Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation

Sommer, Matthias; Brazda, Katharina; Hantel, Michael

doi:10.1016/j.physleta.2011.07.038

Cited by 4 publications

(4 citation statements)

References 22 publications

(50 reference statements)

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“…It should be noted that, apart for notable exceptions (e.g. [17]), the construction of the hydrodynamics Nambu brackets was mainly based on intuition and guessing. The aim of this study is thus to derive the Nambu brackets for hydrodynamics using a geometrical approach, which is based on replacing the Jacobian by a two-form based on the two CLs, as suggested for finite dimensional systems by [18].…”

Section: Introductionmentioning

confidence: 99%

Hydrodynamic Nambu brackets derived by geometric constraints

Blender¹,

Badin²

2015

J. Phys. A: Math. Theor.

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Abstract. A geometric approach to derive the Nambu brackets for ideal twodimensional (2D) hydrodynamics is suggested. The derivation is based on two-forms with vanishing integrals in a periodic domain, and with resulting dynamics constrained by an orthogonality condition. As a result, 2D hydrodynamics with vorticity as dynamic variable emerges as a generic model, with conservation laws which can be interpreted as enstrophy and energy functionals. Generalized forms like surface quasigeostrophy and fractional Poisson equations for the stream-function are also included as results from the derivation. The formalism is extended to a hydrodynamic system coupled to a second degree of freedom, with the Rayleigh-Bénard convection as an example. This system is reformulated in terms of constitutive conservation laws with two additive brackets which represent individual processes: a first representing inviscid 2D hydrodynamics, and a second representing the coupling between hydrodynamics and thermodynamics. The results can be used for the formulation of conservative numerical algorithms that can be employed, for example, for the study of fronts and singularities.

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Section: Introductionmentioning

confidence: 99%

Hydrodynamic Nambu brackets derived by geometric constraints

Blender¹,

Badin²

2015

J. Phys. A: Math. Theor.

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“…[KrMi97], 32.12). Pour cet exemple, on pourra consulter [BiMo91] (voir aussi [SBH11]). Soit g une algèbre de Lie semi-simple de dimension n muni du crochet [., .]…”

Section: Distribution Caractéristiqueunclassified

Structures bihamiltoniennes partielles

Cabau¹,

Pelletier²

2022

Preprint

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“…Thus the Nambu tensor is cyclic and anti-symmetric. Furthermore, it satisfies the generalized Jacobi identity [33,31] N ijk N lpq + N ijq N lkp + N ijp N lqk = 0 (35) in the form required for constant Nambu tensors. Based on (33) a Nambu bracket for arbitrary phase space functions F (x) is written dF dt = {F, C, H}…”

Section: Nambu Structurementioning

confidence: 99%

“…Thus the Nambu tensor is cyclic and anti-symmetric. Furthermore, it satisfies the generalized Jacobi identity [33,31]…”

Section: Nambu Structurementioning

confidence: 99%

Nambu representation of an extended Lorenz model with viscous heating

Blender¹,

Lucarini²

2013

Physica D: Nonlinear Phenomena

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We consider the Nambu and Hamiltonian representations of RayleighBénard convection with a nonlinear thermal heating effect proportional to the Eckert number (Ec). The model we use is an extension of the classical Lorenz-63 model with 4 kinematic and 6 thermal degrees of freedom. The conservative parts of the dynamical equations which include all nonlinearities satisfy Liouville's theorem and permit a conserved Hamiltonian H for arbitrary Ec. For Ec = 0 two independent conserved Casimir functions exist, one of these is associated with unavailable potential energy and is also present in the Lorenz-63 truncation. This Casimir C is used to construct a Nambu representation of the conserved part of the dynamical system. The thermal heating effect can be represented either by a second canonical Hamiltonian or as a gradient (metric) system using the time derivativeĊ of the Casimir. The results demonstrate the impact of viscous heating in the total energy budget and in the Lorenz energy cycle for kinetic and available potential energy.

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Algebraic construction of a Nambu bracket for the two-dimensional vorticity equation

Cited by 4 publications

References 22 publications

Hydrodynamic Nambu brackets derived by geometric constraints

Hydrodynamic Nambu brackets derived by geometric constraints

Structures bihamiltoniennes partielles

Nambu representation of an extended Lorenz model with viscous heating

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