Corresponding author: A. Mohajeri, Earth Systems Science Computational Centre (ESSCC), School of Earth Sciences, The University of Queensland, St. Lucia, QLD 4072, Australia. E-mail: (firstname.lastname@example.org)
 We present a linear instability analysis and numerical simulations describing deformation and melt patterns in pure shear extension of a partly molten rock. Our models implement numerical techniques that enable strong strain localization and are applied to study melt-strain interactions during continental rifting. Our results show that instabilities can initiate with either strain localization or melt localization, followed by a coupled evolution of melt and shear bands driven by a strong melt-viscosity-shear feedback. This indicates that a local increase in melt fraction due to segregation and/or local melting promotes strain localization and may lead to the formation of large shear bands. Melt-shear interactions can therefore enable rifting where tectonic forces are not sufficient to induce melt-free rifting, resulting in lubricated faults, but not necessarily observed volcanism. Finally, our simulations reveal significant asymmetry in melt segregation around localized shear bands, providing new insights into melt distribution across rift boundary faults and other extensional structures.
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 The interplay between shear zones and melt flow is a key unresolved issue in understanding lithospheric-scale tectonic processes such as rifting. While stress-driven melt segregation has been identified as an important factor in tectonic deformation, the enormous scale of processes and the fact they occur deep below the surface limit the availability of field observations and hinder experimental studies on this topic [Stevenson, 1989; Spiegelman, 2003]. Numerical simulations have been used to elucidate the driving mechanism of melt segregation [e.g., Holtzman et al., 2005; Katz et al., 2006], but these were typically tailored to reproduce laboratory experiments and may have therefore overlooked important aspects of tectonic processes. In particular, existing studies of melt localization apply simulations without strong strain localization and therefore poorly represent the effect of fully developed strain structures (e.g., faults) on melt distribution. In addition, the theoretical analysis in these studies typically ignores the effect of compressibility (or changes in bulk viscosity) on the formation of instability and localization [e.g., Butler, 2010, 2012]. Such studies yield limited insights on the coupling between strain localization and melt flow.
 The aims of this paper are to present a numerical model more appropriate for simulating strain-melt interactions and to extend previous theoretical analyses in order to derive implications for tectonic processes of pure shear extension (e.g., rifting and continental extension). Rifting is associated with decompressional melting and thinning of the lithosphere [Schmeling, 2000]. Numerical models of extension of the continental lithosphere (such as the models in this study) are commonly solved using a viscoplastic formulation. In these model studies, the equations of conservation of mass, momentum, and energy are solved for a two-phase (solid-melt) system. Moreover, rifting is modeled by externally prescribing a constant rate of widening with velocities between 2.5 and 40 mm/yr [Schmeling, 2000]. Thus, we apply an established two-phase (rock-melt) viscoplastic formulation augmented by a modified integration technique to simulate the formation of melt bands and fault-like shear features (with highly localized strain). A linear instability analysis is used to validate our numerical algorithm and to derive new insights into the role of melt segregation in facilitating tectonic shear localization. Both numerical and theoretical methods are also used to show the effect of compressibility on material instability. While our numerical models are not tailored to represent specific geological examples, they overcome the limitations mentioned above and provide insights into melt segregation, melt distribution, and the effect of melt localization on deformation patterns in nature. Based on these results, we argue that local melt-shear interactions may play a significant role in rift initiation and continental extension systems that do not display significant volcanism.
2 Melt Segregation and Formation of Melt Bands
 Field observations suggest that melt migrates through the mantle without equilibrating with its surroundings [Kohlstedt and Holtzman, 2009]. One of the main mechanisms for this behavior is stress-driven segregation of melt [Kohlstedt and Holtzman, 2009]. In the viscous upper mantle, small perturbations in melt fraction (porosity) lead to new distributions of viscosity and shear stress (with larger melt fractions corresponding to lower viscosity and vice versa). The new distribution of shear stress results in melt migration from areas with a lower melt fraction to areas with a higher fraction [Kohlstedt and Holtzman, 2009]. Moreover, strain concentrates along melt-rich bands because of the viscosity reduction in these regions. The resulting positive feedback between shear stress and melt fraction enhances melt segregation into melt-enriched pathways of high permeability [Katz et al., 2006; Kohlstedt and Holtzman, 2009]. As a result, partially molten, isotropic, and homogeneous rocks under shear conditions become highly anisotropic and heterogeneous in melt fraction, permeability, viscosity, and stress distribution [Kohlstedt and Zimmerman, 1996; Zimmerman et al., 1999].
2.1 Previous Theoretical and Numerical Studies
McKenzie  introduced a set of equations to describe melt segregation in a partially molten rock. Stevenson  investigated the segregation of homogeneously distributed melt in response to shear stress in a deforming matrix and found that fluid pressure is lower in regions of higher melt fraction. This behavior forms a positive feedback effect whereby melt segregates to melt-rich (low pressure) regions and amplifies the existing variations in melt fraction [Stevenson, 1989]. Spiegelman  investigated the formation of melt-rich bands in simple shear and demonstrated a two-dimensional linear analysis in which melt bands oriented at 45° to the shear plane grew fastest. The analysis also showed that shear bands rotated passively with increasing strain and decayed when rotated to an angle higher than 90° to the shear plane [Spiegelman, 1993, 2003]. This analysis was expanded later to include the effect of strain rate weakening with a power law rheology [Spiegelman, 2003], and numerical simulations applying such rheology were used to explain the difference between analytically predicted band orientations and those observed in laboratory simple shear experiments [Katz et al., 2006].
 Recent studies by Butler [2009, 2010, 2012] examined linear instability and (numerically simulated) melt band formation in pure shear contraction. Using a modified power law equation and considering the effect of buoyancy forces, the author demonstrated that melt bands form parallel to the direction of maximum compression. Butler [2009, 2010] suggested that buoyancy forces may play a dominant role in producing low-angle bands in pure shear compression. While these studies show a good correlation between numerical and analytical pure shear results, the lack of plastic yielding (i.e., faulting) and extreme strain localization (i.e., shear structures) limits the applicability of the simulations for studying large deformation structures in natural pure shear settings (i.e., rifting and lithospheric extension processes).
 The problem of melt-strain interactions was also addressed in many previous studies. Melt and melt segregation have been identified as important contributors to rift processes [Hollister and Crawford, 1986]. Other studies showed how large melt structures can weaken the lithosphere and enable rifting [Buck, 2004, 2006]. While such studies clearly highlight the importance of melt, they describe neither the coupled evolution of melt and strain structures nor the enhancement of shear-melt interactions through dilation of shear structures, modification of bulk viscosity, and the compressibility of partially molten rocks. A better understanding of melt-strain interactions and deformation patterns in compressible material is particularly important in studies of lithospheric extension, where shear zones are expected to dilate [Menéndez et al., 1996; Wu et al., 2000; Finzi et al., 2012] and where magma intrusion is expected to modify the continental lithosphere [Ebinger, 2005; Corti, 2009].
2.2 Governing Equations for Melt Segregation in a Partially Molten Rock
 The governing equations of melt movement in a matrix of partially molten material have been given by McKenzie . The equations were obtained from the conservation of mass, momentum, and energy using expressions from the theory of mixtures. Assuming absence of active melting, from the equations for conservation of melt and matrix mass we can write [McKenzie, 1984]:
where V, v, and ϕ are solid velocity, melt velocity, and porosity, respectively. Combining equations ((1)) and ((2)), the equation governing the difference in the velocity of the melt and matrix is a modification of Darcy's law:
where K is permeability; g is gravity; θ = 2 or 3 for two-dimensional or three-dimensional problems, respectively; δj is the Kronecker delta tensor; ρf is melt density; and ηf and p are the fluid viscosity and melt pressure, respectively. The term (p,j + ρfgδθj) represents the pressure gradient, which is not balanced by gravity and is called excess pore pressure gradient. The permeability in the above equation is dependent on the porosity using the following equation:
where a is defined as the scaling distance between melt inclusion, b is a geometrical parameter in the range of 100–3000, and m = 2 is a power exponent obtained from experiments [McKenzie, 1984]. For incompressible and compressible viscoelastic materials, the solid velocity is linked to melt pressure using the Stokes flow equation [McKenzie, 1984]:
where η and ηB are the shear and bulk viscosity, respectively. The average density is obtained by , where ρs is solid density. Bulk viscosity controls the resistance of mechanical volume change of the solid. For compressible materials, the bulk viscosity is often assumed to be 2 orders of magnitude higher than the shear viscosity. However, for incompressible materials (constant solid density), the ratio ηB/η0 must be infinite. The solid shear viscosity is affected by temperature, strain rate, and porosity. Previous studies have suggested that viscosity drops with an increase in porosity and strain rate [Hirth and Kohlstedt, 1995]:
where is the second invariant of the incompressible component of the strain rate tensor and n and α are constants. Mei et al.  determined that α = 26 corresponds to a diffusion creep regime and α = 31 corresponds to a dislocation creep regime. The power law equation (equation (6)) has been widely used, with the value of n varying from 1 (strain rate–independent viscosity) to a maximum value of 6. Recent studies have suggested other values for n, such as 3.6 and 4.94 for wet and dry dislocation creep of mantle olivine, respectively [Korenaga and Karato, 2008]. The values of bulk and shear viscosities at different melt fractions are important to understanding melting and melt dynamics [Takei and Holtzman, 2009].
 Equations ((1))–((6)) provide a link between solid stresses and fluid pressure. These equations are coupled and should therefore be solved together. The fluid pressure varies with solid deformation and gravity, which drives fluid flow and changes the porosity. Variations in porosity and stress can then change the permeability and viscosity in the constitutive relations. These feedbacks can lead to behaviors such as flow localization and the development of melt bands.
3 Deformation of Partially Molten Rocks in Pure Shear Extension
 In this section, we introduce a linear instability analysis and a numerical model that describe deformation and melt patterns in pure shear extension of a partly molten lithosphere. The linear analysis is used to validate our numerical model and to investigate how compressibility and initial melt fraction affect deformation patterns. The numerical model is then used to study melt-strain interactions in simulations representing lithospheric extension (section 3.2).
3.1 Linear Instability Analysis
 Instability analyses are commonly used in engineering and geomechanics to establish the conditions at which a material will fail and to determine the characteristics of the failure process (e.g., orientation of shear bands). They also provide benchmarks for numerical studies and have been widely used in previous research [Butler, 2010]. In our linear instability analysis, the buoyancy is ignored and the plane wave perturbation is added to the steady constant background state [Muhlhaus et al., 2012]. The derivation is very similar to the one presented by Katz et al.  for the case of simple shear. For more detailed information on the linear instability, see Supporting Information.
 To start, we define the compaction length lcomp and the ratio between reference and bulk viscosity χ:
 The ratio χ appears naturally in the nondimensionalization of the governing equations. The ground state of instability analysis describes the ratio between volumetric strains (D11, D22) as follows [Muhlhaus et al., 2012]:
where Vk,k is the rate of volume change and b1− 3 and a1− 4 are parameters that we define in order to derive the growth rate equation [Muhlhaus et al., 2012].
 For this linear stability analysis, the governing equations are linearized. Concentrating on solutions of the hyperbolic type, we find the following relation (equation (11)) between growth rate ω, the wave number q, and the band orientation β. Figure 1 presents the theoretically predicted orientation of instability (β, for which the deformation band growth rate is maximum) for n = 1–6 and χ = 0.125, φ0 = 0.01, α = 27, and = 0.00025.
 The dimensionless growth rate can be obtained using the following equation:
where is the nondimensional form of ω. The tilde sign indicates the dimensionless form of each parameter. When ϕ0 is infinite, the material is incompressible and equation (11) is simplified to:
where is the dimensionless wave number with respect to the compaction length. If , then the background porosity state is unstable, and this instability grows exponentially with time. Since the porosity of the domain depends on the fastest growing instability, we find the maximum by varying the wave vector (i.e., q and β). The prediction is that the wave vector aligned at 90° to the y direction will have the maximum growth rate:
 Figure 1a shows curves of growth rate as a function of the melt band orientation β for a range of n values (based on equation (13), for incompressible material). While the solutions for strong strain–dependent viscosity (n > 4) display well-defined maxima and relatively high growth rates, the curve for n = 1 has a low (but distinct) peak at β = 90° (noting that the growth rate curves are symmetric about β = 90°). For n values between 2 and 4, the growth rate curves display very broad maxima, indicating a lack of preferred band orientation (i.e., either localization is unlikely or its orientation is not well constrained). Figure 1b shows the growth rate functions for compressible material (based on equation (11)). Unlike in Figure 1a, we recognize distinct maxima for values of n > 2 in Figure 1b, indicating that deformation is expected to localize and to form deformation bands with predictable orientations. The curve for n = 1 displays a small (subtle) maximum, but it is difficult to assess its significance.
 Instability growth rates are much higher when considering a compressible ground state (Figure 1b) compared to those resulting from an incompressible ground state (Figure 1a). These differences indicate that a compressible ground state is much more susceptible to instability and strain localization in deformation bands. In our linear analysis, the relation between compressibility and growth rate arises from equations ((8))–((11)). In these equations, the effect of compressibility is controlled by components of stretching tensor and rate of volume change. For an incompressible ground state, D1,1 = − D2,2 and Vk,k = 0. Inserting these values into equations ((9)) and ((10)) causes equation (11) to result in the growth rate equation for an incompressible ground state, as shown by equation (12). However, for a compressible ground state, D1,1 is connected to D2,2 through equation (8), Vk,k ≠ 0, and the growth rate can be obtained from equation (11). In addition, as the bulk viscosity is a function of ϕ, a higher initial ϕ makes the two-phase system more compressible and therefore more susceptible to localization (see much higher growth rate in Figure 1b compared to Figure 1a). In Figure 1a, the orientation of melt bands is around 60° for n = 6, whereas in Figure 1b, the orientation for n = 6 is 50°. The difference in orientation is even greater for n = 3, with β ≈ 65° for incompressible material (Figure 1a) and β ≈ 50° from a compressible ground state (Figure 1b). Moreover, our analysis reveals great differences between the two ground states for n < 2. While Figure 1a shows that localization from an incompressible ground state with n < 2 occurs at β ≈ 90° (β = 70° − 110°), Figure 1b suggests that localization might not occur for n < 2 (displaying small and probably insignificant maxima at β ≈ 37° for n = 1 and β ≈ 50° for n = 2).
 Finally, Figure 1 suggests that there are two main localization regimes in our results. Compressible ground states exhibit two prevailing band orientations (β ≈ 37° for n < 2 and β ≈ 50° for n > 2; Figure 1b). Incompressible ground states also display two distinct ranges of band orientation (β ≈ 90° for n < 2 and β ≈ 60° for n > 4; Figure 1a), with no preferred instability orientation for mid values (2 < n < 4). Our numerical results, presented in the next section, confirm that these instability regimes emerge in pure shear simulations.
 Figures 1c and 1d demonstrate the significant dependency of growth rate and band orientation on the initial porosity, the strain rate exponent n, and the combination of these two parameters. The very high (and geologically meaningless) n value in these plots (n = 200) induces a greater dependence of viscosity on the strain rate and consequently much higher growth rates of the deformation band. These plots also show the direct dependency of growth rate on porosity, with a higher initial porosity leading to a higher growth rate. From Figures 1c and 1d, we can also see a coupled porosity-n effect on the orientation (β). For both compressible and incompressible ground states, when these two values are higher, the orientation of localization also seems to be higher. The exaggerated high values chosen for n and ϕ in Figures 1c and 1d are used solely to demonstrate the effects of these parameters. As these values are unrealistic, they are not used in the numerical simulations.
 The presented linear analysis is a simplified means to assess deformation instability and does not represent the richness and complexity of natural processes. However, it can still provide important insights on the factors controlling localization. In rift environments, volumetric changes induced by magma intrusion or shear zone dilation will locally alter compressibility and melt fraction and as a result increase the instability growth rate. Furthermore, thermal or material heterogeneities may also induce local melting or melt segregation that may promote instability and the formation of deformation bands. The implications of these factors are further discussed in section 4.
3.2 Numerical Simulation
 To study the mechanical response of the lithosphere to extension, its rheology has to be taken into account. The rheology depends on the fluid content, the thickness of compositional layers, and the boundary conditions. Considering a viscoelastic lithosphere, strain is a function of time but behaves elastically at the instance of stress application and for stress of short duration. The strain depends on viscosity and pressure as described in equation (5). The melt viscosity itself is porosity and strain rate dependent (equation (6)) and the melt pressure is porosity dependent (equation (4)).
 A numerical code was programmed in the finite element method (FEM) environment “escript” [Gross et al., 2008], which is a nonlinear and time-dependent partial differential equation modeling environment. As the primary goal of the present work is to characterize melt and shear localizations, we restrict ourselves to a simple case of a viscous-plastic layer of rock. In our simulations, we neglect the effects of buoyancy, heat, and geochemistry.
 We consider a rectangular domain of dimensions 150 × 500 km in extension submitted to a prescribed velocity boundary condition on the right and left sides and free slip boundaries on the top and bottom edges (Figure 2). The total extension rate used in our study was 10 mm/yr. The model domain is discretized with 150 × 500 elements. Various mesh resolutions have been tested to minimize mesh dependency. The numerical calculations are done in plain strain. An initial random porosity (with perturbations smaller than 0.1 × ϕ0) was introduced in our simulations to facilitate the formation of porosity bands [Katz, 2010; Katz et al., 2006]. In our simulations, compressibility was assumed to be a function of the ratio between bulk viscosity and shear viscosity, allowing us to simulate both compressible and incompressible scenarios.
 In this study, we use the simplest model representation of the Earth (one layer). Therefore, our model is not suitable for reproducing phenomena related to the brittle-ductile transition or changes in fault structures at different depths of the lithosphere. Our models are most relevant for understanding strain localization in the crust and upper mantle, and our interpretation is focused on the interaction between shear and melt structures. This is most relevant in the brittle zones and in strong lower crust and upper mantle regions where melt localization can induce (and enhance) shear localization.
 The chosen time step is sufficiently small to prevent plastic failure of the whole model domain during the first time step when plasticity occurs in the simulation [Mohajeri et al., 2011]. The model shown in Figure 2 is representative of all the simulations (except the ones in section 4). The parameters used in the simulations have been selected in accord with previous studies [Spiegelman, 1993, 2003] and are listed in Table 1.
Table 1. Values of Parameters in our Numerical Models
Bulk to shear viscosity ratio for compressible material
Random perturbation in
Solid (matrix) density
Fluid (melt) density
Power law coefficient
Porosity weakening coefficient
 In order to validate our numerical code, we ran many simulations with a wide range of initial porosities and rheology power law coefficients (n). In the following paragraphs, we compare our numerical results with the relevant linear instability predictions. In Figure 3, the growth rate (top row) and strain rate (bottom row) throughout the incompressible domain are shown for simulations with n = 1 (left column) and n = 6 (right column). As growth rate correlates with enhanced heterogeneity and localization, the banded growth rate pattern in Figure 3 can be interpreted as a tendency to form bands of enhanced and depleted melt fractions (high melt fraction where growth rate is high and low melt fraction where growth rate is low or negative).
 Choosing n = 1 leads to slow growth and somewhat diffused melt bands in the incompressible model. For n = 1 (Figure 3a), diffuse melt bands appear perpendicular (average orientation of 88.5°) to the axis of maximum extension rate separated by melt-depleted bands (in dark blue). The orientation of these simulated melt bands is as expected from the instability analysis for an incompressible Newtonian material (Figure 1a). In contrast, for n = 6, the average orientation of simulated melt bands is 51° (Figure 3b), in accord with analytically predicted values (Figure 1a). Figures 3c and 3d show the strain rates for n = 1 and n = 6. The orientations stated are the average values of all the melt band orientation measurements in Figures 3a and 3b, respectively. Figures 3e and 3f provide more details on the observed variations in band orientation for n = 1 and n = 6. As shown in Figure 3e, the majority (57%) of the melt bands in Figure 3a have an orientation in the range of 87°–90°. Similarly, the majority (58%) of melt bands in Figure 3b display an orientation of 49°–53° (Figure 3f).
 While the orientation of strain bands coalesces with that of melt bands for n = 6 (Figures 3b and 3d), they do not coalesce in simulations with n = 1 (Figures 3a and 3c). This may suggest that where shear bands are well developed (i.e., shear is localized), melt distribution is controlled by the shear band distribution. In the Newtonian case (Figure 3a), where shear is more homogeneous and viscosity is not strain rate dependent, the melt bands are not aligned with the weak shear bands. Furthermore, in simulations with high initial porosity, where instability is more likely to occur, we have observed a complex pattern of near-vertical melt bands cross-cut by inclined shear bands. In this case, both shear and melt bands were diffused and exhibited significant variations in band orientations.
 In simulations with intermediate values of n (1<n<4) and low initial porosity, we did not obtain significant strain or melt localization. This is also supported by the results of the instability analysis for 1<n<4, in which a preferred instability orientation could not be easily determined (maximum growth rate is achieved within a very wide range of angles; 50<β<90). These results might seem in contrast with the fact that rocks, which are typically described by power law rheology with an exponent between 3 and 4, still experience localization. However, we show that volumetric strain and magma intrusion locally modify the lithosphere into a two-phase compressible material that is susceptible to localization and formation of shear-melt bands for n > 2 (Figure 1b). We also note that heat perturbation and preexisting heterogeneities (such as deformation features) would facilitate localization.
 In Figure 4, the orientation of melt bands is shown for a range of n values (1<n<10) based on linear instability predictions and numerical results for an incompressible domain. In simulations with 2<n<4, we observed a relatively wide range of orientations (and very weak localization), whereas for other n values, the simulations result in better defined deformation bands with a narrow range of dominant orientations. This outcome corresponds to the analytical prediction of two instability regimes with distinct band orientations (see also Figure 1a).
 The significant difference between simulated and predicted orientations (particularly for n > 3) is mostly related to the fact that the simulations represent heterogeneous material involving nonlinear processes and strain-melt interactions that are not represented in the instability analysis.
3.3 Simulation of Melt-Shear Interactions in the Presence of Well-Developed Shear Bands
 The numerical results discussed in the previous section do not show extreme strain localizations as typically observed along plate boundaries. Thus, in order to facilitate the formation of localized deformation bands, we introduced a small nucleation point (weakness zone) at the bottom of our model domain and implemented a numerical technique that enables strong localization (“modified integration FEM”) [Hughes, 1987]. The introduction of a weakness zone is commonly used in both numerical (Kaus, 2010) and analogue [Corti et al., 2007] studies in order to facilitate the nucleation of localized strain.
 The nucleation point is a small, square, and weak zone of lowered viscosity, the size of which was selected according to recent studies [i.e., at least 10–20 elements wide but small compared to the model domain and sufficiently far from the side boundaries; Kaus, 2010]. Our initial setup of numerical models aims to reproduce the rheological boundary conditions characterizing the region indicated by stretching of a previously thinned continental lithosphere that contains a “preexisting” weakness zone. Since extension always concentrates in the weakest area of the lithosphere, such processes have a strong influence on the spatial distribution of deformation. As for the location of the nucleation site, we introduced the weak zone in the center of the lower surface of the model, which could represent the center of a rift zone (probably weakened previously by heat), surrounded by stiffer margins. Introducing such a rheological stratification leads to deformation and subsidence in response to mechanical stretching of the lithosphere [Mulugeta and Ghebreab, 2001].
 The modified integration method is commonly used to avoid numerical instability in evaluating the stiffness matrix (second term in equation (5)) in cases where the material is incompressible [Hughes, 1987]. The modified FEM incorporates the selective integration technique, which uses less integration points for evaluating the volumetric part of the stiffness matrix. As such, the method enables the formation of higher strain gradients and hence stronger localization. Even the most detailed simulations (e.g., with a layered lithosphere, buoyancy, and thermal/chemical processes) would not be able to fully represent melt processes near shear structures. They would not identify the ability of melt localization to assist in the formation of shear structures (faults) if they do not incorporate the reduced integration technique (which enables simulation of highly localized shear structures).
 In the modified FEM models, we explored a narrower range of n values (3<n<5) that are thought to represent the realistic behavior of rocks in nature (Korenaga and Karato, 2008). Comparisons of simulations with common FEM and modified FEM are presented in Figures 5a and 5b, showing the distribution of porosity and strain rate. In both cases, introducing the weak zone resulted in the formation of two shear bands with an angle of approximately 50° to the y axis. The strain rate in the shear bands is significantly higher than that in the rest of the domain (indicated by the high density of strain rate contours). This is in accordance with observations of crustal and mantle shear zones [Sibson, 1975; Hobbs et al., 1986; Whitmarsh et al., 2012] and experimental data [Brun and Beslier, 1996; Brun, 1999; Corti, 2012]. It is also supported by the notion that melt processes may further enhance strain localization of lithospheric shear structures [Buck, 2006]. Numerical simulations of shear localization using the common FEM (Figure 5a) yielded relatively wide, noncontiguous melt bands and somewhat diffused shear bands [e.g., Katz et al., 2006; Butler, 2010]. In contrast, in simulations where the modified FEM was employed, shear bands were localized more efficiently, as indicated by the very high density of strain rate contours (Figure 5b).
 In simulations with fully developed (localized) shear bands (Figure 5b), melt seems to segregate along the shear band. This is accompanied by the development of a melt-enriched zone on the upper side of the bands (i.e., in the zone between the two bands) and a melt-depleted zone on the lower side of the shear bands (with a very strong gradient in melt fraction along the shear bands; Figure 5b). These results suggest that melt distribution across dominant shear features (e.g., rift valley boundary faults) may be naturally asymmetric due to local stress and strain patterns near well-developed shear bands. Figure 5c shows that this asymmetric melt distribution is associated with stress-driven volumetric strain (dilation) patterns. Asymmetric melt distribution is recognized in all simulations with highly localized shear structures. Such shear localization is achieved in two types of simulations: using regular FEM simulations with n > 5 and a high initial melt fraction (ϕ0 > 0.1) or in simulations where the modified integration method was applied with n ≥ 3.
 Our simulations display melt enhancements of 1%–5% (compared to the background level) within localized (and narrow) melt bands (i.e., an increase in porosity from 0.010 to 0.0105). This melt localization is found to be very significant for deformation patterns, but the exact geometry of such melt bands in nature cannot be easily determined. The cumulative melt volume in these narrow bands is probably too small to reach the surface as significant volcanism, and the melt bands are possibly too narrow for observation in geophysical imaging. While we found similar melt-strain distributions in simulations with n = 4 and n = 6 (and with a range of initial melt fractions, ϕ0 = 0.01–0.1), further work is required to better understand the sensitivity of this asymmetric pattern to material properties, strain rate, and both shear-induced and buoyancy-induced pressure gradients.
 The improvement of the numerical results due to the implementation of the modified FEM algorithm is demonstrated in Figure 6a. The plot shows profiles of the second invariant of the strain rate tensor () across the middle of the models presented in Figures 5a and 5b. Compared with the normal FEM (solid line in Figures 6a and 6b), the localization is much higher in simulations where the modified FEM was applied (dashed line in Figure 6a).
4 Melt-Shear Interactions in Nature: New Insights Into the Role of Melt in Continental Extension
 Our simulations reproduce results of the linear instability analysis and provide new means to study melt-strain interaction in the vicinity of fully developed shear structures with highly localized strain patterns. In this section, we use the benchmarked numerical code to investigate the formation of shear bands and melt structures in extension of a partially molten rock.
 In natural extensional settings, melt is driven through the solid matrix and into an extensive permeable network of melt-bearing channels. While it is typically impossible to directly observe natural melt segregation and shear zone development, dykes and leucosomes (crystallized felsic melts associated with high-grade metamorphism) provide a snapshot of magma pathways at a particular stage. Leucosomes are commonly found near fractures, folds, and other deformation features [Brown et al., 1995; Davidson et al., 1994; Hollister and Crawford, 1986] and commonly used to study stress-enhanced melt flow and localization [Hollister and Crawford, 1986; Weinberg and Mark, 2008]. Widespread dyke intrusions observed in various rift systems have been suggested as a significant mechanism to heat and weaken the lithosphere prior to rifting [Royden et al., 1980; Buck, 2004]. Buck  argued that dykes fed from a localized source facilitated plate spreading during the initiation of the Afro-Arabian rift system.
 A normal continental lithosphere, according to Buck , is too strong to rift unless heated and weakened by magma (i.e., “magmatic rifting”). On the other hand, in many extensional settings, conjugate crustal faults develop in pure shear to form a typical rift valley morphology without significant volcanism [Corti, 2012]. Furthermore, many rifts worldwide are thought to have been initiated tectonically (without volcanism), with crustal extension forming rift boundary faults [Ebinger, 1989; Ebinger, 2005]. The Newfoundland-Iberia conjugate margins are a particularly well studied example of a non-volcanic rift system [Whitmarsh et al., 2001; Pérez-Gussinyé et al., 2003; Péron-Pinvidic et al., 2007]. In the absence of voluminous magmatism, previous studies have suggested that strain localization was facilitated by the existence of a serpentinized mantle peridotite that acted as a lubricant and focused localization [Lavier and Manatschal, 2006]. Results of analogue and numerical models indicate that other preexisting anisotropies would also act to localize strain and facilitate continental breakup [Corti et al., 2003; Gessner et al., 2007]. In numerical models that considered energy feedback effects and full coupling between the energy, momentum, and continuum equations, it has been shown that shear zones can be localized even in the absence of preexisting weaknesses [Regenauer-Lieb et al., 2006; Weinberg et al., 2007; Rosenbaum et al., 2010].
 Our results suggest that weakening of the lithosphere and strain localization will also be promoted by the existence of a small volume of melt during extension. Indeed, it has been shown that even non-volcanic rifted margins have incorporated a limited volume of magma during rifting [Müntener and Manatschal, 2006]. The origin of this magma could possibly be associated with melt-rich lenses at the base of the lithosphere [Anderson, 2011], which may become incorporated in the rift system even if rifting is driven by far-field tectonics. A recent field study showed that during early rifting stages, distributed and diffused melt may infiltrate the lithosphere only to be entrapped in the shallow lithospheric mantle and later to be deformed by shear and additional (more focused and voluminous) melt infiltration as extension progresses and strain is localized along rift structures [Piccardo, 2010]. This study suggests that the observed melt infiltration may have significantly reduced the total strength of the subcontinental mantle along the rift axial zone even though it resulted in non-volcanic rift margins [Piccardo, 2010].
 Our results also indicate that the susceptibility to form localized instability greatly depends on the bulk viscosity and therefore on the initial melt fraction of the two-phase system (equations ((8))–((12))). This is clearly demonstrated in Figure 1 by the difference in instability growth rates between compressible and incompressible materials and the strong dependence of instability growth rate on the initial melt fraction. It is important to note that accumulation of melt has a direct effect on the growth rate of instabilities and an indirect effect through its effect on the bulk viscosity (and compressibility). This implies that even a local enhancement of porosity (e.g., 1%–5%) may sufficiently promote the formation of localized deformation. These interesting results may shed light on the initial stages of rifting.
 To demonstrate the emerging implications for rift initiation and continental extension processes, we show, in Figure 7, results from a simulation of a relatively melt-rich (ϕ0 = 0.1) lithosphere in pure shear extension (using n = 4 as a mid-range representative of natural rocks) in a domain with a height and a width of 70 and 150 km, respectively. According to our analysis, normal continental lithosphere material (melt deprived and largely incompressible) might not form localized deformation bands in pure shear extension (assuming moderate tectonic forces, n ≈ 3–3.5, and excluding reactivation of weak structures). Instead, distributed (and diffused) melt bands may form in such a lithosphere due to a combination of buoyancy, pressure gradients, and tectonic extensional stresses. Most probably, any local perturbation in heat, composition, or stress distribution would induce melt segregation and local melt accumulation. While the orientation and growth rate of such melt features depend on the strain rate rheology parameter (n), the positive feedback they display in aiding strain localization does not depend on n. Such melt localizations would locally render the system more compressible and more prone to deformation instability, as suggested by our instability analysis (e.g., melt accumulation in the prescribed nucleation site in Figure 7a). Eventually, diffused deformation would become unstable and strain would localize along faults dipping at ~50° (Figures 7a and 7b; similar to those shown in Figure 5b). Melt would then segregate into these shear bands (Figures 7c and 7d) and would enhance localization by means of fault lubrication [Weinberg and Regenauer-Lieb, 2010]. It is important to note that the porosity within simulated shear bands does not exceed 5%–10% and does not display significant along-band flow. Therefore, shear band lubrication by porosity would probably not involve melt flow toward the surface and as such does not necessarily induce volcanism.
 After accumulation of large total strains, lithospheric necking is expected to occur (Figures 5b and 7d) and melt is expected to accumulate also along the emerging rift axis (between the main “boundary faults” in Figures 5b and 7d). While the porosity observed in our simulations (Figure 7) is probably not high enough to produce volcanism or to develop the magmatic dominated final stages of rifting and continental breakup, it would further aid in weakening the lithosphere and promoting localization of deformation and melt flow. These processes could in turn enhance continental thinning and melt supply along the rift axis and eventually [as suggested by Ebinger, 2005 and Corti, 2012] could result in continental breakup and the formation of a spreading center (where extension is predominantly accommodated by magma accretion). These melt-strain patterns could also result in abandonment of extensional detachment faults [Ebinger and Casey, 2001] and rift boundary faults [Corti, 2012] and in formation of rigid blocks separated by zones of enhanced magmatism.
 Recent conceptual models based on experiments and field observations assist us to explain the schematic rifting stages outlined above and captured in Figure 7 [Buck, 2006; Piccardo, 2010; Corti, 2012]. Our work reveals the importance that diffused melt percolation (during initial extensional stages) may play in promoting strain concentration and eventually focused melt intrusion along lubricated shear bands. This outcome provides insights into the formation of observed melt features and shear structures associated with extensional regimes [e.g., in the Voltri Massif, Ligurian Alps, Italy; Piccardo and Vissers, 2007; Piccardo, 2010].
 Furthermore, in order to better explain rift initiation and continental extension of strong lithospheres, our work suggests a “mixed mode” combining characteristics of both the “magmatic extension” model and the “tectonic stretching” model of Buck , where volcanism is not observed and pure magmatic rifting is not supported by observations. We show that even diffuse melt segregation in a normal continental lithosphere could promote instability in the form of strain localization along melt-lubricated faults. The model presents a plausible explanation for non-volcanic rifting without excluding the effect of melt-strain interactions. Such a mechanism could also play an important role in continents undergoing slow extension where the strain rate weakening effect may be small and the accumulation of melt could be the key to achieving instability and localization.
 In the present work, we introduced a simplified dynamic model to study the melt instability problem in a viscoplastic partially molten material. Our two-dimensional FEM models, which consider a random initial porosity and neglect the effects of buoyancy and mass transfer (between solid and liquid phases), show that melt bands form parallel to the direction of maximum stress for n = 1 as observed in laboratory experiments [Takei, 2005]. In simulations with strain rate–dependent viscosity, a coupled shear-melt localization is shown to occur for n > 4, in good agreement with the preferred orientation predicted by the linear instability analysis. In addition, we show that in compressible material, melt bands are more inclined (lower angle to the direction of maximum extension) and their growth rates are higher than those in incompressible material.
 In simulation with a weak nucleation zone, the shear bands seemed to control the distribution of melt. However, to better represent the extreme localization observed in nature, the modified integration method was added to the code, leading to formation of thinner, more localized shear bands in the domain. Where shear localization is unfavorable (n < 4), melt segregation may enable shear localization by increasing the instability growth rates. Once shear bands develop and localize strain, melt distribution is controlled by dilation and stress patterns around these deformation features (Figures 5b and 7). Comparing these results with conceptual rifting models based on laboratory experiments [Corti, 2012], we conclude that melt-strain interactions may play a significant role in rift initiation and in facilitating shear localization along melt-lubricated faults, even where significant volcanism is not observed. While our simulations are too simple to yield structures similar to those observed in rift zones, the localization processes and melt-shear interactions are clearly shown to take an important role in rifting. Our models reveal new insights on the importance of melt-shear interactions in rifting and continental extension. In particular, our results identify the ability of small melt localization to serve as nucleation sites for shear bands. While lithospheric weakening along extension-induced melt bands was suggested before, the tendency of melt concentrations to induce nucleation of shear bands was overlooked.
 Our study demonstrated correlations between our results and geological observations, and ongoing work on this topic will bring more complexity to the current model. By integrating melting, temperature, buoyancy forces, and a realistic lithospheric structure in our models, we expect to faithfully simulate the melt and strain patterns that control geological features such as rifting and continental extension.
 We acknowledge the financial support provided by the Australian Research Council through Discovery Projects DP110103024 (Multiscale processes), DP120102188 (Sand erosion), and DP1094050 (Lithospheric extension). We thank R. Weinberg, L. Gross, and A. S. Virk for their comments and support. The manuscript benefited from comments from two anonymous reviewers.