Small-scale upper mantle convection and crustal … et al 2008.pdfSmall-scale upper mantle convection and crustal dynamics in southern California N. P. Fay, R. A. Bennett, and J. C

Small-scale upper mantle convection and crustal dynamics insouthern California

N. P. Fay, R. A. Bennett, and J. C. SpinlerDepartment of Geosciences, University of Arizona, Tucson, Arizona 85721, USA ([email protected])

E. D. HumphreysDepartment of Geological Sciences, University of Oregon, Eugene, Oregon 97403, USA

[1] We present numerical modeling of the forces acting on the base of the crust caused by small-scaleconvection of the upper mantle in southern California. Three-dimensional upper mantle shear wavevelocity structure is mapped to three-dimensional density structure that is used to load a finite elementmodel of instantaneous upper mantle flow with respect to a rigid crust, providing an estimate of thetractions acting on the base of the crust. Upwelling beneath the southern Walker Lane Belt and SaltonTrough region and downwelling beneath the southern Great Valley and eastern and western TransverseRanges dominate the upper mantle flow and resulting crustal tractions. Divergent horizontal and upwarddirected vertical tractions create a tensional to transtensional crustal stress state in the Walker Lane Belt andSalton Trough, consistent with transtensional tectonics in these areas. Convergent horizontal anddownward directed vertical tractions in the Transverse Ranges cause approximately N–S crustalcompression, consistent with active shortening and transpressional deformation near the ‘‘Big Bend’’ of theSan Andreas fault. Model predictions of crustal dilatation and the forces acting on the Mojave blockcompare favorably with observations suggesting that small-scale upper mantle convection provides animportant contribution to the sum of forces driving transpressional crustal deformation in southernCalifornia. Accordingly, the obliquity of the San Andreas fault with respect to plate motions may beconsidered a consequence, rather than a cause, of contractional deformation in the Transverse Ranges,itself driven by downwelling in the upper mantle superimposed on shear deformation caused by relativePacific–North American plate motion.

Components: 12,968 words, 12 figures.

Keywords: stress; dynamics; crustal deformation; small-scale convection; crust-mantle interaction.

Index Terms: 8164 Tectonophysics: Stresses: crust and lithosphere; 8120 Tectonophysics: Dynamics of lithosphere andmantle: general (1213); 8111 Tectonophysics: Continental tectonics: strike-slip and transform.

Received 12 February 2008; Revised 2 June 2008; Accepted 20 June 2008; Published 12 August 2008.

Fay, N. P., R. A. Bennett, J. C. Spinler, and E. D. Humphreys (2008), Small-scale upper mantle convection and crustaldynamics in southern California, Geochem. Geophys. Geosyst., 9, Q08006, doi:10.1029/2008GC001988.

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1. Introduction

[2] The forces driving active deformation of thelithosphere at a tectonic plate boundary are the sumof those driving global plate motion transmittedthrough the rigid plates from the far-field to theplate boundary [e.g., Atwater, 1970], and thosecreated locally by variations in local density struc-ture [Artyushkov, 1973; Fleitout and Froidevaux,1982; Molnar and Lyon-Caen, 1988]. On a globalscale, the stresses exerted on plates from flowinduced by the mantle’s internal density structure(e.g., sinking slabs) seem to be an importantcontribution to the sum of forces acting on plates[Becker and O’Connell, 2001; Steinberger et al.,2001; Conrad et al., 2004; Ghosh et al., 2006]. Onthe scale of a few 100 km it may be more difficultto determine the importance of small-scale convec-tion on crustal dynamics and deformation becausethe influence in actively deforming regions may beoverprinted by other processes such as plate inter-action. Actively deforming regions have a distinctadvantage, however, in that they provide the nec-essary observables to allow discrimination ofsuperimposed sources of driving forces. In thispaper we aim to resolve the role of small-scaleconvection of the upper mantle in driving defor-mation of the overlying crust in southern Califor-nia. It is clear that present-day deformation insouthern California is strongly influenced by plateinteraction stresses; we show that much of thedeformation that cannot be explained by plateinteraction derives from small-scale convection ofthe underlying upper mantle.

[3] Approximately 50 mm/yr of relative motionbetween the Pacific and North American plates[DeMets andDixon, 1999] is accommodated largelyby strike-slip deformation. The most active struc-ture, the San Andreas fault, accommodates half ormore of the present-day slip budget [e.g., Meadeand Hager, 2005], although its rate depends onlocation. In addition to strike-slip deformationthere is a nontrivial component of nonsimple sheardeformation such as block rotation, shortening anduplift in the Transverse Ranges [e.g., Dibblee,1975; Yeats et al., 1988; Jackson and Molnar,1990; Luyendyk, 1991; Donnellan et al., 1993;Morton and Matti, 1993; Spotila et al., 1998;Onderdonk, 2005; Spotila et al., 2007]. Kinemat-ically, the transpressive deformation in the Trans-verse Ranges can be considered a consequence ofthe ‘‘Big Bend’’ in the San Andreas fault (heredefined as the segment of the San Andreas faultbetween the San Emigdio Bend and San Gorgonio

Bend; see Figure 1), i.e., a large left-step in the right-lateral San Andreas fault system. The San Andreasfault is offset !120 km through the Big Bend(Figure 1), which is !8 times the average seismo-genic depth [Nazareth and Hauksson, 2004],!4 times the average crustal thickness of !30 km[Zhu and Kanamori, 2000; Yan and Clayton, 2007]and greater than the average lithospheric thicknessin southern California of !90 km [Yang andForsyth, 2006a; Humphreys and Hager, 1990].

[4] Dynamically, however, it is not clear why thelithosphere maintains this apparently energeticallyunfavorable geometry [Kosloff, 1977]. Mountainbuilding in the Transverse Ranges, i.e., thrustfaulting and crustal shortening and thickening,requires work against gravity and frictional andviscous resistive forces. There are other activestructures such as the Elsinore-Laguna Salada faultsystem, that if more active and connected with theCerro Prieto fault (Figure 1), would allow relativePacific-North American motion on a more through-going transform system and largely bypass the BigBend geometry. Numerical modeling has shownthat an effect of the present-day geometry of theSan Andreas fault is to promote slip on other slipsystems such as the Eastern California Shear Zoneand offshore faults [Li and Liu, 2006], indicatingthe current San Andreas geometry should not bestable.

[5] However, the San Andreas fault appears to bethe dominant plate boundary fault at presentthroughout most of southern California slipping!20–35 mm/yr [e.g., Sieh and Jahns, 1984;Weldon and Sieh, 1985; Meade and Hager,2005; Bennett et al., 2004; Becker et al., 2005;Fay and Humphreys, 2005]. The Big Bend geom-etry of the San Andreas fault system has existedsince the opening of the Gulf of California (!6 Ma)and likely longer [Wilson et al., 2005], indicatingtoday’s transpressive geometry is a persistent tec-tonic feature of the plate boundary at least over thepast few million years. One possible explanation isthat the San Andreas fault is extremely weak com-pared to the surrounding crust and nearby faults[Zoback et al., 1987; Townend and Zoback, 2000,2004], perhaps owing to its greater accumulatedoffset and structural maturity [Wesnousky, 1988,2005], and the work required to generate a new,straighter fault system is greater than that to driveshortening and mountain building in the TransverseRanges.

[6] Alternatively, upper mantle processes, namelydownwelling beneath the Transverse Ranges, may

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act to draw in the overlying crust and causeshortening superimposed on the shear deformationrelated to plate motion. A number of seismicstudies have shown that the upper mantle velocitystructure beneath southern California is ratherheterogeneous [Raikes, 1980; Humphreys andClayton, 1990; Jones et al., 1994; Kohler et al.,2003; Boyd et al., 2004; Yang and Forsyth, 2006a;Tian et al., 2007]. The dominant upper mantleseismic features are (1) a roughly circular high-velocity body beneath the southern Great Valleyand Sierran foothills adjacent to a low-velocityregion beneath the high elevation of the SierraNevada and southern Walker Lane Belt, (2) lowvelocities beneath the greater Salton Sea area,likely associated with extension and mantle up-welling [e.g., Elders et al., 1972; Lachenbruch et

al., 1985], and (3) high-velocity anomalies beneaththe Transverse Ranges. This latter feature is themost important for this paper and has previouslybeen interpreted as a slab-like feature extending toat least 200 km [Humphreys and Clayton, 1990;Kohler et al., 2003]. Recently, Yang and Forsyth[2006a] have argued that the Transverse Rangevelocity anomalies extend to only !150 km andare separated into two nearly distinct anomaliescentered beneath the eastern Transverse Rangesand offshore near the Channel Islands (Figure 2).

[7] If these velocity anomalies derive largely fromtemperature variations, the velocity structure pro-vides a proxy for the temperature heterogeneity inthe upper mantle and its associated density structure.The high-velocity body beneath the Transverse

Figure 1. Map showing study area and geographic features discussed in the text. Solid lines show fault traces, andthe San Andreas fault (SAF) is indicated with a thick line. The Walker Lane Belt (WLB) extends from the Garlockfault (GAR) to the northwest along the eastern side of the Sierra Nevada Mountains. The Eastern California ShearZone (ECSZ) extends from the Garlock to the southeast. The western Transverse Ranges (WTR) and centralTransverse Ranges (CTR) lie to the west of the SAF, and the eastern Transverse Ranges (ETR) lie to its east.Approximately NW motion of !50 mm/a of the Pacific plate relative to North America [DeMets and Dixon, 1999] isshown. Dashed line shows the approximate projection of the southernmost San Andreas fault into the Great Valley toillustrate the !120 km offset with respect to its central California location. Gray scale gives smoothed elevationabove sea level. SJF, San Jacinto fault; ELS, Elsinore fault; LSF, Laguna Salada fault; CPR, Cerro Prieto fault; SEB,San Emigdio Bend; SBG, San Gorgonio Bend; SS, Salton Sea.

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Ranges has been interpreted as downwelling ofcool, and relatively dense, mantle lithosphere [Birdand Rosenstock, 1984; Sheffels and McNutt, 1986;Humphreys and Hager, 1990]. Houseman et al.[2000] and Billen and Houseman [2004] suggestthe velocity anomalies are a result of a gravity-driven lithospheric instability to account for theirdrip-like structure. Whatever the mechanism offormation, the forces associated with a heteroge-neous distribution of density in the upper mantlemust be balanced by viscous stresses associatedwith gravity-driven flow, or by displacing a hori-zontal density interface such as the Earth’s surface(dynamic topography).

[8] The focus of this paper is to quantify the three-dimensional anomalous upper mantle density struc-ture, the viscous flow induced by this densitystructure, and the effect on the dynamics anddeformation of the overlying crust. We calculatethe instantaneous viscous flow of the uppermostmantle with respect to a rigid crust; this approachallows us to isolate the stresses on the base of thecrust. Three-dimensional upper mantle densitystructure is derived from the seismic tomographymodel of Yang and Forsyth [2006a]. We do notattempt to include any large-scale shear in theupper mantle potentially induced by relative platemotion [e.g., Bourne et al., 1998; Molnar et al.,1999]; our study is restricted to the largely poloidalflow caused by gravity acting on a heterogeneousdistribution of density. The influence of uppermantle flow on crustal dynamics is evaluated bycomparison with regional deformation patternsdetermined from geodetic observations and torquebalance on a crustal block.

2. Three-Dimensional Seismic Velocityand Density Structure of the UppermostMantle

[9] We use the shear wave velocity model derivedfrom surface (Rayleigh) wave tomography of Yang

Figure 2. Upper mantle seismic velocity and densitystructure in southern California. Shear wave velocity(Vs) anomalies from the Yang and Forsyth [2006a]tomography model and inferred density are shown atdepth slices at (a) 70–90 km, (b) 110–130 km, and(c) 150–170 km. The area without stipple, in this andsubsequent figures, shows the well-resolved region ofthe seismic velocity model as defined by Yang andForsyth [2006a]. We use the entire velocity model(which extends beyond the limits of these maps) in ourmodeling as discussed in the text.


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andForsyth [2006a] to estimate the three-dimensionalupper mantle density structure in southern California.Here we only briefly discuss the velocity model.Readers interested in the details of the seismic data,results, and methodology should refer to Yang andForsyth [2006a, 2006b] and Forsyth and Li [2005].We choose the Yang and Forsyth [2006a] modelbecause (1) Rayleigh waves are most sensitive tothe depth range we are most interested in here (e.g.,!50–250 km), (2) they incorporate finite-frequencyeffects in their inversions, and (3) the velocitystructure they find is quite similar to that reportedin previous studies, giving us some confidence it islikely real.

[10] Figures 2a, 2b, and 2c present three represen-tative depth slices through the Yang and Forsyth[2006a] model at depths of 70–90, 110–130 and150–170 km, respectively. Velocity anomalies,dVs, are shown as percent deviations from a 1-Daverage velocity model. Three major fast (blue)structures beneath the southern Great Valley andTransverse Ranges are clear. Slow velocities (red)are strongest at shallowest depths in the southernWalker Lane and at all depths near the Salton Sea.The three dominant fast bodies, and the stressesthey induce on the overlying crust (discussed insection 3) are hereafter referred to as the SierraNevada anomaly, western Transverse Rangesanomaly, and eastern Transverse Ranges anomaly(see Figure 2).

[11] Seismic velocity variations depend primarilyon composition, temperature, partial melt, andanisotropy. In this paper we effectively assumethat the velocity variations are entirely thermal inorigin and adopt a constant scaling factor, g,relating density (r) variations to velocity varia-tions, given by

g " @ ln r=@ lnVs; #1$

[Karato, 1993]. In the upper mantle g is estimatedto be !0.2–0.3 [Karato, 1993; Steinberger andCalderwood, 2006]. We choose g = 0.2 such that,for example, a 5% velocity anomaly maps into a1% density anomaly (dr).

[12] This method of scaling velocity to density iscommonly used in global studies of mantledynamics [e.g., Lithgow-Bertelloni and Silver,1998; Becker and Boschi, 2002; Conrad et al.,2004], although shallow (above !325 km) uppermantle seismic velocity anomalies are oftenexcluded because they may derive from isostaticallycompensated compositional variations associated

with continental cratons and the tectosphere [e.g.,Lithgow-Bertelloni and Silver, 1998; Becker andO’Connell, 2001; Steinberger et al., 2001]. It isunlikely, however, that the velocity variations inthe shallow upper mantle in southern Californiaimaged by Yang and Forsyth [2006a] and othersare dominated by compositional variations becausethe magnitude of the velocity anomalies are muchgreater (!3 times) than that produced by chemicalsegregation of the upper mantle [Humphreys andHager, 1990; Jordan, 1975]. Anisotropy and par-tial melt can also be excluded leaving temperaturevariations as the most likely cause of the velocityanomalies [Humphreys and Hager, 1990]. Ourmodeling results (traction and stress magnitudes,see sections 3 and 4) scale linearly with g andtherefore if any fraction of the seismic velocityanomalies do in fact represent compositional var-iations that are isostatically compensated, we willoverestimate their influence on mantle flow. Wehave, however, chosen g conservatively to mini-mize this effect. In section 4.2 we show that theinfluence on crustal dynamics of mantle flow (withg = 0.2) is of similar magnitude to tectonic plateinteraction, indicating 0.2 is a reasonable value forg in the southern California upper mantle.

[13] Our reference one-dimensional density struc-ture (from iasp91 [Kennett and Engdahl, 1991])increases linearly from 3320 kg/m3 at 35 km to

3490 kg/m3 at 310 km and thus @ dr# $@z

!

!

!

g;dVs

is

positive though typically very small compared todr itself. The corresponding density anomaly struc-ture is shown in Figure 2 with the same color scaleas the seismic velocity anomalies. Density anoma-lies are typically ±10–30 kg/m3, with the largestanomalies in our study area at depths of !50–100 km. The choice of reference density model isnot particularly critical because the numericalmodeling results scale with respect to variationsin background density in the same way as withvariations in g, and the former is likely betterconstrained than the latter.

[14] The density anomaly structure shown inFigure 2 is used as input to our viscous flowcalculations, discussed in the next section. Thebackground density model is not included as itwould create only a lithostatic pressure that doesnot drive differential flow. The nonstippled area inFigure 2 shows the region of the seismic velocitymodel that is well resolved and most reliable [Yangand Forsyth, 2006a]. We use the entire velocitymodel (and inferred anomalous density structure),so as to avoid artificial truncation effects. We


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demonstrate later that restricting the density modelto within the resolved region produces very similarresults.

3. Numerical Modeling

[15] Our primary objective is to resolve the forcescreated by upper mantle density structure thatcontribute to loading the southern California crust.To that end, we calculate the viscous flow of theupper mantle and concomitant tractions on the baseof the crust. Importantly, by driving the flow withgravitational body forces acting on density anoma-lies, their geometry and magnitude known inabsolute value, we are able to predict absolutelevels of stress in the upper mantle and crust. Wesolve the three-dimensional equations of conserva-tion of mass and momentum for incompressibleNewtonian viscous flow using the finite elementcode Gale [Moresi et al., 2003; Landry andHodkinson, 2007; Landry et al., 2008]. We restrictour analysis to very small strains, essentiallyinstantaneous flow, because the seismic tomogra-phy model provides us with only the present-dayvelocity and density structure and we wish toisolate the contribution of sub-crustal densitystructure on crustal stress.

[16] The model domain is a Cartesian grid 1600 km% 1600 km in map view and extends to 1000 kmdepth. The model is centered on southern Californiathat is represented as an oblique-Mercator projec-tion about the Pacific-North American Euler pole[DeMets and Dixon, 1999]. Element spacing is25 km horizontally and 10 km vertically; thiselement resolution was chosen on the basis of theresolution of the seismic tomography model andfiner mesh resolution produces indistinguishableresults. The sides and bottom are held fixed. Wemodel the crust as a highly viscous layer 30 kmdeep that is fixed vertically and unconstrainedhorizontally. This allows the mantle flow to loadthe crust permitting us to monitor crustal stress.Density is assigned to the model by interpolatingthe three-dimensional density structure (Figure 2)to element centroids. The seismic velocity model isgiven in depth slices, each 20 km thick. Thus thedensity structure input into the modeling resemblesa slightly coarser version of Figure 2. Outside theextent of the seismic velocity model (e.g., below250 km), no density anomaly is assigned.

[17] In this paper we are concerned with thestresses acting on the crust from density-drivenupper mantle flow, not the strain in mantle itself.

Therefore, the modeling results we present here areindependent of the absolute value of the viscositychosen for the upper mantle. This is because weprescribe density and compute the flow responseand associated stresses; the stresses necessary tosupport the forces of the density anomalies aredictated by the densities and their spatial distribu-tion. We model cases of a uniform upper mantleviscosity with and without a high-viscosity lid, anupper mantle in which viscosity depends on tem-perature, and a relatively weak lower crust.

3.1. Uniform Upper Mantle Viscosity

[18] Figures 3 and 4 present the vertical andhorizontal tractions on the base of the crust for auniform viscosity upper mantle. Figure 3 gives thevertical normal stress at element centroids. Abovethe three major positive density (negatively buoy-ant) bodies, the western Transverse Ranges, easternTransverse Ranges and Great Valley anomalies,vertical stresses are negative (act to pull the crustdown) with a maximum magnitude of !9 MPaoffshore. The slowest seismic velocities and largestpositive vertical stresses (!14 MPa) occur in thesouthern Walker Lane Belt, just to the east of theSierra Nevada anomaly. dVs of &5 to &6% at 50–70 km depth indicates a likely complete absence oflithospheric mantle there [Yang and Forsyth,2006a]. Relatively warm asthenosphere is thoughtto have passively upwelled to fill the regionvacated by delaminating lithospheric mantle [Zandtet al., 2004; Le Pourhiet et al., 2006]. Yang andForsyth [2006a] show that the region of lowestseismic velocities corresponds well with the locusof Pliocene and Quaternary volcanism. If anypartial melt, which can have strong retardingeffects on shear wave velocities [Hammond andHumphreys, 2000], is retained in the upper mantle,we may be over estimating the inferred densityanomaly and vertical stresses.

[19] These vertical stresses should vertically de-flect the crust. The predicted dynamic topographycan be estimated by dividing the radial stresses bythe deflected density contrast (e.g., 3300 kg/m3)and gravitational acceleration. This gives a maxi-mum upward static deflection of !0.4 km in theeastern Sierra Nevada and southern Walker Lane,and maximum downward deflection of !0.3 kmoffshore in the vicinity of the Channel Islands. Thissimple isostatic calculation overestimates the actualdynamic topography because these relatively shorthorizontal wavelength vertical loads will be partlysupported by flexure of the elastic crust. While the


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total deflected mass must be conserved, flexure ofan elastic plate should broaden and smooth anysubsidence or uplift and thereby decrease localamplitudes. Nonetheless, in the southern SierraNevada active subsidence and sedimentationappears to be burying mountainous topography[Saleeby and Foster, 2004], suggesting at leastsome of the negative dynamic topography pre-dicted by our models may be real, and possiblyincreasing with time. In the remainder of this paperwe focus on the horizontal tractions.

[20] The horizontal tractions at the Moho areshown in Figure 4. Downwelling of the densematerial offshore of the western Transverse Rangesproduces a nearly radial pattern of convergenttractions. Traction magnitudes are typically 1–2 MPa and strongest (!2.7 MPa) onshore to thenorth of the center of the downwelling region.

Tractions beneath the Eastern California ShearZone and western Mojave Desert are convergingon the eastern Transverse Ranges (San BernardinoMountains) and Mojave segment of the SAF. Thepattern here is asymmetric with tractions muchlarger to the northeast of the San Andreas faultthan to the south. To the north of the San Andreasfault tractions have an approximately SW orienta-tion and magnitude of '2.5 MPa with a maximumof 3.2 MPa. Horizontal tractions north of !33.5!Nandwithin the Salton block, between the San Jacintoand San Andreas faults, are nearly parallel to thosetwo faults and directed toward the San BernardinoMountains, and provide some of the force drivingconvergence there [Fay and Humphreys, 2006].Within !100 km of the Salton Sea, horizontaltractions are typically less than 2 MPa and show aclear radial pattern, consistent with horizontal di-vergence of buoyant upwelling.

Figure 3. Vertical normal stresses (colored dots) on the base of crust caused by viscous flow of the upper mantledriven by the three-dimensional density structure in Figure 2. Results are shown at element centroids, and positive/negative indicates upward/downward directed vertical normal stress. The largest positive vertical stress occurs in theWalker Lane Belt above the positively buoyant, slow velocity anomaly there. The largest negative vertical stressoccurs offshore above the negatively buoyant fast anomaly. As in Figure 2, the nonstippled area shows the region ofseismic velocity model that is best constrained, and therefore we focus on this area in our interpretation.


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[21] The horizontal tractions along the California-Nevada border have a maximum magnitude of!3.5 MPa and are directed to the NE. These arecaused by a combination of strong upwelling anddivergence of the buoyant material beneath thesouthern Sierra Nevada and southern Walker LaneBelt, and a relatively deep downwelling in south-eastern Nevada. This latter velocity anomaly liesoutside the well-resolved region of the Yang andForsyth [2006a] velocity model, although a high-velocity body at similar depths in south centralNevada has been previously imaged [Biasi andHumphreys, 1992] and therefore may be real.

[22] The horizontal and vertical tractions in Figures 3and 4 stress the overlying crust and the resultingcrustal stress field is shown in Figures 5 and 6. InFigure 5 we show the principal stresses derivedfrom the horizontal (map view) components (sEE,

sNN, sNE) of the three-dimensional stress tensor at20 km depth. At this depth, the stress field is nearly‘‘horizontal’’ in that one of the principal stresses istypically within !15! from vertical. This isexpected as at the Earth’s surface one principalstress must be vertical and the thickness of thecrustal layer (30 km) is small compared to thewavelength of the tractions acting on its base.Figure 6 portrays the same stress tensor as theorientation of maximum horizontal compressivestress (sHmax, long axis of the bars) and stressregime parameter AY [Simpson, 1997].

[23] It is important to recall that the stress fields inFigures 5 and 6 represent the stress caused only bythe tractions on the base of the crust and not theentire state of stress. To estimate the completestress tensor, as done by Sonder [1990], we mayadd a stress field dominated by approximately

Figure 4. Horizontal tractions at the base of the crust, complement to the vertical tractions in Figure 3, for the caseof a uniform viscosity upper mantle. Traction vectors are shown with their tails located at the finite element meshnodes (25 km spacing). Tractions are convergent in the southern Great Valley and Transverse Ranges and generallydivergent in the southern Walker Lane Belt near the intersection with the Garlock fault and near the Salton Sea.Traction magnitudes are typically ( 3–4 MPa (scale in bottom left corner) and largest near the California-Nevadaborder and in the western Mojave Desert.


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NW–SE shear stress parallel to the relative Pacific-North American plate direction. The 5 MPa scalein the bottom left corner of Figure 5 shows theprincipal stress for such a NW–SE shear-dominatedstress field. Detailed analysis of the total stress fieldin the crust and comparison to stress observations[e.g., Townend and Zoback, 2001, 2004;Hardebeckand Michael, 2004, 2006] is forthcoming in a futurepublication. Here we concentrate on the crustalstress field caused only by small-scale upper mantleconvection.

[24] Principal stress magnitudes in the crust aretypically!5MPa in the Transverse Ranges,!3MPa

in the southern Great Valley and as much as11.5 MPa in the eastern Sierra Nevada and south-ern Walker Lane Belt. These crustal stress magni-tudes are comparable in magnitude to the verticaltractions and generally larger than the horizontaltractions acting at the Moho. This is expected asthe horizontal basal tractions integrate over a muchlarger area than the unit cross-sectional area of thecrust and therefore stress magnitudes in the crustmust be larger to balance the applied force of thebasal tractions. Similar to the basal tractions(Figures 3 and 4), the principal stress magnitudesare linearly dependent on g, the velocity-densityscaling parameter.

Figure 5. State of stress in the crust caused by the vertical and horizontal tractions of Figures 3 and 4. The stresstensor is represented as principal stresses (bars) determined from the stress tensor at 20 km depth. Solid bars indicatecompression, and open bars indicate tension. The length of the bar indicates magnitude, and the scale is given in thebottom left corner. Approximately NE–SW tensional stress dominates in the southern Walker Lane Belt and SaltonSea region. Approximately radial compression (horizontal principal stresses both compressive and approximatelyequal magnitude) exists in the southern Great Valley and the western Transverse Ranges. Approximately NNW–SSEuniaxial compression (one principal stress compressive and much larger in magnitude than the other) dominates in thecentral and eastern Transverse Ranges in the vicinity of the Big Bend segment of the San Andreas fault. The stressstate producing maximum right-lateral shear stress on planes oriented N45W (the approximate orientation of the SanAndreas fault in central California) is shown by the scale in the bottom left corner.


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[25] Radial tension to pure normal stress is pre-dicted in the Sierra Nevada Mountains and south-ern Walker Lane Belt. Magnitudes are largest in thenorth ('11 MPa) and gradually decrease to thesoutheast. Southeast of the Garlock fault the stateof stress transitions to uniaxial tension and strikeslip in the Mojave Desert. This dominantly ten-sional horizontal stress field in the eastern SierraNevada, Walker Lane Belt and Eastern CaliforniaShear Zone is a consequence of the positive

pressure in the crust owing to the large uppermantle buoyancy (Figure 2) and induced verticaltractions on the base of the crust (Figure 3), and theapproximately NE–SW divergent horizontal trac-tions directed toward the downwelling in the GreatValley, the Transverse Ranges, and southernNevada (see Figure 4). Deviatoric tension super-imposed on right lateral shear is consistent with theoccurrence of normal fault earthquakes in thesouthern Sierra Nevada Mountains [Jones and

Figure 6. Crustal stress state (same as Figure 5) shown as the orientation of maximum horizontal compressive stress(sHmax) and stress regime parameter AY, a continuous function of the relative magnitudes of the components of thestress tensor [Simpson, 1997]. As shown by the accompanying arrows, AY = 0 indicates radial tension, AY = 30indicates uniaxial tension, AY = 60 indicates transtension, AY = 90 indicates strike-slip, AY = 120 indicatestranspression, AY = 150 indicates uniaxial compression, and AY = 180 indicates radial compression. The length of the

bar represents the magnitude of the horizontal principal stresses as their vector sum (

""""""""""""""""""""""""""""""

s2H max ) s2

H min

q

). The stress

state is dominated by tension (AY ( 60) and compression (AY ' 120), consistent with largely vertical upwellings anddownwellings in the underlying upper mantle.


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Dollar, 1986], oblique normal-right lateral slip onthe Owens Valley fault [e.g., Beanland and Clark,1994] and regional transtension along the easternmargin of the Sierra Nevada range [Unruh et al.,2003]. A similar system of approximately E–Woriented, essentially uniaxial tension is seen in theSalton Trough region caused by positive (tensional)pressure in the crust and approximately radialdivergence of horizontal tractions (Figure 4).

[26] Uniaxial to radial horizontal compression isseen in the central southern Great Valley, above thenegatively buoyant upper mantle anomaly. Thisstress is a result of the negative pressure in thecrust caused by the downward directed verticalMoho stress and convergent horizontal tractions.The locus of maximum compressive stress (approx.240!E, 36.5!N, !3 MPa) is shifted slightly to theWSW of the center of the upper mantle velocityanomaly indicating that to some degree compres-sion of crust directly overlying and along the E–NE side of the downwelling is overprinted by thetension induced by approximately NE directedhorizontal tractions along the California-Nevadaborder.

[27] The largest compressive principal stresses(!5.7 MPa) occur offshore above the westernTransverse Ranges anomaly and are a result ofthe relatively large negative vertical tractions andconvergent horizontal tractions there. Maximumcompression direction (Figure 6) is on averageN–S, approximately perpendicular to the numer-ous E–W trending reverse and oblique reverse/left-lateral faults such as those that are exposed in thenorthern Channel Islands (Santa Rosa Island fault,Santa Cruz Island fault) and on the mainland (e.g.,Santa Ynez fault, Oak Ridge fault, San Cayetanofault, etc.).

[28] In the central and eastern Transverse Rangesthe stress state is dominated by approximatelyuniaxial compression oriented approximatelyNW–SE in the central Transverse Ranges (SanGabriel Mountains) and approximately NNW–SSE in the eastern Transverse Ranges and MojaveDesert. These stresses are clearly a consequence ofthe negative vertical tractions and the approximatelySSW oriented horizontal tractions, strongest in theMojave Desert. NNW–SSE compressive stress isconsistent with the many approximately E–Wstriking thrust and oblique slip faults accommodat-ing approximately N–S shortening such as theCucamonga fault, Sierra Madre fault, San Gorgo-nio fault zone, North Frontal fault zone, etc. [e.g.,Meisling and Weldon, 1989; Morton and Matti,

1993; Spotila and Sieh, 2000], and N–S contractionthroughout the Mojave [Bartley et al., 1990]. NW–SE compressive stress along the Big Bend of the SanAndreas fault also has the effect of rotating the planeof maximum right-lateral shear stress counterclock-wise with respect to a background stress state causedby plate motions [Sonder, 1990].

[29] We use the entire Yang and Forsyth [2006a]velocity model in the calculations leading to Fig-ures 3–6, including that which falls outside thepolygon defining the well-resolved velocity struc-ture (e.g., Figure 2). To test whether possiblyerroneous velocity and inferred density structureoutside the well-resolved region strongly influen-ces our results, we have performed the sameviscous flow calculations with the density structurerestricted to the well-defined polygonal region ofFigure 2. Figure 7 shows the resulting horizontaltractions and crustal stress field. Except along theCalifornia-Nevada border, where traction magni-tudes decrease by 50% or more, the results arelargely unchanged. Compared to the resultsin Figure 4, horizontal traction orientations arevirtually identical. Magnitudes increase slightly((0.2 MPa) northeast of the San Andreas fault inthe vicinity of the eastern Transverse Rangesdownwelling, decrease slightly ((!0.1 MPa) nearthe offshore downwelling region, and increase((!0.5 MPa) along the eastern side of the SierraNevada anomaly. The crustal stress field (Figure 7b)is also nearly the same with only minor differences.We conclude from this analysis that the computedbasal tractions and crustal stress field in our studyarea are dominated by local, relatively shallow,density structure and not strongly influenced bystructure outside the well-resolved region of thevelocity model. For the remainder of this paper, allmodel comparisons and discussion will refer to theresults shown in Figures 3–6 utilizing the entirevelocity model.

3.2. High-Viscosity Lithospheric Lid

[30] Average lithospheric thickness in southernCalifornia has been estimated to be approximately90 km [Humphreys and Hager, 1990; Yang andForsyth, 2006a], although this does not apply tothe southern Sierra Nevada where lithosphericmantle is likely absent [Yang and Forsyth, 2006a;Savage et al., 2003]. Here we test whether thepresence of a high-viscosity mantle lithosphere lid60 km thick (90 minus average crustal thickness of30 km) significantly changes the predicted stressescompared to the uniform viscosity case in Figure 4.


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All else being equal, a higher viscosity lid shouldsupport a larger fraction of the shallow load, therebyincreasing the tractions on the overlying crust.

[31] Figure 8 shows that this is in fact the case. Weshow the horizontal tractions and crustal stressfield for the case of a lithospheric lid 2 orders of

magnitude higher viscosity than the underlyingmantle. Traction orientations do not change appre-ciably whereas magnitudes increase nearly every-where with maximum increase of !1.2 MPa. Thelargest increase occurs onshore and just offshorenear the western Transverse Ranges anomaly and

Figure 7. (a) Horizontal tractions on the base of the crust (as in Figure 4) and (b) principal stresses in the crust (as inFigure 5) resulting from upper mantle flow driven by density structure restricted to within the well-resolved region ofthe seismic velocity model (nonstippled area). The basal tractions and crustal stress field are very similar inorientation and magnitude to those based on the entire velocity model (Figure 4), implying our results are not stronglydependent upon the velocity and density structure outside the study area.

Figure 8. (a) Horizontal tractions and (b) crustal principal stresses for the case of a high-viscosity lithosphere 90 kmthick and 100 times as viscous as the underlying mantle. The resulting basal tractions (as compared with Figure 4) arevirtually unchanged in orientation and increase in magnitude by ( 25%.


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in the Mojave Desert near the eastern TransverseRanges anomaly to a maximum magnitude of 3.9MPa. The crustal stress field (Figure 8b) is alsolargely unchanged in principal stress orientationsalthough magnitudes increase from !20–50%depending upon location. A similar model (notshown) with only 10 times higher lithospheric lidviscosity produces very similar results indicating 1order of magnitude may be sufficient to signifi-cantly modify how density anomalies are sup-ported in the lithosphere.

3.3. Three-Dimensional ViscosityVariations

[32] The viscosity of the Earth’s mantle is thoughtto be highly temperature dependent, often assumedto follow the Arrhenius relationship where viscos-ity is proportional to exp(Q/RT), where Q is theactivation energy, R is the universal gas constantand T is absolute temperature [e.g., Ranalli, 1995].We assume that the velocity and density anomaliesin the southern California upper mantle arise fromtemperature variations, and therefore should alsoreflect viscosity variations; fast/dense regions arerelatively cool and therefore more viscous, andslow/buoyant regions are relatively warm andtherefore less viscous. To test whether three-di-

mensional viscosity variations influence our esti-mates of tractions on the base of the crust, we havedevised an ad hoc relationship between seismicvelocity and viscosity that mimics the strong de-pendence of viscosity on temperature as in theArrhenius relationship, and avoids the need toestimate temperature anomalies from seismic ve-locity anomalies. Viscosity (h) is given as

h " h010dVs*r# $ #2$

where h0 is the reference viscosity, dVs is thevelocity anomaly (expressed in km/s) and r is aconstant. Figure 9 shows the tractions and crustalprincipal stresses for the case of r = 5, i.e., every0.1 km/s variation in seismic velocity produces afactor of !3.16 (1/2 order of magnitude) change inviscosity, resulting in !2.9 orders of magnitudetotal viscosity variation throughout the entiremodel. Figure 10 shows the case of r = 10, i.e.,every 0.1 km/s variation in seismic velocityproduces a factor of 10 change in viscosity,resulting in over 5 orders of magnitude variationin viscosity throughout the model.

[33] As in the lithospheric lid example in theprevious section, horizontal traction orientationsare largely unchanged and only magnitudes depend

Figure 9. (a) Horizontal tractions and (b) crustal principal stresses for the case of density-dependent (as a proxy fortemperature-dependent) viscosity. Here r = 5 in equation (2), indicating 1/2 order of magnitude increase/decrease inviscosity for every 0.1 km/s increase/decrease in shear wave velocity relative to the layer average. Tractionorientations change little though magnitudes decrease (compared to Figure 4) in the Sierra Nevada, Walker Lane Belt,and western Transverse Ranges. Magnitudes increase in the western Mojave and Eastern California Shear Zone area.The corresponding vertical stresses (not shown) generally increase in magnitude compared to Figure 3, and this ismost pronounced above the dense bodies in the Great Valley and Transverse Ranges.


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on the viscosity distribution, although here theeffect is somewhat less homogeneous. Horizontaltractions on the eastern side of the Sierra Nevadaanomaly decrease in magnitude, owing to theparticularly slow upper mantle seismic velocityand inferred low viscosity. Tractions along theCalifornia-Nevada border directed toward thedownwelling region in southeast Nevada decreasefor the same reason. Traction magnitudes alsodecrease to the north of the offshore downwellingin the western Transverse Ranges, owing to theshallow slow velocities (Figure 2a) and inferredlow viscosity there. Magnitudes generally increasein the Mojave because it is underlain by highvelocities (Figure 2a). For r = 5 (Figure 9), trac-tions increase in the Mojave by (1 MPa to amaximum of !3.3 MPa, and for r = 10 (Figure10), tractions increase by (2.3 MPa to a maximumof !4.5 MPa with typical magnitudes of 2–3.5MPa. The crustal stress fields in Figures 9b and10b change in much the same way with orienta-tions of principal stress largely the same and localvariations in magnitude.

[34] The viscosity-velocity relationship is equation(2), and associated model calculations (Figures 9and 10), are meant only to demonstrate the trend inpredicted stresses if upper mantle viscosity anddensity are both related via temperature; as theviscosity dependence on temperature increases,

basal tractions on the Moho and crustal stress statechange little in orientation and sometimes appre-ciably in magnitude. This effect is strongest to theeast of the downwelling in the southern GreatValley, where tractions and crustal stresses de-crease in magnitude, and in the Mojave Desert tothe northeast of the SAF, where tractions andcrustal stresses increase in magnitude.

3.4. Weak Lower Crust

[35] Thus far we have treated the entire crust asessentially rigid in order to isolate the stressestransmitted across the Moho. This effectivelyassumes the viscosity of the lower crust is (much)greater than the upper mantle, generally consistentwith geodetically inferred estimates of lithosphericviscosity [e.g., Freed et al., 2007; Thatcher andPollitz, 2008]. Nonetheless, it is useful to considerthe case of the lower crust with a reduced viscosity.We have calculated the stresses for the case of thelower crust viscosity an order of magnitude lowerthan the upper mantle. The resulting stress orienta-tions in the crust are very similar to the case of a rigidlower crust (Figure 5). Principal stress magnitudes(vector sum of the horizontal principal stresses)decrease by an average of !0.5 MPa (within thenonstippled area of Figure 5), corresponding to anaverage percent decrease of !12. The relativelysmall decrease in principal stress magnitudes

Figure 10. (a) Horizontal tractions and (b) crustal principal stresses for a heterogeneous viscosity upper mantle withr = 10 in equation (2) (1 order of magnitude increase/decrease in viscosity for every 0.1 km/s increase/decrease inshear wave velocity relative to the layer average).


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reflects the fact that while the relatively low viscos-ity of the lower crust somewhat decreases its abilityto support horizontal shear stresses, the verticalnormal stresses transmitted across the Moho aredecreased by an even smaller amount, resultingin similar stress orientations and only slightlydecreased magnitudes.

4. Comparison With Observations andDiscussion

[36] Do the tractions and crustal stresses in Figures 3–10 contribute in a significant way in driving crustaldeformation in southern California? Unfortunatelyit is not possible to directly answer that question aswe have no means of observing tractions at Mohodepths and only sparse quantitative measurementsof the shallow state of stress in the crust [e.g.,Zoback and Healy, 1992]. We must therefore useother measures as proxies. In this section wediscuss two independent measures of crustal dy-namics: (1) active deformation patterns as revealedby geodetic data and (2) an estimate of the sum offorces acting on a crustal block. Both seem toindicate the stresses on the base of the crust maybe an important contribution to the sum of forcesdriving crustal deformation. For the remainder ofthis paper, we focus on the tractions and crustalstress field for the case of a uniform viscosity uppermantle, shown in Figures 3–6.

4.1. Active Deformation Patterns

[37] An idealized transform plate boundary expe-riences only shear deformation and therefore nochange in surface area with time. Transpression inthe Transverse Ranges and transtension in theWalker Lane Belt and Salton Trough suggests thatthe San Andreas fault system in southern Californiais not such an ideal plate boundary. We illustratethis nonideal plate boundary behavior in activedeformation patterns by estimating the dilatationrate, or rate of increase in surface area, from thedense geodetic velocity field in southern California(Figure 11a). Dilatation rate is the trace of thestrain rate ( _ev) tensor and is coordinate systeminvariant making it a useful measure to comparewith the frame invariant tractions and crustalstresses predicted by our model. We have com-bined the published data sets of Shen et al. [2003]and Kreemer and Hammond [2007] with a newvelocity solution processed at the University ofArizona consisting of existing and new data col-lected in the eastern Transverse Ranges; the latterdata set, including monumentation, data process-

ing, and results, are described by Bennett et al.[2006] and a forthcoming publication (J. C. Spinleret al., manuscript in preparation, 2008). The threedata sets were rotated onto a common stable NorthAmerican reference frame [Blewitt et al., 2005]using common velocities at continuous GPS sitesand the average was taken when more than onevelocity estimate existed at a single point.

[38] To estimate dilatation rate, velocities wereinterpolated to a uniform grid using piecewisecontinuous tri-linear (Delaunay triangulation) in-terpolation, smoothed with a moving-windowGaussian filter (full width half maximum of 120 km)to retain only the relatively long wavelength signal,and differentiated using the finite element method.The results in Figure 11a are similar to those ofHernandez et al. [2005, 2007] who use a differentinterpolation and differentiation scheme. This isnot surprising in that we consider dilatation rate ata wavelength much greater than typical geodeticstation spacing and therefore it is not particularlyimportant how velocity is interpolated betweenstations.

[39] Figure 11a shows two dominant lobes ofnegative dilatation (net area loss) in the westernGreat Valley. The northern lobe may be an artifactof relatively sparse station coverage, althoughsome shortening at high angle to the San Andreasfault is expected in the California Coast Ranges[Wentworth and Zoback, 1989; Argus and Gordon,2001] and very little (and even net extension) isseen along the San Andreas itself (Figure 11a). Thesouthern lobe of negative dilatation is likely realand consistent with contractional structures in thesouthern San Joaquin Valley such as the Pleito andWhite Wolf fault systems [e.g., Keller et al., 1998,2000; Stein and Thatcher, 1981]. In the southeast-ern Great Valley and southern Sierra Nevada, wepredict a minor amount of positive dilatation,consistent with observed extensional faulting[Jones and Dollar, 1986].

[40] Positive dilatation is seen in the southernWalker Lane Belt consistent with active transten-sion along the eastern margin of the Sierra Nevada[Unruh et al., 2003], effectively the westernboundary of the extensional Basin and Rangeprovince [Wernicke, 1992; Wernicke and Snow,1998]. Net area gain is seen in the Mojave Deserteast of the Eastern California Shear Zone and in theeastern Transverse Ranges; the latter is broadlyconsistent with a right step in the transfer of right-lateral slip between the southernmost San Andreasand Eastern California Shear Zone [Savage et al.,


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1993; Johnson et al., 1994; Hudnut et al., 2002].Areal extension is seen near the southern termina-tion of the San Andreas fault, curiously offset tothe NE of the Salton Sea. Negative dilatation isevident in the western Mojave, and central andwestern Transverse Ranges, consistent with geo-logic and active shortening in this region [e.g.,Bartley et al., 1990; Donnellan et al., 1993;Hauksson et al., 1995; Argus et al., 2005].

[41] Some of the dilatation shown in Figure 11amay be a transient effect of postseismic relaxationfollowing major earthquakes [e.g., Nur and Mavko,1974]. Transient surface deformation in the yearsfollowing the 1992 and 1999 Mojave Desert earth-quakes was clearly seen in geodetic data [Pollitz etal., 2001; Freed and Burgmann, 2004; Fialko,2004; Freed et al., 2007], but whether this influ-ences our steady state dilatation estimates, based

on data collected before and after these events, isnot yet clear. The broad agreement between pre-dicted dilatation rates and geologic deformation,and the lack of evidence for postseismic relaxationthroughout all of southern California [Meade andHager, 2005; Argus et al., 2005] allows us toproceed with some confidence that the geodeticallyderived dilatation estimate is generally representa-tive of relatively long wavelength permanent crust-al deformation.

[42] The predicted crustal deformation from ourmodel with a uniform viscosity upper mantle(Figures 3–6) is given in Figure 11b. We calculatethe horizontal dilatation rate from the strain ratetensor at 20 km depth. Extension is reckonedpositive and dilatation rate is equal and oppositeto the vertical strain rate ( _ev) because the crust istreated as an incompressible fluid. Here strain rates

Figure 11. (a) Observed surface dilatation derived from geodetic velocities (black triangles) in southern California.Velocities were interpolated to a grid and smoothed with a moving-window Gaussian filter with wavelength of120 km and differentiated to obtain strain and dilatation rates. Anywhere more than 60 km from a geodetic station isnot shown. Positive dilatation indicates net extension (increase in surface area), and negative dilatation indicates netcompression (decrease in surface area). (b) Predicted relative rates of vertical thinning and horizontal dilatation,determined from the model strain rate tensor at 20 km depth. Results were interpolated from finite element meshnodes to the continuous field shown here. Extension is reckoned positive, and rates are normalized by the maximum.The greatest dilatation rate in both the observed (Figure 11a) and predicted (Figure 11b) fields is at approximately thesame location as Long Valley Caldera (LVC), implying the possibility of a causal relationship between crustalextension, thinning, and volcanism.


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are only relative because absolute value dependson the absolute value of viscosity and this does notenter into our calculations.

[43] Negative dilatation is predicted in the southernGreat Valley resulting from crustal convergenceover the downwelling there. Positive dilatation isseen in the eastern Sierra Nevada and Walker LaneBelt and extends south of the Garlock fault into theeastern Mojave region as well. This is a conse-quence of the upwelling and divergent flow ofbuoyant upper mantle, with the largest divergenceapproximately at the intersection of the WalkerLane Belt, Garlock fault and Eastern CaliforniaShear Zone. Areal extension is also predicted in theSalton Sea area, a result of upwelling in the uppermantle and divergent crustal strain.

[44] Crustal convergence and negative dilatation isseen throughout the Transverse Ranges. The stron-gest area loss is offshore because the upper mantledensity anomaly is largest there and because theinduced horizontal basal tractions are more con-vergent than in the Mojave and eastern TransverseRanges (see Figure 4), resulting in greater netcrustal strain and area loss. This is consistent with

the approximately radially compressive stress inthe offshore/western Transverse Ranges region anduniaxial compressive stress in the Mojave andeastern Transverse Ranges (Figure 5).

[45] Overall, the comparison of observed (Figure 11a)and predicted (Figure 11b) dilatation rate is com-pelling; the model is generally successful in pre-dicting the style and relative rate of crustaldilatation. Negative dilatation is correctly predictedin part of the southern Great Valley and in thewestern Mojave and Transverse Ranges, and pos-itive dilatation is predicted and observed in theWalker Lane Belt, eastern Mojave and Salton Searegions. Notably, the model and observed dilatationfields both show the strongest crustal dilatation andthinning along the eastern margin of the SierraNevada at !241!E, 37.5!N, the location of LongValley Caldera. The observed dilatation could be aresult of a recent shallow magmatic intrusion orexpansion event causing surface uplift and exten-sion [e.g., Newman et al., 2006]. However, ourmodel contains no such mechanism and thereforethe spatial coincidence of observed and predicteddilatation implies a possible causal relationshipbetween upper mantle upwelling and crustal exten-sion and thinning, and the voluminous volcanismand historic seismicity [e.g., Hill, 2006] at LongValley Caldera.

[46] To quantify the correlation between observedand predicted dilatational deformation, we comparethe predicted and observed sign (positive or nega-tive) of dilatation rate at model nodes within ourstudy area (polygon in Figure 11). We find that!70% of the model nodes (25 km spacing) predictthe same sign as observed dilatation at the samepoint, and therefore !70% of the surface area ofsouthern California experiencing significant dilata-tion is correctly predicted by the model. Dilatationrate less than 10 nanostrain/yr is considered insig-nificant and excluded in this calculation. Compar-ing dilatation rates is more difficult because ourmodel does not consider absolute strain rates.

4.2. Forces Acting on the Mojave Block

[47] Thus far we have considered model predic-tions based largely on the orientation of the mantletractions and crustal stress field. Here we considermagnitude by estimating the sum of torques actingon a crustal block. Because the vector sum oftorques on any plate on the surface of the Earthmust be zero, it is useful to consider the set offorces acting on a piece of crust to compare theirrelative magnitudes. Figure 12 shows the Mojave

Figure 12. A simplified Mojave block (double solidline) and the forces acting on it. Basal tractions fromFigure 4 within the block were interpolated to a finergrid. These tractions impart a counterclockwise torqueon the block. Shear stress supported by the left lateralGarlock fault (GAR) also imparts a counterclockwisetorque. Shear stress supported by the right lateral SanAndreas fault (SAF) and Eastern California Shear Zone(ECSZ) impart clockwise torque and may balance thebasal traction and Garlock loads, although we have notincluded stresses acting normal to the block boundariessuch as the excess pressure at depth caused by theelevated San Bernardino Mountains. The torque formthese loads is quantified in the text.


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block, bounded by the dextral San Andreas faultand Eastern California Shear Zone, the sinistralGarlock fault, and the San Bernardino Mountains.For the following simple calculation, it is notnecessary to consider the details of the boundingfaults orientation and the Mojave block is definedas a simple, four-sided polygon. In Figure 12 it canbe seen that the San Andreas fault and EasternCalifornia Shear Zone shear loads impart clock-wise torque (about a hypothetical vertical pole inthe center of the block), and the Garlock shear loadand the basal traction field both impart a counter-clockwise torque, and, qualitatively, their sum iszero.

[48] To quantify these loads, we estimate the torqueon the block from each load by calculating

R

( r* %

F*)dA, where r

*is the radial vector from the center

of the Earth and F*

is the force vector (stress %area) acting on an area element dA. The horizontalbasal tractions were interpolated to a denser grid(10 km spacing) so as to avoid biasing the calcu-lation by a nonuniform distribution of integrationpoints. For the uniform mantle viscosity case inFigure 4, the basal tractions produce a net torque of!1.9 % 1023 mN (magnitude of the torque vector).This value is likely an underestimate because thebasal tractions in the vicinity of the Mojave blockincrease, as compared to the uniform viscositycase, for all scenarios of variable upper mantleviscosity that we have explored. For example, if weuse the traction field from Figure 7a, calculated fora lithospheric lid 2 orders of magnitude higherviscosity than the underlying mantle, the totaltorque is 2.6 % 1023 mN, !25% larger.

[49] To compare with other loads acting on theblock, we use 30 MPa as the depth-averaged, long-term shear stresses supported by block-boundingfaults in southern California [Fay and Humphreys,2006] and assume this stress acts over the approx-imate seismic depth of 15 km. This results in torquemagnitudes of 4.4 % 1023 mN, 5.4 % 1023 mN, and3.9 % 1023 mN for the San Andreas fault, EasternCalifornia Shear Zone, and Garlock fault respec-tively. These torques are approximately a factor of2 larger than the basal traction torque, but clearly ofthe same order of magnitude and thus the basaltractions are likely as relevant to the dynamics ofthe Mojave block as plate interaction stressessupported by active faults. Stresses acting onhorizontal planes such as our basal traction esti-mates (e.g., 3 MPa) may seem small, but it is theintegrated force and torque that influences crustal

dynamics and basal tractions on horizontal planestypically act over a large area.

5. Summary and Conclusions

[50] Crustal deformation in southern California isclearly strongly influenced by shear stress owing torelative motion of the Pacific and North Americanplates. Superimposed on shear deformation is asignificant amount of crustal shortening and moun-tain building in the Transverse Ranges. We haveshown through numerical modeling of the stresseson the crust induced by upper mantle flow, thatmuch of this ‘‘nonideal’’ plate boundary behaviormay be explained by being driven from below bysmall-scale convection of the southern Californiaupper mantle. Favorable comparison with activedeformation determined from geodetic data and thesum of forces acting on crustal blocks suggest thestresses exerted on the crust from small-scaledownwellings and upwellings in the upper mantleare an important component of the sum of forcesdriving crustal deformation in southern California.

[51] Figures 3–10 show the computed tractions onthe base of the crust and the resulting stress fieldswithin the crust. Horizontal tractions are typically3–4 MPa or less in magnitude (for the chosenseismic velocity to density scaling relationship).Traction orientations are largely determined by thedistribution of seismic velocity anomalies [Yangand Forsyth, 2006a] and inferred density structurein the upper mantle, and are largely independent ofviscosity distribution. Three nearly distinct down-welling zones in the southern Great Valley, westernTransverse Ranges and eastern Transverse rangescreate negative vertical tractions and convergenthorizontal shear tractions on the base of the crust.In the southern Great Valley, mantle lithosphereformerly beneath the Sierra Nevada is now sinkingbeneath the southern Great Valley, causing activesurface subsidence [Saleeby and Foster, 2004] andapproximately radially convergent shear tractionson the base of the crust. In the Transverse Ranges,two dominant downwelling zones centered off-shore and beneath the western San BernardinoMountains act to draw in the overlying crust andcause approximately N–S compressive stress(Figure 5), and active shortening (Figure 11).

[52] Crustal deformation driven by small-scaleconvection helps to explain the obliquity of theMojave segment of the San Andreas fault with


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respect to plate motions. Shortening in the Trans-verse Ranges as a consequence of a bend in the SanAndreas fault is a plausible kinematic interpreta-tion, but provides no dynamic explanation for whythe lithosphere maintains this apparently energeti-cally unfavorable plate boundary geometry. Fur-thermore, there does not appear to be any clearrelationship between rock uplift, horizontal short-ening and the obliquity of the San Andreas fault[Spotila et al., 2007]. Ourmodel, which successfullypredicts the majority of the dilatational deformationin southern California without incorporating anyshear strain and oblique geometry of the SanAndreas fault (Figure 11), provides an alternativekinematic interpretation that the bend in the fault isnot a cause but rather a consequence of contractionin the adjacent crust. In this view the fault does notact as a fixed barrier to crustal flow but simply ageographic point attached to the adjacent, deform-ing crust. With respect to a fixed Pacific plate,approximately N–S shortening southwest of theSan Andreas fault in the western TransverseRanges [e.g., Yeats et al., 1988] requires that thetrace of the San Andreas fault move to the south-west. Shortening on the northern side of the SanAndreas fault in the San Bernardino Mountains[e.g., Dibblee, 1975; Spotila and Sieh, 2000]requires that the trace of the San Andreas theremove to the NNE. Both process tend to rotate thetrace of the San Andreas fault counterclockwisewith respect to its previous orientation, and in-crease the obliquity with respect to plate motiondirection.

[53] Thus the dynamic effect of heterogeneousupper mantle density structure and induced flowon crustal deformation in southern California istwofold. First, upper mantle downwelling, stron-gest beneath the Transverse Ranges, drives short-ening of the overlying crust to the southwest andnortheast of the present-day location of the BigBend segment of the San Andreas fault. This has thekinematic consequence of rotating the San Andreascounterclockwise, thereby increasing its obliquitywith respect to plate motions, possibly inducingadditional downwelling of lower crust or mantlelithosphere during convergence. There must be alimit to this positive feedback in that the elevationof the Transverse Ranges is finite and thereforeprovides an independent means of estimating themagnitude of the mantle tractions [Humphreys andHager, 1990; Fay and Humphreys, 2006]. Analysisof a similar feedback process of fault orientationand local topography has also provided constrainton the state of stress within the crust [Fialko et al.,

2005]. Second, the stress field associated with theexcess density beneath the Transverse Ranges(Figures 5 and 6) tends to rotate the plane ofmaximum right lateral shear stress counterclock-wise with respect to the background stress field(caused by plate motions), thereby promoting slipon faults such as the San Andreas oriented coun-terclockwise of the plate motion direction [Sonder,1990].

Acknowledgments

[54] We thank Y. Yang and D. Forsyth for sharing theirtomography model, Walter Landry and the ComputationalInfrastructure for Geodynamics for developing and maintain-ing Gale, C. Kreemer for sharing his GPS data set prior topublication, and two anonymous reviewers for their con-structive reviews. The figures were prepared with GMT[Wessel and Smith, 1998]. This research was supported byNSF grant EAR-0510484 to R.B.

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