Journal Article
Spatial impulse waves: wave height decay experiments at
laboratory scale
Author(s):
Evers, Frederic; Hager, Willi H.
Publication Date:
2016-12
Permanent Link:
https://doi.org/10.3929/ethz-b-000119161
Originally published in:
Landslides 13(6), http://doi.org/10.1007/s10346-016-0719-1
Rights / License:
In Copyright - Non-Commercial Use Permitted
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ETH Library
1
Spatial impulse waves: Wave height decay experiments at laboratory
scale
Frederic M. Evers1 and Willi H. Hager2
Keywords: Displacement wave; Impulse wave; Subaerial landslide; Physical modelling;
Prediction method; Spatial wave propagation.
Abstract
Impulse waves generated by rapid subaerial landslides into water bodies may pose a threat to riparian settlements and infrastructure. Empirically derived prediction equations based on experiments at laboratory scale provide information on key wave characteristics for preliminary hazard assessment. This research discusses existing prediction methods for spatial wave propagation features and compares their results with own impulse wave height decay experiments. While some prediction methods are based on simplified approaches for wave generation such as rigid body slides, others take only limited sets of slide parameters into account, narrowing their range of applicability considerably. The prediction methods are intentionally applied outside their ranges of applicability with the aim to assess their characteristics on an extended parameter range. It is found that a combination of separate terms for wave generation and wave propagation from two different existing prediction methods provides the best representation of the experimental data.
Introduction
Mass wasting events such as extremely rapid subaerial landslides may generate large water waves not only in inland waters including lakes or reservoirs but also along coastlines as well as in bays and fjords. These impulse waves, informally also referred to as mega-tsunamis, involve long-wave characteristics and may run up the shoreline or overtop a dam, thereby endangering adjacent settlements and infrastructure. There is a multitude of accounts of impulse wave events with spatial propagation patterns generated by subaerial landslides. One of the most prominent cases is the Lituya Bay, USA, impulse wave triggered by a rockslide after an earthquake in 1958.
Run-up heights of more than 500 m were recorded at the shoreline opposite the rockslide (Miller 1960; Slingerland and Voight 1979). Further examples in coastal areas include, e.g., the impulse waves generated by a historical rockslide at Knight Inlet, Canada, presumably destroying an indigenous settlement in the sixteenth century (Bornhold et al. 2007) or several earthquake-triggered landslides into Aysén Fjord, Chile, in 2007 (Sepúlveda et al. 2010). Inland waters such as natural lakes surrounded by steep mountains were also subject to landslide tsunamis: In 1971, a rock avalanche caused a catastrophic event at Lake Yanahuin, Peru, with reportedly 400 to 600 fatalities (Plafker and Eyzaguirre 1979); in 1946, a rock avalanche led to an overspill of Landslide Lake beneath Mount Colonel Foster, Canada, inducing a destructive downstream flash flood (Evans 1989); and in 2007, Chehalis Lake and River, Canada, were affected by extensive shoreline damage several kilometers away from the impact location of the landslide (Roberts et al. 2013). Regarding artificial water bodies such as reservoirs, a recent
1 Doctoral Researcher, Laboratory of Hydraulics Hydrology and Glaciology (VAW), ETH Zurich, CH-8093 Zürich, Switzerland, +41 44 633 0877, evers@vaw.baug.ethz.ch
2 Professor, Laboratory of Hydraulics Hydrology and Glaciology (VAW), ETH Zurich, CH-8093 Zürich, Switzerland, +41 44 632 4149, hager@vaw.baug.ethz.ch
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impulse wave event occurred at the Three Gorges Reservoir, China, in 2008 (Huang et al. 2012). Roberts et al.
(2014) compiled a global overview of impulse waves generated by subaerial landslides with 254 registered events in total. Even if the volume of water directly displaced by a slide is comparatively small, it might have a triggering effect on far more devastating processes: glacial lake outburst floods (GLOF) can be released after an impulse wave has substantially eroded the embankment of a moraine-dammed proglacial lake (Richardson and Reynolds 2000). In 1985, the moraine dam of Dig Tsho, Nepal, was eroded due to overtopping after an ice avalanche; within 4 to 5 h, 5 million m3 drained through a breach of 60 m height and 200 m width at an estimated peak discharge of 1600 m3/s causing severe damages in the valley downstream (Vuichard and Zimmermann 1987). Schaub et al.
(2013) predict a future increase of these events in the course of deglaciation.
Starting from a limited slide impact zone, the generated wave train propagates in all directions of the water body.
For assessing whether these impulse waves represent a threat to the shore, it is of crucial importance to understand the processes of spatial impulse wave propagation thereby predicting in particular the rate of wave height decay depending on the propagation angle as the key parameter for run-up height and overtopping volume estimations (Heller et al. 2009). Based on different approaches, numerous hazard assessment studies have been carried out for impulse wave-prone locations worldwide (e.g., Risley et al. 2006; Wieczorek et al. 2007; Ataie-Ashtiani and Malek-Mohammadi 2008; Harbitz et al. 2014; Battaglia et al. 2015), demonstrating the need for reliable prediction methods in engineering practice. The application of equations empirically derived from physical scale models represents one of these prediction methods (Heller et al. 2009). The generation of landslide-generated impulse waves comprises highly dynamic and complex hydraulic processes involving three phases, namely water, air, and slide material. Physical scale models resolve these processes by default and provide a direct causal relation between slide parameters and impulse wave characteristics. In addition, physical scale models are essential for calibration and validation of numerical approaches (e.g., Yavari-Ramshe and Ataie-Ashtiani 2015).
The spatial impulse wave characteristics including the wave height and its decay rate depend on the wave propagation angle relative to the slide direction (Huber and Hager 1997). Previously, two types of studies were conducted with the purpose of investigating spatial impulse wave propagation patterns in physical scale models.
On the one hand, there are site-specific models for assessing risks related to impulse waves generated by subaerial landslides for particular settings (Davidson and McCartney 1975; Chaudhry et al. 1983; Fuchs et al. 2011; Huang et al. 2014; Lindstrøm et al. 2014). On the other hand, there are generic model setups in wave basins for fundamental research and the determination of general mathematical means (Huber and Hager 1997; Panizzo et al. 2005; Liu et al. 2005; Ataie-Ashtiani and Nik-khah 2008; Di Risio et al. 2009; Mohammed and Fritz 2012;
Chen et al. 2015; Heller and Spinneken 2015). The second approach enables the establishment of universal prediction equations based on an empirical evaluation of the measured laboratory data. These equations apply for the assessment of landslide tsunami-related hazards at prototype scale. This study gives an overview of present spatial wave height decay equations including the effect of the wave propagation angle and discusses their limitations by comparing computed results with measurement data obtained from own experiments.
Spatial impulse wave height decay
Empirical formulations of spatial impulse wave generation and propagation processes based on the propagation angle of the wave train were presented Panizzo et al. (2005), Heller et al. (2009), Mohammed & Fritz (2012), and Heller & Spinneken (2015). All empirical formulations include the majority of the slide parameters and wave
3
characteristics shown in the cross section in Fig. 1. Besides still water depth h, these comprise slide impact velocity Vs, slide mass ms, slide thickness s, bulk slide density ρs, bulk slide volume Vs, and slide impact angle α as slide parameters and as wave characteristics amplitudes of the generated wave train´s first crest ac and trough at as well as wave height H at propagation distance r in a similar way as for vertical 2D experiments in wave channels. The top view in Fig. 1 for spatial (3D) wave propagation processes contains, in addition to the propagation distance r, slide width b and wave propagation angle γ relative to the slide direction. All approaches contain the slide Froude number F = Vs/(gh)1/2 at impact onto the water surface, indicating the significance of its governing effect on the impulse wave generation process.
Fig. 1 Cross-section for γ = 0° (left) and top view (right) of governing parameters and characteristics of leading impulse wave crest and trough (adapted from Heller et al. 2009)
Panizzo et al. (2005) conducted experiments in a wave basin with rigid slide bodies that were stopped at the bottom of an inclined sliding rack by a system of springs. The slide’s underwater motion was analyzed in detail and the dimensionless time of the landslide underwater motion
t*s = 0.43 A*w−0.27 F−0.66 (sinα)−1.32 (1)
was identified by analogy with Walder et al. (2003) as a relevant quantity to characterize the generated waves based on the slide Froude number F, the slide impact angle α, and a dimensionless slide front surface A*w = bs/h2. The equation for the decay of the relative wave height by Panizzo et al. (2005) is
H/h = 0.07 t*s−0.3 A*w0.88 (sinα)−0.8 e1.37cos γ (r/h)−0.81 . (2) Compared with free granular slides that keep on advancing on the bottom of the wave basin after having reached the end of the inclined sliding plane, the rigid slide bodies were abruptly decelerated thereby interrupting the wave generation process. Another point to be noted is that an e.g. tenfold increase of the slide width b in Eq. (1) would result in an almost 50% reduction of the dimensionless time of the landslide underwater motion t*s and also a ninefold increase of the calculated wave heights H if all other parameters are kept constant. Furthermore both slide parameters slide thickness s and slide width b are interchangeable, leading to the conclusion that a rigid slide with a s:b ratio of e.g. 1:5 generates impulse waves with the same spatial wave propagation pattern and maximum wave surface envelope as a slide with a ratio of 5:1. Since no information on the slide mass ms is included in the wave height decay formulation, an assessment of impulse waves generated by slides with different bulk slide densities ρs is not accomplishable.
4
The formulation of the spatial wave height decay by Huber & Hager (1997) was adapted by Heller et al. (2009) with the purpose of merging it with the dimensionless impulse product parameter
P = F S1/2 M1/4 {cos([6/7]α)}1/2 (3)
as governing term for the wave generation process. The impulse product parameter P includes the dimensionless quantities slide Froude number F = Vs/(gh)1/2, relative slide thickness S = s/h, relative slide mass M = ms/(ρwbh2), and slide impact angle α and depicts the impulse transferred from the sliding mass to the water body (Heller &
Hager 2010). It is based on 434 experiments conducted in the 2D VAW wave channel involving free granular material and an extensive range of independently varied slide parameters and therefore provides a sound empirical basis. The slide Froude number F features an exponent of 1, indicating its significance for the wave generation process. In combination with decay terms depending on radial propagation distance r and wave propagation angle γ, the relative wave height H/h is given by Heller et al. (2009) as
H/h = (3/2) P4/5 cos2(2γ/3) (r/h)−2/3 . (4)
Eq. (4) is theoretically deduced by replacing the 2D decay term of Heller & Hager (2010) with the 3D decay term presented by Huber & Hager (1997). Note that the slide impact angle α and the slide width b are only included within the impulse product parameter P and thus have no direct additional influence on the wave front decay rate along the wave propagation angle γ at a fixed radial propagation distance r. This implies that impulse waves generated at different slide impact angles α will have the same spatial wave propagation pattern and maximum wave surface envelope as long as they exhibit the identical impulse product parameter P.
While the first two formulations of the spatial wave height decay process only provide singular information on wave height of a maximum wave within the wave train, Mohammed & Fritz (2012) presented a detailed analysis of the first wave crest and trough amplitudes ac and at shown in Fig. 1 as well as the second wave crest amplitude (not considered in this study). Regression analysis resulted in the first relative wave crest and trough amplitudes
ac/h = kac (r/h)naccos γ (5)
at/h = kat (r/h)nat cos γ . (6)
Consequently, a wave propagation angle γ = 90° results in first wave crest and trough amplitudes ac and at equal to zero. Eqs. (5) and (6) include terms for wave generation (kac, kat) and radial decay rate (nac, nat) each, taking separately the governing effect of the main slide parameters including the relative slide length ls/h on the two processes into account. Thus the exponent of the relative radial propagation distance r/h is not fixed, contrary to Eq. (4) by Heller et al. (2009). These generation and decay rate terms are defined as
kac = 0.31 F2.1 S0.6 (7)
kat = 0.70 F0.96 S0.43 (ls/h)−0.5 (8)
nac = −1.2 F0.25 S−0.02 (ls/h)−0.33 (9)
nat = −1.6 F−0.41 B−0.02 (ls/h)−0.14 . (10)
5
Both wave generation functions kac and kat in Eqs. (7) and (8) contain the slide Froude number F as the most influential parameter featuring the largest exponent. Summation of the relative amplitudes yields the relative wave height H/h = (ac/h +at/h) of the first impulse wave. As already stated for Panizzo et al. (2005) also the formulations of the spatial wave height decay by Mohammed & Fritz (2012) does not include the slide mass ms as a governing parameter. As a result calculations of slides with different masses but the same geometric dimensions will predict impulse waves with identical characteristics. Furthermore, the experiments were conducted at a fixed slide impact angle α = 27.1° so that the derived equations do not include this effect.
Heller & Spinneken (2015) conducted experiments with rigid bodies at a fixed slide impact angle α = 45°. In contrast to Panizzo et al. (2005) the geometry of the slide front was wedge-shaped and the sliding mass was not stopped at the end of the inclined sliding plane but underwent a transition of its moving direction parallel to the basin bottom through a circularly bended metal sheet. The wave height decay rate has been determined as
H/h = 2.75 F0.67 S M0.6 (r/h)−1 fγ . (11)
Contrary to Eq. (3) by Heller et al. (2009) as well as Eqs. (7) and (8) by Mohammed & Fritz (2012), the slide Froude number F is not the dimensionless quantity with the highest exponent resulting in a reduced effect on wave height generation. Instead, Eq. (11) contains the relative slide thickness S as the most influential factor. The decay rate is composed of the relative radial propagation distance r/h with a fixed exponent of −1 and a term for the wave propagation angle γ defined as
fγ = cos2{1+exp[−0.2(r/h)]}(2γ/3) . (12)
Table 1 shows slide parameter ranges or limits of applicability, respectively, of the prediction methods described above. No information is given for parameters that were not stated by the authors or determined without assumptions. Note that for the method of Heller et al. (2009) only parameters of the 2D experiments are specified, since the 3D decay terms are not based on joint experiments, but theoretically deduced based on the findings by Huber & Hager (1997).
Table 1 Overview of slide parameter ranges
Method F [-] S [-] M [-] α [°] B [-] ls/h [-]
Panizzo et al.
(2005) 1 - 2.2 0.11 - 0.45 - 16 - 36 0.38 - 1.5 -
Heller et al.
(2009) 0.86 - 6.83 0.09 - 1.64 0.11 - 10.02 30 - 90 0.74 - 3.33 - Mohammed & Fritz
(2012) 1 - 4 0.1 - 0.9 - 27.1 1 - 7 2.5 - 6.8
Heller & Spinneken
(2015) 0.54 - 2.47 0.25 - 0.5 0.25 - 2.49 45 1.2 - 2.4 0.73 - 3.66
6 Experimental setup and program
Landslide generated impulse waves experiments were conducted at VAW in a 4.5 m by 8.0 m wave basin (Fig. 2).
The still water depth h was adjusted to 0.20 m to avoid scale effects (Heller et al. 2008). Mesh-packed slides on an inclinable chute were applied for wave generation. As described by Evers & Hager (2015a) this technique generates impulse wave characteristics comparable to those of free granular slide experiments. The slide’s geometric dimensions were predefined by a release box. Transformation of the slide size during the gravitationally driven acceleration process on the chute was found to be negligible. However, the flexible structure of the mesh- packed slides allowed for deformation after impact onto the water surface and during the underwater slide movement as was observed for free granular slides (Fritz et al. 2003). The slide impact velocity Vs was changed by varying the acceleration distance between release box and still water surface. For slide velocity measurements laser light barriers mounted perpendicular to the sliding plane above the still water surface were used.
Fig. 2 Experimental setup showing positions of measuring points relative to slide impact zone
A videometric measurement technique was applied to obtain a quasi-continuous representation of the free water surface (Evers & Hager 2015b). This approach tracks up to 6,000 point measurements by projecting a grid pattern onto an opaque water surface dyed with titanium dioxide and filming it with 4 spatially referenced cameras (AICON 3D Systems GmbH, Brunswick, Germany) operated at a synchronized frame rate of 24 Hz. For this study 9 interpolated measuring points were selected at radial propagation distances r = 1.0 m, 1.8 m, and 2.6 m from the slide impact location and wave propagation angles γ = 0°, 30°, and 60°, respectively (Fig. 2). The experimental program comprised 6 tests. Their corresponding dimensionless quantities are listed together with the slide impact angle α in Table 2. Besides afore mentioned still water depth h = 0.20 m, slide mass ms = 20 kg and slide width b
= 0.5 m were kept constant. The first four tests at α = 45° and 60° have a similar impulse product parameter P, whereas the last two tests at α = 30° have a lower P, since the inclination of the sliding plane was too small to get a sufficient gravitational acceleration.
7 Table 2 Overview of experimental quantities
Test Symbol F [-] S [-] M [-] α [°] P [-] B [-] ls/h [-]
1 3.11 0.3 1 60 1.34 2.5 2.49
2 2.16 0.6 1 60 1.32 2.5 1.25
3 2.46 0.3 1 45 1.19 2.5 2.49
4 1.78 0.6 1 45 1.22 2.5 1.25
5 1.37 0.3 1 30 0.71 2.5 2.49
6 1.09 0.6 1 30 0.80 2.5 1.25
For test no. 3 a sequence of the wave generation and propagation processes recorded by measurement camera No.
1 (Fig. 2) is shown in Fig. 3. Between t = 0 s and t = 0.25 s the mesh-packed slide impacts the water body and creates an impact crater as well as a splash screen occurring also in 2D experiments (Fritz et al. 2003). Up to t = 0.75 s the impact crater and splash screen collapse and a first wave crest detaches from the impact zone. This first wave crest is not only visually recognizable by the light reflections on the water surface but also at the left border of the grid projection. Compared to the straight left edge at t = 0 s, a propagating first wave crest is observable between t = 0.75 s and 1.25 s. From t = 1.25 s to 1.75 s formation and radial propagation of the second wave crest is apparent. Its shape crest appears to be steeper than that of the first wave crest.
8
Fig. 3 Photographs at various stages of wave generation and spatial propagation process for F = 2.46, S = 0.3, B
= 2.5, M = 1, α = 45° at h = 0.2 m from t = 0 s to t = 1.75 s after slide impact onto still water surface
t = 0.000 s t = 0.125 s t = 0.250 s
t = 0.375 s t = 0.500 s t = 0.625 s
t = 0.750 s t = 0.875 s t = 1.000 s
t = 1.125 s t = 1.250 s t = 1.375 s
t = 1.500 s t = 1.625 s t = 1.750 s
9 Experimental results
The measured wave heights Hmeas of the first impulse waves passing the nine measurement points were extracted from the water surface’s interpolated contour for all tests. Besides still water depth h, measured and predefined slide parameters impact velocity Vs, mass ms, thickness s, width b, and impact angle α (Tab. 2) were taken as input values for the computation methods proposed by Panizzo et al. (2005), Heller et al. (2009), Mohammed & Fritz (2012), and Heller & Spinneken (2015) to predict relative wave heights Hpred/h at the nine measurement points.
The plots shown in Fig. 4 compare measured quantities Hmeas/h with predicted relative wave heights Hpred/h subdivided by the different prediction methods. The different methods were applied for all 6 tests, regardless of their limits of applicability, to analyze their behavior on an extended parameter range.
The wave heights Hpred obtained with the method by Panizzo et al. (2005) underestimate the measured wave heights Hmeas considerably. Most of the predicted wave heights underestimate the actual data by more than 50%. In addition a dependency on relative slide thickness S is evident: slides with S = 0.3 are even more severely underestimated than slides with S = 0.6. The other way round, this implies that the actually measured wave heights Hmeas can be 2 to 4 times larger than Hpred predicted. Since the slide motion was stopped at the end of the inclined sliding plane, the wave generation process was interrupted and the slide’s impulse partly transferred to the decelerating spring system instead of the water column resulting into smaller wave heights.
In general the method by Heller et al. (2009) overestimates the measured wave heights Hmeas. The predicted wave heights scatter in a small range. For smaller relative wave heights with Hmeas/h < 0.3 overestimation partially exceeds +50%. The trend of overestimation tends to decrease for increasing relative wave heights with Hmeas/h >
0.3. This trend applies also for the experiments by Heller & Spinneken (2015). There appears to be no distinct dependency of the predicted wave heights Hpred on slide parameters as relative slide thickness S or slide impact angle α.
The wave height decay equations by Mohammed & Fritz (2012) were derived from experiments at a fixed slide impact angle α = 27.1°. Hence, the experiments with α = 30, 45, and 60° conducted for this study go clearly beyond the method’s limits of applicability and a distinct dependency on the slide impact angle α is evident. Contrary to apparent expectations, the predicted results improve with increased slide impact angle α, yielding an only slight underestimation of up to 25% for α = 60°. The predicted wave heights Hpred for the slide parameter combination of test no. 6 even undercut those computed with the method by Panizzo et al. (2005) by around 50%. No distinct dependency of the predicted wave heights Hpred on relative slide thickness S was noted.
The wave heights Hpred predicted with the method by Heller & Spinneken (2015) scatter evenly in a range of ±50%
around the measured wave heights Hmeas. While the wave heights generated by slides with relative slide thicknesses S = 0.6 tend to be overestimated by up to 50%, wave heights for S = 0.3 are underestimated by up to 50%. For the experiments at slide impact angle α = 45°, the method’s value of applicability, waves generated by the slide with S = 0.6 are well predicted by the method, whereas waves generated with S = 0.3 undercut the measured wave heights Hmeas by 50%. Besides a strong dependency on the slide thickness S, the method generally predicts increased wave heights Hpred for increased slide impact angles α, contrary to its decreasing contribution found by Heller & Hager (2010) in Eq. (3) for 2D experiments.
10
α = 30° (), 45° (), 60° (); S = 0.3 α = 30° (), 45° (), 60° (); S = 0.6 γ = 0° (∙ ∙ ∙ ∙ ∙ ∙), 30° (─ ─ ─), 60° (─ ∙ ∙ ─)
Fig. 4 Measured relative wave heights Hmeas/h versus predicted Hpred/h subdivided by the landslide generated impulse wave height decay equations by Panizzo et al. (2005) (top left), Heller et al. (2009) (top right), Mohammed & Fritz (2012) (bottom left), and Heller & Spinneken (2015) (bottom right); with slide impact angles α = 30°, 45°, 60° and relative slide thicknesses S = 0.6 ( ) and S = 0.3 ( ) for wave propagation angles γ = 0° (∙ ∙ ∙ ∙ ∙ ∙), 30° (─ ─ ─), 60° (─ ∙ ∙ ─)
Due to their single slide impact angle α at the lower bound of the potential range, the wave height decay formulation by Mohammed & Fritz (2012) strongly depends on this slide parameter. Moreover smaller wave heights measured in this study are severely underestimated. Although the experiments of Heller & Spinneken (2015) were also conducted at a single slide impact angle α, their method is more sensitive to slide thickness s. This applies also for the method by Panizzo et al. (2005). Both used rigid bodies preventing any slide deformation. The decay rate term of Heller & Spinneken (2015) reproduces the decay rate of the experiments’ first waves the most appropriate.
Aside from an under- or overestimation based on slide thickness s within the wave generation term, the decay rate along radial propagation distance r fits well to the experimental data.
0.0 0.4 0.8
0.0 0.4 0.8
Hpred/h
Panizzo et al. (2005)
Hmeas/h 0.0 0.4 0.8
0.0 0.4 0.8
Hpred/h
Heller et al. (2009)
Hmeas/h
0.0 0.4 0.8
0.0 0.4 0.8
Hpred/h
Mohammed & Fritz (2012)
Hmeas/h 0.0 0.4 0.8
0.0 0.4 0.8
Hpred/h
Heller & Spinneken (2015)
Hmeas/h
11
The mean relative wave celerity cam/(gh)0.5 of the first wave crest was determined as the average between the measurement points at the radial wave propagation distances r = 1.0 m and 2.6 m for each wave propagation angle γ. The measured values of cam/(gh)0.5 range between 0.9 and 1.1 and match well the theoretical wave celerity of long waves c = (gh)0.5 (Dean & Dalrymple 1991). The wave periods T measured at r = 1.0 m range between 1.3 s and 1.5 s, whereas for r = 2.6 m the range is between 1.7 s and 1.8 s. With the linear wave theory as a simplified assumption, the wave length L can be determined with L = Tc (Dean & Dalrymple 1991), thereby resulting in relative wave lengths L/h of 8.3 to 10.0 for r = 1.0 m and 11.7 to 13.2 for r = 2.6 m. While the wave crest celerities of the generated waves satisfy the condition of long waves, the relative wave lengths do not meet the criterion L/h
> 20 of long waves (Dean & Dalrymple 1991). However, the wave lengths are increasing by 25% to 40% between r = 1.0 m and 2.6 m, and are therefore in transition to long wave dimensions.
Discussion
Depending on the water body type there are various possibilities of mitigation measures in case a sliding mass cannot be prevented from impacting the water surface. One option is an active adjustment of the water body itself:
reservoirs can be lowered to a water level which provides a sufficient freeboard for wave run-up; proglacial lakes can also be drained to a level at which wave induced erosion does not lead to a total failure of the embankment.
For natural lakes and coastal areas early warning systems and evacuation plans need to be developed in addition to the construction of protective structures. The wave height at a certain position along the shore thereby acts as a key parameter for implementation and design of these mitigation measures.
From an engineering point of view the formulation for spatial impulse wave propagation by Heller et al. (2009) is considered the most favorable among these existing: its results are conservative since it tends to overestimate actual wave heights. In addition, its generation term is based on the most comprehensive slide parameter set including slide mass ms and various slide impact angles α. Nevertheless, smaller wave heights with Hmeas/h < 0.3 or, expressed in terms of wave parameters, waves with larger radial propagation distance r and/or propagation angle γ, respectively, are often overestimated by more than 50%. For example, if hydropower reservoir levels are lowered as a mitigation measure, this overestimation might lead to a water level lower than necessary relating to an uneconomical decision.
The formulations of spatial impulse wave propagation by Heller et al. (2009) (Eq. 4) and Heller & Spinneken (2015) (Eq. 11) both feature a similar configuration: there is one part of the equation for wave generation including governing slide parameters, and another part for wave propagation including radial propagation distance r and propagation angle γ. As demonstrated in the previous assessment and analysis, both methods have their specific assets: while Heller et al. (2009) is based on a strong data basis regarding wave generation with free granular slides, the method by Heller & Spinneken (2015) produces the best aligned wave decay predictions with the present experimental data. Fig. 5 shows the measured relative wave heights Hmeas/h versus predicted Hpred/h for a combined approach. Instead of [2.75 F0.67 S M0.6], the combined approach includes [2 P] from Eq. (3) as wave generation term in Eq. (11). Compared to existing methods, this combination allows for an improved prediction of the experimental data. Nevertheless, the empirical basis is rather limited with 6 test. In addition the slide impact angle α is only included within the wave generation term having no direct effect on the spatial wave propagation pattern and maximum wave surface envelope. Furthermore the effect of slide width b is not accounted for in the wave propagation process.
12 α = 30° (), 45° (), 60° (); S = 0.3
α = 30° (), 45° (), 60° (); S = 0.6 γ = 0° (∙ ∙ ∙ ∙ ∙ ∙), 30° (─ ─ ─), 60° (─ ∙ ∙ ─)
Fig. 5 Measured relative wave heights Hmeas/h versus predicted Hpred/h for combined method of Heller et al.
(2009) and Heller & Spinneken (2015); with slide impact angles α = 30°, 45°, 60° and relative slide thicknesses S = 0.6 ( ) and S = 0.3 ( ) for wave propagation angles γ = 0° (∙ ∙ ∙ ∙ ∙ ∙), 30° (─ ─ ─), 60° (─ ∙ ∙ ─) Conclusions
The prediction of wave height decay of landslide generated impulse waves is crucial for implementing appropriate mitigation measures including lowering the water level of a reservoir or designation of evacuation zones. This research analyzes and discusses current empirical prediction methods for wave height decay of spatial impulse waves and compares predictions based on these methods with own experimental data. The main findings are:
• Existing prediction methods for spatial impulse wave generation feature limitations regarding their formulation as well as empirical basis and hence range of applicability;
• A series of 6 tests was conducted with flexible mesh-packed slides in a wave basin and the measured wave heights were compared with wave heights predicted by existing methods;
• Existing wave height decay equations predict measured wave heights with a scatter range exceeding
±50%;
• A combination of two existing methods was found to be suited best to predict the generation and decay of the first impulse wave height with a ±30% scatter for most of the measurement data.
Although the influence of important spatial slide parameters such as the slide impact angle and slide width is not accounted for, the present study provides an indication for the structure of future prediction methods for spatial landslide generated impulse waves.
Acknowledgments
This work was supported by the Swiss National Science Foundation (Project No. 200021-143657/1).
0.0 0.3 0.6
0.0 0.3 0.6
Hpred/h
Hmeas/h
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