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Spacing of faults at the scale of the lithosphere

and localization instability:

1. Theory

Laurent G. J. Montesi1 and Maria T. Zuber2

Massachusetts Institute of Technology, Cambridge, Massachusetts, USA

Received 12 April 2002; revised 21 October 2002; accepted 18 November 2002; published 20 February 2003.

[1] Large-scale tectonic structures such as orogens and rifts commonly display regularlyspaced faults and/or localized shear zones. To understand how fault sets organize with acharacteristic spacing, we present a semianalytical instability analysis of an idealizedlithosphere composed of a brittle layer over a ductile half-space undergoing horizontalshortening or extension. The rheology of the layer is characterized by an effective stressexponent, ne. The layer is pseudoplastic if 1/ne = 0 and forms localized structures if 1/ne < 0.Two instabilities grow simultaneously in this model: the ‘‘buckling/necking instability’’that produces broad undulations of the brittle layer as a whole, and the ‘‘localizationinstability’’ that produces a spatially periodic pattern of faulting. The latter appears onlyif the material in the brittle layer weakens in response to a local perturbation of strainrate, as indicated by 1/ne < 0. Fault spacing scales with the thickness of the brittle layerand depends on the efficiency of localization. Localization is more efficient for morenegative 1/ne, leading to more widely spaced faults. The fault spacing is related to thewavelength at which different deformation modes within the layer enter a resonance thatexists only if 1/ne < 0. Depth-dependent viscosity and the model density offset theinstability wavelengths by an amount aL that we determine empirically. The wavenumber of the localization instability, is kj

L = p( j + aL)(�1/ne)�1/2/H, with H the

thickness of the brittle layer, j an integer, and 1/4 < aL < 1/2 if the strength of the layerincreases with depth and the strength of the substrate decreases with depth. INDEX

TERMS: 8005 Structural Geology: Folds and folding; 8010 Structural Geology: Fractures and faults; 8020

Structural Geology: Mechanics; 8149 Tectonophysics: Evolution of the Earth: Planetary tectonics (5475);

KEYWORDS: faults, folds, buckling, instability analysis, fault spacing, rheology

Citation: Montesi, L. G. J., and M. T. Zuber, Spacing of faults at the scale of the lithosphere and localization instability: 1. Theory,

J. Geophys. Res., 108(B2), 2110, doi:10.1029/2002JB001923, 2003.

1. Introduction

[2] Much of our theoretical understanding of tectonicsstems from continuum mechanics [Turcotte and Schubert,2002]. For instance, certain large-scale patterns of deforma-tion resemble a mode of folding of the strong layers of thelithosphere called buckling in compression and necking inextension [Biot, 1961; Fletcher and Hallet, 1983; Ricardand Froidevaux, 1986; Zuber et al., 1986; Zuber, 1987].The buckling or necking theory predicts a preferred wave-length of deformation that is controlled by the mechanicalstructure of the lithosphere. Hence, recognizing buckling inthe geological records helps to constrain the structure of thelithosphere at the time when buckling occurred.

[3] Faults and shear zones constitute another primaryindicator of the structure of orogens or rifts that can be relatedto continuum mechanics models [Beaumont and Quinlan,1994]. Faults often present a preferred spacing or character-istic 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 fault spacing[Watters, 1991; Brown and Grimm, 1997], but faults actuallyrepresent a localized style of deformation that is not acces-sible using the continuum theories implied in the buckling/necking analysis; deformation occurs mostly, if not entirely,within a narrow band. In this study, we modify the buckling/necking theory to consider explicitly the dynamics of local-ization, and to explore the link between the structure of thelithosphere and patterns of localized shear zones or faults.[4] Buckling and necking produce broad undulations of

the lithosphere in which deformation is distributed. Thepseudoplastic rheology used to model the brittle levels ofthe lithosphere [Fletcher and Hallet, 1983] assumes perva-sively distributed failure or faulting. However, faulting has atendency to localize, to abandon a distributed style of

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2110, doi:10.1029/2002JB001923, 2003

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

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

Copyright 2003 by the American Geophysical Union.0148-0227/03/2002JB001923$09.00

ETG 14 - 1

faulting to concentrate deformation on a few isolated majorfaults [Sornette and Vanneste, 1996; Gerbault et al., 1998].As stress heterogeneities induced by buckling favor faultingin the hinge of large-scale folds [Lambeck, 1983; Martinodand Davy, 1994; Gerbault et al., 1999], localized faultpatterns can be controlled by buckling if they develop afterthe folds have reached sufficient amplitude. However,faulting may occur from the onset of deformation andhence, may develop without the influence of finite-ampli-tude buckling. Indeed, some tectonic provinces displayfaults with a characteristic spacing unrelated to the bucklingwavelength. To cite only examples in compressive environ-ments, faults in the Central Indian Basin [Bull, 1990; VanOrman et al., 1995], in Central Asia [Nikishin et al., 1993],or in Venusian fold belts [Zuber and Aist, 1990] are moreclosely spaced than the wavelength of folds in the sameregion. In these regions, faulting and buckling appear assuperposed deformation styles, each with a characteristiclength scale.[5] Buckling and necking were first studied in Earth

sciences as a mechanism to form folds or boudins inoutcrop-scale layered sequences [Johnson and Fletcher,1994]. While originally derived using a thin plate approx-imations 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 nonlinear rheology, thestress supported by the fluid, s, is related to the secondinvariant of strain rate, _eII, by _eII, / sne, with ne theeffective stress exponent, a measure of the nonlinearity ofthe rock rheology [Smith, 1977; Montesi and Zuber, 2002].Non-Newtonian creep with 1 < ne < 5 is the rheology thatdescribes rocks at sufficiently high temperature to behave ina ductile manner.[6] At low temperature, rocks behave instead in a brittle

manner. The stress that they can support is limited by a yieldstrength, at which failure, faulting, and plastic flow occur.Yielding can be included in thin-plate analyses of folding bylimiting the bending stresses to the yield strength andreducing the apparent flexural rigidity of an elastic orviscous plate accordingly [Chapple, 1969; McAdoo andSandwell, 1985; Wallace and Melosh, 1994]. The thick-plate formulation of the buckling theory is particularly welladapted to an alternative treatment of failure, in which theyielding material is approximated as a highly non-New-tonian fluid in the limit ne ! +1 [Chapple, 1969, 1978;Smith, 1979]. Most lithospheric-scale applications of buck-ling use that approximation [Fletcher and Hallet, 1983;Zuber et al., 1986; Zuber, 1987]. More accurate treatmentsof yielding have been included in buckling/necking theory.Leroy and Triantafyllidis [1996, 2000] and Triantafyllidisand Leroy [1997] studied the necking behavior of a hard-ening elastic-plastic medium at yield using the strain rate-stress rate relations of the deformation theory of plasticity.Localized faulting is predicted only for tectonic stresses inexcess of the critical value for necking and is neverpredicted if the flow theory of plasticity is used [Trianta-fyllidis and Leroy, 1997]. Therefore this model cannotexplain regions that show regularly spaced localized fault-ing. Davies [1990] modeled the buckling of a rigid-plasticlayer with associated flow law surrounded by a rigid base-

ment and a viscous half-space. Faults were forced by a localcusp in the model interface but an initially distributedperturbation remains distributed. Neumann and Zuber[1995] found that a localized perturbation triggers macro-scale shear bands in a power law medium with ne ! +1as well. Fletcher [1998], who also included the effects ofpore fluids and pressure solution on a non-Newtonianporous fluid with ne ! +1, showed that the shear bandsproduced by a localized forcing are ephemeral. He specu-lated that the bands would be stabilized and therefore maycorrespond to faults if strain weakening were included[Fletcher, 1998].[7] All previous treatments of yielding in buckling theory

fail to produce localized deformation; faulting remainsdistributed throughout the material. Hence, any fault patternobserved in nature is expected to develop late in the foldinghistory, with a spacing that is controlled by the buckling ornecking wavelength. However, this is contrary to manygeological observations [Weissel et al., 1980; Nikishin et al.,1993; Krishna et al., 2001]. In order to understand howfault spacing may differ from the buckling/necking wave-length, we study the buckling behavior of simplified litho-sphere models with a rheology that weakens withdeformation upon yielding. The weakening behavior ischaracterized by a negative effective stress exponent. Suchan effective rheology brings a tendency to localize defor-mation and faulting [Montesi and Zuber, 2002].[8] The effective stress exponent, ne, indicates how a

material responds to a perturbation in the deformation field[Smith, 1977]. In this study, deformation is quantified by thesecond strain rate invariant, _eII. When ne < 0, increasing _eIIdecreases the material strength, which ensures localization[Montesi and Zuber, 2002]. The (algebraic) value of ne isdetermined from the physical process that produces local-ization. It incorporates not only the direct effect of perturb-ing the strain rate, but also the possible feedback of internalvariables that control the rheology. For instance, a frictionalmaterial such as the pervasively faulted brittle lithosphererequires a higher stress to deform more rapidly, which wouldresult in positive ne. However, a higher sliding velocity onfaults changes also the physical state of a granular gougeinside the fault and results in apparent weakening andnegative ne [Dieterich, 1979; Scholz, 1998]. In Montesiand Zuber [2002], we derived the conditions for which thisfeedback mechanism and others produce localization, andwe gave values for the corresponding effective stress expo-nents. For frictional sliding, �300 < ne < �50.[9] Neurath and Smith [1982] showed that strain weak-

ening reduces the value of 1/ne in a non-Newtonian fluid,possibly resulting in ne < 0. However, the materials that theystudied never had negative ne. While they recognized thatne < 0 would lead to ‘‘catastrophic failure,’’ or localization,modulated by the interaction with the surroundings of thelayer with negative exponent, Neurath and Smith [1982] didnot present an analysis of buckling with ne < 0.[10] In this paper, we derive the growth spectrum for

simple models of the lithosphere, which indicates how fast agiven wavelength growths in a particular model [Johnsonand Fletcher, 1994]. Buckling and necking occur at wave-lengths at which the growth rate is maximum. We identify anew instability when deformation localizes and call itlocalization instability. Both the localization instability and

ETG 14 - 2 MONTESI AND ZUBER: SPACING OF FAULTS, 1

the buckling/necking instability are associated with partic-ular resonances within the localizing layer. We show howthe wavelengths of the buckling and localization instabil-ities relate to resonances within the model, consideringseveral types of depth-dependent strength profiles thatapproximate the strength profile of a monomineralic litho-sphere. Most of this study is conducted assuming uniformhorizontal shortening, with a generalization to horizontalextension in section 6.2. In a companion paper, we showthat the localization instability is a possible explanation forfault spacing in the Central Indian Basin, which is incom-patible with a buckling instability.

2. Instability Principle

2.1. Solution Strategy

[11] The buckling/necking and localization instabilitiesdevelop in mechanically layered models of the lithosphereundergoing horizontal shortening or extension. In thisstudy, we concentrate on the behavior of a single brittleor plastic layer overlying a ductile half-space under hori-zontal shortening at the �_exx (Figure 1). Horizontal exten-sion is discussed in section 6.2. Each material isincompressible. As the deformation field is constrained tobe two-dimensional, it is completely determined by thestream function j(x, z).[12] The deformation field in each layer is decomposed

into a primary field and a secondary field of much smalleramplitude. The primary field is the flow solution when theinterfaces in the model are perfectly flat and horizontal. It isinvariant in x, with �_ezz ¼ ��_exx, and �_exz ¼ 0. However, anyrelief on the interfaces perturbs the system and drives asecondary flow in the model. In turn, the secondary flowdeforms the interfaces.[13] This analysis determines the rate at which inter-

face perturbations grow in a given model. The flowequations for the secondary deformation field and theboundary conditions are linearized, assuming that theinterface perturbations have a much smaller amplitudethan the layer thickness [Johnson and Fletcher, 1994;Montesi, 2002]. Thus, the Fourier components of aninfinitesimal generic perturbation are decoupled fromone another. Hence, we consider interface perturbationsof the form

xi ¼ x0i exp ikxð Þ; ð1Þ

with x0 the amplitude of the perturbation, k its wavenumber, i an index identifying an interface of the model,and i = (�1)1/2. There are several modes of deformation ineach layer, each characterized by a stream function jj

jj ¼ j0j fj zð Þ exp ikxð Þ; ð2Þ

where fj is the depth kernel, jj0 is the amplitude of jj, and j

is an index identifying each deformation mode. The depthkernel is determined from the equations of Newtonianequilibrium (see Appendix A) and the mode amplitude isdetermined from the stress and velocity boundary condi-tions at each interface [Johnson and Fletcher, 1994;Montesi, 2002]. Because the problem is linearized, jj

0 is alinear combination of {xi}.

[14] The rate at which the interface perturbations growhas a kinematic contribution from the pure shear thicken-ing of the model (primary flow field) and a dynamiccontribution from the secondary flow field [Smith, 1975].We obtain:

dx0idt

¼ �_exxdij þQij

� �xj; ð3Þ

with dij the Kronecker operator and Qij the growth matrix,which depends on k and the strength and density structure ofthe model [Montesi, 2002]. The growth rate Q is theeigenvalue of Q that has the largest real part. The associatedeigenvector describes the fastest growing deformation modeof the model as a whole [Smith, 1975; Zuber et al., 1986].For instance, it determines whether a given layer deforms bybuckling (upper and lower interfaces in phase) or necking(upper and lower interfaces out of phase). The growthspectrum is defined as the function Q(k).

2.2. Effective Rheology and Effective Stress Exponent

[15] The strength of each layer of the lithosphere model isan analytical function of depth. As we address the organ-ization of strain rate within this model, we define theapparent viscosity h by

sij ¼ �pdij þ h_eij; ð4Þ

where s is the stress supported by the material, p thepressure, and _e the strain rate. In general, h depends onthe second invariant of the strain rate, �_eII and on the depthz. In the brittle field, the material strength and therefore the

Figure 1. Schematic of lithosphere models. A layer ofthickness H, viscosity h1(z), and effective stress exponent n1lies over a half-space of different mechanical properties(h2(z), n2 = 3). The model is two-dimensional, incompres-sible, and subjected to pure shear shortening at the rate �_exx.Its density is r. Boundary conditions are no slip at theinterface, no stress at the surface. For the instabilityanalysis, the interfaces are perturbed by an infinitesimalsinusoidal topography of wavelength l and amplitude x1and x2.

MONTESI AND ZUBER: SPACING OF FAULTS, 1 ETG 14 - 3

viscosity may also depend on the strain undergone by thematerial. In this study, we chose to ignore that additionalcomplication, which would also require elasticity to beincluded in the model and make the analysis time-dependent [Neurath and Smith, 1982; Schmalholz andPodladchikov, 1999, 2001][16] In the perturbation analysis, the primary field, which

represents the state of uniform shortening, is denoted byoverbars. It obeys

�sij ¼ ��pdij þ h�_eij; ð5Þ

with h the material viscosity at the externally imposedstrain rate �_eII ¼ _exxj j. It is an analytic function of depth ineach layer. The secondary field, denoted with a tilde,which represents the perturbing flow obeys the apparentrheology:

~sxx ¼ �~pþ hne~_exx; ð6aÞ

~szz ¼ �~pþ hne~_ezz; ð6bÞ

~sxz ¼ h~_exz; ð6cÞ

with ne the effective stress exponent,

1

ne¼ 1þ

�_eIIh

@h@ _eII

�_eIIs

@s@ _eII

: ð7Þ

In deriving equations (6a), (6b), and (6c), we assumed thatthe amplitude of the secondary field is infinitesimal withrespect to the primary field. This approximation is validonly for the onset of the instability.[17] The apparent viscosity of the secondary field is

anisotropic, reduced in the directions of the primary flowfield by a factor ne. As introduced by Smith [1977], theeffective stress exponent is a local measure of the non-linearity of the rheology of a material. We extended thisconcept in the work by Montesi and Zuber [2002] andused ne in a general framework of localization. Whenne < 0, the material is unstable with respect to localperturbations. A negative effective stress exponent indi-cates that the strength is reduced in locations where thestrain rate is enhanced. This situation is unstable andresults in a localized zone of high strain rate [Montesi andZuber, 2002]. The instability analysis of this study showshow these localized deformation areas organize at litho-spheric scale.[18] Beyond the sign of the effective stress exponent,

its algebraic value provides a quantitative measure of theefficiency of localization [Montesi and Zuber, 2002]. If1/ne = 1, the material is effectively Newtonian, and doesnot localize at all. If 0 < 1/ne < 1, the material is welldescribed by non-Newtonian creep. For instance, rocksdeforming by ductile creep have 0.2 < 1/ne < 1. In thisregime, a rock is softening in the sense that its apparentviscosity decreases with increasing strain rate. Localperturbations of strain rate are enhanced by the non-Newtonian behavior, but the material is stable: as its

strength increases with strain rate, there is no dynamicweakening and no localization [Montesi and Zuber,2002]. In the limit 1/ne ! 0+, the material is pseudo-plastic: its strength does not depend on strain rate. Local-ization requires 1/ne < 0. We showed [Montesi and Zuber,2002] that many mechanisms that are associated withlocalized shear zones in the laboratory or in nature havea negative effective stress exponent, often with 1/ne �10�2 to �10�1 in the brittle field.[19] Often, the effective stress exponent is negative only

when an internal feedback process is considered that mayinclude a variable describing damage, or state of a faultgouge. This variable may require a finite time to respond toa local variation of strain rate. This results in an immediatestrengthening response of the system, followed by weaken-ing in the long-time limit. In this study, we ignore thetransient response, arguing that perturbations may be heldfor long enough that steady state is reached, and that theperturbation amplitude is so small that the strengthening‘‘barrier’’ is easily overcome. However, this assumptionshould be relaxed in future work.[20] Most previous studies of lithospheric-scale instabil-

ities treated the brittle upper crust and mantle as pseudo-plastic with 1/ne ! 0+. These instabilities produce bucklingin compression, and necking in extension [Fletcher andHallet, 1983; Ricard and Froidevaux, 1986; Zuber, 1987].In this study, we introduce the solutions for 1/ne < 0, whichproduce regularly spaced shear zones, through a processthat we call localization instability. The more negative 1/neis, the stronger localization is. Intuitively, a more efficientlocalized shear zone can accommodate the deformationfrom a wider nonlocalized region of the lithosphere. Hence,the spacing of localized shear zones should increase when1/ne is more negative.

3. Depth Kernel

3.1. Fundamental Equation

[21] The first step in solving the instability problemdefined above is to determine the expression of the depthkernel, f(z), which gives the depth-dependence of thestream function (equation (2)) and therefore of the defor-mation field for each mode of deformation. Whereas thestrength and the effective viscosity of the lithospheredepend on depth, its effective stress exponent does notnecessarily do so, as it measures the rate at which a rockweakens, scaled by its strength. In fact, ne does notdepend on depth for most processes of localization[Montesi and Zuber, 2002]. Therefore, we make theassumption that ne is independent of depth, z, within eachlayer.[22] We write the equations of Newtonian equilibrium

for the secondary flow, using its apparent rheology(equations (6a), (6b), (6c)), and expressing the strain rateas a function of the stream function. These equations arethen combined to eliminate the pressure term, and sim-plified as their dependence in x is exp(ikx). We obtain(Appendix A)

d4fdz4

þ 2rd3fdz3

� Ak2 � s� � d2f

dz2� Ak2r

dfdz

þ k4 þ k2s� �

f ¼ 0;

ð8Þ


where we used the notation

r zð Þ ¼ dh=dzh

; ð9Þ

s zð Þ ¼ d2h=dz2

h; ð10Þ

A ¼ 4

ne� 2: ð11Þ

[23] Equation (8) admits four solutions for a givenstrength profile h(z), effective stress exponent ne, and wavenumber k. Hence, there are four superposed deformationmodes in each layer, for a given wavelength, each with itsown amplitude that is determined from matching boundaryconditions at each interface [Montesi, 2002].

3.2. Depth Kernel for Exponentialand Constant Viscosity Profiles

[24] The depth kernel can be determined analytically ifthe viscosity varies exponentially with depth. Then, thefunction r does not depend on z, s = r2, and

h ¼ he exp rzð Þ; ð12Þ

with he a constant. Constant viscosity layers are included asthe special case r = 0.[25] For an exponential viscosity profile, the depth kernel

takes the form

f ¼ f0 exp iakz; ð13Þ

with

a ¼ iR

2� 1� 2

ne� R2

4� 4

n2e� 4

ne� R2

� �1=2" #1=2

; ð14Þ

where R = rk [Fletcher and Hallet, 1983].[26] There are four values of the parameter a, each

corresponding to a given deformation mode with spatialdependence: exp[ik (x + az)]. The quantity x + Re(a)z is thephase of the deformation mode. It is conserved along linessloping at an angle arctan (1/Re(a)). Hence, a is referred toas the mode slope. Its imaginary part indicates the rate atwhich the amplitude of the deformation mode decays withdistance away from an interface. Each deformation mode istherefore akin to a damped wave, which leads to theresonances discussed in section 4.2.[27] For constant viscosity layers (r = 0), equation (14)

becomes

a ¼

�iffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�1þ 2 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ne

p� �=ne

q; if 1 < 1=ne;

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 1=ne

p� i

ffiffiffiffiffiffiffiffiffi1=ne

p; if 0 < 1=ne < 1

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1� 2 1�

ffiffiffiffiffiffiffiffiffiffiffiffiffi1� ne

p� �=ne

q; if 1=ne < 0:

8>>>><>>>>:

ð15Þ

By convention, we define a1, a2, a3, and a4, by selecting inequations (14) or (15) the sign combinations (+, +), (+, �),

(�, +), and (�, �). Note that a is real when ne is negative,i.e., when the material localizes. In that case, the streamfunction is correlated along four different slopes, but itsamplitude is constant with depth: interface perturbationsgenerate four wavelike deformation fields in each layer,none of which decays or grows with depth. This makes itimpossible to solve for the behavior of a half-space withnegative ne, which requires the deformation field to vanishat infinity. Hence, the simplest solvable model that includesnegative ne is made of a layer of finite thickness over a half-space with positive ne. Although our formulation can inprinciple handle any number of layers, only this type ofmodel is considered in this paper.[28] The depth kernel for an exponential viscosity pro-

file with r 6¼ 0 is discussed in Montesi [2002]. The domainwhere all a are real is pushed to more negative 1/ne when

R increases, and vanishes altogether at R ¼ 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ

ffiffiffi5

pp.

This is a limit where the viscosity increases rapidlycompared to the perturbation wavelength. However, evenin the large wavelength limit, there are more than twovalues of Re(a), one of which is zero. We will see that thiscondition is sufficient for the localization instability todevelop.

3.3. General Solution

[29] For a nonexponential viscosity profile, equation (8)must be solved numerically. We use a Runge-Kutta inte-gration technique [e.g., Hamming, 1973]. By convention,the depth kernel has a value of 1 at the top of each layer.The four superposed modes of deformation are found bysetting the initial values of the depth-derivatives of f inturn to each solution of f(z) for an exponential viscosityprofile that approximates the actual viscosity profile at thetop of the layer (equation (14)). We verified that thesolutions do not depend on the actual starting schemechosen. The exceptional cases where the solutions aredegenerate are ignored, and do not arise in practice exceptif 1/ne = 1 or 1/ne = 0.[30] In this paper, the numerical solver is used only for

the case where the viscosity depends linearly with depth. Toour knowledge, only Bassi and Bonnin [1988] have con-sidered that case previously. They used a polynomialexpansion of the depth kernel and determined a recurrencerelation between the polynomial coefficients. Althoughtechnically an analytical solution, this scheme is subject tonumerical errors and truncation of the expansion. Wefavored the numerical integration technique as it can handleany viscosity profile such as in the companion paper. Theonly limitation is that h 6¼ 0.

4. Constant Viscosity Analysis

4.1. Growth Spectra

[31] We first consider models where a layer of thicknessH, effective stress exponent n1, and viscosity h1, independ-ent of depth, overlies a half-space of lower viscosity h2 andeffective stress exponent n2 > 0. The exact value of n2matters little. In our reference model, we use n2 = 3 and h2 =h1/10 (Figure 1). This is the simplest approximation of thelithosphere strength structure that displays both localizationand buckling instabilities. The model is shortened at the rate�_exx < 0. For now, the material density is ignored. The


Growth rate Q1000 0.

1

100 10 1

010

2030

4050

Wav

enum

ber

k

b) η=0.

1

η=1

z=0

z=1

z=-∞

a)K

L

KB

/N

Figure

2.

(a)Growth

spectrum

foralayer

ofuniform

viscosity

h 1overlyingahalf-spaceofuniform

viscosity

h 2=h 1/10,

withr=0,n2=3.Solidline:n1=�10;dashed

line:n1=106;dotted

line;n1=100.(b)Viscosity

profile.KB/N�

pandKL

�10forthevalues

ofn1considered.


coordinates x and z are normalized by H, the wave numbersare normalized by H�1, and the stresses are normalized byh1 _eII.[32] We present in Figure 2 the growth spectra for various

values of the effective stress exponent of the layer: 1/n1 =10�2, 10�6, and �10�1. The flow fields for the last twocases are plotted in Figure 3.[33] The cases with positive n1 have been solved before.

The growth rate reaches a local maximum at several wavenumbers separated by the wave number scale KB/N, standingfor wave number scale of the buckling/necking instability[Ricard and Froidevaux, 1986; Zuber, 1987]. The firstgrowth rate maximum, at k � p/2, has usually the highestgrowth rate. It controls the wavelength of the fastest grow-ing instabilities. The wavelengths of the subsequent maximaact like overtones, giving the instability its shape. For n1 = 1,only the first growth rate maximum has a finite growth rate,giving the instability the sinusoidal shape often associatedwith buckling [Johnson and Fletcher, 1994]. For relativelylarger n1 (e.g., n1 = 100 in Figure 2), the growth ratemaxima decay rapidly with wave number. In the pseudo-plastic limit (e.g., 1/n1 = 10�6 in Figure 2), all the growthrate maxima have a similar value, giving the deformationfield a box-car appearance [Neumann and Zuber, 1995].The decay of the growth rate maxima with wave number iscontrolled by Im(a) [Ricard and Froidevaux, 1986].[34] The growth spectrum for the case ne = �10 is best

described as the superposition of a buckling-type spectrumand a sequence of doublets with infinite growth rate(Figure 2). These doublets indicate the localization insta-bility. The difference between the wave numbers of con-secutive doublets defines the wave number scale KL,which is different from KB/N. In Figure 2, KB/N p andKL 10. The reconstructed deformation field (Figure 3b)has the same appearance of two superposed deformationmodes, each having a specific length scale: buckling of thelayer as a whole, and regularly spaced localized shearzones. The infinite growth rate at the localization doublets(Figure 2) is due to the unstable character of localization: alocal perturbation of strain rate weakens the materiallocally, so the strain rate increases further. The strain rateperturbation triggers a positive feedback that results in aninfinite growth rate. The localized shear zones have alarge-scale organization given by the wave number atwhich a divergent doublet is present.

4.2. Resonance

[35] The wave number scales KB/N and KL apparent in thegrowth spectra (Figure 2) correspond to resonances betweenthe superposed deformation modes in the layers of ourmodel. These resonances appear because of the wave-likeform of the deformation modes, already mentioned insection 3.2. Let us consider a secondary deformation fieldwith wave number k. All the distances are scaled by H, thethickness of the layer undergoing the buckling and local-ization instabilities. The deformation field is composed offour deformation modes, each characterized by the modeslope ai, i = 1 to 4 given by equation (14). The phase ofmode a1 changes by Re(aa1

) between the surface z = 1 andthe bottom of the layer z = 0. Similarly, the phase of modea2 changes by Re(aa2

) between z = 0 and z = 1. If twodifferent deformation modes have the same phase at the

surface, their phases are again identical at the bottom of thelayer if their wave number obeys

k ¼ k ja1;a2

¼ 2pj= Re aa1 � aa2ð Þj j; ð16Þ

where j is an integer called the order of the resonance,which indicates how many times the phase of the twomodes is identical below the surface, including the bottomof the layer. If the viscosity of the layer is constant, equation(15) indicates that there are two possible resonances whenn1 > 0 and four possible resonances when n1 < 0.[36] The resonant wave numbers are plotted as a func-

tion of the effective stress exponent in Figure 4. Thepattern of resonances compares well to the buckling andlocalization instabilities. We present in Figure 5 a map ofgrowth rate in the parameter space of 1/n1 � k. Thisrepresentation is akin to a topographic map of the surfaceQ(1/n1, k). The growth spectra shown in Figure 2 representsections through that map taken with constant 1/ne of�0.1, 10�6, and 0.01.[37] The buckling instability is best identified in the 1/ne

> 0 domain by a broad maximum and can be followed intothe negative 1/ne domain (Figure 5). The characteristic wavenumber of this instability, KB/N, varies monotonically with1/ne, following

KB=N ¼ k1;4 ¼ p 1� 1=n1ð Þ�1=2: ð17Þ

The growth rate maxima are located at

kBj ¼ jþ 1=2ð ÞKB=N ; j 2 Z: ð18Þ

The resonance associated with k1,4 involves deformationmodes that propagate into the layer with similar slopes, butdifferent directions (Figure 6a). The combined velocity field

Figure 3. Deformation fields corresponding to the growthspectra in Figure 2, shaded as a function of strain rate. (a)n1 = 106; (b) n1 = �10. The velocity field is computed froma small-amplitude initial perturbation at the surface suchthat x1 / k b, with b a real number. Then, the amplitude ofthe other interface is set to give the eigenvector of thegrowth matrix (equation (3)) with the fastest growth rate.These modes are amplified by the growth rate (limited to anarbitrary value if n1 < 0). The initial spectral shape, theamplification factor, and the limiting growth rate have beenchosen to show clearly (a) the buckling deformation modeand (b) both the buckling and the localization modes.


takes the appearance of convection-like rolls. The growthrate is maximum when these rolls are in phase (bucklingmodes) or out of phase (necking modes) [Ricard andFroidevaux, 1986; Zuber, 1987].[38] In presence of localization, interaction between the

deformation modes is even more crucial because althoughthe layer attempts to generate discontinuities, the ductilesubstrate cannot accommodate them. Therefore, the incip-ient shear zones can form only in a self-consistent network.For that, they use the resonances between deformationmodes. As a consequence, the growth rate is finite at allwave numbers that are not resonant, indicating that themodel is stable and deformation remains distributed (Figure2). Localization is a trait of the material in the layer, but thelarger scale geometry prevents a network of localizeddeformation zones to form at these wavelengths. At reso-nant wave numbers, a network of shear zones can form andthe growth rate is infinite.[39] The resonance involved in the localization instability

is different from that involved in buckling. It is now

KL ¼ k1;2 ¼ p �1=n1ð Þ�1=2: ð19Þ

The growth rate maxima for the localization instability arelocated near:

kLj ¼ jð ÞKL; j 2 Z: ð20Þ

The resonance k1,2 involves modes propagating in the layerwith different slopes but in the same direction (Figure 6).The wave number KL is real only if 1/n1 < 0, as expected ifit arises due to localization.[40] As was pointed out earlier, the effective stress

exponent quantifies the efficiency of the localization proc-ess in the layer. Efficient localization, defined locally and

expressed by 1/ne very negative, produces shear zones thatare able to accommodate the deformation originally spreadover a wide distance. Hence, the wavelength of deformationincreases as 1/ne becomes more negative. In the limit ofperfect localization, all the deformation is localized on asingle fault. Indeed, the wavelength of the localizationinstability goes to infinity in the limit of perfect localization1/ne ! �1 (Figure 5). On the other hand, in the limit ofperfect plasticity (1/ne ! 0), a continuum limit should berecovered. Indeed, in that case, the instability wavelengthgoes to 0; many finely spaced faults are predicted, acontinuum of faults.[41] As the localization instability is characterized by a

doublet of divergent growth rate, centered on the wavenumber given by equation (20), there are actually twopreferred wavelengths of the localization instability for agiven j. The separation of the two branches of a givendoublet is not predicted by the resonance analysis. We notethat the doublets close when several resonances are super-posed (compare Figures 4 and 5), which might indicate theimportance of the derivatives of the stream function.Whereas the resonance analysis uses only the stream func-tion, the boundary conditions include velocity and stresses,which depend on the derivatives of the stream function.Two modes of deformation with different mode slopes havedifferent values of stress and velocity at a given depth, evenif their stream functions at that depth are identical. Thismight suffice to offset slightly the actual resonant wave-lengths. The opening of the localization doublets is usuallysmall enough to be ignored. In addition, the doubletstructure may break down once the nonlinearities in thesystem behavior and transient strengthening are taken intoaccount. These additional aspects of a localizing system areneeded to stabilize localization and probably limit thedivergent growth rate of the localization instability to afinite value. Therefore, applications to the Earth need not

Figure 5. Map of growth rate as a function of effectivestress exponent of the layer and perturbation wave number.Lighter tone indicates high growth rate, with the contoursindicated on the color bar. Model identical to Figure 2.Modes j = 1 and j = 4 of the buckling instability are labeledin white, and modes j = 0, 2, and 5 of the localizationinstability are labeled in black.

Figure 4. Resonant wave numbers for a layer of thicknessH as a function of its effective stress exponent (equation(16)). Thick line: fundamental mode j = 1; Other lines:higher-order resonances. The labels on each branch indicatethe mode slopes a of the involved deformation modes (seeequation (15)).


consider the doublet opening and can replace it conceptuallywith a single peak at the preferred wavelength of theinstability kj

L (equation (20)).

4.3. Effect of Substratum Viscosity

[42] The buckling instability takes its origin in the viscos-ity contrast across the interfaces of the model [Johnson andFletcher, 1994]. Therefore, the growth rate of the bucklinginstability increases with the viscosity contrast. Bucklingrequires that the layer is stronger than the substratum.[43] The localization instability, on the other hand, is

linked to a resonance that is internal to the layer withnegative stress exponent. Therefore, it can grow even ifthe viscosity of the half-space is higher than the viscosity ofthe layer. However, in that case, the spectrum is offset byone half of the characteristic wavelength (Figure 7). Whenthe substrate is more viscous than the layer, the doublets arelocated at

k Lj ¼ jþ 1=2ð ÞKL; j 2 Z: ð21Þ

[44] Physically, the offset is required because the viscoussubstrate cannot follow the localized deformation. At wavenumbers halfway between actual resonances, an incipientshear zone interacts with a negative image of itself. There-fore, these shear zones interact destructively at the interface,which is required by the stronger substratum. When theviscosity of the substrate is small, it provides no resistanceto the localized deformation and the preferred wave numberis exactly at the resonance.[45] The position of the doublet changes continuously

from equation (20) to equation (21) as the substrate vis-cosity increases (Figure 7). In general, we write

kLj ¼ jþ aLð ÞKL; j 2 Z; ð22Þ

with aL an empirical number called the spectrum offset, andKL the wave number scale defined in equation (19). The

spectrum offset is 0 if h2/h1 � 1, aL = 1/4 if h2/h1 � 1, andaL = 1/2 if h2/h1 � 1.[46] For generality, we also define a spectrum offset aB

for the buckling instability

kBj ¼ jþ 1=2� aBð ÞKB=N ; j 2 Z; ð23Þ

with KB/N the wave number scale defined in equation (17).Equations (22) and (23) are written so that aB = aL = 0 formodels where both the layer and the substrate have constantviscosity and the layer is much stronger than the substrate(Figure 2).

4.4. Effect of Model Density

[47] A density contrast at the surface of the modelreduces the growth rate of the buckling instability andincreases its preferred wave number because of therestoring 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 atsmall wave number. If 1/ne ! 0+, the density-inducedreduction of the growth rate at small k can result in themaximum growth rate being at a resonance with j � 1[Neumann and Zuber, 1995]. In addition, there is nogrowth of the buckling mode over half of the wavenumbers, from jKB/N to ( j + 1/2)KB/N at small j. Thisproduces a spectral offset in equation (23) up to aB =�1/4 for j = 1 and large density of the model.[48] If the layer has localizing properties (1/ne < 0), the

model density influences the buckling part of the growthspectrum in the same manner as described above, exceptthat growth rate is enhanced near the divergent doublets ofthe localizing instability. The effect is particularly pro-nounced at small wave number and increases with the

Figure 7. Map of growth rate as a function of substrateviscosity and perturbation wave number. Strength profilesimilar to Figure 2b, except for varying substrate viscosityand n1 = �10. The buckling instability vanishes when thesubstrate is more viscous than the layer, but the localizationinstability persists, although offset by one half of itscharacteristic wave number. Several modes of the bucklinginstability are labeled in black, and modes j = 0, 1, and 2 ofthe localization instability are labeled in white.

Figure 6. Trajectory of lines of equal phase for super-posed deformation modes with a wavelength correspondingto the resonance involved with (a) the buckling instabilityand (b) the localization instability. If ne > 0, jRe(a)j hasonly one value, so that the resonance depicted in (b) is notpossible.


density of the model (Figure 8). However, scaling to Earthconditions indicates that the normalized density rg=h1 _eIIshould be of order 1 to 30 [Zuber et al., 1986], which issmall. For these values, the density has little effect on thelocalization instability, although it does bring a spectraloffset up to aB = �1/4 for the longest preferred wavelengthof the buckling instability ( j = 0).

5. Models With Depth-Dependent Viscosity

[49] The models considered in the previous section areonly crude approximations to the Earth’s structure. Thelayer represents the brittle crust and upper mantle, and thehalf-space represents the hotter ductile rocks leading tothe asthenosphere. The interface between the layer and thehalf-space corresponds to the brittle-ductile transition.[50] Unlike the models in the previous section, the

strength profile of the Earth is continuous across thetransition from brittle to ductile behavior. In idealizedrepresentations of the strength profile of the lithosphere,the brittle-ductile transition occurs at a particular depthwhere the brittle and ductile strengths of rocks are identical[Brace and Kohlstedt, 1980], or is distributed over a depthrange where the brittle and ductile rock strengths arecomparable [Kirby, 1980; Kohlstedt et al., 1995]. In anycase, the strength and therefore the apparent viscosity of thelithosphere varies continuously with depth within eachlayer, due to the pressure dependence of rock strength inthe brittle regime and the temperature dependence of creepin the ductile regime.[51] In the following sections, we change progressively

the simple strength profile of the previous section to a morerealistic strength profile, keeping track of the preferredwavelengths of buckling and localizing instabilities, as wellas their growth rate. Our goal is to derive a simpleprediction of the preferred wavelength of the localization

instability relevant for a layer of rock undergoing a brittle-ductile transition at a specific depth with a strength profilesimilar to the Earth’s. The effect of having the brittle-ductiletransition distributed over a finite depth range or densitycontrasts within the lithosphere is considered in the com-panion paper.[52] The wave number scaling of the instability is still

valid when depth-dependent viscosity is considered. There-fore, we use equations (22) and (23) to describe thepreferred wave numbers of each instability. However, thevalue of the spectral offset parameter is empirically deter-mined for each type of viscosity profile. The viscosity isscaled to 1 at the bottom of the brittle layer. We use ne =�10 as an illustration for localizing behavior. In that case,KL � 10.

5.1. Exponential Decay of Viscosity in the Substrate

[53] Because the stream function for a layer of exponen-tial viscosity profile is proportional to exp[ik (x + az)](equations (2) and (13)), a perturbation with wavelength lpenetrates into a layer to a depth zd l/Im(a). Hence, itsenses a viscosity averaged over zd. Therefore, a bucklinginstability can grow even if the strength profile is continu-ous at the boundary between the layer and the substrate ifthe viscosity of the substrate decreases exponentially withdepth [Fletcher and Hallet, 1983; Zuber and Parmentier,1986]. In fact, most applications of buckling or necking tothe tectonics of terrestrial planets have used a strengthprofile made of a layer of uniform viscosity lying over asubstrate with viscosity profile

h ¼ he exp rzð Þ; ð24Þ

A value of r � 10 is often appropriate [Fletcher and Hallet,1983; Zuber and Parmentier, 1986].[54] There are two differences between the growth spec-

trum of a layer lying over a substrate with exponentiallydecaying viscosity and the previous case of a constant-viscosity half-space, even if the layer is plastic (1/n1 ! 0,buckling instability only, Figure 9). First, the envelope ofthe growth spectrum decreases at high wave number. This isbecause the short wavelength senses only the top of thesubstrate, which has higher viscosity than deeper levels, andtherefore smaller viscosity contrast [Ricard and Froidevaux,1986; Zuber and Parmentier, 1986; Neumann and Zuber,1995]. Second, the instability grows only over one half ofthe range of wave numbers, between j KB/N and ( j + 1/2)KB/N, j 2 Z. Hence, the preferred wave numbers of bucklingbecome

kBj ¼ jþ 1=4ð ÞKB=N ; j 2 Z; ð25Þ

or, using equation (23), the spectrum offset aB is1/4 for thisstrength profile.[55] When the layer is undergoing localization (n1 < 0),

the growth spectrum is described as the superposition of abuckling-like spectrum and a sequence of divergent dou-blets representing the localization instability (Figure 9), asin the model with constant viscosity layers (Figure 2). Atthe smallest wave numbers, the substrate appears veryweak, and the spectral offset aL 0 for j = 0. At largerwave numbers, however, the substrate viscosity is similar

Figure 8. Map of growth rate as a function of modeldensity and perturbation wave number. Strength profilesimilar to Figure 2b, with n1 = �10. The value of thenormalized density rg=h1 _eII is no more than 30 forterrestrial applications. Several modes of the bucklinginstability are labeled to the left, and modes j = 0, 1, and2 of the localization instability are labeled to the right.


Growth rate Q 0.1

100 10 1

010

2030

4050

Wav

enum

ber

k

a)

η=ex

p(10

z)

z=0

z=1

z=-∞

η=1

b)

Figure

9.

(a)Growth

spectrum

foralayer

ofuniform

viscosity

h 1overlyingahalf-spacewithexponentially

decaying

viscosity

h 2=exp(r2z),withr 2

=10,r=0,n2=3.Solidline:

n1=�10;dashed

line:

n1=106.(b)Viscosity

profile.


to that of the layer, so that the spectral offset aL = 1/4 (seebelow equation (22)). In summary, the preferred wave-length of localization follows equation (22) with approx-imately

aL ¼ 0; if j ¼ 0;1=4; if j > 0:

�ð26Þ

Figure 10 shows how the growth spectrum varies as afunction of the decay-depth of the viscosity profile. Notehow the buckling instability vanishes when r < 0 (viscosityof the substrate increasing exponentially with depth),whereas the localization instability is still present. However,the substrate is now stronger than the layer, so that at aL =1/2 at small j.

5.2. Depth-Increasing Strength of the Layer

[56] As the layer corresponds to rocks undergoing brittledeformation, its strength should increase with depth. Wefirst consider models in which the viscosity of the layerincreases exponentially with depth, which is mathematicallymore tractable, and then the more realistic case of aviscosity increasing linearly with depth in the layer. In bothcases, the viscosity is h1 = 1 at the base of the layer. Thesubstrate has a constant viscosity of h2 = 0.1 and a non-Newtonian behavior with n2 = 3.5.2.1. Exponential Viscosity Profile[57] Having an exponential viscosity profile in the

layer reduces its apparent viscosity. Accordingly, thegrowth rate of the buckling instability is reduced com-pared to the constant viscosity case but its preferredwavelength is unchanged (Figure 11). When the materialin the layer is pseudoplastic (1/n1 ! 0+), the envelopeof the growth spectrum does not depend on wavenumber because the depth of penetration into the layerof the perturbation is infinite (Im(a) ! 0): the wholelayer is sampled at all wavelengths. When the layer islocalizing (n1 < 0, Figure 11), the preferred wavelengthof the localization instability is offset by 1/4 of thecharacteristic scale KL. Interestingly, the amplitude of thebuckling mode decreases at the smallest wave numbers if1/n1 < 0 and the layer viscosity increases with depth(Figure 11).[58] A complication arises because the mode slope a

depends on the decay parameter of the viscosity profile(equation (14)). This changes the resonant wave numbers(Figure 12) and therefore the wave number scale of insta-bilities KL and KB/N. Although the localization instabilitystill follows the resonance (Figure 13), there is no longer ananalytical expression for KL or KB/N. However, equation(19) is approximately valid when r � 0, which is the casefor realistic viscosity profiles. Therefore, the preferredwavelengths of localization are given approximately byequation (22) with

aL ¼ 1=4; j 2 Z: ð27Þ

Note that the first localization doublet ( j = 0) is wider thanfor other strength profiles.[59] The long wavelength limit rk > 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2þ

ffiffiffi5

pp, beyond

which not all the solutions of a are real even if 1/ne < 0,does not prevent the growth of a localization instability.

This is because for 1/ne sufficiently negative, one value ofthe mode slope, a, is pure imaginary: there are still twovalues of Re(a), one being zero, and the resonance depictedin Figure 6b is still defined.5.2.2. Linear Viscosity Profile[60] Although an exponential viscosity profile is only a

poor approximation of the linear increase of strength withdepth expected in the brittle layer from friction laws, thereis little difference between the results of the previoussection and the growth spectra obtained with the linearlaw. The amplitude of the buckling mode is reduced, andthe preferred wavelength of the localization instability isoffset by 1/4 of the wavelength scale KL at small wave-lengths. In addition, the resonance wavelength is close to theanalytical value of equation (19) obtained for a constantviscosity layer. This is because the average strength of thelayer is limited to one half of its maximum value when itincreases linearly with depth. The decay parameter r forexponential viscosity profile that produces the same charac-teristics is small. Indeed, the growth spectrum for a layerwith linear viscosity profile is closest to the case r = �2 withan exponential viscosity profile. For these values the reso-nant wave numbers cannot be differentiated from the limitr = 0.

6. Discussion

6.1. Putting It All Together: Growth Spectrumfor Realistic Strength Profiles

[61] A realistic strength profile for application to tectonicshas a plastic or brittle layer with strength increasing linearlywith depth, followed by a layer of half-space of ductilematerial with viscosity decreasing with depth. We learnedfrom the previous sections that there are two superposedinstabilities for a plastic or localizing layer overlying a half-space: the buckling instability that results in broad undu-lation of the layer as a whole when that layer is stronger

Figure 10. Map of growth rate as a function of decaylength of substrate viscosity and perturbation wave number.Strength profile similar to Figure 9b, except for varying r2and n1 = �10. Negative values of r2 indicate that thesubstrate viscosity increases exponentially with depth.Modes j = 0 to 4 of the localization instability are labeled.



1

100 10 1

010

2030

4050

Wav

enum

ber

k

a)

η=ex

p(-5

z)

z=0

z=1

z=-∞

η=0.

1

b)

Figure

11.

(a)Growth

spectrum

foralayer

withviscosity

increasingexponentially

withdepth

h 1=exp(r1z),overlyinga

half-spacewithconstantviscosity

h 2=0.1,withr 1

=�5,r=0,n2=3.Solidline:

n1=�10;dashed

line:

n1=106.(b)

Viscosity

profile.


than the substratum averaged over a wavelength-dependentpenetration depth, and the localizing instability that produ-ces regularly spaced faults or shear zones. Reintroducing H,the thickness of the brittle layer, as a length scale inequations (17), (19), (22), and (23), these instabilities growpreferentially at the wave numbers

kBj H ¼ jþ 1=2� aBð ÞKB=N ; ð28aÞ

kLj H ¼ jþ aLð ÞKL; ð28bÞ

with j an integer, aB and aL spectral offsets that depend onthe strength profile, and KB/N and KL wave number scalesthat correspond to resonances in the brittle layer and aregiven by

KB=N ¼ p 1� 1=n1ð Þ�1=2; ð29aÞ

KL ¼ p �1=n1ð Þ�1=2: ð29bÞ

[62] Depth-increasing viscosity in the layer, depth-decreasing viscosity in the half-space, and buoyancy forcesall decrease the growth rate of the buckling instability(Figures 9 and 11). Hence, the buckling instability showsonly modest growth rates for the most realistic strengthprofile used in this study (Figure 14). Furthermore, theexponentially decaying viscosity in the substrate sup-presses the instability over half of the wave number range(Figure 9), and a surface density contrast cancels theinstability over the other half of the wave number range(Figure 8). It follows that buckling is not a likely expres-sion of shortening in a layered lithosphere (Figure 15)unless the surface density contrast is reduced. Indeed,natural examples of lithospheric-scale buckling are asso-ciated with deformation under a heavy fluid, whichreduces the surface density contrast. In the Central IndianOcean, this fluid stands for the sediments from the Bengal

fan [Zuber, 1987; Martinod and Molnar, 1995]. Manyother regions in which buckling has been documented areunder sedimentary basins [Burov et al., 1993; Cloetingh etal., 1999]. On Venus, the ridge belts grew in the sametime that basaltic floodplains were emplaced [Zuber andParmentier, 1990; Stewart and Head, 2000]. If bucklingdoes grow, the relevant spectral offset is

0 < aB=N < 1=4: ð30Þ

Alternatively, it can be argued that including morerealistic behavior would help buckling even in presentof a relatively high surface density contrast. Schmalholz etal. [2002] show that viscoelasticity changes how thesurface density contrast influences folding. A dynamicsurface redistribution condition [Biot, 1961; Beaumont etal., 1990] would model erosion more accurately than areduced density contrast and would certainly affect oursolution.[63] Neither depth-dependent viscosity nor surface den-

sity contrasts reduce the growth rate of the localizationinstability. Depth-dependent viscosity in the layer and in thesubstrate each offsets the preferred wavelength by about 1/4KB/N. The density of the model also increases the spectraloffset (Figure 15). All things considered, the spectral offsetfor a realistic viscosity profile is

1=4 < aL < 1=2: ð31Þ

[64] A map of growth rate similar to Figure 5 but for arealistic strength profile is presented in Figure 16. It showshow the wave number of the localization instability varieswith the effective stress exponent of the layer. The local-ization instability is seen to follow the resonant wavenumbers, KL. The buckling mode of deformation all butvanishes when depth-dependent viscosity is taken into

Figure 12. Resonant wave numbers for a layer ofthickness H and effective stress exponent n1 = �10 as afunction of the decays depth r of viscosity in the layer.Thick line: fundamental mode j = 1; other lines: higher-order resonances. The j = 1 branches are labeled with themode slopes a of the appropriate deformation modes.Solution derived numerically from equations (14) and (16).

Figure 13. Map of growth rate as a function of decaylength of layer viscosity and perturbation wave number.Strength profile similar to Figure 11b, except for varying r1and n1 = �10. Positive values of r indicate that the layerviscosity decreases exponentially with depth. Modes j = 0to 4 of the localization instability are labeled.



1

100 10 1

010

2030

4050

Wav

enum

ber

k

a)

z=0

z=1

z=-∞

η=ex

p(10

z)b)η=1

+0.1

z

Figure

14.

(a)Growth

spectrum

foralayer

inwhichtheviscosity

increasinglinearlywithdepth

overlyingahalf-spacein

whichtheviscosity

decaysexponentially

withdepth,withr=0,n2=3.Solidline:

n1=�10;dashed

line:

n1=106.(b)

Viscosity

profile.


account. It is visible only near the divergent doublets of thelocalization instability.

6.2. A Note About Extension

[65] Although the previous sections assumed horizontalshortening of the model, the formalism is equally valid forhorizontal extension, for which �_exx > 0. However, thekinematic contribution of the primary flow to the growthof interface perturbations (equation (3)) has the tendency toerase the imposed perturbation under horizontal extension[Smith, 1975]. Hence, only wavelengths with Q > 1 can beobserved.[66] We present in Figure 17 the growth spectra for a

pseudoplastic or brittle layer of uniform viscosity over aweaker half-space under horizontal extension. The corre-sponding deformation fields are plotted in Figure 18. Thespectra are rather similar to the shortening case (Figure 2).In particular, the wave numbers of the growth rate maximafor a pseudoplastic layer (ne = 106) and of the divergentdoublets for the localizing layer (ne = �10) are similar to theshortening case. The major difference between horizontalextension and shortening is the shape of the most unstabledeformation mode over the whole model: the pseudoplasticlayer is necking under extension rather than buckling(Figure 18a). The localization instability gives rise toregularly spaced localized shear zones (Figure 18b).[67] In presence of depth-dependent viscosity and density,

the approach of a spectral offset (equations (23) and (22)) isstill valid. However, the spectral offset for the neckinginstability is �1/4 < aB < 0 if the viscosity of the layerincreases linearly with depth and the viscosity of the half-space decreases exponentially with depth. Hence, the wave-length of necking is generally smaller than the wavelengthof buckling. As was observed in the shortening case, depth-dependent viscosity and model density conspire to reduce

the range of wavelengths where growth of the neckinginstability is possible under horizontal extension, diminish-ing the likelihood that necking be observed in the tectonicrecord, unless the surface density contrast is small. Neckinghas been observed in nature, most prominently in the Basin-and-Range province [Fletcher and Hallet, 1983; Zuber etal., 1986; Ricard and Froidevaux, 1986] and plays animportant role in rifting [Zuber and Parmentier, 1986; Linand Parmentier, 1990]. Grooved terrain on Jupiter’s satel-lite Ganymede may also be formed by necking [Dombardand McKinnon, 2001]. The spectral offset of the local-ization instability in extension is the same as in compres-sion, 1/4 < aL < 1/2.

6.3. Comparison With Numerical Studies

[68] In early studies of fault patterns, faults were eithera posteriori markers of deformation or a priori boundaryconditions. In neither case is faulting a dynamic feature ofthe models or can the self-consistent pattern of faulting bedetermined. The instability of the localization process,which is expressed in our study by the fact that theeffective stress exponent is negative, presents many ana-lytical and numerical challenges. However, recent numer-ical methods have been able to present a continuumapproach to localization, from microscopic scale [Hobbsand Ord, 1989; Poliakov et al., 1994] to global scale[Bercovici, 1995; Tackley, 2000]. With numerical models,it is possible to go beyond the instantaneous patterns offaulting explored in this paper to study how faultingevolves over time [Sornette and Vanneste, 1996; McKin-non and Garrido de la Barra, 1998; Buck et al., 1999;Cowie et al., 2000; Huismans and Beaumont, 2002](Hardacre, K. M., and P. A. Cowie, Controls on strainlocalisation in a 2D elasto-plastic layer: Insight into size-

Figure 16. Map of growth rate as a function of effectivestress exponent of the layer and perturbation wave number.Lighter tone indicates high growth rate, with the contoursindicated on the color bar. Viscosity profile identical toFigure 14b. Note how the buckling instability is reducedcompared to Figure 5. Modes j = 1, 4, and 7 of the bucklinginstability and modes j = 0, 2, and 5 of the localizationinstability are labeled.

Figure 15. Map of growth rate as a function of the modeldensity rg=h1 _eII and wave number. Viscosity profile similarto Figure 14b. Lighter tone indicates high growth rate, withthe contours indicated the color bar. Several modes of thebuckling instability are labeled in white, and modes j = 0 tothe localization instability are labeled in black. As the modeldensity increases, the buckling mode vanishes.


Growth rate Q1000 1

100 10

010

2030

4050

Wav

enum

ber

k

b) η=0.

1

η=1

z=0

z=1

z=-∞

a)

Figure

17.

(a)Growth

spectrum

foralayer

ofuniform

viscosity

h 1overlyingahalf-spaceofuniform

viscosity

h 2=h 1/10,

withr=0,n2=3undergoinghorizontalextension.Solidline:

n1=�10;dashed

line:

n1=106;dotted

line;

n1=100.(b)

Viscosity

profile.


frequency scaling of extensional fault populations, sub-mitted to Journal of Geophysical Research, 2002, herein-after referred to as Hardacre and Cowie, submittedmanuscript, 2002). Our analysis provides a physicalinsight into the origin of the macroscale fault patterns.[69] Using the numerical method FLAC [Cundall, 1989],

Buck and Poliakov [1998], Gerbault et al. [1998, 1999],Cloetingh et al. [1999], and Lavier et al. [2000] exploredthe patterns of faulting in elastic-viscoplastic models fordifferent tectonic environments. Even in the absence ofexplicit weakening, an elastic-plastic rheology is character-ized by a negative stress exponent, or dynamic strain-weakening, because the strain and stress increments uponfailure are not collinear [Montesi and Zuber, 2002]. Local-ization by strain-weakening may behave differently fromthe strain rate-weakening used in our paper. However, theeffective exponent provides a unifying measure of local-ization, and it is relevant to compare the numerical results inpresence of strain-weakening to our model, provided thatwe use �0.1 < 1/ne < �0.01, as appropriate for localizationin elastic-plastic materials [Montesi and Zuber, 2002]. Thefault spacing predicted by our analysis (0.4 < l/H < 2.5) isconsistent with the spacing observed in numerical models.The localization instability (adapted for strain-weakening) isa likely origin of the fault pattern observed in numericalmodels.[70] Explicit strain-softening was shown to enhance fault-

ing in elastic-viscoplastic models and to increase the faultspacing [Gerbault, 1999]. This is again consistent with thelocalization instability, which predicts larger fault spacingsfor more efficient localization. However, if the weakening istoo strong, another transition occurs and a single faultdevelops in numerical models [Lavier et al., 2000]. Theresulting deformation pattern is sometimes asymmetric[Lavier et al., 1999; Huismans and Beaumont, 2002].Frederiksen and Braun [2001] also observed localizationon a single fault and its conjugate in their models thatinclude strain-softening of the viscous, rather than theplastic rheology. They also showed that the fault intensitydepends on the rate of weakening, consistent with local-ization being controlled by the effective stress exponent

rather than only the amount of weakening. Localization to asingle fault is not predicted by our model. It may be due tosecond-order or finite strain effects that we do not addressyet. Sornette and Vanneste [1996] and Cowie et al. [2000]also report on localization of strain over a single fault uponfinite deformation. Rather than following an elastic-plasticrheology, their models are elastic, with fault slip accumulat-ing when a yield criterion is verified. The initial pattern offaulting is dominated by the strong prescribed heterogeneityin these models (Hardacre and Cowie, submitted manu-script, 2002), which prevents a characteristic length-scalefrom developing. As slip on a fault enhances the stress inthe vicinity of the fault tip, the tendency to failure of faultsin that region is enhanced. After finite slip, the fault patternmay localize because of the interaction between neighbor-ing faults [Sornette and Vanneste, 1996; Cowie et al., 2000].The interaction between several active faults may bedescribed with a negative effective stress exponent. Futureimprovement of our model, in particular including a higher-order or time-dependent analysis may address this laterinstability of fault pattern.[71] The numerical studies closest to our study consider

strain rate softening in viscoplastic models [Neumann andZuber, 1995; Montesi and Zuber, 1997, 1999]. They pro-duce regularly spaced faults superposed on either bucklingor necking. Numerical results suggest that faults are mostlyactive in the anticlines of lithospheric-scale folds [Montesiand Zuber, 1997, 1999] or the necks of lithospheric scaleboudins [Neumann and Zuber, 1995]. This interactionbetween the buckling/necking and faulting deformationfields is not predicted in our analysis, for which faults arepresent everywhere (Figures 3 and 18), but may result froman higher-order interaction between the buckling/neckingand localization instabilities. Numerical results indicateseveral faults in the growing anticlines or necks. Theirspacing is consistent with the prediction of our model,showing again a control of the fault pattern by the local-ization instability. As deformation proceeds, some faultscease their activity, and others replace them [Montesi andZuber, 1997, 1999]. Switches in fault patterns are discrete intime. They reuse recently active faults, with new faultsformed in the front of the existing deformation zone,separated from it by the same spacing as within thedeformation zone. Thus, the localization-instability-con-trolled fault spacing is prominent at finite strain.[72] In summary, fault sets produced by numerical models

may show a preferred spacing that is consistent with theprediction of the localization instability. However, a highlevel of initial heterogeneity can prevent a regular spacingto develop [Sornette and Vanneste, 1996]. With finitedisplacement, the fault pattern may collapse on a singlefault [Sornette and Vanneste, 1996; Cowie et al., 2000;Lavier et al., 2000; Frederiksen and Braun, 2001; Huis-mans and Beaumont, 2002] through a process that wecannot address here. All the models undergoing horizontalshortening show regularly spaced fault sets even at finitestrain [Gerbault et al., 1999; Montesi and Zuber, 1999].Accordingly, compressive orogens often display a propagat-ing deformation front with regularly spaced faults [e.g.,Hoffman et al., 1988; Meyer et al., 1998]. However, theimportance of subhorizontal decollements for this behaviorremains to be evaluated.

Figure 18. Deformation fields corresponding to thegrowth spectra in Figure 17, shaded as a function of strainrate. (a) n1 = 106; (b) n1 = �10. Models undergoingextension. Construction otherwise similar to Figure 3.


[73] Other numerical studies focused on the localizationof shear within a fault gouge [Morgan and Boettcher, 1999;Place and Mora, 2000; Wang et al., 2000] or the spatio-temporal localization of slip over a seismogenic fault [BenZion and Rice, 1995; Miller and Olgaard, 1997; Lapusta etal., 2000; Madariaga and Olsen, 2000]. As our study doesnot resolve the temporal evolution of localization andassumes a different geometry than that relevant for seismo-genic and fault gouge processes, we cannot compare ourwork and these studies. Similarly, our analysis cannot beapplied directly to regularly spaced fault sets in strike-slipenvironments [Bourne et al., 1998; Roy and Royden, 2000].However, the use of a negative stress exponent to buildsimple models of localization can be adapted to theseproblems. We hope that future developments of our modelwill address these different geometries as well as theinteraction between the localization and buckling/neckinginstabilities.

7. Conclusions

[74] We have presented new solutions of the perturba-tion analysis of mechanically layered models of the litho-sphere undergoing shortening in which a brittle layer liesover a ductile substrate. In addition to the classicallyrecognized buckling instability, the layer may undergo alocalization instability that results in regularly spacedfaults or shear zones. Localization is possible when theeffective stress exponent of the brittle layer, a generalmeasure of the mechanical response of the material tolocal perturbations, is negative [Montesi and Zuber, 2002].However, localization of deformation produces incipientshear zones in which the deformation field tries to developa discontinuity that is not compatible with coupling with aductile substrate. Therefore, shear zones cannot developunless there is a resonance between several incipient shearzones. This resonance is the basis for a scaling wavenumber, KL, that controls the wavelengths of instability.The resonance that is at the origin of KL exists only if theeffective stress exponent is negative (equation (19)). Theactual wavelength of the instability is linked to KL byequation (22). Efficient localization, corresponding to morenegative 1/ne, leads to longer instability wavelength as alocalized shear zone can accommodate the strain from awider area. In equation (22), the ‘‘spectral offset,’’ aL, is aparameter that indicates how the strength profile of themodel influence the instability wavelength. For a realisticprofile where the strength of the brittle layer increaseslinearly with depth and the strength of the substratedecreases with depth, 1/4 < aL < 1/2. Similar principlesare used to describe the buckling instability except that thescaling wave number, KB/N is rooted in a different reso-nance that does not require a negative effective stressexponent (equation (17)), and that the spectral offset aB(equation (23)) is between 0 and 1/4 in compression andbetween �1/4 and 0 in extension. Model density has onlya minor effect on the localization instability. Buckling ismuch reduced when depth-dependent viscosity is includedand is a likely expression of tectonic deformation only ifthe surface density contrast is reduced, for instancebecause of a high erosion or sedimentation rate.

Appendix A: Derivation of Equation (8)

[75] The conditions of Newtonian equilibrium for thesecondary flow are written

@~sxx@x

þ @~sxz@z

¼ 0; ðA1Þ

@~sxz@x

þ @~szz@z

¼ 0: ðA2Þ

The body forces do not appear in equation (A1) becausethey are balanced by the primary flow. Inserting theexpressions for the stresses (equations (6a), (6b), and (6c))and rearranging, we obtain

@~p

@x¼ 1

2

dhdz

@2j@x2

� 1

ne� 1

2

� �h

@3j@x2@z

� 1

2

dhdz

@j2

@z2� 1

2h@3j@z3

; ðA3Þ

@~p

@z¼ 1

2h@3j@x3

þ 1

ne

dhdz

@2j@x@z

þ 1

ne� 1

2

� �h

@3j@x@z2

: ðA4Þ

The derivatives of equation (A3) with respect to z and ofequation (A4) with respect to x are combined to give

0 ¼ 1

2h@4j@x4

� 1

2

d2hdz2

@2j@x2

þ 2

ne� 1

� �dhdz

@3j@x2@z

þ 2

ne� 1

� �h

@4j@x2@z2

þ 1

2

d2hdz2

@2j@z2

þ dhdz

@3j@z3

þ 1

2h@4j@z4

: ðA5Þ

Using equation (2), the dependence of the stream functionon x can be factored out, leaving equation (8).

[76] Acknowledgments. We thank Oded Aharonson, Mark Behn,Roger Buck, Ray Fletcher, Brad Hager, Greg Hirth, Chris Marone, GregNeumann, Marc Parmentier, Jack Wisdom, and an anonymous reviewer fortheir comments on this paper and previous versions of it. Supported byNASA grant NAG5-4555.

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��L. G. J. Montesi, Department of Geology and Geophysics, Woods Hole

Oceanographic Institution, MS 22, Woods Hole, MA 02543, USA.([email protected].)M. T. Zuber, Department of Earth, Atmospheric, and Planetary Sciences,

Massachusetts Institute of Technology, Cambridge, MA 02139, USA.([email protected])


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