Atmospheric warming has induced melting of many glaciers around the world (WGMS, 2012; IPCC, 2013; Marzeion et al., 2014). The formation of new lakes in de-glaciating high-mountain regions strongly influences landscape characteristics and represents a significant hazard related to climate change (Frey et al., 2010; Rosenzweig et al., 2007; Kattleman, 2003; Richardson and Reynolds, 2000). The glacier-covered area of the Cordillera Blanca range in Peru has decreased from a Little Ice Age peak of 900 km2 to about 700 km2 in 1970, 528 km2 in 2003, and further decreased to 482 km2 in 2010 (UGRH, 2010; Burns and Nolin, 2014). As a consequence of this glacier recession, many glacial lakes have formed or expanded in the Cordillera Blanca that pose various levels of glacial lake outburst flood (GLOF) risk for communities below these lakes (Emmer and Vilímek, 2013).

The steep summits of the Cordillera Blanca are undergoing long-term slope destabilization due to warming and permafrost degradation (Haeberli, 2013). Related ice and rock avalanches are especially dangerous in connection with glacial lakes forming or expanding at the foot of steep mountain slopes because they can trigger large waves in the lakes and potentially lead to GLOFs (Carey et al., 2012; Haeberli, 2013). There are many examples in the Cordillera Blanca of glacier-related incidents and catastrophes (Lliboutry et al., 1977; Carey, 2010; Portocarrero, 2014). A recent example in the Cordillera Blanca is the 2010 event comprised of a nearly 0.5 million m3 ice–rock avalanche from the summit of Nevado Hualcán that fell into Lake 513 and generated waves that overtopped the natural rock dam of the lake, producing flood waves and debris flows that reached the town of Carhuaz (Carey et al., 2012; Schneider et al., 2014). Preventive lowering of Lake 513 by artificial tunnels in the 1990s, creating a freeboard of 20 m, helped avoid a major catastrophe that could have killed many people (Reynolds et al., 1998; Carey et al., 2012; Portocarrero, 2014).

Emmer and Vilímek (2013, 2014) and Haeberli et al. (2010) have recommended that the evaluation of glacial lake hazards be based on systematic and scientific analysis of lake types, moraine dam characteristics, outburst mechanisms, down-valley processes and possible cascades of processes. Changes in climate patterns are likely to increase the frequency of avalanches as a consequence of reduced stability of permafrost, bedrock and steep glaciers in the Cordillera Blanca (Fischer et al., 2012). Under these conditions, avalanches are the most likely potential trigger of GLOFs (Emmer and Vilímek, 2013; Emmer and Cochachin, 2013; Awal et al., 2010; Bajracharya et al., 2007; Richardson and Reynolds, 2000; Costa and Schuster, 1988), acting as the first link in a chain of dependent processes propagating downstream: (1) large avalanche masses reaching nearby lakes, (2) wave generation, propagation, and run-up across lakes, (3) terminal moraine overtopping and/or moraine breaching, (4) flood propagation along downstream valleys, and (5) inundation of riverine populated areas (Worni et al., 2014; Westoby et al., 2014b).

Few studies have attempted to simulate an entire GLOF hazard process chain in a single modeling environment, generally limiting the number of processes considered; e.g., Worni et al. (2014) excluded avalanche simulations from their modeling framework. Worni et al. (2014) and Westoby et al. (2014a) review typical modeling approaches for GLOFs that involve land or ice masses falling into glacial lakes. An approach that separately simulates individual processes predominates, where different processes are connected by using the results of one model as the input for the simulation of the next (e.g., Schneider et al., 2014; Westoby et al., 2014b; Worni et al., 2014). In this paper, this approach was used to produce simulations of each process in the chain from avalanche to inundation, ensuring that the processes were properly depicted. The glacial lake hazard process chain simulated here includes avalanche movement into a lake, wave generation and lake hydrodynamics, wave overtopping and moraine erosion, and downstream sediment transport and inundation.

Physical models of avalanche phenomena have been used to simulate mass movement processes, e.g., snow avalanches, rock slides, rock avalanches or debris flows (Schneider et al., 2010). Rock–ice avalanches exhibit flow characteristics similar to all of these processes, and the choice of an appropriate model is difficult because available models are not able to fully simulate all of the elements of these complex events. Schneider et al. (2010) tested the Rapid Mass Movements – RAMMS – model (Bartelt et al., 2013), a two-dimensional dynamic physical model based on the shallow water equations (SWE) for granular flows and the Voellmy frictional rheology to successfully reproduce the flow and deposition geometry as well as dynamic aspects of large rock–ice avalanches. RAMMS was used here to determine the characteristics of various size avalanches entering the lake, one of the more difficult elements in modeling the process chain.

Impulse waves resulting from the impact of an avalanche with the lake were simulated with a three-dimensional hydrodynamic model, FLOW3D (Flow Science, 2012). Much of the work in impulse wave generation, propagation and run-up has been focused on empirical models that replicate wave characteristics based on laboratory observations (Kamphuis and Bowering 1970; Slingerland and Voight, 1979, 1982; Fritz et al., 2004; Heller and Hager, 2010); numerical simulations have been limited to simplified two-dimensional SWE simulations (Rzadkiewicz et al., 1997; Biscarini, 2010; Cremonesi et al., 2011; Ghozlani et al., 2013; Zweifel et al., 2006). However, the two-dimensional SWE representations do a poor job of modeling wave generation and propagation because vertical accelerations are important and cannot be neglected for slide-generated waves (Heinrich, 1992; Zweifel et al., 2006). Analytical calculations of wave run-up and overtopping typically consider simplified lake geometries (e.g., uniform water depth and constant slope of the terminal moraine) that do not necessarily hold true in natural reservoirs (Synolakis, 1987, 1991; Muller, 1995; Liu et al., 2005). The limitations of empirical and two-dimensional SWE models for simulating lake dynamics leave an opening for a more robust approach. Therefore, a fully three-dimensional non-hydrostatic model (FLOW3D) was used in this paper to simulate the wave generation, propagation and overtopping.

Dynamic modeling of moraine erosion and breaching deals with tradeoffs between reliability, complexity, field data demand, and computational power. Several physical processes converge when natural or artificial dams fail; hydrodynamic, erosive, and sediment transport phenomena, as well as movement of boulders and mechanical or slope failures, interact during dam collapses (Westoby et al., 2014a; Worni et al., 2014). Modeling the erosion of natural and artificial dams has been evolving since the early 1980s, when simple one-dimensional models based on empirical and parametric analyses were developed to represent dam-breach processes, e.g., DAMBRK (Fread, 1988), WinDAM B (Visser et al., 2011), and HR-BREACH (Morris, 2011; Westoby et al., 2015). These models describe breach phenomena by defining the rate of growth of a potential breach and including that breach definition in a hydrodynamic model (Rivas et al., 2015; Fread, 1984). These models are computationally efficient but rely heavily on engineering judgment and analysis of historical failure cases; when the expected breach shape, size and growth rate are unknown, the models have limited ability to predict whether sufficient erosion will occur to produce a breach at a particular site.

Many two-dimensional sediment transport models apply a SWE scheme, in which mobile bed meshes respond to shear stresses from hydrodynamic forces, and use empirical functions of non-cohesive sediment transport to estimate drifting, entrainment, suspended transport, bed-load transport, and deposition of sediment, e.g., IBER (Bladé et al., 2014), Delft3D (operated as a two-dimensional model) (Deltares, 2014) and BASEMENT (Vetsch et al., 2014). These models have the potential to simulate the moraine erosion process considered here, but only BASEMENT is able to account for the important process of slope collapse as erosion occurs and meshes change.

Overtopping waves can cause terminal moraine erosion. Under wave transport conditions, vertical accelerations play an important role in both water and sediment advection, influencing the erosion process and possible moraine failure. Three-dimensional models can efficiently simulate flow phenomena when those vertical accelerations are relevant. However, coupled erosion simulations requiring additional hydro-morphodynamic functions pose additional challenges in three-dimensional modeling. Several models combine three-dimensional numerical schemes with sediment transport formulations, e.g., Delft3D (Deltares, 2014), FLOW3D (Flow Science, 2014) and OpenFoam (Greenshields, 2015). FLOW3D and OpenFoam use a VoF (volume of fluid) method to describe the solid–fluid interface, representing sediment beds as an additional fluid in multi-phase schemes. This approach seems successful for applications where the erodible bed remains submerged throughout the entire simulation or under steady flow conditions, but stability problems arise for cells exposed to drying and wetting periods. Delft3D avoids these stability issues by using a flexible mesh instead of a multi-phase approach to simulate changes due to erosion or deposition; however, it has limitations in representing fluid regions disconnected from boundaries.

Wave overtopping and breach analysis in this paper evolve from the methods reported by Rivas et al. (2015), whose performance evaluation of empirical breach models focused exclusively on hydraulic considerations. That partial perspective sets no physical limit on breach growth, assuming full moraine collapse is possible. This paper goes beyond the previous work to determine the likelihood and potential magnitude of a moraine breach through hydro-morphodynamic simulations of the erosion process. BASEMENT is used to model the dynamic moraine erosion resulting from an overtopping wave. Under this approach, the analysis follows a similar procedure to that applied by Worni et al. (2012, 2014) at Lake Ventisquero Negro, but it reduces the limitations on modeling waves that can cause further moraine collapse by using external results from FLOW3D to calibrate the BASEMENT model. This has allowed exploration of the possibility of full, partial or even null breaches at Lake Palcacocha according to flow characteristics while accounting for soil and morphological properties as well.

The resulting lake outburst floods, after breaching or overtopping of the moraine, comprise highly unsteady flows that are characterized by pronounced changes as they propagate downstream (Worni et al., 2012). Calculating downstream inundation caused by a GLOF event requires the simulation of debris flow propagation, since sediment entrainment can cause the volume and peak discharge to increase by as much as 3 times (Worni et al., 2014; Osti and Egashira, 2009). One-dimensional models based on the St. Venant equations have been used to model the downstream flood wave propagation of a GLOF, e.g., Klimes et al. (2013), who used HEC-RAS (USACE, 2010) to reproduce the 2010 GLOF from Lake 513 in Peru, Cenderelli and Wohl (2003), who used HEC-RAS to reproduce steady-state aspects of GLOFs in the Khumbu region of Nepal, Byers et al. (2013), who used HEC-RAS to model a potential GLOF from Lake 464 in the Hongu valley of Nepal, Meon and Schwarz (1993), who used DAMBRK (Fread, 1988) to model a potential GLOF in the Arun valley of Nepal, and Bajracharya et al. (2007), who used FLDWAV (NWS, 1998) to model a potential GLOF from Imja Lake in Nepal. Two-dimensional SWE models are often used to model downstream impacts of GLOFs, e.g., Worni et al. (2012), who used BASEMENT to model flooding from a GLOF at Shako Cho Lake in India, Schneider et al. (2014), who used RAMMS to model debris flow from an overtopping wave at Lake 513 in Peru, Somos-Valenzuela et al. (2015), who used FLO2D to model downstream inundation from a potential GLOF at Imja Lake in Nepal, and Mergili et al. (2011), who used RAMMS and FLO2D to simulate flooding from Lake Khavraz in Tajikistan. FLO2D (2012) is used here to simulate the downstream inundation.

As an interpretation of downstream consequences, flood hazard denotes potential levels of threat as a function of intensity and likelihood of the arriving inundation (normally probability, but the nature of avalanche events and other processes in the hazard chain restricts one from assigning numerical probabilities). Flood intensity is determined by the flow depth and velocity (García et al., 2003; Servicio Nacional de Geología y Minería, 2007). Likelihood is inversely related to magnitude; i.e., large events are less likely to occur (low frequency) than small events (Huggel et al., 2004). Maps can be prepared that show the level of hazard resulting from the intensity of various likelihood events. This allows communication of the flood hazard at various locations, facilitating planning, regulation, and zoning based on the map, while enhancing communication to the affected community (O'Brien, 2012; USBR, 1988; FEMA, 2003).

This paper describes an analysis of the processes involved in a potential GLOF from Lake Palcacocha in Peru and the resulting inundation downstream in the city of Huaraz. The simulated process cascade starts from an avalanche falling into the lake, resulting in a wave that overtops the damming moraine; the simulation continues with potential erosion due to moraine overtopping and culminates with simulations of the ensuing downstream flooding and inundation in Huaraz. In the following sections, the setting of the problem is presented, followed by descriptions of the physical basis and modeling of each of the processes in the chain. The results of each of the simulated processes are presented, concluding with details of the potential inundation in Huaraz and hazard implications. Mitigation alternatives are investigated through an analysis of several lake-lowering scenarios.

The methodology presented here considers a process chain similar to Worni et al. (2014) depicting an avalanche-triggered GLOF from Lake Palcacocha to assess the potential inundation in Huaraz from such an event (Fig. 2). The simulated avalanche originates from the Palcaraju glacier located directly above the lake. When an avalanche enters the lake, depending on its size and the level of the water surface in the lake, the resulting wave might overtop the damming moraine and possibly initiate an erosive breaching process releasing considerable amounts of water and debris into the Paria River and potentially inundating densely populated areas of Huaraz downstream. The process chain from avalanche to inundation was simulated using four models: potential avalanches were modeled using RAMMS (Christen et al., 2010), lake wave dynamics were modeled with FLOW3D (Flow Science, 2012), the dynamic breaching process was simulated in BASEMENT (Vetsch et al., 2006), and propagation of the flood wave downstream and inundation in Huaraz were simulated in FLO2D (O'Brien, 2003).

The next sections describe each component for the framework used to simulate the hazard process chain: avalanche simulation, wave simulation in the lake, moraine erosion simulation, inundation simulation, and hazard identification.

In non-forested areas, ice–rock avalanches can be generated on slopes of 30–50∘, and in tropical areas the critical slope can be even less (Christen et al., 2005; Haeberli, 2013). Temperate glaciers can produce ice avalanches if the slope of the glacier bed is 25∘ or more, but rare cases with slopes less than 17∘ have occurred (Alean, 1985). The mountains surrounding Lake Palcacocha have slopes up to 55∘; therefore, they have a high chance of generating avalanche events. Nonetheless, it is difficult to forecast when avalanches will occur and where the detachment zone will be located (Evans and Clague, 1988; Haeberli et al., 2010).

The RAMMS avalanche model was used to simulate the progression of avalanches down the mountain to the lake. RAMMS solves two-dimensional, depth-averaged mass and momentum equations for granular flow on three-dimensional terrain using a finite volume method (Christen et al., 2010; Bartelt et al., 2013). The inputs for the model include (1) terrain data (a DEM, described above); (2) fracture height; (3) the avalanche release area; and (4) friction parameters. Descriptions of input parameters (2)–(4) and the criteria used to determine their values are given in the following paragraphs. RAMMS computes the velocity of the avalanche, the distance of the run-out, the pressure distribution, and the height of the avalanche front at different locations below the initiation point.

For the elevation of the Palcaraju glacier above Lake Palcacocha, the potential fracture type is expected to be a slab failure or type I fracture as defined by Alean (1985). Huggel et al. (2004), after Alean (1985), suggest that ice avalanches in slab failures are mainly produced in small and steep glaciers with thicknesses between 30 and 60 m, where they are less frequent in large valley-type glaciers. Alean (1985) shows examples of slab failure with thicknesses ranging from 19 to 35 m and volumes ranging from 1 to 11 million m3. The avalanche above Lake 513 that occurred in 2010 is an example of this type of failure (Schneider et al., 2014). Following these precedents, fracture heights of 25, 35 and 45 m were selected for simulating the small, medium and large avalanches, respectively.

Three avalanche volumes are considered in this work, similar to the avalanche scenarios in Schneider et al. (2014): 0.5 × 106 m3 (small), 1 × 106 m3 (medium) and 3 × 106 m3 (large). These potential avalanche volumes are consistent with the elevations and slopes of the source area. The release area (shown in Fig. 3) was located at an elevation of 5200 m to the northeast of the lake following the main axis of the lake.

The friction parameters required by the RAMMS model are (1) the density of the rock and ice (ρ, in kg m-3), (2) the Coulomb-friction term (μ), and (3) the turbulent friction parameter (ξ) (Bartelt et al., 2013). The Coulomb-friction term with a dry surface friction dominates the total friction when the flow is relatively slow, and the turbulent friction parameter tends to dominate when the flow is rapid, as is the case with the avalanches considered here (Bartelt et al., 2013; Christen et al., 2008, 2010). The friction parameter values used in the RAMMS avalanche model are ξ = 1000 ms-2, μ = 0.12 and ρ = 1000 kg m-3, values similar to those used to model the avalanche into Lake 513 (Schneider et al., 2014).

Lake Palcacocha is very deep near the glacier, with depths up to 72 m, but the last several hundred meters adjacent to the terminal moraine are very shallow, with depths mostly less than 10 m (Fig. 4). This discontinuous lakebed geometry significantly affects wave propagation and run-up, making a hydrodynamic simulation necessary to represent the potential overtopping of the terminal moraine.

To overcome the limitations of analytical methods such as Heller and Hager (2010) in representing wave propagation, run-up and overtopping of the moraine, the FLOW3D three-dimensional hydrodynamic model (Flow Science, 2012) was used to simulate the dynamics of avalanche-generated waves in Lake Palcacocha. The FLOW3D model grid used 400, 150, and 100 grid cells covering distances of 2400, 800, and 650 m in the x, y, and z directions, respectively. The RNG turbulence model with a dynamically computed mixing length and a fully three-dimensional, non-hydrostatic numerical scheme was used in the FLOW3D simulations.

The transfer of mass and momentum from the avalanche to the lake upon impact and the subsequent wave generation and propagation were simulated in FLOW3D by representing the avalanche as a volume of water equivalent to the avalanche volume that flows into the lake from the terrain above. Worni et al. (2014) and Fah (2005) approach the problem in the same way, simulating water instead of avalanche material. The density of the mixture of snow, rock and ice present in an avalanche is very close to the density of water (Schneider et al., 2014). Although the viscosities of the two fluids are different, this approximation of substituting water for the avalanche fluid is handled through adjustments in the model that compensate for any reduction in dissipation of energy due to the lower viscosity of water. To accomplish this, the results of the RAMMS avalanche model were used as calibration parameters; the depth of the avalanche fluid volume and height above the lake at which it is released were iteratively adjusted in FLOW3D until the velocities and depths of the avalanche fluid volume entering the lake matched the characteristics of the avalanche modeled in RAMMS. As long as the mass and momentum of the material hitting the lake in FLOW3D are similar to that of the RAMMS-simulated avalanche, the initial displacement wave should behave similarly as well; the water in the lake is pushed by the incoming avalanche, but the avalanche material does not reach the moraine, and the displaced wave is what propagates across the lake. Differences may arise for reflected waves since the avalanche material might settle in a different way over the lake's bed according to the avalanche properties (water representing avalanche material is more free to flow in the lake than actual rock–ice avalanche material). The primary output from the model is a hydrograph of wave overtopping discharge, if there is any, that is used as input to the downstream inundation model discussed later.

In this paper, BASEMENT was used for hydro-morphodynamic simulations of potential erosion-driven breach failures at Lake Palcacocha. To overcome the two-dimensional SWE limitations of BASEMENT, results of three-dimensional hydrodynamic lake and overtopping wave simulations from FLOW3D were used as calibration parameters. The wave propagation and overtopping of the terminal moraine were simulated in both FLOW3D and BASEMENT. The zone of interest for BASEMENT simulations was at the terminal moraine, where erosion can occur and produce a moraine collapse. However, simulating the wave propagation across the whole lake moves the upstream boundary of the model, favoring a smoother transition at the interface between both models, where flow properties must match.

The BASEMENT model was started in the zone of the lake where wave generation occurs (wave splash zone in Fig. 5), but the method of simulating wave generation was different from that used in FLOW3D because the flow characteristics at the inflow boundary must be artificially altered to compensate for the additional energy loss in the two-dimensional SWE representation of BASEMENT. To facilitate comparison between the FLOW3D and BASEMENT models, hydrographs of results were compared at a common cross section for both models, located at the crest of the terminal moraine (target cross section in Fig. 5). Adjusting the slope of the energy grade line at the upstream boundary (Fig. 5) allowed an iterative increase in momentum inflow until mass and momentum fluxes over the crest of the moraine (target cross section) matched the results from the FLOW3D simulations.

The relevant regions of the FLOW3D model, where fluid motion influences erosion and breach growth, are located near the moraine crest and downstream in the outlet channel. Through a calibration procedure, the BASEMENT model was forced to replicate the hydrodynamic conditions of the FLOW3D wave model. This was achieved by forcing momentum fluxes (that are dissipated further downstream) at the inflow boundary of the BASEMENT model to be unrealistically high. By adjusting energy slopes at the upstream boundary, momentum inflow was iteratively increased until flow properties (mass and momentum fluxes) match the results from fully three-dimensional simulations according to hydrographs of discharge and velocity at the crest of the artificial dam. This procedure aims to guarantee that BASEMENT can properly model mass transport from wave phenomena despite the limitations of the two-dimensional SWE simulations.

BASEMENT applies empirical functions to estimate erosion and deposition rates taking place under the influence of flows from overtopping waves. Erosion resistance comes from soil properties and the morphology of the bed. We have applied a hypothetical set of worst-case soil conditions, intentionally decreasing the erosional strength in the Lake Palcacocha moraine. The logic of this approach is that if breach simulations show no moraine collapse under the worst possible conditions observed in the field, such collapse is unlikely to occur in real settings, where the total soil matrix may contain soil that is more erosion resistant. This approach also seeks to overcome a lack of independent erosion measurements, which makes any attempt at calibration and further refinement of the breach model impossible.

The bed-load transport is modeled with the single-grain Meyer-Peter and Müller (1948) (MPM) model, which automatically discards any erosion resistance from hiding and armoring processes occurring in multi-grain matrixes (Ashida and Michiue, 1971; Wu et al., 2000). Correction factors to account for under- or over-prediction of the rate of bed-load transport in the MPM model range from 0.5 for low transport of sands and gravels to 1.7 for high transport cases (Fernandez and Van Beek, 1976; Ribberink, 1998; Wong and Parker, 2006). A bed-load factor of 2.0 is used here, characterizing high sediment transport conditions. Table 1 displays the set of sediment and slope failure characteristics used to build the Lake Palcacocha hydro-morphodynamic model in BASEMENT. According to field data, coarser soils (d50 ≈ 19 mm) predominate at the walls of the outlet channel left by the 1941 GLOF at Lake Palcacocha (Novotný and Klimeš, 2014) where most of the outburst water would flow in a potential future event. In agreement with the proposed hypothetical worst-case soil condition, a grain size of d50 = 1 mm is assumed, representing characteristics of upper layer soils that may lead to significant underestimation of erosion resistance.

FLO2D (FLO2D, 2012) is used to simulate the flooding downstream of Lake Palcacocha considering debris flow that incorporates sediment characteristics (dynamic viscosity and yield stress) as exponential functions of the sediment concentration by volume. Although the simulation grid in FLO2D is two-dimensional, the flow is modeled in eight directions, solving the one-dimensional continuity and momentum equations in each direction independently using a central, finite difference method with an explicit time-stepping scheme. One of the advantages of FLO2D is that for flows with high sediment concentration the total friction slope can be expressed as a function of the sediment characteristics and the flow depth (FLO2D, 2012; Julien, 2010; O'Brien et al., 1993).

Due to the steepness of the terrain below Lake Palcacocha and low cohesion of the material from the moraine, high velocities and turbulent flows with low dynamic viscosity and low yield stress are expected (Julien and Leon, 2000). Therefore, from the empirical coefficients recommended by FLO2D (2012), these two sets of parameters that describe low yield stress and dynamic viscosity, respectively, are used: α1 = 0.0765, β1 = 16.9, α2 = 0.0648 and β2 = 6.2. Yield stress and viscosity of the flow vary principally with sediment concentration based on empirical relationships where the parameters αi and βi have been defined by laboratory experiment (FLO2D, 2012).

Downstream of Lake Palcacocha the flood will meet huge moraines in a steep canyon. According to Huggel et al. (2004), erosion on the order of 750 m3 m-1 has been found in alpine moraines. In the Andes and Himalaya, erosion cuts can be higher than 2000 m3 m-1, with peak flow concentrations by volume on the order of 60–80 %. Thus, given the uncertainties associated with the calculation of the concentration of sediment, Huggel et al. (2004) recommend using an upper limit for the average flow concentration by volume of 50–60 %. This agrees with Schneider et al. (2014), Julien and Leon (2000) and Rickenmann (1999), who recommend 50 % sediment concentration by volume, which is used in this study.

For the terrain elevation, a DEM was produced for this work (Horizons, 2013). Given the large extent of the domain, running the inundation simulations on this 5 m × 5 m grid was impractical. Therefore, the FLO2D simulations were run on a 20 m × 20 m grid.

Distributed roughness coefficient values were assigned based on land cover in the Paria basin below Lake Palcacocha. Land cover was classified into five categories using the normalized differential vegetation index (NDVI) from a multispectral image of a Landsat 7 image taken on 22 October 2013 after reflectance correction and ISODATA analysis (Chander et al., 2009; Hossain et al., 2009).

Given the lack of detailed information about the buildings and construction materials, an area reduction factor of 20 % was applied to account for the influence of buildings on the flow. Area reduction factors are used in FLO2D to reduce the flood volume storage on grid elements due to buildings or topography (FLO2D, 2012). Although FLO2D allows the inclusion of buildings and obstacles that can affect the inundation trajectory, it was not clear in this work whether the buildings of Huaraz are strong enough to support the impact and, thus, deviate the flow. In some areas, especially near the river, it is highly probable that the flow will destroy the buildings, but further from the river that may be less likely to happen.

Flood intensity is determined by the resulting flow depth and velocity in Huaraz. Various methods of determining the flood intensity from the flood depth and velocity have been developed. The Austrian method (Fiebiger, 1997) uses the total energy of flow as the indicator of intensity. The US Bureau of Reclamation (USBR, 1988) uses a combination of depth and velocity and differentiates these for the impact on adults, cars, and houses. The Swiss method (OFEE et al., 1997) defines intensity, independent of the object subjected to the hazard, as a combination of depth and the product of depth and velocity.

In this work, the Swiss method is adopted to determine flood intensity as adapted for use in Venezuela, where intensity thresholds were calibrated with field data from the 1999 alluvial floods in Venezuela (PREVENE, 2001; García et al., 2002; García and López, 2005). Applying this method requires simulating the different events to predict the spatially distributed maximum depths and velocities for each event, and then transferring these results to GIS where a flood intensity map for each event is created by applying the intensity categorization criteria, low, medium or high (Table 2), to each grid cell in the map.

Flood hazard is a function of intensity and likelihood of an event. In this case, the event is the process chain resulting from an avalanche falling into Lake Palcacocha. The level of water in the lake then determines the resulting wave that may or may not overtop the damming moraine. To determine flood hazard, normally probability would be the term used instead of likelihood, but there are not enough data (i.e., recorded avalanche events) to assign probabilities to the different avalanche events and other processes in the hazard chain; therefore, in keeping with other similar studies (e.g., Huggel et al., 2004), a qualitative probability, or likelihood, is used. Likelihood is inversely related to avalanche magnitude; i.e., as discussed previously, large avalanches are less likely to occur than small avalanches. The flooding intensities for various likelihood events are used to prepare a preliminary hazard map that will allow communication to the affected community of the potential hazard at various locations and can facilitate planning, regulation, and zoning based on the map (O'Brien, 2012).

Following Schneider et al. (2014), Raetzo et al. (2002) and Hürlimann et al. (2006), the debris flow intensities have been classified into three classes, and an intensity-likelihood diagram was used to denote three preliminary hazard levels (Table 3). High hazard – people are at risk of injury both inside and outside buildings; a rapid destruction of buildings is possible. Medium hazard – people are at risk of injury outside buildings. Risk is considerably lower inside buildings. Damage to buildings should be expected, but not a rapid destruction. Low hazard – people are at slight risk of injury. Slight damage to buildings is possible. When multiple scenarios are considered, the highest hazard value for each cell is taken to create the preliminary hazard map (Raetzo et al., 2002).

The three avalanche events (large, medium and small) were simulated in RAMMS. The maximum heights of the avalanche material entering the lake range from 6 m for the small avalanche to 20 m for the large avalanche, and the maximum velocities range from 20 m s-1 for the small avalanche to 50 m s-1 for the large avalanche. The RAMMS model simulation period was 60 s. The avalanches take from 33 to 39 s to reach the lake and the portion of the mass released that reaches the lake within the 60 s simulations ranges from 60 to 84 % (Table 4).

Figure 1 shows the locations of five cross sections downstream of Lake Palcacocha where hydrographs are reported from the FLO2D simulations. Figure 7 and Table 6 show the results of the simulation of the large avalanche with the current lake level. At cross section 1, the hydrograph is still similar to the original hydrograph at the lake, with a high-intensity peak flow that is of relatively short duration. The flow is quickly attenuated as it moves downstream, and the hydrograph at cross section 2, located just upstream of the point where the river canyon narrows and becomes steeper, has a much lower peak than the overtopping hydrograph at the lake, but it is of longer duration. This is expected because the river is relatively wide with gentle slopes between the lake and cross section 2.

Cross section 4 is located at the entrance to the city of Huaraz. The peak discharge of the large avalanche event diminishes about 40 % between cross sections 2 and 4. From the beginning of the large avalanche event it takes the flood wave about 1.3 h to reach cross section 4 (Table 6), and the peak flow arrives shortly after. The peak flow takes about 0.75 h to cross the city to cross section 5 and the peak is attenuated by about 50 % in the crossing. Values for the medium and small avalanche events are shown in Table 6. They take considerably longer to arrive and cross the city, but their peaks are attenuated about 50 % as well. The resulting maximum flood intensities in Huaraz are shown in Fig. 8 for the current lake level and two lake mitigation scenarios (15 and 30 m of lake lowering) and each of the three avalanche scenarios. The highest-intensity areas are near the existing channels of the Quillcay River and the Rio Santa on the southern side of the river.

Preliminary hazard identification uses the flood intensity maps (Fig. 8) and converts them to maps showing the hazard level at different points in the city according to the intensity-likelihood flood hazard matrix shown in Table 3. The resulting hazard is obtained by combining the three avalanche events into a single preliminary hazard map selecting the highest hazard for each cell, which reflects the result of all the possible avalanche combinations (Fig. 9).

The BASEMENT modeling results (see Sect. 4.3.2, Hydro-morphodynamic model) indicate that the overtopping wave generated from the large avalanche event does not cause sufficient erosion to initiate a breach of the moraine and release the lake water, thus rendering a full collapse of the moraine extremely unlikely. The authors consider this scenario nearly impossible given the current understanding of the moraine conditions and the extensive modeling of the moraine using extremely erosive soil characteristics. The decision which scenario to eventually include in a hazard map is not just a scientific question, but also a political one. The results of the breaching scenario are included since they are needed in order to assess the worst-case scenario, something science and engineering must communicate to the decision makers and stakeholders. For the sake of providing complete information, the probable maximum flood as a result of a full breach of the damming moraine at Lake Palcacocha was simulated, assuming this event is the worst possible scenario that could conceivably occur. This probable maximum flood is estimated by modeling the event of a full collapse of the moraine following an overtopping wave generated by a large avalanche that erodes the moraine to the extent that the release of the lake water can maintain the erosion and create a full breach of the moraine. The HEC-RAS breaching model (USACE, 2010) was used to simulate the progression of the breaching process and the resulting breaching hydrograph (Rivas et al., 2015). The inflow hydrograph for downstream simulations of this scenario was created by combining the large avalanche overtopping wave hydrograph under current lake level conditions with the HEC-RAS breach hydrograph.

The flood intensity resulting from this scenario is illustrated in Fig. 10. The flood hazard is not computed since the likelihood of the medium and small avalanches generating waves capable of eroding the moraine to the extent of initiating a breaching process are simply too remote to consider.

A sensitivity analysis of the inundation was performed, and it focused on three components: (1) sediment concentration by volume, (2) rheology of the flow, and (3) roughness.

Sediment concentration: the sediment concentration is an important factor in simulating the inundation in Huaraz because it affects the volume of the flow and consequently the depth of inundation (Somos-Valenzuela, 2014). A potential GLOF will erode the bank along the river, especially where lateral moraines are present (cross section 3), scouring, transporting and depositing soil many times as the flood moves downstream from the lake to the city. FLO2D does not represent this process when using the Mudflow module. Additionally, we did not have field information to perform a study of these effects. Therefore, in this work, a fixed sediment concentration of 50 % by volume was used, which is a good upper limit according to the literature and the FLO2D developers (FLO2D, 2012), but it may be too high if the material available for erosion is not sufficient in the inundation path. Analysis of sensitivity to sediment concentration was performed for the inundation in Huaraz, assessing the effect on velocity and flood stage with sediment concentrations of 0, 20, 30, 40 and 50 % (Somos-Valenzuela, 2014). The flood wave travel times were similar for all cases, and the depths increased with sediment concentration due to the increased volumes (an increase of up to 8 m at cross section 4 for a concentration of 50 % compared to no sediment). Thus 50 % concentration was considered a reasonable value to use, and it gives a conservative result.

Flow rheology: with regard to the possible effects and limitations in the model settings associated with different flow rheologies, we identified two major sources of uncertainty: (1) the physical characteristics of the mixture and (2) the volume of material that will be eroded, transported and deposited again, a process that may happen many times during the trajectory of the flood. FLO2D can simulate the behavior of the mixture, assuming that it will not change throughout the simulation. Consequently, it is not able to consider transformations of the flow rheology; however, changes in concentration by volume can change the dynamic viscosity (η) and yield stress (τy) (O'Brien and Julien, 1988). Additionally, scouring is not simulated in the FLO2D mudflow module, so we prescribe the concentration by volume to be 50 % based on the literature recommendations.

The quadratic rheological model used within FLO2D combines four stress components of hyper-concentrated sediment mixtures: (1) cohesion between particles; (2) internal friction between fluid and sediment particles; (3) turbulence; and (4) inertial impact between particles, where the cohesion between particles is the only parameter that is independent of the mixture concentration or hydraulic characteristics (Julien, 2010; O'Brien and Julien, 1988). According to the few studies of the composition of the Lake Palcacocha moraine (Novotný and Klimeš, 2014), the cohesion can be considered nearly equal to zero, which implies that the resulting mixture would have low yield stress and dynamic viscosity. Consequently, from the list of 10 soils presented in the FLO2D manual (FLO2D, 2012: Table 8, p. 57), we selected parameters that give a low yield stress and dynamic viscosity (Glenwood 2). In addition, a sensitivity analysis was performed using the parameters for the other soils listed in Table 1 (Aspen Pit 2, Glenwood 1, Glenwood 3 with higher dynamic viscosities and yield stresses, and Glenwood 4 with much higher values). The results of the sensitivity analysis (FLO2D simulations) show that the flood arrival time at cross section 4 varies from 1.05 to 1.32 h (compared to 1.32 h with Glenwood 2 parameters; see Table 6 in the original paper). The peak flow varies from 1954 to 3762 m2 s-1 (compared to 1980 m3 s-1 using Glenwood 2). The Glenwood 4 parameters result in the shorter arrival time (somewhat counter-intuitively) and higher peak value. Therefore, the rheology, which is a function of the concentration of the mixture and the soil characteristics, does affect the travel time and the peak flows. The results are not expected to be highly sensitive if the dynamic viscosity were to be lower than what was assumed (Glenwood 2), which is expected from the few soil studies in the area.

The model results show that the flood takes about 45 min to cross the city (travel of front of inundation between cross sections 4 and 5) and the peak flow takes 55 min to cross the city. The inundation spreads through the city, diffusing the peak flow and reducing it considerably. Sensitivity analysis showed that increasing the dynamic viscosity, from Glenwood 2 to Glenwood 4, the flow travels faster, arriving at the city 17 min earlier, crossing the city in 36 min, with the peak flow taking 45 min to cross the city. Glenwood 2 and Glenwood 4 are the lower and higher end, respectively, for the dynamic viscosity parameters used in the sensitivity analysis.

Roughness: the impact of roughness was analyzed in the dissertation of Somos-Valenzuela (2014), who concluded that travel time is sensitive to roughness, increasing by 1.5 h for travel from the lake to cross section 4 if the roughness is increased from 0.1 to 0.4. Also, the peak flow is inversely proportional to the roughness, so lower roughness results in a slightly higher peak (less than 10 % difference in peak flow for 0.1 vs. 0.2 roughness coefficients) (Somos-Valenzuela, 2014). When the roughness within the city is reduced to 0.02, the minimum value recommended for asphalt or concrete (0.02–0.05) (FLO2D, 2012) and the 20 % area reduction factor is removed (so the flood is limited only by the topography), the inundation takes 22 min to cross the city, 50 % of the originally computed time. This is an unrealistic value since it considers the entire land cover of the city to be asphalt with no disturbances, buildings, streets, trees, debris, etc.; however, this can be considered a minimum possible time for the flood to cross the city. If a roughness value of 0.05 is used, then the inundation takes 26 min to cross the city, and if 0.1 is used, a low but more realistic value, the flood takes 36 min to cross the city. Thus, the travel time across the city is more sensitive to changes in roughness values than rheology characteristics.

The relative impacts of the GLOF process components can be seen by analyzing the inundation in the city of Huaraz for each of the scenarios simulated. The avalanche size may have the most significant impact on downstream flood hazard. With the lake at its current level, the affected area in Huaraz for the small avalanche scenario (0.7 km2) is approximately 35 % of the area potentially affected by the large avalanche (2.0 km2). The other process that could significantly influence the flood hazard in the city is the erosion of the damming moraine. Although results from this work indicate that a complete moraine failure is extremely unlikely, the possibility of a catastrophic breach cannot be categorically excluded based on existing evidence. If such a breach were to occur, the inundated area could increase to 4.93 km2, almost 246 % more than the large avalanche–no breach scenario (2 km2). Considering the results of the lake-lowering mitigation scenarios, the reduction in hazard area in Huaraz is mostly in the high hazard zones (see Table 7). There is a 27 and 45 % reduction in the high hazard area (compared to the current lake level) when the lake is lowered 15 or 30 m, respectively.

In this paper, each step in the hazard process chain that could lead to inundation of Huaraz from a GLOF from Lake Palcacocha has been simulated. Of the simulation methods used in this work, the lake hydrodynamics and moraine erosion models are advancements beyond what has been previously reported for GLOF hazard process chain simulations. The use of a fully three-dimensional hydrodynamic model for simulating wave generation, propagation, run-up and overtopping of the damming moraine allows predictive modeling of the process chain through better representation of the physical processes. Other studies (e.g., Schneider et al., 2014) have used a past event to calibrate the models and then used those calibrations for predictive modeling of other scenarios. When data for past events are not available, the three-dimensional model can help overcome the limitations of two-dimensional SWE models. Better representation of the physical processes in the model (i.e., three-dimensional non-hydrostatic) makes the models useful for predictive purposes without a heavy reliance on calibration. Modeling for predictive purposes, such as that presented in this paper, is useful for analyzing potential GLOF impacts and mitigation strategies.

The general lack of field data regarding actual GLOF events leads to many unknowns about the processes, particularly processes related to avalanches, lake dynamics and moraine erosion. Previous simulations of GLOFs have focused on calibrating upper-watershed processes based on post-event observations (Schneider et al., 2014), but there is very little information on avalanche characteristics, magnitude of avalanche-generated waves (Kafle et al., 2016), or erosive capabilities of overtopping waves on which to base validation of these simulated processes. There is still a considerable amount of uncertainty in the three-dimensional modeling approach for avalanche-generated waves. Nonetheless, even post-event field studies of GLOF waves have difficulty accurately characterizing the wave magnitudes. The three-dimensional modeling approach presented in this paper is intended as an alternative to partially overcome the absence of field data from a GLOF event at the location of the study.

Because field data are not available, we attempted to counteract the inability to calibrate the models by using the best available physical representations in our modeling approach. The three-dimensional hydrodynamic model and the hydro-morphodynamic model of moraine erosion can give us a better understanding of the likely outcomes of these processes than models that require extensive calibration (e.g., two-dimensional SWE models and breach simulations such as reported in Rivas et al., 2015). This is not to say that these models are free of significant uncertainties, but as a model provides better mechanisms to represent the underlying physical phenomena, uncertainties move from the model engine to the physical initial and boundary parameters, reducing the number of physical or empirical assumptions. Caution is required in any case because, lacking a means of calibration/validation, these results represent estimations that might deviate from reality without proper analysis or judgment.

Simulations of lake dynamics with a three-dimensional non-hydrostatic model (FLOW3D) and a two-dimensional SWE model (BASEMENT) indicate that the SWE approximation is not adequate to simulate waves generated by avalanches because of the large energy dissipation due to significant vertical accelerations. Two-dimensional hydrostatic models may be adequate for simulating past events where calibration parameters based on field data may be used to overcome the approximations in the SWE model (Schneider et al., 2014), but it is important that calibration be performed at appropriate points in the model to account for energy dissipation as the wave propagates across the lake. The results from the BASEMENT simulations suggest that, without careful setting and adjustment of the model's boundary conditions, two-dimensional models might produce unrealistic results for wave-driven phenomena that underestimate the magnitude of an event. Reference simulations, like those from three-dimensional hydrodynamic models, may help to overcome limitations in the two-dimensional models and turn them into more flexible and efficient tools for erosion and breach failure assessment.

The primary limitation of the lake hydrodynamic model arises from representing an avalanche entering the lake as a volume of water, rather than a combination of rock, ice and snow (Kafle et al., 2016). The wave model calibration method involves controlling the height and depth of the release area in order to influence the fluid height and velocity in the model as the avalanche enters the lake. This helps to overcome the limitations of substituting water for the avalanche fluid mixture, but the water representation does not dissipate the energy in the same way as the true avalanche mixture, and the mixing of the avalanche fluid with the lake is not accurately represented in the model.

The lake model has a considerable amount of uncertainty. The greatest sources of uncertainty are the avalanche characteristics (inputs to the lake model) and the wave generation. The processes associated with wave generation from avalanche impact are poorly understood, and current model limitations do not allow for an avalanche to be simulated with its actual flow characteristics (rheology, density, etc.) in the same environment as the lake dynamics. Therefore, it is difficult to represent wave generation in a fully physical manner. The avalanche characteristics (depth and velocity) have a significant impact on the wave characteristics and moraine overtopping hydrograph. Additionally, the method of representing the avalanche impact boundary condition may overestimate the momentum of the inflow; the result of this may be somewhat larger wave height, but the greatest impact is in the peak flow and total volume of the overtopping wave. The highest estimates of the overtopping wave characteristics are presented in the paper to illustrate a worst-case scenario, but it is likely that the actual magnitude of an avalanche-generated wave may be less than what is reported here.

The moraine erosion simulations used a worst-case approach, depicting the moraine as a structure with very low erosive resistance. Therefore, the resulting moraine erosion is overestimated, i.e., erosion depth, width, length, and growth rate. Thus, the simulations sacrifice accuracy in modeling the erosion process to gain confidence in predicting the potential for moraine breaching and collapse. The erosion simulation results suggest that the Lake Palcacocha damming moraine has adequate stability to resist erosion induced by large waves, since the modeled erosion does not reach from the distal face back to the lake, which would allow the lake water to flow through the breach and accentuate the erosion process and lead to possible moraine failure. The main source of erosive resistance in the simulations is from the morphology of the moraine (e.g., large width to height ratio, long crested dam, and gentle slope of distal moraine face) and not from soil resistance. Previous qualitative assessments of the Lake Palcacocha moraine (Emmer and Vilímek, 2013) and similar structures at other lakes (Worni et al., 2014) assigned very low probabilities of failure of the moraine, but did note its high susceptibility to wave overtopping. This study, however, provides the first quantitative assessment of possible breach failure for the damming moraine at Lake Palcacocha, reinforcing results from the qualitative assessments by using numerical simulations that account for the morphology of both the lake and moraine in a two-dimensional modeling scheme.

The functions in BASEMENT to simulate erosion come from empirical equations of sediment transport developed for fluvial environments. Due to their empirical nature, the equations depend on calibration to achieve accurate results of erosion and deposition rates. Worni et al. (2012) showed that BASEMENT can achieve realistic results using soil parameters that resemble actual moraine properties. The bed-load transport model used in this paper (Meyer-Peter and Müller, 1948) has been derived in different forms since its first release to reverse the model's tendency of overpredicting erosion. Newer bed-load models address this problem by applying a direct reduction factor to resulting transport rates or adding hiding functions to account for multi-grain soil matrixes (e.g., Ashida and Michiue, 1971; Wu et al., 2000). Additionally, the two-dimensional limitation of BASEMENT restricts its application for problems where vertical accelerations are relevant, or vertical flow distribution is not uniform. Under these latter conditions, BASEMENT needs three-dimensional simulations to serve as calibration parameters before applying the model to predict erosion and breach formation.

Even though a prescribed terminal moraine collapse scenario was simulated, it was not included in the preliminary hazard map for two reasons. Firstly, the complete collapse scenario is based on the premise that we should consider a worst-case scenario, but we could not initiate the moraine collapse using our numerical approach; even when a large overtopping wave and highly erosive materials were assumed, the width of the moraine is simply too great, and the erosion does not extend from the distal face of the moraine back to the lake. Therefore, we artificially prescribed and simulated the moraine collapse. Using empirical equations we determined the time that the collapse will take and the hydrograph was calculated following hydrodynamic constraints as indicated in Rivas et al. (2015). Based on these modeling results it is extremely unlikely that the collapse will occur, but it cannot be completely disregarded. Secondly, given the magnitude of the extremely unlikely breach scenario results, it is important to avoid creating confusion as a result of misinterpretation of the results. People in Huaraz should decide whether they want to consider the worst-case scenario in their planning, and this work is limited to informing that decision-making process.

There are still many unknowns about the 1941 event, including the precise lake volume at that time, underlying bathymetry and pre-GLOF moraine morphology, flood volume and discharge hydrograph; aerial images and derived historical maps represent the only sources of information, known to the authors, about the physical characteristics of the 1941 GLOF, providing at least a rough visual estimation of the flood area. In a qualitative comparison with the GLOF from 1941, we used a map published by the Instituto Nacional de Defensa Civil (INDECI, 2003) where three mudflow event extensions are delineated: Aluvion Preincaico, Aluvion Huallac and Aluvion Cojup 1941. In Figure 11 we plot the inundation extension reported in this paper on the map of the 1941 event delineated by INDECI (2003) and confirm that the inundation modeled has reasonable dimensions in comparison with this historical information. The volume at the time was estimated to be on the order of 14 million m3 (Vilímek et al., 2005), which is more than 7 times the volume that we have calculated for the large avalanche (1.8  million m3). This may explain the fact that in our results the inundation does not pass out of the bank from the Cojup River to the Quilcaihuanca River in the area where the rivers are very close together near the entrance to the eastern border of the city. However, these results require caution; a qualitative comparison only describes potential differences between simulated and observed flood areas. Because the moraine failure in 1941 changed the upstream conditions at Lake Palcacocha, historical aerial images of flooded areas constitute no source of information for precise calibration for our model.

According to Vilímek et al.  (2005), the lateral moraine collapse that occurred in 2003 at Lake Palcacocha was due to a wave produced by a landslide on the internal face of the left lateral moraine that was triggered by extensive rainfall precipitation which over-saturated the moraine material. The terminal moraine was eroded, but it did not breach. A downstream flood was produced by the water that overtopped the moraine. While this type of landslide from the lateral moraine is likely to occur again in the future, the work reported here focuses on the potential effects of an avalanche-generated wave because the magnitude of landslides likely to enter the lake are less than the avalanche volumes we have considered, and the effect of a landslide-generated wave may be somewhat mitigated as it propagates diagonally across the lake, whereas an avalanche-generated wave would enter along the longitudinal axis of the lake and is unlikely to be attenuated by reflections off the lateral moraines.