Harris, J. (Jamie) (2013) The kinematics, dynamics and statistics of three-wave interactions in models of geophysical flow. PhD thesis, University of Warwick.

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Abstract

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.

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Kinematics, Fluid dynamics -- Mathematical models, Wave-motion, Theory of, Geophysics
Official Date:	January 2013
Institution:	University of Warwick
Theses Department:	Centre for Complexity Science
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Bustamante, Miguel D.; Connaughton, Colm
Sponsors:	Engineering and Physical Sciences Research Council (EPSRC)
Extent:	vi, 185 leaves : illustrations.
Language:	eng

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