Research

Research interests

My research has focused on the study of nonlinear waves, including:

Among other things, I like to work on problems related to water waves. This is a challenging and fascinating field of research for mathematicians, physicists, oceanographers, engineers, and whether you are a theoretician, experimentalist or computational scientist. On the theoretical side, the water wave problem poses challenging mathematical questions related to nonlinear partial differential equations and dynamical systems. On the practical side, there are direct applications to oceanography and coastal engineering with far-reaching implications for marine biology, weather forecasting and climate modeling. Analytical and numerical methods that have been developed to tackle this problem are also applicable to many other scientific areas involving waves and free boundaries, e.g. in acoustics, optics, plasma physics, material science, general relativity, etc. Quoting the famous Physics Nobel laureate Richard Feynman: “[water waves], which are easily seen by everyone and which are used as an example of waves in elementary courses … are the worst possible example … they have all the complications that waves can have.” But no worries, if you are a student interested in this topic and would like to work with me, your project will be tailored according to your taste and skills.

In this era of data-driven research, on top of the mathematical analysis and numerical simulation, I also try to perform detailed validation against experimental data (from laboratory or field measurements) whenever possible. This requires more work but is quite rewarding.

More recently, my research has also branched out into biomechanics and bone acoustics (see my publications for further details).


Breaking waves:

Fig: Three-dimensional breaking waves at a given instant for two different bottom configurations.

We have developed a numerical model for three-dimensional surface water waves, which solves the full Euler equations for potential flow with a free surface, using a high-order boundary integral/element method and a mixed Eulerian-Lagrangian approach for time integration. This model is applicable to nonlinear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking. In addition to information on the free surface, the velocity and acceleration fields anywhere inside the flow can be accurately computed by this approach.

Further details can be found in Grilli, Guyenne & Dias (2001)Fochesato, Grilli & Guyenne (2005)Guyenne & Grilli (2006)

Wave breaking, generic singularities and scaling laws:

Fig: Apparent contour of the breaking wave on the right above (left panel), time evolution of the overturning region in the direction of wave propagation (middle panel) and along the wave crest (right panel).

By analyzing the singular behavior of solutions to generic nonlinear hyperbolic systems (e.g. inviscid Burgers and pressure-less fluid equations), we have shown how the overturning portion of a breaking wave extends in both horizontal directions, parallel and normal to the wave crest. The boundary of this overturning region, referred to as the “apparent contour” (i.e. the set of all points where the tangent to the water surface is vertical) evolves into a skewed curve. We have found that this curve spreads like a power 3/2 of time in the direction of wave propagation and like a power 1/2 of time along the wave crest. We have also obtained similar power laws for the time evolution of such quantities as the curvature and vertical velocity at the tip of the plunging jet. These theoretical predictions help characterize geometric and kinematic properties associated with the development of wave breaking. They have been compared with direct numerical simulations and laboratory experiments, yielding an excellent agreement in both cases.

Further details can be found in Pomeau, Le Bars, Le Gal, Jamin, Le Berre, Guyenne, Grilli & Audoly (2008)Pomeau, Le Berre, Guyenne & Grilli (2008)

Waves over topography and coastal engineering:

Fig: Solitary wave shoaling on a beach slope.

We have developed a surface spectral method to solve the full Euler equations for potential flow with a free surface and bottom topography. This method can be easily extended to three dimensions and is very efficient using the fast Fourier transform combined with a recursive evaluation of the Dirichlet-Neumann operator. It is applicable to two- or three-dimensional flows as well as to static or moving bottom topography. Extensive tests have been performed to validate this method, including comparison with laboratory experiments of large-amplitude/steep waves over highly varying bottom geometries. In particular, the bottom deformation is not required to be a smooth function of the horizontal coordinates and can describe randomly varying sea beds with sharp features (Craig, Guyenne & Sulem 2009).

Further details can be found in Craig, Guyenne, Nicholls & Sulem (2005)Guyenne & Nicholls (2005)Guyenne & Nicholls (2007)

Random topography and tsunamis:

Fig: Incident KdV soliton over random topography (left panel), Brownian-like motion of the position of the main pulse (middle panel) and long-time attenuation due to cumulative effects of wave scattering (right panel).

Although large-scale features of the ocean floor can be accurately estimated by e.g. shipborne acoustic techniques, this is not so for small-scale features which require prohibitively high-resolution measurements. In wave modeling, this imperfect knowledge of the ocean floor can be accommodated by representing its small-scale features as a stochastic process with suitable statistical properties. Using homogenization theory, we have derived a KdV-type model for long waves propagating over such randomly varying topography. Unlike the case of periodic topography, we have found that the asymptotic solution does not fully homogenize but retains realization-dependent effects. More specifically, its phase has a random component governed by a canonical diffusion process, essentially a Brownian motion, while its amplitude is modulated by a white noise process. These random contributions quantify the degree of uncertainty in the problem, reflecting the fact that limited knowledge of the bathymetry (and of its dynamics) may lead to appreciable errors in the wave forecasting. As an application, our results provide a theoretical basis to explain the systematic mismatch in arrival time that has been reported from large-scale numerical simulations of tsunamis. Additionally, this KdV model exhibits a linear damping term that accounts for wave attenuation due to the cumulative action of wave decoherence and scattering by the bottom inhomogeneities.

Further details can be found in de Bouard, Craig, Diaz-Espinosa, Guyenne & Sulem (2008)Craig, Guyenne & Sulem (2009)

Solitary wave interactions and rogue waves:

Fig: Comparison between numerics (solid line), sum of two KdV solitons (dashed line) and experiments (dots) for the head-on collision of two solitary waves of different amplitudes.

We have used the surface spectral model described above to study solitary wave interactions on constant depth. This model has been tested and validated against laboratory experiments for head-on and overtaking collisions of solitary waves. In both cases, a very good agreement has been found. These experiments were conducted in the W. G. Pritchard Fluid Mechanics Laboratory at Penn State University (J. Hammack & D. Henderson). Our results are interesting for a number of reasons. In particular, they extend stability results for soliton solutions of weakly nonlinear integrable equations to solitary wave solutions of highly nonlinear non-integrable systems. There are also implications for a better understanding of rogue (or freak) waves that can appear all of a sudden in the ocean and reach unusually high amplitudes. A possible mechanism for their formation is the coherent superposition of smaller wave components as examined here.

Further details can be found in Hammack, Henderson, Guyenne & Yi (2005)Craig, Guyenne, Hammack, Henderson & Sulem (2006)

More information on the laboratory experiments can be found on Diane’s webpage

In memory of Joe Hammack with admiration and gratitude

Surface signature of internal waves and quantum mechanics:

Fig: Internal KdV solitons with their surface signature in oceanic conditions corresponding to the Andaman Sea (left panel) and off the Oregon coast (right panel).

Internal waves are not directly visible to the observer but they have a modulating effect on the sea surface, in the form of narrow strips of rough water (also called rips), producing changes in reflectance on aerial and satellite images. To describe these surface-internal wave interactions, we have derived an asymptotic model consisting of a linear Schrodinger equation for surface wavepackets, resonantly coupled with a KdV equation for long internal waves, such that the surface group velocity is equal to the internal phase velocity. In typical oceanic situations where internal waves are observed, we have shown that the surface wave equation is analogous to a semi-classical Schrodinger equation of quantum mechanics, for which the internal waves play the role of a potential well. The surface rips are thus explained in terms of localized bound states associated with this linear Schrodinger equation. Put in other words, surface wave energy is trapped above the internal soliton during its passage, in the same way as electrons would be near an atom nucleus. This analogy is especially intriguing as it relates quantum effects occurring at very small (atomic) scales to fluid flow effects occurring at very large (oceanic) scales.

Further details can be found in Craig, Guyenne & Sulem (2011)Craig, Guyenne & Sulem (2012)Craig, Guyenne & Sulem (2015)

Internal waves and physical oceanography:

Fig: Large-amplitude internal solitary waves in the two-layer model derived by Craig, Guyenne & Kalisch (2004): sequence of wave profiles for varying amplitudes (left panel) and comparison with the KdV soliton plotted in dots (right panel).

We have derived a Hamiltonian formulation for the problem of a dynamic interface with rigid lid boundary conditions, as well as that of a free interface coupled with a free surface. Of particular interest here is the modeling of internal waves which play a key role in many oceanic processes. From this formulation, we have developed a Hamiltonian perturbation theory for the long-wave limit, and we have carried out a systematic analysis of the principal long-wave scaling regimes (Boussinesq, KdV, Benjamin-Ono, Intermediate Long Wave). In addition, we have described a novel class of scaling regimes in which the amplitude of the interface disturbance is of the same order as the mean fluid depth. This has led to the derivation of new evolution equations which exhibit rational dependence in their coefficients of dispersion and nonlinearity. Some of these equations admit solitary wave solutions which have been computed numerically.

Further details can be found in Craig, Guyenne & Kalisch (2004)Craig, Guyenne & Kalisch (2005)Guyenne (2006)

Fluid motion recovery and computer graphics:

Fig: 3D fluid domain with corresponding 2D optical flow on the free surface (left panel) and motion estimation of two vortex patches in a gas flow (right panel).

The optical flow associated with the motion of fluid flows captured on 2D images can be modeled by the Euler equations for free-surface flows, where variations in the image intensity are represented by deformations of the top fluid surface. Previous optical-flow methods in computer vision have only considered geometric features of images to estimate the fluid motion between two consecutive frames. We have developed a physics-based method that uses numerical simulations of the Euler equations, with the advantage that the free-surface dynamics is directly and efficiently resolved thanks to introduction of the Dirichlet-Neumann operator. In particular, we have proposed a series expansion for the Neumann-Dirichlet operator (inverse of the Dirichlet-Neumann operator) to determine the optical flow. Potential applications are numerous, e.g. in computer graphics to create realistic visual effects of nonlinear fluid flows and in oceanography to provide suitable initial conditions for operational forecasting models.

Further details can be found in Xu & Guyenne (2009)Li, Xu, Guyenne & Yu (IEEE CVPR 2010)

Wave turbulence and kinetic theory:

Fig: Computed spectra (solid line) and predicted Kolmogorov power laws (dashed line) in weak wave turbulence.

Weak wave turbulence theory is a suitable tool for the statistical description of large systems dominated by resonant interactions between weakly nonlinear dispersive waves. It can be applied to a wide range of physical problems (oceanography, optics, plasma physics, acoustics, etc.). For example, this kinetic theory can be used to predict the shape and evolution of energy spectra for wind-driven water waves. However, questions have arisen concerning the validity of its predictions, in particular in the presence of localized coherent structures (e.g. solitons, wave collapse). We have developed a model describing weak turbulence in media with two types of interacting waves, for which coherent structures cannot develop. Our numerical results show an excellent agreement with theoretical predictions on both the slope and amplitude of the spectra.

Further details can be found in Dias, Guyenne & Zakharov (2001)Zakharov, Guyenne, Pushkarev & Dias (2001)

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