JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. B8, PAGES 17,747-17,769, AUGUST 10, 1996
Erosion as a driving mechanism of intracontinental mountain growth
J.P. Avouac Laboratoire de Gfophysique,Bruy•res-Le-Ch•ttel, France
E. B. Burov Laboratoirede Gravimfitrieet Gfiodynamique,Institut de Physiquedu Globe de Paris
Abstract. In nature, mountainscan grow and remain as localizedtectonic features over long periodsof time (> 10 m.y.). By contrast,according to currentknowledge of lithosphericrheology and neglectingsurface processes, any intracontinentalrange with a width that exceedsthat which can be supportedby the strengthof the lithosphereshould collapsewithin a few tens of millionsof years.For example,assuming a quartz-dominated crustalrheology, the relief of a range initially 3 km high and 300-400 km wide is reduced by half in about 15 m.y. as a result of lateral spreadingof its crustalroot. We suggestthat surfaceprocesses might actuallyprevent sucha "subsurfacecollapse." Removal of material from topographicheights and depositionin the foreland opposespreading of the crustalroot and could eventuallydrive a net influx of material toward the orogeny.We performeda set of numericalexperiments in order to validate this hypothesis.A sectionof a lithosphere,with a brittle-elasto-ductilerheology, initially loadedby a mountainrange is submittedto horizontalshortening and to surfaceprocesses. If erosionis intense,material is removedmore rapidly than it can be suppliedby crustalthickening below the range, and the topographyis rapidly smoothed.For example,a feature 3 km high and 300-400 kmwide is halvedin heightin about15 m.y. for an erosioncoefficient k - 103 m2/yr(the erosionrate is of the order of a few 0.1 mm/yr). This regime might be called "erosional collapse."If erosionis not activeenough, the crustalroot spreadsout laterallyand "subsurfacecollapse" occurs. In the third intermediateregime, removal of the material by erosionis dynamicallycompensated by isostaticrebound and inward flow in the lower crustso that the range can grow. In this "mountaingrowth" regime the range evolves toward a characteristicgraded shape that primarily dependson the erosionlaw. The erosionrate may be high (e.g., 0.5-0.9 mm/yr), closeto the rate of tectonicuplift (e.g., 0.7-1.1 mm/yr), and few times higher than the rate of topographicuplift (0.15-0.2 mm/yr). These experimentsshow that surfaceprocesses can favor localizedcrustal shortening and participatein the developmentof an intracontinentalmountain. Surfaceprocesses must therefore be taken into accountin the interpretationand modelingof long-term deformationof continentallithosphere. Conversely, the mechanicalresponse of the lithospheremust be accountedfor when large-scaletopographic features are interpreted and modeledin terms of geomorphologicprocesses.
Introduction Tien Shan,which is, exceptfor the Himalayas,the largestand mostactive intracontinental range in the world (Figure 1). The Intracontinentalmountain rangesare generallyinterpreted range is 300-400 km wide in its central portion, with a mean to have resultedfrom localizedcrustal thickening in response elevation of about 3500 m, in a zone of relatively thick and to horizontalshortening. The growthand maintenanceof such tectonizedcrust (Moho depthsfrom 50 to 70 km) [e.g.,Avouac topographicfeatures within continentsmight be taken to indi- et al., 1993].A simpledimensional analysis [Gratton, 1989] as cate that the strength of the crust exceeds the deviatoric well asnumerical simulations [Bird, 1991]show that the topog- stressesassociated with slopesof the topographyand of the raphy of such a range should be reduced by half in a few Moho. Yet, laboratoryexperiments indicate that at the pres- million yearsor at mosttens of millionsof years,depending on sure and temperature conditions of the lower crust, most the assumedrheology of the lower crust. Only the short topo- crustalrocks should flow easily[e.g., Brace and Kohlstedt,1980; Wanget al., 1994]. Irregularitiesof the topographyand of the graphicwavelengths, typically less than a few tens of kilome- Moho shouldtherefore be relaxed by viscoplasticflow in the ters, that can be supportedby the strengthof the upper crust ductilelower crustand decaywith time [Kusznirand Matthews, would be maintainedover geologicalperiods of time. In addi- 1988; Gratton, 1989; Bird, 1991]. Consider for example the tion, surfaceprocesses might be thought to contributeto an even more rapid smoothingof the topography.Altogether, in Copyright 1996 by the American GeophysicalUnion. the absenceof strain localization processesa portion of a Paper number 96JB01344. continental crust submitted to horizontal shortening is ex- 0148-0227/96/96JB-01344509.00 pectedto thickenhomogeneously so that no mountainshould
17,747 17,748 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
78øE • 84øE• 90øE,,•.• -
øN
Velocityrelative tq Eurasia Activefaults •2000m-4000m - 20mm/yr ' ß ß •thrust • •strike-slip•>4000m Figure la. Topographyand scismotcctonicsetting of Tien Shah range.
form. The growth and maintenance of an intracontinental been inferred from gravity modeling [Burovet al., 1990] and mountain range over long periods of time must therefore in- seismicanisotropy [Makeyeva, 1992; Roecker et al., 1993], but volve dynamicalprocesses allowing for long-term localization we contendthat it might not be the key factor. Our point is that of lithosphericstrain below the mountain.Several mechanisms couplingbetween surface processes and flow in the lower crust have been advocated.Intrinsic strain-softeningproperties of could provide an alternative and general explanation. rockscould sustainlocalized thrust faulting at the crustalscale. In this paper we first show that surface and tectonic pro- Alternatively, a range could result from shear stressesat base cessesare not independentprocesses and that they can inter- of the crustinduced by lithosphericunderthrusting or by some act. We suggestin particular that advectionof material at the mantle dynamics[e.g., Beaumont et al., 1994;Ellis et al., 1995]. Earth's surfaceand horizontal flow in the crustmight be cou- Sucha mechanismmay be suggestedfor mountainsassociated pled and that this couplingcould permit mountain growth in with subductionzones or with hotspots[Vogt, 1991] but seems response to horizontal shortening. This mechanism is then inappropriateto explain most intracontinentalmountains. In validated and investigatedon the basis of numerical experi- the case of the Tien Shan, a particular mantle dynamicshas ments in which the evolution of an already mature mountain range is modeled. The model accountsfor the rheological layeringof the lithosphereand for surfaceprocesses. We find that depending of the erosion rate compared to horizontal 5OOO shortening,flow in the lower crust can be "outward" (from N S under the high topography)or "inward" (toward the crustal root of a high topography).When inwardflow occurs,a moun- 4000 tain range can grow and no other mechanism is required to explain localized uplift. Some implicationsabout the role of climate on continentaltectonics and on the geomorphologyof ,-,3000 mountain rangesare then derived.
2OOO Interplays Between Surface and Tectonic Processes
lOOO Tectonic Forcing on Surface Processes Topographiccontrasts are required to set up erosion and I ) -300 -200 -100 0 100 200 300 sedimentationprocesses. Tectonics is therefore a forcingfactor [Km] of surfaceprocesses. Following Ahnert [1970] and Pinet and Figure lb. Topographiccross section. Thick line is average Souriau [1988], Summerfieldand Hulton [1994] have recently elevation along the 50-km-wide,650-km-long swath (box in compiledrates of denudationat the scaleof major river basins. Figure la). Thin lines indicateminimum and maximumeleva- These studiesindicate that denudationis primarily influenced tions, respectively.The differenceof elevationsbetween range by basin topographyso that rates of denudationappear to be and lowland is of the order of 3000 m. systematicallyhigh in areas of active tectonic uplift. Common AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,749
values of mean denudation rates in such areas would be of the with flow in the lower crust, inducedby the topographyof a order of a few 0.1 mm/yr to about 1 mm/yr at the scaleof large rangeof a few thousandmeters high a few hundredkilometers drainagebasins. Such rates are generallyconsistent with esti- wide, is of the order of a few million years.The characteristic matesderived from balancingsediment volumes over geolog- times of erosionaldecay of the topographyof a range and of ical periods of time [Leeder, 1991; SummerfieMand Hulton, lateral collapseof a crustalroot are thus of the sameorder of 1994].Thermochronologic studies indicate, however, local val- magnitude. Since both processesare driven by topographic ues as great as 1 mm/yr (see Leeder [1991] and Molnar and slopes, some coupling may arise. Although it is not often England[1990] for critical reviews).The discrepancybetween pointed out, it has long been recognizedthat this kind of local and basin averaged estimatesis due to the fact that processmight play a major role in elevationchanges within tectonic uplift is probably distributedin brief pulsesover lo- continents(see Westaway[1994] for an review of historical calizeddomains within a drainagebasin [Copelandand Harri- developmentof theseideas). Westaway [1994] made a casefor son, 1990]. In the absenceof any tectonicfeedback, common sucha coupling,with inwardflow, in the contextof extensional valuesof denudationrates shouldlead to the disappearanceof tectonicsin westernTurkey. He proposedthat sedimentload- a major mountainbelt like the Tien Shanin a few millionyears. ing in the sedimentarybasins would have driven flow toward Pinet and Souriau[1988] demonstratedthat denudationleads the uplifted area. This kind of processwas first modeled by to an exponentialdecay of the topographyof a range with a King and Ellis [1990],who modeledcrustal extension using a characteristictime constantof the order of 2.5 m.y. thin elasticplate (uppercrust) overlying an inviscidfluid (low- er crust). Coupling Between Denudation and Tectonic We proposethat this kind of couplingmight also appearin Uplift Due to Isostasy a compressionalcontext. Let us considera portion of a litho- Isostasyprovides an immediate mechanismthat shouldlink sphere, loaded with some initial range in regional isostatic subsurfaceand surface processes.Redistribution of surface balanceand submittedto horizontalcompression (Figure 2a). loads by erosion and sedimentationmust induce tectonic de- Becausea thicker crustand bendingstresses tend to reducethe formation to maintain isostaticbalance. Vertical uplift is ex- strengthof the lithosphere[Burov and Diament, 1995;Ranalli, pectedto partly compensateunloading in the area subjectedto 1994], the lithosphereis weaker beneath the range. At the denudation,while subsidenceshould occur in responseto load- sametime, the stressesresulting from the slopesof the topog- ing by sedimentation.This feedbackmechanism may lead to raphy and of the Moho should drive horizontal flow in the some couplingbetween denudationand tectonicuplift [e.g., lower crust beneath the high topography.In the absenceof Ahnert, 1970]. A first consequenceis that the time needed to horizontal shorteningand erosion the lower crust below the erode a topographicrelief must take into accountremoval of rangewould be extrudedlaterally, as discussedby Bird [1991] the topographicrelief and of the crustalroot. If local isostasy (Figure 2a). Our point is that if erosiontakes place, a regime is assumedand if horizontalstrains are neglected,denudation may be establishedin which horizontal shorteningcould be is dynamicallycompensated by uplift, and the characteristic preferentiallyaccommodated by crustalthickening in the area time of decayof the topographywould then be of the order of below the range (Figure 2b). Surfaceprocesses remove mate- 10 m.y. [Leeder,1991]. It has been argued that some kind of rial from the range and feed the adjacentflexural basins,in- positivefeedback may then arise [Molnar and England, 1990; ducingisostatic imbalance. This imbalanceproduces in turn a Maseket al., 1994].If the slopesof valleyssteepen during river temporary excessof normal stressbelow the foreland basins incision, isostatic readjustment following denudation in a and a deficit below the range,favoring flow in the lower crust mountainrange may resultin a net uplift of the highersummits towardthe crustalroot. Ultimately, this coupledregime might in spiteof the averagelowering of reliefs.Erosion might thus lead to some dynamicequilibrium in which the amount of inducesome uplift of topographicsummits, leading in turn to material removedby erosionwould balancethe material sup- enhancederosion. A rapid uplift of the Himalayanbelt during plied to the range by subsurfacedeformation. the last few million yearsmay have resultedfrom this process [Burbank,1992]. Note, however,that while the peaks might reach higherelevations following isostatic adjustment, the net Modeling: Principles and Numerical effect of erosion in that mode is crustal thinning. Thus this Implementation effectdoes not seemto be a fundamentalingredient in moun- We developeda model in order to validate the coupled tain-buildingprocesses. regimedescribed in the previoussection. For this purposethe modelhas to accountfor (1) surfaceprocesses, (2) the effectof Coupling Between Surface Processesand Horizontal Strains topographicloads and of variationsof crustalthickness on the As mentioned in the introduction, small lateral variations of mechanicalbehavior of the lithosphere,and (3) ductileflow in the crustal thickness should drive horizontal flow in the lower the crust.Therefore we havehad to considera depth and strain crust.Some studies have already pointed out the importanceof dependenttheology of the lithosphere.In this sectionwe pro- sucha processin continentaltectonics [e.g., Lobkovsky, 1988; pose some approximationsand a numerical procedure to ac- Lobkovskyand Kerchman, 1991]. For example, Kruse et al. count for these basic properties.The resultingmodel allows [1991] have shownthat horizontalflow in the lower crusthas computationof the deformation of a two-dimensionalsection regulatedisostatic equilibrium during extensionin the Basin of continental lithospheresubject to horizontal compression and Range. The lower crust would have been extruded from and to surfaceprocesses. under the high topography during that process.Following Westaway[1994], this senseof flow is called"outward." On the Surface Processes other hand, Gregoryand Chase[1994] inferred "inward" flow, We looked for a simple two-dimensionallaw that could toward the crustal root, during the Laramide orogenyof the simulate erosion and sedimentation at the scale of a mountain Frontal Range, Colorado. The characteristictime associated range.The evolutionof a landscaperesults from the combina- 17,750 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
a Noerosion ßSubsurface collapse
b Erosion+ Horizontalcompression ß Mountain Growth ?
,i
Denudation S ed•mentahon' _••.__ entation t'• • -'"•":'•"•'••••'•••••-:•'••••"'...... --'...... s • "•
...... 5•.•:' ...... Y" "•'':•" .• -:• q) •.•.•4:•i.- -.•-•,., "'.,•:.i• '•..'..... i '-
Figure2. Velocityfield in a deformingsection of crustloaded with some range topography and submitted tohorizontal shortening. Weassume that the thick crust beneath the range isrelatively weak and mechanically decoupledfrom the upper mantle. (a) In theabsence ofsurface processes thecrustal root below the range is expectedto spreadout laterally ("outward" flow). (b) Denudationin highlandsand sedimentation in lowlands maydrive "inward" flow in thelower crust: denudation, which removes material from steep and elevated topography,results in upliftdue to isostatic rebound. Sedimentation, which transports eroded material to the lowlands,produces increased subsidence dueto the increased load. Unloading beneath the range and loading beneaththe piedmonts may thus force the lower crust to flowtoward the mountain root. As a result, topographicuplift may be localized below the range, and the range can grow rather than spread laterally.
tion of weatheringprocesses that preparesolid rock for ero- Oh Oqe sion,and transportation byhillslope and stream processes (see o•-= Ox' (2) Carsonand Kirkby [1972] for a review).Although many factors, dependingon the lithologiesand on climate[e.g., Fournier, With constantk, equations(1) and (2) leadto the linear 1960],may control this evolution, quite simple mathematical diffusionequation modelsdescribing the geometricalevolution of the morphol- Oh 02h ogyat the smallscale have been proposed and testedsuccess- fully[e.g., Chorley et al., 1984;Kirkby, 1986]. For example,the O•-= kOx 2. (3) two-dimensional(2-D) evolutionof a scarplikelandform can Thismodel of surfaceprocesses holds only for particular be modeledassuming that the rate of downslopetransport of conditions.The regolithmust form morerapidly than it is debris,qe, is proportionalto the localslope [Culling, 1960; removedby surfacetransport, and slopes must not exceedthe Hankset al., 1984;Avouac, 1993]: frictionalangle of thematerial. Even for scarpsformed in loose Oh alluviumsome complications arise when high scarps are con- qe= -k•xX , (1) sidered.Scarps with height typically in excessof about10 m in aridclimatic zones tend to havesystematically sharper curva- wherek isthe mass diffusivity coefficient, expressed in unitsof turesat crestthan at base[e.g., Andrews and Bucknam, 1987]. areaper time (e.g., m2/yr). In theassumption of mass conser- At thelarger scale, hillslope and stream processes interact and vationalong a 2-D sectionand no tectonicdeformation, h must thesediment transport then depends nonlinearly on the slope obey and on other factorssuch as the slopegradient, the area AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,751 drained above the point, and the distance from the water Structure and Rheology of the Lithosphere divide,so that the simple2-D linear diffusiondoes not applyin Previousinvestigations of the interplaybetween erosion and general [e.g., Gossman,1976]. In spiteof theselimitations, we tectonicshave generallybeen conductedassuming either local have chosen to stick to a linear diffusion law to model erosion isostasy[Ahnert, 1970; Leeder, 1991] or thin plate behavior of in the upland.This model doesnot accuratelymimic the spatial the lithosphere [Beaumont,1981; Flemingsand Jordan, 1990; distributionof denudationin the mountain range, but it leads Beaumontet al., 1992;Masek et al., 1994]. Some authorshave to a sedimentyield at the mountainfront that is roughlypro- actuallyconsidered the possibilityfor ductile flow in the lower portional to the mean elevation of the basin relative to that crustand treated the lower crustas an inviscidfluid overlaidby point (a rough approximationto the sedimentyield resulting a thin elasticplate [Kinget al., 1988;King and Ellis, 1990].The from a changeof elevationh over a horizontal distanced is effects of surface loads and of the variations of crustal thick- k x h/d) and thereforeaccounts for the apparentcorrelation nesson the mechanicalbehavior of the lithospherehave been between elevation and denudation rates observedby Ahnert neglected.The coupled regime describedin the previoussec- [1970], Pinet and Souriau [1988], and Summerfieldand Hulton tion assumesthat strainlocalization below a rangeresults from [1994]. We did not apply the diffusionmodel to the whole weakeningof the lithospheredue to crustal thickening and system,however. We felt that we shouldtake into accountthe bending stresses.In order to accountfor this processwe can major discontinuityin surface processesthat occurs at the treat the lithosphereneither as a one-layer elastic plate with mountainfront. As a river emergesinto the adjacentbasin, its verticallyintegrated properties overlying an inviscidastheno- gradientis sharplyreduced and depositionoccurs. The streams spherenor as a thin viscoussheet [e.g., England and McKenzie, shift from side to side and build up alluvial fans that tend to 1983; Vilotte et al., 1982]. We consider the lithological and form a broad gentlysloping pediment at the baseof the moun- mechanicalrheological layering of the lithosphere(Figure 3). tain range.In addition,a lateral drainageoften developsalong Three lithologicallayers are distinguished:the upper crust,the the foothillsof mountainranges (for example,the Gangealong lower crust,and the mantle. Each layer has specificproperties: the Himalayas,the Paranaalong the Andes,or the Tarim along density,mechanical, and thermal constants(see notation sec- the Tien Shan).Altogether the formationof the pedimentand tion). We assumeno compositionalchanges due to deforma- lateral drainagetend to maintaingentle slopesin the foreland. tion or cooling.The lithologicalboundary between the upper There is therefore a sharpcontrast between river incisionthat and lower crust lies at a fixed depth of 20 km. The bottom of maintainsa ruggedtopography with steepslopes in the moun- the mantle lithosphereis limited by the 1330øCisotherm at a tain range and widespreaddeposition of alluviumin the fore- depth of about 250 km. land. This discontinuityof processesmust be consideredto At small differentialstresses the rocksbehave elastically. In model the sharpbreak in slope at the mountain front that is terms of principal componentsthe relationshipbetween the stress tensor 0- and the strain tensor e can be written generally observedon topographicprofiles acrossmountain belts (Figure lb). In order to simulatethis major changein 0-j= 2/a,eej + ,•(81+ 82+ 83) (j: 1, 2, 3). (5) surface processes,sedimentation in the lowland is modeled assuming"flat deposition":mass is conservedalong the section The parameters X and /ze are Lam•'s constantsrelated to and the sediment at the mountain front is distributed in order Young's modulusE and Poissons'sratio v as to maintain a flat horizontal topographyin the foreland. We arbitrarily set the changefrom diffusionalerosion to flat dep- osition at an elevation of 500 m. (1 + v)(1- 2v);
We consideredvalues for k varyingbetween 103 and 104 E m2/yrthat yield denudation rates of the orderof a few tenths /a'e= 2(1 + v)' of a millimeter per year to i mm/yr for a 200- to 400-km-wide range with a few thousandmeters of relief. Strain weakeningby brittle failure or ductile flow occurs In order to test the sensitivityof our model on the assumed when elastic stresses reach some threshold above which unre- erosion law we also considered nonlinear erosion laws of the coverablestrain growswithout increaseof stress.The condi- form tions of brittle failure are independentof rock type and tem- peraturebut are stronglycontrolled by pressure[Byeflee, 1978]: OhO•-= k* (x, h, •xxOh) O2hOx 2' (4) 0'3 = (0'!- 0'3)/3.9 0'3 < 120 MPa; where 0'3= (0'•- o-3)/2.1 - 100 0'3-> 120 MPa. (6)
This law should be written in terms of effective stresses to k* x, h, = k(x) accountfor the effectof pore pressure.In this studywe do not consider fluids. [e.g., Gossman,1976; Andrews and Bucknam, 1987]. We will Ductile flow in the lithosphereessentially results from ther- refer to the cases with n = 1, 2 as first- and second-order mally activateddislocation creep [e.g.,Kusznir, 1991]: diffusion,respectively. In thesecases we did not introducethe changein regime at the mountain front since the nonlinear • = A* exp (-H*/RT)(0'• - 0'3)n. (7) effectsalready tend to form relatively smooth pediments.It The valuesof the parametersare givenin Table 1. The ratio shouldbe noted that (4) differsfrom that obtainedassuming a of the stress to strain rate defines an effective non-Newtonian nonlineardiffusion coefficient in (1). In that caseconservation viscosity: of masswould lead to an additionalterm (Ok*/Ox)(Oh/Ox) in (4). Idbeff-" (0'1- 0'3)/2• (8) 17,752 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
o
0 , I , I , I , I • I , I • I , I , I , I , I , I , I , I , I , I , I. • 0 ß
0 I-- Z
a o •, "7, 0 • "' ,,- LiD0
I
'• 'o "•"'••':.• , •, ' • ' •E•'•o=•o•o •.••o i i i
, ,, ....
• ' 0'''''' 0 0 0'''''''''''''',',',',',' 0 0 0 0 0 0 0 0 0 0 0 0T• =•ø• •
.
• • 0• • O• o.•
• ' • d ••'•
-
"• ' ,•.' • o• • •,•* •:, .,,., • ••oo • AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,753
Table 1. Parametersof Dislocation Creep for Major Depending on the geotherm and strain rates, the first 30-70 LithosphericRocks and Minerals km of the mantle lithosphereremains elastic.
Mineral/Rock Pa-" s-l kJ mol-i Thermal Model A thermal model is required to define the rheology of the Quarzite(dry) 5 x 10-12 190 3 Diorite(dry) 5.01x 10 • 212 2.4 lithosphereand to fully accountfor the effect of crustalthick- Diabase(dry) 6.31x 10-2ø 276 3.05 ening on the theology of the lithosphere.In this paper the Olivine/dunite(dry)* 7 X 10 14 520 3 geotherm is computedaccording to a half-spaceheat transfer model (for details,see Burov et al. [1993] and Burovand Dia- See equation(7). The data sourcesare Braceand Kohlstedt[1980], Carter and Tsenn [1987], and Kirby and Kronenberg[1987]. merit [ 1995]). Heat transferequations are solvedseparately for *Dislocationclimb at O'l - o'3 -< 200 MPa. For olivine (Dorn's the upper crust, lower crust, and mantle: dislocationglide) at oh - o'3 -> 200 MPa), k = ko exp [-H*(1 - (a• - rr3)/rro)2/RT],where k0 = 5.7 x l0 TMs- l, o'o= 8.5 x 103 MPa; • + uT'x+ vT•,- xiAT= Hd+ H, + vf•, (11) H* = 535 kJ mo1-1. with conditionsof temperatureand heat flux continuityacross the interfaces.The primes indicate differentiationwith respect Although ductile deformation occurseven under low differ- to the spatialcoordinate; X/' is the thermaldiffusivity in the ential stresses,a ductile yield strength can be defined. If consideredlayer (X•.i, X•.2, Xm in notation section). The boundaryconditions are givenin termsof rate of displacement, radiogenicheat H,. is X•.•k•T•lp•.H,exp (-yh•-1) in the upper a averagestrain rate can be derived. This "basic" strain rate is crust.The heatproduction is Hc2C•72• in the lowercrust and usedto define a stressthreshold from (7). Owing to the non- zero in the mantle. The adiabatic temperature gradient in the linearityin (7), if the stressis slightlyless than thislevel, ductile asthenosphere,II, is 0.3øC/km [e.g., Turcotteand Schubert, deformation will processvery slowlyso that most of the im- 1982]. posed deformationwill be absorbedelastically. For example, The boundaryand initial conditionsare T(x, 0, t,) = 0øC for olivine the strain rate decreasesby a factor of 1000 if the (temperatureat the upper surfaceis constantat time t,, where stress lies 10-15% below the "threshold" level. On the other t, is the thermal age); T(x, a, t) = T,,, = 1350øC (a -= 250 hand, the stresslevel cannotsignificantly exceed this threshold km is the depth to the thermal bottom,or thermal thicknessof sinceit would require a strain rate much higher that the one the lithosphere[Burov et al., 1993]); T(x, y, O) = T,, (homo- that is imposedfrom boundaryconditions. This thresholdthus geneoustemperature distributionat the beginning).The pa- definesa ductile yield strength.A temperature about 250ø- rameter t, is defined as the age of the last thermal event 300øC must be exceededfor ductile deformation of quartz, determined from geological data. In this study we used the whereasfor olivine it should be 600ø-700øC[e.g., Brace and geothermthat was obtainedfor the Tarim and Tien Shan area Kohlstedt,1980; Carter and Tsenn,1987]. It resultsin the yield by Burov et al. [1993] that correspondsto a 50-km-thick crust stressenvelope (YSE) being controlledby the conditionsfor with 400 m.y. age. Since we investigatethe evolution of the brittle failure in the shallowcrust and upper mantle and by the systemon a short time span (10-30 m.y.) comparedto the conditionfor ductile failure in the deep crust and deep upper thermal age, we neglect the few percent perturbation to the mantle. Combiningrheological laws (equations(6) and (7)) geotherm due to heat conduction.The thermal model also thusdefine a piecewisecontinuous yield stressenvelope (YSE) neglectsthe screeningeffect of sedimentationdue to the low (Figure3) definedas contours O'f -- O'œ (X, y, t, /}) suchthat conductivityof sediments[England and Richardson,1977] as well as lateral heat advectionand viscousheat dissipation.This rr•: sign(e) min {Io-b[x,y, t, k, sign(e)]l, Irrd(x,y, t, k)l} model might appear oversimplified,but it shouldbe noticed that the simplificationstend to underestimatethe temperature (9) in the crustso that it shouldnot particularlyfavor ductile flow whereo -b [x, y, t, k, sign(e)],rr d (x, y, t, k) arethe "brittle" in the lower crust. and "ductile"yield stressesfrom (6) and (7). Owing to asym- metry of the Byerlee'slaw (6), the yield stressdepends on the Numerical Implementation of the Model modeof deformation:sign(e) - 1 for horizontalcompression In this sectionwe describethe numerical procedure used in and -1 for horizontal extension. the modeling. We distinguishbetween the competent and The differentialstress rr(x, y) at a point is taken to be equal weak layers (Figure 4). The competentlayers [e.g., Smith, to the minimumof o-e andrr f computedas a functionof the 1975;Burov and Diament, 1995] are thosewhich are assumed local strain e = e(x, y, t, k): to contributesignificantly to the flexural strengthof the litho- rr(e) = sign(e) min ([Wl, I•(e)l) (10) sphere.It encompassesthe domainsthat are elasticbut also part of the inelasticallydeforming region. As a convention,a where o-e(e) is the elasticdifferential stress according to (5). If domainis competentif the yield strengthis greaterthan 5% of o-• exceedso f, the materialis consideredas ductileor brittle, the lithostatic pressure[Burov and Diament, 1995]. Only the dependingon which rheologylimits the yield strength.Equa- competent layers are consideredin the computation of the tion (10) impliesthat the lithosphereremains elastic if imposed flexural responseto the appliedvertical and horizontal forces. stressdoes not exceedthe yield stress. The geometryand thicknessof the mechanicallayers depend This formulationaccounts for the lithospheregetting weaker on the lithologicallayering and on the stressfield. Sinceboth when submittedto increasinghorizontal forces (Figure 3a) or evolve during a numerical experiment,the mechanicalstruc- flexuralstresses (Figure 3b) and when the crustgets thicker. ture is computedagain on each time step.Vertical deflections Most of the upper crustremains elastic (at depthsbetween 5 (w) of the competentportions of the crustand mantle litho- and 15-20 km). The crust is mostlyductile below 15-20 km. spheredue to changein the applied stress(in particular, in 17,754 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
400km ?
hc'^ ...... Upper c•ust..p•.•,.tJ•., • i.':•' ornpeten•percrus .,,• ...... •, ..•--.: •, •. ,• .•.•.: •,:.
.Low:ercrust P•z,:::...•P•:-•-• • :'•...... t_ ...... ':.::..•::.•:..:: ...... ,..... :::::::::::::::::::::::::::::::::::::::::::::::: ...... •;•:::•;•:•.•:•:•:::.•:•:•:'•.:•.?:?7577:•:•???::•'•?7%• :•:;•::]•:...... :,...... ,,,, ...... ::
h2 •Um• Upper mantel pm,Pm Competentuppermantle •um
:•. ,, ...... :.-...... • '...... :..''-•?•:•.•.•f.•:•, "•;::::::•::"•"•:...... •...-...--•5:•/•: •;•'•.:..' ...•...... :•:• ...... :•.• ...... ::
h.ydro.sta.. ::..;:. :::: ...... •:. ..AS:TEN C • .....:::::::::::::::::::::::::::::::::::: ...... •.:?,.....::..%::.:.?•:::'•:.:,:•:&:.•::;::...... :.•.:..-•::..,.:,., ...... •,.,.:•.,.,.:•:•.... •,.• .. • ...... y < Lo = 2000 km Figure 4. Initial geometryand boundaryconditions. Initial rangeis a Gaussianmount of---00 km width and 3 km height placed on a Paleozoiclithosphere with initially 35-km-thickcrust. The total width of the region is 2000 km. The 2000-km-longsection is submittedto horizontal shorteningu, correspondingto the average strain rate gxx = u/L. Hydrostatic(Winkler) forcesare assumedat the bottom of the lithosphere.
responseto variationof the thicknessof the viscouschannel in crust.However, owing to both higherviscosity and much lower the lower crust,erosion, and sedimentationat the surface)are thicknessof the strongupper crustallayers, we simplyassume treated as instantaneousdeflections of flexiblelayers (Appen- that v = v(y•3 ) and u = u(y•3) fory _
The followingcontinuity conditions are satisfiedat the in- terfacesbetween the competentlayers and the ductilecrustal channel:
Continuityof verticalvelocity •de ,dh -- + + Topography --> v,.•= vc2;v•-2 = Vm (13a)
Continuityof normal stress Passive marker --> O'...,-- = O-yy,.-t- 2,. O-yy,.2-- = O-yy-t- m (13b) Continuityof horizontalvelocity u;, = uS; u5 =Um+ (3C) Time t +dt Continuityof the tangentialstress I Timet O'xycl---O'xyc2 , O'xy•2: O'xy m (13d) Figure5. Definitionsof tectonicuplift (subsidenceif <0), du; topographicuplift (subsidence if <0), dh' anddenudation Kinematic condition (sedimentationif <0), de. By definition,du = dh + de. O•/Ot= vc•; Ow/Ot= v•5 (13e) Superscriptplus and minus refer to thevalues on theupper from about1 mm/yrto a few centimetersper year.These rates and lowerinterfaces of the correspondinglayers, respectively. largelyspan the rangeof mostnatural large-scale examples of The subscriptsCl, c2, andm refer to the strongcrust ("up- active intracontinentalmountain range. Each experimentis per"),ductile crust ("lower"), and mantle lithosphere, respec- conductedover about 15-20 m.y.with internaltime stepof 20 tively.Power law rheologyresults in self-lubricationand con- years.The geometriesof the differentinterfaces (topography, centration of the flow in the narrow zones of highest uppercrust-lower crust, Moho, basement-sedimentin the temperature,that is, nearthe Moho.Therefore there is little foreland)are computed for eachtime step. We alsodetermine differencebetween assuming no-slip or free slipfor the ductile therate of upliftof thetopography, dh/dt; therate of tectonic crust. upliftor subsidence,du/dt; the rate of denudationor sedi- The spatialsteps used in calculationsare dx = 2 km,d y = mentation,de/dt (Figure5), aswell as stress,strain, and ve- 0.5 km. The requirementof stabilityof integrationof the locityfields in the lithosphere.The reliefof the range,Ah, is diffusionequations (3), (4) (dt < 0.5dx2/k)implies a maxi- defined as the difference between the elevation at the crest mumtime step of 2000years for k = 103m2/yr and of 20years h(0) and in the lowlandsat 500 km from the rangeaxis, for k = l0 s m2/yr.It is lessthan the relaxationtime for the h(500). lowestviscosity value (-50 yearsfor/z = l019Pa s). We chose In the casewhere there are no initial topographicor theo- a time stepof 20 yearsin all computations. logicalirregularities, the medium has homogeneous properties and thereforethickens homogeneously. There are no horizon- tal or verticalgradients of strainso that no mountaincan form. Experiments and Results If the mediumis initiallyloaded with a mountainrange, the Description of Experiments flexuralstresses (300-700 MPa, Figure6a) canbe 3-7 times We conductedseries of experimentsin whicha 2-D section higherthan the excesspressure associated with the weight of of a continentallithosphere, loaded with someinitial range, is the rangeitself (-100 MPa). The deviatoricstresses may be submittedto horizontalshortening (Figure 4). The shortening higherthan the lithostaticpressure up to depthsof 15-20km rate in the crustis assumedequal to that in the mantle(u• - andremain comparable with the pressure up to depthsof order Urn).Our goal was to validatethe coupled regime described in 100km (dependingon the rheology).As a resultthe compe- the first section and to check whether it can allow mountain tent coresare thinnestin the area belowthe range(Figure 6b). growth.Our purpose was not to explainhow some small initial Horizontalshortening of the lithospheretends therefore to be irregularityof the topographymight develop into a mountain absorbedpreferentially by strainlocalized in the weakzone belt.This question is beyondthe scopeof thisstudy because beneaththe range.In all experimentsthe systemevolves rap- fault initiationand propagation were not considered.We only idlyduring the first 1-2 m.y.because the initial geometry is out addressthe problemof the growthand maintenanceof a of dynamicequilibrium. After this reorganization some kind of mountainrange once it has reachedsome mature geometry. dynamicequilibrium settles in whichthe viscous forces due to We thus considera 2000-km-longportion of lithosphere flow in the lowercrust also participate to the supportof the loadedby an initialmajor topographic irregularity. We chosea load. 300- to 400-km-wide"Gaussian" mountain (a Gaussiancurve Case1, No SurfaceProcesses: "Subsurface Collapse" with variancerr = 100km). The rangehas a maximumeleva- tion of 3000 m and is initially regionallycompensated. The In the absenceof surfaceprocesses the lower crustis ex- initialgeometry of the Mohowas computed according to the trudedfrom under the hightopography (Figure 7). The crustal flexuralresponse of thecompetent cores of thecrust and upper rootand the topography spread out laterally (Figure 8a). Hor- mantleand neglecting viscous flow in the lowercrust [Burov et izontalshortening leads to a generalthickening of themedium, al., 1990].In thiscomputation the possibility of internaldefor- but the tectonicuplift belowthe rangeis smallerthan below mationof the mountainrange or of itscrustal root is neglected. the lowlandsso that the relief of the range,Ah, decayswith The sectionis then submittedto horizontal shorteningat rates time(Figure 8b). The systemthus evolves toward a regimeof 17,756 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
II TM
I , I , I ,
o o
• 0 '"'•
o r• tt-• o 1211
o o
I I I I I I I I qldeCl AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,757
agV/gVH.LDN•I:I.LS
o i i i ßI I I I I I . I . I I . I , I , I , I , I , I , I (•
o _o
1
o o
o o
I I I I I I I I T T T T V T' [uJN]qideo 17,758 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
homogeneousdeformation with a uniformly thick crust.In the 5000•I ...... I...... i...... i...... i...... ,i...... I...... i...... i...... i...... particular caseof a 400-km-wideand 3-km-high range it takes about 15 m.y. for the topographyto be reducedby a factor of -5000 - 2. If the medium is submitted to horizontal shortening,the -10000- decay of the topographyis even more rapid due to strain '•'-15000 weakening.These experimentsactually show that assuminga '-'-'-20000 standard rheology for the crust, no intrinsic strain softening --•:-25000 propertiesof the material, and no particular mantle dynamics, a rangeshould collapse in the long term, as a result of subsur- '•-30000 -r'_35oo0 face deformation,even with horizontalshortening of the litho- sphere.This regimein whicha rangedecays by lateralextrusion of -40000 the lower crustal root might be called "subsurfacecollapse." -45000
-50000 Case 2, No Shortening: "Erosional Collapse"
-55000 If erosionis relativelyactive (k of the orderof 104 m2/yr), -500 -400 -300 -200 -100 0 100 200 300 400 500 the topographyof the rangeis washedout rapidly (Figure 9a). Distance [km] Isostaticreadjustment compensates for only a fraction of the denudation, and the elevation in the lowland increases as a Figure 7. Outwardflow pattern in the lowercrust in the caseof result of crustal thickeningessentially due to sedimentation no erosion.The crustalroot below the rangeis extrudedlaterally (Figure 9b). Althoughmechanical collapse of the crustalroot drivenby the shearstresses due to the slopesof the topography also participatesin the processof decay of the range, we may and of the Moho. Velocity ux at y -- y • 3 associatedwith overall call this regime "erosionalcollapse." The time constantasso- compressionis subtracted.In this experiment, the 2000-km- long sectionis submittedto 6.3 mm/yrhorizontal shortening. ciatedwith the decayof the relief in this regimedepends on the massdiffusivity (Figure 9c). For k = 104 m2/yr,denudation rates are of the order of 1 mm/yr at the beginning of the , I , I • I • I • I • I • I , experiment(Figure 9b), and the initial topographyis halved afterabout 5 m.y.For k = 103 m2/yrthe rangetopography is "subsurface" collapse (compression) halved after about 15 m.y. Once the crust and Moho topogra- 6.3 mm/y(Exx--O. ld-15 secm-¾No erosion. phieshave been smoothedby surfaceprocesses and subsurface 3 deformation, the systemevolves again toward the regime of homogeneousthickening.
Case 3, Shortening and Erosion: "Mountain Growth" Mountain growth can occurprovided that denudationsare not too rapid nor too slow comparedto the shorteningrate. 10 We performed a set of experimentsstarting from pointsin the //// \xx,' "subsurfacecollapse" regime, and we then graduallyincreased the intensityof erosion,keeping the shorteningrate constant...... 374• If erosionis not activeenough, the range decaysas a resultof subsurfacedeformation. At somecritical value of k a regimeof
• I f I , I , I I I [ I [ I [ I 4000 ,. I •-=-'•-'--• • -...... i --• • • •z..-•..•: "subsurface" collapse by spreading of the crustal root
3OOO
ß
i "•'2000
O i
O collapse of the crustal root
1000 -500-400 -300 -2•)0 -100 0 100200 300 400 500 _---- 1 - h(O) distance x [km] .... 2.... 2- h(500)
Figure 8a. A caseexample of "subsurfacecollapse" when no i i i I I I I I I I I i i i i erosionis applied.Elevation at crest of the range decaysas a 0 I 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 resultof the subsurfacedeformation as shownin Figure 7. The 2000-km-longsection is submitted to 6.3 mm/yr horizontal time [M. y.] shortening.Evolution (top) of topographyand (bottom) of Figure 8b. Evolutionof elevationh at axisof the range(x = Moho from 0 to 10 m.y. The different curvesare labeled ac- 0 km) and in the lowland(x - 500 km) as a functionof time. cordingto time (0, 1, 2, 3, 4, 5, 10 m.y.). Flexural and flow adjustmentoccurs during the first 1-2 m.y. AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,759
4000 0.7 - "Erosional" collapse (No compression) 0.6 - time= 5 M.y. u - 0.0063mm/y (œzz=O. ld-18 s -•) + rapiderosion (Ks=10000 m2/yr) 3000 .• 0.5- • 0.4-
• 0.2- • 2000 • 0.1- (- LM 0.0- L,' 2 ',3 "0 -0.1 - 1000 ...•-0.2 - "• -0.3- -0 -0.4 -
0 ...•-0.5 - • -0.6 - • -0.7 - , I , I , I , I I I , I , I , I , -0.8 - -0.9 , , , , , . , ., , -500 -400 -300 -200 -100 0 100 200 300 400 500 distance x [km] Figure 9b. Ratesof denudation(deS), surfaceuplift (dhS), •- -10000 and tectonicuplift (du5), t = 5 m.y. with moderateerosion (k -- 104 m2/yr). I 0 I i 0 collapse of the crustal root the topographyflattens. Since the slopesof the topographyand Moho are reducedinward flow in the lower crustcan proceed -20000 -500 -4 • 0 ,-300 i ,-200 i ,-100 i 0i 100i , 200i , 300i 400 500 more rapidly, inducinga reneweduplift. Suchoscillations are particularlyclear in experiment3. Comparisonof the different distance x [km] experimentsindicates that the behavior of the systemis very Figure 9a. A caseexample of "erosionalcollapse" of topog- sensitiveto small changesof parameters.We did not explore raphyin the absenceof horizontalshortening. Evolution (top) the dynamicalbehavior of the systemin the coupledregime, of topographyand (bottom)of the crustalroot with rapid (k = but we suspectthat some chaoticbehavior may arise. 104m2/yr) erosion. The results for different experiments are summarized in Figure 12. As a conventiona given experiment,corresponding to a given mass diffusivityand a given shorteningrate (or dynamicalcoupling can settle in which the relief of the range can grow (Figure 10a). Similarly,we may start from a stateof 4000 "erosional collapse," keep the rate of erosion constant, and graduallyincrease the rate of shortening.We get that for low Erosionalcollapse (Ks=104 & 103rn2/yr) shorteningrates, erosioncan still erasethe range topography, but at some critical value of the shorteningrate a coupled 3000 1 -h(O) (Ks=104) regimecan settle (Figure 10a). In the coupledregime the lower 2- h(500) (Ks= 10 4) crustflows toward the crustalroot (Figure 10b), and the re- 1'- h*(O) (Ks=103) sultingflux exceedsthe amountof material removedfrom the "" ..... 1' 2'.h*(500)(Ks=103) range by surfaceprocesses. Tectonic uplift below the range •'2000 exceedsdenudation (Figure 10c) so that the elevationof the crest increaseswith time (Figure 10d). In this "mountain growth" regime the distribution of deformation remains het- 1 ooo erogeneousin the long term. High strainsin the lower and upper crustare localizedbelow the range allowingfor crustal thickening. The crust in the lowland also thickens owing to sedimentationbut at a smaller rate than beneath the range 0 ';:_'_-----z;-- 2* i i i i i i i i i i i i i i i (Figure 10c). Figure 11 showsthat the rate of growth of the 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 elevationat the crest,dh/dt(x = 0), for differentexperiments in the "mountaingrowth regime." First we seethat for a given time [M, y,] experiment the evolution is far from monotonic in general. Figure 9c. Evolution of the elevation h at the axis of the There are oscillationsprobably related to fluctuationsaround range(x = 0 km) and in the lowland(x = 500 km) with rapid a dynamicequilibrium in which eroded material and inward (k = 104 m2/yr)and moderate(k = 10• m2/yr)erosion. flow in the lower crustrate by a constantratio. Sucha dynamic Fastererosion (k = 104m2/yr) results in morerapid smooth- equilibriumoccurs for experiment1. If denudationis too rapid, ing of the range and increaseof elevationof the forelands. 17,760 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
5 [ I ] I [ t t I , I i I [ I ] I [ I , , I [ I [ I [ I , I , I , I , I , I ,
Localisedstable growth ofthe range (compression) 1.1 time = 3; 5; 10; 13 M.y. u- 44mm/y (Exx--O. 7d-15s't) + linear erosion (Ks=7500 rn2/y) 1.0_
_ 4 lO 0.9 0.8 _ /'- •'"x P--..-i / / 7 x\% •n 0.7
i I 4 ,/ ,/,/,• . m,v',, [ • 0.4 - I /I II/l•lII/1•1 ,••l%• X•X• '• tL • 0.3 d1•.13 ] / II•[I I 0 X•X x x L , d1110 2j] I ,IlyJ ,/•t I : _xm,,I l•X % • [t • 0.2 i i --• I I II•J I • X I•Xx x r i i 1 / /l•J I ', X • • r • 0.0 ...... dh3 i I i • ,/t'tl/ • Xtl/• [ • -0.2 I
• -0.3 I I 0 • -0.4 I
I i , , , , , , , , , i , , , , , , , , , •-o.5-o.6 i i • -0.7
•i • • -o.9-0.8
0 -500-400-300-200-100 0 100 200 300 400 500 -10 distance x [km]
10 Figure 10c. Ratesof denudation(de/dt), surfaceuplift (dh/ dt), and tectonicuplift (du/dt). Notationsdh3, dh5, dh 10, etc., correspondto t = 3, 5, and 10 m.y., etc., respectively. -20 , , . Thick grey line showsinitial topography. -500 -400 -300 -200 -100 0 100 200 300 400 500
distance x [km] averagestrain rate), is definedto be in the "mountaingrowth" Figure 10a. A caseexample of"mountain growth."Coupling regime if the relief of the range increasesat 5 m.y., which between erosional and tectonic processesresults in a slow means that elevation at the crest (x = 0) increasesmore localizedgrowth of a range. In this example,the sectionis rapidlythan the elevationin the lowland(x = 500 km): submittedto 44 mm/yr of shortening,and erosionis relatively intensewith a massdiffusivity of k = 7500 m2/yr.Evolution (top) of topographyand (bottom)of crustalroot. 5000 I I I I I I I I I I I I I I I Localizedstablegrowth ofthe range 5000t...... •...... - u ~ 44 mm/y; _500;•;I...... '...... v.,.:•-•,••...... •..•••..•••?•.___ _•..•., '.._•....,.••.••_••....,.._.__ ...... •7:,• ...... •.... •:•:•. 4OOO - 10000
3OOO •-2o000• -25000 ] . ., L"-'-5• ½...... x•--'"; _ . 2000 1 -h(O) 2 - h(500)
1 ooo -5OOOO
-55000 ,,,',,'.... -,.,', .... '"' .'•.•, '.'*, ,'...... ,,,'...... ,'.,•'*'• -500 -400 -300 -200 -100 0 100 200 3• 400 500
Distance [km] i i i i i i i i i i i i i i Figure 10b. Flow patternin lowercrust. Dashed line in lower 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 c•st correspondsto deflectionof lower crust.Solid line is the time [M. y.] bottomof competentupper c•st. Bendingstresses result in ther- toomechanical"erosion" of bottom of mechanicalupper c•st. A Figure 10d. Evolution of elevation h at the axis of range constanthorizontal velodty, co•esponding the horizontalvelodty (x = 0 km) and in the lowland(x = 500 km) as a function at the bottom of the competentc•st (y •3) is substracted. of time. AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,761
,.i, 0.4 ,,?? I ,.:,,,/ ....."...... ,,& ',...... ø'2l...-?" .... t ..../.? ... .,...:., 1 / ,.,..,.o,,o,,,,,,;,..o.,,,o.,,,.,. '• -0.3 i'"/ • ;.'1 : ' 'J• -0.4 . / ., ,, ,.,..,, ,-- , =-. ß -0.5'V•i// / ;'--" ,"", ! 54 - Ks=6.0K-7 5x 103 m2/yrm2/yr; ßE•x=0.5E -06 x 104;4 s s4.4
-0.7 -0.6Ii• , , i i i i i i i i i i I i i i 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 1 time [M. y.] Figure 11. Time evolutionof surfaceuplift rate dh/dt(x = 0) (at axisof tile ran•½) for differentcases of "mountain•rowth."An experimentis saidto be in the "mountain•rowth" regimeif dh/dt(x = 0) is positive at t = 5 m.y. •1 experimentsshow a rapid reorganizationdu[in• the first ]-2 m.y. Cu•e ] correspondsto a particularstable •rowth. In mostexperimems the •rowth is rather i•m•ular. This is why two experimentswith
dh/dt (0 km) > dh/dt (500 km) at t -- 5 m.y. (14) tion (4)). For a givenshortening rate, experimentsthat corre- spondto similar erosionrates over the range lead to the same Becauseof the complexdynamic shown in Figure 11, very evolution ("erosional collapse," "subsurfacecollapse," or closeexperiments appear to be labeled differently.Neverthe- "mountaingrowth") whatever the erosionlaw. It thus appears less,the "mountaingrowth," "subsurface collapse," and "ero- that the emergenceof the coupledregime doesnot dependon sionalcollapse" domains can be approximatelydrawn on this a particular erosionlaw but rather on the intensityof erosion plot. Higher strain rates lead to a decrease of the effective relative to the effectiveviscosity of the lower crust.By contrast viscosity(/•e•, proportionalto k•/"•) of the non-Newtonianthe graded geometriesobtained in the mountain growth re- lower crustso that a more rapid erosionis neededto allow the feedbackeffect due to surfaceprocesses. The mountaingrowth domainthus corresponds to highermass diffusivities for higher Shortening rate (mm/yr) strainrates (Figure 12). 0.1 1 10 100 10 s Coupled Regime and Graded Geometries I In the coupledregime the topographyof the range can be Erosionalcoilapse ß ß seento developinto a nearlyparabolic graded geometry (Fig- ß ß ß ure 10a). This graded form is attained after 2-3 m.y. and reflectssome dynamic equilibrium with the topographicrate of uplift beingnearly constant over the range(Figure 10c). Rates of denudationand of tectonicuplift can be seen to be also relativelyconstant over the rangedomain (Figure 10c). Geom- etriesfor which the denudationrate is constantover the range are nearly parabolicsince they are definedby de/dt = k d2h/dx2= C2. (15) with C2 being some constant.Integration of this expression SubsUrfacecollapse yieldsa parabolicexpression for h: 10 2 10-•8 10 -17 10 -• 10 -•s 10 -44 C2x2 Strain rate (s-1) h - • + C•x+ Co. (16) Figure 12. Three major regimesof shortening(log-log plot of k-strain rate dependence).Squares correspond to experi- The gradedgeometries obtained in the experimentslightly ments with the "erosional" collapse;triangles correspondto deviate from parabolic curvesbecause they do not exactly that of the "subsurface"collapse. The area (stars)between the correspondto uniformdenudation over the range.This simple triangles and squarescorresponds to regime of "mountain considerationdoes, however, suggest that the overallshape of growth."The box correspondsapproximately to the Tien Shan gradedgeometries is primarily controlledby the erosionlaw. as inferred from the 1-2 cm/yr shorteningrate and the 0.2-0.5 We made computationswith nonlinearerosion laws (equa- mm/yr denudationrate. 17,762 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
4000a: 2-order nonlinearerosion i
Q) 0 ......
. 6000 ...... ' ...... ' ...... ' ...... ' ...... ' ...... I ...... •...... , ...... , ...... ,••'4000 b:1-o 2000
'1 ...... I ...... I ...... I ...... t ...... I ...... I ...... I .... I"
...... I ...... I ...... I ...... I ...... I ...... I ...... I ...... I,,,,,,,,,I,,,,,,,,, ß , 6000
ß ' 4000 c: O-orderlinear '
• 2000 0 ...... I ...... I ...... I ...... I ...... i ...... I ...... I ...... I ...... I ...... -500 -400 -300 -200 -100 0 100 200 300 400 500 Distance [km] Figure 13. Comparisonof different geometriesobtained in the coupled regime for various models of erosion.The sectionis submittedto 44 mm/yr horizontalshortening. Thick grey line showsinitial topography. After rapid evolutionduring the first 2 m.y. the range topographygradually increases with time. (a) Second- ordernonlinear erosion model with asymmetric boundary conditions (vertical force Q,c = 2 x 1012N perunit lengthis appliedon the left edgeof plate). The curvescorrespond to 1, 2, 3, 4, 5, and 10 m.y. (b) First-order nonlinearerosion model. The curvescorrespond to 1, 2, 3, 4, and 5 m.y. (c) Linear diffusionmodel of erosion. The curvescorrespond to 1, 2, 3, 4, and 5 m.y. gime stronglydepend on the erosion law (Figure 13). The quartz rheologyfor the lower crust (Figure 14). Estimatesof first-order diffusionlaw leadsto triangular ranges,whereas the the yield strengthof the lower crust near the Moho boundary second-orderdiffusion leads to plateau-like geometries.It ap- for thermal ages from 0 to 2000 m.y. made by Burov and pears that the graded geometry of a range may reflect the Diament [1995] suggestthat whateverrheology is considered,a macroscopiccharacteristics of erosion. It might therefore be crustthicker than about 40-50 km impliesa low-viscositychan- possibleto infer empirical macroscopiclaws of erosion from nel in the lower crust. The regime that we have evidencedin topographicprofiles across mountain belts providedthat they our numerical simulations could therefore exist for a wide can be assumedto be graded. range of conditions.
Sensitivity to the Rheology and Structure of the Lower Crust Comparison With the Tien Shan The experimentsshown above have been conductedassum- Although the Ticn Shan may not appear as a genericmoun- ing a quartz rheologyfor the whole crust that is particularly tain range, we compare our modeling with that particular favorable for channel flow in the lower crust. In order to show range becauserates of deformationand of erosion have been that the coupled regime of mountain growth can also occur estimated from previousstudies [e.g., Avouac et al., 1993; F. assuminga less ductile lower crust, we also conductedsome M6tivier and Y. Gaudemer, Mass transfer between eastern experimentsassuming more basiclower crustalcompositions Tien Shanand adjacentbasins: Constraints on regionaltecton- (diabase, quartz-diorite) (Table 1). Even with a relatively ics and topography,submitted to GeophysicalJournal Interna- stronglower crust the coupled regime allowing for mountain tional, 1996] and becausethe range has a relativelysimple 2-D growthcan settle(Figure 14). The effectof a lessviscous lower geometry.The Tien Shan is the largestand most activemoun- crust is that for a given shorteningrate, lower rates of erosion tain range in central Asia (Figure la). It extendsfor nearly are required to allow for the growth of the initial mountain. 2500 km betweenthe Kyzil Kum and Gobi deserts,with peak The domain definingthe "mountaingrowth" regime in Figure rising to more than 7000 m. The high level of seismicityin this 12 would thus be shifted toward smaller mass diffusivities when century[Molnar and Deng,1984] and deformationof Holocene a stronger lower crust is considered.The graded shape ob- alluvial formations[Avouac et al., 1993] would indicate a rate tained in this regime doesnot differ from that obtainedwith a of shorteningof the order of 1 cm/yr. In fact, the shortening AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,763
6000
'•' 4000 ._{•2000 (]3 0 ...... I ...... I ...... I ...... I ...... i ...... I ...... I ...... I ...... I ...... -500-400-300-200-100 0 100 200 300 400 500 Distance [km]
6000 ...... ' ...... ' ...... ' ...... ' ...... ' ...... ' ...... ' ...... I ...... I ......
• 4000 b: quartz-diorite , ._{•:D2000
j• ...... I ...... I ...... I ...... I ...... I ...... I ...... I ...... I ...... I ...... -500-400-,300-200-100 0 100 200 ,300 400 500 Distance [km] Figure 14. Influenceof lower crustalrheology (see Table 1 for parameters)for 1.2 cm/yrshortening. After rapid evolution during the first 2 m.y. the range topographygradually increases with time. (a) Quartz- dominatedlower crust;(b) "stronger"quartz-diorite lower crust.The two experimentsare in the mountain growthregime. The curvescorrespond to 0, 1, 2, 3, 4, and 5 m.y.
rate is thoughtto increasefrom a few millimetersper year east al., 1990; Ma, 1987], crustal thickeningbelow the range has of 90øE to about 2 cm/yr west of 76øE [Avouacet al., 1993]. absorbed1.2-4 x 106 km3 (M6tivierand Gaudemer, submit- Clockwiserotation of the Tarim Basin with respectto Dzun- ted manuscript,1996). Crustal thickeningwould thus have garia and Kazakhstanwould be responsiblefor this westward accommodated50-75% of the crustal shorteningduring the increaseof shorteningrate as well as of the increaseof the Cenozoicorogeny, with the remaining25-50% havingbeen fed width of the range(Figure la) [Chene! al., 1991;Avouacet al., back to the lowlandsby surfaceprocesses. Given the geometry 1993]. The gravity studiesby Burov [1990] and Burov et al. of the range,the 0.2-0.5 mm/yr denudationrate impliesa mass [1990] also suggestwestward decrease of the integrated diffusivityof the orderof 2 x 103to 104 m2/yr.These values strengthof the lithosphere.The westwardincrease of the to- actuallyplace the Tien Shanin the "mountaingrowth" regime pographicload and strain rate could be responsiblefor this on the plot in Figure 12. We therefore conclude that the mechanicalweakening. The geologicalrecord suggests a rather localizedgrowth of a range like the Tien Shan could actually smoothmorphology with no great elevation differencesand result from the couplingbetween surface processes and hori- low elevationsin the Tertiary and that the range was reacti- zontal strains.We do not disputethe possibilityfor a complex vated in the middle Tertiary, probablyas a result of the India- mantle dynamicsbeneath the Tien Shan as has been inferred Asia collision [e.g., Tapponnierand Molnar, 1979; Hendrix et by variousgeophysical investigations [Vinnik and Saipbekova, al., 1992]. Fissiontrack agesfrom detrital appatite from the 1984;Makeyeva et al., 1992;Roecker et al., 1993], but we con- northernand southernTien Shanwould placethe reactivation tend that this mantle dynamicshas not necessarilybeen the at about 20 m.y. [Hendrix et al., 1994; E. Sobel and T. A. major drivingmechanism of the CenozoicTien Shan orogeny. Dumitru, Exhumation of the margins of the western Tarim Basin during the Himalayan orogen, submitted to Tectonics, 1995].Such an ageis consistentwith the middle Miocene influx Conclusions and Implications of clasticmaterial and more rapid subsidencein the forelands This study demonstratesthat denudationcould exert some [Hendrixet al., 1992]. The present differenceof elevationof control on the developmentof an intracontinentalmountain about3000 m betweenthe rangeand the lowlands(Figure lb) belt. Dependingon the intensityof the surfaceprocesses, hor- would thereforeindicate a mean rate of uplift of the topogra- izontalcompression of a 2-D sectionof continentallithosphere phy, during the Cenozoic orogeny, of the order of 0.1-0.2 with some initial topographicirregularity can lead either to mm/y. The foreland basinshave collectedmost of the material strain localization below a growing range or to distributed removedby erosionin the mountain.Sedimentary isopachs thickening.Homogeneous thickening occurswhen erosion is indicate that 1.5 +_ 0.5 106 km 3 of material would have been either too strong,in which caseany topographicirregularity is erodedduring the Cenozoicorogeny (F. M6tivier and Y. Gau- rapidlywashed out by surfaceprocesses, or too weak, in which demer,submitted manuscript, 1996), implyingerosion rates of casethe crustalroot below the rangespreads out laterallywith 0.2-0.5 mm/yr on average.The tectonicuplift would thushave formationof a flat "pancake-shaped"topography. In our mod- been of 0.3-0.7 mm/yr on average.On the assumptionthat the eling, mountain growth results from dynamicalcoupling be- range is approximatelyin local isostaticequilibrium [Burov et tween the advectionat the surfaceby surfaceprocesses and at 17,764 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
,.,';//,,/j,•,,,, Himalaya Altyn Tagh Tien Shan ••,...., •,,•,•;,,•,•///////. .½i2 cm/yr TIBET I cm/yri INDIA TARIM 5 cm/yr •'•• STABLEEURASIA ?
Figure 15. Sketch illustratingthe India-Asia collisionand the possibleinfluence of climate on the distribu- tion of strain in central Asia. Although the Himalaya and the Tien Shan rangesare probably submittedto similarhorizontal forces, the Himalaya absorbsabout 2 cm/yr of N-S shortening,while the Tien Shan absorbs only about I cm/yr. The higher strain rates in the Himalaya may be related to the monsoonalclimate that is much more erosive than the climate in the Tien Shan area. depth by flow in the lower crust. Intracontinental orogenies By contrastthe much more erosiveclimatic conditionson the could arisefrom sucha couplingwithout the help of any other southernthan on the northernflank of the Himalaya may have source of strain localization. favored the developmentof systematicallysouth vergent struc- Deformation of a continent may thus be very sensitiveto tures. While the Indian upper crustwould have been delami- surfaceprocesses. The way central Asia has absorbedinden- nated and brought to the surfaceof erosionby north dipping tation of India may somehowreflect this sensitivity.Numerical thrust faults, the Indian lower crust would have flowed below models of continental indentation based on continuum me- Tibet (Figure 15). Surfaceprocesses might therefore have fa- chanics,but that neglect surface processes,predict a broad cilitated injection of lindJan lower crust below Tibet as was zone of crustalthickening, resulting from nearly homogeneous proposed by Zhao and Morgan [1985]. This would explain strain that would have propagated away from the indentor crustalthickening of Tibet with minor horizontal shorteningin [Englandand Houseman,1986; Houseman and England, 1986; the upper crust and minor sedimentation. Vilotteet al., 1982]. In fact, long-livedzones of localizedcrustal We thus suspectthat climatic zonation in Asia has exerted shorteninghave been maintained(in particular,along the Hi- somecontrol on the spatialdistribution of the intracontinental malaya, at the front of the indentor, and the Tien Shan, well strain inducedby the India-Asia collision.The interpretation north of the indentor), and broad zones of thickened crust of intracontinentaldeformation should not be thought of only have resulted from sedimentation rather than from horizontal in terms of boundaryconditions induced by global plate kine- shortening(in particularin the Tarim Basinand to someextent maticsbut alsoin termsof globalclimate. Climate might there- in some Tibetan basinssuch as the TsaYdam).Present kine- fore be consideredas a forcingfactor of continentaltectonics. matics of active deformation in central Asia corroborates a highly heterogeneousdistribution of strain. The 5 cm/yr con- vergencebetween India and stable Eurasia is absorbedby Appendix A: Model of Flexural Deformation lateral extrusionof Tibet and crustal thickeningwith crustal of the Competent Cores of the Brittle-Elasto- thickeningaccounting for about 3 cm/yr [Avouacand Tappon- Ductile Crust and Upper Mantle nier, 1993].About 2 cm/yrwould be absorbedin the Himalayas Deformation of the competentlayers in the crust and man- and 1 cm/yr in the Tien Shan (Figure 15). The indentationof tle, as defined from the BED rheology (Figures 3 and 4) in India into Eurasia has thus induced localized strain below two responseto redistributionof loadsby surfaceprocesses or flow relatively narrow zones of active orogenic processes,while in the lower crust, is computedusing plate equilibrium equa- minor deformation has been distributed elsewhere. Surface tions [Burovand Diament, 1995]: processesmight be partly responsiblefor this highlyheteroge- neousdistribution of deformation.First, activethrusting along the Himalaya and in the Tien Shan may have been sustained •xxO[•xx O(12(1-v E 2)) ( ½e3(0) 02w(x,OX2 t) ) + Tx(•)_ Ow(x, Oxt) l during most of the Cenozoic thanks to continuouserosion. + p_(•)w(x, t) -p+(x, t) = 0 (Ala) Second, the broad zone of thickened crust in central Asia has resulted in part from the redistribution of the sediments eroded from the localizedgrowing reliefs. Moreover, it should Te(•))- = [/•//x(•)L (O2W(OxX,2 t))-l] 1/3 (Alb) be observedthat the Tien Shan experiencesa relatively arid intracontinentalclimate, while the Himalaya is exposedto a M,,(O) = - * very erosivemonsoonal climate. Although rheologicaldiffer- ences might also be advocated, this climatic disparity may '= '= explainwhy the Himalaya absorbstwice as much horizontal shorteningas the Tien Shan,while tectonicforces are probably T,(O)= - • • rr•)(O)dy (Ald) similar. In addition, the nearly equivalentclimatic conditions -- i:1n j:l.,mi l on the northern and southernflanks of the Tien Shan might have favored the developmentof a nearly symmetricalrange. o-•)(0) = sign(Sx•) min [lq, (Ale) AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,765
crustal channel,Ahc2 -- Aho(x, 0) + h + w, will be com- -e(J)trk)= Y,*(rk)02w(x, t))E,(1 - v,2)-• (Alf) pensatedby deflectionof the Moho, whichdepth is hc(x, t) = T•.(x, t) -- Ahc2 q- hc• , and of the upper crust. These where w = w(x, t) is the verticalplate deflection(related to deflectionsare assumedinstantaneous and are computedac- the regional isostaticcontribution to tectonic uplift dui.,. as cordingto AppendixA. Notethat the topo_graphy of the upper du,.,.= w(x, t) - w(x, t - dt)); ck-- {x, y, w, w', w", t}; boundary of the lower crustal channel, h, does not mimic y ispositive downward; y *• = y - y,,•(x); y,• is the depth to the topographicvariation at thesurface, h:•t(x, t) - [t(x,t -dt) = ith neutral(i.e., stress-free, (rxx y•:,, = 0) plane;and y? (x) = du - dye3, wheredye3 - y•3(4), t) - y•3(4), t -- dt) is the y•-, y•+(x) = y•+ are the respectivedepths to the lowerand changeof the depth of the lower boundaryof the elasticcore upperlow-strength interfaces (see Figure 3b). The strengtho'r in the upper crust due to changesof deviatoricstresses (as is definedby equation(9); n - 1 refersto the competentcrust shownin Figure 3b). and n = 2 refers to the competentmantle; m• is the number The equationsgoverning the creepingflow of an incom- of "welded" (continuous(rxx) sublayersin the i th detached pressiblefluid, in Cartesiancoordinates, are, in addition to layer;p_w is a restoringstress (p__ - (Pro-- 9•'•)#) andp + equations(7) and (8), is a sum of surfaceand subsurfaceloads. The most important contributionto p + is from the load of topography,that is, Orxy O(ryy Orxy p + • 9#h(x, t), wherethe topographicheight is h (x, t). The - 0ø'xx+ -+ Fx--0; oy + +Fy = 0 thicknessof the ith competentlayer is y•- - y? = Ahi(x ). The term 02w/Ox2 in (A1) is inverselyproportional to the radiusof platecurvature Rxy • --(c)2W/c)X2)-1 Thus the (rxx= -rxx+ p - -2Ix •xx+ p higher the local curvatureof the plate is, the lower the local (Bla) integratedstrength of the lithosphereis. The integralsin (A1) are definedthrough the constitutivelaws (5)-(7) and (9)-(10) = = relating the stressO-xx and strain exx = exx(Ck)in a given Ou0v) segment{ x, y } of a plate. Note that the valueof the effective elasticthickness Te(ck) holds only for the given solutionfor (ryy= - r•vy+ p = -2Ix •xx+ p plate deflectionw. The nonlinearequations (Ala)-(Alf) are computedusing an iterative approachbased on finite differ- Ou Ov enceapproximation (block matrix presentation) with lineariza- tion by Newton's method [Burov and Diament, 1992]. The 0-•+ •y = 0 (Blb) procedurestarts from calculationof elasticprediction We(X) for w(x), that givespredicted We(X), (OW/OX)e(X), (02W/OX2)e, Where /x is the depth-dependenteffective viscosity, p is usedto derivesolutions for Yi• (d)), yi3 7.(d)), andYn_/(qb) that pressure,u and v are the horizontaland vertical components satisfy(5), (6), (7), and (10). Corrected solutions for Mx and •'• of the velocityv, respectively;F is the bodyforce due to gravity; are deducedand used to obtain T e for the next iteration. At u = O½/Oy is the horizontal componentof velocity of the thisstage we usegradual loading technique to avoidnumerical differential movement in the ductile crust; v = -0 ½/Ox is its oscillations.The accuracyis checkedthrough back-substitution vertical component; and Ou/Oy = •c2o is a component of of the current solutionin (A1) and calculationof the discrep- shear strain rate due to the differential movement of the ma- ancybetween the right and left sidesof (A1). For the boundary terial in the ductile crust (the componentsof the strain rate conditionson the endsof the plate we usecommonly inferred tensor are, consequently,• = 20u/Ox; •2 = Ou/Oy + combinationof plate boundaryshearing force Qx(0), OV/OX;/}22 -- 20v/Oy). Within the low-viscosityboundary layer of the lower crust, the dominant basic process is simple shear on horizontal Qx(ck) - - , (A2) planes,so the principal stressaxes are dipped approximately ,r/2from x andy (hencetryy and trxx are approximately equal). and plate boundarymoment M•(0) (in the case of broken Then, the horizontalcomponent of quasi-staticstress equilib- plate) and w = 0, Ow/Ox = 0 (and h = 0, Oh/Ox = 0) at rium equationdivtr + p# = 0, where tensortr is tr = r - PI (I is identity matrix), can be locally simplifiedyielding thin layer approximation[e.g., Lobkovsky, 1988; Bird, 1991]:
Orxy Op Oryy Appendix B: Model of Flow in the Ductile Crust -- Flow in the low-viscositycrustal channel is computedin a Oy Ox Fx Ox (B2) way similar to the Couette flow model of Lobkovsky[1988], Lobkovskyand Ketchman [1991] (hereafter referred as LK), or An effectiveshear strain rate can be derived kxy = Bird [1991].Our formulationcan allowcomputation of differ- (rxy/21Xeff;therefore, according to constitutiverelation (7), horizontalvelocity u in the lower crust is ent typesof flow ("symmetrical,"Poiseuille, Couette). In the numericalexperiments shown in this paper we consideronly caseswith a mixed Couette/Poiseuille/symmetricalflow, but we u(y) = 2&x 0y + c, also checkedour formulationby comparisonwith that of LK for a Couette flow. The low-viscosity crustal channel is boundedby the competentlayers in the crustat the top and in the mantle at the bottom.Flexure of theselayers will drive flow = 2"A* exp(-H*/RT(y))lrxyl"-'rxy Oy+C,. (B3) in the lowercrust. In turn, changesof the thicknessof the lower 17,766 AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH
Here• = y - Y•3; Y•3 = Y•3((b) is the upper surfaceof the channeldefined from solutionof the system(AI). u = 2"A* exp(-H*/Rr(y))l,xyl"-•,xy Oy "0 hc2-y13 C• is an arbitraryconstant of integrationdefined from the velocityboundary conditions (Figure 4); rxy is definedfrom (B6) depth integrationof (B2). The remote conditionsh - 0, Oh/Ox = 0, w = 0, Ow/Ox - 0 for the strong layers of the Vlø = Ox uOy• Ot lithosphere(Appendix A) are in accordancewith the condition .lhc2-y13(••"0 hc2-y13 (•(•/.ql_W) for ductileflow: x • •c. Uc+2= Uc; Uc2 = Urn;Op/Ox = O, Op/Oy -- hog;P = Po' The later equation gives the variation of thicknessof the The major perturbationto the stressfield is causedby slopes ductile channelin time (equal to the differencebetween the of crustalinterfaces a • Oh/Ox and/3 --• Ow/Ox. These slopes vertical flow at the top and bottom boundaries). are controlledby flexure, isostaticreadjustments, surface ero- The temperature,which primarily controlsthe effectivevis- sion, and modification of the interfacesby stressand temper- cosityof the crust,is muchlower in the uppermostand middle ature (thermomechanicalweakening). In the assumptionof portionsof the upper crust(at 10-15 km depth).As a result, small plate deflectionsthe horizontalforce associatedwith the effectiveviscosity of the middle portionsof the uppercrust variationof the gravitationalpotential energy due to deflection is 2-4 ordershigher than that of thelower crust (10 22 to 1023 of Moho (w) is Pc2gtan (/3) --• P,.2gsint3 --• p,.2gOw/Ox;the Pa s comparedto 10•8 to 1020Pa s, equations(7) and (8)). vertical componentof force is '"'P,-2gcos /3 ---- Pc2g(1 - Therefore we can consider the reaction of the lower crust to Ow/Ox) --• Pc2g.The horizontaland vertical force components deformationof the upper crust as rapid. For numericalrea- due to slopesof the upper walls of the channelare Pc2g tan sons,we cut the interval of variation of the effectiveviscosity at (or)---- pg sin (a) --•Pc2gd•/dx and 0,-29 cos (/3) ----pc2g(1 - 10•9 to 1024 Pa s.Solution for thechannel flow implies that the dh/dx), respectively.The equation of motion (Poiseuille/ channel is infinite in both directions. In our case the channel is Couetteflow) for a thin layerin the approximationof lubrica- semi-infinitebecause of the requirement u = 0 at x - 0. tion theorywill be Velocity boundaryconditions are assumedon the upper and bottom interfaces of the lower crustal channel. Free flow is the Orxy Oryy Op O(h+ w) inherent lateral boundary condition. Oy Ox Ox P,.xq Ox
Appendix C: Analytical Formulation OryyOy + Ox Oy • -p,xq1- Ox (B4) for Ascending Crustal Flow In this appendixwe presentcomputation of the flow under OU c2 0 •Yc2 the range,where the thin channelapproximation of Appendix = o. B may not be valid. In a generalcase of noninertialflow (low Reynoldsnumber) the flowbeneath the mountainrange can be wherep • Po(x) + bcg(Y + Y•.• + h)and b,. is averaged computed,assuming symmetry with respectto the axisof the crustaldensity. In the caseof local isostasy,w and Ow/Ox are range, from the system[Fletcher, 1988; Hamilton et al., 1995] approximately4 times (bc/A(•. - p,,) --• 4) greaterthan h andd[t/dx, respectively. The pressure gradient due to deflec- tion of the Moho is pm•70(h + w)/Ox. Accountingfor the 0 = p,2Fx-•xx+•y 2Ix •yy+•x differenceof densitybetween the crust and mantle, the effec- tive pressuregradient in the crustis (p,, - b,.)gO(h + w)/Ox, withw beingequal to • (pm-- bc)/Pm)'In thecase of regional •-•+•xx2g •yy+•xx (C1) compensation(as computedfrom the system(AI)) the differ- Ou Ov + =0 ence betweenh and w is reduced by a factor 2 to 3. Ox •y Equations(B3) and (B4) imply We define Op/Ox• Op/Ox+ g(p,.20w/Ox+ p•.•O(du)/Ox), Orxy Op O([t+ w) du • d•, andOp/Oy = O•/Oy- 9p,.2,where fl is dynamic, or modified,pressure. The flow is assumedto be Couette/Poiselleflow (Appendix B) awayfrom the symmetryaxis (at a distancea•); a• is equal 0•• P•2g1 -- Ox to 1-2 thicknessesof the channel,depending on channelthick- (B5) ness-to-lengthratio. In practice,at is equal to the distanceat which the equivalentelastic thickness of the crust (as defined by Burovand Diament [1995]) becomesless than 5 km. When the elastic thickness of the crust is so thin, we assumethat the Ov Ou upper boundaryof the ductilelower crustcan be considered Oy Ox free of shearstress. This assumptionmakes the computation easier.Another more realisticboundary condition would prob- Assumingsmall deflections (w/Te << L/Te, h --. 0.2 - 0.5w, ably modify the flow only in a thin zone at top of the low- whereL isthe length of the plate), we get 1 - 0(• + w)/Ox• 1. viscositychannel in the lower crust. Depth integrationof (B5) yieldsthe longitudinaland vertical The remote feedingflux q at x -• _+a l is equal to the value componentsof the velocityin the lower crust.For example,we of flux obtainedfrom depth integrationof the channelsource have (Couetteflow). The flux q is determinedas q --• • u dy (per AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,767
unit length in z direction). This flux feeds the growth of the topographyand deepeningof the crustalroot. Combinationof + • (-1)"y2n+2[(2n+ 2)!]-•V(2"+•)(X) (C4b) n=0 two flow formulationsis completedusing continuity equations [Huppert,1982] G(X) = 3/(s) ds. (C4c) O(h + w) Oq +-- (C2a) uay =o= Ot Ox wheref andT areexpressed in expansion series (f(i), T (i) are q x=a,,x_
p- restoringstress per unit length, Pa/m. Avouac,J.-P., Analysisof scarpprofiles: Evaluation of errorsin mor- h (x, t) surfacetopography, m. phologicdating, J. Geophys.Res., 98, 6745-6754, 1993. Avouac, J.-P., and P. Tapponnier,Kinematic model of activedefor- h (x, t) upper boundaryof ductilechannel, m. mation in central Asia, Geophys.Res. Lett., 20, 895-898, 1993. he, Tc Moho depth, m. Avouac, J.-P., P. Tapponnier, M. Bai, H. You, and G. Wang, Active h•2 lower boundaryof ductile crustal thrustingand foldingalong the northernTien Shan and late Ceno- channel(hc2 <- Tc), m. zoic rotation of the Tarim relative to Dzungaria and Kazakhstan,J. hc•(X, t, w) maximalmechanical thickness of the Geophys.Res., 98, 6755-6804, 1993. upper crust(10-20 km), m. Beaumont, C., Foreland basins,Geophys. J. R. Astron. Soc., 65, 389- 416, 1981. Ahc2(X, t, w, u, v) thicknessof crustalchannel, m. Beaumont, C., P. Fullsack, and J. Hamilton, Erosional control of active Yii depthsto lithologicaland mechanical compressionalorogens, in ThrustTectonics, edited by K. R. McClay, interfaces, m. pp. 1-31, Chapman and Hall, New York, 1992. Pc• densityof upper crust, equal to 2650 Beaumont,C., P. Fullsack,and J. Hamilton, Stylesof crustaldeforma- kg/m3. tion in compressionalorogens caused by subductionof the under- lyinglithosphere, Tectonophysics, 232, 119-132, 1994. Pc2 densityof lower crust, equal to 2900 Beekman, F., Tectonic modellingof thick-skinnedcompressional in- kg/m3. traplate deformation,Ph.D. thesis,Free Univ., Amsterdam,1994. Pm densityof mantle,equal to 3330k•m 3. Bird, P., Lateral extrusionof lower crustfrom under high topography a inclinationof upper bounda• of in the isostaticlimit, J. Geophys.Res., 96, 10,275-10,286,1991. channel(a • Oh/Ox). Brace, W. F., and D. L. Kohlstedt,Limits on lithosphericstress im- posedby laboratoryexperiments, J. Geophys.Res., 85, 6248-6252, D inclinationof lower bounda• of 1980. channel(D • Ow/Ox). Burbank, D. W., Causesof recent Himalayan uplift deducedfrom g accelerationdue to gravity,equal to depositedpatterns in the Gangesbasin, Nature, 357, 680-683, 1992. 9.8 m/s2. Burov, E. B., The short-wavelengthgravity anomaliesand flexure of t, thermal age (• geologicalage), m.y. the continentallithosphere (in Russian),Ph.D. thesis,Inst. of Phys. of the Earth, Acad. of Sci., Moscow, 1990. a thermal thicknessof the lithosphere, Burov, E. B., and M. Diament, Flexure of the continentallithosphere equal to 250 •. with multilayeredrheology, Geophys. J. Int., 109, 449-468, 1992. Tm T at depth a, equal to 1330øC. Burov,E. B., and M. Diament, The effectiveelastic thickness (Te) of X thermaldiffusivity, X = k/pCp,m 2 continentallithosphere: What doesit really mean?,J. Geophys.Res., --1 S . lO0, 3905-3927, 1995. X•.• thermal diffusivity(upper crust),equal Burov, E. V., M. G. Kogan, H. Lyon-Caen, and P. Molnar, Gravity anomalies,the deep structure,and dynamicprocesses beneath the to8.3 x 10 -7m 2S-1 Tien Shan, Earth Planet. Sci. Lett., 96, 367-383, 1990. X•-2 thermal diffusivity(lower crust),equal Burov, E. B., L. I. Lobkovsky,S. Cloetingh, and A.M. Nikishin, to 6.7 x 10 -7 m 2 s-•. Continentallithosphere folding in centralAsia, 2, Constraintsfrom X• thermal diffusivity(mantle), equal to gravityand topography,Tectonophysics, 226, 73-87, 1993. 8.75 x 10 -7 m 2 s-1. Byerlee, J. D., Friction of rocks,Pure Appl. Geophys.,116, 615-626, 1978. k•.• thermal conductivity(upper crust), Carson,M. A., and M. J. Kirkby,Hillslope Form and Processes,475 pp., equalto 2.5 W m-• øK-• CambridgeUniv. Press,New York, 1972. k•.2 thermal conductivity(lower crust), Carter, N. L., and M. C. Tsenn, Flow propertiesof continentallitho- equalto 2 W m-• øK-• sphere,Tectonophysics, 36, 27-63, 1987. Chen, Y., J.-P. Cogne, V. Courtillot, J.-P. Avouac, P. Tapponnier, k• thermal conductivity(mantle), equal E. Buffetaut,G. Wang., M. Bai, H. You, M. Li, and C. Wei, Paleo- to 3.5 W m-• øK-• magneticstudy of Mesozoiccontinental sediments along the north- h• decay scaleof radiogenicheat ern Tien Shan(China) and heterogeneousstrain in centralAsia, J. production(upper crust),equal to 10 Geophys.Res., 96, 4065-4082, 1991. km. Chorley, R. J., S. A. Schumm,and D. E. Sugden,Hillslopes in Geo- morphology,pp. 255-339, Methuen, New York, 1984. H, surfaceradiogenic heat production Colman, S. M., Rock-weatheringrates as functionsof time, Quat. Res., rate per unit mass(upper crust), 15, 50-64, 1981. equalto 9.5 x 10-••øW kg- • Copeland,P., and T. M. Harrison,Episodic rapid uplift in the Hima- H,.2C[•• radiogenicheat (lower crust), equal to layarevealed by 4øAr/3'•Aranalysis of detritalK-feldspar and mus- covite,Bengal fan, Geology,18, 354-357, 1990. 1.7 x 10- •3 øK s-•. Culling, W. E. H., Analyticaltheory of erosion,J. Geol., 68, 336-344, 1960. Davies,G. F., Thcrmomechanicalerosion of the lithosphereby mantle Acknowledgments. E.B. Burov is greatly indebted for numerous plumes,J. Geophys.Res., 99, 15,709-15,722, 1994. discussionsto L. Lobkovsky. G. Houseman, A. Lenardic, and D. Ellis, S., P. Fullsack, and C. Beaumont, Oblique convergenceof the Coblentzprovided detailed and thoroughreviews. We also benefited crust driven by basalforcing: Implications for length-scalesof de- from discussionswith S. Cloetingh and Geoffroy King. This work was formation and strain partitioningin orogens,Geophys. J. Int., 120, funded via contact U4 3009361/IL and U4 5000333/VC with CEA. 24-44, 1995. IPGP publication1381. England, P. C., and G. A. Houseman, Finite strain calculationsof continentaldeformation, 2, Comparisonwith the India-Asia colli- sion zone, J. Geophys.Res., 91, 3664-3676, 1986. References England, P. C., and D. P. McKenzie, A thin viscoussheet model for continentaldeformation, Geophys. J. R. Astron.Soc., 73, 523-532, 1983. Ahnert, F., Functional relationshipsbetween denudation,relief and England,P., and S. W. Richardson,The influenceof erosionupon the uplift in largemid-latitude drainage basins, Am. J. Sci.,268, 243-263, mineral faciesof rocksfrom differentmetamorphic environments, J. 1970. Geol. Soc. London, 134, 201-213, 1977. Andrews,D. J., and R. C. Bucknam,Fitting degradationof shoreline Flemings,P. B., and T. E. Jordan,Stratigraphic modelling of foreland scarpsby a nonlinear diffusionmodel, J. Geophys.Res., 92, 12,857- basins:Interpreting thrust deformation and lithosphererheology, 12,867, 1987. Geology,18, 430-434, 1990. AVOUAC AND BUROV: INTRACONTINENTAL MOUNTAIN GROWTH 17,769
Fletcher, C. A. J., ComputationalTechniques for Fluid Dynamics2, 552 tectonics:Application to geodynamics,Tectonophysics, 199, 343-374, pp., Springer-Verlag,New York, 1988. 1991. Fournier,F., Climat et Erosion:La RelationEntre l'Erosiondu Solpar Ma, X., LithosphericDynamic Atlas of China, China Cartograph.Publ. l'Eau et lesPrecipitations Atmosph6riques, 201 pp., PresseUniv. de House, Beijing, China, 1987. France, Paris, 1960. Makeyeva,L. I., L. P. Vinnik, and S. W. Roecker,Shear-wave splitting Goetze, C., and B. Evans, Stress and temperature in the bending and small scaleconvection in the continentalupper mantle, Nature, lithosphereas constrainedby experimental rock mechanics,Geo- 358, 144-147, 1992. phys.J. R. Astron. Soc., 59, 463-478, 1979. Masek, J. G., B. L. Isacks,T. L. Gubbels,and E. J. Fielding, Erosion Gossman,H., Slope modellingwith changingboundary conditions-- and tectonicsat the margins of continental plateaus,J. Geophys. Effectsof climateand !ithology, Z. Geomophol.,Suppl., 25, 72- 88, 1976. Res., 99, 13,941-13,956, 1994. Gratton, J., Crustal shortening, root spreading, isostasy,and the McAdoo, D.C., Martin, C. F., and S. Polouse, Seasat observationsof growth of orogenicbelts: A dimensionalanalysis, J. Geophys.Res., flexure:Evidence for a stronglithosphere, Tectonophysics, 116, 209- 94, 15,627-15,634, 1989. 222, 1985. Greene, M. T., Geologyin theNineteenth Century: Changing P'iews of a Molnar, P., and Q. Deng, Faulting associatedwith large earthquakes ChangingWorld, 324 pp., Cornell Univ. Press,Ithaca, N.Y., 1982. and the averagerate of deformationin central and easternAsia, J. Gregory, K. M., and C. Chase,Tectonic and climatic significanceof a Geophys.Res., 89, 6203-6228, 1984. late Eocene low-relief,high-level geomorphic surface, Colorado, J. Molnar, P., and P. England,Late Cenozoicuplift of mountainranges Geophys.Res., 99, 20,141-20,160, 1994. and globalclimate change: Chicken or egg,Nature, 346, 29-34, 1990. Hamilton, J. M., J. Kim, and F. Waleffe, Regenerationmechanisms of Molnar, P., and H. Lyon-Caen, Some simple physicalaspects of the near-wall turbulence structures,J. Fluid Mech., 287, 317-348, 1995. support, structure, and evolution of mountain belts, in Processesin Hanks, T. C., R. C. Buckham, K. R. LaJoie, and R. E. Wallace, ContinentalLithospheric Deformation, Spec. Pap. Geol. Soc. Am., Modificationof wave-cutand fault-controlled!andforms, J. Geophys. 218, 179-207, 1988. Res., 89, 5771-5790, 1984. Pinet, P., and M. Souriau, Continental erosion and large-scalerelief, Hansen, E. B., and M. A. Kelmanson,An integral equationjustifica- Tectonics, 7, 563-582, 1988. tion of the boundaryconditions of the driven-cavityproblem, Com- Ranalli, G., Nonlinear flexure and equivalentmechanical thickness of put. Fluids,23(1), 225-240, 1994. the lithosphere,Tectonophysics, 240, 107-114, 1994. Hendrix, M. S., S. A. Graham, A. R. Caroll, E. R. Sobel, C. L. Roecker, S. W., T. M. Sabitova, L. P. Vinnik, Y. A. Burmakov, M. I. McKnight, B. J. Schulein,and Z. Wang, Sedimentaryrecord and Golvanov, R. Mamatkanova, and L. Minirova, Three-dimensional climatic implicationsof recurrent deformation in the Tien Shan: elasticwave velocitystructure of the westernand centralTien Shan, Evidencefrom Mesozoicstrata of the north Tarim, southJunggar, J. Geophys.Res., 98, 15,779-15,795, 1993. and Turpan basins, northwest China, Geol. Soc. Am. Bull., 104, Smith, R. B., Unified theory of the onset of folding, boudinageand 53-79, 1992. mullion structure, Geol. Soc. Am. Bull., 88, 1601-1609, 1975. Hendrix, M. S., T. A. Dumitru, and S. A. Graham, Late Oligocene- Summerfield, M. A., and N.J. Hulton, Natural control on fluvial early Miocene unroofingin the ChineseTian Shan:An early effect denudationrates in major world drainagebasins, J. Geophys.Res., of the India Asia collision,Geology, 22, 487-490, 1994. 99, 13,871-13,883, 1994. Houseman,G., and P. England,Finite strain calculationsof continen- Talbot, C. J., and R. J. Jarvis,Age, budget and dynamicsof an active tal deformation,1, Method and generalresults for convergentzones, salt extrusion in Iran, J. Struct. Geol., 6, 521-533, 1984. J. Geophys.Res., 91, 3651-3663, 1986. Tapponnier,P., and P. Molnar, Activefaulting and Cenozoictectonics Huppert, H. E., The propagationof two dimensionaland axisymmetric of the Tien Shan,Mongolia, and Baykalregions, J. Geophys.Res., 84, gravitycurrents over a rigid horizontalsurface, J. Fluid. Mech., 121, 3425-3459, 1979. 43-58, 1982. Turcotte, D. L., and G. Schubert,Geodynamics: Applications of Con- Kaufman,P.S., and L. H. Royden,Lower crustalflow in an extensional tinuum Physicsto GeologicalProblems, 450 pp., John Wiley, New setting:Constraints from the Halloran Hills region,eastern Mojave York, 1982. Desert, California, J. Geophys.Res., 99, 15,723-15,739, 1994. Vilotte, J.P., M. Daigniares,and R. Madariaga, Numerical modeling King, G. C. P., and S. Ellis, The origin of large local uplift in exten- of intraplate deformation:Simple mechanical models of continental sional regions,Nature, 348, 689-693, 1990. collision,J. Geophys.Res., 87, 10,709-10,728, 1982. King, G. C. P., R. S. Stein, and J. B. Rundle, The growthof geological Vinnik, L. P., and A.M. Saipbekova,Structure of the lithosphereand structuresby repeated earthquakes,1, Conceptual framework, J. asthenosphereof the Tien Shan,Ann. Geophys.,2, 621-626, 1984. Geophys.Res., 93, 13,307-13,318, 1988. Vogt, P. R., Bermuda and Applachian-Labradorrises, common hot- Kirby, S. H., and A. K. Kronenberg, Rheology of the lithosphere: spot processes,Geology, 19, 41-44, 1991. Selectedtopics, Rev. Geophys.,25, 1219-1244, 1987. Wang, J. N., B. E. Hobbs, A. Ord, T. Shimamoto, and M. Toriumi, Kirkby, M. J., A two-dimensionalmodel for slope and streamevolu- NewtonJandislocation creep in quartzites:Implications for the rhe- tion, in HillslopeProcesses, edited by A.D. Abrahams,pp. 203-224, ology of the lower crust,Science, 265, 1204-1206, 1994. Allen and Unwin, Winchester, Mass., 1986. Westaway, R., Evidence for dynamic coupling of surface processes Kohlstedt,D. L., B. Evans,and S. J. Mackwell, Strengthof the litho- with isostaticcompensation in the lower crust during active exten- sphere:Constraints imposed by laboratoryexperiments, J. Geophys. sionof westernTurkey, J. Geophys.Res., 99, 20,203-20,223,1994. Res., 100, 17,587-17,602, 1995. Windley, B. F., M. B. Allen, C. Zhang, Z. Zhao, and G. R. Wang, Kruse, S., M. McNutt, J. Phipps-Morgan,and L. Royden,Lithospheric Paleozoic accretion and Cenozoic redeformation of the Chinese extension near lake Mead, Nevada: A model for ductile flow in the Tien Shan range, central Asia, Geology,18, 128-131, 1990. lower crust,J. Geophys.Res., 96, 4435-4456, 1991. Zhao, W. L., and J. Morgan, Uplift of Tibetan plateau, Tectonics,4, Kusznir,N.J., The distributionof stresswith depth in the lithosphere: 359-369, 1985. Thermo-rheologicaland geodynamicconstraints, Philos. Trans.R. Soc. London A, 337, 95-110, 1991. J.P. Avouac, Laboratoire de G6ophysique,Groupe Tectoniqueet Kusznir, N.J., and D. H. Matthews, Deep seismicreflections and the StructuresLithosph6riques, BP12, 91680, Bruyares-Le-Chatel,France. deformationalmechanics of the continentallithosphere, J. Petrol., (e-mail: [email protected]) Spec.Lithosphere Issue, 63-87, 1988. E. B. Burov, Laboratoire de Gravim6trie et G6odynamique(J.E. Leeder, M. R., Denudation, vertical crustal movements and sedimen- 335), Institut de Physiquedu Globe de Paris,case 89, 4 PlaceJussieu, tary basininfill, Geol.Rundsch., 80(2), 441-458, 1991. 75252 Paris Cedex 05, France. (e-mail: [email protected]) Lobkovsky,L. I., Geodynamicsof Spreadingand SubductionZones, and the Two-LevelPlate Tectonics,251 pp., Nauka, Moscow,1988. (ReceivedJune 26, 1995; revisedMarch 25, 1996; Lobkovsky,L. I., and V. I. Kerchman,A two-levelconcept of plate acceptedApril 29, 1996.)