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Geochemistry Geophysics Geosystems 3 G Article Volume 9, Number 8 12 August 2008 Q08006, doi:10.1029/2008GC001988 AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES Published by AGU and the Geochemical Society ISSN: 1525-2027 Click Here for Full Article Small-scale upper mantle convection and crustal dynamics in southern California N. P. Fay, R. A. Bennett, and J. C. Spinler Department of Geosciences, University of Arizona, Tucson, Arizona 85721, USA ([email protected]) E. D. Humphreys Department 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-scale convection of the upper mantle in southern California. Three-dimensional upper mantle shear wave velocity structure is mapped to three-dimensional density structure that is used to load a finite element model of instantaneous upper mantle flow with respect to a rigid crust, providing an estimate of the tractions acting on the base of the crust. Upwelling beneath the southern Walker Lane Belt and Salton Trough region and downwelling beneath the southern Great Valley and eastern and western Transverse Ranges dominate the upper mantle flow and resulting crustal tractions. Divergent horizontal and upward directed vertical tractions create a tensional to transtensional crustal stress state in the Walker Lane Belt and Salton Trough, consistent with transtensional tectonics in these areas. Convergent horizontal and downward directed vertical tractions in the Transverse Ranges cause approximately N–S crustal compression, consistent with active shortening and transpressional deformation near the ‘‘Big Bend’’ of the San Andreas fault. Model predictions of crustal dilatation and the forces acting on the Mojave block compare favorably with observations suggesting that small-scale upper mantle convection provides an important contribution to the sum of forces driving transpressional crustal deformation in southern California. Accordingly, the obliquity of the San Andreas fault with respect to plate motions may be considered 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 relative Pacific–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 and mantle: 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 crustal dynamics in southern California, Geochem. Geophys. Geosyst., 9, Q08006, doi:10.1029/2008GC001988. Copyright 2008 by the American Geophysical Union 1 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 1. Introduction [2] The forces driving active deformation of the lithosphere at a tectonic plate boundary are the sum of those driving global plate motion transmitted through the rigid plates from the far-field to the plate boundary [e.g., Atwater, 1970], and those created locally by variations in local density structure [Artyushkov, 1973; Fleitout and Froidevaux, 1982; Molnar and Lyon-Caen, 1988]. On a global scale, the stresses exerted on plates from flow induced by the mantle’s internal density structure (e.g., sinking slabs) seem to be an important contribution 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]. On the scale of a few 100 km it may be more difficult to determine the importance of small-scale convection on crustal dynamics and deformation because the influence in actively deforming regions may be overprinted by other processes such as plate interaction. Actively deforming regions have a distinct advantage, however, in that they provide the necessary observables to allow discrimination of superimposed sources of driving forces. In this paper we aim to resolve the role of small-scale convection of the upper mantle in driving deformation of the overlying crust in southern California. It is clear that present-day deformation in southern California is strongly influenced by plate interaction stresses; we show that much of the deformation that cannot be explained by plate interaction derives from small-scale convection of the underlying upper mantle. [3] Approximately 50 mm/yr of relative motion between the Pacific and North American plates [DeMets and Dixon, 1999] is accommodated largely by strike-slip deformation. The most active structure, the San Andreas fault, accommodates half or more of the present-day slip budget [e.g., Meade and Hager, 2005], although its rate depends on location. In addition to strike-slip deformation there is a nontrivial component of nonsimple shear deformation such as block rotation, shortening and uplift 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]. Kinematically, the transpressive deformation in the Transverse Ranges can be considered a consequence of the ‘‘Big Bend’’ in the San Andreas fault (here defined as the segment of the San Andreas fault between the San Emigdio Bend and San Gorgonio 10.1029/2008GC001988 Bend; see Figure 1), i.e., a large left-step in the rightlateral San Andreas fault system. The San Andreas fault is offset !120 km through the Big Bend (Figure 1), which is !8 times the average seismogenic 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 thickness in southern California of !90 km [Yang and Forsyth, 2006a; Humphreys and Hager, 1990]. [4] Dynamically, however, it is not clear why the lithosphere maintains this apparently energetically unfavorable geometry [Kosloff, 1977]. Mountain building in the Transverse Ranges, i.e., thrust faulting and crustal shortening and thickening, requires work against gravity and frictional and viscous resistive forces. There are other active structures such as the Elsinore-Laguna Salada fault system, that if more active and connected with the Cerro Prieto fault (Figure 1), would allow relative Pacific-North American motion on a more throughgoing transform system and largely bypass the Big Bend geometry. Numerical modeling has shown that an effect of the present-day geometry of the San Andreas fault is to promote slip on other slip systems such as the Eastern California Shear Zone and offshore faults [Li and Liu, 2006], indicating the current San Andreas geometry should not be stable. [5] However, the San Andreas fault appears to be the dominant plate boundary fault at present throughout 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 geometry of the San Andreas fault system has existed since the opening of the Gulf of California (!6 Ma) and likely longer [Wilson et al., 2005], indicating today’s transpressive geometry is a persistent tectonic feature of the plate boundary at least over the past few million years. One possible explanation is that the San Andreas fault is extremely weak compared to the surrounding crust and nearby faults [Zoback et al., 1987; Townend and Zoback, 2000, 2004], perhaps owing to its greater accumulated offset and structural maturity [Wesnousky, 1988, 2005], and the work required to generate a new, straighter fault system is greater than that to drive shortening and mountain building in the Transverse Ranges. [6] Alternatively, upper mantle processes, namely downwelling beneath the Transverse Ranges, may 2 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Figure 1. Map showing study area and geographic features discussed in the text. Solid lines show fault traces, and the San Andreas fault (SAF) is indicated with a thick line. The Walker Lane Belt (WLB) extends from the Garlock fault (GAR) to the northwest along the eastern side of the Sierra Nevada Mountains. The Eastern California Shear Zone (ECSZ) extends from the Garlock to the southeast. The western Transverse Ranges (WTR) and central Transverse 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] is shown. Dashed line shows the approximate projection of the southernmost San Andreas fault into the Great Valley to illustrate the !120 km offset with respect to its central California location. Gray scale gives smoothed elevation above 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. act to draw in the overlying crust and cause shortening superimposed on the shear deformation related to plate motion. A number of seismic studies have shown that the upper mantle velocity structure beneath southern California is rather heterogeneous [Raikes, 1980; Humphreys and Clayton, 1990; Jones et al., 1994; Kohler et al., 2003; Boyd et al., 2004; Yang and Forsyth, 2006a; Tian et al., 2007]. The dominant upper mantle seismic features are (1) a roughly circular highvelocity body beneath the southern Great Valley and Sierran foothills adjacent to a low-velocity region beneath the high elevation of the Sierra Nevada and southern Walker Lane Belt, (2) low velocities beneath the greater Salton Sea area, likely associated with extension and mantle upwelling [e.g., Elders et al., 1972; Lachenbruch et al., 1985], and (3) high-velocity anomalies beneath the Transverse Ranges. This latter feature is the most important for this paper and has previously been interpreted as a slab-like feature extending to at least 200 km [Humphreys and Clayton, 1990; Kohler et al., 2003]. Recently, Yang and Forsyth [2006a] have argued that the Transverse Range velocity anomalies extend to only !150 km and are separated into two nearly distinct anomalies centered beneath the eastern Transverse Ranges and offshore near the Channel Islands (Figure 2). [7] If these velocity anomalies derive largely from temperature variations, the velocity structure provides a proxy for the temperature heterogeneity in the upper mantle and its associated density structure. The high-velocity body beneath the Transverse 3 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Ranges has been interpreted as downwelling of cool, and relatively dense, mantle lithosphere [Bird and Rosenstock, 1984; Sheffels and McNutt, 1986; Humphreys and Hager, 1990]. Houseman et al. [2000] and Billen and Houseman [2004] suggest the velocity anomalies are a result of a gravitydriven lithospheric instability to account for their drip-like structure. Whatever the mechanism of formation, the forces associated with a heterogeneous distribution of density in the upper mantle must be balanced by viscous stresses associated with gravity-driven flow, or by displacing a horizontal density interface such as the Earth’s surface (dynamic topography). [8] The focus of this paper is to quantify the threedimensional anomalous upper mantle density structure, the viscous flow induced by this density structure, and the effect on the dynamics and deformation of the overlying crust. We calculate the instantaneous viscous flow of the uppermost mantle with respect to a rigid crust; this approach allows us to isolate the stresses on the base of the crust. Three-dimensional upper mantle density structure is derived from the seismic tomography model of Yang and Forsyth [2006a]. We do not attempt to include any large-scale shear in the upper mantle potentially induced by relative plate motion [e.g., Bourne et al., 1998; Molnar et al., 1999]; our study is restricted to the largely poloidal flow caused by gravity acting on a heterogeneous distribution of density. The influence of upper mantle flow on crustal dynamics is evaluated by comparison with regional deformation patterns determined from geodetic observations and torque balance on a crustal block. 2. Three-Dimensional Seismic Velocity and Density Structure of the Uppermost Mantle [9] We use the shear wave velocity model derived from surface (Rayleigh) wave tomography of Yang Figure 2. Upper mantle seismic velocity and density structure in southern California. Shear wave velocity (Vs) anomalies from the Yang and Forsyth [2006a] tomography model and inferred density are shown at depth slices at (a) 70– 90 km, (b) 110– 130 km, and (c) 150– 170 km. The area without stipple, in this and subsequent figures, shows the well-resolved region of the seismic velocity model as defined by Yang and Forsyth [2006a]. We use the entire velocity model (which extends beyond the limits of these maps) in our modeling as discussed in the text. 4 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics and Forsyth [2006a] to estimate the three-dimensional upper 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 and Forsyth [2006a, 2006b] and Forsyth and Li [2005]. We choose the Yang and Forsyth [2006a] model because (1) Rayleigh waves are most sensitive to the depth range we are most interested in here (e.g., !50–250 km), (2) they incorporate finite-frequency effects in their inversions, and (3) the velocity structure they find is quite similar to that reported in previous studies, giving us some confidence it is likely real. [10] Figures 2a, 2b, and 2c present three representative depth slices through the Yang and Forsyth [2006a] model at depths of 70–90, 110–130 and 150 – 170 km, respectively. Velocity anomalies, dVs, are shown as percent deviations from a 1-D average velocity model. Three major fast (blue) structures beneath the southern Great Valley and Transverse Ranges are clear. Slow velocities (red) are strongest at shallowest depths in the southern Walker Lane and at all depths near the Salton Sea. The three dominant fast bodies, and the stresses they induce on the overlying crust (discussed in section 3) are hereafter referred to as the Sierra Nevada anomaly, western Transverse Ranges anomaly, and eastern Transverse Ranges anomaly (see Figure 2). [11] Seismic velocity variations depend primarily on composition, temperature, partial melt, and anisotropy. In this paper we effectively assume that the velocity variations are entirely thermal in origin and adopt a constant scaling factor, g, relating density (r) variations to velocity variations, given by g ¼ @ ln r=@ ln Vs ; ð1Þ [Karato, 1993]. In the upper mantle g is estimated to be !0.2–0.3 [Karato, 1993; Steinberger and Calderwood, 2006]. We choose g = 0.2 such that, for example, a 5% velocity anomaly maps into a 1% density anomaly (dr). [12] This method of scaling velocity to density is commonly used in global studies of mantle dynamics [e.g., Lithgow-Bertelloni and Silver, 1998; Becker and Boschi, 2002; Conrad et al., 2004], although shallow (above !325 km) upper mantle seismic velocity anomalies are often excluded because they may derive from isostatically compensated compositional variations associated 10.1029/2008GC001988 with continental cratons and the tectosphere [e.g., Lithgow-Bertelloni and Silver, 1998; Becker and O’Connell, 2001; Steinberger et al., 2001]. It is unlikely, however, that the velocity variations in the shallow upper mantle in southern California imaged by Yang and Forsyth [2006a] and others are dominated by compositional variations because the magnitude of the velocity anomalies are much greater (!3 times) than that produced by chemical segregation of the upper mantle [Humphreys and Hager, 1990; Jordan, 1975]. Anisotropy and partial melt can also be excluded leaving temperature variations as the most likely cause of the velocity anomalies [Humphreys and Hager, 1990]. Our modeling results (traction and stress magnitudes, see sections 3 and 4) scale linearly with g and therefore if any fraction of the seismic velocity anomalies do in fact represent compositional variations that are isostatically compensated, we will overestimate their influence on mantle flow. We have, however, chosen g conservatively to minimize this effect. In section 4.2 we show that the influence on crustal dynamics of mantle flow (with g = 0.2) is of similar magnitude to tectonic plate interaction, indicating 0.2 is a reasonable value for g in the southern California upper mantle. [13] Our reference one-dimensional density structure (from iasp91 [Kennett and Engdahl, 1991]) increases linearly from 3320 kg/m3 at 35! km to ! 3490 kg/m3 at 310 km and thus @ ð@zdrÞ! is g;dVs positive though typically very small compared to dr itself. The corresponding density anomaly structure is shown in Figure 2 with the same color scale as the seismic velocity anomalies. Density anomalies are typically ±10–30 kg/m3, with the largest anomalies in our study area at depths of !50– 100 km. The choice of reference density model is not particularly critical because the numerical modeling results scale with respect to variations in background density in the same way as with variations in g, and the former is likely better constrained than the latter. [ 14 ] The density anomaly structure shown in Figure 2 is used as input to our viscous flow calculations, discussed in the next section. The background density model is not included as it would create only a lithostatic pressure that does not drive differential flow. The nonstippled area in Figure 2 shows the region of the seismic velocity model that is well resolved and most reliable [Yang and Forsyth, 2006a]. We use the entire velocity model (and inferred anomalous density structure), so as to avoid artificial truncation effects. We 5 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics demonstrate later that restricting the density model to within the resolved region produces very similar results. 3. Numerical Modeling [15] Our primary objective is to resolve the forces created by upper mantle density structure that contribute to loading the southern California crust. To that end, we calculate the viscous flow of the upper mantle and concomitant tractions on the base of the crust. Importantly, by driving the flow with gravitational body forces acting on density anomalies, their geometry and magnitude known in absolute value, we are able to predict absolute levels of stress in the upper mantle and crust. We solve the three-dimensional equations of conservation of mass and momentum for incompressible Newtonian viscous flow using the finite element code Gale [Moresi et al., 2003; Landry and Hodkinson, 2007; Landry et al., 2008]. We restrict our analysis to very small strains, essentially instantaneous flow, because the seismic tomography model provides us with only the present-day velocity and density structure and we wish to isolate the contribution of sub-crustal density structure on crustal stress. [16] The model domain is a Cartesian grid 1600 km % 1600 km in map view and extends to 1000 km depth. The model is centered on southern California that is represented as an oblique-Mercator projection about the Pacific-North American Euler pole [DeMets and Dixon, 1999]. Element spacing is 25 km horizontally and 10 km vertically; this element resolution was chosen on the basis of the resolution of the seismic tomography model and finer mesh resolution produces indistinguishable results. The sides and bottom are held fixed. We model the crust as a highly viscous layer 30 km deep that is fixed vertically and unconstrained horizontally. This allows the mantle flow to load the crust permitting us to monitor crustal stress. Density is assigned to the model by interpolating the three-dimensional density structure (Figure 2) to element centroids. The seismic velocity model is given in depth slices, each 20 km thick. Thus the density structure input into the modeling resembles a slightly coarser version of Figure 2. Outside the extent of the seismic velocity model (e.g., below 250 km), no density anomaly is assigned. [17] In this paper we are concerned with the stresses acting on the crust from density-driven upper mantle flow, not the strain in mantle itself. 10.1029/2008GC001988 Therefore, the modeling results we present here are independent of the absolute value of the viscosity chosen for the upper mantle. This is because we prescribe density and compute the flow response and associated stresses; the stresses necessary to support the forces of the density anomalies are dictated by the densities and their spatial distribution. We model cases of a uniform upper mantle viscosity with and without a high-viscosity lid, an upper mantle in which viscosity depends on temperature, and a relatively weak lower crust. 3.1. Uniform Upper Mantle Viscosity [18] Figures 3 and 4 present the vertical and horizontal tractions on the base of the crust for a uniform viscosity upper mantle. Figure 3 gives the vertical normal stress at element centroids. Above the three major positive density (negatively buoyant) bodies, the western Transverse Ranges, eastern Transverse Ranges and Great Valley anomalies, vertical stresses are negative (act to pull the crust down) with a maximum magnitude of !9 MPa offshore. The slowest seismic velocities and largest positive vertical stresses (!14 MPa) occur in the southern Walker Lane Belt, just to the east of the Sierra Nevada anomaly. dVs of &5 to &6% at 50– 70 km depth indicates a likely complete absence of lithospheric mantle there [Yang and Forsyth, 2006a]. Relatively warm asthenosphere is thought to have passively upwelled to fill the region vacated by delaminating lithospheric mantle [Zandt et al., 2004; Le Pourhiet et al., 2006]. Yang and Forsyth [2006a] show that the region of lowest seismic velocities corresponds well with the locus of Pliocene and Quaternary volcanism. If any partial melt, which can have strong retarding effects on shear wave velocities [Hammond and Humphreys, 2000], is retained in the upper mantle, we may be over estimating the inferred density anomaly and vertical stresses. [19] These vertical stresses should vertically deflect the crust. The predicted dynamic topography can be estimated by dividing the radial stresses by the deflected density contrast (e.g., 3300 kg/m3) and gravitational acceleration. This gives a maximum upward static deflection of !0.4 km in the eastern Sierra Nevada and southern Walker Lane, and maximum downward deflection of !0.3 km offshore in the vicinity of the Channel Islands. This simple isostatic calculation overestimates the actual dynamic topography because these relatively short horizontal wavelength vertical loads will be partly supported by flexure of the elastic crust. While the 6 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Figure 3. Vertical normal stresses (colored dots) on the base of crust caused by viscous flow of the upper mantle driven 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 the Walker Lane Belt above the positively buoyant, slow velocity anomaly there. The largest negative vertical stress occurs offshore above the negatively buoyant fast anomaly. As in Figure 2, the nonstippled area shows the region of seismic velocity model that is best constrained, and therefore we focus on this area in our interpretation. total deflected mass must be conserved, flexure of an elastic plate should broaden and smooth any subsidence or uplift and thereby decrease local amplitudes. Nonetheless, in the southern Sierra Nevada active subsidence and sedimentation appears to be burying mountainous topography [Saleeby and Foster, 2004], suggesting at least some of the negative dynamic topography predicted by our models may be real, and possibly increasing with time. In the remainder of this paper we focus on the horizontal tractions. [20] The horizontal tractions at the Moho are shown in Figure 4. Downwelling of the dense material offshore of the western Transverse Ranges produces a nearly radial pattern of convergent tractions. Traction magnitudes are typically 1 – 2 MPa and strongest (!2.7 MPa) onshore to the north of the center of the downwelling region. Tractions beneath the Eastern California Shear Zone and western Mojave Desert are converging on the eastern Transverse Ranges (San Bernardino Mountains) and Mojave segment of the SAF. The pattern here is asymmetric with tractions much larger to the northeast of the San Andreas fault than to the south. To the north of the San Andreas fault tractions have an approximately SW orientation and magnitude of '2.5 MPa with a maximum of 3.2 MPa. Horizontal tractions north of !33.5!N and within the Salton block, between the San Jacinto and San Andreas faults, are nearly parallel to those two faults and directed toward the San Bernardino Mountains, and provide some of the force driving convergence there [Fay and Humphreys, 2006]. Within !100 km of the Salton Sea, horizontal tractions are typically less than 2 MPa and show a clear radial pattern, consistent with horizontal divergence of buoyant upwelling. 7 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Figure 4. Horizontal tractions at the base of the crust, complement to the vertical tractions in Figure 3, for the case of a uniform viscosity upper mantle. Traction vectors are shown with their tails located at the finite element mesh nodes (25 km spacing). Tractions are convergent in the southern Great Valley and Transverse Ranges and generally divergent 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-Nevada border and in the western Mojave Desert. [21] The horizontal tractions along the CaliforniaNevada border have a maximum magnitude of !3.5 MPa and are directed to the NE. These are caused by a combination of strong upwelling and divergence of the buoyant material beneath the southern Sierra Nevada and southern Walker Lane Belt, and a relatively deep downwelling in southeastern Nevada. This latter velocity anomaly lies outside the well-resolved region of the Yang and Forsyth [2006a] velocity model, although a highvelocity body at similar depths in south central Nevada has been previously imaged [Biasi and Humphreys, 1992] and therefore may be real. [22] The horizontal and vertical tractions in Figures 3 and 4 stress the overlying crust and the resulting crustal stress field is shown in Figures 5 and 6. In Figure 5 we show the principal stresses derived from the horizontal (map view) components (sEE, sNN, sNE) of the three-dimensional stress tensor at 20 km depth. At this depth, the stress field is nearly ‘‘horizontal’’ in that one of the principal stresses is typically within !15! from vertical. This is expected as at the Earth’s surface one principal stress must be vertical and the thickness of the crustal layer (30 km) is small compared to the wavelength of the tractions acting on its base. Figure 6 portrays the same stress tensor as the orientation of maximum horizontal compressive stress (sHmax, long axis of the bars) and stress regime parameter AY [Simpson, 1997]. [23] It is important to recall that the stress fields in Figures 5 and 6 represent the stress caused only by the tractions on the base of the crust and not the entire state of stress. To estimate the complete stress tensor, as done by Sonder [1990], we may add a stress field dominated by approximately 8 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Figure 5. State of stress in the crust caused by the vertical and horizontal tractions of Figures 3 and 4. The stress tensor is represented as principal stresses (bars) determined from the stress tensor at 20 km depth. Solid bars indicate compression, and open bars indicate tension. The length of the bar indicates magnitude, and the scale is given in the bottom left corner. Approximately NE – SW tensional stress dominates in the southern Walker Lane Belt and Salton Sea region. Approximately radial compression (horizontal principal stresses both compressive and approximately equal magnitude) exists in the southern Great Valley and the western Transverse Ranges. Approximately NNW – SSE uniaxial compression (one principal stress compressive and much larger in magnitude than the other) dominates in the central and eastern Transverse Ranges in the vicinity of the Big Bend segment of the San Andreas fault. The stress state producing maximum right-lateral shear stress on planes oriented N45W (the approximate orientation of the San Andreas fault in central California) is shown by the scale in the bottom left corner. NW–SE shear stress parallel to the relative PacificNorth American plate direction. The 5 MPa scale in the bottom left corner of Figure 5 shows the principal stress for such a NW–SE shear-dominated stress field. Detailed analysis of the total stress field in the crust and comparison to stress observations [e.g., Townend and Zoback, 2001, 2004; Hardebeck and Michael, 2004, 2006] is forthcoming in a future publication. Here we concentrate on the crustal stress field caused only by small-scale upper mantle convection. [24] Principal stress magnitudes in the crust are typically !5 MPa in the Transverse Ranges, !3 MPa in the southern Great Valley and as much as 11.5 MPa in the eastern Sierra Nevada and southern Walker Lane Belt. These crustal stress magnitudes are comparable in magnitude to the vertical tractions and generally larger than the horizontal tractions acting at the Moho. This is expected as the horizontal basal tractions integrate over a much larger area than the unit cross-sectional area of the crust and therefore stress magnitudes in the crust must be larger to balance the applied force of the basal tractions. Similar to the basal tractions (Figures 3 and 4), the principal stress magnitudes are linearly dependent on g, the velocity-density scaling parameter. 9 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 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 the stress tensor [Simpson, 1997]. As shown by the accompanying arrows, AY = 0 indicates radial tension, AY = 30 indicates uniaxial tension, AY = 60 indicates transtension, AY = 90 indicates strike-slip, AY = 120 indicates compression. transpression, AY = 150 indicates uniaxial compression, and AY = 180 indicates radialq ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiThe ffiffiffiffiffiffiffiffilength of the bar represents the magnitude of the horizontal principal stresses as their vector sum ( s2H max þ s2H min ). The stress state is dominated by tension (AY ( 60) and compression (AY ' 120), consistent with largely vertical upwellings and downwellings in the underlying upper mantle. [25] Radial tension to pure normal stress is predicted in the Sierra Nevada Mountains and southern Walker Lane Belt. Magnitudes are largest in the north ('11 MPa) and gradually decrease to the southeast. Southeast of the Garlock fault the state of stress transitions to uniaxial tension and strike slip in the Mojave Desert. This dominantly tensional horizontal stress field in the eastern Sierra Nevada, Walker Lane Belt and Eastern California Shear Zone is a consequence of the positive pressure in the crust owing to the large upper mantle buoyancy (Figure 2) and induced vertical tractions on the base of the crust (Figure 3), and the approximately NE–SW divergent horizontal tractions directed toward the downwelling in the Great Valley, the Transverse Ranges, and southern Nevada (see Figure 4). Deviatoric tension superimposed on right lateral shear is consistent with the occurrence of normal fault earthquakes in the southern Sierra Nevada Mountains [Jones and 10 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics Dollar, 1986], oblique normal-right lateral slip on the Owens Valley fault [e.g., Beanland and Clark, 1994] and regional transtension along the eastern margin of the Sierra Nevada range [Unruh et al., 2003]. A similar system of approximately E–W oriented, essentially uniaxial tension is seen in the Salton Trough region caused by positive (tensional) pressure in the crust and approximately radial divergence of horizontal tractions (Figure 4). [26] Uniaxial to radial horizontal compression is seen in the central southern Great Valley, above the negatively buoyant upper mantle anomaly. This stress is a result of the negative pressure in the crust caused by the downward directed vertical Moho stress and convergent horizontal tractions. The locus of maximum compressive stress (approx. 240!E, 36.5!N, !3 MPa) is shifted slightly to the WSW of the center of the upper mantle velocity anomaly indicating that to some degree compression of crust directly overlying and along the E– NE side of the downwelling is overprinted by the tension induced by approximately NE directed horizontal tractions along the California-Nevada border. [27] The largest compressive principal stresses (!5.7 MPa) occur offshore above the western Transverse Ranges anomaly and are a result of the relatively large negative vertical tractions and convergent horizontal tractions there. Maximum compression direction (Figure 6) is on average N–S, approximately perpendicular to the numerous E–W trending reverse and oblique reverse/leftlateral faults such as those that are exposed in the northern Channel Islands (Santa Rosa Island fault, Santa Cruz Island fault) and on the mainland (e.g., Santa Ynez fault, Oak Ridge fault, San Cayetano fault, etc.). [28] In the central and eastern Transverse Ranges the stress state is dominated by approximately uniaxial compression oriented approximately NW–SE in the central Transverse Ranges (San Gabriel Mountains) and approximately NNW – SSE in the eastern Transverse Ranges and Mojave Desert. These stresses are clearly a consequence of the negative vertical tractions and the approximately SSW oriented horizontal tractions, strongest in the Mojave Desert. NNW–SSE compressive stress is consistent with the many approximately E – W striking thrust and oblique slip faults accommodating approximately N–S shortening such as the Cucamonga fault, Sierra Madre fault, San Gorgonio fault zone, North Frontal fault zone, etc. [e.g., Meisling and Weldon, 1989; Morton and Matti, 10.1029/2008GC001988 1993; Spotila and Sieh, 2000], and N–S contraction throughout the Mojave [Bartley et al., 1990]. NW– SE compressive stress along the Big Bend of the San Andreas fault also has the effect of rotating the plane of maximum right-lateral shear stress counterclockwise with respect to a background stress state caused by plate motions [Sonder, 1990]. [29] We use the entire Yang and Forsyth [2006a] velocity model in the calculations leading to Figures 3–6, including that which falls outside the polygon defining the well-resolved velocity structure (e.g., Figure 2). To test whether possibly erroneous velocity and inferred density structure outside the well-resolved region strongly influences our results, we have performed the same viscous flow calculations with the density structure restricted to the well-defined polygonal region of Figure 2. Figure 7 shows the resulting horizontal tractions and crustal stress field. Except along the California-Nevada border, where traction magnitudes decrease by 50% or more, the results are largely unchanged. Compared to the results in Figure 4, horizontal traction orientations are virtually identical. Magnitudes increase slightly ((0.2 MPa) northeast of the San Andreas fault in the vicinity of the eastern Transverse Ranges downwelling, decrease slightly ((!0.1 MPa) near the offshore downwelling region, and increase ((!0.5 MPa) along the eastern side of the Sierra Nevada anomaly. The crustal stress field (Figure 7b) is also nearly the same with only minor differences. We conclude from this analysis that the computed basal tractions and crustal stress field in our study area are dominated by local, relatively shallow, density structure and not strongly influenced by structure outside the well-resolved region of the velocity model. For the remainder of this paper, all model comparisons and discussion will refer to the results shown in Figures 3–6 utilizing the entire velocity model. 3.2. High-Viscosity Lithospheric Lid [30] Average lithospheric thickness in southern California has been estimated to be approximately 90 km [Humphreys and Hager, 1990; Yang and Forsyth, 2006a], although this does not apply to the southern Sierra Nevada where lithospheric mantle is likely absent [Yang and Forsyth, 2006a; Savage et al., 2003]. Here we test whether the presence of a high-viscosity mantle lithosphere lid 60 km thick (90 minus average crustal thickness of 30 km) significantly changes the predicted stresses compared to the uniform viscosity case in Figure 4. 11 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Figure 7. (a) Horizontal tractions on the base of the crust (as in Figure 4) and (b) principal stresses in the crust (as in Figure 5) resulting from upper mantle flow driven by density structure restricted to within the well-resolved region of the seismic velocity model (nonstippled area). The basal tractions and crustal stress field are very similar in orientation and magnitude to those based on the entire velocity model (Figure 4), implying our results are not strongly dependent upon the velocity and density structure outside the study area. All else being equal, a higher viscosity lid should support a larger fraction of the shallow load, thereby increasing the tractions on the overlying crust. [31] Figure 8 shows that this is in fact the case. We show the horizontal tractions and crustal stress field for the case of a lithospheric lid 2 orders of magnitude higher viscosity than the underlying mantle. Traction orientations do not change appreciably whereas magnitudes increase nearly everywhere with maximum increase of !1.2 MPa. The largest increase occurs onshore and just offshore near the western Transverse Ranges anomaly and Figure 8. (a) Horizontal tractions and (b) crustal principal stresses for the case of a high-viscosity lithosphere 90 km thick and 100 times as viscous as the underlying mantle. The resulting basal tractions (as compared with Figure 4) are virtually unchanged in orientation and increase in magnitude by ( 25%. 12 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Figure 9. (a) Horizontal tractions and (b) crustal principal stresses for the case of density-dependent (as a proxy for temperature-dependent) viscosity. Here r = 5 in equation (2), indicating 1/2 order of magnitude increase/decrease in viscosity for every 0.1 km/s increase/decrease in shear wave velocity relative to the layer average. Traction orientations 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 is most pronounced above the dense bodies in the Great Valley and Transverse Ranges. in the Mojave Desert near the eastern Transverse Ranges anomaly to a maximum magnitude of 3.9 MPa. The crustal stress field (Figure 8b) is also largely unchanged in principal stress orientations although magnitudes increase from !20 – 50% depending upon location. A similar model (not shown) with only 10 times higher lithospheric lid viscosity produces very similar results indicating 1 order of magnitude may be sufficient to significantly modify how density anomalies are supported in the lithosphere. 3.3. Three-Dimensional Viscosity Variations [32] The viscosity of the Earth’s mantle is thought to be highly temperature dependent, often assumed to follow the Arrhenius relationship where viscosity is proportional to exp(Q/RT), where Q is the activation energy, R is the universal gas constant and T is absolute temperature [e.g., Ranalli, 1995]. We assume that the velocity and density anomalies in the southern California upper mantle arise from temperature variations, and therefore should also reflect viscosity variations; fast/dense regions are relatively cool and therefore more viscous, and slow/buoyant regions are relatively warm and therefore less viscous. To test whether three-di- mensional viscosity variations influence our estimates of tractions on the base of the crust, we have devised an ad hoc relationship between seismic velocity and viscosity that mimics the strong dependence of viscosity on temperature as in the Arrhenius relationship, and avoids the need to estimate temperature anomalies from seismic velocity anomalies. Viscosity (h) is given as h ¼ h0 10ðdVs *rÞ ð2Þ where h0 is the reference viscosity, dVs is the velocity anomaly (expressed in km/s) and r is a constant. Figure 9 shows the tractions and crustal principal stresses for the case of r = 5, i.e., every 0.1 km/s variation in seismic velocity produces a factor of !3.16 (1/2 order of magnitude) change in viscosity, resulting in !2.9 orders of magnitude total viscosity variation throughout the entire model. Figure 10 shows the case of r = 10, i.e., every 0.1 km/s variation in seismic velocity produces a factor of 10 change in viscosity, resulting in over 5 orders of magnitude variation in viscosity throughout the model. [33] As in the lithospheric lid example in the previous section, horizontal traction orientations are largely unchanged and only magnitudes depend 13 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 Figure 10. (a) Horizontal tractions and (b) crustal principal stresses for a heterogeneous viscosity upper mantle with r = 10 in equation (2) (1 order of magnitude increase/decrease in viscosity for every 0.1 km/s increase/decrease in shear wave velocity relative to the layer average). on the viscosity distribution, although here the effect is somewhat less homogeneous. Horizontal tractions on the eastern side of the Sierra Nevada anomaly decrease in magnitude, owing to the particularly slow upper mantle seismic velocity and inferred low viscosity. Tractions along the California-Nevada border directed toward the downwelling region in southeast Nevada decrease for the same reason. Traction magnitudes also decrease to the north of the offshore downwelling in the western Transverse Ranges, owing to the shallow slow velocities (Figure 2a) and inferred low viscosity there. Magnitudes generally increase in the Mojave because it is underlain by high velocities (Figure 2a). For r = 5 (Figure 9), tractions increase in the Mojave by (1 MPa to a maximum of !3.3 MPa, and for r = 10 (Figure 10), tractions increase by (2.3 MPa to a maximum of !4.5 MPa with typical magnitudes of 2–3.5 MPa. The crustal stress fields in Figures 9b and 10b change in much the same way with orientations of principal stress largely the same and local variations in magnitude. [34] The viscosity-velocity relationship is equation (2), and associated model calculations (Figures 9 and 10), are meant only to demonstrate the trend in predicted stresses if upper mantle viscosity and density are both related via temperature; as the viscosity dependence on temperature increases, basal tractions on the Moho and crustal stress state change little in orientation and sometimes appreciably in magnitude. This effect is strongest to the east of the downwelling in the southern Great Valley, where tractions and crustal stresses decrease in magnitude, and in the Mojave Desert to the northeast of the SAF, where tractions and crustal stresses increase in magnitude. 3.4. Weak Lower Crust [35] Thus far we have treated the entire crust as essentially rigid in order to isolate the stresses transmitted across the Moho. This effectively assumes the viscosity of the lower crust is (much) greater than the upper mantle, generally consistent with geodetically inferred estimates of lithospheric viscosity [e.g., Freed et al., 2007; Thatcher and Pollitz, 2008]. Nonetheless, it is useful to consider the case of the lower crust with a reduced viscosity. We have calculated the stresses for the case of the lower crust viscosity an order of magnitude lower than the upper mantle. The resulting stress orientations in the crust are very similar to the case of a rigid lower crust (Figure 5). Principal stress magnitudes (vector sum of the horizontal principal stresses) decrease by an average of !0.5 MPa (within the nonstippled area of Figure 5), corresponding to an average percent decrease of !12. The relatively small decrease in principal stress magnitudes 14 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics reflects the fact that while the relatively low viscosity of the lower crust somewhat decreases its ability to support horizontal shear stresses, the vertical normal stresses transmitted across the Moho are decreased by an even smaller amount, resulting in similar stress orientations and only slightly decreased magnitudes. 4. Comparison With Observations and Discussion [36] Do the tractions and crustal stresses in Figures 3– 10 contribute in a significant way in driving crustal deformation in southern California? Unfortunately it is not possible to directly answer that question as we have no means of observing tractions at Moho depths and only sparse quantitative measurements of the shallow state of stress in the crust [e.g., Zoback and Healy, 1992]. We must therefore use other measures as proxies. In this section we discuss two independent measures of crustal dynamics: (1) active deformation patterns as revealed by geodetic data and (2) an estimate of the sum of forces acting on a crustal block. Both seem to indicate the stresses on the base of the crust may be an important contribution to the sum of forces driving crustal deformation. For the remainder of this paper, we focus on the tractions and crustal stress field for the case of a uniform viscosity upper mantle, shown in Figures 3–6. 4.1. Active Deformation Patterns [37] An idealized transform plate boundary experiences only shear deformation and therefore no change in surface area with time. Transpression in the Transverse Ranges and transtension in the Walker Lane Belt and Salton Trough suggests that the San Andreas fault system in southern California is not such an ideal plate boundary. We illustrate this nonideal plate boundary behavior in active deformation patterns by estimating the dilatation rate, or rate of increase in surface area, from the dense geodetic velocity field in southern California (Figure 11a). Dilatation rate is the trace of the strain rate (_ev) tensor and is coordinate system invariant making it a useful measure to compare with the frame invariant tractions and crustal stresses predicted by our model. We have combined the published data sets of Shen et al. [2003] and Kreemer and Hammond [2007] with a new velocity solution processed at the University of Arizona consisting of existing and new data collected in the eastern Transverse Ranges; the latter data set, including monumentation, data process- 10.1029/2008GC001988 ing, and results, are described by Bennett et al. [2006] and a forthcoming publication (J. C. Spinler et al., manuscript in preparation, 2008). The three data sets were rotated onto a common stable North American reference frame [Blewitt et al., 2005] using common velocities at continuous GPS sites and the average was taken when more than one velocity estimate existed at a single point. [38] To estimate dilatation rate, velocities were interpolated to a uniform grid using piecewise continuous tri-linear (Delaunay triangulation) interpolation, smoothed with a moving-window Gaussian 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 of Hernandez et al. [2005, 2007] who use a different interpolation and differentiation scheme. This is not surprising in that we consider dilatation rate at a wavelength much greater than typical geodetic station spacing and therefore it is not particularly important how velocity is interpolated between stations. [39] Figure 11a shows two dominant lobes of negative dilatation (net area loss) in the western Great Valley. The northern lobe may be an artifact of relatively sparse station coverage, although some shortening at high angle to the San Andreas fault is expected in the California Coast Ranges [Wentworth and Zoback, 1989; Argus and Gordon, 2001] and very little (and even net extension) is seen along the San Andreas itself (Figure 11a). The southern lobe of negative dilatation is likely real and consistent with contractional structures in the southern San Joaquin Valley such as the Pleito and White Wolf fault systems [e.g., Keller et al., 1998, 2000; Stein and Thatcher, 1981]. In the southeastern Great Valley and southern Sierra Nevada, we predict a minor amount of positive dilatation, consistent with observed extensional faulting [Jones and Dollar, 1986]. [40] Positive dilatation is seen in the southern Walker Lane Belt consistent with active transtension along the eastern margin of the Sierra Nevada [Unruh et al., 2003], effectively the western boundary of the extensional Basin and Range province [Wernicke, 1992; Wernicke and Snow, 1998]. Net area gain is seen in the Mojave Desert east of the Eastern California Shear Zone and in the eastern Transverse Ranges; the latter is broadly consistent with a right step in the transfer of rightlateral slip between the southernmost San Andreas and Eastern California Shear Zone [Savage et al., 15 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 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 of 120 km and differentiated to obtain strain and dilatation rates. Anywhere more than 60 km from a geodetic station is not shown. Positive dilatation indicates net extension (increase in surface area), and negative dilatation indicates net compression (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 mesh nodes 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 the same location as Long Valley Caldera (LVC), implying the possibility of a causal relationship between crustal extension, thinning, and volcanism. 1993; Johnson et al., 1994; Hudnut et al., 2002]. Areal extension is seen near the southern termination of the San Andreas fault, curiously offset to the NE of the Salton Sea. Negative dilatation is evident in the western Mojave, and central and western Transverse Ranges, consistent with geologic 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 11a may be a transient effect of postseismic relaxation following major earthquakes [e.g., Nur and Mavko, 1974]. Transient surface deformation in the years following the 1992 and 1999 Mojave Desert earthquakes was clearly seen in geodetic data [Pollitz et al., 2001; Freed and Bürgmann, 2004; Fialko, 2004; Freed et al., 2007], but whether this influences our steady state dilatation estimates, based on data collected before and after these events, is not yet clear. The broad agreement between predicted dilatation rates and geologic deformation, and the lack of evidence for postseismic relaxation throughout all of southern California [Meade and Hager, 2005; Argus et al., 2005] allows us to proceed with some confidence that the geodetically derived dilatation estimate is generally representative of relatively long wavelength permanent crustal deformation. [42] The predicted crustal deformation from our model with a uniform viscosity upper mantle (Figures 3–6) is given in Figure 11b. We calculate the horizontal dilatation rate from the strain rate tensor at 20 km depth. Extension is reckoned positive and dilatation rate is equal and opposite to the vertical strain rate (_ev) because the crust is treated as an incompressible fluid. Here strain rates 16 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics 10.1029/2008GC001988 the approximately radially compressive stress in the offshore/western Transverse Ranges region and uniaxial compressive stress in the Mojave and eastern Transverse Ranges (Figure 5). Figure 12. A simplified Mojave block (double solid line) and the forces acting on it. Basal tractions from Figure 4 within the block were interpolated to a finer grid. These tractions impart a counterclockwise torque on the block. Shear stress supported by the left lateral Garlock fault (GAR) also imparts a counterclockwise torque. Shear stress supported by the right lateral San Andreas fault (SAF) and Eastern California Shear Zone (ECSZ) impart clockwise torque and may balance the basal traction and Garlock loads, although we have not included stresses acting normal to the block boundaries such as the excess pressure at depth caused by the elevated San Bernardino Mountains. The torque form these loads is quantified in the text. are only relative because absolute value depends on the absolute value of viscosity and this does not enter into our calculations. [43] Negative dilatation is predicted in the southern Great Valley resulting from crustal convergence over the downwelling there. Positive dilatation is seen in the eastern Sierra Nevada and Walker Lane Belt and extends south of the Garlock fault into the eastern Mojave region as well. This is a consequence of the upwelling and divergent flow of buoyant upper mantle, with the largest divergence approximately at the intersection of the Walker Lane Belt, Garlock fault and Eastern California Shear Zone. Areal extension is also predicted in the Salton Sea area, a result of upwelling in the upper mantle and divergent crustal strain. [44] Crustal convergence and negative dilatation is seen throughout the Transverse Ranges. The strongest area loss is offshore because the upper mantle density anomaly is largest there and because the induced horizontal basal tractions are more convergent than in the Mojave and eastern Transverse Ranges (see Figure 4), resulting in greater net crustal strain and area loss. This is consistent with [45] Overall, the comparison of observed (Figure 11a) and predicted (Figure 11b) dilatation rate is compelling; the model is generally successful in predicting the style and relative rate of crustal dilatation. Negative dilatation is correctly predicted in part of the southern Great Valley and in the western Mojave and Transverse Ranges, and positive dilatation is predicted and observed in the Walker Lane Belt, eastern Mojave and Salton Sea regions. Notably, the model and observed dilatation fields both show the strongest crustal dilatation and thinning along the eastern margin of the Sierra Nevada at !241!E, 37.5!N, the location of Long Valley Caldera. The observed dilatation could be a result of a recent shallow magmatic intrusion or expansion event causing surface uplift and extension [e.g., Newman et al., 2006]. However, our model contains no such mechanism and therefore the spatial coincidence of observed and predicted dilatation implies a possible causal relationship between upper mantle upwelling and crustal extension and thinning, and the voluminous volcanism and historic seismicity [e.g., Hill, 2006] at Long Valley Caldera. [46] To quantify the correlation between observed and predicted dilatational deformation, we compare the predicted and observed sign (positive or negative) of dilatation rate at model nodes within our study area (polygon in Figure 11). We find that !70% of the model nodes (25 km spacing) predict the same sign as observed dilatation at the same point, and therefore !70% of the surface area of southern California experiencing significant dilatation is correctly predicted by the model. Dilatation rate less than 10 nanostrain/yr is considered insignificant and excluded in this calculation. Comparing dilatation rates is more difficult because our model does not consider absolute strain rates. 4.2. Forces Acting on the Mojave Block [47] Thus far we have considered model predictions based largely on the orientation of the mantle tractions and crustal stress field. Here we consider magnitude by estimating the sum of torques acting on a crustal block. Because the vector sum of torques on any plate on the surface of the Earth must be zero, it is useful to consider the set of forces acting on a piece of crust to compare their relative magnitudes. Figure 12 shows the Mojave 17 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics block, bounded by the dextral San Andreas fault and Eastern California Shear Zone, the sinistral Garlock fault, and the San Bernardino Mountains. For the following simple calculation, it is not necessary to consider the details of the bounding faults orientation and the Mojave block is defined as a simple, four-sided polygon. In Figure 12 it can be seen that the San Andreas fault and Eastern California Shear Zone shear loads impart clockwise torque (about a hypothetical vertical pole in the center of the block), and the Garlock shear load and the basal traction field both impart a counterclockwise torque, and, qualitatively, their sum is zero. [48] To quantify these loads, we estimate the Rtorque * on the block from each load by calculating ( r % * * the radial vector from the center F )dA, where r is * of the Earth and F is the force vector (stress % area) acting on an area element dA. The horizontal basal tractions were interpolated to a denser grid (10 km spacing) so as to avoid biasing the calculation by a nonuniform distribution of integration points. For the uniform mantle viscosity case in Figure 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 the basal tractions in the vicinity of the Mojave block increase, as compared to the uniform viscosity case, for all scenarios of variable upper mantle viscosity that we have explored. For example, if we use the traction field from Figure 7a, calculated for a lithospheric lid 2 orders of magnitude higher viscosity than the underlying mantle, the total torque is 2.6 % 1023 mN, !25% larger. [49] To compare with other loads acting on the block, we use 30 MPa as the depth-averaged, longterm shear stresses supported by block-bounding faults in southern California [Fay and Humphreys, 2006] and assume this stress acts over the approximate seismic depth of 15 km. This results in torque magnitudes of 4.4 % 1023 mN, 5.4 % 1023 mN, and 3.9 % 1023 mN for the San Andreas fault, Eastern California Shear Zone, and Garlock fault respectively. These torques are approximately a factor of 2 larger than the basal traction torque, but clearly of the same order of magnitude and thus the basal tractions are likely as relevant to the dynamics of the Mojave block as plate interaction stresses supported by active faults. Stresses acting on horizontal planes such as our basal traction estimates (e.g., 3 MPa) may seem small, but it is the integrated force and torque that influences crustal 10.1029/2008GC001988 dynamics and basal tractions on horizontal planes typically act over a large area. 5. Summary and Conclusions [50] Crustal deformation in southern California is clearly strongly influenced by shear stress owing to relative motion of the Pacific and North American plates. Superimposed on shear deformation is a significant amount of crustal shortening and mountain building in the Transverse Ranges. We have shown through numerical modeling of the stresses on the crust induced by upper mantle flow, that much of this ‘‘nonideal’’ plate boundary behavior may be explained by being driven from below by small-scale convection of the southern California upper mantle. Favorable comparison with active deformation determined from geodetic data and the sum of forces acting on crustal blocks suggest the stresses exerted on the crust from small-scale downwellings and upwellings in the upper mantle are an important component of the sum of forces driving crustal deformation in southern California. [51] Figures 3–10 show the computed tractions on the base of the crust and the resulting stress fields within the crust. Horizontal tractions are typically 3–4 MPa or less in magnitude (for the chosen seismic velocity to density scaling relationship). Traction orientations are largely determined by the distribution of seismic velocity anomalies [Yang and Forsyth, 2006a] and inferred density structure in the upper mantle, and are largely independent of viscosity distribution. Three nearly distinct downwelling zones in the southern Great Valley, western Transverse Ranges and eastern Transverse ranges create negative vertical tractions and convergent horizontal shear tractions on the base of the crust. In the southern Great Valley, mantle lithosphere formerly beneath the Sierra Nevada is now sinking beneath the southern Great Valley, causing active surface subsidence [Saleeby and Foster, 2004] and approximately radially convergent shear tractions on the base of the crust. In the Transverse Ranges, two dominant downwelling zones centered offshore and beneath the western San Bernardino Mountains act to draw in the overlying crust and cause approximately N – S compressive stress (Figure 5), and active shortening (Figure 11). [52] Crustal deformation driven by small-scale convection helps to explain the obliquity of the Mojave segment of the San Andreas fault with 18 of 23 Geochemistry Geophysics Geosystems 3 G fay et al.: upper mantle convection and crustal dynamics respect to plate motions. Shortening in the Transverse Ranges as a consequence of a bend in the San Andreas fault is a plausible kinematic interpretation, but provides no dynamic explanation for why the lithosphere maintains this apparently energetically unfavorable plate boundary geometry. Furthermore, there does not appear to be any clear relationship between rock uplift, horizontal shortening and the obliquity of the San Andreas fault [Spotila et al., 2007]. Our model, which successfully predicts the majority of the dilatational deformation in southern California without incorporating any shear strain and oblique geometry of the San Andreas fault (Figure 11), provides an alternative kinematic interpretation that the bend in the fault is not a cause but rather a consequence of contraction in the adjacent crust. In this view the fault does not act as a fixed barrier to crustal flow but simply a geographic point attached to the adjacent, deforming crust. With respect to a fixed Pacific plate, approximately N–S shortening southwest of the San Andreas fault in the western Transverse Ranges [e.g., Yeats et al., 1988] requires that the trace of the San Andreas fault move to the southwest. Shortening on the northern side of the San Andreas fault in the San Bernardino Mountains [e.g., Dibblee, 1975; Spotila and Sieh, 2000] requires that the trace of the San Andreas there move to the NNE. Both process tend to rotate the trace of the San Andreas fault counterclockwise with respect to its previous orientation, and increase the obliquity with respect to plate motion direction. [53] Thus the dynamic effect of heterogeneous upper mantle density structure and induced flow on crustal deformation in southern California is twofold. First, upper mantle downwelling, strongest beneath the Transverse Ranges, drives shortening of the overlying crust to the southwest and northeast of the present-day location of the Big Bend segment of the San Andreas fault. This has the kinematic consequence of rotating the San Andreas counterclockwise, thereby increasing its obliquity with respect to plate motions, possibly inducing additional downwelling of lower crust or mantle lithosphere during convergence. There must be a limit to this positive feedback in that the elevation of the Transverse Ranges is finite and therefore provides an independent means of estimating the magnitude of the mantle tractions [Humphreys and Hager, 1990; Fay and Humphreys, 2006]. Analysis of a similar feedback process of fault orientation and local topography has also provided constraint on the state of stress within the crust [Fialko et al., 10.1029/2008GC001988 2005]. Second, the stress field associated with the excess density beneath the Transverse Ranges (Figures 5 and 6) tends to rotate the plane of maximum right lateral shear stress counterclockwise with respect to the background stress field (caused by plate motions), thereby promoting slip on faults such as the San Andreas oriented counterclockwise of the plate motion direction [Sonder, 1990]. Acknowledgments [54] We thank Y. Yang and D. 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