Hamiltonian derivation of the Charney–Hasegawa–Mima equation

Tassi, E.; Chandre, Cristel; Morrison, Philip

doi:10.1063/1.3194275

Cited by 24 publications

(22 citation statements)

References 31 publications

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“…For example, if we try to do a numerical search of the quasi-resonant modes with wavenumbers less than or equal to 10000 in size, a computer will need to do a search over 10000 4 possible triads, and evaluate the frequency mismatch. If τ is the time it takes the computer to do one triad, then the total time for this search will be approximately τ × 10 16 . On a medium-sized cluster, using optimised C ++ codes, the effective time for one such computation is about τ ≈ 5 × 10 −8 seconds, which means that the total computing time for all triads is about 15 years!…”

Section: Introductionmentioning

confidence: 99%

Complete classification of discrete resonant Rossby/drift wave triads on periodic domains

Bustamante¹,

Hayat²

2013

Communications in Nonlinear Science and Numerical Simulation

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We consider the set of Diophantine equations that arise in the context of the partial differential equation called "barotropic vorticity equation" on periodic domains, when nonlinear wave interactions are studied to leading order in the amplitudes. The solutions to this set of Diophantine equations are of interest in atmosphere (Rossby waves) and Tokamak plasmas (drift waves), because they provide the values of the spectral wavevectors that interact resonantly via three-wave interactions. These wavenumbers come in "triads", i.e., groups of three wavevectors.We provide the full solution to the Diophantine equations in the physically sensible limit when the Rossby deformation radius is infinite. The method is completely new, and relies on mapping the unknown variables via rational transformations, first to rational points on elliptic curves and surfaces, and from there to rational points on quadratic forms of "Minkowski" type (such as the familiar space-time in special relativity). Classical methods invented centuries ago by Fermat, Euler, Lagrange, Minkowski, are used to classify all solutions to our original Diophantine equations, thus providing a computational method to generate numerically all the resonant triads in the system. Computationally speaking, our method has a clear advantage over brute-force numerical search: on a 10000 2 grid, the brute-force search would take 15 years using optimised C ++ codes on a cluster, whereas our method takes about 40 minutes using a laptop.Moreover, the method is extended to generate so-called quasi-resonant triads, which are defined by relaxing the resonant condition on the frequencies, allowing for a small mismatch. Quasi-resonant triads' distribution in wavevector space is robust with respect to physical perturbations, unlike resonant triads' distribution. Therefore, the extended method is really valuable in practical terms. We show that the set of quasi-resonant triads form an intricate network of connected triads, forming clusters whose structure depends on the value of the allowed mismatch. It is believed that understanding this network is absolutely relevant to understanding turbulence. We provide some quantitative comparison between the clusters' structure and the onset of fully nonlinear turbulent regime in the barotropic vorticity equation, and we provide perspectives for new research.

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Section: Introductionmentioning

confidence: 99%

Complete classification of discrete resonant Rossby/drift wave triads on periodic domains

Bustamante¹,

Hayat²

2013

Communications in Nonlinear Science and Numerical Simulation

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“…More recently there have been several works (Morrison et al 2009;Tassi et al 2009;Chandre et al 2013), following Nguyen & Turski (1999, 2001, that treat the enforcement of the incompressibility constraint of hydrodynamics by Dirac's method of constraints (Dirac 1950). In these works the compressibility constraint was enforced in the Eulerian variable description of the fluid using the noncanonical Poisson bracket of § 3.3 as the base bracket of a generalization of Dirac's constraint theory.…”

Section: Lagrangian-dirac Constraint Theorymentioning

confidence: 99%

“…This we do in § 4.3.1. Alternatively, we can proceed as in Nguyen & Turski (1999, 2001, Tassi et al (2009), Chandre et al (2013) and Morrison et al (2009), starting from the Eulerian noncanonical theory of § 3.3 and directly construct a Dirac bracket with Eulerian constraints. This is a valid procedure because Dirac's construction works for noncanonical Poisson brackets, as shown, e.g.…”

Section: Eulerian-dirac Constraint Theorymentioning

confidence: 99%

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

2020

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The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

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“…In this appendix we present the calculations leading to the beatified Poisson bracket of equation (48). The operators J , D, and D † defined by expressions (7), (41), and (46), respectively, satisfy various identities a few of which we will use. First, the operators J and D † satisfy Leibniz rules, i.e.,…”

Section: A Beatification To Second Ordermentioning

confidence: 99%

A method for Hamiltonian truncation: a four-wave example

Viscondi¹,

Caldas²,

Morrison³

2016

J. Phys. A: Math. Theor.

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A method for extracting finite-dimensional Hamiltonian systems from a class of 2+1 Hamiltonian mean field theories is presented. These theories possess noncanonical Poisson brackets, which normally resist Hamiltonian truncation, but a process of beatification by coordinate transformation near a reference state is described in order to perturbatively overcome this difficulty. Two examples of four-wave truncation of Euler's equation for scalar vortex dynamics are given and compared: one a direct non-Hamiltonian truncation of the equations of motion, the other obtained by beatifying the Poisson bracket and then truncating. arXiv:1509.09247v2 [physics.flu-dyn]

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Hamiltonian derivation of the Charney–Hasegawa–Mima equation

Abstract: The Charney-Hasegawa-Mima equation is an infinite-dimensional Hamiltonian system with dynamics generated by a noncanonical Poisson bracket. Here a first principle Hamiltonian derivation of this system, beginning with the ion fluid dynamics and its known Hamiltonian form, is given.

Cited by 24 publications

References 31 publications

Complete classification of discrete resonant Rossby/drift wave triads on periodic domains

Complete classification of discrete resonant Rossby/drift wave triads on periodic domains

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

A method for Hamiltonian truncation: a four-wave example

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