The Indian Ocean Experiment (INDOEX) had as a primary objective determining the radiative forcing due to anthropogenic aerosols over climatologically significant space and time scales: the Indian Ocean during the winter monsoon, January-March. During the winter monsoon, polluted, low-level air from the Asian subcontinent blows over the Arabian Sea and Indian Ocean. As part of INDOEX, aerosol optical depths were derived from Advanced Very High Resolution Radiometer (AVHRR) data for the cloud-free ocean regions. The AVHRR radiances were first calibrated using the interior zone of the Antarctic and Greenland ice sheets, which proved to be radiometrically stable calibration targets. Optical depths were derived by matching the observed radiances to radiances calculated for a wide range of optical depths and viewing geometry. Optical depths derived with the AVHRR were compared with those derived with NASA's Aerosol Robotic Network (AERONET) CIMEL instrument at the Center for Clouds, Chemistry, and Climate's Kaashidhoo Observatory, as well as with other surface and shipboard observations taken in the INDOEX region. The retrieved and surface-based optical depths agreed best for a new 2-channel, 2- aerosol model scheme in which the AVHRR observations at O·64 and O·84 microns were used to determine relative amounts of marine and polluted continental aerosol and then the resulting aerosol mixture was used to derive the optical depths. Broadband radiative transfer calculations for the mixture of marine and polluted continental aerosols were combined with the 0·64 and 0·84-micron AVHRR radiances to determine the radiative forcing due to aerosols in the INDOEX region. Monthly composites of aerosol optical depth and top of the atmosphere, surface, and atmospheric radiative forcing were derived from calibrated AVHRR radiances for January-March 1996-2000. An inter-annual variability in the magnitude and spatial extent of high value regions is noted for derived optical depths and radiative forcing, with highest values reached in 1999, particularly in the Bay of Bengal which during the IFP was covered by plumes from Indochina. Frequency distributions of the optical depth for 1⁰ x 1⁰ latitude-longitude regions are well represented by gamma distribution functions. The day-to-day and year-to-year variability of the optical depth for such regions is correlated with the long term average optical depth. Interannual variability of the monthly mean optical depths for such regions is found to be as large as the day to day. / Graduation date: 2001

This thesis consists in the study of a periodically forced slow-fast system in both its excitable and oscillatory regimes. The slow-fast system under consideration is the FitzHugh-Nagumo model, and the periodic forcing consists of a train of Gaussian-shaped pulses, the width of which is much shorter than the action potential duration. This system is a qualitative model for both an excitable cell and a spontaneously beating cell submitted to periodic electrical stimulation. Such a configuration has often been studied in cardiac electrophysiology, due to the fact that it constitutes a simplified model of the situation of a cardiac cell in the intact heart, and might therefore contribute to the understanding of cardiac arrhythmias. Using continuation methods (AUTO software), we compute periodic-solution branches for the periodically forced system, taking the stimulation period as bifurcation parameter. We then study the evolution of the resulting bifurcation diagram as the stimulation amplitude is raised. In both the excitable and the oscillatory regimes, we find that a critical amplitude of stimulation exists below which the behaviour of the system is trivial: in the excitable case, the bifurcation diagram is restricted to a stable subthreshold period-1 branch, and in the oscillatory case, all the stable periodic solutions belong to isolated loops (i.e., to distinct closed solution branches). Due to the slow-fast nature of the system, the changes that take place in the bifurcation diagram as the critical amplitude is crossed are drastic, while the way the bifurcation diagram re-simplifies above some second amplitude is much more gentle. In the oscillatory case, we show that the critical amplitude is also the amplitude at which the topology of phase-resetting changes type. We explain the origin of this coincidence by considering a one-dimensional discrete map of the circle derived from the phase-resetting curve of the oscillator (the phase-resetting map), map which constitutes a good approximation of the original differential equations under certain conditions. We show that the bifurcation diagram of any such circle map, where the bifurcation parameter appears only in an additive fashion, is always characterized by the period-1 solutions belonging to isolated loops when the topological degree of the map is one, while these period-1 solutions belong to a unique branch when the topological degree of the map is zero.

The Earth's atmosphere is expected to warm in response to increasing atmospheric concentrations of greenhouse gases (GHG). The response of the Earth's complex and chaotic climate system to the GHG emissions is, however, difficult to assess. In this thesis, two issues of importance for the assessment of this response are studied. The first concerns the magnitude of the natural and anthropogenic emissions of CO2. An atmospheric transport model is used, combined with inventories of anthropogenic CO2 emissions and estimates of natural emissions, to compare modelled and observed variations in the concentration of CO2 at an Arctic monitoring site. The anthropogenic and natural emissions are shown to exert approximately equal influence on Arctic CO2 variations during winter. The primary focus of this thesis is the response of the climate system to the enhanced GHG forcing. It has been proposed that this response may project onto the leading modes of variability. In the present thesis, this hypothesis is tested against the alternative that the spatial patterns of variability change in response to the enhanced forcing. The response of the atmospheric circulation to the enhanced GHG forcing as simulated by a specific coupled global climate model (CGCM) is studied. The response projects strongly onto the leading modes of present-day variability. The spatial patterns of the leading modes are however changed in response to the enhanced GHG forcing. These changes in the spatial patterns are associated with a strengthening of the waveguide for barotropic Rossby waves in the Southern Hemisphere. The Northern Hemisphere waveguide is however unchanged. The magnitude of the global mean responses to an enhanced GHG forcing as simulated by CGCMs vary. Moreover, the regional responses vary considerably among CGCMs. In this thesis, it is hypothesised that the inter-CGCM differences in the spatial patterns of the response to the enhanced GHG forcing are partially explained by inter-CGCM differences in zonal-mean properties of the atmospheric flow. In order to isolate the effect of these differences in the zonal-mean background state from the effects of other sensitivities, a simplified model with idealised forcing is employed. The model used is a global three-level quasi-geostrophic model. The sensitivity of the stationary wave pattern (SWP) to changes in the zonal-mean wind and tropopause height of similar magnitude as those found in response to the enhanced GHG forcing in CGCMs is investigated. The SWP in the simplified model shows a sensitivity of comparable magnitude to the analogous response in CGCMs. These results indicate that the CGCM-simulated response is sensitive to relatively small differences in the zonal-mean background state. To assess the uncertainties in the regional response to the enhanced forcing associated with this sensitivity, ensemble simulations of climate change are of great importance.

In this study we investigate the flow of a Boussinesq fluid contained in a rotating, differentially heated spherical shell. Previous work, on the spherical shell of Boussinesq fluid, differentially heated the shell by prescribing temperature on the inner boundary of the shell, setting the temperature deviation from the reference temperature to vary proportionally with -cos 20, from the equator to the pole. We change the model to include an energy balance equation at the earth's surface, which incorporates latitudinal solar radiation distribution and ice-albedo feedback mechanism with moving ice boundary. For the fluid velocity, on the inner boundary, two conditions are considered: stress-free and no-slip. However, the model under consideration contains only simple representations of a small number of climate variables and thus is not a climate model per se but rather a tool to aid in understanding how changes in these variables may affect our planet's climate. The solution of the model is followed as the differential heating is changed, using the pseudo arc-length continuation method, which is a reliable method that can successfully follow a solution curve even at a turning point. Our main result is in regards to hysteresis phenomenon that is associated with transition from one to multiple convective cells, in a dfferentially heated, co-rotating spherical shell. In particular, we find that hysteresis can be observed without transition from one to multiple convective cells. Another important observation is that the transition to multiple convective cells is significantly suppressed altogether, in the case of stress-free boundary conditions on the fluid velocity. Also, the results of this study will be related to our present-day climate. / UOIT

Nonlinear algebraic equations (NAEs) occur in many areas of science and engineering. The process of solving these NAEs is generally difficult, from finding a good initial guess that leads to a desired solution to deciding on convergence criteria for the approximate solution. In practice, Newton's method is the only robust general-purpose method for solving a system of NAEs. Many variants of Newton's method exist. However, it is generally impossible to know a priori which variant of Newton's method will be effective for a given problem.<p>Many high-quality software libraries are available for the numerical solution of NAEs. However, the user usually has little control over many aspects of what the library does. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis describes a problem-solving environment (PSE) called pythNon for studying the effects (e.g., performance) of different strategies for solving systems of NAEs. It provides the researcher, teacher, or student with a flexible environment for rapid prototyping and numerical experiments. In pythNon, users can directly influence the solution process on many levels, e.g., investigation of the effects of termination criteria and/or globalization strategies. In particular, to show the power, flexibility, and ease of use of the pythNon PSE, this thesis also describes the development of a novel forcing-term strategy for approximating the Newton direction efficiently in the pythNon PSE.

Two immersed boundary methods (IBM) for the simulation of conjugate heat transfer problems with complex geometries are introduced: a finite element (IFEM) and a finite volume (IFVM) immersed boundary methods are discussed. In the IFEM a projection approach is presented for the coupled system of time-dependent incompressible Navier-Stokes equations (NSEs) and energy equation in conjunction with the immersed boundary method for solving fluid flow and heat transfer problems in the presence of rigid objects not represented by the underlying mesh. The IBM allows solving the flow for geometries with complex objects without the need of generating a body-fitted mesh. Dirichlet boundary constraints are satisfied applying a boundary force at the immersed body surface. Using projection and interpolation operators from the fluid volume mesh to the solid surface mesh (i.e., the “immersed” boundary) and vice versa, it is possible to impose the extra constraint to the NSEs as a Lagrange multiplier in a fashion very similar to the effect pressure has on the momentum equations to satisfy the divergence-free constraint. The IFEM approach presented shows third order accuracy in space and second order accuracy in time when the simulation results for the Taylor-Green decaying vortex are compared to the analytical solution. For the IFVM a ghost-cell approach with sharp interface scheme is used to enforce the boundary condition at the fluid/solid interface. The interpolation procedure at the immersed boundary preserves the overall second order accuracy of the base solver. The developed ghost-cell method is applied on a staggered configuration with the Semi-Implicit Method for Pressure-Linked Equations Revised algorithm. Second order accuracy in space and first order accuracy in time are obtained when the Taylor-Green decaying vortex test case is compared to the IFVM analytical solution. Computations were performed using the IFEM and IFVM approaches for the two-dimensional flow over a backward-facing step, two-dimensional flow past a stationary circular cylinder, three-dimensional flow past a sphere and two and three-dimensional natural convection in an enclosure with/without immersed body. The numerical results obtained with the discussed IFEM and IFVM were compared against other IBMs available in literature and simulations performed with the commercial computational fluid dynamics code STAR-CCM+/V7.04.006. The benchmark test cases showed that the numerical results obtained with the implemented immersed boundary methods are in good agreement with the predictions from STAR-CCM+ and the numerical data from the other IBMs. The immersed boundary method based of finite element approach is numerically more accurate than the IBM based on finite volume discretization. In contrast, the latter is computationally more efficient than the former.

Nonlinear algebraic equations (NAEs) occur in many areas of science and engineering. The process of solving these NAEs is generally difficult, from finding a good initial guess that leads to a desired solution to deciding on convergence criteria for the approximate solution. In practice, Newton's method is the only robust general-purpose method for solving a system of NAEs. Many variants of Newton's method exist. However, it is generally impossible to know a priori which variant of Newton's method will be effective for a given problem.<p>Many high-quality software libraries are available for the numerical solution of NAEs. However, the user usually has little control over many aspects of what the library does. For example, the user may not be able to easily switch between direct and indirect methods for the linear algebra. This thesis describes a problem-solving environment (PSE) called pythNon for studying the effects (e.g., performance) of different strategies for solving systems of NAEs. It provides the researcher, teacher, or student with a flexible environment for rapid prototyping and numerical experiments. In pythNon, users can directly influence the solution process on many levels, e.g., investigation of the effects of termination criteria and/or globalization strategies. In particular, to show the power, flexibility, and ease of use of the pythNon PSE, this thesis also describes the development of a novel forcing-term strategy for approximating the Newton direction efficiently in the pythNon PSE.

Recent information theoretical results indicate that dirty-paper coding (DPC) achieves the entire capacity region of the Gaussian multiple-input multiple-output (MIMO) broadcast channel (BC). This thesis presents practical code designs for Gaussian BCs based on DPC. To simplify our designs, we assume constraints on the individual rates for each user instead of the customary constraint on transmitter power. The objective therefore is to minimize the transmitter power such that the practical decoders of all users are able to operate at the given rate constraints. The enabling element of our code designs is a practical DPC scheme based on nested turbo codes. We start with Cover's simplest two-user Gaussian BC as a toy example and present a code design that operates 1.44 dB away from the capacity region boundary at the transmission rate of 1 bit per sample per dimension for each user. Then we consider the case of the multiple-input multiple-output BC and develop a practical limit-approaching code design under the assumption that the channel state information is available perfectly at the receivers as well as at the transmitter. The optimal precoding strategy in this case can be derived by invoking duality between the MIMO BC and MIMO multiple access channel (MAC). However, this approach requires transformation of the optimal MAC covariances to their corresponding counterparts in the BC domain. To avoid these computationally complex transformations, we derive a closed-form expression for the optimal precoding matrix for the two-user case and use it to determine the optimal precoding strategy. For more than two users we propose a low-complexity suboptimal strategy, which, for three transmit antennas at the base station and three users (each with a single receive antenna), performs only 0.2 dB worse than the optimal scheme. Our obtained results are only 1.5 dB away from the capacity limit. Moreover simulations indicate that our practical DPC based scheme significantly outperforms the prevalent suboptimal strategies such as time division multiplexing and zero forcing beamforming. The drawback of DPC based designs is the requirement of channel state information at the transmitter. However, if the channel state information can be communicated back to the transmitter effectively, DPC does indeed have a promising future in code designs for MIMO BCs.

In this dissertation, we explore direct-forcing immersed boundary methods (IBM) under the framework of the lattice Boltzmann method (LBM), which is called the direct-forcing immersed boundary-lattice Boltzmann method (IB-LBM). First, we derive the direct-forcing formula based on the split-forcing lattice Boltzmann equation, which recovers the Navier-Stokes equation with second-order accuracy and enables us to develop a simple and accurate formula due to its kinetic nature. Then, we assess the various interface schemes under the derived direct-forcing formula. We consider not only diffuse interface schemes but also a sharp interface scheme. All tested schemes show a second-order overall accuracy. In the simulation of stationary complex boundary flows, we can observe that the sharper the interface scheme is, the more accurate the results are. The interface schemes are also applied to moving boundary problems. The sharp interface scheme shows better accuracy than the diffuse interface schemes but generates spurious oscillation in the boundary forcing terms due to the discontinuous change of nodes for the interpolation. In contrast, the diffuse interface schemes show smooth change in the boundary forcing terms but less accurate results because of discrete delta functions. Hence, the diffuse interface scheme with a corrected radius can be adopted to obtain both accurate and smooth results. Finally, a direct-forcing immersed boundary method (IBM) for the thermal lattice Boltzmann method (TLBM) is proposed to simulate non-isothermal flows. The direct-forcing IBM formulas for thermal equations are derived based on two TLBM models: a double-population model with a simplified thermal lattice Boltzmann equation (Model 1) and a hybrid model with an advection-diffusion equation of temperature (Model 2). The proposed methods are validated through natural convection problems with stationary and moving boundaries. In terms of accuracy, the results obtained from the IBMs based on both models are comparable and show a good agreement with those from other numerical methods. In contrast, the IBM based on Model 2 is more numerically efficient than the IBM based on Model 1. Overall, this study serves to establish the feasibility of the direct-forcing IB-LBM as a viable tool for computing various complex and/or moving boundary flow problems.

Lean premixed combustors are highly susceptible to combustion instabilities, caused by the coupling between heat release fluctuations and combustor acoustics. In order to predict the conditions under which these instabilities occur and their limit cycle amplitudes, understanding of the amplitude dependent response of the flame to acoustic excitation is required. Extensive maps of the flame response were obtained as a function of perturbation amplitude, frequency, and flow velocity. These maps illustrated substantial nonlinearity in the perturbation velocity - heat release relationship, with complex topological dependencies that illustrate folds and kinks when plotted in frequency-amplitude-heat release space. A detailed analysis of phase locked OH PLIF images of acoustically excited swirl flames was used to identify the key controlling physical processes and qualitatively discuss their characteristics. The results illustrate that the flame response is not controlled by any single physical process but rather by several simultaneously occurring processes which are potentially competing, and whose relative significance depends upon forcing frequency, amplitude of excitation, and flame stabilization dynamics. An in-depth study on the effect of acoustic forcing on the turbulent flame properties was conducted in a turbulent Bunsen flame using PIV measurements. The results showed that the flame brush thickness and the local consumption speed were modulated in the presence of acoustic forcing. These results will not only be a useful input to help improve combustion dynamics predictions but will also help serve as validation data for models.