JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 91, NO. B5, PAGES 4826-4838, APRIL 10, 1986
Extensionof ContinentalLithosphere'
A Model for Two Scalesof Basin and Range Deformation
M.T. ZUBERI AND E.M. PARMENTIER
Dept. of Geological Sciences,Brown University, Providence,Rhode Island
R.C.
FLETCHER
Centerjbr Tectonophysics,TexasA&M University, College Station, Texas
We develop a model for deformationin an extendingcontinentallithospherethat is stratifiedin density
and strength, assuminga rheology consistentwith seismic focal depths and experimentalflow laws. The
model demonstratesthat necking instabilitiesat two wavelengthswill arise due to the presenceof a strong
upper crust and upper mantle separatedby a weak lower crust. The magnitudesof the instabilitiesare
directly related to strengthcontrastswithin the lithosphere,while the dominantwavelengthsof neckingare
controlledmainly by the thicknessesof the stronglayers. The resultsare applied to the Basin and Range
Province of the western United States where two scales of deformation can be recognized, one
correspondingto the spacingof rangesand the other to the width of tilt domains.A Bouguer gravity anomaly and associatedregionaltopographywith a wavelengthcomparableto the width of tilt domainshas also
been recognized.For plausibledensityand strengthstratifications,our resultsshow that the horizontalscale
of shortwavelengthneckingis consistentwith the spacingof individualbasinsand ranges,while that of the
longer wavelengthnecking is consistentwith the width of tilt domains. We thus suggestthat Basin and
Range deformationmay be controlledby two scalesof extensionalinstability.Extensionin the weak lower
crust in this model is laterally displacedfrom regions of upper crustal extension. The resultant horizontal
shearingin the lower crustmay be a mechanismfor the initiationof low-angleextensionaldetachments.
INTRODUCTION
mation occursin the upper crust, where strengthis controlledby
frictional propertiesand the mode of faulting [Byerlee, 1968].
The continentallithosphereis stratifiedin both densityand At greaterdepths,dislocationcreep, which is chiefly a function
strength. In this study, we examine the implicationsof this of strain rate and temperature, is the dominant deformation
stratificationfor large-scaleextensionaldeformationof the litho- mechanism.This resultsin strengththat decreasessharplywith
sphere. In a previous study, Fletcher and Hailer [1983] treated depth. In constructingFigure 1 we assume,as in other studies
the lithosphereas a strongsurfacelayer of uniformstrengthand of continental deformation [Glazner and Bartley, 1985; Bird,
densityoverlyinga weakersubstrateof the samedensity.Their 1978], that flow in the continentalcrustcan be approximatedby
study showedthat unstableextension,or boudinage,resultsin a quartztheologyand in the upper mantleby an olivine theol-
the concentration
of extensioninto regionswith a regularspac- ogy. While this is probablya reasonableassumptionfor the
ing determinedprimarily by the thicknessof the stronglayer. upper mantle, it must be considereda lower limit for the crust.
of the continental
crust,is stronger
A more realistic strengthstratificationof the continentallitho- Feldspar,a majorconstituent
greaterthan 300øC[Tu!!isand Yund,
spherewould consistof a strongupper crust and upper mantle thanquartzat temperatures
separatedby a weak lower crust, with the densityof the crust 1977, 1980]. However, uncertainties in the flow law of feldlessthan that of the mantle. If a strongsurfacelayer necksat a spar, as well as in our knowledgeof the mechanicsof polygiven wavelength,then another strong region at depth may phaseflow and the modal mineralogyof the lower crust, conof
introducea secondwavelengthof necking.In this study, we strain us to assumethat quartz theology is representative
evaluatethe conditionsrequiredfor the growth of two wave- crustal strength.As shown in Figure 1, experimentalresults
lengthsof necking instabilityand apply this hypothesisto the extrapolatedto lower crustal conditionsindicate that olivine is
andtemperatures,
Basin and Range Province, which, as we will presently muchstrongerthanquartzat similarpressures
as is apparentby the sharpincreasein strengthat the crust-mandescribe,exhibitstwo scalesof periodicdeformation.
tle boundary.In fact, despitehighertemperatures
at depth,the
StrengthStratificationof ContinentalLithosphere
uppermantleis even strongerthan the uppercrust, while the
Studiesof the theologyof the continentallithosphere[e.g., lowercrustis a regionof very low strength.
For the conditionsassumedin Figure 1, the uppermantle
Kirby, 1983; Brace and Kohlstedt, 1980] suggesta variationof
strengthwith depth like that illustratedin Figure 1, which was shouldundergobrittledeformation;however,strengthin ductile
to the geothermal
gradientandthe acticonstructedfollowing Brace and Kohlstedt[1980]. Brittle defor- flow is highlysensitive
vationenergyand preexponential
frequencyfactor in the flow
law. A smallchangein any of theseparameterswould shift the
depth of the boundarybetweenbrittle and ductile deformation
INowat Geodynamics
Branch,NASAGoddard
SpaceFlightCenand couldeliminatethe regionof brittle fracturein the mantle.
ter.
Becauseof the uncertainties
in lithosphere
theologywe do not
Copyright1986by the AmericanGeophysicalUnion.
wish to emphasizeabsolutemagnitudes,but rather relative differencesin strength.Brittle deformationin the uppermantleis
Paper number 4B5282.
0148-0227/86/004B-5282505.00
not requiredfor a large strengthcontrastat the Moho, sincethe
4826
ZUBER ET AL ' EXTENSION OF CONTINENTAL
LITHOSPHERIC
•-,soo
ß
-,ooo
ß
ß
-500
ß
E
compress
associated with identified
(MPa)
o
ß
500
ß
ß
4827
lineament patterns; however, in many cases the zones are not
STRENGTH
o':5
LITHOSPHERE
ooo
!
1
tension
structural features.
The spacingsof the rangesand tilt domainsare summarized
in Figure 2. Tilt domain spacings,shown in the top histogram,
were measured between antiformal axes and synformal axes,
normal to strike, from Stewart's [1980] Figure 1. Distances
between domain boundariesrange from as little as 50 km to
almost 500 km, but most cluster around 200 km. This is in con-
trast to the distancesbetween individual ranges, shown at the
bottom of Figure 2. This histogram, reproducedfrom Fletcher
and Hallet [1983], shows that rangesare separatedby an average of about 30 km, or about 1/6 the spacing of the tilt
domains.
In a previous study of large-scaleextensionin the Basin and
Range, Fletcher and Hallet [1983] treated the lithosphereas a
strong plastic layer overlying a weaker viscoussubstrateof the
same density. Using flow laws for a range of rock types, they
found that the lithosphereextendsunstablyproducingregionsof
enhanced and reduced extension. This study showed that the
60
Fig. 1. Strengthof the continentallithospherewith a crustalthickness dominant wavelengthof the necking which arisesdue to unstaof 30 km, •;•x=10-•ss-• anda geothermal
gradient
of 15ø K km• ble extensionis consistentwith the spacingof individual basins
Flow laws were taken from Brace and Kohlstedt {19801. Major contrasts andranges.
in strengthoccur (1) between the brittle upper crust and ductile lower
On the basis of our results for extensionof a strengthand
crust, (2) at the crust-mantleboundary, and (3) within the mantle. Such
density stratified lithosphere, we suggestthat tilt domains may
strengthcontrastsmay lead to the growth of instability during uniform
also be the surfaceexpressionof boundinage-likedeformationin
extension.
the Basin and Range, with a wavelength greater than the
spacingof ranges.We suggestthat the developmentof this secductile strengthof olivine is significantlygreater than that of
ond, longer wavelengthof instability may be related to a strong
quartz at the same PT conditions.
region of the upper mantle separatedfrom the strongupper crust
A zone of low strengthin the continentallithosphereis also
by a weak lower crust. Kinematic models of basin and range
consistentwith seismicresults.In a studyof the distributionof
structure have been proposed which address the relationship
earthquakefocal depths in continentalregions not associated
between the ranges and tilt patterns [Anderson et al., 1983;
with recent subductionzones, Chen and Molnar [1983] found
Zoback et al., 1981; Stewart, 1980, 1978, 1971]. To our
the upper crust and upper mantle to be seismicallyactive and
knowledge, however, no mechanical models have been
the lower crust to be essentiallyaseismic.They interpretedthe
proposedto explain the distribution of large-scale tilted fault
aseismiclower crest as a weak region which deformsby ductile
blocks. In the present study we examine the possibility that
flow. Thus both experimental and observationalevidence indicate that the continental lithospherewill consist of a strong
SPACING OF TILT DOMAINS
uppercrustand mantleseparatedby a weak lower crust.
8
Mean-194Km
N-27
Basin and RangeProvince
As with other regionsof continentalextension,the Basin and
Range exhibits high heat flow, extensive volcanism, regional
uplift, and widespreadnormal faulting. Details of the structure
and geophysicsof this area are summarizedby Eaton [1982],
Zoback et al. [1981], Stewart [1978], Thompsonand Burke
[1974], and others.Modem Basinand Rangetopography,which
is regularly spacedand has a crestto trough amplitudeof about
1 km, initiated at about 13 Ma [Zoback et al., 1981] and trends
generallyN-S in the northernpart of the province and NW-SE
in the south.The area is also markedby a broad-scaleregional
tilt patternof major Cenozoicfault blocks, which strike generally parallel to the basin and range structure. Stewart [1980]
recognizedthese alternatingregionsof consistenttilt directions,
called tilt domains, which are continuous over distances of
50-500 km along strike. The tilt domains are separatedby
"antiformal" and "synformal"axes, which are shown schematically in Figure 8. Within a given tilt domain, fault blocks dip
away from antiformal and toward synformal axes. Transverse
zones, which are characterizedby an absenceof major tilted
blocks and by changesin fault patternsand topographicgrain,
strike parallel to the extensiondirectionand separateregionsof
differing tilt. In some places, transversezones follow regional
4
cr
I00
200
8
3,00
x(Km)
SPACING OF RANGES
6
:::::::::..I..:::::::::::::::::
Mean--:51
Km
:::::::::::::::::::::::::
N 27
4
!ii}ii}i
i 71 erand
Hallet
(1983)
0
'.
,
•:0
,
,•
6b x(Km)
Fig. 2. Spacingsof large-scaledeformationalfeaturesin the Basin and
Range Province. The spacings of tilt domains (top) are distances
between successiveantiformal boundaries and synformal boundaries
measuredfrom Figure I of Sten'art11980]. Spacingsbetweensuccessive
ranges(bottom)are reproducedfrom F/etcherand Ha/let 11983].
4828
ZUBER ET AL.'
EXTENSION OF CONTINENTAL LITHOSPHERE
unstableextensionof continentallithosphereleads to two scales
of deformation, one of which correspondsto the spacing of
ranges and the other to the spacingof tilt domains. We show
that stressesassociatedwith the long wavelength deformation
result in differencesin resolved shear stresson oppositelydipping high-angle normal faults forming individual basins and
ranges. Greater slip on faults with higher resolved shear stress
could result in the observedrotation or tilting of high-anglefault
blocks. Recently, Froidevaux [1985] has suggesteda boudinagelike thermal structureof the upper mantle on the basisof gravity
anomaly patterns. The 200 km wavelength gravity anomaly
identified on the maps of Hildenbrand et al. [1982] coincides
approximatelywith spacingof tilt domainsand thus may also be
relatedto the longerwavelengthof deformation(seeFigure 8).
Instabilityin the lithosphere,extendinghorizontallyat a uniform or mean rate •;xx, occurs due to the amplification or
growth of small, random disturbancesalong initially planar
interfaces.As the disturbancesgrow, the interfacesdeform sinusoidally with an amplitude A. We assume(1) two-dimensional
flow, and (2) the amplitude of the disturbanceto be small compared to its wavelength.Mathematically,this is regardedas the
superposition
of an unstablesecondaryor perturbingflow on the
mean or primary flow which describesthe uniform extensionof
the medium. This linearized formulation describes only the
incipientstagesof unstableextension.
Linearization
In the linearized problem, stressesand strain rates for the
total flow are written
MODEL
FORMULATION
We examine a model of the continentallithospherewhich, as
shown in Figure 3, is composedof three layers and a substrate.
The crust consistsof two layers of thickness h• and h2, each
with a uniform density (@, @•=@2)and strength(x, x•>x2). The
mantle consists of a layer of thickness h3 overlying either a
semi-infinite half-space with uniform strength and density (J
model) or a half-space in which the strengthdecreasesexponentially with depth (C model). In the former case, the strongmantle region is representedas a layer of uniform strength,and the
strengthdecreasesdiscontinuouslyto a lower uniform value in
the substrate.In the latter, mantle strengthis everywherecontinuous but falls to zero at depth. Models with either uniform or
exponentially varying strength were chosen because analytic
solutionscan b• obtained for the flow in each layer, thus avoiding the need for a fully numericalsolution. Neither model is an
exact representationof the rheologicalstructureshown in Figure
1. Variations in the strengthof the lithospherein regionsof uniform compositionare not abrupt, as suggestedby the use of discrete layers. However, the rapid decreasein strengthwith depth
associated with ductile flow results in a relatively rapid
transition from strong to weak. As will be shown later, both
models result in similar dominant wavelengthsand instability
growthrates.
XeU
EXTENSION
(1)
where an overbardesignatesthe mean flow and a tilde the perturbing flow. For thermally activated creep the relationship
between the principal stressesOl and o3 and the strain rate e
has the form
e = A (ol - o3)nexp(-Q/RT)
(2)
whereA is the frequencyfactor, Q is the activationenergy,R is
the gas constant,T is the temperature,and n is the stressexponent. The principalstresses
and the strainrate are relatedby the
the viscosityIt as
o,-o3 = 4½•
(3)
By substituting(3) into (2) to eliminatethe strainrate, the viscositycanbe defined
1/4)A (O1--03)l-n
exp(Q/RT)
(4)
In an isotropicviscousfluid the stress-strain
rate relationships
are written
% = 2[t%- p6,j
MODEL
(5)
!
I
wherep is the hydrostaticstress.By combiningthe expressions
for the normal stressesin (5) and the incompressibilitycondition
(e,,= 0), the strength(x) is defined
x = (Oxx-Ozz)/2 = 2g•:xx
I
(6)
In the presentstudy, Ozz=0 in excessof the hydrostaticstress
since the mean extension is entirely in the x direction. As
described by Fletcher and Hallet [1983], the relationship
betweenthe perturbingstressesand strain rates is obtainedby
substituting(1) and (6) into (5), expanding,and retainingterms
to firstorderin
Oxx= (u/n) (•;xx/•;xx)
- P
Fig. 3. Model of a density and strength-stratifiedlithosphere. The
lithosphereconsistsof a strong upper crust, weak lower crust, and a
strongupper mantle over a weaker mantle substrate.Densitydifferences
occur at the free surface and crust-mantleboundary. Horizontal extensioncausesinitially planarlayersto deform into sinusoidalshapes.
Oz• = (u/n) (•zz/•xx)- P
Ox•:
ß (•xz/•xx)
(7)
ZUBERETAL ' EXTENSION
OFCONTINENTAL
LITHOSPHERE
TABLE
1.
Dimensionless
Parameters
Parameter
law fluid in which the stressexponent nr-->oz. This approximatesa perfectlyplastic material in which deformationcan be
discontinuousacross sliplines, which may representincipient
faults. A perfectlyplasticmaterialmay ,thusbe a representative
rheologicalmodel for a region which deforms by pervasive
faulting [Chappie, 1978]. In this study, we use a value of
Definition
Sl
(•)1-- @o)ghl/%l
S2
(02- @l)ghl/%2
S3
(03- @2)ghl/%3
S4
(04- @3)ghl/•4
R•
x•/x2
R2
x•/x3
R3
Xl/X4
e
•-Jhl
4829
n•= 104 for reasons
discussed
in Appendix
1. Thelowercrust
and the mantle are modeled as uniform power law fluids with
n2= n3- n4= 3, which is characteristicof deformationdue ,todislocationcreep.
To examine the general characterof the deformationin the
strengthjump (J) model, we assumethat the lower crustis 100
times weaker than the upper crust, that the upper mantle is
twice as strongas the uppercrust, and that the mantle substrate
is 50 times weaker than the upper crust. England [1983], in a
A solutionfor the perturbingflow is foundby requiringthat the study of continental extension, also assumed that the upper
perturbing
stresses
and strainratessatisfythe equilibriumequa- mantle made the greatestcontributionto lithosphericstrength.
tionsandtheincompressiblity
andcompatibilityconditions.
Strengthestimates
are basedon experimental
resul•ts
and on the
observedstressreleasedin shallow earthquakes(tens of megaGrowth Rate Factor
pascals).
In thecontinuous
strength
(C) model,themantlelayer
The amplitudeof vertical deformationat an interface has a thicknessof 5 km and the e-folding depth of the substrate
A,(k',t) is proportional
to the exponentialof the meanhorizon- is takento be 6 km in accordancewith Figure 1.
tal strain •;xxtas
A,(k',t) oc exp[(q-1)•;xxt]
(8)
Figure 4 showsthe growthrate factor q as a functionof the
dimensionless wave number k' for three mantle strength
stratifications:curves a and b are for lithosphereswith strong
uppercrustaland uppermantlelayersfor the J and C models,
whereq is the growthrate factorand t is time. The growthrate
factordeterminesthe rate of amplificationof a disturbancewith
dimensionless
wave numberk' (=2jrh•/•,). If q>l, the lithospherewill extendunstably,andthe instabilitywill growfastest
at the wave numberat which q is maximized. At each value of
the wave number there are four values of q (equal to the number of interfaces);however, in generalonly one value will be
respectively,and curve c is the sameas curve a but without the
strong mantle layer. In both curves a and b the grow•thrate
spectrumhas two peaks:that at the larger wave number representsa shortwavelengthinstabilitywhich arisesdue to necking
of the stronguppercrustand that at the smallerwave numberis
a longer wavelengthinstabilitywhich arisesdue to the presence
of the strongregion of the upper mantle. The J and C models
t physicalbehaviorat large wave numbersas dispositiveand thuscontributeto the growthof instability.The havedifferen,
determinationof q requiresa solutionof a linear systemof cussedin Appendix 2 for the simple case of a single strong
equationsto satisfythe velocityand stressmatchingconditions layer. However, as shownin Figure4, both modelspredictsimilar p,hysical
behaviorfor wave numbersof interest(k'<Jr) and
at eachinterfaceasdiscussed
in Appendix1.
The problemin its mostgeneralform is describedby a num- similar dominantwavelengths.The similarity of thesetwo modber of dimensionless
parameterswhich are listed in Table 1. els suggeststhat the existenceof two wavelengthsof instabilty
S•-S4 are ratios of buoyancy to strength at each of the
interfaces.In the presentproblem, S2 and S4 are set to zer9
½
becausethe crust and mantle are of uniform density. In the J
model, R•-R3 are the ratios of the strengthsof the various
regions.In the C model,the strengthat the mantlelayer-substrateinterfaceis continuous
so R3= 1. Here fl0w is determined
by the parameter
or,the ratioof theviscositydecaydepthin the
substrateto the stronglayer thickness.The values of these
parameters
determinewhetherthe lithosphere
will extendunstably, and if so, the dominantwavelength
at whichneckingwill
20o
\
q
Occur.
To consider the implications of strength and density
stratificationin the continentallithosphere,we determinethe
growthrate spectrum
q(k')•for g rangeof modelparameters.
Theresults
arethenapplied,
to discuss
theobserved
wavelengths
ofdeformation
intheBasinfindRange
Province.
o
I
3
--- J model
--
RESULTS
2
C model
Fig. 4. Growthrate of the instabilityq as a functionof dimensionless
We considera strengthstratification
for the continentallitho-
wave number k' for three mantle strength stratifications:curve a, J
with stronguppercrustaland uppermantlelayers(R• = 100,
sphere
likethatshown
inFigure
lYnwhich
a•strong
upper
crUs[model
=0.5, R• = 50); curve b, C model with strongupper crustaland upper
.
andmantle
19yerareseparated
by a weaklowercrust.,Uniform mantlelayers(R• =
100,R, =0.5, •c = !.2): curvec, J modelwith a sin1.2,
density
isassumed
within
thecrus.t
(@c
= 2950kgm-3)andman- gle strongsurfacelayer(R• = 100,R2 =R• = 1). For all modelsS•:
tle (@m=3200'kg
m-3).Theupper
crustis treated
asa po;WerS•=0.05 S-,:S4=0, !l•= l04,and11,=11•=114=3.
4830
ZUBER ET AL.' EXTENSION OF CONTINENTAL LITHOSPHERE
maximum within the strong mantle layer, and its magnitude
decaysexponentiallywith depth. The deformationpenetratesto
a depthcomparableto its wavelength.
Our interpretationof theseresultsin the contextof Basin and
Rangedeformationis that the shorterwavelengthinstabilitymay
control the spacingof individual ranges,and the longer waveMoho
lengthmay controlthe spacingsof long wavelengthtopography
b) long X
and the tilt domains. In the short wavelengthcase, regionsof
enhanced extension and thinning correspondto basins, and
regionsof apparentthickeningcorrespond
to ranges.In the long
wavelengthcase, the peaksand troughsin the long wavelength
c )comb,ned
Moho
displacement
profiledefineregionsin which fault blockschange
tilt directionsas determinedby the senseof horizontal shear.
Regions of alternatingpositive and negative surface gradient
correspondto Stewart'stilt domains.
We will now considerthe influence of lithospherestructure
and theology on the the predictedwavelengthsof deformation
Fig. 5. Displacement
fieldsassociated
with {a) short,(b) long, and (c) focusing primarily on the J model. In an extending layered
combinedwavelengths
of neckinginstability.The shortwavelengthof
medium, an increasein density with depth acrossan interface
deformationclearly showsthe neckingof the brittle uppercrust. Note
reducesthe vertical componentof the perturbingflow and acts
thepresence
of shearingwithinthelowercrustin Figure5c.
to confine deformationto within regions of uniform density.
This effect is most notableat shortwavelengthswhere, as illustrated in Figure 5, the relative magnitudeof deformationat the
does not depend on details of the strengthstratification.The
surface and at the crust-mantleboundary is significantly
growth rate spectrumfor a single strong layer overlying a subdifferent for the long and short wavelengthdeformation.The
strate with uniform strength(Figure 4 curve c) has a magnitude
long wavelengthdeformationhas a large amplitudeat the crusteverywheregreaterthan that for the multilayerJ model, and the
mantle boundaryrelative to that at the surface. In contrast,the
dominant wave number occurs at a point intermediatebetween
short wavelength deformation nearly vanishes at the
the short and long wavelength peaks. This suggeststhat the
crust-mantleboundary.This result, which predictsa flat Moho
presence of a strong subsurface layer acts to damp out
on a scaleof tensof kilometers,holdsfor any reasonablerange
instability over a range of wave numbers with the greatest
of crust-mantle density contrasts (i.e., @.... t<@.... fie). The
damping occurringat the minimum betweenthe short and long
densityincreaseacrossthe crust-mantleboundaryalso acts to
wavelength peaks. The long wavelength instability is thus
decreasethe overall magnitudeof the growth rate spectrum,but
driven by necking of the surface layer rather than by the tenfor the range of physically realistic conditionsthe effect is
dency of the mantle to neck. The mantle layer flexes in small.
responseto necking of the surface layer but does not itself
The amplitudesof the peaksin the growth rate spectrumare
undergonecking. Further discussionof the mechanismof long
directlyrelatedto the strengthsof the stronglayers.The effects
wavelengthdeformationis presentedin Appendix1.
of variations in lithosphericstrength on the behavior of the
Figure 5 shows the crustal displacementfields associated
growthrate functionin the absenceof buoyancyforcesand with
with the short, long, and combined dominant wavelengthsfor
h• = h2= h• = 15 km for the J model are summarized in Table 2.
the J model. These diagrams were constructedby calculating
displacementsat equally spacedpoints on an initially rectangular grid and connectingthe tips of the displacementvectors. In
the linearized theory, the amplitudes of the long and short
a)
W(k z)
b)
W(k',z)
-I
0
wavelength deformation are each proportional to their initial
amplitude. Thus the relative amplitudesare not determinedby
the growth rate factors alone. The amplitudesshown in Figure
crust
5 were chosen to clearly show the character of deformation at
30
mantle
each wavelength. In all cases the mean extension has been
removed, so that areas of apparentcompressionare in reality
areasof minimum extension.In the short wavelengthcase, the
6O
6O
effect of necking of the strong upper crust, as well as the
changein deformationalstyle at the brittle-ductiletransitioncan
be clearly observed.In the longerwavelengthcase, the upward
90
90
deflection of the crust-mantle boundary is the most obvious
deformationalfeature. Flow in both the crust and underlying
CRUSTAL
Illllil!\\\\II
DEFORMATION
llllllllll•\\\\•
FIELD
I
!
Zl
i1•1
I •
mantlecan be represented
in termsof the amplitudeof the vertical velocity W(k',z) at a given wave number. W(k',z), normalized to unity at the surface(z--0), is shownin Figure 6 for the
two
dominant
wave
numbers
in the J model.
For
the short
wavelengthdeformation,the vertical velocity at the base of the
crust and in the underlyingmantle is small, as reflectedin the
small relief on the crustmantle boundaryin Figure 5a. The vertical velocity of the long wavelength deformation reaches a
Fig. 6. Verticalvelocityasa functionof depthfor (a) shortand(b)
long wavelengthsof instability.Amplitudesare normalizedto a surfacevalue of unity. Note that for the short wavelengthcasedeformationis a maximumin midcrustalregions,whilefor the long wave-
lengthcasedeformation
peaksin the lowercrustand strongupper
mantle.
ZUBER ET AL..' EXTENSION OF CONTINENTAL LITHOSPHERE
TABLE 2.
R1
R2
R3
100
100
100
100
100
100
50
50
50
25
25
25
25
0.5
1
1
0.5
2
2
0.5
1
2
0.5
0.5
1
1
50
50
100
100
50
100
50
50
50
25
50
25
50
25
20
20
1
0.5
1
25
50
50
20
20
10
2
1
0.5
50
20
50
10
1
50
10
1
100
10
0.5
100
10
10
10
2
2
1
100
50
10
5
1
5
2
1
2
1
2
1
4831
Effect of StrengthContrastson the DominantGrowthRate Factor
qai
•ai,km
•a2,km
kal'
kd2'
167.19
167.94
167.94
167.19
169.40
169.42
85.42
86.18
87.65
43.43
43.43
44.17
44.17
0.60
0.60
0.60
0.60
0.65
0.65
0.60
0.65
0.70
0.65
0.60
0.70
0.65
2.30
2.30
2.30
2.30
2.30
2.30
2.30
2.30
2.30
2.30
2.30
2.30
2.30
44.17
34.94
35.70
0.70
0.65
0.65
2.30
2.30
2.25
145
41
145
42
51.06
37.23
0.70
2.25
135
42
34.54
35.68
0.70
2.25
135
42
23.89
29.12
32.05
25.59
38.33
33.86
19.20
10.35
0.65
0.70
0.65
0.65
0.70
0.75
0.75
0.80
1.10
2.25
2.25
2.25
2.25
2.15
2.15
2.25
2.15
1.85
145
42
135
42
4.48
17.86
18.61
18.61
17.86
20.13
20.12
18.58
9.91
4.52
2.00
--
1.55
91.13
105.14
117.85
97.13
119.71
140 65
59 38
70 91
82 26
35 64
39 93
41 33
48.90
41.33
35.34
43.34
qd2
157
41
157
157
41
41
157
41
145
41
145
41
157
41
145
41
135
41
145
41
157
41
135
41
145
41
135
41
145
42
145
42
135
44
126
44
126
118
86
61
42
44
51
--
R•, R2,andR3aredefinedin Table1. qd,kd',andkaarethedominantgrowthratefactor,wavenumber,andwavelength,
respectively.
The function is most sensitiveto changesin the strengthof the
uppercrustrelativeto that of the other regions.An increasein
the relative strengthof the upper crust increasesthe amplitudes
of both short and long wavelength peaks but increasesthe
shorterwavelengthpeak to a greaterextent. A relative increase
in the strengthof the strongupper mantle region decreasesthe
amplitudesof both peaks,but the decreaseis lessfor the shorter
wavelength peak. The variation in the dominant wavelengths
due to the differences in assigned layer strength is not
significant.
The presenceof a weak lower crust between a strongupper
crust and mantle is a prerequisitefor a secondwavelengthof
instability. As shown in Figure 4, a single stronglayer over a
uniform viscoussubstratehas a growth rate spectrumwith a single maximum. A strength-stratified
crust overlying a uniformly
strongmantle is an insufficientconditionfor two scalesof instability; the mantle must containa regionof high strength,though
as shownby the growth rate spectrumfor the C model, a discontinuityin strengthwithin the mantleis not required.
If the lower crust and mantle deform by steady state creep
the value of n is probably about 3, but variations within the
range 1-3 have a negligible effect on instability growth rate.
For a single layer embeddedin a weaker medium both with
n---1, the growth rate is less than unity in the absenceof an
unstabledensity stratification[Smith 1977]. On this basis it
might be expectedthat no necking instability would occur in a
strongupper mantle with a linear theology. However, a strong
upper mantle region with n3-I beneatha surface layer with
large n• does show a weak long wavelengthinstability, which
supportsthe contentionthat this instabilityis driven by necking
of the surface layer (see Appendix 1). If brittle deformation
occursin the upper mantle, as suggestedin Figure 1, then flow
in this region may be best describedby a large stressexponent.
Increasingn in the strongpart of the mantle enhancesthe instability of the longerwavelengthof deformation.
Dominant wavelengthsare determined primarily by layer
thicknesses.Smith [1979] showedthat a single layer with large
n embeddedin a weaker medium with n= 1 extends unstably
with a wavelengthto layer thicknessratio of •./h--•4. While our
resultsagree with Smith's for the limiting case, the presenceof
another strong layer (with finite n) and the inclusion of buoyancy forcesat the interfacesmarkedly changethis ratio. Table 3
is a compilationof the dominantwavelengthsof instabilityfor a
rangeof layer thicknesseswith R• = 100, R2= 0.5, and R3= 50 for
the J model. Buoyancyeffectshave been includedhere because
the dimensionlessparametersS•-S4 are a function of layer
thickness.Results show that •./h• ranges from approximately
2.8 to 3 for the shorterwavelength instability and from about 11
to 15 for the longer wavelength instability. However,
•./(h• + h2+ h3)'-•4for the longerwavelength.
Increasing the thicknessof the surface layer, keeping the
thicknessesof the other layersconstant,increasesboth dominant
wavelengths.A decreasein the thicknessof the strong upper
mantle increasesthe longer dominant wavelength but has no
apparenteffect on the shorterwavelength.A decreasein the
thickness of the weak lower crust may slightly decrease the
dominantwavelengthfor the longer wavelengthpeak, but again
the shorterwavelengthpeak is unaffected.These results show
that the lower crustal and upper mantle layers influence the
behavior of the growth rate spectrumto a lesser extent than
doesthe surfacelayer.
APPLICATION TO THE BASIN AND RANGE PROVINCE
We
will
now consider
the observed
structure
of the Basin
and RangeProvincein termsof theseresultsand examinepossi-
4832
ZUBER
ETAL.' EXTENSION
OFCONTINENTAL
LITHOSPHERE
TABLE
3.Dominant
Wavelengths
foraRange
ofLayer
Thicknesses
h•,km
h2,km
h3,km
kdl'
kd2'
kd!,km
2..•'•6•
126
28
12.56
2.80
137
28
•.b6
2.80
ka,,km
kd2/hl
_
5
5
5
o si
10
5
5
0 5o
10
10
5
5
5
10
15
15
15
10
10
5
5
10
10
10
10
15
10
10
5
10
10
5
10
10
10
10
15
15
0 46
224
2 24
052
2 22
121
28
12.08
2.83
0 56
2 26
56
14
11.22
2.78
052
2 26
60
14
12.08
2.78
0 54
2 26
58
14
11.64
2.78
0 5o
2 20
126
28
12.57
2.83
10
20
10
58
14
2.78
0.48
2:20
196
43
13.09
2.86
0.46
2.20
205
43
13.66
2.86
0.50
2.20
188
43
12.57
2.86
0.52
2.22
121
28
12.08
2.83
10
0.48
2.22
131
28
13.09
2.83
15
15
0.50
2.22
126
28
12.57
2.83
15
20
20
20
15
15
15
20
15
15
15
15
15
20
20
0.48
2.20
196
43
13.09
2.86
0.46
2.16
273
58
13.65
2.91
0.44
2.16
286
58
14.28
2.91
0.46
2.16
273
58
13.66
2.91
0.48
2.20
196
43
13.09
2.86
15
20
15
0.46
205
43
13.66
2.88
15
20
20
20
20
10
10
10
25
20
20
10
20
10
10
20
20
20
20
20
10
10
20
20
10
20
20
0.48
2.18
2.18
196
43
13.09
2.88
0.46
2.16
273
58
13.66
2.91
0.46
2.16
273
58
13 66
291
0.42
2.16
299
58
14 96
291
0.50
2.16
251
58
12 57
291
0.52
0.46
2.22
121
28
12 08
283
2.22
137
28
13 66
283
0.50
2.22
126
28
12 57
283
0.46
2.12
341
74
13 66
2 96
25
25
20
0.44
2.12
357
74
14 28
2 96
25
20
20
20
25
20
20
25
25
25
25
25
20
25
25
0.46
2.12
13.09
291
0.44
2.16
i4.28
291
0.46
2.16
2.12
13.66
13.66
291
0.46
74
58
58
58
74
2 96
2.16
341
262
286
273
341
13 66
0.48
2 96
hi, h2, andh3 are the thicknesses
of the stronguppercrust,weaklowercrust,andstronguppermantlelayers.kd' and kd are the dominant
wave numberand wavelength.
ble structuraland geophysicalimplicationsof the style of defor-
tively. The differenceis greaterfor the long wavelengthinsta-
mationi'esulting
fromthe superposition
of the longandshort bility becauselonger wavelengthsof deformationpenetrateto
sensethe differentmantlestrength
wavelengthinstabilities.A 15 km thicknessfor the strongupper greaterdepthandtherefore
crust is reasonableboth from the flow law of quartz [Brace and
Kohlstedt, 1980] and the depth to which earthquakes are
observed in the Basin and Range [Eaton, 1982; Smith, 1978].
stratification in the models. In both models, the scale of the
shortwavelength
disturbance
is overestimated
andthatof the
long wavelength disturbanceis underestimated.Better agree-
For a total crustalthickness
of 30 km [Eaton, 1982;Smith, mentwouldbeobtained
fora thihner
brittlesurface
lfiyeranda
1978; Thompsonand Burke, 1974] the thicknessof the weak
lowercrestis alsoabout15 km. The thickness
of the regionof
highstrength
in theuppermantlefor boththeJ andC modelsis
based on the flow law of olivine [Brace and Kohlstedt, 1980].
In the former case, a 15 km thick layer is chosento representa
region of high strength in the mantle. In the latter case, a
mantle layer thicknessof 5 km and an e-folding depthof 6 km
are chosenon the basisof Figure 1.
Two Scalesof Deformation
thicker strong mantle layer. Strong crustal layer thicknessesof
about 10 and 11 km and mantle layer thicknessesof about 18
and 8 km would be requiredto match the observationsfor the J
and C models, respectively. However, given the major uncer-
tainties
in lithosphere
rheology
discussed
earlier,weemphasize
the existenceof two wavelengthsof instability rather than the
parametervalueswhich providea numericalbest fit. Since contrastsin strengthaffect mainly the amplitudeand not the wavelength of instability, uncertaintiesin our choicesof S values
does
notstrongly
influence
these
estimates
oflayerthickhesses.
The shapeof the growthrate spectrum
depends
not on the
Amplitudesare moredifficultto estimateon the basisof the
actual strengthof the layers but on their strengthratios R•, R2,
and R3. For the variations in strengthassumedin Table 3 and
Figure4 for the J model(R• - 100, R2= 0.5, R3- 50), layerthicknesses of about 15 km correspond to wavelengths of
deformationwhich reasonablymatch the observedspacingsof
ranges and tilt domains. The dominant wave numbersof the
peaksin the growth rate spectrumin Figure 4 correspondto distances of 42 and 40 km between ranges and 170 and 157 km
present
linearized
modelbecause
theydependdirectlyon initial
amplitudes.It is, however,possibleto establisha lower limit of
the dominantgrowthrate factorrequii'edto producethe approximate structuralrelief of the rangesand the largesttilt domains.
For a lowerlimit of totalBasinandRangeext•hsion
of 10%
and initial verticalsurfaceperturbations
in the rangei0-100 m,
dominantgrowthrate factorsof qd--24 and 47 will resultin vertical surface deformation
of the order of 1 km. As summarized
betweenantiformalboundaries
for the J and C models,respec- in Table 2, the minimumlithosphericstrengthcontrastsrequired
ZUBER ET AL.: EXTENSION OF CONTINENTAL LITHOSPHERE
4833
for qa in this range are consistentwith the strength profile
shown in Figure 1. We have not consideredthe effect of fluid
pressure, which would act to reduce the effective frictional
strengthof the upper crust;however, for a hydrostaticvalue the
decreasein average crustal strengthwould not be great enough
to suppress
the growthof eitherwavelengthof instability.
Crustand Mantle Deformation
Figure 5c is a plot of the superpositionof the crustal displacementfields and showsthe combineddeformationfrom the
two wavelengthsof instability. A contrastin the style of deformation between the plastic surface layer and the ductile lower
crust is clearly visible. Regions of maximum extension are of
particularinterestsince faulting shouldlocalize in these areas.
At the surface, localized extension occurs in the short wave-
length basins. The near-surfaceregion at the center of the diagram exhibitsenhancedthinningbecausethe extensionfor both
the long and short wavelengthcomponentsis a maximum there.
This model thus predicts localized highly extended regions
spacedof the order of the long wavelengthinstability. In the
lower crust, areasof greatestextensionoccur primarily beneath
rangesand are horizontally removed from regions of near-surface extension.
Another interestingaspectof Figure 5c is the style of deformation in the vicinity of the brittle-ductiletransition.The superposed long and short wavelength deformation results in
enhancedhorizontal shearingnear the base of the strong upper
crust. Localized shearingat midcrustaldepthscould be a mechanism for the formation of low-angle normal faults. The
location of horizontal shearingwithin the crust in Figure 5c is
consistentwith suggestionsby Andersonet at. [1983] and Wernicke [1981] that these features root far from regions of nearsurface, thin-skinned deformation. In addition, seismic reflection
Fig. 7. Asymmetry of resolved shear stresson conjugate(dip=60 ø)
faults (heavy lines) associatedwith surfacetopographywhich dips down
to the right. The resolvedshear stressis greateron the fault plane which
dips in the direction of the surface topography. See text tk)r implications.
preexistingstructuralweaknesses,such as reactivatedMesozoic
thrusts.Also, the linearizedmodel developedhere can treat only
the initial growth of structuresand cannot accountin detail for
small scale features formed in areas of large horizontal extension. Finally, becausethe model is two-dimensional, it cannot
explain the presence of the transversezones. Stewart [1980]
suggeststhat thesezones mark the northwardmigration of Basin
and Range deformation with time and are likely related to tectonic stressesassociatedwith changes in the plate boundary
geometry of the western margin of North America during the
Cenozoic.
Relation of Gravity Anomalies to Extensional
profiles in the Basin and Range [Allmendingeret at., 1983,
Deformation
1985] and in the Scottish Caledonides [Smythe et at., 1982]
Gravity studies [Eaton et al., 1978] show that most of the
show low-anglereflectorswhich penetrateto midcrustaldepths.
Andersonet al. [1983] have suggestedthat the zone of decou- regional topographyin the Basin and Range is in approximate
pling for detachmentsin regionssuch as the Sevier Desert and isostaticequilibrium. The regional Bouguergravity signature,as
Raft River Valley may be the brittle-ductiletransition. In this depicted in maps of the filtered Bouguer gravity field of the
model the brittle-ductile
transition is not an actual surface of
United States[Hildenbrand et al., 1982], exhibitsa N-S striking
mechanicaldecoupling;instead,shearingis vertically distributed 200 km wavelengthundulation which correlateswith the long
wavelengthregional topographyof the province [Eaton et al.,
in the upperpart of the ductilelower crust.
Shear stressesassociatedwith the long wavelengthdeforma- 1978]. Eaton et al. have suggestedthat these long wavelength
tion provide an explanationfor the tilt domains. In each tilt gravity anomaliesare at least partially due to temperaturevariadomain, high-anglefault blocks associatedwith the short wave- tions in the mantle. Froidevaux [1985] estimates that about 7
lengthdeformationshow a consistentdirectionof tilting or rota- km of relief on the crust-mantleboundary would be required to
tion. Assumethat conjugate,high-anglenormal faults, dipping explain the amplitude of the observed anomalies by crustal
at 550-60ø, form initially in responseto the uniform horizontal thickness variations alone. Since seismic reflection studies in
extension.Shear stressesdue to the growing long wavelength the Basin and Range [Allmendingeret al., 1983, 1985] show a
deformation
will cause differences
in resolved shear stress on
relatively flat Moho, he concludesthat density variations in the
these oppositely dipping faults. As illustrated in Figure 7, a mantle must be responsiblefor the observedlong wavelength
counterclockwiseshear couple on horizontal planes at depth in gravity anomaly and estimatesthat about a 50 km vertical disthe crust occurs in areas where the surface slopes down to the placementof mantle isothermswould be requiredto accountfor
right. Shear stressesof this sense increase the resolved shear the observedanomaly.
The short wavelength deformation is consistentwith large
stresson high-anglefault planesdippingto the right and reduce
it on fault planesdipping to the left. The converseis true if the vertical relief at the surfacecomparedto that at the crust-mantle
surfacedips to the left. Therefore, motion on faults dipping in boundary. However, if the Moho also has relatively small relief
the downslopedirection of the surface is preferred. Thus, in at long wavelengths,ductile extensionalone cannot accountfor
Stewart's [1980] terminology,synformaland antiformalbounda- the amplitude of the observed gravity anomaly. As shown in
ries will coincide with crestsand troughs, respectively,of the Figure 6, the vertical velocity for the long wavelengthdeformation at the Moho is approximatelyequal to the maximum value
long wavelengthsurfacedeformation.
The presentmodel cannotbe appliedto explain all aspectsof in the underlying mantle. Thus the vertical relief on initially
finite deformation, for example, features which form along horizontal surfaceswould be nearly a maximum at the Moho. if
4834
ZUBER ET AL.; EXTENSION OF CONTINENTAL LITHOSPHERE
44•
I1•ø
1 I1• I1•)ø
10'6•
--Syn
sphere, the dominantwavelengthsof necking are most strongly
controlledby the layer thicknesses.The relative strengthsof the
layers controlthe amplitudesof the instabilitiesbut only weakly
affect the dominantwavelengths.
In the lower crust and mantle where dislocationcreep dominates deformation,varying the stress-exponent
in the range of
1-3 has a negligibleeffect on instabilitygrowth rate. In contrast
to the case of a single layer in a weaker viscousmaterial both
with n= 1 which always extends stably, a strong mantle layer
with n3= 1 exhibits a weak long wavelengthinstabilitybecause
mantle deformationis driven by unstableextensionof the upper
crust.
antiformalboundary
synformal boundary
Fig. 8. Tilt domainsand Bouguergravity in the Basin and Range
The two wavelengthsof necking instability that result from
strengthstratificationof the lithospheremay explain the formation of ranges and tilt domains in the Basin and Range Province. For a discontinuousstrength(J) model, thicknessesof the
strongupper crust, weak lower crust, and strongupper mantle
regionsof the continentallithosphereof about 15 km, which are
in agreementwith experimental flow laws and seismic results,
producestructureswith wavelengthsconsistentwith the spacings
of ranges and tilt domains. Regions of localized extension at
depth which arise due to necking of the strongupper crust are
laterally displacedfrom regions of near-surfaceextension. The
resultinglocalizationof horizontal shearingin the upper part of
the weak lower crust predicted by this model may represent
incipientlow-angleextensionaldetachments.
For short wavelength deformation, the model predicts
smaller vertical relief at the crust-mantleboundary than at the
Province. Dashed and solid lines correspondto synformal and antitk•rmal boundaries traced from St(,n'•trt's [1980[ Figure 1. Dotted and
hatched regions correspondto gravity highs and lows, respectively,
taken from the map by Hihh'nbraml ctal. [1982] for all wavelengths
less than 250 km. Anomalies range from approximately +35 to -35
mGals. Note that the longestand most continuoussynlbrmal and anti- surface. This is consistent with the results of recent COCORP
formal boundariestire associaledwith with gravity lows and highs, reflection profiling which show a relatively flat Moho in the
respectively.
the long wavelengthgravity anomalyis due to densityvariations
in the mantle, this suggests,in constrastto the view of Froidevaux [1985], that convective flow generatedby density differencesin the mantle may be requiredto accountfor the magnitude of the anomaly. However, the long wavelength necking
instabilitymay determinethe horizontalscaleat which this convective motion subsequentlyoccurs.
The large-scaletilt pattern, representedby synformaland
antiformal boundaries,and the pattern of long wavelengthgravity anomaliesare shownin Figure 8. Althoughthe surfacestructures are more complicatedthan the gravity field, the longest,
most continuoussynformaland antiformalboundariesoccur in
areas of negativeand positive gravity anomaly respectively.
Seismic reflection studieshelp to define the shallow structure
createdby faulting or brittle deformationbut cannotresolve
deeper ductile deformationwhich producesno well-defined
reflectors. There is thus no direct evidence that tilt domains at
Basin and Range. However, for long wavelength deformation,
the model predicts larger vertical relief at the crust-mantle
boundary than at the surface, and the vertical displacement
which occursat the Moho is nearly equal to the maximum value
that occurs in the mantle. If the long wavelength gravity anomaly in the Basin and Range cannot be attributedto variations
in crustal thickness,then horizontal temperaturevariationsdue
to uplift of isothermsin the necking lithospheremay provide the
requireddensity variations. However, our models predict that if
relief on the Moho is small at long wavelength, unstableextension of the mantle alone cannot account for the amplitude of
deformationrequired. This suggeststhat convective motions in
the mantle, possiblywith a wavelengthcontrolledby the necking instability, may be responsiblefor the long wavelength
gravityanomaly.
APPENDIX l: SOLUTION OF THE PERTURBING
MULTILAYER
FLOW PROBLEM
the surfacecorrespondto deeperductiledeformation.However,
the spatialrelationship
of long wavelengthgravityanomaliesto
We wish to determinethe perturbingflow in an extending
surfacestructuressuggeststhat thesestructuresmay be related mediumconsistingof three viscouslayersover a viscoussubto ductiledeformationat depth.
strate, each with a uniform strength.The equationfor the flow
in a singleviscouslayerwith uniformeffectiveviscosityis
CONCLUSIONS
D4W- 2k2(2/n•-1)D2W+ k4W-0
(AI)
An extendingcontinentallithospherewith a density and
strengthstratification
that is consistent
with seismicityobserva- where D=d/dz, k (=2r[/k) is the wave number, and W is the
tionsand experimentalrock deformationstudiesis unstablewith streamfunctionsatisfiedby
respectto neckingand will deform with two wavelengthsof
cos kx
instability.Shortand long wavelengthsof deformationarisedue
(A2)
to the presenceof the strongupper crust and upper mantle
fi =-k -•Dwsinkx
regionsof the lithosphere.
In simplelayeredmodelsof the litho-
ZUBER ET AL.'
where a and •
EXTENSION OF CONTINENTAL LITHOSPHERE
are the horizontal and vertical components,
respectively,of the perturbingvelocity field. Equation(A1) has
thegeneralsolution[FletcherandHailer, 1983]
A I - Al(0)exp[(q-1)•;xxt],where q is the growth rate factor. By
substitutingthe form of the perturbedinterface and normalizing
with respectto the meanstrainrate •;xx,we rewrite (A7) as
W(k,z) = (A•cos13•kz+ A2sin13•kz)exp(ot•kz)
(-1-q)
+ (Aacos13•kz+ A4sin13•kz)exp(-ot•kz) (A3)
where
ct•= (1/n•)•/2
and13•
= (1 - 1/n•)•/2.
Thissolution
is valid
for all of the layersand the substrate,but eachregionhas a different set of constantsA•-A4 (upper crust), B•-B4 (lower crust),
C•-C4 (upper mantle), and D•-D4 (mantle substrate).In the
/q = [(A•cos[3•kz+ A2sin[3•kz)exp(ct•kz)
+ (A3cos[3•kz+ A4sin[3•kz)exp(-cttkz)]
coskx
A, + '&(0,dl)--0
We wish to solve for the coefficients
(A8)
and the values of the
growth rate parameter,which requiresthe simultaneoussolution
of the matching conditionsand the growth rate equations(A8).
The coupledproblem is solved by the method of static condensation[Bathe, 1982]. We beginwith the 18x18 matrix system
viscous substrate the stresses and velocities must be bounded as
z---->•,thereforeD3 and D4 vanish. The perturbingvelocity and
stresscomponents
within the uppercrustare
4835
Kq) -- qMq)
(A9)
where K is the matrix of matching conditions and vertical
velocities, q is the matrix containing the eigenvalues(growth
rate factors), M is the matrix containingthe coefficientsof the
eigenvalues,and q) is a vector of the coefficientsand amplitudes.This systemcan be rewritten
fi = -ct•{[(A• + 6•A2)cos[3•kz
aaKa•]
I(])l
=qrma
i1I(])
1
+ (A2-6•A•) sin[31kz]exp(ct•kz)
(A10)
ca Kc
d
- [(A3- 6•A4)cos[3•kz
+ (A4 + 6•A3)sin[3•kz]exp(-ct•kz)}
sinkx
L0
wherema is a 4x4 identitymatrix. From (A10)
(A4)
Kcaqba
+ Kccq)c= 0
Ozz= •:•ct•k[(A•cos
[3•kz+ A2sin[3•kz)exp(ct•kz)
-(A3cos [3•kz+ A4sin[3•kz)exp(-ct•kz)]
coskx
(A11)
and
qbc
= -K•-•K•aqb•
Oxz= -x•k{[(A• + 61A2)cos[3tkz
+ (A2- 6•A•) sin[3•kz]exp(ct•kz)
(A12)
Substitutionof (A12) into (AI0) transformsthe system into the
standardeigenproblem
+ [(A3- 6•A4)cos[3•kz
+ (A4 + 6•A3)sin[3•kz]exp(-ct•kz)}
sinkx
(Ka-qMa)q)a - 0
where
6•= (n•- 1)•/2.
(AI3)
where
The linearizationto firstorderin kA requiresthat
Ka= Kaa-KacKcc-•Kca
Oxz(X,dl)= 2'rkAI sin kx
(A14)
(A5)
We solve (AI3) by a QR orthogonalization
algorithm[Dahlquist
where A I is the maximum amplitudeat the ith disturbedinter- and BjOrk, 1974]. To determinethe coefficientsand amplitudes,
face at depth d,. The difference in density acrossan interface we eliminate A• from (AI0) and substitutethe growth rate factor obtained above. The resulting linear system can then be
requires
solvedby standardmethods.
(A6)
(Jzz(X,dl)= (•l-I - Q,)gAicoskx
The magnitudeof the growthrate factorobtainedfrom (A13)
is a strongfunctionof the stressexponentin the surfacelayer.
The normal stresses are continuous, and the shear stress van-
ishes at the free surface, while at the other interfaces both the
stressesand velocitiesare continuous.This provides 14 match-
ing conditions
from whichthecoefficients
canbe obtained.
The rate of amplitudegrowth at each of the interfacesintroducesa systemof four differentialequations
/•1-"'-g2xxAi
+ /q,(0,d•)
In this study, we adopteda large n to approximatea perfectly
plasticmaterial,and it is usefulto examinethe consequences
of
this choice of rheology. Figure A l, which plots qd for both
dominantwavelengthsas a functionof n•, showsthat increasing
n in the stronglayer enhancesthe instabilityof both the long
and short wavelength features. Limiting behavior is not
approached
untiln•103 forthelongwavelength
instability
and
instability,
whichagrees
with
(A7) nl•104 for theshortwavelength
the results of Fletcher and Hallet [ 1983].
where
,• =dA/dt.Thenumber
ofvalues
ofthegrowth
ratefactor
obtainedat a given wave numberequalsthe numberof equations in (A7), though in generalonly one value of q will be
greaterthanone and thuscontributeto the growthof instability.
The shapeof the perturbedinterfacewill changewith time as
A decreasein n• shifts the maxima to longer wavelengths
and decreasesthe magnitudeof the growth rate spectrumat all
wave numbers.The amplitudeof the shorterwavelengthpeak is
decreasedsignificantlyrelative to the longer wavelengthpeak.
As noted in the resultssection,a strongmantle layer stabilizes
4836
ZUBER ET AL.' EXTENSION OF CONTINENTAL LITHOSPHERE
150,
LIMITING
BEHAVIOR
short
•.'
125
IOO
75
longX
25
o' .......
.......
Fig. A1. Magnitudeof the rate growth rate factor q as a functionof
the stressexponentin the surface layer n• for the conditionsS• = 1.2,
S• = 0.05, S_,=S4 =0, R• = 100, R_,= 0.5, andRs= 50. Limiting behavioris
crustallayer, andS• =6, which corresponds
to a layer with finite
strength,andwith 0t= 0.1 andR• = 10.
In the C model, the short wavelength(k' large) components
in the perturbationare greatly dampedrelative to the J model.
For S•--0, the range of instability is 1.4 - k' - 2.9, with q
reachinga maximumof about85 at ka' - 1.9 or X/h= 3.3. Additionalregionsof instabilityare separatedby regionsof damping,
and the magnitudesof subsidiarypeaks decreasesharply. The
effect of finite strengthin the layer is to decreaseinstabilityor
increasedampingat all wave numbers.For S• =6, the region of
instabilityis reducedto a small interval 1.5 --<k' - 1.9 about
the maximum at ka'= 1.6 or X/h=3.9. The maximum in the
growthratefunctionis qa= 20.
In the J model, for the strong layer case, intervals of insta-
bility are centeredat ka'= (n + 1) yr/2, n=0, 1, 2,..., and for all
not reached
untilnr-->104
and l0s for the shortandlongwavelengthpeaksqa= 20. The value R•-- 10 usedwas not expectedto yield
maxima, respectively.
the closestcorrespondence
in qa and ka' with the C model, but
was simply chosenarbitrarily in order to comparethe overall
the systemwith respectto neckingover a limited rangeof wave behaviorof the two models. The characterof the structurespronumbers, with the maximum stabilizationoccurringat the rela- ducedin eachinstabilityintervalof the J model is different:ka'
tive minimum in q. This is suggestedbecause,as illustratedfor in the range 0 - ka' - •t results in a structurewhere vertical
at the surfaceand the layer-substrate
interfaceare
the J model in Figure 4, the growth rate spectrumfor an displacements
extending,strongsurfacelayer over a uniform viscoussubstrate oppositein sign, while ka' in the range •t _ ka' - 2•t results
at the surfaceand the
envelopesand is everywheregreaterthan that for an extending in a structurewhereverticaldisplacements
medium with two strong layers. If the strongupper mantle acts layer-substrate
interfacehave the samesign. The effect of finite
to stabilizethe flow, then both wavelengthsof deformationmust layer strengthin the J model is to reducein the magnitudeof
arise in responseto neckingof the surfacelayer. An exception the instability.Although,as in the C model, the regionof instato this occursif the upper mantle, insteadof the uppercrust, is bility in each interval is shiftedtoward lower k' values(longer
intervals.
characterized by large n. A layered medium in which wavelengths),the effectdiminishesin successive
In the C model, the presence of a single, prominent
nl = n2= n4--3 and n3--->
ochas a single long wavelengthinstability which resultsin pinch and swell deformationof the mantle. maximumover the rangeof wave numbersprovidesan obvious
In the J model, if both the surfaceand upper mantle layer have choice for the wavelengthof maximum instability. In the J
a large n, the growth rate spectrumexhibitsa singleshortwave- model, we have assumed that maxima which fall in the first
length maximumand an inflectionpoint at longerwavelengths. interval of instability, 0 - ka' - •t, correspondto featuresof
We have not been able to identify conditionswhich result in interest,and this requiressome rationalization.From a physical
two discrete maxima. However, in the C model two wavestandpoint,the maxima at large ka' correspondto very short
lengthsof instabilitywill occur if both n• and n3 are large, but wavelengthfeatures, which are not observedin natural strucit is not clear whether deformation in this case is driven by ture. In the multilayeredproblem, the vertical stratificationin
simultaneousneckingof the stronglayersor simply by necking strengthresults in two maxima in the range 0 - ka' - •t.
of the uppercrust.
Analysisof the variationsin physicalpropertiesof the lithosphere (strength, density, stress exponent, layer thickness)
APPENDIX 2: COMPARISON OF MODELS WITH CONTINUOUS
AND DISCONTINUOUS STRENGTH STRATIfiCATION
C-Model
J-Model
In this appendixwe comparemodelsin whichthe strengthof
the substratebeneath a single strong layer (1) decreasesexponentially with depth (C model) or (2) decreasesdiscontinuously
to a lower uniform value at the bottom of the layer (d model).
The variation of strengthwith depth for each model is illustrated
in Figure A2.
Plots of the growth rate factor as a functionof wave number
are shown for the continuous strength (C) model and the
strengthjump (d) model in Figure A3. The C model is charac-
terizedby two parameters,
ct, the ratio of the depth• in which
the effective viscosityin the substratefalls off by 1/e to the
layer thickness
h, andS•--(Player- @o)gh/•;,
the ratio of the lithostaticstressat the base of the layer to the layer strength.The
J model is characterizedby S• and the ratio of layer strengthto
substratestrength, R•. Results for the J model are obtained
from an analytical solutionwhile those for the C model are.
determinednumerically.The two casesexaminedin eachmodel
are for S• =0, which correspondsto the limit of a very strong
Fig. A2. Models of a lithospherewith a continuousdecreasein
strength(C model) and a jump in strength(J model) acrossthe brittleductile transition.
ZUBER ET AL.' EXTENSION OF CONTINENTAL LITHOSPHERE
4837
all behavior of the models is similar within the first (small wave
80
number)intervalof instability.
The falloff in the magnitudesof higher-orderpeaksin the C
model may thus justify the exclusion of these peaks in the J
model. We have used a strengthjump to approximatea continuous decreasein strengthwith depth in the lithosphereand are
therefore interestedin that part of the behavior of the J model
which best approximatesthe behavior of the C model; this
occursin the first instabilityinterval.
C-Model
40
-40
'I
-80
J-Model
Acknowledgments. This research was supported by NASA grant
NSG-7605. Comments by Leigh Royden and George Thompson
improved the paper. We are grateful to Bob Simpson for providing us
with mapsof the Bouguergravityfield in the United States.
IO
q
REFERENCES
o
4
6
e
kø
Allmendinger, R.W., H. Farmar, E. Hauser, J. Sharp, D. Von Tish, J.
Fig. A3. Magnitude
of thegrowthratefactorq asa function
of wave
number k' for the C and J models with (t=().l
and R• = 10. For both
modelsS•=0 (stronglayer) and =6 (finite strengthlayer), S2=0,
ill......= 104,andn,,,/
........
= 1.
describedin the results sectionclearly associatethe lower and
higherwave numberpeaksin this range with the presenceof a
strong upper crust and upper mantle. Only variationsin the
physicalpropertiesof the surfacelayer affect the characterof
the maxima for kd'>J1;. As nlmer---•
oc, the viscous resistance
to deformationdecreasesdramatically, and the disturbanceproducedat an interfacedecaysslowly away from the interfaceand
is partially reflectedat the other interfaces.In this manner,
deformationis controlledby a resonanceeffect [Smith, 1979] in
which the secondaryflow at an interface drives deformationat
the other interfaces.It is this effect which is responsiblefor the
presenceof the higher-orderpeaksin the J model. The magnitudesof successivehigher-ordermaxima decreaseexponentially
when n in the surfacelayer has a finite value. Hence, the large
wave number peaks in the J model are a consequenceof the
assumption
of perfectlyplasticbehaviorin theuppercrust.
It is not possibleto develop a comparativeunderstandingof
the nature
of
the
instabilities
of the two
models
because
a
simple analyticalsolutioncannotbe obtainedfor the C model
[cf. Fletcher and Hailer, 1983]. Therefore, in order to examine
the similarity in behaviorof the C and J models, one must rely
on generalnumericalresultsfor a range of parameters.In both
models, kd/h increasesas S! increasesand falls in the narrow
rangerc -< kd/h • 4 for the C modeland 4 -< kd/h • 8 for the
J model. This ratio has a weak dependenceon c• in the C model
and on Ri in the J model. For both models, the magnitudeof
the growth rate function decreasesas S• increases.In the C
Oliver, and S. Kaufman, Phanerozoic tectonics of the Basin and
Range-ColoradoPlateau transitionfrom COCORP data and geologic
data, in Reflection Seismologyand the Continental Crust: A Global
Perspective, GeodynamicsSet., edited by M. Barazangi and L.
Brown, AGU, Washington,D.D., in press,1985.
Allmendinger, R.W., J.W. Sharp, D. Von Tish, L. Serpa, L. Brown,
S. Kaufman, J. Oliver, and R.B. Smith, Cenozoic and Mesozoic
structure of the eastern Basin and Range Province, Utah, from
COCORP seismic-reflectiondata, Geology, 11,532-536, 1983.
Anderson, R.E., M.L. Zoback, and G.A. Thompson, Implicationsof
selected subsurface data on the structural form and evolution
of some
basinsin the northern Basin and Range Province, Nevada and Utah,
Geol. Soc. Am. Bull., 94, 1055-1072, 1983.
Bathe, K.-J., Finite Element Proceduresin EngineeringAnalysis, pp.
573-586, PrenticeHall, EnglewoodCliffs, N.J., 1982.
Bird, P., Inititiation of intracontinentalsubductionin the Himalaya, d.
Geophys.Res., 83, 4975-4987, 1978.
Brace, W.F., and D.L. Kohlstedt, Limits on lithosphericstressimposed
by laboratoryexperiments,J. Geophys.Res., 85, 6248-6252, 1980.
Byerlee, J.D., Brittle-ductiletransitionin rocks, J. Geophys.Res., 73,
4741-4750,
1968.
Chappie,W.M., Mechanicsof thin-skinnedfold and thrustbelts, Geol.
Soc. Am, Bull., 89, 1189-1198,
1978.
Chen, W.P., and P. Molnar, Focal depthsof intracontinentaland intraplate earthquakesand their implicationsfor the thermal and mechanical
propertiesof the lithosphere,
J. Geophys.Res., 88, 4183-4214, 1983.
Dahlquist,G., and A. Bj6rk, NumericalMethods, pp. 216-217, Prentice-Hall, EnglewoodCliffs, N.J., 1974.
Eaton, G.P., The Basin and Range province: Origin and tectonic
significance,
Annu.Rev. Earth Planet. Sci., 10, 409-440, 1982.
Eaton, G.P., R.R. Wahl, H.J. Prostka, D.R. Mabey, and M.D. Kleinkopf, Regional gravity and tectonic patterns:Their relation to late
Cenozoicepeirogenyand lateral spreadingin the WesternCordillera,
Geol. Soc. Am. Mem. 152, 51-91, 1978.
England, P., Constraintson extension of continental lithosphere,J.
Geophys.Res., 88, 1145-1152, 1983.
Fletcher, R.C., and B. Hallet, Unstableextensionof the lithosphere:A
mechanicalmodel for Basin and Range structure,J. Geophys. Res.,
88, 7457-7466,
1983.
model,
instability
occurs
where
SI • (2Or)
-I, whiletheJ model Froidevaux, C.,
is unstable for all S i. The latter is due to the fact that the
Basin and Range large-scaletectonics:Constraintsfrom
gravityandreflectionseismology,
J. Geophys.Res., in press,1985.
Glazner, A.F., and J.M. Bartley, Evolution of lithosphericstrengthafter
thrusting,Geology, 13, 42-45, 1985.
Goetze, C., and B. Evans, Stressand temperaturein the bending lithosphereas constrainedby experimentalrock mechanics,Geophys.J. R.
strengthcontrastsat both the upper and lower interfacesdrive
the instability, and necking will take place regardlessof the
magnitudeof S•. The magnitudeof the growth rate functionin
Astr. Soc., 59, 463-478, 1979.
the C model dependson both S• and or, while for the J model, Hildenbrand, T.G., R.W. Simpson, R.H. Godson, and M.F. Kane,
q has a weak Si dependenceand a strong R l dependence. Digital colored residualand regional Bouguergravity maps of the
conterminousUnited States with cut-off wavelengthsof 250 km and
Becausethere is no physicalconnectionbetweenR i and or, it is
1000 km, U.S. Geol. Surv. Geophys.Invest. Map, GP-953-A, 1982.
not possibleto obtain a simple relationshipthat describeswhat
S.H., Rheology of the lithosphere, Rev. Geophys., 21,
valuesof theseparametersgive the sameqd or kd. However, Kirby,
1458-1487, 1983.
both modelsyield qd valuesof the sameorderof magnitudefor Proffett, J.M., Jr., Cenozoicgeologyof the Yeringtondistrict, Nevada,
similarR• and ot-I. The abovecomparison
indicates
that
thoughthe natureof the instabilitiesmay be different, the over-
and implicationsfor the nature and origin of Basin and Range faulting, Geol. Soc.Am. Bull., 88, 247-266, 1977.
4838
ZUBER ET AL.: EXTENSION OF CONTINENTAL LITHOSPHERE
Smith, Robert B., Seismicity, crustalstructure,and intraplatetectonics Tullis, J., and R.A. Yund, Hydrolytic weakening of experimentally
of the interior of the western Cordillera, Mem. Geol. Soc. Am., 152,
deformedWesterly granite and Hale albite rock, J. Struct. Geol., 2,
111-144, 1978.
Smith,RonaldB., Formation-of
folds,boudinage,
andmullions
in
non-Newtonian materials, Geol. Soc. Am. Bull., 88, 312-320, 1977.
Smith, RonaldB., The folding of a stronglynon-Newtonianlayer, Am.
J. Sci., 279, 272-287, 1979.
Smythe, D.K., A. Dobinson,R. McQuillan, J.A. Brewer, D.H. Matthews, D.J. Blundell, and B. Kelk, Deep structureof the Scottish
Caledonidesrevealedby the MOIST reflectionprofile, Nature, 299,
338-340,
439-451, 1980.
Wernicke, B., Low-angle normal faults in the Basin and Range Province: Nappe tectonicsin an extendingorogen, Nattire, 291, 645-648,
1981.
Zoback, M.L., R.E. Anderson,and G.A. Thompson,Cainozoicevolution of the state of stressand style of tectonismof the Basin and
Range province of the westernUnited States, Philos. Trans. R. Soc.
London Ser. A, 300, 407-434, 1981.
1982.
Stewart, J.H., Basin and Range structure:A systemof horstsand grabens producedby deep-seatedextension,Geol. Soc. Am. Bull., 82,
1019-1044, 1971.
Stewart, J.H., Basin-rangestructurein western North America: A
review, Mem. Geol. Soc. Am., 152, 1-31, 1978.
Stewart, J.H., Regionaltilt patternsof late Cenozoicbasin-rangefault
blocks, western United States, Geol. Soc. Am. Bull., 91, 460-464,
R. C. Fletcher,Centerfor Tectonophysics,
Texas A&M University,
CollegeStation,TX 77843.
E. M. Parmentier,Departmentof Geological Sciences,Brown University, Providence,R102912.
M. T. Zuber, Geodynamics Branch, Code 621, NASA/Goddard
SpaceFlight Center,Greenbelt,MD 20771.
•980.
Thompson, G.A., and B.B Burke, Regional geophysicsof the Basin
and RangeProvince,Ann. Rev. Earth Planet. Sci., 2,213-238, 1974.
Tullis, J., and R.A. Yund, Experimentaldeformationof dry Westerly
granite,J. Geophys.Res., 82, 5705-5718, 1977.
(ReceivedSeptember20, 1984;
revisedJuly 31, 1985;
acceptedAugust 1, 1985.)