Abstract
Abstract. We are interested in the modelling of wave-current interactions around surf zones at beaches. Any model that aims to predict the onset of wave breaking at the breaker line needs to capture both the nonlinearity of the wave and its dispersion. We have therefore formulated the Hamiltonian dynamics of a new water wave model, incorporating both the shallow water and pure potential flow water wave models as limiting systems. It is based on a Hamiltonian reformulation of the variational principle derived by Cotter and Bokhove (2010) by using more convenient variables. Our new model has a three-dimensional velocity field consisting of the full three-dimensional potential velocity field plus extra horizontal velocity components. This implies that only the vertical vorticity component is nonzero. Variational Boussinesq models and Green–Naghdi equations, and extensions thereof, follow directly from the new Hamiltonian formulation after using simplifications of the vertical flow profile. Since the full water wave dispersion is retained in the new model, waves can break. We therefore explore a variational approach to derive jump conditions for the new model and its Boussinesq simplifications.
Highlights
We will show that the Green–Naghdi equations can be derived from the variational Boussinesq model with a parabolic potential flow profile via an additional approximation to the Hamiltonian
A systematic derivation of a new Hamiltonian formulation for water waves was given starting from the variational principle (2)
It remains an open question to what extent this omission of horizontal vorticity components matters in the shallow water flows investigated here
Summary
Consider an incompressible fluid at time t in a threedimensional domain bounded by solid surfaces and a free surface, with horizontal coordinates x, y, and vertical coordinate z. Through extended Clebsch variables: the velocity potential φ = φ(x, y, z, t), the three-dimensional fluid parcel label l = l(x, y, z, t) and the corresponding Lagrange multiplier vector π = π (x, y, z, t) Such a representation describes a velocity field containing all three components of vorticity ∇ ×U. In an Eulerian variational principle with planar Clebsch variables that only depend on the horizontal coordinates the vertical component of vorticity is retained. This component is constant throughout the whole water column and flows with helicy (Kuznetsov and Mikhailov, 1980) are excluded by construction. Subsequent substitution of one of these variables l, h π , φs or h – rewritten as a functional in Eq (11) – in turn yields (6)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
