Schedule for: 16w5112 - Theoretical and Computational Aspects of Nonlinear Surface Waves

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the 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.

Higher-order models for unidirectional propagation of long-crested water waves will be introduced and their analysis sketched.

Following the method of Bridges (2013, PRSA), it is demonstrated how one may derive PDEs with fifth order dispersion from periodic waves (and, in general, relative equilibrium). Many of the coefficients of the emerging nonlinear approximations are directly related to the system's conservation laws, and those of the dispersive terms are tied to a Jordan chain analysis. Examples illustrating how the theory applies will also be discussed.

It was shown by V. E. Zakharov that the equations for water waves can be posed as a Hamiltonian dynamical system, and that the equilibrium solution is an elliptic stationary point. This talk will discuss two aspects of the water wave equations in this context. Firstly, we generalize the formulation of Zakharov to include overturning wave profiles, answering a question posed by T. Nishida. Secondly, we will discuss the question of Birkhoff normal forms for the water waves equations in the setting of spatially periodic solutions, including the function space mapping properties of these transformations.

We present a general counting result for the unstable eigenvalues of linear operators of the form JL in which J and L are skew- and self-adjoint operators, respectively. Assuming that there exists a self-adjoint operator K such that the operators JL and JK commute, we prove that the number of unstable eigenvalues of JL is bounded by the number of nonpositive eigenvalues of K. As an application, we discuss the transverse stability of one-dimensional periodic traveling waves in the classical KP-II (Kadomtsev-Petviashvili) equation.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

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!

It is shown that standing wave is unstable if we consider four wave nonlinear processes. We compare numerical simulation of instability of weakly nonlinear standing waves on the surface of deep fluid in the framework of the primordial dynamical equations and in a laboratory wave tank experiment. The instability offers a new approach for generation of nearly isotropic spectrum using parametric excitation. Direct measurements of spacial Fourier spectrum confirm existence of the instability in a real life conditions for gravity-capillary surface waves. This is a joint work with Sergei Lukaschuk (The University of Hull, UK).

The linear stability of finite-amplitude capillary waves on inviscid sheets of fluid is investigated. Superharmonic and subharmonic perturbations are considered and a conformal mapping technique is used. The instability results are also checked by time integration of the fully nonlinear unsteady equations. This is joint work with Mark Blyth (UEA).

Nonlinear waves travelling at a constant velocity at the surface of a fluid of finite depth are considered. The fluid is assumed to be incompressible and inviscid and the flow to be irrotational. Gravity and surface tension are taken into account. Both two and three dimensional waves are studied. Classical solutions usually assume that the waves are symmetric. We show that there are in addition an infinite number of branches of non-symmetric waves. These include periodic waves, solitary waves and generalised solitary waves. As time permits extensions to hydroelastic waves will be presented.

A multidomain spectral method with compactified exterior domains combined with stable second and fourth order time integrators is presented for Schrödinger equations. The numerical approach allows high precision numerical studies of solutions on the whole real line or in higher dimensions. At examples for the linear and cubic nonlinear Schrödinger equation, this code is compared to transparent boundary conditions and perfectly matched layers approaches. The code can deal with asymptotically non vanishing solutions as the Peregrine breather being discussed as a model for rogue waves. It is shown that the Peregrine breather can be numerically propagated with essentially machine precision, and that localized perturbations of this solution can be studied

A new method is proposed to relate the pressure at the bottom of a fluid, the shape of the bathymetry, and the surface elevation of a wave for steady flow or traveling waves. Given a measurement of any one of these physical quantities (pressure, bathymetry, or surface elevation), a numerical representation of the other two quantities is obtained via a nonlocal nonlinear equation obtained from the Euler formulation of the water-wave problem without approximation. From this new equation, a variety of different asymptotic formulas are derived. The nonlocal equation and the asymptotic formulas are compared with both numerical data and physical experiments. This is joint work with Vishal Vasan and Daniel Ferguson.

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

We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid subject to self-gravitational force and neglecting surface tension in two space dimensions. We show that for smooth data which are size $\epsilon$ perturbations of an equilibrium state, the solution exists and remains smooth for time of at least $O(\epsilon^{-2})$. This should be compared with the lifespan $O(\epsilon^{-1})$ provided by local well-posedness. The key to the proof is to find a nonlinear transformation of the unknown function and a coordinate change, such that the equation for the new unknown in the new coordinate system has no quadratic nonlinear terms. This is a joint work with Lydia Bieri, Shuang Miao and Sohrab Shahshahani.

The talk is concerned with the incompressible, infinite depth water wave equation in two space dimensions, with gravity and constant vorticity but with no surface tension. We consider this problem expressed in position-velocity potential holomorphic coordinates, and prove local well-posedness for large data, as well as cubic lifespan bounds for small data solutions.

I will provide an overview of recent work, joint with Benjamin Harrop-Griffiths and Mihaela Ifrim, concerning enhanced lifespan bounds for small data gravity waves in the presence of a flat bottom. This work is based on two key ideas, (i) the use of holomorphic coordinates, and (ii) our modified energy method, a quasilinear alternative to normal forms.

We prove that weak solutions of the inviscid surface quasi-geostrophic (SQG) equation are not unique, thereby answering an open problem posed by De Lellis and Szekelihidi. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. This is joint work with T. Buckmaster and V. Vicol.

Existence and admissibility of delta-shock solutions is discussed for hyperbolic systems of conservation laws. One of the systems discussed is fully nonlinear, and does not admit a classical Lax-admissible solution to certain Riemann problems. By introducing complex-valued corrections in the framework of the weak asymptotic method, we show that a compressive delta-shock wave solution resolves such Riemann problems. By letting the approximation parameter tend to zero, the corrections become real valued and the resulting distributions fit into a generalized concept of singular solutions [V. G. Danilov and V. M. Shelkovich, Dynamics of propagation and interaction of delta-shock waves in hyperbolic systems, J. Differential Equations 211 (2005), 333-381]. In this framework, it can be shown that every 2x2 system of conservation laws admits delta-shock solutions. As an example, it is shown that the combination of discontinuous free-surface solutions and bottom step transitions naturally leads to singular solutions featuring Dirac delta distributions in the context of shallow-water flows.

Frequency downshift, i.e. a shift in the spectral peak to a lower frequency, in a train of nearly monochromatic gravity waves was first reported by Lake et al. (1977). Even though it is generally agreed upon that frequency downshifting (FD) is related to the Benjamin-Feir instability and many physical phenomena (including wave breaking and wind) have been proposed as mechanisms for FD, its precise cause remains an open question. Dias et al. (2008) added a viscous correction to the Euler equations and derived the dissipative NLS equation (DNLS). In this talk, we introduce a higher-order generalization of the DNLS equation, which we call the viscous Dysthe equation. We outline the derivation of this new equation and present many of its properties. We establish that it predicts FD in both the spectral mean and spectral peak senses. Finally, we demonstrate that predictions obtained from the viscous Dysthe equation accurately model data from experiments in which frequency downshift occurred.

In the 1960s, Benjamin and Feir, and Whitham, discovered that a Stokes wave would be unstable to long wavelength perturbations, provided that (the carrier wave number) $\times$ (the undisturbed water depth) $> 1.363....$ In the 1990s, Bridges and Mielke studied the corresponding spectral instability in a rigorous manner. But it leaves some important issues open, such as the spectrum away from the origin. The governing equations of the water wave problem are complicated. One may resort to simple approximate models to gain insights. I will begin by Whitham's shallow water equation and the wave breaking conjecture, and move to the modulational instability index for small-amplitude periodic traveling waves, the effects of surface tension and constant vorticity. I will then discuss higher order corrections, extension to bidirectional propagation and two-dimensional surfaces. This is based on joint works with Jared Bronski (Illinois), Mat Johnson (Kansas), Ashish Pandey (Illinois), and Leeds Tao (UC Riverside).

We study the propagation of water waves in a channel of variable depth using the long-wave asymptotic regime. We use the Hamiltonian formulation of the problem in which the non-local Dirichlet-Neumann (DN) operator appears explicitly in the Hamiltonian and due to the complexity of the expressions of the asymptotic expansion associated with this operator in the presence of a non-trivial bottom topography. We perform an ad-hoc modification of these terms using a pseudo differential operator (PDO) associated with the bottom topography. In this talk we propose a Whitham-Boussinesq model for bidirectional wave propagation in shallow water that involves a PDO that consider explicitly the expression for the depth profile. The model generalizes the Boussinesq system, as it includes the exact dispersion relation in the case of constant depth. We will introduce an accurate and efficient numerical method that has been developed to compute this PDO. We present the results for the normal modes and eigen-frequencies of the linearized problem for families of different topographies. We also present some experiments of the evolution of some initial wave profiles over different topographies. Due to the ad-hoc nature of this simplified model we present some comparisons between the full expression of the first term of the asymptotic expansion of the DN operator given by Craig, Guyenne, Nicholls, and Sulem and our PDO approach for some specific topographies.

Starting from the nonlocal Whitham equation with its fully dispersive linear operator, we consider the existence of periodic traveling waves that are not small, albeit connected to the line of vanishing solutions. Of particular interest is the existence of a highest, $C^{1/2}$-cusped, traveling wave solution, which is obtained as a limiting case at the end of the main bifurcation branch of $P$-periodic traveling wave solutions. We prove that this regularity is optimal. Given that the Euler equations admits a highest wave that is not cusped, but Lipschitz continuous, it is an interesting question whether a bidirectional Whitham equation, which carries the full two-way dispersion relation from the Euler equations, could encompass a Lipschitz wave as well. Due to reasons to be explained in the talk, it however turns out that the highest wave for the bidirectional Whitham equation, which we prove to exist, is not Lipschitz --- nor is its optimal regularity described by Hölder or Zygmund spaces. We characterize its behaviour near the wave crest. At the end we outline the first steps towards a more general theory. Our interest is a somewhat general large-amplitude theory for nonlinear dispersive equations. The talk is based on joint works with E. Wahlén, Lund, M. A. Johnson and K. M. Claassen, both Kansas.

The Whitham equation is a nonlocal, nonlinear dispersive wave equation introduced by G. B. Whitham as an alternative wave model equation to the Korteweg-de Vries equation, describing the wave motion at the surface on shallow water. The existence of supercritical solitary wave solutions to the Whitham equation has been shown by Ehrnström, Groves, and Wahlén in 2012. We prove that any such solution decays exponentially, is symmetric and has exactly one crest. Moreover, the structure of the Whitham equation allows to conclude that conversely any classical, symmetric solution constitutes a traveling wave. In fact, the latter result holds true for a large class of partial differential equations sharing a certain structure.

We consider a class of Green-Naghdi type systems and prove the existence of solitary wave solutions. The solutions are identifed as critical points of a scalar functional and we are able to show that there exist minimizers of this functional, under certain constraints. A key component of the proof is the use of the concentration compactness principle. The talk is based upon a work in progress with Erik Wahlen (Lund University) and Vincent Duchene (University of Rennes).

Very long waves running ahead of ships is a recent phenomenon in the Oslofjord in Norway. The waves are triggered when the new very large and relatively fast cruiseferries are running across significant depth changes in the shallow fjord. An asymptotic linear analysis expressing the upstream wave field in terms of a pressure impulse at the depth change is complemented by fully dispersive calculations of the upstream waves. At very shallow positions the local ship speed becomes critical where the effect of nonlinearity is analyzed. A wave length of 700 m, wave height at the shore of 1 m, average depth of the fjord of 35 m, depth change similar to the average depth, ship length of 200 m, moving at subcritical speed, are typical characteristics.

We report on the mathematical and numerical modelling of (non)linear ship motion in (non)linear water waves. We derive a coupled model for the wave-ship dynamics following a variational methodology, in order to ensure zero numerical damping which is important for wave propagation. The final system of evolution equations comprises the classical water-wave equations for incompressible and irrotational waves, and a set of equations describing the dynamics of the ship. The novelty in our model is in the presence of a physical restriction on the water height under the ship, which is enforced through an inequality constraint via a Lagrange multiplier. The model is solved numerically using a variational (dis)continuous Galerkin finite element method with special, new and robust time integration methods. Here we aim to show numerical results for the dynamics of the coupled system in a hierarchy of increasing complexity: linear water-wave and linear ship dynamics, and potentially also fully coupled (non)linear water-wave and nonlinear ship dynamics.

The goal of this talk is to derive some equations describing the interaction of a floating solid structure and the surface of a perfect fluid. This is a double free boundary problem since in addition to the water waves problem (determining the free boundary of the fluid region), one has to find the evolution of the contact line between the solid and the surface of the water. The so-called floating body problem has been studied so far as a three-dimensional problem. Our first goal is to reduce it to a two-dimensional problem that takes the form of a coupled compressible-incompressible system. We will also show that the hydrodynamic forces acting on the solid can be partly put under the form of an added mass-inertia matrix, which turns out to be affected by the dispersive terms of the equations.

This talk will focus on high-order finite difference methods for solving potential flow approximations of nonlinear surface waves interacting with marine structures. Of special interest is the loading and wave-induced response of sailing ships, where it is convenient to work in a reference frame which is translating at constant speed. This introduces non-linear convective terms into the free-surface boundary conditions which are found to require nonlinear numerical schemes to achieve robust and stable solutions. The work builds on the basic numerical solution strategy described in [1, 2]. Inspired by work reviewed for example by Shu [4], we have developed a simplified version of the Weighted Essentially Non-Oscillatory (WENO) scheme which is only slightly more diffusive than the equivalent order centered finite difference scheme. The equations are then put into Hamilton-Jacobi form, and the WENO convective approximations are combined using the Roe-Fix numerical flux proposed by Shu & Osher [5]. In contrast to a simple upwinding strategy, this approach is found to give accurate and stable solutions for all combinations of wave celerity and ship forward speed. To introduce the ship geometry into the numerical solution, we have developed an Immersed Boundary Method based on Weighted Least-Squares difference approximations. This scheme will be described and some preliminary results will be presented. Challenges with respect to tracking the body-free surface intersection line, and treating wave-breaking in a rational way will be raised for discussion.

References:

1. H. B. Bingham and H. Zhang. On the accuracy of finite difference solutions for nonlinear water waves. J. Engineering Math., 58:211-228, 2007.
2. A. P. Engsig-Karup, H. B. Bingham, and O. Lindberg. An efficient flexible-order model for 3D nonlinear water waves. J. Comput. Phys., 228:2100-2118, 2009.
3. A. P. Engsig-Karup, H. B. Bingham, O. Lindberg, and B. T. Paulsen. OceanWave3D; an efficient coastal engineering tool for nonlinear waves., 2015. https://github.com/apengsigkarup/OceanWave3D-Fortran90.
4. C.-W. Shu. High order weighted essentially non-oscillatory schemes for convection dominated problems. SIAM Review, 51(1):82-126, 2009.
5. C.-W. Shu and S. Osher. Essentially Non-Oscillatory and Weighted Essentially Non-Oscillatory Schemes for hyperbolic conservation laws. J. Comput. Phys., 77(2):439-471, 1989.

We have previously, in joint work with Jerry Bona, David Nicholls, and Michael Siegel, demonstrated that truncated series models of gravity water waves exhibit ill-posedness. In joint work with Shunlian Liu, we show that the addition of sufficiently strong dispersion makes such a system well-posed. Physically, this strong dispersion can be relevant, for instance, for hydroelastic waves. The proof uses techniques of paradifferential calculus.

Act 1: We consider periodic traveling waves in a flow with constant vorticity and look in detail at the internal structure of the flow, including particle paths, stagnation points and pressure fields. The results illustrate some of the known theoretical results and also point to new interesting features.

Act 2: Capillary gravity wave packets are described asymptotically by the focussing 2D NLS equation. This equation has solutions that blow up in finite time. We explore the dynamics of blowup initial data under the full Euler equations.

We consider the classical gravity-capillary water-wave problem in its usual formulation as a three-dimensional free-boundary problem for the Euler equations for a perfect fluid. A solitary wave is a solution representing a wave which moves in a fixed direction with constant speed and without change of shape; it is fully localised if its profile decays to the undisturbed state of the water in every horizontal direction. The existence of fully localised solitary waves has been predicted on the basis of simpler model equations, namely the Kadomtsev-Petviashvili (KP) equation in the case of strong surface tension and the Davey-Stewartson (DS) system in the case of weak surface tension. In this talk we confirm the existence of such waves as solutions to the full water-wave problem and give rigorous justification for the use of the model equations. This is joint work with Boris Buffoni and Erik Wahlen.

In this talk, we will report some recent results concerning two-dimensional gravity solitary water waves with hereogeneous density. The fluid domain is assumed be bounded below by an impenetrable flat ocean bed, while the interface between the water and vacuum above is a free boundary. Our main existence result states that, for any smooth choice of upstream velocity and streamline density function, there exists a path connected set of such solutions that includes large-amplitude surface waves. Indeed, this solution set can be continued up to (but does not include) an ``extreme wave`` that possess a stagnation point. We will also discuss a number of results characterizing the qualitative features of solitary stratified waves. In part, these include bounds on the Froude number from above and below that are new even for constant density flow; an a priori bound on the velocity field and lower bound on the pressure; a proof of the nonexistence of monotone bores for stratified surface waves; and a theorem ensuring that all supercritical solitary waves of elevation have an axis of even symmetry. This is joint work with R. M. Chen and M. H. Wheeler.

This talk will examine the nonlinear water waves, and wave-current interactions, which may be prescribed by Gerstner-like exact and explicit solutions to the geophysical $\beta-$plane equations in the equatorial region. In particular, I will present recent work which highlights the role played by the typically-neglected centripetal force terms in describing such physical processes

In this talk we are concerned with the wave length $\lambda$ of smooth periodic traveling wave solutions of the Camassa-Holm equation. The set of these solutions can be parametrized using the wave height $a$. Our main result establishes monotonicity properties of the map $a\mapsto \lambda(a)$ i.e., the wave length as a function of the wave height. We obtain the explicit bifurcation values, in terms of the parameters associated with the equation, which distinguish between the two possible qualitative behaviours of $\lambda(a)$, namely monotonicity and unimodality. The key point is to relate $\lambda(a)$ to the period function of a planar differential system with a quadratic-like first integral, and to apply a criterion which bounds the number of critical periods for this type of systems.

We construct continuous curves of rotating vortex patch solutions to the two-dimensional Euler equations. These curves are large in that, as the parameter tends to infinity, the minimum value on the boundary of the relative angular fluid velocity becomes arbitrarily small. This is joint work with Zineb Hassainia and Nader Masmoudi.

The computation of surface and interfacial waves is a central problem in fluid mechanics. While much has been done, the effect of vorticity on surface and internal wave propagation is still poorly understood. To address this, we first look at shallow-water propagation in density stratified fluids with piecewise linear shear profiles. We show that by allowing for jumps in the shear across the interface, strong nonlinear responses can be generated resulting in phenomena like dispersive shock waves. Thus depth varying currents could play a larger role in interface dynamics than is currently understood. Second, we study the problem of collections of irrotational point vortices underneath a free fluid surface. We present a derivation of a model and numerical scheme which allows for arbitrary numbers of vortices in a shallow-water limit. While we are able to recreate much of the classical results for how surface waves form over two counter-propagating vortices, we go beyond this case and look at an example involving four vortices. Again, our approach allows for any number of vortices to be present, and this lets us provide some hint as to how underwater eddies might generate free surface waves.

This talk concerns the mathematical modeling and numerical simulation of waves in ice sheets as occurring, e.g., in polar regions. A three-dimensional Hamiltonian formulation for ice sheets deforming on top of an ideal fluid of arbitrary depth is presented and nonlinear wave solutions are examined. In certain asymptotic regimes, analytical solutions are derived and compared with fully nonlinear solutions obtained numerically by a pseudospectral method.

The goal of this work is to build on previous results by Parau et al. [1, 2, 3, 4] and produce a more efficient and accurate method for computing solutions to Euler's equations for water waves underneath an ice sheet in three dimensions. As was previously done, we solve the equations via a numerically implemented boundary integral equations method and utilize some high performance computing techniques. In this talk, we give details of the current methods and compare solutions for different models of the ice sheet. This is a joint work with Emilian Parau, Jean-Marc Vanden-Broeck and Paul Milewski.

References:

1. E. I. Parau, and J.-M. Vanden-Broeck Nonlinear two- and three-dimensional free surface flows due to moving disturbances. Eur. J. Mech. B Fluid, Vol 21 (2002), pp. 643-656.
2. E. I. Parau, J.-M. Vanden-Broeck and M. J. Cooker Three-dimensional gravity-capillary solitary waves in water of finite depth and related problems. Physics of Fluids, Vol 17 (2005), pp. 1-9.
3. E. I. Parau, J.-M. Vanden-Broeck and M. J. Cooker Nonlinear three- dimensional gravity-capillary solitary waves. J. Fluid Mech., Vol 536 (2005), pp. 99-105.
4. E. I. Parau, and J.-M. Vanden-Broeck Three-dimensional waves beneath an ice sheet due to a steadily moving pressure. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., Vol 369 (2011), pp. 2973-2988.

I will present an existence and stability theory for solitary waves at the interface between a thin ice sheet and an ideal fluid, which is based on minimising the total energy subject to the constraint of fixed total horizontal momentum. The ice sheet is modelled using the Cosserat theory of hyperelastic shells. Since the energy functional is quadratic in the highest derivatives, stronger results are obtained than in the case of capillary-gravity waves. This is joint work with Mark Groves and Benedikt Hewer.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.