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A semi-implicit, fourth-order time-filtered leapfrog numerical scheme is investigated for accuracy and stability, and applied to several test cases, including one-dimensional advection and diffusion, the anelastic equations to simulate the Kelvin-Helmholtz instability, and the global shallow water spectral model to simulate the nonlinear evolution of twin tropical cyclones. The leapfrog scheme leads to computational modes in the solutions to highly nonlinear systems, and time-filters are often used to damp these modes. The proposed filter damps the computational modes without appreciably degrading the physical mode. Its performance in these metrics is superior to the second-order time-filtered leapfrog scheme developed by Robert and Asselin.
ContributorsBurke, Lee Matthew Moore (Author) / Moustaoui, Mohamed (Thesis director) / Kostelich, Eric (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Mechanical and Aerospace Engineering Program (Contributor) / Barrett, The Honors College (Contributor)
Created2016-05
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
Urban areas produce an urban heat island (UHI), which is manifest as warmer temperatures compared to the surrounding and less developed areas. While it is understood that UHI's are warmer than their surrounding areas, attributing the amount of heat added by the urban area is not easily determined. Current generation modeling systems require diurnal anthropogenic heating profiles. Development of diurnal cycle profiles of anthropogenic heating will help the modeling community as there is currently no database for anthropogenic heating profiles for cities across the United States. With more accurate anthropogenic heating profiles, climate models will be better able to show how humans directly impact the urban climate. This research attempts to create anthropogenic heating profiles for 61 cities in the United States. The method used climate, electricity, natural gas, and transportation data to develop anthropogenic heating profiles for each state. To develop anthropogenic heating profiles, profiles are developed for buildings, transportation, and human metabolism using the most recently available data. Since utilities are reluctant to release data, the building energy profile is developed using statewide electricity by creating a linear regression between the climate and electricity usage. A similar method is used to determine the contribution of natural gas consumption. These profiles are developed for each month of the year, so annual changes in anthropogenic heating can be seen. These profiles can then be put into climate models to enable more accurate urban climate modeling.
ContributorsMilne, Jeffrey (Author) / Georgescu, Matei (Thesis director) / Sailor, David (Committee member) / Brazel, Anthony (Committee member) / Barrett, The Honors College (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / School of Geographical Sciences and Urban Planning (Contributor)
Created2014-05
Description
A numerical study of wave-induced momentum transport across the tropopause in the presence of a stably stratified thin inversion layer is presented and discussed. This layer consists of a sharp increase in static stability within the tropopause. The wave propagation is modeled by numerically solving the Taylor-Goldstein equation, which governs the dynamics of internal waves in stably stratified shear flows. The waves are forced by a flow over a bell shaped mountain placed at the lower boundary of the domain. A perfectly radiating condition based on the group velocity of mountain waves is imposed at the top to avoid artificial wave reflection. A validation for the numerical method through comparisons with the corresponding analytical solutions will be provided. Then, the method is applied to more realistic profiles of the stability to study the impact of these profiles on wave propagation through the tropopause.
ContributorsCole, Alexandra Shea (Author) / Moustaoui, Mohamed (Thesis director) / Kostelich, Eric (Committee member) / Mahalov, Alex (Committee member) / School of International Letters and Cultures (Contributor) / School of Geographical Sciences and Urban Planning (Contributor) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2017-05
Description
Cancer modeling has brought a lot of attention in recent years. It had been proven to be a difficult task to model the behavior of cancer cells, since little about the "rules" a cell follows has been known. Existing models for cancer cells can be generalized into two categories: macroscopic models which studies the tumor structure as a whole, and microscopic models which focus on the behavior of individual cells. Both modeling strategies strive the same goal of creating a model that can be validated with experimental data, and is reliable for predicting tumor growth. In order to achieve this goal, models must be developed based on certain rules that tumor structures follow. This paper will introduce how such rules can be implemented in a mathematical model, with the example of individual based modeling.
ContributorsHan, Zimo (Author) / Motsch, Sebastien (Thesis director) / Moustaoui, Mohamed (Committee member) / School of Mathematical and Statistical Sciences (Contributor) / Barrett, The Honors College (Contributor)
Created2016-12
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 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