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.

No description available.

Geophysical granular flows, such as snow avalanches, pyroclastic density currents, mudslides and debris flows, can be extremely hazardous to local populations, and understanding their complex behaviour remains an important challenge. This project aims to provide insight into these events by exploring different aspects in isolation, using a combination of mathematical theory, numerical simulations and small-scale experiments. Firstly, the effect of lateral confinement is examined by studying granular material moving in an inclined chute. This can have applications to natural releases flowing down confined valleys or conduits, and the relative simplicity of the geometry also provides a useful test case for new theoretical models. One such model is the recent depth-averaged μ(I)-rheology, which, because of the viscous terms introduced into the depth-averaged momentum balance, may be described as an intermediate approach between full constitutive laws and classical shallow-water-type equations for dense granular flows. Here, a generalisation of the new system to two spatial dimensions is described, and the resulting viscous equations are able to capture the cross-slope curvature of the downslope velocity profiles in steady uniform chute flows. This may be regarded as major progress compared to traditional hyperbolic models, which only admit constant velocity solutions. Particle size-segregation in geophysical granular flows is then investigated, which can cause important feedback on the overall bulk properties as it can lead to the development of regions with different frictional properties. A particularly striking example is segregation-induced 'finger' formation, where large particles are segregated to the flow surface and sheared to form a resistive coarse-rich front, which is unstable and spontaneously breaks down into a series of lobate structures. These travel both faster and further than one might anticipate. To model such segregation-mobility feedback effects, the depth-averaged μ(I)-rheology is extended to bidisperse flows by coupling with a depth-integrated model for size-segregation. The system of equations remains mathematically well-posed and is able to qualitatively capture finger formation, with the newly-introduced viscous terms controlling the characteristics of the leveed channels that develop. A more subtle segregation effect is studied in bidisperse roll waves, which form as small irregularities merge and coarsen as they move downslope, eventually growing into destructive large amplitude pulses. Experimental measurements show lateral, as well as vertical, segregation profiles, with the coarser grains accumulating at the fastest moving wave crests. The disturbances that form in mixtures with higher proportions of large particles grow more slowly, leading to smaller amplitude waves that travel at slower speeds, and the new coupled model predicts qualitatively similar behaviour. Finally, the influence of complex topography is investigated. A smooth two-dimensional bump is placed across the width of a chute, which, depending on the initial conditions, can lead to the formation of an airborne jet or granular shock at steady state. A simple depth-averaged model in a curvilinear coordinate system following the topography accurately captures both regimes, and represents a significant improvement on using an aligned Cartesian approach.

This thesis presents different control design approaches for stabilizing networks of quasi-linear hyperbolic partial differential equations. These equations are usually conservative which gives them interesting properties to design stabilizing control laws. Two main design approaches are developed: a methodology based on entropies and Lyapunov functions and a methodology based on the Riemann invariants. The stability theorems are illustrated using numerical simulations. Two practical applications of these methodologies are presented. Netword of navigation channels are modelled using Saint-Venant equations (also known as the Shallow Water Equations). The stabilization problem of such system has an industrial importance in order to satisfy the navigation constraints and to optimize the production of electricity in hydroelectric plants, usually located at each hydraulic gates. A second application deals with the regulation of water waves in moving tanks. This problem is also modelled by a modified version of the shallow water equations and appears in a number of industrial fields which deal with liquid moving parts.

The effects of vertical confinement on a neutrally-buoyant turbulent round jet discharging from a circular nozzle into quiescent shallow water were investigated. The focus was on identifying changes in the mean flow, turbulence characteristics, and large vortical structures of a horizontal water jet at different degrees of vertical confinement. The confinement resulted from the proximity of a lower solid wall and an upper free surface. The jet exit Reynolds number for all cases was 22,500. The depth of the water layer was the principal parameter. The axial and lateral confinements were negligible. Three different degrees of vertical confinement were investigated in addition to the free jet case. For the confined cases, the water layer depth was 15, 10 and 5 times the jet exit diameter. The centreline of the jet was located midway between the solid wall and the free surface. Particle image velocimetry (PIV) was used to investigate the flow behaviour. Measurements were taken on two orthogonal planes along the jet axis; one parallel and one perpendicular to the free surface. For each case, measurements were taken at three locations downstream of the jet exit where the effects of vertical confinement were expected to be significant. All image pairs were acquired at a frequency of 1 Hz using a 2048  2048 pixel camera. This rate was slow enough that the velocity fields were uncorrelated. At each location, two thousand image pairs were acquired in order to extract statistical information about the behaviour of the flow. <p>After completing the cross-correlation analysis of the PIV images and filtering outliers using a cellular neural network with a variable threshold, the statistical quantities such as mean velocities, turbulence intensities, Reynolds shear stress, centreline velocity decay, centreline turbulence intensities, and spread rate were obtained. The proper orthogonal decomposition (POD) technique was applied to the PIV data using the method of snapshots to expose vortical structures. The number of modes used for the POD reconstruction was selected to recover ~40% of the turbulent kinetic energy. An automated method was employed to identify the position, size, and strength of the vortices by searching for closed streamlines in the POD reconstructed velocity fields. This step was followed by a statistical study to understand the effect of vertical confinement on the frequency of vortex occurrence, size, strength, rotational sense, and preferred locations.<p>The results showed that the structure of the flow underwent significant changes because of the vertical confinement. The axial velocity profiles in the vertical plane become almost uniform over the entire depth with a mild peak below the centreline of the jet for the shallowest case, while the axial velocity profiles in the horizontal plane are Gaussian but narrower than the free jet profile. The mean vertical and horizontal velocity profiles show that fluid is drawn from the sides of the jet to its centreline and then diverted upward and downward from the jet axis. The decay rate of the mean centreline velocity becomes slower at downstream locations and the jet width becomes narrower in the horizontal mid-plane compared to the free jet case. The mixing efficiency of the fluid in the vertical plane is significantly inhibited by the confinement while there is a slight effect in the horizontal plane. Also, with increasing vertical confinement, the wall jet characteristics become more dominant. Investigation of the coherent structures revealed that at intermediate distances from the exit the population of vortical structures of either rotational sense is almost identical for all vortex sizes. At downstream locations in the vertical plane, this distribution is changed by the vertical confinement which causes a significant increase in the number of small clockwise vortices. In addition, it was observed that, as the confinement increases, the total number of vortical structures decreases and their sizes increase. This is evidence of the pairing process. Moreover, with increasing confinement the circulation decreases as the flow proceeds downstream on the vertical plane with a corresponding increase in the horizontal plane. This behaviour is consistent with the turbulence intensity results.

The effects of vertical confinement on a neutrally-buoyant turbulent round jet discharging from a circular nozzle into quiescent shallow water were investigated. The focus was on identifying changes in the mean flow, turbulence characteristics, and large vortical structures of a horizontal water jet at different degrees of vertical confinement. The confinement resulted from the proximity of a lower solid wall and an upper free surface. The jet exit Reynolds number for all cases was 22,500. The depth of the water layer was the principal parameter. The axial and lateral confinements were negligible. Three different degrees of vertical confinement were investigated in addition to the free jet case. For the confined cases, the water layer depth was 15, 10 and 5 times the jet exit diameter. The centreline of the jet was located midway between the solid wall and the free surface. Particle image velocimetry (PIV) was used to investigate the flow behaviour. Measurements were taken on two orthogonal planes along the jet axis; one parallel and one perpendicular to the free surface. For each case, measurements were taken at three locations downstream of the jet exit where the effects of vertical confinement were expected to be significant. All image pairs were acquired at a frequency of 1 Hz using a 2048  2048 pixel camera. This rate was slow enough that the velocity fields were uncorrelated. At each location, two thousand image pairs were acquired in order to extract statistical information about the behaviour of the flow. <p>After completing the cross-correlation analysis of the PIV images and filtering outliers using a cellular neural network with a variable threshold, the statistical quantities such as mean velocities, turbulence intensities, Reynolds shear stress, centreline velocity decay, centreline turbulence intensities, and spread rate were obtained. The proper orthogonal decomposition (POD) technique was applied to the PIV data using the method of snapshots to expose vortical structures. The number of modes used for the POD reconstruction was selected to recover ~40% of the turbulent kinetic energy. An automated method was employed to identify the position, size, and strength of the vortices by searching for closed streamlines in the POD reconstructed velocity fields. This step was followed by a statistical study to understand the effect of vertical confinement on the frequency of vortex occurrence, size, strength, rotational sense, and preferred locations.<p>The results showed that the structure of the flow underwent significant changes because of the vertical confinement. The axial velocity profiles in the vertical plane become almost uniform over the entire depth with a mild peak below the centreline of the jet for the shallowest case, while the axial velocity profiles in the horizontal plane are Gaussian but narrower than the free jet profile. The mean vertical and horizontal velocity profiles show that fluid is drawn from the sides of the jet to its centreline and then diverted upward and downward from the jet axis. The decay rate of the mean centreline velocity becomes slower at downstream locations and the jet width becomes narrower in the horizontal mid-plane compared to the free jet case. The mixing efficiency of the fluid in the vertical plane is significantly inhibited by the confinement while there is a slight effect in the horizontal plane. Also, with increasing vertical confinement, the wall jet characteristics become more dominant. Investigation of the coherent structures revealed that at intermediate distances from the exit the population of vortical structures of either rotational sense is almost identical for all vortex sizes. At downstream locations in the vertical plane, this distribution is changed by the vertical confinement which causes a significant increase in the number of small clockwise vortices. In addition, it was observed that, as the confinement increases, the total number of vortical structures decreases and their sizes increase. This is evidence of the pairing process. Moreover, with increasing confinement the circulation decreases as the flow proceeds downstream on the vertical plane with a corresponding increase in the horizontal plane. This behaviour is consistent with the turbulence intensity results.

Sound propagation in very shallow water is one of the issues of the ocean acoustic. Because of close distance to the shore and short range to the bottom, the building of sound propagation model in shallow water is much more difficult than in deep water. Even though, the increasing needs of upper-sea construction engineering and near-shore surveillance make this subject more and more important. This study is to build a high sensitive underwater recording system, use it to collect data and to find out which parameters affect the sound propagation in very shallow water most. The study contains underwater recording system construction, shallow water recording experiment and comparison of OASES simulation results and the collective data. The system is constructed with two ITC6050C hydrophones and data acquisition devices. After several tests of reliability, the system is put in the sea area about 10 m depth. And the two hydrophones were moored 1 m above the bottom and 2.5 m below sea surface separately. The experiment use a moving fishing boat motor noise as sound source and the experimental results are shown as the spectrogram of sound field. The computer simulation uses OASES modules to simulate the experimental area and Pekeris waveguide propagation as the theoretical environment of very shallow water. By comparing the simulation results and the collective data ,the study finds out that the major parameters of sound propagation in the experimental area are the pressure sound speed and the depth of the sound source.

Two kinds of tsunami models are used in this thesis to simulate tsunami propagation in the ocean. One is the linear dispersion tsunami model developed by Port and Airport Research Institute (PARI), Japan. The other is COrnell Multigrid COupled Tsunami model (COMCOT) developed by the School of Civil and Environmental Engineering, Cornell University, that carries on the tsunami run-up computation to the nearshore region. Two kinds of tsunami models have the same mechanism of initial wave profile, which is the vertical seabed displacement as the initial tsunami profile proposed by Mansinha and Smylie (1971). Both models describe the tsunami by the same shallow water equations. At first, the feasibility of the PARI model is established by comparing with the record in Maldives during the South Asia tsunami in December 2004. Then, the COMCOT model in applied to the Pingtong earthquake in December 2006 and is validated by comparing with the tidal station records. Possible submarine fault activities around Taiwan and the Western Pacific ring is simulated by the The PARI model based on moment magnitude scale (¢Ûw) close to the South Asia tsunami. Seven sources are chosen: the Hokkaido, the East Japan, the Ryukyu Islands, the Guishan Island in Taiwan, the Fukien of mainland China, the Luzon Island and the New Guinea. The results suggest the northeast and southwest part of Taiwan have potential tsunami risk. Finally, we simulate the fault activity between Taiwan and Luzon islands by the COMCOT model. The inundation area extends northward to the Tso-Ying and San-Min districts, eastward to the Siao-Gang district and Fengshan city. The Kaohsiung harbor can resist tsunami hazard for moment magnitude scale (¢Ûw) up to 7.58 with maximum wave height of 5.5 meters.

Many problems in shallow water acoustics require accurate predictions of the acoustic field in space and time. The accuracy of the predicted acoustic field depends heavily on the accuracy of the inputs to the propagation model. Oceanographic internal waves are known to introduce considerable temporo-spatial variability to the water column, subsequently affecting the propagation of acoustic waves. As a result, when internal waves are present, errors in model inputs can significantly degrade the accuracy of the predicted acoustic field. Accurate temporo-spatial predictions of the acoustic field in the presence of internal waves therefore depend largely on one's ability to accurately prescribe the water column properties for the acoustic model. This work introduces a data-driven oceanographic model, named the evolutionary propagated thermistor string (EPTS) model, that captures the temporo-spatial evolution of the internal wave field along a fixed track, thereby permitting prediction of temporal fluctuations in the acoustic field. Simultaneously-measured oceanographic and acoustic data from the Office of Naval Research Shallow Water 2006 experiment are utilized in this work. Thermistor measurements, recorded on four oceanographic moorings spaced along the continental shelf, provide the data from which the EPTS model constructs the internal wave field over a 30 km track. The acoustic data were acquired from propagation measurements over a co-located path between a moored source and a vertical line array. Acoustic quantities computed in the model space, such as received level, depth-integrated intensity, and scintillation index are directly compared to measured acoustic quantities to evaluate the fidelity of the oceanographic model. In addition, a strong correlation is observed between the amplitude of the internal wave field and acoustic intensity statistics at a distant receiving array. It is found that the EPTS model possessed sufficient fidelity to permit the prediction of acoustic intensity distributions in the presence of nonlinear internal waves. / text