Spacing of Faults at the Scale of the Lithosphere and ... · Spacing of Faults at the Scale of the Lithosphere and Localization Instability 1: Theory Laurent G. J. Montesi´ 1 Maria

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. , NO. , PAGES 1–16,

Spacing of Faults at the Scale of the Lithosphere and LocalizationInstability 1: Theory

Laurent G. J. Montesi1

Maria T. Zuber2

Massachusetts Institute of Technology, Cambridge, Massachusetts.

Abstract. Large-scale tectonic structures such as orogens and rifts commonly display regularly–spaced faults and/or localized shear zones. Numerical models reproduce this phenomenon. Tounderstand how fault sets organize with a characteristic spacing, we present a semi–analyticalinstability analysis of an idealized lithosphere composed of a brittle layer over a ductile half–space undergoing horizontal shortening or extension. The rheology of the layer is character-ized by an effective stress exponent,ne. The layer is pseudo–plastic if1/ne = 0 and formslocalized structures if1/ne < 0. The tendency for localization is stronger for more negative1/ne. Two instabilities grow simultaneously in this model: thebuckling/necking instabilitythatproduces broad undulations of the brittle layer as a whole, and thelocalization instabilitythatproduces a spatially periodic pattern of faulting. The latter appears only if the material in thebrittle layer weakens in response to a local perturbation of strain rate, as indicated by1/ne <0. Fault spacing scales with the thickness of the brittle layer and depends on the efficiencyof localization. The more efficient localization is, the more widely spaced are the faults. Thefault spacing is related to the wavelength at which different deformation modes within the layerenter a resonance that exists only if1/ne < 0. Depth–dependent viscosity and the model den-sity offset the instability wavelengths by an amount that we determine empirically. The wavenum-ber of the localization instability, iskL

j = π (j + aL) (−1/ne)−1/2

/H, with H the thick-ness of the brittle layer,j an integer, and1/4 < aL < 1/2 if the strength of the layer in-creases with depth and the strength of the substrate decreases with depth.

1. Introduction

Much of our theoretical understanding of tectonics stems fromcontinuum mechanics [Turcotte and Schubert, 2002]. For instance,certain large–scale patterns of deformation resemble a mode offolding of the strong layers of the lithosphere called buckling incompression and necking in extension [Biot, 1961;Fletcher andHallet, 1983;Ricard and Froideveau, 1986;Zuber et al., 1986;Zu-ber, 1987]. The buckling or necking theory predicts a preferredwavelength of deformation that is controlled by the mechanicalstructure of the lithosphere. Hence, recognizing buckling in the ge-ological records helps to constrain the structure of the lithosphereat the time when the structure formed.

Faults and shear zones constitute another primary indicator ofthe structure of orogens or rifts that can be related to continuummechanics models [Beaumont and Quinlan, 1994]. Faults oftenpresent a preferred spacing or characteristic scale [Weissel et al.,1980;Zuber et al., 1986;Davies, 1990;Watters, 1991;Bourne et al.,1998]. The buckling/necking theory might then be used to model

1Now at Woods Hole Oceanographic Institution, Woods Hole,Massachusetts.

2Also at Laboratory for Terrestrial Physics, NASA/Goddard SpaceFlight Center, Greenbelt, MD.

Copyright by the American Geophysical Union.

Paper number .0148-0227/02/$9.00

fault spacing [Watters, 1991;Brown and Grimm, 1997], but faultsactually represent a localized style of deformation that is not acces-sible using the continuum theories implied in the buckling/neckinganalysis; deformation occurs mostly —if not entirely— within anarrow band. In this study, we modify the buckling/necking theoryto consider explicitly the dynamics of localization, and to explorethe link between the structure of the lithosphere and patterns oflocalized shear zones or faults.

Buckling and necking produce broad undulations of the litho-sphere in which deformation is distributed. The pseudo–plastic rhe-ology used to model the brittle levels of the lithosphere [Fletcherand Hallet, 1983] assumes distributed failure or faulting. However,faulting has a tendency to localize, to abandon a distributed style offaulting to concentrate deformation on a few isolated major faults[Sornette and Vanneste, 1996;Gerbault et al., 1998]. As stressheterogeneities induced by buckling favor faulting in the hinge oflarge–scale folds [Lambeck, 1983;Martinod and Davy, 1994;Ger-bault et al., 1999], localized fault patterns can be controlled bybuckling if they develop after the folds have reached sufficient am-plitude. However, faulting may occur from the onset of deformationand hence, may develop without the influence of finite–amplitudebuckling. Indeed, some tectonic provinces display faults with acharacteristic spacing unrelated to the buckling wavelength. Tocite only examples in compressive environments, faults in the Cen-tral Indian Basin [Bull, 1990;Van Orman et al., 1995], in CentralAsia [Nikishin et al., 1993], or in Venusian fold belts [Zuber andAist, 1990] are more closely spaced than the wavelength of foldsin the same region. In these regions, faulting and buckling appearas superposed deformation styles, each with a characteristic lengthscale.

1

2 MONTESI AND ZUBER: FAULT SPACING 1: THEORY

Buckling and necking were first studied in Earth sciences asa mechanism to form folds or boudins in outcrop–scale layeredsequences [Johnson and Fletcher, 1994]. While originally derivedusing a thin plate approximations of viscous and/or elastic materials[Ramberg, 1961;Biot, 1961], the buckling/necking theory was laterdeveloped with a thick plate formulation [Fletcher, 1974;Smith,1975] and was applied to non–Newtonian materials [Fletcher, 1974;Smith, 1977]. For a non–linear rheology, the stress supported bythe fluid,σ, is related to the second invariant of strain rate,εII , byεII ∝ σne , with ne the effective stress exponent, a measure of thenon–linearity of the rock rheology [Smith, 1977;Montesi and Zu-ber, 2002]. Non–Newtonian creep with1 < ne < 5 is the rheologythat describes rocks at sufficiently high temperature to behave in aductile manner.

At low temperature, rocks behave instead in a brittle manner. Thestress that they can support is limited by a yield strength, at whichfailure, faulting, and plastic flow occur. Yielding can be includedin thin–plate analyses of folding by limiting the bending stresses tothe yield strength and reducing the apparent flexural rigidity of aelastic or viscous plate accordingly [Chapple, 1969;McAdoo andSandwell, 1985;Wallace and Melosh, 1994]. The thick–plate for-mulation of the buckling theory is particularly well adapted to analternative treatment of failure, in which the yielding material is ap-proximated as a highly non–Newtonian fluid in the limitne → +∞[Chapple, 1969, 1978;Smith, 1979]. Most lithospheric–scale ap-plications of buckling use that approximation [Fletcher and Hallet,1983;Zuber et al., 1986;Zuber, 1987]. More accurate treatments ofyielding have been included in buckling/necking theory.Leroy andTriantafyllidis [1996, 2000] andTriantafyllidis and Leroy[1997]studied the necking behavior of a hardening elastic–plastic mediumat yield using the strain rate–stress rate relations of the deforma-tion theory of plasticity. Localized faulting is predicted only fortectonic stresses is excess of the critical value for necking and isnever predicted if the flow theory of plasticity is used [Triantafyl-lidis and Leroy, 1997]. Therefore this model cannot explain regionsthat show regularly–spaced localized faulting.Davies[1990] mod-eled the buckling of a rigid–plastic layer with associated flow lawsurrounded by a rigid basement and a viscous half–space. Faultswere forced by a local cusp in the model interface but an initiallydistributed perturbation remains distributed.Neumann and Zuber[1995] found that a localized perturbation triggers macroscale shearbands in a power–law medium withne → +∞ as well. Fletcher[1998], who also included the effects of pore fluids and pressuresolution on a non–Newtonian porous fluid withne → +∞, showedthat the shear bands produced by a localized forcing are ephemeral.He speculated that the bands would be stabilized and therefore maycorrespond to faults if strain–weakening were included [Fletcher,1998].

All previous treatments of yielding in buckling theory fail to pro-duce localized deformation; faulting remains distributed throughoutthe material. Hence, any fault pattern observed in nature is expectedto develop late in the folding history, with a spacing that is controlledby the buckling or necking wavelength. However, this is contrary tomany geological observations [Weissel et al., 1980;Nikishin et al.,1993;Krishna et al., 2001]. In order to understand how fault spac-ing may differ from the buckling/necking wavelength, we study thebuckling behavior of simplified lithosphere models with a rheologythat weakens with deformation upon yielding. The weakening be-havior is characterized by a negative effective stress exponent. Suchan effective rheology brings a tendency to localize deformation andfaulting [Montesi and Zuber, 2002].

The effective stress exponent,ne, indicates how a material re-sponds to a perturbation in the deformation field [Smith, 1977]. Inthis study, deformation is quantified by the second strain rate in-variant, εII . Whenne < 0, increasingεII decreases the materialstrength, which ensures localization [Montesi and Zuber, 2002].

The (algebraic) value ofne is determined from the physical processthat produces localization. It incorporates not only the direct effectof perturbing the strain rate, but also the possible feedback of in-ternal variables that control the rheology. For instance, a frictionalmaterial such as the pervasively faulted brittle lithosphere requires ahigher stress to deform more rapidly, which would result in positivene. However, a higher sliding velocity on faults changes also thephysical state of a granular gouge inside the fault and results in ap-parent weakening and negativene [Dieterich, 1979;Scholz, 1990].In Montesi and Zuber[2002], we derived the conditions for whichthis feedback mechanism and others produce localization, and wegave values for the corresponding effective stress exponents. Forfrictional sliding,−300 < ne < −50.

Neurath and Smith[1982] showed that strain weakening reducesthe value of1/ne in a non–Newtonian fluid, possibly resulting inne < 0. However, the materials that they studied never had negativene. While they recognized thatne < 0 would lead to “catastrophicfailure”, or localization, modulated by the interaction with the sur-roundings of the layer with negative exponent,Neurath and Smith[1982] did not present an analysis of buckling withne < 0.

In this paper, we derive the growth spectrum for simple modelsof the lithosphere, which indicates how fast a given wavelengthgrowths in a particular model [Johnson and Fletcher, 1994]. Buck-ling and necking occur at wavelengths at which the growth rateis maximum. We identify a new instability when the material lo-calizes and call it localization instability. Both the localizationinstability and the buckling/necking instability are associated withparticular resonances within the localizing layer. We show how thewavelengths of the buckling and localization instabilities relate toresonances within the model, considering several types of depth–dependent strength profiles that approximate the strength profileof a mono–mineralic lithosphere. Most of this study is conductedassuming uniform horizontal shortening, with a generalization tohorizontal extension in§6.2. In a companion paper, we show thatthe localization instability is a possible explanation for fault spacingin the Central Indian Basin, which is incompatible with a bucklinginstability.

Layer n1, η1, ρ

Substratum n2, η2, ρ

x

z

ξ1

ξ2H

half-space

λ

εxx

Figure 1. Schematic of lithosphere models. A layer of thicknessH, viscosityη1(z), and effective stress exponentn1 lies over ahalf–space of different mechanical properties (η2(z), n2 = 3). Themodel is two–dimensional, incompressible, and subjected to pureshear shortening at the rateεxx. Its density isρ. Boundary condi-tions are no slip at the interface, no stress at the surface. For theinstability analysis, the interfaces are perturbed by an infinitesimalsinusoidal topography of wavelengthλ and amplitudeξ1 andξ2.

MONTESI AND ZUBER: FAULT SPACING 1: THEORY 3

2. Instability principle

2.1. Solution strategy

The buckling/necking and localization instabilities develop inmechanically layered models of the lithosphere undergoing hori-zontal shortening or extension. In this study, we concentrate onthe behavior of a single brittle or plastic layer overlying a ductilehalf–space under horizontal shortening at the rateεxx (Figure 1).Horizontal extension is discussed in§6.2. Each material is in-compressible. As the deformation field is constrained to be two–dimensional, it is completely determined by the stream functionϕ(x, z).

The deformation field in each layer is decomposed into a primaryfield and a secondary field of much smaller amplitude. The primaryfield is the flow solution when the interfaces in the model are per-fectly flat and horizontal. It is invariant inx, with εzz = −εxx, andεxz = 0 . However, any relief on the interfaces perturbs the systemand drives a secondary flow in the model. In turn, the secondaryflow deforms the interface.

This analysis determines the rate at which interface perturbationsgrow in a given model. The flow equations for the secondary defor-mation field and the boundary conditions are linearized, assumingthat the interface perturbations have a much smaller amplitude thanthe layer thickness [Johnson and Fletcher, 1994;Montesi, 2002].Thus, the Fourier components of an infinitesimal generic perturba-tion are decoupled from one another. Hence, we consider interfaceperturbations of the form

ξi = ξ0i exp(ikx), (1)

with ξ0 the amplitude of the perturbation,k its wavenumber,i anindex identifying an interface of the model, andi = (−1)1/2. Thereare several modes of deformation in each layer, each characterizedby a stream functionϕj

ϕj = ϕ0jφj(z) exp(ikx), (2)

whereφj is the depth kernel,ϕ0j is the amplitude ofϕj , andj is

an index identifying each deformation mode. The depth kernel isdetermined from the equations of Newtonian equilibrium and themode amplitude is determined from the stress and velocity boundaryconditions are each interface [Johnson and Fletcher, 1994;Montesi,2002]. Therefore,ϕ0

j is proportional to{ξi}.The rate at which the interface perturbations grow has a kine-

matic contribution from the pure shear thickening of the model(primary flow field) and a dynamic contribution from the secondaryflow field [Smith, 1975]. We obtain:

dξ0i

dt= (−εxxδij + Qij)ξj , (3)

with δij the Kronecker operator andQij the growth matrix, whichdepends onk and the strength and density structure of the model[Montesi, 2002]. The growth rateQ is the eigenvalue ofQ that hasthe largest real part. The associated eigenvector describes the fastestgrowing deformation mode of the model as a whole [Smith, 1975;Zuber et al., 1986]. For instance, it determines whether a givenlayer deforms by buckling (upper and lower interfaces in phase)or necking (upper and lower interfaces out of phase). The growthspectrum is defined as the functionQ(k).

2.2. Effective rheology and effective stress exponent

The strength of each layer of the lithosphere model is an analyt-ical function of depth. As we address the organization of strain ratewithin this model, we define the apparent viscosityη by

σij = −pδij + ηεij , (4)

whereσ is the stress supported by the material,p the pressure, andεthe strain rate. In general,η depends on the second invariant of thestrain rate,εII and on the depthz. In the brittle field, the materialstrength and therefore the viscosity may also depend on the strainundergone by the material. In this study, we chose to ignore thatadditional complication, which would also require elasticity to beincluded in the model and make the analysis time–dependent [Neu-rath and Smith, 1982;Schmalholz and Podladchikov, 1999, 2001]

In the perturbation analysis, the primary field, which representsthe state of uniform shortening, is denoted by over–bars. It obeys

σij = −pδij + η¯εij , (5)

with η the material viscosity at the externally imposed strain rate¯εII = |εxx|. It is an analytic function of depth in each layer. Thesecondary field, denoted with a tilde, which represents the perturb-ing flow obeys the apparent rheology:

σxx = −p +η

ne

˜εxx, (6a)

σzz = −p +η

ne

˜εzz, (6b)

σxz = η˜εxz, (6c)

with ne the effective stress exponent,

1

ne= 1 +

¯εII

η

∂η

∂εII

=¯εII

σ

∂σ

∂εII

. (7)

In deriving Eq. 6, we assumed that the amplitude of the secondaryfield is infinitesimal with respect to the primary field. This approx-imation is valid only for the onset of the instability.

The apparent viscosity of the secondary field is anisotropic, re-duced in the directions of the primary flow field by a factorne.As introduced bySmith[1977], the effective stress exponent is alocal measure of the non–linearity of the rheology of a material.We extended this concept inMontesi and Zuber[2002] and usedne in a general framework of localization. Whenne < 0, thematerial is unstable with respect to local perturbations. A negativeeffective stress exponent indicates that the strength is reduced inlocations where the strain rate is enhanced. This situation is un-stable and results in a localized zone of high strain rate [Montesiand Zuber, 2002]. The instability analysis of this study shows howthese localized deformation areas organize at lithospheric scale.

Beyond the sign of the effective stress exponent, its algebraicvalue provides a quantitative measure of the efficiency of localiza-tion [Montesi and Zuber, 2002]. If1/ne = 1, the material is effec-tively Newtonian, and does not localize at all. If0 < 1/ne < 1,the material is well described by non–Newtonian creep. For in-stance, rocks deforming by ductile creep have0.2 < 1/ne < 1.In this regime, a rock is softening in the sense that its apparentviscosity decreases with increasing strain rate. Local perturbationsof strain rate are enhanced by the non–Newtonian behavior, but thematerial is stable: as its strength increases with strain rate, thereis no dynamic weakening and no localization [Montesi and Zuber,2002]. In the limit1/ne → 0+, the material is pseudo–plastic:its strength does not depend on strain rate. Localization requires1/ne < 0. We showed [Montesi and Zuber, 2002] that manymechanisms that are associated with localized shear zones in thelaboratory or in nature have a negative effective stress exponent,often with1/ne ∼ −10−2 to−10−1 in the brittle field.

Often, the effective stress exponent is negative only when aninternal feedback process is considered that may include a variable


describing damage, or state of a fault gouge. This variable mayrequire a finite time to respond to a local variation of strain rate.This results in an immediate strengthening response of the system,followed by weakening in the long-time limit. In this study, we ig-nore the transient response, arguing that perturbations may be heldfor long enough that steady–state is reached, and that the perturba-tion amplitude is so small that the strengthening “barrier” is easilyovercome. However, this assumption should be relaxed in futurework.

Most previous studies of lithospheric–scale instabilities treatedthe brittle upper crust and mantle as pseudo–plastic with1/ne →0+. These instabilities produce buckling in compression, and neck-ing in extension [Fletcher and Hallet, 1983;Ricard and Froideveau,1986;Zuber, 1987]. In this study, we introduce the solutions for1/ne < 0, which produce regularly–spaced shear zones, through aprocess that we call localization instability. The more negative1/ne

is, the stronger localization is. Intuitively, a more efficient localizedshear zone can accommodate the deformation from a wider non–localized region of the lithosphere. Hence, the spacing of localizedshear zones should increase when1/ne is more negative.

3. Depth kernel

3.1. Fundamental equation

The first step in solving the instability problem defined above isto determine the expression of the depth kernel,φ(z), which givesthe depth–dependence of the stream function (Eq. 2) and thereforeof the deformation field for each mode of deformation. Whereasthe strength and the effective viscosity the lithosphere depend ondepth, its effective stress exponent does not necessarily do so, as itmeasures the rate at which a rock weakens, scaled by its strength. Infact,ne does not depend on depth for most processes of localization[Montesi and Zuber, 2002]. Therefore, we make the assumptionthatne is independent of depth,z, within each layer.

We write the equations of Newtonian equilibrium for the sec-ondary flow, using its apparent rheology (Eq. 6), and expressing thestrain rate as a function of the stream function. These equationsare then combined to eliminate the pressure term, and simplified astheir dependence inx is exp(ikx). We obtain [Montesi, 2002]

d4φ

dz4+ 2r

d3φ

dz3−

(Ak2 − s

) d2φ

dz2

−Ak2rdφ

dz+

(k4 + k2s

)φ = 0, (8)

where we used the notation

r(z) =dη/dz

η, (9)

s(z) =d2η/dz2

η, (10)

A =4

ne− 2. (11)

Eq. 7 admits four solutions for a given strength profileη(z),effective stress exponentne, and wavenumberk. Hence, thereare four superposed deformation modes in each layer, for a givenwavelength, each with its own amplitude that is determined frommatching boundary conditions at each interface [Montesi, 2002].

3.2. Depth kernel for exponential and constant viscosityprofiles

The depth kernel can be determined analytically if the viscos-ity varies exponentially with depth. Then, the functionr does notdepend onz, s = r2, and

η = ηe exp(rz), (12)

with ηe a constant. Constant viscosity layers are included as thespecial caser = 0.

For an exponential viscosity profile, the depth kernel takes theform

φ = φ0 exp iαkz, (13)

with

α = iR

2±

[1− 2

ne− R2

4±

(4

n2e− 4

ne−R2

)1/2]1/2

,(14)

whereR = r k [Fletcher and Hallet, 1983].There are four values of the parameterα, each correspond-

ing to a given deformation mode with spatial dependence:exp [ik(x + αz)]. Therefore, the amplitude of the stream func-tion varies over a depth scale1/Im(α) and is correlated along lineswith slope−Re(α). Hence,α is referred to as the mode slope.

For constant viscosity layers (r = 0), Eq. 14 becomes

α =

±i

√−1 + 2

(1±√1− ne

)/ne, if 1 < 1/ne,

±√

1− 1/ne ± i√

1/ne, if 0 < 1/ne < 1,

±√

1− 2(1±√1− ne

)/ne, if 1/ne < 0.

(15)

By convention, we defineα1, α2, α3, and α4, by selecting inEq. 14 or 15 the sign combinations(+, +), (+,−), (−, +), and(−,−). Note thatα is real whenne is negative,i.e., when thematerial localizes. In that case, the stream function is correlatedalong four different slopes, but its amplitude is constant with depth:interface perturbations generate four wavelike deformation fieldsin each layer, none of which decays or grows with depth. Thismakes it impossible to solve for the behavior of a half–space withnegativene, which requires the deformation field to vanish at infin-ity. Hence, the simplest solvable model that includes negativene ismade of a layer of finite thickness over a half–space with positivene. Although our formulation can in principle handle any numberof layers, only this type of model is considered in this paper.

The depth kernel for an exponential viscosity profile withr 6= 0

is discussed inMontesi [2002]. The domain where allα are realis pushed to more negative1/ne whenR increases, and vanishes

altogether atR = 2√

2 +√

5. This is a limit where the viscosityincreases rapidly compared to the perturbation wavelength. How-ever, even in the large wavelength limit, there are more than twovaluesRe(α), one of which is zero. We will see that this conditionis sufficient for the localization instability to develop.

3.3. General solution

For a non–exponential viscosity profile, Eq. 7 must be solvednumerically. We use a Runge–Kutta integration technique [e.g.,Hamming, 1973]. By convention, the depth kernel has a value of 1at the top of each layer. The four superposed modes of deformationare found by setting the initial values of the depth–derivatives ofφ

in turn to each solution ofφ(z) for an exponential viscosity profilethat approximates the actual viscosity profile at the top of the layer(Eq. 14). We verified that the solutions do not depend on the actualstarting scheme chosen. The exceptional cases where the solutionsare degenerate are ignored, and do not arise in practice except if1/ne = 1 or 1/ne = 0.


Gro

wth

rat

e Q

1000

0.1

100

10

1

0 10 20 30 40 50Wavenumber k

b)

η=0.1

η=1z=0

z=1

z=-∞

a) KL

KB/N

Figure 2. a) Growth spectrum for a layer of uniform viscosityη1 overlying a half–space of uniform viscosityη2 = η1/10, with ρ = 0, n2 = 3. Solid line:n1 = −10; dashed line:n1 = 106; dotted line;n1 = 100. b) Viscosityprofile. KB/N ≈ π andKL ≈ 10 for the values ofn1 considered.

a)

b)

εII maxmin

Figure 3. Deformation fields corresponding to the growth spectra in Fig. 2, shaded as a function of strain rate. a)n1 = 106; b) n1 = −10. The velocity field is computed from a small–amplitude initial perturbation at the surfacesuch thatξ1 ∝ kb, with b a real number. Then, the amplitude of the other interface is set to give the eigenvector ofthe growth matrix (Eq. 3) with the fastest growth rate. These modes are amplified by the growth rate (limited to anarbitrary value ifn1 < 0). The initial spectral shape, the amplification factor, and the limiting growth rate have beenchosen to show clearly (a) the buckling deformation mode and (b) both the buckling and the localization modes.

We use the numerical solver only for the case where the viscos-ity depends linearly with depth. to our knowledge, onlyBassi and

Bonnin [1988] have considered that case previously. They useda polynomial expansion of the depth kernel and determined a re-currence relation between the polynomial coefficients. Althoughtechnically an analytical solution, this scheme is subject to numeri-

cal errors and truncation of the expansion. We favored the numerical

integration technique as it can handle any viscosity profile such as

in the companion paper. The only limitation is thatη 6= 0.


4. Constant viscosity analysis

4.1. Growth spectra

We first consider models where a layer of thicknessH, effectivestress exponentn1, and viscosityη1, independent of depth, over-lies a half–space of lower viscosityη2 and effective stress exponentn2 > 0. The exact value ofn2 matters little. In our reference model,we usen2 = 3 andη2 = η1/10 (Figure 1). This is the simplest ap-proximation of the lithosphere strength structure that displays bothlocalization and buckling instabilities. The model is shortened atthe rate¯εxx < 0. For now, the material density is ignored. Thecoordinatesx andz are normalized byH, the wavenumbers arenormalized byH−1, and the stresses are normalized byη1εII .

We present in Figure 2 the growth spectra for various values ofthe effective stress exponent of the layer:1/n1 = 10−2, 10−6, and−10−1. The flow fields for the last two cases are plotted in Fig. 3.

The cases with positiven1 have been solved before [Ricardand Froideveau, 1986;Zuber, 1987]. The growth spectrum passesthrough a first maximum at wavenumberk ≈ π/2 that defines thepreferred wavelength of the buckling instability of the layer as awhole (Fig. 3a). A growth spectrum characteristic of the bucklinginstability passes through successive maxima with a wavenumberscaleKB/N —standing for wavenumber of the buckling/neckinginstability. If 1/n1 is finite, the envelope of the growth spectrumdecreases with wavenumber following the decay of the secondaryfield with depth indicated byIm(α) 6= 0 in Eq. 15 [Ricard andFroideveau, 1986;Neumann and Zuber, 1995]. Accordingly, themagnitude of the growth rate does not change withk in the pseudo–plastic limit 1/ne → 0, represented here by1/ne = 10−6, asIm(α) = 0 in that case.

The growth spectrum for the casene = −10 is different fromthe other two (Fig. 2). It is best described as the superposition ofa buckling–type spectrum and a sequence of doublets with infinitegrowth rate. These doublets indicate the localization instability.The difference between the wavenumbers of consecutive doubletsdefines the wavenumber scaleKL, which is different fromKB/N .In Fig. 2, KB/N ∼ π andKL ∼ 10. The reconstructed defor-mation field (Fig. 3b) has the same appearance of two superposeddeformation modes, each having a specific length scale: buckling

-1.5 -1 -0.5 0 0.5 1 1.50

5

10

15

20

α1, α2

α3, α4

α2, α4

α1, α4

α2, α3

α1, α3

Inverse Effective Stress Exponent 1/ne

Wav

enum

ber

k

Figure 4. Resonant wavenumbers for a layer of thicknessH as afunction of its effective stress exponent (Eq. 16). Thick line: Fun-damental modej = 1; other lines: higher order resonances. Thelabels on each branch indicate the mode slopesα of the involveddeformation modes (see Eq. 15).

of the layer as a whole, and regularly–spaced localized shear zones.The infinite growth rate at the localization doublets (Fig. 2) is dueto the unstable character of localization: a local perturbation ofstrain rate weakens the material locally, so the strain rate increasesfurther. The strain rate perturbations trigger a positive feedbackthat results in an infinite growth rate. The localized shear zoneshave a large–scale organization given by the wavenumber at whicha divergent doublet is present.

4.2. Resonance

The wavenumber scalesKB/N andKL apparent in the growthspectra (Fig. 2) correspond to certain resonances between the su-perposed deformation modes in the layer of our model. If thewavelength of the interface perturbation isλ, potential shear zonesare nucleated at the top surface (z = 1, as lengths are normalizedby H) atx = x0 + jλ, j an integer (i.e., j ∈ Z). At each location,

Inverse effective stress exponent 1/n1

-1 -0.5 0 0.5 1

25

20

15

10

5

0

Wav

enum

ber

k10

1

0.1

10-3

103

10-2

102

Grow

th rate Q

Figure 5. Map of growth rate as a function of effective stressexponent of the layer and perturbation wavenumber. Lighter toneindicates high growth rate, with the contours indicated on the colorbar. Model identical to Fig. 2.

2π/KB/N

2π/KL

a

b

Figure 6. Schematic representation of the resonances identifiedwith a) the buckling instability and b) the localization instabil-ity. Incipient shear zones are nucleated at the top boundary witha spacing equal to the perturbation wavelength. They propagatedownward along the four different mode slopesα (Eq. 15). One setis marked by solid lines and the other by dashed lines. A resonanceoccurs when the shear zones from two different nucleation pointsintercept at the bottom of the layer. Ifne > 0, |Re(α)| has onlyone value, so that the resonance depicted in b) is not possible.


four shear zones propagate into the medium, each corresponding toa particular deformation mode, or solution of Eq. 7. A shear zonenucleated atx = x0, z = 1, reaches the bottom of the layer (z = 0)atx1 = x0 − Re(αa1), wherea1 is an index between 1 and 4 thatindicates which mode slopeα, or solution of Eq. 15, is considered.A second shear zone, identical to the first, is generated atx = jλ. Itreaches the bottom of the layer atx2 = x0 + jλ+Re(αa2), wherea2 is another index between 1 and 4, not necessarily the same asa1. A given wavenumberk = 2π/λ is resonant ifx1 = x2, or

kja1,a2 = 2πj/|Re(αa1 − αa2)|, (16)

wherej is the order of the resonance —the number of wavelengthspanned between the surface intercept of the incipient shear zones.The resonant wavenumbers are plotted as a function of the effectivestress exponent in Fig. 4.

At depth, each potential shear zone attempts to generate a dis-continuous deformation field that is incompatible with the responseof the substrate. Therefore, the localized shear zones can developonly if an additional degree of freedom is available, which happenswhen several shear zones interact at the bottom of the layer. Hence,the localized shear zones cannot develop in the model unless theperturbation wavelength is resonant. Indeed, the growth rate is fi-nite at all wavenumbers that are not resonant, indicating that themodel is stable and deformation remains distributed (Fig. 2). Atresonant wavenumbers, there is a self–consistent network of narrowzones of high strain rate and the growth rate is infinite.

The pattern of resonances compares well to the buckling andlocalization instabilities. We present in Fig. 5 a map of growth ratein the parameter space of1/n1–k. This representation is akin toa topographic map of the surfaceQ(1/n1, k). The growth spec-tra shown in Fig. 2 represent sections through that map taken withconstant1/ne of -0.1,10−6, and 0.01.

The buckling instability is best identified in the1/ne > 0domainby a broad maximum and can be followed into the negative1/ne

domain (Fig. 5). The characteristic wavenumber of this instability,KB/N , varies monotonically with1/ne, following

KB/N = k1,4 = π (1− 1/n1)−1/2 . (17)

The growth rate maxima are located at

kBj = (j + 1/2) KB/N , j ∈ Z. (18)

The localization instability is seen as a branch of very highgrowth rate that exists only when1/ne < 0. Its characteristicwavenumber,KL, follows a different resonance from the bucklinginstability

KL = k1,2 = π (−1/n1)−1/2 . (19)

The wavenumberKL is real only if 1/n1 < 0, as expected if itarises due to localization. The growth rate maxima for the localiza-tion instability are located near:

kLj = (j) KL, j ∈ Z. (20)

The relevant resonances are showed schematically in Fig. 6.As was pointed out earlier, the effective stress exponent quanti-

fies the efficiency of the localization process in the layer. Hence,the wavelength of the localization instability should depend onne.The deformation imposed within one wavelength of a given fault islocalized on that fault. Therefore, if localization is efficient (1/ne

more negative), a given fault can accommodate the deformationfrom a wider area than if localization is inefficient (1/n1 less neg-ative). In the limit of perfect localization, all the deformation islocalized on a single fault. Indeed, the wavelength of the localiza-tion instability goes to infinity in the limit of drastic localization

1/ne → −∞ (Fig. 5). In the other case where the localization ismarginal,1/ne → 0−, the instability wavelength goes to 0; manyfinely–spaced faults are predicted.

As the localization instability is characterized by a doublet ofdivergent growth rate, centered on the wavenumber given by Eq.20, there are two preferred wavelengths of the localization insta-bility for a givenj. The separation of the two branches of a givendoublet is not predicted by the resonance analysis. We note that thedoublets close when several resonances are superposed (compareFig. 4 and 5), which might indicate the importance of the derivativesof the stream function. Whereas the resonance analysis uses onlythe stream function, the boundary conditions include velocity andstresses, which depend on the derivatives of the stream function.Two modes of deformation with different mode slopes have differ-ent values of stress and velocity at a given depth, even if their streamfunctions at that depth are identical. This might suffice to offset

Substrate Viscosity η2/η1

0

5

15

30

10

20

25

Wav

enum

ber

k

0.01 0.1 1 10 100

1

104

103

0.1

10

102

Grow

th rate Q

Figure 7. Map of growth rate as a function of substrate viscosityand perturbation wavenumber. Strength profile similar to Fig. 2b,except for varying substrate viscosity andn1 = −10. The buck-ling instability vanishes when the substrate is more viscous than thelayer, but the localization instability persists, although offset by onehalf of its characteristic wavenumber.

0.1 1 10 1000 10000

30

20

10

0

Wav

enum

ber

k

10

1

0.1

10-3

103

10-2

102

Grow

th rate Q

100Model density ρg/η1εII

Figure 8. Map of growth rate as a function of model density andperturbation wavenumber. Strength profile similar to Fig. 2b, withn1 = −10. The value of the normalized densityρg/η1εII is nomore than 30 for terrestrial applications.


slightly the actual resonant wavelengths. The opening of the local-ization doublets is usually small enough to be ignored. In addition,the doublet structure may break down once the non–linearities inthe system behavior and transient strengthening are taken into ac-count. These additional aspects of a localizing system are needed tostabilize localization and probably limit the divergent growth rate ofthe localization instability to a finite value. Therefore, applicationsto the Earth need not consider the doublet opening and can replaceit conceptually with a single peak at the preferred wavelength of theinstabilitykL

j (Eq. 20).

4.3. Effect of substratum viscosity

The buckling instability takes its origin in the viscosity contrastacross the interfaces of the model [Johnson and Fletcher, 1994].Therefore, the growth rate of the buckling instability increases withthe viscosity contrast. Buckling requires that the layer be strongerthan the substratum.

The localization instability, on the other hand, is linked to a res-onance that is internal to the layer with negative stress exponent.Therefore, it can grow even if the viscosity of the half–space ishigher than the viscosity of the layer. However, in that case, thespectrum is offset by one half of the characteristic wavelength (Fig.7). When the substrate is more viscous than the layer, the doubletsare located at

kLj = (j + 1/2) KL, j ∈ Z. (21)

Physically, the offset is required because the viscous substratecannot follow the localized deformation. At wavenumbers halfwaybetween actual resonances, an incipient shear zone interacts witha negative image of itself. Therefore, these shear zones interactdestructively at the interface, which is required by the strongersubstratum. When the viscosity of the substrate is small, it pro-vides no resistance to the localized deformation and the preferredwavenumber is exactly at the resonance.

The position of the doublet changes continuously from Eq. 20 toEq. 21 as the substrate viscosity increases (Fig. 7). In general, wewrite

kLj = (j + aL) KL, j ∈ Z, (22)

with aL an empirical number called the spectrum offset, andKL

the wavenumber scale defined in Eq. 19. The spectrum offset is 0 ifη2/η1 ¿ 1, aL = 1/4 if η2/η1 ≈ 1, andaL = 1/2 if η2/η1 À 1.

For generality, we also define a spectrum offsetaB for the buck-ling instability

kBj = (j + 1/2− aB) KB/N , j ∈ Z, (23)

with KB/N the wavenumber scale defined in Eq. 17. Eq. 22 and23 are written so thataB = aL = 0 for models where both thelayer and the substrate have constant viscosity and the layer is muchstronger than the substrate (Fig. 2). This will change when depth–dependent viscosity is considered in the model layers.

4.4. Effect of model density

A density contrast at the surface of the model reduces the growthrate of the buckling instability and increases its preferred wavenum-ber because of the restoring force on the growing surface topography[Zuber, 1987;Martinod and Molnar, 1995;Neumann and Zuber,1995]. If ne > 0, the growth rate is particularly reduced at smallwavenumber. If1/ne → 0+, the density–induced reduction ofthe growth rate at smallk can result in the maximum growth ratebeing at a resonance withj ≥ 1 [Neumann and Zuber, 1995]. Inaddition, there is no growth of the buckling mode over half of the

wavenumbers, fromjKB/N to (j + 1/2)KB/N at smallj. Thisproduces a spectral offset in Eq. 23 up toaB = 1/4 for j = 1and large density of the model. The wavelength of buckling in theCentral Indian Ocean may be larger to the north of the basin wherethe surface density contrast is reduced by the large sediment supplyof the Bengal fan [Zuber, 1987].

If the layer has localizing properties (1/ne < 0), the model den-sity influences the buckling part of the growth spectrum in the samemanner as described above, except that growth rate is enhancednear the divergent doublets of the localizing instability. The effectis particularly pronounced at small wavenumber and increases withthe density of the model (Fig. 8). However, scaling to Earth con-ditions indicates that the normalized densityρg/η1εII should be oforder 1 to 30 [Zuber et al., 1986], which is small. For these values,the density has little effect on the localization instability, although itdoes bring a spectral offset up toaB = 1/4 for the longest preferredwavelength of the buckling instability (j = 0).

5. Models with depth–dependent viscosity

The models considered in the previous section are only crudeapproximations to the Earth’s structure. The layer represents thebrittle crust and upper mantle, and the half–space represents thehotter ductile rocks leading to the asthenosphere. The interface be-tween the layer and the half–space corresponds to the brittle–ductiletransition.

Unlike the models in the previous section, the strength profile ofthe Earth is continuous across the transition from brittle to ductilebehavior. In idealized representations of the strength profile of thelithosphere, the brittle–ductile transition occurs at a particular depthwhere the brittle and ductile strengths of rocks are identical [Braceand Kohlstedt, 1980], or is distributed over a depth range wherethe brittle and ductile rock strengths are comparable [Kirby, 1980;Kohlstedt et al., 1995]. In any case, the strength and therefore theapparent viscosity of the lithosphere varies continuously with depthwithin each layer, due to the pressure dependence of rock strength inthe brittle regime [Scholz, 1990], and the temperature dependenceof creep in the ductile regime [Kohlstedt et al., 1995].

In the following sections, we change progressively the simplestrength profile of the previous section to a more realistic strength

103

102

10-3

10-2

1

0.1

10

0

50

10

40

30

20

Wav

enum

ber

k Grow

th rate Q

20 10 0 -10 -20Inverse decay depth of the substrate viscosity r2

Figure 9. Map of growth rate as a function of decay length ofsubstrate viscosity and perturbation wavenumber. Strength profilesimilar to Fig. 10b, except for varyingr2 andn1 = −10. Negativevalues ofr2 indicate that the substrate viscosity increases exponen-tially with depth.


Gro

wth

rat

e Q

0.1

100

10

1

0 10 20 30 40 50Wavenumber k

a)

η=exp(10z)

z=0

z=1

z=-∞

η=1

b)

Figure 10. a) Growth spectrum for a layer of uniform viscosityη1 overlying a half–space with exponentially–decayingviscosityη2 = exp(r2z), with r2 = 10, ρ = 0, n2 = 3. Solid line: n1 = −10; dashed line:n1 = 106. b) Viscosityprofile.

profile, keeping track of the preferred wavelengths of buckling andlocalizing instabilities, as well as their growth rate. Our goal isto derive a simple prediction of the preferred wavelength of thelocalization instability relevant for a layer of rock undergoing abrittle–ductile transition at a specific depth with a strength profilesimilar to the Earth’s. The effect of having the brittle–ductile transi-tion distributed over a finite depth range or density contrasts withinthe lithosphere will be considered elsewhere.

The wavenumber scaling of the instability is still valid whendepth–dependent viscosity is considered. Therefore, we use Eq. 22and 23 to describe the preferred wavenumbers of each instability.However, the value of the spectral offset parameter is empiricallydetermined for each type of viscosity profile. The viscosity isscaled to 1 at the bottom of the brittle layer. We usene = −10 asan illustration for localizing behavior. In that case,KL ≈ 10.

5.1. Exponential decay of viscosity in the substrate

Because the stream function for a layer of exponential viscos-ity profile is proportional toexp [ik(x + αz)] (Eq. 2 and 13), aperturbation with wavelengthλ penetrates into a layer to a depthzd ∼ λ/Im(α). Hence, it senses a viscosity averaged overzd.Therefore, a buckling instability can grow even if the strength pro-file is continuous at the boundary between the layer and the substrateif the viscosity of the substrate decreases exponentially with depth[Fletcher and Hallet, 1983;Zuber and Parmentier, 1986]. In fact,most applications of buckling or necking to the tectonics of terres-trial planets have used a strength profile made of a layer of uniformviscosity lying over a substrate with viscosity profile

η = ηe exp(rz), (24)

A value ofr ≈ 10 is often appropriate [Fletcher and Hallet, 1983;Zuber and Parmentier, 1986].

There are two differences between the growth spectrum of alayer lying over a substrate with exponentially–decaying viscosityand the previous case of a constant–viscosity half–space, even if thelayer is plastic (1/n1 → 0, buckling instability only, Fig. 10). First,the envelope of the growth spectrum decreases at high wavenum-ber. This is because the short wavelength senses only the top ofthe substrate, which has higher viscosity than deeper levels, andtherefore smaller viscosity contrast [Ricard and Froideveau, 1986;

Zuber and Parmentier, 1986;Neumann and Zuber, 1995]. Second,the instability grows only over one half of the range of wavenum-bers, betweenj KB/N and(j + 1/2) KB/N , j ∈ Z. Hence, thepreferred wavenumbers of buckling become

kBj = (j + 1/4) KB/N , j ∈ Z, (25)

or, using Eq. 23, the spectrum offsetaB is 1/4 for this strengthprofile.

When the layer is undergoing localization (n1 < 0), the growthspectrum is described as the superposition of a buckling–like spec-trum and a sequence of divergent doublets representing the local-ization instability (Fig. 10), as in the model with constant viscositylayers (Fig. 2). At the smallest wavenumbers, the substrate appearsvery weak, and the spectral offsetaL ∼ 0 for j = 0. At larger

20 10 0 10 200

10

20

30

40

50

Wav

enum

ber

k

Inverse decay depth r

α 1, α 2

α 3, α 4

α1, α4α2, α3

Figure 11. Resonant wavenumbers for a layer of thicknessH andeffective stress exponentn1 = −10 as a function of the decaysdepthr of viscosity in the layer. Thick line: fundamental modej = 1; other lines: higher order resonances. Thej = 1 branchesare labeled with the mode slopesα of the appropriate deformationmodes. Solution derived numerically from Eq. 14 and 16.


Gro

wth

rat

e Q

1000

0.1

100

10

1

0 10 20 30 40 50Wavenumber k

a)

η=exp(-5z)

z=0

z=1

z=-∞

η=0.1

b)

Figure 12. a) Growth spectrum for a layer with viscosity increasing exponentially with depthη1 = exp(r1z),overlying a half–space with constant viscosityη2 = 0.1, with r1 = −5, ρ = 0, n2 = 3. Solid line: n1 = −10;dashed line:n1 = 106. b) Viscosity profile

wavenumbers, however, the substrate viscosity is similar to that of

the layer, so that the spectral offsetaL = 1/4 (see below Eq. 22).

In summary, the preferred wavelength of localization follows Eq.

22 with approximately

aL =

{0, if j = 0,1/4, if j > 0.

(26)

Fig. 9 shows how the growth spectrum varies as a function of the

decay–depth of the viscosity profile. Note how the buckling insta-

bility vanishes whenr < 0 (viscosity of the substrate increasing

exponentially with depth), whereas the localization instability is

still present. However, the substrate is now stronger than the layer,

so that ataL = 1/2 at smallj.

103

102

10-3

10-2

1

0.1

10

0

50

10

40

30

20

Wav

enum

ber

k

Grow

th rate Q

20 10 0 -10 -20Inverse decay depth of the layer viscosity r1

Figure 13. Map of growth rate as a function of decay length of layerviscosity and perturbation wavenumber. Strength profile similar toFig. 12b, except for varyingr1 andn1 = −10. Positive values ofrindicate that the layer viscosity decreases exponentially with depth.

5.2. Depth–increasing strength of the layer

As the layer corresponds to rocks undergoing brittle deformation,its strength should increase with depth. We first consider models inwhich the viscosity of the layer increases exponentially with depth,which is mathematically more tractable, and then the more realisticcase of a viscosity increasing linearly with depth in the layer. Inboth cases, the viscosity isη1 = 1 at the base of the layer. Thesubstrate has a constant viscosity ofη2 = 0.1 and a non–Newtonianbehavior withn2 = 3.

5.2.1. Exponential viscosity profile. Having an exponen-tial viscosity profile in the layer reduces its apparent viscosity.Accordingly, the growth rate of the buckling instability is reducedcompared to the constant viscosity case but its preferred wavelengthis unchanged (Fig. 12). When the material in the layer is pseudo–plastic (1/n1 → 0+) The envelope of the growth spectrum does notdepend on wavenumber because the depth of penetration into thelayer of the perturbation is infinite (Im(α) → 0): the whole layer issampled at all wavelengths. When the layer is localizing (n1 < 0,Fig. 12), the preferred wavelength of the localization instability isoffset by1/4 of the characteristic scaleKL. Interestingly, the am-plitude of the buckling mode decreases at the smallest wavenumbersif 1/n1 < 0 and the layer viscosity increases with depth (Fig. 12).

A complication arises because the mode slopeα depends on thedecay parameter of the viscosity profile (Eq. 14). This changesthe resonant wavenumbers (Fig. 11) and therefore the wavenumberscale of instabilitiesKL andKB/N . Although the localization in-stability still follows the resonance (Figure 13), there is no longer ananalytical expression forKL orKB/N . However, Eq. 19 is approx-imately valid whenr ≈ 0, which is the case for realistic viscosityprofiles. Therefore, the preferred wavelengths of localization aregiven approximately by 22 with

aL = 1/4, j ∈ Z. (27)

Note that the first localization doublet (j = 0) is wider than forother strength profiles.

The long wavelength limitrk > 2√

2 +√

5, beyond which notall the solutions ofα are real even if1/ne < 0, does not preventthe growth of a localization instability. This is because for1/ne


Gro

wth

rat

e Q

1000

0.1

100

10

1

0 10 20 30 40 50Wavenumber k

a)

z=0

z=1

z=-∞

η=exp(10z)

b)

η=1+0.1z

Figure 14. a) Growth spectrum for a layer in which the viscosity increasing linearly with depth overlying a half–spacein which the viscosity decays exponentially with depth, withρ = 0, n2 = 3. Solid line: n1 = −10; dashed line:n1 = 106. b) Viscosity profile.

sufficiently negative, one value of the mode slope,α, is pure imag-inary: there are still two values ofRe(α), one being zero, and theresonance depicted in Fig. 6b is still defined.

5.2.2. Linear viscosity profile. Although an exponential vis-cosity profile is only a poor approximation of the linear increase ofstrength with depth expected in the brittle layer from friction laws,there is little difference between the results of the previous sectionand the growth spectra obtained with the linear law. The amplitudeof the buckling mode is reduced, and the preferred wavelength ofthe localization instability is offset by1/4 of the wavelength scaleKL at small wavelengths. In addition, the resonance wavelengthis close to the analytical value of Eq. 19 obtained for a constantviscosity layer. This is because the average strength of the layer islimited to one half of its maximum value when it increases linearlywith depth. The decay parameterr for exponential viscosity profilethat produces the same characteristics is small. Indeed, the growthspectrum for a layer with linear viscosity profile is closest to thecaser = −2 with an exponential viscosity profile. For these valuesthe resonant wavenumbers cannot be differentiated from the limitr = 0.

6. Discussion6.1. Putting it all together: Growth spectrum for realisticstrength profiles

A realistic strength profile for application to tectonics has aplastic or brittle layer with strength increasing linearly with depth,followed by a layer of half–space of ductile material with viscositydecreasing with depth. We learned from the previous sections thatthere are two superposed instabilities for a plastic or localizing layeroverlying a half–space: the buckling instability that results in broadundulation of the layer as a whole when that layer is stronger thanthe substratum averaged over a wavelength–dependent penetrationdepth, and the localizing instability that produces regularly–spacedfaults or shear zones. ReintroducingH, the thickness of the brittlelayer, as a length scale in Eq. 17, 19, 22, and 23, these instabilitiesgrow preferentially at the wavenumbers

kBj H = (j + 1/2− aB) KB/N , (28a)

kLj H = (j + aL) KL, (28b)

with j an integer,aB andaL spectral offsets that depend on thestrength profile, andKB/N andKL wavenumber scales that corre-spond to resonances in the brittle layer and are given by

KB/N = π (1− 1/n1)−1/2 , (29a)

KL = π (−1/n1)−1/2 . (29b)

Depth–increasing viscosity in the layer, depth–decreasing vis-cosity in the half–space, and buoyancy forces all decrease thegrowth rate of the buckling instability (Fig. 10 and 12). Hence, thebuckling instability shows only modest growth rates for the mostrealistic strength profile used in this study (Fig. 14). Furthermore,the exponentially–decaying viscosity in the substrate suppresses theinstability over half of the wavenumber range (Fig. 10), and a sur-face density contrast cancels the instability over the other half of the

0.01 0.1 1 100

30

20

10

0

Wav

enum

ber

k

10

1

0.1

10-3

103

10-2

102

Grow

th rate Q

10Model density ρg/η1εII

Figure 15. Map of growth rate as a function of the model densityρg/η1εII and wavenumber. Viscosity profile similar to Fig. 14b.Lighter tone indicates high growth rate, with the contours indicatedthe color bar. As the model density increases, the buckling modevanishes.


Gro

wth

rat

e Q

1000

1

100

10

0 10 20 30 40 50Wavenumber k

b)

η=0.1

η=1z=0

z=1

z=-∞

a)

Figure 16. a) Growth spectrum for a layer of uniform viscosityη1 overlying a half–space of uniform viscosityη2 = η1/10, with ρ = 0, n2 = 3 undergoing horizontal extension. Solid line:n1 = −10; dashed line:n1 = 106;dotted line;n1 = 100. b) Viscosity profile.

a)

b)

εII maxmin

Figure 17. Deformation fields corresponding to the growth spectra in Fig. 16, shaded as a function of strain rate. a)n1 = 106; b) n1 = −10. Models undergoing extension. Construction otherwise similar to Fig. 3

wavenumber range (Fig. 8). It follows that buckling is not a likelyexpression of shortening in a layered lithosphere (Fig. 15) unlessthe surface density contrast is reduced. Indeed, natural examples oflithospheric–scale buckling are associated with deformation undera heavy fluid, which reduces the surface density contrast. In theCentral Indian Ocean, this fluid stands for the sediments from theBengal fan [Zuber, 1987;Martinod and Molnar, 1995]. Many otherregions in which buckling has been documented are under sedimen-tary basins [Burov et al., 1993;Cloetingh et al., 1999]. On Venus,the ridge belts grew in the same time that basaltic flood plains wereemplaced [Zuber and Parmentier, 1990;Stewart and Head, 2000].

If buckling does grow, the relevant spectral offset is

0 < aB/N < 1/4. (30)

Alternatively, it can be argued that including more realistic behav-ior would help buckling even in present of a relatively high surfacedensity contrast.Schmalholz et al.[2002] show that including thatconsideration of visco–elastic behavior influence that manner thattheir folding solution depends on surface density contrast. A dy-namic surface redistribution condition [Biot, 1961;Beaumont et al.,


1990] would model erosion more accurately than a reduced densitycontrast and would certainly affect our solution.

Neither depth–dependent viscosity nor surface density contrastsreduce the growth rate of the localization instability. Depth–dependent viscosity in the layer and in the substrate each offsetsthe preferred wavelength by about1/4 KB/N . The density ofthe model also increases the spectral offset (Fig. 15). All thingsconsidered, the spectral offset for a realistic viscosity profile is

1/4 < aL < 1/2. (31)

A map of growth rate similar to Fig. 5 but for a realistic strengthprofile is presented in Fig. 18. It shows how the wavenumber ofthe localization instability varies with the effective stress exponentof the layer. The localization instability is seen to follow the res-onant wavenumbers,KL. The buckling mode of deformation allbut vanishes when depth–dependent viscosity is taken into account.It is visible only near the divergent doublets of the localizationinstability.

6.2. A note about extension

Although the previous sections assumed horizontal shorteningof the model, the formalism is equally valid for horizontal exten-sion, for which¯εxx > 0. However, the kinematic contribution ofthe primary flow to the growth of interface perturbations (Eq. 3)has the tendency to erase the imposed perturbation under horizontalextension [Smith, 1975]. Hence, only wavelengths withQ > 1 canbe observed.

We present in Fig. 16 the growth spectra for a pseudo–plasticor brittle layer of uniform viscosity over a weaker half–space un-der horizontal extension. The corresponding deformation fields areplotted in Fig. 17. The spectra are rather similar to the shorteningcase (Fig. 2). In particular, the wavenumbers of the growth ratemaxima for a pseudo–plastic layer (ne = 106) and of the divergentdoublets for the localizing layer (ne = −10) are similar to theshortening case. The major difference between horizontal exten-sion and shortening is the shape of the most unstable deformationmode over the whole model: the pseudo–plastic layer is neckingunder extension rather than buckling (Fig. 17a). The localization

Inverse effective stress exponent 1/n1

-1 -0.5 0 0.5 1

25

20

15

10

5

0

Wav

enum

ber

k

10

1

0.1

10-3

103

10-2

102

Grow

th rate Q

Figure 18. Map of growth rate as a function of effective stressexponent of the layer and perturbation wavenumber. Lighter toneindicates high growth rate, with the contours indicated on the colorbar. Viscosity profile identical to Fig. 14b. Note how the bucklinginstability is reduced compared to Fig. 5.

instability gives rise to regularly–spaced localized shear zones (Fig.17b).

In presence of depth–dependent viscosity and density, the ap-proach of a spectral offset (Eq. 23 and 22) is still valid. However, thespectral offset for the necking instability is−1/4 < aB < 0 if theviscosity of the layer increases linearly with depth and the viscosityof the half–space decreases exponentially with depth. Hence, thewavelength of necking is generally smaller than the wavelength ofbuckling. As was observed in the shortening case, depth–dependentviscosity and model density conspire to reduce the range of wave-lengths where growth of the necking instability is possible underhorizontal extension, diminishing the likelihood that necking be ob-served in the tectonic record, unless the surface density contrast issmall. Necking has been observed in nature, most prominently inthe Basin–and–Range province [Fletcher and Hallet, 1983;Zuberet al., 1986;Ricard and Froideveau, 1986] and plays an importantrole in rifting [Zuber and Parmentier, 1986;Lin and Parmentier,1990]. Grooved terrain on Jupiter’s satellite Ganymede may also beformed by necking [Dombard and McKinnon, 2001]. The spectraloffset of the localization instability in extension is the same as incompression,1/4 < aL < 1/2.

6.3. Comparison with numerical studies

In early studies of fault patterns, faults were eithera posteriorimarkers of deformation ora priori boundary conditions. In nei-ther case is faulting a dynamic feature of the models or can theself–consistent pattern of faulting be determined. The instabilityof the localization process, which is expressed in our study by thefact that the effective stress exponent is negative, presents manyanalytical and numerical challenges. However, recent numericalmethods have been able to present a continuum approach to local-ization, from microscopic scale [Hobbs and Ord, 1989;Poliakovet al., 1994] to global scale [Bercovici, 1993;Tackley, 2000]. Withnumerical models, it is possible to go beyond the instantaneous pat-terns of faulting explored in this paper to study how faulting evolvesover time [Sornette and Vanneste, 1996;McKinnon and Garrido dela Barra, 1998;Buck et al., 1999;Cowie et al., 2000;Huismans andBeaumont, 2002]. Our analysis provides a physical insight into theorigin of the macroscale fault patterns.

Using the numerical method FLAC [Cundall, 1989], Poliakovand coworkers explored the patterns of faulting in elastic–visco–plastic models for different tectonic environments [Buck and Po-liakov, 1998;Gerbault et al., 1998, 1999;Cloetingh et al., 1999;Lavier et al., 2000]. Even in the absence of explicit weakening, anelastic–plastic rheology is characterized by a negative stress expo-nent, or dynamic strain–weakening, because the strain and stressincrements upon failure are not collinear [Montesi and Zuber, 2002].Localization by strain–weakening may behave differently from thestrain–rate–weakening used in our paper. However, the effective ex-ponent provides a unifying measure of localization, and it is relevantto compare the numerical results in presence of strain–weakeningto our model, provided that we use−0.1 < 1/ne < −0.01, asappropriate for localization in elastic–plastic materials [Montesiand Zuber, 2002]. The fault spacing predicted by our analysis(0.4 < λ/H < 2.5) is consistent with the spacing observed innumerical models. The localization instability (adapted for strain-weakening) is a likely origin of the fault pattern observed in numer-ical models.

Explicit strain–softening was shown to enhance faulting inelastic–visco–plastic models and to increase the fault spacing [Ger-bault, 1999]. This is again consistent with the localization insta-bility, which predicts larger fault spacings for more efficient local-ization. However, if the weakening is too strong, another transition


occurs and a single fault develops in numerical models [Lavier et al.,2000]. The resulting deformation pattern is sometimes asymmetric[Lavier et al., 1999;Huismans and Beaumont, 2002]. Frederiksenand Braun[2001] also observed localization on a single fault andits conjugate in their models that include strain-softening of theviscous, rather than the plastic rheology. They also showed that thefault intensity depends on the rate of weakening, consistent withlocalization being controlled by the effective stress exponent ratherthan only the amount of weakening. Localization to a single fault isnot predicted by our model. It may be due to second order or finitestrain effects that we do not address yet.Sornette and Vanneste[1996] andCowie et al.[2000] also report on localization of strainover a single fault upon finite deformation. Rather than followingan elastic–plastic rheology, their models are elastic, with fault slipaccumulating when a yield criterion is verified. The initial patternof faulting is dominated by the strong prescribed heterogeneity inthese models, which prevents a characteristic length–scale from de-veloping. As slip on a fault enhances the stress in the vicinity of thefault tip, the tendency to failure of faults in that region is enhanced.After finite slip, the fault pattern may localize because of the in-teraction between neighboring faults [Sornette and Vanneste, 1996;Cowie et al., 2000]. The interaction between several active faultsmay be described with a negative effective stress exponent. Futureimprovement of our model, in particular including a higher-orderor time-dependent analysis may address this later instability of faultpattern.

The numerical studies closest to our study consider strain–ratesoftening in visco–plastic models [Neumann and Zuber, 1995;Montesi and Zuber, 1997, 1999]. They produce regularly–spacedfaults superposed on either buckling or necking. Numerical resultssuggest that faults are mostly active in the anticlines of lithospheric–scale folds [Montesi and Zuber, 1997, 1999] or the necks of litho-spheric scale boudins [Neumann and Zuber, 1995]. This interactionbetween the buckling/necking and faulting deformation fields is notpredicted in our analysis, for which faults are present everywhere(Fig. 3 and 17), but may result from an higher-order interaction be-tween the buckling/necking and localization instabilities. Numeri-cal results indicate several faults in the growing anticlines or necks.Their spacing is consistent with the prediction of our model, show-ing again a control of the fault pattern by the localization instability.As deformation proceeds, some faults cease their activity, and othersreplace them [Montesi and Zuber, 1997, 1999]. Switches in faultpatterns are discrete in time. They reuse recently active faults, withnew faults formed in the front of the existing deformation zone,separated from it by the same spacing as within the deformationzone. Thus, the localization–instability–controlled fault spacing isprominent at finite strain. Propagation of a deformation front withcharacteristic fault spacing is also observed in elastic–plastic mod-els [Hardacre et al., 2001], but to our knowledge, the fault spacingin that case has not been studied.

In summary, fault sets produced by numerical models may showa preferred spacing that is consistent with the prediction of thelocalization instability. However, a high level of initial heterogene-ity can prevent a regular spacing to form [Sornette and Vanneste,1996]. With finite displacement, the fault pattern may collapse ona single fault [Sornette and Vanneste, 1996; Cowie et al., 2000;Lavier et al., 2000;Frederiksen and Braun, 2001;Huismans andBeaumont, 2002] through a process that we cannot address here.All the models undergoing horizontal shortening show regularly–spaced fault sets even at finite strain [Gerbault et al., 1999;Montesiand Zuber, 1999]. Accordingly, compressive orogens often displaya propagating deformation front with regularly–spaced faults [e.g.,Hoffman et al., 1988;Meyer et al., 1998]. However, the impor-tance of sub–horizontal decollements for this behavior remains tobe evaluated.

Other numerical studies focused on the localization of shearwithin a fault gouge [Morgan and Boettcher, 1999; Place and

Mora, 2000;Wang et al., 2000] or the spatio–temporal localiza-tion of slip over a seismogenic fault [Ben Zion and Rice, 1995;Miller and Olgaard, 1997;Lapusta et al., 2000;Madariaga andOlsen, 2000]. As our study does not resolve the temporal evolutionof localization and assumes a different geometry than that relevantfor seismogenic and fault gouge processes, we cannot compare ourwork and these studies. Similarly, our analysis cannot be applieddirectly to regularly–spaced fault sets in strike–slip environments[Bourne et al., 1998;Roy and Royden, 2000]. However, the use ofa negative stress exponent to build simple models of localizationcan be adapted to these problems. We hope that future develop-ments of our model will address these different geometries as wellas the interaction between the localization and buckling/neckinginstabilities.

7. Conclusions

We have presented new solutions of the perturbation analysis ofmechanically layered models of the lithosphere undergoing shorten-ing in which a brittle layer lies over a ductile substrate. In additionto the classically–recognized buckling instability, the layer may un-dergo a localization instability that results in regularly–spaced faultsor shear zones. Localization is possible when the effective stressexponent of the brittle layer, a general measure of the mechanicalresponse of the material to local perturbations, is negative [Montesiand Zuber, 2002]. However, localization of deformation producesincipient shear zones in which the deformation field tries to developa discontinuity that is not compatible with coupling with a ductilesubstrate. Therefore, shear zones cannot develop unless there is aresonance between several incipient shear zones. This resonanceis the basis for a scaling wavenumber,KL, that controls the wave-lengths of instability. The resonance that is at the origin ofKL

exists only if the effective stress exponent is negative (Eq. 19). Theactual wavelength of the instability is linked toKL by Eq. 22 whichalso includes a “spectral offset” parameter,aL, that is calibratedempirically in function of the strength profile in the model. Fora realistic profile where the strength of the brittle layer increaseslinearly with depth and the strength of the substrate decreases withdepth,1/4 < aL < 1/2. Similar principles are used to describethe buckling instability except that the scaling wavenumber,KB/N

is rooted in a different resonance that does not require a negative ef-fective stress exponent (Eq. 17), and that the spectral offsetaB (Eq.23) is between 0 and1/4 in compression and between−1/4 and 0in extension. Model density has only a minor effect on the localiza-tion instability. Buckling is much reduced when depth–dependentviscosity is included and is a likely expression of tectonic defor-mation only if the surface density contrast is reduced, for instancebecause of a high erosion or sedimentation rate.

Acknowledgments. We thank Oded Aharonson, Mark Behn, RayFletcher, Brad Hager, Greg Hirth, Chris Marone, Greg Neumann, MarcParmentier and Jack Wisdom for their comments on this paper and previousversions of it. Supported by NASA grant NAG5–4555.

References

Bassi, G., and J. Bonnin, Rheological modelling and deformation insta-bility of lithosphere under extension - II. Depth-dependent rheology,Geophys. J. Int., 94, 559–565, 1988.

Beaumont, C., and G. Quinlan, A geodynamic framework for interpret-ing crustal-scale seismic-reflectivity patterns in compressional orogens,Geophys. J. Int., 116, 754–783, 1994.


Beaumont, C., P. Fullsack, and J. Hamilton, Erosional control of activecompressional orogens, inThrust Tectonics, edited by K. R. McClay, pp.1–18, Chapman and Hall, London, 1990.

Ben Zion, Y., and J. R. Rice, Slip patterns and earthquake populationsalong different classes of faults in elastic solids,J. Geophys. Res., 100,12,959–12,983, 1995.

Bercovici, D., A simple model of plate generation from mantle flow,Geo-phys. J. Int., 114, 635–650, 1993.

Biot, M. A., Theory of folding of stratified viscoelastic media and its impli-cations in tectonics and orogenesis,Geol. Soc. Am. Bull., 72, 1595–1620,1961.

Bourne, S. J., P. C. England, and B. Parsons, The motion of crustal blocksdriven by flow of the lower lithosphere and implication for slip rates ofcontinental strike-slip faults,Nature, 391, 655–659, 1998.

Brace, W. F., and D. L. Kohlstedt, Limits on lithospheric stress imposed bylaboratory measurements,J. Geophys. Res., 85, 6248–6252, 1980.

Brown, C. D., and R. E. Grimm, Tessera deformation and the contemporane-ous thermal state of the plateau highlands, Venus,Earth Planet. Sci. Lett.,147, 1–10, 1997.

Buck, W. R., and A. N. B. Poliakov, Abyssal hills formed by stretchingoceanic lithosphere,Nature, 392, 272–275, 1998.

Buck, W. R., L. L. Lavier, and A. N. B. Poliakov, How to make a rift wide,Phil. Trans. Roy. Soc. Lond., Series A, 357, 671–693, 1999.

Bull, J. M., Structural style of intra-plate deformation, Central Indian OceanBasin: Evidence for the role of fracture zones,Tectonophysics, 184,213–228, 1990.

Burov, E. B., L. I. Lobkovsky, S. Cloetingh, and A. M. Nikishin, Continentalfolding in Central Asia (part II): constraints from gravity and topography,Tectonophysics, 226, 73–87, 1993.

Chapple, W. M., Fold shape and rheology: The folding of an isolatedviscous-plastic layer,Tectonophysics, 7, 97–116, 1969.

Chapple, W. M., Mechanics of thin-skinned fold-and-thrust belts,Geol. Soc. Am. Bull., 89, 1189–1198, 1978.

Cloetingh, S., E. Burov, and A. Poliakov, Lithosphere folding: Primaryresponse to compression? (from central Asia to Paris basin),Tectonics,18, 1064–1083, 1999.

Cowie, P. A., S. Gupta, and N. H. Dawers, Implications of fault array evolu-tion for synrift depocentre development: Insights from a numerical faultgrowth model,Basin Res., 12, 241–261, 2000.

Cundall, P. A., Numerical experiments on localization in frictional materials,Ing. Arch., 59, 148–159, 1989.

Davies, R. K., Models and observations of faulting and folding in layeredrocks, Ph.D., Texas A&M University, 1990.

Dieterich, J. H., Modeling of rock friction 1: Experimental results andconstitutive equations,J. Geophys. Res., 84, 2161–2168, 1979.

Dombard, A. J., and W. B. McKinnon, Formation of grooved terrain onGanymede: Extensional instability mediated by cold, superplastic creep,Icarus, 154, 321–336, 2001.

Fletcher, R. C., Wavelength selection in the folding of a single layer withpower-law rheology,Am. J. Sci., 274, 1029–1043, 1974.

Fletcher, R. C., Effects of pressure solution and fluid migration on initiationof shear zones and faults,Tectonophysics, 295, 139–165, 1998.

Fletcher, R. C., and B. Hallet, Unstable extension of the lithosphere: Amechanical model for Basin-and-Range structure,J. Geophys. Res., 88,7457–7466, 1983.

Frederiksen, S., and J. Braun, Numerical modelling of strain localisationduring extension of the continental lithosphere,Earth Planet. Sci. Lett.,188, 241–241, 2001.

Gerbault, M., Modelisation numerique de la naissance des deformationslocalisees - exemple du flambage lithospherique, These de doctorat,Universite de Montpellier II, 1999.

Gerbault, M., A. N. B. Poliakov, and M. Daignieres, Prediction of fault-ing from the theories of elasticity and plasticity: What are the limits?,J. Struct. Geol., 20, 301–320, 1998.

Gerbault, M., E. B. Burov, A. N. B. Poliakov, and M. Daignieres, Do faultstrigger folding in the lithosphere?,Geophys. Res. Lett., 26, 271–274,1999.

Hamming, R. W.,Numerical Methods for Scientists and Engineers, 2nd ed.,Dover Pub., New York, 1973.

Hardacre, K. M., P. A. Cowie, and F. Nino, Variability in fault size scal-ing due to mechanical heterogeneity: A Finite Element investigation,J. Struct. Geol., submitted, 2001.

Hobbs, B. E., and A. Ord, Numerical simulation of shear band formation infrictional-dilatant material,Ing. Arch., 59, 209–220, 1989.

Hoffman, P. F., R. Tirrul, J. E. King, M. R. St-Onge, and S. B. Lucas, Axialprojections and modes of crustal thickening, Eastern Wopmay Orogen,Nortwest Canadian Shield, inProcesses in Continental Lithospheric De-formation, edited by S. P. Clark, Jr., B. C. Burchfield, and J. Suppe, vol.218 ofGeol. Soc. Am. Sp. Pap., pp. 1–29, Geol. Soc. Am., Boulder, 1988.

Huismans, R. S., and C. Beaumont, Asymmetric lithospheric extension:The role of frictional plastic strain softening inferred from numericalexperiments,Geology, 30, 211–214, 2002.

Johnson, A. M., and R. C. Fletcher,Folding of Viscous Layers, CambridgeUniv. Press, New York, 1994.

Kirby, S. H., Tectonic stresses in the lithosphere: Constraints provided bythe experimental deformation of rocks,J. Geophys. Res., 85, 6356–6363,1980.

Kohlstedt, D. L., B. Evans, and S. J. Mackwell, Strength of the lithosphere:Constraints imposed by laboratory experiments,J. Geophys. Res., 100,17,587–17,602, 1995.

Krishna, K. S., J. M. Bull, and R. A. Scrutton, Evidence for multiphasefolding of the Central Indian Ocean lithosphere,Geology, 29, 715–718,2001.

Lambeck, K., Structure and evolution of the intracratonic basins of CentralAustralia,Geophys. J. Roy. Astron. Soc., 74, 843–886, 1983.

Lapusta, N., J. R. Rice, Y. Ben Zion, and G. Zheng, Elastodynamic analysisfor slow tectonic loading with spontaneous rupture episodes on faultswith rate-and state-dependent friction,J. Geophys. Res., 105, 23,765–23,789, 2000.

Lavier, L. L., W. R. Buck, and A. N. B. Poliakov, Self-consistent rolling-hinge model for the evolution of large-offset low-angle normal faults,Geology, 27, 1127–1130, 1999.

Lavier, L. L., W. R. Buck, and A. N. B. Poliakov, Factors controlling normalfault offset in an ideal brittle layer,J. Geophys. Res., 105, 23,431–23,442,2000.

Leroy, Y. M., and N. Triantafyllidis, Stability of a frictional, cohesive layeron a viscous substratum: Variational formulation and asymptotic solu-tion, J. Geophys. Res., 101, 17,795–17,811, 1996.

Leroy, Y. M., and N. Triantafyllidis, Stability analysis of incipient foldingand faulting of an elasto-plastic layer on a viscous substratum, inAspectsof Tectonic Faulting, edited by F. K. Lehner and J. L. Urai, pp. 109–139,Springer, Berlin, 2000.

Lin, J., and E. M. Parmentier, A finite amplitude necking model of rifting inbrittle lithosphere,J. Geophys. Res., 95, 4909–4923, 1990.

Madariaga, R., and K. B. Olsen, Criticality of rupture dynamics in 3-D,PureAppl. Geophys., 157, 1981–2001, 2000.

Martinod, J., and P. Davy, Periodic instabilites during compression or ex-tension of the lithosphere 2. Analogue experiments,J. Geophys. Res.,99, 12,057–12,069, 1994.

Martinod, J., and P. Molnar, Lithospheric folding in the Indian Ocean andthe rheology of the oceanic plate,Bull. Soc. Geol. Fr., 166, 813–821,1995.

McAdoo, D. C., and D. T. Sandwell, Folding of the oceanic lithosphere,J. Geophys. Res., 90, 8563–8569, 1985.

McKinnon, S. D., and I. Garrido de la Barra, Fracture initiation, growth,and effect on stress field: a numerical investigation,J. Struct. Geol., 20,1673–1689, 1998.

Meyer, B., P. Tapponier, L. Bourjot, F. Metivier, Y. Gaudemer, F. Peltzer,S. Guo, and Z. Chen, Crustal thickening in Gansu-Qinghai, lithosphericmantle subduction, and oblique, strike-slip controlled growth of the TibetPlateau,Geophys. J. Int., 135, 1–47, 1998.

Miller, S. A., and D. L. Olgaard, Modeling seismicity clustering and faultweakness due to high pore pressures,Phys. Chem. Earth, 22, 43–48,1997.

Montesi, L. G. J., Localization instability and the origin of regularly-spacedfaults in planetary lithospheres, Ph.D., Massachusetts Institute of Tech-nology, 2002.

Montesi, L. G. J., and M. T. Zuber, Short wavelength compressional defor-mation in a strong Venusian lithosphere, inAGU Chapman Conferenceon Geodynamics of Venus: Evolution and Current State, Abst., p. 14,Aspen, CO, 1997.

Montesi, L. G. J., and M. T. Zuber, The evolution of fault patterns duringorogeny,EOS Trans. Am. Geophys. Union, 80, S345, 1999.

Montesi, L. G. J., and M. T. Zuber, A unified description of localiza-tion for application to large-scale tectonics,J. Geophys. Res., 107,10.1029/2001JB000,465, 2002.

Morgan, J. K., and M. S. Boettcher, Numerical simulations of granular shearzones using the distinct element method 1. Shear zone kinematics andthe micromechanics of localization,J. Geophys. Res., 104, 2703–2719,1999.

Neumann, G. A., and M. T. Zuber, A continuum approach to the develope-ment of normal faults, inProc. 25th US. Symp. Rock. Mech., edited byJ. J. K. Daemen and R. A. Schultz, pp. 191–198, Balkema, Rotterdam,1995.


Neurath, C., and R. B. Smith, The effect of material properties ongrowth rates of folding and boudinage: Experiments with wax mod-els,J. Struct. Geol., 4, 215–229, 1982.

Nikishin, A. M., S. Cloetingh, L. I. Lobkovsky, E. B. Burov, and A. C.Lankreijer, Continental lithosphere folding in Central Asia (part I): Con-traints from geological observations,Tectonophysics, 226, 59–72, 1993.

Place, D., and P. Mora, Numerical simulation of localisation phenomena ina fault zone,Pure Appl. Geophys., 157, 1821–1845, 2000.

Poliakov, A. N. B., H. J. Herrman, Y. Y. Podladchikov, and S. Roux, Fractalplastic shear bands,Fractals, 2, 567–581, 1994.

Ramberg, H. R., Contact strain and folding instability of multilayered bodyunder compression,Geol. Runds., 51, 405–439, 1961.

Ricard, Y., and C. Froideveau, Stretching instabilities and lithospheric boud-inage,J. Geophys. Res., 91, 8314–8324, 1986.

Roy, M., and L. H. Royden, Crustal rheology and faulting at strike-slip plateboundaries 1. an analytical model,J. Geophys. Res., 105, 5583–5597,2000.

Schmalholz, S. M., and Y. Podladchikov, Buckling versus folding: Impor-tance of viscoelasticity,Geophys. Res. Lett., 26, 2641–2644, 1999.

Schmalholz, S. M., and Y. Y. Podladchikov, Viscoelastic buckle-folding:Maxwell versus Kelvin rheology,Geophys. Res. Lett., 28, 1835–1838,2001.

Schmalholz, S. M., Y. Podladchikov, and J.-P. Burg, Control of foldingby gravity and matrix thickness: Implications for large-scale folding,J. Geophys. Res., 107, 10.1029/2001JB000,355, 2002.

Scholz, C. H.,The Mechanics of Earthquakes and Faulting, CambridgeUniv. Press, New York, 1990.

Smith, R. B., Unified theory of the onset of folding, boudinage, and mullionstructure,Geol. Soc. Am. Bull., 86, 1601–1609, 1975.

Smith, R. B., Formation of folds, boudinage, and mullions in non-Newtonianmaterials,Geol. Soc. Am. Bull., 88, 312–320, 1977.

Smith, R. B., The folding of a strongly non-Newtonian layer,Am. J. Sci.,279, 272–287, 1979.

Sornette, D., and C. Vanneste, Faults self-organized by repeated earthquakesin a quasi-static antiplane crack model,Nonlin. Proc. Geophys., 3, 1–12,1996.

Stewart, E. M., and J. W. Head, III, Evidence for temporal continuity ofdeformation in the Baltis Vallis region of Venus from observations ofCanali topography,Lunar Planet. Sci., XXXI, Abst. 1692, 2000.

Tackley, P. J., Self-consistent generation of tectonic plates in time-dependent, three-dimensional mantle convection simulations 2. Strainweakening and asthenosphere,Geochem. Geophys. Geosyst., 1, Article2000GC000,043, 2000.

Triantafyllidis, N., and Y. M. Leroy, Stability of a frictional, cohesive layeron a viscous substratum: Validity of asymptotic solution and influenceof material properties,J. Geophys. Res., 102, 20,551–20,570, 1997.

Turcotte, D. L., and G. Schubert,Geodynamics: Applications of ContinuumPhysics to Geological Problems, 2nd ed., Cambridge Univ. Press, NewYork, 2002.

Van Orman, J., J. R. Cochran, J. K. Weissel, and F. Jestin, Distribution ofshortening between the Indian and Autralian plates in the Central IndianOcean,Earth Planet. Sci. Lett., 133, 35–46, 1995.

Wallace, M. H., and H. J. Melosh, Buckling of a pervasively faulted litho-sphere,Pure Appl. Geophys., 142, 239–261, 1994.

Wang, Y.-C., X.-C. Yin, F.-J. Ke, M.-F. Xia, and K.-Y.Peng, Numericalsimulation of rock failure and earthquake process on mesoscopic scale,Pure Appl. Geophys., 157, 1905–1928, 2000.

Watters, T. R., Origin of periodically spaced wrinkle ridges on the Tharsisplateau of Mars,J. Geophys. Res., 96, 15,599–15,616, 1991.

Weissel, J. K., R. N. Anderson, and C. A. Geller, Deformation of the Indo-Australian plate,Nature, 287, 284–291, 1980.

Zuber, M. T., Compression of oceanic lithosphere: An analysis of intraplatedeformation in the Central Indian Basin,J. Geophys. Res., 92, 4817–4825, 1987.

Zuber, M. T., and L. L. Aist, The shallow structure of the Martian lithospherein the vicinity of the ridged plains,J. Geophys. Res., 95, 14,125–14,230,1990.

Zuber, M. T., and E. M. Parmentier, Lithospheric necking: A dynamicmodel for rift morphology,Earth Planet. Sci. Lett., 77, 373–383, 1986.

Zuber, M. T., and E. M. Parmentier, On the relationship between isostaticelevation and the wavelengths of tectonic surface features on Venus,Icarus, 85, 290–308, 1990.

Zuber, M. T., E. M. Parmentier, and R. C. Fletcher, Extension of continentallithosphere: A model of two scales of Basin and Range deformation,J. Geophys. Res., 91, 4826–4838, 1986.

Laurent G. J. Montesi, Department of Geology and Geophysics, WoodsHole Oceanographic Institution, MS#22, Woods Hole, MA 02543, [email protected].

Maria T. Zuber, Department of Earth, Atmospheric, and Planetary Sci-ences, Massachusetts Institute of Technology, Cambridge, MA 02139, [email protected].

(Received .)

Documents

Spacing of Faults at the Scale of the Lithosphere and ... · Spacing of Faults at the Scale of the Lithosphere and Localization Instability 1: Theory Laurent G. J. Montesi´ 1 Maria