Sep 10, 1983 - western United States is the regular alternation of fault- bound basins .... shaded relief map of the state of Nevada. ... state of flow by a bar, and.
JOURNAL OF GEOPHYSICAL
RESEARCH, VOL. 88, NO. B9, PAGES 7457-7466, SEPTEMBER 10, 1983
Unstable Extension of the Lithosphere' A Mechanical Model for Basin-and-RangeStructure RAYMOND C. FLETCHER1 U.S. Geological Survey
BERNARD HALLET2 Department of Geology and Department of Applied Earth Sciences, Stanford University To investigate the behavior of the lithosphere undergoingextension, we use a simple rheological model broadly consistent with experimental data on rock creep and with the nature of the brittle/ductiletransition.A plastic surfacelayer overliesa substratethat deformsby power law creep with a stressexponent n = 3 and an effective viscosity that decreaseswith depth. In extension this model shows a strong necking instability, provided that the thermal gradient is sufficiently large; otherwise, stable uniform extensionis indicated. The predicted structuresdisplay uniformly spaced necksor regionsof enhancedextension(basins)alternatingwith regionsof reducedextension(ranges). If the depth to the brittle/ductile transition is roughly 10 km, as suggestedby the maximum depth of seismicfaulting, the model yields spacingsfor the incipient Basin and Range structuresof about 25-60
km, in excellentagreementwith dbservation.
INTRODUCTION
A strikingfeature of the Basin and RangeProvinceof the western United States is the regular alternation of faultbound basins and ranges with a northerly trend and a characteristictransversespacingof 20-50 km (Figures 1 and 2). These structuresoccur acrossa region as much as 600 km wide that has undergone10-30% extension,and locally as much as 50-100% extension [Proffett, 1977; Wernicke et al., 1982], in a direction normal to the structural trend in the last
15-17 m.y. [Stewart, 1971; Thompsonand Burke, 1974]. The regular developmentof Basin and Range structure suggests to us a neckinginstability in the extensionof lithosphere.In extension,a stiff layer of power law fluid with large stress exponent, embeddedin a soft medium, undergoesnecking, resultingin uniformly spaceddomainsof enhancedthinning. The mode with the strongestinstability has a wavelength equal to 4 times the layer thickness [Smith, 1977]. If a stiff surface layer deforming by distributed faulting above the brittle/ductile transition has a thickness of about 10 km, as suggested by the maximum depth of seismicfaulting [Bufe et
al., 1977],a comparable neckinginstabilityin the extension of this layer might then lead to structureswith the spacing observedin the Basin and RangeProvince. This quantitative
correlation motivated thepresent studyof thecharacter and conditionsfor unstableextensionof lithosphere. The lithosphere is treated as a "plastic" surface layer supportedby a substratein which the effective viscosity in extensional flow decreases with depth due to increasing temperatures. This treatment is similar to the conceptual model proposed by Stewart [1971]. An analysis of flow instability in lithosphere with this rheologicalstructure is
carried out and the results are applied to the Great Basin Province. In the case of instability, the secondary flow is initiated by a small initial perturbation in topography and eventually disrupts the uniform extension in the surface layer, giving rise to a regular structure. The analysis applies only to the early stage of this process, during which the pattern of structures is established. The parameters controlling the character and strengthof the necking instability are estimatedon the basis of available experimental data on rock rheology and models for the thermal structure of the region [Lachenbruch and Sass, !978]. Tile
MODEL
We consider a region of extending lithosphere whose horizontal extent, measured in hundreds of kilometers, is large relative to the spacing of the structures of interest, measured
in tens of kilometers.
The extension
is assumed to
be approximately uniform over this region to a depth of the order of 100 km, which includes the structures and the associatedsecondary flow.
The existenceof an instability in extension of lithosphere and its character depend on the vertical variation in the rheologicalbehavior, including its dependenceon temperature and pressure, the density distribution, and the rate of extension. As a first approximation, we consider a uniform density. The density contrast at the earth' s surface will act to stabilizethe flow; internal density contraststendingto either stabilizeor destabilizethe flow are ignored. We shall specify the rheology in a manner consistent with present knowledge of rock rheology but in a form rendering the analysis tractable.
The extension is balanced wholly by thinning of the lithosphere,and the structuresthat may develop are approx• Now at Centerfor Tectonophysics, TexasA&M University. imated as cylindrical with axes normal to the direction of 2 Now at Departmentof Geologyand QuaternaryResearch extension, so that the flow as a whole is plane. Center, University of Washington. In the strong, "brittle" surface layer the rock deforms by This paper is not subject to U.S. copyright. Publishedin 1983 by distributedfaulting. The fault spacingis assumedto be much the American Geophysical Union. smaller than the length scale of the structures, so that the layer may be taken to deform as a continuous medium. A Paper number 3B0909. 7457
7458
FLETCHER AND HALLET:
UNSTABLE EXTENSION
.. •' •
OF LITHOSPHERE
. '":.:•,,• :'"'•:-' '•'.:••.: • '•!i•' ..':-•-•'..•.,.'--•:"•:•,!!i•. .:. "•'.' ..."•. •,•,.. -'•
. ..'
'
.•== .'.... .:-•:-" • ,,.•:-••...... •: ..•=..
•: '•'-'
' "=• ..... • ' •...... • .•":' .:?..••'• .... •:-,,.•. . •=,:.' • -•-•:....• ,.. '•: .:,',•',• ..•..-•
.... •'"' •..... ;..•,•....-..• •.?•.::.• ..:•y •• •'•••', . .-•. .•,.:. .•...,.•.... .... •:•..... .....•:•,.:•....•..'•.. ß •,
;"•:¾7.':':,.-:•,: '7" .
-..•..-•. ..,,,¾•.• ...... : ..... ,
•.•:••.•..•.,,...•. .... ß ".,',::,,./.,
.
,
-- ,.%.,=,, ..•. ,,,,
•
".
,,• ,:•-:"•
Fig. 1. Geologicmapof Nevada[Stewartand Carlson,1977].Alluvialandplayadeposits(light)definebasins.The approximatepositionsof the two traversesfrom which the histogramof rangespacing(Figure2) was constructedare shown as solid lines.
FLETCHER
AND HALLET.'
UNSTABLE
EXTENSION
OF LITHOSPHERE
7459
according to
2,1= [BJ:•"-•)/:]-•
MEAN$1 KM
(2)
whereJ2= ¾n(•r•x - O'zz) 2 + o'•,z 2isthesecond invariantof the deviatodhcstress, and B and the power law exponent n are material properties. In general, B is a function of temperature and a weak function of pressure. Ignoring the latter dependence,
B = B* exp (-Q/RT) O' I(
•
•0
40
i
(3)
whereB* is a constant,Q is an activationenergy,R is the gas constant, and T is the absolute temperature. Incompressibility of the material requires that
50
SPA6IN6 BETWEEN RAN6ES (KM)
Fig. 2. Combinedhistogramof spacingbetween rangesfor the two traverses normal to the trend of ranges on the map shown in Figure 1. Spacingstaken from U.S. Geological Survey 1:500,000 shaded relief map of the state of Nevada.
•
+ ezz= 0
(4)
Here it is appropriate to anticipate the analysis. The substrate and the overlying layer are in a basic state of uniform extension at a rate g•x. First consider the ductile substrate. From (1)
reasonabletheological model would be a perfectly plastic solid with pressure-dependentyield stress. However, to make the analysistractable, we model the surfacelayer as a 6• - 6zz = 4•g• (5) power law fluid with very large exponent nl. Like a plastic solid, this matedhalhas the property of accommodating where we denote quantities applying to the basic state of flow by a bar, and variations in strain rate about the mean rate of horizontal extension, gxx, with negligible alteration of the maximum
shearstress fromits meanvalue,1/2(•xx- &zz)y, wherey
2• = B-• •q•,•,(• - n)= B-l/ngxx-(l- l/n)
(6)
where
denotesthe yield value. In addition, we ignore the pressure dependenceof the yield stress, taking the maximum shear = - az) stressto be independentof depth. While a depthdependence Both s-• and g• are positive for extension, so that absolute can be readily introduced, we felt that in this initial study the values need not be taken. simpler model would be more appropriate. The present Now consider a small flow superimposed on the basic approximationis comparablewith Chapple's[1978]useof a perfectly plastic solid to model the deformation in a fold/th- state, which arisesas followsßLet the surfacehave a slightly rust wedge. It is in strong contrast with Bott's [1976] model wavy topographyßTo satisfy boundary conditionsthere, an for continentalrifting, in which he proposedthat extension additionalflow must be superimposedon the uniform extenof the surface layer is accomplishedby slip on discrete, sion in the surface layer. Then an additional flow in the widely spacednormal faults and that any additionaldeforma- substrateis also required to satisfy the boundary conditions at the interface between the surface layer and the substrate. tion in the coherent blocks between faults is elastic. The elasticpropertiesof the layer are not relevant to our analysis In the analysis of this situation we seek results accurate to of its deformation.If a neckinginstabilitydevelops,faulting first order in the topographic slope. As a first step in the will become more localized. For example, the prominent analysis, we determine the constitutive relations satisfiedby faults boundinggrabens may be supposedto separatethe the additional stresses,denoted &•x, &z•, 6•, and the addinecks, in which extension by distributedfaulting is concen- tional strain rates, denoted gx•,,gzz,g•,z.These are obtained by substitutingthe total stressesr&• = 6xx + &•, ß ß ß , and trated, from regions undergoing much smaller amounts of strain rates e• = g•x + g•, ..., into the constitutive deformation. relations (1) and expanding these to first order in the small We assumea sharp brittle/ductile transition below which additional quantities, using the relations already obtained for the lithospheredeformsby steadystatecreep. At the transi-
tion the shearstressrequiredfor faultingis just equalto that requiredto maintain creep at the specifiedstrain rate. As the
the basic flowß We obtain
&•,• = 2•(1/n) g•,x- 19
temperature gradient increases at a fixed strain rate, the bdhttle/ductiletransition moves toward the free surface; as the strain rate increases at a fixed thermal structure, the transition depth increases. In plane strain the ductile substrate is described by the power law
The analysis is tractable when the depth dependenceof the effective viscosity has the form
rrxx= 2 r/e•x - p
•(z) = Toe•
rrzz= 2rlezz- p
(1)
O'xz= 2
where r&•, ß ß ßare the componentsof the stresstensor, •, ß ß ß are the componentsof strain rate tensor, and p is the pressure. The effective viscosity r/is related to the stress
6'zz= 2il(1/n)&z -/>
(7)
6xz =
(8)
where r/0 is the effective viscosity at the brittle-ductile transition and 7 is the inverse of the viscosity decay length. Supposethe linear temperature distribution T(z) = To-
Oz
(9)
holds in the substrate z < 0. Substitution of this into the
7460
FLETCHER AND HALLET:
z
UNSTABLE EXTENSION
NORMAL STRESS CONTINUOUS SHEAR STRESS ZERO
OF LITHOSPHERE
typical componentin the topographyhas the form z-
tøO
H = •(x) = • cos (hi)
(18)
with amplitude • and wavelength L = 2rr/h; H is the thicknessof the brittle surfacelayer. At the uppersurfacez - H = [, the shear stress vanishes
[%•1• = o p,•,n
and the normal
stress is
[cr•]• = P0g-½/cos (hi)
NORMALSTRESS,SHEARSTRESS AND VELOCITY CONTINUOUS
Fig. 3. Geometry and boundary conditions. The plastic surface layer of thicknessH is shown. In the present first-order analysisthe boundary conditions are applied at the wavy free surface, with amplitude •, and at the horizontal mean interface between the surface layer and the substrate. The deformed shape of this initially horizontal surface has amplitude •'.
(19)
where tz and •, are coordinateslocally normal and tangentto the interface. In the present applicationthe density P0 is
zero; p0 = 1 g/cm3 wouldcorrespond to extensionin a submarineenvironment. Becausethe stressdifference, •xx - •zz, and the density are continuous at the brittle/ductile transition, the boundary conditionsat the interface between
the layer and the substratemay be applied at the plane surfacez = 0. The boundary conditionsthere are
expression (6) using (3), yields
O(z)= [2(B*)Ungxx (• - l/n)]-Iexp[Q/nR(To- Oz)]
(10)
•(•)(x, o) = t•(x,o)
This will be approximated by the form (8) by defining values of r/0 and y in (8), which will, at the brittle/ductile boundary, give values of the viscosity and its rate of change with depth equal to those calculated from (10). A fit of •(0) and [dO/dz]z=O from (10) to the approximate form (8) yields
(20)
axz(•(x,o) = axz(X, o) C•zz(•)(x, o) = •zz(X,o) wheret• and ½ are the horizontalandverticalcomponentsof
rio= [2(B*)l/"gxx(l - l/,0]-I exp(Q/nRTo)
Y= Q/nRTø 2
a;(•(x,0) = a;(x,0)
the velocity. Figure 3 shows the deformed material surface
(11) initially coincidentwith z = 0; its amplitudeis •'.
This approximation decreasesmore rapidly with depth below the brittle/ductile transition than the viscosity defined by (10). In the approximation of the surface layer by a power law fluid with large exponent r/l, (5) definesthe effective viscosity in terms of the specifieduniform strain rate and the yield stress
The solutionfor the perturbingflow is further developedin the appendix. NATURE
OF THE NECKING
INSTABILITY
In the case of strong necking instability, the perturbing flowin the brittle layer hasthe form shownin Figure4a. This doesnot includethe mean, extendingflow, and the absolute scaleof the motion is arbitrary. The deformation involved is
2ry= (•xx(1)- t•zz(l))y -- 4Olgx.r
(12) more readily seen by connecting the ends of the velocity
where the index 1 denotes the surface layer. To first order, the maximum
shear stress is
X•22= ry[1 + (C•xx (1)-- C?zz(l))/2ry] (13) CFxz (l) = 2•/iExz (l)
(14)
Thus
J2(1): ty2
the mean extension has not been included here. If this were
added, the net result would be a much reduced extension in, the range and an enhanced extension in the basin. The
In the limit of very large nl, (7) yields
O'xx (•) = 6'zz (l) = -/5(l)
vectorsto obtain the deformed,initially squaregrid shownin Figure 4b. Thinning of the layer occurs in the "basin" region;the thickeningin the "range" is only apparent,since
deformedgrid showsthat the regionof enhancedor reduced extensionis restrictedto roughlythe upperhalf of the brittle layer. The lower half of the layer bends upward under the
(15)
is unchangedby the introduction of the perturbing flow. At the brittle/ductile
transition
we have
ry = 2r/0g.•
(16)
ry = r* exp (Q/nRTo)
(17)
and by use of (11),
wherer* = (gxx/B*) l/". The boundary value problem for the perturbing flow is illustrated in Figure 3. We have linearized the constitutive relations, and consistentwith this, we linearize the boundary conditions. As a consequence,the problem reduces to the treatment of each Fourier component in the topography and the associated, linearly independent, secondary flow. A
Fig. 4a.
The perturbing flow in the plastic layer over a half
wavelength for thecasen• - 104, n -- 3, •,H = 10,S = 0, andL/H = 4. The magnitudeis arbitrary. Note that the vertical velocity is much larger at the base of the "plastic" layer, even though the present case correspondsto a weightless fluid (S = 0).
FLETCHER AND HALLET: UNSTABLE EXTENSION OF LITHOSPHERE
basin. The large structuralrelief at depth, both within the layer and in the underlying substrate, is a characteristic feature of this kind of instability. Necking instabilityis indicatedby a growth of the topographic amplitude a/, where
d•l/dt = -gxx.4 + •,(•) (0, H) = (-1 + q)gxx at
7461
i
200
-
ioo
(21)
where q = a,(l)(0, H)/gxx•l, evaluatedfrom the solution obtainedin the appendix, is a function of the dimensionless wave numberk - 2•r/(L/H) and of the four parametersn, n•,
o
:
yH, and
S = (p- po)gH/2ry
(22)
For instability,q > 1. In general,q(k) will be positiveover some range in wave number k for only certain ranges of values of n, n•, 7H, and $. An example exhibiting strong instabilityis shownin Figure 5; this correspondsto n• = 10,000,n = 3, 7H = 10, and S = 3. In general,whenq > 0, it is so only for the small range in wave number 1.5 •> k >• 3, corresponding to 2 •> L/H •> 4. The maximumvalueof q, qa, occurs at the dominant wavelengthLa. Althoughthe limit n• -• • bestrepresentsa plasticsurface layer, we treatedthe generalcaseof finite n•. It is of interest to seewherethe limitingbehaviorsetsin. Figure 6 showsthe variationof qawith n• for the casen = 5, 7H = 3, and S = 0. Somewhatsurprisinglythe limiting behavior doesnot set in until about n• = 1000. We might have expectedlimiting behaviorat n• -• 10-20, say, but at thesevaluesthe strength of the instabilityis negligible.Comparablebehavioris shown in the variationof the dominantwavenumberka.In computing resultsfor applicationto Basinand Range structure,we used ni = 10,000. The choice of n for crust and mantle rocks is restricted to a
relatively smallrange. Experimentalstudiesfor monomineralic rocks yield valuesfrom 1.7 to 9.1 [Carter, 1976].On the basisof later argument,the higher values are excluded, and the representativerange is reducedto n -• 2-5. Taking n and ni as essentiallyfixed, the dependenceis
reducedto the two parameters$ and 7H. Large 7H, which we shall see reflects large activation energy Q and small
-ioo
_2oo f i i Fig. 5. Growth factor q as a function of wave number k for n•
-- 104, n = 3, 3,H= 10, and$ = 3. the same way in Figure 7b. Note the small variation of this quantity in the region shown. APPLICATION
OF THE MODEL
TO BASIN AND RANGE
STRUCTURE
If the initiation of Basin and Range structure is to be attributed to a necking instability of the kind considered here, model parametersn, 7H, S, and H chosenon the basis of experimentaldata on rheology and strengthand estimates of extension rate and thermal regime must correspond to a strong necking instability and a dominant wavelength La in the range of the observed spacingof structures. Spacing of structuresranges from 20 to $0 km (Figure 2), with a mean value of about 30 km. Provisionally estimating the thicknessof the brittle layer H from the maximum depth of seismicfaulting in the Great Basin Province, H -• 10-15 km and using the ratio La/H = 3.4-4 obtained from the model, La -• 30-60 km, in good agreementwith the observations. However, the depth to the brittle/ductile transitioncan also be estimated from the data mentioned above; this
stressexponentn, favors instability. Small $, corresponding provides a more stringent test of the model. to large yield stress and large temperature gradient, also A minimum value of the quantity q can be obtained by favors instability. requiring that the necking instability be sufficiently strong to Contoursof qa are plotted in 3,H, S spacefor n• = 10,000 produce the observed structure after an extension of not andn = 3 (Figure7a). The neckinginstabilityrequires3,H> $, the locusqa = 0 beingapproximatelygiven by 3,H = $. Contoursqa = constare roughlystraightlinesparallelto the line 7H = $ and translatedalongthe 7H axis. Contoursof -2.6 the dominantwavelength/thickness ratio LdH are plottedin I
I
I
I
kd k
010oi0• I0z I0• 10 4 Fig. 4b. The ends of the velocity vectors in Figure 4a are connectedto give a deformed, initially square grid over a full wavelength. The mean extension is omitted here, so that the "ranges" show an apparentthickening.
I
I05
IOs
I0•
nI Fig. 6. Variation of qa and ka with the surfacelayer power law exponent n• for the case n = 5, 3,H = 10, and $ = 3. The limiting
behaviorfor n• --> o•is not reacheduntil n• • 103.
7462
FLETCHER AND HALLET: UNSTABLE EXTENSION OF LITHOSPHERE
14
where Ts is the surface temperature, (11) leads to
yH = (QInR)(To- T,)ITo 2
12
(26)
Combining the present-day temperature gradient in the Basin and Range Province, about 26øK/km as estimated from the modelspresentedby Lachenbruch and $ass [1978], and the provisional estimate of a 10-15 km depth to the brittle/ ductile transition yields a transition temperature in the range
I0
$
To• 550ø-700øK. The function(To- TO/To 2, with a surface temperature T, - 300øK, varies over the narrow range 0.82-
0.83 X 10-4 øK-1as Tovariesbetween550øKand700øKand overthesomewhat widerrange0.74-0.83x 10-4 øm-1asTo O0 2
4
6
8
I0
varies between 450øK and 900øK. Thus, over either of these ranges,the function can be taken as essentiallyconstantand,
12
?,'H
Fig. 7a. Contoursof qdplottedin 3•H,S - spacefor the casen• =
usinga valueof 0.8 x 10-4 øK-l, we canwrite
104, n = 3. The straight-line contourqd = 0 is only approximate.
yH = O.096Q/n
(27)
whereQ/nis expressed in unitsof J mol-•. The parameter $, (22), can be written
much greater than 10%, a lower limit for the average extensionin the province. An amplitude a/0 = 10-100 m for the initial perturbationin topographywith a wavelengthof about 30 km would seem not unreasonable. Well-developed Basin and Range structures with amplitude a/= 1 km, or a structural relief of twice this amount, would then require amplificationsa//a/0 = 10-100. The relation betweenamplification and extensional strain is obtained by integrating (21). The approximationobtained when the wavelengthis taken as constantis adequate, and we obtain
S = 13.5(H/ry)
(28)
wherewe have set p = 2700kg m-3 and p0 = 0; H is expressed in km, and ry is expressedin MPa. In the model the yieldstressryis takento be uniformin the layer. Sincery
(23)
enters the parameter $ in the computation through a stress boundary condition at the upper surface, it is appropriateto estimateit here as the yield stressnear the surface. Because of the scale of the motion considered, a value typical of the upper 1 or 2 km is appropriate.We use a value obtainedfrom estimatesof coherent rock strengthby Zoback and Zoback
For an uniform extension of 10%, gxxt- 0.1, and the desired amplification of 10-100 requires
from 9 MPa to as much as 43 MPa and for a depth of 1 km, adding9 MPa to these values, from 18 MPa to as much as 52
In (-•//-•/0) = (qd -- 1)gxxt
MPa. We shallusethe largestadmissiblevalueandtake ry =
qd • 25-50
An adequate strength of instability will be taken to correspond to qd •> 40
[1980].Valuesof ryappropriate for rockat the surfacerange
(24)
This value yieldsan amplificationof 35 for a 10% extension. This constraintis apt to be conservativebecausethe model neglects two effects that would further favor necking, a contributionto the flow, which arisesas a particular solution of the equations of motion if the isotherms in the substrate are treated as nonplanar, and the possiblestrain-softening behavior of the upper layer [Neurath and Smith, 1982]. To model the extendinglithosphere,we oughtto consider a broad range of experimentally determined flow laws. Because these flow laws are generally only available for monomineralicmaterials, we presume that the behavior of any rock lies within the range of behavior of its components.
50 MPa. Estimation of H from (25) requires an estimate for the transition temperature To. From (17),
To= (QInR)/ln(ryl'r*)
(29)
Since (17) and (29) refer to the brittle/ductile transition, it
maybe moreappropriateto usea valuefor rylargerthanthat used to evaluate
S. On the basis of the Mohr-Coulomb
failurecriterion,ry in (29) mightbe taken to be about 100 MPa larger than the near-surfacevalue, assuminga depth to the transition of roughly 10 km. On the other hand, the value of In (rv/r*) is dominatedby the contributionfrom r*.
Some of the materials for which flow laws are available are
not likely candidatesfor the lithosphere, but they are included to broadenthe range of behavior considered.The estimation of yield stressin faultingis carried out independentlyfor a strongrock, for example,granite, undercrustalconditions rather than for the separate rocks for which creep data are
4
,, '
3.6
available.
We now evaluate the parametersyH and $ which, together with n, determine the values of qd and Ld/H. Assuminga uniform thermal gradient in the lithosphere and using H = (To-
Ts)/O
(25)
0
0
2
4
6
8
I0
12
14
¾H
Fig. 7b.
Contours of La/H for the same case. Notice the limited range of this quantity.
FLETCHER AND HALLET'. UNSTABLE EXTENSION OF LITHOSPHERE
TABLE
1.
Model
7463
Fit
B*, (MPa -n
Material Limes tone*
s-•) 199
To,øK
5-I
S
H, km
Ld, km
1.7
n
Q, J mol-• 210
-In (•*/100) 31.0
490
12.3 3.0 3.7 3.9 8.0 8.5 16.3 11.6 10.3 11.6 11.7 8.5 5.7 10.4 6.8 12.3 6.6 7.3 5.0
2.0 1.7 5.4 4.9 2.4 3.3 8.7 5.3 3.5 5.3 4.9 3.9 4.2 6.1 3.7 5.1 6.4 4.3 2.6
7.3 6.2 20.1 18.2 8.8 12.3 32.2 19.8 12.9 19.8 18.1 14.5 15.5 22.7 13.8 19.0 23.7 16.0 9.7
25 24 m -32 45 123 95 47 74 67 54 58 86 52 70 95 61 36
Marble? Dolomite?
3.47 X 10-3 5.04 x 10-•2
8.3 9.1
260 350
8.1 5.5
462 823
Quartzite,dry? Quartzite,naturalH20,
9.35 x 10-• 2.7 X 10-3
6.5 2.0
270 170
6.4 18.9
773 528
2.6 3.2 3.3 2.1 3.2 2.4 3.0
230 540 460 230 390 290 270
17.1 17.8 17.6 20.2 17.8 18.9 15.7
619 1140 814 635 814 771 678
3.95 3.95
230 430
10.1 14.7
704 889
2.55 X 10-4 47.1
3.2 2.6
230 330
12.8 19.4
659 794
Quartzite, wet? Dunitc, dry? Dunitc, dry? Dunitc, wet•' Dunitc, wet? Enstatolite, dry? Enstatolite, wet•'
Albite, naturalH:O, Albite, driers Anorthosite$ Clinopyroxenite$
0.127 2480 4870 0.177 2480 0.838 0.300
2.37 x 10-6 1.88 X 10-4
Clinopyroxeniteõ
2.12 x 10-3
6.4
440
9.0
916
Diabase$
9.24 X 10-4
3.4
260
12.7
717
Aplite$
1.37 X 10-6
3.1
160
11.4
552
*Schmid et al. [1977]. ?Compilationof Carter [1976]. ,G. Sheldon (Brown University, personal communication, 1980). õS. Kirby and A. Kronenberg(unpublishedmanuscript, 1983).
The stress 'r* is a weak function of the extension rate Extension in the Basin and Range Province has amounted to perhaps 10-30% on average [Thompson and Burke, 1974] and perhapslocally up to 50-100% [Proffett, 1977]in the last 17 m.y. These values correspondto average extension rates of 0.2 x 10-15 s-1 to 1.4 x 10-15 s-1. We shall use a value of
Models derived from the creep propertiesof 11 materials: dry and wet dunite (DD', DD", DW', DW"), dry albite (ABD), the first clinopyroxenelisted in Table 1 (CPX'), dry and wet enstatolite(ED, EW), a quartzite with natural water content (QN) and a wet quartzite (QW), and limestone (LS), all lie in the region of strong instability, to the right of the 10-15s-1. Combining thiswithvaluesofB* givenin Table1, contour qa = 40. In addition, the models derived from -In ('r*/100), with 'r* expressedin MPa, rangesfrom about 5 anorthosite (AN) and diabase (DI) are sufficiently close to to 20, with valuesbetween 10 and 20 being typical. The effect this region to be included in the group showingan adequate of increasingthe extension rate by an order of magnitude, strengthof instability. Six models, four of which are derived taking n = 3 as typical, is to increaseIn ('r*/100) by In (10)/3 from materials with creep mechanisms characterized by -• 0.7. Taking ry as 100 MPa, the valuesof the transition large n (marble (MB), dolomite (DOL), dry quartzite (QD), temperature recorded in the next column of Table 1 are and a clinopyroxene (CPX"), together with an albite with obtained. natural water content (ABN) and aplite (APL)), do not A changein the value of ry used changesTo by the satisfy this criterion. Contours of La in Figure 8 show that of the group of 13 fractionalamountATo/To= -In ry/(lnry -- In 'r*). A change in •v by a factorof 2 to 50 or 200MPa increasesor decreases models satisfying the strength of instability criterion, only To by a fraction of the absolute value between 0.03 and 0.06 seven (diabase, anorthosite, wet enstatolite, wet quartzite, for most of the materials and from 0.07 to 0.11 for materials quartzite with natural water content, a wet dunitc, and limestone)have values30 • La • 60 km, comparableto the with larger stress exponents' clinopyroxenite, marble, dry quartzite, and dolomite. The effect on To of an increase or decreasein gx•by a factor of 10 is a decreaseor increaseby the fractional amount -(l/n) In (10)/[In ('r*/100) + In (10)/hi with absolute values ranging from 0.03 to 0.06. Thus the value of the transition temperaturedoes not vary markedly
for variationsin 1-yand g• withinthe ranges50 MPa -< 1-y 200MPa and 10-16 S-1 --
