Glacier de la Plaine Morte is located in the Bernese Alps (46∘23′ N, 7∘30′ E), Switzerland. It is the largest plateau glacier in the European Alps (7.1 km2 in 2019), with 90 % of its surface spanning an elevation range of only 2650–2800 m a.s.l. The ice-marginal Lac des Faverges is located in the upper reaches of the glacier, at its south-eastern margin (Fig. a). According to aerial imagery of the Swiss Federal Office of Topography, the lake started forming in the 1970s and now fills annually during the melt season. Because of the rapid ice loss over the last few years, the basin has enlarged, thus increasing the potential lake volume too . Simultaneously, the maximum lake level lowered due to a significant reduction in ice-surface elevation, and since 2012, the lake water no longer overspills a sediment ridge to the south. Instead of draining superficially towards the Rhône basin, the lake water now drains englacially northwards, into the Rhine basin. The glacier lake outburst floods of Lac des Faverges have occurred annually since 2011 and represent a serious concern for Lenk, a 2300-inhabitant village 10 km downstream of the glacier snout . The lake level and temperature have been monitored in detail since 2012 by Geopraevent AG (https://www.geopraevent.ch/, last access: 10 November 2021) for early warning purposes, and daily images from an automatic camera are also available. An alarm is triggered when the rate of lake-level change reaches a given critical value. projected the future evolution of Glacier de la Plaine Morte for the coming century and also estimated the changes in the lake basin over the next few decades. They concluded that a continuous increase in lake volume is likely, along with an increase in the potential flood hazard for the village of Lenk. In 2018, the lake discharge reached around 80 m3 s-1 causing damage to infrastructure for the first time .

In spring 2019, local authorities decided to limit the maximum lake volume by constructing a supra- and englacial channel to artificially drain the lake water in order to face the increasing threat by floods due to sudden lake drainage. This channel now connects the lake outlet to a permanent large moulin located ∼ 1.3 km westwards and about 20 m lower in elevation (we will refer to this feature as the “Moulin West” in the following; see Fig. a). Past observations have shown that this moulin is in turn connected to the subglacial network and that this connection is often established relatively early during the melt season . In the middle of the channel there is a ∼ 100 m long tunnel (labelled “`micro' tunnel” in Fig. c), which is a remainder of the initial plan to drill a 40 cm diameter englacial tunnel, instead of a surface channel, for part of the distance (only a short section of the tunnel was completed, due to technical issues). The supraglacial channel was dug from the beginning of April until early July 2019. In a first stage, the 4–5 m deep snow cover had to be removed by snowcats. In a second stage, the solid and impermeable ice was cut and removed by an excavator. Because of these artificial interventions, the initial geometry of the channel is well known, with a width of 1 m at the bottom, a 4 to 7 m depth from the ice surface and a length of 1.3 km (see Fig. d). During this second stage, water from ice melt and snowmelt was present in the channel.

The lake is connected via a pre-existing ice-surface canyon and a subsequent natural ice cave to the artificial supraglacial channel (Fig. b). At the beginning of the canyon, where it connects to the lake, the water flows through an englacial siphon for about 30 m.

On 10 July 2019, at 11:00 CEST, the channel spillway elevation was lowered by an excavator to match the lake level and to artificially initiate the lake drainage. At that point, the spillway elevation was 2733.15 m a.s.l., corresponding to a lake volume of 1.49×106 m3 ± 0.11 × 106 m3 and an area of 0.127 km2. The lake water ran only into the first, upper part of the channel (Fig. c and d) and then infiltrated the glacier through a pre-existing moulin located within the micro-tunnel. A dye injection in the channel on 26 July 2019 revealed that the lake water exited the glacier outlet after about 2 h and that there was no significant lake water accumulation within the glacier.

In the following, we limit our attention to the upper part of the channel through which the lake water flowed for 36 d in total (Fig. c and d). This section (termed “channel” henceforth) was located between the cave outlet and the micro-tunnel entrance, was 540 m long, and had an average slope of 0.72 %. We designed our field campaign to monitor the hydraulic and thermodynamic properties of the water flow in the channel and relate that to lake level and volume evolution.

Figure  shows the temporal evolution of lake water volumes between 2012 and 2019 and hourly averaged discharge. The lake water input Qin was only accounted for in the year 2019 since it becomes relevant to take it into account in the calculation of Qout, due to a much smaller value of the latter compared to in previous years. The increasing trends of both maximum volume and peak discharge from 2012 to 2018 are clearly visible. The drainage onset time depends, amongst other things, on the meteorological conditions during the lake-filling phase. In warmer years, the date of complete filling of the lake basin occurs earlier. Also, an early depletion of the winter snow cover is likely to be linked with an early development of the subglacial drainage system, which in turn favours subglacial lake drainage . Since 2014, the lake volume has systematically reached more than 2×106 m3, and the subglacial release of the total lake water occurred within a few days, except for in 2015 when the water drained through a supraglacial channel into a nearby moulin for about 2 weeks.

The 2019 lake drainage pattern is drastically different from in previous years due to the artificial intervention. We distinguish four different phases. The first phase (Phase I) is from 10 July to 1 August 2019, when approximately half of the lake emptied through the supraglacial channel. Phase II is from 2 to 15 August 2019, when the lake level remained roughly constant but lake water was still running in the channel. Phase III is from 15 to 21 August 2019, when lake water stopped running in the channel and the lake level remained constant or slightly increased. Phase IV is from 22 to 27 August 2019, when the second half of the lake volume emptied subglacially, which is similar to the natural drainage mechanism of previous years. Note that the discharge peak of 3.5 m3 s-1 was much lower compared to in other years, e.g. over 20 times lower than in 2018 (Fig. ). In this regard, the technical intervention was very successful.

The channel bottom elevation and its evolution with time at five locations along the channel is presented in Fig. . The incision shows a uniform spatial pattern and is about 8 m during the supraglacial drainage (Phase I and II, 10 July–15 August 2019). Note that the channel slope was not uniform prior to the drainage onset (Fig. d) and that it remained relatively constant after natural adjustments during the first days of drainage. The low slope at the channel segment P5–P3 leads to uniform streamflow, which allowed the formation of clear daily water level cuts (Fig. ). In contrast, the higher slope between P2 and P1 led to more turbulent water flow and subsequent formation of step pools e.g.. Note that meandering did not occur, and, thus, the channel length stayed constant.

Widening is substantial only at P5 where channel width increased from 1 m on 10 July to 3 ± 0.1 m on 8 August 2019. Further downstream, e.g. at P4, the widening is minor (Fig. ). Overall, the channel geometry was mainly driven by vertical incision rather than lateral melting.

The water stage (Fig. ) and the hydraulic slope determine the water discharge. Water stage measurements were challenging during the first days of the supraglacial drainage (i.e. 10 July 2019, beginning of Phase I) since the excavator used to deepen the channel spillway left irregular traces on the channel bottom. Nevertheless, probe measurements (at P5, P4, P3, P2 and P1) together with water pressure sensors (at P5, P3, P2 and P1 only) reveal that the water stage on 10 July 2019 was around 1 m at P5, 0.4 m at P4 (which was close to the spillway location) and 0.3–0.5 m for the other stations. The water stage stabilized at 0.6 m after a few days of drainage (Phase I) at P5 and at around 0.4–0.5 m at the other stations. Daily fluctuations were typically 0.1 m due to the daily melting cycle influencing the lake input. At P1, measurements were soon no longer feasible because of the formation of step pools. The water stage slowly decreased to a value of 0.1–0.2 m uniformly over the channel during Phase III and Phase IV, when lake water no longer drained through the channel.

Figure  presents time series of water temperature, channel discharge and channel bottom elevation at station P3, along with the lake level. Data gaps in discharge and temperature time series are due to disruptions of logging. The daily mean discharge time series between 13 and 24 July 2019 has been calculated using lake-level change and modelled lake inflow (see Sect. ). The peak in channel discharge was reached on 14 July 2019 (Phase I), with a daily average of 1.7 m3 s-1. The good agreement between the different direct discharge measurements (Q^i) and the continuous discharge at P3 indicates that the stage–discharge relationship is valid over the entire drainage duration. The decoupling between Q at P3 and Qout during Phase III and IV (Fig. b) is because the lake no longer drained through the channel at that stage. The temperature and discharge signal of the lake water is clearly visible in the channel until 14 August 2019. This is in contrast with the temperature signal from the glacier's daily melt pattern from 23 August to 4 September 2019 (water temperature close to 0 ∘C at night), when the lake was no longer draining through the channel (beginning of Phase III).

The stable mode of drainage during Phase I is corroborated by the observation that the distance between daily water level cuts in the channel (indicative of channel floor erosion; see Sect. ) corresponds to the rate at which the lake level lowers (Fig. ). At P5, this distance varies between 30 and 60 cm d-1 during the second half of July 2019 and drops on average to 12 cm d-1 for the first half of August 2019.

The streamflow in the channel is highly turbulent during supraglacial drainage. The Reynolds number fluctuates with discharge and is between 2.5 × 105 and 1.5 × 106 (Phase I and II). Note that the transition between laminar flow in the lake and turbulent flow somewhere at the channel entrance is further upstream than P5 (the location is not known exactly).

The Darcy–Weisbach friction factor was calculated for the channel segment between P5 and P3, where accurate calculation of D‾H and thus P‾w was possible. The time series of the inferred friction factor is presented in Table .

Water temperature, measured continuously at P5, P3, P2 and P1, shows an exponential decrease along the channel (Fig. ) and exhibits daily fluctuations (Fig. , e.g. ±0.1 ∘C during Phase I). The relation given in Eq. () is fitted to these temperature observations. Note that the pattern was similar whenever lake water was flowing through the channel.

Figure  presents time series of the Nusselt number Nu calculated from our measurements according to the melt-rate method (Eq. ) and the spatial-cooling-rate method (Eq. ), as well as from the empirical Dittus–Boelter equation (Eq. ) and the Gnielinski correlation (Eq. ). For the Dittus–Boelter equation we use coefficients A, α and β from previous studies (Table ); for the Gnielinski correlation, we use the mean friction factor fD=0.3 (Table ). The heat transfer is dominated by convection, with typical Nu values on the order of 104. The results from the melt-rate and spatial-cooling-rate methods are in good agreement, except for the period 11–16 July 2019 (beginning of Phase I). For this period, the higher Nu values obtained by the melt-rate method compared to the spatial-cooling-rate method could be explained by the very high melt rate at P3 (see Fig. ).

Our values for Nu are significantly higher than the ones derived from the Dittus–Boelter equation using standard coefficients e.g.. We note, however, that the coefficients used by and result in Nu values that lie at the lower and upper edge, respectively, of the uncertainty range of our Nu values. Conversely, the Gnielinski correlation produces values of Nu that are significantly higher than our results, as well as the ones obtained by the alternative methods.

The Nusselt number Nu is dependent on the Reynolds number Re via turbulent mixing. Figure a shows the relation between Nu and Re as determined by our field measurements, along with the previously used parameterizations for Nu. It is noteworthy that our results show a less pronounced dependence of Nu on Re than other assessments. Indeed, fitting the Dittus–Boelter equation (coefficients A and β, Eq. ) to our data yields an exponent of β=0.58, which is low compared to exponents of between 0.75 and 0.93 found by , and (Table ).

The four phases of the lake drainage are interpreted as follows: Phase I is characterized by stable supraglacial lake drainage, i.e. the lake drawdown is controlled by the rate of vertical channel incision. There is a significant difference between the computed lake outflow Qout and the channel discharge Q at P3 during the sub-period 26 July to 30 July 2019 (Fig. b), coinciding with strong rain that ended a heat wave which started on 20 July 2019.

During Phase II, the lake level remained constant (Fig. a), but the lake water was still running in the channel as evidenced by the relatively warm water (Fig. c). This is indicative of the outflow of the lake-channel system behaving like a non-erosive spillway: the channel only received the water from snowmelt and ice melt, which spilled above a constant elevation. The exact location of the spillway is unknown: it could be located either in the englacial siphon between the main lake and the canyon or in the ice cave (Fig. ) although visual inspection gave no evidence for the latter. Phase III is characterized by the stopping of the supraglacial drainage around 15 August 2019. It is likely that the lake surface became too small, such that the lake drawdown became higher than the channel incision. This is especially likely if the channel was disconnected from the main lake by the spillway between the canyon and the main lake or in the case of subglacial leaks. Although no sensors were installed in the channel from 19 to 23 August 2019, the channel incision between 14 and 23 August 2019 was negligible (Fig. ). This indicates that the peak in Qout on 19 August 2019 was due to subglacial rather than supraglacial drainage.

During Phase IV, the lake emptied subglacially, with no influence from the supraglacial channel. The triggering mechanism is presumably similar to in previous years. showed, for example, that the drainage in 2016 was initiated by hydrofracturing.

We calculated the hydraulic friction factor fD of the channel at several instances during Phase I (Table ), when the necessary salt dilution measurements were available. Values for fD range from 0.17 to 0.48, corresponding to a Manning roughness n′ of 0.038 to 0.068 s m-1/3. These values are similar to the ones of , who inferred n′ to be between 0.036 and 0.058 s m-1/3 in supraglacial streams on the Greenland Ice Sheet. Similar observations of revealed strong variability in n′ in time and space, ranging from 0.009 to 0.154 s m-1/3 with a mean value of 0.035 ± 0.027 s m-1/3, similar to ours. The lower end of their range corresponds to a smooth channel, whereas the upper end cannot be explained by ice-channel roughness alone and is an indication not only of slush ice being present in the channel but also of the influence of form friction (as opposed to skin friction) in a complex three-dimensional channel geometry.

Why the friction factors vary so much is unclear from our observations. For instance, there is no correlation with the Reynolds number (Fig. b). Previous studies have already highlighted the need to better quantify the hydraulics of englacial channels e.g. and the need for additional in situ observations to better constrain the parameters that control discharge e.g.. Our work provides a range of accurate, field-based friction factors for a supraglacial stream at different times during a lake drainage and shows its variability. However, the large range of friction factor values reported here and in other studies suggests that using a constant value in modelling studies could be inappropriate. Instead, we suggest that modelling studies should treat fD as a stochastic variable e.g., i.e. that they should use a range of values as opposed to a single one.

In general, the supraglacial drainage of an ice-dammed lake progresses via the incision of the channel by melt. The incision rate determines whether the lake drains gradually, i.e. with an approximately constant discharge over time, or unstably, i.e. with a progressively increasing discharge .

To characterize channel incision, the heat transfer between the advected lake water and the ice channel walls needs to be quantified. This heat transfer is captured by the Nusselt number Nu, which was derived from measurements in this study. Our results (Figs.  and ) are in between the predictions of the Dittus–Boelter equation (Eq. ) using the parameters from , which are higher than our values, and those of , which are lower than our values. This suggests that using the parameterizations of these two studies, which have a cryospheric context as well, gives an interval of Nusselt numbers to consider. Again, as with the hydraulic friction factor, the large range of plausible Nusselt numbers means that it should be a stochastic parameter in modelling studies.

The cause of the discrepancy between the exponent of the Reynolds number in the Dittus–Boelter equation determined in our study (β=0.58) and others (β between 0.75 and 0.93, Table , Fig. a) is not clear. Our water temperature measurements were conducted using CTD sensors which sunk to the channel bottom. Their close proximity to the ice means that they might have measured a temperature below the bulk water temperature, which would lead to an overestimation of Nu using the melt-rate method. Conversely, the results of the spatial-cooling-rate method would likely be less impacted as the temperature e-folding length should not depend on the location of the sensors. Future studies should put an emphasis on accurate water temperature measurements, for instance by using fibre optics methods as in , paying attention to accurately estimating the bulk water temperature.

Longitudinal temperature profiles of supraglacial channels have been studied in both the field and the laboratory but only by one study so far . Water temperature decreases exponentially with distance, which is the basis of estimating Nu in our spatial-cooling-rate method (Eq. ). This method is useful because it is easier to implement and allows a higher sampling rate than the melt-rate method (Eq. ) since it only requires temperature measurements and not the more challenging channel incision rate measurements, as were used in other studies e.g.. If the melt-rate method is chosen, direct measurements of channel incision are needed, and we support the idea of that daily water level cuts can be used to conduct those measurements a posteriori (Fig. ).

Besides the incision rate, the channel aspect ratio plays a major role in the drainage stability of a supraglacial lake: for a given stage, a wider channel has a bigger discharge capacity than a narrower channel and will consequently contribute to stabilizing the drainage by accelerating the lake-level drawdown (, their Eq. 8). Channel width was considered constant in the present field study and was used to calculate the hydraulic diameter and consequently all parameters which are derived from it. The large uncertainties we applied to the width take into account a potential change in width, and the estimated parameters take this fully into account. Moreover, our stage–discharge relation at P3 seems to work well despite a slight widening at that location.

The channel aspect ratio was considered by , who suggest that this ratio is determined by the melt-rate dependence on water depth: if the melt rate is independent of water depth, a very broad channel forms, whereas if it scales linearly, a nearly semi-circular channel forms. To our knowledge, there is no theoretical work available for how melt rates are distributed over the channel perimeter, but extending the study of could shed light on this issue. Future field-based studies could try to quantify the channel aspect ratio, as it would give indications of both lake drainage stability and melt-rate distribution along the channel perimeter.

The maximum lake volume of 1.49×106 m3±0.11×106 m3 reached in 2019 was limited by the constructed supraglacial channel to about two-thirds of its potential volume, and half of the lake water (∼ 0.7 × 106 m3) drained, stably, through the channel once the lake overspilled into it. In this sense, the hazard mitigation of Lac des Faverges was very successful; however construction costs were considerable at CHF 1.7 million. Still, construction costs were lower than the damage costs of CHF 2.5 million in 2018 alone. The relatively small volume of remaining water later drained subglacially during an outburst event (Fig. ). The peak discharge was only ∼ 3.5 m3 s-1 and thus far lower than the ∼ 80 m3 s-1 recorded in 2018.

The intervention at Lac des Faverges thus indicates that a channel dug at the glacier surface prior to the lake filling can successfully limit the maximum lake volume and thus the hazard potential. Indeed, partially or fully filled lakes have been successfully drained in the past via such channels e.g., although not at the scale of the present artificial intervention. Nonetheless, surface lake drainages also have considerable hazard potential e.g., and therefore the prediction of whether a surface drainage proceeds stably or unstably is critical for hazard assessments. proposed a criterion based on lake area and temperature (their Eq. 8). The area and temperature of Lac des Faverges respected this stability criterion when the drainage initiated in 2019 but only barely. However, the past surface drainage of 2015 (Fig. ), which drained stably through a shorter channel (about 400 m), suggests that the 2019 drainage was well within the stable regime; refining this criterion would help improve future hazard assessments.

The artificial channel remained active throughout the summer of 2020, and it was effective in limiting the lake volume and thus the hazard emerging from it. In terms of operations, the substantial amounts of winter snow blown into the channel proved to be challenging as they formed an intermediate blockage. In early August 2020, the winter snow was partially removed by an excavator, and the remaining snow blockage was eroded in a slush-flow-like event, after which the lake drained partially through the supraglacial channel (i.e. corresponding to Phase I in Fig. ) and partially subglacially (i.e. corresponding to Phase IV in Fig. ). Since the slush-flow-like event was relatively difficult to control, additional artificial measures were taken for 2021. These aimed at activating the channel's water flow underneath the extensive snow cover that rebuilds during winter. However, in 2021 the lake drained subglacially when only half full and before lake water could flow through the channel; the lake outlet was situated in the “canyon” region (Fig. b). The drainage occurred in late July 2021 which was early relative to the still extensive snow cover and to the low filling level.