## 1 Introduction

[2] 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.

[3] 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.