Impulse waves generated by mass movement into lakes are common in alpine regions, where avalanches can be large and impact velocities can be high (Heller et al., 2009); hence, avalanches are the most common GLOF triggering mechanism in the Himalayas (Emmer and Cochachin, 2013; Falátková, 2016). For a realistic avalanche-triggered GLOF scenario to be computed, the source and trajectory of an avalanche must first be determined.

Ice and snow cover near Imja Tsho was previously identified by Rounce et al. (2016) with Landsat imagery using a ratio of NIR (near-infrared) and SWIR (shortwave infrared) bands with a threshold of 2.2 (Huggel et al., 2004a). Any ice-covered area with a slope between 45 and 60∘ was considered avalanche-prone (Fig. 1); slopes above this limit are generally too steep to allow for mass accumulation (Alean et al., 1985; Osti et al., 2011). Finally, the areal extent of the initial block of mass to be released was determined using a variable kernel filter, grouping avalanche-prone pixels together if 90 % of the surrounding pixels are also avalanche-prone (Rounce et al., 2016). Ice thickness ranges were determined based on observations in Russia (Huggel et al., 2005), standard values in Switzerland (Huggel et al. 2004b), and estimates in the Chinese Himalaya (Wang et al., 2012). Assumed values fell between 10 and 50 m, such that when combined with areal extents, the total avalanche volume could reach from 2.7 × 104 to 6.7 × 106 m3 (Rounce et al., 2016).

Avalanches were modeled using the Rapid Mass Movement Simulation (RAMMS) Debris Flow module (Bartelt et al., 2013). RAMMS uses the Voellmy–Salm finite volume method to solve the depth-averaged equations governing mass flow in two dimensions, with second-order accuracy (Christen et al., 2010). RAMMS can also model entrained material in a mass flow, which makes it useful for GLOF simulations (Worni et al., 2014). The basic required inputs for RAMMS include a digital elevation model (DEM), the initial avalanche release area and its depth, and parameters for debris density and friction.

The Voellmy-fluid friction model used in RAMMS requires two friction parameters: μ and ξ, the velocity-independent dry Coulomb and velocity-dependent turbulent friction terms, respectively (Bartelt et al., 2013). For the case study presented here, values of μ= 0.12, ξ = 1000 m s-2, and ρ= 1000 kg m-3 were used, which agree with values used in previous GLOF-producing avalanche models (Schneider et al., 2014; Somos-Valenzuela et al., 2016). A sensitivity analysis of these values indicates that they are conservative, since they produce the fastest, farthest traveling, and densest avalanches within accepted standard values (see Bartelt et al., 2013).

A bathymetric survey of the Imja Tsho was conducted on 16–17 June 2016 using an inflatable kayak and a Garmin echoMAP 54dv to measure 4399 points of lake depth. The lake's shoreline was manually delineated using a clear-sky WorldView-2 image (DigitalGlobe, Inc.) from 14 May 2016. The shoreline was converted into point measurements and combined with the depth measurements to interpolate the depth over the entire lake using the Topo to Raster tool in ArcGIS. The lake depth raster was then burned into a regional DEM with a resolution of ∼ 4 m (King et al., 2017). This DEM covered the area between the lake and Dingboche and was used as an input for the models in this study.

Currently, there is no realistic avalanche scenario that can enter the lake; however, the lake is expanding eastwards such that it will be within an avalanche trajectory around 2035 (Rounce et al., 2016; see also Table 5 in Results). Therefore, in order to assess the future hazard, it was necessary to predict the future extent and bathymetry of Imja Tsho. Future lake extents were based on Rounce et al. (2016), which used the average decadal rate of expansion based on lake extents from 2000 to 2015 in conjunction with estimated future overdeepenings identified by GlabTop2 (Linsbauer et al., 2012; Frey et al., 2014). The lake level was assumed to remain constant in future projections, since it has remained relatively constant in the past 15 years (Rounce et al., 2016). The results from the 2016 bathymetric survey were then combined with the overdeepenings identified by GlabTop2 to predict lake bathymetry for future scenarios (Rounce et al., 2016). This future bathymetry was then burned into the DEM.

Although the lake was subjected to a lowering project in the summer of 2016 that reportedly lowered the lake by 3 m (BBC World Service, 2016), this lowering was not accounted for in the GLOF process chain modeling as it is unclear how much the main lake was lowered based on repeat satellite imagery (Fig. 3). Specifically, WorldView-2 (0.5 m; DigitalGlobe, Inc.) images of the lake's outlet complex before and after the lowering project show a clear ring of discoloration and decrease in area around the outlet ponds, but the lack of discoloration near the shore of the main lake suggests that the main lake may not have been lowered to the same extent. Hence, the GLOF process chain was modeled conservatively by not accounting for any lake lowering.

The terminal moraine was also assumed to remain stable in the future. While there is evidence that the moraine has lowered over time (Watanabe et al., 1995), the western shoreline of the lake adjacent to the terminal moraine has remained stable since the late 1980s (Fujita et al., 2009). Furthermore, the moraine's width and gentle slope add to its stability such that degradation of the ice core or piping will not likely pose a major risk, and a wave is more likely to cut through the outlet rather than completely overtop the moraine (Rounce et al., 2016). Weakening of the terminal moraine due to seismic activity was similarly disregarded based on the moraine's width and the lack of appreciable harm it suffered from the 2015 Gorkha earthquake and the earthquake's aftershocks (Byers et al., 2017).

Most methods used to characterize waves generated by avalanches into lakes rely on numerical or empirical models, as analytical methods often cannot capture the complexity of subaerial wave generation (Yavari-Ramshe and Ataie-Ashtiani, 2016). Numerical models generally rely on the 2-D shallow water equations (SWEs) or Boussinesq-type equations, whereas empirical models rely on simplified geometries and are best used as validation for complex numerical simulations (Somos-Valenzuela et al., 2016). While Boussinesq models account for nonlinear effects such as dispersion, their computational cost is higher and their application to real situations often provides no significant benefit over SWE models (Murty and Kowalik, 1993). Conversely, the simplicity of SWE models allows for inclusion of sediment transport, erosion, and deposition without excessive computational time – an advantage of the Basic Simulation Environment for Computation of Environmental Flow and Natural Hazard Simulation (BASEMENT) model (Vetsch et al., 2017). This study used BASEMENT for modeling all phenomena in the GLOF process chain downstream of the avalanche.

The bathymetric survey indicated a maximum depth of the main lake of 157.7 ± 1 m (Fig. 4), a mean depth of 65.2 ± 1 m, a total volume of 88.0 ± 1.4 × 106 m3 (Table 4), and an area of 1.35 km2 ± 0.01 km2. In contrast, a previous survey of the main lake from 2012 had a maximum depth of 116.3 m, a mean depth of 48 m, and a total volume of 61 × 106 m3 (Somos-Valenzuela et al., 2014). The main difference between the two surveys was that the 2012 survey was unable to approach the calving front due to the number of icebergs present at the time and could not accurately measure depths greater than 100 m. This caused the 2012 survey to assume an ice ramp that extended from the middle of the lake to the calving front. The 2016 survey was able to provide more reliable estimates of the lake depth as measurements were made close to the calving front and the Garmin echoMAP 54dv was able to accurately measure the deepest parts of the lake (Fig. 4). The 2016 survey therefore shows a much more abrupt change in depth near the calving front, which results in a greater volume, mean depth, and maximum depth over the whole lake. Furthermore, the eastward expansion of the lake has resulted in a steady increase in volume and depth (Table 4) as the lake expands into overdeepenings of the glacier bed. The bathymetry of the outlet ponds was also measured but not included in the main lake area. The maximum depth was 15.4 ± 1 m, the mean depth was 5.1 ± 1 m, the area was 0.037 ± 0.002 km2, and the total volume was 0.19 ± 0.04 × 106 m3.

Future lake extents were estimated using overdeepenings determined by the GlabTop2 model. The lake expands eastward for the first 20 years before splitting into two arms extending up the Lhotse Shar Glacier (northeast) and Imja Glacier (southeast). Figure 5 illustrates these results, which are superimposed by the deposition of two large avalanches (see Sect. 3.2), one reaching each arm of the future lake.

Avalanche scenarios were computed for two initial starting locations, one to the northeast above Lhotse Shar Glacier and one to the southeast above Imja Glacier (Fig. 5). A small (5 × 104 m3), medium (9 × 105 m3), and large (6.6 × 106 m3) avalanche were considered from both starting locations by varying the areal extent and depth of the initial source released; however, only the large avalanches were able to reach the lake at or before 2045 (Table 5). Specifically, the large avalanche from the northeast reached the lake at the 2025 predicted extent, but mass entry was too small and failed to produce measurable erosion at the terminal moraine unless the predicted 2045 lake was simulated. Large avalanches from both the northeast and southeast reached the lake at the 2045 extent and produced erosion of the moraine. Post-avalanche processes were analyzed for only the two large avalanche scenarios (northeast and southeast), since these were the only avalanches that reached the lake by 2045 and because they were the only ones to cause measurable erosion at the terminal moraine. The resulting mass entry rates within the lake boundary were used as the inflow hydrographs for subsequent BASEMENT modeling (Fig. 5, inset). A comparison of the avalanches from the northeast and southeast showed that the southeast avalanche had a smaller peak discharge into the lake but a larger initial impulse and a steadier decrease in flow. The northeast avalanche had a more variable inflow into the lake, and entered at an angle such that the resulting wave did not propagate directly towards the terminal moraine, which reduced the severity of downstream flooding.

In all scenarios, momentum transfer from avalanches into the lake created waves that ran up the terminal moraine, but only large avalanches from 2045 and beyond resulted in sufficient discharge to cause measurable erosion of the moraine at the lake outlet or flooding at Dingboche. The resulting impulse waves were attenuated in the lake, with a reduction of over 80 % in the first third of the traverse across the lake due to the rapid increase in lake depth. The wave height stabilized as the lake bed slowly sloped upward toward the lake outlet; finally, run-up near the terminal moraine resulted in a slight increase in height (Fig. 6). The 2-D SWE in BASEMENT inherently cause the wave to undergo excessive attenuation. Therefore, wave heights were calibrated by adjusting the inflow hydrographs and boundary widths so that the amplitudes in both BASEMENT and the Heller–Hager empirical model matched at the far end of the initial wave trajectory (far field), after the lake depth begins to slope upwards. Although this results in an abnormally high wave near the avalanche entry in BASEMENT, it creates wave heights that closely match that of the Heller–Hager equations at the terminal moraine, which is the focus of this study. Generally, the time from avalanche entry to terminal moraine run-up and outlet discharge was approximately 3 min; however, the initial trajectory of the wave from the northeast avalanche only lasted approximately 1 min before it ran up the lateral moraine (Fig. 6).

Wave characteristics for the southeast avalanche scenario (Fig. 6a) showed a smaller initial wave height but a similar far-field height relative to the northeast scenario (Fig. 6b), likely because of the direct line of wave propagation from avalanche entry to the terminal moraine. Conversely, avalanche entry from the northeast arm of the lake resulted in an indirect wave propagation that required some refraction (as the wave approached the south lateral moraine at an angle) and reflection (off of the south lateral moraine) before reaching the terminal moraine (see Fig. 5). The resulting loss of energy yielded a smaller run-up at the terminal moraine relative to the southeast avalanche scenario and the Heller–Hager results. For a full list of data and results for all scenarios considered, please see the Supplement for this paper (Lala, 2018).

Erosion and discharge at the terminal moraine were determined for 2000 s (0.5 h) following avalanche entry into the lake, after which discharge from the lake stabilizes. Three cross sections were analyzed: (A) at the lake outlet, where the terminal moraine rises above the lake, (B) at the end of the terminal moraine, and (C) downstream of the terminal moraine within the Imja Khola channel (Fig. 7).

Combined sediment and water discharge at the lake outlet (A) for the large southeast avalanche scenario arrived after about 130 s and peaked at 3140 m3 s-1 for the MPM model and 2904 m3 s-1 for the MPM-Multi model (Fig. 8). Discharge at the outlet showed considerable oscillation due to the leading and trailing waves caused by the avalanche (Heller et al., 2009). After approximately 900 s, discharge stabilized to around 25 m3 s-1 and erosion ceased. The MPM-Multi model had consistently smaller peak discharges than the MPM model, but a similar oscillatory structure, likely due to the smaller volume of debris within the flow. The flood wave arrived at the end of the moraine (B) after 250 s with a peak 290 m3 s-1 for the MPM model and 134 m3 s-1 for the MPM-Multi model. The discharge here showed less oscillation than at the lake outlet (A). The lower and more stable discharge at (B) suggests that the outlet ponds on the terminal moraine (Fig. 2) act as reservoirs that dampen the flood peaks and offer some protection from flooding. After approximately 2000 s, discharge stabilized to around 15 m3 s-1. The MPM-Multi model had consistently smaller peak discharges than the MPM model, again likely because of a lack of sediment transport due to hiding and armoring of the channel, but a similar oscillatory structure and time to stabilization. The flood wave arrived at the Imja Khola channel (C) after 460 s with a peak of 263 m3 s-1 for the MPM model and after 560 s with a peak of 93 m3 s-1 for the MPM-Multi model. The discharge showed less oscillation in the MPM model than at the lake outlet, whereas the MPM-Multi model had dampened all oscillations by this time. After approximately 2000 s, discharge stabilized to around 26 and 20 m3 s-1 for the MPM and MPM-Multi models, respectively. Debris discharge was also analyzed at Dingboche. For the MPM model, the flood reached the village after 3440 s (almost 1 h), with a peak discharge of 160 m3 s-1, and steadily decreased afterwards, dipping below 20 m3 s-1 by 9100 s. The MPM-Multi model showed the flood arriving slightly later (4600 s) but discharges were nearly identical and hence not shown in Fig. 8. In the first 2000 s of the simulation (i.e., before discharge lowers to ∼ 5 m3 s-1), the total volume of water leaving the lake was approximately 251 000 m3 for the MPM simulation and 166 000 m3 MPM-Multi simulation – less than 0.3 % of the total present lake volume (88 million m3). This is notably less than the amount of avalanche material entering the lake (approximately 720 000 m3; Table 5). Moreover, the lake's surface elevation remains slightly above its original elevation at the end of the simulation period (by approximately 0.25 m), suggesting that erosion of the moraine was not sufficient to allow the lake to drain quickly, and may have even allowed the lake to store more water.

As expected, the MPM model resulted in more erosion and higher discharge at the terminal moraine. For all three cross sections, erosion never exceeded 5 m (Fig. 9), which is less than the necessary amount needed to reach the ice core of the moraine and accelerate moraine degradation (Hambrey et al., 2008). The maximum bed erosion at the lake outlet (A) for the MPM and MPM-Multi models was 4.6 and 1.7 m, respectively. The maximum bed erosion at the moraine outlet (B) for the MPM model was 0.75 m and for the MPM-Multi model it was negligible. At the Imja Khola channel (C) there was minimal erosion for the MPM model (< 1 m) and negligible erosion for the MPM-Multi model. In both cases, the moraine was not fully overtopped, and erosion was confined to the outlet channel.

In both the MPM and MPM-Multi scenarios, the flood wave reached the village of Dingboche approximately an hour after the avalanche entered the lake. However, floodwater was confined to the river channel in all cases (Fig. 10). Scouring of and deposition in the channel near the village was negligible. The maximum flow depth remained less than 3 m at Dingboche and flow velocity did not exceed 6 m s-1 (Fig. 10). Hazard was therefore negligible in all parts of the village except the river channel (Fig. 11).

To the best of the authors' knowledge, this study is the first to model an avalanche-induced GLOF process chain at Imja Tsho. Results indicate that Imja Tsho presents little hazard from an avalanche-induced GLOF to downstream communities for the next 3 decades, if current trends of lake expansion continue. This seems to validate the conclusions of some earlier non-dynamic model studies which regarded the terminal moraine as a buffer (Fujita et al., 2009; Watanabe et al., 2009) and suggested the presence of Dingboche and other villages downstream of Imja Tsho likely contributed to undue alarmism in assessing downstream hazard (Watanabe et al., 2009).

While the results from this study indicate no threat to the village of Dingboche, channel flooding still poses a small threat to humans and livestock working or grazing near the river, as well as river crossings further downstream. However, even in the worst-case scenario, flooding reached the village of Dingboche about an hour after the avalanche entered the lake, providing an ample window for warning and evacuation if water levels in the lake and river are monitored.

One reason the results of this study conflict with that of previous GLOF models of Imja Tsho (Somos-Valenzuela et al., 2015; Shrestha and Nakagawa, 2016) is that this study modeled the breach of the terminal moraine based on an avalanche entering the lake as opposed to making assumptions regarding the breach or overtopping. Somos-Valenzuela et al. (2015) modeled the breach of the terminal moraine using a combination of empirical and numerical methods, but assumed the breach would be triggered by piping. While piping is theoretically possible, Imja's wide and gently sloped moraine make this unlikely, especially when one considers Imja's moraine stability compared to other glacial lakes in the region (Fujita et al., 2013). Bajracharya et al. (2007) relied on a similar assumption about internal failure of the moraine and did not consider dynamic causes. The width of the terminal moraine also makes the failure – via overtopping of the moraine – modeled by Shrestha and Nakagawa (2016) unlikely, since even the largest avalanches considered in this study do not fully overtop the terminal moraine.

In contrast to studies that assume dam breaching from internal failures or wave overtopping, studies that relied more on geographic and geomorphic data concluded that Imja Tsho poses little imminent risk and that the lake is currently safe (Fujita et al., 2009; Watanabe et al., 2009; Rounce et al., 2016), but that expansion up-glacier (eastward) must be monitored to continually assess the risk of mass movement into the lake. The results presented in this study indicate that even if eastward expansion continues, the lake will pose little risk for the next 3 decades, although regular monitoring of the terminal moraine and up-glacier mass movement trajectories will be needed to continually reassess downstream hazard.

The use of BASEMENT in this study was a large improvement over previous models. BASEMENT was able to compute debris loads without requiring specification of average or maximum sediment concentrations in the flow, which is necessary for FLO-2D (Somos-Valenzuela et al., 2015) and the method of Shrestha and Nakagawa (2016), respectively. These requirements present a problem, since there are few well-documented extreme flow events from which these parameter values can be estimated (Worni et al., 2014). In contrast, geomorphic parameters needed for BASEMENT can be estimated from field data that are not event-specific. BASEMENT thus reduces the amount of data needed to run a simulation, which is a benefit in data-scarce mountain regions; the extent of necessary sensitivity analysis is also correspondingly reduced. Furthermore, BASEMENT is open-access, making it ideal for stakeholders in developing countries with limited budgets for purchasing commercial software. It also has a user-friendly GUI and can be executed on most modern desktop computers, which facilitates knowledge transfer such that national agencies, with some help from specialists, can adapt the models to new scenarios.

Open-source software such as r.avaflow (Mergili et al., 2017) may contribute to future hazard analysis given the software's ability to model two-phase (i.e., solid debris and water) flow, once calibrated with real-world data. However, impulse wave dynamics are substantially affected by the chosen solid-phase parameters in the underlying model (Pudasaini, 2014), particularly the solids' concentration within the lake, which requires more data and sensitivity analysis than a water-based model. Overall, two-phase models will likely be complementary to, rather than a replacement of, process chain models, since both have advantages for different applications (Worni et al., 2014).

It is important to highlight that this study assessed only avalanche-induced waves as GLOF triggers and their potential for erosion of the terminal moraine and downstream inundation. Mass movement from rockfalls was not assessed, as the lateral moraines of the lake are well-developed and pose little risk of a large slope failure (Rounce et al., 2016). Previous work has considered the possibility of self-destructive moraine failure through piping, seepage, and subsequent erosion (Shrestha et al., 2013; Shrestha and Nakagawa, 2016; Somos-Valenzuela et al., 2015), which historically is the second most common cause of GLOFs in the Himalayas (Emmer and Cochachin, 2013; Falátková, 2016). The melting of buried ice within the moraine could weaken the moraine's ability to withstand the hydrostatic pressure and trigger a self-destructive failure; however, Imja's wide, gently sloped moraine and well-developed outlet complex suggests this is unlikely. Furthermore, the results of this study indicate surface erosion from an overtopping wave will not likely reach the ice core and accelerate melting. Still, seepage at the terminal moraine has been observed on many occasions (Somos-Valenzuela et al., 2015), including the authors' most recent visit (April 2017), and should not be disregarded from future hazard assessments of the lake.

The identification of initial release areas for avalanches is perhaps the largest source of uncertainty in the work reported here. The high altitude of the Himalayas allows for avalanche ice to be frozen to the bedrock, which allows larger volumes to accumulate before release and can lead to avalanches in the millions of cubic meters (Alean, 1985). The thickness of these masses can reach up to 60 m, but a more realistic value would range from 20 to 45 m, based on the method of Wang et al. (2012). The large avalanches used in this study had surface areas of approximately 1.34 × 105 m2, well within the range of large historical avalanches in the Swiss Alps, although these generally had smaller volumes (Alean, 1985). Therefore, the large avalanches used for this study were deemed reasonable and representative of potential extreme events that would represent a worst-case scenario.

One of the major goals of this study was to create a replicable GLOF model that could be applied to other lakes besides Imja Tsho. Results suggest that lakes with larger terminal moraines (such as Thulagi and Lower Barun in Nepal) may be safer than previously assessed, but considerable hazard may still apply to those with smaller or steeper moraines. Future work should apply this model to other lakes with smaller moraines, such as Lumding Tsho, Chamlang North Tsho, Chamlang South Tsho, and Tsho Rolpa (Rounce et al., 2016). Finally, monitoring of Imja Tsho's terminal moraine and expansion should continue, so that assumptions concerning the lake's hazard can be regularly reevaluated.