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Volume 9, Number 8
12 August 2008
Q08006, doi:10.1029/2008GC001988
AN ELECTRONIC JOURNAL OF THE EARTH SCIENCES
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ISSN: 1525-2027
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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
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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
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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
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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
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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.
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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
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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
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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.
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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
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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.
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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
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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.
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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
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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,
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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.
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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%.
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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
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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
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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-
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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.,
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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
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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
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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
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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
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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. Forsyth for sharing their
tomography model, Walter Landry and the Computational
Infrastructure for Geodynamics for developing and maintaining Gale, C. Kreemer for sharing his GPS data set prior to
publication, and two anonymous reviewers for their constructive reviews. The figures were prepared with GMT
[Wessel and Smith, 1998]. This research was supported by
NSF grant EAR-0510484 to R.B.
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