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J Eng Math (2010) 67:3354 DOI 10.1007/s10665-009-9346-3

Summary: J Eng Math (2010) 67:3354
DOI 10.1007/s10665-009-9346-3
Variational water-wave model with accurate dispersion
and vertical vorticity
Colin Cotter Onno Bokhove
Received: 28 February 2009 / Accepted: 30 September 2009 / Published online: 28 October 2009
The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract A new water-wave model has been derived which is based on variational techniques and combines a
depth-averaged vertical (component of) vorticity with depth-dependent potential flow. The model facilitates the
further restriction of the vertical profile of the velocity potential to n-th order polynomials or a finite-element pro-
file with a small number of elements (say), leading to a framework for efficient modelling of the interaction of
steepening and breaking waves near the shore with a large-scale horizontal flow. The equations are derived from
a constrained variational formulation which leads to conservation laws for energy, mass, momentum and vertical
vorticity. It is shown that the potential-flow water-wave equations and the shallow-water equations are recovered in
the relevant limits. Approximate shock relations are provided, which can be used in numerical schemes to model
breaking waves.
Keywords Bores Coastal engineering Variational principles Wave-current interactions
1 Introduction
It is always fascinating to watch waves near the shore line. They approach the shore, steepen as the water becomes
more shallow, break upon further approach, and run up and down the beach or dike. In Fig. 1, wave trains are seen


Source: Al Hanbali, Ahmad - Department of Applied Mathematics, Universiteit Twente


Collections: Engineering