This dissertation presents results for two separate problems, both in the context of variational fracture models. The first problem involved developing and analyzing models of fracture in which we modeled the energy dissipated by crack growth as concentrated on the front of the crack. While many engineering models of fracture are based on a notion of crack front, there had not been a rigorous definition. We present the first work in this area, which includes a natural weak definition of crack front and front speed, a model of fracture whose evolution is described at the crack front, and a relaxation result that shows that these front based dissipations are all effectively equivalent to a Griffith-type dissipation. The second problem involved the computation of stationary points for Mumford-Shah and fracture using a level set method. Our method improves on existing techniques in that it can handle tips in the singular set and can find minimizers that previous techniques are unable to resolve.

Seismic reflection surveying is used to generate acoustic images of the water column. This technique employs conventional multi-channel equipment which is used to image the solid Earth. In the water column, acoustic impedance contrasts are produced by variations in temperature and, to some extent, salinity. Acoustic impulses generated by an array of airguns suspended behind a vessel are reflected from these contrasts and recorded on long cables of hydrophones that are towed below the sea-surface. In this way, two- and three-dimensional images of thermohaline circulation can be generated. Critically, these images have equal vertical and horizontal resolutions of \textit{O}(10)~m. Here, I describe, process, and analyse a calibrated two-dimensional seismic survey from the Bellingshausen Sea of the Southern Ocean and a three-dimensional seismic survey from the Brazil-Falkland Confluence located offshore Uruguay. First, the Bellingshausen survey was designed to image the thermohaline structure across the west Antarctic shelf where warm-core eddies are reported. Processed and calibrated seismic images reveal the detailed thermohaline structure of Circumpolar Deep Water. Many warm-core eddies are observed, which have diameters of 1--12~km and thicknesses of 100--200~m. Pre-stack analysis demonstrates that this eddy field is being advected onto the shelf at speeds of \textit{O}(0.1)~m~s$^{-1}$. An iterative inverse modelling procedure is used to convert reflectivity into temperature and salinity, which confirms that the eddies have anomalously warm centres (i.e. $\sim$1$^{\circ}$C). These results have significant implications for ice shelf melting. Secondly, the Uruguay survey is used to investigate a large-scale frontal system. Although this system has been studied using hydrographic methods, these studies either have limited spatial resolution or have restricted depth penetration. The three-dimensional seismic survey, which was acquired in a `racetrack' pattern, permits the volume to be interrogated. Since the frontal system migrates southwestwards at a speed of \textit{O}(10)~km~day$^{-1}$, this survey is time-lapse in nature. Processed images reveal a band of dipping reflections that extend to depths of $\sim$2000~m. These reflections represent the frontal interface between the Brazil and Falkland currents. Physical oceanographic properties are calculated for images that cross this front. On the warm side of the front, the water mass is characterised by flat and continuous reflectivity. On the cold side of the front, the water mass is characterised by deformed reflectivity on all scales. Pre-stack analysis suggests that near-surface flow at the frontal interface is convergent. Between 0.5 and 1~km depth, a substantial eddy that is 30~km long and 250~m thick is visible on the cold side of the front. Detailed mapping suggests that this eddy grew and decayed over a period of 6~days. Its observed scale and duration are inconsistent with analytical and numerical studies of intra-thermocline eddies. Nevertheless, its duration is consistent with scaling arguments of frictional spin-down. Spatial and temporal distributions of mixing rates (i.e. diapycnal diffusivities) are estimated by spectrally analysing vertical displacements of automatically tracked reflections. Both internal wave and turbulent regimes are identifiable. Recovered diapycnal diffusivities are of \textit{O}($10^{-6}$--$10^{-2.2}$)~m$^{2}$~s$^{-1}$, consistent with hydrographically determined estimates. Mixing is suppressed and enhanced on the warm and cold sides of the front, respectively. Seismic Oceanography has considerable potential to quantify aspects of thermohaline circulation on multiple scales.

Five line transect surveys for marine mammals were conducted offshore of mainland Ecuador and the Galápagos Islands from 2008-2011. These data were used in conjunction with MODIS (Moderate Resolution Imaging Spectroradiometer) observations of ocean color and sea surface temperature (SST) to assess spatial and temporal relationships between surface oceanographic features and cetacean distribution within the Eastern Equatorial Pacific (EEP). Results from this study indicated that oceanographic processes affected cetacean distribution on inter-annual, seasonal, and weekly to monthly time scales. The spatial scales on which these processes affect cetacean distribution are small, the smallest associations being found at 4 km2 bin sizes, as well as 9 km2 and 36 km2 bin sizes. By utilizing ocean color and SST data from the MODIS instrument and analyzing variability of these parameters in addition to average concentration, cetacean distribution within the region was related to the locations of frontal boundaries. Cetaceans were grouped into two categories based on the trophic level and relative depths at which they forage. Cetaceans feeding nearer the ocean surface and lower on the trophic scale were generally found in cooler waters of higher average chlorophyll concentration and elevated variability. Those cetaceans feeding higher on the trophic scale and lower in the water column (mesopelagic and bathypelagic depths) were sighted within relatively warmer waters of reduced temperature variability near areas of high chlorophyll variability (though less variable and lower in average chlorophyll than surface feeding cetaceans), with little spatial and temporal lag between peak surface chlorophyll concentration and cetacean presence. The EEP is a biologically productive region with many competing economic and environmental interests. Ecuador is home to one of the largest artisenal fishing fleets in South America, and entanglement of various cetacean species has been a known issue for several decades (Félix and Haase, 2006; Castro and Rosero, 2010). Seismic exploration, shipping, and tourism are also found on the busy waterways surrounding both mainland Ecuador and the archipelago. The results of this study provide additional insight into the mesoscale processes affecting the distribution and habitat use of cetaceans within the EEP and South American waters and to support ongoing ecosystem management efforts.

Satellite images of chlorophyll concentration in the surface waters of the Gulf of Mexico suggest a high degree of heterogeneity in the phytoplankton biomass. The causes of this variability and the amount of variability in the phytoplankton community structure are not well understood. The physical and chemical conditions of a specific environment can influence phytoplankton community structure by selecting for those phytoplankton species able to survive within that environment. Varying salinity and temperature characteristics give water masses distinct surface water density signatures. This study examined the relationship between phytoplankton biomass, community structure, and different water mass properties by measuring chlorophyll a and algal group concentration across frontal zones. Continuous salinity and temperature measurements were used to calculate continuous density along transects during four cruises on the R/V Gyre between summer 2002 and spring 2004. Frontal zones were identified as areas of sharp density change where σt changed by 1.5 points over a distance of 1 km. Density fronts that coincided with visible temperature fronts (satellite AVHRR images) were selected for biomass and community structure analysis. Discrete water samples were analyzed using fluorometric analysis (total chlorophyll a concentration) and HPLC analysis (photosynthetic pigments). Community composition for discrete samples was determined using CHEMTAX and these values were used to interpolate community composition. Phytoplankton biomass and community structure were examined at a total of 21 density fronts. Unlike previous studies of frontal zones, phytoplankton biomass (measured as chl a concentration) was not significantly higher within frontal zones than in adjacent waters at any of the 21 fronts. Community composition (measured as algal group abundance and diversity) was significantly different between the front and at least one adjacent water mass at front 2 during summer 2002, at front 6 during summer 2003, at front 3 during fall 2003, and at front 3 during spring 2004. Both biomass and community composition were significantly different between fronts at all front pairs during summer 2002. The results of this study suggest that density fronts are not biologically important features in the northern Gulf of Mexico. Lack of high phytoplankton biomass at fronts in the Gulf of Mexico could indicate that unique physical, chemical, or biological processes are occurring.

Addendum inserted onto front end-paper. / Bibliography: p. 261-284. / xv, 284 p. : ill. ; 30 cm. / Title page, contents and abstract only. The complete thesis in print form is available from the University Library. / Provides a thorough analysis of cut-off low phenomena over Australia by using the VHF radar system, operating at a frequency of 54.1MHz / Thesis (M.Eng.Sc.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1999

Thesis (M.Eng.Sc.)--University of Adelaide, Dept. of Physics and Mathematical Physics, 1999. / Addendum inserted onto front end-paper. Bibliography: p. 261-284.

Thesis (Ph. D.)--Florida State University, 1981. / Includes bibliographical references (leaves 90-92).

Thesis (Ph. D.)--York University, 2004. Graduate Programme in Music. / Typescript. Includes bibliographical references (leaves 346-367). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://wwwlib.umi.com/cr/yorku/fullcit?pNQ99266.

The larvae of coastal species interact with nearshore currents that are complex and can alter dispersal. I investigated two sites in southern Oregon with different nearshore hydrodynamics: the first site, Sunset Bay, is a small cove with a topographic front that extends across the mouth during upwelling-favorable winds. Using holey sock drogues at 1.5 m and 5.5 m depths and surface drifters at 10 cm depth, I found that, when the front was present, water at 1.5 m was retained within the bay whereas water was exchanged across the front at a depth of 5.5 m. Surface drifters indicated a surface convergence. Surface plankton tows on either side and within the frontal convergence (a shore-parallel foam line) found significantly higher concentrations of barnacle cyprids, crab megalopae and zoea, polychaete larvae, platyhelminthes juveniles, isopods, amphipods, harpacticoid copepods, and fish eggs. Crustacean nauplii (barnacle, euphausiid, and copepod) and calanoid copepods were not concentrated in the convergence, and when the front and foam line were absent, no taxa were concentrated. Plankton tows taken within the foam line as it dissipated shoreward during a wind-reversal event found that concentrations of cyprids, megalopae, and gastropod veligers remained high as the foam line moved, suggesting that it acts as a moving convergence propagating competent larvae shoreward. I measured settlement of two taxa at Sunset Bay and Shore Acres, an open-coast site <2 km away. Barnacle settlement measured every other day from June-September 2013 was significantly cross-correlated with the maximum daily tidal height at lags of -2 days at Sunset Bay and +2 and +4 days at Shore Acres. Settlement was also significantly negatively cross-correlated with wave height at a lag of -4 days at Sunset Bay. Coralline algae settlement measured during eight 48-hr periods in July-August 2013 was significantly negatively correlated with wave height (n = 8, R2 = 0.76, P = 0.0049) at Shore Acres but not at Sunset Bay. Despite the close proximity of the Sunset Bay and Shore Acres sites, settlement patterns differed between taxa, suggesting that differences in nearshore hydrodynamics might affect the supply of water and larvae to shore. This dissertation includes unpublished co-authored material.

L'objet de cette thèse est l'étude de l'accélération de la propagation dans les équations de réaction-diffusion par un nouveau mécanisme d'échange avec une ligne de diffusion rapide. On répondra à la question de l'influence de ce couplage avec forte diffusivité sur la propagation en généralisant un résultat de Berestycki, Roquejoffre et Rossi de 2013. Le système d'équations étudié a été proposé pour donner une explication mathématique de l'influence des réseaux de transports sur les invasions biologiques. Dans un premier chapitre, on étudiera l'existence et l'unicité de solutions de type ondes progressives via une méthode de continuation. La transition se fait par l'intermédiaire d'une perturbation singulière qui paraît nouvelle dans ce contexte, connectant le système initial à un problème au bord de type Wentzell. Le second chapitre s'intéresse à la vitesse des ondes sus-mentionnées. On y démontre qu'elle croît comme la racine carrée de la diffusivité de l'espèce sur la route, ce qui généralise et démontre la robustesse du résultat de Berestycki, Roquejoffre et Rossi. De plus, on caractérise précisément le ratio de croissance comme unique vitesse admissible pour les ondes d'un système hypoelliptique a priori dégénéré. Enfin dans une dernière partie on s'intéresse à la dynamique. On y montre que ces ondes attirent une large classe de données initiales. En particulier on met en lumière un nouveau mécanisme d'attraction qui permet aux ondes d'attirer des données dont la taille est indépendante de la diffusivité sur la route ; c'est un résultat nouveau au sens où usuellement, l'accélération de fronts de réaction-diffusion se paie en renforçant les hypothèses nécessaires sur la taille des données initiales attirées. / The aim of the thesis is the study of enhancement of propagation in reaction-diffusion equations, through a new mechanism involving a line with fast diffusion. We answer the question of the influence of such a coupling with strong diffusion on propagation by generalizing a result of Berestycki, Roquejoffre and Rossi (2013). The model under study was proposed to give a mathematical understanding of the influence of transportation networks on biological invasions. The first chapter shows existence and uniqueness of travelling waves solutions with a continuation method. The transition occurs through a singular perturbation - new in this context - connecting the system with a Wentzell boundary value problem. The second chapter is concerned with the speed of the waves : we show that it grows as the square root of the diffusivity on the line, generalizing and showing the robustness of the result by Berestycki, Roquejoffre and Rossi. Moreover, the growth ratio is characterized as the unique admissible velocity for the waves of an hypoelliptic a priori degenerate system. The last part is about the dynamics : we show that the waves attract a large class of initial data. In particular, we shed light on a new mechanism of attraction which enables the waves to attract initial data with size independent of the diffusivity on the line : this is a new result, in the sense than usually, enhancement of propagation has to be paid by strengthening the assumptions on the initial data for invasion to happen.