Quadratic invariants for discrete clusters of weakly interacting waves

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Title: Quadratic invariants for discrete clusters of weakly interacting waves
Authors: Harper, Katie L.
Bustamante, Miguel
Nazarenko, Sergey
Permanent link: http://hdl.handle.net/10197/7514
Date: 30-May-2013
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.
Funding Details: University College Dublin
Type of material: Journal Article
Publisher: IOP Publishing
Copyright (published version): 2013 IOP Publishing Ltd
Keywords: Turbulent flows;Nonlinear dynamics;Hamiltonian mechanics;Rossby waves
DOI: 10.1088/1751-8113/46/24/245501
Language: en
Status of Item: Peer reviewed
Appears in Collections:Mathematics and Statistics Research Collection

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