Nonlinear Dispersive Waves

Table of Contents

1. Frontmatter

2. Boundary Value Problems Related to the Muskat Problem

   Joachim Escher, Tony Lyons

   Abstract

   The Muskat problem is a two-phase moving boundary problem describing the motion of the flow of two immiscible fluids separated by a sharp interface. Since Darcy’s law is a widely used simplification for the relevant hydrodynamics under consideration, the corresponding velocity potentials are determined as solutions of elliptic boundary value problems. Of particular interest in applications are cases allowing delta distributions as boundary data. Applying tools from classical analysis, several such scenarios are discussed.

3. Some Flow Characteristics of Stokes Waves via Complex Analysis

   Olivia Constantin

   Abstract

   We present and develop some recent results about flow quantities associated with irrotational periodic travelling waves that propagate at the surface of water over a flat bed.

4. Recovery of Traveling Water Waves with Smooth Vorticity from the Horizontal Velocity on a Line of Symmetry for Various Wave Regimes

   Daniel Böhme, Bogdan-Vasile Matioc

   Abstract

   In the general context of rotational water waves with a given smooth vorticity, it is shown that the wave profile can be recovered from the horizontal component of the velocity field on a line of symmetry. The method, which applies to waves on water of finite and infinite depth, uses only the values of the horizontal velocity of particles located on the line of symmetry that are close to the wave surface. In fact, together with the wave surface, we also recover the velocity field in a suitable surface layer. The explicit recovery formula is valid under the assumption that there are no stagnation points in the fluid for both periodic and solitary waves in each of the three regimes of gravity, capillary–gravity, and capillary waves. The efficiency of this method is illustrated in the context of the explicit solutions provided by Crapper for periodic capillary waves and Gerstner for periodic gravity waves.

5. Numerical Computation of Steady Rotational Waves and Recovery of the Surface Profile from Bottom Pressure Measurements

   Joris Labarbe, Didier Clamond

   Abstract

   This chapter demonstrates how to easily compute rotational gravity waves and solve an inverse problem for the recovery of surface wave profiles from discrete pressure measurements at the seabed. For the first step, we solve the steady Euler equations using a boundary integral method based on the Cauchy integral formula. For the recovery procedure, we reduce the nonlinear equations to an implicit algebraic expression for the free surface and solve it numerically by means of an iterative algorithm. We compare both computed and recovered solutions through a couple of non-trivial numerical examples to illustrate the effectiveness of our approach.

6. Hamiltonian Models for the Propagation of Long Gravity Waves, Higher-Order KdV-Type Equations and Integrability

   Rossen I. Ivanov

   Abstract

   A single incompressible, inviscid, irrotational fluid medium bounded above by a free surface is considered. The Hamiltonian of the system is expressed in terms of the so-called Dirichlet–Neumann operators. The equations for the surface waves are presented in Hamiltonian form. Specific scaling of the variables is selected, which leads to a KdV approximation with higher-order nonlinearities and dispersion (higher-order KdV-type equation, or HKdV). The HKdV is related to the known integrable PDEs with an explicit nonlinear and nonlocal transformation.

7. An Introduction to the Zakharov Equation for Modelling Deep-Water Waves

   Raphael Stuhlmeier

   Abstract

   The Hamiltonian formulation of the water wave problem due to Zakharov and the reduced Zakharov equation derived therefrom have great utility in understanding and modelling water waves. Here we set out to review the cubic Zakharov equation and its uses in understanding deterministic waves in deep water. The background of this equation is developed and several applications are explored. Chief among these is an understanding of dispersion corrections and the energy exchange among modes. It is hoped that readers will be motivated to explore this powerful reformulation of the cubically nonlinear water wave problem for themselves.

8. Rotating Convection and Flows with Horizontal Kinetic Energy Backscatter

   Paul Holst, Jens D.  M. Rademacher, Jichen Yang

   Abstract

   Numerical simulations of large-scale geophysical flows typically require unphysically strong dissipation for numerical stability. Toward energetic balance various schemes have been devised to reinject this energy, in particular by horizontal kinetic energy backscatter. In a set of papers, some of the authors have studied this scheme through its continuum formulation with momentum equations augmented by a backscatter operator, e.g., in rotating Boussinesq and shallow water equations. Here we review the main results about the impact of backscatter on certain flows and waves, including some barotropic, parallel, and Kolmogorow flows, as well as internal gravity waves and geostrophic equilibria. We particularly focus on the possible accumulation of injected energy in explicit medium-scale plane waves, which then grow exponentially and unboundedly, or yield bifurcations in the presence of bottom drag. Beyond the review, we introduce the rotating 2D Euler equations with backscatter as a guiding example. For this we prove the new result that unbounded growth is a stable phenomenon occurring in open sets of phase space. We also briefly consider the primitive equations with backscatter and outline global well-posedness results.

9. Flexural-Gravity Waves Under Ice Plates and Related Flows

   Emilian I. Părău, Claudia Ţugulan, Olga Trichtchenko, Alberto Alberello

   Abstract

   An overview of recent studies of waves propagating at the surface of a fluid covered by floating ice plates is provided in this chapter. Different models for the ice cover are considered, and experimental results with moving loads are discussed. Theoretical and numerical studies of linear solutions, weakly nonlinear models as well as fully nonlinear results are presented. Additionally, the attenuation of waves in the Marginal Ice Zone is briefly addressed.

10. Nonlinear Water Waves and Wave–Current Interactions at Arbitrary Latitude

    Delia Ionescu-Kruse

    Abstract

    We survey some exact solutions in the Lagrangian framework, representing waves at arbitrary latitude that propagate eastward or westward above a flow which accommodates a constant underlying background current, waves that can be both in the direction of the current and in the opposite direction. These waves are linearly unstable to short-wavelength perturbations, if their steepness exceeds a specific threshold. This threshold depends on the latitude and the strength of the underlying current.

11. Hollow Vortices as Nonlinear Waves

    Samuel Walsh

    Abstract

    In this chapter, we discuss recent progress on the existence of steady collections of hollow vortices made by the author with Chen et al. (Desingularization and global continuation for hollow vortices, Preprint, arXiv:2303.03570, 2023). In doing so, we endeavor to highlight the mathematical similarities between the hollow vortex system on the one hand and traveling water waves beneath vacuum on the other. While hollow vortices are not nonlinear waves in the strictest sense, many of the strategies developed in the study of water waves are extremely helpful in their analysis. Several extensions of the results in Chen et al. (Desingularization and global continuation for hollow vortices, Preprint, arXiv:2303.03570, 2023) are also provided, and we outline some important open problems to be considered in future work.

12. Spherical Coordinates for Arctic Ocean Flows

    A. Constantin, R. S. Johnson

    Abstract

    The failure of the conventional geographic coordinates at the poles is a major inconvenience in the study of Arctic Ocean flows. We propose a geometrically elegant solution that consists in rotating the geographical system, thus moving the singularities away from the poles. After presenting the various methods—matrix, axis angle, Euler angles, quaternions—for describing rotations in 3-space, we introduce the new approach, and we outline an application to the Transpolar Drift current.