
from these mountains.A residual belt of negative isostatic
anomaliesparallelsthe trend of the highestpeaksin the Ap-
greaterthan the value determinedfor the AppalachianMountains. It is difficult to explain the reduction in plate thickness
beneaththe Appalachian Mountains in termsof viscousrelaxation of elastic stresses.The ages of both the midcontinent
gravity high (•1100 m.y.) and the Boothia Uplift (-•500 m.y.)
are greater than the age of the Appalachianorogeny (-•200
m.y.). From the model of the Appalachian Mountains presentedhere, it seemsmore likely that the elasticlithosphere
was either mechanicallythinned from above or partially re-
palachians.
Thisgravity
lowcoincides
witha 5•-•km-deep
laxedat thebasebyelevated
temperatures
at thetimeof
crustalroot determinedfrom seismicrefractiondata [Jameset
al., 1968].Given the presentelevationof the mountains,the
sizeof the root is excessive
and overcompensates
the topography. Thus the negative gravity anomaly resultsfrom crustal
structureunrelatedto present-dayisostaticbalance,although
orogenesis.
We face a dilemma in converting the continental compensationmodelsto maximum stressestimates:200-m.y.-old
loads are extremely eroded. The presentstressneeded in the
lithosphereto supportthe mountainswill underestimatethe
it isentirelypossible
thattherootis a remnantfroma period peaklevelsattainedin the past.To minimizethe mitigating
when the Appalachianswere more impressivefeatures.If this effectsof erosion,we shouldconcentrateon extremelyyoung
is the case,the persistenceof the root may indicate that the loads,suchas the Himalaya. Unfortunately,not only are the
6392
MCNUTT: IMPLICATIONSOF GRAVITY FOR STRESS
EASTERN
UNITED
STATES
ISOSTATIC ANOMALY
........
'"":'• > I00
•
WITHIN__.
I00 g.u.
:".:• < -I00
200--g.u. CONTOURS
o
[
500
,
KM
I
Fig. 11. Residualgravity anomaliesin the easternUnited States.
necessarydata unavailable,but the very fact that the mountaliasare youngimpliesthat someof their supportmay be dynamic. From the similarity in elastic plate thicknessin the
continentsand oceans,we suspectthat the continentallithospherecan also supportdeviatoric stressin excessof 100 MPa
in areaswhereit hasnot beenexcessively
thinned,heated,and
fractured.Kirby [1980]suggests,
however,that the weakening
effectsof water may play a more significantrole in the continental granites than in the oceanic basalts.
emplacementof the oldest geologicunits. These bodies are
consideredto be 'one-plateplanets'[Solomon,1978]and lack
the complicated deformation patterns associatedwith the
creationand destructionof lithosphericplates.
Solomon[1977]infers the thermal historyof the moon and
Mercury from the surfacedeformation.Early in a planet's
thermal history the interior warms as core segregationprogresses.
The planetexpands,causingtensionalsurfacefeatures
and volcanism.When differentiationis essentiallycomplete,
the planet coolsand contracts,producingcompressionalsurEXTRATERRESTRIAL
STUDIES
face featuresand cutting off volcanicactivity. For example,
Many of the sametechniques
usedfor analyzingloaddefor- Mars is dominatedby extensionalfeatures,indicating plan-
mationanddetermining
stress
on theearthhavebeenapplied etaryexpansion
overmostof itshistory.Theabsence
of extento the moon and other planetsin the solar system.The two
mostintenselystudiedfrom a stressviewpointare the moon
and Mars. While the remoteness
of theseplanetarybodies
makes it more difficult to interpretthe geologyand obtain
precisemeasurements
of gravity and topography,in at least
sional featureson Mercury placesthe planetary differentiation phase before the formation of the oldest surface
features.The moon lacks extensivecompressionalor extensional features, and therefore its radius cannot have increased
or decreased
by more than 1 km in the past3.8 b.y. The 1-km
one respectthe analysis is simplified. There is no evidence limit is basedon the assumptionthat a changein tangential
from the moon, Mars, or even Mercury that there has been stressof 100 MPa would be sufficientto produceobservable
any horizontal shifting of rigid lithosphericblockssincethe surfacefaulting.Solomon[1977]findsthat acceptable
temper-
MCNUTT:
IMPLICATIONS OF GRAVITY
ature profilesT(z, t) predictinga maximumradiuschangeof
lessthan 1 km over the past 3.8 b.y. have initial profileswith
meltingtemperaturesdown to 200- to 300-km depth and cold
temperaturesin the deep interior. However, the accumulated
thermalstressin the lithospherefrom the acceptablemodelsis
as high as severalthousandmegapascals.In the moon'sinterior the thermal stresses
dissipateby flow on shorttime scales,
FOR STRESS
6393
Solomon[1978] find that the older lunar highlandsare locally
compensated,while the youngermasconmaria are supported
by the strengthof the lithosphere.
it followsthat the lunar
lithospheremust have increasedin thicknessafter the formation of the highland topography.
Olympus Mons, a shield volcano sitgated on the Tharsis
ridge of Mars, has receivedconsiderableattention recently.
but the mechanism and time scale for relaxation of the litho-
With a basalareaof 3 x 105km and a heightof 25 km, Olym-
sphericthermalstressis unknown.The solutionto this problem may have important implicationsfor the stateof stressin
the lithosphereand the thermal evolutionof planets.
Solomonand Head [1979] combine the model for global
stresson the moon with predictedstresses
from lithospheric
flexure in order to explain the spatial and temporal distribution of linearrillesand mareridges.From features3.6-3.8 b.y.
old they infer a relativelythin plate,25-50 km thick. Younger
featuresonly 3-3.4 b.y. old requirea 100-km-thickplate.Note
pus Mons is the largest known volcanic feature in the solar
system[Thurberand Toksoz,1978].The excessmassof Olympus Mons and the other three shieldvolcanoesof the Tharsis
dome is sufficientlylarge to cause a 1.2-km anomaly in the
gravity equipotentialsurfaceand accountfor about 6% of the
planetary oblatenessJ2 [Reasenberg,1977; Kaula, 1979]. De-
that these effective plate thicknessesare several times those
found on earth and imply that the long-termelasticlayer is
much thicker on the moon. Given the dependenceof creep
rate on temperature(equation(10)), a thick elasticlayer may
be the consequence
of a more graduallyslopinggeothermfor
the moon relative to the earth. The observation that the com-
pensationfor youngerlunar featuresrequiresa thicker lithospheremight alsobe consistentwith a viscoelastic
plate mechanism,but in this casea viscoelastic
explanationis unlikely.
Unlessthe initial plate thicknessat time zero is much greater
than 100 km, it is difficultfor loadsmore than 3 b.y. old but
only 0.1-0.8 b.y. apart in age to differ in effectiveplate thicknessby 50%.From gravity and topographicdata, Thurberand
bate has centered on whether
or not the Tharsis dome volca-
noes are compensated.Phillipsand Saunders[1975] conclude
that while older regionsof Mars are locally compensated,the
topographyof the Tharsisregionis youngand mostly uncompensated.Their observationcould be explained by the scenario invoked by Solomonand Head [1979], in which the lithospherethickensbetweenthe time of formation of young and
old topography.
Thurberand Toksoz[1978] directly model the compensation
of Olympus Mons usingflexure theory. For an elasticplate
thicknessas small as 100 km, the model predicts500 MPa for
the maximum
extensional
surface stress. Thurber
WAVELENGTH (Kt4)
I
WESTERN
0
and Toksoz
see no evidence of surface faulting in responseto the high
stress,nor do they observean arch of the order of 4 km high
that would be produced by plate flexure. Their preferred
model, an elasticplate approximately 200 km thick, predicts
I
I
I
I
I
I
I
i
i
u.s.
EASTERN U.S. & CANADA
-3.2
-2.8
-2.4
-I
LOG WAVE NUMBER K (KI4
- 2.0
-I.6
)
Fig. 12. Isostaticresponse
functionsfrom the easternand westernUnited States.
-
6394
MCNUTT:
•
I
i
IMPLICATIONS OF GRAVITY FOR STRESS
I
I
I
I
WESTERN
U.S.
_
z• km._•__•_.
I
I
EASTERN
I
I
I
U.S. 8• CANADA
25
25
Compensationdepth in km
25 km
ompensation
d
30 km
35 km
20
2O
40
km
45
km
1.5 %
15
20 %
__zo_
yo___
.
I0
5O %
50 %
t 35
km.,
• . . 66
%
80 %
-
/
I
i016
I
I
i018
I
I
I
1020
I
i022
I
i024
FLEXURAL RIGIDITY (Nm)
I
I
i016
I
I
1018
I
I
1020
I
1022
I
I
1024
FLEXURAL RIGIDITY ( Nm )
Fig. 13. One-norm
misfitbetween
observed
response
functions
in Figure14andQ fromelastic
platemodels.
Plateparameters
areflexuralrigidityD andcompensation
depthZc.Dashedlinesindicatetheprobability
thattheobserved
misfitis
causedby random errors in the data.
an arch amplitudeof only a kilometeror two, in agreement Sjogren[1979]fails to detectany low-densitycompensating
with the topographicdata, and producesmaximum surface massfor OlympusMons. The gravity anomalyover the feastresses
lessthan 100 MPa. At depth,however,the maximum ture is about 20% greater than expectedfrom the observed
stressunderthe load wouldbe greaterthan 100MPa. Using volumeusinga Bouguertheoryand suggests
that even denser
Jeffreys'[1924]calculationof stressunderlyinga triangular materialliesbeneaththe volcano.Sjogrenconcludes,
'This esload restingon an elastichalf spaceas a very roughestimate, sentially uncompensated600-km feature produceskilobar
the maximumdeviatoricstressoccursat 150-kindepth and stresses
that demanda rigid, thick lithosphere,or somerather
equals about 200 MPa.
uniquescenarioabout very youngtopographyobtainedon a
Thurberand Toksoz's[1978]modelassumes
that Olympus presentlyseismicallyinert planet.'
Mons is supportedonly by the strengthof the Martian lithoSmithet al. [1979]note a high correlationbetweengravity
sphere.Phillipset al. [1980]arguethat the impliedstresses
are and topographyon the planet Venus. Their data are best extoo largeand that the Tharsisregionmustbe at leastpartially plained by a regionalcompensationmechanismwith D -- 5 x
supportedby dynamic forcesfrom below.
1023
N m. The corresponding
40-kinplatethickness
issosimiThere is even an indication that stress estimates based on a
lar to that determinedfrom terrestrialstudiesthat we mightbe
volumemeasureof excess
massfrom OlympusMons may be temptedto concludethat the two planetshave comparable
too low. From Viking Orbiter2 high-resolution
gravitydata, strengthand rheologicalpropertiesin their surfacelayers.The
observationthat the temperatureon the surfaceof Venus is
/DEFORMED
LAYER
400øC
complicates
the
Watts
[1978]
has
proposed that
the base
of interpretation.
the elastic layer
in the
earth
correspondsto approximatelythe 450øCisotherm.At thisstagethe
limitations in the data allow only speculation;if the same
model appliesto Venus, there are at least two possibleex-
ELASTIC
LAYER
planations
for the compensation.
Temperature
may be a very
slowingincreasingfunction of depth beneath the Venusian
surface,or the chemistryof the rocksmight allow greater
strengthat higher temperatures.
CONCLUSIONS
At this point we must in someway answerthe question,
What magnitudestresscan the lithospheresupporton geoFig. 14. Schematicdiagramshowingtwo stylesof tectonicdeformation.(a) Shallowcompressional
forcesdeformweak sedimentary logic time scales?Owing to the fundamental differencesin
layer overlyingmore competentelasticplate. (b) Vertical or tensional oceanic and continental tectonic settingsthe oceanic loads
forcesdisruptentire thicknessof competentlayer.
tend to provide a good estimate of maximum stresslevel,
MCNUTT: IMPLICATIONS
OF GRAVITYFOR STRESS
whilethe continentsbetterindicatethe durationof stresssupport. For example,from seismicrefraction,surfacegravity,
and bathymetrystudies,the spatialextentand magnitudeof
lithosphericstrain causedby the Hawaiian load is rather well
6395
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Dorman,L. M., andB. T. R. Lewis,Experimental
isostasy,
1, Theory
documented. There exists nowhere on the continents such a
of determinationof the earth'sisostaticresponseto a concentrated
load,J. Geophys.
Res.,75,3357-3365,1970.
classicexample of lithosphericflexure. The continual overGaposchkin,
E.
M.,
and
K. Lambeck,Earth'sgravityfieldto sixteenth
printing of continentalorogenic,volcanic,and epeirogenic
degreeand stationcoordinates
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ruleout relaxationthroughoutthethickness
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tion on the continents
on billion-yeartime scales.While it is
dangerousto apply oceanicstressestimatesto the continents
or assumethat continentaltime scalesare appropriateto the
Gunta:R., A quantitativeevaluationof the influenceof the lithosphereon the anomaliesof gravity,J. FranklinInst., 236, 373-396,
1943.
oceans,
it appears
thata mechanical
lithosphere,
30-km
orso Hanks,
T.C.,TheKuriltrench'-Hokkaido
risesystem:
Large
shallow
thick, sustainsstresses
to about 200 MPa over billion-year
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Geophys.
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timescales.
The modelresultsarenot unique,in partowingto
and
the ambiguityof gravitydataand in part to the uncertaintyin Hanks,T. C., Earthquakestressdrops,ambienttectonicstresses,
rheology, but it is unlikely that refinement in observations
coulddrasticallychangetheseestimates.
In particular,thereis
no conflictbetweenempiricalflow laws extrapolatedfrom
laboratorydata and observedgeophysicaldeformationdata.
Acknowledgments.
I Wish
tothank
T. C.Hanks,
R.L.Parker,
A.
McGarr,J. COckran,andan anonymous
reviewerfor suggesting
several improvements
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