Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T14:47:24.275Z Has data issue: false hasContentIssue false
  • English
  • Français

Differential formulation of the gyrokinetic Landau operator

Part of: Solved and Unsolved problems in Plasma Physics Focus on Fusion

Published online by Cambridge University Press:  05 January 2017

Eero Hirvijoki ,
Alain J. Brizard  and
David Pfefferlé
Eero Hirvijoki*
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA
Alain J. Brizard
Affiliation:
Department of Physics, Saint Michael’s College, Colchester, VT 05439, USA
David Pfefferlé
Affiliation:
Princeton Plasma Physics Laboratory, Princeton, NJ 08543, USA
*
†Email address for correspondence: ehirvijo@pppl.gov
Get access Rights & Permissions [Opens in a new window]

Abstract

Subsequent to the recent rigorous derivation of an energetically consistent gyrokinetic collision operator in the so-called Landau representation, this paper investigates the possibility of finding a differential formulation of the gyrokinetic Landau collision operator. It is observed that, while a differential formulation is possible in the gyrokinetic phase space, reduction of the resulting system of partial differential equations to five dimensions via gyroaveraging poses a challenge. Based on the present work, it is likely that the gyrocentre analogues of the Rosenbluth–MacDonald–Judd potential functions must be kept gyroangle dependent.

Keywords

fusion plasmaplasma nonlinear phenomenaplasma simulation
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 1 , February 2017 , 595830102
Copyright
© Cambridge University Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Abel, I. G., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2008 Linearized model Fokker–Planck collision operators for gyrokinetic simulations. I. Theory. Phys. Plasmas 15 (12), 122509.CrossRefGoogle Scholar
Brizard, A. J. 2004 A guiding-center Fokker–Planck collision operator for nonuniform magnetic fields. Phys. Plasmas 11 (9), 4429–4438.Google Scholar
Brizard, A. J., Decker, J., Peysson, Y. & Duthoit, F.-X. 2009 Orbit-averaged guiding-center Fokker–Planck operator. Phys. Plasmas 16 (10), 102304.Google Scholar
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421–468.CrossRefGoogle Scholar
Burby, J. W., Brizard, A. J. & Qin, H. 2015 Energetically consistent collisional gyrokinetics. Phys. Plasmas 22, 100707.CrossRefGoogle Scholar
Cary, J. R. & Brizard, A. J. 2009 Hamiltonian theory of guiding-center motion. Rev. Mod. Phys. 81, 693–738.Google Scholar
Catto, P. J. & Tsang, K. T. 1977 Linearized gyro-kinetic equation with collisions. Phys. Fluids 20 (3), 396–401.Google Scholar
Goncharov, P. R. 2010 A note on the Coulomb collision operator in curvilinear coordinates. Plasma Phys. Control. Fusion 52, 102001.CrossRefGoogle Scholar
Hager, R., Yoon, E. S., Ku, S., D’Azevedo, E. F., Worley, P. H. & Chang, C. S. 2016 A fully non-linear multi-species Fokker–Planck–Landau collision operator for simulation of fusion plasma. J. Comput. Phys. 315, 644–660.CrossRefGoogle Scholar
Hirvijoki, E., Brizard, A. J., Snicker, A. & Kurki-Suonio, T. 2013 Monte Carlo implementation of a guiding-center Fokker–Planck kinetic equation. Phys. Plasmas 20 (9), 092505.Google Scholar
Landau, L. D. 1937 The kinetic equation in the case of Coulomb interaction (translated from German). Zh. Eksp. Teor. Fiz. 7, 203–209.Google Scholar
Li, B. & Ernst, D. R. 2011 Gyrokinetic Fokker–Planck collision operator. Phys. Rev. Lett. 106, 195002.CrossRefGoogle ScholarPubMed
Littlejohn, R. G. 1982 Hamiltonian perturbation theory in noncanonical coordinates. J. Math. Phys. 23, 742–747.Google Scholar
Madsen, J. 2013 Gyrokinetic linearized Landau collision operator. Phys. Rev. E 87, 011101.Google Scholar
Mishchenko, A. & Könies, A. 2007 A many-particle approach to the gyro-kinetic theory. J. Plasma Phys. 73 (5), 757–772.Google Scholar
Pataki, A. & Greengard, L. 2011 Fast elliptic solvers in cylindrical coordinates and the Coulomb collision operator. J. Comput. Phys. 230 (21), 7840–7852.CrossRefGoogle Scholar
Rosenbluth, M. N., Macdonald, W. M. & Judd, D. L. 1957 Fokker–Planck equation for an inverse-square force. Phys. Rev. 107, 1–6.CrossRefGoogle Scholar
Sugama, H., Watanabe, T.-H. & Nunami, M. 2009 Linearized model collision operators for multiple ion species plasmas and gyrokinetic entropy balance equations. Phys. Plasmas 16 (11), 112503.Google Scholar
Taitano, W. T., Chacón, L., Simakov, A. N. & Molvig, K. 2015 A mass, momentum, and energy conserving, fully implicit, scalable algorithm for the multi-dimensional, multi-species Rosenbluth–Fokker–Planck equation. J. Comput. Phys. 297, 357–380.CrossRefGoogle Scholar
Xu, X. Q. & Rosenbluth, M. N. 1991 Numerical simulation of ion-temperature-gradient-driven modes. Phys. Fluids B 3 (3), 627–643.CrossRefGoogle Scholar