Transient waves generated by a moving bottom obstacle: a new near-field solution

Abstract

We consider the classical problem of a single-layer homogeneous fluid at rest and a low, slowly varying, long and positive bottom obstacle, which is abruptly started from rest to move with a constant speed V. As a result a system of transient waves will develop, and we assume that locally in the region over the obstacle dispersion can be ignored while nonlinearity cannot. The relevant governing equations for the nearfield solution are therefore the nonlinear shallow water (NSW) equations. These are bidirectional and can be formulated in terms of a two-family system of characteristics. We analytically integrate and eliminate the backward-going family and achieve a versatile unidirectional single-family formulation, which covers subcritical, transcritical and supercritical conditions with relatively high accuracy. The formulation accounts for the temporal and spatial evolution of the bound waves in the vicinity of the obstacle as well as the development of the transient free waves generated at the onset of the motion. At some distance from the obstacle, dispersion starts to play a role and undular bores develop, but up to this point the new formulation agrees very well with numerical simulations based on a high-order Boussinesq formulation. Finally, we derive analytical asymptotic solutions to the new equations, providing estimates of the asymptotic surface levels in the vicinity of the obstacle as well as the crest levels of the leading non-dispersive free waves. These estimates can be used to predict the height and speed of the leading waves in the undular bores. The numerical and analytical solutions to the new single-family formulation of the NSW equations are compared to results based on the forced Korteweg-de Vries/Hopf equation and to numerical Boussinesq simulations.

Dispersive shock waves and modulation theory

“…The outlined GS analytical framework for the study of transcritical dispersive hydrodynamic flows proved to be quite general (see, e.g., [239,240]). In the fluid mechanics context, it was successfully applied to the description of resonant generation of atmospheric internal waves (the Morning Glory, see Fig.…”

Lagrangian modelling of nonlinear viscous waves generated by moving seabed deformation

“…[see 17 , 23]. If the neighbourhood of the i-th target particle is sparsely populated, det(D i ) ≃ 0, which would make (10) indeterminate. In such cases, the stabilising matrix D i is substituted by the identity matrix.…”

“…Related work on gravity waves generated by MSD in inviscid liquid has a long, distinguished tradition. For example, [8], [9] and [10] studied the generation of weakly nonlinear solitary waves by an MSD translating at speed u in shallow water of constant depth h much smaller than the typical wavelength λ such that µ 2 = (2πh/λ) 2 ≪ 1. The wave amplitude ζ considered by the foregoing was also much smaller than λ, i.e.…”

Weakly nonlinear theory for dispersive waves generated by moving seabed deformation

“…Several landslide tsunami models have been developed using non-dispersive formulations for long waves, similar to those used for earthquake tsunamis (e.g. see Liu & Mei 2003; Wang, Liu & Mei 2010) and moving obstacles (Madsen & Hansen 2012). However, from a fluid dynamics point of view, landslide tsunamis are different from earthquake tsunamis because of their peculiar generation mechanism.…”