This thesis presents Wronskian and Gram solutions to both the Korteweg-de Vries and Kadomtsev-Petviashvili equations, which are then scalable to arbitrarily large numbers of interacting solitons. Through variable transformation and use of the Hirota derivative, these nonlinear partial differential equations can be expressed in bilinear form. We present both Wronskian and Gram determinants which satisfy the equations. N=1,2,3 and higher order solutions are presented graphically; parameter tuning and the resultant behavioral differences are demonstrated and discussed. In addition, we compare these solutions to naturally occurring shallow water waves on beaches.

This thesis examines chaotic mixing in wavy-type channels and two-layer shallow water flow. For wavy-type channels, the equations of motion for vortices and fluid particles are derived assuming two-dimensional irrotational, incompressible flow. Instantaneous positions of the vortices and particles are determined using Lagrangian tracking, and are conformally mapped to the physical domain. Unsteady vortex motion is analysed, and vortex-induced chaotic mixing in the channels studied. The dynamics of mixing associated with the evolution of the separation bubble, and the invariant manifolds are examined. Mixing efficiencies of the different channel configurations are compared statistically. Fractal enhancement of productivity is identified in the study of auto-catalytic reaction in the wavy channel. For the two-layer shallow water model, an entropy-correction free Roe type two-layer shallow water solver is developed for a hyperbolic system with non-conservative products and source terms. The scheme is well balanced and satisfies the C-property such that smooth steady solutions are second order accurate. Numerical treatment of the wet-dry front of both layers and the loss of hyperbolicity are incorporated. The solver is tested rigorously on a number of 1D and 2D benchmark test cases. For 2D implementation, a dynamically adaptive quadtree grid generation system is adopted, giving results which are in excellent agreement with those on regular grids at a much lower cost. It is also shown that algebraic balancing cannot be applied directly to a two-layer shallow water flow due to the lack of simultaneous referencing for the still water position for both layers. The adaptive two-layer shallow water solver is applied successfully to flow in an idealised tidal channel and to tidal-driven flow in Tampa Bay, Florida. Finally, chaotic advection and particle mixing is studied for wind-induced recirculation in two-layer shallow water basins, as well as Tampa Bay, Florida.

Although the spectral technique is frequently used for the horizontal discretization in global atmospheric models, it is not common in limited area models (LAMs) because of the non-periodic boundary conditions. We apply the Haugen-Machenhauer extension technique to a regional three-layer shallow-water model based on double Fourier series. The method extends the time-dependent boundary fields into a zone outside the integration area in a way that periodic fields are obtained. The boundary fields necessary for the regional model simulations are calculated in advance by a global simulation performed. In contrast to other studies, we use exactly the same numerical model for the global and the regional simulation, respectively. The only difference between these simulations is the model domain. Therefore, a relatively objective measure for errors associated with the extension technique can be obtained. First, we compare an analytic stationary non-linear and non-periodic solution of the governing model equations with the spectral LAM solution. Secondly, we compare the time evolution of pressure and fiow structures during a westerly fiow across an asymmetric large-scale topography in the global and regional model domains. Both simulations show a good agreement between the regional and the global solutions. The rms-errors amount to about 2 m for the layer heights and 0.2 ms-1 for the velocity components in the mountain fiow case after a 48 h integration period. Finally, we repeat this simulation with models based on 2nd and 4th order finite differences, respectively, and compare the errors of the spectral model version with the errors of the grid point versions. We demonstrate that the high accuracy of global spectral methods can also be realized in the regional model by using the Haugen-Machenhauer extension technique. / Obwohl spektrale Techniken häufig zur horizontalen Diskretisierung in globalen Atmosphärenmodellen genutzt werden, sind sie aufgrund der nicht-periodischen Randbedingungen in Regionalmodellen nicht üblich. Wir verwenden das Erweiterungsverfahren von Haugen und Machenhauer in einem Flachwassermodell mit drei Schichten, das auf doppelten Fourier-Reihen basiert. Das Verfahren setzt die zeitabhängigen Randfelder so in einen Bereich außerhalb des Integrationsgebiets fort, daß man periodische Randbedingungen erhält. Die für die Simulationen mit dem Regionalmodell benötigten Randfelder werden mittels einer zuvor durchgeführten globalen Simulation berechnet. Im Gegensatz zu anderen Untersuchungen verwenden wir genau das gleiche Modell für die globale und die regionale Simulation. Der einzige Unterschied zwischen den beiden Simulationen ist das Modellgebiet. Dadurch erhält man ein relativ objektives Maß für die Fehler, die durch die Anwendung des Erweiterungsverfahrens entstehen. Als ersten Test vergleichen wir zunächst eine analytische, stationäre, nicht-lineare und nicht-periodische Lösung der Modellgleichungen mit der spektralen Lösung des Regionalmodells. Zweitens vergleichen wir die zeitliche Entwicklung von Druck- und Strömungsmustern während einer westlichen Strömung über eine unsymmetrische, großskalige Topographie im globalen bzw. regionalen Modellgebiet. Beide Simulationen zeigen eine gute Übereinstimmung der globalen und regionalen Lösungen. Die rms-Fehler betragen ungefähr 2 m für die Schichthöhen und 0.2 ms-1 für die Geschwindigkeitskomponenten bei der Bergüberströmungssimulation nach einer Integrationszeit von 48 h. Darüberhinaus wiederholen wir diese Simulation mit auf Finiten Differenzen 2. bzw. 4. Ordnung basierenden Modellen und vergleichen die Fehler der spektralen und der Gitterpunktversionen. Wir zeigen, daß die hohe Genauigkeit der globalen spektralen Methoden durch die Anwendung des Erweiterungsverfahrens von Haugen und Machenhauer auch auf das regionale Gebiet übertragen werden kann.

Esta tese apresenta uma nova metodologia para animação de objetos líquidos em imagens. Contrariamente às técnicas existentes, este método é baseado em um modelo físico, o que proporciona efeitos realísticos. A perspectiva da imagem é obtida com a intervenção do usuário, por um esquema simples de calibração da câmera, o qual permite a projeção da camada da imagem a ser animada sobre um plano horizontal no espaço tridimensional. As equações de águas rasas conduzem a simulação e as informações de altura são projetadas de volta ao espaço da imagem utilizando traçado de raios. Além disso, efeitos de refração e iluminação são aplicados durante este estágio, resultando em animações realísticas e convincentes / This work presents a new methodology for animating liquid objects depicted in a still image. In contrast to existing techniques, the proposed method relies on a physical model to accomplish the animation, resulting in realistic effects. Image perspective is handled through a simple user assisted camera calibration scheme which allows one to project the image layers to be animated onto the horizontal plane in the three-dimensional space. Shallow-Water equations drive the simulation and the resulting height field is projected back to the image space via ray-tracing. Refraction and lighting effects are also accomplished during the ray-tracing stage, resulting in realistic and convincing animations

Hydraulic structures such as dykes and groynes are commonly used to help control river flows and reduce flood risk. The present research aims to develop an idealized model of the hydrodynamics in the vicinity of a large river bend, and the advection and mixing processes where groynes are located. In this study a curvilinear model of shallow water equations is applied to investigate chaotic advection of particles in a river bend similar in dimensions to a typical bend in the River Danube, Hungary. First, a curvilinear grid generator is developed based on Poisson-type elliptic partial differential equations. The grid generator is verified for benchmark tests concerning a circular domain and for distorted grids in a rectangular domain. It is found that multi-grid (MG) and conjugate gradient (CG) methods performed better computationally than successive over-relaxation (SOR) in generating the curvilinear grids. The open channel hydrodynamics are modelled using the shallow water equations (SWEs) derived by depth-averaging the continuity and Navier-Stokes momentum equations. Both Cartesian and curvilinear forms of the shallow water equations are presented. Both sets of equations are discretized spatially using finite differences and the solution marched forward in time using fourth-order Runge-Kutta scheme. The shallow water solvers are verified and validated for uniform flow in the rectangular channel, wind-induced set up in rectangular and circular basins, flow past a sidewall expansion, and Shallow flow in a rectangular channel with single groyne. A Lagrangian particle tracking model is used to predict the trajectories of tracer particles, and bilinear interpolation is used to provide a representation of the continuous flow field from discrete results. The particle tracking model is verified for trajectories in the flow field of a single free vortex and in the alternating flow field of a pair of blinking vortices. Excellent agreement is obtained with analytical solutions, previously published results in the literature. The combined shallow flow and Lagrangian particle tracking model is then used to simulate particle advection in the flow past a side-wall cavity containing a groyne and reasonable agreement is obtained with published experimental and alternative numerical data. Finally, the combined model is applied to simulate the shallow flow hydrodynamics, advection and mixing processes in the vicinity of groynes in river bend, the dimensions representative of a typical bend in the Danube River, Hungary.

The marine shallow-water hydrothermal vent system of Tutum Bay, Ambitle Island, PNG discharges hot, acidic, arsenic-rich, chemically reduced fluid into cool, alkaline, oxygenated seawater. Gradients in temperature, pH, total arsenic (TAs) and arsenic species, among others, are established as the two aqueous phases mix. Hydrous ferric oxides (HFO) are precipitated around focused venting, and coat the surrounding sediments visibly to 150 m away. HFO coatings, mechanical transport and weathering of volcanoclastic sediments, as well as dissolution of carbonate sediments nearer to venting, combine to alter sediment chemistry substantially. Tutum Bay surface sediments have a mean As concentration of 527 mg/kg. Arsenic at concentrations up to 50 mg/kg (mean = 19.7 mg/kg) was extracted from the easily extractable fraction of surface sediments. Arsenic is elevated in surface seawaters (8 µg/L) directly over hydrothermal vents, and As(III) is substantially enriched in both surface and bottom seawater throughout Tutum Bay. Surprisingly, aqueous As(V) far exceeded aqueous As(III) at almost all distances and depths investigated for Tutum Bay pore waters. These data indicate that throughout Tutum Bay, chemical disequilibria among As species provides potential metabolic energy for arsenite oxidizing and arsenate reducing microorganisms, and that As is bioavailable from two major environments: 1) easily-exchangeable As from surface sediments, and 2) in surface seawaters, which may allow for biological uptake and trophic transfer through plankton. The soft coral Clavularia sp., the calcareous algae Halimeda sp., and the tunicate Polycarpa sp. were collected and analyzed to assess bioaccumulation and biotransformation patterns. All organisms collected from the hydrothermal area displayed higher (2 to 20 times) TAs. Concentrations of arsenic species in their tissues were also elevated compared to the control site. Increased concentrations were observed near focused venting. Distinct arsenic speciation patterns in Clavularia and Polycarpa collected from near hydrothermal venting suggest rapid methylation/detoxification of arsenic, with enhanced bioaccumulation of dimethylarsenate and arsenobetaine as products of the organisms metabolic pathways. Elevated concentrations of As(III) in Halimeda suggest that this organism is not as efficient at methylating inorganic arsenic. The presence of arsenobetaine in Halimeda suggests the biomethylation pathway for calcareous algae is different from commonly studied seaweeds.

Flux gradient terms and source terms are two fundamental components of hyperbolic systems of balance law. Though having distinct mathematical natures, they form and maintain an exact balance in a special class of solutions, which are called steady-state solutions. In this dissertation, we are interested in the construction of well-balanced schemes, which are the numerical methods for hyperbolic systems of balance laws that are capable of exactly preserving steady-state solutions on the discrete level. We first introduce a well-balanced scheme for the Euler equations of gas dynamics with gravitation. The well-balanced property of the designed scheme hinges on a reconstruction process applied to equilibrium variables---the quantities that stay constant at steady states. In addition, the amount of numerical viscosity is reduced in the areas where the flow is in (near) steady-state regime, so that the numerical solutions under consideration can be evolved in a well-balanced manner. We then consider the shallow water equations with friction terms, which become very stiff when the water height is close to zero. The stiffness in the friction terms introduces additional difficulty for designing an efficient well-balanced scheme. If treated explicitly, the stiff friction terms impose a severe restriction on the time step. On the other hand, a straightforward (semi-) implicit treatment of the stiff friction terms can greatly enhance the efficiency, but will break the well-balanced property of the resulting scheme. To this end, we develop a new semi-implicit Runge-Kutta time integration method that is capable of maintaining the well-balanced property under the time step restriction determined exclusively by non-stiff components in the underlying equations. The well-balanced property of our schemes are tested and verified by extensive numerical simulations, and notably, the obtained numerical results clearly indicate that the well-balanced property plays an important role in achieving high resolutions when a coarse grid is used. / acase@tulane.edu

The biodiversity of two distinct marine environments was observed to describe the biogeocomplexity of these extreme ecological systems. A shallow-water hydrothermal vent in Papua New Guinea served as a study of a thermophilic ecosystem influenced by arsenic rich vent fluids while a 60 m deep offshore primarily anoxic karst sink served as a study of an anaerobic sulfur-influenced habitat. Both environments support unique biological communities that are influenced by the physical and chemical pressures imposed on them by the harsh conditions of these systems. In Tutum Bay, Ambitle Isle, Papua New Guinea, a transect was created from a shallow hydrothermal vent that extended 120 m away from the vent. Previous studies have shown that the geochemistry of the system is heavily influenced by arsenic which is toxic to most organisms. In this study, macro- and meiofauna were collected and scored and combined with bacterial sequence data collected along the length of the transect. It was found that near vent sites harbored biological communities more similar than sites further from the vent. Many species were found only at sites near the hydrothermal vent. Near-vent communities were less diverse than those away from the vent, and biodiversity generally increased as distance from the vent increased. Distinct correlations between thermophilic organisms and temperature were observed. The metabolic repertoire of the microbial communities suggests that many strategies are used to obtain energy and carbon. The relative abundance of bacteria containing genes to reduce arsenic was comparable to those able to reduce sulfur compounds. Primary production appeared to be a mix of chemo- and phototrophy. Food webs and association analysis suggest a complex interplay between macrofaunal, meiofaunal and bacterial communities. While the system is heavily influenced by arsenic, no specific correlation between the relative abundance of arsenic metabolizing organisms and the amount of arsenic in the system could be drawn. This is likely due to the fact that most of the arsenic produced by the system is readily adsorbed onto iron oxyhydroxides, reducing the arsenic's bioavailability. The anoxic conditions at Jewfish sink provide a different hurdle than the hot arsenic conditions found in Papua New Guinea. The anoxic conditions are shared by other pit features found in karst geography, but the metabolic processes between Jewfish sink and these other karst habitats are different. The blue holes and black holes of the Bahamas are some of the most well-studied of these karstic pits. In these features, which are large circular pits with diameters of over 300 m, light and sulfur are used as a means of energy acquisition. Jewfish sink, having an opening only 6 m in diameter, is light restricted compared to these systems. As a result, the strategy of organisms dwelling in the anoxic conditions of the sink is different than those found at the well-studied holes in the Bahamas. Geochemical measurements were recorded over two time periods spanning a combined total of 6 years. The anoxic bottom waters of Jewfish sink remain stable and contained high levels of sulfide throughout most of the seasons studies. Sequence analysis of prokaryotes within the sink showed that sulfur reducers had the highest relative abundance compared to other functional guilds. To monitor the changes of the microbial communities within the sink, bacterial communities were examined at 4 depths within the sink at 9 different intervals over a period of 685 days. Denaturing Gradient Gel Electrophoresis (DGGE) was used to fingerprint 16s rRNA bacterial communities and dissimilatory sulfite reducing communities by targeting the 16s rRNA bacterial gene and the dsr gene associated with dissimilatory sulfite reducing bacteria and archaea. The lowest depth studied within the sink (40 m) remained stable chemically and biologically until a turnover event occurred within the second winter of the study. This turnover event disrupted the biological communities at 40 m and led to a reestablished community comprised of different species that those found prior to the event. Upper waters within the sink show that clines establish themselves seasonally and partition zones that confine bacterial communities that are more similar to each other within these zones while excluding bacterial communities that are outside of these zones. Oxygenated water was shown to not contain prokaryotes containing the dsr gene. As the oxycline changed seasonally, dissimilatory sulfite reducing prokaryotes containing the dsr gene remained in the anoxic zone and required time to reestablish themselves whenever oxygenated water displaced them.

Nous étudions dans ce document des problèmes mathématiques de la mécanique des fluides. Ce travail s'articule principalement autour de deux thèmes : les modèles bi-couches à surface libre ou à toit rigide, l'analyse de problèmes définis dans des domaines à frontière libre.

On aborde des questions liées à l'étude de quelques aspects de la modélisation mathématique en mécanique des fluides et des structures. Cette étude s'articule principalement autour de deux thèmes : le couplage fluid-structure, l'analyse de problèmes d'écoulements peu profonds à surface libre.