Lidar has demonstrated its utility in meteorological studies, wind resource assessment, and wind farm control. More recently, lidar has gained widespread attention for autonomous vehicles.

The first part of the dissertation begins with an application of a coherent Doppler lidar to wind gust characterization for wind farm control. This application focuses on wind gusts on a scale from 100 m to 1000 m. A detecting and tracking algorithm is proposed to extract gusts from a wind field and track their movement. The algorithm was implemented for a three-hour, two-dimensional wind field retrieved from the measurements of a coherent Doppler lidar. The Gaussian distribution of the gust spanwise deviation from the streamline was demonstrated. Size dependency of gust deviations is discussed. A prediction model estimating the impact of gusts with respect to arrival time and the probability of arrival locations is introduced. The prediction model was applied to a virtual wind turbine array, and estimates are given for which wind turbines would be impacted.

The second part of this dissertation describes a Time-of-Flight lidar simulation. The lidar simulation includes a laser source module, a propagation module, a receiver module, and a timing module. A two-dimensional pulse model is introduced in the laser source module. The sampling rate for the pulse model is explored. The propagation module takes accounts of beam divergence, target characteristics, atmosphere, and optics. The receiver module contains models of noise and analog filters in a lidar receiver. The effect of analog filters on the signal behavior was investigated. The timing module includes a Time-to-Digital Converter (TDC) module and an Analog-to-Digital converter (ADC) module. In the TDC module, several walk-error compensation methods for leading-edge detection and multiple timing algorithms were modeled and tested on simulated signals. In the ADC module, a benchmark (BM) timing algorithm is proposed. A Neyman-Pearson (NP) detector was implemented in the time domain and frequency domain (fast Fourier transform (FFT) approach). The FFT approach with frequency-domain zero-paddings improves the timing resolution. The BM algorithm was tested on simulated signals, and the NP detector was evaluated on both simulated signals and measurements from a prototype lidar (Bhaskaran, 2018).

The dynamics of a fluid flow inside 2D square and 3D cubic cavities

under various configurations were simulated and analyzed using a

spectral code I developed.

This code was validated against known studies in the 3D lid-driven

cavity. It was then used to explore the various dynamical behaviors

close to the onset of instability of the steady-state flow, and explain

in the process the mechanism underlying an intermittent bursting

previously observed. A fairly complete bifurcation picture emerged,

using a combination of computational tools such as selective

frequency damping, edge-state tracking and subspace restriction.

The code was then used to investigate the flow in a 2D square cavity

under stable temperature stratification, an idealized version of a lake

with warmer water at the surface compared to the bottom. The governing

equations are the Navier-Stokes equations under the Boussinesq approximation.

Simulations were done over a wide range of parameters of the problem quantifying

the driving velocity at the top (e.g. wind) and the strength of the stratification.

Particular attention was paid to the mechanisms associated with the onset of

instability of the base steady state, and the complex nontrivial dynamics

occurring beyond onset, where the presence of multiple states leads to a

rich spectrum of states, including homoclinic and heteroclinic chaos.

A third configuration investigates the flow dynamics of a fluid in a rapidly

rotating cube subjected to small amplitude modulations. The responses were

quantified by the global helicity and energy measures, and various peak

responses associated to resonances with intrinsic eigenmodes of the cavity

and/or internal retracing beams were clearly identified for the first time.

A novel approach to compute the eigenmodes is also described, making accessible

a whole catalog of these with various properties and dynamics. When the small

amplitude modulation does not align with the rotation axis (precession) we show

that a new set of eigenmodes are primarily excited as the angular velocity

increases, while triadic resonances may occur once the nonlinear regime kicks in.

A continuously and stably stratified fluid contained in a square cavity subjected to harmonic body forcing is studied numerically by solving the Navier-Stokes equations under the Boussinesq approximation. Complex dynamics are observed near the onset of instability of the basic state, which is a flow configuration that is always an exact analytical solution of the governing equations. The instability of the basic state to perturbations is first studied with linear stability analysis (Floquet analysis), revealing a multitude of intersecting synchronous and subharmonic resonance tongues in parameter space. A modal reduction method for determining the locus of basic state instability is also shown, greatly simplifying the computational overhead normally required by a Floquet study. Then, a study of the nonlinear governing equations determines the criticality of the basic state's instability, and ultimately characterizes the dynamics of the lowest order spatial mode by the three discovered codimension-two bifurcation points within the resonance tongue. The rich dynamics include a homoclinic doubling cascade that resembles the logistic map and a multitude of gluing bifurcations.

The numerical techniques and methodologies are first demonstrated on a homogeneous fluid contained within a three-dimensional lid-driven cavity. The edge state technique and linear stability analysis through Arnoldi iteration are used to resolve the complex dynamics of the canonical shear-driven benchmark problem. The techniques here lead to a dynamical description of an instability mechanism, and the work serves as a basis for the remainder of the dissertation.

This honors thesis explores and models the flow of air around a cylindrical arrow that is rotating as it moves through the air. This model represents the airflow around an archery arrow after it is released from the bow and rotates while it flies through the air. This situation is important in archery because an understanding of the airflow allows archers to predict the flight of the arrow. As a result, archers can improve their accuracy and ability to hit targets. However, not many computational fluid dynamic simulations modeling the airflow around a rotating archery arrow exist. This thesis attempts to further the understanding of the airflow around a rotating archery arrow by creating a mathematical model to numerically simulate the airflow around the arrow in the presence of this rotation. This thesis uses a linearized approximation of the Navier Stokes equations to model the airflow around the arrow and explains the reasoning for using this simplification of the fully nonlinear Navier Stokes equations. This thesis continues to describe the discretization of these linearized equations using the finite difference method and the boundary conditions used for these equations. A MATLAB code solves the resulting system of equations in order to obtain a numerical simulation of this airflow around the rotating arrow. The results of the simulation for each velocity component and the pressure distribution are displayed. This thesis then discusses the results of the simulation, and the MATLAB code is analyzed to verify the convergence of the solution. Appendix A includes the full MATLAB code used for the flow simulation. Finally, this thesis explains potential future research topics, ideas, and improvements to the code that can help further the understanding and create more realistic simulations of the airflow around a flying archery arrow.

Two urban flows are analyzed, one concerned with pollutant transport in a Phoenix, Arizona neighborhood and the other with windshear detection at the Hong Kong International Airport (HKIA).

Lagrangian measures, identified with finite-time Lyapunov exponents, are first used to characterize transport patterns of inertial pollutant particles. Motivated by actual events the focus is on flows in realistic urban geometry. Both deterministic and stochastic transport patterns are identified, as inertial Lagrangian coherent structures. For the deterministic case, the organizing structures are well defined and are extracted at different hours of a day to reveal the variability of coherent patterns. For the stochastic case, a random displacement model for fluid particles is formulated, and used to derive the governing equations for inertial particles to examine the change in organizing structures due to ``zeroth-order'' random noise. It is found that, (1) the Langevin equation for inertial particles can be reduced to a random displacement model; (2) using random noise based on inhomogeneous turbulence, whose diffusivity is derived from $k$-$\epsilon$ models, major coherent structures survive to organize local flow patterns and weaker structures are smoothed out due to random motion.

A study of three-dimensional Lagrangian coherent structures (LCS) near HKIA is then presented and related to previous developments of two-dimensional (2D) LCS analyses in detecting windshear experienced by landing aircraft. The LCS are contrasted among three independent models and against 2D coherent Doppler light detection and ranging (LIDAR) data. Addition of the velocity information perpendicular to the lidar scanning cone helps solidify flow structures inferred from previous studies; contrast among models reveals the intramodel variability; and comparison with flight data evaluates the performance among models in terms of Lagrangian analyses. It is found that, while the three models and the LIDAR do recover similar features of the windshear experienced by a landing aircraft (along the landing trajectory), their Lagrangian signatures over the entire domain are quite different - a portion of each numerical model captures certain features resembling those LCS extracted from independent 2D LIDAR analyses based on observations. Overall, it was found that the Weather Research and Forecast (WRF) model provides the best agreement with the LIDAR data.

Finally, the three-dimensional variational (3DVAR) data assimilation scheme in WRF is used to incorporate the LIDAR line of sight velocity observations into the WRF model forecast at HKIA. Using two different days as test cases, it is found that the LIDAR data can be successfully and consistently assimilated into WRF. Using the updated model forecast LCS are extracted along the LIDAR scanning cone and compare to onboard flight data. It is found that the LCS generated from the updated WRF forecasts are generally better correlated with the windshear experienced by landing aircraft as compared to the LIDAR extracted LCS alone, which suggests that such a data assimilation scheme could be used for the prediction of windshear events.

Earth-system models describe the interacting components of the climate system and

technological systems that affect society, such as communication infrastructures. Data

assimilation addresses the challenge of state specification by incorporating system

observations into the model estimates. In this research, a particular data

assimilation technique called the Local Ensemble Transform Kalman Filter (LETKF) is

applied to the ionosphere, which is a domain of practical interest due to its effects

on infrastructures that depend on satellite communication and remote sensing. This

dissertation consists of three main studies that propose strategies to improve space-

weather specification during ionospheric extreme events, but are generally applicable

to Earth-system models:

Topic I applies the LETKF to estimate ion density with an idealized model of

the ionosphere, given noisy synthetic observations of varying sparsity. Results show

that the LETKF yields accurate estimates of the ion density field and unobserved

components of neutral winds even when the observation density is spatially sparse

(2% of grid points) and there is large levels (40%) of Gaussian observation noise.

Topic II proposes a targeted observing strategy for data assimilation, which uses

the influence matrix diagnostic to target errors in chosen state variables. This

strategy is applied in observing system experiments, in which synthetic electron density

observations are assimilated with the LETKF into the Thermosphere-Ionosphere-

Electrodynamics Global Circulation Model (TIEGCM) during a geomagnetic storm.

Results show that assimilating targeted electron density observations yields on

average about 60%–80% reduction in electron density error within a 600 km radius of

the observed location, compared to 15% reduction obtained with randomly placed

vertical profiles.

Topic III proposes a methodology to account for systematic model bias arising

ifrom errors in parametrized solar and magnetospheric inputs. This strategy is ap-

plied with the TIEGCM during a geomagnetic storm, and is used to estimate the

spatiotemporal variations of bias in electron density predictions during the

transitionary phases of the geomagnetic storm. Results show that this strategy reduces

error in 1-hour predictions of electron density by about 35% and 30% in polar regions

during the main and relaxation phases of the geomagnetic storm, respectively.

The tools developed for the use of investigating dynamical systems have provided critical understanding to a wide range of physical phenomena. Here these tools are used to gain further insight into scalar transport, and how it is affected by mixing. The aim of this research is to investigate the efficiency of several different partitioning methods which demarcate flow fields into dynamically distinct regions, and the correlation of finite-time statistics from the advection-diffusion equation to these regions.

For autonomous systems, invariant manifold theory can be used to separate the system into dynamically distinct regions. Despite there being no equivalent method for nonautonomous systems, a similar analysis can be done. Systems with general time dependencies must resort to using finite-time transport barriers for partitioning; these barriers are the edges of Lagrangian coherent structures (LCS), the analog to the stable and unstable manifolds of invariant manifold theory. Using the coherent structures of a flow to analyze the statistics of trapping, flight, and residence times, the signature of anomalous diffusion are obtained.

This research also investigates the use of linear models for approximating the elements of the covariance matrix of nonlinear flows, and then applying the covariance matrix approximation over coherent regions. The first and second-order moments can be used to fully describe an ensemble evolution in linear systems, however there is no direct method for nonlinear systems. The problem is only compounded by the fact that the moments for nonlinear flows typically don't have analytic representations, therefore direct numerical simulations would be needed to obtain the moments throughout the domain. To circumvent these many computations, the nonlinear system is approximated as many linear systems for which analytic expressions for the moments exist. The parameters introduced in the linear models are obtained locally from the nonlinear deformation tensor.

Coherent vortices are ubiquitous structures in natural flows that affect mixing and transport of substances and momentum/energy. Being able to detect these coherent structures is important for pollutant mitigation, ecological conservation and many other aspects. In recent years, mathematical criteria and algorithms have been developed to extract these coherent structures in turbulent flows. In this study, we will apply these tools to extract important coherent structures and analyze their statistical properties as well as their implications on kinematics and dynamics of the flow. Such information will aide representation of small-scale nonlinear processes that large-scale models of natural processes may not be able to resolve.

This thesis shows analyses of mixing and transport patterns associated with Hurricane Katrina as it hit the United States in August of 2005. Specifically, by applying atmospheric velocity information from the Weather Research and Forecasting System, finite-time Lyapunov exponents have been computed and the Lagrangian Coherent Structures have been identified. The chaotic dynamics of material transport induced by the hurricane are results from these structures within the flow. Boundaries of the coherent structures are highlighted by the FTLE field. Individual particle transport within the hurricane is affected by the location of these boundaries. In addition to idealized fluid particles, we also studied inertial particles which have finite size and inertia. Basing on established Maxey-Riley equations of the dynamics of particles of finite size, we obtain a reduced equation governing the position process. Using methods derived from computer graphics, we identify maximizers of the FTLE field. Following and applying these ideas, we analyze the dynamics of inertial particle transport within Hurricane Katrina, through comparison of trajectories of dierent sized particles and by pinpointing the location of the Lagrangian Coherent Structures.

Using weather data from the Weather Research and Forecasting model (WRF), we analyze the transport of inertial particles in Hurricane Katrina in order to identify coherent patterns of motion. For our analysis, we choose a Lagrangian approach instead of an Eulerian approach because the Lagrangian approach is objective and frame-independent, and gives results which are better defined. In particular, we locate Lagrangian Coherent Structures (LCS), which are smooth sets of fluid particles which are locally most hyperbolic (either attracting or repelling). We implement a variational method for locating LCS and compare the results to previous computation of LCS using Finite-Time Lyapunov Exponents (FTLE) to identify regions of high stretching in the fluid flow.