Introduction

Large rock-ice avalanches are geophysical mass flows composed of a mixture of rock and ice. They can be extremely hazardous due to their extremely high velocity and long runout1,2,3. In earthquake-prone regions globally changing climate appears to be exacerbating these hazards4. Scientists have long recognized rapid global climate change favors the instability of mountains in glacial and periglacial areas5,6,7. Nevertheless, to our knowledge, no hazard risk assessment considers how the impact of climate change contributes to their destructive potential, especially in terms of runout and flow regime transitions, such as the formation of hazardous air blasts.

One phenomenon of climate change is frequent snowfall anomalies in high-altitude regions. In the past decades, the duration of snowfall decreased, while snowfall intensity showed an increasing trend8,9,10. The thick snow cover arising from snowfall anomalies is an important mass source of the avalanche core. This snow entrainment process amplifies the avalanche volume11,12, lubricates the avalanche movement13, and exacerbates the formation of a rock-ice-snow powder avalanche14. This type of avalanche often generates powerful air blasts capable of causing damage and human fatalities far beyond the reach of avalanche core15.

Another feature of the ongoing climate change problem is warming16,17, moreover, how changing snow and air temperatures will change avalanche flow dynamics. In existing rock-ice avalanche models, the sliding mass is treated as a thermally insulated system, ignoring the effect of ambient environment18,19. Frictional shearing20, entrainment process13, and particle collisions21 are heat energy sources that change the avalanche temperature and produce meltwater. It is important to note that avalanche snow (ice) exists near its melting point and frictional heating can easily supply the necessary energy input needed to produce meltwater. The porous-medium structure of avalanches and the dispersive movement of granular particles allow the intake and outburst of ambient air14. This interaction with the ambient environment is inevitable during avalanche movement and can either enhance or hinder the heating process. The temperature difference between the avalanche and ambient air leads to a heat exchange that greatly influences meltwater production and the flow regime of the avalanche core.

One striking example is the rock-ice avalanche of Langtang (2015), which was triggered by the Gorkha earthquake in a warm season, releasing several ice masses well above the snowline22. The thick snow cover amplified the avalanche volume and formed a rock-ice-snow powder avalanche. Fujita et al.23 attributed the massive destruction caused by the avalanche directly to a snow cover anomaly. The air blast nearly destroyed the Langtang village and flattened a forest on the valley counter-slope24.

In this paper, we reconstruct the evolution of the Langtang avalanche and generated air blasts based on documented field measurements and numerical modeling. We further conduct simulations with different snow cover depths and air temperatures to investigate the impact of snow entrainment and ambient environment, indicating how seasonal and climatic influences may affect the danger of rock-ice avalanches. The study focuses exclusively on changes in runout and flow regimes under different snow cover and air temperature conditions, rather than variability in avalanche occurrence due to climate change. Our primary goal is to quantify the danger arising from rock-ice-snow avalanches, containing both a dense flow core and dust cloud, in different mountain conditions that can be eventually associated with changing climate scenarios.

Langtang avalanche

On 25 April 2015, the Gorkha earthquake (Mw7.9) triggered a large rock-ice avalanche in the Langtang Valley, Nepal. Details of the Langtang avalanche are well recorded in existing literature. Thus, here we briefly introduce the essential information concerning this disastrous event. The avalanche was initiated as a multi-source ice avalanche (initiated in several release areas) of ~3.5 × 106 m3, and the release areas are located over 6000 m a.s.l. (Fig. 1a, b). Satellite images (April–May 2015) indicated an evident snowline at 4000–4500 m a.s.l. Four anomalous snowfall events occurred during the previous winter (October 2014–April 25, 2015), resulting in a thick snow cover on the mountain surface23. Field investigation reveals that the released ice mass entrained snow (over 4500 m a.s.l.) and debris cover (below 4500 m a.s.l.) along the travel path (Fig. 1c)23. According to pre- and post-event digital surface models25, we knew that the avalanche involved a total volume of 14.38 × 106 m3 of rock, ice, and snow mass. Of this, a total of 6.95 × 106 m3 accumulated in the Langtang Valley25. The Langtang village is located on the valley floor (Fig. 1d, e) and was not struck directly by the avalanche core. The air blast generated by the avalanche destroyed large parts of the village and caused over 350 deaths. This is confirmed by field observations showing that many houses constructed of stone slabs were flattened or destroyed by the air blast24 (Supplementary Fig. 1a). Furthermore, the air blast impacted an area of 1 km up and down the valley and flattened a forest on the opposite mountain (Fig. 1e, Supplementary Fig. 1b), as described by Kargel et al.22.

Fig. 1: Overview of the Langtang Avalanche.
figure 1

a The release area from Langtang Lirung. The size of released materials helps estimate the initial volume. b The direction from where additional ice was released from around Langtang II. Both are shown in detail in the right part of the figure. c The plateau where the rock entrainment occurred. d The headwall above the village (height ~500 m). e Shows the view angle of the photo as an inset where destroyed houses (e1), blown-over trees (e2), and deposited ice–debris mix on the far side of the valley (e3) are shown. Images taken from the helicopter by D.F. Breashears/GlacierWorks.

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Two local meteorological stations record the air temperature: Kyanjing station at 3862 m a.s.l. (6.3 km away from the Langtang village) and Yala Base camp station at 5058 m a.s.l. (10.2 km from the Langtang village)26. The average measured temperature on 25 April was −0.65 to 9.05 °C at Kyanjing station (2012–2014, 2016–2018) and −8.10 to 0.18 °C at Yala Base camp station (2013, 2016–2018). Another pluviometer at 4831 m a.s.l. recorded an air temperature of −0.4° to 3.6° within 30 min before the avalanche occurred, indicating a warm environment23. The Gorkha earthquake occurred at 11:56 Nepal Standard Time, therefore a high temperature (9 and 0 °C at 3862 and 5058 m a.s.l., arising from the recorded max value) is applied for the following analysis, indicating an air temperature gradient of ~0.75 °C per 100 m elevation difference. According to Fujita et al.23, the snow depth at Yala station was ~1.5 m. Using this snow depth value, combined with the location of the snow line, we derived a preliminary estimate of the snow cover gradient. Based on this estimate, the snow depth at the release area was calibrated, ensuring that the entrained and total avalanche volumes match the measured data.

Results

Langtang avalanche and the generated air blast

Modeling results of the Langtang avalanche are presented in Fig. 2. The snow depth at the release area is estimated to be 3 m, and a gradient of −0.15 m per 100 m is determined with this depth value and the location of the snow line. The modeled released ice volume is 3.65 × 106 m3, and the sliding mass entrained snow and rock materials of 11.20 × 106 m3 during the movement process (simulated total volume of 14.85 × 106 m3). The error between the simulated and actual volumes (initial volume of 3.50 × 106 and total volume of 14.38 × 106 m3) is within 5%. Two main deposit areas are observed in the simulation: the platform at 4500 m a.s.l. and the Langtang Valley (Fig. 2a). The depth of deposits in the Langtang Valley is over 30 m, as observed by Fujita et al.23. The simulated avalanche volume and the deposit area match these observations. The calculated avalanche core reached the maximum velocity of over 90 m s−1 at 5000–5500 m a.s.l. When the avalanche passed the Langtang Valley, immediately before hitting the toe of the opposite mountain, it was traveling at 57 m s−1 (Fig. 2b), matching the velocity estimated by the run-up equation (63 m s−1, Kargel et al.22).

Fig. 2: Modeling results of the Langtang avalanche and the generated air blast.
figure 2

a Final deposit distribution of the avalanche core, matching the observations (red lines). b Maximum velocity of the avalanche core. c Maximum mean pressure of the air blast. d and e Powder pressure profile at the Langtang village (point A in c) and the forest at the mountain toe (point B in c). The satellite image arises from Planet (Images ©2021 Planet Labs PBC).

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Figure 2c shows the dynamic pressure of the Langtang avalanche-induced air blast. The air blast shows an impact area far beyond the avalanche core, covering the whole Langtang village and the forest on the opposite mountainside. In the Langtang village, the dynamic pressure of the air blast exceeds 15 kPa but decreases gradually to 6 kPa as it moves over the village (a rule of thumb used by avalanche engineers is that a 1 kPa air blast will violently blow a human to the ground at impact). The turbulent fluctuations greatly magnify the air blast pressure. At a specific location (simulation point) near Langtang village (point A in Fig. 2c), the maximum pressure reaches 28 kPa (Fig. 2d), nearly double the pressure arising from the mean velocity. Such high pressures are capable of destroying houses, as observed. On the opposite mountain slope, the pressure of the air blast reaches a mean of 10 kPa and a maximum of 18 kPa at the mountain toe (Fig. 2e) but decreases substantially as it climbs up the mountain. Tree-breakage calculations follow the method proposed by Feistl et al.27. Trees on the opposite mountain face are Abies and Rhododendron species with an average diameter of 0.16 m23. Using a bending strength of 72 MPa, modeling results indicate a tree-breakage area of about 0.8 km2, extending 1 km up and down the valley from the deposit center and 550 m up the mountain (Fig. 2c). The calculated tree-breakage area essentially matches the observations22.

Impact of snow entrainment

We set up scenarios to investigate the impact of snow cover on the dynamics of rock-ice avalanches and air blasts. The snow cover at the release area is set at 0–3 m in depth with a constant gradient of −0.15 m per 100 m. The air temperature remains fixed. In the RAMMS model, the snow cover is adjusted based on the slope angle and curvature along with the elevation gradient to produce a realistic snow distribution on mountainous terrain. Snow cover distributions are shown in Fig. 3a–d.

Fig. 3: Impacts of snow entrainment on the destructive potential of the Langtang avalanche and generated air blasts.
figure 3

a–d Snow cover distribution in different scenarios. Red dotted line represents the calculated domain. e–h Impacts of snow cover on the avalanche extent. i–l Impacts of snow cover on the air blast dynamics. m–p Pressure profile at the Langtang village (Point A in i) in cases of different snow covers. The satellite image arises from Planet (Images ©2021 Planet Labs PBC).

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Modeling results indicate a longer runout distance and higher mobility in the case of a thick snow cover (Fig. 3e–h). When the snow cover at the release area reaches 3 m, which corresponds to the modeling results in Fig. 2, the calculated avalanche dynamics match the observed conditions well. In the case without considering snow entrainment (snow cover = 0 m), the rock-ice avalanche stops before hitting the opposite mountain. The sliding mass primarily deposits at the 4500 m a.s.l. platform, with only a little mass reaching the Langtang Valley. For the air blast hazards, the cloud dynamics are again related to the thickness of the snow cover. Both the impact area and dynamic pressure of the air blast decrease in the case of a thin snow cover (Fig. 3i–p). When there is no snow cover, the mean dynamic pressure at the Langtang village is only 2.5 kPa, and only a few trees are damaged on the opposite mountain (Fig. 3l, p), showing minor destruction compared with the actual event. This implies that the entrained snow facilitates the formation of a dispersed powder avalanche and is a primary factor that led to the Langtang disaster. Our numerical results strongly support the hypothesis of Fujita et al.23.

Impacts of ambient temperature

Further scenarios of different air temperatures are designed to investigate the impact of temperature variation on the danger arising from rock-ice avalanches, as presented in Fig. 4. The air temperature at 3862 m a.s.l. (Kyanjing meteorological station) is set −1 to 19 °C with a constant gradient of 0.75 °C per 100 m. Here the snow cover depth remains fixed. Modeling results indicate that the avalanche core shows higher mobility in the case of a warm environment. When the air temperature at 3862 m a.s.l. reaches −1 °C, the heat transfer with the cold air restricts the melting of snow and ice. The meltwater is 74,000 t at the end of the event (Fig. 4g), and the maximum water content in the avalanche core is ~600 mm m−3 (Fig. 4d). In this scenario, relatively small amounts of material deposit in the Langtang Valley (Fig. 4a) compared with the actual conditions. When the air temperature is 19 °C at 3862 m a.s.l., a warm environment, the produced meltwater reaches 170,000 t (Fig. 4i), more than twice the amount calculated for a cold environment. The maximum water content in the avalanche core reaches over 1800 mm m−3 (Fig. 4f), a high value. The rheological relationship applied here is an ever-decreasing Coulomb friction resistance with increasing water content13:

$${\mu }_{w}={\mu }_{\min }+({\mu }_{\Phi }-{\mu }_{\min })\exp \left[\frac{{-m}_{{\rm {w}}}}{{m}_{0}}\right]$$
(1)

where \({\mu }_{\Phi }\) is the dry friction coefficient, \({\mu }_{\min }\) is the fully lubricated friction coefficient, representing the lowest friction, \({m}_{{\rm {w}}}\) is the water height in the flow and \({m}_{0}\) is the reference height. The friction decreases from \({\mu }_{\Phi }\) (dry avalanche) to \({\mu }_{\min }\) (fully lubricated flow) with the increase in water content. In the case of a warm environment (Fig. 4f), the abundant amount of meltwater at the avalanche front provides lubrication (decreases the friction, Supplementary Fig. 2) leading to a fluid-like flow regime in the Langtang Valley and thus a long runout distance (Fig. 4c). The avalanche moved over 1.5 km downstream of the Langtang Valley. For the Langtang avalanche, due to the low temperature of the released materials, the meltwater appears at ~20 s after the avalanche initiation and accumulates in the frontal lobe of the avalanche deposits (Fig. 4e, h).

Fig. 4: Impacts of air temperature on the avalanche extent and meltwater production.
figure 4

a–c Impacts of air temperature on the avalanche extent. d–f Impacts of air temperature on the water content distribution within the avalanche core. g–i Impacts of air temperature on meltwater production. Here, the air temperature is set as −1, 9, and 19 °C at 3862 m a.s.l. with a constant temperature gradient. The satellite image arises from Planet (Images ©2021 Planet Labs PBC).

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Discussion and implications

Scientists suggest that ongoing climate change favors the initiation of rock-ice avalanches, but its impacts on avalanche formation and flow are rarely quantified3,6,28. With this purpose, using the Langtang avalanche, we numerically investigate two important phenomena arising from climate change to the destructive potential of rock-ice avalanches: snowfall anomalies and global warming.

Snowfall anomalies

For a rock-ice avalanche, snowfall anomalies will exacerbate the avalanche mobility and lead to an air blast problem. Anomalous snowfall and the associated thick snow cover favor snow entrainment in the mountain environment. This process can magnify the avalanche volume and change the flow behavior. Compared with ice and rock mass, snow is easily entrained because of the low shear stress threshold29. Snow is a material of low friction and can decrease the equivalent Coulomb frictional coefficient (in Eq. (13)) of the avalanche core. One primary consequence of the snow entrainment is the increment of flow height, which increases the normal and shear stress acting on the ground surface. Combined with the entrainment process12, the additional shearing work attributable to the shear stress increment exacerbates the heat energy production within the avalanche core, therefore favoring meltwater production. The lubrication effect of the meltwater greatly enhances the avalanche mobility. Furthermore, the flowing resistance is a function of fluctuation energy \({R}_{\Phi }\) (Eq. (13)), again related to the shearing work and the flow height. Chute experiments30 indicate that friction decreases in proportion to the increase in the gravitational work rate or the shearing work rate. Without considering the entrainment process, there is no mass source for avalanches in motion. For a finite-sized avalanche, shear gradients within the avalanche core cause the flow height to decrease. The shearing work rate and production of fluctuation energy subsequently decline, and friction increases. The avalanche begins to starve and decelerate, eventually stopping on the slope.

As for the air blast hazard, the process of snow entrainment facilitates the formation of a rock–ice–snow avalanche with a dense granular core consisting of rock and ice fragments and a dust cloud of suspendable particles. The initial mass and momentum of the air blast are known to arise from the air movements caused by displacing air in the avalanche core14. The entrained snow is, therefore, an important mass/momentum source to transfer mass and momentum to the powder cloud, magnifying its impact area and dynamic pressure. In the case without snow entrainment, the Langtang avalanche-induced air blast hardly destroys the Langtang village and the forest (Fig. 3l), principally different from the actual conditions. This implies air blast hazards could show a higher destructive force after an intense snowfall event. We suggest the previous snowfall anomalies and the snow entrainment were a primary factor that caused the Langtang disaster.

Heat exchange with air temperature

For rock-ice avalanches, the generation of heat energy and the phase change processes are of paramount interest due to the unique properties of ice and snow18,31. The components of snow and ice reside close to their melting points, and the frictional heat generated during the avalanche propels these materials toward a phase transition. This process is tempered by the ambient air temperature. Recorded videos3 and our detailed investigations into avalanche evolution provide compelling evidence of the physical interaction between the avalanche core and the ambient air temperature. The avalanche core entrains the ambient air during the movement process and outbursts the dust-mixed air to generate the powder cloud. The Reynolds number of natural rock-ice avalanches32 reaches 104−106. Such a turbulent structure facilitates the mixture of the granular particles and entrained air, therefore exacerbating a heat exchange between the avalanche core with the ambient environment. Our modeling results of the Langtang avalanche suggest a warm air temperature amplifies the meltwater production and lubricates the flowing mass.

For simplification and engineering applications, we used an experimentally based heat transfer relationship for sphere particles33. Though the impacts of particle shape and porosity are ignored, this relationship is sensitive to the particle size. More precisely, the particle surface area. According to the heat transfer relationship (Eq. (12)):

$${\dot{q}}_{\Phi \to \Lambda }=H{A}_{{\rm {s}}}\left({T}_{\Phi }-{T}_{\Lambda }\right)=\frac{{{\rm {Nu}}}\cdot {k}_{{\rm {a}}}}{2r}\cdot \frac{{3\rho }_{\Phi }{h}_{\Phi }A}{4{\rho }_{g}r}\left({T}_{\Phi }-{T}_{\Lambda }\right)=\frac{{3\rho }_{\Phi }{h}_{\Phi }A{k}_{{\rm {a}}}}{8{\rho }_{{\rm {g}}}}\cdot \frac{{{\rm {Nu}}}}{{r}^{2}}$$
(2)

The heat transfer is proportional to r−1.5–r−2.0. In this study, due to the lack of accurate particle size distribution, particle sizes for snow (7 cm), ice (10 cm), and rock (30 cm) are determined based on our practical experience with numerous historical avalanches and the existing literature34,35. This implies heat energy exchange between snow particles with air reaches 8.9–18.4 times rock particles of the same volume. For the Langtang avalanche, the volume of the entrained snow accounts for >70% of the total volume23. The process from avalanche initiation to deposition lasts ~3 min, and the air temperature in the Langtang Valley reaches ~14 °C. The small-sized snow particles, prolonged movement, and warm air temperature intensify the heat exchange and meltwater production, resulting in a flow-like movement in the Langtang Valley. Sensitivity analysis indicates a longer runout distance, more meltwater production, and a highly fluid flow regime of avalanches in a warm environment. The impacts of heat transfer with the ambient air on meltwater production and avalanche runout can be even higher than the contribution of snowpack temperature (see extra simulations presented in Supplementary Fig. 3).

Notably, the Gorkha earthquake and Langtang avalanche occurred at noon, in a warm environment. If the avalanche occurs at night, a cold environment hinders the meltwater production and leads to a smaller runout and impact area that is similar to the scenario depicted in Fig. 4a. We are not able to predict when a catastrophic earthquake will occur, but our analysis of air temperature impacts indicates that the destructive potential of rock-ice avalanches is not only related to large-scale global warming but also to short-term diurnal temperature.

Risk assessments of rock-ice avalanches are a primary concern in high-altitude mountainous regions. Confronted with the problem of extreme and rare events and the lack of historical documentation, avalanche dynamics modeling will play a key role in assessing the safety of settlements, transportation routes, and hiking trails. To our knowledge, the impacts of climate change on the destructive potential of rock–ice avalanches are multifaceted, but rarely considered. We suggest the snowfall anomalies and warm air temperature greatly contributed to the Langtang disaster.

Conclusions

At noon on April 25, 2015, the Mw 7.9 Gorkha earthquake precipitated a large rock–ice avalanche in the Langtang Valley. The resultant air blasts obliterated the Langtang village, a nearby forest, and led to the tragic loss of over 350 lives. Through meticulous field investigations and advanced numerical modeling, we have identified two principal factors contributing to the severity of the Langtang disaster: anomalous snowfall and elevated temperatures. The substantial snow cover facilitated snow entrainment, promoting the development of a dispersed powder avalanche. This entrained snow significantly increased the avalanche volume, acted as a lubricant for the flowing mass, and amplified the destructive power of the air blast. Furthermore, the warm air temperature, in conjunction with the turbulent structure of the avalanche core, intensified heat exchange between the granular particles and the entrained air. This thermal process enhanced meltwater production, consequently altering the avalanche’s mobility. These mechanisms were crucial in intensifying the Langtang disaster. We emphasize the critical finding that incorporating the effects of snow cover and air temperature factors heavily influenced by seasonal and climatic changes—is essential when performing hazard analyses for high-altitude rock-ice avalanches. Our results reveal the inherent fragility of natural systems, particularly the mobility and destructive potential of rock-ice avalanches. Minor variations in temporal factors, climatic conditions, or the antecedent state of an event can either intensify or mitigate disastrous outcomes. This insight compels us to ponder how many disasters have been averted or caused by seemingly insignificant changes in the natural environment.

Methods

Dynamic model of rock–ice avalanche

General framework

We apply a depth-averaged model to simulate the flow dynamics of the rock-ice avalanche and the generated air blast19,36. The avalanche core Φ and the air blast Π are described by two distinct depth-averaged layers. The core layer comprises a granular ensemble of rock, ice, and snow particles capable of dispersion and compression, changing the interstitial air space of the core. The core movement is strongly influenced by the surface terrain and allows for both air intake and outburst. During the outburst process, the avalanche core transmits mass and momentum to the cloud Π and creates a turbulent structure in the cloud14. The cloud is, therefore, modeled as a turbulent flow, including suspensions of ice and rock dust that are transferred from the avalanche core. The governing equations for both the avalanche core and dust cloud are solved using well-established finite volume schemes within the software rapid mass movement simulation (RAMMS)36.

Avalanche core Φ

Here we follow the avalanche core model19,36. The model involves some essential processes of an avalanche flow, including entrainment of path materials37 and meltwater production19. In the model, the movement of the avalanche core Φ is described by three state valuables: the so-called co-volume height \({\hat{h}}_{\Phi }\), the dispersed or flowing height \({h}_{\Phi }\) and the depth-averaged velocity \({\vec{u}}_{\Phi }\) that parallel to the slope surface. The co-volume represents the densest packing of ice/rock granules found in the deposit area, corresponding to a co-volume density \({\hat{\rho }}_{\Phi }\). Because the avalanche core is a multiphase flow of mixed components, the co-volume height \({\hat{h}}_{\Phi }\) is defined as

$${\hat{h}}_{\Phi }=\frac{{\rho }_{{\rm {i}}}}{{\hat{\rho }}_{\Phi }}{\hat{h}}_{{{\rm{i}}}}+\frac{{\rho }_{{\rm {s}}}}{{\hat{\rho }}_{\Phi }}{\hat{h}}_{{{\rm{s}}}}+\frac{{\rho }_{{\rm {r}}}}{{\hat{\rho }}_{\Phi }}{\hat{h}}_{{{\rm{r}}}}+\frac{{\rho }_{{\rm {w}}}}{{\hat{\rho }}_{\Phi }}{\hat{h}}_{{{\rm{w}}}}$$
(3)

where \({\rho }_{{\rm {i}}}\), \({\rho }_{{\rm {s}}}\), \({\rho }_{{\rm {r}}}\), \({\rho }_{{\rm {w}}}\), \({\hat{h}}_{{{\rm{i}}}}\), \({\hat{h}}_{{{\rm{s}}}}\), \({\hat{h}}_{{{\rm{r}}}}\), \({\hat{h}}_{{{\rm{w}}}}\) are the material density and co-volume height of ice, snow, rock, and water, respectively. The model assumes both constant density and velocity profiles of each material. The primary governing equations are as follows:

$${\left({\hat{h}}_{\Phi }\right)}_{t}+{{{div}}}\left({\hat{h}}_{\Phi }{\vec{u}}_{\Phi }\right)=\frac{{\rho }_{\Sigma }}{{\hat{\rho }}_{\Phi }}{\dot{M}}_{\Sigma \to \Phi }-\frac{{\hat{\rho }}_{\Pi }}{{\hat{\rho }}_{\Phi }}{\dot{M}}_{\Phi \to \Pi }$$
(4)
$${\left({h}_{\Phi }\right)}_{t}+{{\rm {div}}}\left({h}_{\Phi }{\vec{u}}_{\Phi }\right)={\mathbb{D}}\left(t,{k}_{z},{\dot{k}}_{z},{\ddot{k}}_{z}\right)$$
(5)
$$\left\{\begin{array}{c}{\left({\hat{h}}_{{{\rm{r}}}}\right)}_{t}+{{\rm {div}}}\left({\hat{h}}_{{{\rm{r}}}}{\vec{u}}_{\Phi }\right)=\frac{{\rho }_{\Sigma }}{{\rho }_{{\rm {r}}}}{\eta }_{{\rm {r}}}{\dot{M}}_{\Sigma \to \Phi }-{\upsilon }_{{\rm {r}}}{\dot{M}}_{\Phi \to \Pi {{\rm{r}}}}\\ {\left({\hat{h}}_{{{\rm{i}}}}\right)}_{t}+{{\rm {div}}}\left({\hat{h}}_{{{\rm{i}}}}{\vec{u}}_{\Phi }\right)=\frac{{\rho }_{\Sigma }}{{\rho }_{{\rm {i}}}}{\eta }_{{\rm {i}}}{\dot{M}}_{\Sigma \to \Phi }-\frac{{\dot{Q}}_{{\rm {m}}}}{{\rho }_{{\rm {i}}}L}-\frac{{\theta }_{{\rm {i}}}}{{\theta }_{{\rm {i}}}+{\theta }_{{\rm {s}}}}{\upsilon }_{{\rm {i}}}{\dot{M}}_{\Phi \to \Pi {{\rm{i}}}}\\ {\left({\hat{h}}_{{{\rm{s}}}}\right)}_{t}+{{\rm {div}}}\left({\hat{h}}_{{{\rm{s}}}}{\vec{u}}_{\Phi }\right)=\frac{{\rho }_{\Sigma }}{{\rho }_{{\rm {s}}}}{\eta }_{{\rm {s}}}{\dot{M}}_{\Sigma \to \Phi }-\frac{{\dot{Q}}_{{\rm {m}}}}{{\rho }_{{\rm {s}}}L}-\frac{{\theta }_{{\rm {s}}}}{{\theta }_{{{\rm{i}}}}+{\theta }_{{\rm {s}}}}\frac{{\rho }_{{{\rm{i}}}}}{{\rho }_{\rm {{s}}}}{\upsilon }_{{\rm {i}}}{\dot{M}}_{\Phi \to \Pi {{\rm{i}}}}\\ {\left({\hat{h}}_{{{\rm{w}}}}\right)}_{t}+{{\rm {div}}}\left({\hat{h}}_{{{\rm{w}}}}{\vec{u}}_{\Phi }\right)=\frac{{\rho }_{\Sigma }}{{\rho }_{\rm {{w}}}}{\eta }_{{\rm {w}}}{\dot{M}}_{\Sigma \to \Phi }+\frac{{\dot{Q}}_{{\rm {m}}}}{{\rho }_{{\rm {w}}}L}\end{array}\right.$$
(6)
$${\left({\hat{h}}_{\Phi }{\vec{u}}_{\Phi }\right)}_{t}+{{\rm {div}}}\left({\hat{h}}_{\Phi }{\vec{u}}_{\Phi }\bigotimes {\vec{u}}_{\Phi }+{p}_{\Phi }I\right)=\vec{G}{\hat{h}}_{\Phi }-\frac{{\vec{u}}_{\Phi }}{{||}{\vec{u}}_{\Phi }{||}}{S}_{\Phi }-\frac{{\hat{\rho }}_{\Pi }}{{\hat{\rho }}_{\Phi }}{\dot{M}}_{\Phi \to \Pi }{\vec{u}}_{\Phi }$$
(7)
$${\left({\hat{h}}_{\Phi }{R}_{\Phi }\right)}_{t}+{{\rm {div}}}\left({R}_{\Phi }{{\hat{h}}_{\Phi }\vec{u}}_{\Phi }\right)={\alpha }_{\Phi }{\dot{W}}_{\Phi }-{\dot{M}}_{\Phi \to \Pi }{R}_{\Phi }-{\beta }_{\Phi }{\hat{h}}_{\Phi }{R}_{\Phi }$$
(8)
$$\left({{{\hat{h}}_{\Phi }E}_{\Phi }}\right)_{t} +{{\rm {div}}}\left({E}_{\Phi }{\hat{h}}_{\Phi }{\vec{u}}_{\Phi }\right)=\left[1-{\alpha }_{\Phi }\right]{\dot{W}}_{\Phi } \\ +{\beta }_{\Phi }{\hat{h}}_{\Phi }{R}_{\Phi }+{{\rho }_{\Sigma }}\left[c_{\Sigma }{T}_{\Sigma }+\frac{1}{2}||{\vec{u}}_{\Phi }||^{2}\right]{\dot{M}}_{\Sigma \to \Phi }{-\dot{Q}}_{{\rm {m}}}{-\dot{q}}_{\Phi \to \Lambda }$$
(9)
$${\left({m}_{\Phi }\right)}_{t}+{{\rm {div}}}\left({m}_{\Phi }{\vec{u}}_{\Phi }\right)=\frac{{\dot{Q}}_{{\rm {m}}}}{\left[{\rho }_{{\rm {w}}}L\right]}$$
(10)
$${\int }_{0}^{\triangle t}{\dot{Q}}_{{\rm {m}}}{{\rm {d}}t}={\rho }_{{\rm {w}}}{c}_{\Phi }{h}_{\Phi }\left[{T}_{\Phi }-{T}_{{\rm {m}}}\right]$$
(11)

Equations (4) and (7) represent the mass and momentum balances of the avalanche core, which involves the avalanche entrainment \({\dot{M}}_{\Sigma \to \Phi }\) and mass/momentum transfer to the cloud \({\dot{M}}_{\Phi \to \Pi }\). Equation (5) describes the dispersive movement of the avalanche core. The term \({\mathbb{D}}(t,{k}_{z},{\dot{k}}_{z},{\ddot{k}}_{z})\) represents the variation of core height due to dispersive pressure effects; we employ the model presented in Buser and Bartelt38. Equation (6) is the mass balance of each phase in the core, including rock, ice, snow, and water. The mass change of each phase arises from the entrainment process, air blast initiation, and phase change. Equations (8) and (9) show the balance of fluctuation energy and heat energy. Equation (10) describes the production and transport of meltwater in the avalanche core. The meltwater arises from both the ice melting and snow melting. The model assumes the mean core temperature \({T}_{\Phi }\) never exceeds the melting temperature of \({T}_{{\rm {m}}}\) until all the ice and snow melt. The heat energy applied for meltwater production is presented in Eq. (11). The parameters included in Eqs. (4)–(11) are well described by Bartelt et al.19 and Munch et al.39 and are listed in Table 1.

Table 1 Description of variables in the rock–ice avalanche-air blast model
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In the heat energy balance of Eq. (9), the heat transfer from avalanche to the ambient, entrained air is newly involved, defined as

$${\dot{q}}_{\Phi \to \Lambda }=H{A}_{{{\rm{s}}}}({T}_{\Phi }-{T}_{\Lambda })$$
(12)

where \({A}_{{{\rm{s}}}}=n\pi {r}^{2}\), \(n=\frac{{3\rho }_{\Phi }{h}_{\Phi }A}{4{\rho }_{g}\pi {r}^{3}}\) is the particle number, \({\rho }_{{{\rm{g}}}}\) is the particle density, r is the particle radius, \({T}_{\Phi }-{T}_{\Lambda }\) represents the temperature difference between the avalanche core \({T}_{\Phi }\) and the ambient air \({T}_{\Lambda }\). \(H=\frac{{{{\rm{Nu}}}}\cdot {k}_{{{\rm{a}}}}}{2r}\) is the experimentally based heat transfer coefficient33, \({k}_{{{\rm{a}}}}=\) 0.0257 W m−1 K−1 is the thermal conductivity, \({{{Nu}}}=2+0.6{{\mathrm{Re}}}^{\frac{1}{2}}{\Pr }^{\frac{1}{3}}\) is the dimensionless Nusselt number, \({Re}=\frac{2{ur}}{{\nu }_{{{\rm{a}}}}}\) is the Reynolds number, \(Pr =\frac{{c}_{{{\rm{a}}}}{\nu }_{{{\rm{a}}}}}{{k}_{{{\rm{a}}}}}\) is the Prandtl number of the air, \({c}_{{{\rm{a}}}}\) is the specific heat capacity of the air.

A fundamental feature of the avalanche core model is the partitioning of a primary dissipative process, shearing, into the production of heat \({E}_{\Phi }\) (internal energy) and non-directional kinetic energy \({R}_{\Phi }\) (granular temperature). The work done by shearing \({\dot{W}}_{\Phi }={S}_{\Phi }{{\rm{||}}}{\vec{u}}_{\Phi }{{\rm{||}}}\) is divided into microscopic and macroscopic random energy (heat and granular fluctuations) by parameter \({\alpha }_{\Phi }\) 38. Shearing is controlled by the process-based Voellmy-type rheology40:

$${S}_{\Phi }={\mu }_{\Phi }({R}_{\Phi },{T}_{\Phi },{m}_{{{\rm{w}}}}){\rho }_{\Phi }{gh}\cos \theta +{\rho }_{\Phi }g\frac{{u}_{\Phi }^{2}}{{\xi }_{\Phi }({R}_{\Phi },\,{T}_{\Phi },\,{m}_{{{\rm{w}}}})}$$
(13)

where (\({\mu }_{\Phi }\), \({\xi }_{\Phi }\)) are the Coulomb and turbulent friction coefficients, respectively, defined as functions of the fluctuation energy \({R}_{\Phi }\), temperature \({T}_{\Phi }\) and water content \({m}_{{{\rm{w}}}}\). The flow friction \({S}_{\Phi }\) is dependent on the dispersive properties of the random energy \({R}_{\Phi }\), avalanche mobility, and, therefore, the formation of the powder cloud is strongly influenced by the shearing process.

Powder cloud Π

A similar set of partial differential equations is proposed to describe the movement of the powder cloud Π. To track the mass changes of different materials in the cloud, an improvement of the model is to suggest the powder cloud as a mixture of a rock powder cloud \({\Pi }_{{{\rm{r}}}}\) and an ice powder cloud \({\Pi }_{{\rm{{i}}}}\). Therefore, the cloud density depends on the volumetric ratio of rock and ice within the cloud. We begin by presenting the mass balance equations:

$${\left({\hat{h}}_{\Pi {{\rm{r}}}}\right)}_{t}+{{{\rm{div}}}}\left({\hat{h}}_{\Pi {{\rm{r}}}}{\vec{u}}_{\Phi }\right)={\dot{M}}_{\Phi \to \Pi {{\rm{r}}}}$$
(14)
$${\left({\hat{h}}_{\Pi {{\rm{i}}}}\right)}_{t}+{{{\rm{div}}}}\left({\hat{h}}_{\Pi {{\rm{i}}}}{\vec{u}}_{\Phi }\right)={\dot{M}}_{\Phi \to \Pi {{\rm{i}}}}$$
(15)
$${\left({\hat{h}}_{\Pi }\right)}_{t}+{{{\rm{div}}}}\left({\hat{h}}_{\Pi }{\vec{u}}_{\Phi }\right)={\frac{{\hat{\rho }}_{\Pi {{\rm{r}}}}}{{\hat{\rho }}_{\Pi }}\dot{M}}_{\Phi \to \Pi {{\rm{r}}}}+\frac{{\hat{\rho }}_{\Pi {{\rm{i}}}}}{{\hat{\rho }}_{\Pi }}{\dot{M}}_{\Phi \to \Pi {{\rm{i}}}}$$
(16)
$${\left({h}_{\Pi }\right)}_{t}+{{{\rm{div}}}}\left({h}_{\Pi }{\vec{u}}_{\Pi }\right)={\frac{{\rho }_{{{\rm{r}}}}-{\hat{\rho }}_{\Pi {{\rm{r}}}}}{{\rho }_{{{\rm{r}}}}-{\rho }_{\Lambda }}\dot{M}}_{\Phi \to \Pi {{\rm{r}}}}+{\frac{{\rho }_{{{\rm{i}}}}-{\hat{\rho }}_{\Pi {{\rm{i}}}}}{{\rho }_{{{\rm{i}}}}-{\rho }_{\Lambda }}\dot{M}}_{\Phi \to \Pi {{\rm{i}}}}+{\dot{M}}_{\Lambda \to \Pi }$$
(17)

Equations (14) and (15) represent the mass balance of the rock powder cloud \({\Pi }_{{{\rm{r}}}}\) and ice powder cloud \({\Pi }_{{{\rm{i}}}}\), respectively, and Eq. (16) described the mass balance of the total powder cloud Π. Similar to the core, (\({\hat{h}}_{\Pi {{\rm{r}}}}\), \({\hat{h}}_{\Pi {{\rm{i}}}}\), \({\hat{h}}_{\Pi }\)) represents the initial height of \({\Pi }_{{{\rm{r}}}}\), \({\Pi }_{{{\rm{i}}}}\) and Π, respectively, and are given by the initial cloud density (\({\hat{\rho }}_{\Pi {{\rm{r}}}}\), \({\hat{\rho }}_{\Pi {{\rm{i}}}}\) and \({\hat{\rho }}_{\Pi }\)) before blowing out from the avalanche core \(\Phi\). The true cloud height \({h}_{\Pi }\) is affected by clouds ejected from the core (\({\dot{M}}_{\Phi \to \Pi {{\rm{r}}}}\), \({\dot{M}}_{\Phi \to \Pi {{\rm{i}}}}\)) and ambient air entrainment \({\dot{M}}_{\Lambda \to \Pi }\). Due to this air entrainment, the cloud height increases during the propagation process and the density decreases to \({\rho }_{\Pi }\), satisfying \({\rho }_{\Pi }={\rho }_{{{\rm{i}}}}\frac{{\nu }_{i}{\hat{h}}_{\Pi {{\rm{i}}}}}{{h}_{\Pi }+{\nu }_{i}{\hat{h}}_{\Pi {{\rm{i}}}}+{\nu }_{r}{\hat{h}}_{\Pi {{\rm{r}}}}}+{\rho }_{{{\rm{r}}}}\frac{{\nu }_{r}{\hat{h}}_{\Pi {{\rm{r}}}}}{{h}_{\Pi }+{\nu }_{i}{\hat{h}}_{\Pi {{\rm{i}}}}+{\nu }_{r}{\hat{h}}_{\Pi {{\rm{r}}}}}+{\rho }_{\Lambda }\frac{{h}_{\Pi }}{{h}_{\Pi }+{\nu }_{i}{\hat{h}}_{\Pi {{\rm{i}}}}+{\nu }_{r}{\hat{h}}_{\Pi {{\rm{r}}}}}\), \({\rho }_{{{\rm{i}}}}\) = 971 kg m−3 is the ice density, \({\rho }_{{{\rm{r}}}}\) = 2500 kg m−3 is the rock density, \({\rho }_{\Lambda }\) = 1.225 kg m−3 is the air density (\({\nu }_{{{\rm{i}}}}=\frac{{\hat{\rho }}_{\Pi {{\rm{i}}}}-{\rho }_{\Lambda }}{{\rho }_{{{\rm{i}}}}-{\rho }_{\Lambda }}\), \({\nu }_{{{\rm{r}}}}\) = \(\frac{{\hat{\rho }}_{\Pi {{\rm{r}}}}-{\rho }_{\Lambda }}{{\rho }_{{{\rm{r}}}}-{\rho }_{\Lambda }}\)) represents the solid fraction in the initial ice and rock powder cloud.

The momentum balance of the powder cloud is

$${\left({\hat{h}}_{\Pi }{\vec{u}}_{\Pi }\right)}_{t}+{{{\rm{div}}}}\left({\hat{h}}_{\Pi }{\vec{u}}_{\Pi }\bigotimes {\vec{u}}_{\Pi }+{p}_{\Pi }I\right)={\dot{M}}_{\Phi \to \Pi }{\vec{u}}_{\Phi }+\frac{\left({\hat{\rho }}_{\Pi }-{\rho }_{\Lambda }\right)}{{\hat{\rho }}_{\Pi }}\vec{G}{\hat{h}}_{\Pi }-\frac{{\vec{u}}_{\Pi }}{{||}{\vec{u}}_{\Pi }{||}}{S}_{\Pi }$$
(18)

The mixed ice/rock powder cloud moves with a mean velocity vector \({\vec{u}}_{\Pi }\). The cloud is driven by the initial momentum transferred from the avalanche core \({\dot{M}}_{\Phi \to \Pi }{\vec{u}}_{\Phi }\) and the gravity \(\frac{\left({\rho }_{\Pi }-{\rho }_{\Lambda }\right)}{{\rho }_{\Pi }}\vec{G}\).

Another important improvement of the proposed model is the inclusion of turbulence. The instantaneous air blast velocity \({\widetilde{u}}_{\Pi }\) is written as the sum of a mean \({\vec{u}}_{\Pi }\) and a fluctuating component \({\vec{u}{\prime} }_{\Pi }\). The energy associated with the fluctuation of the granules \({R}_{\Pi }\left(x,y,z,t\right)\) can be written as41

$${R}_{\Pi }\left(x,y,z,t\right)=\frac{{\hat{\rho }}_{\Pi }}{2}\left[{{{u}_{x}}^{{\prime} }}^{2}\left(t\right)+{{{u}_{y}}^{{\prime} }}^{2}\left(t\right)+{{{u}_{z}}^{{\prime} }}^{2}\left(t\right)\right]=\frac{{\hat{\rho }}_{\Pi }}{2}{{u{\prime} }^{2}}_{\Pi }(t)$$
(19)

The fluctuation energy is divided into three orthogonal components in the x, y, z directions. Here the velocity fluctuation is assumed to be isotropic, and thus \({R}_{\Pi ,x}={R}_{\Pi ,y}={R}_{\Pi ,z}=\frac{1}{3}{R}_{\Pi }\). The balance equation of the fluctuation energy can be written as:

$${\left({\hat{h}}_{\Pi }{R}_{\Pi }\right)}_{t}+{{{\rm{div}}}}\left({R}_{\Pi }{\hat{h}}_{\Pi }{\vec{u}}_{\Pi }\right)={\dot{M}}_{\Phi \to \Pi }{R}_{\Phi }+{\dot{W}}_{\Pi }+\frac{1}{2}{\rho }_{\Lambda }{\dot{M}}_{\Lambda \to \Pi }{u}_{\Pi }^{2}-{\beta }_{\Pi }{\hat{h}}_{\Pi }{R}_{\Pi }$$
(20)

We suggest the fluctuation energy has three sources: fluctuation energy that is created in the avalanche core and transported to the cloud \({\dot{M}}_{\Phi \to \Pi }{R}_{\Phi }\), internal shearing \({\dot{W}}_{\Pi }=\left[{S}_{\Pi }\right]{{\rm{||}}}{\vec{u}}_{\Pi }{{\rm{||}}}\) and air entrainment \(\frac{1}{2}{\rho }_{\Lambda }{\dot{M}}_{\Lambda \to \Pi }{u}_{\Pi }^{2}\). The fluctuation energy \({R}_{\Pi }\) of the cloud has a short lifetime and its dissipation is controlled by a decay coefficient \({\beta }_{\Pi }\) 15.

In the air blast model, the air entrainment \({\dot{M}}_{\Lambda \to \Pi }\) and drag resistance \({S}_{\Pi }\) are defined as a contribution of both laminar and turbulent parts. Air entrainment is suggested as a function of the turbulent velocity42,43, the square root of the turbulent energy \({R}_{\Pi }\): \({\dot{M}}_{\Lambda \to \Pi }={\alpha }_{{{\rm{L}}}}\left({\rho }_{\Pi }-{\rho }_{\Lambda }\right)+{\alpha }_{{{\rm{T}}}}\sqrt{{R}_{\Pi }{\hat{h}}_{\Pi }}\left({\rho }_{\Pi }-{\rho }_{\Lambda }\right)\). The drag resistance is suggested as a direct function of the average velocity and turbulent energy: \({S}_{\Pi }={\rho }_{\Pi }({\mu }_{{{\rm{L}}}}{u}_{\Pi }+{\mu }_{{{\rm{T}}}}{R}_{\Pi }{\hat{h}}_{\Pi })\). The parameters (\({\alpha }_{{\rm{{L}}}}\), \({\alpha }_{{{\rm{T}}}}\)) and (\({\mu }_{{\rm{{L}}}}\), \({\mu }_{{{\rm{T}}}}\)) are sets of laminar/turbulent parameters controlling air entrainment and drag resistance. These parameters, along with the turbulent decay parameter \({\beta }_{\Pi }\) control the magnitude of the avalanche air blast.

The vertical profiles of the velocity, density, and dynamic pressure, which are greatly influential to assess the air blast hazard, refer to Zhuang et al.15. The velocity profile follows a parabolic form and is determined using the boundary turbulent velocity values and the mean value. The density profile follows a linear profile decreasing from the bottom to the top of the cloud. Therefore, the vertical profile of the total pressure is written as \(P\left(z\right)=\frac{1}{2}\cdot {\rho }_{\Pi }\left(z\right)\cdot {{u}_{\Pi }\left(z\right)}^{2}=\frac{1}{2}\cdot {\rho }_{\Pi }\left(z\right)\cdot {[{\bar{u}}_{\Pi }(z)+{u{\prime} }_{\Pi }\left(z\right)]}^{2}\). Here, we assume the worst case is that the fluctuation velocity is always in the same direction as the laminar mean velocity. In this study, we applied DEM arising from SPOT satellite images to do the simulation, referring to Ragettli et al.44.

Tree-breakage calculation

The assessment of air blast-induced tree breakage follows the method proposed by Feistl et al.27. The tree bending stress arising from the air blast loading is

$${\sigma }_{\Pi }={c}_{\Pi }{\rho }_{\Pi }\frac{8{\widetilde{u}}_{\Pi }^{2}}{\pi {d}^{3}}w{H}^{2}\cos \gamma$$
(21)

where \({c}_{\Pi }\) is the drag coefficient, \({\widetilde{u}}_{\Pi }\) is the instantaneous air blast velocity, which is the sum of the mean velocity \({\vec{u}}_{\Pi }\) and turbulent velocity \(u{\prime}\), \({\rho }_{\Pi }\) is the cloud density, (\(w\), \(H\), \(d\)) represents the stem diameter, height and effective crown width of trees, respectively, \(\gamma\) is slope angle. The tree breakage occurs when the bending stress \({\sigma }_{\Pi }\) exceeds the experienced tree strength.