Unsteady undular bores in fully nonlinear shallow-water theory

Details Unsteady undular bores in fully nonlinear shallow-water theory Unsteady undular bores in fully nonlinear shallow-water theory

2005-07-14

arxiv.org/abs/nlin/0507029v2

Phys Fluids, 18 (2006) Art No 027104

Nonlinear Sciences

Pattern Formation and Solitons

Revised version accepted for publication in Phys. Fluids, 51 pages, 9 figures

Scientific paper

10.1063/1.2175152

We consider unsteady undular bores for a pair of coupled equations of Boussinesq-type which contain the familiar fully nonlinear dissipationless shallow-water dynamics and the leading-order fully nonlinear dispersive terms. This system contains one horizontal space dimension and time and can be systematically derived from the full Euler equations for irrotational flows with a free surface using a standard long-wave asymptotic expansion. In this context the system was first derived by Su and Gardner. It coincides with the one-dimensional flat-bottom reduction of the Green-Naghdi system and, additionally, has recently found a number of fluid dynamics applications other than the present context of shallow-water gravity waves. We then use the Whitham modulation theory for a one-phase periodic travelling wave to obtain an asymptotic analytical description of an undular bore in the Su-Gardner system for a full range of "depth" ratios across the bore. The positions of the leading and trailing edges of the undular bore and the amplitude of the leading solitary wave of the bore are found as functions of this "depth ratio". The formation of a partial undular bore with a rapidly-varying finite-amplitude trailing wave front is predicted for ``depth ratios'' across the bore exceeding 1.43. The analytical results from the modulation theory are shown to be in excellent agreement with full numerical solutions for the development of an undular bore in the Su-Gardner system.

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