Julien Touboul

Fast, convergent local-mode expansions of nonlinear water waves are discussed for the representation of the velocity and stream function. Subsequently, the representations are used to derive and study a novel nonlinear coupled-mode system of differential equations on the horizontal plane, with respect to unknown horizontal velocity modal amplitudes and free-surface elevation. The coupled-mode system, in conjunction with the convergence properties of the local-mode series, facilitates the numerical solution of the water wave propagation problem over general bottom topography. The efficiency of the present method is demonstrated through various examples, including the simulation of periodic waves in a constant depth and over trapezoidal bar test cases. The results show the robustness and accuracy of the coupled-mode system in capturing the complexities of wave transformations over non-uniform bathymetric features. Moreover, truncating the modal expansions of the wave velocity field by keeping only the first mode leads to a low-cost, single-mode nonlinear wave model with enhanced dispersion characteristics that is useful for engineering applications.

The wave–current–seabed interaction problem is studied by using a coupled-mode system developed for modeling wave scattering by non-homogeneous, sheared currents in variable bathymetry regions. The model is based on a modal series expansion of wave velocity based on vertical eigenfunctions, dependent on local depth and flow parameters, including propagating and evanescent modes. The latter representation is able to accurately satisfy the wave flow continuity condition and the no-entrance boundary condition on the sloping parts of the seabed. A new derivation of a simplified nonlinear system is introduced using decomposition to a mean flow and a perturbative wave field. To force the system to consider incoming waves at the inlet, boundary knowledge of periodic, travelling nonlinear water waves over a flat bottom is required. For this purpose, specific solutions are derived using the semi-analytical method based on the stream function formulation, for cases of water waves propagating above linearly and exponentially sheared currents. Results obtained by the application of the CMS concerning the propagation of waves and currents—in particular, examples characterized by depth inhomogeneities—are presented and discussed, illustrating the applicability and performance of the method.

Wave conditions from the open sea to the coast, provide necessary information, the good understanding and modeling of coastal dynamics, the design of coastal engineering structures coastal engineering structures, for navigation or the flooding risks evaluation. It is also a potential way to access information present below the surface. Refraction, diffraction and reflection of waves are not only forced by variations in bathymetry but also by the presence of currents. However, the effects of current in propagation models have long been limited to the consideration of homogeneous current in the water column. Nevertheless, currents are usually observed to be sheared vertically by wind, tides or waves. When wave propagate in the presence of currents their celerity is modified. It is also affected by the vertical structure of the current. This work proposes and discusses methods for reconstructing current fields from wave data based on synchronous analysis of wave spectra at different points in space. We consider here the 2D case of progressive or partially stationary waves in the presence of homogeneous currents or with a vertical sheared profile. The study is based on data from experiments carried out at the Bassin de Génie Océanique FIRST(BGO), for progressive or partially stationary waves in the framework of the ANR project MORHOC'H 2. These test campaigns allowed us to test the sensitivity of the wave’s phase evolution during its propagation in order to estimate the feasibility of reconstructing either constant or sheared currents in the water column. Under the assumption of a progressive wave, the study of the phase evolution shows a significant influence of the current, allowing to reach the intensity of a uniform current. The calculated phase evolution in the presence of sheared current is consistent with the theory, but for small values of shear, the phase velocity changes are much smaller, making the method more sensitive to "noise". Furthermore, for a partially stationary wave, a significant impact of its phase evolution is observed in the propagation direction of the incident wave even for weak reflections, making it necessary to include this parameter in the reconstruction of currents from synchronous wave measurement data.Acknowledgements:The DGA is thanked for funding the AID thesis grant of Alexandra CUEVAS, as well as for the ANR grant: ANR-21-ASM1-003.

The recent numerical study of Barthelemy et al. (J. Fluid Mech., vol. 841, 2018, pp. 463–488) investigated the local properties of two-dimensional (2-D) and three-dimensional (3-D) nonlinear unsteady gravity wave packets in deep and uniform intermediate depth water. Their study focused on the breaking inception transition zone separating maximum recurrence and marginal breaking, and reported that a suitably normalized energy flux localized at the steepest crest in the packet provides a robust breaking threshold parameter. Our present study uses the fully nonlinear boundary integral element method solver developed by Touboul & Kharif (Nat. Haz., vol. 84, issue 2, 2016, pp. 585–598) to investigate breaking inception of 2-D deep water nonlinear water wave packets propagating in the presence of a background current that varies linearly with depth. We seek to validate whether the proposed generic breaking inception threshold holds for the case of constant background vorticity. Results are presented for different packet bandwidths and background vorticity levels.

For various experimental reasons, the measurement of water waves propagating in shallow water environments such as surf zones or coastal areas is a difficult task. Deploying surface measuring instruments can be inconvenient, dangerous, or simply expensive. Thus, such measurements are often performed using bottom mounted pressure sensors. Unfortunately, the problem of reconstructing surface elevation based on a single point pressure sensor is an ill-posed problem.Indeed, the pressure data collected should be inverted to provide the related water elevation. However, the transfer function traditionally used to perform this inversion is subject to question. When considering very long waves, like tides and tsunamis, the pressure is hydrostatic as long as dispersive effects can be neglected and recovering surface elevation from the bottom pressure does not imply any particular difficulty. Yet, for steeper waves propagating in such depth conditions, nonlinearity might play a significant role (Didenkulova et al., 2021).In coastal areas, the propagation of water waves is more complex, and often involves dispersion or nonlinearity. In such areas, one may find wind waves, which are strongly dispersive, even in the coastal zone. Using linear theory might be helpful, in such cases, but is also subject to questions (Touboul & Pelinovsky, 2018). Besides, other corrections related to their dispersive behaviour might play a significant role. Various phenomena, such as partially standing waves (Touboul & Pelinovsky, 2014), or the superimposition of current, might also play a significant role.In this work, we investigate the performance of classical reconstruction techniques, but also more recent approaches (Oliveras et al., 2012, Clamond & Constantin, 2013, Bonneton et al., 2018), by confronting their prediction to field data collected in the central part of the Bay of Biscay using current meters mounted with pressure and acoustic surface tracking sensors . These data are obtained in various depth conditions, often in extreme conditions and provide pressure records, current velocity, and direct measurement of the water elevation. Thus, the use of methods presenting various degrees of sophistication allows us to analyze in details the respective roles played by the current, the dispersion, and the nonlinearity.The joint French-Russian grant No. 19-55-15005 is acknowledged.[1] E. Didenkulova, E. Pelinovsky & J. Touboul, Long-wave approximations in the description of bottom pressure, Wave Motion, vol. 100, No. 1, 102668 (2021)[2] J. Touboul & E. Pelinovsky, "On the use of linear theory for measuring surface waves using bottom pressure distribution", Eur. J. Mech. B: Fluids, 67, 97–103, (2018).[3] J. Touboul & E. Pelinovsky, "Bottom pressure distribution under a solitonic wave reflecting on a vertical wall", Eur. J. Mech., B. Fluids, 48, p. 13-18, (2014).[4] K.L. Oliveras, V. Vasan, B. Deconinck, D. Henderson, Recovering the water wave profile from pressure measurement, SIAM J. Appl. Math. 72 (3) 897–918 (2012).[5] P. Bonneton, D. Lannes, K. Martins., H. Michallet, A nonlinear weakly dispersive method for recovering the elevation of irrotational surface waves from pressure measurements, Coastal Engineering 138, 1–8 (2018).[6] D. Clamond, A. Constantin, Recovery of steady periodic wave profiles from pressure measurements at the bed, J. Fluid Mech. 714, 463–475 (2013).

There is growing interest for water-wave flows through arrangements of cylinders with application to the performance of porous marine structures and environmental flows in coastal vegetation. For specific few cases experimental data are available in the literature concerning the modification of the dispersion equation for waves through a dense array of vertical cylinders. This paper presents a numerical study of the porosity effects on the dispersion relation of water waves through such configurations. To this aim, the sloshing problem in a tank full of vertical cylinders intersecting the free surface is studied using the finite element method, and the influence of the porosity on the wave number is quantified. On the basis of numerical results, a new modification of a dispersion relation for porous medium is suggested based on a wide range of collected data. Moreover, the domain of validity of this new dispersion relation is examined considering the number of cylinders and the extrapolation to the infinite medium.