The model is set in a glass box (70 cm long, 70 cm wide and 5 cm deep) (Fig. 1). A 5 cm thick, flat, horizontal, permeable and erodible substratum, made of sand (d50=100 µm) saturated with pure water and compacted to ensure homogeneous values for its density (ρbulk=2000 kg m-3), porosity (Φ=41 %) and permeability (K=10-4 m s-1), rests on the box floor. The ice sheet portion is modelled with a 3 cm thick layer of viscous (η=5×104 Pa s) and transparent but refractive (n=1.47) silicon putty placed on the substratum. The model is not designed to simulate an entire ice sheet. The silicon layer is circular in top view (radius = 15 cm) to avoid lateral boundary effects on silicon flow. Subglacial meltwater production is simulated by injection of water with a punctual injector, 4 mm in radius, placed at a depth of 1.8 cm in the substratum and connected to a pump (Fig. 1). The injector is located below the centre of the silicon layer to be consistent with the circular geometry of the experiment. The water discharge is constant (1.5 dm3 h-1) over the duration of the experiment and generates water flow at the silicon–substratum interface and within the substratum. Water discharge is calculated beforehand so that water pressure exceeds the combined weight of the sand and silicon layers. The injection of water starts when the silicon layer reaches the dimensions we fixed for every experiment (15 cm radius and 3 cm thickness) and a perfect transparency. Once injected, water flow is divided into a Darcy flow within the substratum and a flow at the silicon–substratum interface. The water flowing at the silicon–substratum interface originates from a pipe forming at the injector once water pressure exceeds the cumulative pressure of the silicon and sand layers. The ratio between the Darcy flow and the flow at the silicon–substratum interface is inferred from computations of the water discharge flowing through the pipe based on the substratum properties and the input discharge. We estimate that 75 % of the input discharge in the substratum and 25 % of the input discharge along the silicon–substratum interface is transferred as Darcy flow.

In order to monitor the development of landforms on the substratum, we use six synchronized cameras equidistant from the experiment centre (Fig. 1) taking photographs of the experiment every 5 s. Two cameras (orange in Fig. 1) cover the whole extent of the experiment and four cameras (blue in Fig. 1) focus on specific regions to obtain higher-resolution images. These cameras take simultaneous pictures with differing positions and orientations. Digital elevation models of the silicon surface and of the substratum are derived from these images by photogrammetry. The ultimate stage of the experiment is to remove distortions due to light refraction through the silicon putty and apply corrections to the substratum topography. This treatment is achieved using a custom algorithm able to evaluate the gap between the measured altitude and the real altitude of each pixel of the DEM (see detailed post-treatment methods in Lelandais et al., 2016). Tests performed on previously known topographies show that the vertical precision of the retrieved digital elevation models is better than 10-1 mm.

The flow velocity of the silicon layer is monitored near its base (Vbase), at mid-depth (Vmid) and at its surface (Vsurface), with an additional camera placed over the centre of the experiment (green in Fig. 1). For that purpose, the camera records the position in pictures taken at regular time intervals in ultraviolet (UV) light of 180 UV paint drops (1 mm in radius) placed 1 mm above the base, at mid-depth and at the surface of the silicon layer (Figs. 1, S1 in the Supplement). The monitoring of every UV marker position through time was used to produce velocity and vertical displacement maps. Vertical displacement maps are interpolated from the subtraction of the DEM at time t with the DEM generated from the photographs taken a few seconds before the injection. Velocity maps are interpolated from the subtraction of the position of every marker at time t with the position of the same markers at the previous stage. These passive markers are transparent at visible wavelengths and do not alter pictures of the substratum taken through the silicon cap. They represent less than 0.5 % of the silicon layer in volume and tests have shown that they do not affect its overall rheological behaviour. Uncertainties in the measured position of markers on images are less than 1 pixel in size (i.e. less than 10-1 mm); thus uncertainties in the derived velocities are comprised between 5×10-4 and 2×10-3 mm s-1, depending on the time interval between photographs.

In this study, we focus our attention on the relations among subglacial water flow, subglacial erosion and ice flow using an experiment approach. Considering that in our model meltwater is simulated by an injection of water, the rules of a classical scaling in which the model is a perfect miniaturization of nature are not practical (Paola et al., 2009). In this perspective, we base the scaling of our model on the displacement of the natural ice and experimental silicon margins through time. We use a unit-free speed ratio between the silicon and ice margin velocity and the incision rate of experimental and natural tunnel valleys. In this way, the complexity of the relations among subglacial hydrology, subglacial erosion and ice flow, which is one of the main issues in numerical modelling, is included in the velocity values. The scaling attests that the value of the ratio between margin velocity and incision rate of tunnel valleys in the experiment falls within the field validity defined by the range of natural settings (full details in Lelandais et al., 2016). The main scaling limit regards the viscosity ratios among glacier ice, silicon putty and water. The size of the experimental ice stream, being partly controlled by the high silicon viscosity, may be underestimated compared to the size of modelled tunnel valleys.

Considering that our model is a simplification of nature, we cannot simulate the real complexity of the natural processes. In contrast to ice, the commercial silicon putty we use (Dow Corning, SGM36) is impermeable, Newtonian, isotropic, and its viscosity is nearly independent of temperatures between 10 and 30 ∘C. Therefore, rheological softening with strain rate, temperature, anisotropy, and meltwater content (e.g. Bingham et al., 2010) cannot be reproduced. The silicon putty cannot reproduce the ice–water phase transition either, requiring the use of punctual water injection in the experiment. This punctual injection does not simulate the mosaic of meltwater production regions existing beneath glaciers or the episodic input from supraglacial–englacial meltwater reservoirs. Experimental meltwater routing is predominantly controlled by the water discharge we inject in our system and therefore differs from parameters controlling hydrology in glacial systems. Subglacial meltwater routing is indeed controlled by the ice surface slope, the bed topography, and the glacier mass balance (Rothlisberger and Lang, 1987). The ice surface slope controls potentiometric surfaces, generally guiding subglacial water flow parallel to ice sheet surfaces (Glen, 1954; Shreve, 1972; Fountain and Walder, 1998). Finally, the substratum we use is homogeneous, flat and composed of a well-sorted mixture of sand-sized grains. This model, designed to decipher the interaction between subglacial hydrology and ice dynamics, hinders the influence of bed topography and geology (especially the influence of subglacial till) (Winsborrow et al., 2010). The deformation of the subglacial till and its complex rheological behaviour is known to promote ice streaming (Alley et al., 1987), modify the subglacial hydrology, and alter the size of tunnel valleys. The development of an analogue material scaled to reproduce subglacial till characteristics is extremely difficult so we did not try to include the equivalent of a till layer in the experiment. We thus assume that the velocity contrasts observed in the experiment are likely to be amplified in natural ice sheets by the complex rheological behaviour of ice and till. This may lead to the development of narrower ice streams with higher relative velocities and sharper lateral shear margins in natural ice sheets than in the experiment (Raymond, 1987; Perol and Rice, 2015).

This experiment was repeated 12 times with identical input parameters (a 30 mm thick silicon layer of 150 mm radius; constant water input of 1.5 dm3 h-1 during 1800 s). After an initial identical state, a six-stage ice stream life cycle linking outburst flooding, transitory ice streaming, and tunnel valley development has been observed for all these simulations (Figs. 2, 3).

Initial state (Fig. S2). As long as no water is injected in the substratum, the silicon layer spreads under its own weight and displays the typical parabolic surface profile of an ice sheet. It increases in diameter and decreases in thickness with time, thus producing a radial pattern of horizontal velocities, which increase in magnitude from the centre (Vsurface<3×10-3 mm s-1) to the margin (Vsurface=8×10-3 mm s-1) (Fig. S2). Vbase is close to 0 over the full extent of the silicon layer (VbaseVsurface∼0 %), indicating coupling with the substratum. The silicon flow pattern changes when meltwater production is simulated by injecting water at a constant discharge (1.5 dm3 h-1), beneath the silicon layer.

Stage 1 (Figs. 2a, 3a). A water pocket grows below the centre of the silicon layer and raises its surface by 2 mm. Above the water pocket, the silicon accelerates (Vsurface≥35×10-3 mm s-1) and is decoupled from the substratum (VbaseVsurface=75 to 80 %). Below the rest of the silicon layer, lower velocities (Vsurface=8×10-3 mm s-1, VbaseVsurface=40 to 50 %) indicate higher basal friction. These results are consistent with inferences that meltwater ponding can form pressurized subglacial water pockets associated with basal decoupling, surface uplift, and ice flow acceleration in natural ice sheets (e.g. Hanson et al., 1998; Elsworth and Suckale, 2016; Livingstone et al., 2016). In the experiment, however, these effects are restricted to an approximately circular region and are not sufficient to produce channelized ice streaming.

Stage 2 (Figs. 2b, 3b). The water pocket expands and migrates towards the margin of the silicon layer. The lack of channels incised in the substratum indicates that this displacement occurs as distributed water drainage without any basal erosion. In the silicon layer, the region of surface uplift, basal decoupling and acceleration (Vsurface=18×10-3 mm s-1, VbaseVsurface=75 to 85 %) expands and migrates downstream with the water pocket. Similar migrations of pressurized subglacial water pockets have been observed or inferred under modern and ancient ice sheets (Fricker et al., 2007; Carter et al., 2017), sometimes associated with migrations of regions of ice surface uplift and ice flow acceleration (Bell et al., 2007; Stearns et al., 2008; Siegfried et al., 2016). The experiment indicates that the migration of water pockets at the ice–bed interface can contribute to the emergence of ice streams.

Stage 3 (Figs. 2c, 3c). When the water pocket reaches the margin of the silicon layer, it drains suddenly as a sheet flow. This marginal outburst flood is still fed by distributed drainage and conveys sand particles eroded from the substratum towards a low-angle marginal sedimentary fan (up to 40 mm long, 30 mm wide and 0.3 mm thick; Fig. S3). Simultaneously, the silicon flow focuses in a stream (200 mm wide at the margin) that propagates upstream from the silicon margin to the water injection area. This stream immediately peaks in velocity (Vsurface=80×10-3 mm s-1, 16 times higher than the surrounding silicon) and is entirely decoupled from its substratum (VbaseVsurface>90 %). Although similar relations between outburst floods and ice flow accelerations have been suspected in modern (Alley et al., 2006; Bell et al., 2007; Stearns et al., 2008) and former (Livingstone et al., 2016) ice sheets, they have been documented for valley glaciers only (e.g. Anderson et al., 2005). In these regions, they can produce sudden meltwater discharges that exceed the capacity of distributed subglacial meltwater drainages and promote basal decoupling and ice flow acceleration (e.g. Magnússon et al., 2007). The experiment confirms that outburst floods can promote basal decoupling and trigger ice streaming in ice sheets (Fowler and Johnson, 1995).

Stage 4 (Figs. 2d, 3d). The distributed subglacial drainage system starts to channelize: two valleys (TV1 and TV2) appear below the margin of the silicon layer and gradually expand by regressive erosion of the substratum. At this stage, TV1 is 30 mm long, 12 mm wide and 0.5 mm deep; TV2 is 80 mm long, 10 mm wide and 0.5 mm deep. These valleys, with their constant widths, undulating long profiles and radial distribution, are analogue to natural tunnel valleys in their dimensions, shapes, and spatial organization (Lelandais et al., 2016; Fig. 4). They are fed by distributed water drainage. The sand eroded from the substratum transits through these valleys and accumulates in high-angle marginal sedimentary fans, higher in elevation than the valley floors (TV1 fan is up to 27 mm long, 30 mm wide and 0.5 mm thick; TV2 fan is up to 20 mm long, 24 mm wide and 1 mm thick; Fig. S3). In response to progressive channelization of the water drainage into the expanding valleys, the silicon stream narrows and slows down (120 mm wide at the margin; Vsurface=24×10-3 mm s-1). The silicon stream, still channelized, still flows 8 times faster than the rest of the silicon layer and is still decoupled from the substratum (VbaseVsurface>85 %). These results are consistent with inferences that channelization of hitherto distributed subglacial water drainage systems can occur and reduce ice flow velocity after outburst floods (Kamb, 1987; Retzlaff and Bentley, 1993; Magnússon et al., 2007), and it can be responsible for narrowing and deceleration of ice streams (Raymond, 1987; Retzlaff and Bentley, 1993; Catania et al., 2006; Beem et al., 2014; Kim et al., 2016). At this stage of the experiment, this transition, which corresponds to the initiation of tunnel valleys, is not sufficient to stop ice streaming, however.

Stage 5 (Figs. 2e, 3e). A new transient water pocket grows below the silicon layer, migrates and drains as an outburst flood, thus forming a new low-angle marginal sedimentary fan with a lateral offset of 4 cm with respect to TV1. This induces the activation of a second stream (Vsurface=40×10-3 mm s-1) decoupled from its substratum (VbaseVsurface=80 %) and the initiation of a new radial valley (TV3), in a hitherto slow-moving region of the silicon cap. Simultaneously, the first silicon stream switches off (Vsurface=10×10-3 mm s-1) and recouples to its substratum (VbaseVsurface=30 %), but water and sand still flow through TV1 and TV2. At this stage, TV1 is 100 mm long, 8 mm wide, and 0.7 mm deep and its fan is up to 21 mm long, 40 mm wide, and 1.1 mm thick; TV2 is 80  mm long, 7.5 mm wide, and 0.6 mm deep and its fan is up to 20 mm long, 28 mm wide, and 1.6 mm thick. This result is consistent with inferences that natural ice streams can switch on and off, surge, and jump in location in response to changes in subglacial water drainage reorganization (Catania et al., 2012; Le Brocq et al., 2013; Beem et al., 2014; Hulbe et al., 2016). The experiment further indicates that this complex behaviour is controlled by the growth and migration, in various possible directions, of transient pressurized subglacial water pockets that form successively as long as the discharge capacity of tunnel valleys systems is not sufficient to efficiently drain the available meltwater.

Stage 6 (Figs. 2f, 3f). Since their initiation, TV1, TV2, and TV3 have progressively increased in width, depth and length. At this stage TV1 is 100 mm long, 17 mm wide, and 1.2 mm deep and its fan is 28 mm long, 4 mm wide, and 1.5 mm high at the maximum; TV2 is 80 mm long, 10 mm wide, and 0.8 mm deep and its fan is up to 16 mm long, 23 mm wide and 1.6 mm thick; TV3 is 60 mm long, 11 mm wide, and 0.55 mm deep and its fan is up to 14 mm long, 23 mm wide, and 0.7 mm thick. Their overall volume and discharge capacity have thus increased (Fig. 5). In response to this increased drainage efficiency, the second stream gradually decays (Vsurface=5×103 mm s-1) and recouples to its substratum (VbaseVsurface=35 %). The silicon layer ultimately recovers a radial flow pattern (Fig. 3f). This result is consistent with the inference that ice streams may decelerate and even switch off in response to reduction of subglacial water pressures when efficient subglacial water drainage systems develop (Retzlaff and Bentley, 1993; Beem et al., 2014; Livingstone et al., 2016; Kim et al., 2016). In the experiment, this development is governed by the expansion of tunnel valley networks. Large glaciotectonic thrust masses at the ice margin near tunnel valley fans are generally assumed to be field evidence of a fast ice flow stage prior to drainage through tunnel valleys (Hooke and Jennings, 2006).

This experiment has been reproduced 12 times with identical input parameters. We always observe the same processes and events acting in a similar chronological order: (1) water pocket forms, (2) water pocket migrates, (3) water pocket drains (outburst flood) and silicon stream switches on, (4) tunnel valleys form in response to channelization, (5) silicon stream slows down, and (6) it finally switches off in response to the increase in drainage efficiency during tunnel valley development. However, despite this consistency in the progress of all simulations, some variability has been detected. We measured different migration rates for the water pocket ranging from 30 to 80 s that may result from small changes in subglacial topography and in the dynamics of silicon–bed decoupling. Considering a constant water discharge and the characteristics of the experiment, a longer period of migration implies a longer period of water storage and a bigger water volume released at the silicon margin during the pocket drainage. We therefore recorded peak velocities for water pocket drainage ranging from 6.10-3 to 12.10-3 mm s-1. In response to variations in the water volume drained at the margin and peak discharge, the maximum width of the silicon stream varies from 120 to 250 mm. The magnitude of the outburst flood triggered during water pocket drainage also influences the number of tunnel valleys that subsequently form during the channelization stage. A high-magnitude outburst flood generates a wider erosion beneath the silicon that will be suitable for the development of multiple tunnel valleys. Hence, the number of tunnel valleys at the end of the experiments ranged from one to five with one to three tunnel valleys formed simultaneously during the initiation of the channelization stage. These valleys range from 40 to 120 mm long, 3 to 18 mm wide, and 0.3 to 1.8 mm deep. During tunnel valley development, the evolution of drainage efficiency varies among the experiments. A relatively inefficient system of tunnel valleys induces upstream water pocket formation. As observed in Fig. 3e, the drainage of this belated water pocket may provoke water rerouting beneath the silicon layer and subsequent lateral migration of the silicon stream. We counted zero to two events of silicon stream migration for single experiments. Finally, the time required to reach the phase of ice stream decay highly depends on the number of tunnel valleys formed during the experiments and their progressive development. We observed a lifetime for the silicon stream ranging from 500 to 1700 s, correlated with the evolution of the drainage efficiency during tunnel valley development.