A two-dimensional numerical simulation of lithospheric shortening shows the formation of a stable crustal-scale shear zone due to viscous heating. The shear zone thickness is controlled by thermomechanical coupling that is resolved numerically inside the shear zone. Away from the shear zone, lithospheric deformation is dominated by pure shear, and tectonic overpressure (i.e., pressure larger than the lithostatic pressure) is proportional to the deviatoric stress. Inside the shear zone, deformation is dominated by simple shear, and the deviatoric stress decreases due to thermal weakening of the viscosity. To maintain a constant horizontal total stress across the weak shear zone (i.e., horizontal force balance), the pressure in the shear zone increases to compensate the decrease of the deviatoric stress. Tectonic overpressure in the weak shear zone can be significantly larger than the deviatoric stress at the same location. Implications for the geodynamic history of tectonic nappes including high-pressure/ultrahigh-pressure rocks are discussed.
 Shear zones are prominent deformation structures in many orogens such as the European Alps. The Alps are made of tectonic nappes, and many of them are considered a result of ductile shearing [e.g., Steck, 1990; Escher and Beaumont, 1997; Merle, 1998]. Understanding the pressure, stress, and temperature evolution in crustal shear zones is essential for the reconstruction of the tectonic history of nappes exhibiting high-pressure (HP) and ultrahigh-pressure (UHP) rocks, such as the Adula nappe in the central Alps [e.g., Nagel, 2008], or exhibiting a Barrovian metamorphic overprint, such as the nappes of the Lepontine dome in the central Alps [e.g., Todd and Engi, 1997; Berger et al., 2011]. The generation and exhumation of nappes with HP-UHP rocks [e.g., Burov et al., 2001; Jolivet et al., 2003; Ford et al., 2006; Beaumont et al., 2009] and the source of heat for the low-pressure high-temperature Barrovian metamorphism postdating nappe formation and observable in many orogens [e.g., Jamieson et al., 1998; Hartz and Podladchikov, 2008; Berger et al., 2011] are two major issues of mountain building still under debate.
 Crustal-scale shear zone formation during lithospheric shortening is studied with a two-dimensional (2-D) thermomechanical numerical model. A representative simulation is presented to quantify the relationship between deviatoric stress and pressure inside and around the shear zone. The crustal-scale shear zone develops due to shear heating [e.g., Ogawa, 1987; Crameri and Kaus, 2010; Thielmann and Kaus, 2012], and its thickness is controlled by thermomechanical coupling that is resolved numerically inside the shear zone. Based on the numerical results, a conceptual model for the pressure and stress relationship in crustal-scale shear zones, developing during continental collision, is proposed.
 The 2-D model is based on the finite element method and uses the Million a Minute (MILAMIN) solver [Dabrowski et al., 2008] and the mesh generator Triangle [Shewchuk, 2002]. The algorithm solves the force balance equations including gravity, the mass conservation equation for incompressible materials, and the heat equation including radiogenic heat production, A, and viscous heating (Table 1). The rheology is viscoelastoplastic using a Mohr-Coulomb yield criterion. Earlier versions of the algorithm are described in more detail in Burg and Schmalholz  and Schmalholz et al. . Erosion is not considered here. The applied viscous flow law is a linear combination of a linear and a power law viscous flow law. Using the square root of the second invariant of the deviatoric stress tensor, , with x and y being the horizontal and vertical directions, respectively, provides a viscosity, η:
where μ is the linear viscosity for τII ≪ τp and T = Tc, n is the power law stress exponent, T is the temperature, τp is the stress at the transition between linear and power law viscous behaviors, Q is the activation energy, and R is the gas constant. All equations have been solved in dimensionless form. For the nondimensionalization, four characteristic scales were applied: the initial thickness of the crust, hc; the initial temperature at the Moho, Tc; the initial lithostatic pressure at the Moho, σc; and the reference viscosity at the base of the lower crust, μc (all at the left model side; Figure 1). All presented parameters and results are dimensionless so that the results can be scaled to various natural conditions (Table 1).
aw = width, k = thermal conductivity, ρ = density, c = specific heat, G = shear modulus, α = thermal expansion coefficient, b = cohesion. All other parameters are explained in the text.
2.4 × 10−2
7.4 × 10−4
1.3 × 10−3
5 × 104
 The boundary conditions for the velocities are as follows: The left, right, and bottom boundaries are free slip, and the top boundary is a free surface (Figure 1). The free surface is stabilized using the algorithm of Kaus et al. . A horizontal velocity is applied at the left and right boundaries and is modified every time step to maintain a constant background shortening rate, EB. The boundary conditions for the temperature are as follows: The heat flow through the left and right boundaries is zero, and the temperatures at the top and bottom boundaries are constant (Figure 1). The bottom temperature in the right half, T2, is a factor 1.05 larger than the temperature in the left half, T1, to generate a small temperature asymmetry. The model has three layers representing the upper and lower crusts and the lithospheric mantle (Table 1). A circular weak inclusion with a linear viscous flow law is in the middle of the lower crust.
 The free surface and the layer boundaries are defined by contour lines consisting of finite element nodes (2001 nodes on each boundary). The numerical mesh consists of about 1.2 million seven-node triangular finite elements, and if the mesh becomes too distorted, a new mesh is generated but the positions of the contour nodes are not modified.
 Previously published systematic simulations showed that during shortening, the lithospheric deformation is dominated by either thickening, folding or shearing (due to viscous heating) [e.g., Schmalholz et al., 2009]. Here, parameters have been applied that cause a shearing deformation mode due to viscous heating (Table 1) [e.g., Kaus and Podladchikov, 2006; Crameri and Kaus, 2010].
 At 15% shortening, a crustal-scale shear zone has developed that is characterized by high values of the square root of the second invariant of the strain rate tensor, EII, and shows a thrust-type shear sense (Figure 2c; movies of the results are available as auxiliary material). T in the shear zone is elevated due to viscous heating (Figure 2d). The τII is small in the shear zone (Figure 2a). In contrast, to the left and right of the shear zone are horizontal bands of high τII that corresponds to τII at the plastic yield strength. The tectonic overpressure, Po, is the difference between the pressure, P (mean stress), and the lithostatic pressure [e.g., Mancktelow, 2008]. Around the horizontal bands of high τII, also, Po is high. This agrees with the results of Petrini and Podladchikov , which showed that for a pure shear dominated deformation of the lithosphere, Po ~ τII in plastically deforming regions for angles of internal friction θ ~ 30°, as applied here (Table 1). However, the results show that Po is also high inside the shear zone where τII is significantly smaller (Figure 2b). Po increases from the top of the shear zone toward the crust-mantle boundary, indicating a more or less constant overpressure gradient along the shear zone (Figure 2b). No significant Po occurs in the shear zone in the mantle because compressive stresses around the shear zone are lacking (Figure 2b). Simulation 3 in the auxiliary material shows that significant Po also occurs around the weak inclusion within the weak lower crust during the initial stages of shear zone formation. During these initial stages, Po is largest in and around the shear zone.
 The relation between Po and τII can be investigated by considering the horizontal variation of the depth-averaged values of the horizontal total stress, σxx = −P + τxx; of the horizontal deviatoric stress, τxx; of P; and of Po (Figure 3). Depth-averaged σxx represents the horizontal force per unit length and is constant in the horizontal direction if external horizontal stresses acting at the top and bottom of the lithosphere are negligible [e.g., Molnar and Lyon-Caen, 1988], which is the case for the presented simulation (Figure 3a). Depth-averaged P shows an increase around the position of the shear zone (Figure 3a). Around the shear zone, depth-averaged Po differs significantly from depth-averaged τxx. Depth-averaged τxx is smallest around the shear zone because the shear zone is weak due to viscous heating. To maintain constant depth-averaged σxx horizontally across the shear zone (i.e., the horizontal force balance), P must increase to compensate the small τxx. Therefore, the depth-averaged Po can increase when depth-averaged τxx decreases and vice versa in and around the shear zone (Figure 3b). Maximal Po and τxx vary similarly along the horizontal direction (Figure 3c). The highest Po is higher than the highest τxx (Figure 3c; around X/hc = −1.5) because deviatoric stresses are limited by the yield strength whereas pressure is unlimited. The depth at which maximal Po and τxx occur is generally similar, but this depth can vary significantly around the shear zone (Figure 3d). The maximal Po has generally the same order of magnitude as the maximal τII within the model.
 The main results are summarized in a conceptual model (Figure 4). In the dominantly by pure shear deforming regions, the deviatoric stresses can be high and overpressure is directly related to deviatoric stress [Petrini and Podladchikov, 2000]. The high deviatoric stresses transmit the deformation imposed at the lateral boundaries. The thrust-type simple shear deformation inside the weak shear zone is driven by the pressure gradient along the shear zone (Figure 2b), and the small deviatoric stresses actually resist the shearing. A constant depth-averaged horizontal total stress (i.e., horizontal force balance) requires an increase of the pressure if the deviatoric stresses decrease due to the weakening inside the shear zone.
 The maximal values of Po/σc are ~1.4 to the left of the shear zone (Figure 3c; X/hc ~ −1.5) and ~0.7 inside the shear zone (Figure 3c; X/hc ~ 0.3). The σc is typical of the order of 1 GPa (e.g., 2800 kg m−3 × 9.81 m s−2 × 35 km), and Po can reach such values also if τII at the same location is significantly smaller (Figure 3). The maximal values of T/Tc in the crust inside the shear zone are ~1.4, providing for initial temperatures at the Moho Tc = 500–600°C values of T = 700–840°C, which indicate a temperature increase due to viscous heating in agreement with previously published studies [e.g., Burg and Gerya, 2005]. Such increased values of T and Po can affect the P-T-time trajectories of deformed rocks considerably. The results have, therefore, considerable implications for the geodynamic interpretation of HP-UHP rocks found in tectonic nappes because many studies assume that P is always identical to the lithostatic pressure and that Po is negligible [e.g., Jolivet et al., 2003]. A typical argument against tectonic overpressure is that many rock units in which HP-UHP rocks are found are mechanically weak (i.e., low effective viscosity and τII) and, therefore, significant values of Po are not possible [e.g., Brace et al., 1970; Schreyer, 1995]. Several studies [e.g., Gerya et al., 2008; Li et al., 2010] presented numerical results where significant Po (up to 100% of lithostatic values) is common in the lithosphere for subduction scenarios. However, the significant Po in these simulations is indeed restricted to strong rocks, while weak rocks usually exhibit low Po. In particular, rock samples that returned to the near surface from great depth within the weak subduction channel do not record significant Po [Li et al., 2010]. These numerical studies, therefore, support the abovementioned argument that weak rocks do not record significant Po. The results presented here also show that no significant Po occurs in the mantle part of the shear zone (Figure 2b). Our results do, however, restrict the context of applicability of this argument and show that during continental collision, significant Po can occur in weak crustal shear zones (i.e., rocks exhibiting τII ≪ Po). Many natural nappes are considered as result of thrust-type ductile shearing in crustal rocks. In the central Alps, such nappes often preserve their original Mesozoic-Tertiary sedimentary cover and exhibit considerable lithological consistency [e.g., Steck, 1990; Nagel, 2008]. Rooting of these crustal nappes in a weak subduction channel in the deep mantle is entirely due to conversion of pressure to depth based on the assumption of negligible Po that, as we show here, may not be universally applicable to all weak shear zones. A considerably simpler interpretation of high pressure recorded by crustal thrust-type weak shear zones is due to development of significant Po in order to satisfy the horizontal force balance during the continental collision.
 A crustal shear zone developing during lithospheric shortening due to viscous heating is a weak zone exhibiting decreased deviatoric stresses and effective viscosities. The pressure increases inside this weak shear zone to maintain a constant depth-averaged horizontal total stress and to fulfill the horizontal force balance across the shear zone. The pressure in the weak shear zone can be significantly larger than the lithostatic pressure, and this tectonic overpressure can be significantly larger than the deviatoric stress at the same location.
 The presented results indicate that pressure in natural thrust-type crustal shear zones and in tectonic nappes can be significantly larger than the lithostatic pressure, even if the rock units in the shear zone are weak. This has fundamental implications for the geodynamic reconstruction of tectonic nappes including HP-UHP rocks, because, for such reconstructions, the tectonic overpressure is usually neglected.
 The results show the importance of numerically resolving and understanding the rheological properties inside a shear zone and the mechanism of strain localization.
 This work is supported by SNF projects 200021_131897 and 200021_144250 and the University of Lausanne. We thank Taras Gerya and two anonymous reviewers for their constructive comments.
 The Editor thanks David Yuen, W. Buck, and an anonymous reviewer for their assistance in evaluating this paper.