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Description
Nonlinear dispersive equations model nonlinear waves in a wide range of physical and mathematics contexts. They reinforce or dissipate effects of linear dispersion and nonlinear interactions, and thus, may be of a focusing or defocusing nature. The nonlinear Schrödinger equation or NLS is an example of such equations. It appears as a model in hydrodynamics, nonlinear optics, quantum condensates, heat pulses in solids and various other nonlinear instability phenomena. In mathematics, one of the interests is to look at the wave interaction: waves propagation with different speeds and/or different directions produces either small perturbations comparable with linear behavior, or creates solitary waves, or even leads to singular solutions. This dissertation studies the global behavior of finite energy solutions to the $d$-dimensional focusing NLS equation, $i partial _t u+Delta u+ |u|^{p-1}u=0, $ with initial data $u_0in H^1,; x in Rn$; the nonlinearity power $p$ and the dimension $d$ are chosen so that the scaling index $s=frac{d}{2}-frac{2}{p-1}$ is between 0 and 1, thus, the NLS is mass-supercritical $(s>0)$ and energy-subcritical $(s<1).$ For solutions with $ME[u_0]<1$ ($ME[u_0]$ stands for an invariant and conserved quantity in terms of the mass and energy of $u_0$), a sharp threshold for scattering and blowup is given. Namely, if the renormalized gradient $g_u$ of a solution $u$ to NLS is initially less than 1, i.e., $g_u(0)<1,$ then the solution exists globally in time and scatters in $H^1$ (approaches some linear Schr"odinger evolution as $ttopminfty$); if the renormalized gradient $g_u(0)>1,$ then the solution exhibits a blowup behavior, that is, either a finite time blowup occurs, or there is a divergence of $H^1$ norm in infinite time. This work generalizes the results for the 3d cubic NLS obtained in a series of papers by Holmer-Roudenko and Duyckaerts-Holmer-Roudenko with the key ingredients, the concentration compactness and localized variance, developed in the context of the energy-critical NLS and Nonlinear Wave equations by Kenig and Merle. One of the difficulties is fractional powers of nonlinearities which are overcome by considering Besov-Strichartz estimates and various fractional differentiation rules.
ContributorsGuevara, Cristi Darley (Author) / Roudenko, Svetlana (Thesis advisor) / Castillo_Chavez, Carlos (Committee member) / Jones, Donald (Committee member) / Mahalov, Alex (Committee member) / Suslov, Sergei (Committee member) / Arizona State University (Publisher)
Created2011
Description
This thesis outlines the development of a vector retrieval technique, based on data assimilation, for a coherent Doppler LIDAR (Light Detection and Ranging). A detailed analysis of the Optimal Interpolation (OI) technique for vector retrieval is presented. Through several modifications to the OI technique, it is shown that the modified technique results in significant improvement in velocity retrieval accuracy. These modifications include changes to innovation covariance portioning, covariance binning, and analysis increment calculation. It is observed that the modified technique is able to make retrievals with better accuracy, preserves local information better, and compares well with tower measurements. In order to study the error of representativeness and vector retrieval error, a lidar simulator was constructed. Using the lidar simulator a thorough sensitivity analysis of the lidar measurement process and vector retrieval is carried out. The error of representativeness as a function of scales of motion and sensitivity of vector retrieval to look angle is quantified. Using the modified OI technique, study of nocturnal flow in Owens' Valley, CA was carried out to identify and understand uncharacteristic events on the night of March 27th 2006. Observations from 1030 UTC to 1230 UTC (0230 hr local time to 0430 hr local time) on March 27 2006 are presented. Lidar observations show complex and uncharacteristic flows such as sudden bursts of westerly cross-valley wind mixing with the dominant up-valley wind. Model results from Coupled Ocean/Atmosphere Mesoscale Prediction System (COAMPS®) and other in-situ instrumentations are used to corroborate and complement these observations. The modified OI technique is used to identify uncharacteristic and extreme flow events at a wind development site. Estimates of turbulence and shear from this technique are compared to tower measurements. A formulation for equivalent wind speed in the presence of variations in wind speed and direction, combined with shear is developed and used to determine wind energy content in presence of turbulence.
ContributorsChoukulkar, Aditya (Author) / Calhoun, Ronald (Thesis advisor) / Mahalov, Alex (Committee member) / Kostelich, Eric (Committee member) / Huang, Huei-Ping (Committee member) / Phelan, Patrick (Committee member) / Arizona State University (Publisher)
Created2013
Description
The atomization of a liquid jet by a high speed cross-flowing gas has many applications such as gas turbines and augmentors. The mechanisms by which the liquid jet initially breaks up, however, are not well understood. Experimental studies suggest the dependence of spray properties on operating conditions and nozzle geom- etry. Detailed numerical simulations can offer better understanding of the underlying physical mechanisms that lead to the breakup of the injected liquid jet. In this work, detailed numerical simulation results of turbulent liquid jets injected into turbulent gaseous cross flows for different density ratios is presented. A finite volume, balanced force fractional step flow solver to solve the Navier-Stokes equations is employed and coupled to a Refined Level Set Grid method to follow the phase interface. To enable the simulation of atomization of high density ratio fluids, we ensure discrete consistency between the solution of the conservative momentum equation and the level set based continuity equation by employing the Consistent Rescaled Momentum Transport (CRMT) method. The impact of different inflow jet boundary conditions on different jet properties including jet penetration is analyzed and results are compared to those obtained experimentally by Brown & McDonell(2006). In addition, instability analysis is performed to find the most dominant insta- bility mechanism that causes the liquid jet to breakup. Linear instability analysis is achieved using linear theories for Rayleigh-Taylor and Kelvin- Helmholtz instabilities and non-linear analysis is performed using our flow solver with different inflow jet boundary conditions.
ContributorsGhods, Sina (Author) / Herrmann, Marcus (Thesis advisor) / Squires, Kyle (Committee member) / Chen, Kangping (Committee member) / Huang, Huei-Ping (Committee member) / Tang, Wenbo (Committee member) / Arizona State University (Publisher)
Created2013
Description
It is possible in a properly controlled environment, such as industrial metrology, to make significant headway into the non-industrial constraints on image-based position measurement using the techniques of image registration and achieve repeatable feature measurements on the order of 0.3% of a pixel, or about an order of magnitude improvement on conventional real-world performance. These measurements are then used as inputs for a model optimal, model agnostic, smoothing for calibration of a laser scribe and online tracking of velocimeter using video input. Using appropriate smooth interpolation to increase effective sample density can reduce uncertainty and improve estimates. Use of the proper negative offset of the template function has the result of creating a convolution with higher local curvature than either template of target function which allows improved center-finding. Using the Akaike Information Criterion with a smoothing spline function it is possible to perform a model-optimal smooth on scalar measurements without knowing the underlying model and to determine the function describing the uncertainty in that optimal smooth. An example of empiric derivation of the parameters for a rudimentary Kalman Filter from this is then provided, and tested. Using the techniques of Exploratory Data Analysis and the "Formulize" genetic algorithm tool to convert the spline models into more accessible analytic forms resulted in stable, properly generalized, KF with performance and simplicity that exceeds "textbook" implementations thereof. Validation of the measurement includes that, in analytic case, it led to arbitrary precision in measurement of feature; in reasonable test case using the methods proposed, a reasonable and consistent maximum error of around 0.3% the length of a pixel was achieved and in practice using pixels that were 700nm in size feature position was located to within ± 2 nm. Robust applicability is demonstrated by the measurement of indicator position for a King model 2-32-G-042 rotameter.
ContributorsMunroe, Michael R (Author) / Phelan, Patrick (Thesis advisor) / Kostelich, Eric (Committee member) / Mahalov, Alex (Committee member) / Arizona State University (Publisher)
Created2012
Description
The OMFIT (One Modeling Framework for Integrated Tasks) modeling environment and the BRAINFUSE module have been deployed on the PPPL (Princeton Plasma Physics Laboratory) computing cluster with modifications that have rendered the application of artificial neural networks (NNs) to the TRANSP databases for the JET (Joint European Torus), TFTR (Tokamak Fusion Test Reactor), and NSTX (National Spherical Torus Experiment) devices possible through their use. This development has facilitated the investigation of NNs for predicting heat transport profiles in JET, TFTR, and NSTX, and has promoted additional investigations to discover how else NNs may be of use to scientists at PPPL. In applying NNs to the aforementioned devices for predicting heat transport, the primary goal of this endeavor is to reproduce the success shown in Meneghini et al. in using NNs for heat transport prediction in DIII-D. Being able to reproduce the results from is important because this in turn would provide scientists at PPPL with a quick and efficient toolset for reliably predicting heat transport profiles much faster than any existing computational methods allow; the progress towards this goal is outlined in this report, and potential additional applications of the NN framework are presented.
ContributorsLuna, Christopher Joseph (Author) / Tang, Wenbo (Thesis director) / Treacy, Michael (Committee member) / Orso, Meneghini (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor)
Created2015-05
Description
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.
ContributorsWake, Christian (Author) / Tang, Wenbo (Thesis director) / Moustaoui, Mohamed (Committee member) / Kostelich, Eric (Committee member) / Barrett, The Honors College (Contributor) / College of Liberal Arts and Sciences (Contributor)
Created2012-12
Description
The goal of this project was to examine the separatricies that define regions of distinct flow behaviors in realistic time-dependent dynamical systems. In particular, we adapted previously available methods for computing the Finite-Time Lyapunov Exponent (FTLE) to a set of measured wind velocity data in order to visualize the separatricies as ridges of the FTLE field in a section of the atmosphere. This visualization required a number of alterations to the original methods, including interpolation techniques and two different adaptive refinement schemes for producing more detailed results. Overall, there were two computations performed with the wind velocity data: once along a single spherical surface, on which the separatricies could be visualized as material lines, and then along a three-dimensional section of the atmosphere, for which the separatricies were material surfaces. The resulting figures provide an image of the Antarctic polar vortex from the wind velocity data, which is consistent with other data gathered on the same date.
ContributorsUpton, James Thomas (Author) / Tang, Wenbo (Thesis director) / Moustaoui, Mohamed (Committee member) / Barrett, The Honors College (Contributor) / School of International Letters and Cultures (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor)
Created2014-05
Description
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.
ContributorsDeibel, Angelica Rae (Author) / Tang, Wenbo (Thesis director) / Moustaoui, Mohamed (Committee member) / Kostelich, Eric (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Department of Physics (Contributor)
Created2013-05
Description
The physics of waves control most of the world, in multiple forms, such as electromagnetic waves. Mathematicians and physicists have developed equations which describe the patterns in which waves evolve over time, while moving through space. Due to their partial differential form, solutions to these equations must be approximated. This study introduces a new numerical scheme to perform the approximation which is highly stable and computationally efficient. This numerical scheme is formulated with respect to Maxwell’s equations, employing spatial and temporal staggering to implement a fourth-order phase accuracy. It is then compared to the traditional Yee scheme and the Runge-Kutta 3 scheme in one-dimensional applications, revealing a similar accuracy to the Runge-Kutta 3 scheme while requiring less computations per time step. Simulations are then performed in the two-dimensional case. First, no boundary conditions are implemented, causing reflection at the edge of the spatial domain. Next, the simulation is conducted while employing absorbing boundary conditions, simulating wave propagation over an infinite spatial domain. These results are compared to the results of a large domain simulation, in which the wave propagation does not reach the boundaries. Comparing the simulations, it is concluded that the numerical scheme is stable and highly accurate when employing absorbing boundary conditions. Finally, the scheme is tested in two dimensions with wave propagation through nonlinear media, as opposed to the prior simulations which were performed as if in a vacuum. After performing spectral analysis on the resulting waves after a long-time domain simulation, the resulting angular frequencies match those expected from theory. Therefore, the scheme is concluded to be powerful in one-dimensional, two-dimensional, and nonlinear simulations, all while being computationally efficient.
ContributorsKirvan, Alex Ander (Author) / Moustaoui, Mohamed (Thesis director) / Kostelich, Eric (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05
Description
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.
ContributorsCholinski, Christopher John (Author) / Tang, Wenbo (Thesis director) / Herrmann, Marcus (Committee member) / Mechanical and Aerospace Engineering Program (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2019-05