New Directions in Water Waves, Workshop and Summer School

Prime functions and quadrature domains: new mathematical tools for water waves

This short course will present an overview of two mathematical objects: the prime function associated with a multiply connected domain, and the notion of a quadrature domain. The lectures will show their relevance to the theme of this LMS-Bath Symposium, the theory of water waves, providing a novel vantage point that helps to understand some recent results in the area, and pointing the way to new ones. These mathematical ideas are relevant not just to water waves, however, and are likely to prove valuable in many other research problems

Exponential asymptotics with applications to fluid dynamics and water waves

Steady periodic water waves with vorticity in two and three dimensions

My lectures will centre on existence theory for steady periodic water waves, modelled by the inviscid Euler equations with a free surface. This topic has a centuries-old history, but is still the subject of active research. More precisely, I will focus on waves with non-zero vorticity, which is for example of relevance in modelling wave-current interactions. While a specific family of waves has been known as far back as 1802 and small-amplitude waves were constructed in the 1930’s, a large-amplitude theory started to emerge only around 20 years ago. I will cover some of the progress in the last two decades, and in particular recent work on overhanging waves with interior stagnation points. This is a phenomenon which does not occur in the irrotational setting. The three-dimensional theory is much less developed and it is only recently that some small-amplitude results have started to appear for doubly periodic waves with vorticity. I will try to explain why and describe some of these results. Mathematical methods which will be used include local and global bifurcation theory based on Lyapunov-Schmidt reduction and degree theory, as well as a priori estimates and maximum principle arguments for elliptic equations. The lectures will hopefully provide a solid background for some of the talks in the workshop.

Stability of bilinear shear currents with a free surface

The linear stability of homogenous shear flows between two rigid walls is a classical problem that goes back to Rayleigh (1880). Among other things, Rayleigh was able to show that a shear flow with no inflection points is linearly stable. The generalisation of this stability criterion to the free-surface setting is not straightforward and was established much later by Yih (1971) (under certain restrictions) and, more recently, Hur & Lin (2008).
In the case when a shear flow with a free surface is modelled by constant vorticity layers, no stability criterion is known. As a first step in this direction we consider the stability analysis of a bilinear shear current and establish a criterion for the stability of the flow. The effect of density stratification on the stability of the flow will also be investigated.

Defects and wave jumps in shallow water hydrodynamics

Waves in the open ocean are rarely perfect and defects arise due to interaction between two or more wave fields and interaction with the shoreline, for example. This talk will focus on the role of Whitham modulation theory (WMT) in the modelling of defects. In one space dimension defects can be modelled as a shock wave connecting two fields of Stokes waves of differing wavenumber. The theory of Sprenger & Hoefer (2020) is reviewed where WMT is used to construct these shocks in the fifth-order KdV equation. Then implications and generalisations are discussed. A related topic, feeding into the above theory is defects at the Benjamin-Feir instability transition. The Whitham (1967) modulation equations for this problem are reviewed, new properties discussed, and re-modulation is introduced at the transition point to find a dispersive higher-order modulation equation due to Ratliff (2017). Emerging from this latter equation are wave jumps that in turn generate frequency downshifting.

Internal solitary wave shoaling

Internal solitary waves (ISWs) are finite amplitude waves of permanent form that travel along density interfaces in stably stratified fluids. They owe their existence to an exact balance between non-linear wave steepening effects and linear wave dispersion. They are common in all stratified flows especially coastal seas, straits, fjords and the atmospheric boundary layer. Whilst ISWs can travel considerable distance over a flat bottom without change of form, under certain conditions, such as when shoaling, their form can change considerably. As they do so, dissipation produced by the motion of breaking waves, both in the benthic boundary layer and the pycnocline, is identified as a key process in the global cascade of energy from global-scale mechanical forcing to dissipation. In this presentation a combined experimental and numerical study will illustrate the effect of stratification form on the shoaling characteristics of ISWs propagating over a smooth, linear topographic slope. It is found that the form of stratification affects the breaking type associated with the shoaling wave. In a thin tanh stratification (homogeneous upper and lower layers separated by a thin pycnocline), good agreement is seen with past studies. Waves over the shallowest slopes undergo fission. Over steeper slopes, the breaking type changes from surging, through collapsing to plunging with increasing wave steepness Aw/Lw for a given topographic slope, where Aw and Lw are incident wave amplitude and wavelength, respectively. In a surface stratification regime (linearly stratified layer overlaying a homogeneous lower layer), the breaking classification differs from the thin tanh stratification. Plunging dynamics are inhibited by the density gradient throughout the upper layer, instead collapsing-type breakers form for the equivalent location in parameter space in the thin tanh stratification. In the broad tanh profile regime (continuous density gradient throughout the water column), plunging dynamics are likewise inhibited and the near-bottom density gradient prevents the collapsing dynamics as well. Instead, all waves either fission or form surging breakers. As wave steepness in the broad tanh stratification increases, the bolus formed by surging exhibits evidence of Kelvin–Helmholtz instabilities on its upper boundary. In both two- and three-dimensional simulations, billow size grows with increasing wave steepness, dynamics not previously observed. If time, shoaling mode-2 ISWs will be also considered. Features of wave shoaling include (i) formation of an oscillatory tail, (ii) degeneration of the wave form, (iii) wave run up, (iv) boundary layer separation, (v) vortex formation and re-suspension at the bed and (vi) a reflected wave signal. In shallow slope cases, the wave form is destroyed by the shoaling process; the leading mode-2 ISW degenerates into a train of mode-1 waves of elevation and little boundary layer activity is seen. For steeper slopes, boundary layer separation, vortex formation and re-suspension at the bed are observed. The boundary layer dynamics are shown (numerically) to be dependent on the Reynolds number of the flow.

Resurgence, exponential asymptotics, and pathologies in toy water-wave problems

Exact solutions for steadily travelling water waves with submerged
point vortices

Blow-up in infinite time for the Keller-Segel system

We study the Keller-Segel system in the plane with an initial condition with sufficient decay and critical mass 8 pi. We find a function u0 with mass 8 pi such that for any initial condition sufficiently close to u0 and mass 8 pi, the solution is globally defined and blows up in infinite time. This proves the non-radial stability of the infinite-time blow up for some initial conditions, answering a question by Ghoul and Masmoudi (2018). This is joint work with Manuel del Pino (U. of Bath), Jean Dolbeault (U. Paris Dauphine), Monica Musso (U. of Bath) and Juncheng Wei (UBC).

Dynamics of concentrated vorticities in 2d and 3d Euler flows

A classical problem that traces back to Helmholtz and Kirchhoff is the understanding of the dynamics of solutions to the Euler equations of an inviscid incompressible fluid when the vorticity of the solution is initially concentrated near isolated points in 2d or vortex lines in 3d. We discuss some recent results on these solutions' existence and asymptotic behavior. We describe, with precise asymptotics, interacting vortices, and traveling helices, and extension of these results for the 2d generalized SQG. This is research in collaboration with J. Dávila, A. Fernández, M. Musso and J. Wei.

Embedded Mode-2 Internal Solitary Waves

Inside stratified fluids, regions of rapid density variation with respect to depth (pycnoclines) act as waveguides for horizontally propagating internal waves. In this talk we shall examine internal waves of the second baroclinic mode (mode-2), by computing travelling wave solutions to a simplified three-layer model. We will be presenting numerical solutions to both the full Euler system, and a reduced model called the three-layer Miyata-Choi-Camassa (MCC3) equations. Mode-2 waves (typically) occur within the linear spectrum, and are hence associated with a resonant mode-1 oscillatory tail. However, in line with recent results for the MCC3 system by Barros, Choi & Milewski (JFM, 2020), we will present numerical evidence that these oscillations can be found to have zero amplitude, resulting in truly localised solutions. We relate large amplitude solutions to the so-called conjugate states of the system, where the limiting solutions of many of the solution branches are a heteroclinic orbit between conjugate states (i.e. wavefront solutions).

Resonant free-surface water waves in a circular cylinder

The resonant interaction of nonlinear free-surface gravity waves has been studied extensively over the past two centuries, with a particular focus on domains that are large compared to the characteristic wavelength (such as oceans). However, resonant wave interactions arising in confined three-dimensional geometries have received relatively little attention, despite being a central consideration in the design of man-made structures arising in hydrological engineering, such as industrial-scale fluid tanks. Here I will present the results of a combined theoretical and computational investigation into the onset and dynamics of resonant free-surface flows in a circular cylinder, paying particular attention to the behaviour of low-order resonances.

Global bifurcation for corotating vortex pairs

The existence of a local curve of corotating vortex pair solutions to the two-dimensional Euler equations was proven by Hmidi and Mateu via a desingularization of a pair of point vortices. In this talk, we construct a global continuation of these local curves. The proof relies on an adaptation of the powerful analytic global bifurcation theorem due to Buffoni and Toland, which allows for the singularity at the bifurcation point. Along the global curve of solutions, either the angular fluid velocity vanishes or the two patches self-intersect. This is a joint work with Claudia García (University of Barcelona).

Theoretical and numerical investigations of extreme waves through oblique soliton interactions

Extreme water-wave motion is investigated analytically and numerically by considering two-soliton and three-soliton interactions on a horizontal plane. We successfully determine numerically that soliton solutions of the unidirectional Kadomtsev-Petviashvili equation (KPE), with equal far-field individual amplitudes, survive reasonably well in the bidirectional and higher-order Benney-Luke equations (BLE). A well-known exact two-soliton solution of the KPE on the infinite horizontal plane is used to seed the BLE at an initial time, and we confirm that the KPE-fourfold amplification approximately persists. More interestingly, a known three-soliton solution of the KPE is analysed further to assess its eight- or ninefold amplification, the latter of which exists in only a special and difficult to attain limit. This solution leads to an extreme splash at one point in space and time. Subsequently, we seed the BLE with this three-soliton solution at a suitable initial time to establish the maximum amplification: it is approximately 7.8 for a KPE amplification of 8.4. In our simulations, the computational domain and solutions are truncated approximately to a fully periodic or half-periodic channel geometry of sufficient size, essentially leading to cnoidal-wave solutions. Moreover, special geometric (finite-element) variational integrators in space and time have been used in order to eradicate artificial numerical damping of, in particular, wave amplitude.

Long weakly-nonlinear ring waves and their relatives in stratified fluids with currents

In this talk I will give an overview of the recent developments in the studies of long weakly-nonlinear ring waves and hybrid solutions consisting of a part of a ring wave and two tangent plane waves. The waves propagate in a stratified fluid over a parallel shear flow. There exists a linear modal decomposition (separation of variables) in the far-field set of Euler equations with the boundary conditions appropriate for oceanographic applications, more complicated than the known decomposition for plane waves. The modal equations constitute a new spectral problem and lead to the need to construct a singular solution of a nonlinear first-order ODE responsible for the adjustment of the speed of the ring wave in different directions. The usual surface ring waves propagating over a parallel current have elongated wavefronts. It transpired that in some cases interfacial ring waves propagating over the same current could be squeezed. We consider a large family of currents which could be used to model wind-generated currents, river inflows and exchange flows in straits. Joint work with X. Zhang, C. Hooper and R. Grimshaw.

Some problems in exponential asymptotics

The role of exponentially small terms in some nonlinear models will be highlighted.

Benjamin and Lighthill conjecture for steady water waves with vorticity

Leapfrogging for Euler equations, and the vortex filament conjecture

We consider the Euler equations for incompressible fluids in 3-dimension. A classical question that goes back to Helmholtz is to describe the evolution of vorticities with a high concentration around a curve. The work of Da Rios in 1906 states that such a curve must evolve by the so-called "binormal curvature flow". Existence of true solutions whose vorticity is concentrated near a given curve that evolves by this law is a long-standing open question that has only been answered for the special case of a circle travelling with constant speed along its axis, the thin vortex-rings. In this talk I will discuss the construction of helical filaments, associated to a translating-rotating helix, and of two vortex rings interacting between each other, the so-called leapfrogging. The results are in collaboration with J. Davila (U. of Bath), M. del Pino (U. of Bath) and J. Wei (U. of British Columbia).

Stability of waves on fluids with constant vorticity

The stability of periodic travelling waves on fluid of infinite depth is examined in the presence of a constant background shear field. The effects of gravity and surface tension are ignored. The base waves are described by an exact solution that was discovered recently by Hur and Wheeler (J. Fluid Mech., 2020). Linear growth rates are calculated using both an asymptotic approach valid for small-amplitude waves and a numerical approach based on a collocation method.

Temporally periodic water waves with small surface-tension

We consider the simplest inviscid and irrotational formulation of a time-dependant water wave, with the inclusion of the effects of both gravity and surface tension. Solutions are sought that are both periodic in time and space from the perspective of the lab frame.

For zero surface tension, two types of solutions satisfying this criteria have been extensively studied: standing waves (which oscillate vertically), and travelling waves (which may be steady in a co-moving frame). Recently, Wilkening (2021) demonstrated the numerical existence of temporally periodic solutions that display a mixture of these standing and travelling components, which may be classified with an appropriate bifurcation parameter.

We extend this formulation to also include the effect of surface tension. Since the limit of small surface tension is singularly perturbed, the resultant structure of solutions differs significantly from that found in the absence of interfacial tension. The resultant solutions contain high-frequency ``parasitic ripples", whose amplitude is seen to be exponentially-small in the surface tension. The connections between these numerical results and the analytical techniques of exponential asymptotics will also be discussed.

Singularities in Water Waves.

Capillary waves, interfacial waves and waves with vorticity

Water waves have been studied for over 200 years and many important results have been obtained. However there are still open questions for nonlinear waves. In this talk we will follow branches of solutions and focus on the limiting configurations (i.e. the waves of maximum amplitude which can be reached on each branch). We shall assume the fluids to be incompressible and inviscid. We will include various effects in the dynamic boundary condition and show how the limiting configurations differ for capillary waves, gravity waves interfacial waves (i.e. waves travelling at the interface between two layers of fluid of different constant densities) and waves with constant vorticity. These problems are mathematically difficult because of the nonlineartity and the presence of free surfaces. They are solved by a combination of analytical and numerical methods. These include boundary integral equation methods and series truncation methods. Both periodic and solitary waves are considered in frames of reference moving with the waves. New branches of solutions for interfacial waves will be presented. They usually emerge as branches bifurcating from known branches but they do not have an equivalent in the classical problem of pure gravity surface waves. Some unexpected connections between capillary waves and waves with constant vorticity will also be explored in the talk. As time permits asymmetric waves and other types of waves will be discussed.

Some of these recent works are joint with Xin Guan, Zhan Wang, Vera Hur, Frédéric Dias, Tao Gao and Alex Doak

Orbital stability of internal waves

In this talk, we discuss recent work on the nonlinear stability of capillary-gravity waves propagating along the interface dividing two immiscible fluid layers of finite depth. The motion in both regions is governed by the incompressible and irrotational Euler equations, with the density of each fluid being constant but distinct. A diverse collection of small-amplitude solitary wave solutions for this system have been constructed in the case of strong surface tension (as measured by the Bond number) and slightly subcritical Froude number. We prove that all of these waves are (conditionally) orbitally stable in the natural energy space. Moreover, the trivial solution is shown to be conditionally stable when the Bond and Froude numbers lie in a certain unbounded parameter region. For the near critical surface tension regime, we prove that one can infer conditional orbital stability or orbital instability of small-amplitude traveling waves solutions to the full Euler system from considerations of a dispersive PDE model equation.

These results are obtained by reformulating the problem as an infinite-dimensional Hamiltonian system, then applying a version of the Grillakis–Shatah–Strauss method developed with Varholm and Wahlén. This is joint work with Robin Ming Chen.

On new global bifurcation results for the travelling periodic water wave problem

First, the talk outlines recent progress in the two-dimensional case, where the problem is equivalently cast into the form “identity plus compact”. The main advantages (and the novelty) of this new reformulation are that no simplifying restrictions on the geometry of the surface profile and no simplifying assumptions on the vorticity distribution (and thus no assumptions regarding the absence of stagnation points or critical layers) have to be made. Second, I discuss steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. The corresponding problem can be formulated as an elliptic free boundary problem in terms of Stokes’ stream function. Here, a certain change of variables helps to overcome the generic coordinate-induced singularities.
This is joint work with André Erhardt and Erik Wahlén.