This thesis considers the properties of MHS equilibria formed through non-resistive MHD relaxation of analytical non-potential magnetic field models, which contain two null points connected by a generic separator. Four types of analytical magnetic fields are formulated, with different forms of current. The magnetic field model which has a uniform current directed along the separator, is used through the rest of this thesis to form MHS equilibria and to study reconnection. This magnetic field, which is not force-free, embedded in a high-beta plasma, relaxes non-resistively using a 3D MHD code. The relaxation causes the field about the separator to collapse leading to a twisted current layer forming along the separator. The MHS equilibrium current layer slowly becomes stronger, longer, wider and thinner with time. Its properties, and the properties of the plasma, are found to depend on the initial parameters of the magnetic field, which control the geometry of the magnetic configuration. Such a MHS equilibria is used in a high plasma-beta reconnection experiment. An anomalous resistivity ensures that only the central strong current in the separator current layer is dissipated. The reconnection occurs in two phases characterised by fast and slow reconnection, respectively. Waves, launched from the diffusion site, communicate the loss of force balance at the current layer and set up flows in the system. The energy transport in this system is dominated by Ohmic dissipation. Several methods are presented which allow a low plasma-beta value to be approached in the single-separator model. One method is chosen and this model is relaxed non-resistively to form a MHS equilibrium. A twisted current layer grows along the separator, containing stronger current than in the high plasma-beta experiments, and has a local enhancement in pressure inside it. The growth rate of this current layer is similar to that found in the high plasma-beta experiments, however, the current layer becomes thinner and narrower over time.

Observations of the surface of the Sun reveal multi-scaled, mixed magnetic features that carpet the entire solar surface. Not surprisingly, the global magnetic fields extrapolated from these observations are highly complex. This thesis explores the topology of the Sun's global coronal magnetic fields. The magnetic skeleton of a magnetic field provides us with a way of examining the magnetic field and quantifying its complexity. Using specialised codes to find the magnetic skeletons which were written during the course of this work, we first examine potential field extrapolations of the global solar coronal magnetic field determined from observed synoptic magnetograms from the Heliospheric Magnetic Imager on the Solar Dynamics Observatory. The resolution of the PFSS models is found to be very important for discovering the true nature of the global magnetic skeleton. By increasing the maximum number of harmonics used in the potential field extrapolations and, therefore, the grid resolution, 60 times more null points may be found in the coronal magnetic field. These high resolution fields also have a large global separator network which connects the coronal magnetic field over large distances and involves between 40 % and 60 % of all the null points in the solar atmosphere. This global separator network exists at both solar minimum and solar maximum and has separators that reach high into the solar atmosphere (> 1R☉) even though they connect null points close to the solar surface. These potential field extrapolations are then compared with magnetohydrostatic (MHS) extrapolations of the coronal magnetic field which also provide us with information about the plasma in the corona. With a small component of electric current density in the direction perpendicular to the radial direction, these MHS fields are found to have a plasma beta and pressure typical of the corona. As this small component of electric current density grows, the heliospheric current sheet is warped significantly and the magnetic field, plasma beta and pressure become unphysical. Torsional spine reconnection is also studied local to a single null point. First using a dynamical relaxation of a spiral null point under non-resistive magnetohydrodynamics (MHD) to a MHS equilibrium is form in which a current layer has built up around the spine lines. Then the reconnection under resistive MHD in this current sheet is studied. The current about the spine lines is dissipated and the magnetic energy is mainly converted into heat directly as the field lines untwist about the spine line.

The deployment of dedicated short-range communications (DSRC) roadside units (RSUs) allows a connected or automated vehicle to acquire information from the surrounding environment using vehicle-to-infrastructure (V2I) communication. However, wireless communication using DSRC has shown to exhibit null points, at repeatable distances. The null points are significant and there was unexpected loss in the wireless signal strength along the pathway of the V2I communication. If the wireless connection is poor or non-existent, the V2I safety application will not obtain sufficient data to perform the operation services. In other words, a poor wireless connection between a vehicle and infrastructure (e.g., RSU) could hamper the performance of a safety application. For example, a designer of a V2I safety application may require a minimum rate of data (or packet count) over 1,000 meters to effectively implement a Reduced Speed/Work Zone Warning (RSZW) application. The RSZW safety application is aimed to alert or warn drivers, in a Cooperative Adaptive Cruise Control (CACC) platoon, who are approaching a work zone. Therefore, the packet counts and/or signal strength threshold criterion must be determined by the developer of the V2I safety application. Thus, we selected an arbitrary criterion to develop an empirical method of ascertaining the null points from a DSRC RSU. The research motivation focuses on developing an empirical method of calculating the null points of a DSRC RSU for V2I communication at a highway on/off-ramp. The intent is to improve safety, mobility, and environmental applications since a map of the null points can be plotted against the distance between the DSRC RSU and a vehicle's onboard unit (OBU). The main research question asks: 'What is a more robust empirical method, compared to the horizontal and vertical laws of reflection formula, in determining the null points from a DSRC RSU on a highway on/off ramp?' The research objectives are as follows: 1. Explain where and why null points occur from a DSRC RSU (Chapter 2) 2. Apply the existing horizontal and vertical polarization model and discuss the limitations of the model in a real-world scenario for a DSRC RSU on a highway on/off ramp (Chapter 3 and Appendix A) 3. Introduce an extended horizontal and vertical polarization null point model using empirical data (Chapter 4) 4. Discuss the conclusion, limitations of work, and future research (Chapter 5). The simplest manner to understand where and why null points occur is depicted as two sinusoidal waves: direct and reflective waves (i.e., also known as a two-ray model). The null points for a DSRC RSU occurs because the direct and reflective waves produce a destructive interference (i.e., decrease in signal strength) when they collide. Moreover, the null points can be located using Pythagorean theorem for the direct and reflective waves. Two existing models were leveraged to analyze null points: 1) signal strength loss (i.e., a free space path loss model, or FSPL, in Appendix A) and 2) the existing horizontal and vertical polarization null points from a DSRC RSU. Using empirical data from two different field tests, the existing horizontal and vertical polarization null point model was shown to contain limitations in short distances from the DSRC RSU. Moreover, the existing horizontal and vertical polarization model for null points was extremely challenging to replicate with over 15 DSRC RSU data sets. After calculating the null point for several DSRC RSU heights, the paper noticed a limitation of the existing horizontal and vertical polarization null point model with over 15 DSRC RSU data sets (i.e., the model does not account for null points along the full length of the FSPL model). An extended horizontal and vertical polarization model is proposed that calculates the null point from a DSRC RSU. There are 18 model comparisons of the packet counts and signal strengths at various thresholds as perspective extended horizontal and vertical polarization models. This paper compares the predictive ability of 18 models and measures the fit. Finally, a predication graph is depicted with the neural network's probability profile for packet counts =1 when greater than or equal to 377. Likewise, a python script is provided of the extended horizontal and vertical polarization model in Appendix C. Consequently, the neural network model was applied to 10 different DSRC RSU data sets at 10 unique locations around a circular test track with packet counts ranging from 0 to 11. Neural network models were generated for 10 DSRC RSUs using three thresholds with an objective to compare the predictive ability of each model and measure the fit. Based on 30 models at 10 unique locations, the highest misclassification was 0.1248, while the lowest misclassification was 0.000. There were six RSUs mounted at 3.048 (or 10 feet) from the ground with a misclassification rate that ranged from 0.1248 to 0.0553. Out of 18 models, seven had a misclassification rate greater than 0.110, while the remaining misclassification rates were less than 0.0993. There were four RSUs mounted at 6.096 meters (or 20 feet) from the ground with a misclassification rate that ranged from 0.919 to 0.000. Out of 12 models, four had a misclassification rate greater than 0.0590, while the remaining misclassification rates were less than 0.0412. Finally, there are two major limitations in the research: 1) the most effective key parameter is packet counts, which often require expensive data acquisition equipment to obtain the information and 2) the categorical type (i.e., decision tree, logistic regression, and neural network) will vary based on the packet counts or signal strength threshold that is dictated by the threshold criterion. There are at least two future research areas that correspond to this body of work: 1) there is a need to leverage the extended horizontal and vertical polarization null point model on multiple DSRC RSUs along a highway on/off ramp, and 2) there is a need to apply and validate different electric and magnetic (or propagation) models. / Ph. D. / The deployment of dedicated short-range communications (DSRC) roadside units (RSUs) allows a connected or automated vehicle to acquire information from the surrounding environment using vehicle-to-infrastructure (V2I) communication. However, wireless communication using DSRC has shown to exhibit null points, at repeatable distances. The null points are significant and there was unexpected loss in the wireless signal strength along the pathway of the V2I communication. If the wireless connection is poor or non-existent, the V2I safety application will not obtain sufficient data to perform the operation services. In other words, a poor wireless connection between a vehicle and infrastructure (e.g., RSU) could hamper the performance of a safety application. For example, a designer of a V2I safety application may require a minimum rate of data (or packet count) over 1,000 meters to effectively implement a Reduced Speed/Work Zone Warning (RSZW) application. The RSZW safety application is aimed to alert or warn drivers, in a Cooperative Adaptive Cruise Control (CACC) platoon, who are approaching a work zone. Therefore, the packet counts and/or signal strength threshold criterion must be determined by the developer of the V2I safety application. Thus, we selected an arbitrary criterion to develop an empirical method of ascertaining the null points from a DSRC RSU. The research motivation focuses on developing an empirical method of calculating the null points of a DSRC RSU for V2I communication at a highway on/off-ramp. The intent is to improve safety, mobility, and environmental applications since a map of the null points can be plotted against the distance between the DSRC RSU and a vehicle’s onboard unit (OBU). The main research question asks: “What is a more robust empirical method, compared to the horizontal and vertical laws of reflection formula, in determining the null points from a DSRC RSU on a highway on/off ramp?” The research objectives are as follows: 1. Explain where and why null points occur from a DSRC RSU (Chapter 2) 2. Apply the existing horizontal and vertical polarization model and discuss the limitations of the model in a real-world scenario for a DSRC RSU on a highway on/off ramp (Chapter 3 and Appendix A) 3. Introduce an extended horizontal and vertical polarization null point model using empirical data (Chapter 4) 4. Discuss the conclusion, limitations of work, and future research (Chapter 5). The simplest manner to understand where and why null points occur is depicted as two sinusoidal waves: direct and reflective waves (i.e., also known as a two-ray model). The null points for a DSRC RSU occurs because the direct and reflective waves produce a destructive interference (i.e., decrease in signal strength) when they collide. Moreover, the null points can be located using Pythagorean theorem for the direct and reflective waves. Two existing models were leveraged to analyze null points: 1) signal strength loss (i.e., a free space path loss model, or FSPL, in Appendix A) and 2) the existing horizontal and vertical polarization null points from a DSRC RSU. Using empirical data from two different field tests, the existing horizontal and vertical polarization null point model was shown to contain limitations in short distances from the DSRC RSU. Moreover, the existing horizontal and vertical polarization model for null points was extremely challenging to replicate with over 15 DSRC RSU data sets. After calculating the null point for several DSRC RSU heights, the paper noticed a limitation of the existing horizontal and vertical polarization null point model with over 15 DSRC RSU data sets (i.e., the model does not account for null points along the full length of the FSPL model). An extended horizontal and vertical polarization model is proposed that calculates the null point from a DSRC RSU. There are 18 model comparisons of the packet counts and signal strengths at various thresholds as perspective extended horizontal and vertical polarization models. This paper compares the predictive ability of 18 models and measures the fit. Finally, a predication graph is depicted with the neural network’s probability profile for packet counts =1 when greater than or equal to 377. Likewise, a python script is provided of the extended horizontal and vertical polarization model in Appendix C. Consequently, the neural network model was applied to 10 different DSRC RSU data sets at 10 unique locations around a circular test track with packet counts ranging from 0 to 11. Neural network models were generated for 10 DSRC RSUs using three thresholds with an objective to compare the predictive ability of each model and measure the fit. Based on 30 models at 10 unique locations, the highest misclassification was 0.1248, while the lowest misclassification was 0.000. There were six RSUs mounted at 3.048 (or 10 feet) from the ground with a misclassification rate that ranged from 0.1248 to 0.0553. Out of 18 models, seven had a misclassification rate greater than 0.110, while the remaining misclassification rates were less than 0.0993. There were four RSUs mounted at 6.096 meters (or 20 feet) from the ground with a misclassification rate that ranged from 0.919 to 0.000. Out of 12 models, four had a misclassification rate greater than 0.0590, while the remaining misclassification rates were less than 0.0412. Finally, there are two major limitations in the research: 1) the most effective key parameter is packet counts, which often require expensive data acquisition equipment to obtain the information and 2) the categorical type (i.e., decision tree, logistic regression, and neural network) will vary based on the packet counts or signal strength threshold that is dictated by the threshold criterion. There are at least two future research areas that correspond to this body of work: 1) there is a need to leverage the extended horizontal and vertical polarization null point model on multiple DSRC RSUs along a highway on/off ramp, and 2) there is a need to apply and validate different electric and magnetic (or propagation) models.