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Quadratic invariants for discrete clusters of weakly interacting waves

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Author(s)
Harper, Katie L. 
Bustamante, Miguel 
Nazarenko, Sergey 
Uri
http://hdl.handle.net/10197/7514
Date Issued
30 May 2013
Date Available
17T10:12:30Z February 2016
Abstract
We consider discrete clusters of quasi-resonant triads arising from a Hamiltonian three-wave equation. A cluster consists of N modes forming a total of M connected triads. We investigate the problem of constructing a functionally independent set of quadratic constants of motion. We show that this problem is equivalent to an underlying basic linear problem, consisting of finding the null space of a rectangular M × N matrix A with entries 1, −1 and 0. In particular, we prove that the number of independent quadratic invariants is equal to J ≡ N−M∗ N−M, where M∗ is the number of linearly independent rows in A. Thus, the problem of finding all independent quadratic invariants is reduced to a linear algebra problem in the Hamiltonian case. We establish that the properties of the quadratic invariants (e.g., locality) are related to the topological properties of the clusters (e.g., types of linkage). To do so, we formulate an algorithm for decomposing large clusters into smaller ones and show how various invariants are related to certain parts of a cluster, including the basic structures leading to M∗ < M. We illustrate our findings by presenting examples from the Charney–Hasegawa–Mima wave model, and by showing a classification of small (up to three-triad) clusters.
Sponsorship
University College Dublin
Other Sponsorship
Russian Federation
Engineering and Physical Sciences Research Council (EPSRC)
Type of Material
Journal Article
Publisher
IOP Publishing
Journal
Journal of Physics A: Mathematical and Theoretical
Volume
46
Start Page
1
End Page
32
Copyright (Published Version)
2013 IOP Publishing Ltd
Keywords
  • Turbulent flows

  • Nonlinear dynamics

  • Hamiltonian mechanics...

  • Rossby waves

DOI
10.1088/1751-8113/46/24/245501
Web versions
http://mathsci.ucd.ie/~miguel/My_Papers/Quadratic_Invariants_Clusters_JPA_2013.pdf
Language
English
Status of Item
Peer reviewed
This item is made available under a Creative Commons License
https://creativecommons.org/licenses/by-nc-nd/3.0/ie/
Owning collection
Mathematics and Statistics Research Collection
Scopus© citations
12
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Jan 28, 2023
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