JOURNAL OF GEOPHYSICAL
Finite
2.
Strain
RESEARCH, VOL. 91, NO. B3, PAGES 3664-3676, MARCH
Calculations
of Continental
10, 1986
Deformation
Comparison With the India-Asia Collision Zone
PHILIP
ENGLAND
Department of Geological Sciences,Harvard University, Cambridge, Massachusetts
GREGORY
HOUSEMAN
ResearchSchool of Earth Sciences,Australian National University, Canberra
Numerical experiments on a thin viscous sheet model for deformation of continental lithosphere
subjectedto an indenting boundary condition yield distributionsof crustalthickness,of stressand strain
rate, and of latitudinal displacementsthat may be compared with observationsin the India-Asia collision
zone. A simpleindenting boundary condition applied to initially laterally homogeneoussheetsobeyinga
power law rheology producesresults that are in broad agreementwith the observations,provided that
the power law exponentis three or greater and the sheetcan support vertically integrated stressdifferencesof 2 x 1013(-t-5 X 1012)N m -• in regionsin front of the indenter.Under theseconditions,the
calculated deformation shows accommodation of convergenceprimarily by crustal thickening, to produce a plateau in front of the indenter. Palaeomagneticdata from India and Tibet, and the observed
distribution of topography, suggestthat much of the post-Eoceneconvergenceof India with Asia has
been taken up by deformation within Asia that involved crustal thickening. The principal difference
between
calculation
and observation
is the absence from
the calculated
strain
rate fields of east-west
extension of the plateau in front of the indenting boundary. The calculations show that once such a
plateau is formed, the buoyancy force associatedwith the crustal thicknesscontrast inhibits further
thickening and the plateau strains at less than half the rate of its immediate surroundings.Seismically
determined regional strain rates exhibit a similar distribution, with the Tibetan plateau straining at about
one quarter the rate of the Tien Shan and Ningxia-Gansu regions. Calculated principal compressive
stress orientations and regional strain rates agree with the seismically determined quantities in the
Mongolia-Baikal, Tien Shan, Tibet, and Ningxia-Gansu regions of Asia, to within the uncertainty of the
latter. The vertically integrated stressesthat are calculated for the viscous sheet are comparable with
those that can be supported by a rheologically stratified continental lithosphere obeying laboratorydetermined flow laws. We suggestthat the thin viscous sheet model, described in this paper and its
companion, gives a simple and physically plausible description of the observed deformation in central
Asia; in this description the predominant mechanism of accommodation of continental convergenceis
diffuse crustal thickening, with shear on vertical planes playing a subsidiary role once large crustal
thickness contrasts have been established.
1.
INTRODUCTION
The collision of India with Asia probably occurred during
the major decrease in India's northward velocity in the
Eocene, and since then, India has moved approximately
northward by between 1800 and 2500 km [McKenzie and
Sclater, 1971; Molnar and Tapponnier, 1975]. Although some
of this convergencehas undoubtedly been taken up by deformation of the northern margin of the Indian plate (see, for
example, Gansser [1966] and summary by Lyon-Caen and
Molnar [1983]), most estimates of this deformation leave over
1500 km of convergence to be accounted for; the difference
has usually been explained in one of two ways: thrusting of
the Indian continental lithosphere beneath Asia or deformation
within
Asia.
Underthrusting of India was suggestedby Argand [1924]
and has been advocated more recently by Powell and Conaghan [1973] and Barazangi and Ni [1982]. In contrast, several
authors have proposed that diffuse deformation within Asia
has been the primary means of accommodatingIndia's northward motion [e.g., Dewey and Burke, 1973; Molnar and Tapponnier, 1975; England and McKenzie, 1982; Vilotte et al.,
1982]. As there is at present no general agreement on the
relative importance of the two mechanisms,it is worth summarizing the arguments here.
Copyright1986by theAmericanGeophysical
Union.
Paper number 5B5548.
Much of the seismicevidence that bears on this question
has been discussedby Molnar [1984]. There is clear evidence
for underthrusting of India beneath the Himalaya at present
[e.g., Ni and Barazangi, 1984; Baranowskiet al., 1984], but
this seismicityextendslessthan 150 km north from the Himalayan front. To the north of the High Himalaya the seismicity
is diffuse, and in the Tibetan plateau, fault plane solutionsof
most earthquakes show large components of normal and
strike-slipfaulting, indicating a combination of east-westextensionand north-south shorteningof the region [Molnar and
Tapponnier,1975, 1978; Ni and Yorke, 1978; Molnar and
Chen, 1983; Molnar and Deng, 1984]. We shall discussthe
diffuse seismicityfurther in section3, but note here that if a
major underthrustdoes exist beneath the Tibetan plateau, it
has no seismicexpression.
Barazangi and Ni [1982] argue on the basis of efficient
propagationand high velocity of P, and S, acrossTibet that
the plateau must be underlainby a shieldlikestructure,which
they infer to be the Indian continental lithosphere,and conclude that their observations
are inconsistent
with the forma-
tion of the Tibetan plateau by the diffuse deformation of Asia.
This conclusion may be disputed on two grounds. First, regardlessof how the Tibetan crust has doubled, it seemsto
have done so in a time that is short compared with the thermal time constant of the lithosphere, and we should expect,
therefore,that temperaturesat the Moho have not been greatly perturbedfrom their precollisionvalues;it is not legitimate,
then, to argue againstdiffusedeformation within the Tibetan
plateau from the absenceof low P, or S, velocitiesthere.
0148-0227/86/005B-5548505.00
3664
ENGLANDAND HOUSEMAN:FINITE STRAINCALCULATIONS
OF CONTINENTALDEFORMATION,
2
Second,as Molnar and Chen [1984] argue, even if Sn velocities
are "shieldlike" beneath much of Tibet, the thickness of the
layer in which thesevelocitiesoccur is unconstrained.Molnar
and Chen [1984] interpret $-P residualsobtained from earthquakes in Tibet as indicating that the mean crust and upper
mantle S wave velocity beneath central Tibet is 4-8% lower
than beneaththe Himalaya.
Palaeomagneticdata strongly support the idea that most of
the convergence of India with Asia was accommodated by
shorteningwithin Asia. The measuredinclinations on the Cretaceous Takena formation and the Eocene Lingzizong volcanics of southern Tibet show that the southern margin of
Tibet has moved north by around 2000 km since the collision
of India with Asia [Zhu et al., 1977; Molnar and Chen, 1978;
Achache et al., 1984].
For these reasons, it seemslikely that much of the convergence between India and Asia is taken up by deformation
within Asia rather than at the suture zone [-Molnar and Tapponnier, 1975]. The purpose of this paper is to compare the
results of numerical experimentson a continuum model (presented by Houseman and England [this issue], hereafter re-
ferred to as paper 1) for continentaldeformation with observations made in the India-Asia
collision
zone. In this model
the
continental lithosphere is treated as a homogeneousviscous
layer whosedeformationcan be calculatedby consideringthe
verticalintegralsof the stresses
actingon it and of its rheology
(which is taken to be that of a power law fluid). For a given
choiceof boundary conditions,the calculateddeformation depends on two parameters: the dimensionlessArgand number
Ar, which is a measure of the lithospheric strength, and the
power law exponent n of the rheology (seepaper 1, section 2).
We shall compare the results of calculations using this model
with observations in the India-Asia collision zone of topography, present stress,and strain rate fields and with measures of
Tertiary deformation provided by palaeomagnetic data. The
comparison provides constraints on the effective values of n
and Ar in a collision zone and elucidates the physical mechanisms that control
the distribution
of deformation
within
3665
between predominantly plane strain and three-dimensional
deformation is an important one, which determines not only
the configuration of deformation but also its dynamics [England and McKenzie, 1982; paper 1].
In order to assessthe importance of crustal thickeningin
the Cenozoic convergenceof India with Asia, we make a
simple mass balance calculation. Following Cochran [1982],
we require any column of continentallithosphereto be in
isostatic balance with a reference mid-ocean ridge column,
consistingof a thicknesshw of seawater,density Pw, and a
thicknessof ht, of crust, density pt,, which are underlain by
asthenosphereof density Pa' A column of continental lithosphereof thicknessL, crustalthicknessS, averagecrustaldensity Pc, and average mantle density p,, has an elevation e,
given by
e = 1/Pa[S(pc -- Pro)- L(Pm-- Pa)- hw(Pa-- Pw)- ht,(Pa-- Pt,)]
(1)
We assume that before the collision of India with Asia, the
presently elevated region of Asia was at an average elevation
eo, whose value we allow to vary between 0 and 500 m above
sea level, and that changesin elevation have been generated
by isostaticallycompensatedcrustal thicknesschangeswithin
a constant thicknesslithosphere.
Consider a region originally of area Ao, but now of area A,
as a result of deformation during the India-Asia collision. If
volume is conserved,
(e- eo)
dA- So•a 1-
(Ao
-
(2)
where So is the crustal thicknesscorresponding to elevation eo.
If the area of Asia has been reduced by an amount of convergence C between India and Asia along a front of width W,
cw = (A - Ao)
Asia.
-
-
(3)
A
2.
TERTIARY VERTICAL STRAIN OF ASIA
The first quantitative approach to the deformation of Asia
in continuum mechanics terms was made by Molnar and Tapponnier [1975] and Tapponnier and Molnar [1976-1, who applied the results of slip line theory to the indenting of Asia by
India. They were careful to point out the instantaneousand
infinitesimal nature of slip line field theory and that their assumed plane strain deformation could only be justified in the
presenceof a vertical load on the lithosphere, which could be
provided by variations in crustal thickness,such as the present
plateau. Clearly, finite amplitude continental deformation
cannot be treated usinga plane strain model unlessthe strikeslip motion is much greaterthan the vertical strain involvedin
forming topographic and crustal thicknesscontrasts.Several
interpretationsof the integratedhistory of continental deformation have been in terms of purely plane strain finite displacements,and Tapponnieret al. [1982] have suggestedthat
convergenceof India with Asia has been taken up almost
entirelyby the lateral motion of portionsof Asia on strike-slip
The integral in equation (3) is estimated from the 10 x 10
arc min topographic averagesof the Navy Fleet Numerical
OceanographyCenter. The area A is chosento be the region
of Asia lying between 65ø and 110øE and between 50øN and
the Himalayan front (see footnote to Table 1); the quantity
•Ae da is 2.24x 107km3. The width W of the convergent
front is given a value of 2200 km; this is the distancebetween
the points 1 and 2 in Figure 1, which we take as being the
west and east ends of the Himalayan front. From the equation
(1),
So•aa1--
= eo
+L(pro-1)
+hw(1•-•a•)+
ht,(1--•aa)
(4)
we use Cochran's choice of values for hw(2.5 km), ht,(5 km), Pw
(1.03 Mg m-3), and pt, (2.8 Mg m-3) for the coefficientof
thermal expansion and the 0øC density of mantle material
faults.
(3.4 x 10-5 øC-• and 3.33 Mg m -3) and for the astheno-
It is our contention, based on the results of calculations
using the thin sheet model (England and McKenzie [1983],
En•tland et al. [1985], paper 1, section 3, and below) that the
large crustal thicknessand topographic contrasts in Asia were
produced by the collision of India with Asia and that they
sphere temperature (1333øC). Thus, once a value for e0 is
chosen,the convergenceC calculatedfrom equation (3) depends on the choice of L and Pm(equation (4)); C increases
with decreasingPmor L. We considerlithospherethicknesses
L of 100-150 km and average lithospheric mantle densities
of 3.21-3.23, consistentwith Moho temperaturesof 800øC and
representa large fraction of the total strain. The distinction
3666
ENGLANDANDHOUSEMAN:
FINITESTRAINCALCULATIONS
OFCONTINENTAL
DEFORMATION,
2
TABLE 1. Convergence
Calculated
FromAreaIntegralof
Elevation
eo,
m
0
100
300
500
; edA
--eoA
,
A
km 3
2.24 x
2.14 x
1.95 x
1.78 x
10?
107
10?
107
in Asia
C,
km
C + C1,
km
3100-2100
2900-2000
2500-1700
2200-1500
3300-2300
3100-2200
2700-1900
2400-1700
Seetext for symbols.
The integralIIA e dA is evaluated
for the
region,A, boundedby the lines65øE,50øN,110øE,and (30øNfrom
65ø to 85øE,28øNfrom80øEto 95øE,and25øNfrom95ø to 110øE).
Thehigherandlowervaluesof calculated
convergence
correspond
to
the combinations
(L- 100 km; Pm= 3.21Mg m-3) and (L- 150
km; pm- 3.23Mg m-3), respectively.
Moore, 1971] the estimatesof convergencein Table 1 are
increasedby approximately 200 km, to 3300-2300 km for
e0 = 0 and to 2400-1700km for e0 = 500 m.
The estimateof 1700-3300km for the convergence
that is
accommodatedby crustal thickeningin Asia should be compared with the estimate of about 1800-2500 km of conver-
gencesincethe major decrease
in India'snorthwardvelocity
[McKenzie and Sclater,1971; Molnar and Tapponnier,1975;
Molnar et al., 1981] and with the estimatesbasedon palaeomagnetic data [Zhu et al., 1977; Molnar and Chen, 1978;
Achacheet al., 1984] that there has been around 2000 km of
shorteningnorth of the Tsangposuturezone sincethe early
Tertiary. Note that the latter figure does not include 300-500
km of underthrusting
at the Tsangposuturezoneand southof
it [Gansser,1966;Lyon-CaenandMolnar, 1983].
The greatestsourceof uncertaintyin this calculationlies in
estimatingthelithospheric
thickness
in Asia;interpretations
of
surfacewave dispersionacrossTibet [Romanowicz,1982;
400øC, respectively.Table 1 shows the range of calculated
convergence
for differentvaluesof e0. Thesecalculationsgive
an estimateof 3100-2100 km for the amount of convergence Jobertet al., 1985] suggest
that the top of the low-velocity
betweenIndia andAsiaalongthe Himalayanfrontif the orig- zone is at a depth of around 100 km, but the relation of this
observation to the isostatic calculation is uncertain. Note that
inal elevation were zero. This figure reducesto 2200-1500 km
for an original elevation of 500 m.
we have assumedthat the lithospherehas the samethickness
The above estimatesignore material that has been eroded
beforeand aftercrustalthickening.If we had assumedthat the
from Asia since the collision; a conservative estimate of the
lithospherethickenedalong with the crust, we shouldhave
volumeof this materialmay be obtainedby taking doublethe estimatedgreaterconvergencefrom the sameelevationdistri-
volumeof about107km3 of Neogenesediment
in the Bengal bution;equally,if thelithosphere
thinnedduringcrustalthick-
fan [Curray and Moore, 1971]. The additional convergence ening,lessconvergencewould be calculated.
implied is
The calculationabove suggeststhat most of the convergencehasbeenabsorbedby crustalthickeningand that there
x 107
P.•s
km
C1--2W'So
Pm
(5)
wherePsis the sedimentdensity.For Pc= 2.8 Mg m-3 [Cochran, 1982] (seeabove)and Ps= 2.2 Mg m-3 [Curray and
is littleconvergence
that stillneedsto be accounted
for by the
plainstrainmechanisms
advocated,
for example,by Tapponnieret al. [1982]; the resultsof this calculationare certainly
inconsistent
with the hypothesis
that horizontalplanestrain
A
40 ø N
D
TARIM
UASIN
/V FRON
80 • E
I00 ø E
Fig. 1. Sketchof regions
referredto in thetext.The 2000-and4000-mcontours
aremarked,andtheareasabove4000
m arestippled.
Pointsmarked"1" and"2" arereferred
to in section
2, andthesixareaswhoseseismically
determined
strainrates[-MolnarandDeng,1984]are discussed
in section3.3 are shownby thicklinesand the lettersA-F: A,
Mongolia-Baikal;
B,TienShan;C, TibetanPlateau;D, Ningxia-Gansu;
E, Xianshuihe
andwestern
Sichuan;
F, Yunnan.
ENGLAND AND HOUSEMAN:FINITE STRAIN CALCULATIONSOF CONTINENTALDEFORMATION,2
has been the dominant means of accommodating convergence
between India and Asia in the last 40-50 m.y.
3.
COMPARISONS OF RESULTS OF THIN
WITH
SHEET CALCULATIONS
OBSERVATIONS IN ASIA
Below we compare the results of calculations using the
model discussedin paper 1 with the principal observablesthat
relate to the strain history of the India-Asia collision zone.
These are the crustal thickness distribution, the seismically
estimated stress and strain rate fields, and the latitudinal dis-
placementsinferred from palaeomagneticdata. The original
shapeof and the boundaryconditionson the thin sheetthat
we have used are shown in Figure lb of paper 1. The calculations discussedin this paper do not addressthe rheology of
the indenter, and in comparingthe calculationswith the observations in Asia, no distinction has been made between the
mechanicalpropertiesof material of the Indian plate that has
deformed since the collision and material north of the Indus-
Tsangpo suture zone. Except where indicated, we use the dimensionalconstantsgiven in Table 1 of paper 1 and choose
the axis of symmetry in our calculationsto be coincidentwith
a line running N20øE through the middle of the Himalaya, at
about
85øE.
3.1. Crustal Thicknessand SurfaceElevation
The calculated crustal thicknessdistribution after a given
amount of convergenceis governed by the stress-strainrate
exponentn and by the Argand numberof the sheet(seepaper
3667
2 or for a greater initial crustal thicknessthan the 35 km used
in the calculations.
Crustal thicknessis related to surfaceelevation by equation
(1), so the above seismicobservationscan be used to estimate
the average crustal density of the plateau. The average elevation of land within the 5-km contour in Asia is 5.4 km (Navy
Fleet Numerical Oceanography Center)' if we assume that
crustal thicknessesin this area fall within the range quoted
above (65-80 km) and assumealso that crust whose elevation
is zero is between 30 and 35 km thick, then the range of
crustaldensityfor Pm= 3.27 Mg m-3 is 2.70-2.93 Mg m-3
(the lower limit being for S - So = 30 km, and the upper limit
for S- So = 50 km).
To quantify the comparison between observed and calculated elevation differences, we show in Figure 2 the hypsometric curves for the relevant part of Asia and for the numerical calculations.
The
elevation
distribution
in Asia
is taken
from the Navy Fleet Numerical Oceanography Center
10 x 10 arc min compilations over the area 45ø-125øE and
15ø-60øN and illustrated by plotting the area that lies above a
given elevation as a function of that elevation (thick curves,
Figure 2). Provided that the area covered by Figure 1 is
always included, this curve is only sensitiveto choice of geographic area at elevationslessthan 1500 m.
The correspondingelevation distribution for the numerical
experiments
is obtainedusingPc= 2.8 Mg m-3, eo= 0, and
So = 35 km in equation (1). The uncertainty in calculated elevation correspondingto the range in crustal densitycalculated
above is shown for selectedpoints in Figure 2 (see caption).
1, Figure 4)Increasing (decreasing)Pc correspondsto extension (compresEstimates of crustal thicknessin Tibet have been made by
sion) of the elevation axis relative to the hypsometric curve. In
Chen and Molnar [1981] using surface wave dispersion and
addition, uncertainty in eo (not shown in Figure 2) allows the
the travel times of Pn and Sn phases,by Romanowicz [1982]
calculated curves to be translated parallel to the elevation
and Joberr et al. [1985] using surfacewaves and by Hirn et al.
axis, and the curves can be further modified by changing the
[1984a, b] using wide-angle reflection profiling in the Lhasa
total convergence in the experiment (e.g., the case n = 3,
block. The estimatesof averagecrustal thicknessof the plaAr = 1 in Figures 2a and 2c).
teau lie in the range 65-75 km, and Chen and Molnar [1981]
The important featuresof the Asian hypsometriccurve are a
considerthat 80 km is an upper limit.
Several of the calculations yield plateau crustal thicknesses break in slope at approximately 2 km elevation and others at
that are within the range of 60-80 km (Table 2), the maximum approximately 4.5 and 5.5 km elevation. Thus those calculated
profiles that show a relatively uniform slope that is steeper
plateau crustal thickness being 74 km for n--10, Ar--1.
Note, however, that greater plateau thicknessesare calculated than the Asian curve (n= 1, Ar=O in Figure 2b' n=3,
for more convergencethan the 2000 km (40 Ma) usedin Table Ar = 3 and n = 3, Ar = 10 in Figure 2a; and n = 10, Ar = 10
in Figure 2b) are unsatisfactory,regardlessof the choice of Pc
or eo.
TABLE 2.
Calculated Plateau Crustal Thickness and Driving
Forces
Plateau
Crustal f
Thickness,km
Argand
n
Number
3
3
5
5
5
10
10
10
0
1
1
2
3
1
2
3
Axis
64
59
66
62
59
74
67
63
Average
64
59
67
62
60
74
67
64
(al -- {73)dz,
Pc/Pm
0.81
0.78
0.83
0.80
0.78
0.86
0.83
0.81
10•2 N m-•
...
24.8
33.2
17.8
12.6
38.8
22.4
16.4
Plateau crustal thickness: "Axis" gives maximum crustal thickness
on the y axes after 40 m.y. "Average" is average crustal thickness
within region C (seesection3.3). Vertical integral of (a• -- a3) is evaluated at the center of the indenter boundary. Dimensional stressesare
calculatedfrom the model parametersof paper 1, Table 1, exceptthat
Pc/Pmis chosen so that the average plateau crustal thicknesscorresponds to a surface elevation of 5.4 km. Dimensional stressesare
undefined for zero Argand number (see paper 1, section 2) (also see
Englandand McKenzie [1982, p. 301]).
Figure 2c shows the calculated elevation distributions for
four combinationsof n, Ar and convergentdisplacement(between 1800 and 2250 km)' these agree to within the uncertainty with the observeddistribution, except at 3 km, where
the observedvalues lie about 200 m outside the uncertainty
interval of thosecalculationswith Ar > 0. Except for the case
n = 3, Ar- 0, these curves diverge systematicallyfrom the
observed elevation distribution, in having greater area between2 and 3.5 km elevation-thisis consistentwith the qualitative observation that may be made from Figures 3 and 4
that the observedtopographydropsmore steeplyaway from
the plateauthan doesthe calculatedtopography.
The distributionof elevationis illustratedin Figures3 and 4
for the four calculationswhosehypsometricdistributionsare
shownin Figure2c, usingPc- 2.7 Mg m-3 (beingthe lower
end of the aboveestimates).In eachof the casesillustrated,the
convergenceresultsin a plateaulike region of approximately
the same horizontal
dimensions as the indenter. The knots of
very high topography (very thick crust) at the cornersof the
indenter result from the no-slip condition on the indenter
boundary' a boundary condition that permitted slip along
strike would probably diminish these contrasts, as would the
3668
ENGLAND
AND
HOUSEMAN'
FINITE
STRAIN
CALCULATIONS
OFCONTINENTAL
DEFORMATION,
2
action of erosionin nature.The calculatedplateauelevation
outsidetheseknotsis above5 km for thecases
(n = 3, Ar = 0;
n= 10, Ar=2) and above 4 km in the other two cases;
choosinga valueof 2.9 Mg m-3 for pc would reducethese
elevationsto 4 and 3 km, respectively.
Table 2 indicatesthe
12
I
O
I
Ar=O
x
n=3
Ar=
ß
n=3
n=3
Asio
Ar=3
Ar=10
•
•
valuesof Pc/Pro
that wouldbe requiredto yieldaverage
pla-
I
n:3
I
teau elevations of 5.4 km for different combinations of n and
Ar discussed
in thispaper.
EnglandandHouseman
•1985] showthat lateralheterogeneitiesin the lithosphere
caninfluence
the lengthscaleof the
deformation
asprofoundly
asdo differences
in the valueof n;
theTarim Basinmayreflectan anomalously
strongregionin
the India-Asiacollisionzone•e.g.,Molnarand Tapponnier,
1981], and thin sheet calculations that contain such a hetero-
o
(D
geneityshowmuchsteeper
slopeswheretheplateauabutsthe
strongregionthando the calculations
reportedhere•seeVilotte et al., 1984; Englandand Houseman,1985]. In view of
theseconsiderations,
we regardany effectivepowerlaw exponentn between3 and 10 (andpossibly
greaterthan 10)as
0
x xj
I
I
.
2
3
4
5
6
7
8
Surfoce Height, H km
consistent with the observations.
Although the comparisonof Figure 2 seemsto rule out
valuesof Ar greaterthan about 1 for n = 3, or 3 for n = 10,
numericalexperiments
that includea strengthheterogeneity
representing
the Tarim Basin[Vilotte et al., 1984;Englandand
k '
'
I • .•. •
[ h.
• " •• •x
"IOI-
'
'
'
I
I
q
' n=10
Ar=10 /
I k • •'
Houseman,1985] may allow theselimits on Ar to be increased
'
o n:l Ar=O
• x n=10
n=10At=
3
Ar=
2
by a factor of approximately3. In the next sectionwe usethe
four calculations
of Figure2c to investigate
the influenceof
varying n in comparisonsbetween calculated and observed
stress and strain rate fields.
3.2.
Stress Orientation and Fault Plane Solutions
Molnar et al. [1973], in a studyof the focalmechanisms
of
large earthquakes in Asia, noted that the P axes of these
eventswerealignedbetweenN-S and NE-SW.Althoughthe
viscous
sheetmodeldoesnotcontaindiscontinuities,
it ispos-
•
b)
0
sible to compare the stressfields calculatedfrom it with the
I
seismic
observations,
if it is assumed
that thebrittlelayerof
X•x x
2
3
4
5
6
Surfcce Height, H km
7
i
i
i
6
7
8
the lithosphereaccommodatesthe samestrain rate field as its
viscous
substrate
(paper1,section
3.3andAppendix
A).
In Figures4a and4c we showtheorientations
of theprin-
12
i
i
cipal compressivestressdirections calculatedfor the thin vis-
coussheetwith n = 5, Ar = 2, after 40 m.y. of deformation
andfor n -- 3, Ar = 1 after45 m.y.'thesearesuperimposed
on
O
n = 3
Ar=O
x
n = 3
Ar=
F1
n = 5
Ar= 2
•
n = I0
Ar=2
•
Asio
I0
contours
of thecrustalthickness.
It canbeseenthattheprincipal compressive
stressaxesin generallie parallelto the
steepest
gradients
of crustalthickness
(andtopography).
Very
similarresultsare obtainedfor the casewheren = 10,Ar = 2
•
I
and wheren - 3, Ar = 0. Figure 4b showsthe orientationsof
the P axesfor earthquakes
analyzedby Tapponnier
and
Molnar [1977, 1979],Molnar and Chen[1983] and Molnar
andDeng[1984]' we haveincludedthoseearthquakes
from
TableA1 of Tapponnier
andMolnar[1977],Table1 of Tapponnier and Molnar [1979], Table 1 of Molnar and Chen
[1983],andTable1 of MolnarandDeng[1984],whoseP axes
plungeat lessthan60øandarewell-constrained
(_+30ø)bythe
datapublished
by theseauthors.
In interpreting
thesedata,it
must be rememberedthat it is often difficult to constrainthe
(D
o
c)
I
I
I
I
2
3
4
5
8
Surfoce Height, H km
Fig. 2. Comparison of hypsometric curves for Asia with those for
axesof eventsin Asiato betterthan _+30ø,particularly
for
thin sheetcalculations
usingPc= 2.8 Mg m-3. Error bars through
thrustor normalfaults[MolnarandChen,1982].Neverthe- the
symbolsrepresentuncertainty in elevation attributable to uncerless,the earthquake
datashowa strikingsimilarity
withthe tainty in Pc; left end of bar calculatedusingPc- 2.9 Mg m-s and
are shownfor
calculations
shownin Figures4a and4c,particularly
in the right end,Pc= 2.7 Mg m-s. For clarity,uncertainties
relationof theP axisorientation
to thegradient
of topogra- only a few pointsin eachfigure.(a) Thin sheetcalculationswith n - 3,
phy.
It appears, then, that the stressorientations inferred from
present day seismicity in Asia are consistentwith the defor-
after 40 m.y. (b) Thin sheet calculations with n- 10, and n- 1,
Ar = 0, after 40 m.y. (c) Thin sheet calculationswith n = 3, Ar = O,
n = 5, Ar - 2, after 40 m.y.; n - 10, Ar = 2, after 36 m.y.; and n = 3,
Ar = 1, after 45 m.y. (seeFigure 4c).
ENGLAND
ANDHOUSEMAN:
FINITE
STRAIN
CALCULATIONS
OFCONTINENTAL
DEFORMATION,
2
3669
b
Fig.3. Distributions
ofelevation
calculated
fromthethinsheet
model
andobserved
inAsia.
Contours
ofelevation
are
at 1 km.Observations
in AsiaarefromNavyFleetNumerical
Oceanography
Center,
smoothed
witha Gaussian
filterof
width 1ø. Calculated
crustalthicknesses
are converted
to elevationusingequation(1) with e0= 0, So= 35 km, and
Pc= 2.7Mg m-3, Pm=3.27Mg m-3. (a)Elevations
calculated
forthinsheet
withn= 10,Ar= 2 after36m.y.of
convergence.
(b)Elevations
inAsia.(c)Elevations
calculated
forthinsheet
withn = 3,Ar = 0 after40m.y.ofconvergence.
mation of a continuous sheetwhose tectonics are governed by
an indentingboundaryconditionsimilarto that assumedhere.
3.3.
Present-Day Rates
We can continue the line of argument of the previous sec-
tion to comparethe calculated
strainrate fieldswith the observed distribution of strain rate in Asia. Molnar and Deng
the calculationsdiscussedin this paper. Areas in the calculations that correspondto the areas in Asia consideredby
Molnar and Deng were determinedby overlayingthe finite
elementgrid for eachcalculationon an appropriatelyscaled
map(seegeographical
andmodelcoordinates
in Figures3 and
4).
For comparisonwith the finite element calculationsthe
[1984] summedthe momenttensorsfor this century'slarge strain rate tensorsgiven by Molnar and Deng [1984] are reearthquakes
in nineregionsof Asiaand calculated
the strain solvedinto symmetricand antisymmetriccomponents.Table
rate tensorfor each region.We considersix of their regions, 3 lists the azimuths of the horizontal projection of the prinstrain rate axis,the valuesof the second
which are outlined in Figure 1. The only region of major cipal compressive
activity(strainratesgreaterthan0.5%m.y.-x)notconsideredinvariant of the symmetrical component of the strain rate
istheirregionH (Himalaya);thisregionhasno counterpart
in tensorfor each region and the equivalentquantitiesfrom the
3670
ENGLAND
ANDHOUSEMAN:
FINITESTRAIN
CALCULATIONS
OFCONTINENTAL
DEFORMATION,
2
Fig. 4. Comparisonof the trendsof P axesdetermined
for earthquakes
in Asia(seetext)with the orientations
of the
principalstress
directions
in thinsheetcalculations.
(a)Contoursof elevation
at 1-kmintervals
(thinlines)andorientations
of horizontal
principalstresses
(crosses)
for a thinsheetcalculation
withn = 5, Ar = 2 after40 m.y.of convergence.
Thick
armsto crosses
indicatecompressional
deviatoricstress;thinnerarmsindicateextensional
deviatoricstress.
(b)Horizontal
projections
of P axesof largeearthquakes
in Asia(thicklines,seetext)andcontours
of elevationat 1-kmintervals(thin
lines)(seeFigure3b).(c) Sameas Figure4a exceptfor calculation
with n = 3, Ar = 1 after 45 m.y. (2250km) of
convergence.
four finite elementcalculationsof Figure 2c (n = 3, Ar = 0, 1;
n= 5, Ar=
2; n= 10, Ar=2).
The orientations of the calculated and observedprincipal
compressivestrain rate axesagreeto within 22ø for regionsA,
B, and C and to within 35ø for regions D and E, while the
observedvariationsfrom region B to regionE is nearly 100ø.
The exceptionis regionF (Yunnan):herecalculatedprincipal
compressiveand extensionalstressdirections are approximatelyat right anglesto thoseinferredby Molnar and Deng,
who expressstrongreservationsabout the extent to which the
historical seismicdata reflect the long-term strain in this
region. In view of this uncertainty and of the fact that calcu-
lated deformationin regionsE and F is very sensitiveto the
boundaryconditionsthat we useat the cornersand alongthe
sidesof the indenter,which may not be appropriatefor regionsE and F, we consideronly areasA-D for the rest of this
dicussion.
The magnitudesof the observedstrain rates (expressedin
Table3 by thesecond
invariantof thesymmetrical
strainrate
tensor;seepaper1,equation(5))varyby a factorof 4.5,with a
patternoflowstrainrateswithintheplateau
(region
C),strain
ratesabout4 times
higher
in regions
bordering
theplateau
(B,
Tien Shan;D, Ningxia-Gansu),
and lowerstrainratesto the
northandeastintheMongolia-Baikal
region
(A).
ENGLAND
ANDHOUSEMAN:
FINITESTRAINCALCULATIONS
OFCONTINENTAL
DEFORMATION,
2
3671
TABLE3. Comparison
Between
Seismically
Determined
StrainRatesin AsiaandStrainRatesFrom
Thin Sheet Calculations
Region
A
P Axis /•
Molnarand
Den•] [ 1984]
Thinsheet,
Region
B
P Axis /•
Region
C
Region
D
Region
E
P Axis /•
P Axis /•
P Axis /•
Region
F
T Axis
40ø
1.25 349ø
3.0
359ø
0.7
66ø
3.2
86ø
1.2
84ø
0.5
23ø
1.0
356ø
3.4
21ø
1.3
35ø
2.9
55ø
3.2
155ø
1.6
24ø
1.6
6ø
2.6
20ø
1.4
34ø
2.3
58ø
2.6
159ø
1.2
26ø
1.6
3ø
2.6
20ø
1.6
31ø
2.5
58ø
2.1
157ø
1.1
25ø
1.0
0ø
1.7
21ø
2.4
32ø
2.2
53ø
2.3
155ø
1.2
n-- 10, Ar=2
Thinsheet,
n=5,
Ar=2
Thinsheet,
n=3,
Ar=l
Thinsheet,
n=3,
Ar--O
Trends
ofP axes(plunges
arelessthan5ø)calculated
fromsymmetrical
portions
ofobserved
regional
moment
tensors
[MolnarandDeng,1984]andfromtheaverage
strainratetensor
ofcomparable
regions
in thecalculations
(plunges
arezero);in thelattercase
they axisisassumed
to pointN20øE.
Forreg!on
F the T axesarecompared,
asobserved
P axisis nearvertical.
Second
invariant
of strainrateE is
calculated
fromstrainratetensors
givenbyMolnarandDeng[1984]or fromregionally
averaged
onesin
thinsheet
calculations;
unitsarepercent
permillionyearsor 3.17x 10-•6 s- •
It is notoriouslydifficultto relateshort-termseismicdeformationto deformationovergeologicaltimespans,and Molnar
and Dengcautionthat thereis an uncertainty
of a factorof 2
in their estimatesbasedon 80 yearsof data; thus given the
uncertainties,
the long-termstrainratesin regionsA-D might
all be the same. Neverthelessthe presentdistribution of seismic deformation in Central Asia is consistentwith that of the
calculations. In the calculations, the build up of the crustal
thicknessin front of the indentingboundarycausesthe region
around their edgesand to an enhancementof strike-slipover
compressionaldeformationin their vicinities[e.g., Vilotte et
al., 1984; Englandand Houseman,1985]. It is possiblethat the
inclusion of such features in a model would provide closer
agreementbetweencalculationand observation,but we do
not feel that proliferation of parameterswill, at present,lead
to greater understanding.
The slightlycloseragreement
between
the calculated
and
of mostintensedeformationto migrateawayfrom thisbound-
observedstrainratesfor the sheetwith n = 10,comparedwith
thosefor n = 3 or 5, appearsto favor the higherpowerlaw
ary (seepaper1). After40 m.y.of convergence,
the mostin-
exponent.
However,themostrapidlystraining
regions
(Tien
tense deformation in all the cases described here, except for
Shan and Ningxia-Gansu)each appear to border on tec-
n - 3, Ar = 0, is in an arcuateregionthat encircles
the plateau
and corresponds
to regionsB and D in Figure 1.
tonically
stableblocks
(theTarimBasinandtheOrdosmassif,
respectively).
Thusn = 3 or 5 mayin factgiveequallyaccept-
The calculatedstrain rates in regions A, B, and D agree
with the observedones to within 25% for n = 10, Ar - 2 and
n -- 5, Ar = 1; to within30% for n = 3, Ar - 1; and to within
40% for n - 3, Ar- 0. The calculatedstrainratesin regionC
(theplateau)are all higherthanthoseobserved;
for the cases
ableresultsin thepresence
of lateralheterogeneities.
However,
thecomparisons
of thissection
suggest
thata sheetwitharbitrarilyhighstrength
(Ar -- 0),for whichgravityhasno influenceon the deformation,doesnot yield present-daystrainrate
fields that are consistentwith observation,even though the
distribution
of topography
may be acceptable
(section3.1).
For a sheetin whichAr > 0, gravityactsto inhibit further
oncea criticalcrustalthickness
is reached,
vations,and it is just outside(a factor of 2.5) for the case crustalthickening,
n = 10 and n = 5 the difference is within the uncertainty of a
factor of 2 that Molnar and Den•l[1984] attach to their obser-
n- 3, Ar-- 1; however,in eachcase,the differencebetween
calculationand observationin regionC is many timesgreater
than that in the other three regions (A, B, D). For the case
n = 3, Ar = 0 the calculatedstrainrate in regionC is 3.4 times
that observedand, furthermore,is greater than the strain rate
in regionsB and C; thus we do not regardthe casen--3,
Ar = 0 as adequatelyfitting the observations.
and to transferthe deformationto the outer edgesof the
plateau
(seeFigure5 of paper1);thisdoesnot occurfor
Ar = 0 withanyvalueof n usedin ourcalculations
(paper1,
Figure 3).
3.4.
Palaeoma•lnetic Data
Determinationsof palaeomagnetic
pole positionsin the
A consistent difference between calculation and observation,
India-Asia collisionzone are sparse,and becausemost of the
with the exception
of regionB (TienShan),is that the seismic
data show a greatercontributionto the strain rate tensors
from strike-slipmotion than do the numericalexperiments.
This differenceis mostpronouncedin Tibet (regionC), where
reliable data come from near the Tertiary suture zone, they
the calculationsshow predominantlycrustal thickening in
casesshownin Figure2c, and with the boundaryconditionsof
mainly constrainthe total amount of shortening,and not its
distributionwithin Asia. In Figure 5 we comparethe available
observations with the thin sheet calculations. For two of the
contrastto the predominantly
strike-slipmotionwith crustal Figurelb (paper1),thesolidlinesin Figure5 showthecalcuextension
shownby theseismic
data[MolnarandDen•l,1984]. lated y directiondisplacements,
after 40 m.y. of convergence,
We attribute this differenceto the no-slip condition that we
haveappliedon the boundaryof the indenter.Allowingflow
tangentialto the boundarymaypermiteast-west
extension
to
occur in the calculations.
As in the case of the crustal thickness distribution (section
for materialpointslying on linesparallelto the y axis,with x
coordinates0 (on the axisof symmetry),D/8 (halfwaybetween
the axis and the cusp), D/4 (through the cusp), and 3D/8
(halfwayacrossthe sideboundaryof theindenter).
Also shown are the observationsof Klootwijk and Radhak-
[1981] from the westernsyntaxis,of Achacheet
3.1),lateralheterogeneities
can appreciably
affectthe strain rishnamurthy
ratefield;in particular,theycanleadto stressconcentrations al. [1984] fromtheLhasablock,andof Zhu et al. [1981] from
3672
ENGLAND
ANDHOUSEMAN'
FINITESTRAIN
CALCULATIONS
OFCONTINENTAL
DEFORMATION,
2
a) xt = 0
2000
I
• Iøøø
I
E
0•
-
I
•
k
b) xt:O. 125
Ladakh
I C)
X
•:0.25
-"'" 3000.o_
o
ß
d) x½--0.375
Lhasa
2000-
•[ Outer
Pamlr
I000
-
BIOI "•----"•---L
o
0.,5
1.0
0.5
1.0
Y-Coordinate
Fig. 5. Y directiondisplacements
after40 m.y.of convergence,
fromtwo thin sheetcalculations.
Uppercurvescalculated for n -- 3, Ar -- 1, lower curvesfor n = 10, Ar = 2; curvesfor n - 5, Ar-
2 lie betweenthesecurves.Horizontal axis
is y coordinate
at 40 m.y.;B is theboundary
of theindenter.
Solidlinesshowtotaly direction
displacement
for points
(Figure5a)on theaxisof symmetry
in themodel:x - 0; (Figures5b-5d)on linesparallelto they axiswithx coordinates
respectively
D/8, D/4, and 3D/8. Data pointsfromKlootwijkandRadhakrishnarnurthy
[1981] (diamonds),
Zhu et al.
[1981](opensquares),
andAchache
et al. [1984](solidsquares)
arethedifferences
between
theobserved
palaeolatitudes
andthoseexpected
for pointsof thesamegeographical
coordinates
attached
to a rigidEurasian
plate;theseareconverted
to Y directiondisplacement
by multiplication
by sec(20ø)to take accountof our adoptedorientationof N20øEfor the
symmetry
axisof themodels.Onlydatafromrocksof Cretaceous
or youngerageareplotted.
the Lhasa block and Tibet; these latter data must be con-
tionsto matchthe topography,Ar •< 1 for n = 3 and Ar •< 3
sideredas the leastreliable,as the resultswereobtainedusing
AF demagnetizationonly (seeAchacheet al. [1984] for discussion).Thesedata are shownin Figure 5; for comparison
with the calculateddisplacements,the data from the western
syntaxis [Klootwijk and Radhakrishnarnurthy,
1981] and for
Zhamo [Zhu et al., 1981] are plotted by the profile for
for n = 10,whenthereareno significant
lateralheterogeneities
in lithospheric
strength.
The dimensional
valuesof thequantity I0t'(a•- (r3)dzevaluated
at the middleof the indenter
x - D/4, which correspondsto a line through the corner of
the indenter,and the remainingdata are plotted by the profile
for x--D/8 correspondingto lines approximately 1100 km
from the axis of symmetry of the indenter.
Keeping in mind the sparsityof the data, the uncertaintyof
magnetization ages,and the differencesbetween the motion of
India and the motion of the indenterin the numericalexperiments, these data are not definitive tests of the model. How-
ever, with the exception of the point at Zhamo, there is no
obvious disagreement,and they point to the need for more
extensivepalaeomagneticmeasurementsthroughout Central
Asia.
isostaticallycompensatedcrustal thicknessof 65-80 km for
theTibetanplateau(about7 x 10•2 N m-• [e.g.,Frank,1972;
P. Molnar and H. Lyon-Caen,unpublishedmanuscript,
1985]); they reflectonly the stressrequiredto drive the deformation of the viscous sheet.
Althougha verticallyaveragedpower law rheologyhas
beenusedthroughoutthisstudy,the discussion
of powerlaws
with n > 3 hasimpliedconsideration
of rheologies
otherthan
the powerlaw creepof silicates:in particular,it takesaccount
of thepossibility
thatthelithospheric
strength
iscontrolled
by
a combination
of frictionon faultsandsteadystatepowerlaw
creep.
In what followswe comparethe valuesof n and Ar indicated by the numericalexperimentswith valuesestimatedfor a
stratifiedrheologybasedon laboratorymeasurements.
Sucha
4.
4.1.
boundary at 40 m.y. are listed in Table 2. These valuesare
considerably
greaterthan are necessary
to supporta static,
Rheology
DISCUSSION AND CONCLUSIONS
comparison
is boundto be provisional
because
of the large
uncertaintiesin extrapolatinglaboratory determinationsof
rheologyto geologicallengthand time scales.We makeuseof
a summaryof the rheologyof the lithosphereby Braceand
Kohlstedt[1980];theydividethe continental
lithosphere
into
In the numericalexperimentspresentedin this paper, and in
paper 1, the vertically averaged rheology of the continental
lithosphere is described by two parameters: the stress-strain threeregimes:
an uppercrustthatfailsby faulting,andobeys
exponent n and the Argand number. In this section we discuss Byerlee's law, underlain by a ductile lower crust whose
the ranges of these parameters that, from the comparisonsof propertiesare governed
by steadystatecreepof quartz,which
section 3, seem to be appropriate for the deforming litho- in turn is underlainby an uppermantlewhosestrengthis
spherein Asia. The numericalexperimentsshow that accept- governedby the laboratory-determined
propertiesof dry oliable distributionsof deformationand topographycan be cal- vine [Goetze, 1978].
culated for some choiceof At, provided n is greater than 3,
Using this stratifiedrheology,we calculatethe vertically
and there is no reason,from the comparisonsof section3, to
integratedstressdifferenceunder specifiedconditionsof temprefer any one value of n between 3 and 10. For the calcula- peratureand biaxialstrainrate and comparethe resulting
ENGLANDANDHOUSEMAN'
FINITE STRAINCALCULATIONS
OF CONTINENTAL
DEFORMATION,
2
stress-strainrate relations with the model parameters, Ar and
n (paper 1, equations(3) and (6)). For details of this procedure,
see Sonder and England [1986].
The calculatedstrain rate in the region correspondingto the
3673
sphere from the crust. We let 7 vary between 5 and 15 K
km-1, corresponding
to a heat flow at the Moho of between
15 and 45 mW m-2 for a conductivityof 3 W m-1 K -1
Figure 6a illustratesthe rangeof Moho temperaturefor which
Tibetan plateauis about 3 x 10-16 s-1 (Table 3), and the
FL liesbetween1.5 and 2.5 x 1013N m-1 for the rheologi-
deformation is approximately plane strain with y axis compression balanced by vertical thickening. The vertically integrated stressdifferencecalculatedfor the three casesthat give
acceptablefits to the hypsometryand strain rate observations
(n=3, Ar= 1;n=5, Ar=2;andn=
10, Ar=2) isbetween
cally stratified continental lithospherethickening at a strain
1.5and 2.5 x 1013N m- 1(Table2).We calculatethe quantity
FL =
(01 -- 15'3)
dz
(6)
at a strain rate of 3 x 10-•6 s-• for a rheologicallystratified
lithosphere obeying the deformation laws summarized by
Brace and Kohlstedt [1980]. The al and cr3 are the greatest
and least principal stresses;the governing parameters are the
Moho temperature T2u,the thermal gradient in the mantle 7,
the depth to the brittle-ductile transition ZBD,and the ratio of
pore fluid to lithostatic pressurein the crust 2; 2 is assumedto
be zero in the mantle.
Molnar and Chen [1983] determined the focal depths of 16
crustal earthquakes in Tibet to be less than 15 km, and we
choosea value of 15 km for ZBD and let pore fluid pressure
vary from zero (2- 0) to lithostatic pressure (2 = 1); in the
latter casethere is no contribution to the strength of the litho-
A
rate of 3 x 10-16 s-1. It can be seen that these conditions are
reached for Moho temperatures between 360ø and 650øC.
Thesetemperaturesare uncertainby at least _+100øC,owing
to the uncertainty in the laboratory values of the material
constantssummarized by Brace and Kohlstedt (see,for example, discussionby Sonderand England[1986]).
With Moho temperaturesin the lower end of this range, the
upper mantle can support an integrated stressdifference between 1.5 and 2.5 x 10•3 N m -1, without contribution from
the crust,but at the highertemperatures,
the stressis supported by a combination of ductile deformation of olivine and
friction in the upper crust; when 2- 0, the crustal contribution to the strengthin Figure 6a is between35% (curvefor
S•= 0, F• = 2.5 x 1013N m-1) and 60% (for ,• = 0, F• = 1.5
X 1013N m-1).
Sonder and England [1986] show that over a strain rate
rangeof • 10-13to 10-•7 s-• the verticallyintegrated
rheology of sucha combinedbrittle and ductilelithospheremay, to
a closeapproximation,be describedby a power law, suchas
has beenusedin this paper and in paper 1. This is illustrated
in Figure 6b, where curve A approximatesthe relation between F• and strain rate for the conditionsof Figure 6a under
which friction on faults contributesmost to the verticallyintegratedrheology (point A' 7--15 K km-1, F• = 1.5
X 1013N m-1, S•= 0) and curvesB and C showthisrelation
for caseswhen friction on faults contributesnothing to the
integral (•, = 1). In each case, over the range of strain rates
7
E
illustrated(10-13 to 10-17 s- 1),the verticallyaveragedrheology is closely approximated by a power law, with n about 17
for curve A, about 8 for curve B, and about 7 for curve C; the
400
a)
500'B
• 600
Moho Temperature oc
700
differentvaluesof n in the latter two casesindicate varying
contributions to the vertically averagedrheology from Dorn
law plasticity [Brace and Kohlstedt, 1980]. The Argand
number is calculated from the value of F L at the reference
strainrate of 1.4 x 10-1,•s-1 (paper1, Table 1, equations
(3)
and (6) and Sonderand England [1986])' it lies between 1 and
2 for rheologiesthat give a value of 1.5 to 2.5 x 1013N m-1
for the driving forceFt, in Figure 6a.
13
A
n=17
The range of effectivepower law exponentn that is calculated from verticallyintegratedlaboratoryrheologyshouldbe
B
14
compared with the range of n for which the thin sheetcalcula-
tionsyieldresultsconsistent
with observations
in Asia (section
3). The comparisonof stratifiedrheologyundercompression
with the thin viscoussheetrheologyis presentedas a possible
15
explanation for n values larger than 3. However, there are
16
perhaps other stratified rheologiesthat would give similar
valuesof n and support driving forcesof the order of 1013N
17
12.8
b)
13.2
1:5.6
14.0
Iog•o(FL N m-• )
Fig. 6. (a) Conditionsof Moho temperatureand sub-Moho thermal gradient 7, for which the quantity F L (equation (6)) lies between
25 and 15 x 1012N m -• at a strain rate of 3 x 10-•6 s-•. Using
Brace and Kohlstedt's[1980] choicesof deformation law. Hatched
region showsintervals of 3' and T•t that satisfy these requirements;
values of 2 given by each line are for the crust only: 2 = 0 in the
mantle. A, B, and C refer to conditions illustrated in Figure 6b. (b)
Plots of log•o (F0 againstlog•o (•) for the three setsof conditions(A,
B, C) picked out in Figure 6a. Solid symbolsindicate Ft• evaluated at
strain rates between 10-13 and 10-•7 s-• and the straight lines are
pickedby eye.For eachline, n = d(log•og)/d[log•o(F0].
m-1. It shouldalso be noted that within the brittle layer,
Byerlee'slaw predictsthat thrust,normal, and strike-slipfaults
have differentstresses
on them at failure.Thus the depthaveraged
rheologyof the thin viscous
sheetshouldnot simply
depend on the second invariant of the strain rate tensor as
assumed in the thin viscous sheet calculations but should be a
more complicatedfunctionof the tensor,dependingon the
relativemagnitudeof thedifferentcomponents
andtheirsigns.
4.2.
Tectonic History
Previous interpretations of the India-Asia collision have
tended to emphasize the role of strike-slip faulting in the tec-
3674
ENGLAND AND HOUSEMAN'FINITE STRAIN CALCULATIONSOF CONTINENTALDEFORMATION,2
ary conditions.This discrepancydoes not invalidate the thin
sheet model, which can produce net extensionalstrain with
purely compressiveboundary conditions[England and McKenzie, 1983]. We do not regard England and McKenzie's
that is in quantitativeagreementwith the major featuresof the boundary conditionsas appropriate for the deformation in
present-daydeformationfield and requirescrustalthickening Asia (see paper 1, section3.2), but it seemspossiblethat apas the major means of accommodatingthe convergencebe- propriate boundary conditionsmay exist that could result in
tween India and Asia. We feel obliged to attempt some expla- extensional strains on the plateau once a sufficient elevation
nation for the discrepancybetweenthesetwo viewpoints.
contrast had been built up.
tonic evolution of the region (e.g., Molnar and Tapponnier
[1975], Tapponnierand Molnar [1976], and Tapponnieret al.
[1982], but see Dewey and Burke [1973] and England and
McKenzie [1983]). In contrast, we have presented a model
The thin sheet calculations with n between 3 and 10 and Ar
between 1 and 3 can reconcile these differing views to some
extent becausethe sheetdeformsfirst by thickeningand then,
4.3.
Conclusions
This paper compares the results of numerical experiments
as the resultingbuoyancyforcesincrease,a plateauis formed on a thin viscous sheet model for continental deformation
in front of the indenter and strike-slip deformation becomes with observationsof topography, stressand strain rate fields
more important(paper 1, Figures6 and 8). However,the cal- and palaeomagneticlatitudinal displacementsin Asia. It is not
culationsdo not show the degreeof strike-slipmotion that is in the nature of the simple model discussedhere that it could
seen in the focal mechanisms[Molnar and Deng, 1984] and
certainly not the apparent predominanceof suchmotion that
is inferred from satellite photographs [e.g., Tapponnier and
Molnar, 1976].
The earlier emphasison strike-slip deformation may result
partly from the fact that strike-slip faults are, by their very
nature, easier to detect from satellite photographs than are
thrust faults; quantitative studiesof the seismicdeformation in
Asia have concluded that at least half of the present-day
northward convergenceis accommodatedby crustal thickening [Chen and Molnar, 1977; Molnar and Deng, 1984].
The results of our calculations are dependent on our assumptionsof homogeneityof strengthand initial crustalthickness distribution. As has been remarked by Vilotte et al.
[1984] and England and Houseman [1985], the presence of
lateral heterogeneitiescan lead to enhancementof strike-slip
deformation around their edgeswhen compared with the laterally homogeneous models discussedhere. Alternatively,
horizontal
motions
not
accounted
for
in
our
calculations
could have occurredif, for example, the Tibetan plateau (or a
large fraction of it) existed before the collision; the related
buoyancy forceswould then presumablyinhibit further thickening and result in an enhanced contribution of strike-slip
deformation to the Tertiary deformation. England and Searle
[1986] consider the influence on the deformation of such an
elevation contrast along the southern boundary of the Lhasa
block in the late Cretaceous and early Tertiary, but the implication of their study, and of the present one, would be that if
such an elevation
contrast
existed over the whole area of the
present plateau, there should be evidence of widespread late
Cretaceous/early Tertiary compressive deformation around
the edge of the plateau. The hypothesisof a precollision plateau of comparable extent to the present one is hard to reconcile with reports of Cretaceousmarine sedimentson the western Qiang Tang block [Norin, 1946].
A prominent feature of the deformation in Tibet is the
crustalthinning expressedin the north-southgrabens[Tapponnier and Molnar, 1976; Molnar and Tapponnier, 1978; Tapponnier et al., 1981] and in fault plane solutions [Molnar and
Chen, 1983; Molnar and Deng, 1984]. This is a relatively
recent phenomenonand accountsfor only a small fraction of
the accumulated strain in Asia [Armijo et al., 1982]. It seems
likely that, as Molnar and Tapponnier[1978] suggest,the extension is driven by the extensional deviatoric stressesassociated with the elevated plateau. However, although there is
some east-west extension in the north of the plateau in some
of our calculations, it is less than the north-south shortening,
and net extension of the plateau is not present in our results
nor, so far as we are aware, in the published results of other
laboratory or numerical experimentswith comparablebound-
reproduce in great detail the features of the India-Asia collision zone nor was it our desire to do so. For instance, we
have ignored lateral heterogeneitiesin the lithosphere [-see
Vilotte et al., 1984;England and Houseman,1985-],erosion and
faulting. We carry out our calculationsin terms of the vertical
integralsof stressesacting on the thin sheet,and the observations with which we comparethe resultsare of featureson the
scale of hundreds
to thousands
of kilometers.
On this scale the results of the thin sheet calculations
are
consistent with the observations in Asia, to within the uncertainties discussedin section 3. Where discrepanciesexist, we
believe they could generallybe removed by adding complexity
to the model (see,for instance,the influence of lateral heterogeneities on the topography and strain rate distribution I-Vilotte et al., 1984' England and Houseman, 1985]).
The thin sheet calculations yield results that are consistent
with the observations,provided that the sheetsare capable of
supporting vertically-integratedstressdifferencesof 1.5 to 2.5
x 1013 N m -1 at a strain rate of 3 x 10 -•6
s-•
and have
power law exponent n greater than 3. The vertical integrals of
stress difference
at this strain
rate
calculated
for the conti-
nental lithosphere from laboratory deformation laws [Brace
andKohlstedt,1980] fall within the range1.5 to 2.5 x 10•3 N
m- • for Moho temperatures
between360øand 650ø(_+100øC)
and, under these conditions, the vertically integrated deformation laws exhibit a power-law-like dependenceof strain
rate on driving force with effective power law exponents between 6 and 17 (section 4.1; Figure 6b; see also Sonder and
England [1986]). Thus the parameter range for the thin sheet
that provides an acceptablefit to the observationsin Asia is
also consistentwith laboratory-determined rheologiesfor the
continental lithosphere.
The orientation of the principal stressesin Asia at the present day appears to be determined primarily by the configuration of the continental collision, but the magnitude and distribution of present-day strain rates is determined not only by
this configuration, as would be the casefor plane-strain deformation [e.g., Tapponnier et al., 1982] but depends also on the
past history of crustalthickeningand the consequentinfluence
of gravitational forces (section 3). Those thin sheet calculations that yield elevated plateaux comparable to the Tibetan
plateau show that the major accommodation of convergence
is by thickening of the sheet.The distribution of topography
in the India-Asia collision zone (sections2 and 3.1) indicates
that most of the convergencebetween India and Asia is expressedin crustal thickening,and we regard the present-day
prominence of strike slip deformation as a relatively recent
result of this thickening and not as a dominant mode of accommodation of convergencethroughout the Tertiary I-En•Ilandand McKenzie, 1982, 1983' England,1982' paper 1].
ENGLANDANDHOUSEMAN:
FINITE STRAINCALCULATIONS
OFCONTINENTAL
DEFORMATION,
2
3675
Acknowledgments.
The development
of this paperhas benefited Jobert, N., B. Journet, G. Jobert, A. Hirn, and Sun K., Deep structure
from discussionswith Dan McKenzie; we are grateful for reviews
from Peter Bird and from Peter Molnar, who provideda reviewthat
wentbeyondthecall of duty.Thisworkwassupported
by National
ScienceFoundationgrantsEAR81-07659and EAR84-08352and by
NASA grant NAS5-27229.
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G. Houseman, ResearchSchool of Earth Sciences,Australian National University,Canberra,ACT 2601, Australia.
(ReceivedMarch 25, 1985;
revisedNovember 5, 1985;
acceptedNovember7, 1985.)