Hostname: page-component-586b7cd67f-gb8f7 Total loading time: 0 Render date: 2024-11-23T11:25:57.428Z Has data issue: false hasContentIssue false
  • English
  • Français

Fluctuation-dissipation relations for a plasma-kinetic Langevin equation

Part of: Collisions in Collisionless Plasmas

Published online by Cambridge University Press:  05 September 2014

A. Kanekar ,
A. A. Schekochihin ,
W. Dorland  and
N. F. Loureiro
A. Kanekar*
Affiliation:
Department of Physics, University of Maryland, College Park, MD 20742-3511, USA
A. A. Schekochihin
Affiliation:
Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK
W. Dorland
Affiliation:
Department of Physics, University of Maryland, College Park, MD 20742-3511, USA
N. F. Loureiro
Affiliation:
Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, 1049-001 Lisbon, Portugal
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

A linearised kinetic equation describing electrostatic perturbations of a Maxwellian equilibrium in a weakly collisional plasma forced by a random source is considered. The problem is treated as a kinetic analogue of the Langevin equation and the corresponding fluctuation-dissipation relations are derived. The kinetic fluctuation-dissipation relation reduces to the standard “fluid” one in the regime where the Landau damping rate is small and the system has no real frequency; in this case the simplest possible Landau-fluid closure of the kinetic equation coincides with the standard Langevin equation. Phase mixing of density fluctuations and emergence of fine scales in velocity space is diagnosed as a constant flux of free energy in Hermite space; the fluctuation-dissipation relations for the perturbations of the distribution function are derived, in the form of a universal expression for the Hermite spectrum of the free energy. Finite-collisionality effects are included. This work is aimed at establishing the simplest fluctuation-dissipation relations for a kinetic plasma, clarifying the connection between Landau and Hermite-space formalisms, and setting a benchmark case for a study of phase mixing in turbulent plasmas.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 1 , January 2015 , 305810104
Copyright
Copyright © Cambridge University Press 2014 

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

REFERENCES

Armstrong, T. P. 1967 Numerical studies of the nonlinear Vlasov equation. Phys. Fluids 10, 1269.Google Scholar
Barnes, A. 1966 Collisionless damping of hydromagnetic waves. Phys. Fluids 9, 1483.CrossRefGoogle Scholar
Bershadskii, A. and Sreenivasan, K. R. 2004 Intermittency and the passive nature of the magnitude of the magnetic field. Phys. Rev. Lett. 93, 064501.Google Scholar
Celnikier, L. M., Harvey, C. C., Jegou, R., Moricet, P. and Kemp, M. 1983 A determination of the electron density fluctuation spectrum in the solar wind, using the ISEE propagation experiment. Astron. Astrophys. 126, 293.Google Scholar
Celnikier, L. M., Muschietti, L. and Goldman, M. V. 1987 Aspects of interplanetary plasma turbulence. Astron. Astrophys. 181, 138.Google Scholar
Chen, C. H. K., Bale, S. D., Salem, C. and Mozer, F. S. 2011 Frame dependence of the electric field spectrum of solar wind turbulence. Astrophys. J. Lett. 737, L41.Google Scholar
Dorland, W. and Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5, 812.CrossRefGoogle Scholar
Foote, E. A. and Kulsrud, R. M. 1979 Hydromagnetic waves in high beta plasmas. Astrophys. J. 233, 302.CrossRefGoogle Scholar
Fried, B. D. and Conte, S. D. 1961 The Plasma Dispersion Function. New York: Academic Press.Google Scholar
Goswami, P., Passot, T. and Sulem, P. L. 2005 A Landau fluid model for warm collisionless plasmas. Phys. Plasmas 12, 102109.Google Scholar
Gould, R. W., O'Neil, T. M. and Malmberg, J. H. 1967 Plasma wave echo. Phys. Rev. Lett. 19, 219.Google Scholar
Grant, F. C. and Feix, M. R. 1967 Fourier-Hermite solutions of the Vlasov equations in the linearized limit. Phys. Fluids 10, 696.Google Scholar
Hammett, G. W., Beer, M. A., Dorland, W., Cowley, S. C. and Smith, S. A. 1993 Developments in the gyrofluid approach to tokamak turbulence simulations. Plasma Phys. Control. Fusion 35, 973.CrossRefGoogle Scholar
Hammett, G. W., Dorland, W. and Perkins, F. W. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4, 2052.Google Scholar
Hammett, G. W. and Perkins, F. W. 1990 Fluid moment models for Landau damping with application to the ion-temperature-gradient instability. Phys. Rev. Lett. 64, 3019.Google Scholar
Hatch, D. R., Jenko, F., Bañón Navarro, A. and Bratanov, V. 2013 Transition between saturation regimes of gyrokinetic turbulence. Phys. Rev. Lett. 111, 175001.CrossRefGoogle ScholarPubMed
Hedrick, C. L. and Leboeuf, J.-N. 1992 Landau fluid equations for electromagnetic and electrostatic fluctuations. Phys. Fluids B 4, 3915.CrossRefGoogle Scholar
Hnat, B., Chapman, S. C. and Rowlands, G. 2005 Compressibility in solar wind plasma turbulence. Phys. Rev. Lett. 94, 204502.Google Scholar
Kanekar, A., Schekochihin, A. A., Hammett, G. W., Dorland, W. and Loureiro, N. F. 2014 Passive advection in a collisionless plasma. in preparation.Google Scholar
Kramers, H. A. 1927 La diffusion de la lumiere par les atomes. Atti Cong. Intern. Fisici (Transactions of Volta Centenary Congress), Como, vol. 2.Google Scholar
Kronig, R. 1926 On the theory of dispersion of X-rays. J. Opt. Soc. Am. 12, 547.Google Scholar
Kubo, R. 1966 The fluctuation-dissipation theorem. Rep. Prog. Phys. 29, 255.CrossRefGoogle Scholar
Landau, L. 1946 On the vibration of the electronic plasma. J. Phys. USSR 10, 25.Google Scholar
Lenard, A. and Bernstein, I. B. 1958 Plasma oscillations with diffusion in velocity space. Phys. Rev. 112, 1456.Google Scholar
Lithwick, Y. and Goldreich, P. 2001 Compressible magnetohydrodynamic turbulence in interstellar plasmas. Astrophys. J. 562, 279.CrossRefGoogle Scholar
Loureiro, N. F., Schekochihin, A. A. and Zocco, A. 2013 Fast collisionless reconnection and electron heating in strongly magnetized plasmas. Phys. Rev. Lett. 111, 025002.Google Scholar
Marsch, E. and Tu, C.-Y. 1990 Spectral and spatial evolution of compressible turbulence in the inner solar wind. J. Geophys. Res. 95, 11945.CrossRefGoogle Scholar
Ng, C. S., Bhattacharjee, A. and Skiff, F. 1999 Kinetic eigenmodes and discrete spectrum of plasma oscillations in a weakly collisional plasma. Phys. Rev. Lett. 83, 1974.Google Scholar
Parker, S. E. and Carati, D. 1995 Renormalized dissipation in plasmas with finite collisionality. Phys. Rev. Lett. 75, 441.Google Scholar
Passot, T. and Sulem, P. L. 2004 A Landau fluid model for dispersive magnetohydrodynamics. Phys. Plasmas 11, 5173.CrossRefGoogle Scholar
Passot, T. and Sulem, P. L. 2007 Collisionless magnetohydrodynamics with gyrokinetic effects. Phys. Plasmas 14, 082502.Google Scholar
Plunk, G. G. 2013 Landau damping in a turbulent setting. Phys. Plasmas 20, 032304.CrossRefGoogle Scholar
Plunk, G. G. and Parker, J. T. 2014 Irreversible energy flow in forced Vlasov dynamics. arXiv:1402.7230.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Plunk, G. G., Quataert, E. and Tatsuno, T. 2008 Gyrokinetic turbulence: a nonlinear route to dissipation through phase space. Plasma Phys. Control. Fusion 50, 124024.CrossRefGoogle Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. and Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182, 310.Google Scholar
Schekochihin, A. A., Kanekar, A., Hammett, G. W., Dorland, W. and Loureiro, N. F. 2014 Stochastic advection and phase mixing in a collisionless plasma. in preparation.Google Scholar
Snyder, P. B., Hammett, G. W. and Dorland, W. 1997 Landau fluid models of collisionless magnetohydrodynamics. Phys. Plasmas 4, 3974.CrossRefGoogle Scholar
Watanabe, T.-H. and Sugama, H. 2004 Kinetic simulation of steady states of ion temperature gradient driven turbulence with weak collisionality. Phys. Plasmas 11, 1476.Google Scholar
Zocco, A. and Schekochihin, A. A. 2011 Reduced fluid-kinetic equations for low-frequency dynamics, magnetic reconnection, and electron heating in low-beta plasmas. Phys. Plasmas 18, 102309.CrossRefGoogle Scholar