Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-25T06:45:20.417Z Has data issue: false hasContentIssue false
  • English
  • Français

Dispersion relation for electrostatic waves in plasmas with isotropic and anisotropic Kappa distributions for electrons and ions

Published online by Cambridge University Press:  05 October 2017

L. F. Ziebell Open the ORCID record for L. F. Ziebell [Opens in a new window] ,
R. Gaelzer Open the ORCID record for R. Gaelzer [Opens in a new window]  and
F. J. R. Simões Jr
L. F. Ziebell*
Affiliation:
Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970, Porto Alegre, RS, Brazil
R. Gaelzer
Affiliation:
Instituto de Física, Universidade Federal do Rio Grande do Sul, 91501-970, Porto Alegre, RS, Brazil
F. J. R. Simões Jr
Affiliation:
Instituto de Física e Matemática, Universidade Federal de Pelotas, 96010-900, Pelotas, RS, Brazil
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Velocity distribution functions which feature extended tails with power-law dependence have been consistently observed in the solar wind environment and are frequently modelled by the so-called Kappa distributions. Different forms of Kappa distributions are commonly employed in analytical studies, and despite their similarities, they can produce different effects on the dispersion properties that occur in a plasma. We consider two different and widely used forms of Kappa distributions, in both isotropic and anisotropic cases, and systematically discuss their effects on the dispersion relations of Langmuir and ion-sound waves. It is shown that in the case of Langmuir waves, one of the forms leads to the expression for the Bohm–Gross dispersion relation, valid for plasmas with Maxwellian velocity distributions, while the other form of Kappa functions leads to a dispersion relation with significant difference regarding the Maxwellian case, particularly in the case of small values of the kappa index. For ion-sound waves, the dispersion relations obtained with the different forms of Kappa distributions are different among themselves, and also different from the Maxwellian case, with difference which increases for small values of the kappa index. Some results obtained from numerical solution of the dispersion relations are presented, which illustrate the magnitude of the perceived differences. Some results obtained with relativistic particle-in-cell simulations are also presented, which allow the comparison between the dispersion relations obtained from analytical calculations and the frequency–wavelength distribution of wave fluctuations which are observed in numerical experiments.

Keywords

plasma wavesspace plasma physics
Type
Research Article
Information
Journal of Plasma Physics , Volume 83 , Issue 5 , October 2017 , 905830503
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

Abdul, R. F. & Mace, R. L. 2014 A method to generate kappa distributed random deviates for particle-in-cell simulations. Comput. Phys. Commun. 185 (10), 2383–2386.CrossRefGoogle Scholar
Gaelzer, R. & Ziebell, L. F. 2014 The dispersion relations of dispersive Alfvén waves in superthermal plasmas. J. Geophys. Res. 119 (12), 9334–9356.CrossRefGoogle Scholar
Gaelzer, R. & Ziebell, L. F. 2016 Obliquely propagating electromagnetic waves in magnetized kappa plasmas. Phys. Plasmas 23 (2), 022110.CrossRefGoogle Scholar
Gaelzer, R., Ziebell, L. F. & Meneses, A. R. 2016 The general dielectric tensor for bi-kappa magnetized plasmas. Phys. Plasmas 23 (6), 062108.CrossRefGoogle Scholar
Hapgood, M., Perry, C., Davies, J. & Denton, M. 2011 The role of suprathermal particle measurements in CrossScale studies of collisionless plasma processes. Planet. Space Sci. 59, 618–629.CrossRefGoogle Scholar
Hasegawa, A., Mima, K. & Duongvan, M. 1985 Plasma distribution function in a superthermal radiation-field. Phys. Rev. Lett. 54 (24), 2608–2610.CrossRefGoogle Scholar
Hau, L. N. & Fu, W. Z. 2007 Mathematical and physical aspects of Kappa velocity distribution. Phys. Plasmas 14 (11), 110702.CrossRefGoogle Scholar
Hau, L. N., Fu, W. Z. & Chuang, S. H. 2009 Response to ‘Comment on “Mathematical and physical aspects of Kappa velocity distribution”’ [Phys. Plasmas 16, 094701 (2009)]. Phys. Plasmas 16 (9), 094702.CrossRefGoogle Scholar
Hellberg, M. & Mace, R. 2002 Generalized plasma dispersion function for a plasma with a kappa-Maxwellian velocity distribution. Phys. Plasmas 9 (5, 1), 1495–1504.CrossRefGoogle Scholar
Hellberg, M. A., Mace, R. L., Baluku, T. K., Kourakis, I. & Saini, N. S. 2009 Comment on ‘Mathematical and physical aspects of Kappa velocity distribution’ [Phys. Plasmas 14, 110702 (2007)]. Phys. Plasmas 16 (9), 094701.CrossRefGoogle Scholar
Hellberg, M., Mace, R. & Cattaert, T. 2006 Effects of superthermal particles on waves in magnetized space plasmas. Space Sci. Rev. 121, 127–139.CrossRefGoogle Scholar
Lazar, M. 2012 The electromagnetic ion-cyclotron instability in bi-Kappa distributed plasmas. Astron. Astrophys. 547, A94.CrossRefGoogle Scholar
Lazar, M., Fichtner, H. & Yoon, P. H. 2016 On the interpretation and applicability of $\unicode[STIX]{x1D705}$ -distributions. Astron. Astrophys. 589, A39.CrossRefGoogle Scholar
Lazar, M., Pierrard, V., Poedts, S. & Schlickeiser, R. 2012 Modeling space plasma dynamics with anisotropic Kappa distributions. Astrophys. Space Sci. Proc. 33, 97–107.CrossRefGoogle Scholar
Lazar, M. & Poedts, S. 2009a Firehose instability in space plasmas with bi-kappa distributions. Astron. Astrophys. 494, 311–315.CrossRefGoogle Scholar
Lazar, M. & Poedts, S. 2009b Limits for the firehose instability in space plasmas. Solar Phys. 258, 119–128.CrossRefGoogle Scholar
Lazar, M. & Poedts, S. 2014 Instability of the parallel electromagnetic modes in Kappa distributed plasmas – II. Electromagnetic ion-cyclotron modes. Mon. Not. R. Astron. Soc. 437 (1), 641–648.CrossRefGoogle Scholar
Lazar, M., Poedts, S. & Schlickeiser, R. 2011 Proton firehose instability in bi-Kappa distributed plasmas. Astron. Astrophys. 534, A116.CrossRefGoogle Scholar
Leubner, M. P. 2002 A nonextensive entropy approach to Kappa-distributions. Astrophys. Space Sci. 282 (3), 573–579.CrossRefGoogle Scholar
Leubner, M. P. 2004 Core-halo distribution functions: a natural equilibrium state in generalized thermostatistics. Astrophys. J. 604, 469–478.CrossRefGoogle Scholar
Leubner, M. & Schupfer, N. 2000 Mirror instability thresholds in suprathermal space plasmas. J. Geophys. Res. 105 (A12), 27387–27391.CrossRefGoogle Scholar
Leubner, M. & Schupfer, N. 2001 A general kinetic mirror instability criterion for space applications. J. Geophys. Res. 106 (A7), 12993–12998.CrossRefGoogle Scholar
Li, B. & Cairns, I. H. 2014 Fundamental emission of type III bursts produced in non-Maxwellian coronal plasmas with Kappa-distributed background particles. Solar Phys. 289, 951–976.CrossRefGoogle Scholar
Livadiotis, G. 2015 Introduction to special section on origins and properties of Kappa distributions: statistical background and properties of Kappa distributions in space plasmas. J. Geophys. Res. 120 (3), 1607–1619.CrossRefGoogle Scholar
Livadiotis, G. & McComas, D. J. 2009 Beyond kappa distributions: exploiting Tsallis statistical mechanics in space plasmas. J. Geophys. Res. 114, A11105.CrossRefGoogle Scholar
Livadiotis, G. & McComas, D. J. 2011 Invariant Kappa distribution in space plasmas out of equilibrium. Astrophys. J. 741 (2), 88.CrossRefGoogle Scholar
Livadiotis, G. & McComas, D. J. 2013 Understanding Kappa distributions: a toolbox for space science and astrophysics. Space Sci. Rev. 175, 183–214.CrossRefGoogle Scholar
Mace, R. L. 2003 A Gordeyev integral for electrostatic waves in a magnetized plasma with a kappa velocity distribution. Phys. Plasmas 10, 2181.CrossRefGoogle Scholar
Mace, R. L. & Hellberg, M. A. 1995 A dispersion function for plasmas containing superthermal particles. Phys. Plasmas 2 (6), 2098–2109.CrossRefGoogle Scholar
Mace, R. L. & Hellberg, M. A. 2009 A new formulation and simplified derivation of the dispersion function for a plasma with a kappa velocity distribution. Phys. Plasmas 16 (7), 072113.CrossRefGoogle Scholar
Mace, R. & Hellberg, M. 2003 Generalized Langmuir waves in a magnetized plasma with a Maxwellian–Lorentzian distribution. Phys. Plasmas 10 (1), 21–28.CrossRefGoogle Scholar
Maksimovic, M., Pierrard, V. & Riley, P. 1997 Ulysses electron distributions fitted with Kappa functions. Geophys. Res. Lett. 24 (9), 1151–1154.CrossRefGoogle Scholar
Olbert, S. 1968 Summary of experimental results from MIT detector on IMP-1. In Physics of the Magnetosphere: Based Upon the Proceedings of the Conference held at Boston College June 19–28, 1967 (ed. Carovillano, R. L., McClay, J. F. & Radoski, H. R.), pp. 641–659. Springer.CrossRefGoogle Scholar
Omura, Y. & Matsumoto, H. 1993 Computer Space Plasma Physics: Simulation Techniques and Software. Terra Scientific Publishing Company.Google Scholar
Podesta, J. J. 2015 Small-amplitude Langmuir pulse excited by a planar grid electrode in a flowing plasma. J. Plasma Phys. 8 (5), 905810503.Google Scholar
dos Santos, M. S., Ziebell, L. F. & Gaelzer, R. 2014 Ion firehose instability in plasmas with plasma particles described by product bi-kappa distributions. Phys. Plasmas 21, 112102.CrossRefGoogle Scholar
dos Santos, M. S., Ziebell, L. F. & Gaelzer, R. 2015 Ion-cyclotron instability in plasmas described by product-bi-kappa distributions. Phys. Plasmas 22 (12), 122107.CrossRefGoogle Scholar
dos Santos, M. S., Ziebell, L. F. & Gaelzer, R. 2016 Ion firehose instability in a dusty plasma considering product-bi-kappa distributions for the plasma particles. Phys. Plasmas 23 (1), 013705.CrossRefGoogle Scholar
Silva, R., Plastino, A. & Lima, J. 1998 A Maxwellian path to the q-nonextensive velocity distribution function. Phys. Lett. A 249 (5–6), 401–408.CrossRefGoogle Scholar
Summers, D. & Thorne, R. M. 1991 The modified plasma dispersion function. Phys. Fluids B 3 (8), 1835–1847.CrossRefGoogle Scholar
Thorne, R. M. & Summers, D. 1991 Landau damping in space plasmas. Phys. Fluids B 3 (8), 2117–2123.CrossRefGoogle Scholar
Tsallis, C. 1988 Possible generalization of Boltzmann–Gibbs statistics. J. Stat. Phys. 52 (1–2), 479–487.CrossRefGoogle Scholar
Tsallis, C., Mendes, R. & Plastino, A. 1998 The role of constraints within generalized nonextensive statistics. Phys. A 261 (3–4), 534–554.Google Scholar
Vasyliunas, V. M. 1968 A survey of low-energy electrons in evening sector of magnetosphere with OGO 1 and OGO 3. J. Geophys. Res. 73 (9), 2839.CrossRefGoogle Scholar
Ziebell, L. F. & Gaelzer, R. 2017 On the influence of the shape of kappa distributions of ions and electrons on the ion-cyclotron instability. Phys. Plasmas 24 (9), 102108.CrossRefGoogle Scholar