Hostname: page-component-76fb5796d-22dnz Total loading time: 0 Render date: 2024-04-26T12:03:56.425Z Has data issue: false hasContentIssue false
  • English
  • Français

A Hamiltonian gyrofluid model based on a quasi-static closure

Part of: Hamiltonian Methods in Plasma Physics

Published online by Cambridge University Press:  24 August 2020

E. Tassi Open the ORCID record for E. Tassi [Opens in a new window] ,
T. Passot  and
P. L. Sulem Open the ORCID record for P. L. Sulem [Opens in a new window]
E. Tassi*
Affiliation:
Université Côte d'Azur, CNRS, Observatoire de la Côte d'Azur, Laboratoire J.L. Lagrange, Boulevard de l'Observatoire, CS 34229, 06304Nice CEDEX 4, France
T. Passot
Affiliation:
Université Côte d'Azur, CNRS, Observatoire de la Côte d'Azur, Laboratoire J.L. Lagrange, Boulevard de l'Observatoire, CS 34229, 06304Nice CEDEX 4, France
P. L. Sulem
Affiliation:
Université Côte d'Azur, CNRS, Observatoire de la Côte d'Azur, Laboratoire J.L. Lagrange, Boulevard de l'Observatoire, CS 34229, 06304Nice CEDEX 4, France
*
†Email address for correspondence: etassi@oca.eu
Get access Rights & Permissions [Opens in a new window]

Abstract

A Hamiltonian six-field gyrofluid model is constructed, based on closure relations derived from the so-called ‘quasi-static’ gyrokinetic linear theory where the fields are assumed to propagate with a parallel phase velocity much smaller than the parallel particle thermal velocities. The main properties captured by this model, primarily aimed at exploring fundamental problems of interest for space plasmas such as the solar wind, are its ability to provide a reasonable agreement with kinetic theory for linear low-frequency modes, and at the same time to ensure a Hamiltonian structure in the absence of explicit dissipation. The model accounts for equilibrium temperature anisotropy, ion and electron finite Larmor radius corrections, electron inertia, magnetic fluctuations along the direction of a strong guide field and parallel Landau damping, introduced through a Landau-fluid modelling of the parallel heat transfers for both gyrocentre species. Remarkably, the quasi-static closure leads to exact and simple expressions for the nonlinear terms involving gyroaveraged electromagnetic fields and potentials. One of the consequences is that a rather natural identification of the Hamiltonian structure of the model becomes possible when Landau damping is neglected. A slight variant of the model consists of a four-field Hamiltonian reduction of the original six-field model, which is also used for the subsequent linear analysis. In the latter, the dispersion relations of kinetic Alfvén waves and the firehose instability are shown to be correctly reproduced, relatively far in the sub-ion range (depending on the plasma parameters), while the spectral range where the slow-wave dispersion relation and the field-swelling instabilities are precisely described is less extended. This loss of accuracy originates from the breaking of the condition of small phase velocity, relative to the parallel thermal velocity of the electrons (for kinetic Alfvén waves and firehose instability) or of the ions (in the case of the field-swelling instabilities).

Keywords

plasma instabilitiesspace plasma physics
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 4 , August 2020 , 835860402
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press

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

Alexandrova, O., Chen, C. H. K., Sorriso-Valvo, L., Horbury, T. S. & Bale, S. D. 2013 Solar wind turbulence and the role of ion instabilities. Space Sci. Rev. 178, 101–139.CrossRefGoogle Scholar
Bale, S. D., Kasper, J. C., Howes, G. G., Quataert, E., Salem, C. & Sundkvist, D. 2009 Magnetic fluctuation power near proton temperature anisotropy instability thresholds in the solar wind. Phys. Rev. Lett. 103, 211101.CrossRefGoogle ScholarPubMed
Basu, B. & Coppi, B. 1982 Field-swelling instability in anisotropic plasmas. Phys. Rev. Lett. 48, 799–801.CrossRefGoogle Scholar
Basu, B. & Coppi, B. 1984 Theory of field-swelling instability in anisotropic plasmas. Phys. Fluids 27, 1187.CrossRefGoogle Scholar
Boldyrev, S. & Perez, J. C. 2012 Spectrum of kinetic Alfvén turbulence. Astrophys. J. Lett. 758, L44.CrossRefGoogle Scholar
Bourouaine, S. & Chandran, B. D. G. 2013 Observational test of stochastic heating in low-$\beta$ fast-solar-windstreams. Astrophys. J. 774 (2), 96.CrossRefGoogle Scholar
Bowen, T. A., Badman, S., Hellinger, P. & Bale, S. D. 2018 Density fluctuations in the solar wind driven by Alfvén wave parametric decay. Astrophys. J. Lett. 854 (2), L33.CrossRefGoogle Scholar
Brizard, A. 1992 Nonlinear gyrofluid description of turbulent magnetized plasmas. Phys. Fluids 4, 1213–1228.CrossRefGoogle Scholar
Bruno, R. & Carbone, V. 2013 The solar wind as a turbulence laboratory. Living Rev. Sol. Phys. 10, 2.CrossRefGoogle Scholar
Chen, C. H. K. & Boldyrev, S. 2017 Nature of kinetic scale turbulence in the Earth's magnetosheath. Astrophys. J. 842, 122.CrossRefGoogle Scholar
Chen, C. H. K., Klein, K. G. & Howes, G. G. 2019 Evidence for electron Landau damping in space plasma turbulence. Nat. Comm. 10, 740.CrossRefGoogle ScholarPubMed
De Camillis, S., Cerri, S. S., Califano, F. & Pegoraro, F. 2016 Pressure anisotropy generation in a magnetized plasma configuration with a shear flow velocity. Plasma Phys. Control. Fusion 58 (4), 045007.CrossRefGoogle Scholar
Del Zanna, L., Velli, M. & Londrillo, P. 2001 Parametric decay of circularly polarized Alfvén waves: multidimensional simulations in periodic and open domains. Astron. Astrophys. 367, 705–718.CrossRefGoogle Scholar
Dorland, W. & Hammett, G. 1993 Gyrofluid turbulence models with kinetic effects. Phys. Fluids B 5, 812–835.CrossRefGoogle Scholar
Franci, L., Cerri, S. S., Califano, F., Landi, S., Papini, E., Verdini, A., Matteini, L., Jenko, F. & Hellinger, P. 2017 Magnetic reconnection as a driver for a sub-ion-scale cascade in plasma turbulence. Astrophys. J. Lett. 850, L16.CrossRefGoogle Scholar
Grasso, D., Borgogno, D., Tassi, E. & Perona, A. 2020 Asymmetry effects driving secondary instabilities in two-dimensional collisionless magnetic reconnection. Phys. Plasmas 27, 012302.CrossRefGoogle Scholar
Grasso, D. & Tassi, E. 2015 Hamiltonian magnetic reconnection with parallel heat flux dynamics. J. Plasma Phys. 81, 495810501.CrossRefGoogle Scholar
Hammett, G. W., Dorland, W. & Perkins, F. W. 1992 Fluid models of phase mixing, Landau damping, and nonlinear gyrokinetic dynamics. Phys. Fluids B 4, 2052.CrossRefGoogle Scholar
Hammett, G. W. & Perkins, F. W. 1990 Fluid moment models for Landau damping with application to the ion-temperature gradient instability. Phys. Rev. Lett. 64, 3019–3022.CrossRefGoogle ScholarPubMed
Hellinger, P., Trávníĉek, P., Kasper, J. C. & Lazarus, A. J. 2006 Solar wind proton temperature anisotropy: linear theory and WIND/SWE observations. Geophys. Res. Lett. 33, L09101.CrossRefGoogle Scholar
Hoppock, I. W., Chandran, B. D. G., Klein, K. G., Mallet, A. & Verscharen, D. 2018 Stochastic proton heating by kinetic-Alfvén-wave turbulence in moderately high-$\beta$ plasmas. J. Plasma Phys. 84, 905840615.CrossRefGoogle ScholarPubMed
Huang, J., Kasper, J. C., Vech, D., Klein, K. G., Stevens, M., Martinović, M. M., Alterman, B. L., Ďurovcová, T., Paulson, K., Maruca, B. A., et al. 2020 Proton temperature anisotropy variations in inner heliosphere estimated with the first Parker solar probe observations. Astrophys. J. Suppl. 246, 70.CrossRefGoogle Scholar
Hunana, P., Tenerani, A., Zank, G. P., Goldstein, M. L., Webb, G. M., Velli, M. & Adhikari, L. 2019 An introductory guide to fluid models with anisotropic temperatures. Part II. Kinetic theory, Padé approximants and Landau fluid closures. J. Plasma Phys. 85, 205850603.CrossRefGoogle Scholar
Keramidas Charidakos, I., Waelbroeck, F. L. & Morrison, P. J. 2015 A Hamiltonian five-field gyrofluid model. Phys. Plasmas 22, 112113.CrossRefGoogle Scholar
Kulsrud, R. M. 1983 MHD description of plasma. In Handbook of Plasma Physics (ed. Galeev, A. A. & Sudan, R. N.), vol. 1, pp. 115–145. North-Holland Publishing Company.Google Scholar
Kunz, M. W., Abel, I. G., Klein, K. G. & Schekochihin, A. A. 2018 Astrophysical gyrokinetics: turbulence in pressure-anisotropic plasmas at ion scales and beyond. Phys. Plasmas 84, 715840201.CrossRefGoogle Scholar
Kunz, W. M., Schekochihin, A. A., Chen, C. H. K., Abel, I. G. & Cowley, S. C. 2015 Inertial-range kinetic turbulence in pressure-anisotropic astrophysical plasmas. J. Plasma Phys. 81, 325810501.CrossRefGoogle Scholar
Kuznetsov, E. A., Passot, T. & Sulem, P. L. 2012 On the mirror instability in the presence of electron temperature anisotropy. Phys. Plasmas 19, 090701.CrossRefGoogle Scholar
Madsen, J. 2013 Full-F gyrofluid model. Phys. Plasmas 20, 072301.CrossRefGoogle Scholar
Mandell, N. R., Dorland, W. & Landreman, M. 2018 Laguerre–Hermite pseudo-spectral velocity formulation of gyrokinetics. J. Plasma Phys. 84, 905840108.CrossRefGoogle Scholar
Marsch, E. 2012 Helios: evolution of distribution functions 0.3–1 AU. Space Sci. Rev. 172, 23–39.CrossRefGoogle Scholar
Matteini, L., Landi, S., Velli, M. & Matthaeus, W. H. 2013 Proton temperature anisotropy and magnetic reconnection in the solar wind: effects of kinetic instabilities on current sheet stability. Astrophys. J. 763, 142.CrossRefGoogle Scholar
Miloshevich, G., Passot, T. & Sulem, P. L. 2020 Modeling imbalanced collisionless Alfvén wave turbulence with nonlinear diffusion equations. Astrophys. J. 888, L7.CrossRefGoogle Scholar
Morrison, P. J. 1998 Hamiltonian description of the ideal fluid. Rev. Mod. Phys. 70, 467–521.CrossRefGoogle Scholar
Passot, T. & Sulem, P. L. 2006 A fluid model with finite Larmor radius effects for mirror mode dynamics. J. Geophys. Res. 111, A04203.CrossRefGoogle Scholar
Passot, T. & Sulem, P. L. 2007 Collisionless magnetohydrodynamics with gyrokinetic effects. Phys. Plasmas 14, 082502.CrossRefGoogle Scholar
Passot, T. & Sulem, P. L. 2015 A model for the non-universal power-law of the solar wind sub-ion scale magnetic spectrum. Astrophys. J. Lett. 812, L37.CrossRefGoogle Scholar
Passot, T. & Sulem, P. L. 2019 Imbalanced kinetic Alfvén wave turbulence: from weak turbulence theory to nonlinear diffusion models for the strong regime. J. Plasma Phys. 85, 905850301.CrossRefGoogle Scholar
Passot, T., Sulem, P. L. & Tassi, E. 2017 Electron-scale reduced fluid models with gyroviscous effects. J. Plasma Phys. 83, 715830402.CrossRefGoogle Scholar
Passot, T., Sulem, P. & Tassi, E. 2018 Gyrofluid modeling and phenomenology of low-$\beta _e$ Alfvén wave turbulence. Phys. Plasmas 25, 042107.CrossRefGoogle Scholar
Pokhotelov, O. A. & Onishchenko, O. G. 2014 Role of finite ion temperature in the generation of field swelling instability. Geomagn. Aeron. 54, 20–22.CrossRefGoogle Scholar
Sahraoui, F., Goldstein, M. L., Belmont, G., Canu, P. & Rezeau, L. 2010 Three-dimensional anisotropic $k$ spectra of turbulence at subproton scales in the solar wind. Phys. Rev. Lett. 105, 131101.CrossRefGoogle ScholarPubMed
Schep, T. J., Pegoraro, F. & Kuvshinov, B. N. 1994 Generalized two-fluid theory of nonlinear magnetic structures. Phys. Plasmas 1, 2843–2851.CrossRefGoogle Scholar
Scott, B. 2010 Derivation via free energy conservation constraints of gyrofluid equations with finite-gyroradius electromagnetic nonlinearities. Phys. Plasmas 17, 102306.CrossRefGoogle Scholar
Shi, Y., Lee, L. C. & Fu, Z. F. 1987 A study of tearing instability in the presence of a pressure anisotropy. J. Geophys. Res. 92, A12171–A12179.CrossRefGoogle Scholar
Snyder, P. B. & Hammett, G. W. 2001 A Landau fluid model for electromagnetic plasma microturbulence. Phys. Plasmas 8, 3199–3216.CrossRefGoogle Scholar
Snyder, P. B., Hammett, G. W. & Dorland, W. 1997 Landau fluid models of collisionless magnetohydrodynamics. Phys. Plasmas 4, 3974–3985.CrossRefGoogle Scholar
Štverák, Š., Trávníček, P., Maksimovic, M., Marsch, E., Fazakerley, A. N. & Scime, E. E. 2008 Electron temperature anisotropy constraints in the solar wind. J. Geophys. Res. 113, A03103.CrossRefGoogle Scholar
Szegö, G. 1975 Orthogonal Polynomials. American Mathematical Society.Google Scholar
Tassi, E. 2015 Hamiltonian fluid reductions of electromagnetic drift-kinetic equations for an arbitrary number of moments. Ann. Phys. 362, 239.CrossRefGoogle Scholar
Tassi, E. 2019 Hamiltonian gyrofluid reductions of gyrokinetic equations. J. Phys. A: Math. Theor. 52, 465501.CrossRefGoogle Scholar
Tassi, E., Grasso, D., Borgogno, D., Passot, T. & Sulem, P. L. 2018 A reduced Landau-gyrofluid model for magnetic reconnection driven by electron inertia. J. Plasma Phys. 84, 725840401.CrossRefGoogle Scholar
Tenerani, A., Velli, M. & Hellinger, P. 2017 The parametric instability of Alfvén waves: effects of temperature anisotropy. Astrophys. J. Lett. 851 (2), 99.CrossRefGoogle Scholar
Waelbroeck, F. L., Hazeltine, R. D. & Morrison, P. J. 2009 A Hamiltonian electromagnetic gyrofluid model. Phys. Plasmas 16, 032109.CrossRefGoogle Scholar
Waelbroeck, F. L. & Tassi, E. 2012 A compressible Hamiltonian electromagnetic gyrofluid model. Commun. Nonlinear Sci. Numer. Simul. 17, 2171.CrossRefGoogle Scholar