+publisher:"University of Colorado" +contributor:("Keith Julien")

University of Colorado

1. Grooms, Ian. Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow.

Degree: PhD, Applied Mathematics, 2011, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/14

► This thesis documents three investigations carried out in pursuance of a doctoral degree in applied mathematics at the University of Colorado (Boulder). The first… (more)

▼ This thesis documents three investigations carried out in pursuance of a doctoral degree in applied mathematics at the University of Colorado (Boulder). The first investigation concerns the properties of rotating Rayleigh-Benard convection – thermal convection in a rotating infinite plane layer between two constant-temperature boundaries. It is noted that in certain parameter regimes convective Taylor columns appear which dominate the dynamics, and a semi-analytical model of these is presented. Investigation of the columns and of various other properties of the flow is ongoing. The second investigation concerns the interactions between planetary-scale and mesoscale dynamics in the oceans. Using multiple-scale asymptotics the possible connections between planetary geostrophic and quasigeostrophic dynamics are investigated, and three different systems of coupled equations are derived. Possible use of these equations in conjunction with the method of superparameterization, and extension of the asymptotic methods to the interactions between mesoscale and submesoscale dynamics is ongoing. The third investigation concerns the linear stability properties of semi-implicit methods for the numerical integration of ordinary differential equations, focusing in particular on the linear stability of IMEX (Implicit-Explicit) methods and exponential integrators applied to systems of ordinary differential equations arising in the numerical solution of spatially discretized nonlinear partial differential equations containing both dispersive and dissipative linear terms. While these investigations may seem unrelated at first glance, some reflection shows that they are in fact closely linked. The investigation of rotating convection makes use of single-space, multiple-time-scale asymptotics to deal with dynamics strongly constrained by rotation. Although the context of thermal convection in an infinite layer seems somewhat removed from large-scale ocean dynamics, the asymptotic methods generalize directly to the second investigation which simply adds large spatial scales – the transition from convectively unstable to convectively stable dynamics does not change the mathematical framework. The rotating Navier-Stokes equations in the Boussinesq approximation and the equations derived from them asymptotically in the investigation of rotating convection include dispersive and dissipative linear terms that are stiff, i.e. that hinder numerical solution by explicit methods. A variety of methods which purport to alleviate this difficulty have been derived, and have been tested on and applied largely to problems with purely dissipative linear terms. But it was heretofore unfortunately quite difficult to judge and compare how effectively these methods achieve their goal when the stiff linear term is both dissipative and dispersive. The third investigation therefore introduces a visual, analytical method for comparing the linear stability properties of the various methods (the linear stability properties being a proxy for their ability to… Advisors/Committee Members: Keith Julien, Bengt Fornberg, Baylor Fox-Kemper.

Subjects/Keywords: Fluid Dynamics; Dynamic Systems

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APA (6^th Edition):

Grooms, I. (2011). Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/14

Chicago Manual of Style (16^th Edition):

Grooms, Ian. “Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow.” 2011. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/14.

MLA Handbook (7^th Edition):

Grooms, Ian. “Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow.” 2011. Web. 27 Feb 2021.

Vancouver:

Grooms I. Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow. [Internet] [Doctoral dissertation]. University of Colorado; 2011. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/14.

Council of Science Editors:

Grooms I. Asymptotic and Numerical Methods for Rapidly Rotating Buoyant Flow. [Doctoral Dissertation]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/14

University of Colorado

2. Nixon, Sean David. Development and Applications of Soliton Perturbation Theory.

Degree: PhD, Applied Mathematics, 2011, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/18

► This thesis examines the effects of small perturbation to soliton solutions of the nonlinear Schrödinger (NLS) equation on two fronts: the development of a… (more)

▼ This thesis examines the effects of small perturbation to soliton solutions of the nonlinear Schrödinger (NLS) equation on two fronts: the development of a direct perturbation method for dark solitons, and the application of perturbation theory to the study of nonlinear optical systems including the dynamics of ultra-short pulses in mode-locked lasers. For dark soliton solutions of the NLS equation a direct perturbation method for approximating the influence of perturbations is presented. The problem is broken into an inner region, where core of the soliton resides, and an outer region, which evolves independently of the soliton. It is shown that a shelf develops around the soliton which propagates with speed determined by the background intensity. Integral relations obtained from the conservation laws of the NLS equation are used to determine the properties of the shelf. The analysis is developed for both constant and slowly evolving backgrounds. A number of problems are investigated including linear and nonlinear dissipative perturbations. In the study of mode-locking lasers the power-energy saturation (PES) equation is a variant of the nonlinear NLS equation, which incorporates gain and filtering saturated with energy, and loss saturated with power (intensity). Solutions of the PES equation are studied using adiabatic perturbation theory. In the anomalous regime individual soliton pulses are found to be well approximated by soliton solutions of the unperturbed NLS equation with the key parameters of the soliton changing slowly as they evolve. Evolution equations are found for the pulses’ amplitude, velocity, position, and phase using integral relations derived from the PES equation. It is shown that the single soliton case exhibits mode-locking behavior for a wide range of parameters. The results from the integral relations are shown to agree with the secularity conditions found in multi-scale perturbation theory. In the normal regime both bright and dark pulses are found. Here the NLS equation does not have bright soliton solutions, and the mode-locked pulse are wide and strongly chirped. For dark pulses there are two interpretations of the PES equation. The existence and stability of mode-locked dark pulses are studied for both cases. Soliton strings are found in both the constant dispersion and dispersion-managed systems in the (net) anomalous and normal regimes. Analysis of soliton interactions show that soliton strings can form when pulses are a certain distance apart relative to their width. Anti-symmetric bi-soliton states are also obtained. Initial states mode-lock to these states under evolution. Advisors/Committee Members: Mark J. Ablowitz, Keith Julien, Gregory Beylkin.

Subjects/Keywords: Mode-locked lasers; Nonlinear Schrodinger equation; Nonlinear waves; Perturbation theory; Solitons; Applied Mathematics; Optics

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APA (6^th Edition):

Nixon, S. D. (2011). Development and Applications of Soliton Perturbation Theory. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/18

Chicago Manual of Style (16^th Edition):

Nixon, Sean David. “Development and Applications of Soliton Perturbation Theory.” 2011. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/18.

MLA Handbook (7^th Edition):

Nixon, Sean David. “Development and Applications of Soliton Perturbation Theory.” 2011. Web. 27 Feb 2021.

Vancouver:

Nixon SD. Development and Applications of Soliton Perturbation Theory. [Internet] [Doctoral dissertation]. University of Colorado; 2011. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/18.

Council of Science Editors:

Nixon SD. Development and Applications of Soliton Perturbation Theory. [Doctoral Dissertation]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/18

University of Colorado

3. Byrne, Erin. The Post-Fragmentation Probability Density for Bacterial Aggregates.

Degree: PhD, Applied Mathematics, 2011, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/19

► The post-fragmentation probability density of daughter flocs is one of the least well-understood aspects of modeling flocculation. This dissertation addresses the problem of determining… (more)

Subjects/Keywords: Applied Mathematics

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APA (6^th Edition):

Byrne, E. (2011). The Post-Fragmentation Probability Density for Bacterial Aggregates. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/19

Chicago Manual of Style (16^th Edition):

Byrne, Erin. “The Post-Fragmentation Probability Density for Bacterial Aggregates.” 2011. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/19.

MLA Handbook (7^th Edition):

Byrne, Erin. “The Post-Fragmentation Probability Density for Bacterial Aggregates.” 2011. Web. 27 Feb 2021.

Vancouver:

Byrne E. The Post-Fragmentation Probability Density for Bacterial Aggregates. [Internet] [Doctoral dissertation]. University of Colorado; 2011. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/19.

Council of Science Editors:

Byrne E. The Post-Fragmentation Probability Density for Bacterial Aggregates. [Doctoral Dissertation]. University of Colorado; 2011. Available from: https://scholar.colorado.edu/appm_gradetds/19

University of Colorado

4. Halko, Nathan P. Randomized Methods for Computing Low-Rank Approximations of Matrices.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/26

► Randomized sampling techniques have recently proved capable of efficiently solving many standard problems in linear algebra, and enabling computations at scales far larger than… (more)

▼ Randomized sampling techniques have recently proved capable of efficiently solving many standard problems in linear algebra, and enabling computations at scales far larger than what was previously possible. The new algorithms are designed from the bottom up to perform well in modern computing environments where the expense of communication is the primary constraint. In extreme cases, the algorithms can even be made to work in a streaming environment where the matrix is not stored at all, and each element can be seen only once. The dissertation describes a set of randomized techniques for rapidly constructing a low-rank ap- proximation to a matrix. The algorithms are presented in a modular framework that first computes an approximation to the range of the matrix via randomized sampling. Secondly, the matrix is pro- jected to the approximate range, and a factorization (SVD, QR, LU, etc.) of the resulting low-rank matrix is computed via variations of classical deterministic methods. Theoretical performance bounds are provided. Particular attention is given to very large scale computations where the matrix does not fit in RAM on a single workstation. Algorithms are developed for the case where the original matrix must be stored out-of-core but where the factors of the approximation fit in RAM. Numerical examples are provided that perform Principal Component Analysis of a data set that is so large that less than one hundredth of it can fit in the RAM of a standard laptop computer. Furthermore, the dissertation presents a parallelized randomized scheme for computing a reduced rank Singular Value Decomposition. By parallelizing and distributing both the randomized sampling stage and the processing of the factors in the approximate factorization, the method requires an amount of memory per node which is independent of both dimensions of the input matrix. Numerical experiments are performed on Hadoop clusters of computers in Amazon's Elastic Compute Cloud with up to 64 total cores. Finally, we directly compare the performance and accuracy of the randomized algorithm with the classical Lanczos method on extremely large, sparse matrices and substantiate the claim that randomized methods are superior in this environment. Advisors/Committee Members: Per-Gunnar Martinsson, Keith Julien, David M. Bortz.

Subjects/Keywords: hadoop; mahout; mapreduce; out of core; randomized sampling; singular value decomposition; Mathematics

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APA (6^th Edition):

Halko, N. P. (2012). Randomized Methods for Computing Low-Rank Approximations of Matrices. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/26

Chicago Manual of Style (16^th Edition):

Halko, Nathan P. “Randomized Methods for Computing Low-Rank Approximations of Matrices.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/26.

MLA Handbook (7^th Edition):

Halko, Nathan P. “Randomized Methods for Computing Low-Rank Approximations of Matrices.” 2012. Web. 27 Feb 2021.

Vancouver:

Halko NP. Randomized Methods for Computing Low-Rank Approximations of Matrices. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/26.

Council of Science Editors:

Halko NP. Randomized Methods for Computing Low-Rank Approximations of Matrices. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/26

University of Colorado

5. Larremore, Daniel Benjamin. Critical Dynamics in Complex Excitable Networks.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/28

► We study the effect of network structure on the dynamical response of networks of coupled discrete-state excitable elements to two distinct types of stimulus.… (more)

▼ We study the effect of network structure on the dynamical response of networks of coupled discrete-state excitable elements to two distinct types of stimulus. First, we consider networks which are stochastically stimulated by an external source. Such systems have been used as toy models for the dynamics of some human sensory neuronal networks and neuron cultures. The collective dynamics of such systems depends on the topology of the connections in the network. Here we develop a general theoretical approach to study the effects of network topology on dynamic range, which quantifies the range of stimulus intensities resulting in distinguishable network responses. We find that the largest eigenvalue of the weighted network adjacency matrix governs the network dynamic range. Specifically, a largest eigenvalue equal to one corresponds to a critical regime with maximum dynamic range. This result appears to hold for random, all-to-all, and scale free topologies, and is robust to the inclusion of time delays and refractory states. We gain deeper insight into the effects of network topology using a nonlinear analysis in terms of additional spectral properties of the adjacency matrix. We find that homogeneous networks can reach a higher dynamic range than those with heterogeneous topology. Second, we consider networks stimulated only once at a single node, with dynamics allowed to evolve without additional stimulus. Each realization of such a process will create a cascade of activity of varying duration and size. We analyze the distributions of cascade size and duration in complex networks resulting from a single nodal excitation, finding that when the largest eigenvalue is equal to one, so-called ``critical avalanches'' are power-law distributed in size, with exponent -3/2, and power-law distributed in duration, with exponent -2. We employ techniques from dynamical systems to recover the differences among avalanches started at different network nodes, also deriving distributions for avalanches in subcritical and supercritical regimes. Advisors/Committee Members: Juan G. Restrepo, James D. Meiss, Keith Julien.

Subjects/Keywords: avalanche; complex network; criticality; neural network; nonlinear dynamics; Applied Mathematics

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APA (6^th Edition):

Larremore, D. B. (2012). Critical Dynamics in Complex Excitable Networks. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/28

Chicago Manual of Style (16^th Edition):

Larremore, Daniel Benjamin. “Critical Dynamics in Complex Excitable Networks.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/28.

MLA Handbook (7^th Edition):

Larremore, Daniel Benjamin. “Critical Dynamics in Complex Excitable Networks.” 2012. Web. 27 Feb 2021.

Vancouver:

Larremore DB. Critical Dynamics in Complex Excitable Networks. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/28.

Council of Science Editors:

Larremore DB. Critical Dynamics in Complex Excitable Networks. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/28

University of Colorado

6. Chen, Yan. Asymptotic Series Solutions to One-Dimensional Helmholtz Equation.

Degree: MS, Applied Mathematics, 2014, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/54

► We present a variation of the method of Berry & Howls, which eliminates some of the inherent error in their approach. Using Dingle's change… (more)

Subjects/Keywords: Airy function; Asymptotics; Hyperasymptotics; Modeling; Oscillation; Series; Applied Mathematics

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APA (6^th Edition):

Chen, Y. (2014). Asymptotic Series Solutions to One-Dimensional Helmholtz Equation. (Masters Thesis). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/54

Chicago Manual of Style (16^th Edition):

Chen, Yan. “Asymptotic Series Solutions to One-Dimensional Helmholtz Equation.” 2014. Masters Thesis, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/54.

MLA Handbook (7^th Edition):

Chen, Yan. “Asymptotic Series Solutions to One-Dimensional Helmholtz Equation.” 2014. Web. 27 Feb 2021.

Vancouver:

Chen Y. Asymptotic Series Solutions to One-Dimensional Helmholtz Equation. [Internet] [Masters thesis]. University of Colorado; 2014. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/54.

Council of Science Editors:

Chen Y. Asymptotic Series Solutions to One-Dimensional Helmholtz Equation. [Masters Thesis]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/54

University of Colorado

7. Stotsky, Jay Alexander. Mathematical and Computational Studies of the Biomechanics of Biofilms.

Degree: PhD, 2018, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/107

► Bacterial biofilms are communities of bacteria growing on a surface to which they have adhered, typically in an aqueous environment. The motivation to understand biofilm… (more)

Subjects/Keywords: a posteriori error analysis; biofilms; immersed boundary method; mathematical biology; point processes; Applied Mathematics; Biomechanics

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APA (6^th Edition):

Stotsky, J. A. (2018). Mathematical and Computational Studies of the Biomechanics of Biofilms. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/107

Chicago Manual of Style (16^th Edition):

Stotsky, Jay Alexander. “Mathematical and Computational Studies of the Biomechanics of Biofilms.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/107.

MLA Handbook (7^th Edition):

Stotsky, Jay Alexander. “Mathematical and Computational Studies of the Biomechanics of Biofilms.” 2018. Web. 27 Feb 2021.

Vancouver:

Stotsky JA. Mathematical and Computational Studies of the Biomechanics of Biofilms. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/107.

Council of Science Editors:

Stotsky JA. Mathematical and Computational Studies of the Biomechanics of Biofilms. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/107

University of Colorado

8. Maiden, Michelle. Dispersive hydrodynamics in viscous fluid conduits.

Degree: PhD, Applied Mathematics, 2019, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/141

► Viscous fluid conduits provide an ideal system for the study of dissipationless, dispersive hydrodynamics. A dense, viscous fluid serves as the background medium through… (more)

Subjects/Keywords: Fluid Dynamics; Partial Differential Equations

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APA (6^th Edition):

Maiden, M. (2019). Dispersive hydrodynamics in viscous fluid conduits. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/141

Chicago Manual of Style (16^th Edition):

Maiden, Michelle. “Dispersive hydrodynamics in viscous fluid conduits.” 2019. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/141.

MLA Handbook (7^th Edition):

Maiden, Michelle. “Dispersive hydrodynamics in viscous fluid conduits.” 2019. Web. 27 Feb 2021.

Vancouver:

Maiden M. Dispersive hydrodynamics in viscous fluid conduits. [Internet] [Doctoral dissertation]. University of Colorado; 2019. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/141.

Council of Science Editors:

Maiden M. Dispersive hydrodynamics in viscous fluid conduits. [Doctoral Dissertation]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/141

University of Colorado

9. Nieves, David Joseph. Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows.

Degree: PhD, Applied Mathematics, 2016, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/70

► This thesis is a collection of studies concerning an asymptotically reduced equation set derived from the Boussinesq approximation describing rotationally constrained geophysical flow. The… (more)

▼ This thesis is a collection of studies concerning an asymptotically reduced equation set derived from the Boussinesq approximation describing rotationally constrained geophysical flow. The first investigation is concerned with a statistical identification of coherent and long-lived structures in rotationally constrained Rayleigh-Bénard convection. Presently, physical laboratory limitations challenge experimentalists while spatio-temporal resolution requirements challenges numericists performing direct numerical simulations of the Boussinesq equations. These challenges prevent an exhaustive analysis of the flow morphology in the rapid rotating limit. In this study the flow morphologies obtained from simulations of the reduced equations are investigated from a statistical perspective. Auto- and cross-correlations are computed from temporal and spatial signals that synthesize experimental data that may be obtained in laboratory experiments via thermistor measurements or particle image velocimetry. The statistics used can be employed in laboratory experiments to identify regime transitions in flow morphology, capture radial profiles of coherent structures, and extract transport properties belonging to these structures. These results provide a foundation for comparison and a measure for understanding the extent to which rotationally constrained regime has been accessed by laboratory experiments and direct numerical simulations. A related study comparing the influence of fixed temperature and fixed heat flux thermal boundary conditions on rapidly rotating convection in the plane layer geometry is also investigated and briefly summarized for the case of stress-free mechanical boundary conditions. It is shown that the difference between these thermal boundary conditions on the interior geostrophically balanced convection is asymptotically weak. Through a simple rescaling of thermal variables, the leading order reduced system is shown to be equivalent for both thermal boundary conditions. These results imply that any horizontal thermal variation along the boundaries that varies on the scale of the convection has no leading order influence on the interior convection, thus providing insight into geophysical and astrophysical flows where stress-free mechanical boundary conditions are often assumed. The final study presented here contrasts the previous investigations. It presents an investigation of rapidly rotating and stably stratified turbulence where the stratification strength is varied from weak (large Froude number) to strong (small Froude number). The investigation is set in the context of the asymptotically reduced model which efficiently retains anisotropic inertia-gravity waves with order-one frequencies and highlights a regime of wave-eddy interactions. Numerical simulations of the reduced model are performed where energy is injected by a stochastic forcing of vertical velocity. The simulations reveal two regimes: one characterized by the presence of well-formed, persistent and thin turbulent… Advisors/Committee Members: Keith Julien, Ian Grooms, Bengt Fornberg, Mark Hoefer, Jeffrey Weiss.

Subjects/Keywords: geophysical flows; geostrophic turbulence; Quasi-geostrophic flows; rotating flows; stratified flows; Atmospheric Sciences; Geophysics and Seismology

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APA (6^th Edition):

Nieves, D. J. (2016). Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/70

Chicago Manual of Style (16^th Edition):

Nieves, David Joseph. “Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows.” 2016. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/70.

MLA Handbook (7^th Edition):

Nieves, David Joseph. “Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows.” 2016. Web. 27 Feb 2021.

Vancouver:

Nieves DJ. Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/70.

Council of Science Editors:

Nieves DJ. Investigations of Reduced Equations for Rotating, Stratified and Non-hydrostatic Flows. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/appm_gradetds/70

University of Colorado

10. Biagioni, David Joseph. Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/29

► This thesis consists of two parts. In Part I, we describe an algorithm for approximating the Green's function for elliptic problems with variable coefficients… (more)

▼ This thesis consists of two parts. In Part I, we describe an algorithm for approximating the Green's function for elliptic problems with variable coefficients in arbitrary dimension. The basis for our approach is the separated representation, which appears as a way of approximating functions of many variables by sums of products of univariate functions. While the differential operator we wish to invert is typically ill-conditioned, its conditioning may be improved by first applying the Green's function for the constant coefficient problem. This function may be computed either numerically or, in some case, analytically in a separated format. The variable coefficient Green's function is then computed using a quadratically convergent iteration on the preconditioned operator, with sparsity maintained via representation in a wavelet basis. Of particular interest is that the method scales linearly in the number of dimensions, a feature that very desirable in high dimensional problems in which the curse of dimensionality must be reckoned with. As a corollary to this work, we described a randomized algorithm for maintaining low separation rank of the functions used in the construction of the Green's function. For certain functions of practical interest, one can avoid the cost of using standard methods such as alternating least squares (ALS) to reduce the separation rank. Instead, terms from the separated representation may be selected using a randomized approach based on matrix skeletonization and the interpolative decomposition. The use of random projections can greatly reduce the cost of rank reduction, as well as calculation of the Frobenius norm and term-wise Gram matrices. In Part II of the thesis, we highlight three practical applications of sparse and separable approximations to the analysis of renewable energy data. In the first application, error estimates gleaned from repeated measurements are incorporated into sparse regression algorithms (LASSO and the Dantzig selector) to minimize the statistical uncertainty of the resulting model. Applied to real biomass data, this approach leads to sparser regression coefficients corresponding to improved accuracy as measured by k-fold cross validation error. In the second application, a regression model based on separated representations is fit to reliability data for cadmium telluride (CdTe) thin-film solar cells. The data is inherently multi-way, and our approach avoids artificial matricization that would typically be performed for use with standard regression algorithms. Two distinct modes of degradation, corresponding to short- and long-term decrease in cell efficiency, are identified. In the third application, some theoretical properties of a popular chemometrics algorithm called orthogonal projections to latent structures (O-PLS) are derived. Advisors/Committee Members: Gregory Beylkin, Alireza Doostan, Peter Graf, Gunnar Martinsson, Keith Julien.

Subjects/Keywords: Curse of dimensionality; Direct Poisson solver; High dimensional partial differential equations; Numerical analysis; Randomized canonical tensor decomposition; Separated representations; Applied Mathematics

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APA (6^th Edition):

Biagioni, D. J. (2012). Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/29

Chicago Manual of Style (16^th Edition):

Biagioni, David Joseph. “Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/29.

MLA Handbook (7^th Edition):

Biagioni, David Joseph. “Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations.” 2012. Web. 27 Feb 2021.

Vancouver:

Biagioni DJ. Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/29.

Council of Science Editors:

Biagioni DJ. Numerical construction of Green’s functions in high dimensional elliptic problems with variable coefficients and analysis of renewable energy data via sparse and separable approximations. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/29

University of Colorado

11. Hammond, Jason Frank. Analysis and Simulation of Partial Differential Equations in Mathematical Biology: Applications to Bacterial Biofilms and Fisher's Equation.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/31

► In this dissertation, we investigate two important problems in mathematical biology that are best modeled using partial differential equations. We first consider the question… (more)

Subjects/Keywords: biofilm; fisher's equation; fluid mechanics; immersed boundary method; painleve; partial differential equations; Applied Mathematics; Biomechanics and Biotransport

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APA (6^th Edition):

Hammond, J. F. (2012). Analysis and Simulation of Partial Differential Equations in Mathematical Biology: Applications to Bacterial Biofilms and Fisher's Equation. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/31

Chicago Manual of Style (16^th Edition):

Hammond, Jason Frank. “Analysis and Simulation of Partial Differential Equations in Mathematical Biology: Applications to Bacterial Biofilms and Fisher's Equation.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/31.

MLA Handbook (7^th Edition):

Hammond, Jason Frank. “Analysis and Simulation of Partial Differential Equations in Mathematical Biology: Applications to Bacterial Biofilms and Fisher's Equation.” 2012. Web. 27 Feb 2021.

Vancouver:

Hammond JF. Analysis and Simulation of Partial Differential Equations in Mathematical Biology: Applications to Bacterial Biofilms and Fisher's Equation. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/31.

Council of Science Editors:

Hammond JF. Analysis and Simulation of Partial Differential Equations in Mathematical Biology: Applications to Bacterial Biofilms and Fisher's Equation. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/31

University of Colorado

12. Bachman, Scott Daniel. A Diagnostic Suite of Models for the Evaluation of Oceanic Mesoscale Eddy Parameterizations.

Degree: PhD, Atmospheric & Oceanic Sciences, 2012, University of Colorado

URL: https://scholar.colorado.edu/atoc_gradetds/29

► The practice of modeling geophysical fluid flows has grown tremendously in concert with recent advances in computing power. To study the climate models must… (more)

▼ The practice of modeling geophysical fluid flows has grown tremendously in concert with recent advances in computing power. To study the climate models must simulate centuries of real time, a difficulty made worse by the need to capture fine-scale (eddy) activity. Turbulence at scales ranging from 10 km to 250 km, whose coherent structures are colloquially referred to as mesoscale eddies, is of particular interest because of its ability to transport and mix water masses, and because it dominates the oceanic kinetic energy budget. As of the writing of this dissertation, it also happens to represent the cutting edge in OGCM resolution, hence the need for skillful parameterizations. Calibration and evaluation of such parameterizations is the focus of this work. An "eddy parameterization challenge suite" is being developed, consisting of a set of high-resolution tracer experiments designed to assist in parameterizing subgridscale processes in ocean models. In each experiment, multiple tracers are initialized in a frontal spindown simulation designed to mimic the stirring effect of mesoscale eddies. Diagnosis of an eddy transport tensor is performed by inverting a matrix of passive tracer gradients, each of which is assumed to satisfy an identical linear flux-gradient relationship. Aspects of the matrix inversion are explored, including the implications of overdetermining the linear relationship using a large number of tracers. Two sets of simulations, featuring Eady-like and exponential stratification, allow us to investigate scaling laws and vertical structures of the eddy transport tensor. The diagnosed tensor reproduces the horizontal transport of an active tracer (buoyancy) to within ±7% and the vertical transport to within ±12%. The derived scalings are shown to be close in form to the Gent and McWilliams (1990) and Redi (1982) diffusivity tensors, with a magnitude that varies in space and time. The parameterization suite is also used to evaluate an extant scheme (Ferrari et al., 2010) and to recommend improvements. We also attempt a local scaling for the along-isopycnal diffusivity and argue that it is unlikely that any such scaling can be written as a simple function of the velocity, stratification, or eddy variances. Advisors/Committee Members: Baylor Fox-Kemper, Frank Bryan, Jeffrey Weiss, Keith Julien, Nicole Lovenduski.

Subjects/Keywords: eddy parameterization; transport tensor; oceanic kinetics; Oceanography

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APA (6^th Edition):

Bachman, S. D. (2012). A Diagnostic Suite of Models for the Evaluation of Oceanic Mesoscale Eddy Parameterizations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/atoc_gradetds/29

Chicago Manual of Style (16^th Edition):

Bachman, Scott Daniel. “A Diagnostic Suite of Models for the Evaluation of Oceanic Mesoscale Eddy Parameterizations.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/atoc_gradetds/29.

MLA Handbook (7^th Edition):

Bachman, Scott Daniel. “A Diagnostic Suite of Models for the Evaluation of Oceanic Mesoscale Eddy Parameterizations.” 2012. Web. 27 Feb 2021.

Vancouver:

Bachman SD. A Diagnostic Suite of Models for the Evaluation of Oceanic Mesoscale Eddy Parameterizations. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/atoc_gradetds/29.

Council of Science Editors:

Bachman SD. A Diagnostic Suite of Models for the Evaluation of Oceanic Mesoscale Eddy Parameterizations. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/atoc_gradetds/29

University of Colorado

13. Villavert, John. The Analysis of Some Nonlinear Partial Differential Equations.

Degree: PhD, Applied Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/33

► The underlying theme of this dissertation centers on the development of novel mathematical tools used in the analysis of some important types of nonlinear… (more)

▼ The underlying theme of this dissertation centers on the development of novel mathematical tools used in the analysis of some important types of nonlinear partial differential equations. Namely, this dissertation examines semilinear elliptic systems and hyperbolic conservation laws and inviscid techniques for their regularization. The existence theory for a class of global, semilinear elliptic systems, including Hardy-Littlewood-Sobolev and stationary Schrödinger type systems, is developed by combining the shooting method with topological degree theory. First, a `target' map is defined which aims the shooting method, then non-degeneracy conditions are established which guarantee the continuity of the map. The continuity of the target map allows for the application of tools from degree theory to show the map is onto. Then the existence result follows from this surjectivity property of the target map along with a non-existence result for the corresponding Navier boundary value problem. Next, an extension of the Hardy-Littlewood-Pólya inequality, which may be regarded as the discrete analogue of the Hardy-Littlewood-Sobolev inequality, is established along with an accurate estimate of the best constant for this inequality. The other class of problems examines a shock-regularization method for hyperbolic conservation laws that applies a spatial averaging of the nonlinear terms in the partial differential equations. A central motivation is to promote the idea of applying a recently developed filtering technique, rather than viscous regularization, as an alternative to the simulation of shocks and turbulence for inviscid flows. On the other hand, the study presented here also generalizes and unifies past mathematical and numerical analysis of the method applied to the one-dimensional Burgers' and Euler equations. This examination primarily concerns the analysis of this technique and addresses two fundamental issues. The first is the global existence and uniqueness of classical solutions for the regularization technique under the more general setting of quasilinear, symmetric hyperbolic systems in higher dimensions. The second issue examines scalar conservation laws and describes the sufficient conditions that guarantee this inviscid regularization technique captures the unique entropy or physically relevant solution of the original, non-averaged problem as filtering vanishes. Advisors/Committee Members: Congming Li, Kamran Mohseni, Keith Julien, Stephen Preston, Harvey Segur.

Subjects/Keywords: inviscid regularization; shooting method; spatial averaging; Topological degree theory; Applied Mathematics; Mathematics; Other Mathematics

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APA (6^th Edition):

Villavert, J. (2013). The Analysis of Some Nonlinear Partial Differential Equations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/33

Chicago Manual of Style (16^th Edition):

Villavert, John. “The Analysis of Some Nonlinear Partial Differential Equations.” 2013. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/33.

MLA Handbook (7^th Edition):

Villavert, John. “The Analysis of Some Nonlinear Partial Differential Equations.” 2013. Web. 27 Feb 2021.

Vancouver:

Villavert J. The Analysis of Some Nonlinear Partial Differential Equations. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/33.

Council of Science Editors:

Villavert J. The Analysis of Some Nonlinear Partial Differential Equations. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/33

University of Colorado

14. Baldwin, Douglas Eugene. Dispersive shock wave interactions and two-dimensional ocean-wave soliton interactions.

Degree: PhD, Applied Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/35

► Many physical phenomena are understood and modeled with nonlinear partial differential equations (PDEs). Unfortunately, nonlinear PDEs rarely have analytic solutions. But perturbation theory can… (more)

▼ Many physical phenomena are understood and modeled with nonlinear partial differential equations (PDEs). Unfortunately, nonlinear PDEs rarely have analytic solutions. But perturbation theory can lead to PDEs that asymptotically approximate the phenomena and have analytic solutions. A special subclass of these nonlinear PDEs have stable localized waves – called solitons – with important applications in engineering and physics. This dissertation looks at two such applications: dispersive shock waves and shallow ocean-wave soliton interactions. Dispersive shock waves (DSWs) are physically important phenomena that occur in systems dominated by weak dispersion and weak nonlinearity. The Korteweg-de Vries (KdV) equation is the universal model for phenomena with weak dispersion and weak quadratic nonlinearity. Here we show that the long-time asymptotic solution of the KdV equation for general step-like data is a single-phase DSW; this DSW is the `largest' possible DSW based on the boundary data. We find this asymptotic solution using the inverse scattering transform (IST) and matched-asymptotic expansions; we also compare it with a numerically computed solution. While multi-step data evolve to have multiphase dynamics at intermediate times, these interacting DSWs eventually merge to form a single-phase DSW at large time. We then use IST and matched-asymptotic expansions to find the modified KdV equation's long-time-asymptotic DSW solutions. Ocean waves are complex and often turbulent. While most ocean-wave interactions are essentially linear, sometimes two or more waves interact in a nonlinear way. For example, two or more waves can interact and yield waves that are much taller than the sum of the original wave heights. Most of these nonlinear interactions look like an X or a Y or two connected Ys; much less frequently, several lines appear on each side of the interaction region. It was thought that such nonlinear interactions are rare events: they are not. This dissertation reports that such interactions occur every day, close to low tide, on two flat beaches that are about 2,000 km apart. These interactions are related to the analytic, soliton solutions of the Kadomtsev-Petviashvili equation. On a much larger scale, tsunami waves can merge in similar ways. Advisors/Committee Members: Mark J. Ablowitz, Keith Julien, Bengt Fornberg, Willy Hereman, Patrick Weidman.

Subjects/Keywords: Applied Mathematics

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APA (6^th Edition):

Baldwin, D. E. (2013). Dispersive shock wave interactions and two-dimensional ocean-wave soliton interactions. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/35

Chicago Manual of Style (16^th Edition):

Baldwin, Douglas Eugene. “Dispersive shock wave interactions and two-dimensional ocean-wave soliton interactions.” 2013. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/35.

MLA Handbook (7^th Edition):

Baldwin, Douglas Eugene. “Dispersive shock wave interactions and two-dimensional ocean-wave soliton interactions.” 2013. Web. 27 Feb 2021.

Vancouver:

Baldwin DE. Dispersive shock wave interactions and two-dimensional ocean-wave soliton interactions. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/35.

Council of Science Editors:

Baldwin DE. Dispersive shock wave interactions and two-dimensional ocean-wave soliton interactions. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/35

University of Colorado

15. Fox, Adam Merritt. Destruction of Invariant Tori in Volume-Preserving Maps.

Degree: PhD, Applied Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/36

► Invariant rotational tori play an important role in the dynamics of volume-preserving maps. When integrable, all orbits lie on these tori and KAM theory… (more)

Subjects/Keywords: KAM theory; Greene's residue criterion; near-critical conjugacies; Applied Mathematics

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APA (6^th Edition):

Fox, A. M. (2013). Destruction of Invariant Tori in Volume-Preserving Maps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/36

Chicago Manual of Style (16^th Edition):

Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/36.

MLA Handbook (7^th Edition):

Fox, Adam Merritt. “Destruction of Invariant Tori in Volume-Preserving Maps.” 2013. Web. 27 Feb 2021.

Vancouver:

Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/36.

Council of Science Editors:

Fox AM. Destruction of Invariant Tori in Volume-Preserving Maps. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/36

University of Colorado

16. Reynolds, Matthew Jason. Nonlinear approximations in tomography, quadrature construction, and multivariate reductions.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/37

► This thesis consists of contributions to three topics: algorithms for computing generalized Gaussian quadratures, tomographic imaging algorithms, and reduction algorithms. Our approach is based… (more)

▼ This thesis consists of contributions to three topics: algorithms for computing generalized Gaussian quadratures, tomographic imaging algorithms, and reduction algorithms. Our approach is based on using non-linear approximations of functions. We develop a new algorithm for constructing generalized Gaussian quadratures for exponentials inte- grated against a non-sign-definite weight function. These quadratures integrate band-limited exponentials to a user-defined accuracy. We also introduce a method of computing quadrature weights via l∞ minimization. Second, we develop a new imaging algorithm for X-ray tomography. This algorithm, Polar Quadrature Inversion, uses rational approximations to approximate tomographic projections with a near optimal number of terms for a given accuracy. This rational signal model allows us to augment the measured data by extending the tomographic projection's domain in Fourier space. As the extended data from all the projections fill a disk in the Fourier domain, we use polar quadratures for band-limited exponentials and the Unequally Spaced Fast Fourier Transform to obtain our image. We demonstrate that the resulting images have significantly improved resolution without additional artifacts near sharp transitions. Finally, we develop an extension of existing reduction algorithms for functions of one variable to functions of many variables. By reduction, we understand an approximation (to a user-supplied accuracy) of a linear combination of decaying exponentials by a representation of the same form but with a minimal number of terms. While for functions of one variable there is an underlying theory based on the analysis of functions of one complex variable, no such theory is available for the multivariate case. Our approach is a first step in the development of such theory. We demonstrate our algorithm on two examples of multivariate functions, a suboptimal linear combination of real-valued, decaying exponentials, and that of complex-valued, decaying exponentials. Advisors/Committee Members: Gregory Beylkin, Gunnar Martinsson, Keith Julien, Francois Meyer, Rafael Peistun.

Subjects/Keywords: Applied Mathematics

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APA (6^th Edition):

Reynolds, M. J. (2012). Nonlinear approximations in tomography, quadrature construction, and multivariate reductions. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/37

Chicago Manual of Style (16^th Edition):

Reynolds, Matthew Jason. “Nonlinear approximations in tomography, quadrature construction, and multivariate reductions.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/37.

MLA Handbook (7^th Edition):

Reynolds, Matthew Jason. “Nonlinear approximations in tomography, quadrature construction, and multivariate reductions.” 2012. Web. 27 Feb 2021.

Vancouver:

Reynolds MJ. Nonlinear approximations in tomography, quadrature construction, and multivariate reductions. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/37.

Council of Science Editors:

Reynolds MJ. Nonlinear approximations in tomography, quadrature construction, and multivariate reductions. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/37

University of Colorado

17. Rasca, Anthony P. Modeling Solar Wind Mass-Loading Due to Dust in the Solar Corona.

Degree: PhD, Applied Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/44

► Collisionless mass-loading was first discussed to describe interactions between the solar wind and cometary atmospheres. Recent observations have led to an increased interest in… (more)

Subjects/Keywords: Dust; Fluid Modeling; Mass-Loading; Shocks; Solar Wind; Sungrazing Comets; Applied Mathematics; Fluid Dynamics; Instrumentation

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APA (6^th Edition):

Rasca, A. P. (2013). Modeling Solar Wind Mass-Loading Due to Dust in the Solar Corona. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/44

Chicago Manual of Style (16^th Edition):

Rasca, Anthony P. “Modeling Solar Wind Mass-Loading Due to Dust in the Solar Corona.” 2013. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/44.

MLA Handbook (7^th Edition):

Rasca, Anthony P. “Modeling Solar Wind Mass-Loading Due to Dust in the Solar Corona.” 2013. Web. 27 Feb 2021.

Vancouver:

Rasca AP. Modeling Solar Wind Mass-Loading Due to Dust in the Solar Corona. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/44.

Council of Science Editors:

Rasca AP. Modeling Solar Wind Mass-Loading Due to Dust in the Solar Corona. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/44

University of Colorado

18. Nelson, Nicholas James. Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos.

Degree: PhD, Astrophysical & Planetary Sciences, 2013, University of Colorado

URL: https://scholar.colorado.edu/astr_gradetds/28

► Solar-like stars exhibit a rich variety of magnetic activity, which is driven by dynamo action in the stellar interior. In the Sun, strong dynamo… (more)

▼ Solar-like stars exhibit a rich variety of magnetic activity, which is driven by dynamo action in the stellar interior. In the Sun, strong dynamo action creates global-scale magnetic fields which undergo cyclic reversals as well as smaller-scale dipolar active regions which have global-scale organization. Dynamo action is a highly nonlinear process which is enabled by the interplay of turbulent convection, rotation, and stratification. Seeking to explore the convective origins of magnetism in sun-like stars, we have used 3D MHD simulations with the Anelastic Spherical Harmonic (ASH) code to model elements of these dynamos. Previous simulations have demonstrated that large-scale "wreaths" of toroidal magnetic field can be achieved in the convection zone without a tachocline of shear at its base, as was thought to be necessary, and that these wreaths can yield reversals in global magnetic polarity. We find that cyclic reversals of global magnetic polarity in wreath-building dynamos can be achieved by increasing the level of turbulence in solar-like simulations. By decreasing the effective diffusion we demonstrate that large-scale magnetic wreaths can persist in simulations where explicit diffusion has been decreased to levels at which it no longer plays a significant role in the key dynamical balances required to achieve wreath-building dynamo action. Magnetic reversals are attained when resistive diffusion of the poloidal magnetic fields becomes too small to prevent turbulent magnetic induction from generating opposite polarity poloidal fields. In order to attain even less diffusive simulations, we explore more a dynamic Smagorinsky model. Using the dynamic Smagorinsky model, we achieve a dynamo simulation capable of building buoyant magnetic loops which rise coherently through our simulated domain. These loops ascend via a combination of magnetic buoyancy and advection by convective giant cells. These buoyant loops originate within sections of the magnetic wreaths in which turbulent flows amplify the fields to much higher values than is possible through laminar processes. We measure statistical trends in the polarity, twist, and tilt of these loops. Loops are shown to preferentially arise in longitudinal patches somewhat reminiscent of active longitudes in the Sun, although broader in extent. We show that the strength of the axisymmetric toroidal field is not a good predictor of the production rate for buoyant loops or the amount of magnetic flux in the loops that are produced. Finally, we explore the effects of a new upper boundary condition on ASH simulations. Previous simulations have employed an impenetrable upper boundary condition, which imposed an unphysical viscous boundary layer in the upper layers of the convection zone. We have implemented and tested an alternative boundary condition which imposes small-scale convective plumes on the upper boundary, mimicking the small-scale convective motions from the near-surface layers. We find that for suitable choices of plume parameters we can… Advisors/Committee Members: Juri Toomre, John Bally, Keith Julien, Mark Miesch, Mark Rast.

Subjects/Keywords: computational physics; Convection; magnetohydrodynamics; solar physics; turbulence; Astrophysics and Astronomy; Physical Processes; Plasma and Beam Physics

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APA (6^th Edition):

Nelson, N. J. (2013). Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/astr_gradetds/28

Chicago Manual of Style (16^th Edition):

Nelson, Nicholas James. “Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos.” 2013. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/astr_gradetds/28.

MLA Handbook (7^th Edition):

Nelson, Nicholas James. “Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos.” 2013. Web. 27 Feb 2021.

Vancouver:

Nelson NJ. Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/astr_gradetds/28.

Council of Science Editors:

Nelson NJ. Magnetic Wreaths, Cycles, and Buoyant Loops in Convective Dynamos. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/astr_gradetds/28

University of Colorado

19. Sen, Amrik. A Tale of Waves and Eddies in a Sea of Rotating Turbulence.

Degree: PhD, Applied Mathematics, 2014, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/46

► In this thesis, we investigate several properties of rotating turbulent flows. First, we ran several computer simulations of rotating turbulent flows and performed statistical… (more)

Subjects/Keywords: Hamiltonian Dynamics; Nonlinear waves; perturbation Theory; Rotating Turbulence; Symmetries and reduction; Applied Mathematics; Geophysics and Seismology; Physics

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APA (6^th Edition):

Sen, A. (2014). A Tale of Waves and Eddies in a Sea of Rotating Turbulence. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/46

Chicago Manual of Style (16^th Edition):

Sen, Amrik. “A Tale of Waves and Eddies in a Sea of Rotating Turbulence.” 2014. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/46.

MLA Handbook (7^th Edition):

Sen, Amrik. “A Tale of Waves and Eddies in a Sea of Rotating Turbulence.” 2014. Web. 27 Feb 2021.

Vancouver:

Sen A. A Tale of Waves and Eddies in a Sea of Rotating Turbulence. [Internet] [Doctoral dissertation]. University of Colorado; 2014. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/46.

Council of Science Editors:

Sen A. A Tale of Waves and Eddies in a Sea of Rotating Turbulence. [Doctoral Dissertation]. University of Colorado; 2014. Available from: https://scholar.colorado.edu/appm_gradetds/46

University of Colorado

20. Webb, Adrean Andrew. Stokes Drift and Meshless Wave Modeling.

Degree: PhD, Applied Mathematics, 2013, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/48

► This dissertation is loosely organized around efforts to improve vertical ocean mixing in global climate models and includes an in-depth analysis of Stokes drift,… (more)

▼ This dissertation is loosely organized around efforts to improve vertical ocean mixing in global climate models and includes an in-depth analysis of Stokes drift, optimization of a new global climate model wave component, and development of a meshless spectral wave model. Stokes drift (hereafter SD) is an important vector component that appears often in wave-averaged dynamics. Mathematically, SD is the mean difference between Eulerian and Lagrangian velocities and intuitively can be thought of as the near-surface ocean current induced from wave motion. Increasingly, spectral wave models are being used to calculate SD globally. These models solve a 5D wave action balance equation and typically require large computational resources to make short to medium-range forecasts of the sea state. In the first part, a hierarchy of SD approximations are investigated and new approximations that remove systematic biases are derived. A new 1D spectral approximation is used to study the effects of multidirectional waves and directional wave spreading on SD. It is shown that these effects are largely uncorrelated and affect both the magnitude and direction of SD in a nonlinear fashion that is sensitive with depth. In the second part, efforts to add a wave model component to the NCAR Community Earth System Model are discussed. This coupled component will serve as the backbone to a new Langmuir mixing parameterization and uses a modified version of NOAA WAVEWATCH III (a third-generation spectral wave model). In addition, the governing wave action balance equation is reviewed and several variations are derived and formulated. In the third part, construction of a monochromatic spectral wave model using RBF-generated finite differences is described. Several numerical test cases are conducted to measure performance and guide further development. In kinematic comparisons with WAVEWATCH III, the meshless prototype is approximately 70–210 times more accurate and uses a factor of 12 to 17 less unknowns. Advisors/Committee Members: Baylor Fox-Kemper, Keith Julien, Mark Ablowitz, Natasha Flyer, Bengt Fornberg.

Subjects/Keywords: Langmuir mixing; meshless method; rbf-generated finite differences; spectral wave model; Stokes drift; unstructured nodes; Applied Mathematics; Oceanography

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APA (6^th Edition):

Webb, A. A. (2013). Stokes Drift and Meshless Wave Modeling. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/48

Chicago Manual of Style (16^th Edition):

Webb, Adrean Andrew. “Stokes Drift and Meshless Wave Modeling.” 2013. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/48.

MLA Handbook (7^th Edition):

Webb, Adrean Andrew. “Stokes Drift and Meshless Wave Modeling.” 2013. Web. 27 Feb 2021.

Vancouver:

Webb AA. Stokes Drift and Meshless Wave Modeling. [Internet] [Doctoral dissertation]. University of Colorado; 2013. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/48.

Council of Science Editors:

Webb AA. Stokes Drift and Meshless Wave Modeling. [Doctoral Dissertation]. University of Colorado; 2013. Available from: https://scholar.colorado.edu/appm_gradetds/48

University of Colorado

21. Greer, Benjamin J. Exploring the Dynamics of Near-Surface Solar Convection with Helioseismology.

Degree: PhD, Astrophysical & Planetary Sciences, 2015, University of Colorado

URL: https://scholar.colorado.edu/astr_gradetds/38

► I present a new implementation of local helioseismology along with observations of near-surface solar convection made with this method. The upper 5% of the… (more)

▼ I present a new implementation of local helioseismology along with observations of near-surface solar convection made with this method. The upper 5% of the solar radius (35 Mm) is known as the Near-Surface Shear Layer (NSSL) and is characterized by strong rotational shear. While the physical origin of this layer remains unknown, current theories point to convective motions playing an important role. In this thesis I investigate the properties of convection in the NSSL using a newly-developed high-resolution ring-diagram analysis. I present measurements of the speeds and spatial scales of near-surface flows and from these infer that the degree of rotational constraint on convective flows varies significantly across this layer. In-depth analysis of the convective patterns reveals the pervasive influence of coherent downflow plumes generated at the photosphere. These structures link the convective pattern of supergranulation seen in surface observations with the deeper motions found within the NSSL and further hint at the importance of rotation in this layer. These observations of transient, small-scale convective motions are enabled by the use of improved local helioseismic techniques. Local helioseismology relies on observations of the solar wavefield to produce measurements of plasma flows beneath the surface. In general, this has the capability to map out the subsurface convective flows in three-dimensions, but is often limited in accuracy, resolution, and depth range by the specifics of the analysis procedure. Here, I focus on a particular implementation of local helioseismology called ring-diagram analysis that involves analyzing small patches of the solar surface to build up three-dimensional maps. I will present a new analysis scheme for ring-diagram helioseismology that produces maps of the subsurface flow fields with higher fidelity and vastly higher resolution than previously possible. This is achieved through a combination of novel tools including a robust nonlinear fitting procedure and a highly efficient linear inversion technique. I present these new methods and demonstrate how they enable a new class of high-resolution helioseismic observations. The scientific results made possible with these methods display the power of the new techniques and aid our understanding of near-surface solar dynamics. Advisors/Committee Members: Bradley Hindman, Juri Toomre, Mark Rast, Benjamin Brown, Keith Julien.

Subjects/Keywords: Helioseismology; Solar Physics; near-surface shear layer; convective motion; solar dynamics; Physical Processes; The Sun and the Solar System

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APA (6^th Edition):

Greer, B. J. (2015). Exploring the Dynamics of Near-Surface Solar Convection with Helioseismology. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/astr_gradetds/38

Chicago Manual of Style (16^th Edition):

Greer, Benjamin J. “Exploring the Dynamics of Near-Surface Solar Convection with Helioseismology.” 2015. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/astr_gradetds/38.

MLA Handbook (7^th Edition):

Greer, Benjamin J. “Exploring the Dynamics of Near-Surface Solar Convection with Helioseismology.” 2015. Web. 27 Feb 2021.

Vancouver:

Greer BJ. Exploring the Dynamics of Near-Surface Solar Convection with Helioseismology. [Internet] [Doctoral dissertation]. University of Colorado; 2015. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/astr_gradetds/38.

Council of Science Editors:

Greer BJ. Exploring the Dynamics of Near-Surface Solar Convection with Helioseismology. [Doctoral Dissertation]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/astr_gradetds/38

University of Colorado

22. Wong, Tony E. The Impact of Stable Water Isotopic Information on Parameter Calibration in a Land Surface Model.

Degree: PhD, Applied Mathematics, 2016, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/72

► The single largest uncertainty in climate model energy balance is the surface latent heating over tropical land. Furthermore, the partitioning of the total latent… (more)

▼ The single largest uncertainty in climate model energy balance is the surface latent heating over tropical land. Furthermore, the partitioning of the total latent heat flux into contributions from surface evaporation and plant transpiration offers acute insight into the hydrological and biogeochemical behaviors of an ecosystem, but is notoriously difficult to establish directly. Evapotranspiration (ET) partitioning relies heavily on knowledge of the relative pathways by which water moves from the soil to the atmosphere. These pathways are parameterized by ecosystem resistances, which may not be known with great certainty in practical situations. Resolving these issues requires the development of statistical methods to maximize the use of limited information to best improve models. First, we introduce a commonly-used land surface model, the Community Land Model version 4 (CLM4). We describe an approach to calibrating select model parameters to observational data in a Bayesian estimation framework, requiring Markov chain Monte Carlo sampling of the posterior distribution. We demonstrate the ability of this Bayesian framework to constrain land-atmosphere exchanges of moisture and heat in CLM4, and yield an estimate of ET partitioning which is informed by data. Next, an isotopically-enabled version of CLM4 (iCLM4) is described in detail and validated using site-level and global observations. By leveraging the unique signatures of evaporation and transpiration on the ratios of stable water isotopes, additional constraint on the ET partitioning may be obtained. Finally, an extensive set of isotopic, meteorological and hydrological data from Erie, Colorado, USA is assimilated to calibrate land-atmosphere fluxes and state variables in iCLM4. It is demonstrated that the inclusion of water isotopic data in the assimilation step provides additional constraint on the estimated ET partitioning, and the benefits of these water isotopic datasets relative to common, non-isotopic datasets is quantified. Advisors/Committee Members: William Kleiber, David C. Noone, Keith Julien, Jem Corcoran, David Lawrence.

Subjects/Keywords: Bayesian; calibration; isotope; Markov; model; uncertainty; Applied Mathematics; Hydrology

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APA (6^th Edition):

Wong, T. E. (2016). The Impact of Stable Water Isotopic Information on Parameter Calibration in a Land Surface Model. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/72

Chicago Manual of Style (16^th Edition):

Wong, Tony E. “The Impact of Stable Water Isotopic Information on Parameter Calibration in a Land Surface Model.” 2016. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/72.

MLA Handbook (7^th Edition):

Wong, Tony E. “The Impact of Stable Water Isotopic Information on Parameter Calibration in a Land Surface Model.” 2016. Web. 27 Feb 2021.

Vancouver:

Wong TE. The Impact of Stable Water Isotopic Information on Parameter Calibration in a Land Surface Model. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/72.

Council of Science Editors:

Wong TE. The Impact of Stable Water Isotopic Information on Parameter Calibration in a Land Surface Model. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/appm_gradetds/72

University of Colorado

23. Martin, Bradley Pifer. Application of Rbf-Fd to Wave and Heat Transport Problems in Domains with Interfaces.

Degree: PhD, Applied Mathematics, 2016, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/79

► Traditional finite difference methods for solving the partial differential equations (PDEs) associated with wave and heat transport often perform poorly when used in domains… (more)

Subjects/Keywords: finite differences; heat equation; interfaces; mesh free; RBF; wave equation; Applied Mathematics

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APA (6^th Edition):

Martin, B. P. (2016). Application of Rbf-Fd to Wave and Heat Transport Problems in Domains with Interfaces. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/79

Chicago Manual of Style (16^th Edition):

Martin, Bradley Pifer. “Application of Rbf-Fd to Wave and Heat Transport Problems in Domains with Interfaces.” 2016. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/79.

MLA Handbook (7^th Edition):

Martin, Bradley Pifer. “Application of Rbf-Fd to Wave and Heat Transport Problems in Domains with Interfaces.” 2016. Web. 27 Feb 2021.

Vancouver:

Martin BP. Application of Rbf-Fd to Wave and Heat Transport Problems in Domains with Interfaces. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/79.

Council of Science Editors:

Martin BP. Application of Rbf-Fd to Wave and Heat Transport Problems in Domains with Interfaces. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/appm_gradetds/79

University of Colorado

24. Satkauskas, Ignas V. Numerical Calculus of Probability Density Functions.

Degree: PhD, Applied Mathematics, 2017, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/93

► In this thesis we construct novel functional representations for the Probability Density Functions (PDFs) of random variables and develop efficient and accurate algorithms for… (more)

▼ In this thesis we construct novel functional representations for the Probability Density Functions (PDFs) of random variables and develop efficient and accurate algorithms for computing the PDFs of their sums, products and quotients, again in the same representation. We consider two important cases of random variables: non-negative random variables and random variables taking both positive and negative values. For the first case, we use approximations by decaying exponentials with complex exponents, while for the second case we develop a Gaussian-based multiresolution analysis (GMRA). The need to represent distributions of products and quotients of random variables appear in many areas of theoretical and applied sciences. However, there are currently only limited number of numerical techniques for computing such products and quotients and this thesis presents new numerical methods for this purpose. Current methods for computing the product and quotients typically rely on a Monte Carlo type approach, where the PDFs of the product or quotient are sampled individually and the histogram of the resulting PDF is obtained from computed products or quotients of the individual samples. Although Monte Carlo methods are easy to implement, they suffer from slow convergence and therefore are not well suited for achieving high accuracy. Another method for computing the PDFs of the products and ratios of positive independent random variables relies on the Mellin transform and we describe such methods in greater detail in the thesis. Although mathematically appealing, techniques based on the Mellin transform lack in robust and stable numerical algorithms for computation of the inverse Mellin transform, hence making them not universally applicable. Our novel representations and associated numerical algorithms produce a general framework for computing of PDFs of random variables which we call numerical calculus of PDFs in functional form. The new fast algorithms of this thesis allow user to select computational accuracy; the speed of algorithms only weakly depends on such selection. We demonstrate the performance of new algorithms on multiple examples using accuracies that are well beyond the reach of Monte Carlo based methods. Advisors/Committee Members: Gregory Beylkin, Alireza Doostan, Michael Sprague, Bengt Fornberg, Keith Julien.

Subjects/Keywords: multiresolution analysis; probability density function; product of random variables; Applied Mathematics; Probability

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APA (6^th Edition):

Satkauskas, I. V. (2017). Numerical Calculus of Probability Density Functions. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/93

Chicago Manual of Style (16^th Edition):

Satkauskas, Ignas V. “Numerical Calculus of Probability Density Functions.” 2017. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/93.

MLA Handbook (7^th Edition):

Satkauskas, Ignas V. “Numerical Calculus of Probability Density Functions.” 2017. Web. 27 Feb 2021.

Vancouver:

Satkauskas IV. Numerical Calculus of Probability Density Functions. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/93.

Council of Science Editors:

Satkauskas IV. Numerical Calculus of Probability Density Functions. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/93

University of Colorado

25. Mitchell, Rebecca Amelia. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.

Degree: PhD, Applied Mathematics, 2017, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/94

► Mixing of a passive scalar in a fluid flow results from a two part process in which large gradients are first created by advection and… (more)

Subjects/Keywords: area-preserving maps; chaotic mixing; dynamical sytems; Perron-Frobenius; Applied Mathematics

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APA (6^th Edition):

Mitchell, R. A. (2017). Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/94

Chicago Manual of Style (16^th Edition):

Mitchell, Rebecca Amelia. “Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.” 2017. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/94.

MLA Handbook (7^th Edition):

Mitchell, Rebecca Amelia. “Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps.” 2017. Web. 27 Feb 2021.

Vancouver:

Mitchell RA. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. [Internet] [Doctoral dissertation]. University of Colorado; 2017. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/94.

Council of Science Editors:

Mitchell RA. Designing a Finite-Time Mixer: Optimizing Stirring for Two-Dimensional Maps. [Doctoral Dissertation]. University of Colorado; 2017. Available from: https://scholar.colorado.edu/appm_gradetds/94

University of Colorado

26. Plumley, Meredith J. Investigations of Asymptotic Models for Convection-Driven Flows in Geophysical and Astrophysical Fluid Systems.

Degree: PhD, 2018, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/103

► Geophysical flows, such as the turbulent motion within natural systems, are characterized by a wide range of spatial and temporal scales. Due to this… (more)

▼ Geophysical flows, such as the turbulent motion within natural systems, are characterized by a wide range of spatial and temporal scales. Due to this complexity, approaches that rely on solving the full Navier-Stokes equations are limited to values of parameters that are far from the extreme values characteristic of geophysical regimes. Therefore, results from these simulations must be extrapolated over orders of magnitude to apply to Earth's core. In this thesis, I'll present an alternative strategy for modeling these flows, which relies on deriving reduced models using asymptotic techniques. These models are investigated using efficient scientific computing methods and yield results that are within geophysically relevant parameter regimes. This thesis presents asymptotically reduced models for rapidly rotating convection in a plane layer geometry and highlights a few applications of these reduced models. Specifically, the effect of different boundary conditions on key dynamics is investigated to facilitate comparison with experimental studies. The main scientific question is if these boundary conditions are passive in geophysical parameter regimes and how the rate of rotation influences the heat flux in the systems. While direct numerical simulation and laboratory experimental results can only examine this for moderate values of rotation, the asymptotic model is applied to determine an empirical scaling for the impact of these boundary layers within the rapidly rotating regime. The dynamics of electrically conducting fluids and self-sustained magnetic fields are also investigated with the use of asymptotic models. The presence of electromagnetic fields presents added complexity in the dynamics and the use of asymptotic analysis allows us to investigate a geo/astrophysically relevant limit that is not attainable in current DNS simulations. The asymptotic reduction leads to multiple timescales that require new numerical strategies for solving and two multiscale numerical methods are tested on this new model for both single mode and multimode cases. The model is used to characterize the influence of dynamo action on convection and explore the large scale structure of the flows, especially in comparison to both non-magnetic convective results of rapidly rotating asymptotic models as well as results from dynamo simulations that are not in the geophysical parameter space. We find that dynamo action leads to fundamental changes in the dynamics, compared to non-magnetic convection, and current work is aimed at understanding the implications for natural dynamos. Advisors/Committee Members: Keith Julien, Michael A. Calkins, Ian Grooms, James Meiss, Peter Hamlington.

Subjects/Keywords: asymptotic models; fluid systems; dynamics; geophysical; astrophysical; Applied Mathematics; Geophysics and Seismology

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APA (6^th Edition):

Plumley, M. J. (2018). Investigations of Asymptotic Models for Convection-Driven Flows in Geophysical and Astrophysical Fluid Systems. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/103

Chicago Manual of Style (16^th Edition):

Plumley, Meredith J. “Investigations of Asymptotic Models for Convection-Driven Flows in Geophysical and Astrophysical Fluid Systems.” 2018. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/103.

MLA Handbook (7^th Edition):

Plumley, Meredith J. “Investigations of Asymptotic Models for Convection-Driven Flows in Geophysical and Astrophysical Fluid Systems.” 2018. Web. 27 Feb 2021.

Vancouver:

Plumley MJ. Investigations of Asymptotic Models for Convection-Driven Flows in Geophysical and Astrophysical Fluid Systems. [Internet] [Doctoral dissertation]. University of Colorado; 2018. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/103.

Council of Science Editors:

Plumley MJ. Investigations of Asymptotic Models for Convection-Driven Flows in Geophysical and Astrophysical Fluid Systems. [Doctoral Dissertation]. University of Colorado; 2018. Available from: https://scholar.colorado.edu/appm_gradetds/103

University of Colorado

27. Lipinski, Douglas Martin. Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications.

Degree: PhD, Applied Mathematics, 2012, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/56

► Lagrangian coherent structures (LCS) are recently defined structures used to analyze transport in dynamical systems with general time dependence. LCS techniques have seen increasing… (more)

Subjects/Keywords: dynamical systems; fast algorithms; Lagrangian coherent structures; Applied Mathematics

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APA (6^th Edition):

Lipinski, D. M. (2012). Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/56

Chicago Manual of Style (16^th Edition):

Lipinski, Douglas Martin. “Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications.” 2012. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/56.

MLA Handbook (7^th Edition):

Lipinski, Douglas Martin. “Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications.” 2012. Web. 27 Feb 2021.

Vancouver:

Lipinski DM. Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications. [Internet] [Doctoral dissertation]. University of Colorado; 2012. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/56.

Council of Science Editors:

Lipinski DM. Efficient Ridge Tracking Algorithms for Computing Lagrangian Coherent Structures in Fluid Dynamics Applications. [Doctoral Dissertation]. University of Colorado; 2012. Available from: https://scholar.colorado.edu/appm_gradetds/56

University of Colorado

28. Martin, Ruth Ann. Toward a General Solution of the Three-Wave Resonant Interaction Equations.

Degree: PhD, Applied Mathematics, 2015, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/67

► The resonant interaction of three wavetrains is the simplest form of nonlinear interaction for dispersive waves of small amplitude. Such interactions arise frequently in… (more)

▼ The resonant interaction of three wavetrains is the simplest form of nonlinear interaction for dispersive waves of small amplitude. Such interactions arise frequently in applications ranging from nonlinear optics to internal waves in the ocean through the study of the weakly nonlinear limit of a dispersive system. The slowly varying amplitudes of the three waves satisfy a set of integrable nonlinear partial differential equations known as the three-wave equations. If we consider the special case of spatially uniform solutions, then we obtain the three-wave ODEs. The ODEs have been studied extensively, and their general solution is known in terms of elliptic functions. Conversely, the universally occurring PDEs have been solved in only a limited number of configurations. For example, Zakharov and Manakov (1973, 1976) and Kaup (1976) used inverse scattering to solve the three-wave equations in one spatial dimension on the real line. Similarly, solutions in two or three spatial dimensions on the whole space were worked out by Zakharov (1976), Kaup (1980), and others. These known methods of analytic solution fail in the case of periodic boundary conditions, although numerical simulations of the problem typically impose these conditions. To find the general solution of an nth order system of ordinary differential equations, it is sufficient to find a function that satisfies the ODEs and has n constants of integration. The general solution of a PDE, however, is not well defined and is usually difficult, if not impossible, to attain. In fact, only a small number of PDEs have known general solutions. We seek a general solution of the three-wave equations, which has the advantage of being compatible with a wide variety of boundary conditions and any number of spatial dimensions. Our work indicates that the general solution of the three-wave equations can be constructed using the known general solution of the three-wave ODEs. In particular, we try to construct the general solution of the three-wave equations using a Painleve-type analysis. For now, we consider a convergent Laurent series solution (in time), which contains two real free constants and three real-valued functions (in space) that are arbitrary except for some differentiability constraints. In order to develop a full general solution of the problem, the two free constants must also be allowed to have spatial dependence, and one more function must be introduced. That is, a full general solution of the problem would involve six of these real-valued functions. Advisors/Committee Members: Harvey Segur, James Meiss, Diane Henderson, Keith Julien, Bengt Fornberg.

Subjects/Keywords: Nonlinear; Painleve; PDEs; Waves; resonant interaction; three wavetrains; nonlinear interaction; dispersive system; Special Functions

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APA (6^th Edition):

Martin, R. A. (2015). Toward a General Solution of the Three-Wave Resonant Interaction Equations. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/67

Chicago Manual of Style (16^th Edition):

Martin, Ruth Ann. “Toward a General Solution of the Three-Wave Resonant Interaction Equations.” 2015. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/67.

MLA Handbook (7^th Edition):

Martin, Ruth Ann. “Toward a General Solution of the Three-Wave Resonant Interaction Equations.” 2015. Web. 27 Feb 2021.

Vancouver:

Martin RA. Toward a General Solution of the Three-Wave Resonant Interaction Equations. [Internet] [Doctoral dissertation]. University of Colorado; 2015. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/67.

Council of Science Editors:

Martin RA. Toward a General Solution of the Three-Wave Resonant Interaction Equations. [Doctoral Dissertation]. University of Colorado; 2015. Available from: https://scholar.colorado.edu/appm_gradetds/67

University of Colorado

29. Maiden, Michelle Dorothy. Dispersive Hydrodynamics in Viscous Fluid Conduits.

Degree: PhD, 2019, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/151

► Viscous fluid conduits provide an ideal system for the study of dissipationless, dispersive hydrodynamics. A dense, viscous fluid serves as the background medium through… (more)

Subjects/Keywords: dispersive hydrodynamics; dispersive shock waves; nonlinear waves; soliton fission; solitons; Applied Mathematics

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APA (6^th Edition):

Maiden, M. D. (2019). Dispersive Hydrodynamics in Viscous Fluid Conduits. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/151

Chicago Manual of Style (16^th Edition):

Maiden, Michelle Dorothy. “Dispersive Hydrodynamics in Viscous Fluid Conduits.” 2019. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/151.

MLA Handbook (7^th Edition):

Maiden, Michelle Dorothy. “Dispersive Hydrodynamics in Viscous Fluid Conduits.” 2019. Web. 27 Feb 2021.

Vancouver:

Maiden MD. Dispersive Hydrodynamics in Viscous Fluid Conduits. [Internet] [Doctoral dissertation]. University of Colorado; 2019. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/151.

Council of Science Editors:

Maiden MD. Dispersive Hydrodynamics in Viscous Fluid Conduits. [Doctoral Dissertation]. University of Colorado; 2019. Available from: https://scholar.colorado.edu/appm_gradetds/151

University of Colorado

30. Bao, Lei. Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models.

Degree: PhD, Applied Mathematics, 2016, University of Colorado

URL: https://scholar.colorado.edu/appm_gradetds/73

► This thesis presents the ongoing work on the numerical aspects of designing a numerical frame- work on which to build a high-resolution atmospheric model… (more)

▼ This thesis presents the ongoing work on the numerical aspects of designing a numerical frame- work on which to build a high-resolution atmospheric model using the discontinuous Galerkin (DG) methods. As the horizontal resolution exceeds the hydrostatic limit (1/10◦ or 10 km), which is usu- ally referred as the non-hydrostatic (NH) scale, the compressible Euler system must be employed to characterize the motion of the air flow. To simulate this system numerically, we consider the DG method for the spatial discretization and cubed-sphere grid system. The High-Order Method Modeling Environment (HOMME) is a highly scalable hydrostatic dynamical core based on spectral element and/or DG methods. It utilizes cube-sphere geometry and shows great scalability. Our goal is to extend HOMME-DG model to the non-hydrostatic scale. We use the global shallow water equations to study the influence of the full conservative equation sets in conserving integral invariants is rigorously compared against the vector-invariant form. Several important components, such as the horizontal discretization and numerical diffusion are also discussed briefly. The terrain-following height-based coordinate transform is adopted to handle the orography. For the time discretization, we consider a Horizontally explicit and Vertically implicit operator splitting based on Strang-splitting approach. HEVI treats the vertical component implicitly and the horizontal component explicitly. As a consequence, the maximum allowed time-step size is only constrained by the horizontal grid spacing, which is usually several orders of magnitude higher than the vertical. We compare HEVI operator splitting with Implicit-explict (IMEX) linear-nonlinear splitting ideas. We also perform the linear stability study of various IMEX Runge-Kutta schemes. HEVI-Strang splitting shows large stability region in the well-resolved scale and only requires one implicit solve compared with other IMEX-RK schemes. This study is the first time testing the DG scheme with the dimensional splitting approach. The HEVI-Strang scheme shows qualitatively comparable results at a more lower computational cost. The efficiency of the linear solver resulting from the Newton’s method is also investigated. A right preconditioner is suggested to improve the convergence of the GMRES iterative solver. Numerical results show that the preconditioned GMRES and the direct solvers are both viable options to solve the vertical implicit component. The global 3D DG-NH model is constructed by vertical stacking of the horizontal cubes- sphere layers. The 3D global advection problem is tested using two DCMIP test cases. We also present some preliminary results for the non-hydrostatic inertia gravity wave test utilizing the HEVI-Strang time integration scheme. The results of 3D DG-NH model based on HEVI-Strang time integration scheme are qualitatively in line with other non-hydrostatic models. The time-step size of the HEVI-Strang scheme is not affected as the vertical grid spacing varies… Advisors/Committee Members: Henry M. Tufo, Thomas A. Manteuffel, Keith Julien, Ram D. Nair, Congming Li.

Subjects/Keywords: Cubed-sphere; Discontinuous Galerkin; Non-hydrostatic; numerical framework; atmospheric model; air flow; Atmospheric Sciences; Numerical Analysis and Computation

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APA (6^th Edition):

Bao, L. (2016). Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models. (Doctoral Dissertation). University of Colorado. Retrieved from https://scholar.colorado.edu/appm_gradetds/73

Chicago Manual of Style (16^th Edition):

Bao, Lei. “Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models.” 2016. Doctoral Dissertation, University of Colorado. Accessed February 27, 2021. https://scholar.colorado.edu/appm_gradetds/73.

MLA Handbook (7^th Edition):

Bao, Lei. “Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models.” 2016. Web. 27 Feb 2021.

Vancouver:

Bao L. Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models. [Internet] [Doctoral dissertation]. University of Colorado; 2016. [cited 2021 Feb 27]. Available from: https://scholar.colorado.edu/appm_gradetds/73.

Council of Science Editors:

Bao L. Efficient Time-Integration Schemes for Discontinuous Galerkin Non-Hydrostatic Atmospheric Models. [Doctoral Dissertation]. University of Colorado; 2016. Available from: https://scholar.colorado.edu/appm_gradetds/73

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Open Access Theses and Dissertations