Spacing of Faults at the Scale of the Lithosphere and Localization Instability: 2. Application to the Central Indian Basin Laurent G

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 108, NO. B2, 2111, doi:10.1029/2002JB001924, 2003

Spacing of faults at the scale of the lithosphere and localization instability: 2. Application to the Central Indian Basin Laurent G. J. Monte´si1 and Maria T. Zuber2 Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA Received 12 April 2002; revised 21 October 2002; accepted 10 December 2002; published 20 February 2003.

[1] Tectonic deformation in the Central Indian Basin (CIB) is organized at two spatial scales: long-wavelength (200 km) undulations of the basement and regularly spaced faults. The fault spacing of order 7–11 km is too short to be explained by lithospheric buckling. We show that the localization instability derived by Monte´si and Zuber [2003] provides an explanation for the fault spacing in the CIB. Localization describes how deformation focuses on narrow zones analogous to faults. The localization instability predicts that localized shear zones form a regular pattern with a characteristic spacing as they develop. The theoretical fault spacing is proportional to the depth to which localization occurs. It also depends on the strength profile and on the effective stress exponent, ne, which is a measure of localization efficiency in the brittle crust and upper mantle. The fault spacing in the CIB can be matched by ne 300 if the faults reach the depth of the brittle– ductile transition (BDT) around 40 km or ne 100 if the faults do not penetrate below 10 km. These values of ne are compatible with laboratory data on frictional velocity weakening. Many faults in the CIB were formed during seafloor spreading. The preexisting faults near target locations separated by the wavelength of the localization instability were preferentially reactivated during the current episode of compressive tectonics. The long- wavelength undulations may result from the interaction between buckling and localization. INDEX TERMS: 8010 Structural Geology: Fractures and faults; 8020 Structural Geology: Mechanics; 8159 Tectonophysics: Evolution of the Earth: Rheology—crust and lithosphere; KEYWORDS: faults, Central Indian Basin, fault spacing, folds, strength envelope, diffuse plate boundary

Citation: Monte´si, L. G. J., and M. T. Zuber, Spacing of faults at the scale of the lithosphere and localization instability: 2. Application to the Central Indian Basin, J. Geophys. Res., 108(B2), 2111, doi:10.1029/2002JB001924, 2003.

1. Introduction: Tectonics of the Central Indian Capricorn plate rotates anticlockwise with respect to the Basin (CIB) Indian plate about a pole of rotation located near the Chagos Bank [Royer and Gordon, 1997], compressing the CIB in a [2] The CIB is the region SE of India delimited by the roughly N-S direction. The deformation area was originally Ninetyeast ridge to the east, the Chagos-Laccadive ridge to identified from a relatively high earthquake activity [Guten- the west, and the SE and central Indian ridges to the berg and Richter, 1954; Sykes, 1970; Stein and Okal, 1978; south. To the north, the basin is covered by the Bengal fan Bergman and Solomon, 1985] and is also recognized in the that accommodates sediments from the High Himalayas gravity field as a region of E-W trending linear anomalies (Figure 1). [Stein et al., 1989]. [3] The best-studied oceanic diffuse plate boundary is [4] Previous studies showed that shortening in the CIB is locatedintheCIB[Wiens et al.,1985;Gordon et al., expressed at two length scales: 1990]. This intraplate deformation area is part of the Indo- 1. The basement is folded at 200 km wavelength. The Australian plate. Following the nomenclature of Gordon undulations, of amplitude up to 2 km, are visible both from [2000], the composite Indo-Australian plate is made of three long seismic reflection profiles [Weissel et al., 1980] and as component plates: India, Capricorn, and Australia. The E-W lineations of the gravity field [Stein et al., 1989]. 2. Reverse faults (Figure 2) cut through the sediment 1Now at Department of Geology and Geophysics, Woods Hole cover and the crystalline basement [Eittreim and Ewing, Oceanographic Institution, Woods Hole, Massachusetts, USA. 1972; Weissel et al., 1980; Bull and Scrutton, 1990, 1992; 2 Also at Laboratory for Terrestrial Physics, NASA Goddard Space Chamot-Rooke et al., 1993], delimiting 5–20 km wide Flight Center, Greenbelt, Maryland, USA. crustal blocks [Neprochnov et al., 1988; Krishna et al., Copyright 2003 by the American Geophysical Union. 2001]. The average fault spacing is 7–11 km [Bull, 1990; 0148-0227/03/2002JB001924$09.00 Van Orman et al., 1995].

ETG 15 - 1 ETG 15 - 2 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

Previous studies have focused on the geometry of these faults in order to constrain the magnitude of the N-S shortening [Eittreim and Ewing, 1972; Bull and Scrutton, 1992; Chamot-Rooke et al., 1993; Van Orman et al., 1995]. Faults may be divided into two separate populations, one north verging and the other south verging [Bull, 1990; Chamot-Rooke et al., 1993] and groups of faults may have been active at different times [Krishna et al., 1998, 2001]. Numerical models have shown that faults may be important in allowing buckling to develop, in particular by lowering the stress required for buckling [Wallace and Melosh, 1994; Beekman et al., 1996; Gerbault, 2000]. However, what controls the fault spacing has not been thoroughly addressed. [7] While the basement undulations can be explained by buckling of the lithosphere, the fault pattern cannot: fault spacing is much less than the spacing of basement undulations, and faulting, being a localized rather than distributed deformation mode, is not accounted for in the usual buckling analysis. However, Monte´si and Zuber [2003] modified the buckling analysis to account for the fact that brittle rocks have a tendency to localize deformation. In that case, two superposed instabilities grow simultaneously in lithosphere models undergoing horizontal shortening. One is the buckling instability [Fletcher, 1974; Zuber, 1987], resulting in broad undulations of the lithosphere as a whole. The other, that we call the localization instability, results in a network of localized shear zones that we interpret as Figure 1. Satellite-derived bathymetry of the CIB region faults. The developing fault network has a characteristic from ETOPO5 [Smith and Sandwell, 1997]. ANS, Afanazy- fault spacing, which scales with the thickness of the brittle Nikishin seamounts. Shaded region, intraplate deformation layer and is a function of the efficiency of localization. area [Gordon et al., 1990]. [8] Because fault spacing is seldom discussed, we first review the evidence for regularly spaced faults in the CIB, paying special attention to the occurrence of fault reactiva- [5] The long-wavelength deformation may correspond to tion. Then, we apply the localization instability analysis buckling of a thick plastic plate [Zuber, 1987]. Buckling derived by Monte´si and Zuber [2003] to the CIB, compar- requires that the density contrast between the lithosphere ing the theoretical wavelength of instability with the fault and the overlying fluid be small [Zuber, 1987; Martinod spacing. We explore different assumptions about the and Molnar, 1995]. With a high density contrast, the growth strength profile of the lithosphere at the brittle–ductile rate of buckling becomes vanishingly small. Hence, the transition (BDT), focusing on the manner that they influ- Bengal fan may be instrumental in allowing buckling as the ence fault spacing, and the efficiency of localization that mobile fan sediments provide a heavier ‘‘fluid’’ than ocean matches the observation. Finally, we discuss the localization water. Zuber [1987] also documented an increase in the mechanism implied for the CIB, the depth of localization, wavelength of basement undulations toward the north, and how faulting and buckling may be related. Our theory which is consistent with increased sediment supply but also provides a unified model for creating basement undulations with the northward increase of lithospheric age. The Whar- and regularly spaced faults in the CIB. ton basin, immediately to the east of the Ninetyeast ridge (Figure 1), displays similar gravity lineations that may represent basement undulations beyond the reaches of the 2. Faults in the CIB Nicobar fan [Cloetingh and Wortel, 1986; Tinnon et al., 2.1. Geometry and Activity 1995]. The Wharton basin is seismically active [Robinson et [9] Although some faults in the CIB are visible in the al., 2001; Abercrombie et al., 2003], and fracture zones in it basement, their clearest expression is often a tight fold in have been recently reactivated [Deplus et al., 1998]. In the the sedimentary sequence. The faults are subvertical in the absence of fan sediments, it is not clear how buckling sediments, but dip around 40 in the basement [Bull, 1990; developed in the southern parts of the Wharton basin. Chamot-Rooke et al., 1993]. They are sometimes imaged to Folding of a thin elastic plate [McAdoo and Sandwell, depths of 6 km, below which the available data lose 1985] is an alternative origin for the basement undulations. resolution. It is thought that the faults penetrate into the However, it requires that the flexural rigidity of the litho- upper mantle [Bull and Scrutton, 1992; Chamot-Rooke et sphere be reduced, possibly by yielding or faulting [McA- al., 1993]. doo and Sandwell, 1985; Wallace and Melosh, 1994]. [10] The faults originated in the basement and propagated [6] The origin of the pattern of reverse faults in the CIB into the sediment cover [Bull and Scrutton, 1992]. Geo- has received less attention than the lithospheric folds. physical evidence shows that folding in the CIB was MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2 ETG 15 - 3

Figure 2. Multichannel seismic reflection profile from the CIB [Bull and Scrutton, 1992]. Finely spaced faults with a spacing of 7 km are superposed on a long-wavelength basement undulation. episodic, with individual pulses of activity marked by [13] The first measure (1a) is the usual fault spacing unconformities in the sediments [Krishna et al., 1998]. measure. It indicates the width of the block delimited by Faults terminate at the uppermost unconformity in a given consecutive faults. The second measure (1b) represents the area. The earliest unconformity was dated as late Miocene distance between the midpoints of two consecutive blocks. (7.5 Ma) [Cochran et al., 1989] and the latest as late If the deformation accommodated on a fault i was distrib- Pleistocene (0.8 Ma) [Krishna et al., 1998]. Krishna et al. uted over a region surrounding it, the width of that region [2001] find that the center of activity shifts over time. would be roughly 2i. The spacing distribution is well [11] No earthquake has been associated unequivocally described as lognormal, which might be expected, as spac- with an observed fault. It is possible that the faults are no ing is constrained to be positive. As Bull and Scrutton longer active or that the rate of seismicity too low for the [1992] already recognized, the cumulative frequency dia- existing record. However, the absence of a small aperture gram is not fractal. The second measure, which corresponds seismic network in the region prohibits relocation of events to a moving average of the first, gives a tighter distribution to the depth range where the faults are observed. The late of spacing (Figure 3 and Table 1). Pleistocene unconformity spans the region that has been the [14] A possible physical origin for the significant spread most seismically active, indicating a relation between seis- in the distribution of spacing is as follows. Starting from a mogenic faults and surface faulting [Krishna et al., 2001]. set of regularly spaced faults, the location of each fault is In addition, many earthquakes can be correlated with sur- perturbed randomly. Spacing is measured on the perturbed face features, in particular ancient fracture zones [Stein and Okal, 1978; Bergman and Solomon, 1985]. Therefore, it is possible that the rupture planes of earthquakes down to 40 km are somehow related to the shallower faults, although no direct link between the shallow and deep faults can be proven at this point. 2.2. Fault Spacing [12] Van Orman et al. [1995] have compiled the location of faults along several seismic reflection profiles running roughly N-S across the intraplate deformation area. We use two different schemes to compute fault spacing, , from their data sets:

1i ¼ xiþ1 xi; ð1aÞ

x x ¼ iþ1 i1 ; ð1bÞ 2i 2 where xi is the distance from the fault i to a reference point, projected onto the N-S direction. The actual shortening direction is less than 10 from the projection direction [Cloetingh and Wortel, 1986; Tinnon et al., 1995; Coblentz Figure 3. Histograms of fault spacing determined using et al., 1998], which introduces a systematic error in our the measures (a) 1 and (b) 2 (1) from fault locations spacing estimate of less than a few percent. Histograms of reported by Van Orman et al. [1995] on two seismic lines in fault spacing are presented in Figure 3 and spacing statistics the CIB. Black 78.8 line, gray 81 line. Insets represent are gathered in Table 1. schematically the measure of spacing. ETG 15 - 4 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

Table 1. Statistics of Fault Spacing (km)a Position (Number of Faults) Measure Mean Median Standard Deviation Interquartile Scale Coefficient of Skew

78.8E (127) 1 6.54 4.56 6.32 5.57 2.57 2 6.51 4.90 4.87 4.40 2.12 81E (49) 1 8.38 6.90 6.01 7.45 1.38 2 8.40 8.24 3.87 4.84 1.10 aFault positions (J. Van Orman, personal communication, 2000). fault location. We conducted Monte Carlo simulations of fault spacing because a significant fraction of faults are not this process and found that the skewness and median of the reactivated features [Bull, 1990; Chamot-Rooke et al., simulated distributions are compatible with the data of Van 1993]. Therefore, preexisting abyssal hill-bounding faults Orman et al. [1995] when the amplitude of the perturbation were only partially reactivated. Accordingly, several base- of fault location is at least half the original fault spacing. As ment faults have been identified in the CIB that do not affect the mean is conserved by this process, we favor using mean the sedimentary cover. They are probably nonreactivated values in each data set as the ‘‘characteristic fault spacing.’’ normal faults [Bull and Scrutton, 1992]. Alternative explanations for the distribution of fault spacing can be devised and should be explored in future study. 3. Origin of ‘‘Regularly Spaced’’ Fault Sets [15] Our mean spacing values (6.5 and 8.4 km) are consistent with the results of Van Orman et al. [1995], [19] As the 7–11 km spacing of reverse faults in the CIB who found a 7 km fault spacing using the same data set, is probably not inherited from the pattern of preexisting and of Bull [1990], who determined a 6.6 km fault spacing normal faults, we follow the hypothesis that it developed in that region. However, data from seismic lines more to the over the last 7.5 Ma, in the current compressive environ- east of the CIB give longer average fault spacing, between ment. As the geophysical evidence suggests that many 9.0 and 11.6 km (Krishna, personal communication, 2001). reverse faults are reactivated preexisting faults, we suppose An eastward increase of fault spacing may be related to that reactivation occurred preferentially in the vicinity of higher strain in the eastern CIB [Van Orman et al., 1995; target locations. These target locations would have a char- Royer and Gordon, 1997]. acteristic spacing between 7 and 11 km, controlled by one [16] In summary, we take the fault spacing in the CIB to of the two instabilities that grow in mechanically layered be 7–11 km, depending on the location. Variations in fault lithospheres: the buckling instability and the localization spacing cannot yet be related to differences of tectonic instability [Monte´si and Zuber, 2003]. These deformation history. At any given location, fault spacing is rather modes can be discriminated upon using the wavelength of broadly distributed around the mean value. deformation that they predict. First, we present how each instability develops and then show that only the localization 2.3. Reactivation instability is consistent with the observed fault spacing. [17] Many of the observed reverse faults are probably reactivated normal faults that formed during seafloor 3.1. Buckling and Localization Instabilities spreading 65–90 Myr ago [Bull and Scrutton, 1990; Cha- [20] A lithosphere undergoing horizontal shortening can mot-Rooke et al., 1993]. The strongest argument in favor of produce regularly spaced faults in at least two different reactivation is the strike of the faults, parallel to magnetic ways. lineations and perpendicular to fracture zones [Eittreim and 1. Buckling instability: a strong layer such as the upper Ewing, 1972; Bull, 1990]. lithosphere can buckle, producing broad undulations with [18] As many of the observed faults are reactivated, it is wavelength lB [Biot, 1961; Fletcher, 1974; Zuber, 1987]. conceivable that the observed spacing is inherited from the When the undulations reach sufficient amplitude, associated original distribution of normal faults. However, as we show stress heterogeneities can force faults to achieve a spacing below, the observed fault spacing is not consistent with the of lB if faults develop at the crests of anticlines, or lB/2 if expected configuration of preexisting faults. The reactivated faults also develop at the hinges of the folds [Lambeck, faults probably originated as abyssal hill-bounding faults. 1983; Gerbault et al., 1999]. Studies of modern spreading centers indicate that the width 2. Localization instability: when a material has certain of abyssal hills decreases with increasing spreading rate properties (defined below and by Monte´si and Zuber [Goff et al., 1997]. As the half-spreading rate of the Mid- [2002]), local perturbations are unstable and produce Indian Ridge was about 10 cm yr1 at the time of formation localized shear zones akin to faults. These localized shear of the CIB [Sclater and Fisher, 1974; Patriat and Segoufin, zones organize themselves with a characteristic spacing, lL, 1988], the width of the abyssal hills expected in that region that is controlled by the mechanical layering of the is around 2 km, similar to the modern East Pacific Rise lithosphere and the efficiency of localization [Monte´si and [Goff, 1991; Goff et al., 1997]. Faults themselves could be Zuber, 2003]. more closely spaced, as in the slower-spreading Mid-Atlan- [21] The preferred wavelength of the localization insta- tic Ridge [Goff et al., 1997]. In addition, volcanism might bility and the buckling instability are derived as follows. The mask some faults in morphological observations [Macdon- lithosphere is idealized as a sequence of horizontal layers of ald et al., 1996]. In any case, the preexisting fault spacing given mechanical properties. As long as the interfaces derived from abyssal hills cannot be more than 2 km, a between these layers are flat and horizontal, the model factor of 3–5 less than observed. Moreover, the spacing of deforms by pure shear shortening. However, any topography reactivated structures is probably larger than the 7 km mean on the model interfaces drives a secondary flow, which in MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2 ETG 15 - 5 turn deforms the interfaces. Assuming that the secondary faults [Bull and Scrutton, 1990] and reactivation probably deformation field and the interface perturbations have small leads to localization by frictional velocity weakening, we amplitude, each wavelength of deformation grows self- use 300 < ne < 50 as an a priori range of admissible ne. similarly at a rate, q, which depends on the perturbation [27] The buckling instability develops when a plastic layer wavelength l. The wavelengths that have high growth rates lies over a weaker ductile substrate [Smith, 1979; Fletcher are more likely to be expressed in the tectonic record. and Hallet, 1983]. The localization instability requires ne <0 Maxima of the growth spectra q(l) are identified with the and arises regardless of the strength structure of the litho- preferred wavelengths of lithospheric-scale instabilities. sphere [Monte´si and Zuber, 2003]. It has a preferred wave- [22] The type of instability that develops in a given length only if there is a substrate with ne > 0 beneath a layer mechanical structure and its preferred wavelength depend with ne < 0. A reconstruction of the deformation field of on the effective stress exponent, ne, of the brittle layer. The layers undergoing the buckling and the localization insta- effective stress exponent is a general measure of the non- bilities is shown by Monte´si and Zuber [2003]. linearity of the apparent rheology of a material [Smith, 3.2. Instability Scaling 1977]. Although ne can be defined for any rheology [Monte´si and Zuber, 2002], we assume in this study that [28] The growth spectra of the buckling and localization the strength of a material, s, depends only on the second instabilities are characterized by successive growth rate invariant of strain rate, e_II. Then, the effective stress maxima defining their preferred wavelengths [Monte´si and exponent has the form Zuber, 2003]. For each wavelength, there are four super- imposed deformation modes in each layer of the model. The s de_II ne ¼ : ð2Þ instabilities appear at wavelengths where these deformation e_II ds modes are resonant. Using the theoretical dependence of the resonant wavelengths to the effective stress exponent n [23] The strength of a material increases with the strain e [Monte´si and Zuber, 2003], the preferred wavelengths of the rate when ne > 0. However, the apparent viscosity of a buckling and localization instabilities, lB and lL,are material with ne > 1 decreases at a location where the strain rate is anomalously high. Although this leads to enhanced approximately deformation at that location if the stress is further increased, 2 1=2 deformation is otherwise stable: localized shear zones do lB=H ¼ Â ðÞ1 1=ne ; ð3Þ not appear spontaneously but must be triggered by a j þ 1=2 aB localized forcing. An example of such a material is a rock deforming by dislocation creep, for which ne 3–5. 2 1=2 [24] For localized shear zones to appear spontaneously, lL=H ¼ ÂðÞ1=ne ; ð4Þ the material must be characterized by a weakening behavior j þ aL [Hobbs et al., 1990; Monte´si and Zuber, 2002]. If not only the apparent viscosity but also the strength of the material with H the thickness of the plastic or localizing layer and j decreases when the strain rate is anomalously high, a local an integer that gives the order of the resonance involved in perturbation of strain rate grows unstably to form a local- the instability. The parameters aB and aL are called the ized deformation zone, even if the material was loaded spectral offsets and depend on the strength profile of the uniformly. Such dynamic weakening is characterized by lithosphere. Assuming that the strength increases with depth in the localizing layer, and that it decreases with depth in a ne <0[Monte´si and Zuber, 2002]. [25] In addition to being a criterion for localization, the substrate with ne > 0, as on Earth, we find [Monte´si and effective stress exponent provides a quantitative measure of Zuber, 2003] localization efficiency. Efficient localization produces only a few faults, as individual localized shear zones can accom- 0 < aB < 1=4 ð5aÞ modate the deformation originally distributed over a wider area. Localization is more efficient for more negative 1/ne. 1=4 < a < 1=2: ð5bÞ We refer to the limit 1/ne !1as drastic localization and L 1/ne ! 0 as marginal localization. In the limit 1/ne ! 0, a material is regarded as perfectly plastic. [29] The instabilities are best observed in nature at their [26] The value of the effective stress exponent is loosely longest wavelength, given by (3) or (4) with j = 0. The related to the localization mechanism. Localization by fric- smaller instability wavelengths ( j 1) may also be tional velocity weakening leads to 300 < ne < 50, expressed in the tectonic record, but are difficult to differ- whereas localization by cohesion loss upon failure or non- entiate from noise in the longer-wavelength deformation associated elastic-plastic flow give 50 < ne < 10 [Mon- field. Therefore, they will be ignored in what follows. With te´si and Zuber, 2002]. In this study, we assume that ne is the approximation 1/ne 0 (5a) and j = 0, (3) further constant in the brittle layer; we do not account for a possible reduces to: change of localization mechanism with depth, although only 4 < lB=H < 8: ð6Þ the shallow faults can be reactivated structures, and their deeper extension must have formed during the current The relations between predicted instability wavelength and shortening episode. Hence, the effective stress exponent effective stress equations (3) and (4) are plotted in Figure 4. inferred from our models is the value appropriate for an [30] We use (6) and (4) to estimate how thick a layer equivalent layer in which a single localization process is undergoing the buckling or the localization instability must active. As we know that reactivation occurred in the shallow be if the instability has the 7–11 km wavelength observed ETG 15 - 6 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

Table 3. Model Parameters 12345 Crust no no no no yes S (MPa) 1 300a 300b 11 Tw (C) 111150 1 zloc (km) 44 50 11 10 44 n1 for 7 km 333 700 200 100 333 n1 for 11 km 140 350 100 50 140 a ne < 0 in layer with saturated strength. b 6 ne =10 in layer with saturated strength.

50, the inferred depth of faulting varies from 6 to 50 km (Table 2). The smallest H corresponds to the minimum depth to which faults are observed, 6 km whereas the largest H corresponds to the depth of BDT predicted by the thermal age of the CIB, 65–90 Ma [Sclater and Fisher, 1974; Patriat and Segoufin, 1988; Krishna and Gopala Rao, 2000]. [33] In summary, (3) and (4) indicate that only the localization instability is compatible with the observed spacing Figure 4. Ratio between instability wavelength l and and depth penetration of faults in the CIB. In the next layer thickness H as a function of the effective stress section, we present complete growth spectra for different exponent. Gray field, buckling instability. Black field, lithosphere models that strengthen this statement by avoid- localization instability. ing assumptions regarding the value of the spectral offset aL. 4. Fault Spacing as a Function of the Strength in the CIB. The estimated thicknesses are compiled in Profile of the Lithosphere Table 2, where we use four different relations between l 4.1. Model Setup and H. The first two are the upper and lower bounds of the buckling instability (6). The other two are for the local- [34] To better constrain the range of lithosphere structures and effective stress exponents that can match the fault ization instability, using ne = 300 and ne = 50. We also indicate H appropriate for explaining the wavelength of the spacing in the CIB, we now present growth spectra for five basement undulations. The layer of thickness H implied in models of the lithosphere that follow alternative assump- each instability is interpreted as a strong surface layer for tions on the strength profile and material stability at the the buckling instability and as a layer in which deformation BDT (Table 3). By presenting the full growth spectra, we localizes into faults for the localization instability. They are avoid using the approximate value of the instability wave- not necessarily the same physical layer. length (3) and (4) and assuming a value for the spectral [31] From this simple analysis, we reject the buckling offset (5). instability as controlling the fault spacing in the CIB. A 4.1.1. Strength Profile layer that buckles with a 7–11 km wavelength is at most 3 [35] Each model is composed of several horizontal layers. km thick (Table 2). Only the predeformation sediments have Each layer is given an analytical viscosity profile, an the appropriate thickness in the CIB, but because they are effective stress exponent, and a density (Figure 5). weaker than the basement, they would not develop buck- [36] The viscosity profile follows the lithospheric strength ling. Moreover, faults are observed in the basement, which profile, which depends on the dominant mechanism of would be surprising if the fault pattern was controlled by the deformation at each depth (see Appendices A and B). We sediment cover. As faults penetrate to at least 6 km, l/H is consider that the strength is controlled by the weakest of the less than one, which is not compatible with the preferred resistance of frictional sliding or dislocation creep, with the wavelength of buckling (Figure 4). strength possibly limited by a saturation value, S. A new layer [32] On the other hand, the localization instability can is defined each time that the dominant deformation mecha- explain the spacing of faults in the CIB. For 300 < ne < nism, the density, or the effective stress exponent changes. [37] Rocks deform viscously at high temperature by dislocation creep [Evans and Kohlstedt, 1995]. The strength Table 2. Thickness of Brittle Layer, H, Needed to Match the sd in this regime depends on the rate of deformation of the Instability Wavelength la rock and the temperature. In all our models, we assume a 15.5 1 Buckling (6) Localization (4) strain rate of 10 s given by plate reconstructions 1 n =±1 n =±1 n = 300 n = 50 [Gordon,2000],andalineargeothermof15Kkm e e e e appropriate for a 65–90 Ma old lithosphere as in the CIB aB =0 aB =1/4 aL =1/2 aL =1/4 b [Parsons and Sclater, 1977]. The temperature field saturates 7 1.8 0.9 30 6.1 at 1350C. Because the earliest evidence of faulting and 11b 2.8 1.4 48 9.7 200c 50 25 870 180 folding is only 7.5 Ma old whereas the age range of aAll distances in km. lithosphere in the CIB spans 30 Ma, a more precise estimate bFault spacing. of the geotherm is not justified. With the assumed thermal cWavelength of basement undulations. profile, the crust is brittle throughout, and is therefore MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2 ETG 15 - 7

Figure 5. Schematic of lithosphere models. A sequence of layers undergoes shortening at the rate e_xx = 15.5 1 10 s under an inviscid fluid of density rs. The strength profile follows the weakest of frictional resistance, ductile strength, or a saturation strength. Each layer i is characterized by a density ri, a thickness Hi, an effective stress exponent ni, and a viscosity profile hi(z) that correspond to the dominant deformation mechanism in that depth range (see Appendices A and B). A substratum with rheology corresponding to the lowest level of the model is included for mathematical convenience. The shading differentiates 6 between localizing layers (darkest shade, ni < 0), plastic layers (medium shade, ni =10 ), and ductile layers (lightest shade, ni = 3.5). The same shading is used in the strength profile in the next figures. ignored in models 1–4. As the oceanic upper mantle is regime, models 2 and 3 include an intermediary layer with probably dry because of water extraction during formation strength S = 300 MPa [Kohlstedt et al., 1995]. As it is of the crust at mid-oceanic ridges [Hirth and Kohlstedt, uncertain whether localization occurs in the semibrittle 6 1996], we use the flow law for dry olivine of Karato et al. regime or not, we use ne < 0 in model 2 and ne =10 in [1986]. Neglecting thermal or grain size feedback processes, model 3. In addition, friction may become velocity strength- the effective stress exponent is 3.5 in this regime [Monte´si ening above a critical temperature Tw [Stesky et al., 1974; and Zuber, 2002]. Tse and Rice, 1986; Blanpied et al., 1998]. Hence, in model [38] At lower temperature and pressure, rocks are brittle. 4, the strength in the brittle regime increases with depth Because of the presence of preexisting faults, the brittle until it exceeds the ductile strength (S !1), but we use ne 6 strength of lithosphere is controlled by frictional sliding and =10 if T Tw = 150C. We did not study models in which is a universal function of pressure, sf [Byerlee, 1978]. At saturation of the strength envelope and a transition to ne = 6 low temperature, sf decreases with sliding velocity [Scholz, 10 occur at different depths. We report in Table 3 the 1990] with 300 < ne < 50 [Monte´si and Zuber, 2002]. It maximum depth of localization, zloc, as well as the effective is possible that at depth, the brittle lithosphere behaves more stress exponent needed in the localizing layers to produce a plastically, with 1/ne ! 0. fault spacing of 7 or 11 km in each model. [39] Because the brittle strength of the lithosphere [40] The density contrast between a model and the over- increases with depth and its ductile strength decreases with lying fluid is important for the development of buckling. depth, the brittle and ductile strengths are equal at a Here, we assume that the overlying fluid is sediment with 3 particular depth, the depth of the BDT [Brace and Kohl- density rs = 2300 kg m , which favors buckling. However, stedt, 1980]. Model 1 (Figure 6) follows such a simple only the high-density mantle is considered in models 1–4, strength profile, with only two layers, one brittle (n1 <0), which reduces buckling. The crust, being entirely brittle for the other ductile (n2 = 3.5), separated by the BDT. The the imposed geotherm, has only a minor effect on the strength depth of the BDT is 44 km in that model. However, the envelope. However, it is included as a new layer in model 5 transition between the brittle and the ductile deformation because it reduces the surface density contrast and introduces may be spread over a large depth range because of the a density contrast within the model. Model 5 is otherwise suppression of microcracking at high pressure [Kirby, 1980] similar to model 1, with the BDT occurring at a single depth. and the progressive onset of ductility for the different 4.1.2. Solution Strategy minerals composing mantle rocks [Kohlstedt et al., 1995]. [41] As described by Monte´si and Zuber [2003] and As the rock strength is roughly constant in the transition Monte´si [2002], we derive the growth rate of perturbations ETG 15 - 8 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

Figure 6. (a) Growth spectrum and (b) strength profile for model 1. Solid line n1 = 333, dashed line 6 n1 = 166, dotted line n1 =10. The gray-shaded area in (a) marks the range of fault spacing observed in the CIB. The order of the localization instability, j, is indicated for the case n1 = 333. of the models as a function of the perturbation wavelength. because the different minerals that constitute a rock become First, we solve for the shape of individual deformation modes ductile at different depths [Kirby, 1980; Kohlstedt et al., in each layer using the depth-dependent viscosity profile and 1995]. Therefore, we consider models where the strength is its depth derivatives. The amplitude of each deformation limited at 300 MPa, as advocated by Kohlstedt et al. [1995]. mode is found by matching stress and velocity boundary Then, the strength envelope is saturated at 11 km depth. conditions at each interface. This links the velocity field [45] In model 2 (Figure 8), we retain the effective stress within the model to the amplitude of interface perturbations, exponent of frictional sliding in the region where strength is thereby determining the growth rate of the perturbation. saturated (11–48 km deep). The localization instability requires n1 = 700 for a 7 km fault spacing and n1 = 4.2. Lithosphere Strength Models: Results 350 for a 11 km fault spacing (Table 3). To match the fault 4.2.1. Model 1: Localization Everywhere in the spacing in the CIB, localization in model 2 must be less Brittle Regime efficient than any mechanism considered by Monte´si and [42] Model 1 assumes that there is no limiting strength to Zuber [2002]. This may indicate that the effective stress the failure law (S !1) and that localization occurs when- exponent varies with depth, with the material being more ever the Byerlee criterion for frictional sliding is reached (Tw plastic in the BDT zone. !1). Hence, the BDT is idealized as a point [Brace and [46] Therefore, we present an alternative treatment of the Kohlstedt, 1980], which is at a depth of 44 km for the assumed broad BDT in which the region where strength reaches the 6 geotherm, strain rate, and mantle flow law (Figure 6). saturation value is plastic, with ne =10 (model 3). In [43] In this model, the localization instability occurs at Figure 9, we present the growth spectra for model 3 with n1 7kmifn1 = 333 and 11 km if n1 = 166 (Figure 6 and = 200 and n1 = 100, which produce fault spacings of Table 3), if we consider the longest wavelength of the 7 and 11 km, respectively (Table 3). Hence, limiting local- instability ( j = 0). These values of ne are within the bounds ization to the depth where the strength increases with depth of experimental data (300 < ne < 50) although on the brings n1 back to the range of experimental values. low efficiency of localization side. The tradeoff between the [47] To produce the same fault spacing, l, localization efficiency of localization, expressed by n1, and the preferred must be more efficient (1/n1 more negative) in model 3 than spacing of faults is shown in Figure 7 by a map of the in model 1. This is needed to compensate the smaller growth rate as a function of n1 and the wavelength of thickness of the localizing layer, H1, in model 3. Indeed, perturbation. Higher growth rates are in white and corre- the deformation imposed over a given area can be accom- spond to the localization instability. Additional divergent modated by a smaller number of faults if there is a greater peaks at smaller wavelengths mark harmonics of the local- tendency for localization, resulting in larger l/H(4). ization instability ( j 1) (Figures 6 and 7). If present, they [48] The tradeoff between the limiting rock strength, S, are harder to recognize in nature, because the expression of and the effective stress exponent is shown in Figure 10, the fundamental mode ( j = 0) is noisier than our idealized where we assume that the layer where strength is saturated model predicts. The high growth rate at long wavelengths (l is plastic, as in model 3. If the limiting strength, S,isso 100 km) (Figure 6) may be associated with the buckling small that the brittle layer is less than 20 km thick, the instability. It will be discussed in section 5.3. effective stress exponent needed to produce a 7 km fault 4.2.2. Models 2 and 3: Limiting Strength spacing and S are correlated: decreasing H1 requires more [44] The strength of the lithosphere may be limited at the efficient localization. However, if the brittle layer is approach of the BDT because of cataclastic flow and between 20 and 40 km thick, ne 350, regardless of S. MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2 ETG 15 - 9

Figure 7. Map of growth rate as a function of effective stress exponent of brittle layer, n1, and perturbation wavelength for the strength profile of Figure 6. Higher growth rates correspond to lighter tones. The branches j = 0 and j = 1 of the localization instability traverse this portion of the parameter space. They indicate the predicted fault spacing as a function of n1.

Increasing H1 in that range still requires less efficient 4.2.3. Model 4: Temperature-Limited Localization localization, but it also changes the shape of the strength [49] In model 4, the brittle strength follows Byerlee’s law profile significantly, decreasing the importance of the layer of friction down to the ideal BDT, as in model 1. However, in which the strength does not depend on depth. In (4), localization is limited to temperatures less than Tw in model changing the shape of the strength envelope influences the 4, to model the transition from velocity-weakening to 6 spectral offset aL. This effect roughly compensates the velocity-strengthening friction. For T > Tw, ne =10. Early change of H1 in the 20–40 km depth range. Hence, the experiments by Stesky et al. [1974] indicated that Tw could effective stress exponent needed to match a 7 km fault be as low as 150C for olivine-rich rocks. However, these spacing is roughly constant for H1 in that depth range. experiments contradict the distribution of intraplate earth-

Figure 8. (a) Growth spectrum and (b) strength profile for model 2. Solid line n1 = n2 = 700, dashed 6 line n1 = n2 = 350. There is no positive growth rate for n1 =10. The gray-shaded area in (a) marks the range of fault spacing observed in the CIB. The order of the localization instability, j, is indicated for the case n1 = 700. ETG 15 - 10 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

Figure 9. (a) Growth spectrum and (b) strength profile for model 3. Solid line n1 = 200, dashed line 6 n1 = 100. There is no positive growth rate for n1 =10 . The gray-shaded area in (a) marks the range of fault spacing observed in the CIB. The order of the localization instability, j, is indicated for the case n1 = 200. quake focal depths, which show a cutoff temperature of at [50] If we consider that localization is limited to 150C least 600C in the mantle [Chen and Molnar, 1983; Wiens (10 km) but that the brittle strength always increases with and Stein, 1983]. Unfortunately, no other experimental depth, effective stress exponents of 100 to 50 are needed study to our knowledge has explored how the velocity to explain fault spacings of 7 or 11 km, respectively (Figure dependence of friction changes with temperature for oli- 11 and Table 3). These values are within the range of vine-rich rocks. We explore with model 4 how temperature- laboratory data on frictional sliding, although more compat- limited localization influences the localization instability. ible with localization by cohesion loss.

Figure 10. Map of growth rate for a 7 km wavelength as a function of saturation value of the strength profile, S, converted into a maximum depth of localization in the upper axis, and the effective stress exponent of brittle layer, n1. The strength profile is similar to model 3 (Figure 9b) with variable S. High growth rate (lighter tones) indicates the region of the parameter space where the localization instability is at 7 km. The map shows what n1 is needed to explain a 7 km fault spacing as a function of the limiting strength (or depth extent) of localization. Localization is more efficient toward the top of the figure. MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2 ETG 15 - 11

Figure 11. (a) Growth spectrum and (b) strength profile for model 4. Solid line n1 = 100, dashed line 6 n1 = 50, dotted line n1 =10. The gray-shaded area in (a) marks the range of fault spacing observed in the CIB. The order of the localization instability, j, is indicated for the case n1 = 100.

[51] To match the same fault spacing, localization must be exponents, ne, that produce a fault spacing of 7 or 11 km. more efficient in model 4 than in the case where localization Except for model 2 (layer with saturated strength and occurs everywhere in the brittle regime (model 1), because negative ne at the BDT) the values of ne required to match of the smaller scaling depth H1. Localization must also be a 7–11 km fault spacing are compatible with friction more efficient in model 4 than model 3, where the strength velocity weakening. However, the constraints on the local- saturates in the plastic regime, because of the different types ization mechanism provided by the value of ne are suffi- of strength profile and corresponding spectral offset aL(4). ciently broad that no localization mechanism can be [52] We show in Figure 12 how the effective stress ascertained from our analysis. In addition, the mechanism exponent required for a 7 km fault spacing correlates with may change with depth, which would result in the inferred the depth limit of localization. In contrast with Figure 10, ne being an average of the actual depth-dependent ne. the value of ne needed to match the 7 km fault spacing is [55] Velocity weakening localizes deformation in the correlated with the depth of localization, H1, for all H1. This following manner [Monte´si and Zuber, 2002]. Consider that is because the strength envelope is identical whether there is the lithosphere is riddled with preexisting faults. The faults localization or not; there is no depth range where the change are activated when the frictional resistance is overcome (i.e., of H1 changes the shape of the strength envelope and the when the stress exceeds a critical value). However, in the spectral offset. case of velocity weakening, the coefficient of friction 4.2.4. Model 5: Effect of the Crust decreases with sliding velocity. If a fault slides slightly [53] As the crust was ignored in models 1–4, there was faster than its surrounding ones, it becomes weaker, so that only one density contrast, at the surface of the model. In it accommodates more shortening. Its sliding rate increases, model 5, the crust is present. It reduces the surface density which further weakens the fault, and so on. contrast and introduces a second density contrast within the [56] An important aspect of this mechanism is that faults brittle layer. The model is otherwise identical to model 1 are present before the deformation begins. This is indeed (S ! + 1 and Tw ! + 1). The density stratification pertinent to the tectonics of the CIB, as many of the produced by the crust has only a minor effect on the seismically imaged thrust faults are reactivated structures preferred wavelength of the localization instability, as is [Bull and Scrutton, 1990]. Thus, the following history of shown by comparing Figures 13 and 6. However, the crust faulting can be deduced. Faults were generated during may be important for the development of the buckling seafloor spreading, 65–90 Myr ago, dependent on the instability, as will be discussed in section 5.3. location in the current CIB. These faults were probably abyssal hill-bounding normal faults, with a spacing of order 2 km. Starting 7.5 Ma, the faults, by then buried under the 5. Discussion sediment cover, were reactivated in a reverse sense by N-S 5.1. Localization Mechanism and History of Faulting compression. Because of the intensity of localization asso- [54] Using the scaling relations developed by Monte´si ciated with fault reactivation, a 7–11 km fault spacing and Zuber [2003], we have shown that the fault spacing in developed in the region, constrained by the requirement of the CIB is compatible with a localization instability, using preexisting faults. Thus, preexisting faults in the whereas it cannot be explained by a buckling instability. vicinity of target locations separated by 7–11 km were We have used different lithospheric strength profiles to preferentially reactivated. This explains why not all faults estimate more accurately the range of effective stress were reactivated, and also the high noise level in the fault ETG 15 - 12 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

Figure 12. Map of growth rate for a 7 km wavelength as a function of limiting temperature of localization, Tw, converted into a maximum depth of localization in the upper axis, and effective stress exponent of localizing layer, n1. The strength profile is similar to model 4 (Figure 11b) with variable Tw. High growth rate (lighter tones) indicates the region of the parameter space where the localization instability is at 7 km. The map shows what n1 is needed to explain a 7 km fault spacing as a function of the limiting temperature (or depth extent) of localization. Localization is more efficient toward the top of the figure. spacing histograms. Occasionally, the reactivated fault pat- indicate an average effective stress exponent for the CIB tern was complemented with a new fault. The faults lithosphere. propagated upward into the sediment cover and downward because of the deepening of the BDT between the time of 5.2. Depth of Localization fault creation and reactivation. Fault propagation requires [57] Even without considering that the actual temperature the creation of new fault surfaces, which may result in profile or background strain rate of the Central Indian different ne than reactivation. Again, our analysis can only Ocean could be different from our models, there is a wide

Figure 13. (a) Growth spectrum and (b) strength profile for model 5. Solid line n1 = 333, dashed line 6 n1 = 166, dotted line n1 =10. The gray-shaded area in (a) marks the range of fault spacing observed in the CIB. The order of the localization instability, j, is indicated for the case n1 = 333. MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2 ETG 15 - 13 range of localization depths that are consistent with the observed fault spacing. We summarize in Figure 14 how varying the thickness of the localizing layer is compensated by varying the efficiency of localization, expressed by the effective stress exponent ne. The faults may penetrate no deeper than imaged (6–8 km) if ne 50 or reach the predicted depth of BDT around 40 km if ne 300. This makes it impossible to evaluate whether the assumed geotherm of 15 K km1 is realistic. Heat flow data in the CIB indicate higher surface heat flow than expected for a lithosphere of this age, but the most likely explanation is a shallow temperature perturbation, as might be induced by deformation or serpentinization [Weissel et al., 1980; Stein and Weissel, 1990; Verzhbitsky and Lobkovsky, 1993]. [58] Another piece of information pertinent to the depth of faulting in the CIB comes from the focal depth of earthquakes. Although the depth of shallow earthquakes Figure 15. Histograms of earthquake centroid depths cannot be confidently assessed from teleseismic data, binned every 5 km. (a) Harvard CMT catalogue (14 events). information is available for deeper events. Histograms of (b) Events relocated by Bergman and Solomon [1985] and centroid depth from two different data sets (Figure 15) Petroy and Wiens [1989] (13 events). show broad maxima between 10 and 30 km, and no earthquakes below 45 km. Although earthquake data were Therefore, we argue that among the models presented not used in choosing the flow law, geotherm, and strain herein, model 1, in which the strength increases to an rate of our models, the distribution of focal depths and our idealized BDT around 40 km, and localization is not strength profiles are remarkably consistent. As velocity limited with depth, is most compatible with the data in weakening is needed for earthquake generation [Rice, the CIB. The continuous range of focal depths also argues 1983; Dieterich, 1992] and is the most likely localization against models that explain the long-wavelength deforma- mechanism for the faults in the CIB, earthquake data tion of the CIB by folding of the putative elastic core of imply that localization reaches 40 km depth. In the the lithosphere [McAdoo and Sandwell, 1985]. absence of a plastic layer around the BDT, models with [59] However, the faults that were reactivated during the saturated strength envelope imply ne out of the range current shortening episode did not reach 40 km depth, as compatible with velocity-weakening friction (Table 3). they formed during seafloor spreading and rocks deeper than a few kilometers are ductile in mid-ocean ridge settings [Hirth et al., 1998]. Fault reactivation in the CIB was probably accompanied by formation of new faults in the 6–40 km depth range. Therefore, the most realistic model may have two different values of ne, both being negative, above and below an 6 km depth horizon. We have not explored this family of models due to the poor constraints on the values of ne. The delay between the onset of shortening indicated from plate reconstruction [Royer and Gor- don, 1997] and fault activation [Curray and Munasinghe, 1989; Krishna et al., 2001] might reflect the need to build enough stress to break the deeper levels of the lithosphere [Gerbault, 2000]. 5.3. Origin of Long-Wavelength Deformation [60] Although we focused in this study on the short wavelength of deformation in the CIB, expressed by regularly spaced faults, the long wavelength of deformation, thought to represent buckling of a strong plastic layer, gives important Figure 14. Tradeoff between efficiency of localization constraints on the lithospheric structure [McAdoo and Sand- (effective stress exponent) and depth of localization. well, 1985; Zuber, 1987; Martinod and Molnar, 1995]. Localization is more efficient toward the top of the figure. [61] The 200 km wavelength deformation is compatible The solid lines mark the localization instability for a with buckling of a strong plastic layer 25–50 km thick ((6) strength profile as in model 3 (strength-limited localization) and Table 2), the same thickness that is compatible with the (Figures 9 and 10) and the dashed lines use a strength fault spacing. The growth spectra presented above have a profile as in model 4 (temperature-limited localization) significant growth rate in the wavelength range 100–300 (Figures 11 and 12). The localization instability passes km. However, that portion of the growth spectrum does not between these two lines for intermediate strength profiles correspond to pure buckling but to an interaction between (shaded areas). Darker shade, 7 km fault spacing. Lighter localization and buckling. If only the buckling instability 6 shade, 10 km fault spacing. can develop, as when n1 =10, the combination of depth- ETG 15 - 14 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

assumes that the lithosphere is homogeneous, there are a many preexisting structures in the region, such as the Ninetyeast ridge or the Afanazy-Nikishin seamounts (Figure 1). Karner and Weissel [1990] proposed that seamounts triggered a flexural response of the lithosphere. It is conceivable that a buckling could be triggered in the same manner. However, there is some debate concerning how big a load the seamounts would provide, and what their relation to the intraplate deformation area and other features, in particular a proposed 85 ridge. Clearly, more work must be done to address the importance of preexisting structures on the buckling and localization instabilities.

6. Conclusions

[64] The pattern of faulting in the CIB probably developed from a combination of fault reactivation and localization instability. Localization is defined as the process by which deformation in a homogenous lithosphere focuses on specific regions or localized shear zones that we identify with faults. This process is driven by local perturbations of the deformation rate, and develops a lithospheric-scale Figure 16. Map of growth rate as a function of effective pattern through what we call the localization instability stress exponent and perturbation wavelength for the strength [Monte´si and Zuber, 2003]. The preferred wavelength of profile of model 1. High growth rates at long wavelengths the localization instability can be matched to the 7–11 km come from the complex trajectory of the mode j = 0 of the spacing of fault in CIB. Alternative controls on fault localization instability and large opening of the branches of spacing, such as plastic buckling or elastic folding, cannot the divergent doublet that marks the instability. explain the observed fault spacing. [65] The preferred wavelength of the localization instability depends on the effective stress exponent, ne,a dependent strength profile and the density of the lithosphere parameter that measures the efficiency of localization. prevents significant growth of the buckling instability. The values of effective stress exponent needed to explain [62] The high growth rate at 100–300 km wavelength a fault spacing of 7–11 km vary from 300 to 100 arises from a combination of localization and buckling depending on the exact strength profile assumed in the instability. We show in Figure 16 a growth rate map for lithosphere. They are consistent with experimental data on model 1, similar to Figure 7, but for a larger parameter frictional sliding. Localization during frictional sliding is range. The localization instability is characterized by a also relevant for the CIB tectonics as many faults formed divergent doublet in the growth rate spectrum; there are during seafloor spreading and were reactivated during the two peaks associated with one instance of the localization current N-S shortening episode. The preexisting faults in instability. The wavelength of the doublet is given by (4), the vicinity of locations separated by the wavelength of but we have no control on the opening of the doublet, which the localization instability were preferentially reactivated. is defined as the difference between the values of n1 This can explain the significant spread in the fault spacing required to have either peak at a given wavelength [Monte´si distributions. and Zuber, 2003]. In Figure 16, we see that the opening [66] Earthquake data indicate faulting to the BDT, around increases with perturbation wavelength, to be maximum at 40 km, leading us to favor a model in which localization wavelengths close to that of the buckling instability. We occurs to the BDT with ne 300 and the strength of the observed this phenomenon in every model that we consid- brittle layer increases continuously with depth. However, ered. This interaction between localization and buckling is the deepest faults must have formed during the current responsible for high growth rate in the range of wavelength compressive tectonics whereas the shallowest structures observed in basement undulation in the CIB. Faults were are reactivation of spreading-related normal faults. This found to help buckling in finite element models [Wallace probably results in different localization processes active and Melosh, 1994; Beekman et al., 1996; Gerbault et al., in different depth range, each with a particular effective 1999], which is also consistent with the interaction between stress exponent. Our analysis gives only an average effec- localization and buckling being necessary for development tive stress exponent for the whole lithosphere. of basement undulations. Localization-aided buckling can [67] The long wavelength of deformation, expressed by explain why buckling is observed in the Wharton Basin broad undulations of the CIB lithosphere, is most likely the beyond the reaches of the Nicobar fan, in spite of the large result of an interaction between buckling and localization. surface density contrast in that region. However, it should Significant growth in the 100–300 km wavelength range is be noted that undulations have develop without associated observed in the spectra of models with negative ne only. faulting in some areas of the CIB [Krishna et al., 2001]. However, the nature of this interaction as well as the effect [63] Alternatively, the buckles may be aided by hetero- of preexisting geological features must be further explored geneities in the CIB lithosphere. Although our model before this explanation can be ascertained. MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2 ETG 15 - 15

Appendix A: Viscosity Profile for Brittle Failure and T is the absolute temperature. With the axially symmetric configuration of the experimental studies, the [68] Expressed in terms of stress invariants instead of the second invariants of the strain rate and stress tensors are usual form of normal and shear stress resolved on a related to e_ and s by particular fault, Byerlee’s law of friction states a

3=2 e_II ¼ 2 e_a; ðB2Þ sII ¼ C þ f sI; ðA1Þ with C the cohesion term, f = sin(f) the coefficient of 1=2 sII ¼ 2 s: ðB3Þ friction with f the friction angle, and sI and sII the first and second invariant of stress, respectively. C and f are related to the parameters S and m of Byerlee [1978] by These relations are used to express (B1) as a function of e_II and s . pffiffiffiffiffiffiffiffiffiffiffiffiffi II [71] On the other hand, our models assume a pure shear C ¼ S= 1 þ m2; ðA2aÞ geometry. Hence, we express e_II and sII as a function of the horizontal normal strain rates and deviatoric stress, e_xx and pffiffiffiffiffiffiffiffiffiffiffiffiffi txx: f ¼m= 1 þ m2: ðA2bÞ e_ ¼ 31=2 Â 21=2e_ ; ðB4Þ We use only the high pressure branch of Byerlee’s law, with II xx S = 50 MPa and m = 0.6.

[69] For a model undergoing horizontal shortening, we 1=2 1=2 sII ¼ 3 Â 2 txx: ðB5Þ assume that szz equals the weight of the overburden rocks, p . Along the horizontal axis, we have ov They are related by the apparent viscosity, h,by

sxx ¼p þ 2he_xx; ðA3Þ e_xx ¼ 2htxx; ðB6Þ with p the pressure and h the apparent viscosity. The stress invariants become from which we obtain

1=n1 sI ¼pov þ 2he_xx; ðA4aÞ h ¼ Be_II exp Q=nRT; ðB7Þ

with s ¼ hje_ j: ðA4bÞ II xx pffiffiffi pffiffiffi 1=n B ¼ 3 2 3=A : ðB8Þ Introducing these relations in (A1), we obtain

C fpov Depth dependence of this viscosity profile comes from the h ¼ : ðA5Þ geotherm, T(z), herein taken as linear. je_xxj2f e_xx

In each layer, the overburden pressure varies as [72] Acknowledgments. We are grateful to Jim Van Orman and Jim Cochran for access to their fault data sets, K. S. Krishna and Jon Bull for a preprint of their work, John Goff for discussion of abyssal hill spacing, and pov ¼ pt rgzðÞ zt ; ðA6Þ Chris Robinson and Rachel Abercrombie for sharing their studies of the 1999 Wharton Basin earthquake. Comments by Oded Aharonson, Jon Bull, Brad with pt and zt the pressure and depth at the top of the layer Hager, Greg Hirth, Marc Parmentier and Jeff Weissel helped us improve this and r the density of the material in that layer. Finally, we manuscript significantly. Supported by NASA grant NAG5-4555. obtain References Abercrombie, R. E., M. Antolik, and G. Ekstro¨m, The June 2000 M 7.9 C fpt f rgzt f rg w h ¼ þ z: ðA7Þ earthquakes south of Sumatra: Deformation in the India-Australia Plate, je_xxj2f e_xx je_xxj2f e_xx J. Geophys. Res., 108(B1), 2018, doi:10.1029/2001JB000674, 2003. Beekman, F., J. M. Bull, S. Cloetingh, and R. A. Scrutton, Crustal fault reactivation facilitating lithospheric folding/buckling in the central Indian Ocean, in Modern Developments in Structural Interpretation, Validation and Modelling, Geol. Soc. Spec. Publ., vol. 99, edited by P. G. Buchanan and D. A. Nieuwland, pp. 251–263, Geol. Soc., London, 1996. Appendix B: Viscosity Profile for Ductile Flow Bergman, E. A., and S. C. Solomon, Earthquake source mechanisms from body-waveform inversion and intraplate tectonics in the northern Indian [70] Dislocation creep flow laws have the form Ocean, Phys. Earth Planet. Inter., 40, 1–23, 1985. Biot, M. A., Theory of folding of stratified viscoelastic media and its n implications in tectonics and orogenesis, Geol. Soc. Am. Bull., 72, e_a ¼ As expðÞQ=RT ; ðB1Þ 1595–1620, 1961. Blanpied, M. L., C. J. Marone, D. A. Lockner, J. D. Byerlee, and D. P. where e_ is the axial shortening rate of a sample, s is the King, Quantitative measure of the variation in fault rheology due to a fluid–rock interactions, J. Geophys. Res., 103, 9691–9712, 1998. differential stress applied on a sample, n is the stress Brace, W. F., and D. L. Kohlstedt, Limits on lithospheric stress imposed by exponent, Q is the activation energy, R is the gas constant, laboratory measurements, J. Geophys. Res., 85, 6248–6252, 1980. ETG 15 - 16 MONTE´ SI AND ZUBER: SPACING OF FAULTS, 2

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