Hostname: page-component-5d59c44645-k78ct Total loading time: 0 Render date: 2024-02-27T21:00:35.567Z Has data issue: false hasContentIssue false

Two-dimensional gyrokinetic turbulence

Published online by Cambridge University Press:  19 October 2010

G. G. PLUNK*
Affiliation:
Department of Physics and IREAP, University of Maryland, College Park, MD 20742, USA Department of Physics, University of California, Los Angeles, CA 90095, USA Wolfgang Pauli Institute, University of Vienna, A-1090 Vienna, Austria
S. C. COWLEY
Affiliation:
EURATOM/CCFE Association, Culham Science Center, Abington OX1 3DB, UK Plasma Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, UK
A. A. SCHEKOCHIHIN
Affiliation:
Plasma Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, UK Rudolf Peierls Centre for Theoretical Physics, University of Oxford, Oxford OX1 3NP, UK Institut Henri Poincaré, Université Pierre et Marie Curie, 75231 Paris CEDEX 5, France
T. TATSUNO
Affiliation:
Department of Physics and IREAP, University of Maryland, College Park, MD 20742, USA
*
Email address for correspondence: gplunk@umd.edu

Abstract

Two-dimensional gyrokinetics is a simple paradigm for the study of kinetic magnetised plasma turbulence. In this paper, we present a comprehensive theoretical framework for this turbulence. We study both the inverse and direct cascades (the ‘dual cascade’), driven by a homogeneous and isotropic random forcing. The key characteristic length of gyrokinetics, the Larmor radius, divides scales into two physically distinct ranges. For scales larger than the Larmor radius, we derive the familiar Charney–Hasegawa–Mima equation from the gyrokinetic system, and explain its relationship to gyrokinetics. At scales smaller than the Larmor radius, a dual cascade occurs in phase space (two dimensions in position space plus one dimension in velocity space) via a nonlinear phase-mixing process. We show that at these sub-Larmor scales, the turbulence is self-similar and exhibits power-law spectra in position and velocity space. We propose a Hankel-transform formalism to characterise velocity-space spectra. We derive the exact relations for third-order structure functions, analogous to Kolmogorov's four-fifths and Yaglom's four-thirds laws and valid at both long and short wavelengths. We show how the general gyrokinetic invariants are related to the particular invariants that control the dual cascade in the long- and short-wavelength limits. We describe the full range of cascades from the fluid to the fully kinetic range.

Type
Papers
Copyright
Copyright © Cambridge University Press 2010

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

Abel, I. G., Barnes, M., Cowley, S. C., Dorland, W. & Schekochihin, A. A. 2008 Linearized model Fokker–Planck collision operators for gyrokinetic simulations. Part I. Theory. Phys. Plasmas 15, 122509.Google Scholar
Antonsen, T. M. & Lane, B. 1980 Kinetic equations for low frequency instabilities in inhomogeneous plasmas. Phys. Fluids 23, 12051214.Google Scholar
Batchelor, G. K. 1959 Small-scale variation of convected quantities like temperature in turbulent fluid. Part 1. General discussion and the case of small conductivity. J. Fluid Mech. 5, 113133.Google Scholar
Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, II233II239.Google Scholar
Bernard, D. 1999 Three-point velocity correlation functions in two-dimensional forced turbulence. Phys. Rev. E 60, 61846187.Google Scholar
Boffetta, G., Lillo, F. De & Musacchio, S. 2002 Inverse cascade in Charney–Hasegawa–Mima turbulence. Europhys. Lett. 59, 687693.Google Scholar
Brizard, A. J. & Hahm, T. S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421468.Google Scholar
Catto, P. J. & Tsang, K. T. 1977 Linearized gyrokinetic equation with collisions. Phys. Fluids 20, 396401.Google Scholar
Charney, J. G. 1971 Geostrophic turbulence. J. Atmos. Sci. 28, 10871095.Google Scholar
Dorland, W. & Hammett, G. W. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5, 812835.Google Scholar
Fjørtoft, R. 1953 On the changes in the spectral distribution of kinetic energy for two-dimensional non-divergent flow. Tellus 5, 225.Google Scholar
Frieman, E. A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502508.Google Scholar
Frisch, U. 1995 Turbulence: The Legacy of A. N. Kolmogorov. Cambridge University Press.Google Scholar
Görler, T. & Jenko, F. 2008 Multiscale features of density and frequency spectra from nonlinear gyrokinetics. Phys. Plasmas 15, 102508.Google Scholar
Hasegawa, A. & Mima, K. 1978 Pseudo-three-dimensional turbulence in magnetized nonuniform plasma. Phys. Fluids 21, 8792.Google Scholar
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2006 Astrophysical gyrokinetics: basic equations and linear theory. Astrophys. J. 651, 590614.Google Scholar
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2008 A model of turbulence in magnetized plasmas: implications for the dissipation range in the solar wind. J. Geophys. Res. 113, A05103.Google Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in the locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 1618.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 14171423.Google Scholar
Krommes, J. A. 2002 Fundamental statistical descriptions of plasma turbulence in magnetic fields. Phys. Rep. 360, 1352.Google Scholar
Krommes, J. A. & Hu, G. 1994 The role of dissipation in the theory and simulations of homogeneous plasma turbulence, and resolution of the entropy paradox. Phys. Plasmas 1, 32113238.Google Scholar
Lesieur, M. & Herring, J. 1985 Diffusion of a passive scalar in two-dimensional turbulence. J. Fluid Mech. 161, 7795.Google Scholar
Obukhov, A. M. 1941 a On the distribution of energy in the spectrum of turbulent flow. Dokl. Akad. Nauk SSSR 32, 2224.Google Scholar
Obukhov, A. M. 1941 b Spectral energy distribution in a turbulent flow. Izv. Akad. Nauk SSSR Ser. Geogr. Geofiz. 5, 453466.Google Scholar
Rutherford, P. H. & Frieman, E. A. 1968 Drift instabilities in general magnetic field configurations. Phys. Fluids 11, 569585.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Plunk, G. G., Quataert, E. & Tatsuno, T. 2008 Gyrokinetic turbulence: a nonlinear route to dissipation through phase space. Plasma Phys. Control. Fusion 50, 124024.Google Scholar
Schekochihin, A. A., Cowley, S. C., Dorland, W., Hammett, G. W., Howes, G. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. 182, 310377.Google Scholar
Sugama, H., Okamoto, M., Horton, W. & Wakatani, M. 1996 Transport processes and entropy production in toroidal plasmas with gyrokinetic electromagnetic turbulence. Phys. Plasmas 3, 23792394.Google Scholar
Tatsuno, T., Barnes, M., Cowley, S. C., Dorland, W., Howes, G. G., Numata, R., Plunk, G. G. & Schekochihin, A. A. 2010 a Gyrokinetic simulation of entropy cascade in two-dimensional electrostatic turbulence. J. Plasma Fusion Res. Ser. 9, 509.Google Scholar
Tatsuno, T., Dorland, W., Schekochihin, A. A., Plunk, G. G., Barnes, M., Cowley, S. C. & Howes, G. G. 2009 Nonlinear phase mixing and phase-space cascade of entropy in gyrokinetic plasma turbulence. Phys. Rev. Lett. 103, 015003.Google Scholar
Taylor, J. B. & Hastie, R. J. 1968 Stability of general plasma equilibria. Part I. Formal theory. Plasma Phys. 10, 479494.Google Scholar
Taylor, J. B. & McNamara, B. 1971 Plasma diffusion in two dimensions. Phys. Fluids 14, 14921499.Google Scholar
Watanabe, T.-H. & Sugama, H. 2004 Kinetic simulation of steady states of ion temperature gradient driven turbulence with weak collisionality. Phys. Plasmas 11, 14761483.Google Scholar
Yaglom, A. M. 1949 On the local structure of a temperature field in a turbulent flow. Dokl. Akad. Nauk. SSSR 69, 743.Google Scholar