An adaptive mesh refinement algorithm for shallow water equations is presented. The algorithm uses upwind scheme that is Godunov type and which approximately solves the Riemann problem using Roe's technique. A highly accurate solution is achieved by using the adaptive mesh refinement technique of Berger and Oliger for mesh refinement algorithm. The numerical method is second-order accurate and approximately max-min preserving by using van Leer limited-slope technique. One-dimensional nesting algorithm has been implemented successfully. Numerical results on a test problem verify the second order accuracy of the algorithm. The nested grid results yield the equivalent solution to that of the corresponding fine grid solution.

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Significant effort has been made to quantify and predict roll damping of vessels in the past. Similarly, efforts have been made to provide effective methods for calculating the roll gyradius of vessels. Both the damping and the gyradius of a vessel are traditionally quantified through the use of a sally test. Experience with the USS Midway showed that shallow water has significant effect on the rolling period and thus the experimentally determined roll gyradius. To date, little effort has been directed to the problem of the effect of shallow water on roll damping and roll period except when trying to match model and full scale experimental data. No clear guidelines exist for the boundary between deep and shallow water or the amount of overprediction of roll period that is likely for a given water depth. In order to provide greater understanding of the effects of shallow water on roll period and roll damping, this thesis performed experiments in varying scale water depths for 5 models: 4 box barges and a model of the USS Essex. The following conclusions were reached: As water depth to draft ratio, d/T, approaches 1 the roll period can increase as much as 14%. The boundary between deep and shallow water is a water depth somewhere between 4 and 7 times the vessel draft depending on the particulars of the vessel's hull form. Vessels with a larger beam to draft ratio will experience shallow water effects in relatively deeper water, that is to say the depth to draft ratio will be greater at the upper limit of deep water. Additionally, vessels with a higher beam to draft ratio will experience larger shallow water effects for a given depth to draft ratio. Finally, for vessels of very fine hull forms, the boundary between deep and shallow water will occur a relatively shallower depths, in other terms, the boundary will occur at a lower depth to draft ratio. / Master of Science

A importância do estudo de aqüíferos livres e rasos se evidencia quando se leva em conta que tais aqüíferos são altamente vulneráveis a contaminação e que em geral se encontram em regiões densamente povoadas. Com o objetivo de melhor compreender a dinâmica do fluxo de água subterrânea na zona saturada destas formações aqüíferas, foi construído e calibrado um detalhado modelo tridimensional e transiente para representar o fluxo subterrâneo em uma porção de um aqüífero livre e raso que ocorre no Parque Ecológico do Tietê, no Município de São Paulo. Este modelo, baseado em dados de cargas hidráulicas de 81 poços de monitoramento instalados em uma área de 320m2 medidas diariamente, em valores diários de precipitação na área e em 75 determinações de condutividades hidráulicas, foi construído e executado através do software Visual MODFLOW e calibrado por meio do software PEST, através de um método de regressão não linear. Os resultados gerados por este modelo para um evento de recarga rápida indicam que a distribuição de cargas hidráulicas na área de estudo é governada pela recarga regional, pela distribuição das condutividades hidráulicas horizontais e pelas vazões específicas das camadas superiores da formação, mas é praticamente independente da intensidade e distribuição temporal da recarga local, que influencia apenas os níveis mais rasos da formação. Além disso, verifica-se o caráter extremamente dinâmico do fluxo, que responde rapidamente ao evento de recarga regional, apresentando alterações significativas de intensidade e direção no intervalo de apenas alguns dias. Tais resultados, verificados apenas por meio de um modelo tão detalhado como o produzido neste trabalho, sugerem que estratégias comuns de avaliação podem não ser capazes de avaliar suficientemente bem o comportamento deste tipo de aqüífero. / The importance of studying unconfined and shallow aquifers is verified when taking into account that these aquifers are highly vulnerable to contamination and that they are often located under heavily populated regions. A detailed three-dimensional and transient model, which represents the groundwater flow in an area of an unconfined and shallow aquifer at the Parque Ecológico do Tietê, in the Municipality of São Paulo, Brazil, was created and calibrated with the objective of enhancing the comprehension of the dynamics of the groundwater flow in the saturated zone of these formations. This model was based on hydraulic head data of 81 monitoring wells, which were installed in an area of 320 m2 and measured on a daily basis, on daily values of precipitation occurring in the area and 75 determinations of hydraulic conductivities. The model was created and run using the software Visual MODFLOW and calibrated using the software PEST, based on a non-linear regression method. The results generated by this method for an event of rapid recharge indicate that the distribution of the horizontal hydraulic conductivities and by the specific yield of the upper layers of the formation, but it is mostly disconnected from the intensity and temporal distribution of the local recharge, that only influences the shallowest levels of the formation. Furthermore, the extremely dynamic character of the flow can be verified, which responds rapidly to the regional recharge event, presenting significant changes of intensity and direction within only a few days. These results, which were only observable through a very detailed model such as the one produced and presented in this study, suggest that commonly employed evaluation strategies may not be capable of evaluating this kind of aquifer in a satisfactory way.

acase@tulane.edu / Shallow water models are widely used to describe and study fluid dynamics phenomena where the horizontal length scale is much greater than the vertical length scale, for example, in the atmosphere and oceans. Since analytical solutions of the shallow water models are typically out of reach, development of accurate and efficient numerical methods is crucial to understand many mechanisms of atmospheric and oceanic phenomena. In this dissertation, we are interested in developing simple, accurate, efficient and robust numerical methods for two shallow water models --- the Saint-Venant system of shallow water equations and the two-mode shallow water equations. We first construct a new second-order moving-water equilibria preserving central-upwind scheme for the Saint-Venant system of shallow water equations. Special reconstruction procedure and source term discretization are the key components that guarantee the resulting scheme is capable of exactly preserving smooth moving-water steady-state solutions and a draining time-step technique ensures positivity of the water depth. Several numerical experiments are performed to verify the well-balanced and positivity preserving properties as well as the ability of the proposed scheme to accurately capture small perturbations of moving-water steady states. We also demonstrate the advantage and importance of utilizing the new method over its still-water equilibria preserving counterpart. We then develop and study numerical methods for the two-mode shallow water equations in a systematic way. Designing a reliable numerical method for this system is a challenging task due to its conditional hyperbolicity and the presence of nonconservative terms. We present several numerical approaches---two operator splitting methods (based on either Roe-type upwind or central-upwind scheme), a central-upwind scheme and a path-conservative central-upwind scheme---and test their performance in a number of numerical experiments. The obtained results demonstrate that a careful numerical treatment of nonconservative terms is crucial for designing a robust and highly accurate numerical method for this system. / 1 / Yuanzhen Cheng

This thesis examines the structure, stability and interaction of geophysical vortices. We do so by restricting our attention to relative vortex equilibria, or states which appear stationary in a co-rotating frame of reference. We approach the problem from three different perspectives, namely by first studying the single-vortex, quasi-geostrophic shallow-water problem, next by generalising it to an (asymmetric) two-vortex problem, and finally by re-visiting the single-vortex problem, making use of the more realistic, although more complicated, shallow-water model. We find that in all of the systems studied, small vortices (compared to the Rossby deformation length) are more likely to be unstable than large ones. For the single-vortex problem, this means that large vortices can sustain much greater deformations before destabilising than small vortices, and for the two-vortex problem this means that vortices are able to come closer together before destabilising. Additionally, we find that for large vortices, the degree of asymmetry of a vortex pair does not affect its stability, although it does affect the underlying steady state into which an unstable state transitions. Lastly, by carefully defining the "equivalence" between cyclones and anticyclones which appear in the shallow-water system, we find that cyclones are more stable than anticyclones. This is contrary to what is generally reported in the literature.

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Coastal and shallow lakes are often subjected to eutrophication due to nutrients from catchment farming activities. Lake Ellesmere (Te Waihora) is a hyper-eutrophic lake which has gained recent attention because of concerns over its ecological health and fishery status. This study investigated the benthic ecology of the lake by extensive spatial and temporal sampling. Eight littoral sites were sampled on a single occasion, and 20 benthic sites were sampled once per season for one year. Water chemistry conditions, substrate and invertebrate communities varied significantly around the lake. Salinity, pH, DO and seston were primarily affected by freshwater inputs from inflow streams and salt water intrusion due to the lake opening to the sea. On these occasions, salinity reached 32 ‰ at the lake outlet. The lake invertebrate community was depauperate, comprising of only two species of invertebrate predators restricted to the littoral zone and eight benthic invertebrate taxa, dominated by oligochaetes, amphipods and chironomids. Benthic invertebrate abundances also reflect the dominant local substrate, where oligochaetes and chironomids preferred areas of silt substrate, whereas Potamopyrgus preferred harder substrate. Stable isotope and gut analysis determined that the primary food sources within the lake were phytoplankton and algae. Macrophytes provided a minimal contribution to the food web, possibly relating to the change in status from a clear water, macrophyte dominated lake to a turbid, phytoplankton dominated condition since the Wahine Storm in 1968. Isotope analysis also showed that the lake food web was markedly different in its carbon values from food webs of its inflow streams and nearby marine source. However the lake food web did show a marine-derived carbon signature. A mesocosm experiment testing the effect of common lentic predators on the abundance of the lake chironomid Chironomus zealandicus, showed that if invertebrate predators were present in the lake they could markedly reduce the abundance of the pest prey species. This study highlights that the frequent re-suspension of bottom sediments, lake level fluctuation resulting in wetting and drying of littoral zones, and the management of the lake opening to the sea all have an effect on the benthic ecology of Te Waihora.

Shallow lakes in Muskoka-Haliburton have been largely ignored in previous limnological and paleolimnological studies, as many are considered to be less desirable for cottage development and other cultural activities. Nonetheless, shallow lakes offer important habitat for many animals. 30 oligo- and mesotrophic, shallow lakes were chosen for a paleolimnological study to assess the impacts of multiple stressors on cladoceran invertebrates. Fossil cladoceran remains preserved in the sediment samples of the study lakes were used to evaluate the nature and magnitude of any changes in assemblages from both modern and pre-industrial times. Relationships between present-day assemblages and key environmental variables were investigated using redundancy analysis, which identified that lake area (p<0.01) and Secchi depth (p<0.05) were significant predictors of assemblage composition in the shallow lakes. Secchi depth was not correlated to water clarity measures, as it usually is, but rather to macrophyte abundance. The modern-day assemblages were compared to the pre-industrial assemblages using the snapshot “top-bottom” paleolimnological approach. The top and bottom assemblages were compared using an ANOSIM which was not significant (p=0.2), confirming that modern assemblages were similar to pre-industrial assemblages in shallow lakes. Full core analysis from three shallow lakes also showed only subtle changes in littoral assemblage composition through time, further supporting results from the top/bottom study. The changes recorded were mainly in the pelagic taxa, with the littoral taxa relatively very stable through time. These small, shallow lakes were then compared and combined with a deep lake set from the same region. The results show some striking differences between shallow and deep lakes. For example, pelagic taxa appear to be driving changes within assemblages across a depth gradient, with most of the changes in the deeper lakes. Water chemistry has an increasingly important role in structuring cladoceran assemblage as lakes become deeper, although lake morphometry does play an important role in defining cladoceran assemblages in all the study lakes. Cladoceran assemblages in shallow lakes appear to be more stable than deeper lakes, despite being exposed to the same regional stressors, such as acidification, calcium decline, and climate change. / Thesis (Master, Biology) -- Queen's University, 2012-12-31 13:41:52.662