Despite a large diversity of borehole pressure records, a few general patterns are easy to identify. The most common is the contrast between an inactive winter regime and an active summer period. During winter, most sensors show stable, high (near overburden) water pressures, interrupted only during a 2–4 month period of summer activity starting in June–July (Fig. ). The onset of the active summer period (or “spring event”) occurs during rapid thinning of the snowpack under high summer temperatures. After the spring event, 20 % of sensors show a drop in diurnal running mean pressure, and most start displaying diurnal oscillations.

Pressure records alone do not allow us to determine the characteristics of water flow at the bed, and visual observations at the bottom of boreholes often fail due to the high turbidity of the water after drilling. However, in a few exceptional cases, we were able to observe water flow at the bed directly. We will describe the two most clear-cut cases.

On 28 July 2013, while installing a sensor at the bottom of a borehole, strong periodic pulls were felt through the sensor cable, revealing a conduit with turbulent, fast water flow in the bottom 50 cm of the borehole. This borehole was also the only one in which there was an audible sound of flowing water. The location of the hole is marked as “Fast-Flow” in Fig. , it was drilled at the very end of the field operations, and no further detailed on-site investigation was conducted.

The fast-flow borehole was 93 m deep and drained at a depth of 87 m during drilling. On the first recorded diurnal pressure peak, the water reached a pressure of about 5.2 m (6 % of ice overburden). A water sample retrieved from the bottom showed moderate turbidity. Two pressure sensors were installed in this borehole, 10 and 80 cm above the bed, the upper one with a snubber and the lower one without.

Panel (c) of Fig.  shows the pressure recorded in the fast-flow borehole for the first 33 days after installation, and panel (d) shows the pressure records in three boreholes along the same line across the glacier at 15 m spacing. Note the lack of similarity between the fast-flow hole pressure record and those from other nearby boreholes. This lack of similarity contrasts with the typical behaviour of boreholes exhibiting diurnal pressure oscillations. Such boreholes usually share a similar pattern of pressure oscillations with one or more neighbouring boreholes, forming a cluster that extends some distance laterally across the glacier (see Sect. ).

However, in the case of the fast-flow borehole, somewhat similar temporal pressure patterns were observed down-glacier and at much larger distances than the 15 m lateral borehole spacing, as shown in panels (e) and (g), and less so in panel (h). By contrast, a set of boreholes exhibiting very different variations close to those in panel (d) is shown in panel (f). For reference, panel (i) shows the remaining pressure time series recorded in the same area, highlighting the diversity of pressure patterns observed. No systematic time lags were found between peaks in the fast-flow borehole and pressure peaks of boreholes displayed in panels (e) and (g).

The grouping of boreholes into panels in Fig.  was done on the basis of spatial proximity in panel (c), and on the basis of a commonality of diurnal pressure variations in the remaining panels. In particular, we have clustered the records on the basis of commonality in how the amplitude of diurnal pressure variations changes in time. For instance, the similarity between the records in panel (g) should be obvious. However, note that there can be subtler similarities: panels (c), (e) and (g) at least partially share a period of larger diurnal amplitudes leading up to 3 August, a hiatus lasting until 10 August punctuated by a diurnal pressure peak late on 6 August, and a period of renewed diurnal oscillations lasting until 17 August; this differs from the pattern of diurnal oscillations seen in panel (h). Grouping boreholes in this way is partially a subjective measure, and we will present a more systematic clustering method in a separate paper, which has helped to guide the groupings here; see also . All borehole groupings presented in the following figures were manually selected using the same criteria as described for Fig. .

Several features stand out in the pressure record from the fast-flow borehole: sharp diurnal pressure peaks and a small time lag between peak surface temperatures and diurnal water pressure maxima (1–3 h), as well as the general similarity between the temporal variations in pressure and temperature. The correlation between the two, computed over a moving window, stayed above 0.7 for several days (Fig. c, grey shading). This high correlation was more pronounced late in the season, also coinciding with the water pressure dropping to atmospheric values at night.

A contrasting observation of water flow was made on 23 July 2014, when a clear water sample was retrieved from the bottom of a borehole (“Slow-flow” in Fig. ) and the borehole camera was deployed. The resulting borehole video (see Supplement) reveals a slowly flowing, thin layer of turbid water at the borehole bottom overlain by clear water, an unusual condition that allowed the observation, as the water in a bed-terminating hole is usually highly turbid due to the basal sediments disturbed by the drill jet.

The slow-flow borehole was 62 m deep, and the first recorded diurnal pressure peak reached 48 m (85 % of ice overburden). One pressure transducer with snubber was installed 6 cm above the bed. Figure c shows the pressure recorded in the slow-flow borehole (black line). Pressure records from three other boreholes in the same across-glacier line and one sensor downstream are shown in red in the same panel, while the record of a fifth borehole in the same line is shown in panel (d). Note that there are four virtually indistinguishable records in panel (c) during 23–25 July (see also Fig. ). After a data gap caused by a corrupted compact flash card, the records have become more dissimilar by 2 August, but continue to exhibit common pressure variations. The pressure time series from the borehole that is part of the line immediately below the slow-flow hole by contrast has significantly higher mean water pressure and the diurnal pressure variations have a much smaller amplitude. We have included it in panel (c) because it is the only one in that lower line that matches one of the other pressure records in panel (c) well, if we remove their means and scale them to have unit variance.

Most boreholes showing diurnal pressure oscillations share the general features displayed by the slow-flow borehole, specifically (1) smooth pressure peaks and troughs, (2) pressure patterns well differentiated from the atmospheric temperature pattern, (3) mean pressures during periods with diurnal oscillations that lie between 55–120 % of the overburden ice pressure (much higher than in the fast-flow hole), (4) peak pressures that typically lag peak temperatures by 2–8 h and (5) patterns of temporal pressure variations that are often similar to neighbouring boreholes both in the along- and across-glacier direction.

On average, during summer, 71 % of sensors showed the behaviour observed in the slow-flow borehole at some point, as assessed visually from the presence of smooth diurnal pressure oscillations. Only eight boreholes (3 % of the total, shown as red markers in Fig. ) exhibited water pressures dropping to atmospheric pressure, one of the key characteristics of the fast-flow borehole. Six of them were found during the three years with the highest cumulative positive degree day count in the dataset (2013: 437 ∘C days, 2009: 386 ∘C days, 2015: 297 ∘C days).

However, these figures may not be representative as drilling was concentrated in some areas. For this reason, we have selected 70 boreholes in two across-glacier profiles (blue markers in Fig. ). Among those, 81 % show a behaviour qualitatively similar to the slow-flow borehole, and 4 % that of the fast-flow one. However, note that even these statistics remain biased, as borehole spacing along these lines is concentrated in areas that were of interest due to likely drainage activity, and crevassed areas are under-represented as sensor signal cables typically have a short life span there.

We emphasize that the borehole in which fast flow was observed initially displayed a relatively smooth diurnal cycle, and the statistics above are based on the identification of diurnal pressure oscillations reaching atmospheric pressure at night: it is thus possible that more boreholes intersect conduits with fast-flowing water, without the observed pressure records indicating as much.

The remaining 26 % (or 15 % in the two cross-glacier lines in Fig. ) of boreholes do not show any significant diurnal pressure oscillations at any point during the year. These “disconnected” boreholes usually show year-round mean pressures between 90–120 % of ice overburden. Disconnected boreholes frequently show a near-constant pressure signal, but not always, with some exhibiting difficult-to-interpret temporal variability. In 2016 there were 55 disconnected boreholes, allowing us to treat their behaviour statistically. However, despite only slight differences in mean pressures between winter and summer there is a slow but statistically significant decrease in water pressure during summer, starting around the spring event and amounting to about 6 % of the overburden pressure in total (Fig. ).

When the whole dataset is viewed over a given time window during summer, it is often possible to identify multiple clusters of boreholes, each exhibiting a specific pattern of temporal pressure variations. Often, these patterns are defined by the way in which the amplitude of diurnal oscillations changes over time. While boreholes in a given clusters will share the pattern of temporal variability, this will differ significantly from the pattern of temporal variability in the other clusters.

One example of this phenomenon comes from the boreholes in Fig. f, where we can see a group of boreholes that display a very coherent signal but with a distinctive two-day period. However, those boreholes in (f) are directly adjacent to those in (e). The latter by contrast show a very different pattern of diurnal pressure variations (that we have associated with the fast-flow borehole, along with panels c and g).

Less clear-cut, though indicative of the same phenomenon is Fig. , where we see boreholes in panels (d)–(f) that exhibit quite different diurnal pressure variations from those observed in panel (c) (the group associated with the slow-flow borehole). Figure 3 of also shows an example of the same phenomenon during July and August 2011: borehole B in that figure is, in fact, one of a group of 5 that exhibit almost identical diurnal water pressure oscillations that are quite distinct from those in boreholes A1–A6 in the same figure.

The spatial patterning of the drainage system into distinct clusters becomes much clearer when a dense borehole array with good spatial coverage is available. During the summer of 2015, there were 88 boreholes with active sensors on the plateau indicated in Fig. , and 66 further downstream. The corresponding pressure records are presented in Fig. . Between 26 June and 27 August, 42 boreholes on the plateau (panel c) and 11 boreholes downstream (panel d) showed a highly coherent pressure signal that was qualitatively different from the atmospheric temperature signal (panel b) and the majority of other boreholes in the two areas (panel f). There was no consistent time lag between sensors in the plateau and downstream. However, there was a clear drop in amplitude of diurnal oscillations (panels c and d), where the latter showed amplitudes around 15–30 % of those seen in the plateau.

Five of the sensors on the plateau were capable of conductivity measurements (panel g). We emphasize that, in general, the spatial patterning was recognizable only in the pressure records, and pressure oscillations were not associated with conductivity changes. Although all five sensors showed very similar temporal variations in pressure (panel c), the conductivity time series bear far less similarity to each other, with only a handful of abrupt conductivity changes common to three of the sensors (S2 and S4 marked in panel g).

The group of 14 boreholes in panel (e) also shares common diurnal pressure variation patterns, though this is not immediately clear as the mean pressures and amplitude of pressure variations varies significantly. For that reason we have highlighted one line in black that shows these variations clearly. Notably, these variations are “inverted” versions of the pressure variations seen in panel (b), with peaks becoming troughs and vice versa. These anti-correlated boreholes, in contrast to those in panels (c) and (d), have smaller diurnal oscillations amplitudes, and the oscillations are superimposed on a signal with near-constant running mean and high mean pressure, usually close to overburden. Therefore, if diurnal variations were filtered out, these pressure records would resemble the winter regime. At 15 m spacing between boreholes, we do not observe sequences of boreholes smoothly transitioning from correlated to anti-correlated, in the sense that there appears to be no continuous change in phase and amplitude from borehole to borehole: we observe a sharp boundary between correlated and anti-correlated boreholes, or correlated boreholes and boreholes exhibiting no diurnal oscillations. Note that one of the records in panel (f) of Fig.  also anti-correlates strongly with the record in panel (e) during the later part of the time window shown, and the record in panel (d) anti-correlates strongly with the record from the adjacent borehole S4 during August; anti-correlation of this kind is a common feature of the dataset, often but not always involving boreholes in close proximity.

There is typically another set of boreholes that show very similar diurnal variations in water pressure super-imposed on near-constant or slowly changing diurnal running mean values. The diurnal variations for this set have very small amplitude (typically 0.2–0.6 m, exceptionally up to 6 m), and resemble a square-wave with super-imposed high-frequency variations. Matching oscillations can be observed in multiple boreholes spread over large distances, both along and across the glacier, and both diurnal and much higher frequency features in the pressure signal are preserved between these boreholes. An example from 2011 involving boreholes across the width of the study area, and recorded by different data loggers, is shown in Fig. . Clearly, the oscillations can be both correlated or anti-correlated with each other. Not shown in Fig.  is the longer-term evolution of water pressure in the same boreholes. While they share short-time-scale temporal variability, their long-term pressure variations are generally not well correlated.

The pressure observations primarily give us a two-dimensional picture of the drainage system. The drilling process itself as well as borehole camera investigation provides additional information on englacial connections . 37 % of all the boreholes drained completely or partially during the drilling process, as did 39 % of those in the cross-glacier lines marked as blue symbols in Fig. . For simplicity, we will give statistics for the entire dataset in running text below, and the corresponding figure for the cross-glacier lines in parentheses. Of the boreholes that drained during drilling, only 14 % (0 %) drained when reaching the bed, and the remaining 86 % (100 %) drained at some point during the drilling process, suggesting connections to englacial conduits or voids. Such connections were also observed on multiple occasions using the borehole camera. Drainage events occurred at all depths during drilling, but with a slight preference for greater depths, with 60 % (59 %) happening in the lower half of the boreholes. This remains true for the 2012 drilling campaign, where the first sensors were installed before the spring event and observations are likely to reflect winter conditions. Unfortunately, water level change and duration of drainage events were not recorded.

During the borehole re-freezing process, 29 % (11 %) of the boreholes showed a pressure spike typically about 1.3 times overburden pressure, suggesting that freezing happened in a confined space. In total, 62 % (73 %) of these initially confined boreholes showed diurnal oscillations during the first week, suggesting that some degree of connection was developed with a drainage system experiencing diurnally varying water input.

In 2014 and 2015, three one-year-old boreholes were re-drilled, and the sensors were recovered (boreholes A, B and C in Fig. ). During this process, we found that holes A and C had sections about 8–12 m long near the bed that had remained unfrozen for the entire year, suggesting that boreholes, as well as natural englacial conduits close to the bed, could remain open through the winter. In borehole A, contact with the bed had erroneously been assumed after the initial drilling based on highly turbid water. However, borehole video footage taken after re-drilling showed that the original borehole had terminated at an isolated rock. From the depth of nearby boreholes, we estimate that the sensor was installed approximately 4 m above the bed. Nonetheless, the diurnal water pressure oscillations recorded in borehole A continued to mimic other nearby bed-terminating boreholes that were drilled in 2014 and 2015, indicating a persistent connection. Figure i shows the pressure record in borehole A and the point at which the re-drilling took place.

We have described the apparent spatial patterning of the drainage system above. However, this patterning is not fixed but evolves over time. In Fig. c, it is clear that all 42 boreholes show very coherent temporal pressure variations at the start of the observation period. During late July and August, the pressure variations in some of the boreholes become more distinct until, by late August, there is no longer a common signal and all boreholes show dissimilar temporal pressure variations.

In Fig. c, this emerging patterning is evident only in the more disordered appearance of the plot for later times. In Fig. , we show the 42 boreholes of Fig. b separated into subgroups. Within these groupings, it is clear that boreholes can switch from having closely correlated pressure records to behaving independently and, less frequently, to being strongly correlated again (panels d–h in particular); For simplicity, we refer to boreholes as being “connected” while they exhibit the same temporal pattern of pressure variations, and as “disconnected” otherwise. For each grouping, we have computed a mean pressure displayed in black, including only the boreholes that are connected at a given time; in some cases, no boreholes were connected to each other, and we still used the last borehole to exhibit diurnal oscillations to define the set of connected boreholes. In that case, the black mean curve obscures the corresponding, coloured borehole pressure time series. These mean curves for each panel are re-plotted in the corresponding borehole marker colour in panel (c).

The major dichotomy in Fig.  is between the groupings in panels (d)–(g) and (i) on one hand and panel (k) on the other. The distinctions between panels (d)–(g) in particular are more subtle, and generally relate to the absence or subdued nature of certain diurnal peaks in them: for instance, panels (d)–(f) all show diurnal oscillations on 1 and 2 August, while the larger group in (g) does not; there are other examples close to the end of the summer season. For all groupings in panels (d)–(g), it appears that the early season records resemble each other more closely than those late in the season, as was already evident in Fig. b, and that fewer boreholes have disconnected early in the season.

At a much smaller scale, a similar fragmentation of the drainage system is shown in Figs. c and , where we see four boreholes that are initially very well connected during late July having become much less well connected in August, although with the diurnal pressure oscillations still showing some similarities between several of the boreholes. Interestingly, one borehole (S2, yellow) has ceased to exhibit oscillations by August, but is straddled by two that still do (S1 and S3, 15 m to either side), suggesting a relatively fine-scale structure to the drainage system locally.

In addition to spatial patterning, Figs.  and hint at an overall evolution towards lower mean water pressures and larger diurnal oscillations. The seasonal evolution of the drainage system may be evident not only in its spatial extent, but also in the evolution of mean water pressure and its response to surface melt input. Perhaps the simplest measure of sensitivity to surface melt input is what we term the relative amplitude of pressure to temperature oscillations: we compute standard deviations of pressure time series from boreholes that exhibit diurnal oscillations at some point of the season, and also standard deviations of positive air temperatures (the maximum of air temperature and 0 ∘C). We compute these standard deviations over one-day running windows, and define the ratio of the two running standard deviations to be the relative amplitude of pressure to temperature variations. Taking the running standard deviation of air temperature as a marker of surface melt rate variability (see Sect. S2 in the Supplement for a discussion about this assumption), the relative amplitude defined in this way gives an indication of how sensitive the drainage system is to variations in water input.

In Fig. , we see that the running standard deviation in pressure only vaguely tracks its temperature counterpart. However, the relative amplitude systematically increases during much of the season (Fig. c), except during an interval of colder weather and surface snow around the beginning of August, while the mean water pressure also decreases (Fig. d).

Above, we have seen that boreholes can become disconnected from each other, going from a state in which they undergo synchronous and virtually identical pressure variations over time to a state in which borehole pressure appears to evolve independently. The reverse change also happens, though less frequently (except during the spring event). The change from connected to disconnected and its reverse can take different forms. In a few cases, disconnection is gradual, with the boreholes continuing to exhibit similar diurnal pressure oscillations that progressively become more dissimilar in amplitude, phase, and mean water pressure. The record from H1 in Fig. c (yellow line) is one such example. However, in most cases the transition is abrupt, and the same is true of boreholes connecting with each other: a rapid change in water pressure can occur over the course of a few hours or less as a connection is established. We term such abrupt transitions “switching events”, following .

Figure shows multiple examples in the boreholes labelled S1–S4 in Fig. a, spaced 15 m apart. Perhaps unsurprisingly, the majority of switching events involving new connections seem to occur while water pressure is increasing or after a recent increase, while disconnections tend to occur as water pressure is falling (see Fig. c, d, f for several obvious examples), though the two are rarely symmetric, with disconnection usually occurring at a lower water pressure than the original connection. The record from sensor D1 in Fig. c is one such example, where arrows labelled S1–S7 mark multiple switching events. The borehole originally disconnected from the main group on 11 July, but reconnects on several occasions during periods of high water pressure in the active drainage system, disconnecting when water pressure subsequently drops. Note that for the first two reconnections, S2 on 21 July and S4 on 24 July, the switching events are clearly associated with large drops in conductivity as seen in panel (g), suggesting an inflow of meltwater that has spent less time in contact with the bed .

Towards the end of the melt season, most boreholes have become disconnected from each other, and water pressure in them typically rises again towards overburden, remaining nearly constant through the winter. However, in some cases, we observe quasi-periodic pressure variations in winter as previously reported in . Figure  shows the winter pressure record for the two sensors installed in the fast-flow borehole, extending the summer record shown in Fig. c. As in other boreholes, we see water pressures rising at the end of the summer season. This is briefly interrupted during early September, when surface snow cover temporarily disappears, and a drop in water pressure occurs in the borehole, accompanied by the resumption of diurnal oscillations. This is followed once again by the termination of diurnal oscillations and a sharp rise in water pressure towards overburden once the surface becomes snow-covered again. However, unlike in most other boreholes, that rise towards overburden is interrupted by oscillations lasting from 2 to 12 days. During these oscillations, water pressure can drop rapidly to as little as a quarter of the overburden, followed by a slower rise in pressure back towards overburden, stabilization, and a renewed rapid drop.

As observed in Fig. , there is large interannual variability in positive air temperatures and hence, presumably, in surface melting, both in terms of onset and intensity. In addition, we expect that differences in the snow-pack can also affect water delivery to the englacial system; presumably, a thicker snow-pack can store or refreeze surface meltwater, and leads to higher average surface albedo during the summer. Figure  shows a view of the study area from an automated camera on 19 July in 2012–2015. These images illustrate very significant differences in surface snow cover at the height of the summer melt season: the visible snow cover in each image is part of the remaining winter snowpack.

Alongside the interannual variability in temperature and snow cover, there are also significant season-to-season differences in the water pressure records. Differences in drilling objectives from season to season make year-to-year comparisons difficult except in one part of the study area. Figure  shows a compilation of pressure records from a set of boreholes drilled in almost the same locations every year in the lower plateau area from 2012 to 2015, as indicated by two red polygons in Fig. . There are four boreholes (surrounding the fast-flow hole in Fig. ) included here for 2013–2015 that were not drilled in 2012, and four that were drilled in 2012 but not in later years. Alongside air temperatures and snow cover, we also indicate the total PDD count prior to 15 June, and at the end of September. Also shown are the median of the dates on which diurnal oscillations appear and disappear over all boreholes with functioning sensors (red lines). The latter are clearly a crude measure of drainage system evolution as they are biased by borehole locations and drilling dates. Despite these differences, the borehole pressure records clearly indicate some systematic differences, with a relative absence of diurnal pressure oscillations in 2012 and 2013, though accompanied by very low water pressures associated with the fast-flow borehole in 2013 (see also Fig. ), and a larger number of “connected” boreholes with large-amplitude diurnal oscillations in 2015. We also note that we drilled new boreholes in 2014–2015 in the location of the 2013 fast-flow hole without encountering more evidence of turbulent water flow.

The timing of spring events and speed at which the evolution of the drainage system occurs appears to be linked systematically to the availability of meltwater. Cool, snowy summers are most obviously linked to a poorly developed drainage system with weak diurnal cycles (2012) and poor correlations between boreholes, as well as the absence of a sharp spring event (Fig. ).

The spatial structure of the drainage system also varies from year to year. The plateau area reliably has drainage activity, though upstream area pattern in Fig.  does not directly agree with the observed drainage structure (Fig. ), but is merely suggestive. Channel formation is influenced by pressure gradients controlled by surface and bed topography. However, changes in water supply geometry and the instability inherent in channel growth and competition between emerging channels implies that channels need not form in the same location every year. This is consistent with our failure to find in 2014 and 2015 a channel at the 2013 fast-flow borehole location.

Boreholes do not only disconnect from or reconnect to each other during the summer, a significant number of boreholes never connect at all. Others disconnect from the drainage system as it becomes more focused and fragmented into subsystems during stage 2. Some boreholes even disconnect and reconnect multiple times (Figs. , and ).

There is typically a very clear distinction between connected holes showing a similar response to the diurnal input, and disconnected ones that do not. Within a given drainage subsystem, there is typically no gradual phase shift or diminution of oscillation amplitudes from borehole to borehole, as would be expected if the drainage system were a diffusive system with a finite diffusivity . Effectively, our data suggest that, if the distributed system is diffusive, its diffusivity is very high, or zero where the system has become disconnected.

This observation contrasts with the interpretation of data in , who identify a gradual phase shift and decay in amplitude of diurnal pressure oscillations away from an inferred subglacial channel location. However, in our view, the phase lag in their Fig. 5 can also be interpreted as showing diurnal switching of their borehole 40 from being well-connected with their boreholes 29 and 35 to being disconnected. The latter interpretation would be consistent with , and with Fig.  here also showing an example of switching events with similar characteristics on South Glacier.

, suggest that the bed substrate at their study site is composed of glacial till of varying grain size distributions, acknowledging that “a network of small channels” on a hard bed could also account for their observations. However, in terms of hydrology, till and a distributed drainage system at the ice-bed interface share many characteristics: we expect both to give rise to a diffusive model for water pressure if water storage in the distributed system is an increasing function of water pressure. The primary difference is in how the permeability of that system evolves. In the “hard-bed” view, the permeability evolves over time in response to changes in effective pressure, whereas for a granular till, porosity, and thus permeability are simply functions of effective pressure and thus respond instantly to changes in it . The main inconsistency of appealing to drainage through continuous till layer as the main pathway for water flow is that we would expect to see more standard diffusive behaviour, and certainly no sharp switches between connected and disconnected portions of the bed. In addition, till with a sufficient coarse-grained fraction of cobbles and boulders would probably be capable of supporting the formation of cavities in the lee of those larger grains. In short, if till is capable of creating cavities, or is interspersed with bedrock bumps or somehow capable of supporting switching events by other means, then our interpretation would not be affected by assuming a hard or granular bed.

Pressure measurements at South Glacier thus suggest that the distributed parts of each drainage subsystem are hydraulically well-connected, with all connected boreholes showing almost identical pressure variations. However, the limited electrical conductivity and turbidity measurements also indicate that relatively little water might actually flow in the distributed system . Unlike in the data in , there are no diurnal variations in electrical conductivity. With a hydraulically well-connected system, this has to correspond to a low water storage capacity, where substantial variations in water pressure do not require similarly large changes in stored water. Alternatively, storage capacity could be relatively localized, so that water does not need to flow everywhere. show how water pressure variations with no corresponding change in conductivity can be observed over an impermeable bed. However, the proposed mechanism requires the boreholes to be disconnected, and would not operate on hydraulically connected boreholes as in this case.

While there are typically insignificant cross-glacier differences in diurnal pressure response within well-defined drainage subsystems, the same is not true in the down-glacier direction, even where we believe a hydraulic connection can be identified. The pressure time series along the inferred channel system in Fig.  (panels c, e, and g) are merely suggestive of a hydraulic connection, but hardly identical. The amplitude of pressure variations decreases markedly downstream from the fast-flow borehole, which would be consistent with a diffusive system, though it is unclear whether the change in amplitude occurs along the length of the channel, or within a putative distributed system flanking the channel, as the holes further down-glacier most likely did not sample the channel directly. However, importantly, there is no systematic phase lag accompanying the decrease in amplitude, as would be predicted by a diffusion model . However, it is conceivable that additional water input from surface sources along the flow path can have a significant effect on the phase of the pressure signal.

We have referred to boreholes that cease to exhibit diurnal pressure variations as having disconnected. Connection and disconnection typically manifest themselves very abruptly in time (Fig. , see also Fig. 5 of ). This transition usually takes from a few dozen minutes to a few hours. However, the initiation of the transition, often identified as a clear change in the rate of change of pressure with respect to time, can in many cases have the appearance of an instantaneous phenomenon, even at our shortest sampling interval of 1 min. Therefore, it is unclear if these time scales can be associated with the connection or disconnection process, as they might only represent how fast the system responds to a perhaps instantaneous switch between connected and disconnected states.

Usually, disconnection occurs during a drop in water pressure in the subsystem, and reconnection during an increase (Figs. and ). This is consistent with connection or disconnection resulting from viscous creep closing connections between individual cavities within the distributed system , or presumably with elastic gap opening or closing if sufficiently rapid. Disconnection could also be the result of cavities shrinking while remaining connected, if the borehole simply terminates on an ice-bed contact area between connected cavities and those contact areas are systematically larger than the ∼10 cm diameter of our boreholes. This process has been observed previously by . However, it seems unlikely that this effect, which should be random, would lead to a recognizable spatial structure of narrow drainage regions flanked by increasing large disconnected regions. Instead, we would expect a random distribution of apparently connected and disconnected boreholes.

The anti-correlated signals we observe in our data (Fig. e) have previously been explained by a mechanical load transfer mechanism, where the ice around a pressurized conduit redistributes normal load, reducing the normal stress over neighbouring areas of the bed. Unconnected water pockets in those areas would thus experience a drop in water pressure . A three-dimensional Stokes flow model supports this interpretation, and suggest that the anti-correlation pattern depends on the bed slope, which can be one of the factors affecting the observed distribution of borehole displaying this behaviour. Boreholes exhibiting anti-correlated pressures must be effectively disconnected, so that a change in normal stress mainly causes changes in the pressurization of the borehole rather than water exchange. The load transfer mechanism is consistent with our observations.

An alternative explanation suggests that such signals are associated to enhanced cavity opening due to basal sliding changes . However, it is unlikely that a variation in sliding would precisely mimic the local water pressure variations in the adjacent drainage subsystem, as suggested by Fig.  e: the force balance that determines sliding velocities should be affected by changes in basal shear stress across a larger portion of the bed.

Note that we observe anti-correlated signals in boreholes that are not immediately adjacent to boreholes showing a correlated signal (purple and blue markers in Fig. ). It would be difficult to explain the anti-correlated signal in these boreholes by normal load transfer over larger distances, when other disconnected boreholes nearby show no such behaviour. This suggests that the connected drainage system can contain fine structure (either as channels or narrow regions of distributed drainage) with lateral extents smaller than the ∼15 m borehole spacing. The same is indicated by the formation of disconnected “islands” in lines of otherwise connected boreholes at the same spacing as seen in Fig.  for the August observation period (see also , for analogous observations).

We have referred to a borehole as disconnected when observations show that pressure variations on a diurnal time scale are not communicated to a borehole. However, the evolution of the mean water pressure in disconnected boreholes is consistent with a residual amount of water leakage into the connected drainage system: during the summer, that mean pressure gradually decreases. The end of the monotonic increase in water pressure of disconnected boreholes observed in Fig. , coincides with the spring event, followed by a slow decrease. The large sample obtained in summer 2016 supports this trend up to a 99 % confidence despite the large variability of the observations.

As in , such a slow evolution could be accounted for by flow through a relatively impermeable till aquifer underlying a much more effective but less pervasive interfacial drainage system, and the magnitude of that leakage could have a significant impact on basal sliding rates if disconnected areas act as sticky spots.

Widespread hydraulic isolation of the bed in winter is supported by high recorded water pressures and the marked pressure drop at the spring event observed in 20 % of boreholes. In contrast, theories based on a remnant “distributed” system would ordinarily suggest relatively low water pressures in winter . Although it is possible that some boreholes do not connect because they were not properly drilled to the bed, we believe that the existence of persistently disconnected areas is robust. Non-spatially biased samples suggest that up to 15 % of the bed could remains unconnected year round. The existence of such unconnected holes, and the possibility of dynamic connection and disconnection, represents a challenge to existing drainage models, which typically assume pervasive connections at the bed.

In addition to conduits at the bed interface, englacial conduits are known to exist inside temperate glaciers . However, it is unclear whether they allow mostly vertical water transport, or if horizontal water transport over significant distances is also possible through them. Frequent drainage events during drilling (also observed by ) suggest the existence of a large number of englacial conduits, and borehole re-drilling observations show that upward conduits can remain open through the winter season in a layer extending several metres above the bed. However, we have no evidence of significant along-glacier drainage in winter, while we know that englacial connections can remain. This suggests that the englacial connections remain isolated from each other during winter. It is unclear if they can connect in summer and establish an englacial drainage system capable of supporting significant down-glacier drainage. The persistence of conduits through winter is most likely related to the basal layer of temperate ice , and hydraulic isolation preventing creep closure. The apparent ubiquity of englacial conduits suggests a need to assess their role in downstream water transport in the future; if significant, this represents another area of improvement for drainage models.

Strong correlations over long distances were observed in boreholes displaying all the features of disconnected boreholes except a superimposed low-amplitude diurnal pressure variations with high-frequency variations (Fig. ). From their wide spatial distribution, it appears impossible for them to be connected by hydraulic conduits. As such conduits would need an extremely high diffusivity to preserve the observed high-frequency features over large distances (>500 m as seen in Fig. ). Moreover, a high diffusivity is at odds with the diverging evolution of temporally smoothed borehole water pressures in the same holes.

These signals do not seem to be instrumental artifacts, and in many cases were recorded by independent data loggers. We have also considered effects due to induction on non-twisted signal cable coils, temperature, or solar irradiation. However, in those cases, such signals should also be superimposed on records from distributed drainage systems, contrary to our observations. Possible explanations must be related to periodic large-scale stress changes in the ice compressing disconnected boreholes the volume of which must remain constant, thereby eliciting an instant water pressure response. The most likely cause of such large-scale stress changes would appear to be the occurrence of periodic diurnal basal slip events as suggested by .

We generally assume that the sensors at the bottom of boreholes measure the water pressure at the bed. However, this may not always be the case if the sensor becomes encased in ice, is connected to an englacial conduit, or if the borehole did not reach the bed or has penetrated into the basal till.

It is likely that with time, some sensors can become encased in ice, as suggested by the fact that older sensors are less likely to show diurnal oscillations (for that reason, sensors in old disconnected boreholes were often decommissioned before they ceased to produce a signal), and the observations in doubly instrumented boreholes (see Sect. S1 of the Supplement). Digital confinement data suggest that in some cases, as in Fig. , the termination and initiation of diurnal oscillations is associated with an increase and decrease in confinement. This observation would also be consistent with ice encapsulation of the sensor during winter.

Although the upper end of the boreholes typically freezes shut within a few days, the abundance of englacial conduits opens the possibility that the sensors could connect to an englacial conduit through the lower portion of the borehole while it is still open. In such a case, the pressure record could at least partially reflect the evolution of englacial conduits instead of the basal drainage system.

Alternatively, in the absence of englacial connections, a sensor in a borehole that fell short of the bed would appear as disconnected, even if the underlying bed is not. However, we believe this is not a common situation due to the strict procedures followed to assess whether a borehole reached the bed or not (see Sect. ).

Observations by using wireless pressure sensors installed across the basal till layer in a glacier in Norway showed that, while a sensor at the ice-till interface shows clear diurnal variations, another one placed a short distance away inside the till layer can show a signal very similar to our disconnected boreholes. This could be a problem affecting some of our sensors, as borehole drilling could eventually penetrate the till. Nevertheless, the lifespan of a sensor buried in the till ought to be short if there is differential motion between ice and the sensor placement in the till e.g., causing the signal cable to tear. Indeed, one sensor that was accidentally installed directly on the bed with limited (1 m) cable slack, rather than suspended just above the bed, survived for only just over a month, and showed uncharacteristic high-frequency noise superimposed on a smooth diurnal oscillation (see the lowest curve in Fig. g).

Calibration drifts may affect in situ sensors over time, and differences in measured water pressure may not be reflective of an actual pressure gradient between two boreholes (Supplement Sect. S1); consequently, we have taken similarity in response to diurnal forcing as our indicator of connections, rather than looking directly at the evolution of pressure gradients.

We assume an arbitrary network of conduits connecting nodes labelled by a single index i; the edge connecting nodes i to j is identified by the double index ij. The basic set-up of the model, with a handful of alterations identified below, proceeds as in .

Along each network edge ij, we assume there are nc conduits connecting node i to node j: one “R”-conduit that can behave either as a Röthlisberger (R) channel or a cavity, as in , with average cross-section SR,ij, and nc-1 “Kamb” (K) conduits that behave only as cavities, and are not subject to enlargement by melting. This configuration mimics the sheet of and avoids the pitfall of having to resolve every basal conduit. We denote their average cross-sectional area by SK,ij. The conduits evolve according to dSR,ijdt=c1QR,ijΨR,ij+uhR(1-SR,ij/SR0)-c2SR,ij|Pe,ij|n-1Pe,ij,dSK,ijdt=uhK(1-SK,ij/SK0)-c2SK,ij|Pe,ij|n-1Pe,ij. Here QR,ij is discharge from node i to j in the R-conduit, and ΨR,ij the hydraulic gradient along the R-conduit, u is sliding velocity, hR the size of bed obstacles supporting cavity formation, and SR0 is the cavity-size cut-off at which further conduit enlargement drowns out bed obstacles. Pe,ij is the effective pressure driving conduit closure (related to Ni as described by Eq.  below), and c1, c2 are the same constants as in . Subscripts K refer to equivalent quantities for the K-conduits.

We associate a nominal effective pressure Ni with each node, defined as overburden minus basal water pressure. Hydraulic potential Φi at each node and hydraulic gradient Ψ along the network edges are given by Φi=Φ0,i-Ni,ΨR,ij=Φi-ΦjLij,ΨK,ij=Φi-ΦjTLij, where Φ0,i=ρigsi+(ρw-ρi)gbi is the geometrical contribution to hydraulic potential, ρi and ρw being the densities of ice and water, g acceleration due to gravity, si and bi ice surface and bed elevation at the node. Lij is the length of the network edge and T≥1 is the tortuosity of the K-conduits, relative to the R-conduits. In a departure from previous models, we assume a percolation cut-off for flow along the conduits and write QR,ij=c3max⁡(SR,ij-SPR,0)α|ΨR,ij|β-2ΨR,ij,QK,ij=c3max⁡(SK,ij-SPK,0)α|ΨK,ij|β-2ΨK,ij. Here c3 is the same constant as in , and α and β are constant exponents as in , while SPR and SPK are the constant thresholds that conduit sizes must reach before water can flow in the conduits. For linked cavities, such a threshold is easy to justify: while SK may be the average cross-sectional area of cavities along the network edge, the local cavity size will naturally vary as bed obstacles are uneven, and it is natural to expect that cavities with non-zero size may fail to connect. We apply the threshold equally to the R-conduit as it can act as either a channel or a cavity, and a cut-off must apply self-consistently in its cavity state. A node that is connected to others purely by conduits that are all below the percolation threshold is then hydraulically disconnected from the drainage system.

At each node, water can be stored in englacial void space connected to the node, with volume storage capacity per unit water pressure Vp see also. Water can also be supplied externally to each node at a locally defined rate mi, and flows along network edges through R- or K-conduits, or possibly through a permeable porous substrate if the conduits are closed. To account for conservation of mass, we also associate half the volume of water stored in a conduit between two nodes with each node, and likewise, account for half the water created by wall melting in an R-conduit as water supply to each node. Consequently, we impose mass conservation in the form VpdNidt+∑jQR,ij+(nc-1)QK,ij+kleakΨij=∑jρi2ρwc1QR,ijΨR,ijLij+mi, where kleak represents the possibility of a substrate (till) with non-zero permeability. Sums over j are taken over nodes connected to node i.

To close the model, we need to relate the conduit effective pressure Pe,ij to the nominal effective pressures Ni at network nodes. We write this in the form Pe,ij=∑kGijkNk, where the sum is over all node indices k in the network, and Gijk is a suitable positive averaging kernel that satisfies ∑kGijk=1; in our network model below we put Gijk=1/2 if k=i or k=j and Gijk=0 otherwise for simplicity; however, this is a surprisingly key assumption (see also Sect. S3 of the Supplement). Suppose we have kleak=0 and no hydraulic connection at all between adjacent nodes. This can lead to arbitrarily large effective pressure gradients. The usual assumption of cavity formation models breaks down, namely that adjacent cavities are subject to the same nominal effective pressure, defined as overburden pressure (or far field normal stress) minus a common water pressure. The rate of opening or closing of a cavity is unlikely to be a function of its own nominal effective pressure alone, and is likely to be affected by stresses around other nearby cavities and thus dependent on their nominal effective pressure. The observation of anti-correlated water pressure records in our dataset indicate a load transfer of overburden onto highly pressurized parts of the bed also supports this assumption. We try to capture this load transfer effect by the averaging kernel G above.

More practically, if conduit opening and closing were driven by a local effective pressure variable alone, then the generalization of our model to a distributed sheet would result in disconnected parts of the bed potentially never reconnecting. In order for a disconnected region to reconnect, sheet thickness in the disconnected region needs to change. On the absence of leakage through the substrate, the only way that can happen in a manner that is driven by the hydrology of the connected regions is through a non-local sheet closure term, or through the sliding velocity u. We expand on this in Sect. S3 of the Supplement, but note here that conduit closure must involve a non-locally defined effective pressure in order for our percolation model to function as intended, allowing for expansion as well as the contraction of a connected region at the bed.

A key component that the model above continues to miss is the ability to open conduits due to overpressurization of the system . While not necessary to explain switching events during the main melt season, this is likely to be key in establishing a drainage system at the start of the melt season: unlike existing sheet models, in which a distributed system always exists and can simply be expanded through water supply in the spring, the percolation limit model above allows the system to shut down completely, and rapid re-establishment through a spring event is likely to require overpressurization. We discuss the extension of the approach in and further in Sect. S3 of the Supplement.

Network-based models for drainage channelization (e.g. ) have been used previously to model the seasonal evolution of drainage systems. In particular, they have been used to model the evolution of a channelized system from a more spatially extensive one as in stages 1 and 2 identified in this paper, and the subsequent shut-down of the system as in stage 3. See Fig. 3 of , Fig. 5 of , and Fig. 12 of for examples of seasonal drainage evolution. What these models are missing is the ability to capture the formation of disconnected regions at the glacier bed and the subsequent ability of parts of the bed to connect and disconnect to the drainage system, which is what we focus on here.

Instead of attempting to model a full seasonal cycle, we focus here on the effect of time-dependent water input into a fully channelized drainage system, in order to test whether our modification of existing models can capture the qualitative behaviour of the switching events such as those in Fig.  (see in particular panel g). In other words, we focus on the behaviour of the drainage system in stage 2. Our simulation is an idealized run, not based on the specific geometry or properties of the South Glacier field site and does not claim to reproduce observations beyond their generic features. Work to use proxies for surface melt rates and likely surface water supply routing in an inverse model for the drainage system is currently underway and will be reported elsewhere. Our aim here is simply to study the qualitative features of our forward model, modified from existing ones found in the literature, and to compare them equally qualitatively with our borehole records.

The rectangular domain is 5 km long and 1 km wide. The ice and bed surfaces are used with contour lines in panels (c) and (g) of Fig. , where black lines are surface contours at 100 m intervals, and grey lines are bed contours at the same intervals. Zero inflow is prescribed at the sides and top of the domain, and zero effective pressure at the lower end of the domain y = 0. The network geometry is the same as indicated in Fig. 1 of the Supplement to , with a total of 201×201 nodes.

We allow water to be supplied in 40 discrete locations (effectively, moulins). Each moulin undergoes a diurnal cycle the amplitude of which varies over several days, with mean water supply rates also varying over several days; the dominant period of the cycle is the same for each moulin but the ratio of diurnal amplitude to mean water supply is chosen randomly (while maintaining positive water supply rates at all times), and we have allowed for slight phase shifts between moulins. The time series of water supply to all moulins are shown in Fig. i.

We show two different simulations. Both use the parameter values shown in Table , except that the percolation cut-offs SPR and SPK have been set to zero in one of the simulations (to identify the effect a percolation limit has on our results), and uhR and uhK are also set to one-tenth the value given in Table  in the same simulation (without a percolation limit, cavities of the same size will permit larger discharge, so we reduce the cavity opening rates uhR and uhK in order to limit cavity size).

Figure  shows one set of panels for each simulation, identified by the numbers 1 (no percolation cut-off) and 2 (finite percolation cut-off) in the panel labels. When referring to a specific panel for both simulations at the same time, we will identify it by the letter in the panel label. Both simulations start from a fully channelized steady state computed with moulin water supply set to constant values. Diurnal oscillations are subsequently superimposed on those constant water supply values. The system is run for several days to account for transients before the detailed results shown in Fig.  are computed.

The channelized configuration of the system does not change during the simulation (compare panels a and e in Fig. ), although it differs slightly between the two simulations. Pressure oscillations result from the time-dependent water supply. These pressure variations are confined to the connected drainage system (compare panels c and g of Fig. , and see also the Supplement movie #2).

In simulation 2 (with non-zero percolation cut-offs), the extent of connected drainage system also evolves, shown as blue areas in panels (d2) and (h2) of Fig.  and the Supplement movie #2. While pockets of water can move down-glacier essentially without connection to the channelized system (see the left-hand side of the main drainage axis in Supplement movie #2), the main feature here is the expansion of the connected system at times of large water supply and low effective pressures. The larger connected system in panel (h2) of Fig.  corresponds to peak water supply, the smaller system in panel (d2) to the minimum water supply in the cycle shown. This is at least qualitatively consistent with our observation of switching events, that establish connectivity during periods of increasing water supply.

In Fig. , we focus on what an array of boreholes would observe. The borehole array is located in the black rectangle shown in panel (d2) of Fig. . Panels (a2)–(c2) in Fig.  show the evolution of the connected parts of the bed within the borehole array, while panels (d2)–(g2) show pressure time series, again grouped subjectively.

The presence of at least two distinct drainage subsystems is immediately obvious (circles and diamonds in panels (a2)–(c2), corresponding to the time series shown in panels d2 and e2 respectively). These two subsystems correspond to two different drainage channels. The grouping of boreholes in panel (f2) (triangles) is intermittently connected to the diagonal channel of panel (e2) (diamonds), with different boreholes connecting and disconnecting at different times, through connection and disconnection are again typically favoured by low and high effective pressures, respectively. This is in qualitative agreement with our actual borehole data (see Fig. g). In addition, there is an additional grouping of persistently disconnected boreholes (Fig. g2, crosses), although two of these become very poorly connected later in the cycle, permitting an excursion in effective pressure without obvious diurnal cycling.

One important aspect of the synthetic borehole records in panel (f2) of Fig. , is the relatively minimal attenuation of amplitude and minimal phase lags observed within that distributed system relative to the channel (panel e2) to which the distributed system connects, and the abrupt switching to nearly constant effective pressures on disconnection. Compared with borehole data from South Glacier, we do not reproduce the tendency of disconnected boreholes to experience rising water pressure (i.e. falling effective pressure), which we believe is related to the dynamics of disconnected boreholes incised upwards into the ice being squeezed by anisotropic stresses in the ice, an effect this drainage model is not designed to capture.

Removing the percolation cut-off for simulation 1 increases the ability of the distributed system to drain water (Eqs.  and ). To account for this we simultaneously lower conduit opening rates uhR and uhK to 3.47×10-7 m2 s-1, keeping all other parameters the same. In order to create comparable drainage structures in both simulations, simulation 1 was started from the same initial state as simulation 2. The percolation cut-offs and conduit opening rates were then gradually changed with water supply rates held constant until a new steady state was achieved, before imposing the same diurnal oscillations as in simulation 2 again (panel i of Fig. ).

A small change in channel configuration results from removing the percolation cutoffs: two of the main drainage channels along the centre of the glacier in Fig.  panels (a2) and (e2) collapse onto a single channel in panels (a1) and (e1), as they are no longer isolated from each other by the percolation cut-off. However, the main difference in model results is the much larger region over which the effect of oscillatory water input is felt away from the channels (compare Fig.  panels c1 and g1 with panels c2 and g2). This is a natural consequence of enforcing connectivity in the drainage system everywhere (see Fig.  panels d1 and h1).

On the left-hand side of Fig.  we use the same groupings of synthetic boreholes as for the simulation with a percolation cut-off (right-hand side of the same figure). The boreholes marked as diamonds produce an almost identical pressure time series as in the first simulation (compare panels e1 and e2 in Fig. ), indicating that the behaviour of channelized drainage is not substantially affected by dispensing with the percolation cut-off. By contrast, the boreholes marked as circles in Fig.  experience higher mean effective pressures and bigger oscillations in panel (d1) than (d2). This is the result of the two channels on the left-hand edge of the domain in Fig.  panels (a2)–(c2) having been merged into a single channel in panels (a1)–(c1): the percolation cut-off allows subsystems to co-exist separately in closer proximity, in accordance with our observations at South Glacier.

The biggest difference is in the behaviour of the boreholes within the distributed system surrounding the channels (Fig. , panels f and g). Unlike in the case of the model with a percolation cut-off, the results in panels (f1) and (g1) no longer exhibit switching events, and there are no persistently disconnected boreholes. Instead, we see evidence of typically diffusive behaviour away from the channels: a reduction in the amplitude especially of the higher frequency (diurnal) forcing components with an attendant phase shift, and the absence of a sharp division between drainage subsystems. This behaviour mimics that in Fig. 8 of , but contrasts with our field observations. Those observations indicate minimal variations in amplitude and phase shifts within drainage subsystems, with sharp boundaries separating different subsystems. The inability to explain those features of our field observations motivates the model modification we have proposed here.

That modification comes with one major drawback, which we do not attempt to resolve here. While the model is able to open drainage connections spontaneously, this is a slow process driven by viscous deformation, controlled by the non-local effective pressure term in Eq. (). When the drainage system is subject to a rapid increase in water supply, the physics by which drainage connections are established may involve either elastic hydrofracture driven by overpressurization (e.g. ) or the large-scale uplift of ice at flotation as described in . As we discuss in Sect. S3 of the Supplement, the latter is not straightforward to incorporate into our modified model, as is the former (which requires a blending of elastic and viscous effects). We identify this as an important area for future research.