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The inherently three-dimensional nature of magnetized plasma turbulence

Part of: Present Achievements in SpacePlasmas

Published online by Cambridge University Press:  27 November 2014

Gregory G. Howes
Gregory G. Howes*
Affiliation:
Department of Physics and Astronomy, University of Iowa, Iowa City, IA, 52242, USA
*
†Email address for correspondence: gregory-howes@uiowa.edu
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Abstract

It is often asserted or implicitly assumed, without justification, that the results of two-dimensional investigations of plasma turbulence are applicable to the three-dimensional plasma environments of interest. A projection method is applied to derive two scalar equations that govern the nonlinear evolution of the Alfvénic and pseudo-Alfvénic components of ideal incompressible magnetohydrodynamic (MHD) plasma turbulence. The mathematical form of these equations makes clear the inherently three-dimensional nature of plasma turbulence, enabling an analysis of the nonlinear properties of two-dimensional limits often used to study plasma turbulence. In the anisotropic limit, k⊥ ≫ k∥, that naturally arises in magnetized plasma systems, the perpendicular 2D limit retains the dominant nonlinearities that are mediated only by the Alfvénic fluctuations but lacks the wave physics associated with the linear term that is necessary to capture the anisotropic cascade of turbulent energy. In the in-plane 2D limit, the nonlinear energy transfer is controlled instead by the pseudo-Alfvén waves, with the Alfvén waves relegated to a passive role. In the oblique 2D limit, an unavoidable azimuthal dependence connecting the wavevector components will likely cause artificial azimuthal asymmetries in the resulting turbulent dynamics. Therefore, none of these 2D limits is sufficient to capture fully the rich three-dimensional nonlinear dynamics critical to the evolution of plasma turbulence.

Type
Research Article
Information
Journal of Plasma Physics , Volume 81 , Issue 2 , April 2015 , 325810203
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Copyright
Copyright © Cambridge University Press 2014 

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References

REFERENCES

Batchelor, G. K. 1969 Computation of the energy spectrum in homogeneous two-dimensional turbulence. Phys. Fluids 12, 233.Google Scholar
Belcher, J. W. and Davis, L. 1971 Large-amplitude Alfvén waves in the interplanetary medium, 2. J. Geophys. Res. 76, 3534–3563.Google Scholar
Biskamp, D. 2003 Magnetohydrodynamic Turbulence, Cambridge University Press: Cambridge.CrossRefGoogle Scholar
Boldyrev, S. 2006 Spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 96 (11), 115 002.Google Scholar
Che, H., Goldstein, M. L. and Viñas, A. F. 2014 Bidirectional energy cascades and the origin of kinetic Alfvénic and Whistler turbulence in the solar wind. Phys. Rev. Lett. 112 (6), 061 101.CrossRefGoogle ScholarPubMed
Cho, J. and Lazarian, A. 2004 The anisotropy of electron magnetohydrodynamic turbulence. Astrophys. J. Lett. 615, L41–L44.CrossRefGoogle Scholar
Cho, J. and Lazarian, A. 2009 Simulations of electron magnetohydrodynamic turbulence. Astrophys. J. 701, 236–252.CrossRefGoogle Scholar
Cho, J. and Vishniac, E. T. 2000 The anisotropy of magnetohydrodynamic Alfvénic turbulence. Astrophys. J. 539, 273–282.CrossRefGoogle Scholar
Dobrowolny, M., Mangeney, A. and Veltri, P. 1980a Fully developed anisotropic hydromagnetic turbulence in interplanetary space. Phys. Rev. Lett. 45, 144–147.CrossRefGoogle Scholar
Dobrowolny, M., Mangeney, A. and Veltri, P. 1980b Properties of magnetohydrodynamic turbulence in the solar wind. Astron. Astrophys. 83, 26–32.Google Scholar
Donato, S., Greco, A., Matthaeus, W. H., Servidio, S. and Dmitruk, P. 2013 How to identify reconnecting current sheets in incompressible Hall MHD turbulence. J. Geophys. Res. 118, 4033–4038.Google Scholar
Donato, S., Servidio, S., Dmitruk, P., Carbone, V., Shay, M. A., Cassak, P. A. and Matthaeus, W. H. 2012 Reconnection events in two-dimensional Hall magnetohydrodynamic turbulence. Phys. Plasmas 19 (9), 092 307.CrossRefGoogle Scholar
Drake, D. J., Schroeder, J. W. R., Howes, G. G., Kletzing, C. A., Skiff, F., Carter, T. A. and Auerbach, D. W. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. IV. Laboratory experiment. Phys. Plasmas 20 (7), 072 901.CrossRefGoogle Scholar
Elsasser, W. M. 1950 The hydromagnetic equations. Phys. Rev. 79, 183.CrossRefGoogle Scholar
Galtier, S., Nazarenko, S. V., Newell, A. C. and Pouquet, A. 2000 A weak turbulence theory for incompressible magnetohydrodynamics. J. Plasma Phys. 63, 447–488.CrossRefGoogle Scholar
Ganguli, G., Rudakov, L., Scales, W., Wang, J. and Mithaiwala, M. 2010 Three dimensional character of whistler turbulence. Phys. Plasmas 17 (5), 052 310.CrossRefGoogle Scholar
Gary, S. P., Saito, S. and Li, H. 2008 Cascade of whistler turbulence: particle-in-cell simulations. Geophys. Res. Lett. 35, 2104.CrossRefGoogle Scholar
Gary, S. P., Saito, S. and Narita, Y. 2010 Whistler turbulence wavevector anisotropies: particle-in-cell simulations. Astrophys. J. 716, 1332–1335.Google Scholar
Goldreich, P. and Sridhar, S. 1995 Toward a theery of interstellar turbulence II. Strong Alfvénic turbulence. Astrophys. J. 438, 763–775.CrossRefGoogle Scholar
Greco, A., Valentini, F., Servidio, S. and Matthaeus, W. H. 2012 Inhomogeneous kinetic effects related to intermittent magnetic discontinuities. Phys. Rev. E 86 (6), 066 405.CrossRefGoogle ScholarPubMed
Haynes, C. T., Burgess, D. and Camporeale, E. 2014 Reconnection and electron temperature anisotropy in sub-proton scale plasma turbulence. Astrophys. J. 783, 38.Google Scholar
Higdon, J. C. 1984 Density fluctuations in the interstellar medium: evidence for anisotropic magnetogasdynamic turbulence. I. Model and astrophysical sites. Astrophys. J. 285, 109–123.CrossRefGoogle Scholar
Howes, G. G. 2015 Kinetic turbulence. In: Magnetic Field in Diffuse Media, Springer: New York.Google Scholar
Howes, G. G., Drake, D. J., Nielson, K. D., Carter, T. A., Kletzing, C. A. and Skiff, F. 2012 Toward astrophysical turbulence in the laboratory. Phys. Rev. Lett. 109 (25), 255 001.Google Scholar
Howes, G. G. and Nielson, K. D. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. I. Asymptotic solution. Phys. Plasmas 20 (7), 072 302.Google Scholar
Howes, G. G., Nielson, K. D., Drake, D. J., Schroeder, J. W. R., Skiff, F., Kletzing, C. A. and Carter, T. A. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. III. Theory for experimental design. Phys. Plasmas 20 (7), 072 304.CrossRefGoogle Scholar
Howes, G. G., Tenbarge, J. M. and Dorland, W. 2011 A weakened cascade model for turbulence in astrophysical plasmas. Phys. Plasmas 18, 102 305.CrossRefGoogle Scholar
Howes, G. G., TenBarge, J. M., Dorland, W., Quataert, E., Schekochihin, A. A., Numata, R. and Tatsuno, T. 2011 Gyrokinetic simulations of solar wind turbulence from ion to electron scales. Phys. Rev. Lett. 107, 035 004.Google Scholar
Iroshnikov, R. S. 1963 The turbulence of a conducting fluid in a strong magnetic field. Astron. Zh. 40, 742. (English Translation: Sov. Astron. 7, 566 (1964).)Google Scholar
Kraichnan, R. H. 1965 Inertial range spectrum of hydromagnetic turbulence. Phys. Fluids 8, 1385–1387.Google Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10, 1417–1423.Google Scholar
Kraichnan, R. H. and Montgomery, D. 1980 Two-dimensional turbulence. Rep. Prog. Phys. 43, 547–619.Google Scholar
Markovskii, S. A. and Vasquez, B. J. 2010 The effect of spectral anisotropy of fast magnetosonic turbulence on the plasma heating at the proton kinetic scales. Phys. Plasmas 17 (11), 112 902.Google Scholar
Markovskii, S. A. and Vasquez, B. J. 2011 A short-timescale channel of dissipation of the strong solar wind turbulence. Astrophys. J. 739, 22.Google Scholar
Markovskii, S. A. and Vasquez, B. J. 2013 Magnetic helicity in the dissipation range of strong imbalanced turbulence. Astrophys. J. 768, 62.Google Scholar
Maron, J. and Goldreich, P. 2001 Simulations of incompressible magnetohydrodynamic turbulence. Astrophys. J. 554, 1175–1196.CrossRefGoogle Scholar
Matthaeus, W. H. and Goldstein, M. L. 1982 Measurement of the rugged invariants of magnetohydrodynamic turbulence in the solar wind. J. Geophys. Res. 87, 6011–6028.CrossRefGoogle Scholar
Montgomery, D. 1982 Major disruptions, inverse cascades, and the Strauss equations. Phys. Scr. T2A, 83.Google Scholar
Montgomery, D. and Turner, L. 1981 Anisotropic magnetohydrodynamic turbulence in a strong external magnetic field. Phys. Fluids 24, 825–831.Google Scholar
Narita, Y., Comisel, H. and Motschmann, U. 2014 Spatial structure of ion-scale plasma turbulence. Frontiers Phys. 2, 13.Google Scholar
Narita, Y., Gary, S. P., Saito, S., Glassmeier, K.-H. and Motschmann, U. 2011 Dispersion relation analysis of solar wind turbulence. Geophys. Res. Lett. 38, L05 101.Google Scholar
Nielson, K. D., Howes, G. G. and Dorland, W. 2013 Alfvén wave collisions, the fundamental building block of plasma turbulence. II. Numerical solution. Phys. Plasmas 20 (7), 072 303.CrossRefGoogle Scholar
Osman, K. T., Matthaeus, W. H., Greco, A. and Servidio, S. 2011 Evidence for inhomogeneous heating in the solar wind. Astrophys. J. Lett. 727, L11.CrossRefGoogle Scholar
Parashar, T. N, Salem, C., Wicks, R., Karimabadi, H., Gary, S., Chandran, B. and Matthaeus, W. H. 2014a Turbulent dissipation challenge – problem description. ArXiv e-prints, 1405.0949.Google Scholar
Parashar, T. N., Vasquez, B. J. and Markovskii, S. A. 2014b The role of electron equation of state in heating partition of protons in a collisionless plasma. Phys. Plasmas 21 (2), 022 301.CrossRefGoogle Scholar
Parashar, T. N., Servidio, S., Breech, B., Shay, M. A. and Matthaeus, W. H. 2010 Kinetic driven turbulence: structure in space and time. Phys. Plasmas 17 (10), 102 304.Google Scholar
Parashar, T. N., Servidio, S., Shay, M. A., Breech, B. and Matthaeus, W. H. 2011 Effect of driving frequency on excitation of turbulence in a kinetic plasma. Phys. Plasmas 18 (9), 092 302.Google Scholar
Parashar, T. N., Shay, M. A., Cassak, P. A. and Matthaeus, W. H. 2009 Kinetic dissipation and anisotropic heating in a turbulent collisionless plasma. Phys. Plasmas 16 (3), 032 310.Google Scholar
Perrone, D., Valentini, F., Servidio, S., Dalena, S. and Veltri, P. 2013 Vlasov simulations of multi-ion plasma turbulence in the solar wind. Astrophys. J. 762, 99.Google Scholar
Roberts, O. W., Li, X. and Li, B. 2013 Kinetic plasma turbulence in the fast solar wind measured by cluster. Astrophys. J. 769, 58.CrossRefGoogle Scholar
Robinson, D. C. and Rusbridge, M. G. 1971 Structure of turbulence in the zeta plasma. Phys. Fluids 14, 2499–2511.Google Scholar
Sahraoui, F., Goldstein, M. L., Belmont, G., Canu, P. and Rezeau, L. 2010 Three dimensional anisotropic k spectra of turbulence at subproton scales in the solar wind. Phys. Rev. Lett. 105 (13), 131101.CrossRefGoogle ScholarPubMed
Saito, S., Gary, S. P., Li, H. and Narita, Y. 2008 Whistler turbulence: particle-in-cell simulations. Phys. Plasmas 15 (10), 102 305.Google 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, 310–377.Google Scholar
Servidio, S., Carbone, V., Primavera, L., Veltri, P. and Stasiewicz, K. 2007 Compressible turbulence in Hall Magnetohydrodynamics. Planet. Space Sci. 55, 2239–2243.CrossRefGoogle Scholar
Servidio, S., Dmitruk, P., Greco, A., Wan, M., Donato, S., Cassak, P. A., Shay, M. A., Carbone, V. and Matthaeus, W. H. 2011 Magnetic reconnection as an element of turbulence. Nonlinear Proc. Geophys. 18, 675–695.Google Scholar
Servidio, S., Matthaeus, W. H., Shay, M. A., Cassak, P. A. and Dmitruk, P. 2009 Magnetic reconnection in two-dimensional magnetohydrodynamic turbulence. Phys. Rev. Lett. 102 (11), 115 003.Google Scholar
Servidio, S., Matthaeus, W. H., Shay, M. A., Dmitruk, P., Cassak, P. A. and Wan, M. 2010 Statistics of magnetic reconnection in two-dimensional magnetohydrodynamic turbulence. Phys. Plasmas 17 (3), 032 315.Google Scholar
Servidio, S., Valentini, F., Califano, F. and Veltri, P. 2012 Local kinetic effects in two-dimensional plasma turbulence. Phys. Rev. Lett. 108 (4), 045 001.Google Scholar
Shebalin, J. V., Matthaeus, W. H. and Montgomery, D. 1983 Anisotropy in mhd turbulence due to a mean magnetic field. J. Plasma Phys. 29, 525–547.Google Scholar
Sridhar, S. and Goldreich, P. 1994 Toward a theory of interstellar turbulence. 1: weak Alfvenic turbulence. Astrophys. J. 432, 612–621.CrossRefGoogle Scholar
Strauss, H. R. 1976 Nonlinear, three-dimensional magnetohydrodynamics of noncircular tokamaks. Phys. Fluids 19, 134–140.Google Scholar
TenBarge, J. M. and Howes, G. G. 2012 Evidence of critical balance in kinetic Alfvén wave turbulence simulations. Phys. Plasmas 19 (5), 055 901.Google Scholar
Tronko, N., Nazarenko, S. V. and Galtier, S. 2013 Weak turbulence in two-dimensional magnetohydrodynamics. Phys. Rev. E 87 (3), 033 103.Google Scholar
Verscharen, D., Marsch, E., Motschmann, U. and Müller, J. 2012 Kinetic cascade beyond magnetohydrodynamics of solar wind turbulence in two-dimensional hybrid simulations. Phys. Plasmas 19 (2), 022 305.Google Scholar
Wan, M., Matthaeus, W. H., Karimabadi, H., Roytershteyn, V., Shay, M., Wu, P., Daughton, W., Loring, B. and Chapman, S. C. 2012 Intermittent dissipation at kinetic scales in collisionless plasma turbulence. Phys. Rev. Lett. 109 (19), 195 001.Google Scholar
Wan, M., Matthaeus, W. H., Servidio, S. and Oughton, S. 2013 Generation of X-points and secondary islands in 2D magnetohydrodynamic turbulence. Phys. Plasmas 20 (4), 042 307.CrossRefGoogle Scholar
Wan, M., Oughton, S., Servidio, S. and Matthaeus, W. H. 2010 On the accuracy of simulations of turbulence. Phys. Plasmas 17 (8), 082 308.CrossRefGoogle Scholar
Woltjer, L. 1958a A theorem on force-free magnetic fields. Proc. Natl. Acad. Sci. 44, 489–491.Google Scholar
Woltjer, L. 1958b On hydromagnetic equilibrium. Proc. Natl. Acad. Sci. 44, 833–841.Google Scholar
Wu, P., Perri, S., Osman, K., Wan, M., Matthaeus, W. H., Shay, M. A., Goldstein, M. L., Karimabadi, H. and Chapman, S. 2013a Intermittent heating in solar wind and kinetic simulations. Astrophys. J. Lett. 763, L30.Google Scholar
Wu, P., Wan, M., Matthaeus, W. H., Shay, M. A. and Swisdak, M. 2013b von Kármán energy decay and heating of protons and electrons in a kinetic turbulent plasma. Phys. Rev. Lett. 111 (12), 121 105.Google Scholar
Zweben, S. J., Menyuk, C. R. and Taylor, R. J. 1979 Small-scale magnetic fluctuations inside the macrotor tokamak. Phys. Rev. Lett. 42, 1270–1274.Google Scholar