Author(s): Ben Constance; Valentin Heller
Linked Author(s): Valentin Heller
Keywords: Kadomtsev–Petviashvili equation; Korteweg–De Vries equation; Landslide-tsunamis; Shallow water waves; Wave decomposition techniques
Abstract: When a large-scale mass movement disturbs a significant body of water in a reservoir, lake or the sea, landslide-tsunamis are generated. This threat must be assessed and, if possible, mitigated. Up to now, landslide-tsunami prediction and characterisation is often centred on the use of empirical equations predicting the tsunami characteristics such as the wave height and amplitude. Such equations fail to appropriately capture the underlying physics and dispersive nature of landslide-tsunamis consisting of the superposition of several wave components. Spectral decomposition techniques such as the Fourier, wavelet or Hilbert-Huang transforms were sometimes applied with the aim to provide physically more meaningful results. However, these techniques do not adequately account for many landslide-tsunami characteristics. These methods can give a useful snapshot of a landslide-tsunami’s composition. However, they are to varying degrees unable to differentiate between the basic processes behind landslide-tsunami behaviour, and in particular the effect of the water depth. The water depth is important as landslide-tsunamis are intermediate- to shallow-water waves. Crucially, the dominant constituent modes of tsunamis, which might prove most significant and potentially devastating, are not reliably identified by existing techniques. Mathematical methods exist to understand a landslide-tsunami’s composition and to predict how it will propagate. The Korteweg–De Vries (KdV) differential equation (in one dimension) and its sister differential equation Kadomtsev–Petviashvili (KP) (in two dimensions) have long provided a description of non-linear shallow-water waves. Application of the Inverse Scattering Transform (IST) to these equations has been documented in the technical literature, being successfully used to describe gravity water waves. We are developing software placing these methodologies in the hands of engineers to accurately anticipate the impending nature of tsunamis. The validation of this MATLAB codebase is the subject of this article, where we show how our KdV IST accurately describes laboratory wave-flume experiments. This KdV analysis is seen to fall short when applied to wave basin experimental results producing directional (two dimensional) spreading of the wave energy. However, by applying our KP IST analysis we obtain impressive agreement with the basin data. We also discuss our roadmap to offer a turn-key engineering tool for the analysis of landslide-tsunamis.