Abstract
Under conditions when the particles’ thermal gyro-radii are smaller than the Debye length \(\lambda _D\), the magnetic field influence on the collision dynamics and associated transport processes becomes significant and has to be considered. In this paper, derivation of the kinetic equation based on the Fokker–Planck approach is reviewed for uniform magnetized plasmas with \(\rho _{the}<\lambda _D<\rho _{thi}\), where \(\rho _{the}\) and \(\rho _{thi}\) are the electron and ion thermal gyro-radii, respectively. Generally speaking, current tokamak plasmas lie in the above studied parameter range. Serving as an example for the application of the derived magnetized kinetic equation and demonstrating the modifications to the transport coefficients from a strong magnetic field, the electron-ion temperature relaxation is reviewed. The relaxation time exhibits a noticeable magnetic field dependence.
Similar content being viewed by others
References
N. Rostoker, M.N. Rosenbluth, Test particles in a completely ionized plasma. Phys. Fluids 3, 1–14 (1960)
N. Rostoker, Kinetic equation with a constant magnetic field. Phys. Fluids 3, 922–927 (1960)
M.H.A. Hassan, C.J.H. Watson, Magnetized plasma kinetic theory. I. Derivation of the kinetic equation for a uniform magnetized plasma. Plasma Phys. 19, 237–247 (1977)
A.H. Øien, Anomalous temperature relaxation and particle transport in a strongly non-uniform, fully ionized plasma in a strong magnetic field. J. Plasma Phys. 53, 31–48 (1995)
Y.L. Klimontovich, Kinetic Theory of Nonideal Gases and Nonideal Plasmas (Pergamon Press, New York, 1982), pp. 189–198
P.H. Yoon, Collisional relaxation of bi-Maxwellian plasma temperatures in magnetized plasmas. Phys. Plasmas 23, 072114 (2016)
A.G. Peeters, D. Strintzi, The Fokker–Planck equation, and its application in plasma physics. Ann. Phys. 17, 142–157 (2008)
M.N. Rosenbluth, W.M. MacDonald, D.L. Judd, Fokker–Planck equation for an inverse-square force. Phys. Rev. 107, 1–6 (1957)
J. Hubbard, The friction and diffusion coefficients of the Fokker–Planck equation in a plasma. Proc. R. Soc. London Ser. A 260, 114–126 (1961)
J. Hubbard, The friction and diffusion coefficients of the Fokker–Planck equation in a plasma II. Proc. R. Soc. London Ser. A 261, 371–387 (1961)
A.G. Sitenko, C. Yu-mai, Coefficients of dynamical friction and diffusion in a plasma. Sov. Phys.-Tech. Phys. 7, 978–984 (1963)
S. Ichimaru, M.N. Rosenbluth, Relaxation processes in plasmas with magnetic field. Temperature relaxations. Phys. Fluids 13, 2778–2789 (1970)
T.B. Kaiser, Comments on “Relaxation processes in plasmas with magnetic field”. Phys. Fluids 22, 593–594 (1979)
K. Matsuda, Fokker–Planck equation for a plasma in a magnetic field with electrostatic fluctuations. Phys. Fluids 26, 1508–1515 (1983)
A.A. Ware, Electron Fokker–Planck equation for collisions with ions in a magnetized plasma. Phys. Rev. Lett. 62, 51–54 (1989)
C.E. Newman, A generalization of the equations governing the evolution of a particle distribution in a random force field. J. Math. Phys. 14, 502–508 (1973)
C. Dong, W. Zhang, D. Li, Fokker–Planck equation in the presence of a uniform magnetic field. Phys. Plasmas 23, 082105 (2016)
C. Dong, D. Li, C. Jiang, Electron-electron collision term describing the reflections induced scattering in a magnetized plasma. Chin. Phys. Lett. 36, 075201 (2019)
C. Dong, W. Zhang, J. Cao, D. Li, Derivation of the magnetized Balescu-Lenard-Guernsey collision term based on the Fokker–Planck approach. Phys. Plasmas 24, 122120 (2017)
Y.M. Aliev, V.P. Silin, Rate of equalization of “longitudinal” and “transverse” plasma temperatures. Nucl. Fusion 3, 248–250 (1963)
D. Montgomery, G. Joyce, L. Turner, Magnetic field dependence of plasma relaxation times. Phys. Fluids 17, 2201–2204 (1974)
C. Dong, H. Ren, H. Cai, D. Li, Effects of magnetic field on anisotropic temperature relaxation. Phys. Plasmas 20, 032512 (2013)
V.P. Silin, On relaxation of electron and ion temperatures of fully ionized plasma in a strong magnetic field. Sov. Phys. JETP 16, 1281–1285 (1963)
P. Ghendrih, A. Samain, J.H. Misguich, Magnetic field dependence of the energy-equipartition frequency and the resistivity. Phys. Lett. A 119, 354–358 (1987)
C. Dong, H. Ren, H. Cai, D. Li, Temperature relaxation in a magnetized plasma. Phys. Plasmas 20, 102518 (2013)
Y.M. Aliev, A.R. Shister, Transport phenomena in a plasma in a strong magnetic field. Sov. Phys. JETP 18, 1035–1040 (1964)
S. Ichimaru, T. Tange, Theory of classical and anomalous diffusion of a plasma across a magnetic field. J. Phys. Soc. Jpn. 36, 603–609 (1974)
A.H. Sørensen, E. Bonderup, Electron cooling. Nucl. Instrum. Methods Phys. Res. 215, 27–54 (1983)
H.B. Nersisyan, C. Toepffer, G. Zwicknagel, Interactions Between Charged Particles in a Magnetic Field: A Theoretical Approach to Ion Stopping in Magnetized Plasmas (Springer, Heidelberg, 2007)
L.I. Men’shikov, New directions in the theory of electron cooling. Phys. Usp. 51, 645–680 (2008)
H.B. Nersisyan, G. Zwicknagel, Cooling force on ions in a magnetized electron plasma. Phys. Rev. ST Accel. Beams 16, 074201 (2013)
E.S. Evans, S.A. Cohen, D.R. Welch, Particle-in-cell studies of fast-ion slowing-down rates in cool tenuous magnetized plasma. Phys. Plasmas 25, 042105 (2018)
S. Cohen, E. Sarid, M. Gedalin, Collisional relaxation of a strongly magnetized ion-electron plasma. Phys. Plasmas 26, 082105 (2019)
K. Matsuda, Anomalous magnetic field effects on electron-ion collisions. Phys. Rev. Lett. 49, 1486–1488 (1982)
M. Psimopoulos, D. Li, Cross field thermal transport in highly magnetized plasmas. Proc. R. Soc. London Ser. A 437, 55–65 (1992)
D.H.E. Dubin, T.M. O’Neil, Cross-magnetic-field heat conduction in non-neutral plasmas. Phys. Rev. Lett. 78, 3868–3871 (1997)
S. Chandrasekhar, Stochastic problems in physics and astronomy. Rev. Mod. Phys. 15, 1–89 (1943)
The closest distance \(L_c\) between the two colliding particles is identical to the usual impact parameter when the particles’ actual relative trajectory deviates slightly from its unperturbed one during the collision process. However, for the e-e collisions with reflections, the relative parallel velocity of the two electrons reverses during the collision and the relative motion changes notably compared to the unperturbed case. Under this condition, \(L_c\) and the impact parameter are not equivalent. Thus, \(L_c\) is used instead of the impact parameter to characterize the collision
J.L. Hurt, P.T. Carpenter, C.L. Taylor, F. Robicheaux, Positron and electron collisions with anti-protons in strong magnetic fields. J. Phys. B At. Mol. Opt. Phys. 41, 165206 (2008)
H.B. Nersisyan, G. Zwicknagel, Binary collisions of charged particles in a magnetic field. Phys. Rev. E 79, 066405 (2009)
D.H.E. Dubin, Parallel velocity diffusion and slowing-down rate from long-range collisions in a magnetized plasma. Phys. Plasmas 21, 052108 (2014)
H.B. Nersisyan, G. Zwicknagel, C. Toepffer, Energy loss of ions in a magnetized plasma: Conformity between linear response and binary collision treatments. Phys. Rev. E 67, 026411 (2003)
M. Greenwald, A. Bader, S. Baek, M. Bakhtiari, H. Barnard, W. Beck, W. Bergerson, I. Bespamyatnov, P. Bonoli, D. Brower et al., 20 years of research on the Alcator C-Mod tokamak. Phys. Plasmas 21, 110501 (2014)
Acknowledgements
This work was supported by the National MCF Energy R&D Program under Grant No. 2018YFE0311300, the National Natural Science Foundation of China under Grant Nos. 11875067, 11835016, 11705275, 11675257, and 11675256, the Strategic Priority Research Program of Chinese Academy of Sciences under Grant No. XDB16010300, the Key Research Program of Frontier Science of Chinese Academy of Sciences under Grant No. QYZDJ-SSW-SYS016, and the External Cooperation Program of Chinese Academy of Sciences under Grant No. 112111KYSB20160039.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Dong, C., Li, D. Strong Magnetic Field Effects on the Collision Term and Electron-Ion Temperature Relaxation. J Fusion Energ 39, 390–400 (2020). https://doi.org/10.1007/s10894-020-00280-3
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10894-020-00280-3