PROCEEDINGS
OF THE SEVENTEENTH
LUNAR AND PLANETARY
SCIENCE CONFERENCE,
PART 2
JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 92, NO. B4, PAGES E541-E551, MARCH 30, 1987
Constraintson the LithosphericStructureof Venusfrom Mechanical Models and
Tectonic
Surface Features
MARIA
t. ZUBER
Geodynamics
Branch,NASA/Goddard SpaceFlight Center,Greenbelt,Maryland
Radar imagesof the surfaceof Venusshow numerousstructuresthat are interpretedas havingformed
dueto horizontalcompression
and extensionof the lithosphere.Many of thesefeaturesexhibitcharacteristic
scales(widths or spacings)of deformation,one of 10-20 km and another of 100-300 km. For a range
of simplemodels,we test the hypothesisthat theselengthscalesare controlledby dominantwavelengths
arisingfrom unstablecompression
or extensionof the Venuslithosphere.Resultsshowthat the existence
of tectonicfeaturesthat exhibit both length scalescan be explainedif, at the time of deformation,the
lithosphereconsistedof a crust that was relativelystrong near the surfaceand weak at its base and
an uppermantlethat wasstrongerthan or of nearlycomparablestrengthto the uppercrust.The spacings
of thesefeaturesimply crustal
thicknesses
in the approximaterange 5-30 km and a thermal gradient
1
not greaterthan 25 K km-. Featuresthat exhibit only the smallerscaleof deformationcan be explained
by either a lithospherewith a thick crust that overliesa weak mantle or a lithospherewith a strong
mantle but with small internal strengthcontrasts.For a broad range of parameters,the modelsrequire
not only that smallerscalecompressional
and extensionalfeatureshave similar spacings,but also that
the widthsand spacings
of largerscalecompressional
featuresbe greaterthan thoseformedin extension.
This is consistentwith observeddifferencesin the characteristic
lengthscalesof larger scalesurfacefeatures,
as well as the similaritiesin spacingsof observedsmallerscalefeatures.
INTRODUCTION
Radar imagesof the surfaceof Venusobtainedby PioneerVenus[Masursky et al., 1980; Pettengillet al., 1980], earthbasedobservations
[Campbellet al., 1983,1984],and the Venera
15/16 spacecraft
[Barsukovet al., 1986;Basilevsky
et al., 1986]
reveala varietyof featuresof presumed
tectonicorigin.Although,
as shownin Figure 1, thesestructuresare globally distributed
over the surface of the planet, many exhibit distinct
morphologicalsimilaritiesthat suggesta commonmechanism
of formation. Specifically,numerousfeaturesdisplay linear
trends and characteristicwidths and/or spacingsthat are
suggestive
of deformationcontrolledby dominantwavelengths
arisingfrom the growth of compressional
and/or extensional
instabilitiesin the lithosphere.Previousanalyseshave shown
how deformation at preferred wavelengthscan arise when a
model lithospherethat contains one or more rheologically
competentlayersis horizontallystressed
[Fletcherand Hallet,
1983; Zuber et al., 1986; Zuber and Parmentier, 1986; Ricard
and Froidevaux, 1986; Zuber, 1986a].Since thesewavelengths
are primarily controlledby the thicknesses
of the competent
layers and the strength and density stratification of the
lithosphere,observations
of the geometries
of tectonicfeatures
arisingfrom unstabledeformationprovidea direct indication
of the underlyinglithosphericstructure.Unstabledeformation
has been invoked to explain large-scaleextensionaltectonic
featuresin the Basinand RangeProvince[Fletcherand Hallet,
1983; Froidevaux, 1986; Zuber et al., 1986; Ricard and
Froidevaux, 1986] and rift zones[Zuber and Parmentier,1986]
and intraplate compressional
deformationstructureson the
seafloor[Zuber, 1986b]. This mechanismmay also explain
periodicanomaliesin the earth'sgravity field related to long
wavelengthdeformationof the uppermantle[Froidevaux,1986;
Zuber et al., 1986; Ricard et al., 1986; Zuber, 1986b]. If the
regular development of tectonic features on Venus is a
Copyright
1987bytheAmerican
Geophysical
Union.
Paper number 6B7315.
0148-0227/ 87/ 006B-7315505.00
E541
consequenceof unstable deformation, then the widespread
occurrenceof thesestructuresprovidesdirect evidencefor the
lithosphericstructureof this planet on a globalscale.
In this study, we investigate the stability of the Venus
lithospherein extensionand compression.The purposeis to
assessthe conditions for which unstable compressionaland
extensionaldeformationcan develop,and to relatethe predicted
dominant wavelengthsto the geometriesof observedtectonic
featuresand the rheologicalstructureof the lithosphere.We
begin with a discussionof evidence for the extensional or
compressionalorigin of some prominent surface features,
followed by a descriptionof morphologicalcharacteristicsof
features
that
are consistent
with
formation
due to unstable
deformation. Subsequently, we show how variations in
temperature, pressure, and composition with depth in the
lithospherecan result in strengthcontraststhat may be of
sufficientmagnitudeto allow unstabledeformationto develop.
Finally, we discussimplications for the compositionaland
rheologicalstructureof the Venus lithospherein terms of the
relationshipsbetweenthe characteristicwidthsand spacingsof
observedfeaturesand various model parameters.We conclude
that unstable compressionand extension constitute viable
mechanisms for the formation
of tectonic features on Venus.
On the basisof observationsand model results,we suggestthat
the Venuslithosphere,at least in someregions,containstwo
mechanicallycompetentlayersthat we infer to correspondto
the upper crustand upper mantle.
TECTONIC
FEATURES
Beta Regio Rift
One of the most conspicuous
tectonicfeatureson Venusis
the Beta Regio Rift (see Figure 2). First interpreted as an
extensionalfeature by Masursky et al. [1980], the rift trends
approximately north-south and runs for over 1000 km. As
observedin 2-km-resolutionArecibo radar images[Campbell
et al., 1984], the rift consistsof a central depressionwith a
aepth of over 1 km flankedby topographicuplifts.The average
E542
ZUBER:VENUSLITHOSPHERIC
STRUCTURE
270 ø
0o
90 ø
Dali and Diana
180 ø
•re,,,•
••• SHTAR
••,.•
T•
Chasmata
Two prominent ridge-and-trough systems within the
equatorial highlands of Aphrodite Terra, Dali and Diana
Chasmata,were interpretedas rift structuresby Pettengillet
belts
30 ø
al. [1979] and Masurskyet al. [1980]. On the basisof PioneerVenus altimetry, thesefeatureshave lengthsof over 1000 km
0 o [Ehmannand Head, 1983],widthsof 75-100 km, depthsgreater
than 2 km, and raisedrimswith heightsof 0.5-2.5 km [Schaber,
_30 ø sma -30ø
1982]. McGill et al. [1983] noted that the terrain within the
D•ana
Chasma
/
central
troughs of the chasmatais of a complex nature, but
_60 ø I
I
I
I
d -60ø
agreedwith the interpretationof a tectonicorigin.Smallerscale
270 ø
0o
90 ø
180 ø
lineationssuchas thosefound within the Beta Regio Rift are
Fig. 1. Sketchmap of Venusshowingthe locationsof someprominent not observed in the chasmata; however, at Pioneer-Venus
surfacefeaturesof presumedtectonic origin that exhibit one or two resolutionsuchfeatures,if they are present,are not resolvable.
60ø
- Akna
Montes
•
•
60 ø
PLANITIA•__•
ndge
APHRODITE
}•/TERRA
I
lengthscalesof deformation.Featuresdescribed
in thisstudythat exhibit
smaller scale(10-20 km) spacingsare located in Beta Regio, Akna
and Freyja Montes, and in the ridge belts.Featuresthat exhibit larger
scalesof deformationare locatedin Beta Regio, AphroditeTerra, and
the ridge belts.
flank-to-flank width of the rift is in the range 100-200 km.
McGill et al. [1981] have noted that this feature is morphologicallysimilar to the earth'sEast African Rift. Within the
centraldepressionof Beta Regio are alternatingradar-bright
Banded
Terrain
Banded terrain is located in the mountains surrounding
LakshmiPlanumin the IshtarTerra regionof Venus.As revealed
in 3-km-resolutionAreciboradar images[Campbellet al., 1983],
thesefeaturesconsistof linear bands of alternatinggreaterand
lesserbackscatterthat are generallyalignedin the directions
of the topographiccontoursthat define the mountain ranges.
Bands are continuous for several hundreds of kilometers
and -dark lineations,which have characteristicspacingsof 1020 km and are continuousalongstrikefor up to severalhundreds
of kilometers. On the basis of the Arecibo radar data, which
are sensitiveto variationsin surfaceroughnessat centimeterto-meter scales,the lineations have been interpretedas faults
and
have characteristicspacingsof 10-20 km. On the basisof their
parallel and continuousnature and their relationshipto the
topography, Campbellet al. [1983] suggestedthat the bands
are most likely of tectonic origin. Solomon and Head [1984]
showedthat a numberof simpleextensionaland compressional
by Campbellet al. [1984]. Characterizationand mappingof modelsare compatiblewith the spacingsof bandsfor plausible
combinedArecibo and Venera data setsled Stofan et al. [1986] rangesof physicalpropertiesof the Venuslithosphere.However,
to suggest
that thesefaultswereproducedby extensionrelated theseauthors suggestthat the linearity and continuity of the
bands combined with evidence for band closure in Maxwell
to the formation of the centralrift depression.
275 ø
286 ø
37øN
'
I
\
/
I
--
Breght
.....
Dark
.-"
•
Bregh! hnear
edge of br,ght area
tt,
!
'
o
)oN
Fig. 2. Topographicprofiles(right) and sketchmap of bright and dark linearfeatures(left) as determinedfrom Arecibo
radar of the Beta Regio Rift Zone [from Campbellet al., 1984]. The lineationswithin the centraldepressionof the rift
are interpretedas faults. The width of the rift and the spacingof the lineationsdefinetwo lengthscalesof apparent
extensional
deformation.
ZUBER: VENUS LITHOSPHERIC STRUCTURE
E543
1986c].Within the ridgebelts,systemsof subparallelridgesand
groovestrend generallyparallel to the belts. The ridges and
groovesare continuousfor distances
of 100-200km and have
regular spacingsof 10-20 km [Basilevskyet al., 1986]. On the
basisof their similarityto the bandedterrainin Ishtar Terra,
the ridge belts have been interpreted as a product of
compressional
deformationby Basilevsky
et al. [1986].
Relationshipsof TectonicFeaturesto LithosphericStructure
All of these features have linear and often parallel strikes,
regular spacings,and/or characteristicwidths. Other tectonic
features that exhibit these characteristicsare discussedby
Basilevsky
et al. [1986].Table1 showsthat the widthsand
oOL•,
0%
o
o0•,l
oOE
o0•
o091,
Fig. 3. Sketchmap showingthe distributionof ridge belts (shaded)
in part of northernhemisphereof Venusas determinedfrom Venera
15 and 16 radar images[after Barsukovet al., 1986].Within the ridge
belts,smallerscaleridgesand groovesstrikeapproximatelyparallelto
the belts.The spacingsof the ridge beltsand the ridgesand grooves
definetwo lengthscalesof apparentcompressional
deformation.
Montesare mostconsistent
with foldingor faultingin response
to regional-scalecompression.
spacingsof thesefeaturesfall broadly into two ranges,one of
10-20 km and anotherof approximately100-300 km. The Beta
Regio Rift and the ridgebeltsexhibit both scalesof deformation,
while the bandedterrain showsonly the smallerscale.For the
chasmataonly the largerscaleof deformationis resolvablewith
presentdata, so it is unknown whethera smallerscaleexists.
Solomon and Head [1984] examined a range of tectonic
modelsfor the banded terrain and concludedthat the regular
spacingof the bandswascontrolledby a brittle or high-viscosity
surface layer that they interpreted as the Venus elastic
lithosphere.In studiesappliedto the Basinand RangeProvince
of the western U.S., Zuber et al. [1986] and Ricard and
Froidevaux [1986] showed that two scales of periodic
deformation can arise in an extendingmedium that contains
two strohglayersseparatedby a weaker layer. For the earth's
continentallithosphere,thestronglayerscorrespond
to theupper
crustand that part of the mantlethat makesa major contribution
to lithospheric
strength,whiletheintermediate
layercorresponds
to the weak part of the ductile lower crust [Froidevaux, 1986;
Zuber et al., 1986;Ricard and Froidevaux, 1986].By analogy,
Figure3 showsa sketchmap of a groupof prominentfeatures it is suggestedthat the two scalesof tectonicfeatureson Venus
termedridgebeltsthat wererevealedby Venera15and 16imaging can also be explained by a lithospherethat, as illustratedin
radar [Barsukovet al., 1986;Basilevskyet al., 1986].The Venera the following section,consistsof a crust that is weaker at its
radar system is sensitive to surface slopes and therefore basethan the underlyingmantle.
topography. On the basisof the Venera data, the ridge belts
MODEL DEVELOPMENT
have been describedas north-southtrendinglinear ridgesthat
are continuousalong strike for distancesof up to 1000 km The VenusLithosphere
[Basilevskyet al., 1986].Direct measurements
from mapsderived
from the Venera data show the ridge belts to have a regular
Figure 4 showsa possiblerangeof maximum principalstress
spacingperpendicular
to strikeof approximately300km [Zuber, differencesthat can be supportedby the Venuslithospherein
Ridge Belts
TABLE
1.
Scales of Tectonic Surface Deformation
on Venus
Long
LongWavelength
Short
Short Wavelength
Wavelength Width or Spacing Wavelength Width or Spacing
Probable Style
Component
(km)
Component
(km)
Data
Beta Regio
Rift Zone
Extensional
Arecibo
Rift Zone
Width
Extensional
PioneerVenus
Tectonic
Feature
Dali and
Rift Zone
Diana
Width
100-200
Lineations
75-100
Not resolvable
10-20
if present
Set
Ref.*
C1
Chasmata
BandedTerrain
Ridge Belts
Belt spacings
300
Bands
10-20
Compressional
Arecibo,
Venera 15/ 16
Ridges-and-
10-20
Compressional
Venera 15/ 16
grooves
*C1 = Campbellet al., [1984]; S = Schaber[1982];C2 = Campbellet al. [1983]; B = Basilevskyet al. [1986].
C2, B
E544
ZUBER: VENUS LITHOSPHERIC STRUCTURE
the Moho, and the weakerlayer correspondsto the lower crust.
Later resultswill show that in an extendingor compressing
modellithospherewith two stronglayersseparatedby a weaker
layer two wavelengths
of deformationmay develop.While the
modelsdo not requirethat the layersand substratecorrespond
to thecrustalandmantleregionsdiscussed
above,thisrheological
stratificationis appealingin its simplicityand plausibility.In
(MPa)
0' 1 - 0' 3
' '.::!!::..
lO
the absence of better data from which to estimate Venus' near-
surfacecompositionand rheology,we will proceedunder the
assumption
that the lithosphereis stratifiedin this manner.
•, 20
N
LithosphereModels
.,=.,
30 -- •xx-- 10'•5s'•
Horizontal
Horizontal
Extension
Compression
40
800
600
400
200
0
200
400
Fig. 4. Strength,definedas differentialstress(o• - o3), as a function
of depthin the Venuslithospherefor horizontalcompression
(left) and
extension(right). The strengthenvelopewas constructedfor a surface
temperature
of 700K, a thermalgradientof 15K km-•, a strainrate
of 10-•5s
-•, andzeroporepressure.
Thelithosphere
consists
of a diabase
crust and a dry olivine mantle. An arbitrary crustalthicknessof 10
km is assumed.The brittle strength of the crust and mantle was
determinedfrom Byedee[1968], and the ductile strengthsof diabase
and olivine were taken from Shelton and Tullis[1981] and Braceand
Kohlstedt[ 1980],respectively.
Note for both compression
and extension
that the lower crust is weakerthan the upper crustand upper mantle.
Such a strength stratificationmay lead to unstabledeformation in a
compressingor extendinglithosphere.
The Venus lithosphereis assumedto consistof a crust and
mantle with density contrastsat the surfaceand at the base
of the crust. As shown in Figure 5, the crust is modeled as
a strong layer overlying a weaker layer, each with uniform
strength,and the mantle consistsof a uniformly stronglayer
that overliesa halfspace.Two different strengthstratifications
are consideredfor the mantle. In the strengthjump (J) model,
strengthis discontinuousat the base of the layer and falls to
a loweruniformvaluein the substrate.In the continuousstrength
(C) model, strengthis continuousat the baseof the layer and
decreasesexponentiallywith depth in the substrate.Both of
thesestrengthstratificationsare considerablysimplerthan that
which is likely to exist on Venus(cf. Figure 4). However, the
nearly analytical solutionsfor the flow obtained in each case
permit insightinto the physicalnature of the deformationthat
cannot be gained from fully numerical solutions.While the
growth rates of the instabilitiespredictedby the modelsvary
somewhat,the dominant wavelengthsare in good agreement
to the degreethat they can be compared[Zuber et al., 1986].
Idealizedrheologiesare chosento approximatethe response
of the Venus lithosphere to imposed deviatoric stresses.
Experimentshave shownthat a nonlinearviscousmaterialwith
compression and extension as predicted by laboratory
experiments on rock theology. Where stress increases
approximatelylinearly with depth, deformationoccursin a a steady-state
constitutive
law of the form e = on (wheree is
brittle manner. Brittle strengthis determinedby the frictional the strain rate, o is stress,and n -• 3 is the stressexponent)
resistanceto sliding at fracture surfacesand is essentially representsa mediumin which deformationoccursprimarily by
insensitive
to strainrate, temperature,and mineralogy[Byedee, ductilecreep[e.g., Weertmanand Weertman,1975].A perfectly
1968].At greaterdepthsdeformationoccursby ductileflow plastic theology, which is a continuum representationof a
and strengthis dependenton strainrate, composition,and most
medium that undergoesdeformation by faulting, is approxicriticallyon temperature[e.g., Weertmanand Weertman,1975].
mated by a stressexponent of n = • in the steady-statecreep
The mode of deformation at a given depth is determinedby
relationship. Both perfectly plastic and nonlinear viscous
the lesserof the brittle and ductilestrengths.
theologiesare considered
for the uppercrustaland uppermantle
In Figure 4 it is assumedthat mantle flow is governedby layers,and nonlinearviscousbehavioris assumed
for the lower
a dry olivinetheologyon the basisof the hypothesized
similarity
crustallayer and mantle substrate.
in bulk compositionof Venusand the earth [BasalticVolcanism
Study Project, 1981,pp. 682-685]. A diabasecrustis assumed Stylesof UnstableDeformation
on the basis of Venera lander results that show surface rocks
In a layered medium that is extendedor compressedat a
in a number of rolling plains and lowlandssitesto be similar
in chemistryand mineralogyto tholeiitic and alkaline basalts mean horizontal strain rate, •xx, deformation developsunder
[Surkov et al., 1984].A comparisonof relativestrengthsat the conditions for which small amplitude (much less than layer
Moho shows that diabase is weaker than olivine at the same
thicknesses)perturbations along the free surface or layer
P-T conditionsfor both compressionand extension;thus the interfacesamplify with time [cf. Biot, 1957, 1960].The nature
upper mantle is considerablystrongerthan the lower crust. A of theseperturbationsin some instancesdeterminesthe style
crust that is thicker than that shown would be even weaker
of deformationthat develops.In an extendingor compressing
at its baseasthe ductilestrengthof the lowercrustprogressively mediumin whichinitial perturbationsare distributedrandomly,
decreaseswith increasingdepth. For the conditionsassumed deformationdevelopsperiodicallyat the dominantwavelength,
in Figure4, the lowercrustwill be weakerthantheuppermantle Xd, in a directionnormal to the applied stress[e.g., Fletcher,
1974; Smith, 1975]. For a compressingmedium with a single
at the Moho for crustalthicknesses
up to about 30 km.
or foldingmodeof deformation
Figure4 illustratesa scenariofor whichthe Venuslithosphere competentlayerthe asymmetric
couldcontaintwo relativelystronglayersseparatedby a weaker is preferred,while for an extendingmedium the symmetricor
layer. As for the earth'scontinentallithosphere,the stronglayers pinch-and-swellmode is most likely. These are illustratedin
of a singlestronglayeroverlying
correspond
to the uppercrustandthe uppermantlejust beneath Figure6 for a mediumconsisting
ZUBER: VENUS LITHOSPHERIC STRUCTURE
E545
Po
,1
1:)4,n4,"r4 I
I
•- A(0-1' O'3)
ne'olFIT
J Model
C Model
Fig. 5. Models of a Venuslithospherecontaininga strongupper crust and upper mantle separatedby a weak lower
crust. The wider and narrower shadingindicate the crust and mantle, respectively.The diagram at the left showsthe
parametersthat describethe layersand substrate.Parametersp, n, and r are the density,stressexponentin the stressstrainrate relationship,andstrength,respectively.
In the relationshipbetweenstrainrate,gxx,and stress,o, Q is the activation
energy,R is the gas constant,T is temperature,and A is the frequencyfactor. The strengthis definedby the product
of the viscosityand the horizontalstrain rate in the basicstate of uniform extensionor compression.
The diagramsto
the centerand right schematically
show the strengthstratificationsexamined,where the length of each sectionrepresents
the relativestrengthof the medium.In the J modelstrengthin the mantleis discontinuous,
while in the C modelstrength
is continuousand decreases
exponentiallywith depthin the substrate.Both modelspredictsimilardominantwavelengths
for compressional
and extensionalinstabilities.
a weakersubstrate.If the lithospherecontainsa thermalanomaly
or a structuralweaknessat depth, an initial perturbationthat
isspatiallylocalizedmaybeappropriate.In an extendinglayered
mediumin which a layer thicknessperturbationlocalizesat the
base of the competentlayer, deformation nucleatesat the
perturbation.The patternof deformation,whichconsistsof a
centraldepression,
beneathwhichthelayerthinsdueto necking,
and upliftedflanks,is morphologicallyanalogousto a rift zone
where D = d/dz, k(=2rr/h) is the wave number, and W is
the stream function. Solutions of (2) in the form shown are
given by Fletcher and Hallet [1983], while those for a layer
of uniform viscosityor strengthin which • -• • are given by
Zuberet al. [ 1986].Thesesolutionsyieldthe perturbingvelocities
and stressesin the layers and substrate.The amplitude of a
perturbationat the ith interfaceat time t, Ai (k,t), for a medium
in compression
can be written
[Zuber and Parmentier,1986].The characteristic
width of the
deformation is controlledby the dominant wavelengthof the
extensionalinstability.The dominantwavelengtharisingfrom
the growth of either random or localizedperturbationsis a
Ai(k,t) -- Ai(k,O
) exp [(qcomp
+ 1) •xxt]
(3)
and for a medium in extension
function of the mechanical structure of the medium, and in
particular the thicknessof the competent layer. Unstable
deformationat the dominantwavelengthcan thereforeexplain Compressional Instability
both the periodicspacingsof extensionaland compressional
Extensional Instability
features and the characteristic width of a rift.
Theory
For a power-lawviscouslayerin whichviscosity/•decreases
with depthz as
/• = •oe'z/C
(1)
Fig. 6. Stylesof unstabledeformationin a mediumconsistingof a
layer that is more competentthan the underlyingsubstrateand for
where• istheviscositydecaydepthand •o is a referenceviscosity, which initial disturbancesalong interfacesare distributedrandomly.
For compression
an antisymmetricor folding mode of deformation
the governingequationfor the perturbingflow is
develops
whilefor extension
deformation
occursin a symmetric
or pinchand-swell mode. The parameter A refers to the amplitude of the
perturbingflow at an interface.The total flow in the mediumconsists
D4W+ 2•-lD3W+ [C-2- 2k2(2/n-l)]
(2) of the perturbingflow plus a pure shear componentresultingfrom
the basicstateof uniform compression
or extensionillustratedby the
D2W- 2k2/;
-] (2/n - 1)DW+k2(k2+ •-2)W-0
arrows.
E546
ZUBER:VENUS LITHOSPHERICSTRUCTURE
Compression
32 16
I
!
8
4
2
!
i
I
32 16
Extension
x/h
4
2
8
I
400
250
350
300
200
250
150
2OO
150
100
lOO
5O
5o
-5o
,
!
I
I
I
I
I
2
I
3
-50
I
I
I
I
2
i
3
k • - 2"iT h•
Fig. 7. Growth rate spectrafor model lithospheresin compressionand extension.The dominant wavelengthoccursat
the maximumof the growthrate, q. The growthrate is nondimensionalized
by the meanhorizontalstrainrate, gxx,while
the wave number and wavelengthare nondimensionalized
by the thicknessof the surfacelayer, h•. The upper curves,
constructedfor a mediumwith a singlestronglayer (shaded),have one peak and indicatedeformationcharacterizedby
one dominant wavelength.The lower curves,constructedfor a medium with two stronglayers separatedby a weaker
layer,havetwo peaksandindicatedeformation
withtwo dominantwavelengths.
Extensional
andcompressional
instabilities
with one and two dominantwaavelengths
may expla•_n
the lengthscalesof many tectonicsurfacefeatureson Venus.For
both singleand multiple layer modelsS• = 1, S2 = 0, a = 0.2 and 0.6 for compression
and extension,respectively,
n•
= 104,andn4: 3. Forthemultilayer
modelS3= 0.01,S2= S4: 0, R] -- 100,R2-- 0.5,andn2-- n3-- 3.
(4)
Ai(k,t) -- Ai(k,0) exp [(qcxt- 1)•xxt]
whereAi(k,0) is the initialamplitudeandqcomp
andqextarethe
compressionaland extensionalgrowth rate factors.The value
of unity in eachof the exponentialtermsrepresents
the kinematic
distortion
of the
medium
due
to
uniform
extension
or
wavelength,)k/h1 : 2rr/k', on the top. Both are normalized
to the thicknessof the strong surfacelayer. The upper curve
in each case,which correspondsto a mediumwith one strong
layer, contains a single peak. This indicates deformation
characterizedby a singledominantwavelength,which is about
four layerthicknesses
for boththe compressional
andextensional
casesshown. The lower curve in each case correspondsto a
mediumthat containstwo stronglayersseparatedby a weaker
layer. These growth rate spectraexhibit two maxima, which
indicate that deformation develops with two dominant
compression.
Instabilitieswill amplify with time for conditions
in whichthe exponentialsaregreaterthan unity. If thiscondition
is not met, initial perturbationswill decay and an extending
or compressing
mediumwill thin or shortenuniformly.
wavelengths.
The growth rate factor reflectsthe relativecontributionsto
In an unstablemediumwith two stronglayers,the positions
dynamicinstabilitygrowth of the driving force,which depends
and amplitudesof the shorterand longerwavelengthpeaksin
on the slopeof the perturbedinterface,andtheviscousresistance
the growth rate spectrumare controlled to the greatestextent
of the medium to deformation. In an unstable medium, the
by the propertiesof theselayers.However, becausethe strong
wave number at which q is a maximum definesthe wavelength
layersarecoupledto theweaklayer andthe substrate,the growth
at which a disturbancewill grow most rapidly and eventually
rates of the instabilitiesand the dominant wavelengthsare
dominatethe flow. This isthe dominantwavelength[Biot, 1961].
In thisformulation,thedominantwavelengthisdeterminedfrom
the eigenvaluesolution of the system of equationsfor the
perturbingflow in eachlayer,the boundaryconditionsrequiring
the continuity of stressesand velocitiesat each interface, and
the ratesof perturbationgrowth at eachinterface.The method
of solution for a multilayeredmedium is discussedby Zuber
et al. [1986].
RESULTS
determinedby the mechanicalpropertiesof the entire medium.
Consideran extendingmodel lithospherewith a plasticupper
crustallayer and a nonlinear(n = 3) viscouslower crustallayer,
mantle layer, and mantle substrate.Although,as illustratedin
Figure 7, two wavelengthsof deformation can develop,the
mantle layer is not itself unstable,but deformspassivelyin
response
to the unstabledeformationof the uppercrustallayer.
The longerwavelengtharisesbecausethe mantle layer resists
deformation,and suppresses
over a rangeof wave numbersthe
instabilityinducedby the uppercrustallayer. The maximum
suppression
occursat the relativeminimumbetweenthe peaks
WavelengthSelection
in the growthrate spectrum[Zuber et al., 1986].If the upper
Figure 7 showsexamplesof growth rate spectrafor the C mantle contains a region in which deformation occurs
model in compressionand extension.The growthrate is shown predominantlyby brittle (seeFigure4) or ductile[cf. Chappie
as a function of wave number, k', on the bottom axis and and Forsyth, 1979] faulting rather than ductileflow, then a
ZUBER: VENUS LITHOSPHERIC STRUCTURE
TABLE
2.
Dimensionless
Parameters
Parameter
E547
to the verticallyaveragedstrengthof the layersand substrate.
The limiting caseof a very strong layer in which buoyancy
Definition
forces have no effect on the characteristics
S•
(p, - p.) gh,
S2
(p• - p•) gh•
7'1
the flow.
The parameterS•, which is the ratio of the buoyancyforce
due to the densitycontrastat the free surfaceto the strength
of the surfacelayer, has the greatesteffectof the S parameters
on the dominant wavelengths.The relationshipsbetweenthe
dominant wavelengthsand S• for the multilayeredC model
are representedin Figure 8. For compression,the effect of a
stabilizingdensitycontrast(P0< Pt) is to decreasethe longer
dominant wavelengthfor a range of S•, while for extension
the opposite holds. For compression,the longer dominant
wavelengthis independentof the influencesof buoyancyand
strengthfor S• < 0.1, then decreases
rapidly in the range 0.1
< S• < 10. For extension,Xd/h markedly increasesfor 0.1 <
S• < 10andisrelativelyindependentof thisparameterotherwise.
Figure 8 exemplifiesthe effectsof the couplingof the strong
layersby illustratinghowthelongerdominantwavelength,which
arisesdue to the presenceof the strongmantle layer, is affected
by the physicalpropertiesof the near-surface.In contrast,the
shorterdominant wavelength,which is to the greatestextent
controlledby the surfacelayer, is not significantlyaffectedby
T2
S3
(p, - p•) gh,
S4
(P,I - PO gh,
of deformation
correspondsto S = 0. Large S correspondsto a weak layer
in whichbuoyancyforcesplay a dominantrole in determining
T3
7'4
R•
7'2
7'3
R3
7'4
hi
plasticrheologymay be appropriatefor the uppermantlelayer.
An extendingmodellithosphere,in whichboth the strongupper
crustaland mantle layersare plasticand the weak lower crustal
and mantlesubstrateare nonlinearviscous,deformsin response
to instabilitiesinducedby each of the strong layers [Ricard
and Froidevaux, 1986]. This illustratesthat two wavelengths variationsin S•.
If, as on the earth, the density contrast at the crust-mantle
of deformation can develop if the lithospherecontains two
boundary
of Venus is stabilizing, then for compressionand
competentlayers separatedby an incompetentlayer; it is not
extensionincreasingS3will decreaseand increase,respectively,
necessary
for both of the competentlayersto be unstable.
the longerdominant wavelength.However, over a broad range
of S3 valuesthe changesin the longer dominant wavelengths
Effectsof BuoyancyForcesand StrengthStratification
are not significant.As densityvariationswithin the crust and
If the stressesrequired for deformation are large, then mantle would be expectedto be less than that at the Moho,
buoyancy forces arising from density contrasts within the the effectsof variations of S2 and S4 should be even lessthan
lithospherewill havea negligibleeffecton the patternof unstable thoseof S3.Thus the wavelengthsof tectonicfeaturesobserved
flow. However, buoyancyforceswill dominatewhen deviatoric at the surface cannot provide meaningful constraintson the
stresses
in thelithosphereareinsufficientto dynamicallysupport subsurfacedensitystructure.
the topography arising from unstable deformation. In this
The dominant wavelengthsand growth rates can also be
formulation, the parametersSi, listed in Table 2, relate the consideredin termsof the internalstrengthstratificationof the
buoyancyforcesarisingfrom densitycontrastsat eachinterface lithosphere.The parametersRi, which are definedin Table 2,
lOO
i
1
'
i
lO(
i
i
i
i
Compression
i
Extension
o.125
50
I
0.125
5t
l
0.25
0.25
!
o.5 k d
0.5
Xa/h1 10
1,
-
kd2
•
I
10'4
I
10-2
--
I
1
!
kd2
2
I
100
Sl- (P•'Po)gh•
"1'1
102
0'4
10'2
100
102
(I9 •'PO) ghl
Sl=
'T1
Fig. 8. Relationships
betweendominantwavelength,Xd, dominantwave number,kd', and the dimensionless
ratio, S],
of the buoyancyforceat the surfaceto the averagestrengthof the surfacelayer for compressional
and extensionalinstabilities
in a multilayeredmedium.The dominantwavelengths
are normalizedto the thicknessof the strongsurfacelayer h•. The
subscripts1 and 2 referto the longerand shorterdominantwavelengths,
respectively.
E548
ZUBER: VENUS LITHOSPHERIC STRUCTURE
representthe strengthsof the subsurfacelayers and substrate
in terms of the strengthof the surfacelayer. In the C model
the strengthin the mantlesubstratevarieswith depth.For this
model the substraterheology is describedby a, the ratio of
the decay depth of the substratestrengthto the thicknessof
the surfacelayer. Zuber et al. [ 1986]and Zuber and Parmentier
[1986] have investigatedthe influenceof theseparameterson
the unstableflow in layeredmedia. Thesestudiesshowedthat
if the mediumhasa plasticsurfacelayer and is nonlinearviscous
otherwise,changesin the internal strengthin most casesdo
not significantlyaffect the dominant wavelengths.Therefore
uncertaintiesin the strength stratification do not markedly
influence estimationsof the dominant wavelengths.However,
sinceinstabilitygrowthisdrivenby differences
in strengthacross
interfacesin the medium, the growth rate factors do depend
on thestrengthstratification.
The effectof increasing
thestrength
contrast at an interface is to increasethe growth rate factor
and vice-versa.In other words, increasingthe relative strength
of a competentlayer with respectto an incompetentlayer
increasesthe degreeof instability.For modelsthat exhibit two
dominant wavelengths,increasingthe relative strengthof the
strongsubsurfacelayer with respectto the strongsurfacelayer
resultsin an enhancedlongerwavelengthof instability.Similarly,
increasingthe relativestrengthof the strongsurfacelayer with
respectto the strongsubsurfacelayer increasesthe growth rate
of the shorterwavelengthof instability. If the relative strength
of the surfacelayer is more than a few timesgreaterthan that
of thestrongsubsurface
layer,thelongerwavelength
of instability
can be suppressed.
The longerwavelengthcan alsobe suppressed
if strength contrastselsewherein the lithosphereare small
becauseof the couplingof strong layersdescribedpreviously.
In contrast,if the relativestrengthof the mantlelayer is many
times greaterthan that of the surfacelayer, both wavelengths
of instability will still occur. The shorter wavelengthis not
suppressed
in this casebecausethe growth rate factor varies
directlywith the power-lawexponentsof the stronglayers.The
plasticsurfacelayercanbeunstableevenif it ismarkedlyweaker
than the strongsubsurfacelayer, as long as both are stronger
than the intermediatelayer and substrate.
Compressionaland ExtensionalGrowth Rates
The model lithosphere in Figure 7 is more unstable in
compressionthan extension,as evidencedby the fact that the
peak in the growth rate functionof the former is greaterthan
that of the latter.This is a generalcharacteristic
of hydrodynamic
instabilitygrowth [Smith, 1975, 1977]. For the range of cases
examinedin this study, a model lithospherewith at least one
strong layer is always unstable (i.e., always exhibits at least
one dominant wavelength) in compression,while a model
lithospherein extensioncan be stablefor largeS• and/or small
internal strengthcontrasts.Extensionalinstabilitycan also be
suppressed
if the surfacelayer is characterizedby a viscous(1
•<n• •<3) ratherthanplastic(n• = o•)rheology.The predominance
of viscousbehavior near the surfacewould be expectedfor a
lithospherewith a sufficientlyhigh thermal gradient.
Whether a theoretically unstable medium is unstable in
practicedependson the dominant growth rate, the amplitude
of an initial perturbation,Ai(k,0), andthe meanhorizontalstrain,
gxxt.As illustratedby equations(3) and (4), if any of the above
are sufficientlysmall, the amplification of the instability will
be negligible.
APPLICATION TO VENUS LITHOSPHERE
The previoussectionillustrateshowvariousmodelparameters
affectthe dominantwavelengthsand growthratesand describes
in generaltermshow unstabledeformationcanoccurfor a range
of conditionsin an extending or compressingdensity- and
strength-stratifiedmedium. By consideringthe resultsin the
contextof the wavelengths
and characteristic
widthsof observed
surfacefeatureson Venus, we can place broad constraintson
the structureof the lithosphere.
SurfaceLayer Thicknessand Strength
The widths and spacingsof tectonicfeaturessummarizedin
Table 1 and the relationships
betweenha/h and S• in Figure
8 allow constraintson the thicknessof the strongsurfacelayer.
In intervalswhereha/h variesmarkedly with S•, the thickness
of this layer is constrainedby its strength.This criterionis met
for a rangeof S• for the longerwavelengthof instability.
To estimatethe thicknessof the surfacelayerfrom the longer
wavelengthfeatures,first observefrom Table 1that thesefeatures
have spacingsor widths of approximately300 and 150 km,
respectively,
for compression
and extension.For the conditions
assumedin Figure 4 it can be seenthat the strengthof the
uppercrustdoesnot exceeda few hundredMPa. Compressional
and extensionalstrengthsof 200 and 100 MPa, respectively,
are assumed,bearing in mind the considerableuncertainties
associated
with thesevalues.If parametervaluesin thenumerator
of S• of p• - 3.0g cm-3,/90 -- 0.0 g cm-3,andg -- 887cm
-2
s are appropriate for Venus, then the compressionaland
extensionallong wavelengthrelationshipsimply surfacelayer
thicknessestimatesof about 16 and 8 km, respectively.
It is also possibleto estimateh• from the spacingsof the
shorter wavelengthstructures;however, becausethe dominant
wavelength does not vary significantly with S• in either
compressionor extension, the layer strength provides no
constraint.The ratio ha/h for both compressionand extension
fallsin the range2.7-4, whichfor a featurewith a 10-kmspacing
yieldsan upper crustallayer thicknessin the range 2.5 < h•
< 3.7 km. For a 20-km spacing,5 • h• • 7.4 km. The best
agreementbetweenthe short and long wavelengthresultsoccurs
if for compressionS• • 0.5 and for extensionS• • 1, but the
constraintsprovidedby the layer strengthare violated. Given
the uncertaintiesin the strength,however,a broaderrange of
stresses than considered above cannot be ruled out. For a wide
range of Ri, Si, and a, the models predict h• to be between
approximately2 and 22 km. The upperpart of thisrange,which
corresponds
to the limit of large S• for the longerwavelength
compressional
features,isinconsistent
with therangedetermined
from the shorterwavelengthspacings.More refined estimates
of h• will requirebetter knowledgeof the flow behaviorof the
crust and mantle, the regional strain rates and geothermal
gradients,and morerealisticmodelsof the rheologicalstructure
of the lithosphere.For the broadrangeof parametersexamined
in this study,most of the rangeof h• is consistentwith earlier
resultsof Solomon and Head [1984]. In their models,which
did not considera strongsubsurfacelayer, this parameterwas
found to be between 1 and 10 km.
Figure8 illustratesanotherinterestingoutcomeof themodels.
For the shorterwavelengthinstability,ha/h is approximately
the samefor compression
and extensionover the rangeof S•.
Thus for similar surface layer thicknesses,the dominant
ZUBER: VENUS LITHOSPHERIC STRUCTURE
wavelengths
for theseextensionaland compressional
instabilities
should be comparable.From Table 1 it can be seenthat the
shortwavelengthspacings
of featuresinterpretedas extensional
and compressional
fall in the samerange.Alternatively,for S•
< 1 and a given surfacelayer thickness,the longer dominant
wavelengthfor compressionis considerablygreater than that
for extension.On Venusthe ridge belts,which are thoughtto
haveformedin compression,
havea spacingof approximately
300 km. In contrast,rifts,whichformedin extension,havewidths
that do not exceed200 km. If the thicknessesof the strong
surfacelayers in regionsin which these features are present
are not significantlydifferent, then the predicteddominant
wavelengthsfor unstableextensionand compressionshownin
Figure 8 may explain the differencesin scalesof the longer
wavelengthfeaturesand the similaritiesin the spacingsof the
shorterwavelengthfeatures.However,on the basisof the models
alone, it is not possibleto distinguishbetweenan extensional
and compressional
originfor surfacefeaturesthat exhibit only
the shorterwavelengthspacing.
E$49
mantle layer; the spacingof thesefaults controlsthe width of
the rift. In terrestrialrifts wheregood quality gravity, seismic,
and heatflow data exist,evidencefor a coherentdown-dropped
mantle block is not apparent[cf. Rambergand Morgan, 1984].
If through-goingnormalfaultsasimpliedby the Vening-Meinesz
model exist in the Venus mantle and control
the widths of rift
zones,then subsurfacedeformation associatedwith rifts on this
planet must be fundamentallydifferentthan that characteristic
of the best-studied terrestrial rift zones.
Crustal Rheology and Thickness
In the Resultssection,we notedthat in a multilayeredmedium
the longer wavelengthof instability can be suppressed
if the
mantle layer is weaker than the upper crustallayer. If tectonic
features that exhibit two dominant wavelengthsformed in
responseto unstabledeformation,then their presencein a given
regionimpliesthat the upper crustin that area is either weaker
or at least not significantlystrongerthan the upper mantle.
This is consistentwith the depth distributionof lithospheric
strengthshownin Figure 4. Unfortunately,this doesnot better
Mantle Rheology
constrainthe compositionof the Venus crust, as a range of
For the arbitrarily chosencrustalthicknessshownin Figure plausiblecrustalmaterialssatisfythis requirement.
4, deformation in the mantle just below the base of the crust
The existenceof two scalesof deformation requires strong
occursin a brittle manner.However,if the crustweresufficiently crustaland mantle layersthat are separatedby a weaker lower
thick, deformationeverywherein themantlewouldbedominated crust.For anupperlimitof thethermal
gradient
of 25 K km-•,
= 10-•5s
-1,anda diabase
composition,
theminimum
crustal
by ductileflow. For the parametervaluesassumedin Figure •-xx
4, this would occurfor crustalthicknesses
greaterthan 17 and thickness for which a weak lower crust occurs is about 5 km.
11 km for extensionand compression,respectively.Previous A lower strain rate or a dominant crustal mineral with a weaker
studieshave shown that the growth rates of extensionaland ductilestrength(e.g.,feldspar)wouldpermita somewhat,though
compressional
instabilities
varydirectlywith the stressexponents not significantly,smallervalue. An approximateupper limit
of the strong layers [Smith, 1977; Fletcher and Hallet, 1983; of the crustalthicknessin a regionthat containstwo wavelengths
Zuberet al., 1986].Thereforea modellithospherewith a plastic of deformation is 30 km, given an estimatedlower limit for
(brittle) mantle layer is more unstablethan that with a power- thethermal
gradient
of 10K km-• andtheconditions
assumed
law viscouslayer.The resultsshownthusfar havebeencalculated above. For a thicker crust the lithospheredoes not contain a
assuminga nonlinearviscous(n3- 3) mantlelayer;substitution regionof uppermantlestrength.Variationsin thermalgradient,
of a plastic layer would slightly decrease the dominant strain rate, and compositionwould alter this value in a manner
wavelengthof the longerwavelengthof instabilityand increase similar to that describedabove. For example, for dT/dz = 25
is about15 km. The
the dominant growth ratesof both long and short wavelength K km-•, the maximumcrustalthickness
instabilities.The implicationsof this are that the lithosphere overallrange of crustalthicknesses
in regionsthat exhibit two
would be even more unstablethan suggestedby the resultsup scalesof deformation as determinedfrom the presentmodels
to this point. To producemultiple wavelengthsof lithospheric is lessthan the thicknesspredictedon the basisof petrological
deformationin the presentmodel, the upper crust and upper argumentsregardingthe depth of partial meltingin the Venus
mantle mustbe strongerthan the lower crust,but the dominant mantle [Anderson, 1980], but is in general agreementwith
style of deformationil• the strongpart of the mantle can be estimates from thermal [Morgan and Phillips, 1983] and
either brittle or ductile. For the strain rate and compositions lithosphericstress[Banerdt,1986]models.
The existenceof a single,shorterwavelengthof deformation
assumedin Figure 4, thermal gradientsup to approximately
25K km-• willaccommodate
a strong
mantleregionofsufficient in a region can be explainedeither by a lithospherein which
thickness(only a few kilometers)to permit the development the crustis thick enoughthat the underlyingmantleis too weak
of a long wavelengthof instability. This range of thermal to allow the growth of a long wavelengthinstability, or a
gradientsis consistentwith estimatesderived from thermal lithospherethat containsa strongupper mantle but in which
The latter
modelsthat assumeboth purelyconductive[Solomonand Head, the longerwavelengthinstabilityhasbeensuppressed.
1982] and hot spot [Morgan and Phillips, 1983] heat loss on is possibleif strengthcontrastsin the lithosphereare small.
Venus.
Of the tectonicfeaturesdiscussed
in this study,only the banded
The style of deformation in the Venus upper mantle as terrain in Ishtar Terra exhibits a single, short wavelengthof
suggested
by the geometriesof tectonicfeatureshas also been deformation; however, in radar maps of the Venus northern
addressedby Banerdt and Golombek[1986], who applied the hemisphereproducedfrom the Venera 15/16 data [Barsukov
wedgesubsidencemodel of Vening-Meinesz[1950] to explain et al, 1986; Basilevskyet al., 1986], many ridge-and-groove
the widths of rifts. In their study, extensionof a model Venus patterns that display only short wavelength spacings are
lithospherecontaining a brittle mantle layer results in the apparent.Morgan and Phillips[ 1983]determinedthat a crustal
formationof a simplegrabenin themantle.The grabenisdefined thicknessof up to 60 km is required in Ishtar Terra on the
by normal faults that bound a down-droppedsectionof the basisof isostaticcompensationmodels,but noted that in other
E550
ZUBER: VENUS LITHOSPHERIC STRUCTURE
areasa thick crust is not required by the gravity data. A thick
crustin Ishtar hasalsobeensuggested
by Banerdtand Golombek
[1986] in an applicationof the elastic-plasticbucklingmodel
of McAdoo and Sandwell[1985]to the bandedterrain.In other
regionsof Venus that exhibit only the shorterwavelengthof
deformation, the modelsdescribedin this study suggestthat
detailsof the lithosphericstrengthstratificationcan explainthe
absenceof longer wavelengthtectonicfeatures.An underlying
thick crustcannotbe ruled out, but is not required.
CONCLUSIONS
We have invoked unstable deformation in a density- and
strength-stratified
lithosphere
to explainthecharacteristic
widths
and spacings of many tectonic features on Venus. For
compression
and extensionwe haveshownthat two wavelengths
of deformationcan occurif the lithospherecontainstwo strong
brittle or ductile layers, one at the surface and the other at
depth,that are separatedby a weakerductilelayer.We interpret
the strong layers as the upper crust and upper mantle, and
the intermediatelayer asthe lower crustin the Venuslithosphere.
The upper crustal layer primarily controls the geometriesof
featureswith spacings
in the range10-20 km suchasthe banded
and ridge-and-grooveterrains,while the strongmantle region
primarily controls the geometriesof featureswith widths or
spacingsof 100-300 km, suchas rift zonesand ridgebelts.
The modelspredictthat the smallerscalecompressional
and
extensionalfeaturesshould exhibit comparablewavelengths.
This is consistent with the observation
that surface features in
and the style(s)of tectonicsare to be discerned[e.g., Kaula
and Phillips, 1981;Head et al., 1981]. Further understanding
of the lithosphericstructureof Venusthroughmodelssuchas
thosepresentedin this paper and future high resolutionradar
imagingof surfacefeatureswill be requiredto helpachievethese
goals.
Acknowledgments.I thankMarc Parmentierfor helpfuldiscussions,
Claude Froidevaux for a constructivereview, and Jim Garvin for useful
commentson an early draft of the manuscript.This work wasinitiated
while the author was with the Departmentof GeologicalSciencesat
Brown Universitywheresupportwas providedby NASA grant NSG7605. Subsequentsupportwas providedby a National Academyof
Sciences-National
ResearchCouncilAssociateship.
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(ReceivedMay 23, 1986;
revised November 18, 1986;
acceptedDecember9, 1986.)