Applied Mathematics, fluid mechanics, nonlinear waves, physical oceanography, turbulence, Hamiltonian systems, free boundary problems, differential equations, scientific computing, numerical methods for nonlinear PDEs.
My main research is on nonlinear waves:
Fig: Three-dimensional wave breaking over a sloping ridge.
We have developed a numerical model for three-dimensional surface water waves. The model 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. The model is applicable to nonlinear wave transformations from deep to shallow water over complex bottom topography up to overturning and breaking.
Further details can be found in Grilli, Guyenne & Dias (2001), Fochesato, Grilli & Guyenne (2005), Guyenne & Grilli (2006)
Fig: 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.
Further details can be found in Craig, Guyenne, Nicholls & Sulem (2005), Guyenne & Nicholls (2005), Guyenne & Nicholls (2007)
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. The 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. The experiments were conducted in the W. G. Pritchard Fluid Mechanics Laboratory at Penn State University (J. Hammack, D. Henderson).
Further details can be found in Hammack, Henderson, Guyenne & Yi (2004), 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
Fig: Large-amplitude internal solitary waves in the two-layer model derived by Craig, Guyenne & Kalisch (2004): (left) sequence of wave profiles for varying amplitudes, (right) comparison with the KdV soliton (dots).
We have derived a Hamiltonian formulation of the problem of a dynamic interface with rigid lid boundary conditions, as well as that of a free interface coupled with a free surface. From this formulation, we have developed a Hamiltonian perturbation theory for the long-wave limits, 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 novel 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)
Fig: Computed spectra (solid line) and predicted Kolmogorov power laws (dashed line) in weak wave turbulence.
Weak wave turbulence theory is an efficient tool for the statistical description of systems dominated by resonant interactions between small-amplitude dispersive waves. It can be applied to a wide range of physical problems (oceanography, optics, plasma physics, acoustics, etc.). For example, this 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 turbulence in media with two types of interacting waves, for which coherent structures cannot develop. Our numerical results show a very good agreement with theoretical predictions.
Further details can be found in Dias, Guyenne & Zakharov (2001), Zakharov, Guyenne, Pushkarev & Dias (2001)