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The significance of the impulse product parameter P is reviewed, which is believed to be the most universal parameter for subaerial landslide tsunami (impulse wave) prediction. This semi-empirical parameter is based on the streamwise slide momentum flux component and it was refined with a multiple regression laboratory data analysis. Empirical equations based on P allow for a simple prediction of wave features under diverse conditions (landslides and ice masses, granular and block slides,

An important class of tsunamis is caused by mass movements including landslides, rock falls, underwater slumps, glacier calving, debris avalanches or snow avalanches [

Irrespective of where these waves are caused, they are a considerable hazard and the total cumulative death toll of Unzen (1792), Ritter Island (1888), Vajont (1963) and Papua New Guinea (1998) alone is likely to exceed 22,100 [

Subaerial landslide tsunamis are particularly challenging to predict because the mass, initially located above the water surface, impacts the water body and may entrain a large amount of air. Slide velocities of up to 128 m/s were estimated based on slide deposits [

A common method for the assessment of this hazard is to conduct a physical model study in the laboratory environment. Generic physical model studies [

Herein, the significance of the impulse product parameter P is reviewed, which is believed to be the most universal parameter for generic landslide tsunami (impulse wave) predictions. Useful analytical derivations based on P are deduced and it is illustrated how P greatly simplifies hazard assessment through real-world predictions.

Definition sketch of subaerial landslide tsunami (impulse wave) generation (adapted from Heller and Hager [

The semi-empirical impulse product parameter P was developed by Heller and Hager [^{1/2} ^{1/4}{cos[(6/7)^{1/2}

Equation (1) includes the slide Froude number F = _{s}^{1/2}, the relative slide thickness _{s}_{w}b_{s}h^{2}) and the hill slope angle _{s}_{s}_{w}_{s}_{s}_{s}_{s}Q_{s}V_{s}^{1/2} ≈ (_{s}sb_{s}V_{s}^{2}cos^{1/2} = _{s}^{1/2}^{1/2}_{s}^{1/2}_{s}^{1/2}

The expression on the right hand side in Equation (2) was further refined with a multiple regression data analysis based on hundreds of 2D experiments, resulting in the establishment of P [_{s}^{1/2}^{1/2} resulting in the smallest data scatter. The relative effect of _{s}^{1/2} in the slide mass _{s}_{s}_{s}^{1/4} found to be less pronounced in the data analysis than predicted in Equation (2).

_{M}

Example of prediction diagram based on P: Relative maximum wave height _{M}_{M}^{4/5} (^{2} = 0.82, _{M}

Empirical equations based on P derived in 2D physical model studies by Heller and Hager [_{M}_{M}_{M}_{M}_{s}^{1/2}_{s}_{s}_{s}_{s}_{s}

Wave parameter | Heller and Hager [ |
Heller and Spinneken [ |
||
---|---|---|---|---|

Slide type | granular | block | ||

Maximum amplitude | _{M}^{4/5} |
(^{2} = 0.88) |
_{M}_{s}^{1/2}]^{9/1}° |
(^{2} = 0.88) |

Streamwise distance at _{M} |
_{M} =^{1/2} |
(^{2} = 0.23) |
− | − |

Maximum height ( |
_{M}^{4/5} |
(^{2} = 0.82) |
_{M}_{s}^{1/4}]^{9/1}° |
(^{2} = 0.93) |

Maximum period | _{M}^{1/2}(^{1/2} |
(^{2} = 0.33) |
_{M}_{s}^{1/2}]^{1/4}(^{1/2} |
(^{2} = 0.24) |

Amplitude evolution | ^{−1/3}]^{4/5} |
(^{2} = 0.81) |
^{−1/3}_{s}^{3/4}]^{9/1}° |
(^{2} = 0.85) |

Height evolution | ^{−1/3}]^{4/5} |
(^{2} = 0.80) |
^{−1/3}_{s}^{1/2}]^{9/1}° |
(^{2} = 0.89) |

Period evolution | ^{5/4}]^{1/4}(^{1/2} |
(^{2} = 0.66) |
^{5/4}_{s}^{1/3}]^{1/4}(^{1/2} |
(^{2} = 0.53) |

_{M}_{M}_{M}_{M}^{2} > 0.80.

The parameter P allows for the derivation of theoretical aspects which are as universally applicable as P itself. The slide centroid impact velocity _{s}_{s}_{sc}(1 − tan^{1/2}

The parameter Δ_{sc}_{sc})^{1/2}^{1/2}[_{s}_{w}b_{s}^{1/4}/^{3/2} and f(^{1/2}{cos[(6/7)^{1/2} in
^{1/2}{cos[(6/7)^{1/2} =

_{M}_{M}_{M}_{max}_{max}

f(_{max}

The values for _{max}^{2}

For the typical value _{max}_{max}

The well documented 1958 Lituya Bay case [_{g}^{3} sliding from a maximum altitude of 914 m above sea level on a slope of _{s}_{s}_{s}^{9} kg. These result in a slide Froude number F = 2.66, a relative slide thickness

Artist’s impression of 1958 Lituya Bay rockslide generating a tsunami of ~162 m in height destroying forest up to maximum run-up height of 524 m (adapted from Heller and Hager [

Fritz

The prediction of _{M}_{M}_{M}_{M}_{M}

The following examples show that empirical predictions based on P are often conclusive enough to replace expensive prototype specific physical or numerical model studies, or at least to provide well founded recommendations on whether a more expensive investigation is required. Heller

Whereas P was developed for subaerial landslide tsunamis (impulse waves), it does not apply to submarine landslide tsunamis [

The parameter P was thus far mainly tested for wave channel (2D) rather than for wave basin (3D) geometries. Even though 2D geometries can reflect real-world cases (e.g. narrow reservoirs, lakes or fjords), the wave propagation is commonly of 3D nature. Transformation methodologies of results from 2D to 3D were presented by Huber and Hager [

The relevance of the semi-empirical impulse product parameter P was reviewed, which is believed to be the most universal parameter to predict subaerial landslide tsunamis (impulse waves). The parameter P includes all relevant slide parameters affecting the wave generation process in wide test ranges such as densities heavier and lighter than water or slide impact angles between 30° and 90° (_{M}_{M}_{M}

Despite the fact that P was derived under idealized conditions, it is considered a useful and effective parameter for estimates in real-world cases. This was demonstrated for the 1958 Lituya Bay case, where a good agreement between the “observed” and predicted wave heights resulted. Four further real-world studies conducted by other authors involving rock falls at Lake Lucerne, Switzerland; snow avalanches in the planned Kühtai reservoir in Austria; rock falls at Lake Como, Italy; and potential slope failures in Mitchell Pit Lake in Canada, provided evidence that first estimates based on P are often conclusive enough to replace expensive prototype specific physical or numerical model studies, or at least provide well founded recommendations on whether a more expensive investigation is required.

The parameter P applies to subaerial and potentially to partially submerged landslide tsunamis (impulse waves); however, it does not apply to submarine slides. A further limitation of P is its derivation for wave channel (2D) tests. Equations for waves propagating in 3D were proposed based on 2D to 3D transformation methods including P, and real-world applications showed that these equations result in realistic values. In the light of this success, the application of P to more complex water body geometries is the subject of an ongoing research effort.

The authors thank H.M. Fritz and A. Zweifel for having provided their experimental data. J. Spinneken is acknowledged for comments on an earlier version of this review. The work was supported by the Swiss National Science Foundation (Grant No. 200020-103480/1) and an Imperial College London Junior Research Fellowship.

Conceived and designed the experiments: VH of experiments at Imperial College London, WHH of experiments at ETH Zurich. Performed the experiments: VH. Analyzed the data: VH of data obtained at Imperial College London, VH WHH of data obtained at ETH Zurich. Wrote the review: VH. Commented on and improved the review: WHH.

The authors declare no conflict of interest.

The 2D experiments were conducted in two prismatic wave channels, namely the granular slide tests in an 11 m (L) × 0.500 m (W) × 1 m (H) wave channel at ETH Zurich [

Limitations of P: Parameter ranges of physical model studies of Heller and Hager [

Name | Symbol | Dimension | Heller and Hager [ |
Heller and Spinneken [ |
---|---|---|---|---|

Slide model type | − | − | granular | block |

Channel width | (m) | 0.500 | 0.600 | |

Still water depth | (m) | 0.150–0.675 | 0.300, 0.600 | |

Slide thickness | (m) | 0.050–0.249 | 0.120 | |

Grain diameter | _{g} |
(mm) | 2.0–8.0 | − |

Streamwise distance | (m) | 0–8.90 | 0–17.7 | |

Slide impact velocity | _{s} |
(m/s) | 2.06–8.77 | 0.59–3.56 |

Bulk slide volume | (m^{3}) |
0.0167–0.0668 | 0.0373 | |

Bulk slide density | _{s} |
(kg/m^{3}) |
590–1,720 | 1,534 |

Slide mass | _{s} |
(kg) | 10.09–113.30 | 57.23 |

Slide width | _{s} |
(m) | 0.500 | ~0.588, ~0.578, 0.526 |

Slide front angle | (°) | not systematic investigated | 30, 45, 60, 90 | |

Transition type | (−) | none | none and circular shaped | |

Hill slope angle | (°) | 30–90 | 45 | |

Slide Froude number | F | (−) | 0.86–6.83 | 0.34–2.07 |

Relative slide thickness | (−) | 0.09–1.64 | 0.20–0.40 | |

Relative slide mass | (−) | 0.11–10.02 | 0.27–1.21 | |

Relative streamwise distance | (−) | 0–59 | 0–40 | |

Impulse product parameter | P | (−) | 0.17–8.13 | 0.16–1.19 |

Number of tests | (−) | 434 | 144 |

The granular slide material in the 434 tests at ETH Zurich (Zurich, Switzerland) was accelerated in a box with up to 8 bar air pressure with a pneumatic landslide generator [_{s}_{4}) and polypropylene (PP), consisted of four cylindrically shaped grains of diameter of _{g}_{s}^{3}) and lighter (_{s}^{3}) than water. Mixtures of different grain diameters and densities were also included in the test program. However, the grain diameter and grain size distribution were found to have a negligible effect on the wave features and are not included in P. A small fraction of the tests was conducted at a small water depth

The 144 block model slide experiments at Imperial College London (London, UK) involved a hill slope ramp of constant front angle _{s}_{s}