We study the dynamics, kinematics and statistics of resonant and quasiresonant three-wave interactions appearing in models of geophysical flow. In these dispersive wave systems, the phenomenon of nonlinear resonance broadening plays a significant role across all three different branches of wave turbulence theory: from the statistical, to the discrete, and even the mesoscopic, formed as an intermediate regime between the two. The principal aim of this thesis is to understand the processes by which resonance broadening can induce a transition between each of these three different regimes. Beginning with the discrete case, we study two variants of the isolated triad: one with a constant additive forcing term; and the other in the presence of detuning. We provide a detailed analysis of both of these systems, covering their integrability and boundedness properties, showing that for almost all initial conditions the motion remains quasi-periodic and periodic respectively. Interestingly, we show that moderate amounts of detuning can actually promote energy exchange, increase the period and in rare instances cease to be periodic at all; each of these statements are contrary to what was previously thought. This motivates a more detailed study into the kinematics of resonance broadening. By analysing how the set of quasi-resonant modes develops under increased broadening, we show that a percolation-like transition exists, independent of the dispersion relationship used. At critical levels of broadening, we see the emergence of a single quasi-resonant cluster that begins to dominate the entire system. We argue that the formation of this cluster provides a way of characterising the turbulent state of the system, distinguishing between the discrete and statistical regimes. Through direct numerical simulation of the Charney-Hasegawa-Mima equation, we then assess whether this view is truly representative of the underlying dynamics. Here we find that the generation of quasi-resonantly excited modes can be detected through the statistical measures of total correlation and mutual information. We conclude by suggesting that these techniques have an incredible potential to infer the signature of both resonant and quasi-resonant clusters in fully realised turbulent systems, and yet are also subtle enough to detect qualitative changes in the underlying dynamics between different interacting modes.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:589833 |
| Date | January 2013 |
| Creators | Harris, Jamie |
| Publisher | University of Warwick |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://wrap.warwick.ac.uk/58419/ |
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