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http://hdl.handle.net/10197/6702
Mon, 25 Oct 2021 13:41:15 GMT2021-10-25T13:41:15ZNonlinear wave interactions : beyond weak nonlinearity
http://hdl.handle.net/10197/11661
Abstract: An important aspect of the dynamics of nonlinear wave systems is the effect of finite amplitude phenomena — that is, phenomena which can only manifest beyond the limit of weak nonlinearity. The work in this thesis aims to bridge the gap between the phenomenology of finite amplitude effects in nonlinear wave systems and the existing theories describing these systems. We describe the phenomenon of precession resonance, a manifestly finite amplitude phenomenon characterised by a balance between the linear and nonlinear timescales of the system. We then investigate numerically the region of convergence of the normal form transformation to understand if precession resonance can be described with tools commonly used to study nonlinear wave systems. We find that the boundary of the region of convergence of the transformation closely matches the values which lead to precession resonance, giving us an understanding of where precession resonance lies with the general theory of wave turbulence. We further investigate the phenomenon of precession resonance by considering a more general system, where two nonresonant triads interact. It is found that precession resonant behaviour exists between two nonresonant triads, and can be found in quasiresonant regimes when the linear frequencies of the triads are close in value. The scaling amplitude required to trigger precession resonance in these quasiresonant regimes is small, demonstrating the manifestation of precession resonance in weakly nonlinear systems. We continue this investigation of precession resonance in weakly nonlinear systems by extending our study to five-wave quasiresonances. We apply this to the case of deep gravity water waves propagating in one dimension and find that precession resonant behaviour is present in the system for quasiresonant quintet interactions. Finally, we investigate the effect of finite amplitudes on the wave turbulent energy cascade in the Charney-Hasegawa-Mima equation. It is found that, at intermediate nonlinearity, the anisotropy from the weakly nonlinear limit and the presence of precession resonance from the finite-amplitude effects combine to allow for the most efficient energy transfers to zonal scales. Overall, precession resonance presents itself as a natural extension of the concept of resonances to finite-amplitude regimes. In the limit of weak nonlinearity, precession resonance can be reduced to exact wave resonances. In the case of quasiresonances, precession resonance corresponds to an interaction that maximises the efficiency of energy transfers in the system. Scaling beyond the case of weak nonlinearity we recover the original definition of precession resonance.
Title: Nonlinear wave interactions : beyond weak nonlinearityWed, 01 Jan 2020 00:00:00 GMThttp://hdl.handle.net/10197/116612020-01-01T00:00:00ZAssessing late-time singular behaviour in models of three dimensional Euler fluid flow
http://hdl.handle.net/10197/10623
Abstract: The open question of regularity of the fluid dynamical equations is considered one of the most fundamental challenges of mathematics and physics [C. L. Fefferman. Existence and smoothness of the Navier-Stokes equation. The millennium prize problems, pages 57-67 (2000)]. While the viscous Navier-Stokes equations have more physical relevance, the inviscid Euler equations present the greatest challenge and exhibit the most extreme behaviours. For this reason, the numerical study of possible finite-time blowup is typically concerned with these inviscid equations. Extensive numerical assessment of finite-time blow up of 3D Euler has been carried out, albeit with conflicting yes and no conclusions with regard to the existence of finite time singularity.
The fundamental difficulty of this important problem is the lack of analytic solutions or any a priori knowledge of asymptotic behaviour. A secondary obstacle is that the spatial collapse associated with intense vortex stretching results in numerical solutions becoming unresolved beyond a certain time. It is therefore imperative to devise a framework with nontrivial blowup dynamics and where analytic solutions are known in order to validate and compare various numerical methods, for the purposes of accurately solving the system and diagnosing blowup.
In this regard, I have proposed investigating the issue of Euler finite-time blowup using a novel approach where the original system of equations is bijectively transformed to a new mapped system which is globally regular in time [M. D. Bustamante. 3D Euler equations and ideal MHD mapped to regular systems: Probing the finite-time blowup hypothesis. Physica D: Nonlinear Phenomena, 240(13):1092-1099 (2011)]. Since no known analytical solution for the full 3D Euler equations exist, I have studied the robustness of the proposed novel approach using the one-dimensional Burgers equation and a proposed new one-parameter family of models of the 3D Euler equations on a 2D symmetry plane. The proposed 2D symmetry plane model equations were motivated by the work on stagnation-point-type exact solution of 3D Euler equations by Gibbon et al. [J. Gibbon, A. Fokas, and C. Doering. Dynamically stretched vortices as solutions of the 3D Navier-Stokes equations. Physica D: Nonlinear Phenomena, 132(4):497-510 (1999)].
I have shown that the mapped system’s numerical solution leads to more accurate estimates of the blowup quantities compared with the original system. I also established that only by using the mapped system can certain late-time behaviours be observed and asymptotic trends be established.
Title: Assessing late-time singular behaviour in models of three dimensional Euler fluid flowFri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10197/106232016-01-01T00:00:00ZFourier Phase Dynamics in Turbulent Non-Linear Systems
http://hdl.handle.net/10197/10622
Abstract: The research presented in this thesis examines in detail the role of triad Fourier phase dynamics across a range of turbulent fluid systems. In 1D Burgers, we see a clear link between the Fourier space triad phases and real-space shocks, the key dissipative structures of the dynamics. This link is evident also in the intermittency statistics, where time periods of high phase synchronisation contribute the majority of the extreme events that characterise intermittent behaviour. The reduction of degrees of freedom is also explored, with Fractal Fourier decimation used to remove modes across all scales of the system. We find that the phase synchronisation mechanism is extremely sensitive to such changes, and coherence is quickly lost as degrees of freedom are suppressed. We further extend these phase dynamics concepts by examining the forward enstrophy cascade in 2D Navier Stokes. Again the importance of the triad Fourier phases is clear, with strong preference for values that contribute to the forward cascade. We will see that at a snapshot in time, only a subset of the Fourier modes are responsible for the formation of small-scale vorticity filament structures that govern the total enstrophy dissipation of the ow. The final stage is to expand the definition of the triad Fourier phase to a non-scalar field in 3D Navier-Stokes. Utilising helical mode decomposition, we show the differing behaviour of the helical triad interaction classes and once again how helical triad phases play a vital role in the efficiency and directionality of energy flux in 3D turbulence. In a similar fashion to the 2D Navier-Stokes enstrophy cascade, we again find only a small energetic subset of the Fourier modes are important contributors to the flux toward small scales, and thus to the intermittent bursts of dissipation that characterise these chaotic flows. Finally we discuss how these exciting new results could be applied to other turbulent systems and how such coherent phase dynamics may lead to a better understanding of the mechanism behind Intermittency in Turbulence.
Title: Fourier Phase Dynamics in Turbulent Non-Linear SystemsMon, 01 Jan 2018 00:00:00 GMThttp://hdl.handle.net/10197/106222018-01-01T00:00:00ZNumerical modelling and statistical emulation of landslide induced tsunamis: the Rockall Bank slide complex, NE Atlantic Ocean
http://hdl.handle.net/10197/8603
Abstract: This thesis studies submarine sliding and tsunami generation at the Rockall Bank, NE Atlantic Ocean through numerical and statistical modelling. Two numerical codes are used to perform the simulations from the submarine sliding to tsunami generation, propagation and inundation. The landslide model is VolcFlow and the tsunami model is VOLNA. Some of the basic rheological regimes used to model submarine landslides are briefly discussed, with a comparison in the case of the Rockall Bank. The latest version of VOLNA is validated against an analytical solution. The brief geological history of the area under study is also given. The numerical simulations explore different scenarios of failure in the area, and assess their tsunamigenic potential and the impact of the tsunamis on the current topography of the Irish shoreline. The results of the simulations exhibit a great variability that derives from the parameters used as input in the landslide model. There is a need to quantify this uncertainty. To do so, a Bayesian calibration of the parameters is initially performed, which leads to the posterior distributions of the input parameters. A statistical emulator, which acts as a surrogate of the numerical process is then built. The emulator can lead to predictions of the process in excessively fast (when compared to the simulations) computational speeds. For the examined case, the emulator propagates the uncertainties in the distributions of the input parameters resulting from the calibration, to the outputs. As a result, the predictions of the maximum free surface elevation at specified locations are obtained.
Title: Numerical modelling and statistical emulation of landslide induced tsunamis: the Rockall Bank slide complex, NE Atlantic OceanSun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/10197/86032017-01-01T00:00:00ZSpectral properties of nonnegative matrices
http://hdl.handle.net/10197/8597
Abstract: The spectral properties of nonnegative matrices have intrigued pure and applied mathematicians alike, beginning with the classical works of Oskar Perron and Georg Frobenius at the start of the twentieth century. One question which stems naturally from this area of research is that of the "Nonnegative Inverse Eigenvalue Problem", or NIEP. This is the problem of characterising those lists of complex numbers which are "realisable" as the spectrum of some entrywise nonnegative matrix. This thesis explores the NIEP, as well as one of its variants, the "Symmetric Nonnegative Inverse Eigenvalue Problem", or SNIEP, which considers realisability by a symmetric nonnegative matrix.The question of determining which operations on lists preserve realisability is pertinent in the NIEP, since such operations can allow us to construct more complicated lists from simple building blocks. We present some new results along these lines. In particular, we discuss how to replace parts of realisable lists by longer lists, while preserving realisability.In those cases where a realising matrix is known to exist, one can consider studying the properties of this matrix. We focus our attention on the problem of characterising the diagonal elements of the realising matrix and achieve a complete solution in the case where every entry in the list (apart from the Perron eigenvalue) has nonpositive real part. In order to prove this result, we derived complex analogues of Newton's inequalities, which are of independent interest.In the context of the SNIEP, we unify a large body of research by presenting a recursive method for constructing symmetrically realisable lists and showing that essentially all previously know sufficient conditions are either contained in, or equivalent to the family we introduce. Our construction also reveals several interesting properties of the family in question and allows for an explicit algorithmic characterisation of the lists that lie within it.Finally, we construct families of symmetrically realisable lists which do not satisfy any previously known sufficient conditions.
Title: Spectral properties of nonnegative matricesFri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/10197/85972016-01-01T00:00:00ZA new implementation of the elliptic curve method of integer factorization using Edwards and Hessian curves
http://hdl.handle.net/10197/8540
Abstract: In this thesis, three main ideas characterise a new implementation of the Elliptic Curve Method (ECM) of integer factorization. The first idea is the use of Edwards/Hessian curves for all elliptic curve computations, including stage 2. The second idea is the use of families of elliptic curves with larger torsion subgroup over quartic number fields, in particular $\Z/4\Z\oplus\Z/8\Z$ for a family of Edwards curves, and $\Z/6\Z\oplus\Z/6\Z$ for a family of Hessian curves. The third idea is the generation of respectively hundreds/a few thousand of such curves from families given by Jeon/Kim/Lee, which have small parameters and a point of infinite order having small projective coordinates, leading to improved efficiency in scalar multiplication. The curves are generated using SAGE. The FFT continuation for stage 2 is implemented. The performance of the software is analysed and compared to the leading implementations, in terms of effectiveness/speed/memory usage. The implementation is tested on ICHEC's Fionn cluster. The use of Hessian curves in an implementation of ECM appears new. A new discovery is that EECM-MPFQ's `good curves' for ECM having torsion $\Z/2\Z\oplus\Z/8\Z$ are a subset of the Jeon-Kim-Lee $\Z/4\Z\oplus\Z/8\Z$ family, yielding many hundreds more good curves than the $100$ from two torsion families produced for EECM-MPFQ, not to mention several thousand curves from the $\Z/6\Z\oplus\Z/6\Z$ family with small parameters. This remediates one drawback of EECM-MPFQ - lack of good curves. Another discovery is that, for small-parameter Hessian curves, the speed of the addition formula is particularly fast, allowing Hessian scalar multiplication without windowing to compete with Edwards scalar multiplication with windowing. Since windowing is not required, the associated higher memory cost is not incurred. This has a beneficial consequence for ECM in memory-constrained environments such as when implemented on GPUs. This in turn may benefit the sieving stage of the number field sieve.
Title: A new implementation of the elliptic curve method of integer factorization using Edwards and Hessian curvesThu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10197/85402015-01-01T00:00:00ZSpatial and spatio-temporal modelling of Sitka spruce tree growth from forest plots in Co. Wicklow
http://hdl.handle.net/10197/8539
Abstract: Individual tree growth in forest plots is spatially dependent, changes overtime and the magnitude of spatial dependence may also change over time,particularly in stands subjected to thinning. Models for tree growth in theliterature have been mainly restricted to either spatial models or temporalmodels. Spatial models have been mostly restricted to those that haveGaussian variograms with comparisons at single time points while dynamicmodels ignore tree competition caused by close spatial proximity. Spatio-temporalmodels were therefore developed to represent the individual treegrowth of Sitka spruce (Picea sitchensis (Bong.) Carr.) based on data fromthree long-term, repeatedly measured, experimental plots in Co. Wicklow,Ireland.The initial thinning treatments for the three plots were: unthinned, 40%thinned and 50% thinned. Tree growth was defined as the difference inthe measured diameter at breast height (DBH) (cm) at regular intervals.Thinned and unthinned plots were modelled separately as they were notadjacent. A model for tree growth over all locations in a plot and all timepoints was fitted using a sum-metric spatio-temporal variogram. Negativespatial correlation at small distances (due to competition) is evident atseparate time points while at larger distances it is positive and this isadequately modelled with a wave function. The correlation of a singletree over time also followed a wave variogram while the spatio-temporalanisotropy parameter captured the changing spatial wave intensity.Models with fixed effects of age, number of neighbours and polygon areawere also considered. Predicted values for models were computed usingregression-kriging and mean squared error of prediction was used tocompare models and thinning strategies. Both thinned plots clearly outperformedthe unthinned plot in terms of total individual tree DBH growthand also at a stand level. Spatio-temporal bootstrap methods were usedto assess the precision of the spatio-temporal model parameter estimates.The models indicate, once fixed effects are accounted for, that spatialvariability and correlation is more important than temporal. The modelsprovide insights into the nature of tree growth and it is seen that modellingspatial dependence is important in the understanding of managementstrategies and silvicultural decision making.
Title: Spatial and spatio-temporal modelling of Sitka spruce tree growth from forest plots in Co. WicklowThu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/10197/85392015-01-01T00:00:00ZMathematical modeling and optimization of wave energy converters and arrays
http://hdl.handle.net/10197/7898
Abstract: The aim of this work is to develop methodologies and understand the dynamics of waveenergy energy converters (WECs) in some problems of practical interest. The focus is ona well known WEC - the Oscillating Wave Surge Converter (OWSC). In the first work, amathematical model is described to analyze the interactions in a wave energy farm comprising of OWSCs. The semi-analytical method uses Green’s integral equation formulation and Green’s function, yielding hyper-singular integrals which are later solved using the Chebyshev polynomial of the second kind. A new methodology for the optimization of large wavefarms is then presented and the approach includes a statistical emulator, an active learning approach (Gaussian Process Upper Confidence Bound with Pure Exploration) and a genetic algorithm. The modular concept of the OWSC, which has emerged to address some of the shortcomings in the original design of the OWSC, is also described and investigated using a semi analytical approach for cylindrical modules. In another work, the dynamics of the OWSC near a straight coast is analyzed and for a particular case, a significant enhancement in the performance of the OWSC is observed. This interesting result motivated the following study, where it is investigated if a breakwater can artificially enhance the performance of the OWSC. Lastly, a new approach is presented to analyze the interactions between two different kind of WECs (an OWSC and a Heaving Wave Energy Converter), performing different modes of motion.
Title: Mathematical modeling and optimization of wave energy converters and arraysSat, 01 Aug 2015 00:00:00 GMThttp://hdl.handle.net/10197/78982015-08-01T00:00:00Z