Nonlinear wave interactions : beyond weak nonlinearity

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Title: Nonlinear wave interactions : beyond weak nonlinearity
Authors: Walsh, Shane
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Date: 2020
Online since: 2020-11-04T08:00:08Z
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.
Funding Details: Irish Research Council
Type of material: Doctoral Thesis
Publisher: University College Dublin. School of Mathematics and Statistics
Qualification Name: Ph.D.
Copyright (published version): 2020 the Author
Keywords: Wave turbulenceNonlinear dynamicsDynamical systemsNumerical simulations
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Theses

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