Gravitational Potential As A Source of Earthquake Energy. Larry Barrows and C.J. Langer. Tectonophysics, 76 (1981) 237-255

8/14/2019 Gravitational Potential As A Source of Earthquake Energy. Larry Barrows and C.J. Langer. Tectonophysics, 76 (1981

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Tectonophy sics,7 6 (1 98 1) 23 7 25 5ElsevierScient i f ic Publ ishingCompany, Amsterdam -237Printed n The Nether lands

GRAVITATIONAL POTENTIAL AS A SOURCEOF EARTHQUAKEENERGY

LARRY BARROWS * an d C.J. LANGEROffice of Earthquake Studies, U.S. Geological Suruey, Bo x 25A46, Denuer Federal Center,Denuer, Colo. 80225 (U.S.A.)(ReceivedApr i l 18, 1980;revised version accepted November 10, 1980)

ABSTRACTBarrows, L. and Langer, C.J., 1981. Gravitat ional potent ial as a source of earthquakeenergy.Tectonophys ics , 6 : 237 255.

Some degree of tectonic s tress within the earth originates from gravity acting upondensity structures. Th e work performed by this "gtavitational tectonics stress" must haveformerly existed as gravitational potential energy contained in the stress-causing ensitystructureAccording to the elast ic ebound theory (Reid, 1910), the ener gy of e arthquakescomesfrom an elastic strain field built up by fairly continuous elastic deformation in the periodbetween events. For earthquakes resulting from gravitational tectonic stress, he elasticrebound theory requires the transfer of energy from the gravitational potential of thedensity structures into an elastic strain f ield prior to the event.An alternate theory involves partial gravitational collapse of the stress-causing ensitystructures. The earthquake energy comes directly from a net decrease n gravitationalpotential energy. The gravitational potential energy releasedat the time of th e earthquakeis split between the energy released by the earthquake, including wor k done in the faultzone, and an increase n stored elastic strain energy. The stressassociatedwith this elasticstrain f ield should oppose further fault slip.

INTRODUCTIONThe distribution of density within the earth's crust and upper mantle ishighly heterogeneous.To a first approximation, the earth's density increasesv\rith depth, the effect of which is a splrerically concentric density distribu-tion. Deviations from this averagei::dens[ty ersus depth relation result in

lateral density variations. These ateral density variations are here referred toas density structures.The existenceof such density structures s demonstratedby measured variations in'gravitational attraction or gravity anomalies.

* Now at Sandia National Laboratories, Albuquerque, N.M. 87115, U.S.A.0040-1951/81/0000-O000/$ 02.50 O f ggf ElsevierScient i f ic Publ ishingCompany


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238The gravitational potential energy of the earth would be minimized if theIayered material were arranged in a purely concentric density distribution,where density increaseswith depth. The presenseof lateral density variations,or density structures, increases he gravitational potential energy above thisminimum value. Because the earth is in a state of increased gravitational

potential energy, stress must exist which tends to deform the materialtowards the state of minimum energy. This stress s here referred to as gravi-tational tectonic stress.The role of gravity in generating tectonic stress s commonly recognized.A familiar example of gravity-induced tectonism is the rise of a salt diapirinto a more dense overburden. Other examples are discussed n the works ofRamberg (1968), Van Bemmelen(1976), Jacoby (1970), DeJongand Schol-ten (1973), and Artyushkov (1973).It is not our intention to review the nature of gravitational tectonic proces-ses.Rather, in this report, we assume hat tectonic stressoriginates from grav-ity acting on the earth's density structures. We then examine the conse-quencesof this assumption on earthquake mechanicsand energetics.If the stress originates from gravity acting on density structures, thenwork done by this stress must once have existed as gravitational potentialenergy of the density structures. This work should include the energyby those earthquakeswhich resulted from t[is stress.Earthquakes are one result of tectonic stress. According to the elasticrebound theory (Reid,1910) the energy of earthquakescomes from an elasticstrain field built up by fairly continuous tectonic deformation in the periodbetween earthquakes. For earthquakes resulting from gravitational tectonicstress, the elastic rebound theory requires the transfer of energy from thegravitational potential of the stress-causing ensity structures into an elasticstrain field prior to the event.This report describes a process in which earthquake energy may comedirectly from the gravitational potential of the density structures. In the firstsection, the process s demonstrated with some simple fini te+lement models.The second section describes two actual events for which the observedchanges in surface elevation indicate the release of significant amounts ofgravitational potential energ"y.The last section contains general expressionsof the energy changes.Details of the computational techniques and a moreextensivediscussionof the mechanicsare given n Barrows (1978).A GRAVITY.DRIVEN HRUSTFAULTOrigin of the sfress

Consider a regional variation in surface elevation such as encounteredbetween a mountainous uplift and the adjacent lowlands. Gravity and seis-mic measurements have shown that, to a first approximation, the increasedload of the mountains is compensated by roots of low density material


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A e l v .239

H i g h l o n d s

C r u s td e n st v = O

\c r u s t

p. .u r t < f mont teU p p e r M o n t l ed e n s i f y =

f m o n t t e

Fig. 1. Density cross-section hrough an isostatically-compensatedchange n regional eleva-t ion (not to scale) .

extending into the more densesubstratum (Airy isostasy). Fig. 1 is a hypo-thetical cross-section hrough such a change n regional elevation.Note that there is more gravitational potential energy in this crustal andupper-mantle density structure than woufd be present if the same materialwere arranged in flat layers with the lighter crust on top. Energy can beextracted from the system by lowering crustal material from the highlands tothe lowlands or by exchanging crustal material from the roots of the high-lands with upper-mantle material at the base of the crust beneath the low-lands. Because he configuration is unstable and in a state of higher potential,stress must exist which is driving the system towards the state of lowerpotential. This stress s inherent in the density structure. Of course, stressfrom external forces or internal processesother than gavity may also be pre-sent but these are not included in the following model.Artyushkov (1973) analytically evaluated the tectonic stressdue to grav-it y acting on such a crustal thickness nhomogeneity. He found that its mag-nitude is of the order of the additional loads associatedwith regional reliefof the crustal surface and concluded that such stress is the cause of rockdeformation and earthquakes. In the following development finite-elementtechniques are used to find the orientation and magnitude of the principalstressaxes in a two-dimensional model. This stress s then employed to drivea high-angle thrust fault.Thrust and fold belts that develop along the flanks of uplifts have oftenbeen related to gravitat ional preading f the neighbouringmountainousarea.This mechanismwas favored by Price (197L, 1973) for the Forelandthrustand fold belt of the CanadianRocky Mountains and by Milici (1975), and byDennison (1976) for the Valley and Ridge Province of the AppalachianMountains. The gravitational mechanics involved in emplacing such thrust

L o w l o n d s


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240sheets are described by Eltiott (1976). The following model differs in thatit includes the effect of crustal roots implied by Arry isostasy n addition tothe mountainous uplifts and does not include a sedimentary surface layer.However, the model is consistent with their general concept of the origin ofthe tectonic forces which formed the thrust and fold belts.Material properties and program background

Crustal and upper-mantle density structures persist throughout the dy-namic lifetime of regional tectonic features. During the stmctural develop-ment of regional tectonic features, rock bodies permanently deform intonew configurations. As a first approximation, this deformation is viscous andthe bulk volume of material remains constant. For the following stresscalcu-lations the crust and upper mantle are assumed o behave as ncompressible,Newtonian (linear) viscous fluids. The assumption of a purely viscous rheol-ogy may seem inconsistent with the visco-elastic behavior of actual rocks.However, a visco-elasticproblem can be modeled with a purely viscousrheol-ogy provided the deviatoric stresses change very slowly with time (Bullen,1963, p. 37). It is also assumed that the model thickness is much greaterthan its width so a plane-shain-rate approximation cut be applied. Theappropriate NavierStokes field equation describing the material is then:Ftti, i - P, i * PEi= 0where 2p = viscosity, ui = velocity vectot, p = pressure (the negative of meanstress) p = density , Ei = gravitational acceleration and the indices followinga cornma imply differentiation,The solution to a particular problem is the velocity field (both horizontaland verbical components) and pressure field which simultaneously satisfyboth the Navier.Stokes field equation throughout the material and theboundary conditions. Strain rates, stresses, fid other quantities of interestare derived from these velocity and pressure ields.In finite-element analysis, the unknown fields are assumed to vary in aknown manner between a finite number of locations (nodal points) withinthe material and on its surface. The values of the fields at these nodal pointsare then the unknowns. Some function of the unknown fields (usually inter-nal energy or energy dissipation) is expressed n terms of the finite number ofnodal point values. Minimizing this function with respect to the nodal pointvalues yields a set of simultaneous equations which are solved for the actualvalues. Strain rates, stresses,and other quantities of interest are then calcu-lated from the manner in which the unknown fields were assumedto varybetween nodal points. The basic concepts of the finite-element technique aregiven n the texts by Zienki.ewicz 1971) and Desai and Abel (1972).The Navier fietd equation for static incompressible Hookean elastic defor-mation is equivalent to the Navier.Stokes field equation for slow incompres-sible Newtonian viscous flow except that u; becomes a displacement vector


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241,and p the elastic shearmodulus. Thus, with proper substitutions, a finite-ele-ment program for elastic deformation can be used for problems in viscousflow. Commonly available finite-element programs for plane-strain analysisof compressibleelasticmaterialsuse two displacement components (horizon-tal and vertical) as the unknown fields. These programs fail for incompres-sible materials for which three fields are required (two displacement com-ponents and pressure).To solve his problem, Hetrmann (1965) reformulatedthe elastic field equations into a form valid for both compressible and incom-pressible materials. Hughes and Allik (1969) used his formulation to developthe corresponding finite-element techniques employing three unknown fieldsat each node. The techniques of Hughes and Allik have been incorporatedinto the computer program of Desai and Abel (1972) and, noting the equiv-alence of the incompressible elastic and incompressible viscous field equa-tions, it is used for the following two-dimensional viscousmodel. In the pre-sent program, it is necessary o approximate incompressibility with a viscousPoisson's ratio (not be con fused with the elastic Poisson's ratio) of 0 .499instead of 0.5 to avoid instabilities n the solution equations.However, this isnot thought to significantly affect the results.,Sfress nalysis

Fig. 2 is a finiteclement model of an isostatically compensatedchange nregional elevation. The highlands are one kilometer above the adjacent ow-lands. The cmstal density is 2.9 Elcm3 and the upper mantle density is 3.3g/cm3. The increased oad of the highlands is exactly compensatedby rootsof lower density cmstal material extending 7 .25 km in to the upper mantle.

Fig.2. Finite-element model of a 1-km, isostatically-cornpensated hange n regional eleva-t ion.

L o w l o n d s H i g h l o n d s

C r u s te = 2.9 gm/cc

U p p e r M o n t l et : 3.3 gm/cc


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242Material viscosity is uniform throughout. The sides of the model are con-strained horizontally and free to move vertically. The base is constrainedvertically and free to move horizontally. Nodal points are located at thecorners and centroids of each quadrilateral element.The model is loaded with a constant gravity field and the finite-elementprogram eomputes the stressat the centroid of each element. This stress s acombination of both the tithostatic overburden and the gravitational tectonicstressesnherent in the density structure. The gravitational tectonic stress sfound by subtracting the lithostatic overburden at each element centroidfrom the total stress.The orientation and magnitude of the principal gravita-tional tectonic stress l(es associatedwith a 1-km change n regional elevationare plotted in Fig. 3.This stress s predominately extensional in the highlands and compressivein the lowlands. The maximum value of stress s approximately 100 bars inthe crustal section below the elevation change and its magnitude decreases\vith distance from the elevation change. The derived gravitational tectonicstress s independent of the absolute value of the material viscosity becauseof the lack of forces, other than gravity, and of velocities in the boundarycondit ions.The extensional stress n the highlands is an example of less-thanJithostaticIateral confining stress.Normal faults, fornled in response to this stresssys-tem, should result in local surface subsidenceand the releaseof gravitationalpotential energy. In the subsequent discussion it will be shown that a high-

L o w l o n d s f - l m p l i e dT h r u s t F o u l t

t H i s h l o n d s

l l i I I r, * . a ' . -

x x t-.a * +

I \ . ' . - A * +, I + +

I O O b o r S t r e s s e s\ . 2 - - - C O m P r e S S i O n.2\.' ' - - e x fe n s i o n

Fig.3. Gravitational tectonic stress nherent to a 1-km, isostatically-compensatedchangein regional elevation assuming uniform viscosity.


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243angle thrust fault formed in response to the compressive stress n the low-lands also releasesgravitational potential energy.Modeled faulting

The implied fault on Fig. 3 is along one of the two planes of marimumshearstressoriented at 45 degreesbetween the principal stressaxes.The grav-itational tectonic stress s compatible with thrusting of the highlands up andover the lowlands along this fault. This sense of faulting is consistent withisostatic uplift of the highlands in response o a lowering of the cmst-mantleinterface. The following method was used to model the displacements andenergy releaseassociatedwith gravity-driven motion along such a thrust fault.(1) A new finite-element model of an isostatically-compensated,1-kmchange in regional elevation is shown in Fig.4. The material configuration isthe same as before except that a zone of fault material 70-m thick is addedin the location of the proposed fault.(2) The fault material is initially assigned he same material properties asthe surrounding cmst. The model is loaded with a uniform gravity field andthe finiteclement program derives a set of baselinenodal velocities.(3) The viscosity of the fault material is reduced to one percent of that ofthe surrounding crust and mantle. The model is again loaded with gravityand the finite-element program derivesa new set of nodal velocities.(4 ) The nodal velocities associatedwith fault motion are the differencebehveen the new set and the baselineset.The finite-element model, including the material properties, are in dimen-

H i g h l o n d sL o w l o n d s

Fig. 4. Finiteelement model of a gravity-driven, high-angle hrust fault.


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2 4 4sionlessunits (length, melss, nd time). The model resultsor "nodal velocities"are in the same units and may be interpreted as either particle velocities in aNewbonian viscous-flow pioblem or displacements in an incompressibleHookean elasticdeformation problem. In the crseof the latter problem,length is in kilometers and mass s in grams. For reasonsgiven later, we adoptthe elasticdeformation interpretation for the fault-model results. We alsoassumean elastic Young's modulus of 1012dyn.lcm'. The model results arethen simply displacements. These surface and fault-plane nodal displace-ments are plotted in Figs. 5 and 6.Energy releaseand elastic strain

The modeled fault displacement was driven by the gravitational tectonicstresscharacterized in Fig. 3. This stressresulted from gravity acting on thecntstal thickness inhomogeneity. It follows that the energy released duringthe fault displacement must have corne from a decrease n the gravitationalpotential energy of this crustal and upper mantle density shucture. Thechange in gravitational potential energy (for 60 cm of dip-slip fault displace-ment) was found by multiplying the weight of each element times the verticalchange in its center of gravity associatedwith the fault displacement. Thesum over all elements n the model was a dqcreaseof 0.2 . LO22 rg . This sumapplies o a l-km-thick model. For a 10-km-long fault (neglectingedgeeffecils)the decrease is 2.0 '7022 erg. If half this energ'y is converted into seismicwavestheresultis a magnitude- 6.8earthquake(from log,oE= 11.8 + 1.5 m6).As previously noted, the finite-element results can be interpreted as eitherparticle velocities in a Newtonian viscous-flow problem or as displacementsin an incompressible Hookean elasticdeformation problem. The displace-

o. 26o01

- ' = r ]t f bE g uo :N ( D' ; Jo ) oI

ri 20o ^ lc ) s c: 3q r A

300 400

Fig. 5. Surface deformation accompanying 6O-cm slip along the gravity-driven, high-anglethrust fault. '

20 0o lo n g(k m)


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245

S u r f o c e N o d e s

d i s p l o c e m e n l

F A U L T P L A N E

Bollom Nodes j- '- '- ' '-

N O D A LD I S P L A C E M E N T S

\

S u r f o c e N o d e s F o u l l M o l e r i o l

I i l\- \,/f\ /-/iI I Borom"{:41,/t'.,/ II I li(2\lI I l/ \1,/ 'rtD E T A I L S FF A U L T O N E

Fig. 6. Fault-plane nodal displacementsaccompanying 6O-cmalong the gravity-driven, high-angle hrust fault. slip (a t th e ground surface)


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246ments caused by faulting did not significantly alter the original crustal thick-ness inhomogeneity. If incompressible Newtonian viscosity completelydescribed the mechanical behavior of the material, then the gravitational-tec-tonic stress in Fig. 3 would still be acting on the fault zone and additionaldisplacements would recur in subsequent time increments. However, theviscous interpretation was based on the behavior of rocks undergoing perma-nent tectonic deformation and it is well established that rocks behave elastic-ally in the shorter periods of seismic waves. In the elastic interpretation, theshear shess acting on the fault zone in the postfaulting configuration is insig-nificant because the fault zone material was assigned a negligible shearstrength. The gravitational tectonic shess on the fault is then opposed orbalanced by stress resulting from the elastic deformation accompanying thefaulting. This elastic strain field would then viscously dissipate during theperiod leading up to the next earthquake.A shorbcoming of the fault model is that it applies to incompressible elas-tic deformation and earth materials are elastically compressible. The gravita-tional tectonic stress(Fig. 3) calculated from the viscous model is the sameas the shess which caused the elastic deformation plotted in Figs. 5 and 6.The elastic deformation of a viscoelasticmodel with a finite elastic compres-sibility should be similar, but not identical, to that given here. However, theexpressions of general energetics, presentqd in a later section, would un-doubtedly be more complex.EARTHQUAKEOBSERVATIONSThe 1964 Alashan earthquake

The 1964 Alaskan earthquake probably resulted from low angle thrustingbehreen the subducting Pacific plate and the oveniding subduction complexof the Norbh American plate (Plafker, t972a). Fig. 7 (toom Stacy, 1969, p.79) is a contour map of the vertical changes n surface configuration whichaccompanied the earthquake.These vertical displacementswere determined from systematic field studiesof displaced shorelines, a comparison of pre- and post-earbhquake ide-gaugemeasurements, depth soundings, and first-order survey level lines (Plafker,L972a, p. 906). In addition, qualitative information on the general offshoreextent and the relative amount of uplift was obtained from analysis of therezulting tsunami waves(Plafker,L972a, p. 914).A broad area along the western margin of the Gulf of Alaska, whichincludes Kodiak Island and the Kenai Peninsula, subsided as a result of theearthquake. The total volume of this subsidencewas estimated from the mapby Plafker (1972b) to be approximately 10r? cm3.A similar areaon the con-tinental shelf and slope was also uplifted. Note that the area of subsidence sat a higher mean elevation than is the area of uplift. Simultaneous subsidenceof highlands and uplift of lowlands implies that there was a release of gravita-tional potential energy.


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4, ../- lrVl' ./-^-z ,/'(.^.,t(. l1/:Y"*'^,ffil-!

'r';''o'.

, , : \,'l,$83

- - t -

x oncters

Fig. 7. Vertical deformation accompanying the 1964 Alaskan earthquake. The contoursare in meters and the edge of th e continental shelf is indicated by the dotted line (fromSt acy ,1969 , p . 79 ) .

The amount of gravitational potential energy released during the earth-quake can be estimated by assuming hat:(1) The subsidedvolume equals he uplif td volume (10tt .*').(2) The mean elevation difference between the land area that subsided(Kodiak Island and Kenai Peninsula) and the area that was uplifted (conti-nental shelf and slope) is 1 km(3) The mean density of the subsidedand uplifted materiat is 2.5 g/cm3.(4 ) The deformed area was partially subaerialand partially submarine. Forthese calculations we adopt the conservative assumption that the entire iueais underwater (effective density reduced to 1.5 g/cm3).Then from:

AGp=gpvh

.tI\..- t.'.""i'


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248where : A'GP = the change in gravitational potential energ"y(ergs); g - thegravitational athaction (980 dynes/gram-force); p = the average effectivedensity (1.5 g/cm3); V= the volume of material (10t t .* t ) ; h= the meanelevation difference (1 km = 10s cm); there was a net decrease n gravita-tional potential energy of L.5 '102s ergs.This value is five times the amountof seismic energy release (3 ' 1024 ergs) calculated by Press and Jackson(L972) for the main shock. This estimate, of course, is dependent on theabove assumptions and does not allow for possible intemal density changesor any elevation changesbeyond the extent of the map. However, it is reason-able to assume that the observed vertical changes n surface configuration doinfer the releaseof gravititional potential energy.The indicated decrease in gravitational potential energ:y can be betterunderstood tectonically through a model of low-angle subduction zonethrusting proposed by Hamilton (1973) and by Dickenson (1977). In describ-ing the space and time relations between stratigraphic and orogenic belts tiedto plate consumption at subduction zones,Dickenson states:

"As subduction tends to drag the toe of the slope landward, gravitationalcollapse of the growing subduction complex compensatesby inducing d6col-lement that detaches weak materials from the upper surfaceof the descend-ing plate. This processallows the toe of the slope to slip relatively seawardabove the surface of d6collement as the desgendingplate slides beneath thesubduction complex. Hamilton (1973) aptly cornpares he subduction com-plex, his "melange wedge", to a standing wave that spreadsgravitationaltyforward at the same speed with which it is dragged back by the subductingplate". Also, Va n Bemmelen (L976, p. 172) discusses similar concept forthrust-fault earthquakes along the Japanese rench.The decrease in gravitational potential energy, implied by the observedvertical changes in surface configuration and the gravitational spreadingmechanism, suggest hat the 1964 Alaskan earthquake was more like a grav-itational collapse event than an elastic rebound event. If it was a gravitationalcollapse event, then, as will be shown in the next section, t must be accom-panied by an increase, rather than release,of elastic strain energy. The stres-ses of this new elastic strain field would oppose the gravitational tectonicstresses hich caused he earthquake.The Disie Valley-Fatruiew Peak earthquahes

The importance of including gravitational potential energy in earthquakeenergetics s demonstrated by Meister et al.(1968). Four earthquakesoccur-red during 1,954 in the Dixie Valley-Faiwiew Peak area, Nevada. Theseevents probably resulted from oblique slip along steeply dipping normalfaults. The accompanying sqrface deformation was measured with pre- andpostcarthquake leveling and triangulation suryeys conducted by the U.S.Coast and Geodetic Survey. Meister et al . (1968) calculated both the changein gravitational potential energy inferred by the leveling data and the mini-


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249mum change n elastic strain energy inferred by the triangulation data.The change in gravitational potential energy was determined from the ob-served owering of the ground surface and the assumption that this subsidenceresulted from volumeteric compression of the upper 15 km of the crust. Theminimum change in elastic strain energy was calculated from the obseruedsurface strains, a coefficient of rigidity of 4 .LOrr dyn.lcm', and the assump-tion that the surface strain extended through the upper 15 km of the cmst.They found a decrease n gravitational potential energy of 4.3 . 1023ergs anda minimum change in elastic strain energy of 1.3 '1023 ergs.The total elasticwave energy for the earthquake was estimated from the Gutenberg Richterrelat ion to be L.2 ' 1023ergs.The assumptionwasmadeby Meisteret al. (1968, p. 5991) that the changein elastic strain energy represented a decrease n the energy value. However,it is geologically reasonable to attribute normal faulting to gravity forces. Ifthe earthquakes were gravitationally driven, then, as shown in the next sec-tion, the change in elastic strain energy would express an increased energyvalue ffid, the increaseof the elastic strain energy and the seismic wave energywould come from the decrease n gravitational potential energy. The valuesof the energy changescalculated by Meister and others are consistent withthis mechanism.A relation between-seismicity and gravitational tectonic stress s also sug-gested by correlations of epicenters with steep gradients in the regional Bou-guer gravity field. Simmons et al. (1979) noted such a correlation in Washing-ton and attributed it to the static stresses aused by the anomalous masses.Goodacre and Hasegawa (1980) found a similar conelation between epi-centers and gravity anomalies along the lower St. Lawerence valley. Theyalso suggest that the gravitationally induced stressesmay be a contributingfactor to the production of earthquakes.GENERAL ENERGETICS

In finite-element analysis, a continuum is approximated by an assemblageof small bodies or volumes. Each volume contains severalnodal points usuallylocated at the verticies. The unknown fields are assumed o vary in a knownmanner between the nodal points. The values of the unknown fields at thenodal points, or "nodal displacements," are the unknowns in the problem.A functional of the unknown fields (usually energy or energydissipation)is expressed n terms of the nodal displacements. Variational procedures areused to minimize this functional with respect to the nodal displacements andobtain an equilibrium equation for the modeled continuum. For static analy-sisof a simple elastic body this equilibrium equation has the form:lKl {z}= Unwhere: IK] is the stiffnessmatrix , {u} is avector of the nodal displacement,{U} is a vectorof the nodal oads.


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250The elasticstrain energy n the body is:sE =| {u}r Kl {z}ffid, substituting {u} - [K-t l {U}, then:

sE=+{wlK-' l{u}Assuming hat the only loads acting upon the body are hosedue to grav-ity, then the gravitationalpotential energy s:Gp = -{u}, {U}= {u}, K-' ] u}

An earthquake is modeled as the development of a fault surface with negli-gible shear strength. In finite-element notation this is represented by a changein the stiffness matrix:lK i l * [Kr ]where the subscripts i and f refer to initial and final states, respectively. Sucha change in stiffness is accompanied by a change n displacements:{u , } - [K i ' ] {U} -> {u r } = [K i ' ] {U}There is a decrease n gravitational potential energy:AGP - {U} , ( [K i ' ] - tK i r ) {U}and an increase n elastic strain energy:AsB +| U} ' ( [K i ' ] - tKa' ) {U}

Thus, in the simplest case,a gravitationally-driven earthquake is aceompa-nied by a decrease n gravitational potential energy and an increase n elasticstrain energy. The increase n elastic strain energy is only half the decrease ngravitational potential energy leaving unaccounted-for residual energy. If thisresidual energy goes into transient seismic waves, then the seismic waveenergy equals the increase n elastic strain energy. With appropriate assump-tions, sirnilar results may be obtained from the expressions or general ener-geticsgiven by Dahlen (7977) and Savageand Walsh (1978).Dahlen (7977, p.253) derived a generalexpression or the energy releasedby an earthquake in a rotating, self-gravitating body. This expression is givenby Savageand Walsh in the form:

AE I bt,.?) u; s,.,/ \.Lwhere: oli is the ambient stress ensor before faulting, o;i is the stress ensorassociated with faulting in an otherwise unstressed body, u; is the displace-ment vector, dSi ir a unit normal vector to an element of fault surface, andthe integal is over the fault surface. Energy effects of rotation and self-gravi-


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25 1tation are implicit in the above expression through their contribution to theambient stress ensor before faulting.Savage and Walsh also give general expressions for the change in internalelastic energy and for the change n gravitational potential energy. The changein gravitational potential energy is:AGP= { f iu idu

r= - J o ' i ' 1u ;d ' s iEwhere f; is the gravitational body force vector and the integral of the firstexpression is over the volume of the body. The stress ensor oi i is a solutionto the linear elastic equations:ij = A-tut,i nu,i) the strain tensoroii = C;ip1m Hooke's lawo','i,! fi = O the equilibrium condition,zubject to the stress ree boundary condition on the free surface.The change n internal elastic strain energy is:

n / n . . \AsE I (o' , ,*+)u;dS;I f ;u,d,u; \ L ' uSubsti tut ing f iuid.u=- I o' i l iuidsj , then:

r / o , , \A,SE I I oii * -t - o',!i/ , 0",: \ z /LJIf all the prefaulting ambient stress s gravitationally induced, then:o i = -oiiIf earthquakes result from the creation of a fault surface with negligibleshearstrength, then in the post-faulting configuration:oii = oliGiven the preceding assumirtions the energy expressions are:

- fLE- , I o , i u ;dS;;- fAISE=i J oi iu;d,51E

fAGP=- l o i i u i f f iJt


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252Again the earthquake is associatedwith a decrease n gravitational potentialenergy. This decrease in gravitational potential is equally split between theenergy released by the earthquake and an increase n elastic shain energy.Our assumption that al l prefaulting ambient stress s gravitationally induceddepends on the material properties. In our finite-element models we assumedincompressibility. For incompressible elastic media, the gravitational stress na flat-Iayered earth is a lithostatic pressure.For a compressible elastic media,lithostatic pressure is a combined effect of both the gravitational stressand alocked-in elastic stress (Collette, 1976). The energetics of gravitationally-induced faulting in a compressible elastic media are more complex thanthose presented here. For further considerations see Savage and Walsh(L978). Another assumption, probably notrealized at depth within the earth,is that stress s linearly related to strain.SPECULATIONS

The modeled density structure and the two examples cited may appear tobe of only local significance. However, in the broadest sense, ectonic stresswithin the earth may result from gravity acting on density structures. Moun-tain ranges can be isostatically uplifted in response to complex densitychangesdeep within the crust and upper mantle. Global convection can resultfrom the gravitydriven rise of hot, low deirsity material and gravitydrivensinking of cool, high density material. The density contrasts result from ther-mal expansion but the stressand resulting motion are due to gravity.Continuing along the same ine, plate tectonics may represent a form ofgravitydriven convection in which the plates themselvesdetermine the flowpattern. The driving force may be a combination of new plates sliding later-alty off an uplifted wedge of asthenosphere beneath the ridges and of oldplates sinking back into the mantle at subduction zones. Jacoby (1970)expands on this mechanism and demonstrates the adequacy of the totalenergy budget.Oceanic ridges and subduction zones are regions o f anomalous crustal andupper-mantle density structures. They are also seismically active. This ob-servation is consistent with the proposed gravity collapsemechanism. Trans-form faulting does not fit easily into this mechanism and may represent clas-sical elastic rebound.The general equations of earthquake energetics Dahlen,7977; Savage ndWalsh, 1978) do not distinguish between the gravity-collapse and elasticrebound mechanisms.Both appear easibleand both result in similar, but notidentical, displacements Barrows, 1978). The important distinction is thesource of the energy.The elastic rebound mechanism nvolves he release felasticstrain energ:ywhich was built up by slow tectonic deformation betweenevents. The gravity collapse mechanism releases gravitational potentialenergy directly from the stress-causing ensity structure. To the extent thattectonic stress originates from gravity acting on density structures, the grav-itational collapse mechanism appears simplier.


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253CONCLUSIONS

At least some tectonic forces originate from gravity acting on crustal andupper-mantle density structures. The energy of those earthquakes whichoccur in response to these gravitational tectonic forces ultimately comesfrom the gravitational potential of the density structures. This earthquakeenergy may come directly from this gravitational potential rather thanindirectly through an intermediate elastic-strain field. The proposed mecha-nism involves the partial eollapse along a fault of a gravitationally-unstabledensity structure. The sudden deformation is accompanied by the releaseofgravitational potential energy and the increase of elastic strains. Further faultmovement is stopped by the stressof these elastic strains.Releaseof gravitational potential energy is implicit in the ground-surfacedeformations which accompanied the 7964 Alaskan earthquake and theL954 Dixie Valley-Fairview Peak earthquakes. In each instance more energywas released from the gravitational potential of the surface configurationthan was radiated as transient seismicwaves.The proposed mechanism differs from the more generally accepted theoryof elastic rebound originally formulated by Reid (1910). However, bothmechanisms involve a change in elastic strain energ:y. n the elastic reboundmodel, the elastic strain energy, which was slowly accumulated betweenearthquakes, is released into seismic wave3. In the gravity-collapse model adecrease n gravitational potential energy is apportioned between the genera-tion of seismic waves and an increase n the elastic strain energy. The gravity-collapse model does not require the buildup of significant elastic shainsbefore the earthquake.

ACKNOWLEDGMENTSAppreciation is extended to the Mobil Oil Corporation for their supportof the initial study of the finite-element applications to gravitational tectonics.The Colorado School of Mines is gratefully acknowledged for their financialassistance.We would parbicularly like to acknowledge Joe Andrews and James Savagefor their consultations and helpful criticisms of the manuscript. Alvaro Espi-nosa and Maurice Major also contributed support to this research effort.

REFERENCESArtyushkov, E.V., 7973. Stresses n the lithosphere caused by crustal thickness inhomo-gen i t ies . . Geophys.Res. 78 : 767 5-7 708.Barrows, L., 1978. Gravitat ional Tectonic and Stat ic Seismi c Model ing with Fini te Ele-ments. Ph.D. Thesis,Colorado School of Mines, Golden, Colo., T 1983: 198 pp.Bullen, K.8., 1963. An Introduction to the Theory of Seismology. Cambridge UniversityPress,Cambridge.


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2 5 4Collet te,8.J. , 19?6. Normal state of st ress n the l i thosphere. Pure Appl.Geophys., 1L4:285-286.Dahlen, F.A., I977. The balance of energy in earthquake faulting. Geophys. J.R. Astron.Soc., 48: 239-261-.DeJong, K.A. and Scholten, R. , 19?3. Gravity an d Tectonics. Wiley, Ne w York,502 pp .York, 502 pp.Dennison, J.M., 1976. Gravity tectonic removal of the Blue Ridge anticlinorium to formValley and Ridge province. Geol . Soc. Am. Bul l . , 87: L470-1476.Desai, C.S. and Abel, J.F., t972. Introduction to th e Finite Element Method. Van No-strand/Reinhold, Ne w York, 477 Pp .Dickinson, W.R., L977, Tectono-stratigtaphic evolution of subductiontontrolled sedi-mentary assemblages. n: M. Talwani and W.C. Pitman III (Editors), Island Arcs, DeepSea Tlenches an d Back-Arc Basins. Am. Geophys. IJnion, Washington, D.C., PP.33-4 0 .El l iot t , D., 1976. The motign of thrust sheets. . Geophys. Res., 81: 949-963.Goodacre, A,K. and Hase[&", H.S., 1980. Gravitationally induced stressesat structuralboundaries. Can. J. Earth Sci., 17: 1286-129L.Hamil ton, W., 19?3. Teetonics of the Indonesian region. Geol . Soc. Malaysia Bul l . ,6:3-10.Herrmann, L.R., 1965. Elasticity equations for incompressible and nearly incompressible

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255Simmons, G., Ti l lson, D., Murphy, V.and WanE, H., !979. Earthquakes, gravi ty, andstress n Washington (abstr.). Earthquake Notes, 50 (4): 59-{0.Stacey,F.D., 1969. Physicsof the Earth. Wiley, New York.Van Bemmelen, R.W., 1976. Plate tectonics and the undat ion model: a comparison.Tectonophysics, 3 2 L45-182.Zienkiewicz, O.C., 1971. The Finite Element Method of Engineer ingScience.McGraw-Hil l , New York, 521 pp.

Documents

Gravitational Potential As A Source of Earthquake Energy. Larry Barrows and C.J. Langer. Tectonophysics, 76 (1981) 237-255