Journal of Engineering Mathematics

, Volume 67, Issue 1–2, pp 33–54 | Cite as

Variational water-wave model with accurate dispersion and vertical vorticity

  • Colin Cotter
  • Onno Bokhove
Open Access


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 profile 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.


Bores Coastal engineering Variational principles Wave-current interactions 



The photograph “sun-on-bore”, as the late Professor Howell D. Peregrine (DHP) called it, in Fig. 3a featured most prominently on the poster “maths makes waves” (Fig. 4) in the campaign “World Mathematical Year 2000, Posters in the London Underground”. Howell Peregrine was always keen that his photographs were used for good, whether it be in the sciences or (fine) arts. O.B. was a postdoctoral research associate with DHP (and Professor Andrew Woods) in 1998 and 1999 at the School of Mathematics in Bristol, U.K. Thereafter, our scientific, geological and walking tours continued, till March 2007. We thank Sander Rhebergen and Shavarsh Nurijanyan for proofreading several sections of the manuscript. Finally, it is a pleasure to thank the organizers of two workshops, Geometric and Stochastic Methods in Geophysical Fluid Dynamics in Bremen 2008 (a.o., Marcel Oliver) and Numerical Modelling of Complex Dynamical System at the Lorentz Center in 2008 (a.o., Jason Frank), for providing the seeding opportunities of the research presented.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


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Copyright information

© The Author(s) 2009

Authors and Affiliations

  1. 1.Department of AeronauticsImperial CollegeLondonUK
  2. 2.Department of Mathematics, Institute of Mechanics, Processes and Control, TwenteUniversity of TwenteEnschedeThe Netherlands

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