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Home , Detachment fault, Fault mechanics, Plate theory

, VOL. 7, NO. 5, PAGES959-973, OCTOBER1988

FLEXURAL ROTATION OF NORMAL FAULTS

W. RogerBuck

Lamont-DohertyGeological Observatory of ColumbiaUniversity, Palisades, New York

Abstract. A conceptualmodel is proposedfor the flat-lying abandonednormal (or "detachment") generationof low-anglenormal faults in Metamorphic below slicesof upperplate rocks and sedimentary infill; Core Complexes. The model is based on three (2) a strongcontrast in metamorphicgrade acrossthe assumptions:(1) the isostaticresponse to normalfault abandoned"detachment"; (3) rapid movementof lower motionis of regionalextent; (2) when a fault segment plate rocksfrom midcrustaldepths to shallowerdepths. is significantlyrotated from the optimumangle of slip, These results are qualitatively in agreement with relativeto the crustalstress field, it is replacedby a new geologicobservations for corecomplexes. planarfault orientedin the optimumdirection; and (3) the fault cuts the entire upper crust and fault motion alwaysnucleates in the sameregion at the baseof the INTRODUCTION uppercrust. The stressfield is consideredto be uniform throughthe crustand the regionalisostatic response of There has been much recent discussion about the the crust to loads is computedusing the thin plate origin of "Metamorphic Core Complexes." Faults flexureapproximations. Active low-angle normal faults within these complexes appear to have undergone a are difficult to reconcile with theories, normal sense of motion but are gently dipping or earthquakefocal mechanism studies, and geochronologic subhorizontal[e.g, Profett, 1977; Davis, 1980; Miller resultsindicating rapid coolingof corecomplex rocks. et al., 1983]. This is particularly interesting since The model does not require active fault slip on seismicallyactive normal faults are generallyobserved low-angle faults. The flexural response to normal to be dipping at steeperangles (>30 ø) [e.g., Jackson, faulting is shown to be significantly affected by 1987]. anelasticbehavior of the crust and by loading due to More than20 corecomplexes have been recognized sedimentation.The anelasticresponse to large in the of western North stressesresults in significantreduction in the effective America, all of Tertiary age [Crittendenet al. 1980]. elasticrigidity of the uppercrust. This can explainthe Severalcharacteristics of metamorphiccore complexes observedshort wavelengthof topographicresponse to are illustratedin Figure 1. The fundamentalrelation- normal fault loads. The model results in: (1) a nearly ship in core complexes is that low-grade or un- metamorphosedrocks overlie high-grademetamorphic Copyright1988 "core" rocks [Davis, 1983]. The boundary between by the AmericanGeophysical Union. rocksof contrastingmetamorphic grade, which is often subhorizontal,has been variously called a "low-angle Papernumber 8T0340. normal fault," a "detachment," or a "decollement." 0278-7407/88/008T-0340 $10.00 Although these terms imply that fault motion took 960 Buck: FlexuralRotation of NormalFaults

(a) Harcuvar Mountains

WSW ENE

Miocene Sedimentary pC Gneiss & VolcanicRocks & _%•• • _ _ Granite

Mylonitic IZll III I11 II

"Detachment"

(b)

SW NE Older& VolcanicTertiary RocksSedimentary YoungerTertiary Sedimentary & Volcanic Rocks

tic "Detachment" on Metamorphic & Intrusive Rocks i i i i i i 0 km 10

Fig.1. Interpretativecross sections oftwo metamorphic corecomplexes showing features common tomany core complexes:(a)the Harcuvar Mountains inArizona [after Rehrig and Reynolds, 1980] and (b) the Whipple Mountains in Californiawith inferred doming removed [after Davis, 1980]. placeon a nearly flat-lyingfault, thereis considerable thatthey were deformed in a temperaturerange where debateon the origin of suchstructures. Here the term quartz flowed but where feldspars fractured. "detachment"is usedin a nongeneticsense to describe High-temperaturerock mechanicsexperiments show themajor discontinuity in a corecomplex. that this could occur between about 400ø and 500 øC A normal senseof offset motion on the detachment [Caristan,1982]. The lower plate mylonitesoften is inferred from strain indicatorsand structuralrelations showoverprinting of brittledeformation on the ductile [e.g., Davis et al. 1983, 1986]. Upperplate rocks, fabric[e.g., Crittenden et al., 1980]. thoseabove the detachment, have typically undergone Geochronologicstudies of lowerplate rocks from onlybrittle while lowerplate rocks have several core complexesindicate rapid uplift of experiencedconsiderable ductile deformation [e.g., midcrustalrocks [Davis, 1988]. Dallmeyer et al. [1986] Miller et al. 1983; Dokka et al., 1986]. The lower use40Ar/39Ar dating of lowerplate rocks from the platerocks frequently show mylonitization, indicating RubyMountain metamorphic core complex in eastem Buck: Flexural Rotation of Normal Faults 961

350 which are also thought to be greatly extended, are typically buried under deep piles of sediment. Mid-oceanridges are the majorareas of platedivergence, 300 - but becauseof the difficulty in working in deep water they are are not well mappedon a fine scale. Also, there is little or no erosionof mid-oceanridges, making it difficult to infer cross sectional relations among 250-coolingpath envelope faults. The basic problems to be explained in core complexes are twofold: (1) how to produce a • 200- subhorizontaldetachment surface separating rocks of • practicallimit of very differentmetamorphic grade and (2) how to rapidly • coolingpath envelope bring myloniticrocks from 10-15 km depthup closeto the surface. In this papera new model for the origin of detachmentsand associatedstructures is proposed. Before developingthis model a brief review of other 100-- ideasfor the originof detachmentsis given.

PREVIOUS MODELS / Several models have been proposedto explain observationsin metamorphiccore complexes. They fall broadlyinto two categories,according to how 0 4 8 12 16 20 24 28 are broughtto the surface. The proposedmechanisms Time (rvla) for this are: (1) motion on a high-anglefault or set of Fig. 2. Tcml•ramr½-dm½curve inferred for rockswithin (UD) high-anglefaults and (2) motionon a low-anglenormal andbelow (LD) theRuby Mountains"detachment" in fault. For reasons discussedbelow, 30 ø is used here as b•e,.,don fissiontrack d•Ling [afiicr DoJd•.• ½t •.l., 1986]. the cutoffbetween low-angle and high-angle faults. In the first model the detachment is assumed to mark a brittle-ductile transition which starts out at a depth of 7-10 km depth in the crust [Profett, 1977; Nevada to infer rapid cooling from above 500øC to Rehrig and Reynolds,1980; Miller et al., 1983; Gans et below 300ø. In a fissiontrack studyof myloniticrocks al., 1985]. The crust above the detachmentdeforms as a from the same area, Dokka et al. [1986] find that set of closely spacedparallel normal faults, initially cooling from about 300øC to below 70øC occurred slippingat a high angle. As the faults slip, they must between about 25 and 23 Ma [see Figure 2]. If the be rotated to a shallower dip simply to conserve geothermalgradient is assumedto have been 25øC/km, volume. The detachment is considered to be the surface thenthis wouldgive a rate of upwardmotion of greater wherefaults end and more ductile(pure) shearbegins. than 4 km/m.y. Mylonites from the Whipple When the set of upperplate faults has been rotated to a Mountain core complex in California have also been shallowangle so that they canno longerslip, a new set studiedusing 40Ar/39Ar and fission track methods of high-anglefaults cuts them. [Davis, 1988;Dokka andLingrey, 1979]. Thesestudies This model is in accord with simple rock indicateupward rates of motionfor mylonitesbelow the mechanicstheory and seismicobservations discussed Whipple Mountain detachmentwhich are as great or below. However, it is difficult to produce the greaterthan for the Ruby Mountaindetachment. sharpnessof the contactbetween metamorphic grades Pseudotachylite,a rock which is thought to be which is observed.It is difficult to understandwhy the formedonly duringearthquake faulting, has been found detachmentshould continue to be a boundarybetween in association with detachments. This has led some areasdeforming differently as it is broughtclose to the authors to claim that low-angle fault slip may be surface.If the detachmentsurface began moving in the seismogenic[John, 1987]. depth range of the brittle-ductiletransition, then, as it Core complexeshave had a great influence on moves up, it should move out of the brittle-ductile discussionsof extension,since they are foundin one of transition. This is true even at extension rates of the few accessibleareas which appear to have been several cm/yr (see thermal calculationsof Buck et al. extended by large amounts. Continental margins, [1988]). Even if the detachment were to remain a 962 Buck: Flexural Rotation of Normal Faults boundarybetween faulting and more distributedpure Jackson[1987] reportson a worldwide survey of strain, there should be a gradational contact fault plane solutions for large continental normal between metamorphicgrades across the detachment. faultingearthquakes. He showsthat the overwhelming Simple shear along the upper plate set of faults and majorityof suchearthquakes nucleate in the depthrange rotation of the faults is equivalentto pure shearon a 6-15 km on faults dipping between 30 ø and 60 ø. grossscale [Jackson,1987]. of a crustal Among all earthquakesstudied whose faulting was columnwill lead to thinningof that column,but rocks observedto break the groundsurface, allowing one of from differentlevels will not be juxtaposed.Finally, to the nodalplanes of the faultplane solution to be pull a flat-lying detachmentsurface up from the bottom unambiguouslychosen as the fault plane,no event with of the upper crust and keep it relatively continuous a dip lessthan 30 ø wasobserved. wouldrequire extremely uniform extension of the upper The active low-angle normal fault model has crust. Active faultsobserved today in extendingregions difficulty explainingthe geochronologicdata from core tend to be associatedwith topographywhich reflects complexesdiscussed above. Figure3 showsthat a fault movementalong widely separatedfaults. Twenty-five dippingat 10ø would have to slip at a rate of at least2.5 kilometersis a typical spacingbetween faults in the cm/yr to explainthe rate of coolingof lower platerocks Basinand Range [Fletcher and Hallet, 1983]. To bring inferredfrom fissiontrack studies. The present-dayrate a detachmentup from depthwould require a muchcloser of extensionof the entirewidth of the Basinand Range spacingof faults [only a few kilometers]and extremely provinceis estimatedfrom geodeticmeasurements to be uniform slip on these faults. A special mechanismis less than about 1 cm/yr [D. E. Smith, personal neededto accountfor the closespacing of primaryfaults communication,1988]. To matchthe geochronologic requiredby thismodel. data, several times the extension rate for the entire Basin The second model assumes that the detachment and Range would have to be concentratedon a single surfacewas an activelow-angle normal fault [Wemicke, low-anglefault. 1981]. This does explain the sharp contactbetween In a study which did not specificallylook at the rocksof very differentmetamorphic grade across a fault surface. It is alsoin accordwith observationsof large lateral offsets between structural markers across a detachment[e.g., Davis, 1988]. In this modelthe faults in the upperplate (seeFigure 1) abovethe detachment 15 FaultSlip Rate -• 450 are consideredto be due to structuralcollapse of the hangingwall of a low-anglefault. The major problem with this model is that low-anglenormal fault motion FaultDip is not favoredin an extendingbrittle layer [e.g., Jaeger = o and Cook, 1959] and that low-angle normal fault earthquakesare seldomobserved seismically [Jackson 3OO and McKenzie, 1983; Jackson,1987]. Simple fault mechanicstheory [Anderson, 1951] predictsthat the preferredorientation for activenormal faults is for dips between 45 ø and 70 ø from the horizontal. This theoryis basedon the assumptionthat themaximum compressive C•l is verticaland the minimumcompressive stress c• 3 is horizontal.This 150 assumptionis in accordwith stressmeasurements made ß in boreholesaround active faults [e.g., Zoback and Model Dependent ..' Healy, 1984]. Sibson [1985] points out that for a typical staticrock frictioncoefficient of 0.75, frictional Path...... '" normal slip failure cannotoccur on planesdipping less than 37 ø. Bradshaw and Zoback [1988] consider the I I { effect of pore pressureon frictional sliding of normal 10 8 6 4 2 0 faults. Evenwith overpressuring,they find that for slip Time after start of fault motion(m.y.) to occur on faults dipping less than 20ø requires Fig. 3. Depth-timeand temperature-timecurve for rocksjust frictional sliding coefficients close to zero. below an active fault which is slippingat 0.5 cm/yr for two Overpressuringshould be difficult to achieve on a different dip angles. Temperatureis related to depth by normal fault, since any secondarynormal faults in the assuminga constantgeothermal gradient of 30 øC/km. See upperplate should provide an escapepath for . text for discussion. Buck: Flexural Rotation of Normal Faults 963 problemof metamorphiccore complex origin a different normalfaulting model is considered.Buck et al. [ 1988] calculatethe effects of assuminglocal isostaticcom- Unfracturedrock/ / pensationof topographyproduced by movementon a / / Existing planar normal fault. This results in lower plate I • (footwall)rocks, originally adjacent to the activenormal fault, being exposedat the surface. Mylonites formed by strain at midcrustallevels would be carried to the surfaceif this model were correct. The essentiallyflat fault surfaceis "abandoned"in that it no longer moves when it is rotatedinto a horizontalposition. For a high NormalStresss O'N normalfault dip angle(>30 ø) a moderateslip rate on the fault would causelower plate rocksto be uplifted and Fig. 5. Plot of shearstress x versusnormal stress cr n for cooled at a rate compatiblewith geochronologicdata frictional sliding on a fault (Byedee'slaw) and for fracture discussedabove (see Figures 2 and 3). However, the (Coulombfailure) of rock. The Mohr'scircle showsthat when the angleof the fault surfaceto the minimumprincipal stress local compensationmodel doesnot explain how upper hasrotated/50 from the optimumvalue, 0, thenfailure of the plate rocks could end up on top of an abandoned rock can occur. Here it is assumed that the mimimum stress detachmentsurface. It alsoignores the fact that thereis is horizontal. no reasonto expectzero flexural strengthof the crustin an extendingregion. Evans, 1979]. Extensional stressesshould reduce the CONCEPTUAL PHYSICAL MODEL flexural rigidity of the crust,as describedbelow and by Bodine et al. [1981]. High crustaltemperatures should The results of Buck et al. [1988] prompted the lead to lower flexural strength.The Basin andRange is presentstudy, which is a first attempt to considerthe undergingextension, and heat flow there is abouttwice possibleeffects of regional isostaticresponse to large that of averagecontinental crust [Lachenbruch and Sass, offsets on a normal fault. The model is based on three 1977]. However, analysisof the correlationof gravity assumptions: and topographyin the Basin andRange by Bechteland 1. The isostaticresponse to normal fault motion is Forsyth [1987] indicates a finite flexural strengthfor of regionalextent. that region. 2. If a fault segmentis significantlyrotated from As noted earlier, fault mechanicstheory predicts the optimumangle of slip relative to the crustalstress that faults have a range of optimal orientationsto the field, then a new planar fault orientedin the optimum principal stresses. According to the Mohr-Coulomb directionwill replaceit. criterion [Coulomb, 1773], shear failure will occur once 3. The fault cuts the entire upper crust (Figure 4) the shearstrength and internalfriction of a material are and continuesto cut throughthe samearea of the base overcome[Jaeger and Cook, 1959]. Rock mechanics of the upper crust. Fault motion is always considered experimentsshow that when a fault is rotated away to nucleatein the sameregion at the baseof the upper from the optimal angle for shear, frictional sliding crust. alongthat fault eventuallybecomes more difficult than Theseassumptions have a basisin observationsand creatinga new fault throughunfractured rock (seeNur et theory. First, basedon understandingof the matedais al. [1986] for further discussion). Laboratorystudies that makeup the crust,one would expectthat the crust showthat the shearstress •; requiredfor motion on an shouldhave a finite flexural strength[e.g., Goetze and existingfracture is controlledby the cohesivestrength S and the normalstress o n actingon the fracture[e.g., Byedee, 1978]:

x =s+po n (1) Elastic Upper where I.t is the coefficient of . This relation is plotted on a Mohr circle diagram in Figure 5 for two Inviscid Lower Crust valuesof the cohesivestrength. S 1 is the cohesionfor an existingfault and SO is for unfracturedrock. As Fig. 4. Schematicof a normalfault cuttingthe entireupper long as a normal fault remains close to the optimum crust. The baseof the uppercrust is alwaysat a fixed depth, dip angle for frictional sliding 0, which for this as shownby the dashedline. example is about 60ø, no new faults will be cut. 964 Buck: Flexural Rotation of Normal Faults

However, if the fault rotates so much that the Mohr of parametersdetermined by laboratorymeasurements. circle comes to intersectthe line representingshear Extrapolationfrom laboratorystrain rates to geologic stressrequired to cut a new fracture,a new fault will be strain rates is not well quantified [Goetze and Evans, cut. The amountof rotation,•0, requiredfor thecutting 1979]. Uncertaintiesin the averagecomposition of the of a new fault dependson threevalues: (1) the difference crustmay lead to largeuncertainties in estimatesof the betweenthe cohesivestrength of the unfracturedrock, averagestrength of partsof the crust[e.g., Kusnir and So,and the cohesive strength of preexistingfaults, S1; Park, 1987]. Offset of a primary fault may lead to (2) the coefficient of friction g=tant•, and; (3) the motionon otherfaults. Thesesecondary faults may not absolute magnitude of the horizontal and vertical be easily described using . stresses(Ol and03). Theangle of rotationis Erosionand sedimentationact as loadsdeforming the crust. It is appropriateto first considersimplified calculations for crustal deformation. If the results of =_ cos-1[ 2 (So- S 1) cosO ] (2) thisstudy are considered interesting then more complex 2 (o•-o 3) andrigorous models should be constructed.

For the example shown in Figure 5 this rotation is Approximations about 15ø. The value of g and S1 for an existing fracture shouldbe relatively independentof rock type Two majorapproximations are made in formulating [Byeflee, 1978]. The cohesivestrength of unfractured numerical calculations. First, the is rock, however, depends strongly on rock type. consideredto be uniform throughthe crust with the Therefore •0 must be considered a variable in the model. maximumprincipal compression vertical. Second,the For most estimatesof g for crustalrocks the preferred regional isostatic response of the crust to loads is angle of slip is about60 ø [Sibson,1985]. computedusing the thin plate flexure approximations The idea that a normal fault could cut the entire [e.g.,Timoshenko and Woinowsky-Krieger, 1959]. upper crustand not continuethrough the lower crustis A uniform stress field is assumed so that the also basedon theory and observation.The seismically Mohr-Coulomb failure criteria, describedabove, can be determinedfocal depth of continentalnormal faulting usedto predict when an active fault will be abandoned earthquakesare stronglyclustered in the range6-15 km and replaced by a new fault. Stresses due to [Jackson, 1987]. Gabbros and diorites, which are topography,plate bending, and secondaryfaults may thought to constitutemuch of the lower crust, should contribute to a nonuniform stress field. These effects flow at lower crustal temperatures and make the determinationof a preferredfault orientation [Caristan, 1982]. At geologic strain rates, little extremelycomplex. deviatoric stress could be stored in lower crustal rocks. Many authorshave modeled regional compensation Fault motion at midcrustal depths may produce by treating the lithosphereas a thin elastic plate mylonites. A combination of grain size, mineral floating on an inviscid [e.g., Vening Meinesz, assemblages,and crystalographicfabrics is likely to 1941; Walcott, 1970; Watts, 1978]. VeningMeinesz make a midcrustal zone weaker than the [1950] used thin plate theory to approximatethe surroundingrock [e.g., White, 1976]. This may cause responseof the lithosphereto normalfaulting. Several fault motion to nucleaterepeatedly in a localized area. different approximationsare implied by the the thin Also, localized intrusion of magma may make a plate theory [Timoshenko and Woinowsky-Krieger, particular area of crust weaker than other areas. The 1959]. Among them are that the plate is thin compared correlation in time and space between nonbasaltic to the wavelength of its deformation, vertical volcanism and extension within core complexeshas displacementsare small compared to the flexural beennoted by severalauthors [e.g., Glazner and Bartley, wavelength,vertical normal stresses within the plate are 1984; Gans, 1987]. small comparedto plate bendingstresses, and the plate is perfectly elastic. Small vertical displacementsare NUMERICAL MODEL required for the derivation of a simple differential relationbetween loads p(x) and verticaldisplacements Quantificationof this conceptualmodel is a very w(x): difficult task. The stresses and strains within crust undergoing large normal fault offset may be quite (3) complex. Recent investigationsin rock mechanics V2(DV2w) + p gw =p suggestthat the crust may be divided into several regionseach with a unique flow law [Scholz, 1987]. whereD(x) is the flexuralrigidity of the lithosphere,p Eachtype of deformationis describedby a differentset is the crustaldensity, g is the accelerationof gravity, Buck: Flexural Rotation of Normal Faults 965 and V2= 32/3x2 + 32/3y2 [Timoshenko and The problem of the responseto large offset of a Woinowsky-Krieger,1959]. This relationdepends on normal fault has recently been treated by several the neglectof membranestresses and strainsand on the authors. C. P. G. King et al., (The growth of use of the small angleapproximation. Equation (3) is geological structuresby repeated 1; solvedfor pointson a uniformlyspaced grid usingan Conceptual framework, submitted to Journal of implicit numericalmethod. Geophysical Research, 1988) and R. C. Stein et al., Comparisonsbetween predictions of thin plate (The growth of geological structuresby repeated theoryand more general theories of elasticplate bending earthquakes2; Field examplesof continentaldip slip have been made by several workers. Comer [1983] faults,submitted to Journal of GeophysicalResearch, solvedthe generalproblem of the responseof an elastic 1988; hereinafter referred to as RCS [1988]) use the plate of finite thicknessto harmonicloads, assuming formulation of Rundle [1982] to model the vertical small vertical displacements.His resultsshow that the deflectionof elasticupper crust to fault motionand thin thin plate resultsare accurateapproximations of thick plateflexure approximations to describeits responseto plate resultswhen the flexural wavelengthis greater sedimentloading. They modelfault offsetsas great as 5 than about6 timesthe elasticplate thickness.Savage km and requireeffective flexural rigiditiesas low as 2 and Gu [1985] usedthe thin plate approximationin km to matchthe observedshape of crustallayers. J.K. modeling a normal fault using the same geometry Weisseland G. D. Kamer, (Flexuraluplift of flanks assumedhere. They show that the surfacedeflection due to tectonicdenudation of the lithosphereduring approximatedby thin plate theorycompares well with extension, submitted to Journal of Geophysical the deformationpredicted by a two-dimensionalsolution Research,1988) usethin plate flexureto modelnormal for small displacementsof an elastic layer over a fault offset. They successfullymodel both oceanicand viscoelastichalf-space [Rundle, 1982]. The lengthof continental normal faults. The treatment of the normal time after faultingwas chosento be long comparedto fault problemused by all of these authorsis very the viscousrelaxation time of the half-spaceso that it similarto that usedin this paper. wouldapproximate the behavior of an inviscidlayer. In the presentmodel the thin plate theory is not Thin plate theory has been appliedto geologic rigorouslyjustdied since finite rotations are required for problemseven in caseswhere some of the thin plate abandonmentof faults. It is the purposeof this study approximationsare not rigorouslyvalid. The bending to make only qualitativecomparisons with data on the stressesat subductionzones are sufficientlylarge that geometryof core complexstructures. One shouldnot part of the lithospheremay deformanelastically [e.g., view the numericalresults as quantitative predictions of Goetzeand Evans,1979; Chappleand Forsyth,1979]. the conceptualmodel. Rather, this is the first step in Thus the approximationof perfect breaks going beyond the "artist's conception" phase of down. When anelasticbehavior is treated,the rigidity interpretinga conceptualmodel. of the lithospherecan be shownto dependon plate curvature(e.g., Bodineand Watts [1979]; and discussion Model Formulation in the next section). When variationsin rigidity as a functionof curvatureare included,agreement between The model is describedby three stepswhich are observedand predicted topography at subductionzones repeatedin time. Thesesteps are illustratedin Figure6. is good [McAdoo et al., 1978; Chappleand Forsyth, The first stepis the offsetof a high-anglenormal fault 1979; Bodine and Watts, 1979; McNutt and Menard, due to a relativeseparation of the two sidesof the fault. 1982]. Several reference frames can be adopted without In the laccolith intrusion problem vertical changingthe modelresults. For simplicity a reference displacements are large compared to flexural frame is chosen in which the footwall is fixed. The wavelength. Pollard and Johnson[1973] and Jackson hangingwall side of the normalfault movesparallel to and Pollard [1988] use thin plate theory to treat the the fault anddownward, as shownin Figure6a. This is bendingof sedimentaryrock layersby the intrusionof a the sameframe of referenceused by Vening-Meinesz laccolith. They considerdata for Mr. Holmes in the [1950] and Savageand Gu [1985]. The next stepis the HenryMountains, Utah, wherethe inferredtopographic isostaticresponse to the fault offset. The treatmentof offset is 1.2 km. The wavelength of bending is the flexural effect of normal faulting is given by the approximatelyof 5 km. In this case,several of the thin thin plateequation (equation (3)). The flexuralresponse plate approximations are not strictly valid such as causesrotation of the fault at shallow depths while perfect elasticity,the small-angleapproximation, and deeperparts of the fault remain straight[Figure 6b]. the neglectof membranestresses. However, the simple The f'malstep is the cuttingof a new upperpart of the theory providesgood quantitativeagreement with the normal fault at the optimumdip angle, as given by shapeof deformedbeds [Jackson and Pollard, 1988]. Mohr-Coulombtheory (Figure 6c). 966 Buck: Flexural Rotation of Normal Faults

2 ETe (a) Offset D= (5) • x 12( 1-v 2 )

Footwall

or Young'sModulus E is takento be5 x 1010N/m 2 and Hanging Wall Lower Plate Poissson's ratio v is 0.5. The term effective elastic or Upper Plate thicknessTe [e.g.,Bodine et al., 1981] is usedsince the physical propertiesof the lithospheremay vary with depth.T e is thethickness of a perfectlyelastic plate, with given elastic constants, which has the same (b) Flex rigidity as the lithosphere. The wavelength of the flexuralresponse toa pointload is about 5(Te)3/4 in kilometersfor Te in kilometers.In thispaper the only parameterused to describethe flexural responseof the crustwill be Te, with the understanding thatit isrelated to D throughequation (5). Derivation of strains through the upper crust (c) Break New Fault requires a corollary approximation. The flexural responseto loadsis consideredto produceonly vertical displacements.Vertical columnsof crust are assumed to slip freely past each other. The only horizontal displacementsare due to offset of the fault. Points representingthe surface,the active fault, and pastfault tracesare followedthrough successive fault offsets. Fig. 6. Model formulation. (a) shows offset along a When fault offset occurs, the normal fault should high-anglenormal fault; (b) showsflexural responseto the topographicload causedby the offset. The flexure causes be rotated as shown in Figure 6. If small fault rotation of the high angle normal fault over a flexural displacementsare considered,then both the constant wavelength.In (c) the rotated fault is abandonedand a new stressfield andthin plate approximationsare valid. In extension of the fault, which is rooted in the lower crust, is this model it is assumedthat the responseto largefault cut throughthe top of the uppercrust. displacementsis qualitativelysimilar to the resultsfor small displacements.Large fault offsetsare requiredto producefault rotationswhich allow part of the fault to be abandoned. The fault offset representsa load which must be The activefault dipsinitially at an angle0. Fault isostaticallycompensated. In the chosenframe of offset is increased until the active fault has rotated a set referencethe hangingwall movesdown parallel to the amount fi0, at which time a new fault is cut. See fault duringoffset. The downwardcomponent of offset Figure 5c. As noted before, this is consistentwith at any positionx is fly(x). The load,p(x), due to this Mohr-Coulombtheory for a uniform stressfield. The offsetis a negativeload sinceit tendsto pull the surface new fault dipsat the originalangle 0 andintersects with up. For the hangingwall side,p(x) is given by the old fault at a specifieddepth, the nucleationdepth. The depth of nucleationis taken to be 15 km in all the p(x) =-fiy(x)p g (4) calculations.Varying this depth a few kilometershas little effect on the results. where p is the crustal density. Note that when the surfaceis horizontalthe fault offsetresults in p[x] being the same for all x positions. When the surfaceis not horizontal, the load due to fault offset is a function of x RESULTS position. The thin plate flexure equation (3) is used to Constant Flexural Rigidity describe vertical displacementsdue to loads p(x). Flexure tendsto spreadthe displacementsover a wide The results of one calculation for extension with an region even when the load is concentrated. The effectiveelastic thickness Te of 0.5 km areshown in wavelength of the responsedepends on the flexural Figure 7. For this calculation,0 is taken to be 60ø and rigidity D. The rigidity D canbe relatedto the effective fi0 is taken to be 5 ø. Model results for different elasticthickness of thelithosphere Te: amountsof extensionare shown in the figure. In the Buck:Flexural Rotation of NormalFaults 967 earlystage of extension,inactive faults are only slightly (a) Te= 1.0km rotatedand a topographichigh is developedon the footwall side of the active fault. As extensioncon- E 0 tinuesfaults at the surfaceare rotatedto a constantangle of about30 ø . The abandonedfaults progressively flatten •10 with depth,eventually becoming horizontal. The •20 footwallcontinuously moves up on a high-anglefault. The rocksbelow the detachmentcome up from greater depthsas the extensioncontinues. Fault-bounded (b) Te = 0.5km blockson top of the detachmentbecome smaller as extension continues. E 0 Theeffect of changingthe effective flexural rigidity of the crustis illustratedin Figure8. The plotsshow •10 thefaulting geometry for effective elastic thicknesses of Q20 1.0 km, 0.5 km, and 0.25 km. All lengthsin the calculatedfault geometries scale approximately with the Te to the3/4 power. This scaling is notsurprising (C) Te= 0.25km sincethe wavelength of theflexural response depends on E 0 Te3/4. Thescaling would be exact if thedepth of nucleationwere not at a finitedepth. The hanging wall •10 topographyisunreasonably large unless Te isextremely small. • 20 • • 20 km

Fig.8. Sameas Figure 7 fordifferent flexural rigidities: Te = 1 km,T e = 0.5km, Te = 0.25 km.

Te=0.5km 0 =60 ø 60 =5 ø

(a) Variable Flexural Rigidity • o Thestrong bending of theelastic lithosphere should •o leadto a loweringof T e. Basedon laboratory studies of the deformationof mineralswhich make up the crust c•2o andupper mantle [e.g., Goetzeand Evans,1979], bendingstresses should cause part of theelastic upper (b) crustto departfrom perfectelasticity. This should occurbecause stresses within the plate are limited by • o thestrength of thelithosphere. A plotof theestimated maximum stressdifference which can be maintainedin •o theupper crust (the yield stress envelope) is shownin c•2o Figure9a. Detaileddiscussion of theconstruction of thistype of figureare given by Braceand Kohlstedt [1980]. At shallowdepths and low temperaturesthe (c) frictionalresistance along rock fracture planes controls E 0 thestrength required for deformation. At depthswhere temperaturesare high, rocks deform by ductilecreep. El0 The maximumductile strength may be describedby a temperatureand composition dependent flow law. For r320 I I 20 km Figure9a the crustis takento be describedby flow constantsfor diabase[Cadstan, 1982], and a linear Fig. 7. Topographyand positions of activeand abandoned temperaturegradient of 25ø/kmis assumed. faultsresulting from a calculationwith a faultangle 0 of 60ø, Using the conceptthat lithosphericstresses are aneffective flexural rigidity of 0.5km, and a rotationangle •50 of 5ø. A lineat 2 km depthis alsoplotted. Horizontal offsets limitedby a yield stressenvelope, a relationbetween of approximately(a) 15 km, (b) 30 km, and(c) 60 km are platecurvature and plate rigidity can be derived [Chapple shown. There is no verticalexaggeration. andForsyth, 1979; Bodine and Watts, 1979; McNutt 968 Buck: Flexural Rotation of Normal Faults

(a) Deviatoric Stress (MPa) relation between curvature and effective elastic thickness -100 0 100 20'0 300 of a plateT e. Sincecrustal composition and thermal state are uncertain, this treatment is warranted. The Cataclastic Appendixgives this derivation. Figure 9c showsthe By-Hyd By-Hyd variation of the effective elastic thicknessT e and 5 Elastic curvature w" for the yield stressenvelope shown in Figure9b. Whencurvatures are greater than 10 '6 m'l,

Ductile the platerigidity goesdown markedly. The lowering of rigidity by plate bending is incorporatedinto the calculation of the responseto normal fault offset. After a fault offset of 200 m the vertical responseto the new load is calculatedusing equation(3). The curvatureat eachpoint is calculated andthe local reduction in Te is givenby equation(A5) (in the Appendix). The grid spacingis takento be 200 (b) Deviatoric Stress (MPa) m. To insurenumerical stability, the valuesof Te are -200 - 1o o o 1oo Obd averagedusing Gaussian interpolation with a half width of 1 km. The width of the Gaussian distribution used for interpolationhas litfie effect on the modelresults as long as it is greater than the minimum flexural wavelengthresolved by the grid spacingbut not more than several times this distance. Using the new distributionof Te, theresponse to faultoffset loading is recomputed.For each offset the distribution of Te and curvatureis iterated to convergence(until changesin vertical offset are less than 40 m between iterations). The fault offset is increaseduntil the rotation of the top of the fault exceeds/50,as in earlier calculations.

(c) Te Constant (a) O.8 • Effective Te (w") • Elastic

Te (0) 0.4 O.0.26 ,.•ness I I I I I I I 10-10 10-8 10-6 10-4 10-2 =10

Plate Curvature,w"(m -•)

Fig. 9. (a) Yield stressenvelope showing the stresswhich (b) Te (w") causes rock failure as a function of depth for a given temperaturestructure and strain rate within the crustal lithosphere.(b) a simplified yield stressenvelope used to • 0 calculatethe approximatedependence of the effectiveflexural rigidity and the curvatureof the lithospheregiven in the Appendix. (c) the relation betweenthe effective elastic lithosphericthickness Te asa functionof platecurvature, w" • 10 for the yield stressenvelope shown in (b) as givenby equation •) 20 E0 t- I i (A5). Te is normalizedby itsvalue for zerocurvature. 20 km Fig. 10. Two treatmentsof offseton a singlefault required to and Menard, 1982]. For the purposeof this paperthe produce5 ø of rotationof the fault. (a) showsresults with yield stress envelope can be approximatedby a constantflexural rigidity and (b) showsthat when T e is taken diamond-shapedenvelope, as shown in Figure9b. This to be a functionof platecurvature a muchsmaller fault offset simplifiedshape allows the derivationof an analytic is requiredto produce5ø of rotationof thefault. Buck: Flexural Rotation of Normal Faults 969

The effectiverigidity of the crustwill be reducedby (a) 50 = 5 ø even relatively small fault offsets. This is illustratedin 5 Figure 10 for two caseswhere fault offset is continued until a 5ø rotationof the top of the fault is achieved. In thecase where T e is takento be a constantthickness of 0 5 km, a fault offset of 20 km is neededto producethe rotation.In the casewhere T e is initially5 km thick but is thinnedby bending,only 6 km of fault offset is required. Also, the wavelengthof the flexuralresponse to fault offset is dimimishedby more than a factor of 10 by the end of this calculation. :•1o The results of one calculation with repeated •2o abandonmentof fault segmentsare shownin Figure 11 with Te = 5 km forcrust of zerocurvature. T e dropsto very low valuesin the regionadjacent to activefaulting. This has the effect of reducingthe model topographic ,se = lO ø (b) relief since much smaller fault offset is required to 5 produce a given rotation of the active normal fault. This treatmentof the problem still leadsto very large topographicslopes, especially on the hangingwall side of the fault. 0

Calculation with Sedimentation •- o Largetopographic slopes should lead to erosionand :•1o structuralcollapse of the hangingwall block. Erosion ratesdepend on rock type andenvironmental factors in a •20 • i 20 km complexway. Collapseof the hangingwall block by faulting is also difficult to model. Both processes Fig. 12. Results of variable rigidity calculationsin which should act to fill in the topographic low over the sediment has been added to fill in all levels below 1000 m depth. Two casesof assumedangle of fault rotationbefore fault abandonment are shown: •50 = 5 ø and •50 = 10 ø.

o =60':' 8e =5 ø actively slipping fault. in the interest of simplicity, only the effect of sediment loading is treated here. Sedimentis madeto fill in all depthsbelow a depthof 1 km. The sedimentis assignedthe samedensity as the restof the crust. The modelsediment can be thoughtof as coming from out of the plane of the calculation, since no erosionis accountedfor in this plane. The local plate rigidity is again taken to be a function of platecurvature through equation (A5). Figure12 shows the results of two calculationswith sedimentloading •1o added to the normal fault loading. The calculations differ only in the amount of fault rotation before •20 I ' i abandonment •0. 20 km The effect of the model sedimentation is to reduce the topographicslopes resulting from fault motion. Fig. 11. Topographyand positions of activeand abandoned This actsto makethe stressfield morenearly uniform, faults from a calculationin which the rigidity dependson curvatureof theupper crust. Active fault dip 0 = 60ø and•50 = justifyingthe use of Mohr-Coulombtheory to calculate fault orientation. The inf'fll results in the abandoned 5ø . The upper line showsthe effective elastic lithospheric thicknessTe for positionsacross the fault system. T e = 5 km detachmentbeing buried under wedges of the sediments, for areas with zero curvature. ratherthan being exposed at the surface. 970 Buck: Flexural Rotation of Normal Faults

DISCUSSION AND CONCLUSIONS the Basin and Range primarily seems to produce high-anglenormal fault structures[Fletcher and Hallet, Severalfeatures of metamorphiccore complexes are 1983; Jackson, 1987]. These calculations do not qualitatively reproducedby this model. The model addressthis problem, but the model results do have predicts(1) a nearly flat-lying abandonednormal fault implicationsfor it. In this model, crustalextension is below slicesof upperplate rocks and sedimentary infill, accommodatedby lowercrustal material being pulled up (2) a strongcontrast in metamorphicgrade acrossthis and cooled. Were the Moho pulledup by repeatedfault abandoned"detachment," and (3) rapid movementof motion it might becomeenergetically more favorable lower plate rocks from midcrustaldepths to shallower for faultingand extension to occurin an adjacentarea of depths. unthinnedcrust [Artyushkov, 1973]. Viscousflow in In the flexural rotation model the major slip the lower crustwould tend bring materialinto the area motion takes place on a high-angle fault. Mylonites of crustalthinning and flattenMoho. The rate of flow could be formed by strain in the midcrustand carded in the lower crustis directly proportionalto rapidly upward in the footwall of a high-anglefault. there. The lower crust must flow sufficientlyin the For example,the model shownin Figure 12b predicts time between major faulting events to maintain a that material from 15 km depth would be broughtto relativelyflat Moho. High crustaltemperatures could within 3 km of the surface with 10 km of extension. lower the viscosity enough for this to happen [e.g., Observationsin core complexesshow that the material Kusnir and Park, 1987]. The seismicallydetermined below the detachment(at 0-5 km depth) originatedin Moho in the Basin and Range is nearly flat the middle of the crust (at 10-15 km depth) and was [Allmendingeret al., 1987], indicatingthat the lower broughtup in as little as 2 m.y. [Dokka et al., 1986; crust can flow fast relative to other areas where there is Davis; 1988]. Extension rates as low as 0.3 crn/yr measurableMoho topography. When core complexes could do this. Active high-angle fault motion is were formed,the local crustaltemperatures may have consistent with seismic observations of normal fault beengreater than today. Severalworkers have noted the earthquakes.It is alsoconsistent with the occurrenceof correlation of nonbasaltic volcanism with extension in earthquakegenerated pseudotachylites being associated core complexes [Glazner and Bartley, 1984; Gans, with detachments. 1987]. This type of volcanismmay reflectmelting of The results of calculations with sediment infill the lower crust. Temperatureshigh enoughto cause suggestthat suchloads may be an importantcontrol on meltingshould make the averageviscosity of the lower detachment geometry. Structural collapse of the crustquite low. Thusextension might be concentrated hanging wall block would also act to fill in the in the area of a single fault when the crust there is topographiclow over the actively slipping fault, as anomalouslyhot. would volcanics. All infilling material should New geologicdata could be usedto testpredictions contributeto the wedgeswhich bury the detachment. of the flexuralrotation model. This modelpredicts that The infill would be essentiallyunmetamorphosed, while fault blocksare formed sequentially, becoming younger the rocksbelow part of the detachmentwould be highly toward the hanging wall. In the altemative model of metamorphosed. One can speculatethat differences rotating sets of high-angle faults [e.g., Miller et al. betweencore complexes may be dueto differenterosion 1983] the fault blocks are formed contemporaneously. rates and to the resistance of the hanging wall to The activelow-angle model [Wemicke, 1981] makesno collapse. predictionof timing of block faulting. Radiometric A low effective elastic thickness of the crust is dating of volcanic strataformed at the time of active required to producea geometryof abandonedfaults fault motion in core complexes may allow de- which are of about the same dimensions as observed for terminationof the sequenceof faulting. In the Death core complexes. Extremely high crustal temperature Valley area, which may be a nacentcore complex,the gradientscould lead to such low elastic rigidity. A sequenceof block faulting is consistentwith the betterexplanation is that plate bendingstresses and the flexural rotation model [J. D. Walker, personalcom- finite yield strengthof the crustlower rigidities. RCS munication, 1988]. (1988), in studying high-angle normal faults with More work is needed to test whether the displacementsup to 5 km, also require low flexural approximationsused here reasonablydescribe how the rigidities. The relation between plate bending and crust respondsto normal faulting. Two-dimensional rigidity canalso explain their results(see Figure 10). numericalcalculations of the effect of largeoffsets of a A major question about metamorphic core normalfault in a crustwith brittle and ductileregions complexesis why they are only found in a relatively must be done to confirm these results. The calculations small number of places. The present-dayextension of will be complexand dependon many physicalpara- Buck: Flexural Rotation of Normal Faults 971 meters.However, the similaritybetween predictions of this simple model and field observationsshould be 2 enoughto warrantfurther work on the isostaticresponse of the crust in extension. Te (x) = Tec (x) { [2Zbd +Tec(X) 2Zbd] } • [ Tec (x)]

The relationbetween T e andw" is shownin Figure9c Consideringa simplifiedyield stressenvelope as for Obd/ Zbd= 0.33kbar/krn. shownin Figure 9b, a relationbetween curvature of a lithosphericplate and its effectiveelastic thickness is derived following Bodine et al., [1981] and McNutt Acknowledgments. Thanks to Jeff Weissel for [1984]. The effectiverigidity of the lithosphere,D, is discussingthe treatment of the flexural responseto relatedto the bendingmoment of the plate, M, through normal faulting and to Mike Steckler for explaining the curvatureof the plate: subtle points of yield stress envelopes. Fernando Martinez and Robin Bell patientlydiscussed this model D(x)- M(x) (A1) as it was being formulated. Reviewsby JamesJackson, w"(x) Amos Nur, Bemard Coakley, Jim Cochranand Sharon Quayle led to improvementsin the text. Supportedby The moment is calculatedby integratingthe horizontal NSF grant OCE 8610213 and by an ARCO research stressesOxx through the plate: scholarship.Lamont contributionnumber 4324.

M(x)=f O'xx(x,z)z dz (A2) REFERENCES

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