Snow avalanches break, uproot and overturn trees causing damage to forests.
The extent of forest damage provides useful information on avalanche
frequency and intensity. However, impact forces depend on avalanche flow
regime. In this paper, we define avalanche loading cases representing four
different avalanche flow regimes:

Forest damage caused by avalanches reveals the
complex and variable nature of avalanche flow. Avalanches cut through forests
leaving paths of broken and fractured tree stems, overturned root plates and
torn branches

The protective capacity of forests depends on the ability of trees and tree
clusters to survive avalanche loading. Forests can stop small and frequent
avalanches because the trees are not destroyed

Destroyed and surviving trees after avalanche impact in Täsch.

The tree-breaking threshold depends on both the avalanche loading and tree
strength. Trees fall if (1) the bending stress exerted by the moving snow
exceeds the bending strength of the tree stem

The primary goal of this paper is to introduce tree breaking into avalanche
dynamics calculations in order to toggle between different frictional
processes acting on the avalanche. We therefore define four loading cases
representing four different avalanche flow regimes. These are

Avalanches exert a pressure

A dry avalanche is divided into a dense flowing core (subscript

Denotation for the different flow regimes:

Schematic illustration of an avalanche with velocity

A wet snow avalanche contains only a dense flowing core and exerts a pressure

Denotation to calculate the loading.

Because trees grow vertically, and the avalanche applies pressure in the slope
parallel flow direction, the force

The total force

The bending stress is calculated from the maximum torque, which is located at
the stem base. We assume a fixed support and a homogenous distribution of
mass and velocity (see Fig.

Various studies have investigated avalanche impact pressures on obstacles

The pressure formula Eq. (

The pressure exerted by the powder cloud is

The loading width of the tree depends on the location in the forest.
In dense forest stands tree crowns tend to be narrower than if they stand
alone. Additionally the loading width

The total force

The bending stress of a powder cloud is then

The pressure exerted by an individual snow granule is

The average density

We solve this equation for

The pressure per unit area that a dense flowing avalanche exerts on a tree is
calculated similarly to the powder cloud loading except that now we consider
the avalanche core

Granules of the intermittent layer with density

We apply the CPM developed by

A second method to calculate glide snow pressure was developed by

The quasi-static pressure of the avalanche in this case is therefore

Schematic illustration of a wet snow accumulation behind a tree. The
volume of the snow depends on the opening angle

The bending stress equation for powder avalanches

The avalanche exerts the pressure not only on the stem but also on
low branches. Trees at the stand edge usually have branches close to the
ground surface as light conditions are favorable. This effect is especially
pronounced for evergreen trees such as spruces. We therefore assume that the
stem diameter is a poor measure for the effective width at stand edge. For
trees with low-lying branches at the forest edge, we assume a magnification
factor between 1.5 (leafless trees) and 2.5 (evergreen trees),

Woody debris carried by the avalanche increases the impact pressure when
hitting a tree. Similar to the additional point load exerted by single snow
granules

The self weight of the tree when bending just before breaking is an
additional load that increases the bending stress on the lower parts of the
stem. This effect can even be higher if snow is loaded on branches and
increases the tree mass. Second-order bending effects are thereby introduced
into the problem

For flexible structures such as trees, the inertial response of the tree
must be considered

We assumed in all four flow regimes that the avalanche flows close to
the ground. However, an avalanche flowing over a deep snow cover will hit the
tree higher up the stem. This effect can be included in the analysis by
simply increasing the moment arm of applied force. Whereas the increase in
momentum can be high at stand edges where deep snow covers occur, the effect
is negligible where dense canopy suppresses snow accumulation on the ground.
The increase of the exerted pressure is 50 % if an avalanche with flow
height

Bending strength of different tree species according to

Trees break if the bending stress exerted by the avalanche exceeds the
bending strength of the tree

We only calculate the stress that is sufficient to break trees not to uproot
trees. Previous studies proved that pressures required for uprooting are in
the same range or higher than for stem breakage

To test the performance of Eq. (

Bending stresses exerted on trees of specific forest areas are calculated for
each grid cell from flow height, density, velocity and slope angle.
Additionally for each forest area an average stem diameter has to be
specified. The bending strength of the predominant tree species is taken from
literature (Table

We applied the new model approach to back calculate avalanche events with forest damage in Switzerland and Germany. Calculated forest damage was compared to the actual observed damage. Stem diameters in 1.3 m height, tree species and exact tree location were documented for two avalanche events with forest destruction in Monbiel and Täsch, Switzerland. In total we documented 324 broken, 173 uprooted trees and 710 trees that withstood the avalanche loading. Six avalanches with forest destruction, that released in an avalanche cycle in winter 2009 in Germany were additionally simulated to test on the model performance.

A large wet snow avalanche (release volume approx. 150 000

Simulated dynamic impact pressure ^{©}, DV 033594, 2014). Note that the
calculated velocity and runout distance agreed well with the observations.

Bending stress

The calculated forest destruction with the new RAMMS module (red
areas) in comparison to the observed forest stand (blue and yellow dots) and
deposition area in Täsch. The simulated extent of the avalanche is
underlaid with white. We chose the following parameter values to calculate
this avalanche: ^{©}, DV 033594, 2014).

Simulation results for the dense flowing core for six avalanches
with forest damage in Germany. The avalanches released at Wendelstein (1),
Ahornalpe (2), Hochries (3), Spinnergraben (4), Frillensee (5) and
Zwillingswand (6). In ^{©} Bayerische
Vermessungsverwaltung, Bayerisches Landesamt für Umwelt).

A cold powder snow avalanche released on 4 March 2014 in Täsch in Wallis,
Switzerland. The road and the rail tracks to Zermatt were buried in deep
snow. Aerial pictures and the insight of our visit to the site the next day
allowed us to reconstruct the avalanche release volume (approx.
80 000

Approximately 10 m high deflecting dams were built along the north side of
the avalanche path to protect the village Täsch from being hit by extreme
avalanche events. These dams worked well in the upper part of the avalanche
path where the avalanche hardly overflew the dam. In the lower part, however,
the dam lost its deflecting effect because of avalanche depositions of
earlier events. The channel was almost filled up to the dam crown, and the
avalanche from 4 March partly went straight down the slope. A young forest,
consisting mainly of birches and larches was destroyed, and the avalanche hit
the road north of the gallery, went through the river bed and hit the rail
tracks on the other side (see Fig.

Velocities, flow height and the powder cloud diffusion were modeled with the
extended RAMMS version. We assume snow entrainment along the track, with
0.5 m entrained snow in an elevation of 2500 m decreasing by 10 cm every
100-height meters. The velocity driven entrainment law was applied with

We calculated the forest destruction and compared the results with the actual
observed tree damage (Fig.

We simulated six avalanches that released in Germany end of February 2009 and
tested the new model approach (Fig.

Using the standard dynamic calculation formula for

The avalanches in Täsch and Germany consisted of cold snow and were
accompanied by large powder clouds. At Täsch the powder cloud was higher
than the trees (see Fig.

For the Monbiel case study, the standard dynamic pressure formula
underestimated the forest damage over the entire slope, including the
transition and runout zones. We therefore tested the three proposed
approaches to calculate the wet snow pressure on the trees: (1) the sliding
block model (SBM),

To investigate how snow avalanches destroy forests, we developed
four flow regime dependent impact formulas. These are

Dry, powder-snow avalanches exert dynamic pressures on the tree stem and the crown. Although the applied impact pressures can be small (less than 3 kPa), bending stresses in the stem can be large due to the torque action of the blast. The impact pressure, cloud height and impact area must be taken into account to predict forest destruction. Destructive bending stresses can easily be reached even if the density of the snow-air mixture is low. The destructive potential depends on the crown area that is affected by the snow blast and not only on the velocity and density of the powder cloud. The crown area varies with tree position in the forest and on the foliation. Single evergreen trees are exposed to the full avalanche blast and bending stresses are higher in comparison to leafless trees sheltered in clustered forest stands. The presence of deciduous conifer (larch) and broadleaf (birch) tress in an avalanche track is a possible indication of powder snow activity. These tree types can survive powder avalanche blasts because their effective crown areas are small, in contrast to evergreen trees.

The impact pressure formula for the intermittent case was derived by considering individual granule impacts. Interestingly, if we assume a homogeneous velocity and density distribution in the avalanche core, the formula is the same as the standard impact pressure formula used in practice. The density of the flow, however, is considerably smaller in the intermittent case. Therefore, the intermittent case seldom provides pressures higher than the dense case for dry avalanches. However, the intermittent impact formula should be modified to include the effect of particle clusters and uneven velocity distributions. This case could also be modified to include impacts caused by entrained woody debris or rocks. More real examples where these effects could be documented are needed to test a modified formula.

Destructive pressures of wet snow avalanches were back-calculated using two
quasi-static modeling approaches, the “sliding block model” (SBM) and the
“creep pressure model” (CPM). The results were compared to the standard
dynamic approach used for dry snow avalanches. Dynamic models severely
underestimated the applied pressures and cannot reproduce the bending
stresses required for tree breakage. The sliding block model provided the
highest bending stresses and the more realistic results. The creep pressure
model underestimated the applied loading in the Monbiel case study. To apply
the sliding block model, engineers must estimate the avalanche volume length

A primary goal of our work is to underscore the importance of field surveys to document forest damage by avalanches. These surveys can provide valuable information on avalanche characteristics and return periods that are needed to formulate hazard scenarios. Broken trees serve as valuable sensors that record avalanche intensity. Our results, however, indicate that impact pressures on trees depend on the avalanche flow regime. For wet snow avalanches, the pressure additionally depends on the distribution of trees in the forest and on terrain features. By differentiating between four different avalanche flow regimes, a more inclusive and reliable analysis of forest damage is possible.

This work was funded by the Bavarian Environment Agency. We would especially like to thank Armin Fischer from the Bavarian avalanche service who provided the data of the avalanche events in Bavaria. Leo Jörger and Christian Rüsch from the forest department in Randa and Klosters provided us with valuable information on the forest cover in Monbiel and Täsch and made the field work possible. Edited by: M. Parise Reviewed by: three anonymous referees