2010 program of study : swirling and swimming in turbulence

Balmforth, N. J.; Thiffeault, Jean-Luc

doi:10.1575/1912/5058

Cited by 2 publications

(7 citation statements)

References 14 publications

(55 reference statements)

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“…We note, this model is derived by means of matched asymptotic expansions of a Hamiltonian 2+1 mean field theory near a marginally stable equilibrium, and also by comparison with the results of numerical simulations [2]. Finally, in Sec.…”

Section: Introductionmentioning

confidence: 92%

“…The presence of the continuous spectrum, which is responsible for Landau damping on the linear level, causes conventional perturbation analyses to fail because of singularities that occur at all orders of perturbation. However, in [2], it was shown by a suitable matched asymptotic analysis, how the single-wave model emerges from the large class of 2+1 theories of Sec. III of [1].…”

Section: Commentary: Degeneracy and Nonlinearitymentioning

confidence: 99%

“…An in-depth description of he SW model is beyond the scope of the present contribution, but we comment that in addition to the normal form that aligns with the CHH bifurcation there is also a degenerate from associated with the CSS bifurcation. We refer the reader to [2] (notably Sec. V) for further details.…”

Section: Commentary: Degeneracy and Nonlinearitymentioning

confidence: 99%

“…In Sec. V we present the idea that a certain mean field Hamiltonian system, the single-wave model [2,20,21], is a nonlinear normal form for the CHH bifurcation that describes the eventual nonlinear saturation of the resulting instability near criticality.…”

Section: Introductionmentioning

confidence: 99%

“…A careful counting of eigenvalues, with and without symmetry, is undertaken, leading to the definition of a degenerate continuum steady state (CSS) bifurcation. It is described how the CHH and CSS bifurcations can be viewed as linear normal forms associated with the nonlinear single-wave model described in [2], which is a companion to the present work and that of [1].…”

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confidence: 99%

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Continuum Hamiltonian Hopf Bifurcation II

Hagstrom¹,

Morrison²

2013

Nonlinear Physical Systems

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Building on the development of [1], bifurcation of unstable modes that emerge from continuous spectra in a class of infinite-dimensional noncanonical Hamiltonian systems is investigated. Of main interest is a bifurcation termed the continuum Hamiltonian Hopf (CHH) bifurcation, which is an infinite-dimensional analog of the usual Hamiltonian Hopf (HH) bifurcation. Necessary notions pertaining to spectra, structural stability, signature of the continuous spectra, and normal forms are described. The theory developed is applicable to a wide class of 2+1 noncanonical Hamiltonian matter models, but the specific example of the Vlasov-Poisson system linearized about homogeneous (spatially independent) equilibria is treated in detail. For this example, structural (in)stability is established in an appropriate functional analytic setting, and two kinds of bifurcations are considered, one at infinite and one at finite wavenumber. After defining and describing the notion of dynamical accessibility, Kreȋn-like theorems regarding signature and bifurcation are proven. In addition, a canonical Hamiltonian example, composed of a negative energy oscillator coupled to a heat bath, is treated and our development is compared to pervious work in this context. A careful counting of eigenvalues, with and without symmetry, is undertaken, leading to the definition of a degenerate continuum steady state (CSS) bifurcation. It is described how the CHH and CSS bifurcations can be viewed as linear normal forms associated with the nonlinear single-wave model described in [2], which is a companion to the present work and that of [1]

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Section: Introductionmentioning

confidence: 92%

Section: Commentary: Degeneracy and Nonlinearitymentioning

confidence: 99%

Section: Commentary: Degeneracy and Nonlinearitymentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

mentioning

confidence: 99%

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Continuum Hamiltonian Hopf Bifurcation II

Hagstrom¹,

Morrison²

2013

Nonlinear Physical Systems

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Continuum Hamiltonian Hopf Bifurcation I

Morrison¹,

Hagstrom²

2013

Nonlinear Physical Systems

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Hamiltonian bifurcations in the context of noncanonical Hamiltonian matter models are described. First, a large class of 1 + 1 Hamiltonian multi-fluid models is considered. These models have linear dynamics with discrete spectra, when linearized about homogeneous equilibria, and these spectra have counterparts to the steady state and Hamiltonian Hopf bifurcations when equilibrium parameters are varied. Examples of fluid sound waves and plasma and gravitational streaming are treated in detail. Next, using these 1 + 1 examples as a guide, a large class of 2 + 1 Hamiltonian systems is introduced, and Hamiltonian bifurcations with continuous spectra are examined. It is shown how to attach a signature to such continuous spectra, which facilitates the description of the continuous Hamiltonian Hopf bifurcation. This chapter lays the groundwork for Kreȋn-like theorems associated with the CHH bifurcation that are more rigorously discussed in our companion chapter [1].

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2010 program of study : swirling and swimming in turbulence

Cited by 2 publications

References 14 publications

Continuum Hamiltonian Hopf Bifurcation II

Continuum Hamiltonian Hopf Bifurcation II

Continuum Hamiltonian Hopf Bifurcation I

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