Geodesic
Hamiltonian reduced fluid model for plasmas with temperature and heat
Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 3, pp. 477-496 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

For an arbitrary number of species, we derive a Hamiltonian fluid model for strongly magnetized plasmas describing the evolution of the density, velocity, and electromagnetic fluctuations and also of the temperature and heat flux fluctuations associated with motions parallel and perpendicular to the direction of a background magnetic field. We derive the model as a reduction of the infinite hierarchy of equations obtained by taking moments of a Hamiltonian drift-kinetic system with respect to Hermite–Laguerre polynomials in velocity–magnetic-moment coordinates. We show that a closure relation directly coupling the heat flux fluctuations in the directions parallel and perpendicular to the background magnetic field provides a fluid reduction that preserves the Hamiltonian character of the parent drift-kinetic model. We find an alternative set of dynamical variables in terms of which the Poisson bracket of the fluid model takes a structure of a simple direct sum and permits an easy identification of the Casimir invariants. Such invariants in the limit of translational symmetry with respect to the direction of the background magnetic field turn out to be associated with Lagrangian invariants of the fluid model. We show that the coupling between the parallel and perpendicular heat flux evolutions introduced by the closure is necessary for ensuring the existence of a Hamiltonian structure with a Poisson bracket obtained as an extension of a Lie–Poisson bracket.
Keywords: fluid reduction, plasma, noncanonical Poisson bracket, drift kinetics. 
@article{TMF_2016_188_3_a8,
     author = {E. Tassi},
     title = {Hamiltonian reduced fluid model for plasmas with temperature and heat},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {477--496},
     year = {2016},
     volume = {188},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2016_188_3_a8/}
}
TY  - JOUR
AU  - E. Tassi
TI  - Hamiltonian reduced fluid model for plasmas with temperature and heat
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2016
SP  - 477
EP  - 496
VL  - 188
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2016_188_3_a8/
LA  - ru
ID  - TMF_2016_188_3_a8
ER  - 
%0 Journal Article
%A E. Tassi
%T Hamiltonian reduced fluid model for plasmas with temperature and heat
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2016
%P 477-496
%V 188
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2016_188_3_a8/
%G ru
%F TMF_2016_188_3_a8
E. Tassi. Hamiltonian reduced fluid model for plasmas with temperature and heat. Teoretičeskaâ i matematičeskaâ fizika, Tome 188 (2016) no. 3, pp. 477-496. http://geodesic.mathdoc.fr/item/TMF_2016_188_3_a8/

[1] A. Brizard, Phys. Fluids B, 4:5 (1992), 1213–1228 | DOI | MR

[2] B. Scott, Phys. Plasmas, 17:10 (2010), 102306 | DOI

[3] T. Passot, P. L. Sulem, Phys. Plasmas, 11:11 (2004), 5173–5189 | DOI | MR

[4] T. Passot, P. L. Sulem, Phys. Plasmas, 14:8 (2007), 082502, 14 pp. | DOI

[5] T. Passot, P. L. Sulem, P. Hunana, Phys. Plasmas, 19:8 (2012), 082113 | DOI

[6] P. B. Snyder, G. W. Hammett, W. Dorland, Phys. Plasmas, 4:11 (1997), 3974–3985 | DOI | MR

[7] P. B. Snyder, G. W. Hammett, Phys. Plasmas, 8:7 (2001), 3199–3216 | DOI

[8] H. Sugama, T.-H. Watanabe, W. Horton, Phys. Plasmas, 8:6 (2001), 2617–2628 | DOI | MR

[9] Y. Sarazin, G. Dif-Pradalier, D. Zarzoso, X. Garbet, Ph. Ghendrih, V. Grandgirard, Plasma Phys. Control. Fusion, 51:11 (2009), 115003, 14 pp. | DOI

[10] P. J. Morrison, Phys. Lett. A, 80:5–6 (1980), 383–386 | DOI | MR

[11] J. E. Marsden, A. Weinstein, Phys. D, 4:3 (1982), 394–406 | DOI | MR | Zbl

[12] B. A. Kupershmidt, Yu. I. Manin, Funkts. analiz i ego pril., 12:1 (1978), 25–37 | DOI | MR | Zbl

[13] J. Gibbons, Phys. D, 3:3 (1981), 503–511 | DOI | MR | Zbl

[14] J. Gibbons, D. D. Holm, C. Tronci, Phys. Lett. A, 372:7 (2008), 1024–1033 | DOI | MR | Zbl

[15] J. E. Marsden, P. J. Morrison, A. Weinstein, “The Hamiltonian structure of the BBGKY hierarchy equations”, Fluids and Plasmas: Geometry and Dynamics (University of Colorado, Boulder, Colorado, USA, July 17–23, 1983), Contemporary Mathematics, 28, ed. J. E. Marsden, AMS, Providence, RI, 1984, 115–124 | DOI | MR | Zbl

[16] A. A. Chesnokov, M. V. Pavlov, Acta Appl. Math., 122:1 (2012), 367–380 | DOI | MR | Zbl

[17] M. Perin, C. Chandre, P. J. Morrison, E. Tassi, Ann. Phys., 348 (2014), 50–63 | DOI | MR | Zbl

[18] M. Perin, C. Chandre, P. J. Morrison, E. Tassi, J. Phys. A: Math. Theor., 48:27 (2015), 275501, 24 pp. | DOI | MR | Zbl

[19] E. Tassi, J. Phys. A: Math. Theor., 47:19 (2014), 195501, 25 pp. | DOI | MR | Zbl

[20] E. Tassi, Eur. Phys. J. D, 68:7 (2014), 196–206 | DOI

[21] E. Tassi, J. Phys.: Conf. Ser., 561:1 (2014), 012018, 9 pp. | DOI

[22] E. Tassi, Ann. Phys., 362 (2015), 239–260 | DOI | MR | Zbl

[23] R. D. Hazeltine, J. D. Meiss, Plasma Confinement, Dover, New York, 2003

[24] D. Grasso, E. Tassi, J. Plasma Phys., 81:5 (2015), 495810501, 22 pp. | DOI

[25] P. J. Morrison, Rev. Modern Phys., 70:2 (1998), 467–522 | DOI | MR

[26] J. E. Marsden, T. S. Ratiu, Introduction to Mechanics and Symmetry. A Basic Exposition of Classical Mechanical Systems, Texts in Applied Mathematics, 17, Springer, New York, 2010 | DOI | MR

[27] H. J. de Blank, “Kinetic electrons in drift-Alfv'en current-vortex filaments”, Proceedings of the 28th Conference on Controlled Fusion and Plasma Physics (Funchal, Madeira, Portugal, 18–22 June, 2001), Europhysics Conference Abstracts (ECA), 25A, EPS, France, 2001, 1709–1712

[28] F. L. Waelbroeck, E. Tassi, Commun. Nonlinear Sci. Numer. Simul., 17:5 (2012), 2171–2178 | DOI | MR | Zbl

[29] E. Tassi, P. J. Morrison, D. Grasso, F. Pegoraro, Nucl. Fusion, 50:3 (2010), 034007, 8 pp. | DOI

[30] E. Tassi, P. J. Morrison, F. L. Waelbroeck, D. Grasso, Plasma Phys. Control. Fusion, 50:8 (2008), 085014, 29 pp. | DOI

[31] F. L. Waelbroeck, R. D. Hazeltine, P. J. Morrison, Phys. Plasmas, 16:3 (2009), 032109 | DOI

[32] F. L. Waelbroeck, P. J. Morrison, W. Horton, Plasma Phys. Control. Fusion, 46:9 (2004), 1331–135 | DOI

[33] R. D. Hazeltine, C. T. Hsu, P. J. Morrison, Phys. Fluids, 30:10 (1987), 3204–3211 | DOI | Zbl

[34] D. Dagnelund, V. P. Pavlenko, Phys. Scr., 71:3 (2005), 293–297 | DOI | Zbl

[35] A. Weinstein, Phys. Fluids, 26:2 (1983), 388–390 | DOI | MR | Zbl

[36] B. N. Kuvshinov, F. Pegoraro, T. J. Schep, Phys. Lett. A, 191:3–4 (1994), 296–300 | DOI

[37] J.-L. Thiffeault, P. J. Morrison, Phys. D, 136:3–4 (2000), 205–244 | DOI | MR | Zbl

[38] E. Cafaro, D. Grasso, F. Pegoraro, F. Porcelli, A. Saluzzi, Phys. Rev. Lett., 80:20 (1998), 4430–4434 | DOI