In this thesis a high order finite difference scheme is derived and implemented solving the shallow water equations using the SBP-SAT method. This method was tested against various benchmark problems were convergence was verified. The shallow water equations were also solved on a multi-block setup representing a tsunami approaching a shoreline from the ocean. Experiments show that a bottom topography with many spikes provides a dispersing effect on the incoming tsunami wave. Higher order convergence is not guaranteed for the multi-block simulations and could be investigated further in a future study.

Ocean acidification may cause biodiversity loss, alter ecosystems and impact food security, yet uncertainty over ecological responses to ocean acidification remains considerable. Most work on the impact of ocean acidification on foraminifera has been short-term laboratory experiments on single species. To expand this, benthic foraminiferal assemblages were examined across shallow water CO2 gradients in the Gulf of California, off the islands of Ischia and Vulcano in Italy and off Papua New Guinea. Living assemblages from the Gulf of California did not appear to show a response across a pH range of 7.55 – 7.88, although the species assemblage was impoverished in all locations and the dead assemblage was less diverse at the lowest pH sites where there was evidence of post mortem dissolution. At Vulcano, the small macroalga, Padina pavonica, did not protect calcareous foraminifera from the adverse effects of ocean acidification. Calcareous taxa disappeared from the assemblage and were replaced by agglutinated foraminifera as mean pH reduced from 8.19 to 7.71. Settlement of benthic foraminifera onto artificial collectors off Vulcano was adversely affected in the acidified water, with few species as pCO2 increased and evidence of post-mortem dissolution. The foraminiferal tests, collected off Papua New Guinea, had lower δ11B as mean pH decreased from 7.99 – 7.82 for small (250 – 500 µm) Amphistegina lessonii, but not for A. lessonii or Calcarina spengleri >500 µm. In the larger foraminifera, photosynthetic activity by symbionts may begin to dominate the boron isotopic signature. Overall, the responses of foraminiferal assemblages to ocean acidification are complex, but there was an overall reduction in species diversity in infaunal, epifaunal and epiphytic assemblages as pCO2 increased. This raises serious concerns for the survival of shallow water calcareous benthic foraminifera as the oceans continue to acidify, with implications for benthic ecosystems and inorganic carbon cycling.

Flood propagation over urban areas can cause an interaction between the free-surface flow and large underground pipe networks used for storm drainage and sewage, causing outflows and inflows at the bed. The associated waves may collide with each other and the surface waves. In this thesis the shallow water equations are used to model this type of wave interaction over dry or wet beds with bathymetry gradients and friction terms. The proposed shallow water scheme is solved based on finite volume high-resolution Godunov-type methods. The solver is well-balanced and can accurately balance the source terms and flux-gradients for the steady-state solutions. The solver also utilises a new type of Riemann wave speed to provide depth-positive results over nearly dry beds and dry states. Additionally a new type of source term is introduced in the continuity equation to model pipe inflow and outflow conditions at bed connections. For the standard one-dimensional shallow water equations the numerical results are validated with analytical solutions or other reference solutions provided in the literature. This includes the incipient Riemann problems for nearly dry and dry-states, steady flow over a hump in a rectangular channel and the wave propagation problem. Eventually, the generation of dry bed in the middle, over discontinuous topography is considered. Close agreement is achieved between the shallow water scheme and analytical or reference solutions for the above test cases. For the shallow water problems with influx/efflux source terms comparisons are made with STAR-CD, a commercial Navier-Stokes solver for general fluid flow prediction. The shallow water model is first used to simulate vertical flows through finite gaps in the bed. Next, the interaction of the vertical flows with a dam-break flow is considered for both dry and wet beds. An efflux number, En, is defined based on the vertical efflux velocity and the gap length. A parameter study is undertaken to investigate the effect of the one-dimensional approximation of the present model, for a range of non-dimensional efflux numbers. It is found that the shallow flow model gives sensible predictions at all times provided En<0.5, and for long durations for En>0.5. Dam break flow over an underground connecting pipe is also considered for the one-dimensional efflux problems. To solve two-dimensional problems the shallow water scheme uses the dimensional-splitting method which solves each one-dimensional Riemann problem in the x- and y-directions separately. The cross-derivative terms for second-order accuracy are incorporated by solving another Riemann problem in the orthogonal direction. For two-dimensional problems first the dam-break problems are considered over wet and dry beds. Then, flood propagation over complex terrain is demonstrated. Next, efflux discharge is modelled in isolation over a dry bed and then with dam-break interaction, comparing with STAR-CD results. Again very good agreement is shown between the two-dimensional shallow water model and STAR-CD for the efflux numbers of En<0.5. For modelling the inundation problem over an underground pipe network the solver is coupled with the general underground pipe network solver to calculate the efflux discharge as the flood waves pass through the pipe network. For analysing the pipe network with unknown effluxes an additional set of equations is incorporated into the solution of a general pipe network solver. The shallow water solver coupled to an underground pipe network is then used to simulate dam-break interaction with pipe networks with 9 and 25 nodes to demonstrate the versatility of the method.

The MRS/MISS of the Makgabeng Formation encompasses sand cracks, wrinkle marks, mat fragments, mat chips and roll-ups and those of the Magaliesberg formation are wrinkle marks, petees/petee ridges, sand cracks, and multi-directional ripples. The sedimentary process that moderated the formational mechanism of the MISS of the Makgabeng Formation is (descriptively allochthonous) of high energy (inter-dune depositional setting) that eroded, transported and re-deposited mat bound sediments. The genetic mechanism of the MISS of the Magaliesberg Formation is descriptively authochthonous because of enhanced resistance of biostabilized sediments to being reworked. XRF (major and trace) and XRD analysis (qualitative and quantitative) was done on MISS bearing sedimentary rock layers (A) and underlying sedimentary sections (B) of Magaliesberg and Makgabeng samples. Result show high quartz content of all the analyzed samples compared to average sandstones. This premise suggests a relation of microbes (e.g. cyanobacteria) to phototrophy and/photoautotrophy because of the conduction properties of translucent quartz. Also plausible inference is that the intense chemical weathering that produced the quartz arenite was positively influenced by microbes, as noted in some Proterozoic basins. There is higher concentration of Ba in all A samples compared to B (Makgabeng and Magaliesberg) which might be emblematic of biogenicity. The Magaliesberg analyzed samples (MAG 101, 102, 103) exhibit homogeneity by the higher concentration of Al2O3, TiO2, K2O, and P2O5, and lower concentration of SiO2 in the A compared to the B subsamples of a particular sample. Also, Magaliesberg analyzed samples (MAG 101, 102, 103) exhibit homogeneity by the lower concentration of quartz and higher concentration of muscovite in the A compared to the B subsamples. This exact established negative correlation between the duo of SiO2 and quartz, and the quartet of Al2O3, TiO2, K2O, and P2O5, and muscovite as in Magaliesberg samples pertains also to a Makgabeng sample (MKG 102; roll-up). MKG 101 (mat fragment) deviates from this mineralogical and geochemical trend. Each of the A samples of MAG 101, 102, 103, are uniformly of higher concentration in Ce, Cr, Nb, Th, V, Y, Zn, Zr compared to the B version of that sample. MKG 101 and 102 are uniformly of lower concentration of Ce, Cr, Nb, Th, V, Y, Zn, Zr in A compared to the B version of that sample. The A of each of the samples MAG 101, 102, and 103 has higher concentration of Hf and Rb compared to its B; a character that is also exhibit in MKG 102, and MKG 101 is vice versa. Microscopy shows that A of all the samples is of smaller grain size compared to B, espousing affinity of microbes to fine-medium grained sandstones. Microscopy of the Magaliesberg Formation samples show Pseudo petee ridges and pseudo cross lamination which reflect biostabilization, and microscopy of the Makgabeng Formation show roll-ups, mat chips and composite mat chips. The MISS genetic difference of the two formations is related to energy, water residence time (emergence and inundation), Ph, and similarity is related to mutuality in shallow water environment. Mat types are inferred to be biologically, physically and chemically moderated adaptations of microbial communities to specific cum peculiar locally prevailing environmental conditions; factors that are premised on taphonomy and ecology. / Dissertation (MSc)--University of Pretoria, 2014. / tm2015 / Geology / MSc / Unrestricted

Accurate prediction of tsunami waves requires complete boundary and initial condition data, coupled with the appropriate mathematical model. However, necessary data is often missing or inaccurate, and may not have sufficient resolution to capture the dynamics of such nonlinear waves accurately. In this thesis we demonstrate that variational data assimilation for the continuous shallow water equations (SWE) is a feasible approach for recovering both initial conditions and bathymetry data from sparse observations. Using a Sadourny finite-difference finite volume discretisation for our numerical implementation, we show that convergence to true initial conditions can be achieved for sparse observations arranged in multiple configurations, for both isotropic and anisotropic initial conditions, and with realistic bathymetry data in two dimensions. We demonstrate that for the 1-D SWE, convergence to exact bathymetry is improved by including a low-pass filter in the data assimilation algorithm designed to remove scale-scale noise, and with a larger number of observations. A necessary condition for a relative L2 error less than 10% in bathymetry reconstruction is that the amplitude of the initial conditions be less than 1% of the bathymetry height. We perform Second Order Adjoint Sensitivity Analysis and Global Sensitivity Analysis to comprehensively assess the sensitivity of the surface wave to errors in the bathymetry and perturbations in the observations. By demonstrating low sensitivity of the surface wave to the reconstruction error, we found that reconstructing the bathymetry with a relative error of about 10% is sufficiently accurate for surface wave modelling in most cases. These idealised results with simplified 2-D and 1-D geometry are intended to be a first step towards more physically realistic settings, and can be used in tsunami modelling to (i) maximise accuracy of tsunami prediction through sufficiently accurate reconstruction of the necessary data, (ii) attain a priori knowledge of how different bathymetry and initial conditions can affect the surface wave error, and (iii) provide insight on how these can be mitigated through optimal configuration of the observations. / Thesis / Candidate in Philosophy

The focus of this dissertation is analyzing and manipulating acoustic wave propagation in metamaterials, which can be used to assist the design of acoustic devices. Metamaterials are artificial materials, which are arranged in certain patterns at a scale smaller than the wavelength and can exhibit properties beyond those naturally occurring materials. With metamaterials, novel phenomena, such as focusing, super absorption, cloaking and localization of ultrasound, are theoretically proposed and experimentally verified. In recent years, a planar version of metamaterials, often called meta-surfaces, has attracted a great deal of attention. Meta-surfaces can control and manipulate the amplitude, phase, and directions of waves. In this dissertation, we conducted a systematic study by deriving the effective medium theories (EMTs), and developing the theoretical and numerical models for our proposed designed metamaterial. Very recently, acoustic meta-surfaces have been used in the design of acoustic lenses, which can achieve various functionalities such as focusing and collimation. In the designs of acoustic lenses, impedance is an important issue because it is usually difficult to make the impedance of the lens equal to that of the environment, and mismatched impedance is detrimental to the performance of the acoustic lens. We developed an EMT based on a coupled-mode theory and transfer matrix method to characterize the propagation behavior and, based on these models, we report two designs of acoustic lenses in water and air, respectively. They are rigid thin plates decorated with periodically distributed sub-wavelength slits. The building block of the acoustic lens in water is constructed from coiling-up spaces, and that of the acoustic lens in air is made of layered structures. We demonstrate that the impedances of the lenses are indeed matched to those of the background media. With these impedance-matched acoustic lenses, we demonstrate acoustic focusing and collimation, and redirection of transmitted acoustic energy by finite-element simulations. In the framework of the hidden source of the volume principle, an EMT for a coupled resonator structure is derived, which shows that coupled resonators are characterized by a negative value of the effective bulk modulus near the resonance frequency and induce flat bands that give rise to the confinement of the incoming wave inside the resonators. The leakage of sound waves in a resonance-based rainbow trapping device prevents the sound wave from being trapped at a specific location. Based on our EMT, we report a sound trapping device design based on coupled Helmholtz resonators, loaded to an air waveguide, to effectively tackle the wave leakage issue. We show that a coupled resonators structure can generate dips in the transmission spectrum by an analytical model derived from Newton’s second law and a numerical analysis based on the finite-element method. We compute the transmission spectra and band diagram from the effective medium theory, which are consistent with the simulation results. Trapping and the high absorption of sound wave energy are demonstrated with our designed device.

The research detailed in this study focuses on the determination of an appropriate objective function to aid parameter estimation when simulating areas influenced by tidally varying flows. Three objective functions that are measures of how well the model results match field data at several locations and times were tested. A set of test cases is developed to represent tidally influenced systems and allow for the testing of the objective functions. These objective functions were tested by computing their values and comparing them for the various estimated parameters. Based on results of the first method of testing a further analysis was performed using PEST, an automatic parameter estimation tool. A weighted least squares of the velocity and water surface values with a weight function on the velocity term based on the shallow water equations is found to be a reasonable objective function at this point in the research.

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Riemann problems are of prime importance in computational fluid dynamics simulations using finite elements or finite volumes discretizations. In some applications, billions of Riemann problems might need to be solved in a single simulation, therefore it is important to have reliable and computationally efficient algorithms to do so. Given the nonlinearity of the flux function in most systems considered in practice, to obtain an exact solution for the Riemann problem explicitly is often not possible, and iterative solvers are required. However, because of issues found with existing iterative solvers like lack of convergence and high computational cost, their use is avoided and approximate solvers are preferred. In this thesis work, motivated by the advances in computer hardware and algorithms in the last years, we revisit the possibility of using iterative solvers to compute the exact solution for Riemann problems. In particular, we focus on the development, implementation, and performance comparison of iterative Riemann solvers for the shallow water and Euler equations. In a one-dimensional homogeneous framework for these systems, we consider several initial guesses and iterative methods for the computation of the Riemann solution. We find that efficient and reliable iterative solvers can be obtained by using recent estimates on the Riemann solution to modify and combine well-known methods. Finally, we consider the application of these solvers in finite volume simulations using the wave propagation algorithms implemented in Clawpack.