Seasonal-scale abrasion and quarrying patterns from a two-dimensional ice-flow model coupled to distributed and channelized subglacial drainage

Highlights

• Hydrologically-coupled ice-flow model was used to calculate seasonal erosion patterns.

• Process-specific erosion laws were implemented for abrasion and quarrying.

• Channelized drainage leads to up-glacier migration of abrasion rates.

• Multi-day variations in meltwater input are more important than diurnal or seasonal.

• Quarrying patterns with Iverson's (2012) law differ from abrasion patterns.

Abstract

Field data and numerical modeling show that glaciations have the potential either to enhance relief or to dampen topography. We aim to model the effect of the subglacial hydraulic system on spatiotemporal patterns of glacial erosion by abrasion and quarrying on time scales commensurate with drainage system fluctuations (e.g., seasonal to annual). We use a numerical model that incorporates a dual-morphology subglacial drainage system coupled to a higher-order ice-flow model and process-specific erosion laws. The subglacial drainage system allows for a dynamic transition between two morphologies: the distributed system, characterized by an increase in basal water pressure with discharge, and the channelized system, which exhibits a decrease in equilibrium water pressure with increasing discharge. We apply the model to a simple synthetic glacier geometry, drive it with prescribed meltwater input variations, and compute sliding and erosion rates over a seasonal cycle. When both distributed and channelized systems are included, abrasion and sliding maxima migrate ~ 20% up-glacier compared to simulations with distributed drainage only. Power-law sliding generally yields to a broader response of abrasion to water pressure changes along the flowline compared to Coulomb-friction sliding. Multi-day variations in meltwater input elicit a stronger abrasion response than either diurnal- or seasonal variations alone for the same total input volume. An increase in water input volume leads to increased abrasion. We find that ice thickness commensurate with ice sheet outlet glaciers can hinder the up-glacier migration of abrasion. Quarrying patterns computed with a recently published law differ markedly from calculated abrasion patterns, with effective pressure being a stronger determinant than sliding speeds of quarrying rates. These variations in calculated patterns of instantaneous erosion as a function of hydrology-, sliding-, and erosion-model formulation, as well as model forcing, may lead to significant differences in predicted topographic profiles on long time scales.

Introduction

Erosion rates measured or inferred in highly glaciated catchments vary over four to five orders of magnitude (Hallet et al., 1996, Koppes and Montgomery, 2009). Accordingly, glaciated environments can be highly erosive (e.g., Molnar and England, 1990, Hallet et al., 1996, Shuster et al., 2005, Valla et al., 2011) or can protect the pre-glacial landscape (e.g., Thomson et al., 2010, Ferraccioli et al., 2011). Since the Pliocene, most mountain ranges worldwide have undergone large modifications because of the action of glaciers and ice sheets. The extent and rate of these modifications are still highly debated, however (e.g., Molnar and England, 1990, Brozovic et al., 1997, Koppes and Montgomery, 2009, Herman et al., 2010, Montgomery and Korup, 2010, Thomson et al., 2010, Steer et al., 2012). The formation of large erosional features (greater than kilometer-scale, e.g., overdeepenings and tunnel valleys) is still poorly explained, although recent comprehensive descriptions of such features (Dürst Stucki et al., 2010, Preusser et al., 2010, Cook and Swift, 2012) support the original hypothesis that subglacial water is a key control on the initiation and evolution of an overdeepening (Hooke, 1991) or a tunnel valley (Dürst Stucki et al., 2010). Using a numerical model of coupled ice and water flow (Pimentel and Flowers, 2011) and equipped with erosion laws (Hallet, 1979, Iverson, 2012), we make a preliminary assessment of the relative rates and patterns of glacial erosion on subseasonal to annual time scales.

It is well established that glaciers and ice sheets erode through three main mechanisms: abrasion, quarrying (or plucking), and the action of subglacial streams (e.g., Boulton, 1974, Boulton, 1979, Hallet, 1979, Alley et al., 1997, Cohen et al., 2006). Abrasion takes place when clasts dragged by the overriding ice carve grooves in the bedrock. This process is controlled primarily by the sliding velocity (Hallet, 1979). Quarrying results from the lee-side failure of bedrock obstacles caused by stress gradients imparted by sliding (e.g., Iverson, 1991, Hallet, 1996). When ice overrides an obstacle, a lee-side cavity can form and the deviatoric stresses in the bedrock increase on the stoss side. This stress gradient can be accentuated by water pressure fluctuations in the cavity. Increasing water pressure enlarges the cavity, by lifting the roof and by enhancing sliding, thus reducing the contact area; when the water pressure drops, local deviatoric stresses in the bed then increase dramatically (Iverson, 1991, Hallet, 1996, Cohen et al., 2006). If, in addition, cracks are present on the stoss side or on top of the obstacle, the obstacle will experience fatigue and may eventually fracture. A positive feedback can arise if the cracks are sufficiently small that their hydraulic conductivity is low compared to that of the subglacial drainage system. In such a case, cracks can sustain high water pressures even when the cavity is drained, enhancing the fatigue of the step (Iverson, 1991, Cohen et al., 2006).

The role of subglacial streams is often treated as being limited to flushing sediments from the ice–bedrock interface (e.g., Humphrey and Raymond, 1994, Alley et al., 1997, Riihimaki et al., 2005). These streams, however, are known to have a significant effect on the dissolution of highly water-soluble bedrock (Walder and Hallet, 1979). Moreover, they may have an even greater erosive power and higher flushing efficiency than subaerial rivers owing to the high water pressures that can be achieved subglacially (Alley et al., 1997). Creyts et al. (2013) apply sediment transport equations to a subglacial drainage system formulated as a macroporous sheet (Creyts and Schoof, 2009) and show that strong diurnal cycles in water input enhance sediment transport efficiency. Despite the potential importance of subglacial streams to overall glacial erosion, we reserve their treatment for future study and focus exclusively on abrasion and quarrying here.

To date, studies employing numerical models of glacial erosion almost exclusively use the abrasion law developed by Hallet (1979), in which the abrasion rate is proportional to the sliding velocity raised to a power equal to or greater than one (e.g., MacGregor et al., 2000, Tomkin and Braun, 2002, Anderson, 2005, Herman and Braun, 2008, Egholm et al., 2009). Some efforts have been made to add a parameterization of the bedrock slope to the aforementioned law in order to simulate quarrying (e.g., MacGregor et al., 2009, Egholm et al., 2012), but few studies include a process-based plucking law (e.g., Hildes et al., 2004). In nearly all the literature, therefore, both abrasion and quarrying are proportional to sliding speed.

Sliding of an ice mass is highly influenced by the basal water pressure (e.g., Müller and Iken, 1973, Lliboutry, 1976, Iken, 1981), which in turn is jointly controlled by the rate of water influx to the bed and the morphology of the drainage system (e.g., Iken, 1981, Nienow et al., 1998). When the drainage system is ‘distributed’ (e.g., Fountain and Walder, 1998), water flow is relatively slow and the water pressure rises with increasing input. Different formulations have been proposed to describe a distributed drainage system, including flow through subglacial till (e.g., Clarke, 1987), a network of linked cavities (e.g., Lliboutry, 1976, Kamb, 1987) or a macroporous sheet with laminar (e.g., Flowers and Clarke, 2002) or turbulent flow (e.g., Creyts and Schoof, 2009). In contrast, a ‘channelized’ drainage system transports water through a network of Röthlisberger (R)-channels, Nye (N)-channels, or canals (Röthlisberger, 1972, Nye, 1976, Walder and Fowler, 1994). The most commonly used conceptual model is that of a network of R-channels, which at equilibrium exhibits a decrease in water pressure with increasing discharge. The transition between the distributed and channelized drainage systems is responsible for the up-glacier migration and eventual termination of the ‘spring speed-up event’ observed at the onset of the melt season in many glaciers (e.g., Iken and Bindschadler, 1986, Mair et al., 2003, Anderson et al., 2004, Bartholomaus et al., 2007, Fischer et al., 2011).

Numerical modeling of the dynamic transition between distributed and channelized subglacial drainage systems is still nascent (Flowers et al., 2004, Kessler and Anderson, 2004, Flowers, 2008, Schoof, 2010, Pimentel and Flowers, 2011, Hewitt et al., 2012, Hewitt, 2013, Werder et al., 2013) with the coupling to ice flow rarely included (c.f. Arnold and Sharp, 2002, Flowers et al., 2011, Pimentel and Flowers, 2011, Hewitt, 2013). Attempts to include the role of subglacial hydrology in erosion models have thus far been limited to treatments of distributed drainage (MacGregor et al., 2009, Herman et al., 2011, Egholm et al., 2012), largely because of the complexity of combining drainage system types and the computational burden of representing processes varying on highly disparate time scales. However, the models above have been successful in demonstrating a role for hydrology in the formation of overdeepenings (Herman et al., 2011, Egholm et al., 2012) as originally suggested by Hooke (1991), even with simple representations of the drainage system.

Here we take a first step in exploring the potential influence of a dual-morphology subglacial drainage system on modeled patterns of glacial erosion. We employ the hydrologically coupled, two-dimensional model of Pimentel and Flowers (2011) — which integrates a higher-order flowband representation of ice dynamics (Blatter, 1995, Pattyn, 2002) — with distributed and channelized subglacial drainage (Flowers, 2008). The model returns the sliding and water pressure patterns, which enable us to calculate erosion rates through process-specific laws: abrasion as formulated by Hallet (1979) and quarrying as recently derived by Iverson (2012). We restrict our investigation to examining instantaneous erosion rates with a stationary glacier geometry in order to isolate the influence of the melt season and glacier hydrology on seasonal patterns of erosion. In a first set of experiments, we explore the influence of the representation of basal drainage and sliding on calculated erosion by comparing (i) distributed-only to distributed and channelized drainage systems, and (ii) Coulomb-friction sliding to power-law (Weertman-type) sliding. In a second set of experiments, we examine the model response to changes in the frequency, amplitude, and total volume of meltwater forcing, with the aim of determining whether subannual fluctuations in glacier hydrology and dynamics are important for erosion modeling (c.f. Herman et al., 2011). We also touch on the role of glacier size to highlight differences between glacier- versus ice sheet-scale problems.