Schedule for: 24w5207 - Nonlinear Water Waves: Rigorous Analysis and Scientific Computing

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Meet and Greet at BIRS Lounge (PDC)

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

We present a framework to compute and study two-dimensional water waves that are quasi-periodic in space and/or time. This means they can be represented as periodic functions on a higher-dimensional torus by evaluating along irrational directions. In the spatially quasi-periodic case, we consider both traveling waves and the general initial value problem. In both cases, the nonlocal Dirichlet-Neumann operator is computed using conformal mapping methods and a quasi-periodic variant of the Hilbert transform. We obtain traveling waves either as a generalization of the Wilton ripple problem or through bifurcation from large-amplitude periodic waves. In the temporally quasi-periodic case, we devise a shooting method to compute standing waves with 3 quasi-periods as well as hybrid traveling-standing waves that return to a spatial translation of their initial condition at a later time. Many examples will be given to illustrate the types of behavior that can occur.

In this talk we describe stable High-Order Spectral algorithms for the numerical simulation of Dirichlet-Neumann operators (DNOs) which arisein boundary value and free boundary problems from a wide variety of applications (e.g., fluid and solid mechanics, electromagnetic and acoustic scattering). More specifically, we consider DNO defined on domains inspired by the simulation of ocean waves subject to quasi-periodic boundary conditions. We have recently shown that the DNO, when perturbed from a flat interface configuration, is parametrically analytic (as a function of deformation height/slope) for profiles of finite smoothness. The method of proof suggests a stable and high-order method of numerical simulation that we now present.

We present a framework for computing and studying two-dimensional spatially quasi-periodic gravity-capillary water waves of finite depth. Specifically, we adopt a conformal mapping formulation of the water wave equation and represent quasi-periodic water waves by periodic functions on a higher-dimensional torus. We will present numerical examples of traveling quasi-periodic water waves and the time evolution of water waves over quasi-periodic bathymetry. We will also discuss an approach to extend this study to three dimensions

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

I will give some background and overview on fractional and nonlocal equations appearing either directly in, or associated to, the water wave problem. Originally arising from the nonlocal coupling of the free interface and the fluid bulk in the Euler equations, the interest for nonlocal effects and operators have increased substantially over the last decade or so – and this interest can be seen in other areas of PDE as well. The talk will focus on some instances of such equations; the differences between the local and nonlocal cases, when present; and techniques and results that are available, or open. Since this is a masterclass, I will not delve too deep into technical details, but hope the talk shall be accessible for everyone.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

In this talk, I will present a global bifurcation result for doubly periodic gravity-capillary water waves with vorticity. Specifically, I consider the case that the velocity field is a Beltrami field; that is, the velocity and vorticity fields are collinear. The result is based on a reformulation of the problem as ‚identity + compact‘ and analytic global bifurcation theory. This is joint work with Giang To and Erik Wahlén (both Lund).

The Camassa-Holm-KP equation (CH-KP equation) is a two-dimensional extension of the Camassa-Holm equation, similarly to how the regular KP equation is a two-dimensional extension of the KdV equation. In the talk I will outline how to prove existence of periodically modulated waves, i.e, steady solutions which have a solitary wave profile in the x-direction and a periodic profile in the y-direction. This is achieved through reformulating the problem as a dynamical system for a perturbation of the line solitary wave solutions, where the periodic direction takes the role of time, then applying an infinite dimensional version of the Lyapunov centre theorem. This talk is based on a joint work with Douglas Svensson Seth (NTNU) and Yuexun Wang (Lanzhou University)

We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local well-posedness in the Hadamard sense, i.e., local existence, uniqueness, and the first proof of continuous dependence on the data, all in low regularity Sobolev spaces; (ii) Enhanced uniqueness: Our uniqueness result holds at the level of the Lipschitz norm of the velocity and Holder 1/2 regularity of the free surface; (iii) Stability bounds: We construct a nonlinear functional which measures, in a suitable sense, the distance between two solutions (even when defined on different domains) and we show that this distance is propagated by the flow; (iv) Energy estimates: We prove refined, essentially scale invariant energy estimates for solutions, relying on a newly constructed family of elliptic estimates; (v) Continuation criterion: We give the first proof of a sharp continuation criterion at the same level as the Beale-Kato-Majda criterion for the boundaryless case; (vi) A novel proof of the construction of regular solutions. Our entire approach is in the Eulerian framework and can be adapted to work in more general fluid domains

Periodic travelling waves that solve the capillary-gravity Whitham equation have been fully characterised in the case of small and even waves. This characterisation is complemented by the work presented in this talk dealing with small asymmetric periodic travelling waves. Such asymmetric waves are far more scarce than the even ones and can only be constructed in certain cases for weak surface tension. The method also generalizes in a straightforward way to a class of similar equations for which we either can prove the existence of or non-existence of asymmetric solutions. However, the proof relies on some technical calculations that are different for each equation. We discuss how this can be done for the Babenko equation, which is equivalent to the full water wave problem, to determine the existence of small amplitude Capillary-Gravity Waves.

The gravity current is the flow of one fluid within another caused by the density difference between the two fluids. Benjamin (J. Fluid Mech., vol.31, 1968, pp.209-248) derived an approximate solution on the assumption that the interface moves in permanent form with constant speed and a corner flow is created near the separation point. This work evaluates the accuracy of Benjamin's approximate solution, and improves it using local flow analyses near some critical points of gravity currents. The results show that the secondary singularities at the critical points are essential for gravity currents

The resonant interaction of waves is a significant energy-transfer mechanism in our oceans, and is associated with instabilities such as the Benjamin-Feir (or modulational) instability. Provided there are sufficiently many conserved quantities the resonant interaction equations can be integrated explicitly in terms of elliptic functions. This approach of integration, pursued at least since the 1960s in the context of water waves, leads to the somewhat cumbersome task of classifying the roots of a polynomials which depend on often rather complicated coefficients. It also threatens to obscure the connection between a resonant interaction and the associated (linear) instability, particularly as the latter is typically approached via the assumption of small modal amplitudes. In this talk, I will suggest that reducing resonant interaction to two variables (a planar dynamical system) is preferable to reduction to a single variable (integration). By suitable choice of dynamic phase and energy variables, the classification of solutions can be simplified greatly, and it is often possible to identify physically interesting bifurcation parameters. The phase plane allows for a transparent view of the associated dynamics, and a clear role for the linear stability analysis. Moreover, avoiding elliptic functions makes the presentation more pedagogically accessible. I will illustrate the above with a variety of examples from water waves and hydrodynamics. I aim to show how new insight can be gained, and new interpretations of classical results found, from the dynamical systems perspective.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Historically, most research on water waves has been based on the assumption of irrotational flow, but in the last decades there has been a lot of progress on waves with vorticity. This is for example useful for modelling wave-current interactions. In my talk I will give an overview of recent analytical work on steady (or travelling) waves with vorticity, focusing on the periodic situation. In the two-dimensional setting this includes results on large-amplitude gravity waves, where one finds overhanging waves in addition to the classical peaked highest waves which also appear in the irrotational case. The three-dimensional problem is more challenging, but I will describe some existence results obtained in the last years.

Nearshore hydrodynamics involves complex interactions across multiple spatial and temporal scales, driven by various wave processes. Among these, wave breaking and the associated energy dissipation are particularly important. Despite advances in understanding wave breaking, the precise mechanisms that trigger it remain unresolved. This study explores the use of the Lagrangian downward acceleration of fluid particles near the wave crest as a dynamic criterion for identifying wave breaking in shallow waters. The idea of using downward acceleration as an indicator traces back to Longuet-Higgins (1963), who theoretically showed that the acceleration near the crest of a regular wave is −0.5g for the highest wave in deep water. While several threshold values have been proposed through theoretical and experimental investigations, no general consensus has been reached. Using stereo imaging of tracer particles in the surf at Sylt, Germany, coupled with careful data analysis and visual assessment of approximately 100 wave events, we demonstrate that this criterion accurately classifies breaking and non-breaking waves in over 91% of cases. This work is in collaboration with H. Kalisch, M. Buckley, M. Bjørnestad, T. Gronemann, K. Holand, J. Horstmann, M. Streβer, and S. Fromenteau

The cylindrical KdV equation can be derived as a long wave approximation for the description of radially symmetric waves for a number of 2D dispersive systems. In this short note we discuss a few aspects related to the validity question of this formal approximation. For a 2D Boussinesq equation we present an approximation result. This is joint work with D. Pelinovsky.

The Camassa-Holm equation models the unidirectional propagation of waves in shallow water. The stability of its solitary traveling wave solutions with respect to perturbations in the direction of propagation has been extensively studied. In this short talk, I will focus on transverse stability. To this end we consider a two-dimensional generalisation of the Camassa-Holm equation, which is similar to the way the KP equation extends the famous KdV equation to two dimensions. To prove transverse stability we study the spectrum of an operator which arises after linearisation around the perturbation in suitably weighted spaces. We show that the double eigenvalue of the linearized equations related to the translational symmetry breaks under a transverse perturbation into a pair of the asymptotically stable resonances and that the continuous spectrum is located in the left half-plane. Moreover, we prove that small-amplitude solitary waves are linearly stable with respect to transverse perturbations by performing careful resolvent estimates and making use of an asymptotic reduction of CH to KdV. The talk is based on joint work with Yue Liu and Dmitry Pelinovsky.

In this short talk I briefly present some recent results where we derive a higher-order KdV equation (HKdV) as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also includes Coriolis forces. We show that there is an explicit transformation connecting our derived HKdV with integrable equations of a similar type, namely KdV5, Kaup-Kuperschmidt equation, Sawada-Kotera equation, Camassa-Holm and Degasperis-Procesi equations. This is joint work with Rossen Ivanov and Zisis Sakellaris.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

The Birkhoff-Rott integral gives the fluid velocity on a vortex sheet, and is the basis of some versions of well-posedness theory for water waves. The Birkhoff-Rott integral has two usual forms, one for flows which decay at horizontal infinity, and one for flows which are horizontally periodic. We give a new formula for the Birkhoff-Rott integral which unifies and extends these: it is a single formula which applies in both the decaying and periodic cases, and which applies more generally.

I will provide an overview of relevant results (analytical, asymptotic, computational) in the literature, both classical and more recent on the instabilities of water waves. I will emphasize especially results for steep waves, near the extreme 120 degree wave.

We examine the stability of two-dimensional waves on the surface of inviscid fluid of infinite depth in the presence of a constant vorticity shear current. The basic state solution corresponds to a periodic travelling wave that is moving at constant speed. The branch of periodic travelling waves is remarkable in that it can be described by an exact solution (Hur & Wheeler 2020). Our concern is with the linear stability of these travelling waves. We study this first for small amplitude waves and we identify resonant triad interactions leading to instability for non-zero wave amplitude. Stability for arbitrary amplitude waves is determined numerically using a Floquet-Fourier-Hill approach and a collocation method. The effect of weak gravity is also considered.

I will discuss the ‘global’ nonlinear asymptotic stability of the traveling front solutions to the Korteweg-de Vries–Burgers equation, and other dispersive-dissipative perturbations of the Burgers equation. Earlier works made strong use of the monotonicity of the profile, for relatively weak dispersion effects. We exploit the modulation of the translation parameter, establishing a new stability criterion that does not require monotonicity. Instead, a certain Schrodinger operator in one dimension must have exactly one negative eigenvalue, so that a rank-one perturbation of the operator can be made positive definite. Counting the number of bound states of the Schrodinger equation, we find a sufficient condition in terms of the ’width’ of a front. We analytically verify that our stability criterion is met for an open set in the parameter regime including all monotone fronts. Our numerical experiments, revealing more stable fronts, suggest a computer-assisted proof. Joint with Blake Barker, Jared Bronski, and Zhao Yang

In 1981, McLean discovered via numerical methods that Stokes waves, even those of small amplitude, are unstable with respect to transverse perturbations of the initial data. A proof of this fact has been missing ever since. In this talk, I present the first ever proof of the existence of transverse instabilities of Stokes waves. In particular, it will be shown that the linearization of the full water wave equations about the Stokes waves admit solutions that grow exponentially in time. The proof is joint work with Huy Nguyen at the University of Maryland and Walter Strauss at Brown University.

We consider the two-dimensional water-wave problem with a general non-zero vorticity field in a fluid volume with a flat bed and a free surface.The nonlinear equations of motion for the chosen surface and volume variables are expressed with the aid of the Dirichlet-Neumann operator and the Green function of the Laplace operator in the fluid domain. Moreover, we provide new explicit expressions for both objects. The field of a point vortex and its interaction with the free surface is studied as an example. In the small-amplitude long-wave Boussinesq and KdV regimes, we obtain appropriate systems of coupled equations for the dynamics of the point vortex and the time evolution of the free surface variables. This is joint work with Rossen Ivanov.

We study the modulation of short surface gravity waves interacting with long internal waves in a two-layer system. A second-order nonlinear system derived from the two-layer Euler equations is studied numerically and it is shown that the modulation is much pronounced under a near-resonance condition. The results are compared with laboratory experiments and possible reduced models are discussed.

We consider a three-fluid system confined between two rigid walls. It is well known that the Gardner equation, when adopted to describe long waves in this physical system, is able to predict the coexistence of internal solitary waves of opposite polarity, for some physical parameters. This is so because, across the parameter space, the cubic nonlinearity coefficient for the Gardner equation may change sign. We shall test the validity of these results by analysing the predictions of the strongly, and fully, nonlinear theories. References: [1] Doak, A., Barros, R., Milewski, P.A. 2022 Large mode-2 internal solitary waves in three-layer flows. J. Fluid Mech. 953, A42 [2] Barros, R., Choi, W., Milewski, P.A. 2020 Strongly nonlinear effects on internal solitary waves in three-layer flows. J. Fluid Mech. 883, A16.

1. Kat Philips: Title: Filling the Gap on Droplet Dynamic. Abstract: It is known that in order for the droplet to rebound and not coalesce with the bath, a layer of air, often millimetric thin, must be present throughout the entire impact. In the absence of this air layer, or if it becomes too thin and destabilises, the attractive forces within the liquids will cause the drop to coalesce into the bath. We are interested in the dynamics at play when considering millimetric droplets in the small deformation limit - specifically ensuring that the drop will rebound. Current linear models of this system focused on computational efficiency often neglect the role of air during the dynamics, instead imposing immiscible conditions so the air may be omitted entirely whilst ensuring that coalescence cannot occur. These models, whilst efficient, miss the nuanced dynamics that can arise within the air layer. Our work aims to `fill the gap' on both the knowledge within the literature, but also between the droplet and bath, and dynamically incorporate the air layer into a model for the drop-air-bath dynamics. We utilise classical fluid dynamic techniques to consider each region of fluid independently, then carefully couple the three regions through the air-bath and drop-bath interface, resolving the air layer during impact using lubrication theory to obtain the pressure within the layer which acts upon the bath and fluid 2. Ellie Byrnes. High-Amplitude Stokes Waves in a Fluid of Fixed Finite Depth. Abstract: We apply a Newton-conjugate residual method to the conformal formulation of the one-dimensional Euler Water Wave Problem in terms of the horizontal spatial variable $x$ and the vertical spatial variable $y$ to find Stokes waves in a particular conformal depth. We then find Stokes waves traveling in a fluid of a fixed physical depth through the use of a root-finding method. We compute the poles of these waves viewed as a function of the complex variable $z=x+iy$, identify any bifurcation points that occur as these wave increase in amplitude, and analyze the evolution of various physical quantities (energy, speed, steepness) as a function of amplitude through the lens of the infinite-depth asymptotic analysis done by Longuett-Higgins and Fox. 3. Nils Gutheil. Pulse solutions to a class of quasilinear symmetrisable hyperbolic evolutionary equations over exponentially long time scales. Abstract: This project deals with the construction of generalised pulse solutions to a class of analytic quasilinear symmetrisable evolutionary equations of the form \[ u_{\xi} = A^{\varepsilon}(u)u_{\tau} + B^{\varepsilon}(u) \] whose linearisation exhibits a 1:1 resonance coupled with an infinite number of purely imaginary eigenvalues. Here a generalised pulse solution means a solution that is localised (but not necessarily evanescent) in \(\xi\) and \(2\pi\)-periodic in \(\tau\). The problem is formulated as an infinite-dimensional dynamical system with finitely many stable and unstable and infinitely many neutral directions. By transforming part of the system into a normal form with an exponentially small remainder term, we regard it as a perturbation of the system without remainder term, which has an explicit pulse solution. A suitable iteration scheme for the infinite-dimensional part of the system, combined with energy estimates for quasilinear symmetrisable hyperbolic systems, yields the existence of small-amplitude pulse solutions on time scales which are exponentially large compared to the magnitude of the pulse. 4. Levent Batakci. Preliminary work: The Instabilities of 2-D Periodic Traveling Water Waves. Abstract: Using the framework due to Nicholls and Reitich, we compute 2-D traveling solutions to the classical water wave problem. The exact reformulation of the classical surface water wave problem due to Ablowitz, Fokas, and Musslimani then provides a convenient framework to examine the spectral stability of these waves. We consider zero-average perturbations and linearize about the computed solutions, in line with the works of Deconinck and Oliveras on 1-D waves. The generalized Fourier-Floquet-Hill method allows us to efficiently and accurately compute the stability spectra of these waves. We examine the instabilities of 2-D waves in both shallow and deep water, comparing our findings to the results for longitudinal and transverse perturbations of 1-D waves.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will survey the results on the local and global well-posedness of the water-wave problems

We are concerned with low regularity solutions for the water wave equations in two space dimensions. Our focus here is on both local and global solutions. Such solutions have been proved to exist earlier but in much higher regularity. Our goal is to improve these results and prove local and global well-posedness under minimal regularity and decay assumptions for the initial data. One key ingredient here is represented by a newly introduced method which we call ``the balanced cubic estimates''. Another is the nonlinear vector field Sobolev inequalities, an idea first introduced together with Tataru in the context of the Benjamin-Ono equations

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In this talk, I will describe recent results related to the interactions of ocean waves with floating objects. This problems are motivated in particular by application to marine renewable energies. Depending on the physical model used to describe the waves (water waves or asymptotic models such as nonlinear shallow water or Boussinesq equations), but also on the dimension, and on the geometry of the object and its motion (fixed, constrained or freely floating), this problem raises different new mathematical questions (corner singularities, initial boundary value problems, free boundary problems with nonkinematic boundary condition, etc.)

I will present recent results on the surface wave determination from measurements of the pressure at the seabed. In particular, I will discuss the effects of surface pressure (wind, surface tension, etc.) and of noise in the measurements

Recently, many exact solutions have been found for surface waves travelling upon a fluid with vorticity and submerged vortices. The simplest of these, with either constant vorticity or a single point vortex within the periodic domain, possesses the same exact solution for capillary waves originally obtained by Crapper. More complicated exact solutions have vorticity and multiple vortices within the periodic domain. These exact solutions were found via conformal mapping and analytic function theory. However, they are likely to be unstable, so the study of their instability and subsequent time evolution is of interest. In this talk, I will discuss a potential-flow formulation of this problem, the study of solution stability, and a 1D surface formulation of the 2D problem in which the vortex evolution conditions within the fluid are still satisfied.

Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

In the early 1900's several authors presented expansions to approximate ripples with harmonic resonance at Bond numbers where Stokes' expansion is singular. These waves are now called Wilton ripples. Their procedure is effectively a Lyapunov-Schmidt reduction, and has been employed numerous times to asymptotically approximate, numerically compute, and prove existence of these waves. Historically solvability conditions in the expansion are satisfied using corrections to the wave speed and the ratio of the two modes. In the classic expansion, the ratio of the two modes can take only special values. Recently another expansion has been used to study the bifurcation of small amplitude bimodal ripples, which shows that waves bifurcate with almost all ratios of the two modes provided the problem is expanded in Bond number. In this talk the history of both expansions will be presented and the connection between the two will be explained

Waves sloshing in a container of rectangular cross-section can behave very differently than those in a circular cylinder. This talk covers one aspect of how geometry affects the nonlinear evolution of waves. Nonlinear resonance is a mechanism by which energy is continuously exchanged between a small number of wave modes, and is common to many nonlinear dispersive wave systems. In the context of free-surface gravity waves such as ocean surface waves, nonlinear resonances have been studied extensively over the past 60 years (and were mentioned in the award of the 2021 Nobel Prize in Physics to Klaus Hasselmann) almost always on domains that are large compared to the characteristic wavelength. In this case, the dispersion relation dictates that only quartic (4-wave) resonances may occur. In contrast, nonlinear resonances in confined three-dimensional geometries have received relatively little attention, where, perhaps surprisingly, stronger 3-wave resonances do occur. We will present results completely characterizing the configuration and dynamics of resonant triads in cylindrical basins of arbitrary cross sections, demonstrating that these triads are ubiquitous, with a rectangular cross section being an exception where they do not occur.

In this talk we will show how the conformal mapping technique can be used to accurately model surface wave generation due to seabed deformations. Our main contribution to this topic is that we are able to specify the seabed deformation in the so-called physical domain instead of an ad hoc parametrization in the canonical domain. This allows us to reproduce laboratory conditions accurately and deal with relatively 'rough' and 'fast' seabed deformations. With this technique at hand we revisit the classical laboratory results obtained by Hammack in the 70's and explore the differences between active and passive tsunami generation.

In this talk, we will present some recent results on the finite-time blowup of hollow vortices. These are solutions of the two-dimensional Euler equations with the fluid domain being the complement of finitely many Jordan curves $\Gamma_1, …, \Gamma_M$. The flow is irrotational and incompressible, but with a nonzero circulation around each boundary component. The ``vortex core'' bounded by each $\Gamma_k$ is modeled as a bubble of ideal gas: the pressure is constant in space and inversely proportional to the area of the vortex. This can be thought of as the isobaric approximation. Our results come in two parts. First, for all m > 3, we construct a family of solutions taking the form of a near-circular m-fold symmetric hollow vortex that collapses self-similarly into the origin in finite time. This represents an implosion in the sense that the pressure in the vortex core simultaneously diverges to infinity. We also obtain a rigidity result: for m = 2 and m=3, the purely circular collapsing vortices are locally unique among all collapsing vortices with uniform velocity at infinity. The second part concerns configurations of multiple hollow vortices. The existence of collapsing configurations of point vortices is classical. We prove that generically, these can be desingularized to yield a families of hollow vortex configurations that exhibit self-similar finite-time implosion. Specific examples of an imploding trio and quartet of hollow vortices are given. This is joint work with Ming Chen (University of Pittsburgh) and Miles Wheeler (University of Bath)

Usually steady water waves are constructed by perturbing (laminar) flows with a flat surface. In the setting of axisymmetric capillary water waves, we go in the somewhat opposite direction, starting at static configurations with constant mean curvature surfaces, so-called unduloids. As an interesting interplay between water waves, geometry, and elliptic integrals, we show rigorously that to any such configuration there connects a global continuum of non-static configurations, which confirms previous numerical observations. Here, we allow for arbitrary vorticity and swirl. This is joint work with Anna-Mariya Otsetova (Aalto) and Erik Wahlén (Lund).

Fully localised three-dimensional solitary waves are steady water waves which are evanescent in every horizontal direction. In this talk I present an existence theory for such waves under the assumptions that the relative vorticity and velocity fields are parallel (`Beltrami flows', which include the special case of irrotational flows), that the free surface of the water takes the form $\{z=\eta(x,y)\}$ for some function $\eta: {\mathbb R}^2\rightarrow{\mathbb R}$, and that the influence of surface tension is sufficiently strong. The governing equations are formulated as a single equation for $\eta$, which is then shown by an argument of Lyapunov-Schmidt type to be locally equivalent to a perturbation of the KP-I equation $$\partial_x^2\left( \partial_x^2\zeta - \zeta - 3\zeta^2\right)-\partial_y^2\zeta=0.\hspace{1in}(\star)$$ It has recently been shown by Liu, Wei and Yang that $(\star)$ has an infinite family $\{\zeta_k\}$ of symmetric ‘lump’ solutions of the form $$\zeta_k(x,y)=-2\partial_x^2\log\tau_k(x,y),\qquad k=1,2,\ldots,$$ where $\tau_k$ is a polynomial of degree $k(k+1)$ given by an explicit formula. They also show that $\zeta_1$ and $\zeta_2$ are nongenerate and announce that the same is true for $\zeta_k$, $k \geq 3$ (details to be published in a later paper). Using these results, one can apply a suitable variant of the implicit-function theorem to construct fully localised three-dimensional solitary waves which are perturbations of scalings of the Liu-Wei-Yang lump solutions to the KP-I equation.

There has been recent interest in the study of surface waves in cylindrical domains, with a particular focus on axisymmetric solitary waves and periodic waves. Examples include waves travelling along the length of a horizontal cylinder of fluid, as well as axisymmetric waves (i.e., targets or spots) on the upper surface of a vertical cylinder of fluid. In each case, the rotational symmetry reduces the problem to a quasi- two-dimensional PDE with spatially-dependent terms and continuity conditions at the origin. While restricting to axisymmetric solutions should represent a simplification of the full three-dimensional problem, much of the analytic framework required to study such solutions is not well-established in polar coordinates. In this talk we will explore an analytic framework for radial functions, utilising non-autonomous differential operators and novel functions spaces, and discuss their application in the context of surface waves. We will also discuss vector calculus, flattening transformations and the Stokes stream function, exploring how each topic relates to our framework.

A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

In polar regions, floating ice on frozen lakes is often used as a road to access isolated communities. Under certain conditions, the ice can be modeled as a thin plate, and studying hydroelastic waves propagating at the ice-water interface is essential for the safe use of these ice roads. A better understanding of wave-ice interactions in the Marginal Ice Zone (MIZ) is also important in the context of climate change. The MIZ is the area between the open ocean and the continuous ice cover in the Arctic and Antarctic regions. In this talk, I will provide an overview of recent results in these areas of research, with a focus on the effects of nonlinearity.

5-day workshop participants are welcome to use BIRS facilities (TCPL ) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 11AM.

We construct gravity water waves with constant vorticity having the approximate form of a disk joined to a strip by a thin neck. This is the first rigorous existence result for such waves, which have been seen in numerics since the 80s and 90s. Our method is related to the construction of constant mean curvature surfaces through gluing, and involves combining three explicit solutions to related problems: a disk of fluid in rigid rotation, a linear shear flow in a strip, and a rescaled version of an “exceptional domain” discovered by Hauswirth, Hélein, and Pacard. This is joint work with Juan Dávila, Manuel del Pino, and Monica Musso.

Everyone has seen the v-shaped Kelvin wake-pattern visible in the wake of a moving object on the surface of water. These patterns are a rare example of a fluid dynamics phenomena well-known to scientists and layman alike. However, the wake is undesirable for a number of reasons; it can cause erosion to river banks and cause wave-drag, thus reducing the fuel efficiency. Therefore, the design of a moving body that can reduce or even eliminate these waves is important for sustainability. A typical approach is to model the boat by an imposed pressure distribution in the free-surface Bernouilli condition. In this talk we show, using a simple mathematical argument, that by a judicious choice of a pressure distribution, wave-free solutions are possible in the context of a model system; the forced Kadometsev-Petviashvili equation. Strikingly, we show that these solutions are stable, so they could potentially be visualised in a physical experiment.

Confining a liquid to a partially filled basin can lead to the spontaneous formation of resonant triads of gravity waves, in which three standing wave modes interact and continuously exchange energy. The existence of these triads depends critically on the dimensions of the container and the depth to which it is filled. Furthermore, the internal resonance induces waves of relatively large magnitude, representing a potential hazard in many industrial and geophysical settings, from the transport of liquid cargo to the flooding of lakes induced by wind-driven seiches. Although recent progress has been made in understanding the onset and evolution of these resonant triads for small-amplitude waves in simple geometries, there remain a number of challenges in the realms of both rigorous analysis and scientific computation. In this talk, I will briefly outline some of the key open questions in developing a complete understanding of the existence and evolution of resonant triads in confined basins, and their mitigation in industrial settings.

In periodic wave motion, particles beneath the wave undergo drift in the direction of wave propagation, a phenomenon known as Stokes drift. While extensive research has been conducted on Stokes drift in water wave flows, its counterpart in electrohydrodynamic flows remains relatively unexplored. In this talk, I will discuss our numerical investigation of Stokes drift beneath periodic traveling irrotational waves in a dielectric fluid under the influence of normal electric fields. Through numerical simulations utilizing conformal mapping, we compute particle trajectories and analyze the resultant Stokes drift behavior beneath periodic traveling waves. Our findings indicate that variations in the electric field impact particle velocities while maintaining trajectory shapes. Moreover, the kinetic energy associated with a particle depends on its depth location and is a non-decreasing convex function in a moving frame and constant in a laboratory frame.

The talk will discuss the solitary-wave solutions of the so-called abcd-Boussinesq system, which is a model of two equations that can describe the propagation of small-amplitude long waves in both directions in water of finite depth and was derived by Bona-Chen-Saut. If the system is Hamiltonian, where the parameters b and d satisfy b = d > 0, small solutions of the system in the energy space are globally defined. Then, a variational approach is applied to establish the existence and nonlinear stability of the set of solitary-wave solutions for generalized abcb-Boussinesq systems. The main idea of the proof is to show that the traveling-wave solutions of the generalized abcb-Boussinesq system converge to nontrivial solitary-wave solutions of the generalized Korteweg-de Vries equation. (This is a joint work with R. de A. Capistrano-Filho and J. R. Quintero)

Wave action in the surf zone is the main factor in setting up current and circulation patterns, including cross-shore and long-shore currents, exit and entrance flows and vortex motion. These wave-driven currents have a significant impact on sediment transport, beach erosion, as well as the distribution of marine microorganisms and pollutants. Understanding the nature of surfzone circulation patterns is also important from the point of view of beach safety in particular in the context of rip currents. In the present study, we employ the Boussinesq Ocean and Surf Zone model (BOSZ) developed by Volker Roeber to explore circulation patterns, focusing particularly on the role of nearshore vortices. More precisely, the emphasis is on understanding how wave parameters such as waveheight, mean period, mean direction as well as environmental conditions such as tidal elevation and bathymetry, influence the nature of the resulting surfzone circulation.

We present some recent results concerning exact solutions to the equations of motion in cylindrical coordinates. Furthermore, we give some criteria for the stability of the exact solutions as well as wave-breaking results.