Seminars (TWW)

The lecture gives an overview of a series works done with Alexey Slunyaev. We develop a new paradigm of how to describe linear and weakly nonlinear dynamics of waves on jet currents with a particular emphasis on new mechanisms of freak wave formation.

From numerous seamen accounts and insurer records it has been known for long time that rogue waves events are quite frequent on certain currents, e.g. on notorious in this respect Agulhas current. The theoretical explanation of this fact is still lacking, which is due to our overall poor understanding of wave nonlinear evolution on currents.

We develop a new systematic asymptotic theory of waves dynamics on jet currents which does not rely on the WKB approximation. First, solutions to the general linear 2D boundary-value problem for jet currents with arbitrary lateral profiles are found in terms of an asymptotic series in natural small parameters. In essence, we employ approximate separation of variables, which is always justified for the oceanic conditions.

There are three types of waves on the current: passing through, reflected and trapped. The key role in the new approach is played by the trapped modes. The trapped modes themselves (rather than comprising them harmonic components) participate in the nonlinear interactions. A general weakly nonlinear theory of trapped mode evolution is being developed. The properties of resonant interactions are qualitatively different from those between the waves in the absence of a current. In particular, three-wave interactions are always allowed in deep water and may play an important role in wave field evolution. There are three main advantages of the developed approach: (i) it is systematic and can identify and address the situations where the commonly adopted paradigm is not applicable; (ii) the current could be almost arbitrary, i.e. weak/strong, or smooth/with sharp edges; (iii) the initially 3-D problem is reduced to solving 1D evolution equations with the lateral and vertical dependence being prescribed by the corresponding modal structure. The modes can participate in both three and four- interactions. The prevalence of one type over another depends on parameters of waves and currents. A rich variety of possible regimes is being explored.

From the perspective of rogue wave occurrence we have identified several new mechanisms which have no analogues in the absence of currents.

For length scales of a few cm or less, where both gravity and surface tension are important, a rich variety of locally confined nonlinear waves of permanent form can propagate on the surface of shallow and deep water. Such gravity-capillary solitary waves (in one dimension) and lumps (in two dimensions) have been studied extensively in the last two decades theoretically, by analytical and numerical techniques, as well as in laboratory experiments. This body of work will be surveyed and open questions for future work will be highlighted.

The talk will overview of results and open problems related to the Nash-Moser theory of periodic standing and spatial water waves. The topics include the Nash-Moser implicit function theorem, the descent method, and the paradifferential change of variables.

A solitary wave is a localized coherent structure that maintains its shape while it travels at constant speed. Solitary waves are caused by a cancellation of nonlinear and dispersive effects in the medium. In this talk, we discuss a special type of solitary wave: wavepacket solitary waves. In contrast to the celebrated KdV soliton, wavepacket solitary waves have intrinsic length scale and feature oscillatory decaying tails. Two examples, capillary-gravity waves and flexural-gravity waves, will be presented to illustrate the importance of wavepacket solitary waves, the numerical methods for constructing these waves and their dynamics.

We compute new families of time-periodic and quasi-periodic solutions of the free-surface Euler equations involving standing waves, standing-traveling waves and collisions of solitary waves of various types. In the collision case, similarities notwithstanding, the new solutions are found to be well outside of the KdV and NLS regimes. A Floquet analysis shows that many of the new solutions are linearly stable to harmonic perturbations. Evolving such perturbations (nonlinearly) over tens of thousands of cycles suggests that the solutions remain nearly time-periodic forever. Parts of the talk will serve as an introduction to later talks in the session.

In this talk, I will discuss the spectral stability for the periodic traveling wave with constant vorticity. We will discuss relationships between stability of the traveling wave with respect to long-wave perturbations and the structure of the bifurcation curve for small amplitude solutions.

Turbulent flow over unsteady monochromatic and Stokes waves has been investigated for various wave steepness ka and wave age $c_r/U_*$ using a Differntial Second-Moment (DSM) closure turbulence model. Also presented is DSM simulation of flow over steady monochromatic waves of different wave steepness for comparision with Direct Numerical Simulation (DNS) reported by Sullivan et al. (2000). It is shown that there is a good agreement between results obtained from DNS and DSM simulations. For the simulations of unsteady waves, the growth factor $kc_i$ is derived from a quadratic secular equation in complex phase speed c, obtained at the air-sea interface, whose solution leads to two wave speeds $c_1$ and $c_2$. The wave speed $c_1$ corresponds to free-surface waves damped by viscous stresses in the boundary layers at the interface, whilst the wave speed $c_2$ is associated with Tollmien-Schlichting instabilities in the shear airflow over the surface wave. It is shown that the wave growth through the complex phase speed $c_1$ is finite and waves do not become sharp-crested, therefore waves are unlikely to break. However, when the complex wave speed $c_2$ is adopted, for growth of Stokes waves, the waves rapidly peak and become sharp-crested. As the wave becomes relatively steep a secondary vortical motion is induced around the wave crest with a region of high pressure behind and low pressure ahead of the crest. This pressure asymmetry will very likely to cause the wave to break. It is also observed that as the wave grows and becomes steeper the critical layer elevates from the inner to the middle layer. Consequently, the cat's-eye become more and more asymmetrical and dramatically affects the flow field above the wave. Furthermore, at relatively high steepness, flow separation is observed in the lee of the wave. Also, in simulations reported here, the effect of non-separating sheltering is seen to be pronounced since the boundary layer is perturbed and thickeness on the leeside of the crest due to turbulent shear stress in the inner region. It is therefore argued that the wave growth is due to combined interactions between the elevated critical layer, the non-separated sheltering, and turbulence. From numerical results obtained, a new parametrization for the energy-transfer parameter is derived which agrees well with that obtained by Belcher & Hunt (1993) for slow moving waves and has the salient features of those evaluated numerically and asymptotically for intermidiate as well as fast moving waves. From the colective accounts of results of numerical simulations reported here, it is concluded that the critical layer has dynamical effect, and plays a crucial role, in shear flows over unsteady water waves.

An approximation method based on a 'relaxed' variational principle is presented. The advantages of this relaxed formulation are illustrated with various examples in shallow and deep waters. A candidate model for tsunamis modelling over significant bathymetry variations is also presented.

In fluid dynamics, a "splash" singularity occurs when a locally smooth interface self-intersects in finite-time. It is now well-known that solutions to the water waves equations (and a host of other one-phase fluid interface models) has a finite-time splash singularity. By means of elementary arguments, we prove that such a singularity cannot occur in finite-time for vortex sheet evolution (or two-fluid interfaces). This means that the evolving interface must lose regularity prior to self-intersection. We give a proof by contradiction: we assume that such a singularity does indeed occur in finite-time. Based on this assumption, we find precise blow-up rates for the components of the velocity gradient which, in turn, allows us to characterize the geometry of the evolving interface just prior to self-intersection. The constraints on the geometry then lead to an impossible outcome, giving the contradiction. This is joint work D. Coutand.

The existence of solitary-wave solutions to the three-dimensional water-wave problem with is predicted by the Kadomtsev-Petviashvili (KP) equation for strong surface tension and Davey-Stewartson (DS) equation for weak surface tension.The term solitary wave describes any solution which has a pulse-like profile in its direction of propagation, and these model equations admit three types of solitary waves. A line solitary wave is spatially homogeneous in the direction transverse to its direction of propagation, while a periodically modulated solitary wave is periodic in the transverse direction. A fully localised solitary wave on the other hand decays to zero in all spatial directions. In this talk I outline mathematical results which confirm the existence of all three types of solitary wave for the full gravity-capillary water-wave problem in its usual formulation as a free-boundary problem for the Euler equations. Both strong and weak surface tension are treated. The line solitary waves are found by establishing the existence of a low-dimensional invariant manifold containing homoclinic orbits. The periodically modulated solitary waves are created when a line solitary wave undergoes a dimension-breaking bifurcation in which it spontaneously loses its spatial homogeneity in the transverse direction; an infinite-dimensional version of the Lyapunov centre theorem is the main ingredient in the existence theorem. The fully localised solitary waves are obtained by finding critical points of a variational functional.

The talk links together the paper of mine from 2010 on the flip-through impact and recent research on ringing loads on monopile substructures caused by near-breaking and breaking waves. We have made experiments in a collaborative project with the Danish Hydraulics Institute and computations with a free-surface Navier-Stokes solver coupled to a potential flow model of the outer wave field.

Some numerical methods for water waves, such as the Craig-Sulem method, involve expanding terms in the water wave evolution equations as series, truncating those series, and then simulating the resulting equations. For one such scheme, we present analytical evidence that the truncated system is in fact ill-posed; this involves further reducing the evolution equations to a model for which we can prove ill-posedness. We then present numerical evidence that the full truncated system is ill-posed, showing that arbitrarily small data can lead to arbitrarily fast growth. We present this numerical evidence for multiple levels of truncation. We are able to prove that by adding a viscosity to the system, we instead arrive at a well-posed initial value problem. This is joint work with Jerry Bona and David Nicholls.

In this lecture, we will discuss the irrotational water wave problem. We will place the problem in the larger context of vortex sheets; the vortex sheet is the interface between two irrotational fluids, allowing for a jump in the tangential components of velocity across the interface. We will present the equations of motion in both 2D and 3D, for the water wave problem both with and without surface tension. We will present the numerical method of Hou, Lowengrub, and Shelley (HLS) for computing solutions of the initial value problem in 2D, and how the HLS ideas can be used to prove short-time well-posedness of the initial value problem. The corresponding numerical method and well-posedness proofs for the three-dimensional problem will also be discussed. If time allows, we will go beyond the initial value problem, and discuss how the vortex sheet formulation with the HLS ideas can be extended to treat other problems, such as the traveling wave problem. This talk includes joint work with Nader Masmoudi, Michael Siegel, Svetlana Tlupova, and possibly others.

A class of water wave problems concerns the dynamics of the free interface separating an inviscid, incompressible and irrotational fluid (water), under the influence of gravity, from a zero-density region (air). In these lectures I will present some recent methods and ideas developed concerning the local and global wellposedness of these problems, where the fluid region has either no fixed boundary, or has a fixed vertical wall. The emphasis will be on the understanding of the mathematical structure that leads to the results on global existence of small and smooth waves, local existence of arbitrary smooth waves, the persistence of water waves with angled crests, and in the case where there is a vertical wall, the interaction of the free interface with the wall. These lectures will be accessible to graduate students and post-doctors.

The lectures will provide an introduction to the theory of "modulation" and its role in the derivation of model equations, such as the KdV equation, Boussinesq equation, KP equation, and Whitham modulation equations, and their role in the theory of water waves. The classical theory of modulation, such as Whitham modulation theory, will be reviewed, and a new approach will be introduced. The new approach is based on modulation of background flow. Methodology that is key to the theory is symmetry and conservation laws, relative equilibria, Hamiltonian and Lagrangian structures, multiple scale perturbation theory, and elementary differential geometry. By basing the theory on modulation of relative equilibria, new settings are discovered for the emergence of KdV and other modulation equations. For example, it is shown that the KdV equation can be a valid model for deep water as well as shallow water. The lectures are introductory, and no prior knowledge is assumed.