TECTONICS, VOL. 11, NO. 2, PAGES303-315,APRIL 1992
ISOSTATIC
TECTONIC
VISCOUS
LAYERED
this unloadingprovidesimportant constraintson the
REBOUND
DUE
TO
DENUDATION:
A
FLOW
MODEL
OF A
LITHOSPHERE
mechanicalbehavior of continental lithosphereduring
extension.
Spencer[1982,1984]andHowardet al. [1982a]
ShimonWdowinskix and Gary J. Axen
Department of Earth and Planetary Sciences,Harvard
University, Cambridge, Massachusetts
Abstract. A four-layer model of the upper 150 km of
the Earth is used to calculate the viscousresponseof
continentalcrust and the underlyingmantle to tectonic
denudation. The model comprisesa strong upper crustal
layer, a weak lower crustal layer, a very strong mantle
lithospherelayer, and a weak mantle asthenospherelayer,
which is in accord with experimental constraints on
strength-depth profilesfor continental lithosphere. The
strength of each layer is representedby its effective
viscosity.Flow in the crust and mantle is driven by
proposedthat isostaticreboundand domingof
Cordilleran metamorphic core complexesoccurreddue to
tectonic denudationof the crystallinecoresby
detachmentfaults that boundthem. Spencer[1982]
suggestedthat the large ratio of amplitude to diameter
(~ 0.1) of the uplifted domesindicatesnonelastic
behavior. By assumingAiry compensation,Spencer
[1984]modeledupliftsresultingfrom variousstrain
distributionsin the upper plate of detachmentfaults,
reproducingobservedgeometriesof metamorphiccore
complexes
in the southernBasinand Rangeprovince.
Buck[1988]modeledthe casein whichcrustalstrength
buoyancyforces,which arisefrom the unloadingof an
allochthonalong a detachmentfault by a seriesof
is boundedby a simple elastic-plasticfailure envelope
and extensionof an elastic-plasticcrustal layer abovean
inviscidlayer is accommodated
by steepnormalfaulting
of the upper layer coupledwith rapid flow of the inviscid
layer. He showed,for areasof high crustal curvature
where bending stressesin the elastic-plasticlayer exceed
the strength of crustal rocks, that the effectiveelastic
instantaneous
displacements
(earthquakes
or rapid creep
events).Numericalsolutions,obtainedby usinga finite
thickness of the crust decreases to a fraction of a
kilometer from the 5 km initial thickness assumed.
element technique, predict footwall uplift, Moho
deflection,and surfacetopographythat are consistent
with observationsfrom the Basin and Range provinceof
modeledflexure due to extensionof an elasticplate
the western United
States.
The calculated
curvature
of
the footwall uplift is also similar to that observedand is
sensitiveto the geometry of the detachmentfault. Such
bendingneed not be elasticallycontrolled;hencethe
curvaturesof footwall domesdo not clearly place limits
on the effectiveelastic thicknessof the extending crust.
The upward deflection of the Moho and the surface
topographyare sensitiveto the viscositystructure and
enableus to bound the range of the variousviscosities.
By matching observationsfrom the Basin and Range
province, which indicate no Moho deflectionand low
magnitudeof surfacetopography(_•3-5 km), we estimate
the upper crustal, lower crustal, and mantle lithospheric
viscosities
in the ranges102x-1023
Pa s, 10x9-102•Pa s,
and 10•-10 •3 Pa s, respectively.
INTRODUCTION
Movementon gently dipping normal faults
(detachments)
is an importantmeansof accommodating
continentalcrustalextension[Anderson,1971;
Armstrong, 1972; Wernicke, 1981; Wernickeand
Burchfiel,1982].Suchfaults commonlyunroofmidcrustal
levels,and the resultingbuoyancyforcesin the
lithospherecauseisostatic adjustment by uplift of the
footwall(Figure1). The response
of the lithosphere
to
•Nowat Scripps
Institution
of Oceanography,
La
Jolla, California.
Blockand Royden[1990]and King and Ellis [1990]
overlyingan inviscidfluid. Blockand Royden[1990]
consideredthe caseof detachmentfaulting and lateral
removalof upper crustal material, and King and Ellis
[1990]modeledsteepnormalfaultingof the elasticlayer.
Both concludedthat for elastic parameters
experimentally derived for upper crustal rocksthe
effectiveelastic thicknessof the upper crust must be in
the range 0.5-4 km in order to producecurvatures
similarto thoseobservedin the Basinand Range
province. To keep the upper crust in the elastic
deformation
field,King and Ellis [1990]usedan effective
Young'smodulusthat is lower than experimentally
determined valuesby a factor of ~60.
Wernickeand Axen [1988]proposedcombinedelastic
and nonelasticbehaviorof the lithosphereduring
extension,arguingfor significantelasticstrengthin the
extendinglithosphere,which supportslocally
uncompensatedtopographicloadsof the Basin and
Rangeprovince[e.g.,Eatonet al., 1978].They suggested
that a maximum topographicload is reached,after which
nonelasticresponsesdominate the deformation.
Blockand Royden[1990]andWernicke[1990]
consideredthe elevationsof metamorphiccore complexes
relativeto their unextendedsurroundings.Both
concludedthat if corecomplexesare locally isostatically
supportedthen the compensatingfluid is of crustal
densityand not of mantle density.
The observations
and modelsdiscussed
abovesuggest
that isostaticreboundof tectonicallydenudedterrains
proceedssteadily throughnonelasticprocesses
over
geologically
measurable
times() 10,000years),oncethe
elasticstrengthof the extendinglithosphereis exceeded.
Ductileshearzones(mylonitezones),tensto thousands
Copyright1992by the AmericanGeophysical
Union.
Paper number 91TC02341.
of metersthick are ubiquitousin metamorphiccore
complexes.These shearzonesrecordsignificantductile
0278-7407/92/91TC-02341$10.00
flow in the footwalls of detachments.
Such nonelastic
304
Wdowinski and Axen: Denudationand Isotasy--A Flow Model
Beaver Dam Mts., UT
a.
Present
day'
breakaway
CastleCliff Detachment
WSW
/
I
o
I
-
b. Whipple Mts., CA
//
To unrotated
Colorado
Plateau
ENE
'
•
.
o
_.
stranaea
km
o
allochthon
trace of
Whipple
detachment
metamorphic
I
i ection
Restored: .......
.....
•
__ --?
/./
..... .
ISot:•hd:nd//
3•o
./,,.
•z•z!kraal
• N
T
.
....
...
..-
To breakaway
5 km
//
c. Chemehuevi Mts., CA
stranded
,,..,,
metamorphic
core > stranded
..•__
allochthon_
.•.--To
breakaway
.•._allochthon__•
Chemehuevi detachment
WSW
fault
km%TS•,pC .-'•' -----. 5g
p•
0
Mohave Wash
To unrotated
allochthon
ENE
Tsvp•grn
TsvKgTsvTsvp(:!km
Kg
5
km
Fig. 1. Geometriesof flexuresand domesrelatedto tectonicdenudation.(a) Present-dayand
restoredcrosssectionthroughthe breakawayzoneof the Castle Cliff detachmentin the Beaver
Dam Mountains,Utah, at the westedgeof the ColoradoPlateau,showingthe flexurethere [after
Wernickeand Axen, 1988].The sectionis parallelto transportdirection.The westernmost
exposed
footwall(at pointA) restores
to a paleodepth
of 7 km (A'), indicatingtectonic
redistribution of at least that thicknessof upper crust. About 200 of rotation of the detachment
hasoccurreddueto postdenudation
uplift. (b) Structurecontourmap of the Whipple
detachment
fault (a metamorphiccorecomplexdome)in the WhippleMountains,California
[afterFrost,1981].Contoursarelabeledin hundreds
of feet. Thereis about1.2km of relief(4000
feet) on the domeacross
roughly20 km of exposure
of the core.(c) Crosssectionof the
Chemehuevidetachmentsystemin the ChemehueviMountains,California[after John,1987].
Sectionis parallel to the transport direction. The detachmentis domedabout I km across15 km
of exposure,but hasbeenupliftedmorethan that amount.Upendedblocks("stranded
allochthon")left behindby the migratingallochthonare up to 13 km thick measured
perpendicular
to the Tertiary strata [Howardet al., 1982b].This is a minimuminitial depthto
the detachment. Both Chemehueviand Whipple detachmentfaults are rooted to the northeast,
beneath the unrotated Hualapai Mountains and adjacentrangesthat mergewith the Colorado
Plateau. ThereforeFigureslb and lc representonly the top of muchlarger domicalfault surfaces
[e.g.,Howardet al., 1987,Figure3]. Units are Ts¾,Tertiarysediments
and volcanicrocks;Mz,
Mesozoicstrata; Kg, Creteceous
intrusiverocks;Pz, Paleozoicstrata; pC,Precambriancrystaline
rocks;and pCrn,mylonitized Precambriancrystallinerocks.
o
Wdowinskiand Axen: Denudationand Isotasy--A Flow Model
processesapparently dominate much of the deformation
history of metamorphic core complexes.Here we
investigatea nonelasticresponseof the lithosphereto
tectonic denudation above a detachment fault by
modeling the upper 150 km of the Earth as four layers of
different viscosities.Each layer in the model corresponds
to a thermally and/or compositionally
definedlayer in
305
breakaway
I ,- allochthon
•d•m'•-Jupper
crus[
•
lower
crus[
\ Moho
mantle
lithosphere
the Earth.
THE
MODEL
mantle
Areas that have experiencedtectonic denudation,such
as Cordilleran metamorphic core complexes,show
significantextensionof upper crustal rocks above the
detachmentsurface. Controversyover the distribution of
extension in the lower crust and mantle lithosphere
centerson whether upper crustal extensionis
accommodatedby extension of the lower crust and
mantlelithospheredirectlybelowcorecomplexes("in
situ") [e.g.,Miller et al. 1983],or whetherlow-angle
normal faults connectregionsof upper crustal extension
with regionsof lower crustal or mantle lithospheric
extensionthat are laterallyremoved[e.g.,Wernicke,
1985;Jones,1987].In situ mantlelithosphericextension
results in net subsidence of the area that would be a
metamorphic core complex, in stark contrast to the
observationof relatively high-standingcore complexes
domes[e.g.,Blockand Royden,1990].We assumethat
no in situ extensionoccursin deeper levels of the
lithosphere;the crust and mantle respondonly to
unloading due to lateral removalof an allochthonin the
upper crust.
The above assumptionreducesthe denudationproblem
to a relaxation problem, but not to a simple one, because
the durationof theunloading
(106-107years)coincides
with the timescaleof lithospheric
relaxation(10•-106
years).As a result,the relaxationproblemmustbe
treated within the framework of the unloadingprocess.
We approachit by assumingthat unloadingoccursby a
seriesof instantaneoussmall displacementsof the
allochthon that mimic deformation by earthquakesor
rapid creepeventsin the brittle upper crust and that
relaxation occursbetween displacementevents. The short
timeintervals(103-104years)between
the displacement
events do not allow the crust and the mantle
to reach a
full relaxation during the duration of the unloading.
We considera region of continentalcrust overlying
mantle, with upper crustal extensionaccommodatedby
displacementof an allochthonalong a detachmentfault
(Figure2). The allochthonis assumed
to move
horizontally by a seriesof small instantaneous
displacements.Geologicalstudiesindicate that
allochthon displacementscan reach a horizontal distance
on the orderof 50 km in a time scaleof 2-10 m.y. [e.g.,
Reynolds and Spencer, 1985; Davis and Lister, 1988;
Wernickeet al., 1989]. Detachmentfaults apparentlydip
gently(0ø-30ø) in the middlecrust,as evidenced
by
seismicprofiles[e.g.,Allmendingeret al., 1986]and the
lack of large metamorphicgradientsfor tens of kilometers
in the direction of transport in the footwall of
detachments
[Crittendenet al., 1980].In the uppercrust
the faults may steepentoward the breakawayfault
[Howardet al., 1982a]or be gentlydippingat depthsof
asthenosphere
I
I
<--
250
km
Fig. 2. Schematicdiagram of the model showinga
three-layer lithosphere comprisedof upper crust, lower
crust, and mantle lithosphere,overlyingthe
asthenosphere,
with the upper crust extendingby
tectonicdenudation.The allochthon(longdashedlines)
movesfrom the breakawayat a constantvelocity along
the detachmentfault. By assuminga laterally uniform
thermal gradient, the transitionsbetweenthe upper and
lowercrust(dottedline), and betweenthe mantle
lithosphereand asthenosphere
(shortdashedline), which
are temperature dependent,becomedepth dependentas
well.
The
transition
between
the lower crust and the
mantlelithosphere(Moho) is a compositional
boundary.
only a few kilometers[Wernickeet al., 1985, 1989;Axen
et al., 1990].In the model,we considersimple
detachmentfault geometries,which are characterizedby
two parameters:the allochthonthickness
(h0) and the
detachment
ramp angle(0), whichis the dip of the
segmentof the detachmentbetweenthe surface(the
breakaway)and the horizontalmidcrustalpart
(Figure3).
Crustal responseto tectonic denudationextends over a
regionthat is wider than the denudedterrain. However,
at a sufficientdistancefrom the breakaway
(100-150km)
the crust and the mantle are assumednot to respondto
the unloading. Reflectionseismicprofilesfrom the
northernBasinand Rangeprovince(COCORP 40øN
seismic-reflection
transect)showthat the Mohois fiat
acrossregionswith large gradientsof upper crustal strain
[Klemperer
et al., 1986;Hauseret al., 1987],suggesting
that the flow occursmostly in the middle to lower crust.
Thus we assumethat the responseto denudationdecays
with depth, and at a sufficientdepth, 150 km or more,
the mantle asthenosphere
doesnot deformsignificantly.
The amount of deformation
within the crust and
mantledependson their strength.We usea four-layer
modelcomprisinga strongupper crustallayer, a weak
lowercrustallayer, a very strongmantle lithosphere
layer, and a weak mantle asthenosphere
layer, whichis in
accordwith experimentalconstraintson strength-depth
profilesfor continentallithosphere[e.g.,Braceand
Kohlstedt,1980].Alternationbetweenweakand strong
layers tends to concentrate the deformation within the
weak layers, especiallywithin the lower crust, becauseof
its proximity to the denudedregion. By assuminga
laterally uniform geothermalgradient, the transitions
306
Wdowinski and Axen: Denudationand Isotasy--A Flow Model
u= = 0
•,, = p.gh(x,t)
/
traction
P•
/
/
no•/
free
•
P•
/
/
•
/
/
slip//
/
/
/
/
no
/
• slip
P•
/
•
/
/
/
/'////////////////////////
/
no
slip
Fig.3. Schematic
diagram
of[hemodelshowing
[heparameters
andboundary
conditions
of•he
flowwi[hin [he denuded
li[hosphere.
Parameters
are•c, uppercrustalviscosity;
•, lower
crus[alviscosi[y;
•m•,manfieli[hospheric
viscosi[y;
•m•,manfieas•henospheric
viscosi[y;
p•,
c•us[aldensi[y;
pro,manfiedensi[y;
d, alloch[hon
displacement;
u0,denudation
veloci[y;
•,
normalsiress;
h0,alloch[hon
•hickness;
O,rampangle;g, •heacceleration
due•o gravi[y;andUx,
horizon•a•
vdoci•y.The alloch•hon
•hickness
(h(x,•)) a• anygiven•ime(•) andhorizontal
position
(x) is calculated
by•sumingcons[an•
horizontal
displacemen•
ra•eof[healloch[hon
(d= u0•)andconservation
of•heanoch•hon
m•s ½he[wodolled
•%ions
a•eequa•
ina•ea).
betweenthe upperandlowercrust(dottedline in Figure
2), and betweenthe mantlelithosphere
and
asthenosphere
(shortdashedline in Figure2), whichare
(higher
flowvelocities)
andforGr• cx>
(very
low-viscosity),
the deformation
is veryfast.The
characteristic
parameters
(Table1) yieldvaluesof the
temperature dependent,becomedepth dependentas well.
Grashofnumber in the range 1-1000.
The transition
between the lower crust and the mantle
lithosphere(Moho)is a compositional
boundary.
Parameter Ranges
Mathematical
There are four types of parametersthat are neededto
be specifieda priori: the thicknesses,
the viscosities,
and
the densitiesof the the four layers(uppercrust,lower
crust,mantlelithosphere,
andmantleasthenosphere)
and
the allochthon
geometry(Figure3). In orderto reduce
the numberof free parameters,we chooseto keepsomeof
them constantthroughoutall calculations.Our free
parametersare the viscositystructureof the fourlayers
and the allochthongeometry.The upper and lowercrust
are assumedto be 20 km thick, the mantle lithosphereis
assumedto be 40 km thick, and the portion of the
mantle asthenosphere
that is considered
in the modelis
70 km thick(Table2; noticethat the paremeters
in
Table 2 are nondimensionilized).The allochthon
Formulation
The continental
crust and the mantle are assumed to
behaveover long time intervalsas incompressible
viscous
fluids. As a first approximation,we neglect along-strike
variations and use two-dimensionalvertical plane strain
calculations. We use the plane strain formulation of
Wdowinskiand O'Connell[1990],whichaccounts
for
viscousand buoyancyforces. The governingequationsare
basicallythe Navier-Stokes
equations(withoutinertial
terms)that dependon the dimensionless
Grashofnumber
Gr= gpøxø2
r/ouo
(1)
whereg is the accelerationdue to gravity, and x0, u0, r/0,
and p0 are the characteristiclength,velocity,viscosity,
and density,respectively.The Grashofnumber is the
ratio of buoyancyforcesto viscousforcesand determines
the ability of a viscousfluid to respondto buoyancy
TABLE 1. Values of the Characteristic Parameters That
Are Used to Evaluate the Grashof Number in the
Calculations
forces[Turner,1973].It hasa similareffectto that of the
Argandnumber[Englandand McKenzie,1982]in thin
viscoussheet calculations.In this study, we keep g, x0,
u0, and p0 constant,and as a result Gr is a functiononly
of the characteristic
viscosity(r/0). As Gr-• 0,
correspondingto very high characteristicviscosity,the
viscous
fluid deformsveryslowly(lowvelocities)in
responseto a given buoyancyforce. As the Grashof
number increases,correspondingto lower viscosity,the
deformation due to the same buoyancyforce is faster
Parameter
g
gravitational
acceleration
x0
characteristiclength
u0
characteristic
velocity
p0
r/0
characteristic
density
characteristic
viscosity
Gr
Grashof number
Value
10m s-2
100 km
10mm yr-•
3220kgm-a
102•-1024
Pa s
0.1-1000
Wdowinski and Axen: Denudationand Isotasy--A Flow Model
TABLE
2.
Values of the Dimensionless Parameters Used
in the Calculations.
Sue
Sic
Sml
h0
0
r/u½
Parameter
Value
upper crust thickness
lower crustal thickness
mantle lithosphere thickness
allochthon thickness
detachmentramp angle
upper crust viscosity
0.2
0.2
0.4
0.15
15ø-450
1
r/l½
lowercrustviscosity
r/ml
mantle lithosphereviscosity
r/ma
mantleasthenosphere
viscosity
pc
Pm
crustal density
mantle density
1-10-3
1-10
1-10-3
0.8
1.0
307
Buoyancyforcesare dominated by the upper surface
topography,which changesduring the allochthon
displacement.
Hencesmalldensityvariations(< 5%)
lower in the crust are negligibleand are omitted from the
model.
Boundary Conditions
In this study, we investigatethe crustal and mantle
responseto tectonic denudationcausedby extensionof
the upper crust; the lower crust and the mantle are
assumedto deform only in responseto the unloading in
the upper crust. In addition, we assumethat tectonic
denudationis a local phenomenonand that at a sufficient
distancefrom the breakaway(100-150km), the crustand
the mantle do not respond to the unloading. Thus we
imposea conditionof no slip alongthe vertical and the
bottomboundaries(Figure3).
geometryis definedby its thickness,whichis assumedto
be 15 km, and by the detachmentramp angle, which
rangesfrom 15ø to 45ø.
The model is most sensitiveto the upper crustal
viscosity,which determinesdownwardpropagationof the
deformationdue to the unloading. If the upper crustal
viscosityis very high (r/he-• •), the uppercrustwill not
deform in responseto the unloading, and, consequently,
no deformation will occur below the upper crust. Lower
values of upper crustal viscosity allow deformation in the
upper crust, which leads to deformation in the lower crust
and mantle. We choosethe upper crustal viscosityas the
characteristic
viscosity(r/0) and henceneedto specify
only the viscosityratios, which representthe relative
strengthsof the layers. We are aware of no estimatesof
the effectiveviscosityof the upper crust, but by assuming
that the upper crustal effectiveviscosityis lower than
that of oceaniclithosphereand higherthan that of the
asthenosphere,we can bound a range of possiblevalues.
The effective viscosityof oceaniclithospherehas been
Although the Earth's surfaceis a traction-free
boundary, the detachment surface beneath the
allochthon,which is excludedfrom our calculations,is
subjectedto tractions. We imposethe allochthonload, as
a normal stressk.oundary conditionalongthe detachment
fault. The normalstress(•rzz(x,t) = pcgh(x,t)) is
proportionalto the heightof the allochthon(h), where
the height of the allochthonis a function of horizontal
position(x) and time (t) (seebelow). This implicitly
assumesnegligibleflexural strengthfor the allochthon.
The allochthon is assumedto move by a seriesof
instantaneous
displacements
(earthquakes
or creep
events),without displacingthe detachmentsurface.This
is in accordwith our assumptionthat below the
detachment surface the crust and the mantle respond
only to the unloading, without experiencingin situ
extension.Thus along the detachmentsurfacewe impose
zero horizontal velocity and a condition on the normal
stress(azz), whichallowsthis surfaceto movevertically
and to sustain tractions. Elsewherealong the upper
boundary we imposetraction-free boundary conditions.
estimatedto be 1023-10
24Pa s fromthe flexuralresponse
to long-termloads[Walcott,1970].The effectiveviscosity
Method of Solution
of the asthenospherehas been estimated from studiesof
We use a finite element techniquewith eight-noded
quadrilateralisoparametricelementsto solvenumerically
for the velocity field. In addition, a penalty function
formulation is used to replace the pressureterm for
incompressibilefluid and solvedby a selectivereduced
postglacial
reboundto be ~1021Pa s [Cathies,1975;
Peltierand Andrews,1976].The aboveestimatesbound
the characteristic
viscosityto be in the range1021-1024
Pa s (Table 1). The dimensionless
lowercrustalviscosity
(r•½)andmantleasthenospheric
viscosity(r/ma)are
chosen
to be in the range1-10-3 , because
the lowercrust
and the mantle asthenosphereare weakerthan the upper
crust(Table 2). The dimensionless
mantlelithospheric
viscosity(r/ml),whichis assumedto be strongerthan the
integrationtechnique[Zienkiewicz,
1977].We conducted
variousnumerical experimentswith 200-800 elementsand
100-400 time stepsto ensurethat the solutionsare gridand time-step independent. As well, variouspatch tests
have been conducted
to ensure that the code is free of
upper crust, is chosento be in the range 1-10. For
comparison,in a study of a neckinginstability in the
zero-energyand propagatingspuriousmodes
Basinand Range,Zuberet al. [1986]assumed
that the
time-independent, we solvefor the velocity field at
successive
time steps. At the initial stage the allochthon
has not been displaced,the buoyancyforces are zero, and
there is no flow within the lithosphere. The allochthon is
assumedto move by a seriesof instantaneous
displacements.We introduce the unloading by changing
lower crust is 100 times weaker than the upper crust, the
mantle lithosphere is twice as strong as the upper crust,
and the mantle asthenosphereis 50 times weaker than
the upper crust.
We considera simple caseof a buoyant crust overlying
[Zienkiewicz,
1988].Becausethe governing
equationis
a densermantle;the crustaldensity(pc) is assumedto be
the verticalstressboundaryconditions
(az•(x,t)), which
20%lessthan the mantledensity(pro)(2670kg m-3
versus3320kg m-3 [BlockandRoyden,1991]).
would arise from horizontal displacementof the
allochthon,along the detachmentsurface. The stress
308
Wdowinski and Axen: Denudationand Isotasy--A Flow Model
boundary conditionat any given time and horizontal
RESULTS
position(•rzz(x,t)) is calculatedby assuming
a constant
horizontal displacementrate of the allochthonand
conservationof the allochthon mass. These assumptions
correspondto the end member caseof an internally
unextended allochthon. Mathematically, the stressis
calculatedby assumingthat the two dotted regionsin
Figure 3 are equal in area; the area of the dotted
Figure 4 showsthe velocityfield that is generated
within the crust and mantle in responseto tectonic
denudation.The generalflow pattern is upwelling
beneath the breakawayand downwellingbetweenthe
breakawayand the side boundaries. This resultsfrom the
no-slipboundaryconditionsalongthe sideand bottom
boundaries,which enforcethe subsidence
of the upper
rectangle(d x h0) and the heightof the dottedtriangle
(d x tan 0) determinethe lengthof the triangle
(2h0/tan 0) (or of the trapezoidin later stagesof the
denudation).The velocityfieldwithin the crustand the
surface near the side boundaries in order to conserve the
upliftedmassnear the breakaway.The magnitudeof the
flow velocitydependson the lateral extent of the
denudedregionand on the Grashofnumber(Gr), which
mantle, due to the unloading, is calculatedfor each time
step. In order to calculatethe elementgrid of the next
time step, we assumeconstant velocity between two
successivetime steps. This assumptionis valid for small
time stepsin which the time increment between stepsis
smaller than 50,000 years. We use 200-400 time steps to
calculate the lithospheric and asthenosphericresponseto
a denudationalevent occurringduring 2-6 Ma.
1.o
50
determinesthe ability of the viscousfluid to respondto
buoyancyforces.The velocityof the flow decreases
with
depthwherethe velocitygradientis dependenton the
vertical viscositystructure. A very high viscositylayer
(almostrigid), whichreduces
or mayevenhalt the
downwardpropagationof the deformation,can decrease
significantly
the flow in that layerand belowit. In
km
2.00
,
50
km
,
(b)
Fig. 4. The flow field within the crust and mantle in responseto tectonicdenudation.The
velocityscaleis given by the arrow belowthe lowerleft corner,which is scaledto the
characteristic
velocity(u0). (a) Velocityfieldfor a constant
viscosity
(Gr = 500,
Tuc-- Tic-- Trnl-- Trna-- 1). The mantleparticipates
in the overallflow. (b) Velocityfieldfor a
morerealisticviscosity
structure((Gr = 500, Tu½
= 1; Tic----0.01; Tml-- 2; Tma-- 0.01). The flow
is concentratedwithin the crust and forms a low-viscositychannelflow.
Wdowinskiand Axen: Denudation
and Isotasy--AFlow Model
contrast, a low-viscositylayer, which accommodatesmost
of the deformation, tends to amplify the flow in the layer;
the magnitude of the flow increaseswith lower viscosity
I
•o •m
I
I<-
309
•0 •m
(•)
•o ,•
(•)
->1
values.
For depth-invariant
viscosity(Figure4a), the vertical
velocity gradient is small and the mantle participates in
the overall flow. For a possiblymore realistic viscosity
structure, where the lower crustal and mantle
asthenosphericviscositiesare 2 ordersof magnitude less
than the upper crustal and mantle lithosphericviscosities
15' ramp
(Figure4b), the flowfieldis confinedto the crust,with
negligibleflow in the mantle. By generatinga channel
flow in the low-viscositylower crust, rather than by
deformingthe strong mantle lithosphere,the lower crust
movesmore effectivelyinto the denudedarea. Similarly,
:30' reu•p
•o •
if the mantlelithosphere
is almostrigid (veryhigh
viscosity)the unloadingaffectsonly the crust. The
low-viscosityof the mantle asthenospherehas no
influence on the flow, becausemost of the flow is
concentratedwithin the crust, and the very strongmantle
lithosphere does not transmit deviatoric stressdownward.
d-0
km
..
45' ramp
Fig. 6. Simulation of geometricalrelations in the crust
for variousdetachmentgeometriesafter 50 km of
allochthon displacement. The time-stepping procedureis
the same as for Figure 5, with u0 = 1.0, Gr = 500,
r/,c = 1, r/to= 0.01, r/,t = 2, r/,a = 0.01. The
detachment
ramp dips(a) at 15ø, (b) at 30ø, and (c) at
45 ø . See text for discussion.
We try to simulate observedgeometricalrelations
betweenvariousrockunits (Figure1) by tracingthe
shapesof markerthat initially are horizontal(Figure5a).
Figure 5b showsan exampleof thesemarkersafter being
deformedin responseto 50 km of allochthon
50
displacement
during5 m.y. (calculatedin 250 time
steps).The isostaticreboundin response
to the
unloadingcausescrustal uplift beneath the denuded
km
d =
50
km
allochthon and subsidenceof the upper and the
detachmentsurfacesin order to conservethe uplifted
mass. Horizontal markers that were initially located
beneaththe allochthon(at depthof 15 km) are uplifted
to the surface and form a dome-shapedstructure similar
to that of metamorphiccorecomplexdomes(Figure1).
Although the two-dimensionalformulationindicatesa
cylindrical structure, the term dome is usedbecauseof
the similarity to the observedmetamorphicdomes. The
domegeometryis determinedby the allochthonthickness
(h0) andthe horizontaldisplacement
(d); the allochthon
thicknessdeterminesthe amplitude of the dome, and the
displacement
determines
the diameter(wavelength)of
the dome. There are three important geometrical
,
50
krn
,
Fig. 5. Simulation of geometricalrelationsbetweenrock
units (with u0 = 1.0, Gr = 500,
•uc•- 1, •lc -- 0.01, •ml •- 2, •ma•- 0.01). (a) At the
initial stage, the undeformedhorizontal markers are 2.5
km apart. (b) After 50 km of allochthondisplacement,
during5 m.y. (calculated
in 250timesteps).The short
dashedline indicatesthe originalshapeof the
detachment.
The markers are deformed and show
footwalluplift, Moho deflection,and surfacetopography.
featuresthat are producedby th• model:flexureand
footwall uplift near the breakaway,Moho deflection,and
surfacetopography.
Figure 6 showsthat for a given viscositystructure,
various curvaturesof the uplifted footwall are produced
with variousdetachmentgeometries.Furthermore,for a
givendetachmentgeometrythe curvatureof the uplifted
footwall changeslittle for a large rangeof the other
parameters(Figures7 and 8). Thus the geometryof the
flexure is sensitiveto the detachmentgeometrybut
relatively insensitiveto the strength of the variouslayers.
310
Wdowinski
andAxen:Denudation
andIsotasy--A
FlowModel
•
50 k_m
,
•-
50
deflectionis verylow (< 1 km), and belowthat valuethe
-•
amplitude is practically zero.
Our model assumes that the crust and the mantle
1.00
Mohouplift = 6 km
I<-
•o,,•
(•)
behaveover long periodsof time as viscousfluids. Any
topographicrelief that is not dynamicallysupportedwill
decaywith a sufficientamountof time. However,during
the finite period of time of a denudationevent the crust
and mantle cannotfully relax, allowingsomerelief to
exist.The short-wavelength
(up to about20 km) relief
follows the shape of the denudedallochthon. The
long-wavelength
relief(50-100km) is supportedby the
•}], -- O.10
Yoho uplift = 3 km
•-
50
km
-•
(•)
flow field that is induced by the denuded allochthon and
has a dome shape that is centered around the
topographiclow next to the detachmentat the surface.
This may causethe metamorphic core and nearby areas
to havehigherelevationsthan their surroundings
(Figure
6). Detachmentgeometrydetermines
the short
•}l• '- 0.01
1/ohouplift < 1 km
•_
6O km
_•
wavelengthsurfacerelief, but can alsoaffect long
wavelengthrelief, becauseit determinesthe sizeand the
distribution of buoyancyforces;in terms of the model it
determinesthe magnitudeand distributionof the stress
boundary conditions.For a given detachmentgeometry,
the magnitude of the relief dependsmostly on the
Grashofnumber(Gr), whichdetermines
the relaxation
•}l, = 0.10;
Z/m•= 20.0
Yoho uplift < 1 km
time of the upper crust. However, the relief can vary
significantlyduring the early stagesof a denudation
event, when the wavelengthof the relief is too short to
Fig. 7. Calculationsof Moho deflectionas a function of
the dimensionless
lowercrustalviscosity(r/to)with other
parametersheld constant(u0 = 1.0,Gr = 500,
%c- 1; •]ml: 2; •]ma: 0.01). The time stepping
procedure
is the sameasfor Figure5. (a) r/l½= 1.0, (b)
•/1½
= 0.1, (c) • = 0.01, and (d) •/• = 0.1 and •]ml: 20.
,
50 krn
,
00 km
•-
-•
100
Moho uplift is measuredbetweenthe highestand lowest
point on the Moho. See text for discussion.
max. relief
30
= 7 km
km
I,- -•
The resultscomparefavorablywith severalfootwall
uplifts in the Basin and Rangeprovincein termsof
amplitude,curvature,and wavelengthof the synformal
part nearthe breakaway(Figurela and Wernickeand
200
overallflow,and the Mohois deflected
upward(Figure
7a). The amplitudeof the deflectiondecreases
as the
lowercrustalviscositydecreases
(Figures7b and7c), or
asthe mantlelithospheric
viscosity
increases
(Figure7d).
For both detachmentgeometryand Grashofnumberheld
constant,the amplitude of Moho deflectionis determined
by the ratio betweenthe lowercrustaland the mantle
lithosphericviscosities.When this ratio is lower than
0.01 (for Gr in the range100-1000)the amplitudeof the
max.
relief
= 4 km
3O kin
•_ _.
Axen [1988,Figures2 and 3]).
In contrast,the Moho deflectionand the topography
are sensitiveto the viscositystructure and to the Grashof
number, which enablesus to place boundson someof
theseparameters.The upward deflectionof the Moho is
causedby an upwellingflow within the mantle, beneath
the breakaway.As shownin Figure 4, the magnitudeof
flow in the mantle is controlledby the lower crustal and
mantle lithosphericviscosities.When the lower crustal
viscosityis the sameas the upper crustaland mantle
lithosphericviscosities,the mantle participatesin the
(b)
Ccr =
500
2 Ma after
denudation
ceased
Ccr =
500
(•)
max. relief
= 2 km
30 km
I<- ->l
max. relief
< !km
Fig. 8. Calculationsof surfacerelief as a functionof the
Grashofnumber(Gr) after 30 km of allochthon
displacement,
during3 m.y. (calculatedin 300 time
steps);the otherparameters
areheldconstant(u0- 1.0,
r/he= 1; r/l½= 0.01; r/ml= 2; r/ma= 0.01). (a) Gr = 100,
(b) Gr = 200, (c) Gr: 500,and (d) no surfacerelief,2
Ma after the denudation
ceased(with Gr = 500).
Wdowinski and Axen: Denudationand Isotasy--A Flow Model
have any significanteffecton the viscousforces.In later
stages,when the wavelengthof the denudedregionis
longer,the relief is determinedby the Grashofnumber.
Figure 8 showsexamplesof surfacetopographyafter 30
311
::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
km of allochthon displacement,as a function of the
:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::
Grashofnumber(calculatedin the samewayasFigure5).
For Gr = 100 the maximumreliefreaches4 km (Figure
8a), whereasfor Gr = 500 the maximumreliefis lower
and reachesonly 2 km (Figure8c). After denudation
ceases,the modeledlithospherecanfully relax (in 1-2
m.y.), flatteningthe surfacereliefbut preservingthe
geometryof the upliftedfootwalldome(Figure8d).
IIIIIIIIIIIllll•ll[11111111111111111111llllllllllllllllllllllllllll
DISCUSSION
Fig. 9. Finite strain in the footwallof the detachment.
Several aspectsof this model agree well with
geometriesobservedin areas where large-magnitude
upper crustal extension has been concentratedon
low-anglenormal faults. The model reproducesthe
curvature and uplift observedin detachment terrains
After 30 km of allochthon displacement, during 3 m.y.
(a) Beforedeformation
eachsquare
is2.5x 2.5km2. (b)
(calculated
in 300time steps)the squares
are deformed.
Notice that near the breakawaythe surfacesquareshave
experiencedbreakaway-side-down
simple shear along
vertical planes.
(e.g., Figure6) withoutelasticflexureof the crust.
Therefore elastic behavior need not have played an
important role in the formation of suchuplifts, as
postulatedby Wernickeand Axen [1988].This agrees
The geometry of the synform near the breakawayis
largely a function of the geometryof the detachmentfault
well with elastic flexure models that require the bulk of
there(seeprevioussectionand Figure6). Curvatureand
the crustin suchareas(all but a fractionof a kilometer)
final dip of initially horizontal markers in the breakaway
synform increasewith initial dip of the detachmentramp.
This agreeswith footwall geometriesobservedin the
Mormon Mountains region below detachmentsthat had
different initial dips, and where initially subhorizontal
to have behaved nonelasticallyin order to reconcilethe
magnitudesof fiber stresseswith the strength of crustal
materials[Buck,1988;Blockand Royden,1990,King
and Ellis, 1990].
The topographicrelief produced by our model would
be ephemeral if the crust were truly viscous. However,
the upper crust clearly has a substantial elastic thickness
even when in extension,as indicated by normal-fault
earthquakefoci to depthsof 10-15km [e.g.,Jackson,
1987]andby locallyuncompensated
topography
[e.g.,
Eaton et al., 1978].Effectiveelasticthickness
deduced
from the curvatureof footwalldomes(i.e., the part of
flexed crust in whichstrengthis greaterthan fiber
stresses)
doesnot represent
the the thickness
of the
uppercrustthat elasticallysupportssuchtopographic
loads[e.g.,Wernicke
andAxen,1988].Because
the
boundarybetweenupper and lowercrustis principallya
thermalone,viscouslyupwellinglowercrustis in effect
transferredto the upper crustif coolingis rapid relative
to uplift. In that casethe thicknessof the uppercrustal
layerthat storesthe elasticstressre]easedin earthquakes
(ourhigh-viscosity
uppercrust)neednotbe substantially
changed
and maysupporttopographic
reliefproduced
throughnonelasticmechanisms.
Suchan effectmay be representedby the westernedge
of the ColoradoPlateau, where a topographichigh of
0.3-1.0 km abovethe regionalplateau elevationsis
parallelto the breakaway
zone[Wernicke,
1985].Locally
this relief is spatia]lycoincidentwith a large-scalefold,
but elsewhereuplift was accommodatedlargely by
subverticalfaults [Moore,1972;Wernickeand Axen,
1988].Thesefaultsarelargeenough
(to ~ 4 km offset)
that they probablypenetratedthroughthe entirestrong
uppercrustallayer. Suchbreakaway-side-down
shearis
layeringis present[Axenet al., 1990].
In metamorphic core complexesit is generally
impossibleto recognizeinitially horizontal markers like
thosein Figure 6, making the strain pattern shownthere
difficult to apply directly. The dome typically is defined
by mylonitic foliation that may have formed in a dipping
shear zone. For example, the last mylonites probably
form parallel to the base of the detachment ramp in the
ductile midcrustal. Such initially dipping markers will
record a different strain path than originally horizontal
ones[e.g.,Axen and Wernicke,1991].
Seismicreflectionprofilesin the Basin and Range
province show a relatively fiat, featurelessMoho
[Klempereret al., 1986;Hauseret al., 1987].This occurs
in our model
when the ratio of lower crustal to mantle
lithosphericviscositiesis lessthan 0.01, causingMoho
uplift over a broad area in responseto local unroofing.
AlthoughMohouplift is minorin our models(0-7 km),
this is mostly due to the modest amount of extension: 50
km in 250 km length, or 20% extension.Much larger
amountsof extensionhave been documentedregionally
(e.g., 200-500%in the Basinand Rangeprovincenear
LasVegas,Nevada[Wernicke
et al., 1988]).Flowin the
middle to lower crust may be an important mechanism
for couplingupper crustal extensionalstrain with uplift
of the upper mantle.
Relief on the Moho is also minimized by local
shorteningand thickeningof lower and middle crust
belowdenudedterrains(Figure9). This is importantin
consistentwith the strain path predictedby our mode]
deeply eroded regions,where such a zone may be
mistakenly interpreted as an evidenceof continental
(Figure9; seediscussion
belowandAxenandWernicke
orogeny.
[19911).
The model predicts geometricalrelations such as
312
WdowinskiandAxen:Denudation
andIsotasy--AFlow Model
footwalluplift, Moho deflectionand surfacetopography,
which can be comparedwith the observationsand used to
bound the range of the free parameters.First we estimate
the characteristicviscosityby evaluatingthe rangeof
possiblevaluesof the Grashofnumber.Topographicrelief
createdin our viscousflow modelof the crustduringa
denudationevent would decayrapidly after cessationof
movementon the detachment(in 1-2 m.y.).
In activelyextendingareasof the Basinand Range
province,such as around Death Valley, the maximum
relief is about 3 km, with regionallyaveragedtopographic
relief more typically in the range 1-1.5 km. Here we
neglecteffectsof erosionand of the load due to density
differences
betweenbasinfill and bedrock,whichhas the
sameeffectasslightlyincreasing
loadsdueto topographic
relief. The sensitivityof the topographyto the Grashof
numberdeterminesGr in the range 100-1000,whichwill
supportmaximumtopographyof about 3 km throughout
the entire time-steppingcalculations,dependingon the
detachmentgeometry.In our formulationthe Grashof
numberis a functiononly of the characteristic
viscosity
becauseall the other characteristicparametersare fixed.
Thus Gr in the range 100-1000 yieldsan upper bound of
1021-1022
Pa s for the characteristic
viscosity,
whichis
the effectiveviscosityof the upper crust. As shownin
Figure 4, the lower crustal viscositydeterminesthe lower
crustalvelocityfield. By assumingthat the velocityof
After 30 km of allochthondisplacement
in 3 m.y., the
grid has been deformed,and eachquadrilateral
representsthe net localfinite strain. Near the breakaway,
the finite strain in the footwallis principallyrepresented
by breakaway-side-down
subverticalsimpleshear.
Breakaway-side-down
faultingis almostubiquitousalong
the westernmargin of the ColoradoPlateau on faults
from outcropto map scalewith displacements
measured
in millimetersto kilometers[e.g.,Moore, 1972;Smithet
el., 1987;Wernickeand Axen, 1988],and agreeswell with
the aboveresult. Bartleyet el. [1990]showedthat the
strain path followedby the footwallof the centralMojave
metamorphic core complex is consistentwith a flexural
failuremechanismof uplift, akin to that modeledby
Buck [1988].However,ambiguitiesexistwhichalsoallow
for an interpretation in which distributed subvertical
simpleshearaccountsfor the strainpath [Axenand
Wernicke,1991].For example,muchof the footwalluplift
there apparentlyoccurredalongsubvertical
hanging-walbside-up
structures
[Bartleyet el., 1990],
consistentwith the latter interpretation.
The finite strain of the denuded footwall
near the
allochthon(Figure9) is somewhatdifficultto applydue
to the lack of appropriate strain markers there. For
example, the unroofedgrid elementsdirectly adjacentto
the allochthon(Figure9b) are nearlysquare,implying
little net finite strain. In fact, these elementshave
flow in the lower crust cannot exceed more than 10
experienced
(1) earlyhanging-wall-side-down
subvertical
cm/yr, whichis 10 timesfasterthan the unloadingrate,
simpleshear as they passedout from under the baseof
we estimate the lower crustal viscosityto be 10-100
times lower than the upper crustalviscosity.Thus our
subverticalsimple shear, as they passedout from under
lowercrustalviscosityestimateis 10•9-102•Pa s. Our
estimateagreeswith that of Kruseet el. [1991],who
likely be obliteratedby subsequentmylonitizationas the
estimated the effectiveviscosityof the lower crust to be
footwallis translatedup the deeper(ductile)part of the
10•8-10•ø Pa s in the ColoradoPlateauadjacentto the
ramp. Hence the strain markers most suitable for
Basin and Rangeprovince.For a shorterlength scaleof
recordingthe the secondstrainevent(2) wouldbe
channelflow (150 km) the viscositycanbe higher,about
initially dipping mylonites. See Axen and Wernicke
10• Pa s [Kruseet el., 1991].Figure7 shows
that no
[1991]for further discussion.
the hangingwall ramp, then (2) hanging-wall-side-up
the hangingwall. The earlierstrainhistory(1) would
Moho deflection occurs when the ratio of lower crustal to
mantle lithosphericviscositiesis lowerthan 0.01; thus we
estimatethe viscosityof the mantle lithosphereto be in
LIMITATION
the range10•-10 •a Pa s. We cannotestimatethe mantle
This simple model ignoresalong-strikevariationsand
assumestwo-dimensionalflow. However,the domal
structureof denudedfootwallssuggeststhat
three-dimensionalgeometryand flow are important. It is
possiblethat the transport-parallelwarpingof
asthenosphericviscositybecauseour model is not
sensitiveto this parameter. The averagestrengthof the
lithosphereis predominantlydeterminedby the strength
of the very strongmantlelithosphere[Englandand
McKenzie,1982].This suggests
that the effective
lithospheric
viscosity
shouldbe 1021-10
•a Pa s. Indeed,
England[1986]arguedthat the averageviscosity
of the
lithospheremight needto be as low as 1022Pa s for
extensionalregionsto deformas rapidly as they do.
The variousmodelspredictfinite-straingeometriesand
strain historiesthat can be comparedwith field
observations.For elasticallycontrolledflexural failure
models[e.g.,Buck,1988;King andEllis, 1990]
OF THE
MODEL
detachment
surfaces
(e.g.,Figurela) is dueto variations
alongstrike in thicknessof the allochthon,rather than
later foldingor an initially broadly corrugatedfault
surface. At present, a three-dimensionalmodel is difficult
to formulatebecausethe geometry,boundaryconditions,
andvariousparameters
(viscosity
anddensitystructure)
are poorly constrained.Investigationsof two-dimensional
modelscan imposesomeconstraintson the problem,
whichwill help to constrainfuture investigations
of more
layer-parallelshorteningand elongationare predictedfor
the concaveand convexareasof flexures,respectively.
realistic three-dimensional
This is complicatednear the allochthon becausethe
lowercrust actingin responseto tectonicdenudation,
rather than attempt a formulationthat representsthe
crust itself and the wide variety of deformation
mechanismsthat may be in play. The continental
lithospheredeformsby variousprocesses,
suchas, brittle
failure, power-law creep, and cataclasticflow. Other
footwallthere is first bent and then unbent,and
discussion
of ,*,hestrain path followedby suchan
elasticallyflexingfootwallis beyondthe scopeof this
paper[seeAxenandWernicke,1991].Figure9 showsa
grid that is initially formedof 2.5 km by 2.5 km squares.
models.
Here we model the processof flow in the middle and
Wdowinski and Axen: Denudationand Isotasy--A Flow Model
studies of continental
deformation
have assumed that the
313
lithospherebehavesas perfectlyplasticmaterial[e.g.,
Tapponierand Molnar, 1976],as a viscoelastic
material
probablynot a first-ordereffect(e.g., comparethe results
of Buck [1988]for experimentswith and without
sedimentation).Futuremodelsshouldaddressthe
[e.g.,Vilotteet al., 1986],or asa powerlawfluid [e.g.,
strength as well as the load of the allochthonand
Englandand McKenzie,1982].In our calculations,
we
assumethat the crust and the mantle behaveoverlong
period of time as Newtonianfluids,which significantly
sediments.
simplifies the model and the calculations. Kruse et al.
In the model, extension is limited to the upper crust,
whereas the lower crust and mantle deform only in
responseto the unloading. This is in accord with the
[1991]showthat linearviscous
rheologysatisfactorily
small
approximates deformation by power law creep in the case
of channelizedflow, as modeled here for the weak lower
crust. Therefore the linear viscousflow assumptionneed
not necessarilycorrespondto any specific deformation
metamorphic core complexesand contrasts with the
much larger amountsof subsidencethat would be
mechanism
indicates that the locus of mantle lithospheric extension
related to metamorphic core complex formation is
laterally removed from the denuded terrains in the upper
deformation
in order to relate the rate and locus of
to the stresses derived from tectonic
unroofing.The effectiveviscositiesrepresentaverage
strengthsof each of the four layers, each of which
deformsby the variousmechanisms
(e.g., brittle failure,
creep,and cataclasticflow).
Temperature variations, which are neglectedin the
model, mostly affect the strength of rocks. We have taken
into account the first-order effect of the temperature by
separatingthe upper 150 km of the Earth into strong and
weak layers. Lateral temperature variations, which are
second-ordereffects,may changethe shape of the layers,
or locally affect the strength of rocks,but shouldnot
changethe general results. Calculationswith more
realistic and lesssimplistic constitutive relations will
probably produce somewhat different results and should
be approachedin future studies.
Although the model createstopographyduring the
deformation event, the topography decaysrapidly after
cessation
of movementon the detachment(in 1-2 m.y.).
amount
of the overall
subsidence
observed
in
expectedif in situ mantleextensionoccurred[White and
McKenzie,1988]. We feel that the bulk of the data
crust[e.g.,Wernicke,1985;Jones,1987].However,
examples exist in which in situ mantle extension is
indicated(e.g., the Rio GrandeRift [Olsenet al., 1979;
Prodehland Lipman, 1989]). Futuremodelingshould
consider
this scenario
also.
In our model,as well as thoseof Buck [1988],Block
and Royden[1990],and King and Ellis [1990],footwall
uplift is accommodatedentirely by bending. Although
this may be the casein a number of places, it also
appearsthat uplift can be accommodatedalong discrete
zonesof failure (suchas steepfaults), as wasapparently
the casein the North Virgin Mountains[e.g.,Wernicke
and Axen, 1988].This impliesthat the mechanical
spectrum of footwall responseto tectonic denudation is
broader
than has been considered
to date in continuum
modeling. Such responsesshould be consideredin future
studies.
Long-livedtopographyin areas where rapid extension
ceased10 m.y. or moreago(e.g., WhippleMountains,
[Davisand Lister, 1988],Nopahand RestingSpring
ranges[Wernickeet al., 1989]suggests
that the crusthas
CONCLUSIONS
significantelastic strength and that this strength
supportsthe topographicloads. In the real world, the
viscousresponseof the lithosphereto the unloadingis
stressdependent. Thus during a denudation event, when
the stressesare high, the deformation and the
topography are dominated by the viscousbehavior of the
crust. After cessationof denudationthe stressesdecrease,
used to calculate the viscousresponseof continental crust
and elastic behavior
dominates
the crustal
deformation.
Visco-elasticconstitutive relations may be more suitable
for the purposeof explainingtopography.However,
viscoelasticrelations demand specifyingadditional free
parameters which will complicate the model.
In order to keep a simpletime-steppingprocedure,the
allochthon and sedimentary basin loads were excluded
from our calculations. We consideredonly the allochthon
load and ignored any strength or deformationwithin the
allochthon. In reality, deformation within the allochthon
changesthe load distribution, but this is not expectedto
be a first-ordereffect[cf. Spencer,1984].Internal
extensionof the hanging wall leedsto asymmetrical arch,
with gentler dips on the hanging wall side. The
allochthonis probably divided into two strength
domains:the unextendedpart probably has strength
similar to the bulk of the upper crust, and the extended
part may account for little strength. Similarly, the effects
of basin-fill loads need to be analyzed, although this is
A four-layermodelof the upper 150 km of the Earth is
and the underlyingmantle to tectonicdenudation.The
model comprisesa strongupper crustallayer, a weak
lower crustallayer, a very strongmantle lithosphere
layer,and a weakmantleasthenosphere
layer,whichis in
accordwith experimentalconstraintson strength-depth
profilesfor continentallithosphere.The strengthof each
layeris represented
by its effectiveviscosity.The flowin
the crust and mantle is driven by buoyancyforces,which
arisefrom the unloadingof an allochthonalong a
detachmentfault by a seriesof instantaneous
displacements
(earthquakes
or rapid creepevents).
Numerical solutions,obtained by using a finite element
technique,predict footwall uplift, Moho deflection,and
surfacetopography.The curvatureof the footwall uplift
is similar to that which has been observedin regionsthat
experiencetectonicdenudation(e.g., the Basinand
Rangeprovince),and is determinedby the geometryof
the detachment fault. The upward deflection of the Moho
and the surfacetopographyare sensitiveto the viscosity
structure and enable us to bound the range of the various
viscosities.By matching observationsfrom the Basin and
Range province,which indicate little Moho deflectionand
low magnitudeof surfacetopography(_<3-5km), we
estimate the upper crustal, lower crustal, and mantle
lithosphericviscosities
in the ranges102•-1023Pa s,
314
Wdowinskiand Axen: Denudationand Isotasy--AFlow Model
1019-10
•1 Pa s, and 10•1-1023Pa s, respectively.
Melosh[1990]showedthat extensionof an elasticlayer
upper crust that locally differ from those that would be
produced,for example,by elasticallycontrolledflexure,
suchas that proposed
by Buck[1988](seealsoAxenand
Wernicke[1991]).Field studiesin appropriately
chosen
areasshouldbe able to provide thesetests.
overlyinga viscouslayer couldresult in initial
detachmentgeometriessimilarto thosemodeledhere: a
steeperportion that transectsthe strong,upper crustal
layer, and a fiat portion at depth wherestrengthhas
decreased.Although our modeldoesnot includeelastic
strengthof the upper crustallayer, it doesaccountwell
for deformationthat would be expectedafter the elastic
limit of the upper crust has been reached.More
comprehensive
viscoelasticmodelsseemto be required,
in order to reproduceinitial fault trajectories,
deformationhistories,and resultingtopography.
An important feature of any model of lithospheric
behavior is that it be testable. The modelspresented
here suggeststrain historiesand strain statesin the
Acknowledgments.
This work wassupportedby NASA
grant NAG5-840, and NSF grant EAR-8903912awarded
to R. J. O'Connell, and NSF grantsEAR 8451181and
EAR 8917227awardedto B.P. Wernicke.Axen'ssalary
wasprovidedby the NSF GraduateFellowship
Program.
We thank LeslieSonderand RogerBuckfor helpful
reviews.We gratefully acknowledgehelpful discussions
with John Bartley,BruceBuffett, RichardJ. O'Connell,
Mark Linker, J. K. Snow,Joann Stock, and Brian P.
Wernicke.
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(ReceivedJuly 2, 1990;
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