Variations in runout of rock avalanches controlled by fragmentation, not basal friction

Rock avalanches are large rockslides consisting of highly fragmented materials that display exceptionally long runouts, which are found to correlate with their volume. Such volume-dependent runouts are conventionally attributed to dynamic lowering of the effective basal friction. However, even for similar volumes, the runouts are seen to span several orders of magnitude suggesting additional controlling factors. Here, we perform analogue models of fragmenting rockslides and compare them to natural rock avalanches. We show that for a given low basal friction, the runout of rock avalanches varies over 5 two orders of magnitude and is determined by their degree of fragmentation. The fragmentation is observed to cause spreading, but also to increased mechanical interactions between fragments. Consequently, the runout’s dependence on fragmentation appears to be determined by the competition between spreading and internal friction. This shows that variation in degree of fragmentation can explain the large variation of runout of rock avalanches. Copyright statement. TEXT 10


Introduction
With volumes larger than 10 6 m 3 , and speeds reported at over 150 km/h (Campbell, 1989), the destructive power of rock avalanches is unprecedented. They are exceptional hazards produced when very large rockslides disintegrate during transport (Hungr et al., 2013). The travel distance of the deposit front, or runout, is an important measure for their hazard assessment (Vaunat and Leroueil, 2002) and is generally found to be longer than their fall height (Hsü, 1975). This suggests low effective 15 basal frictions, which is usually attributed to a reduced normal stress at the base, for which various mechanisms have been proposed (e.g. Kent, 1966;Shreve, 1968;Hsü, 1975;Melosh, 1979;Campbell, 1989). For convenience, the reduction in basal friction can be considered through an effective coefficient of friction, µ ef f , which emerges naturally when considering energy conservation of the simple case of a rigid block of mass M sliding down a planar slope where H and L is the vertical and horizontal displacement of the block, respectively. Unlike a block, however, a rock avalan-20 che may deform and spread. Additionally, field observations of the displacement of rock avalanches are often given by vertical (H) and horizontal (L) distance from the deposit's front to the top of the main scarp. The resulting ratio is known as the Heim's ratio (Heim, 1882, as cited in Hsü, 1975, or the apparent coefficient of friction (Manzella and Labiouse, 2012), and serves as a proxy for µ ef f .
One of the best established, but perhaps least understood observations of rock avalanches, is the dependence of the Heim's 25 ratio on volume: rockslides below a size of approximately 10 6 m 3 all have a relatively constant Heim's ratio of ∼0.4-0.7, but for larger rockslides it decreases with volume, reaching values <0.1 for volumes larger than 10 9 m 3 (Lucas et al., 2014). This suggests a scale dependent mechanism of decreasing apparent friction with volume that is only relevant for large volumes (Davies and McSaveney, 1999). Whether the effective friction also depends on volume is uncertain, though analysis by Lucas et al. (2014) suggests that this is the case.

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The uncertainty lies in the fact that the runout is defined by the front of the deposits, and therefore contains the combined effect of both translation and spreading of the rock mass. The additional travel distance caused by spreading can have a profound effect on the runout (Staron and Lajeunesse, 2009), especially if the effective basal friction is low. Though often not considered, the process of fragmentation can play a crucial role in the spreading of rock avalanches (Bowman et al., 2012).
Firstly, one may expect that the finer the material, the more flow-like the behavior, allowing the rock mass to spread more 35 easily (Locat et al., 2006). Secondly, models of fragmenting rockslides suggest that dynamic fragmentation actively increases the spreading (Bowman et al., 2012;De Blasio and Crosta, 2015). However, fragmentation has also been shown to consume energy (Haug et al., 2016;Zhao et al., 2017), potentially at a cost to the runout length. The effect of fragmentation on the spreading, and therefore on the final runout of rock avalanches remains largely unknown.
We perform analogue models of dynamically fragmenting rock avalanches ( Figure 1). While we do not know its origin, we 40 hypothesize that there exists some mechanism that causes a low, but constant effective coefficient of friction. This entails that any variation in Heim's ratio is the result of variations in the degree of fragmentation. We describe the dependence observed between the runout and the degree of fragmentation in the form of a scaling law. Finally, we compare our experimental results to a set of natural Data and discuss their relevance to natural systems. All data underlying this study as well as additional data will be published open access in Haug et al. (2020).

Methods
In the experiments, a block of height h and length l 0 (width = l 0 ) of rock analogue material is accelerated down a plate held at an angle of 45 • to the horizontal (Figures 1). After 1 m of travel, the sample impacts a horizontal plate causing it to fragment. Once the sample has moved onto the horizontal plate, the fragmented material spreads and finally comes to rest. We expect the effective friction coefficient to always be larger than the apparent one, i.e. µ ef f > µ ap . Since the apparent friction 0.35 m, and data is captured by a camera with a frame rate of 500 Hz (see Haug et al., 2020, for movies of these experiments).

Observations
The Heim's ratio for experiments with different degrees of fragmentation (m c ) is plotted in Figure 2a. The two series of experiments follow the same trend (Haug et al., 2016), and no distinction is made between them in this figure. It shows that the Heim's ratio decreases rapidly for low to intermediate degrees of fragmentation, reaching a minimum at m c ≈ 5 of about 0.2.

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With further increase of fragmentation a gentle increase in the Heim's ratio is observed. A similar behavior is observed for the length of the deposits (Figure 2b), which increase until m c ≈ 5 before it decreases for higher values. Figure 3 presents snapshots from the two additional experiments captured at a higher temporal resolution. As expected, the sample with the highest cohesion ( Figure 3a) is observed to fragment less than the one with a lower cohesion (Figure 3b).
However, these images also reveal that the fragments of the stronger sample spreads with limited interaction between the 80 different fragments, whereas the fragments from the weaker sample collide and slide next to each other. This shows that a higher amount of internal deformation is experienced with increased fragmentation.

Interpretation of experimental results
The rapidly decreasing Heims's ratio for m c < 5 observed in Figure  . However, here we show that the Heim's ratio is not simply decreasing with the degree of fragmentation, but that it displays an optimum for m c ≈ 5. Importantly, the lowest apparent basal friction, equivalent to the lowest Heim's ratio, is close to the implemented basal friction (i.e. friction coefficient of 0.15 between samples and glass). Therefore, all processes operating in our models (e.g. fragmentation, internal 90 friction between fragments) tend to consume energy and thereby reduce runout from its optimum (Haug et al., 2016). Considering the increased internal deformation observed with the degree of fragmentation (Figure 3), the reduction of runout with m c > 5 appears to be the result of the increased energy dissipation through internal friction. Consequently, the minimum of the Heim's ratio observed in Figure 2a is the result of a competition between spreading and internal friction.
This competition can be formalized into a scaling law by considering the conservation of energy (see Appendix A) where α, β, and γ are constants. This equation illustrates the competition between spreading (βe −mc/γ ) and the increasing energy dissipation (αlog(m c )) with m c . A best fit of this function is presented in Figure 2a

Comparison to a natural data set
We compare our experimental results ( Figure 2) with data from nine rock avalanches reported by Locat et al. (2006), which show no clear volume dependence of runout. This makes this data set ideal to test whether a scale-independent process is operating besides dynamic basal weakening. However, not all the rock avalanches reported in (Locat et al., 2006) are comparable to our experimental setup by means of material properties and geometries (Figure 1). Based on slope geometry, the Queen 105 Elizabeth slide is discarded because of its run-up on the opposite valley wall. Also discarded is the Charmonétier slide because of the sudden free fall stage at the end of its transport. Additionally, the Arvel slide was observed to bulldoze soft material in front of it, and such complexities are not considered in our models so this one is also neglected.  (Figure 2b) suggests that the reduction of spreading indicated by the experiments for m c > 5 continues for even higher degrees of fragmentation. The Heim's ratios of the neglected slides are all, as expected, higher than the selected data set for their respective degrees of fragmentation, illustrating the importance of topography (e.g. opposite valley wall) and processes such as bulldozing.
The similarity seen between experimental and natural data suggests that the rock avalanches considered here have a close to constant effective friction of about 0.15. This implies that over a range of two orders of magnitude (from 2 · 10 6 to 90 · 10 6 m 3 ), the effective coefficient of friction of rock avalanches is independent of volume. Consequently, our results suggest that the 120 variation seen in Heim's ratio for these rock avalanches are not caused by a different basal friction, but by differing degrees of fragmentation. This shows that fragmentation plays a governing role in the runout of rock avalanches and should be included in hazard assessments.
Data availability. The data for this paper will be available as an open access data publication (Haug et al., 2020).
Video supplement. Videos for this paper will be available as an open access data publication (Haug et al., 2020).

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Appendix A: A mathematical description of the dependence on fragmentation of Heim's ratio Generally, the conservation of energy of a sliding mass M requires that where g is the gravitational acceleration, H is the vertical fall height, L p is the entire travel path of the slide and W is the sum of any other energy dissipating terms. Here, we have assumed Coulomb friction at the base. For the geometry of our experimental setup (i.e. Figure 1), and also roughly for the set of selected rock avalanches, the L p can be expressed in terms of 130 the horizontal runout L as where L s is the length and θ the angle of the slope, and l 0 is the initial length of the slide. It is assumed that the additional travel length due to spreading is equal to half the deposit length (L spread ). Since l 0 is expected to be very small compared to the other terms, it is neglected in the further analysis. Inserting this expression into Equation A1 and solving for L gives where it is emphasized that both L spread and W is expected to be functions of the basal friction, µ, internal friction, φ, the 135 degree of fragmentation, m c , as well as a possible non-linear dependence between L spread and W . With this expression for the runout, the Heim's ratio is suggests that W/M gh can be described with a logarithmic function of m c . Additionally, based on the shape of both the Heim' ratio and the L spread plotted in Figure 2, it appears that it can be reasonably described by an exponential function of m c . This 140 leads to the approximation that such that the Heim's ratio can be expressed as Author contributions. TEXT

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