A double-layer Boussinesq-type model for highly nonlinear and dispersive waves

28 pages, 5 figures. Soumis à Proceedings of the Royal Society of London A.We derive and analyze in the framework of the mild-slope approximation a new double-layer Boussinesq-type model which is linearly and nonlinearly accurate up to deep water. Assuming the flow to be irrotational, we formulate the problem in terms of the velocity potential thereby lowering the number of unknowns. The model derivation combines two approaches, namely the method proposed by Agnon et al. (Agnon et al. 1999, J. Fluid Mech., 399 pp. 319-333) and enhanced by Madsen et al. (Madsen et al. 2003, Proc. R. Soc. Lond. A, 459 pp. 1075-1104) which consists in constructing infinite-series Taylor solutions to the Laplace equation, to truncate them at a finite order and to use Padé approximants, and the double-layer approach of Lynett & Liu (Lynett & Liu 2004, Proc. R. Soc. Lond. A, 460 pp. 2637-2669) allowing to lower the order of derivatives. We formulate the model in terms of a static Dirichlet-Neumann operator translated from the free surface to the still-water level, and we derive an approximate inverse of this operator that can be built once and for all. The final model consists of only four equations both in one and two horizontal dimensions, and includes only second-order derivatives, which is a major improvement in comparison with so-called high-order Boussinesq models. A linear analysis of the model is performed and its properties are optimized using a free parameter determining the position of the interface between the two layers. Excellent dispersion and shoaling properties are obtained, allowing the model to be applied up to deep water. Finally, numerical simulations are performed to quantify the nonlinear behaviour of the model, and the results exhibit a nonlinear range of validity reaching deep water areas

Assessment and validation of bathymetric inversion techniques using synthetic waves

Bathymetric inversion using video images is a new and promising technique in order to monitor beach morphology. In this TFG, a Boussinesq model called FunWave and a linear wave propagation routine will be used to create synthetic wave propagation over different bathymetries. Two bathymetric inversion methodologies will be subsequently applied to retrieve the bathymetry, comparing their results and analysing their sensitivity to different aspects, in order to understand and improve their performance

Finite volume and pseudo-spectral schemes for the fully nonlinear 1D Serre equations

After we derive the Serre system of equations of water wave theory from a generalized variational principle, we present some of its structural properties. We also propose a robust and accurate finite volume scheme to solve these equations in one horizontal dimension. The numerical discretization is validated by comparisons with analytical, experimental data or other numerical solutions obtained by a highly accurate pseudo-spectral method.Comment: 28 pages, 16 figures, 75 references. Other author's papers can be downloaded at http://www.denys-dutykh.com

Cross-shelf circulation and momentum and heat balances over the inner continental shelf near Martha's Vineyard, Massachusetts

Submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy at the Massachusetts Institute of Technology and the Woods Hole Oceanographic Institution September 2007.The water circulation and evolution of water temperature over the inner continental shelf are investigated using observations of water velocity, temperature, density, and bottom pressure; surface gravity waves; wind stress; and heat flux between the ocean and atmosphere during 2001-2007. When waves are small, cross-shelf wind stress is the dominant mechanism driving cross-shelf circulation. The along-shelf wind stress does not drive a substantial cross-shelf circulation. The response to a given wind stress is stronger in summer than winter. The cross-shelf transport in the surface layer during winter agrees with a two-dimensional, unstratified model. During large waves and onshore winds the cross-shelf velocity is nearly vertically uniform, because the wind- and wave-driven shears cancel. During large waves and offshore winds the velocity is strongly vertically sheared because the wind- and wave-driven shears have the same sign. The subtidal, depth-average cross-shelf momentum balance is a combination of geostrophic balance and a coastal set-up and set-down balance driven by the cross-shelf wind stress. The estimated wave radiation stress gradient is also large. The dominant along-shelf momentum balance is between the wind stress and pressure gradient, but the bottom stress, acceleration, Coriolis, Hasselmann wave stress, and nonlinear advection are not negligible. The fluctuating along-shelf pressure gradient is a local sea level response to wind forcing, not a remotely generated pressure gradient. In summer, the water is persistently cooled due to a mean upwelling circulation. The cross-shelf heat flux nearly balances the strong surface heating throughout midsummer, so the water temperature is almost constant. The along-shelf heat flux divergence is apparently small. In winter, the change in water temperature is closer to that expected due to the surface cooling. Heat transport due to surface gravity waves is substantial.My last three years of thesis work were supported by National Aeronautics and Space Administration Headquarters under the Earth System Science Fellowship Grant NNG04GQ14H, and by WHOI Academic Programs Fellowship Funds. I also benefited from the freedom of a Clare Boothe Luce Fellowship during my first year in the Joint Program, which allowed me more time than is usual to explore different research topics before choosing an advisor. This research was also funded by the National Aeronautics and Space Administration under grant NNG04GL03G and the Ocean Sciences Division of the National Science Foundation under grants OCE-0241292 and OCE-0548961. The Martha's Vineyard Coastal Observatory is partly funded by the Woods Hole Oceanographic Institution and the Jewett/EDUC/Harrison Foundation. The ADCP deployments at CBLAST site F were funded by National Science Foundation Small Grant for Exploratory Research OCE-0337892. Ship time for deployment and recovery of the F ADCP was provided by Robert Weller through Office of Naval Research contracts N00014-01-1-0029 and N00014-05-10090 for the Low-Wind Component of the Coupled Boundary Layers Air-Sea Transfer Experiment

Physical Mechanisms of the Rogue Wave Phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin-Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrodinger equation, the Davey - Stewartson system, the Korteweg - de Vries equation, the Kadomtsev - Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon

Comparison of sea-ice freeboard distributions from aircraft data and cryosat-2

The only remote sensing technique capable of obtain- ing sea-ice thickness on basin-scale are satellite altime- ter missions, such as the 2010 launched CryoSat-2. It is equipped with a Ku-Band radar altimeter, which mea- sures the height of the ice surface above the sea level. This method requires highly accurate range measure- ments. During the CryoSat Validation Experiment (Cry- oVEx) 2011 in the Lincoln Sea, Cryosat-2 underpasses were accomplished with two aircraft, which carried an airborne laser-scanner, a radar altimeter and an electro- magnetic induction device for direct sea-ice thickness re- trieval. Both aircraft flew in close formation at the same time of a CryoSat-2 overpass. This is a study about the comparison of the sea-ice freeboard and thickness dis- tribution of airborne validation and CryoSat-2 measure- ments within the multi-year sea-ice region of the Lincoln Sea in spring, with respect to the penetration of the Ku- Band signal into the snow