Bedrock stresses induced through exhumation and tectonic processes play a key role in the topographic evolution of alpine valleys. Using a finite difference model combining the effects of tectonics, erosion, and long-term bedrock strength, we assess the development of near-surface in situ stresses and predict bedrock behavior in response to glacial erosion in an Alpine Valley (the Matter Valley, southern Switzerland). Initial stresses are derived from the regional tectonic history, which is characterized by ongoing transtensional or extensional strain throughout exhumation of the brittle crust. We find that bedrock stresses beneath glacial ice in an initial V-shaped topography are sufficient to induce localized extensional fracturing in a zone extending laterally 600 m from the valley axis. The limit of this zone is reflected in the landscape today by a valley “shoulder,” separating linear upper mountain slopes from the deep U-shaped inner valley. We propose that this extensional fracture development enhanced glacial quarrying between the valley shoulder and axis and identify a positive feedback where enhanced quarrying promoted valley incision, which in turn increased in situ stress concentrations near the valley floor, assisting erosion and further driving rapid U-shaped valley development. During interglacial periods, these stresses were relieved through brittle strain or topographic modification, and without significant erosion to reach more highly stressed bedrock, subsequent glaciation caused a reduction in differential stress and suppressed extensional fracturing. A combination of stress relief during interglacial periods, and increased ice accumulation rates in highly incised valleys, will reduce the likelihood of repeat enhanced erosion events.
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 Glacial erosion has long been recognized as one of the most important processes countering the tectonically driven growth of mountain belts and as such plays an integral role in controlling the past, present, and future development of mountainous landscapes. Although recent studies provide essential data on both the effects [e.g., Briner and Swanson, 1998; Duhnforth et al., 2010; Goehring et al., 2011; Häuselmann et al., 2007; Koppes and Montgomery, 2009; Muttoni et al., 2007; Preusser et al., 2010; Shuster et al., 2005] and timing [e.g., Berger et al., 1993; Johnsen et al., 2001; Lisiecki and Raymo, 2005; Muttoni et al., 2003; Penck and Brückner, 1909; Petit et al., 1999; Schlüchter et al., 2004; Zachos et al., 2001] of major alpine glacial events, questions remain regarding the connection between erosional processes and characteristic elements of formerly glaciated landscapes (e.g., U-shaped and overdeepened valleys, polished and striated bedrock surfaces, and surface-parallel extensional “exfoliation” or “sheeting” fracture systems) [e.g., Anderson et al., 2006; Bahat et al., 1999; Braun et al., 1999; Egholm et al., 2009; Hallet, 1979; Harbor et al., 1988; Herman et al., 2011; Herman and Braun, 2008; Martel, 2006; Pedersen and Egholm, 2013; Pelletier et al., 2010; Sternai et al., 2013; Tomkin, 2007]. As a result, the processes driving both the past and future development of alpine valleys remain enigmatic.
 This shortcoming may be addressed by better understanding the role rock mechanics plays in regulating the interaction of stresses imposed by tectonic and erosional processes. Investigations combining both geomorphological and geomechanical aspects of the Earth system are rare in the literature, although existing studies offer unique insights into the contribution of various processes to landscape change [e.g., Augustinus, 1995; Gerber and Scheidegger, 1969; Kinakin and Stead, 2005; Martel and Muller, 2000; Miller and Dunne, 1996; Molnar, 2004; Savage and Swolfs, 1986; Whalley, 1974]. Most recently, [Leith et al., 2013] evaluated the effect of tectonics and exhumation on near-surface fracture generation in a range of hypothetical topographic and tectonic settings. They discussed the effects of long-term bedrock strength and highlighted path-dependent and threshold-limited aspects of bedrock behavior during exhumation, illustrating the relevance of their approach by addressing the mechanics of extensional fracture generation within a generic glacial valley. Here we test the significance of relationships identified by [Leith et al., 2013] as they affect processes driving the evolution of real glacial landscapes. By comparing insights derived from numerical modeling to patterns of spatial and temporal erosion evident within the Alpine landscape (here “Alpine” or “Alps” refers specifically to the central European Mountain belt, while corresponding common nouns “alpine” or “alps” are used as a general reference to mountainous regions), we are then able to identify key relationships influencing the development of a prominent Alpine Valley (the Matter Valley, Canton Valais, Switzerland).
2.1 Physics of Glacial Erosion
 Processes of glacial erosion include abrasion, quarrying (or plucking), incision of meltwater channels, and chemical weathering or dissolution. While morphological evidence suggests that abrasion and quarrying are essential elements of U-shaped valley formation, recent studies using cosmogenic nuclides to date bedrock surfaces have determined that glacial abrasion rates are relatively low, on the order of ~0.04 mm/yr to ~0.4 mm/yr [Briner and Swanson, 1998; Duhnforth et al., 2010; Goehring et al., 2011]. Total erosion through abrasion during the previous (Würmian) Alpine glacial cycle is thus likely to have been less than 30 m. When considered in the context of the Alpine glacial record, which identifies five major Pleistocene glacial cycles since the end of the mid-Pleistocene revolution (MPR) at marine isotope stage (MIS) 22 [e.g., Berger et al., 1993; Johnsen et al., 2001; Lisiecki and Raymo, 2005; Muttoni et al., 2003; Penck and Brückner, 1909; Petit et al., 1999; Schlüchter et al., 2004; Zachos et al., 2001], it is unlikely abrasion alone was responsible for the development of deep (~1000 m) U-shaped glacial valleys in strong bedrock, and additional processes (such as quarrying) likely account for up to 85% of alpine valley erosion. Measurements of sediment grain size distributions in modern glacial lakes and meltwater channels support these interpretations, indicating that quarrying accounts for roughly 50%–70% of recent subglacial sediment production [Hallet et al., 1996; Loso et al., 2004; Riihimaki et al., 2005]. However, bedrock surfaces polished through abrasion are characteristic of formerly glaciated landscapes, and the dominance of quarrying, at least during the last glacial cycle, remains controversial.
 Notable studies by Iverson , Hallet , Cohen et al. , and Iverson  help address this inconsistency by providing insight into the mechanics of glacial quarrying. These empirical and numerical studies suggested that while total stresses beneath glacial ice are generally not high enough to create new bedrock fractures, basal water pressure fluctuations can affect the distribution and magnitude of stresses applied to preexisting fractures. They have explained how these fluctuations can drive subcritical crack growth, producing subvertical fractures on the lee side of bedrock steps. The contribution of these glacier-induced fractures to the quarrying process is uncertain, however, and although the bedrock fracture pattern has been shown to play an important role in controlling the transition from abrasion to quarrying (a decrease in fracture spacing from ~3.3 m to ~1.1 m is considered sufficient to force a system toward quarrying) [Briner and Swanson, 1998; Duhnforth et al., 2010; Hooyer et al., 2012], Hooyer et al.  found that preexisting fracture networks strongly control glacial quarrying in a range of environments. They proposed that destruction of intact rock bridges within preexisting joints is the principal subglacial fracturing mechanism. Aside from this debate, it is likely that subglacial fracturing defines sediment availability [Hallet, 1996] and is thus the limiting factor for both controlling quarrying rates and the development of deep glacial valleys in strong rock.
 Despite the importance of subglacial fracture development, most numerical models seeking to predict large-scale patterns of glacial erosion have not explicitly incorporated fracture mechanisms in their formulation [Iverson, 2012]. They instead assumed that erosion rates scale with ice-sliding velocity and/or subglacial hydrology and integrated an empirically derived linear or power law factor to account for bedrock erodibility [e.g., Anderson et al., 2006; Braun et al., 1999; Egholm et al., 2009; Hallet, 1979; Herman et al., 2011; Herman and Braun, 2008; Pedersen and Egholm, 2013; Pelletier et al., 2010; Sternai et al., 2013; Tomkin, 2007]. This erodibility factor aims to capture certain aspects of bedrock behavior, suggesting that weaker rock is more likely to fracture and aid glacial erosion, although field evidence seems to contradict the assumption, and glacial valleys in stronger bedrock tend to be deeper and narrower [Augustinus, 1995 and references therein; Matthes, 1930].
2.2 Spatial and Temporal Patterns of Glacial Erosion
 Large-scale glacial erosion models are capable of producing quasi-parabolic valley cross sections from initial V-shaped topographies, with a notable caveat—they do not readily reproduce the shoulder morphology often seen marking the transition from V-shaped to U-shaped valleys several hundred meters below the level of maximum ice occupation in postglacial landscapes (Figure 1). Harbor  reproduced this feature by limiting the maximum ice elevation to the shoulder level or adding a more erodible material to the center of the valley [Harbor et al., 1988], while Braun et al.  used a “constriction factor” to limit erosion in the region of greatest curvature close to the valley axis. Harbor et al.  noted that the introduction of a more erodible layer in the center of their model caused a marked change in the erosional pattern, as linear hillslopes near the ridges progressively lowered while a broad shallow trough formed at the valley axis. The requirement for these exceptional rules to reproduce common morphologies suggests that an important element of the erosion process is missing from these models, an inference strengthened when comparing model results to evidence of temporal glacial erosion in mountain regions.
 The oldest sediments in overdeepened Alpine Valleys commonly date to the end of the MPR (Figure 2b) or at least indicate repeated occupation and excavation of infill during most glaciations since the MPR [Preusser et al., 2010], suggesting valley formation may have taken place early in the glacial record and subsequent glaciations had limited effect on the landscape. Sedimentary studies on the southern side of the Alps [Muttoni et al., 2003] complement these observations, as a shift from central to southern Alpine dominated provenance at 0.87 Ma (the end of the MPR) is associated with a marked coarsening of material and the appearance of turbidite deposits (Figure 2e). Muttoni et al.  interpreted this change to reflect the progradation of glacially sourced alluvial fans, referencing studies in the Venice area and the Bengal Fan, where coarsening and rapid increases in the deposition rate of glacially derived sediments at MIS 22 are similarly unique in the sedimentary record. Higher-resolution studies by Shuster et al.  and Häuselmann et al. [2007, 2008] constrained between 1000 m and 2000 m of alpine relief production to short periods in the regional glacial histories, during which erosion rates in the valleys (the Klinaklini Valley, Alaska, and Aare Valley, Switzerland) exceeded 5 mm/yr and 1.2 mm/yr, respectively (Figure 2c). In each of these locations, a prominent shoulder is present on the valley walls, well below the maximum ice elevation. Neither study conclusively identified a subsequent period of rapid relief formation, mirroring the aforementioned pattern of early relief production inferred from overdeepened valley sediments throughout the Alps.
 These studies documented exceptional periods of rapid erosion coincident with a transition to pronounced glaciation following a long period of subaerial landscape development. Although such temporal patterns are often not the focus of glacial erosion models, this record of widespread nonuniform relief development cannot be replicated by existing models, which commonly predict an up valley propagation in the erosion front while feedbacks between erosion, relief, and ice accumulation tend to keep the sediment flux relatively constant over time [e.g., Sternai et al., 2013]. As with the problem of modeling valley shoulder formation, it seems unlikely that periods of unique or exceptionally strong alpine-wide glacial erosion will be resolved without incorporating additional physics to describe subglacial processes.
 We propose that both the distinct valley shoulder morphology, as well as increasing evidence for a sharp onset of glacial erosion, indicate threshold-limited and/or path-dependent behaviors may control deep U-shaped glacial valley formation. [Leith et al., 2013] associate the generation of sheeting or exfoliation joints in highly stressed near-surface bedrock with such behaviors, suggesting that they develop in a manner similar to spalling and rock burst in deep excavations (see Figures 3a–3c). These phenomena are now well-understood examples of macroscopic extensional fracture growth [Diederichs, 2003, 2007; Germanovich and Dyskin, 2000; Hoek, 1968; Kaiser and Kim, 2008], and drawing parallels to stress changes resulting from tunneling and mining activities [Leith et al., 2013] proposes that natural extensional fractures result from threshold-limited microcrack development in intact rock as erosion and deglaciation reduces confining stresses on critically stressed bedrock. They find that favorable conditions for such fracture formation are encountered close to the axis (i.e., the inner ~20%) of alpine valleys when the principal tectonic strain is oriented along valley or throughout the landscape when tectonic strain is oriented perpendicular to the valley. [Leith et al., 2013] propose that an initial phase of glacial erosion penetrating a mantle of weathered bedrock can provide the required stress path to drive active subglacial extensional fracture formation, although stress relaxation during interglacial periods inhibits subsequent new fracture growth during later glacial periods.
 Drawing from the principles developed by [Leith et al., 2013], we assess short- and long-term bedrock behavior in response to stresses imposed by glacial and tectonic processes within an Alpine valley. We use the Matter Valley as an example to both derive realistic parameter values for our model initialization and evaluate the contribution of proposed behaviors to the operation of geomorphic processes within the valley. Considering these processes in a more regional context then allows us to identify key feedbacks and controls on spatial and temporal patterns of Alpine-wide glacial erosion. In particular, we draw attention to threshold-limited bedrock behavior that may have driven the production of a distinctive valley shoulder morphology 700 m below the Last Glacial Maximum (LGM) trimline (Figure 1), and positive feedbacks favoring an initial period of rapid U-shaped glacial valley development.
3 Study Area
3.1 Tectonic History
 The Alpine Mountain belt is the most prominent marker of a broad region of deformation associated with north-south collision between the Apulian and European Plates. Our study area in the Matter Valley belongs primarily to the Penninic nappes, the intermediate units of a series of stacked nappes within the Alpine chain. As intermediate terranes (including the Piemont-Liguria oceanic basin, Briançonnais microcontinent, Valais Ocean, and European continental crust below the Adriatic Plate) were progressively subducted beneath the Apulian Plate between 65 Ma and 35 Ma, an excessively thick accretionary wedge developed above the subduction zone. Accommodation of collision within this thick accretionary wedge led to the development of a northward dipping retro-thrust (the “Insubric Line”) to the south of our study region, while a series of low-angle backthrusts propagated northward. As these thrusts became more persistent and extended to the surface, they defined a doubly vergent wedge structure, leading to increased exhumation and coinciding with a gradual reduction in continental convergence up until approximately 7 Ma [Schmid et al., 1996] (Figure 2a).
 The most recent geological evidence of late-stage collision across the Alps is associated with the cessation of tectonic activity across the Jura Mountains in the late Miocene (ca. 3.3 Ma, see Figure 2a) [Petit et al., 1996], although ongoing Africa-Eurasia collision within the Mediterranean can be observed from GPS measurements and earthquake fault plane solutions [Serpelloni et al., 2007]. Despite this long history of regional collision, structural and tectonic evidence suggests that the principal tectonic regime for the central (Swiss) region of the Alps throughout perhaps the last 30 Ma is associated with extensional or transtensional strain. The Penninic nappes experienced localized extension throughout much of this period (Figure 2a) [Sue et al., 2007]. Decreasing convergence rates around 7 Ma were associated with an increase in normal-mode tectonics within the nappes, driving a transition from boundary to body forces as isostacy/buoyancy began to take over [Sue et al., 2007]. Extensional strains are ongoing today, evident in both inner Alpine seismic and GPS records.
3.2 Lithology and Structure
 Most of our study area is contained within two subzones of the Penninic nappes: the Grand St. Bernard multinappe system (namely the Siviez-Mischabel nappe) and the Zermatt-Saas Fee zone (Figure 4) [Schmid et al., 2004, 1996]. The Siviez-Mischabel nappe is composed of Permian-Carboniferous crystalline rocks derived from the Briançonnais microcontinent, including orthogneisses and paragneisses, mica-rich schists, and quartzites [Bussy et al., 1996; Escher et al., 1993; Genier, 2007]. The Zermatt-Saas Fee zone contains ultramafic rocks from the Piemont-Liguria basin and is located at the head of the Matter Valley. Foliation generally dips gently toward the west or southwest following the main lithologic contacts, and both foliation-parallel and foliation-perpendicular fractures are common, while large-scale normal faults with lengths of up to several hundred meters occur with high frequency (Figure 4) [Girod, 1999; Sue et al., 2007; Willenberg, 2004; Yugsi, 2010]. The Rhone Fault approximates the present-day Basal Penninic Thrust and strikes subparallel to the Rhone Valley immediately north of our field area, providing a distinct lithologic, structural, and tectonic northern boundary to the study region.
3.3 Extensional Fracturing
 Exfoliation joints display features consistent with tensile (Mode 1) fracture, and as such, can readily be identified by characteristic aspects of their morphology, orientation, and generation [Bahat et al., 1999; Bucher and Loew, 2009]. Development of macroscopic extensional fractures is strongly dependent on both intact and rock mass properties and is favored in massive, isotropic, homogeneous materials. Stress concentrations within more variable rock masses increase internal shear stresses and encourage increasing Mode 2 or Mode 3 (sliding or tearing) fracture development. We observe exfoliation joints within the foliated orthogneiss and paragneiss regions of the Matter Valley where the rock mass is more homogeneous or displays minimal preexisting tectonic fracturing. However, the distribution of these macroscopic fractures is limited within the valley, as weak mica-rich shear zones and heterogeneities in the intact gneiss fabric tend to inhibit persistent exfoliation fracture development, favoring alternative fracture modes. Additional insight into the behavior of the paragneiss and orthogneiss materials can be derived from a drainage diversion tunnel located on the western side of the valley opposite the village of Randa (Figure 4), near the location of our modeled cross section [Girod, 1999]. Although much of the tunnel is either beyond the limit of natural exfoliation fracture development (i.e., >200 m depth) or the rock mass is not conducive to the generation of exfoliation-type fractures (particularly at tunnel portals), recent extensional fractures in the form of spalling are present within massive orthogneiss units on the middle to upper valley side of the tunnel approximately 600 m below ground.
3.4 Glacial History
 As part of the Alpine Mountain belt, the Matter Valley has been subject to a long history of repeated glacial and interglacial cycles (Figure 2b). Local topographic relief is on the order of 2500 m, among the greatest in the Alps, and the strong contrast between presently glaciated mountain peaks and temperate inner U-shaped valleys is evidence for the complex and spatially variable glacial imprint on the region (Figure 1). Glacially smoothed or striated bedrock surfaces and remnant moraine deposits are common at most elevations within the area, from the ~1300 m valley floor to >3000 m where glacial erosion is ongoing.
 Long-term (~65 Ma) records of Alpine glacial and interglacial activity can be inferred from the global climatic record, which has been resolved through high-resolution studies of glacial ice and ocean sediment cores (Figure 2b) [e.g., Johnsen et al., 2001; Lisiecki and Raymo, 2005; Petit et al., 1999; Zachos et al., 2001]. This long and largely coherent global temperature record supports fragmentary evidence preserved in superposed moraine sequences [e.g., Ivy-Ochs et al., 2009; Penck and Brückner, 1909], stratigraphy of outwash deposits [e.g., Muttoni et al., 2003; Preusser et al., 2011], and glacial trimlines [e.g., Kelly et al., 2004] to describe a complex history of repeated Alpine glacial cycles dating back to the late Pliocene.
 The mid Miocene marks the gradual transition from a long warm period to ~15 Ma of progressive cooling, increasing climatic variability, and likely associated increase in global ice volumes (Figure 2b) [Zachos et al., 2001]. Strong glacial/interglacial cyclicity began during the mid Pliocene with a climatic shift associated with the onset of major Northern Hemisphere glaciation at 2.7 Ma; this was followed by repeated ~41 ky period glacial/interglacial cycles [Raymo, 1994]. Evidence suggests that the Alps experienced only minor glaciation during this time with gradually increasing sediment yields linked to an efficient fluvial system [Muttoni et al., 2003; Willett, 2010]. The mid-Pleistocene revolution [Berger et al., 1993] between 0.94 Ma and 0.89 Ma then marked an increase in global ice volumes and periodicity to the present duration of ~100 ky [Imbrie et al., 1993; Raymo et al., 1997]. Five of these longer-duration climatic oscillations, associated with marine isotope Stages 22, 16, 12, 6, and 5d-2, produced significant ice volumes in the Alps (Figure 2b). The tendency for subsequent glaciations to overprint former phases, however, means that little is known about the effect of these early, extensive glaciations or subsequent cycles on the evolving landscape prior to the Würmian glaciation (MIS 5d-2). During the LGM (24–21 ka BP) [Schlüchter, 1988, and references therein] ice lobes fed by major Alpine Valleys such as the Rhine and Rhone extended north to the Jura and beyond Lake Constance, while a more temperate climate and lower forefield elevations on the southern side of the Alps restricted ice extents to the range front [Bini et al., 2004; Florineth and Schlüchter, 1998; Ivy-Ochs et al., 2006; Kelly, 2003]. At this time the Matter Valley was occupied by the ~1400 m thick Valais icefield, a major contributor to the LGM Rhone Glacier (Figure 1) [Kelly, 2003]. Deglaciation of the Alps was initiated at ~18 ka BP [Ivy-Ochs et al., 2006; Schlüchter, 1988, and references therein].
4 Modeling Strategy
4.1 Trilinear Fracture Envelope
 To assess the relationship between in situ stresses and erosional processes in glacial landscapes, we use material behavior described by a trilinear fracture envelope to analyze the bedrock response to stress changes within the glacially and tectonically active Matter Valley. Initially applied by Diederichs  to evaluate the potential for spalling in deep tunnels, the trilinear fracture envelope accounts for microcrack initiation and extensional fracture propagation under various principal stress conditions (Figure 5). The envelope consists of three key behavioral thresholds: (1) the microcrack initiation threshold, (2) the extensional fracture limit, and (3) the empirical Hoek-Brown shear failure envelope. Unlike Diederichs et al. , however, who apply the envelope to determine plastic yield limits for numerical models, we use the envelope to assess likely modes of bedrock fracture in a model limited by long-term bedrock strength criteria: namely tensile fracture, new microcrack development, and Byerlee's empirical friction law [see Anderson [1905, 1951]; Byerlee ; Brace and Kohlstedt ; Leith et al. .
 The microcrack initiation threshold (σCI, Figure 5) describes the maximum principal stress required to induce tensile failure of intergranular bonds in rock under compressive loading. This threshold marks the onset of inelastic strain in intact rock [Lajtai and Lajtai, 1974; Tapponnier and Brace, 1976] and can limit long-term stress magnitudes in the near surface [Leith et al., 2013]. Equation ((1)) correlates the microcrack initiation stress with a rock's unconfined compressive strength (UCS) and the minimum principal (or confining) stress (σ3) by the relationship:
where A and B are empirically derived constants [Diederichs, 2007].
 Individual microcracks provide potential nucleation sites for macroscopic fractures, while well-developed microcrack fabrics create preferential pathways for fracture propagation [Hoek and Bieniawski, 1965; Nasseri and Mohanty, 2008]. Stable macroscopic fracture lengths are a function of the principal stress ratio, an increase in which leads to a nonlinear increase in crack propagation from initial flaws [Hoek and Bieniawski, 1965]. The extensional fracture limit (σsp) describes the minimum stress required to strongly increase stable fracture lengths for a given material (represented by the spalling constant, msp) and confining stress (equation ((2)) and Figure 5). Although the length of stable cracks is dependent on the initial flaw length and orientation [Fairhurst and Cook, 1966], stresses in excess of this limit will promote crack extension, interaction, and coalescence [Diederichs, 2003, 2007].
 Since the extensional fracture limit is a product of confinement, even relatively low surface-parallel stresses at the ground surface (where the confining stress is zero) can generate long extensional fractures as the inhibiting stress tends toward zero. [Leith et al., 2013] suggest that new microcrack development in bedrock where stresses exceed both the microcrack initiation threshold and the extensional fracture limit can trigger unstable macroscopic extensional fracturing as the initiating stress can be significantly above that required for stable fracture propagation. This mechanism is analogous to both spalling in underground excavations and exfoliation or sheeting joint formation in natural landscapes.
 The modified Hoek-Brown failure criterion [Hoek et al., 2002] is incorporated in the trilinear fracture envelope to describe the shear strength of fractured rock masses (Figure 5). Although we anticipate that ongoing fracture development will lead to a reduction in rock mass strength, natural variability in the nucleation, orientation, and interaction of these fractures means that it is difficult to constrain the relationship. For simplicity, we therefore assume the majority of our model is competent rock mass with a Geological Strength Index (GSI) >80 and discuss the likely effect of reduced shear strength as a result of near-surface fracturing in section 6.4.
4.2 Intact Rock and Rock Mass Properties
 We use representative rock mass properties based on past investigations in the study area [Girod, 1999; Girod and Thelin, 1998; Willenberg, 2004; Yugsi, 2010], as well as available literature values for the microcrack initiation and extensional fracture limits in intact crystalline rock [Diederichs et al., 2007; Leith et al., 2013]. These properties are summarized in Table 1 and are used to define the three behavioral thresholds required to plot a material-specific trilinear fracture envelope (Figure 6) for the orthogneiss (Randa augengneiss) dominant in our chosen cross section (see section 4.1).
 While present-day topography provides an important control to evaluate gravitational stresses, the combination of ongoing tectonic strain and developing bedrock stresses during exhumation can play an equally or more important role in defining the in situ stress state of the brittle crust [Leith et al., 2013]. Here we review the evolution of tectonic and erosional processes in the Matter Valley in order to set the initial conditions for our in situ stress model.
4.3.1 Regional Stress Analysis
 The Penninic units of the central Alps have undergone local extension throughout at least the late Tertiary stages of Africa-Eurasia collision [Sue et al., 2007] (Figure 2a). This is most clearly expressed in normal and strike-slip faults associated with “late Alpine faulting,” which dominate the Alpine arc and overprint ductile compressional structures associated with formation of the nappes [Champagnac et al., 2003; Grosjean et al., 2004]. The majority of these faults indicate orogen-parallel extension prior to the Miocene, with strike-slip oblique followed by normal faulting perpendicular to the axis of the Alpine belt taking over in the late Miocene to Pliocene. Extension is, however, likely to be younger than transcurrent strains within our study area, as normal-mode faults overprint earlier strike-slip structures in some locations [Champagnac et al., 2006; Sue et al., 2007].
 While some extensional structures indicate possible deformation below the brittle-ductile transition, most are post metamorphic and cross-cut folds, schistocity, and nappe boundaries. A wide range of fault mineralization and structural properties indicate that normal faulting has taken place under diverse conditions (temperature, depth, strain rate, fluid pressure, and composition), suggesting a long extensional history for the brittle crust [Champagnac et al., 2004; Girod and Thelin, 1998]. These fault structures indicate that the minimum principal stress (σ3) in the brittle crust below Valais is oriented horizontally and has remained consistent for the changing Tertiary stress regimes (azimuth ~065° in southern Valais and ~025° in the north of the study region), reflecting a single phase of deformation [Champagnac et al., 2006]. The change from transcurrent to extensional strain mentioned above is likely to have resulted from the regional maximum horizontal stress (σH) reducing below the magnitude of the vertical stress (σv, i.e., reducing from the maximum principal (σ1) to intermediate principal (σ2) stress), possibly reflecting a reduction in regional strain rates [Champagnac et al., 2003, 2006]. Earthquake locations and fault plane solutions support the above structural observations, and predominantly normal-mode earthquakes south of the Rhone Valley indicate a minimum principal stress orientation of ~010°, with brittle strain largely restricted to the upper 10 km of the crust [Kastrup et al., 2004; SED, 2011]. Interpolation of regional GPS measurements indicates that NNE-SSW extension is ongoing at a present-day rate of approximately 0.3 nanostrain/yr, with a similar or lesser component of ESE-WNW compression [Tesauro et al., 2005].
 Information on long-term (Ma to ka) exhumation and erosion rates can be derived from high-resolution thermochronometric and sedimentological studies. Fission track thermochronometry for sites across the western Alps indicates that erosion rates have increased nonlinearly from a value of ~0.3 mm/yr at 14 Ma to 0.6 mm/yr or more at 2 Ma [Vernon et al., 2008] (Figure 2c). This inference is supported by low-temperature thermochronometry from the central Swiss Alps [Valla et al., 2012], although the more recent study suggests that fission track rates may include two short pulses of very rapid erosion (up to 2 mm/yr) near the end of the Miocene (until ~10–8 Ma) and during the Pliocene (between ~2 and 0 Ma), and the mean long-term rates of Vernon et al.  may therefore be slightly high. Recent denudation rates derived from cosmogenic 10Be concentrations in riverborne sediment from the Matter Valley are consistent with both interpretations, suggesting rates of 1.13 ± 0.46 mm/yr for the last ~2 ka [Wittmann et al., 2007] (Figure 2c).
 Fission track data for the Randa Orthogneiss indicate that surficial bedrock passed through the brittle-ductile transition (estimated at 10 km beneath the Valais) [Ellis and Stockhert, 2004] during a period of rapid exhumation at around 30 Ma [Bussy et al., 1996; Steck and Hunziker, 1994], at which time the Penninic nappes had already entered into a period of ongoing extension, possibly reflected in a slowing of the exhumation rate (Figure 2a). As exhumation reached a minimum at approximately 13 Ma, the upper ~4 km of the present-day crust had passed through the brittle-ductile transition, and intrinsic forces begun to control the formation of the orogen. Higher-resolution thermochronometry data [e.g., Valla et al., 2012; Vernon et al., 2008, and references therein] then constrain the remaining ~6 km of exhumation, during which rates generally increased as body forces and the effects of a cooling and more variable climate begun to dominate orogenesis. Comparing thermochronometric data to the global climatic record, we can account for approximately 9 km of exhumation between the termination of the Mi-1 Glaciation and the MPR, enough to negate any effects of this earlier glaciation on present-day crustal bedrock (Figure 2d).
4.3.2 Conceptual Crustal Stress Model
 Based on the regional stress analysis, highlighting a history of ongoing tectonic extension and continued exhumation under relatively stable climatic conditions, we adopt a conceptual stress model similar to that of [Leith et al., 2013]. We assume that stresses in the brittle crust at the onset of major glaciation (the initial stage of our model) are in equilibrium with Byerlee's law and long-term intact rock strength (see section 4.1) and reflect a stress field consistent with a topography in which relief has been constant through time. The regional stress history supports principally strike-slip or normal fault development during which the minimum principal stress (σh) was oriented parallel to the valley axis. Steeply dipping N-S trending faults representing this stress regime are prominent structural elements within the Matter Valley (Figure 4), and given the above description, likely represent primary tectonic structures throughout the brittle crust.
 As discussed by [Leith et al., 2013], the elastic response of laterally confined bedrock to exhumation leads to a reduction of all three principal stresses, although horizontal components will be retained unless relieved through faulting or sufficiently rapid tectonic extension. This preservation of horizontal stresses will lead to an increase in the ratio of horizontal to vertical stress and therefore increase normal stresses across steeply dipping normal faults. If extensional tectonic strain rates are relatively low, these increasing normal stresses will tend to lock up fault structures, preventing slip. Moderate seismic activity, with normal-mode earthquakes throughout the Valais region [Maurer et al., 1997], indicates that this is not the case and that the long-term N-S tectonic strain rate exceeds that of similarly oriented exhumation-induced strains. This means that the subvertical and ~ NNE-SSW trending horizontal stresses throughout the majority of the crust are likely to be limited by slip on critically stressed, optimally oriented normal faults, and we therefore use Byerlee's law to define initial valley-parallel (i.e., out-of-plane) stresses (Table 2). A consistent ESE-WNW orientation for both the historical maximum horizontal stress and present-day convergent strain, combined with a lack of similarly trending reverse faults, indicates that exhumation-induced stresses oriented across the valley are unlikely to be relieved through faulting. As in [Leith et al., 2013], we therefore assume that stresses perpendicular to the primary tectonic strain orientation (i.e., in-plane) are limited by long-term intact rock strength (Table 2).
Table 2. FDM Regional Stresses and Geometry
Tectonic stress ratio
Maximum ice elevation
Width of U-shaped valley
Depth of U-shaped valley
Valley curvature exponent
Width of river incision
Depth of river incision
Lower model boundary
−10, 000 m
4.4 Finite Difference In Situ Stress Modeling
 We use an elastoplastic, plane-strain Finite Difference Model (FDM) [Itasca Consulting Group, Inc, 2011], initialized using representative regional stresses and bedrock strength to estimate the stress field beneath one half of a recreated V-shaped cross section of the Matter Valley (Figure 7 and Table 2). The valley geometry is derived from the western ridge profile of the valley, assuming that this is representative of a subaerial Miocene landscape (Figure 8, Stages 1 and 2) and the valley form is symmetrical beyond the model boundaries. Slopes on the eastern side of the valley are frequently affected by deep-seated landsliding and are less likely to represent the preglacial topography.
 The path-dependent nature of brittle bedrock behavior means that stress redistribution during U-shaped valley incision depends on the dynamics of local bedrock erosion and glacial ice development [Leith et al., 2013]. Although glaciation and deglaciation are likely to take place over a period of thousands of years, there are currently almost no data to constrain the rate at which a rock mass responds to external stress changes over long timescales [Damjanac and Fairhurst, 2010], making justification of relative response rates difficult. In order to illustrate aspects of ice accumulation, glacial erosion, and bedrock behavior, we therefore adopt a staged modeling procedure, representing changes from short-term elastic bedrock response to long-term brittle response by switching between elastic and elastoplastic constitutive behavior (Figure 8). We allow the maximum ice elevation to extend up to the level of the LGM trimline (Figures 7 and 8, Stages 3 and 4) and adopt elastic conditions for ice accumulation (Stages 3, 21, and 27–31) or downwasting (Stages 15–19 and 23), assuming that this was relatively rapid. We simulate erosion of the present-day inner U-shaped valley by manually removing slices of bedrock (Stages 5–14), alternating between elastic and elastoplastic constitutive behaviors where long-term strength-limited stress conditions are assumed to keep pace with valley development. We consider this latter assumption to be conservative, as bedrock stresses are allowed to relax following each erosional stage. If elastic stresses were maintained during U-shaped valley formation, stresses would concentrate around the valley, increasing the intensity of predicted fracturing during deglaciation. During full interglacial or glacial conditions (Stages 2, 14, 20, 22, 24, 26, and 32), we allow stresses in the model to relax to long-term strength-limited conditions.
 Late model stages include the removal of a 100 m deep V-shaped wedge of bedrock from the valley center (Stages 25–26), allowing us to test the effect of interglacial river incision on in situ stress conditions in a generic alpine valley. We complete the model run by reintroducing ice in the valley to simulate a subsequent glaciation (Stages 27–31), maintaining elastic conditions until LGM-equivalent ice cover is reached, before switching to elastoplastic conditions to allow stresses to reduce to long-term strength-limited levels (Stage 32). As ice during these final stages primarily acts to load the model surface and reduce differential stresses, plastic strains in the final model stage are negligible.
5 Model Results
5.1 In Situ Stresses
 In situ stresses in our model cross section are derived from the tectonic and erosional history of the Alpine region. These stresses are redistributed beneath the V-shaped topography during initial equilibration stages, while both elastic and brittle-plastic material behaviors regulate their magnitude. The resulting stress field is generally consistent with the specified regional NNE-SSW extensional tectonic regime, although stresses in the near-surface are more strongly influenced by the combined effects of exhumation, long-term bedrock strength, and topography. These effects are most evident in a large region of long-term strength-limited stresses extending as much as 3 km below the valley axis (Figure 9). Plastic strain in this region causes displacement toward the valley axis, relieving crustal stresses beneath the ridge and increasing both horizontal and vertical stresses beneath the valley. The elastic response to this strain reduces both the in-plane and out-of-plane subhorizontal stresses at the ridge, and under plane-strain conditions can develop out-of-plane tensile stresses (Figure 9). We consider this to be an artifact of the 2-D model; topographic relief perpendicular to the model plane would normally facilitate valley-parallel strains, preventing significant tensile stresses from developing in this region.
5.2 Near-Surface Fracture Distribution
 By comparing modeled stresses to behavioral thresholds described by the material-specific trilinear fracture envelope (Figure 6), we can assess the probable distribution and modes of both microscopic and macroscopic bedrock fracture (Figure 10). As microcrack initiation (i.e., long-term bedrock strength) defines the maximum allowable differential stress in elastoplastic model stages, model solutions converge to exclude stresses in excess of this threshold. We therefore illustrate the distribution of bedrock at the microcrack initiation threshold (colored blue) and verging on brittle fracture limits (within 10% of the fracture threshold, colored gray). As mentioned above, strain toward the valley axis following model initialization creates an anomalous region of tensile fracture beneath the ridge (shaded yellow, Figure 10a). This is underlain by a region of macroscopic fracture propagation (orange) and bounded by a zone of rapid new extensional fracture development near the valley axis (red).
 Glacial ice confinement in subsequent model stages suppresses microcrack development (Figure 10b), limiting the zone of critically stressed bedrock to the valley axis. Staged erosion under elastic conditions then concentrates stresses below the valley floor, promoting microcrack development, before the model is switched to elastoplastic constitutive behavior and the zone of critical differential stresses drops beneath the valley floor while surface stresses reduce to subcritical levels (Figure 10c). Lowering the glacier surface under elastic conditions then allows rock slopes to strain toward the valley, increasing tensile stresses beneath the ridges, and promotes rapid extensional fracture formation by decreasing the confining stress within the valley (Figure 10d). Maximum and minimum principal stress trajectories near the valley floor are respectively oriented parallel and normal to the ground surface (Figures 10d and 10e), and resulting extensional fracture orientations should therefore align with the valley walls. Long-term strength-limited interglacial conditions are represented in Figure 10e, and we illustrate the effects of subsequent glaciation in Figure 10f. Glacial reoccupation both confines bedrock (strongly reducing predicted microscopic and macroscopic fracture formation) and forces the valley walls apart, suppressing tensile stresses above the valley shoulders, producing a generally more stable stress state.
5.3 Inner Valley Fracture Distribution
 The in situ stress field close to the axes of our modeled V-shaped and U-shaped valleys is consistent with that required to support the development of new microcracks and subsequent unstable propagation of macroscopic extensional fractures in near-surface bedrock (see section 4.1). Stresses in the steady state V-shaped valley support microcrack development and macroscopic fracture propagation up to ~600 m from the valley axis and to ~150 m depth (Figure 11a). Transitioning to a U-shaped valley under elastoplastic model conditions and constant ice loading then decreases stresses beneath the developing valley shoulders, evident as an increase in local tensile failure from Figures 11b to 11c. Each elastic stage of this erosion process concentrates stresses and increases the stress gradient beneath the valley floor, drawing the zone of microcracking toward the surface (Figures 11b–11d). Fracture distributions in the final U-shaped valley stages change dramatically between elastic and elastoplastic models, indicating that significant microcrack generation is required to relieve stresses beneath such deep U-shaped valleys (cf. Figures 11c and 11d). Although rapid macroscopic extensional fracturing is suppressed by the LGM-equivalent ice cover during these stages, reducing ice thickness during valley incision will favor macroscopic fracture development, as the ratio of stored bedrock stress to normal stresses derived from ice loading the valley walls increases. The extent of fracturing will be limited by the distribution of new microcrack growth (Figure 11e). Figure 11f illustrates the predicted fracture distribution following a reduction of stresses to long-term limits within the valley, indicating that macroscopic extensional fracture propagation may take place up to ~100 m beneath the valley floor. This is similar to depths of exfoliation fracturing reported in literature [i.e., Bucher and Loew, 2009; Jahns, 1943].
 Stream incision, earthquake shaking, and rock mass degradation may all act to reduce near-surface in situ stresses during interglacial periods. Figure 12a presents a simple example where fluvial incision at the valley axis draws stress trajectories down below the U-shaped valley floor. Subsequent loading by glacial ice will then confine destressed bedrock, suppressing macroscopic fracture development, and reduce the extent of fracturing (cf. Figures 12a and 12b). Finally, a return to LGM-equivalent glacial conditions will suppress fracturing and have a strong stabilizing effect on bedrock throughout the valley (cf. Figures 12c and 11d).
 Understanding the influence of tectonic and erosional processes on bedrock fracture development provides us with important new insights into the interaction of these processes in alpine landscapes. While our model does not allow coupled quantitative analysis of feedbacks between fracture generation, glacial erosion, and the resultant in situ stress field, our results do support a qualitative assessment of potential feedbacks and resulting landforms.
6.1 Enhanced Glacial Erosion
 In this work, we suggest that the extensional fracture mechanism identified by [Leith et al., 2013] can lead to a process of enhanced quarrying, as the rate of fracture development and therefore bedrock quarrying may be much greater than in the classical quarrying model of Hallet . Both models of enhanced and classical quarrying favor thinning ice, though for different reasons. Iverson  and Hallet  define ice loading as the maximum principal stress (σ1) on meter-scale bedrock steps which may be isolated from the valley-scale stress field, finding that thinning ice increases stress concentrations due to lee-side cavity water pressure fluctuations. [Leith et al., 2013] suggest that ice loading is the minimum principal stress (σ3) in strong bedrock forming the valley, and reducing ice thickness therefore promotes fracturing by decreasing the confining stress on bedrock while maintaining high surface-parallel stresses (equation ((2))). Although glacier-bed shear stress and flow velocity are likely to be important controls on the glacial erosion rate, our models indicate that enhanced quarrying will be ubiquitous across a region close to the axis of both the initial V-shaped and final U-shaped valleys. Pore pressure-induced extensional fracture may additionally aid the process, as elevated pressures within highly stressed rock are likely to reduce the effective stress magnitudes by equal amounts [i.e., Haimson and Chang, 2002], causing an increase in the effective stress ratio and therefore drive stresses above the extensional fracture limit (equation ((2))). This concept is supported by Holzhausen  who documents quarrymen producing horizontal extensional fractures up to 10,000 m2 in size by injecting compressed air over several days to increase pore pressures in rock maintaining high surface-parallel stresses.
6.2 Valley Shoulder
 The valley shoulder in the present-day Matter Valley corresponds with the distal extent of critically stressed bedrock in our initial V-shaped model (Figure 11a). We suggest that this feature delimits the border between relatively low stress upper rock slopes subjected to abrasion or classical quarrying and the deep U-shaped glacial valley formed through enhanced quarrying. While the valley shoulder is 700 m below the level of LGM ice cover, we propose that the sharp nature of this landform (Figure 13) reflects the threshold-limited, rather than incremental, mechanical behavior of critically stressed bedrock in our models. Extensional fractures in the form of exfoliation joints may be evident on rock slopes outside of the U-shaped valley, preserved from an earlier topography with different in situ stress conditions [e.g., Lidmar-Bergström, 1997]. Multiple phases of joint development may also be present below the valley shoulder, possibly formed through periods of staged ice unloading or valley incision [e.g., Bucher and Loew, 2009].
6.3 Path Dependency
 The effect of glacial ice cover on near-surface stresses within an evolving landscape is unique in terms of the loading rate (through ice development), residence time, breakdown, and the potential for erosion under confined conditions. The interaction of glacial processes with other tectonic and erosional processes can create changes in in situ stresses, landscape morphology, and bedrock strength over a wide range of time scales and may therefore be strongly path-dependent. Glaciers perform a number of critical tasks in facilitating macroscopic extensional fracture development: (1) they remove weathered surficial material while both maintaining normal stress on the substrate and limiting the progression of weathering, (2) they can erode into highly stressed bedrock under high total but low differential stress conditions, and (3) they control the generation of new macroscopic fractures in near-surface rock through normal stress variations at the glacier bed. In order to illustrate path-dependent effects and potential feedbacks between glaciation and bedrock fracturing, we ignore geometric effects and consider two simple cases where glaciers either develop on a fractured subaerial bedrock landscape with no topography or begin downwasting (and therefore reducing confining stress) on highly stressed intact bedrock. In the first case (Scenario A), we assume that a well-developed fracture network resulting from decreasing confining stresses during exhumation permeates near-surface bedrock and is underlain by critically stressed rock limited by the microcrack initiation threshold. The depth to intact bedrock can be estimated by determining the confining stress at which the extensional fracture limit and the microcrack initiation threshold intersect (i.e., combining equations ((1)) and ((2)) where σsp = σCI). Assuming that σ3 is the confining stress provided by overlying bedrock (i.e. σ3 = ρrockg z), the depth to critically stressed intact rock (z) can be written as
 Given the parameters listed in Table 1 and assuming that the density of fractured rock is equal to that of intact rock, equation ((3)) yields a depth of approximately 170 m. The massive rock and high differential stresses predicted beneath this depth can provide optimal conditions for rapid subglacial extensional fracturing, although in situ differential stresses must either be maintained or reduced during overburden removal for glacial ice to reach this rock without promoting further fracturing. Erosion must therefore occur under constant or increasing confinement, and considering the relative densities of rock and ice (Table 1), the glacier thickness should then increase at least 3 times as fast as bedrock erosion, and ~500 m of ice will be required to maintain confinement on rock assumed in this example. Higher ice accumulation rates will tend to shift stresses to the right on a principal stress diagram (see Figure 5), away from the fracture envelope. Conversely, sufficiently high erosion rates will elevate in situ stresses above the microcrack or extensional fracture limits, promoting brittle strain through rock mass degradation and maintaining, or possibly reducing, the in situ stress gradient. Unless critical stresses are maintained in the subglacial bedrock, new microcrack development, and therefore rapid macroscopic fracture development, may not occur until ice confinement reduces below that initially provided by eroded bedrock.
 In the second case (Scenario B), we assume that differential stresses within massive subglacial bedrock equilibrate to a significant thickness of overlying ice, either as a result of glacial erosion into critically stressed rock or tectonic adjustment to a persistent glacial presence (i.e., an ice cap) in the landscape (although the potential for this second case is debated) [cf. Carlsson and Olsson, 1982; Pascal et al., 2010]. Rock at the glacier-bedrock interface can then maintain critical long-term stresses, and in contrast to the previous example, which requires ice pressure to reduce below the original confining stress, any reduction in ice thickness will favor microcrack propagation and possibly induce macroscopic extensional fracturing.
6.4 Positive Erosional Feedback
 The observations in this study highlight an intriguing positive feedback, which may affect the timing and magnitude of U-shaped valley incision in alpine regions. As described by [Leith et al., 2013], deglaciation can promote extensional fracturing by increasing differential bedrock stresses to critical levels. In section 6.1 we discuss how this active extensional fracture mechanism can then enhance the quarrying process, increasing the rate of glacial erosion. This can generate a positive feedback in which increasing differential stresses, resulting from erosion into more highly stressed bedrock, continually propagate an active fracture front beneath the incising glacier (see section 6.5 for further comment). Progression of the fracture front will depend on the dynamics of the glacial system, and in particular, rates of ice unloading, glacial erosion, and stress relief though brittle strain in the underlying bedrock. As discussed in section 6.3, the process may temporarily stall if the rate of ice accumulation significantly exceeds that of erosion (and fracture development is suppressed) or be halted if stress relief through bedrock fracturing permanently reduces differential stresses near the valley floor below the microcrack initiation threshold.
6.5 Cross-Sectional Form
 Our suggestion that enhanced glacial quarrying regulates sediment supply, and therefore controls rates of U-shaped valley erosion, challenges the assumptions of existing glacial erosion models, which favor the effect of variable glacier sliding velocities in regulating sediment supply. Although the classical description of quarrying is partially dependent on sliding velocity [Hallet, 1996; Iverson, 1991, 2012], we propose that the curved form of deep, inner U-shaped glacial valleys may more likely result from enhanced glacial erosion and efficient sediment transport. Valley form is thereby a product of the interaction between in situ stresses and glacial processes. Broad, shallow, glacial valleys may be more strongly influenced by ice flow velocity profiles, with associated abrasion and quarrying of weak rock extending to the level of ice occupation, while deeper inner U-shaped troughs may owe their origin to enhanced erosion within the limit of the valley shoulders.
 The eventual depth of U-shaped valleys may be limited by either the shear strength of bedrock close to the valley floor or the periodicity of glacial cycles. Bedrock incision beneath ice increases shear stresses close to the valley axis (evident from the compression of lateral stress coefficient contours close to the valley axis in Figure 9d), while at the same time high differential stresses promote microcrack development and macroscopic fracture propagation in the same region (see contours of differential stress in Figure 9c). Although shear stresses are unlikely to grow to levels that exceed the strength of intact rock, fracture generation associated with incision and glacial unloading can reduce rock mass strength near the valley axis, possibly leading to the development of local shear failure. This will act to reduce differential stresses throughout the valley and suppress enhanced glacial erosion. Alternatively, if stresses are maintained throughout a glacial cycle, the resulting valley depth may simply reflect the erosional capacity of the system until deglaciation.
 Rock mass strength reduction or interglacial river incision can alter the in situ stress field during interglacial periods and decrease critical near-surface stresses within a zone close to the valley axis (Figures 12a and 12b). In the case of river incision, stress trajectories are drawn down beneath the channel, creating a low-stress zone at the former valley floor and sharply reducing the extent of critically stressed surficial bedrock. Abraded bedrock surfaces observed above highly incised river channels may be an indication that interglacial conditions have reduced near-surface in situ stresses and that glaciers can no longer efficiently erode this bedrock.
6.6 Implications for Alpine Valley Development
 The active bedrock fracturing mechanism described here for the Matter Valley may be an essential aspect of glacial geomorphology, aiding the production of characteristic erosional landforms by providing a continuous supply of fractured bedrock beneath glacial ice. This feedback requires specific environmental conditions in order to first develop sufficient differential stresses to induce fracturing and then expose rock to erosion while maintaining these stresses. In tunneling this sequence is easily achieved, as the excavation process instantaneously reorients the maximum principal stress parallel to the tunnel wall while simultaneously reducing the minimum principal stress to zero. Rock maintaining high in situ stresses on the Earth's surface is however relatively rare, as generally slow subaerial erosion rates enable chemical and physical weathering to relieve stress. Glacial erosion provides a unique set of conditions where bedrock excavation can take place under confinement, while the cold, stable environment limits chemical weathering of bedrock [Prestrud Anderson et al., 1997] and helps maintain stress. As in Scenario A in section 6.3, periods of significant erosion with thin to moderate (100–500 m) ice cover are likely to favor this feedback process, and therefore rapid down cutting of alpine valleys, providing the stress path required to form a propagating subglacial fracture front.
 The requirement that destressed bedrock be removed prior to the onset of enhanced quarrying may be illustrated from observations of glacial and interglacial landforms in shield regions. Such areas currently maintain high in situ stresses (Figure 3a) and exhibit old, widely spaced, extensional fractures in the form of sheeting joints, which in some cases date back to exposure of a Cambrian landscape [Lidmar-Bergström, 1997]. Striations and characteristic glacial landforms indicate that flowing ice was present, though largely ineffective as an erosional agent in these regions, and total erosion following the formation of sheeting joints has been exceptionally low, commonly on the order of tens of meters throughout the Pleistocene [Lidmar-Bergström, 1997; Sugden, 1976]. As topography in these landscapes is typically more subdued than that of alpine regions, channelized ice flow is likely to be weaker during the onset of glaciation, and similar to Scenario A in section 6.3 and Stages 31–32 of our glacial erosion model, if glaciers are unable to erode significantly into the sparsely jointed bedrock during ice accumulation, they will tend to reduce differential stresses in the underlying bedrock and suppress enhanced erosion throughout the period of glaciation.
 We suggest that exhumation and subaerial weathering during warm periods [van Husen, 2004, and references therein] prior to the onset of glaciation is likely to have produced a thick mantle of fractured overburden in the Matter Valley. This mantle provided confining stress on underlying bedrock (Figures 14a and 14b), until the onset of major glaciation initiated erosion of this surface cover (Figure 14c) and exposed highly stressed bedrock under confined conditions (Figure 14d). A favorable in situ stress state, in which stresses exceeded both the microcrack initiation threshold (equation ((1))) and the extensional fracture limit (equation ((2))), then led to the generation of macroscopic extensional fractures at the ice-bedrock interface (Figure 14e), promoting enhanced glacial erosion. A positive feedback developed from this process, and further erosion exposed additional critically stressed bedrock, driving fracture development and deepening the valley (Figure 14f). In situ stresses high on the inner valley walls declined as a result of incision, decreasing the elevation of the critically stressed bedrock threshold and leading to slower erosion in these areas. In accordance with dating evidence (see section 2.2), we propose that this is likely to have occurred at or soon after the MPR, when large valley glaciers had the opportunity to erode weak overburden and access intact rock.
 Following an initial erosive cycle, perturbation of in situ stresses by fluvial incision or rock mass degradation limited the extent of critically stressed bedrock to near the valley floor. Reinitiation of enhanced quarrying would then require erosion of destressed rock under ice confinement; for a 40 ky glaciation (typical of the 100 ky period mid-Pleistocene climate cycles) and 170 m of destressed overburden (see section 6.3), this would have required an average incision rate on the order of ~4 mm/yr. This value is an order of magnitude greater than typical rates of glacial abrasion in massive, unweathered granite or granodiorite (see section 6.1) and suggests that subsequent glacial cycles following this initial period of enhanced erosion may not have resulted in significant topographic modification.
 In situ stresses within alpine landscapes result from ongoing interactions between tectonic and erosional processes, regulated by the mechanical behavior of bedrock. Evidence for these interactions and behavioral thresholds may be preserved in the geomorphology and geochronology of evolving alpine regions.
 High regional in situ stresses are likely to have led to the development of critically stressed bedrock near the axis of the Matter Valley (Canton Valais, Switzerland), promoting new microcrack development and unstable propagation of extensional fractures where differential stresses exceed both the microcrack initiation threshold and extensional fracture limit of the trilinear fracture envelope. The presence of these fractures enhanced the efficiency of quarrying and therefore total erosion near the axis of the valley. Our FDM results suggest that macroscopic extensional fractures may occur to a depth of ~150 m beneath the initial V-shaped valley and up to 600 m laterally from the axis. This extent of predicted fracturing is coincident with the present-day valley shoulder, and the feature may therefore reflect the maximum lateral extent of enhanced erosion during an initial period of U-shaped valley incision. In situ stresses in this region will cause fractures to develop parallel to the maximum principal stress and normal to the minimum confining stress, generating joint surfaces parallel to topography. The orientation of extensional fractures in the vicinity of the valley shoulder may conform to a relict topography, while fractures developed during later phases of valley evolution will reflect the more recent valley form. Near-surface differential stresses in the valley floor can be modified by local topographic perturbations (e.g., incising stream channels) or limited by the development of shear failures during interglacial periods, both of which will reduce the potential for enhanced erosion during subsequent glaciations.
 Depending on the progression of glaciation, subglacial fracturing may either take place in response to downwasting glacial ice or be suppressed by increasing ice cover. If the glacial residence time is long and landscape stresses adjust to the ice load, then excess differential stresses accumulated in underlying bedrock may be sufficient to induce fracturing and enhanced erosion in response to any degree of ice thinning. Otherwise, critical differential stresses may not develop until the confining stress provided by glacial ice reduces below that formerly provided by bedrock eroded during the glacial cycle. Once fracturing begins beneath ice cover, a progressive fracture front can precede in subglacial bedrock, providing a positive feedback and enhancing erosion rates.
 Recent evidence suggests that glacial erosion rates are strongly controlled by the glacier's ability to quarry bedrock, which is in turn closely tied to the nature or degree of bedrock fracturing. We suggest that active subglacial fracturing in response to high differential in situ stresses will generate a more erodible substrate, enhancing quarrying and promoting down cutting of alpine valleys. Development of the classical U-shaped form may therefore derive from a combination of both glacier dynamics and the in situ stress state of underlying bedrock. Moderate ice cover (~500 m) during glacial periods can erode weathered surface material while maintaining sufficient confinement on highly stressed underlying bedrock. In contrast, stress relief facilitated by ongoing brittle strain or weathering and rock mass strength degradation during interglacial periods is not conducive to the generation of high near-surface stresses unless rates of in situ stress development (i.e., through erosion) exceed the rate of stress reduction (currently an unknown parameter).
 Initial glacial erosion into the former Miocene Alpine landscape is likely to have provided large valley glaciers a unique opportunity to cut into critically stressed bedrock under confined conditions. Aided by enhanced quarrying, erosion during early glacial cycles following the MPR (0.87 Ma) may have led to the rapid development of spectacular U-shaped glacial valleys preserved in the landscape today.
 We would like to thank K. Evans for the insightful dialog and S. Sikaneta for the patient discussion and thoughtful comment. J. Sanders provided useful feedback on an earlier version of the manuscript, and V. Gischig and J. Hansmann were extremely helpful in overcoming modeling-related challenges during the development of this project. Comments from D. Stead, the associate editor of the Journal of Geophysical Research – Earth Surface, A. Densmore, and an anonymous reviewer significantly improved the clarity of the manuscript. This research was supported by ETH Research grant ETH-29 09-2.