Initial boundary value problems for hyperbolic systems, and dispersive perturbations

D. Lannes, Lecture notes volume of the Winter School on Fluid Dynamics, Dispersive Equations and Quantum Fluids (Bressanone, december 2018), to appear. Download

The goal of these notes is to point out similarities and differences between two kinds of initial boundary value problems in dimension one. The first one concerns hyperbolic systems (such as the nonlinear shallow water equations) while the second one concerns dispersive perturbations of such systems (such as Boussinesq systems). In the absence of a boundary, that is, for the initial value problem, the link between both classes is quite obvious but in the presence of a boundary, the situation is more complex and dispersive boundary layers must be understood if one wants to understand the links between both classes of problems. After reviewing several types of initial boundary value problems (some of which being free boundary problems) arising in the study of waves in shallow water, we sketch the general theory for hyperbolic initial boundary value problems developed in [18] and that encompasses all of the above examples that involve hyperbolic systems. Such a general theory does not exist for dispersive perturbations of hyperbolic systems, but we treat two important examples involving Boussinesq systems. In the first one, we show that the nature of the initial boundary value problem shares little in common with the hyperbolic configuration. For instance, the problem has the structure of an ODE and no higher order compatibility conditions on the data are required to have solutions of high regularity. These differences naturally raise the questions of the control of the time of existence and of the dispersionless limit; they are addressed in a second example motivated by a wave-structure interaction problem. We explain the approach developed in [13] to treat this problem, pointing out the role played by dispersive boundary layers.

Relation between orbital velocities, pressure, and surface elevation in nonlinear nearshore water waves.

K. Martins, P. Bonneton, D. Lannes, and H. Michallet, Journal of Physical Oceanography 51 (2021), 3539–3556. Download

The inability of the linear wave dispersion relation to characterize the dispersive properties of nonlinear shoaling and breaking waves in the nearshore has long been recognized. Yet, it remains widely used with linear wave theory to convert between subsurface pressure, wave orbital velocities, and the free surface elevation associated with nonlinear nearshore waves. Here, we present a nonlinear fully dispersive method for reconstructing the free surface elevation from subsurface hydrodynamic measurements. This reconstruction requires knowledge of the dispersive properties of the wave field through the dominant wavenumbers magnitude k, representative in an energy-averaged sense of a mixed sea state composed of both free and forced components. The present approach is effective starting from intermediate water depths—where nonlinear interactions between triads intensify—up to the surf zone, where most wave components are forced and travel approximately at the speed of nondispersive shallow-water waves. In laboratory conditions, where measurements of k are available, the nonlinear fully dispersive method successfully reconstructs sea surface energy levels at high frequencies in diverse nonlinear and dispersive conditions. In the field, we investigate the potential of a reconstruction that uses a Boussinesq approximation of k, since such measurements are generally lacking. Overall, the proposed approach offers great potential for collecting more accurate measurements under storm conditions, both in terms of sea surface energy levels at high frequencies and wave-by-wave statistics (e.g., wave extrema). Through its control on the efficiency of nonlinear energy transfers between triads, the spectral bandwidth is shown to greatly influence nonlinear effects in the transfer functions between subsurface hydrodynamics and the sea surface elevation.

Ability of the nonlinear method to reconstruct an extreme wave (below in green, plot of the nonlinear correction)

Freely floating objects on a fluid governed by the Boussinesq equations

G. Beck, D. Lannes, Ann. IHP/Analyse non linéaire, to appear Download

We investigate here the interactions of waves governed by a Boussinesq system with a partially immersed body allowed to move freely in the vertical direction. We show that the whole system of equations can be reduced to a transmission problem for the Boussinesq equations with transmission conditions given in terms of the vertical displacement of the object and of the average horizontal discharge beneath it; these two quantities are in turn determined by two nonlinear ODEs with forcing terms coming from the exterior wave-field. Understanding the dispersive contribution to the added mass phenomenon allows us to solve these equations, and a new dispersive hidden regularity effect is used to derive uniform estimates with respect to the dispersive parameter. We then derive an abstract general Cummins equation describing the motion of the solid in the return to equilibrium problem and show that it takes an explicit simple form in two cases, namely, the nonlinear non dispersive and the linear dispersive cases; we show in particular that the decay rate towards equilibrium is much smaller in the presence of dispersion. The latter situation also involves an initial boundary value problem for a nonlocal scalar equation that has an interest of its own and for which we consequently provide a general analysis.

Waves interacting with an object moving vertically

Influence of dispersion on the oscillations of an object released from an out of equilibrium position

Wave-Structure Interactions and Wave Energies

Siam News Volume 54 | Number 04 | May 2021 Online article

Some issues on the mathematical analysis of wave-structure interaction are presented, and their relevance for the development of wave energies are reviewed.

Creation of the institute « Mathematics of the Planet Earth »

ICIAM Dianoia, Volume 9, Issue 2, April 2021 Online article

Interview with Laure Saint-Raymond and Arnaud Guillin, directors of the Institute of Mathematics of the Planet Earth

Interview: Building interdisciplinarity around Ocean sciences

ICIAM Dianoia, Volume 9, Issue 1, January 2021 Online article

The UN general assembly has proclaimed the Decade of Ocean Science for Sustainable Development (2021-30). The French National Centre for Scientific Research (CNRS) is among the world’s leading research institutions. Its scientists explore the living world, matter, mathematics, the universe, and the functioning of human societies in order to meet the major challenges of today and tomorrow. How does this institution plan to address the specific issue of the ocean, and what role may mathematics play?

Anne Corval, advisor to the scientific director of the CNRS, coordinated a Task Force Ocean to prepare the institution for this Decade, and Fabrizio D’Ortenzio, oceanographer at the CNRS, was one of the animators of the meetings organized to gather contributions from all sciences. In this interview, they address this question.

Modeling shallow water waves

D. Lannes, Nonlinearity 33 (2020), R1 Download

We review here the derivation of many of the most important models that appear in the literature (mainly in coastal oceanography) for the description of waves in shallow water. We show that these models can be obtained using various asymptotic expansions of the ‘turbulent’ and non-hydrostatic terms that appear in the equations that result from the vertical integration of the free surface Euler equations. Among these models are the well-known nonlinear shallow water (NSW), Boussinesq and Serre–Green–Naghdi (SGN) equations for which we review several pending open problems. More recent models such as the multi-layer NSW or SGN systems, as well as the Isobe– Kakinuma equations are also reviewed under a unified formalism that should simplify comparisons. We also comment on the scalar versions of the various shallow water systems which can be used to describe unidirectional waves in horizontal dimension d = 1; among them are the KdV, BBM, Camassa–Holm and Whitham equations. Finally, we show how to take vorticity effects into account in shallow water modeling, with specific focus on the behavior of the turbulent terms. As examples of challenges that go beyond the present scope of mathematical justification, we review recent works using shallow water models with vorticity to describe wave breaking, and also derive models for the propagation of shallow water waves over strong currents.

Normal mode decomposition and dispersive and nonlinear mixing in stratified fluids

B. Desjardins, D. Lannes, J.-C. Saut, Water Waves (2020), 1-40 Download

Motivated by the analysis of the propagation of internal waves in a stratified ocean, we consider in this article the incompressible Euler equations with variable density in a flat strip, and we study the evolution of perturbations of the hydrostatic equilibrium corresponding to a stable vertical stratification of the density. We show the local well-posedness of the equations in this configuration and provide a detailed study of their linear approximation. Performing a modal decomposition according to a Sturm–Liouville problem associated with the background stratification, we show that the linear approximation can be described by a series of dispersive perturbations of linear wave equations. When the so-called Brunt–Vaisälä frequency is not constant, we show that these equations are coupled, hereby exhibiting a phenomenon of dispersive mixing. We then consider more specifically shallow water configurations (when the horizontal scale is much larger than the depth); under the Boussinesq approxima-tion (i.e., neglecting the density variations in the momentum equation), we provide a well-posedness theorem for which we are able to control the existence time in terms of the relevant physical scales. We can then extend the modal decomposition to the nonlinear case and exhibit a nonlinear mixing of different nature than the dispersive mixing mentioned above. Finally, we discuss some perspectives such as the sharp stratification limit that is expected to converge towards two-fluid systems.

Field data-based evaluation of methods for recovering surface wave elevation from pressure measurements

A. Mouragues, P. Bonneton, D. Lannes, B. Castelle, and V. Marieu, Coastal Engineering, 150:147 – 159, 2019 Download

We compare different methods to reconstruct the surface elevation of irregular waves propagating outside the surf zone from pressure measurements at the bottom. The traditional transfer function method (TFM), based on the linear wave theory, predicts reasonably well the significant wave height but cannot describe the highest frequencies of the wave spectrum. This is why the TFM cannot reproduce the skewed shape of nonlinear waves and strongly underestimates their crest elevation. The surface elevation reconstructed from the TFM is very sensitive to the value of the cutoff frequency. At the individual wave scale, high-frequency tail correction strategies associated with this method do not significantly improve the prediction of the highest waves. Unlike the TFM, the recently developed weakly-dispersive nonlinear reconstruction method correctly reproduces the wave energy over a large number of harmonics leading to an accurate estimation of the peaked and skewed shape of the highest waves. This method is able to recover the most nonlinear waves within wave groups which some can be characterized as extreme waves. It is anticipated that using relevant reconstruction method will improve the description of individual wave transformation close to breaking.

Location map with the field site of La Salie indicated by the black circle. (b) Unmanned aerial vehicle photo of the field site at mid-tide during the experiment. A video system was installed on the pier shown in the lefthand side of the image. The yellow star and the red star show the location of the video system and the instrument, respectively, during the experiment.

Waves interacting with a partially immersed obstacle in the Boussinesq regime

D. Bresch, D. Lannes, G. Métivier, to appear in Analysis & PDE, Analysis & PDE, 14 (2021), 1085-1124 Download

This paper is devoted to the derivation and mathematical analysis of a wave-structure interaction problem which can be reduced to a transmission problem for a Boussinesq system. Initial boundary value problems and transmission problems in dimension d= 1 for 2 x 2 hyperbolic systems are well understood. However, for many applications, and especially for the description of surface water waves, dispersive perturbations of hyperbolic systems must be considered. We consider here a conguration where the motion of the waves is governed by a Boussinesq system (a dispersive perturbation of the hyperbolic nonlinear shallow water equations), and in the presence of a fixed partially immersed obstacle. We shall insist on the differences and similarities with respect to the standard hyperbolic case, and focus our attention on a new phenomenon, namely, the apparition of a dispersive boundary layer. In order to obtain existence and uniform bounds on the solutions over the relevant time scale, a control of this dispersive boundary layer and of the oscillations in time it generates is necessary. This analysis leads to a new notion of compatibility condition that is shown to coincide with the standard hyperbolic compatibility conditions when the dispersive parameter is set to zero. To the authors’ knowledge, this is the rst time that these phenomena (likely to play a central role in the analysis of initial boundary value problems for dispersive perturbations of hyperbolic systems) are exhibited.