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The geometrical gyro-kinetic approximation
Post-edited
Authors : Frénod, Emmanuel (Author of the conference)
CIRM (Publisher )
58D10 - Spaces of imbeddings and immersions
58J37 - Perturbations; asymptotics
58J45 - Hyperbolic equations
58Z05 - Applications to physics
70H05 - Hamilton's equations
82D10 - Plasmas
CIRM (Publisher )
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| tokamak gyrokinetic hamiltonian system canonical coordinates Darboux coordinate system Lie coordinate system Lie sum gyrokinetic approximation Questions |
Abstract :
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].)
MSC Codes : 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].)
58D10 - Spaces of imbeddings and immersions
58J37 - Perturbations; asymptotics
58J45 - Hyperbolic equations
58Z05 - Applications to physics
70H05 - Hamilton's equations
82D10 - Plasmas
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Information on the Video
Film maker : Hennenfent, GuillaumeLanguage : English Available date : 13/08/15 Conference Date : 11/08/15 Subseries : Research talks arXiv category : Analysis of PDEs ; Differential Geometry ; Mathematical Physics Mathematical Area(s) : PDE ; Mathematical Physics Format : QuickTime (.mov) Video Time : 01:01:53 Targeted Audience : Researchers Download : https://videos.cirm-math.fr/2015-08-11_Frenod.mp4 
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Information on the Event
Event Title : CEMRACS: Coupling multi-physics models involving fluids / CEMRACS : Couplage de modèles multi-physiques impliquant les fluidesEvent Organizers : Frénod, Emmanuel ; Maitre, Emmanuel ; Rousseau, Antoine ; Salmon, Stéphanie ; Szopos, Marcela Dates : 20/07/15 - 28/08/15 Event Year : 2015 Event URL : http://conferences.cirm-math.fr/1278.html
Citation Data
DOI : 10.24350/CIRM.V.18803503Cite 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 |
Bibliography
- [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
