TECTONICS,
EFFECTS
OF DENSITY
ORIENTATION
RELATION
VOL. 9,
NO. 4,
CONTRASTS
OF STRESSES
TO PRINCIPAL
THE TRANSVERSE
1990
ON THE
IN THE
STRESS
PAGES 761-771, AUGUST
LITHOSPHERE:
DIRECTIONS
IN
RANGES, CALIFORNIA
Leslie J. Sonder
Departmentof Earth Sciences,DartmouthCollege,
Hanover,New Hampshire
Abstract.The influenceof stresses
arisingfrom horizontal
densitycontrasts
on theorientation
andrelativemagnitudes
of
principalstresses
in an otherwiseunifo•xnlithosphericstress
field is investigated.
A simplemodelis constructed,
in whicha
local deviatoricstressdueto a densityanomaly,embedded
within orjust below thelithosphere,anda regionallyconstant
deviatoricstressfield areeachapproximated
by biaxialtensors.
The net stress field is obtained from the sum of the two. Both
therelativemagnitudes
of principalstresses
andthemagnitude
of theangulardifferencein principalstressdirectionof the
summedtensorcomparedwith thatobtainedin theabsence
of
buoyancyforcesdependon two parameters.
The first is the
ratio 'c/'c',where'cis a measureof the magnitudeof the
regionaldeviatoricstressand'c'is themagnitudeof thestress
arisingfrom buoyancyforcesassociated
with thedensity
anomaly.The secondparameteris theanglebetweenthetrend
of the densityanomalyandthe directionof maximum
compressional
stressthatobtainsin the absenceof any
perturbation
by thelocalbuoyancyforces.The directions
of
theprincipalaxesof thetotalstressfield arefoundto differby
up to 90øfrom thoseof thereferencestressfield. The modelis
appliedto the TransverseRanges,California,wherethe
observed23ø differencein orientationof principalhorizontal
compressive
stresscomparedwith theprincipalcompressive
stressdirectionin centralCaliforniaconstrainsthe predicted
value of 'c/'c'to be approximately-0.4. This is consistent
with an independently
calculatedrangeof 'c/'c'in which'c'is
inferredfrom seismological
constraints
on themagnitudeof
densityvariationsunderneath
theTransverseRangesand'c is
inferredfrom observations
of heatflow alongthe SanAndreas
Copyright 1990
by the AmericanGeophysicalUnion.
Paper number 90TC00369.
0278-7407/90/90TC-00369510.00
fault in centralCalifornia.The agreementbetweenthe two
estimates
of 'c/'c'supports
thehypothesis
thattheobserved
differences
in horizontalprincipalstressorientationin
Californiacanbeexplained
by thecombined
influence
of a
localnegative
buoyancy
forceundertheTransverse
Ranges
and
a regionalstress
fieldassociated
withtranscurrent
deformation
withinthePacific-NorthAmericanplateboundaryzone.The
observed
counterclockwise
angulardifferencein principal
horizontalstressdirectionin the TransverseRanges compared
with centralCaliforniaimpliesthatthe planeof maximum
rightlateralshearstress
is alsorotatedcounterclockwise
relativeto that in centralCalifornia. This supportsthe
possibility
thatthe"bigbend"in theSanAndreas
faultmaybe
a consequence
of thenegative
buoyancy
fomesactingin the
Transverse
Ranges,andnotthecauseof Transverse
Ranges
formation, as has often been assumed.
1. INTRODUCTION
It is well known that densitycontrastswithin the
lithospherecanleadto horizontaldifferencesin the magnitude
of lithosphericstress[e.g.,Bott andDean, 1972;Artyushkov
1973; Fleitout and Froidevaux, 1982]. It should also be
evidentthat the buoyancyforcesdue to thesedensitycontrasts
may alsoaffect the orientationof the principalstresses,
yet
comparativelylittle noticehasbeentakenof this. In this paper
I developa simplemodelto examinetheeffectsof a density
anomalyon the stateof stressin the lithosphere,with
particularattentionpaid to changesin the orientationof the
principalstresses.
The model is testedby comparisonwith
observations
from the TransverseRanges,California,where
thereis goodseismicevidencefor a wedge-shaped
regionof
anomalouslyhigh densityin the uppermantle[Humphreyset
al., 1984] and wherethereare significantdifferencesin the
orientationof the stressfield comparedwith surrounding
regions[e.g., Zoback et al., 1987].
762
Sonder:Effectsof DensityContrasts
on StressDirections
2. METHOD
Althoughtherearelnanyfactorsthatlnayaffectthe stateof
stressin the lithosphere,suchas drivingandresistingforcesat
theedgesof plates.or differences
in thestrengthof the
lithosphere,the subjectof interesthereis variationsin the
stressfield arisingfroin horizontaldensitycontrasts.
Hence,
for simplicity,the stateof stressis approximatedas that
resultingfi'omthe linearsuperposition
of two statesof stress.
Source of stress
Extent
Large-scale sources
of stress (e.g., plate
boundary interactions)
Regional
Geometry
The first, taken as a reference state, is assumedto include all
othersourcesof stressexceptdensityvariationsnearor at the
baseof the lithosphere,whicharedescribedby the second
stressstate.Sincethe purposeof this paperis to examinethe
extentto whichbuoyancyforcesalonecanmodifythe
orientationof the principalstresses,
the referencestateis taken
to be horizontallyandverticallyconstant;it lnay be thus
describedby a singletensorwith constantcomponents.
Althoughtheseassumptions
representan over-silnplification,
the simplemodelthatcanbe derivedfrom themprovides
usefulinsightsintothe effectsof buoyancyforceson the
orientationof stresseswithin the lithosphere.
I make the further assumptionthat the referencestressstate
T is biaxial, with one principalaxis vertical,and represents
one of eithercompressional,
tensional,or strike-slipstatesof
Buoyancy forces arising
from
Local
horizontal
density contrasts
i•/T •ø•
Total
stress
Local
stress:
Compression:
-,•
0
0
0
0
0
0
0
'r
0
0
0
0
'r
0
0
0
-'•
0
0
0
'r
0
0
0
0
T=
Fig. 1. Schematicdescriptionof modelpresentedin section2,
showingorientationof regionalstresstensor(T), stresstensor
duetodensity
anomaly
(T'),andtotalstress
tensor
(Ttøt),
and
definitionof 0, 0', and coordinatesystems.Double mTows
indicatedirectionsof principalhorizontalcomponents
of T,
T', andTtøt.
Tension:
T=
(lb)
AppendixA showsthat, for the regionabovethe density
anomaly,the near-surfacedeviatoricstressfield is
approximatelyconstantandmay be expressedasa biaxial
tensorin the (x'-y'-z) coordinatesystem:
Strike-slip:
-'r
T=
0
(lc)
All stressesare deviatoric,tensionalstresses
have positive
sign,and'r is alwayspositive.The coordinate
frameis shown
in Figure 1; the z axis is verticalandthe x axis is takento be
parallelto thedirectionof maximumhorizontalcompression.
(Note that in the extensionalcase, the x-axis is parallel to the
interrnediate
principalstressdirection,sincethe maximum
compressional
stressis vertical.)
The deviatoricstressfield arisingfrom a local density
anomalydependson the shapeandmagnitudeof the anomaly.
Definingan x'-y'-zcoordinate
systemsothatz is verticaland
x' andy'are parallelandperpendicular,
respectively,
to the
longhorizontalaxisof the anomaly,I considerherea density
anomalyof constantmagnitudeAp andthicknessL andwhose
extentin the x' directionis muchgreaterthanits width 2w, so
thatvariationsin densityin the x' directioncanbe neglected.
0
0
0
'r'
0
0
0
,
T=
(2)
-'•
•:', the magnitudeof the components,
is approximately
0.3 Apg- 0.4 Apg, whereg is the accelerationdueto gravity.
If'r' is negative,thistensorrepresents
a negativebuoyancy
fome whichwouldlead to a stateof horizontalcompression
perpendicular
to the trendof the anomaly;conversely,if'r' is
positive, horizontaltensionis implied.
Thecoordinate
frame(x'-y'-z)inwhichT' isdefined
doesnot
generallycoincidewith the (x-y-z) frame in whichT is defined.
Asshown
inFigurel, toexpress
T' in thesamecoordinate
frameasT, T' mustberotated
through
anangle0 about
a
verticalaxis.Thus, 0 is the anglebetweenthe trendof the
densityanomaly(the x'-axis)andthe x-axis.
Thenetstress
tensor,
Ttot,isthetensor
sumofT andT'
(Equations1 and2), andin the x-y-z referenceframemaybe
written
as:
Sonder:Effectsof DensityContrastson StressDirections
763
Compression:
The two solutions(Equations(4) and (5)) differ only by the
factor of two in front of the term 'c/'c'.
-'c+'c'sin20 'c'sin0cos0
T tot= 'c'sin 0 cos 0
0
0
(3a)
'c'cos2 0
0
0
'r - "c'
'c'sin0 cos0
0
Ttot._17'
sin0cos
0 'C+ 'C'
cos
20
0
0
0
RESULTS
3.1. Changein Orientationof Horizontal Principal Stresses
Tension:
'c'sin2 0
3. GENERAL
-'r-
(3b)
The angularamount0' by which the orientationof the
horizontalprincipalstresseschangesin the presenceof
buoyancyforcesdependssimplyon two factors.One is the
ratio of the magnitudeof the regionalstresses
and the
Strike-slip
"c'
Strike-slip:
60
-'c+'c'sin20 'c'sin0cos0 0
Ttot= '1;'
sin0COS
0 'C
+'C'
cos
20 0
0
0
(3C)
45
30
'r'
The principaldirectionsof this summedtensorwill in most
casesdiffer by rotationabouta verticalaxisfrom theprincipal
directions
of bothT andT'. Thex"-y"-zreference
frameis
o
-15
-30
definedto coincidewith the principalaxesof Trotsothatthe
x" axiscorresponds
to thedirectionof maximumhorizontal
compressional
stress.The angle0' is definedasthe angle
betweenthe directionsof the x axis andthe x" axis (Figure 1).
Thus0' is the angleby whichthepresenceof a lineardensity
anomalycausesthedirectionsof thehorizontalprincipal
o.s
-45
-60
-75
-90
-90-75-60-45-30-15
stresses of the summed stress tensor Trot to differ from the
principalstressdirectionsof thereferencestate(whenthereis
no densityanomaly).
relative
60
45
o
ltan-•[
sin20
]+•
-2'c/ 'c'- cos20
75
15
-2'c/'c'-cos 20
(4)
2
Compression/extension
90
3o
1tan-•
[ sin20
]
2
-15
-30
2
-45
-60
to the x axis.
WhenT represents
eithera compressional
or extensional
stateof stress,theseanglesare
-75
-90
-90-75-60-45-30-15
1tan-•[.
sin20
]
2
ltan-•[
sin20
]+•_
- x / x'-cos 20
0
15 30 45
60 75 90
0 (degrees)
- 'c/ 'c'-cos 20
(5)
2
15 30 45 60 75 90
0 (degrees)
Theorientation
of theprincipalaxesof T tøtarefoundin the
usualmanner,thatis, by transforming
T' into the x"-y"-z
referenceframe by rotatingit throughthe angle0', settingthe
off-diagonalcomponents
to zero,andsolvingfor 0'. For the
caseof Equation(lc), whenthereferencestateis strike-slip,
the principalaxesare at anglesof
0
2
Fig. 2. Changein orientation
(0') of principalhorizontal
compressive
stress
asa functionof 0. Numbers
oncurvesrefer
to valuesof 'c/'c'.Stipplingindicates
fieldsthatarephysically
inaccessible.
(a) Strike-slipreference
state(T). (b)
Compressional
andextensional
reference
states(T).
764
Sonder:Effectsof DensityContrastson StressDirections
Strike-slip
magnitudeof the stressarisingfrom the densityanomaly,
expressedas'c/'c'.The otherfactor,0, characterizes
the
geometryof the system.Figure2 showshow 0' dependson
thesetwo parameters.Considerfirst the curve with 'c/'c'=
+ oo.In this casethe buoyancyforcesdue to the density
anomalyare negligible,so that Trot is identicalto T and 0' is
reference
state
2.5
1.5
normal
0.5
zero.
-0.5
Conversely,as 'c/'c'--> 0, stressesdue to the buoyancy
forcesdominate,so that the principalhorizontalstresses
of
Trotbecomeparallelto the x' andy' axes(seeFigure1). The
sign of 'c'deterrninesthe axis to which x" is parallel.When 'c'
is negative,the principalstressnormalto the trendof the
densityanomalyis compressive,so that x" is parallelto the y'
axis and 0' differsfi'om0 by 90ø. When'c'is positive,the
greatesthorizontalcompressivestressdirectionis parallelto
the trendof the densityanomaly,so 0'= - 0.
For intermediate
values of 'c/'c' the orientation
thrust
ri•
a,,, s.t.
-1.5
-2.5
Extensional
reference
state
0.5
normal
0.0
of the
thrust
-0.5
principalaxesof Trot generallydiffersfrom thatof bothT and
T'. In general,
theeffectof a negative
density
anomaly
('½< 0)
is to rotatethe principalhorizontalcompressive
stress
directiontowardthe normalto the trendof the density
anomaly,andthatof a positivedensityanomaly('½> 0) is to
rotatetheprincipalhorizontalcompressional
stressdirection
-1.5
away from the norrnal.
Discontinuitiesin the curves(for example,for the strike-slip
case, when 'c/'c'= + 0.5 and 0 = + 90ø or when 'c/'c'= - 0.5
and 0 = 0) occurwhen the two horizontalprincipalstresses
of
-2.5
-2.0
b
Trot beomeequal,sothat neithercanbe uniquelyidentifiedas
the greatestor leastprincipalhorizontalcompressive
stress.
2.5
reference state. On the other hand, when 'c/'c'is small, the
densityanomalyhasa large effect, so the signof
'c/'c'determines
the faulting style.Thus, when 'c/'c'is small and
negative,the verticalstressis alwaysthe leastcompmssive,so
thrustfaulting is expected;when'c/'c'is small andpositive,the
vertical stressis alwaysthe mostcompressivestress,hence
normalfaultingis predicted.
For intermediatevaluesof 'c/'c',the type of faultingdepends
on the relative orientation
of the reference stress state and the
trendof the densityanomaly.Small changesin the relative
magnitudesof 'cand'c'can changethe styleof deforrnation
betweenstrike-slipand normalfaulting,or strike-slipand
reference
state
thrust
20
3.2. Style of Deformation
The styleof deformation,as well asthe orientationof the
horizontalstresses,
can changeas the resultof the
superposition
of the stressdueto buoyancyforcesandthe
referencestress.Accordingto Anderson's
[ 1951] faulting
criteria,the relativemagnitudesof the principalstresses
deterrninethe faultingregime:dependingon whetherthe
verticalstressis the greatest,interrnediate,
or leastprincipal
stress,norrnal,strike-slip,or thrustfaulting,respectively,is
predicted.Figure3 showspredictedfaultingregimesbasedon
Anderson'scriteriaandprovidesusefulinsightinto how
variationsin buoyancyforcescanaffectthe tectonicsettingof
a region.For eachreferencestateandorientationof thedensity
anomaly,eachone of the threefaultingregimesis possiblein
somerangeof 'cand '½.When the absolutevalue of 'c/'c'is
large, the influenceof the densityanomalyis small, so that
the predictedfaultingstyleremainsthe sameasthatof the
Compressional
1 5
•,,
•
,
,
1 0
05
normal
O0
ßC
thrust
-0 5 ...................................
-90-75-60-45-30-150
15 3045
60 75 90
e(degrees)
Fig. 3. Expectedfaultingstyle,accordingto Anderson's
[1951] theory,as a functionof 'c/'c'and 0. Note that no failure
criterionwas considered,so plotsindicatethe faultingregime
assumingthatfailurehasoccmTed.(a) Strike-slipreference
state.(b) Extensionalreferencestate.(c) Compressional
reference state.
thrustfaulting [cf. DalmayracandMolnar, 1981;Froidevaux
andIsacks,1984; Molnar andLyon-Caen, 1988].
Alternatively, if the relativemagnitudesof the stresses
remain
constant,but 0 changesover time (for example,by a changein
the orientationof the regionalstressfield or by finite
deformationthat resultsin a rotationof the densityanomaly)
thenthe styleof subsequent
deformationcanbe greatly
changed.
Sonder:Effectsof DensityContrastson StressDirections
In consideringFigure 3, it is importantto note that whether
or not faultingwith the indicatedstyleoccurswill dependon
the magnitudesof the stressesin relationto the failure strength
of the crust.If the uppercrustis stronganddetbl'msby slip
accordingto Byedee'sLaw, thenlargechangesin the
magnitudesof the greatestand leastprincipalstresses
are
neededto move acrossthe boundarybetweenthe normal
faultingandthrust•hultingfields.This is representedin Figure
3 by heavylines betweenthesefields. However, for transitions
betweenstrike-slipand normalthulting,or strike-slipand
thrustfaulting,the changesin magnitudeof stresscomponents
can be small.
765
4.
COMPARISON
OF
THE
MODEL
TO
THE
TRANSVERSE RANGES, CALIFORNIA
4.1. Tectonicsof the TransverseRanges
Most geologicalandphysiographical
featuresof California
southof the Mendocinotriple junction trend northwesterlyand
can be relatedto regionalnorthwest-southeast
shearing
associated with relative motions of the Pacific and North
Americanplates.An obviousexceptionis theTransverse
Rangesprovince,whichtrendseast-westacrossthe strikeof
mostotherstructures(Figure4a). The provincealsodiffers
40 ø
a
38 ø
36 ø
34 ø
32 ø
124ø
122ø
120ø
118ø
116ø
Fig. 4. (a) Principalstressdirectiondata in California.Shortlines indicatethe orientationof the
maximumhorizontalcompressive
stress.Heavyline outlinesTransverseRangesProvince.Adaptedfrom
Zobacket al. [1987]. (b) Rosediagramsshowingprincipalhorizontalcompressive
stressdirectionsand
meanazimuths(squares).Radii of circlesrepresent20% of the data;N is the sizeof eachdataset.Diagram
for central California includes data located north of latitude 35.5 ø N and west of the Sierra Nevada
physiographic
province(seeFigure4a). Data usedin theTransverseRangesplot are locatedbetween
latitudes33.5ø N and34.5ø N and westof longitude115.5ø W. Data arefrom Figure4a andM.D.
Zoback (privatecommunication,1989).
114ø
766
b
Sonder:Effectsof Density Contrastson StressDirections
Central
California
Transverse
Ranges
approximatelyperpendicularto it. Figure4b showsrose
diagramsfor two data subsetsof Zobacket al. [ 1987, also
M.D. Zoback, private communication,1989]. The mean
directionsfor centralCaliforniaandtheTransverseRanges,
listed in Table 1, are statisticallydistinctfrom eachother. In
addition, within the uncertainties, the mean direction
determinedfor centralCalifornia is statisticallyindistinctfrom
that of N 43.9ø + 1.9ø dete•xninedby Mount and Suppe[1987]
from a different subsetof the Zoback et al. [ 1987] data
compilation.
N=99
N=51
Fig. 4 (continued)
TABLE 1. Mean PrincipalHorizontal
CompressiveStressDirections
Region
Na
Mean
Azimuth b
from otherlocationsalongthe SanAndreasfault in thateastwesttrendingthrustfaultsand folds with east-westaxesare
observed,in contrastto thenorthwest-trending
faultsandfold
axesfoundin the centralCaliforniaCoastRangesandin areas
southof theTransverseRanges.
The SanAndreasfault,whichpresentlyaccommodates
about
two-thirdsof thePacific-NorthAmericarelativeplatemotion
of5 to6 cmyr-1[DeMets
etal.,1986;
Minster
andJordan,
1984, 1987], trends about N 35ø W in central California, but
whereit cutsthroughtheTransverseRangesit trendsmore
westerlyby about20ø to 30ø, resultingin the "big bend"
(Figure4a). The mostwidelyacceptedhypothesis
for the
originof the late Cenozoicupliftof theTransverse
Rangesis
thatmovementalongtheright-lateralSanAndreasfault caused
compression
andhorizontalshortening
alongthelengthof the
left-stepping"bigbend" in thefault (e.g.,Crowell[1979] and
manyothers).However,thisexplanationis not entirely
satisfactory
because
it requiresthe"bigbend"to existpriorto
thecompression
andhencedoesnotaddress
theproblemof
howor why the "bigbend"originated.
The modelpresented
in
thispaper,whenappliedto the tectonicsof theTransverse
Ranges,providesnot only insightinto the possibleoriginsof
the variationsin orientationof principalstresses
in California,
butan alternativeexplanation
for theoriginof the"bigbend".
To comparethe predictionsof the modelto the tectonicsof
theTransverse
Ranges,I usestructural
andgeophysical
observationsto constrain 0 and 0', and use these to estimate
the rangeof x/x' appropriate
to theTransverse
Ranges.The
modelis testedby comparingthe predictedvaluesof 'c/x'with
independently
derivedestimates
of 'candx'.
4.2. Observations
of Sn'essOrientationsin California
Zoback et al. [ 1987] summarizedmost of the available stress
directiondatafor California,includingboreholedata,young
(< 2 Ma) volcanicalignments,and earthquakefocal
mechanisms.Figure 4a, from Zoback et al. [ 1987], suggests
that principalhorizontalcompressivestressdirectionsare
northeast-southwest
in centralCalifornia and the regionsouth
of the TransverseRanges,but in the TransverseRanges
provincethe maximumcompressivestressdirectionsare
orientedmore northerly. Also notethat, as the trendof the
San Andreasfault changesas it passesthroughtheTransverse
Ranges,the maximumcompressional
stressdirectionsremain
Radius of 95%
Confidence
Interval
b(deg)
Central
Califomia
99
N 37 E
8
Transverse
51
N 14 E
12
Ranges
aNumber
of observations.
bDetermined
usingmethods
fromMardia[1972].
In termsof the modelfor stressrotationpresentedearlier
(sections2 and 3 andFigure 1), thesedataallow estimatesto
be madeof 0 and0'. The averagedirectionof themaximum
horizontalcompressivestressin centralCalifornia is N 37ø E
(Table 1) andthetrendof thedensityanomalyunderthe
TransverseRangesis aboutN 90ø E [Humphreyset al., 1984],
so0 shouldbe approximately
53ø.In theTransverse
Ranges
the meandirectionof the maximumhorizontalcompressional
stressis N 14ø E, so 0' is approximately23ø. The 95%
confidenceintervalslistedin Table 1 providesomemeasureof
uncertaintyin 0 and0'. If thetrendof theTransverse
Ranges
anomalyis assumedto be uncertainby +10%thentherangeof
0 is between35ø and71o; thepermissible
rangeof 0' is
between3ø and43ø.Theserangesareplottedin Figure5a and
showthat the rangeof x/x' is constrainedto be between0 and
-4, with a valueof-0.4 mostprobable,corresponding
to the
mean values of 0 and 0'.
It is not the purposeof thispaperto offer an explanationfor
theobservationthatmaximumhorizontalcompressive
stresses
are perpendicularto the San Andreasfault; Zobacket al. [ 1987]
andMount andSuppe[1987] haveproposed
thatthismaybe
attributed to the mechanical weakness of the San Andreas fault.
Instead, I concentrate on the observation that both the stresses
and the San Andreas fault differ in orientation within the
TransverseRanges,comparedwith theirorientationsin central
Califomia.I showbelowthatthemodelpresented
abovecan
account for the orientation of the stresses in the Transverse
Ranges;in addition,if, for whateverreason,within the
TransverseRangesthe San Andreasfault maintainsthe same
anglewith respectto the stressesas it doesin central
California,the the modelalsoofferssomeinsightinto the
causeof the "big bend"in the SanAndreasfault.
Sonder:Effectsof DensityContrastson StressDirections
767
4.4. Evidencefor a DensityAnomalyunder the Transverse
Rangesand b•dependentEstimateof r'
6O
45
30
15
o
-15
-45
-60
-75
-90
-90-75-60-45
-30-15
0
15 30 45 60 75 90
Ranges.
Using
values
fordVp/dT
of -0.5ms-1deg-1
and
0p/•Tof-0.11kgm-3deg-1
(at1250øC)fromAnderson
and
e' (degrees)
2.5
b
1.5
0.5
Severalanalysesof earthquaketravel-timedatahaveshown
that arrival times for P waves travelling throughthe
TransverseRangesprovinceare 0.5 to 1 secondearly compared
with thosefor waveswith pathsoutsidethe TransverseRanges
[HadleyandKanamori,1977; Raikes, 1980;Walck and
Minster, 1982], suggestingthe presenceof higherdensities
underneaththe TransverseRanges.This hasbeenconfirmedby
a seismictomographystudyby Humphreyset al. [1984] that
revealsa vertical,wedge-shaped
regionextendingto a depthof
100 to 250 km which is 2-3% faster than its surroundings.
The regioncorrelateswell with the locationof the Transverse
Rangesand is approximately300 km (E-W) by 100 km (N-S)
in horizontalextent.If this high velocity region is interpreted
as beingthe resultof cold temperatures,it follows that there
shouldbe a significantdensityanomalyunderthe Transverse
Bass [1984], whereVp is P wave velocity,T is temperature,
andpisdensity,
gives
avalue
fordVp/dp
of4.4m4 s-1kg-1.
Combinedwith the observedvelocityanomalyof 2-3% under
the TransverseRanges[Humphreyset al., 1984], this suggests
ß
adensity
anomaly
ofapproximately
40- 60kgm-3,or
normal
1 -2%.
ß
::.
.....:.:...'i
"-":''-'•:...
:.... :.'•ii•!i:.•i
•ii•
-0.5
thrust
the magnitudeof the massanomaly ApL is around0.9 - 1.5
-1.5
-2.5
Taking the top of the densityanomalyat the Moho
(30 km depth),if it extendsto 250 km, thenL = 220 km, so
ii""'---'-"-"••ii?.:...:•i
_-
x 107kgm-2,fortherange
ofApdetermined
above.
If the
, , strike
sip..............
•'"'"•"""••:••,..
-90-75-60-45-30-15
0
15 30 45 60 75 90
e (degrees)
Fig. 5. (a) As Figure2a.Boxindicates
rangeof OandO'
deterrninedfrom observationsin California. (b) As Figure 3a.
densityanomalyextendsonly to 150 km depth,as in the
westernTransverseRanges,the massanomalyis less,between
4.8and8.4x 106kgm-2.Figure
A2shows
that,forthehalfwidth of the densityanomaly(about50 km) equal to its depth,
the surfacemagnitudeof the deviatoricstressesis
approximatelyApgL/3. Therefore,'c'can be as large as49
Boxesindicatetherangesdetermined
fromconstraints
on 0 and
0' (lightstippling)asdescribed
in text,andfromobservational
MPa,if L = 220kmandAp= 70kgm-3,orassmall
as
16MPa,if L = 120kmandAp= 40kgm-3.
constraints
on 'candL (dark stippling).Dots showvalues
obtainedfromthe meanprincipalstressdirections.
'c/'c'between-0.16 and-1.1, which includes the mean value
4.3. Independentestimateof v
The lack of a localizedheatflow anomalyassociated
with
Taking theseconstraintson 'cand '( gives a rangeof
of -0.4 predictedearlier andmatcheswell with the field that
wasobtainedusingthe observational
limits on 0 and0' (see
Figure 5a). The agreementsuggeststhat the simplemodel
presentedhere can accountfor the observedvariationin
principalstressdirectionsbetweenthe TransverseRangesand
otherpartsof the Pacific-NorthAmerica plateboundaryzone
in Califomia.
the San Andreas fault has been used to constrain the frictional
shear stress on the San Andreas fault to be less than 20 MPa
5. DISCUSSION
[Lachenbruch and Sass, 1980; 1981].
Both lower and upperboundsfor 'ccanbe estimatedusinga
slightlydifferentreasoning[SonderandEngland,1986]. A
regionin centralCalifornia,75-100 km wide andcenteredover
theCoast
Ranges,
hasa mean
heatfluxthatis20-40mWm2
greaterthanthat foundto the east[e.g.,LachenbruchandSass,
1980, Figure 15]. If this anomalyresultsentirelyfrom shear
dissipationwithin the lithospherethen'cmust be between8
and 18 MPa for ambient shear strain rates around 9-10 x 10-15
s-1 [Savage
etal.,1981'Thatcher,
1975,1983]anda
lithospherethicknessof 120 km.
Severaldifferenthypotheses
havebeenproposedfor the
originof the velocityanomalyunderneath
theTransverse
Ranges. Sheffelsand McNutt [1986] suggestedthat it was the
resultof intracontinental
subductiondue to compression
across
the TransverseRanges.However,if the downdiplengthof the
velocity anomalyis takenas indicativeof the amountof
convergence,then theremusthave beenas much as 200 km of
convergence
acrossthe centralTransverseRanges.Weldonand
Humphreys[1986] point out that the volumeof crustinvolved
in thisamountof convergence
far exceedsthepresent-day
768
volume of crustin the TransverseRanges.In addition,the
intracontinental
subduction
hypothesis
doesnot explainwhy
or whencompression
developedacrosstheTransverse
Ranges
in the first place.
Humphreyset al. [1984] proposethatthe seismicallyfast
regionundertheTransverse
Rangesis a negativelybuoyant
convectiveinstability.Sucha featuredoesnot requirepreexistingcompression
to develop;the time-dependent
and
chaoticnatureof convectionin the earth'smantlepermitsthe
developmentof convectiveinstabilitiesanywherealongthe
baseof the thermalboundarylayer. If the hypothesisthat the
densityanomalyis a convectiveinstabilityis tentatively
adopted,thena speculativebut highly intriguingconsequence
arises.The instabilitymay havebeensmallerin the past,and,
if •: hasremainedrelativelyconstant,•:/•:'wouldhavebeen
morenegativein earliertimes.Figure5 showthatthe presentday meansof 0 and'c/•:'plot in the field wherethrustingis
expectedto dominate.However,if'c/'c'wasmorenegative,the
meanvaluesof 0 and •:/•:'wouldbe morelikely to plot in the
strike-slipfield. Thusthe growthof a convectiveinstability
providesa mechanismby whichthe styleof deformationin the
Transverse
Rangesmayhavechangedfromdominantlystrikeslip to dominantlythrusting.Supportfor this speculationmay
be providedby evidencein the SanBemadinomountainsthat
earliest deformation on thrust faults occurred between 9.5 and
4.1 Ma, at a time whenstrike-slipdeformationwasalready
activeon the SanGabrielfault; compressional
deformation
mayhavebegunevenlaterin theSanGabrielmountains
[MeislingandWeldon, 1989].
This simplemodelfor the orientationof principalstresses
hasan importantimplicationfor therelationship
betweenthe
deformation
producing
theTransverse
Rangesandthe"big
bend" of the San Andreas fault. Rather than the Transverse
Rangesbeingformedby compression
acrossa spatiallyfixed
"bigbend",ashasbeencommonlyassumed,
boththe
Transverse
Rangesandthe "bigbend"canbe understood
to be
consequences
of thenegativebuoyancyforceunderthe
TransverseRanges.If theSanAndreasfault maintainsthe
samehighangleto themaximumhorizontalcompressional
stressas it doesin centralCalifornia [Mount and Suppe, 1987;
Zobacket al., 1987], thenprogressive
counterclockwise
changein directionof themaximumhorizontalcompressive
stresswouldimply a corresponding
countemlockwise
rotation
of the trendof the SanAndreasfault, thusproducinga bendin
the fault as it passesthroughtheTransverseRanges.
Sonder:Effectsof DensityContrastson StressDirections
styleof deformationfor example,in the instanceof the
CaliforniaTransverseRanges,from dominantlystrike-slipto
dominantlythrusting).The senseandmagnitudeof these
changesdependon two parameters:first, the relative
magnitudesof the regionaldeviatoricstressandthe deviatoric
stressdue to the localbuoyancyforces,andsecond,the trendof
the densityanomalyin relationto the directionof the principal
horizontalcompressivestressof the regional,unperturbed,
stress
tensor.
The principalhorizontalstresses
in the TransverseRanges
are orientednortherly,in contrastto the northeasterly
directions observed in central California
and Califomia
south
of the TransverseRanges.Also, thereis goodseismological
evidencefor a wedge-shaped
regionof highdensityundemeath
the TransverseRanges.From the trendof the densityanomaly
andthe 23ø differencein orientationof principalstress
directionswithin andoutsidetheTransverseRanges,theratio
of the magnitudesof the regionalstressassociated
with strike-
slipdeformation
andthelocalstress
dueto thedensity
anomalyis predictedto be approximately
-0.4, witha rangeof
0 to -4. This rangeis consistentwith that estimated
independently
from consideration
of the differencesin seismic
velocityundertheTransverse
Rangesandtheirsurroundings,
andthelackof a largeheatflow anomalyassociated
with
deformationalongthe SanAndreasfault in centralCalifornia.
Hence,themodelpresented
here,althoughcertainlynotthe
only possibleexplanationof the stressorientationdata,does
providea simpleunderstanding
of theobservedvariationsin
stressstatealongthe San Andreasfault in California. The
success
of themodelwhenappliedto Californiasupports
the
speculation
thatboththe uplift andcompression
in the
Transverse
Rangesandtheformationof the"bigbend"of the
SanAndreasfaultareattributable
to thesamecause,namely
theformationandgrowthof a convective
downwelling
under
theTransverse
Ranges[Humphreys
et al., 1984].
APPENDIX
A: STRESSES
RESULTING
BURIED
DENSITY
ANOMALY
FROM
A
The densityanomalyis assumedto haveconstantmagnitude
Ap, andto extendovera width2w in they' directionandover
a depthrangeL (Figure A1 a). Its lengthin the x' directionis
assumedto be muchgreaterthanits dimensionsin eitherthe
y'-directionor z direction,so that variationsof densityin the
x'-directioncanbe neglectedandthe equationsof motionare
reduced to their two-dimensional form. In addition, in order to
6. CONCLUSIONS
I haveuseda simplemodelto investigatetheeffectof a
densityanomalyon an otherwiseregionallyconstantstress
field, and haveemphasizedchangesin the orientationand
relativemagnitudesof the principalstresses.
If thedensityof
the anomalyexceedsthat of its surroundings
(that is, produces
a negativebuoyancyforce),the effecton the stateof stress
within the lithosphereis to increasethe anglebetweenthe
maximumhorizontalcompressional
stressdirectionandthe
trendof the lineardensityanomaly.The angledecreases
if the
buoyancyforcesare positive(densityof the anomalyis less
thanits surroundings).
The relativemagnitudes
of the principal
stresses
can alsobe affected,potentiallyalteringthe expected
simplifythe solutionof the equationsof motion,thedensity
anomalyis approximatedby expressingit as a constantmass
anomalyAm = ApL, collapsedontothe surfacez = d, with
width 2w in the y' direction(Figure Alb). This allowsthe
solutiondomainto be divided into two regions,an upperlayer
anda lower half-space,bothof uniformdensityandviscosity,
separated
by the anomaloussurfacemassdistributionat z = d.
A Fourier seriessolutionto the equationsof motionis
described below; thus the anomalous mass distribution Am
mustbe madeperiodic.Figure Alc showsthisperiodic
extensionof themassdistribution;the wavelength23•is taken
to be muchgreaterthan the width 2w of the massanomaly,so
that the stressdistributionin the region ly'l < w approximates
that of a single, isolatedmassanomaly.For an anomalyAm
Sonder:
Effectsof DensityContrasts
onStress
Directions
769
layer (seeFigureA1) the generalsolutionto (A2) is
v=sin
nmd'
[_(A+B+B
)en•Z/X
(•__)
naz
+(C-D
+Dnaz)e-•Z/•]
(A3)
w=cøs
nmd'
I(A+
Bn/l:z
)en•k
(__•__)
+(C+D
nnz
/e-'nzl]
where A, B, C, and D are constantsof integration.
Boundaryconditions
on z = 0 arezeroshearstress
(Zy,
z= 0)and
zero
vertical
velocity
(w=0).Velocities
are
boundedas z --> oo.At the interfacez = d betweenthe layers,
velocitiesand shearstressare continuousand there is a step
(c)
changeg•m in verticalstress,
dueto themassanomaly.For
theseboundaryconditions,
6p L
•.-w
%+ W
w
ml =
Fig. A1. (a) Coordinatesystemandparameters
usedto describe
a regionof anomalousdensityAp, width 2w, andthicknessL.
(b) Densityanomalyis collapsedontothe horizontalsurface
z = d; themassanomalyper unit areais ApL. (c) Periodic
B•= - &ng%
e_md•
4finn
extension
of themassanomaly.Widthis 2w, wavelength
2%,
magnitude+__ApL.
C1--&ng
(1+n•d]
4rln•:%
3. I e-•
of constant
magnitudeApL, with width2w andwavelength
2%,theFourierseriesandcoefficients
•m aregivenby:
D•= - &ng%
e.r•
4finn
(A4)
Am= •L,•rnncos(n•'/%)
A2 =0
n=l
(A1)
B2=0
(•rnn
=/ 0
neven
For each term in the Fourier series, the stressdistribution is
4qnx
3. I
3. I
132
=-&ng•_
emrd•
+e.md•)
4finn
found,following the methoddescribedby, amongmany
others,Fleitout and Froidevaux[1982, AppendixA1 ]. The
total stressfield is foundby superposition
of eachFourier
wheresubscripts
1 and2 referto theupperlayerandlowerhalf
component.
In the absenceof inertial forces,the equationsof motion for
a uniform incompressibleNewtonianviscousfluid can be
space,respectively.
In the upperlayer, the deviatoricstresses
are
written
1;y'y'=
gApL
Z [1-(-1)n•
cos
O:v O:v
y,2
n0y,
32w32w13p
-I-
ay,2 M
sinn:trw
e.md
•
n--1
(A2)
--
rl
wherev andw arethe y' andz compronents
of velocity,p is
the nonhydrostatic
pressure,
andrl is viscosity.Within each
[e•z/•(z-d)/%-e
-mz•
(z+d)/%]
l:zz
=-gApL
• [1-(-1)nl
cos
{n½)
sin
(n--•-)
e-md•
n=l
(i5)
ß[emrzA
(z-d)/•-e
'=z-A
(z+d)/•,]
770
Sonder:
Effects
of DensityContrasts
onStress
Directions
gApL
E [1-{-1)n•
sin
Anderson,E. M., TheDynamicsof Faulting,Oliver and
Boyd, Edinburgh,1951.
Artyushkov,E. V., Stresses
in thelithosphere
causedby
crustalthicknessinhomogeneities,
J. Geophys.Res.,78,
sinnmve.r•c•
n=l
ß[enaz/•'
(z-d)/•
+e'mzA
(z+d)/•]
7675-7708, 1973.
Atthesurface
theshear
stress
'ry,
z iszero,
andforz < w/2,
themagnitudeof the shearstressis alwayslessthan10% of
the magnitudeof the normalstresses.
Thus the near-surface
deviatoricstressfield is, to a goodapproximation,
biaxial.
It remains
toshow
howthemagnitude
of'rzz(or'ry,y,)
depends
on thewavelength
• of theperiodicextension
of the
massanomaly,andhow it varieswith distance,y',
perpendicular
to thetrendof theanomaly.FigureA2 shows
'rzz at z = 0 asa functionof horizontaldistancey'. For • >
5w, 'rzzdoesnotdepend
strongly
on•,/w.In theregionabove
Bott, M. H. P. andD. S. Dean, Stresssystemsat young
continentalmargins,Nature, 235, 23-25, 1972.
Crowell,J. C., The SanAndreasFaultsystemthroughtime,
J. Geol. Soc. London, !36, 293-302, 1979.
Dalmayrac,B., andP. Molnar, Parallelthrustand normal
faultingin Peruandconstraintson the stateof stress,Earth
Planet. Sci. Lett., 55, 473-481, 1981.
DeMets, C., R. G. Gordon,S. Stein,D. Argus,andD.
Woods,Pacific-NorthAmericaspreading
ratein theGulf of
California, EOS Trans. AGU, 67, 905, 1986.
Fleitout,L., andC. Froidevaux,
Tectonics
andtopography
for
a lithosphere
containing
densityheterogeneities,
Tectonics,
the centerof the massanomaly(I y'l < w/2), the magnitudeof
the stressvaries by less than 10% of its maximum value, so
may be consideredto be approximatelyconstant.The stressis
greatestabovethe centerof the massanomaly(y' = 0) and
decreaseswith horizontaldistance.For l y'l > 4w/3, the stress
is alwaysless than 20% of its maximum value, so that the
solutioncloselyapproximatesthat for a singleisolatedmass
anomalyandis not stronglyinfluencedby the periodic
extensionsof the massanomaly that were introducedto obtain
1, 21-56, 1982.
Froidevaux, C., and B. L. Isacks, The mechanical state of the
the Fourier
Lachenbruch,
A. H., andJ. H. Sass,
Heatflowandenergetics
of theSanAndreas
faultzone,J. Geophys.
Res.,85, 6185-
series solution.
lithospherein theAltiplano-Punasegmentof theAndes,
Earth Planet. Sci. Lett., 7!, 305-314, 1984.
Hadley, D., and H. Kanamori,Seismicstructureof the
TransverseRanges,California, Geol. Soc.Am. Bull., 88,
1469-1478, 1977.
Humphreys,E., R. W. Clayton,andB. H. Hager,A
tomographicimageof mantlestructurebeneathsouthern
California, Geophys.Res. Lett., !!, 625-627, 1984.
6222, 1980.
0.4
Lachenbruch,A. H., andJ. H. Sass,Correctionsto 'Heatflow
and energeticsof the SanAndreasfault zone'andsome
d--w
0.3
0.2
additional commentson the relation between fault friction
0.1
andobserved
heatflow, J. Geophys.
Res.,86, 7171-7172,
1981.
0
-0.1
10
-0.2
-0.3
-0.4
0.0
0.5
1.0
1.5
2.0
2.5
Y '/W
Fig. A2. Vertical deviatoricstressat z = 0 as a functionof
y'/w for the massanomalydistributionshownin Figure Alc.
Numbers on curves refer to values of )•/w.
Mardia, K. V., Statisticsof DirectionalData, Academic,San
Dieg, Calif., 1972.
Meisling, K. E., and R. J. Weldon, Late Cenozoictectonicsof
the northwestern
SanBernardinoMountains,southern
California, Geol. Soc.Am. Bull. , !01, 106-128, 1989.
Minster,J. B., andT. H. Jordan,Vectorconstraints
on
Quaternarydeformation
of thewesternUnitedStateseastand
westof the SanAndreasfault, in Tectonicsand
Sedimentation
alongtheCalifornia
Margin,edited
byJ.K.
CrouchandS. B. Bachman,
pp. 1-16,Soc.Econ.Paleontol.
Mineral., Pac. Sect., 1984.
Minster,J. B., andT. H. Jordan,Vectorconstraints
on
Acknowledgments.
While partof thiswork wascarriedoutI
wassupported
by a ChaimWeizmannPostdoctoral
Fellowship
at Caltech.
WhileI wasthereB. Hagergenerously
provided
fundsfortravelandmiscellaneous
costs(NASAgrantNAG5842).I thankP. England
andE. Humphreys
forhelpful
discussions,
andC. Jones,S. King, M.D. Zoback,B.
Sheffels,R. Westaway,
andananonymous
reviewerfor
commenting
on earlierversions
of themanuscript.
I amalso
gratefulto M.D. Zobackforpermission
to usehisfigureand
datacompilationin thispaper.
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(ReceivedOctober11, 1989;
revisedJanuary29, 1990;
accepted
February2, 1990.)