Seamount Loading and Stress in the Ocean Lithosphere

JOURNAL OF GEOPHYSICAL RESEARCH,VOL. 85,NO. BI I, PAGES 6403-6418,NOVEMBER 1980

SeamountLoading and Stressin the OceanLithosphere

KuRr Leunecr eNo S. M. N¡Kl¡oclu'

Research School oJ Earth Sciences,Auslralian National University, Canberra 2600 Australia

One of the principal arguments for large stress differences in the lithosphere comes from the modeling of the deflection of the ocean lithosphere under seamount loads. Most published models indicate maximum stress differençes of 2-3 kba¡ per kilometer of deflection, and maximum values approach l0 kbar. Geophysical support for the elastic plate theory comes mainly from gravity, which requires a broad negative anomaly on the flanks of theactual seamount, although these anomalies are not very sensitive to the density structure ofthe crust or to the density ofthe sediment fiIl-in or the extent ofthis fill-in. The stress in the plate, however, is sensitive to these parameters, and in consequence, gravity is not a reliable in- dicator of the stress stâte. The lithospheric flexure model has been examined in some detail to evaluate this stress dependence on crustal densities and rheology. The approximations inherent in the linear theory have also been investigated. The main conclusions are that the stress differences can be signiflcantly reãuced (l) by adopting a lower density for the sediment fill-i¡ than the usual 2.8 g cm-3, (2) by introducing a depth-dependent nonelastic rheology, and (3) by introducing large-deflection theory for the larger loads. For large loads, súch as Oahu Island discussed by Watts (1978), the maximum stress under the load can be reduced to about 2 kbar or less near the upper surface of the plate, and the maximum stressdiffe¡ences oû - ozzneed not exceed I kbar. The introduction of the nonelastic rheology results in a stress,temperature, and load duration dependence ofthe'apparent'flexu¡al rigidity; large loads, loads of long duration, or loads on a high-temperature lithosphere all result in a thinner apparent plate thickness than small loads, loads of short duration, or loads on a cool lithosphere, all.other factors being equal.

INTRoDUCTIoN Evidence for a high finite strength of the lithosphere in a o The Hawaiian Archipelago and other seamoutt chaits geologicalsetting has come mainly from flexure calculations il,h¡vc,since the work by VeningMeinesz [1941], usually been of seamountloading and of the bending of the oceanplate at the ií.lntcrprete¿as being regionally rather than locally com- the trenches,but we will be concernedhere only with flexure model is usually tested f,pcnsatedfeatures, and thei¡ existencehas often been invoked former. The validity of the gravity Vening Meinesz, l94l; ¡,.l.rsevidence that the oceanlithosphere is capableof supporting agai¡st observationsof le.g., iìdeviatoricstresses that reach severalkilobars le.g., Walcott, Waus, 19781orgeoid height [e.g.,Cazenave et ø1.,1979, 1980¡' il1976l.The regional compensationis usually modeled as the Waus, 19791,but both observationsare poor indicators of de= it,elnsticsupport of the load by an elasticplate overlyi-nga fluid formation and evenpoorer i¡dicators of the stressstate of the f. mcdium[Gunn, 1943].These models typically indicate stress lithosphere. For example, the gravity anomaly over a sea- l:diflerencesof about 2 kbar per kilometerof deflection(e.g., mourt can be equally well explainedby a regional as by a lo- iì.llalcott [976]; seebelow), while the total deflectionmay at- cal compensationmodel ir which the depth of compensation ;1'Ininseveral kilometers. Watts et al. ll975l suggesta maximum exceedsthat of the regional model [e.g., Vening Meinesz, ;ii'dcflectionof about 2.5 km below the Great Meteor seamount, l94l], but the maximum stressdi-fferences in the two models i::whileWalcott U97Ol suggests deflections exceedi¡g 7 km be- will differ by as much as an order of magnitude. Only from i low the Hawaiian Archipelago.Hence stressdifferences of up other geophysicalobservations can a clear distinction be made I to l0 kbar are not uncommonin thesemodels and will occur betweenthese two models.There is usually little bathymetric at both the lower and upper surfacesof the elasticplate, which evidencefor the deflection,and this is attributed to any such lypicallyhave a thicknessof 20-30 km. If theseestimates, par- depressionbeing at least Partly filled in by volcanic material ticularly those at the lower boundary, where the plate is i-n and sediments.One exception to this is the flexure of the lension,are correct,they haveimportant implicationson both oceanlithosphere under the Hawaiian chain, where the posi- , the brittle and flow behaviorof the lithosphere.Laboratory tion of the outer arch is usedas a measureof the flexural pa- measurements,employing large strai-nrates and short loading rameter / deûnedbelow fWalcott, 19701.Another exceptionis . cycles,indicate that brittle fractureor signiflcantflow will not the observationof uplift of islands on or near the outer arch ' occur in the lithosphereuntil stressdi-fferences reach several and Menard, 1978].In both thesecases the deduced ' [McNuu kilobars,but the dìfficultiesof extrapolatiogfrom laboratory quantity is /, but the quantity of interestis the flexural rigidity ' to geologicalconditions are geûerally well understood [e.g., D, which is proportional to /4, and neither measurementswill Paterson,1976, lgigl. In particular, any deformation mecha- give a partiCuhily precisedetermination of D unlessthe den- nlsmoperative in the laboratory at a given temperaturewill sity structureof the load and crust is well defined.Seismic ob- lr probably become impofant at much lower temperaturesin servationsof the i¡creasing depth of the Moho le.g-,Furumoto ' gcologicalsituations: and anv additional mechanisms.not evi- et al.,19681or ofthe increasingthickness ofthe oceanlayer 2 dcnt under the laboratory ctnditions but which becomesig- fWootlard, 19701under the Hawaiian load have also been nificant in geologicalsituations, will only reducethe strength used as evidencefor regional compensation,although Suyen- , ot the material. aga ll979l has interpreted theseresults as indicating a com- t:.ç- ' is partly local and partly regional. On leave pensatiot model that : from University of Queensland, Department of Survey- its tng, St. Lucia 4067. Australia. In this paper we explore further the plate theory and geologicalapplication rather than add to an expandinglitera- Copyright @ 1980 by the American Geophysical Union. . ture on flexural rigidity estimates'The elastic plate theory " Paper g0B0549. 6403 . number o¡ 48-0227/80/o8oB_0549$o r.oo fr fi:. 6404 LAMBECK AND NAKIBOGLU: STRESSES TN THE OCEAN LTTHOSPHERE goesback at least as far as the work of Hertz in 1884lHertz, the point of loading, the deformation w(r) can be readily ex- 18951,and early applicationswere by VeningMeinesz |9411 pressedin terms of zero-orderKelvi¡-Bessel functions as and by Gunn ll943l. The theory is well developedin the engi- neering mechanicsliterature le.g., Reisner, 1944,1946; Fred- -#"^lî) erick, 1956; Timoshenkoand Woinowsky-Krieger, 1959; Fung, ,(i): Qa) 19651and in the recentgeophysical literature by Brotchieand Silvester[969] and Brotchie[971]. Most geophysicalappli- where /o : D/p"g, / is the'radius of relative sti,ffness,'or the 'flexural cationshave usedthe thi¡ plate, small-deflectiontheory, even parameter.'The solution (2a) is equivalent to that though the computed deflectionsreach a signi,ûcantfraction given by Hertz ll895l. At the origin r : 0, of the elastic plate thickness.In the subsequentsections we (2) will addressourselves to (l) the thin plate theory, the w(A): -ql2/8D (2b) large-deflectiontheory, (3) the geophysicalapplications of this (4) the stress theory, the consequencesof seamountloadilg on The solution (2a) behavesas a damped sinusoidwith the sign in light statein the oceancrust, and (5) somepublished results of w changingabout every 4/ but with a rapid decreasei¡ the of the above analysis. maximum amplitudesbetween successive zeros. With the typical parametersgiven in Table I the maximum negative w THIN PLATETHeony value, occurring at about 5/, is only abottt l.5Voof w(0), and The basicequation expressingthe deformationw(x, y) of an the next maximum, occurring at about 9.5/, is only O.4Voof elasticplate of flexural rigidity D, loadedat its upper surfacez u(0). Thus for a l-km downward deflectionat z : 0 the maxi- : -H/2 by q(x, y) is mum upward deflectionwill be only of the order of 15 m and will occur at about 5/ from the point of loading. In evaluating DYYw + p"gw: q (l) the Kelvi¡ functions for point or axisymmetricloads (seebe- where low) somecare needs to be taket in evaluatingthem when the argumentsare large, and we have found it convenientto use ü (t- when r,// exceedsabout 7. V: -* the asymptoticapproximations dx- dY Solutionsfor (l) take a particularly simple form for simple (see, Brotchieand Silvester The vertical coordi¡ate z is measureddownward from the axisymmetricalloads for example, density p, height å, and middle plane of the undeformedinûnite plate whosesurfaces [1969]).For a uniform disc load of < is are given by z : +H/2. Thus positive w indicatesa depres- radiusl the deflectionat r I sion, and negativew an upfft of the plate. Parameterp" is the the underlying fluidlike and g is gravity. density of medium, ,: * ., t* + c,B"t (3a) This equation expressesthe force balancebetween the surface nú[, (i (;)] load q, the elasticresponse ofthe plate, and the reaction p¿w of the substratumor the base of the plate, assumi¡g that a andatr>lis static situation has been attained. Apart from assumptionsof homogeneouselastic properties and an initial hydrostatic equilibrium state,equation (l) assumesthat (l) the efects on bending of shearingstresses and normal pressureson planes parallel to the x-y plane have been ignoredand (2) deflections are small in comparisonwith the plate thickness.These as- will investigated sumptions be below. The constantsÇ and C,.'are evaluatedfrom the appropriate Equation (l) is solvedfor the speci.ûedload q(x, y) an.dap- boundary conditions l-5. With conditions l-4 and writing g: propriate boundary conditions. For a load of ûnite dimen- r/1, a : A/1, sions, speci.ûedby a boundary r : A(x, y), frequently used boundary conditionsare the followiag: condition l, w and dw/ Ct: aKer' a dr are finite at r: 0; condition 2, w a¡d dw/dr arc continuous at r : Ai condition 3, momentsand shearsare continuousat r Cz: -a Kel' a : A; and condition 4, w and r-t /dw/dr vanishat infi:rity. This cr':0 last condition follows from the requirementthat the moments (3c) vanish at infinity. cr' :0 In somegeophysical problems it may be desi¡ableto restrict Ct' : aBet' a the loadhg deformationto occur only over a û¡ite radius r : A, ) Ar. This could be the casewhere the seamount,a result C¿' : -a Be7' a of an endogenicthermal process,sits on a part of the plate that may be of higher averagetemperature and is therefore The algebrafor the constantsC' and Ci subjectto condi- weaker than the plate further away. Rather than treat the tions l-3 and condition 5 is more tedious,but the effectof the complicatedproblem of loading of an inhomogeneousplate, change in boundary conditions caû be readily seenby com- an order of magnitudesolution may be obtainedby modifying paring the appropriatepoint load solutions.For the load q at r the boundary conditions. The boundary conditions would :0andforr<,B. then be conditions l-3 and w : dw/dr: 0 at r : I (condi- nI2 tion 5). w:- Cl Ber{+ GBeiÐ (4a) For a point load and a responsethat is symmetricalabout #D(Kei{+ LAMBECK AND NAKTBoGLU: STRESSEStN THE OCEAN LrrHospHERE 6405

TABLE 1. Nominal LithosphericParameters Adopted in the The surfaceloads on the right-hand side of these equations Deflection and StressCalculations contain the constantwater load pogft*which, sinceit resultsin Value, a constantdeflection, can be subtractedout. Then within the Parameter cgs units load, where h* : h - w, h being the observedheight of the load relative to the deformed seafloor, RigiditY¡r 3.67x 10" Laméconstant À 9.45x l0rt Young'smodulus E 1.0x l0t2 DVYw * (p"- pògw: (p - pòSh (6a) poisson'sratio z : X/z(X + p.) 0.36 Thicknessof lithospherefl 1.76x 106 and outsidethe load, Flexural rigidity D : (p/3) {t(À + ùH3l/(^ + 4')} 0.5x l03o Radiusof relativestiffness / 5.37x 106 DYYw + (p, - pùgw:O (6å) Densityof substratump, 3.3 EffectivedensitY ofload p 1.8 In theseformulations the deflectionof the plate is resistedby Densityof sedimentfrIl-in 2.8 the buoyancy force p"gw, and this implies that when w be- comesnegative, this force exertsa tension,as is appropriatein with this problem. In the seamount-loadingproblems there is usually little U: U/, bathymetric evidence for large depressionsof the seafloor C, : (Bei b Kei' b - Be7'b Kei b)/a about the foot of the load, and either the flexural rigidity is (4b) high or it is assumedthat the deflectedlithosphere is ûlled -Kei' Cr: (Kei bBe{ b bBer b)/a with sedimentsor with the volcanic rock that also makes up p¡ and is completeup to o : (Ber bBel' b - Ber'å Bei å) the load. If this ûIl-in has a density the originally undeformed surfacez : 0, (l) becomes,within At r: 0, the loaded aîea r =,4 and subtractingout the constant load Pogh*, ,ror: -ir,) (4c) s{r DWw + (p" - ÐCtz',:(p - pògh* (7a) in comparisonwith (2å). For å : 2, for example(B :21), the Outsidethe load the solution consistsof two parts, one where reduction in the maximum deflectionis about 40Vo,ar^d fot b w is positive, = 4 this reduction is about l07o.In the followi-ngdiscussion we DYYw+(p,- pìsw:O (7b) will restrictourselves to the boundary conditions l-4, keeping in mind that they will result in upper limits to the deflection. and one where w is negative, an axisymmetricparabolic load of radiusA at z: -H/ For DVYv +(p"- pùgw:0 (7c) 2, the solution with the boundary conditions l-4 for r < A ß In this casethe flexural parameter wi-ll have three di-fferent - ro; ,: - * c, Bers+c,Bei€] (5¿) definitions:where r < A, I : ID/ (p" p)gl, wherer > A and *[(t ä w is positive, I : ÍD/(p" - pìgl'/o; and where r > A a¡d w is : - po)g)'/. The solution in eachdomai¡ is andforr>Ais negative,I lD/(p" ofthe form (3a) or (3å), but now the constantsare most readily evaluatednumerically, subject to the appropriate bound- -: Kerg + c4Kei 6l (så) *[c, ary conditions l-5 and subjectto the continuity conditions at x : a and at the point where the deflectionchanges sign. The with simplest solution follows when p : pr, an assumption of mathematical convenience rather than of physical reality, : geometry c, z(rer'- I*"r' ') si¡ce it obviatesthe need to k¡ow the complete of the load; only the load h* abovethe meanseafloor is required. c,:z(ne,'- Thenatr<1, I""t'') DYVw + (p, - ùsw : (p - pògh* (8¿) Analytical solutions for more complex axisymmetricloads and at r > A and positive w, can be obtained by approximatiagthe load by a number of discsand integrating analytically,but for most configurations DYYw+(p"- p)gw:0 (8ó) In the a numerical approach will be found to be simpler. and thesolution is asfor (1) exceptthat the flexuralparameter p loading the above solutions, is the density of the material is now deflned by plate, while p,.is the density of the substratum.If the load is below water of density p6 and of height å* above the unde- f : D/(p" - ùe (e) formed surface,(l) becomes,within the load, For negativew the equation is that given by (7c).The assump- up to the originally unde- DWp + p"gw: p(h* + w)g tion that the ûll-in has occurred formed surfaceis strictly an ad hoc one, for if large deposition and outside the load, both where deflection is positive and rates-haveoccurred, the frll-in would not have stoppedat the negative, level correspondingto this surfacebut would have gone up to the level correspondingto the maximum uplift of the plate DYYw + (p" - pòBw: p"gh* near x = 5/ before spilling over onto the surroundi¡g seafloor. LAMBECK AND NAKIBOGLU: STRESSESIN THE OCEAN LITHoSPHERE

Crustol rpdel

1 0 O* *\

E E J js J p.z,g g..-3 lo)", Z I È x j8 (l ;;;F*.' l''-' t\ E En & 3 ó 0 ? g i9 .v .82 E ëo b ô ¡ I

! x Ã J ar c{ -lb b- at ^ r .eL I b' o

= .õ gI' 3 5 .E &

ECL P o o

Fig. l. (Top) Deflections,(middle) stress,atd (bottom) gravity perturbationsfor disc and equivalentparabolic loads of density equal to the fill-in density(left-hand side).The right-hand side is for a disc load without sedimentâll-in, and the moat is filled with water. Both loads arecovered with water. The gravity perturbationshave beencomputed for the disc loadsonly, and the contributionsto gravity from the topographyand deflectionof the Moho are shownas well asthe total perturbations.

If the additional load is of uniform thicknessAå in relation to the solution of the equations becomesparticularly simple' the undeformedsurface, the equationsare thosegiven by (7ø), sincewithin the load, (8a) is valid and (8å) is valid eve.rywhere (7b), and (7c), with the further terms Aåpg, Lhp¿, and Ahp"g, beyond the loadedarea. In the subsequentdiscussion we con' respectively,added to the right-hand sideand-where l¡* is now sider (8) to be valid everywhereso that effectivelywe assume measuredwith respectto z: -Ah.If we takep.: p¡: p, then fill-in to have occuned up to the point of maximum uplift. If LAMBECK.ÀND NAKIBoGLU: STRESSEStN THE OCEÄN LrrHospHERE 6407

complete, as appearsto be the case for the Hawaiian seamounts, or the ûll-in is of a density di-fferentfrom the load, iterative solutionswill generallybe required. Solutionsfor ar- bitrary levelsof fill-in are discussedby Lambeck [980a]. Figure I illustratesthe deflectionsfor both disc and para- J bolic loadsusing the parametersgiven irr Table l. The param- c etersI and å for the latter have been selectedsuch that the two loadshave the samesurface area and both exert the same o mean pressureon the plate. Both loads are underwater, and two casesare considered,where the depressionshave and where they have not been filled with materials of the same density as the load itself. The resultsof the disc and equivalent parabolicloads are quite comparable,and for most of the Fig.2. Deflections ofthe elastic plate offlexural rigidity D due to presentillustrative examplesit will sufficeto cónsideronly the a disc load of variable radius I and height å : 5 km. The depressed disc load. In the fill-i-n models the maximum deformation is is assumed to be filled with sediments surface of density equal to that increasedby a factor of about 3 in relation to the water ûll-in of the load. model. The maximum amplitude and the v¡idth of the deflection are also much increasedin the ûrst model. In both casesthe this is not the case,then the deflection and stresseswill be maximum upliflt valuesfor w remain small,less than 100m for marginally overestimated.As far as matchiag the computed a maximum deflectionof 5 km, and this conflrms the validity gravity and geoid anomalieswith observations,this approxi- of neglectingthe small complicationarising in the solution for mation is of no consequence.If the ûll-in is only partially negativew. Dependingon the degreeof fill-in and on the den-

Distance(knr) Distance(km)

Distance(km)

ts Þ'

Fig. 3. (l-eft) Deflections,stress, and g-ravityanomalies for the discload on a plate ofvariable flexural rigidity (curve l, D:5x lGEdyncm;curve2,D:5 x ldedyncm;curve3,D:5x l03odyn"å¡.eUotherparameterr"rã"rinTablel, and the load geometryis asin Figure l.-(RighÐ Deflections,stress, and gravity for the samediìc load with variable density - Ãp = p^ p (cur1e l, Âp : $.2 g cm-3; curve 2, Ap : 0.Sg cm-3; curve 3, Áp : 0.e g cm-3). 6408 LAMBECK AND NAKIBOGLU: STRESSESIN THE OCEAN LITHoSPHERE sity of the flll-in the actual solutionswill generallylie between thesetwo limits. !s As the radius of the load increases,the maximum deflec- !¡ tions occurri¡g under the centerofthe load approachthe iso- jo static limits of h(p - pù/b" - p) for the sedimentfill-in case, (kn) although owi-ngto the oscillatory behavior of solution (3) for o w(r) the maximum deflection attained is somewhat greater than this limit (Figure 2). The use of elasticplate models for loads of large areal extent will thereforegive a responsethat 6 di-ffersfrom a local isostaticresponse in only the edgeeffects. ¡È2 Apart from the densityof the load and of the fill-in, key pa- ! rameters defining the magnitude and wavelengthof the de- o o6 flection are the flexural rigidity D and the densitycontrast p - .n4 ¿. For a uniform plate, D is defined as

D:+#h*:d_øo (l0a) where p, À are the Lamé parameters,fI is the thick¡ess of the plate, E is Young's modulus, and z is Poisson'sratio. Table I a summarizessome typical parametersfor the ocean litho- o sphere.If ¡r and À (or ,E and r) are variablewith depth, ã= P!x.+-tt) I o: q [''' ,, o, (10å) tl _H/2 ^+ ¿lr The usual and convenientprocedure is to assumethat D is uniform, resulting in an equivalent flexural rigidity or an equivalent elasticthickness fI. The D andH can subsequently be interpretedi¡ terms of more realisticplate properties.Fig- ure 3 illustratesthe deflectionfor di.fferentD. In the limit D'+ 0, / + 0 and w approachesthe isostariclimit å(p - po)/(p,- p). As D decreases,the wavelengthofthe deflectiondecreases, and the magnitude o[ the negative deflectionsincreases. The density contrastp, - p affectsboth the magnitudeof the deflection and the wavelength; the larger the contrast, the smaller / and the smaller the magnitudeof the deflection(Fig- ure 3). STRESSSrer¡ or rHE PLATE The stressesin the x-y plane for axisymmetricloads follow from [e.g., Timoshenkoand Yloinowsky-Krieger,1959] lzD lûw v dw\ o,,: 'læ * (l la) H3p ¿æl l2D lûw ldw\ ooo: t * (l lå) Ipp \' o{ I æl gr- 422) . (rrc) ".":# ftl# i #) Ã O,e:0

The stress components o,. and oa¿reach maxima at z : +H/2. The vertical stress is le.g., Frederick, 19561

, Ll3]| o": o';n'+ (q'- " -lHl^l"l I (l ld) t')lTH I l.I Fig. a. (a) Maximum stresseso,/ and. plate where : -pg(h* + os6occurring in a 4, w) is the load on the upper surfaceof the loaded by discsof differentradii R and height ,¡. All parametersare -pgw a deformedplate and Qz: is the load on the lower sur- as in Table l. (å) Maxinum stressesd,, o6'efor platesof di-fferentD. face of the plate. The maximum value for o,, is reachedat the upper surface z : -H/2. The remai¡i¡g stress-tensorele- of which never exceed I kbar for the examples considered ments o.,, or, are small everywhere,and o,- o6s,o>z caî be con- here. sideredprincipal stressesnear the surfaceofthe plate. On the With the deflection solutions (equations (3a), (3b), (5a), middle plane, o,, and o," are the only nonzeroelements, both (5á), and (llø)-(llc), analytical expressions for the stresses LAMBECK AND NAKIBoGLU: STRESSESIN THE OCEAN LITHoSPHERE

/" - 2 4 ,1030 ' to3o I ,,2tt ,\rz.l \--..-.-.,/' /,, l - l 28 3.7, lO I .t II ,/ 3'ro2e Ø !lt oo4 ,lil itll ltl

D = 3.7' 1028

Fig.4. (continued)

arereadily obtained with the aid of the variousproperties and order of 4/. The maximum deviatoric stressesnow occur to- recursiverelations of the Kelvin functions le.g., Abramowitz ward the edge of the load. Seamountsare generally much andStegun,19651. Figure I illustratesresults for the disc and smaller than this, and the maximum stressdi-fferences occur- parabolicloads discussedpreviously for the caseof ûll-in and ring in the plate are given by (12), assumingthàt other aspects no frtl-in. For the parametersof Table I and for small l, both of the model are valid. As D decreases,the maximum stress o,, ar'do0o reach maxima below the centerof the load that sig- di-fferencesalso decrease,and the maximum stressesoccur nifrcantlyexceed ozz, aîd. the maximum stressdi-fference in the near the edge of the load. Figure 4 givessôme examples. plate is essentially given by o,,(x : 0, z : !H) or o¿¿(x: 0, = tIÐ at x : 0. with (3) and (ll) at x:0, z RIGOROUSPLATE THEORY The thin plate, small-deflectiontheory used so far is appli- 3D p - po o..lma*: toot^ *: h*0 + v,t{ret, 4 cable when (l) the deflectionsand the slope of the deflected tttf U_ p t t surfaces are small, (2) there is mi¡imal stretching of the l1\2 ,4 A middle plane durilg deformation so that the plate deforms - 3sl (, - po)l¡*(l* ,) Kei' ,l î 7 02) into a developablesurface, and (3) the in-plane forces acting t", on the middle plane are negligible. The deformationsin the For the parametersgiven in Table l, a disc of 60-km radius, above examples are sufficiently large in comparison to the and for the caseof sedimentflll-i¡ a ûominal estimateof the plate thicknessthat the effect of some of these assumptions maximum stressis needsfurther investigation, while the eflect of any regional stresson the plate, not associatedwith the loading, may also o^^*=2.6h* kbar (l3a) have an important bearing on the deformation. plate in both horizontal and vertical wherethe height of the load h* is in kilometers.For the water The equilibrium of the Fung, frll-in case. directions is given by Yon Karman's equations le.g., 19651.For axisymmetric loading and boundary conditions o*.* = 1'3å* kbar (r3å)theseequations reduce to For the parabolic load with sedimentfill-in, ldó ûw I t\' DVYw*(p"-pìgw:S*;; -* L++ .4a\ orr¡max: od'*.* : 6Sl - ps)h*(l + v) ar rdrdr Hl b \--, I vvq: -; 4*', (r4b) [".'14ì+ Zre¡ l+ì - 3-l 0z') .ar ar L 1t¡ a [r, a-) andfor A : 60 km andother parameters as in Table I, whereó is the stressfunction which is relatedto the mean horizontal stressresultants N, and N, by o-^* = l'9å* kbar ' (l3c) For the sameparabolic load and the water ûll-i¡ case, Lo!:N,: l''' o,,d, o-.* = l'0å* kbar (t3d) r ar tl -øtz

As the radiusI increases,o),,-* and orr,** under the centerof o66dz the load reach the isostaticlimit of þ - pòghwhen I is of the ffi:*,: Inu',', LAMBECK Á,ND NAKIBoGLU: STRESSESIN THE ocEAN LITHoSPHERE

È 5 J

I a ì

o -o J

¡ a E ; Þ

5 xfo28 sxto2e sx to3o 3 4 5 6 D (dyne-cm.) tt (t-)

Fig. 5. . . Comparisonof (top) maximum deflectionsand (bottom) stresso¡, basedon the linear theory (dashedtine) and the completetheory (solid lines).The resultson the left-hand sideare for variable D, and the resultsonihà right-hand side are for a variable height of the disc load. All other parametersare as in Table L

If the horizontal displacementof a point on the middle plane lend themselvesto an analytic and closedform solution, while is denoted by u, then from Hooke's law, numerical methodshave seriousdi-fficulties for the caseof an infi¡ite plate (seethe appendix). r Iûa v dó\ We have followed a perturbation schemefor solving the ":El*-;*l equationsby noting that both the right-hand side of(15ó) and Introducilg the dimensionlessquantities the secondterm on the right of (l5a) are small, evenif the deflections are of the same order as the thicknessof the plate. Hencethe solution of (15ø)can be obtained by perturbing the s:;"wö-r 4:; €:- tt)l correspondingsmall-deflection problem (see the appendix). The nature perturbation (l4a) and (14å)reduce to of the schemeis to overestimatethe correctionsto the linear theory by not more lhan 207ofor the td results given here. Figure 5 summarizessome results for the Wf+f:a+ (ll') (l5a) EdË comparisonof the rigorous and linear theoriesapplied to the disc load with fill-in and with the plate parametersgiven in ((') (rsb)Table L The large-deflectiontheory doesnot ¡esult in a signif- icant changein the maximum deflections,but the linear theory can lead to an overestimationof the stressesor,1max, oe¡lmxt wherea :0 when { > a; otherwise,c i= p: -12(l - SP/D, by several kilobars. The slope of the deflected surface is û) (l/IÐ2. The boundary condition for a clampededge at r: smaller and the compensationis more regionalthan predicted .B is z : 0 at this distance,while for an infinite plate with a by the linear free edge, theory. As D decreases,that is, as the plate gets weaker, the membrane stressesbegin to dominate, and the small-deflectiontheory underestinatesthe stresses(Figure 5). ß^+: nm4:o For most of the seamountproblems'the *_ af *_ dr tendencywill be to overestimatethe stressbelow the centerof the load and to un- The boundary value problem is of the jury type'with a strong derestimateit somewhatnear the edgesof the load. When the singularity at the origin. Equations (15ø) and (15å) do not magnitudeof the load is increased,the discrepancyincreases LAMBECK AND NAKIBoGLU: STRESSES IN THE OCEAN LITHoSPHERE 64ll

(seeresults on the left-handside ofFigure 5); and for a 6-km The elliptic integralscan be expandedinto seriesfor 0 < m < ioad, as is approachedby the larger Paciûc volcanic islands, | [e.g.,Abramowitz and Stegun,1965, p. 591],and the gravity the linear theory would ¡esult in stressestimates that are i¡ er- perturbation is integrated numerically for di-fferentshapes of ror by nearly 50Vo.For the nominal parametersin Table I and the boundary z : z(r). The geoid heights,for the plane ap- the load parametersin Figure I the linear stressestimates are proximation, follow from too high by about 20Vo.OrJy fo¡ loads of height less than 3 km is the simple theory adequate. õgrdr d0 about .:*l:"1,::"" glt/2 rth From theseexamples it appearsthat the linear theory will [ro'+ f- 2rtr cos yield better estimatesfor the deflectionthan for the uenerally or, becauseôg is function of r onlY, Itress,and in problems where the deflectionis deducedfrom gravity,the resulting linear theory stressesmay be over- 2 f* K(n/z,n) ôt tr:- ¡ _=-1-rir (r7a) estimatedby as much as 50Vofor the extremeloading condi- T8 l-o r+ro tions.If the estimatesof flexural rigidity ani crustal stressare :4rro/t gravity basedon the computationof the distanceto the peripheral where n + r)'?. Strictly, the anomaly Ag, de- bulge,the liaear theory will lead to an overestimationof both fi¡ed as deflectionand stress,since this distanceis a function of the of the deflectedplate. as:ôs -T * slope .a( GRnvITy AND GEoID ANoMALIES should be i¡troduced into the above integral instead of the The principal geophysicalevidence for the regional com- gravity perturbation ôg. Either an iterative procedurecan be p€nsationof seamountshas been gravþ, and estimatesfor adopted to solve for N from ô9, or the geoid height term is flexural rigidity of the plate have beendeduced mainly by the simply ignored, since its contribution is small in the present matchilg the model with gravity observationsin either the problems:il does not exceedabout l0 m and Ag - ôg = l spatial domain Watts and Cochran, 19741or the fre- [e.9., mGal. Since Iim-,"K --+ oo, the inner zone efects of the quencydomai¡ llatts, 1978].More recently,it hasbeen [e.9., above integralsare evaluatedby a limiting process possibleto compare these models with observationsof the geometricform of the geoid Cazenaveet al.,1979, 1980; [e.g., r,,,.\ eôg(ro), 2 frcz'K6grdr lírans, 1979;Lambeck, l980al. ¡vlfnl: T-, (r7b) I rBlo r+ro The gravity and geoid anomaliesover seamountsare a con- sequenceofseveral lateral densityanomaly sources associated wheree is the sizeof the inner zoÍte alea.Equations (17 a) ar.d - with (1) the load itself, ofeffective density p po (in realistic (l7b) arc integratednumerically using trapezoidaland Rom- modelsthe density structure will be considerablymore com- berg rules. plex), (2) the fill-in, of densityp¡ in the sedimentfill-in caseor Figure I illustratesthe gravity anomaliesover the nominal of density po il the water fill-in case,(3) the deformation of disc loads.In the sedimentûll-i¡ model with p¡: p, the grav- surfacesof equal density within the plate due to the bending, ity perturbation is the sum of the short-wavelengthpositive particularly the interfacebetween ocean crustal layers2 and 3 Bouguercorrection and the broad-wavelengthnegative anom- and layer 3 and the maûtle, (4) compressionof material under aly associatedwith the density contrastswithin the crust and the load, and (5) an eventual density contrast at the base of with the crust-mantle interface. The resultant proflles are the elasticplate, although the baseof the plate will usually be quite typical of gravity anomaliesseen over many seamounts consideredas an isotherm so that there will be no density dis- and provide the main observationalevidence in supportof the continuity here. The density of the flll-in is usually very simi- elasticplate model. The water fi.ll-i¡ casegives a similar grav- lar to that of the upper oceaû crust, si¡ce seismicrefraction ity anomaly exceptthat the negativeanomaly is due to (l) the work has not found a discontinuity there. AIso compression density contrastin the.crustand at the crust-mantleinterface, effectscan probably be ignored, so the mai¡ contributions to effectsthat no\r are lessthan before,since the defolmation of the gravity or geoid anomaliescome from sourcesI and 3. the plate is reducedin both magnitude and wavelength,and The axisymmetricloads used above lend themselvesto rela- (2) the nearer-the-surfacecrust-water interface. The total tively simple analytical expressionsfor the perturbations il anomaly is not very different from the fill-in case. gravity and in geoid height. Consider a homogeneousmass Figure 3 (left) iJlustratessome examples of the gravity boundedby a surfacez: z(r) andthe planez: 0. The gravity anomaly for the samedisc load and crustalmodel as i-nFigure perturbation at Pe(ro,0 : 0, zù due to a masselement of this l, and it should be immediately noted that gravity varies load at P(r, 0, z')is nearly 100times slower than D and about 4 times slower than stress.An increasein D by 2 ordersof magnitudeincreases the lø lt2¡ lzt¡t (zo-z)dzrdrd0 og:-rtp ttt amplitude ofthe total gravity perturbation ôg by about 1007a, I I I - (zs - tl :O I 0-O ! z-o lP + ro2 2rro cos 0 + z)213/2 i¡creasesthe width of the negativeanomaly by about 200 km, Integrating, first over z and then over á, and decreasesthe magnitude of the negativeanomalies at the flanks by less than 40 mGal. The best discriminant betweet f- Ktu/2.m) lnn different values of D would be the width of the negative ôe:4Gp o, (16) l_"tai;J;; a:;frñ\=", anomaly,since the magnitude of the anomaliesis also a function of uncertaintiesin the density structr¡reof load and crust. whereK(n/2, nr) is a completeelliptic integral of the ñrst kind The effect on gravity of a change in density contrast Ap be: with modulus tween the load and mantle is lessmarked (Figure 3 (right))' A decreasein Ap results in a signiûcant increasein the deflec- ^:offi:;¡ tion, but the overall change i-n gravity due to two competing 6412 LAMBECK AND NAKIBOGLU: STRESSES IN THE OCEAN LITHOSPHERE

À (km) r (km) Fig. 6. Powerspectra of the gravity anomaliescorresponding to the modelsin Figure 3 for (a) variable D and (å) variable A,p.

effects-the increasein the deflection and a decreasei¡ den- tional and geologicalnoise of the data is also considered. sity contrast-is slight. This result, plus the insensitivity of However, as is also indicated in Figure 4, the stressesdi-ffer gravity to D, means that errors in the density estimatescan quite signi-ûcantlyfrom one model to the next, conûrmiagthat contribute significantly to both stressand gravity anomalies gravity is an unsatisfactoryindicator not only of deformation and that these will make accuratedeterminations of D very (i.e., of w) but also of the stressstate of the lithosphere. difficult. The characteristicsof the gravity anomaliessuggest that a comparisonin the frequencydomain may be more helpful in PHYSICALPRoPERTIES oF THE Ocr,nN LIruospHenr, discriminating between the various models. Insofar as the Insofar as the previous results have shown that gravity bathymetry is an invariant in the presentmodels, we consider anomaliesare not very sensitiveto severalphysical parame- only the Fourier spectrumof the gravity anomalies(Figure 6). ters but that stressis, a more detailed evaluation of the con- The spectrafor the disc loadsare not very sensitiveto changes sequencesof chosencrustal models is appropriate. in either D or Ap except at the long-wavelengthpart of the spectrum. However, in real situations this part of the spectrum Rheology will also be contaminated by contributions from other long- The flexural rigidity deûned by (l0a) is a function of the wavelength gravity anomalies not associatedwith the sea- elasticparameters À and p ot E arLdy, which are normally de- mount. duced from seismicvelocity observations.The deformation Theseresults, explored further below, indicatethat the dis- e(t) of the lithosphereat time r can be consideredas being due criminatory power of gravity anomalies for estimating the flexural rigidity is rather limited, in particular if the observa- E (dvne-.niz) ¡1-vz¡ rott rd'? pjtlo \ 0.5 E

2,/

Fig: 7. Schematicvariation with depth of the normalized 'apparent'shear modulus¡rt (equation(19)). Ifstress is increasedthe lowe¡ Fig. 8. (Top) Variation of E/Q - l¡ witn depth according to part ofcurve 2 movesto the left (curve l) and intersectst-e elasticre- (23a) aú, (bottom) variation of the stress factor oo with depth. Two sponsecurve nearer the surfacæ,resulting in a stress-dèpendentflex- models are shown: the uniform plate (curves l) and the plate in which ural rigidity and elasticplate thickness. the elastic parameters follow a parabolic depth dependence (curves 2). LAMBECK AND NAKIBOGLU: STRESSES IN THE OCEAN LITHOSPHERE 6413

2.O 2.O 2.O 2.2 2.2

5.O7 5.O4 4.61 4.1 4.4 Loyer 2

6.69 6.73 6.12 ó.4 ó.8 (3')

Loyer 3

7.1

7.5 (3 1

8.2 8.15 8J5 8J3 8.2

Fig. 9. Summaryof oceancrustal modelsdiscussed in text. (ø) Seismicv, velocities.(å) Densities.

If this is comparedwith (10ø)and the sameelastic parameters loading is also òbvious from (19); with time, D decreasesac- are adopted,the plate with variable rigidity will be somewhat cording to /-'(t).Typically, IQ) n r, where a variesfrom 0.25 thicker than the uniform plate. \Mith (23d), to 0.3,or /(t) obasalts,as determinedin the laboratory, are of pendenceof the effectiveflexural rigidity on the duration of the order 5.3-5.8 km s-t, and densitiesare of the order of 6414 LAMBECK AND NAKIBOGLU: STRESSES IN THE OCEAN LITHOSPHERE to an initial elasticresponse followed by creepor, very gener- lished, the trend of¡r* in Figure 7 can be anticipatedto reflecl ally, the actual conditions in the lithosphere. The variation of the bulk modulus with depth can be ex- (l8a) pectedto be much lessthan that of ¡1,so the function F = ¿¡11 <Ð: a) - i,+*t z'z)will exhibit a similar variation with depth as¡r*. Mathe- matically, this depth dependenceis convenientlyrepresented where o is the applied deviatoric stressand 4 the viscosity, by the error function which itself may be stressdependent; 4 will also be a function pressure. place a - of temperatureand Si¡ce loading takes over F : Fo exp {-al(z zù/zol2} (2ra) fiaite time, typically of the order of 106years, the relaxed modulus rather than the unrelaxedmodulus, measured at seis- where, as before, z is measuredwith respectto the midplane of the undeformedequivalent plate and zo is the value of z mic frequencies,is an appropriatechoice for ¡1,and Anderson at and Minster, in an earlier draft of their paper in this issue which the maximum ,F'ois attained. With (lOa) and (2la) the flexural rigidity becomes lMinster and Anderson, 19801,suggested that a reduction of abour 50Voof the seismicvalues may be appropriate.The cor- z>zo F^ | - respondingelastic parametersgiven in Table I are based on D: å I t'exp {-al(z z)/zol2} dz (Ztb) rL I -stz this assumption.As the bulk modulusis little affectedby these dispersion effects, both À and z will i¡crease. In the plate where the integral is evaluated down to a depth where the model usedi¡ the abovecalculations the viscouselement is ig- strength of the upper mantle is very low. If the maximum nored, and strai¡ is describedby value of F is assumedto occur at a predetermineddepth and we equate(21å) with the observedflexural rigidity, we obtai¡ e: o/4t* (18å) an estimateof ,F'o.Alternatively, if .Fois fixed, zo can be deter- where ¡r* is an apparent or equivalent rigidity. Equati:rg this mined; and once a choiceof theseparameters has been made, with (l8a) gives the stressesfollow directlyfrom (lla) and (llå). That is, with (: w/l and { : r/1, o,, can be written as t¡* : ------t-- (re) - I + (þ/ùl$) o.,:-os(z)l# -'Efr) In the upper parts of the crust the responsewill be domi¡ated by the elasticterm of (l 8ø),and ¡r* can be expectedto i¡crease and similarly for the other stresses,with with depth owing to the efects of (1) an increasingconfi-ning pressure,(2) possibly pore water, - -z - a decreasein porosity and oo@): o,l exp {-a[(z zò/zol2] (22a) and (3) compositional changes,particularly in the passage from the oceancrustal layer 2 to layer 3. At somedepth, ow- as comparedwith ing to increasingtemperature the flow term will becomedomi- nant, and oo?) : -Fo'Q/l) (22b)

p-*: n/f (t) where Fs'is the depth-independentvalue for the equivalent plate. The implications of (22) are immediately obvious i.n The viscosity will be pressure(P) 4 temperature(Ð and de- that the maximum stressdifferences associated with the load- pendent through the generalcreep law ing can be conûned to the upper and cooler part ofthe plate è: Ad exp [-(ã* + PV*)/RTI: o/n (20) and they can be signi-ûcantlyreduced at the lower depths, where temperaturesare higher and the material is less resis- where volume, E* is the activation energy,/* is the activation tant to flow deformation. The analytical evaluationof Qlb) and I and n are constants.Thus the apparentrigidity ¡r* will and the solution for the parametersis cumbersome,and for be depth dependent.A schematicvariation of ¡^r*with depth is the present purposeswe approximate (zla) by a parabolic illustrated in Figure 7, where an ad hoc transitionfrom elastic function to plastic flow has been introduced. Laboratory experiments suggestthat flow will be the dominant deformation mecha- r: r"le. '(h)-. nism at temperaturesexceeding about one-half the melting (äl e3a) point temperature,but in geologicalsituations, flow is likely to With (l0a), occur at much lower temperaturesle.g., Carter, 1976].Also many of the experimentalresults are basedon geologicalma- terials not appropriate for the oceaniccrust and perhaps not ,: +*(+-#l (23b) even for the upper mantle. The most signifrcantexgerimental resultsare basedon olivine single crystalsor on dunite aggre- Figure 8 illustratesa nominalexample in which E:0.758o at gates,but while Othello would like the earth to consist of a thesurface z: -H/2 and E: Esatz: -H/4.The constants single perfect crystal, the actual earth is undoubtedly more ate complex. Recent results on Websteriteindicate that at rela- 23e1 tively shallow depths (<60 km) the equivalentviscosities are A: B:-n C:-1 considerably less than those of dry dunite lAvé Lallemant, u 19771.The problems of scaling from the laboratory measure- and mentsto the geologicalproblems are suchthat the experimen- tal resultscan, for the presentpurposes, only be consideredas D: 0.61,=rrEo - ,. Ip grossupper limits; but while the scaleof ¡r* cannot be estab- rzlL vo') LAMBECK AND NAKIBoGLU: STRESSES IN THE OCEAN LTTHOSPHERE 64t5

I 65-2.75g cm-3,but the i¡ situdensities will be lessthan this' are quite different from density model I than for model 3, 2.3-2.5g cm-3.Ocean layer 3 is believedto be o,.,-o*ranging from nearly 8 kbar for the former to about 12 i)--mainly ,n" rung" of a mafic compositionof metagabbroand gabbro kbar for the latter, for both the disc and parabolic loads.With For a load of density *hose seismicvelocities range from about 6.5 to 7.2 km s-t densitymodel 4, o,,t-", is evenhigher. g fill-in density of 2.4 cm-3,the maximum stress andwhose densities range from about2.8 to 3.0[e.g., Christen- 2.8 cm-3 and to of l0 kbar for theseparticular load geometries. sen,19781. The mantle velocitiescorrespond approximately is of the order part Figure l0 illustratesthe gravity anomalies thevelocity of peridotiteswith densitiesin the range3.25-3.40 The lower of density models.The Bouguer g cm-r. To simplify the gravity calculations,layers I and 2 and geoid heights for the two for all four density models consideredhere are al- ñave been combi¡ed into an equivalent layer of the same anomalies other i¡ the spatial domai¡, rhicknessbut with a depth-weighteddensity. Likewise, the most indistinguishablefrom each margiaally (seealso Figures two layersmaking up layer 3 in models4 and 5 have been and their spectraalso di-fferonly probably be very mi¡or in real combi¡ed.Figure 9å illustratesthe adopteddensity models. 3 and 6): any di-fferenceswould geometries,variations i¡ Density model I correspondsto the averagç model of Raitt situations of noisy data, irregular to that deducedfrom the other averaged densitywithin the load, and the presenceof long-wavelength [1963]and is similar These results in- models.Model 2 correspondsto oneof the regionalmodels of anomalies of other than seamountorigia. Christensenand Salisbury[1975] (model 5 of Figure9a). When dicateonce again that the gravity anomaliesare not very sen- averagedin the abovementioned manner, this modeldoes not sitive to the crustal densitiesbut that the stressesare and that differ greatly from density model l. Added to this is model 3, a significantstress reduction can be obtainedifthe densityof tleducedfrom figuresgiven by Wattsand Cochran[1974] and filt-in is taken to be of the order 2.3-2.4g cm-3 i¡stead of the used by them i¡ their analysesof the Hawaiian-Emperor usualbut improbable2.8 g cm-3.The introductionof an i¡- seamounts,and model 4, usedby Watts[1978] in a more te- elastic responsefor the lower part of the layer reducesthe cent study of these seamounts.These models differ from the stressesnear the lower surfaceofthe plate,and a very signiû- othersin that layer 2 has been given a higher density than cant reduction can be achievedwhere it is most necessaryto would be expectedfrom the seismicevidence and the total do so. As this also introducesa stress,temperature, and dura- thicknessof the crust is less.Watts' model also usesa some- tion of loading dependenceof the response,there is some what highermantle densitythan the other models. hope that careful studiesof isostasyof seamountsmay lead to Finally, an estimateof the density of the load is required. a better understandingof the inelastic responseof the litho- An appropriate choice seemsto be in the range 2.60-2.75tn sphereon the geophysicaltime scale.This needscloser inves- that the volcano is presumablysimilar i-ncomposition to the tigation. basalticlayer 2 but freer from the various factors that appear In order to quantify someof the aboveresults and observa- to decreasethe density of the in situ layer 2 materials.A num- tions we will considerthe Hawaiian seamountswhose isostatic ber of gravity studies,however, have i¡dicated lower densities, statehas beenmuch studied si¡ce the work of VeningMeinesz about2.5 g cm-3 [Woollard,1954] and 2.3 g cm-3 lMalahof I l94l] and Gunnll943l. Most recently, IV'atts[1978] computed and Woollard, l97ll (see also Strangeet al. 11965l).Possibly, the deflectionsfor theseseamounts along a number of profiles the central region of the volcanic load is of higher density for which both gravity and bathymetry have been observed. than the remainder of the load, and clearly, the averageden- The largesttopographic load analysed,Oahu, can be approxi- sity may exceedthat of the crust upon which it immediately mated by an equivalent disc load similar to that used i¡ the rests.Thus the common assumptionthat the density of the abovecalculations, namely, of radius 60 km at its baseand of fill-in equalsthe density of the load may not be valid, and it height about 4 km above the averageseafloor. With the den- would lead to an overestimationof deformation and stress. sity model 4 of Figure 9, Watts obtained,from two proûles,Ð - 2.66 x l03o dyn cm, and this is larger than the nominal value usedin the abovecalculations by a factor of 5. The cor- DIScUSSIoN respondi-ngmaximum deflectionfor Watts' estimateof D and Figure l0 illustrates the deflections, stress o,., Bouguer grav- his densitymodel will be about 2.2km (seeFigure 3). If the ity perturbation, and total geoid anomaly for the constant ge- fill-i¡ is of a density of about 2.3 rather than 2.8, the deflec- ometry (A : 60 km, h :4 km) disc loads with density p equal tion will be reduced to about 70Voof this value, or lessif the to that oflayer 2. The adopted flexu¡al rigidity is 5 x l02s dyn fill-in is not complete,as appearsto be the casefor the Hawai- cm, and the li¡ear theory has been used. The maximum de- ian chai¡. The evidencefor this is the outer arch, which rises flection for these models is determi-ned by p^ - p, and the two some500 m abovethe mean seafloor,and this cannotbe at- illustrated examples are for density models I and 3, with tributed to the load-associatedelastic uplift alone, since all model I resulting i¡ deflections that are about 50Voof those of models i¡dicate values signiûcantly lessthan 100 m. The in- model 3, which was used by lYatts and Cochran [974] in their terpretation of the bathymetry is further complicatedby the study of the Hawaiian seamounts. Density models I ard 2 broad regional swell characteristicof this part of the Pacific, give very similar results, while model 4, used by lVatts ll978l and if this surfaceis taken as the referencesurface, then there in a study of the same seamouûts, gives deflections that are is almost no need for any fill-in at all, and the resultson the about 807oof those of model 3. If in model I the mantle den- right-hand side of Figure I would be a more appropriate de- sity is i-ncreasedto 3.4 g cm-3, deflections will be decreased by scription. As much as I km of unfllled moat would remain, about 107o.If the volcanic load, includi-og the part below the and the total deflection need not exceedabout 1.5 km below surface z : 0, is of higher density than the fill-in, the deflec- Oahu. tions will lie between those correspohdingto models I and 3. Seismicrefraction observationsfor a thickening of layer 2 Similar results for the deflectionsand stresscomponent o,, are about the Hawaiian seamountsremains inconclusive. Models illustrated for the equivalent paraboìic load. for the crust below Oahu i¡dicate a rapid i¡crease of crustal As expected from the earlier results, the stressesin the plate thickness [Malahoff and l(oollard, l97ll, but the seismic LAMBECK AND NAKIBOGLU: STRESSES IN THE OCEAN LITHoSPHERE

with estimatesby Crittenden}9671from the Bonneville uplift that stressdi-fferences below the conti¡ental crust do not ex_ ef-#*- ,,// .l--#+- ceeda few tensofbars. ;,v/ :'// In summary,rules such as (13) may grosslyoverestimate the å^l/' x,!/-l maximum stressdifferences that occur i¡ the plate below the I seamounts,and actual valuesneed not exceedI or 2 kbar. If compensationis partly local, as suggestedby Furumoto et al. [971] and Suyenaga[979], ther rhestress differences will not 10 10 differ much from the simple pgå formula for local isostatic compensation.These stresses are valid for loadsthat have per- j5 is sistedfor up to 108years, and they are not inconsistentwith ! ! I-ambeck\ õo [980á] value of about 500 bars fs¡ ¿ minimrrm ss¡- mate of the strength of the crust based on the analysis of global isostasy,ia which loading times may be much longer.

APPENDIX The Poisson-Kirchoff,or small-deflection,theory is adequate for problemswhere (l) the deflectionsand the slopesof the deflectedsurfaces are small, (2) the middle plane doesnot stretchby significantamounts while the plate is deforming,so ri:T that the plate deforms into a developablesurface, and (3) the Ël300 æ0 100 ef \Pcriod kn in-plane forcesacting middle plane on the are negligible. É\3 :61- \ Von Karman's formulation (equations(l4a) and (14ó)) al- þ t-\\ rl\\ lows theseconditions to be relaxed,although the solutionsfor = 3F '\\ 8t \\ simple loadsdo not ol\\, lend themselvesto the analytic and closed forms. We haveused two di-fferentapprbaches to solving these Fig. 10. (Lêft) Deflections,stress o,. Bouguergravity per- equations.In the first approachthe eight initial conditions at turbation,and geoidheight for crustaldensity models I and 3 sub- x : 0 are specifredin terms of four eigenvaluesobtai¡ed from jected to a disc load and fill-in of densityequal to load density. the power seriesexpansion of the solutionsabout the origin. (Right) Deflectionsand stresso¡¡ for the parabolicload. (Lower right) The boundary conditionsat infinity are similarþ expressedin Powerspectra ofthe Bouguergravity anomalies for discload. the terms of parametersderived from the asymptoticexpansions of the dimensionlessdeflection K: w/ Ð and of its derivatives. model of Furumoto et al. 11968lis more indicative of local Thesedi-fferential equations are integratedfrom both ends us- compensationthan of regional compensation.A signi-ûcant ing Merson'smethod, and the two solutions,corresponding to thickening of the crust, up to l0 km, is generallynoted below the i¡terior (tF x/I) < ø) and exterior (€ > ¿) regions, are the major islandslFurumoto et al., l97ll, but the resultsdo not compared at the edge of the load Ë : a. Using Newton's appear to be very consistentfrom one ref¡actionprofile to the shooting method, an improved set of parameters is esti- .next. The thickening of the ocean layer 2 is generally much mated. It is found, however,that the resulting solutions are lessthan this, only about 2-3 km below Oahu, and Woollørd very much dependenton where we take the edgeof the plate [1970] has suggestedthat this is a more appropriateindicator to be, and this is perhapsbest explained by noting that if we of the amount of flexure that has occurred.Furumoto et al. cut the plate at Ë: na, wheren - 5-7, then we are practically [971] and, morerecently, Suyenaga [1979] have attempted to treating a free-edgedfinite plate which deformsi¡to a devel- reconcile these two di-fferentseismic results by a model con- opable surfaceunder the usual loading conditions.The com- taining elementsof both local and regionalcompensation. parison of such solutionswith the Poisson-Ki¡chof theory for Using Watts' parameters,the i¡ferred maximum stressbe- an infinite plate would thereforebe misleading.On the other low the center of the load is of the order 5 kbar (equation hand, by increasingthe value of n the accumulationof errors (12')), but if density model I is adopred,implying that the in the numerical integrationscan becomesevere; and for this gravity data are insensitiveto the crustalmodel (seeFigure 6), reason and since a large number of models were treated, we ' this stressis reducedto about 3.5 kbar. Use of the small-de- have opted for the following perturbation scheme. flection theory results in a further overestimationof the When 41= þ/D):0, (l5a) and (15å)reduce to the small- stressesby about 20Vo(see Figure 5). A further reduction to deflection case with known analytical solutions for certain about 2-2.5 kbar resultsif the sedimentfill-in is incomplete. symmetrical loading and boundary conditions.Vy'e will con- Finally, if the apparentshear modulus is depth dêþendentac- sider here only the caseof a disc load. The dimensionlessde- cording to (23), the maximum stresseso,6 o66etthe upper sur- flection fo obtained usiag the linear theory is given by (3), or face are further reduced to less than 2 kbar (Figure 8). The * (Al) vertical stresso,, (equation (l ld)) is of the order of about I fo: a(l C, Ber{ + C, Bei$ Ê= a kbar at this surface,and all stresses are compressive.Hence fo: a(G Ker { + C¿Kei 6) {> a (42) the maximum stressdifference on the upper surfacebeneath the load need only be of the order of I kbar. The maximum For small valuesof { u/ecan write theseexpressions in a series stressdiflerences Dear the base of the plate can be reduced form: much more and need not exceed a few hundredbars or even fo:ao + ar{2+ aot * ... 0< g< a (A3) less,in keeping with basal shear stressesof the order of 100 : bars that have been suggestedfor driving plate tectonicsand fo åo+ bzËz+ å, ln{ + åo{,Ia{ + ... a< { - I (A4) LAMBECK AND NAKIBOGLU: STRESSES IN THE OCEAN LITHOSPHERE 6417

with where ko: ao * 2ac2d, 4o: d(l + Cr) (A8) kz: az* a(\crda- *cdr) given az: qcz/4 Hence corrections to the corresponding coefficients of fo by the lilear theory will be

-aCt/64 a¿: 6ao: 2orrO, (45) a": 2- y)c,-îr,f 6a,:otd#-+l-- - " [co I t At the origin the deflection is . tzr C. - I b,: al i.C,+ #(t * ln2 y)I Lro + J wl,-s: ksl and Euler's constanty equal to 0.5772"' . Substituting (43) The stresseso,n o66ãtx : 0 follow from (lla) and (llå), or (15å) and integrating into Yield l1\2 a,,t,-u: ooet,-o: l2(l + v)pgl;l lk, \: do* dz* + d4{ + "'+ Alnx * Bx2ln'x (46) t", resultantsìf,, Nu should be fi-niteeverywhere, Sincethe stress and the correspondi-ogerror in the stressescomputed with the g,A: B:0 for{< ø.Hence d¡anddrarc includingat{: Poison-Kirchofftheory is the unknown coefficientsto be determinedfrom the continuity condition at Ë: a. The coefficientdo is ôol,-o:(o-- o,Ð1.-o: 12(r - O*(*)'^(# ' d4: C22a2P/256 +l The solution outsidethe load but around the edgeÉ = a is ob= tained by substituti-ng(44) into (15å),and since only the û¡st and secondderivatives of4 are required at x: a, it will suf- REFERENCES frceto integrate up to 4'. That is, Abramowitz, M., and I. A. Stegun,Handbook of MathematicalFunc' ,¡ons,Dover, New York, 1965. Avé Lallemant, H. G., Experimentaldeformation of clino-pyroxene B2 B3 lnË 4,: (8Bo- l0,Br+ nnr) , , (abstract),Eos Trans.AGU, 58, 513, l9'l'7. fi+ TÈ- 4 Ë Brotchie, J. F., Flexure of a liquid-filled sphericalshell ir a radial gravity freld, Mod. Geol.,J, 15-23, 1971. B^ On crustalflexure, Geophys.Res-, - (A7) Brotchie,J. F., and R. Silvester, "/. tÉttË+(2h-582) *^t 74,5240-5252,1969. Carter, N. L., Steady state flow of rocks, Rev. Geophys.Space Phys., ÊÈ3 d3 14,30t-360, t9'16. * triln2 + Bo h'{ * Studiesof the Geos3 al- { 16- , Cazenave,4., K. Dominh, and K. Lambeck, timeter derived geoid over seamountsin the Indian Ocean,in The Geodesyand Geodynamics,vol. 2, ed- where IJseof Artficial Satellitesfor ited by G. Veis, pp. 478499, National Téchnical University of Athens, Greece,1979. Bo: + b4)(Zbz+ 3b4) B(2b2 Cazenave,A., B. Lago, K. Dominh, and K. Lambeck, On the re- B,:8Bbo(br+ba) sponseof the oceanlithosphere to seamountloads from Geos 3 sat- ellite radar altimetry observations,Geophys. J. Roy.Astron. Soc.,in Br:2ßbtbo press,1980. . Christensen,N. I,, Ophiolites, seismicvelocities and oceanic crustal B,: -8b,2 structure,Tectonophysics, 47, l3l-157, 1978. Christensen,N. I., and M. H. Salisbury,Structure and constitutionof Bo: 4ßboz the lower oceaniccrust, Rev.Geophys. Space Phys., I 3,5'I-86,1975. Crittenden,M. D., Viscosityand ñnite strengthof the mantle as deter- the å, being the coefficientsdefined by (45) and d, being the mined from water and ice loads, Geophys.J. Roy. Astron. Soc., 14, 26r-2'Ì9. t967. thi¡d unknown coefficientto be determinedtogether with d6 Frederick, D., On some problemsin bending of thick circular plates and d from the continuity conditionsat Ë: a. on an elasticfoundation, J. Appl. Mech.,23,195-200,1956. In most seamountloadiag problems,a = l, but if we want Fung, Y. C., Foundationsof Solid Mechanics,525pp., Prentice-Hall, to treat situationswhere ¿ > l, the sameprocedure as outlined EnglewoodClitrs, N. J., 1965. Hussong, above can be followed provided that additional terms are in- Furumoto, A. S., G. P. Woollard, J. F. Campbell,and D. S. in the thickness of the crust in the Hawaiian Archi- cluded in the expansions(A3) Variations and (44). That this can be done pelago,in The Crustand UpperMantle of the Pacifc Area, Geophys. is obviousfrom the convergerceof the Kelvin functions in the Monogr. Ser.,vol. 12, edited by L. Knopof, C. L. Drake, and P. J. domain with which we are coûcertredhere. Hart, pp.94-lll, AGU, Washington,D. C., 1968. ^t Ë: a, N, and ìf, are continuous,and this givestwo alge- Furumoto, A. S., J. F. Campbell,and D. M. Hussong,Seismic refrac- ridge, Kauai to Midway Island, braic.equationsin d, ard dr. Onced, is determined,(46) and tion surveysalong the Hawaiian Bull. Seismol.Soc. Amer., 61,147-166, 197I. (43) are substitutedinto (l5a), and the result is integratedto Gunn, R., A quantitative study of isobaric equilibrium and gravity give anomaliesin the Hawaiian Islands,J. Franklin Inst., 238,373-396, 1943. f:ko+k'{+"' Herta H., Über das Gleichgewichtschwim¡nender elastischer Platten' 6418 LAMBECK AND NÁ,KrBocLU: STRESSESrN THE OCEAN LrruospsBne

in GesammelteY[/erke von Heinrich Hertz, vol. l, pp. 2gg-294, J. over the Medeira atea, Proc. Kon. Ned. Akad. Wetensch.,44, l_ll, Barth,Leipzig, 1895. r94t. Lambeck, K., Flexure of the ocean lithospherefrom island uplift, tilafcgtt, R. I., Flexureof the lithosphereat Hawaii, Tectonophysícs,g, bathymetry,and geoid height observations,submitted to Geoþhys. 435-446.1970. J. Roy.Astron. Soc.,l9$0a. Walcott, R. I., Lithosphericflexure, analysis of gravity anomaliesand Lambeck,K., Estimates of stressdifferences in the crustfrom isostatic the propagationof seamountchains, in The Geophysicsof the pal considerations,J. Geophys. gJ, Res., this issue,l9g0å. cfic OceanBasin and Its Mar_gi1,Geophys. Monogr..Ser, vol. 19, Malahofi 4., and G. P. Woollard, Geophysicalstudies of the Hawai- edited by G. H. Sutton, M. H. Manghnani, and R. Moberþ, pp. ian Ridge and Murray Fracture in The Sea,vol. 4, part 2, ed- 431438,AGU, Washington,D. C., 1976.'- ited by A. E. Maxwell, "o.te, yórk, pp. 73-131,Interscience, New l9?1. Watts, A. 8., An analysisof isostasyin the world's oceans,I, Hawai- McNutt, M., and H. W. Menard, Lithospheric flexure and uplifted ian-Emperor seamount chain, "L Geophys..Res., 8j, 59g9-6004, atolls, "/. Geophys..Res.,8J, 1206-1212,-197g. 1978. Minster, J. 8., and D. L. Anderson,Dislocations and nonelasticproc- Watts, A. 8., On geoid heights derived from Geos 3 altimeter dara essesin the mantle, J. Geophys.,Res.,85, this issue,1980. {o,lg ]!g Hawaiian-Emperor seamount chain, J. Geophys.Res., g4, Paterson,M. S., Somecurrent aspectsof experimentalrock deforma- 3817-3826,t979. tion, Pl¡¡I. Tians. Roy. r#aìts, _ Soc.London, Ser.A, 2g3,163_172,1976. A. 8., and J. R. Cochry'n,Gravity anomaliesand flexureof the Paterson,M. S., The mechanicalbehaviour of roclisunder crustal and Iithospherealong the Hawâiian-Emperorseamount chain, Geophys. mantle conditions,in TheEarth; Its Origin,Structure and Evolution, J. Roy. Astron. Soc.,38, ll9-141,1974. edited by M. W. McElhinny, york, pp. 469499, Academic,New Watts, A. 8., J. R. Cochran, and G. Selze¡ Gravity anomaliesand t979. flexure of the lithosphere:A three-dimensionalstudy of the G¡eat Raitt, R. W., The crusral rocks, in The Sea,vol. 3, edited by M. N. MeteorSeamount, J. Geophys.Res.,80, l39l-1398, 1975. Hill, pp. 85-102,John york, Wiley, New 1963. Woollard, G. P., Crustalstructure beneath oceanic islands, proc. Roy, Re]y9r, phys., F.: 9n !!. theory of bendingof elasticplates, ,/. Math. Soc., Ser.A, 222, 361-387,1954. 23,184-t9t, 1944. Woolla¡d, G. P,, Evaluation of the isostaticmechanism and role of Reisner, E., Stresses and small displacementsof shallow spherical mineralogic transformationsfrom seismicand gravity data, phys. shells,."LMath. Phys.,25,279-300. 1946. Earth Planet.Interiors, 3,484493, 1970. Strange,W. 8., G. P. Woollard, and J. C. Rose,An analvsisof the Woollard, G. P., The interrelationshipsof crustal and upper mantle gravity field over the Hawaiian Islandsin terms parameter ofcrustaÍ structure, values in the Pacific, Rev.Geophys. Space pþs , t 3, g7_ Pac. Sc¿,/9,381-389, 1965. t37,1975. Suyena¡ia,W., Isostasyand flexure of the lithosohereunder the Ha- __waiiln islands,"/. Geophys.Res., g4,55gg-56(i4, lg7g Timoshenko, S. P., and S. Woinowsky-Krieger,iheory of platesand (ReceivedOctober 22, 1979: Såel/s,580 pp., McGraw-Hill, Ner¡ryork, 1959. revisedMarch 31, 1980; Vening Meinesz,F. 4., Gravity over the Hawaiian Archipelago and acceptedApril 16, 1980.)