As the second largest river in the U.S., the entrance to the Columbia River is home to some of the most extreme wave conditions on the Pacific Coast. Winter storms commonly generate waves 6-8 m in height, which in combination with strong tidal currents, can produce dangerous navigation conditions. To improve understanding of the wave dynamics in this complex setting, the SWAN model is applied; 2 hindcasts are conducted and an operations forecast is developed. The model is forced with offshore wave heights obtained from a buoy located in 134 m water depth (for the hindcasts) and a specialized WaveWatchIII forecast (for the forecast). In both cases tidal currents are obtained from SELFE, a circulation model of the Columbia River. The hindcasts are validated through measurements obtained from an inshore buoy located in 25 m water depth, a 4-week field experiment and remote sensing methods. The model performs best at the location of the buoy, with a normalized root-mean-squared error (NRMSE) of 11%, primarily because it is outside the area of strong tidal currents. Within the river mouth, the model is able to predict the changes in the wave field due to currents, but its performance is limited by errors in velocity estimates and strong shears in the tidal current profile. From the modeling work, it is evident that wave transformations at the mouth of the river are dominated by the tidal currents. The forecast has been operational since August 2011 and provides 45-hours of predictive wave information. In comparison with measured wave heights at the buoy, the forecast performs well, with a NRMSE of 16%. The majority of errors are caused by errors in the input conditions, since they themselves are forecasted. Additional errors arise from phase-resolved properties in the wave field that the model is unable to produce; these errors are also present in the hindcasts. Despite the limitations, this forecast provides valuable information to bar pilots since it includes the effects of the tidal currents. / Graduation date: 2013

Fluid-structure interaction (FSI) is an interesting and challenging interdisciplinary area comprised of fields such as engineering- fluids/structures/solids, computational science, and mathematics. FSI has several practical engineering applications such as the design of coastal infrastructure (such as bridges, levees) subjected to harsh environments from natural forces such as tsunamis, storm surges, etc. Development of accurate input conditions to more detailed and complex models involving flexible structures in a fluid domain is an important requirement for the solution of such problems. FSI researchers often employ methods that use results from physical wave basin experiments to assess the wave forces on structures. These experiments, while closer to the physical phenomena, often tend to be time-consuming and expensive. Experiments are also not easily accessible for conducting parametric studies. Alternatively, numerical models when developed with similar capabilities will complement the experiments very well because of the lower costs and the ability to study phenomena that are not feasible in the laboratory. This dissertation is aimed at contributing to the solution of a significant component of the FSI problem with respect to engineering applications, covering accurate input to detailed models and a numerical wave basin to complement large-scale laboratory experiments. To this end, this work contains a description of a three-dimensional numerical wave tank (3D-NWT), its enhancements including the piston wavemaker for generation of waves such as solitary, periodic, and focused waves, and validation using large-scale experiments in the 3D wave basin at Oregon State University. Performing simulations involving fluid dynamics is computational-intensive and the complexity is magnified by the presence of the flexible structure(s) in the fluid domain. The models are also required to take care of large-scale domains such as a wave basin in order to be applicable to practical problems. Therefore, undertaking these efforts requires access to high-performance computing (HPC) platforms and development of parallel codes. With these objectives in mind, parallelization of the 3D-NWT is carried out and discussed in this dissertation. / Graduation date: 2011

[Formulae and special characters can only be approximated here. Please see the pdf version of the abstract for an accurate reproduction.] Wave run-up is the vertical uprush of water when an incident wave impinges on a free- surface penetrating body. For large volume offshore structures the wave run-up on the weather side of the supporting columns is particularly important for air-gap design and ultimately the avoidance of pressure impulse loads on the underside of the deck structure. This investigation focuses on the limitations of conventional wave diffraction theory, where the free-surface boundary condition is treated by a Stokes expansion, in predicting the harmonic components of the wave run-up, and the presentation of a simplified procedure for the prediction of wave run-up. The wave run-up is studied on fixed vertical cylinders in plane progressive waves. These progressive waves are of a form suitable for description by Stokes' wave theory whereby the typical energy content of a wave train consists of one fundamental harmonic and corresponding phase locked Fourier components. The choice of monochromatic waves is indicative of ocean environments for large volume structures in the diffraction regime where the assumption of potential flow theory is applicable, or more formally A/a < Ο(1) (A and a being the wave amplitude and cylinder radius respectively). One of the unique aspects of this work is the investigation of column geometry effects - in terms of square cylinders with rounded edges - on the wave run-up. The rounded edges of each cylinder are described by the dimensionless parameter rc/a which denotes the ratio of edge corner radius to half-width of a typical column with longitudinal axis perpendicular to the quiescent free-surface. An experimental campaign was undertaken where the wave run-up on a fixed column in plane progressive waves was measured with wire probes located close to the cylinder. Based on an appropriate dimensional analysis, the wave environment was represented by a parametric variation of the scattering parameter ka and wave steepness kA (where k denotes the wave number). The effect of column geometry was investigated by varying the edge corner radius ratio within the domain 0 <=rc/a <= 1, where the upper and lower bounds correspond to a circular and square shaped cylinder respectively. The water depth is assumed infinite so that the wave run-up caused purely by wave-structure interaction is examined without the additional influence of a non-decaying horizontal fluid velocity and finite depth effects on wave dispersion. The zero-, first-, second- and third-harmonics of the wave run-up are examined to determine the importance of each with regard to local wave diffraction and incident wave non-linearities. The modulus and phase of these harmonics are compared to corresponding theoretical predictions from conventional diffraction theory to second-order in wave steepness. As a result, a basis is formed for the applicability of a Stokes expansion to the free-surface boundary condition of the diffraction problem, and its limitations in terms of local wave scattering and incident wave non-linearities. An analytical approach is pursued and solved in the long wavelength regime for the interaction of a plane progressive wave with a circular cylinder in an ideal fluid. The classical Stokesian assumption of infinitesimal wave amplitude is invoked to treat the free-surface boundary condition along with an unconventional requirement that the cylinder width is assumed much smaller than the incident wavelength. This additional assumption is justified because critical wavelengths for wave run-up on a fixed cylinder are typically much larger in magnitude than the cylinder's width. In the solution, two coupled perturbation schemes, incorporating a classical Stokes expansion and cylinder slenderness expansion, are invoked and the boundary value problem solved to third-order. The formulation of the diffraction problem in this manner allows for third-harmonic diffraction effects and higher-order effects operating at the first-harmonic to be found. In general, the complete wave run-up is not well accounted for by a second-order Stokes expansion of the free-surface boundary condition and wave elevation. This is however, dependent upon the coupling of ka and kA. In particular, whilst the modulus and phase of the second-harmonic are moderately predicted, the mean set-up is not well predicted by a second-order Stokes expansion scheme. This is thought to be caused by higher than second-order non-linear effects since experimental evidence has revealed higher-order diffraction effects operating at the first-harmonic in waves of moderate to large steepness when k < < 1. These higher-order effects, operating at the first-harmonic, can be partially accounted for by the proposed long wavelength formulation. For small ka and large kA, subsequent comparisons with measured results do indeed provide a better agreement than the classical linear diffraction solution of Havelock (1940). To account for the complete wave run-up, a unique approach has been adopted where a correction is applied to a first-harmonic analytical solution. The remaining non-linear portion is accounted for by two methods. The first method is based on regression analysis in terms of ka and kA and provides an additive correction to the first-harmonic solution. The second method involves an amplification correction of the first-harmonic. This utilises Bernoulli's equation applied at the mean free-surface position where the constant of proportionality is empirically determined and is inversely proportional to ka. The experimental and numerical results suggest that the wave run-up increases as rc/a--› 0, however this is most significant for short waves and long waves of large steepness. Of the harmonic components, experimental evidence suggests that the effect of a variation in rc/a on the wave run-up is particularly significant for the first-harmonic only. Furthermore, the corner radius effect on the first-harmonic wave run-up is well predicted by numerical calculations using the boundary element method. Given this, the proposed simplified wave run-up model includes an additional geometry correction which accounts for rc/a to first-order in local wave diffraction. From a practical view point, it is the simplified model that is most useful for platform designers to predict the wave run-up on a surface piercing column. It is computationally inexpensive and the comparison of this model with measured results has proved more promising than previously proposed schemes.

[Truncated abstract] Internal waves redistribute energy and momentum in stratified lakes and constitute the path through which the energy that is introduced at the lake scale is cascaded down to the turbulent scales where mixing and dissipation take place. This research, based on intensive field data complemented with numerical simulations, covers several aspects of the energy flux path ranging from basin-scale waves with periods of several hours to high frequency waves with periods of few minutes. It was found that, at the basin-scale level, the horizontal shape of the lake at the level of the metalimnion controls the period and modal structure of the basin-scale natural modes, conforming to the dispersion relationship of internal waves in circular basins. The sloping bottom, in turn, produces local intensification of the wave motion due to focusing of internal wave rays over near-critical slopes, providing hot spots for the degeneration of the basin-scale waves due to shear instabilities, nonlinear processes and dissipation. Different types of high-frequency phenomena were observed in a stratified lake under different forcing conditions. The identification of the generation mechanisms revealed how these waves extract energy from the mean flow and the basin-scale waves. The changes to the stratification show that such waves contribute to mixing in different ways . . . Detailed field observations were used to develop a comprehensive description of an undocumented energy flux mechanism in which shear-instabilities with significant amplitudes away from the generation level are produced in the surface layer due to the shear generated by the wind. The vertical structure of these instabilities is such that the growing wave-related fluctuations strain the density field in the metalimnion triggering secondary instabilities. These instabilities also transport energy vertically to the thermocline where they transfer energy back to the mean flow through interaction with the background shear.

O conhecimento de zonas potencialmente instáveis no fundo do mar é de fundamental importância para o desenvolvimento das estruturas marinhas, pois permite posicionar estruturas offshore em áreas mais seguras, reduzindo-se possíveis danos, custos e eventual poluição ambiental. Nesse contexto, o objetivo do presente trabalho é investigar, através de uma abordagem analítico-numérica, a estabilidade de maciços submarinos submetidos ao carregamento cíclico da onda. O efeito das ondas de água sobre o leito submerso é descrito pela propagação de uma onda de pressão ao longo de sua superfície, empregando-se a teoria linear de Stokes. São considerados maciços com superfície superior horizontal e inclinada, constituídos por material coesivo (argilas) e material granular (areias). Em maciços constituídos por solos finos, a capacidade resistente do material é modelada pelo critério de Tresca não-homogêneo e a análise da estabilidade é desenvolvida em condição não drenada. Por outro lado, em leitos granulares, a resistência do meio depende explicitamente do valor da poropressão, sendo descrita classicamente pelo critério de Coulomb sem coesão. A análise de estabilidade é então desenvolvida em tensões efetivas e o gradiente de poropressão atua como uma força volumétrica sobre o esqueleto, caracterizando o modo de carregamento principal deste material. Em razão da tendência a se densificar quando submetido a um estado de tensões desviadoras cíclicas, ocorre, em geral, a acumulação de excesso de poropressão no maciço granular. Consequentemente, o acoplamento entre o comportamento do material e o carregamento cíclico tem fundamental importância sobre o cálculo do excesso de poropressão desenvolvido. Para determinação das forças de percolação, considera-se uma abordagem simplificada baseada na partição das deformações em contribuições reversível e irreversível, que permite desacoplar o cálculo da pressão intersticial induzida pela onda. Aplicando-se conceitos da teoria da Análise Limite é possível formular limites inferiores e superiores da máxima amplitude segura do carregamento da onda. Finalmente, os efeitos da declividade da superfície do leito e da espessura de camada de solo sobre a estabilidade são analisados. / The knowledge of potentially unstable areas on the seabed is of fundamental importance to the development of marine structures, because it allows to install offshore structures in safer areas, reducing possible damages, costs and eventual environmental pollution. In this context, the objective of the present work is to investigate the stability of submarine soil masses subjected to the wave cyclic loading through analytical-numerical approaches. The effect of water waves on the submerged bed is described by the propagation of a pressure wave along its surface, using the Stokes’s linear theory. Soil masses with horizontal and sloped upper surface, composed of cohesive material (clays) and granular materiais (sands) are considered in this study. In soil masses constituted of fine soil, the material strength capacity is modeled by the non-homogeneous Tresca criterion and the stability analysis is carried out in undrained condition. On the other hand, in granular beds, the strength explicitly depends on the pore pressure value, being classically described by the Coulomb criterion without cohesion. Then, the stability analysis is developed in effective stress and the pore pressure gradient acts as a volumetric force on the skeleton, characterizing the main charging mode of this material. Due to the tendency to densify when subjected to a cyclic deviatoric stress state occurs, in general, the build-up of pore pressure excess in the granular mass. Consequently, the coupling between the material behavior and the cyclic loading has fundamental importance in the calculation of the pore pressure excess generated. In order to define the seepage forces, a simplified approach based on the partition of deformations in reversible and irreversible contributions is considered, which allows to decouple the wave-induced pore pressure calculation. Applying the concepts of the limit analysis theory it is possible to formulate upper and lower boundaries of the maximum safe amplitude of the wave loading. Finally, the effects of the seabed surface steepness and of the soil layer thickness on the stability are analyzed.

O conhecimento de zonas potencialmente instáveis no fundo do mar é de fundamental importância para o desenvolvimento das estruturas marinhas, pois permite posicionar estruturas offshore em áreas mais seguras, reduzindo-se possíveis danos, custos e eventual poluição ambiental. Nesse contexto, o objetivo do presente trabalho é investigar, através de uma abordagem analítico-numérica, a estabilidade de maciços submarinos submetidos ao carregamento cíclico da onda. O efeito das ondas de água sobre o leito submerso é descrito pela propagação de uma onda de pressão ao longo de sua superfície, empregando-se a teoria linear de Stokes. São considerados maciços com superfície superior horizontal e inclinada, constituídos por material coesivo (argilas) e material granular (areias). Em maciços constituídos por solos finos, a capacidade resistente do material é modelada pelo critério de Tresca não-homogêneo e a análise da estabilidade é desenvolvida em condição não drenada. Por outro lado, em leitos granulares, a resistência do meio depende explicitamente do valor da poropressão, sendo descrita classicamente pelo critério de Coulomb sem coesão. A análise de estabilidade é então desenvolvida em tensões efetivas e o gradiente de poropressão atua como uma força volumétrica sobre o esqueleto, caracterizando o modo de carregamento principal deste material. Em razão da tendência a se densificar quando submetido a um estado de tensões desviadoras cíclicas, ocorre, em geral, a acumulação de excesso de poropressão no maciço granular. Consequentemente, o acoplamento entre o comportamento do material e o carregamento cíclico tem fundamental importância sobre o cálculo do excesso de poropressão desenvolvido. Para determinação das forças de percolação, considera-se uma abordagem simplificada baseada na partição das deformações em contribuições reversível e irreversível, que permite desacoplar o cálculo da pressão intersticial induzida pela onda. Aplicando-se conceitos da teoria da Análise Limite é possível formular limites inferiores e superiores da máxima amplitude segura do carregamento da onda. Finalmente, os efeitos da declividade da superfície do leito e da espessura de camada de solo sobre a estabilidade são analisados. / The knowledge of potentially unstable areas on the seabed is of fundamental importance to the development of marine structures, because it allows to install offshore structures in safer areas, reducing possible damages, costs and eventual environmental pollution. In this context, the objective of the present work is to investigate the stability of submarine soil masses subjected to the wave cyclic loading through analytical-numerical approaches. The effect of water waves on the submerged bed is described by the propagation of a pressure wave along its surface, using the Stokes’s linear theory. Soil masses with horizontal and sloped upper surface, composed of cohesive material (clays) and granular materiais (sands) are considered in this study. In soil masses constituted of fine soil, the material strength capacity is modeled by the non-homogeneous Tresca criterion and the stability analysis is carried out in undrained condition. On the other hand, in granular beds, the strength explicitly depends on the pore pressure value, being classically described by the Coulomb criterion without cohesion. Then, the stability analysis is developed in effective stress and the pore pressure gradient acts as a volumetric force on the skeleton, characterizing the main charging mode of this material. Due to the tendency to densify when subjected to a cyclic deviatoric stress state occurs, in general, the build-up of pore pressure excess in the granular mass. Consequently, the coupling between the material behavior and the cyclic loading has fundamental importance in the calculation of the pore pressure excess generated. In order to define the seepage forces, a simplified approach based on the partition of deformations in reversible and irreversible contributions is considered, which allows to decouple the wave-induced pore pressure calculation. Applying the concepts of the limit analysis theory it is possible to formulate upper and lower boundaries of the maximum safe amplitude of the wave loading. Finally, the effects of the seabed surface steepness and of the soil layer thickness on the stability are analyzed.

Thesis (Ph.D.)--Joint Program in Applied Ocean Science and Engineering (Massachusetts Institute of Technology, Dept. of Ocean Engineering; and the Woods Hole Oceanographic Institution), 2000. / Includes bibliographical references (p. 299-312). / by Sean Patrick McKenna. / Ph.D.

Ce travail s'inscrit dans l'étude mathématique d'écoulements de fluides environnementaux. Nous en abordons deux aspects, à travers deux contextes distincts d'application.En lien avec la simulation des écoulements en milieux poreux, on s'intéresse dans une première partie à la discrétisation d'opérateurs de diffusion anisotropes hétérogènes par des méthodes de volumes finis sur des maillages généraux. Dans le but d'obtenir des solutions approchées qui respectent les bornes physiques des modèles, notre attention se porte sur la conservation du principe du maximum pour les opérateurs elliptiques. Nous présentons des mécanismes généraux permettant de corriger tout schéma volumes finis afin de garantir un principe du maximum discret tout en préservant certaines de ses propriétés principales. On étudie en particulier les propriétés de coercivité et de convergence des schémas corrigés.La deuxième partie est consacrée à la construction de modèles approchés pour la propagation des vagues en eaux peu profondes et sur des topographies irrégulières. A cet effet, nous proposons tout d'abord une adaptation de la démarche d'étude classique à des écoulements bidimensionnels sur des topographies polygonales. Dans un cadre plus général, nous développons ensuite une démarche formelle qui débouche sur des alternatives non locales à quelques modèles classiques (équations de Saint-Venant, équations de Serre, système de Boussinesq). Ces nouveaux modèles contiennent des termes régularisants pour les contributions du fond. / This work investigates two research questions associated with environmental flows and their mathematical modeling.The first part is devoted to the development of finite volume methods for anisotropic and heterogeneous diffusion operators arising in models of porous media flows. To ensure that the approximate solutions lie within physical bounds, we aim at maintaining a discrete analogous of the maximum principle for elliptic operators. Starting from any given cell-centered finite volume scheme, we present a general approach to devise non-linear corrections providing a discrete maximum principle while retaining some main properties of the scheme. In particular, we study the coercivity and convergence properties of the modified schemes.The second part of this work focuses on the derivation of approximate models for shallow water wave propagation over rough topographies. In the particular case of one-dimensional polygonal bottom profiles, we first propose an adaptation of the usual derivation method using complex analysis tools. We then develop a formal approach to account for more general topographies. We propose nonlocal alternatives to some classical models (namely Saint-Venant equations, Serre equations and Boussinesq system). All these alternative models only involve smoothing contributions of the bottom.

Cette thèse présente quelques contributions à l'étude qualitative de solutions d'équations aux dérivées partielles non linéaires dans des modèles issus de l'optique et de la mécanique des fluides. Nous nous intéressons plus précisément à l'existence de solutions et à leur stabilité temporelle. Le Chapitre 1 est consacré à l'équation de Lugiato-Lefever, qui est une variante de l'équation de Schrödinger non linéaire et qui a été dérivée dans plusieurs contextes en optique. En utilisant des outils de la théorie des bifurcations et des formes normales, nous procédons à une étude systématique des solutions stationnaires de cette équation, et prouvons l'existence de solutions périodiques et localisées. Dans le Chapitre 2, nous présentons un critère simple d'instabilité linéaire pour des ondes non linéaires. Nous appliquons ce résultat aux équations de Lugiato-Lefever, de Kadomtsev-Petviashvili-I et de Davey-Stewartson. Ces deux dernières équations sont des équations modèles dérivées en mécanique des fluides. Dans le Chapitre 3, nous montrons un critère d'instabilité linéaire pour des solutions périodiques de petite amplitude, par rapport à certaines perturbations quasipériodiques. Ce résultat est ensuite appliqué à l'équation de Lugiato-Lefever. / In this thesis we present several contributions to qualitative study of solutions of nonlinear partial differential equations in optics and fluid mechanics models. More precisely, we focus on the existence of solutions and their stability properties. In Chapter 1, we study the Lugiato-lefever equation, which is a variant of the nonlinear Schrödinger equation arising in sereval contexts in nonlinear optics. Using tools from bifurcation and normal forms theory, we perfom a systematic analysis of stationary solutions of this equation and prove the existence of periodic and localized solutions. In Chapter 2, we present a simple criterion for linear instability of nonlinear waves. We then apply this result to the Lugiato-Lefever equation, to the Kadomtsev-Petviashvili-I equation and the Davey-Stewartson equations. These last two equations are model equations arising in fluid mechanics. In Chapter 3, we prove a criterion for linear instability of periodic solutions with small amplitude, with respect to certain quasiperiodic perturbations. This result is then applied to the Lugiato-Lefever equation.

Cette thèse est consacrée à l'analyse mathématique de l'équation d'Euler incompressible à surface libre. On se concentre sur la propriété dispersive et sur la théorie de Cauchy à faible régularité. Une grande part de la thèse est consacrée à l'étude de l'équation des ondes de gravité-capillarité. On établit des critères d'explosion et la persistance de régularité dans les espaces de Sobolev. En démontrant les estimations de Strichartz pour les solutions à faible régularité, on obtient des théories de Cauchy pour les données initiales dont la vitesse peut être non-lipschitzienne. Dans une autre part de la thèse, on étudie la propriété dispersive des équations des ondes de surface. Plus précisément, on s'intéresse aux estimations de Strichartz. On démontre que, pour les solutions raisonnablement régulières, les équations des ondes de surface non linéaires obéissent aux mêmes estimations de Strichartz comme dans le cas des équations linéarisées. / This dissertation is devoted to the mathematical analysis of the water waves systems. We focus on the dispersive property and the Cauchy problem for rough initial data. One of the main objects of study is the gravity-capillary water waves system. We establish blow-up criteria and the persistence of Sobolev regularity. By proving Strichartz estimates for rough solutions, we obtain Cauchy theories for non-Lipschitz initial velocity. In another part of the dissertation, we study the dispersive property of the fully nonlinear water waves systems. More specifically, we are interested in Strichartz estimates. We prove for sufficiently smooth solutions that the nonlinear systems obey the same Strichartz estimates as their linearizations do.