Hamiltonian variational principles have provided, since the 1960s, the means of developing very successful wave theories for nonlinear free-surface flows, under the assumption of irrotationality. This success, in conjunction with the recognition that almost all flows in the sea are not irrotational, raises the question of extending Hamilton’s principle to rotational free-surface flows. The Euler equations governing the bulk fluid motion have been derived by means of Hamilton’s principle since the late 1950s. Nevertheless, a complete variational formulation of the rotational water-wave problem, including the derivation of the free-surface boundary conditions, seems to be lacking until now. The purpose of the present work is to construct such a missing variational formulation. The appropriate functional is the usual Hamilton’s action, constrained by the conservation of mass and the conservation of fluid parcels’ identity. The differential equations governing the bulk fluid motion are derived as usually, applying standard methods of the calculus of variations. However, the standard methodology does not provide enough structure to obtain the free-surface boundary conditions. To overcome this difficulty, differential-variational forms of the aforementioned constraints are introduced and applied to the boundary variations of the Eulerian fields. Under this transformation, both kinematic and dynamic free-surface conditions are naturally derived, ensuring the Hamiltonian variational formulation of the complete problem. An interesting feature, appearing in the present variational derivation, is a dual possibility concerning the tangential velocity on the boundary; it may be either the same as in irrotational flow (no condition) or zero, corresponding to the small-viscosity limit. The deeper meaning and the significance of these findings seem to deserve further analysis.

1. Introduction 2. An overview of the fully non-linear Hamiltonian Coupled-Mode Theory 2.1. Classical differential and variational formulation 2.2. The Hamiltonian Coupled-Mode Theory 3. A concise description of the numerical implementation 3.1. Numerical solution of the kinematical substrate problem 3.2. Numerical solution of the Hamiltonian evolution equations 4. Propagation of a solitary wave over some typical bathymetries. Validation and limitations of the present method 4.1. Propagation over a shelf 4.2. Propagation over a step 4.3. Comparison and assessment of the findings of Sec. 4.1 and 4.2 4.4. Propagation over a trench 5. Propagation of a solitary wave over two trenches and reflection at a vertical wall 5.1. Propagation over the trenches 5.2. Run-up on the wall 5.3. Maximum force exerted on the vertical wall Conclusions A numerical study of the run-up and the force exerted on a vertical wall by a solitary wave propagating over two tandem trenches and impinging on the wall AbstractThe propagation and transformation of water waves over varying bathymetries is a subject of fundamental interest to ocean, coastal and harbor engineers. The specific bathymetry considered in this paper consists of one or two, naturally formed or man-made, trenches. The problem we focus on is the transformation of an incoming solitary wave by the trench(es), and the impact of the resulting wave system on a vertical wall located after the trench(es). The maximum run-up and the maximum force exerted on the wall are calculated for various lengths and heights of the trench(es), and are compared with the corresponding quantities in the absence of them. The calculations have been performed by using the fully nonlinear water-wave equations, in the form of the Hamiltonian coupled-mode theory, recently developed in Papoutsellis et al (Eur. J. Mech. B/Fluids, Vol. 72, 2018, pp. 199-224). Comparisons of the calculated free-surface elevation with existing experimental results indicate that the effect of the vortical flow, inevitably developed within and near the trench(es) but not captured by any potential theory, is not important concerning the frontal wave flow regime. This suggests that the predictions of the run-up and the force on the wall by nonlinear potential theory are expected to be nearly realistic.The main conclusion of our investigation is that the presence of two tandem trenches in front of the wall may reduce the run-up from (about) 20% to 45% and the force from 15% to 38%., depending on the trench dimensions and the wave amplitude. The percentage reduction is greater for higher waves. The presence of only one trench leads to reductions 1.4 -1.7 times smaller.Keywords: nonlinear water waves; wave-trench interaction; solitary wave over varying bathymetry; run-up on vertical wall; force on vertical wall; submerged breakwater

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