H 2 The geometrical gyro-kinetic approximation

Auteurs : Frénod, Emmanuel (Auteur de la Conférence)
CIRM (Editeur )

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tokamak gyrokinetic hamiltonian system canonical coordinates Darboux coordinate system Lie coordinate system Lie sum gyrokinetic approximation Questions

Résumé : At the end of the 70', Littlejohn [1, 2, 3] shed new light on what is called the Gyro-Kinetic Approximation. His approach incorporated high-level mathematical concepts from Hamiltonian Mechanics, Differential Geometry and Symplectic Geometry into a physical affordable theory in order to clarify what has been done for years in the domain. This theory has been being widely used to deduce the numerical methods for Tokamak and Stellarator simulation. Yet, it was formal from the mathematical point of view and not directly accessible for mathematicians.
This talk will present a mathematically rigorous version of the theory. The way to set out this Gyro-Kinetic Approximation consists of the building of a change of coordinates that decouples the Hamiltonian dynamical system satisfied by the characteristics of charged particles submitted to a strong magnetic field into a part that concerns the fast oscillation induced by the magnetic field and a other part that describes a slower dynamics.
This building is made of two steps. The goal of the first one, so-called "Darboux Algorithm", is to give to the Poisson Matrix (associated to the Hamiltonian system) a form that would achieve the goal of decoupling if the Hamiltonian function does not depend on one given variable. Then the second change of variables (which is in fact a succession of several ones), so-called "Lie Algorithm", is to remove the given variable from the Hamiltonian function without changing the form of the Poisson Matrix.
(Notice that, beside this Geometrical Gyro-Kinetic Approximation Theory, an alternative approach, based on Asymptotic Analysis and Homogenization Methods was developed in Frenod and Sonnendrücker [5, 6, 7], Frenod, Raviart and Sonnendrücker [4], Golse and Saint-Raymond [9] and Ghendrih, Hauray and Nouri [8].)

Codes MSC :
58D10 - Spaces of imbeddings and immersions
58J37 - Perturbations; asymptotics
58J45 - Hyperbolic equations
58Z05 - Applications to physics
70H05 - Hamilton's equations
82D10 - Plasmas

    Informations sur la Vidéo

    Langue : Anglais
    Date de publication : 13/08/15
    Date de captation : 11/08/15
    Collection : Partial Differential Equations ; Mathematical Physics
    Sous collection : Research talks
    Domaine : PDE ; Mathematical Physics
    Format : QuickTime (.mov) Durée : 01:01:53
    Audience : Chercheurs ; Doctorants , Post - Doctorants
    Download : https://videos.cirm-math.fr/2015-08-11_Frenod.mp4

Informations sur la rencontre

Nom de la rencontre : CEMRACS: Coupling multi-physics models involving fluids / CEMRACS : Couplage de modèles multi-physiques impliquant les fluides
Organisateurs de la rencontre : Frénod, Emmanuel ; Maitre, Emmanuel ; Rousseau, Antoine ; Salmon, Stéphanie ; Szopos, Marcela
Dates : 20/07/15 - 28/08/15
Année de la rencontre : 2015
URL Congrès : http://conferences.cirm-math.fr/1278.html

Citation Data

DOI : 10.24350/CIRM.V.18803503
Cite this video as: Frénod, Emmanuel (2015). The geometrical gyro-kinetic approximation. CIRM. Audiovisual resource. doi:10.24350/CIRM.V.18803503
URI : http://dx.doi.org/10.24350/CIRM.V.18803503


  • [1] Littlejohn, Robert G. (1979). A guiding center Hamiltonian: A new approach. Journal of Mathematical Physics, 20(12), 2445-2458 - http://dx.doi.org/10.1063/1.524053

  • [2] Littlejohn, Robert G. (1981). Hamiltonian formulation of guiding center motion. Physics of Fluids, 24(9), 1730-1749 - http://dx.doi.org/10.1063/1.863594

  • [3] Littlejohn, Robert G. (1982). Hamiltonian perturbation theory in noncanonical coordinates. Journal of Mathematical Physics, 23(5), 742-747 - http://dx.doi.org/10.1063/1.525429

  • [4] Frénod, E., & Lutz, M. (2014). On the Geometrical Gyro-Kinetic Theory. Kinetic and Related Models, 7(4), p.621-659. - https://hal.archives-ouvertes.fr/hal-00837591

  • [5] Frénod, E., Raviart, P.A., & Sonnendrücker, E. (2001). Two-scale expansion of a singularly perturbed convection equation. Journal de Mathématiques Pures et Appliquées, 80(8), 815-843 - http://dx.doi.org/10.1016/S0021-7824(01)01215-6

  • [6] Frénod, E., & Sonnendrücker, E. (1998). Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field. Asymptotic Analysis, 18(3-4), 193-214. - https://hal.inria.fr/inria-00073462

  • [7] Frénod, E., & Sonnendrücker, E. (2000). Long time behavior of the two dimensionnal Vlasov equation with a strong external magnetic field. Mathematical Models & Methods in Applied Sciences, 10(4), 539-553 - http://dx.doi.org/10.1142/S021820250000029X

  • [8] Frénod, E., & Sonnendrücker, E. (2001). The Finite Larmor Radius Approximation. SIAM Journal on Mathematical Analysis, 32(6), 1227-1247 - http://dx.doi.org/10.1137/S0036141099364243

  • [9] Ghendrih, P., Hauray, M., & Nouri, A. (2010). Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions. Kinetic and related models, 707-725 - http://dx.doi.org/10.3934/krm.2009.2.707

  • [10] Golse, F., & Saint-Raymond, L. (1999). The Vlasov-Poisson system with strong magnetic field. Journal de Mathématiques Pures et Appliquées, 78(8), 791-817 - http://dx.doi.org/10.1016/S0021-7824(99)00021-5

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