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Kinetic Alfvén wave generation by velocity shear in collisionless plasmas

Published online by Cambridge University Press:  23 April 2020

Teresa Maiorano
Affiliation:
Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy
Adriana Settino
Affiliation:
Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy
Francesco Malara*
Affiliation:
Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy
Oreste Pezzi
Affiliation:
Gran Sasso Science Institute, Viale F. Crispi 7, I-67100 L. Aquila, Italy INFN/Laboratori Nazionali del Gran Sasso, Via G. Acitelli 22, I-67100 Assergi (AQ), Italy
Francesco Pucci
Affiliation:
Centre for Mathematical Plasma Astrophysics, Department of Mathematics, KU Leuven, Celestijnenlaan 200B, 3001Leuven, Belgium
Francesco Valentini
Affiliation:
Dipartimento di Fisica, Università della Calabria, 87036 Rende (CS), Italy
*
Email address for correspondence: francesco.malara@fis.unical.it

Abstract

The evolution of a linearly polarized, long-wavelength Alfvén wave propagating in a collisionless magnetized plasma with a sheared parallel-directed velocity flow is here studied by means of two-dimensional hybrid Vlasov–Maxwell (HVM) simulations. The unperturbed sheared flow has been represented by an exact solution of the HVM set of equations of (Malara et al., Phys. Rev. E, vol. 97, 2018, 053212), thus avoiding spurious oscillations that would arise from the non-stationary initial state and inevitably affect the dynamics of the system. We have considered the evolution of both a small and a moderate amplitude Alfvén wave, in order to separate linear wave–shear flow couplings from kinetic effects, the latter being more relevant for larger wave amplitudes. The phase mixing generated by the shear flow modifies the initial perturbation, leading to the formation of small-scale transverse fluctuations at scales comparable with the proton inertial length/Larmor radius. By analysing both the polarization and group velocity of perturbations in the shear regions, we identify them as kinetic Alfvén waves (KAWs). In the moderate amplitude run, kinetic effects distort the proton distribution function in the shear region. This leads to the formation of a proton beam, at the Alfvén speed and parallel to the magnetic field. Such a feature, due to the parallel electric field associated with KAWs, positively compares with solar wind observations of suprathermal ion populations, suggesting that it may be related to the presence of ion-scale KAW-like fluctuations.

Type
Research Article
Copyright
© Cambridge University Press 2020

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References

Araneda, J. A., Marsch, E. & Viñas, A. F. 2008 Proton core heating and beam formation via parametrically unstable Alfvén-cyclotron waves. Phys. Rev. Lett. 100 (12), 125003.CrossRefGoogle ScholarPubMed
Belcher, J. & Davis, L. 1971 Large-amplitude Alfvén waves in the interplanetary medium. J. Geophys. Res. 76 (16), 35343563.CrossRefGoogle Scholar
Bale, S., Kellogg, P., Mozer, F., Horbury, T. & Reme, H. 2005 Measurement of the electric fluctuation spectrum of magnetohydrodynamic turbulence. Phys. Rev. Lett. 94 (21), 215002.CrossRefGoogle ScholarPubMed
Bale, S., Badman, S. T., Bonnell, J. W., Bowen, T. A., Burgess, D., Case, A. W., Cattell, C. A., Chandran, B. D. G., Chaston, C. C., Chen, C. H. K. et al. 2019 Highly structured slow solar wind emerging from an equatorial coronal hole. Nature 576 (21), 237242.CrossRefGoogle ScholarPubMed
Bruno, R. & Carbone, V. 2013 The solar wind as a turbulence laboratory. Living Rev. Sol. Phys. 10, 2.CrossRefGoogle Scholar
Califano, F., Chiuderi, C. & Einaudi, G. 1990 Nonresonant resistive dissipation of incompressible magnetohydrodynamic waves. Astrophys. J. 365, 757763.CrossRefGoogle Scholar
Califano, F., Chiuderi, C. & Einaudi, G. 1992 Nonresonant resistive dissipation of compressible magnetohydrodynamic waves. Astrophys. J. 390, 560566.CrossRefGoogle Scholar
Carbone, V. & Veltri, P. 1990 A shell model for anisotropic magnetohydrodynamic turbulence. Geophys. Astrophys. Fluid Dyn. 52 (1–3), 153181.CrossRefGoogle Scholar
Carbone, V., Malara, F. & Veltri, P. 1995 A model for the three-dimensional magnetic field correlation spectra of low-frequency solar wind fluctuations during Alfvénic periods. J. Geophys. Res. 100 (A2), 17631778.CrossRefGoogle Scholar
Cerri, S. S., Kunz, M. W. & Califano, F. 2018 Dual phase–space cascades in 3D hybrid Vlasov–Maxwell turbulence. Astrophys. J. Lett. 856, L13.CrossRefGoogle Scholar
Chen, C., Boldyrev, S., Xia, Q. & Perez, J. 2013 Nature of subproton scale turbulence in the solar wind. Phys. Rev. Lett. 110 (22), 225002.Google ScholarPubMed
Dasso, S., Milano, L. J., Matthaeus, W. H. & Smith, C. W. 2005 Anisotropy in fast and slow solar wind fluctuations. Astrophys. J. 635 (2), L181L184.CrossRefGoogle Scholar
Davila, J. M. 1987 Heating of the solar corona by the resonant absorption of Alfvén waves. Astrophys. J. 317, 514521.CrossRefGoogle Scholar
Décamp, N. & Malara, F. 2006 Electron acceleration in turbulent coronal loops by kinetic Alfvén wave. In SOHO-17. 10 Years of SOHO and Beyond, vol. 617, p. 26. ESA Publications Division.Google Scholar
Gary, S. P. & Nishimura, K. 2004 Kinetic Alfvén waves: linear theory and a particle-in-cell simulation. J. Geophys. Res. 109, A02109.CrossRefGoogle Scholar
Ghosh, S., Matthaeus, W. H., Roberts, D. A. & Goldstein, M. L. 1998a The evolution of slab fluctuations in the presence of pressure-balanced magnetic structures and velocity shears. J. Geophys. Res. 103 (A10), 2369123704.CrossRefGoogle Scholar
Ghosh, S., Matthaeus, W. H., Roberts, D. A. & Goldstein, M. L. 1998b Waves, structures, and the appearance of two-component turbulence in the solar wind. J. Geophys. Res. 103 (A10), 2370523716.CrossRefGoogle Scholar
Goldstein, B. E., Neugebauer, M., Zhang, L. D. & Gary, S. P. 2000 Observed constraint on proton–proton relative velocities in the solar wind. Geophys. Res. Lett. 27 (1), 5356.CrossRefGoogle Scholar
Goodrich, C. & Lazarus, A. 1976 Suprathermal protons in the interplanetary solar wind. J. Geophys. Res. 81 (16), 27502754.CrossRefGoogle Scholar
Hamlin, N. D. & Newman, W. I. 2013 Role of the Kelvin–Helmholtz instability in the evolution of magnetized relativistic sheared plasma flows. Phys. Rev. E 87, 043101.Google ScholarPubMed
Hellinger, P., Trávníček, 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
Hellinger, P. & Trávníček, P. 2011 Proton core-beam system in the expanding solar wind: hybrid simulations. J. Geophys. Res. 116, A11101.CrossRefGoogle Scholar
Hellinger, P. & Trávníček, P. 2013 Protons and alpha particles in the expanding solar wind: hybrid simulations. J. Geophys. Res. Space Phys. 117, 54215430.CrossRefGoogle Scholar
Hollweg, J. V. 1987 Resonance absorption of magnetohydrodynamic surface waves physical discussion. Astrophys. J. 312, 880885.CrossRefGoogle Scholar
Hollweg, J. V. 1999 Kinetic Alfvén wave revisited. J. Geophys. Res. 104 (A7), 1481114819.CrossRefGoogle Scholar
Howes, G. G., Dorland, G. W., Cowley, S., Hammett, G. W., Quataert, E., Schekochihin, A. A. & Tatsuno, T. 2008a Kinetic simulations of magnetized turbulence in astrophysical plasmas. Phys. Rev. Lett. 100 (6), 65004.CrossRefGoogle Scholar
Howes, G. G., Cowley, S. C., Dorland, W., Hammett, G. W., Quataert, E. & Schekochihin, A. A. 2008b A model of turbulence in magnetized plasmas: implications for the dissipation range in the solar wind. J. Geophys. Res. 113, A05103.CrossRefGoogle Scholar
Kappraff, J. & Tataronis, J. 1977 Resistive effects on Alfvén wave heating. J. Plasma Phys. 18 (02), 209226.CrossRefGoogle Scholar
Kaghashvili, E. K. 1999 On the acceleration of the solar wind: role of the inhomogeneous flow. Astrophys. J. 512 (2), 969974.CrossRefGoogle Scholar
Kiyani, K. H., Chapman, S. C., Sahraoui, F., Hnat, B., Fauvarque, O. & Khotyaintsev, Y. V. 2013 Enhanced magnetic compressibility and isotropic scale invariance at sub-ion larmor scales in solar wind turbulence. Astrophys. J. 763 (1), 10.CrossRefGoogle Scholar
Landi, S., Velli, M. & Einaudi, G. 2005 Alfvén waves and shock wave formation at an x-point magnetic field configuration. Astrophys. J. 624 (1), 392401.CrossRefGoogle Scholar
Lee, M. & Roberts, B. 1986 On the behavior of hydromagnetic surface waves. Astrophys. J. 301, 430439.CrossRefGoogle Scholar
Lysak, R. L. & Song, Y. 2011 Development of parallel electric fields at the plasma sheet boundary layer. J. Geophys. Res. 116, A00K14.CrossRefGoogle Scholar
Malara, F., Veltri, P., Chiuderi, C. & Einaudi, G. 1992 Incompressible disturbances in nonuniform media-formation of small scales. Astrophys. J. 396, 297310.CrossRefGoogle Scholar
Malara, F., Primavera, L. & Veltri, P. 1996a Compressive fluctuations generated by time evolution of Alfvénic perturbations in the solar wind current sheet. J. Geophys. Res. 101 (A10), 2159721617.CrossRefGoogle Scholar
Malara, F., Primavera, L. & Veltri, P. 1996b Formation of small scales via Alfvén wave propagation in compressible nonuniform media. Astrophys. J. 459, 347364.CrossRefGoogle Scholar
Malara, F., Petkaki, P. & Veltri, P. 2000 Dissipation of Alfvén waves in force-free magnetic fields: competition between phase mixing and three-dimensional effects. Astrophys. J. 533, 523534.Google Scholar
Malara, F., De Franceschis, M. F. & Veltri, P. 2003 Alfvén wave propagation and dissipation in a 3D-structured compressible plasma. Astron. Astrophys. 412 (2), 529539.CrossRefGoogle Scholar
Malara, F., De Franceschis, M. F. & Veltri, P. 2005 Dissipation of Alfvén waves in complex 3D coronal force-free structures. Astron. Astrophys. 443 (3), 10331046.CrossRefGoogle Scholar
Malara, F., De Franceschis, M. F. & Veltri, P. 2007 Alfvén wave dissipation and topological properties of 3D coronal force-free magnetic fields. Astron. Astrophys. 467 (3), 12751284.CrossRefGoogle Scholar
Malara, F., Pezzi, O. & Valentini, F. 2018 Exact hybrid Vlasov equilibria for sheared plasmas with in-plane and out-of-plane magnetic field. Phys. Rev. E 97, 053212.Google ScholarPubMed
Malara, F., Nigro, G., Valentini, F. & Sorriso-Valvo, L. 2019 Electron heating by kinetic Alfvén waves in coronal loop turbulence. Astrophys. J. 871, 66.CrossRefGoogle Scholar
Marsch, E., Mühlhäuser, K.-H., Schwenn, R., Rosenbauer, H., Pilipp, W. & Neubauer, F. 1982 Solar wind protons: three-dimensional velocity distributions and derived plasma parameters measured between 0.3 and 1 AU. J. Geophys. Res. 87 (A1), 5272.CrossRefGoogle Scholar
Marsch, E. 2006 Kinetic physics of the solar corona and solar wind. Living Rev. Solar Phys. 3, 1.CrossRefGoogle Scholar
Matteini, L., Landi, S., Velli, M. & Hellinger, P. 2010 Kinetics of parametric instabilities of Alfvén waves: evolution of ion distribution functions. J. Geophys. Res. 115, A09106.CrossRefGoogle Scholar
Matthaeus, W. H., Goldstein, M. L. & King, J. 1986 An interplanetary magnetic field ensemble at 1 AU. J. Geophys. Res. 91 (A1), 5969.CrossRefGoogle Scholar
Matthaeus, W. H., Goldstein, M. L. & Roberts, D. A. 1990 Evidence for the presence of quasi-two-dimensional nearly incompressible fluctuations in the solar wind. J. Geophys. Res. 95 (A12), 2067320683.CrossRefGoogle Scholar
Matthaeus, W. H., Servidio, S., Dmitruk, P., Carbone, V., Oughton, S., Wan, M. & Osman, K. T. 2012 Local anisotropy, higher order statistics, and turbulence spectra. Astrophys. J. 750 (2), 103.CrossRefGoogle Scholar
McLaughlin, J., Hood, A. & De Moortel, I. 2011 Review article: MHD wave propagation near coronal null points of magnetic fields. Space Sci. Rev. 158 (2–4), 205236.CrossRefGoogle Scholar
Milano, L. J., Matthaeus, W. H., Dmitruk, P. & Montgomery, D. C. 2001 Local anisotropy in incompressible magnetohydrodynamic turbulence. Phys. Plasmas 8 (6), 26732681.CrossRefGoogle Scholar
Mok, Y. & Einaudi, G. 1985 Resistive decay of Alfvén waves in a non-uniform plasma. J. Plasma Phys. 33 (01), 199208.CrossRefGoogle Scholar
Nariyuki, Y., Umeda, T., Suzuki, T. & Hada, T. 2014a Ion acceleration by parallel propagating nonlinear Alfvén wave packets in a radially expanding plasma. Nonlinear Process. Geophys. 21 (1), 339346.CrossRefGoogle Scholar
Nariyuki, Y., Hada, T. & Tsubouchi, K. 2014b Collisionless damping of circularly polarized nonlinear Alfvén waves in solar wind plasmas with and without beam protons. Astrophys. J. 793 (2), 138.CrossRefGoogle Scholar
Ofman, L. & Aschwanden, M. 2002 Damping time scaling of coronal loop oscillations deduced from transition region and coronal explorer observations. Astrophys. J. Lett. 576 (2), L153.CrossRefGoogle Scholar
Oughton, S., Matthaeus, W. H., Wan, M. & Osman, K. T. 2015 Anisotropy in solar wind plasma turbulence. Phil. Trans. R. Soc. Lond. A 373 (2041), 20140152.Google ScholarPubMed
Oughton, S., Priest, E. R. & Matthaeus, W. H. 1994 The influence of a mean magnetic field on three-dimensional magnetohydrodynamic turbulence. J. Fluid Mech. 280, 95117.CrossRefGoogle Scholar
Petkaki, P., Malara, F. & Veltri, P. 1998 Topological formation of small scales in magnetohydrodynamics: a fast dissipation mechanism. Astrophys. J. 500 (1), 483491.CrossRefGoogle Scholar
Pezzi, O., Parashar, T. N., Servidio, S., Valentini, F., Vásconez, C. L., Yang, Y., Malara, F., Matthaeus, W. H. & Veltri, P. 2017a Revisiting a classic: the Parker–Moffatt problem. Astrophys. J. 834, 166.CrossRefGoogle Scholar
Pezzi, O., Parashar, T. N., Servidio, S., Valentini, F., Vásconez, C. L., Yang, Y., Malara, F., Matthaeus, W. H. & Veltri, P. 2017b Colliding Alfvénic wave packets in magnetohydrodynamics, Hall and kinetic simulations. J. Plasma Phys. 83, 905830105.CrossRefGoogle Scholar
Pezzi, O., Malara, F., Servidio, S., Valentini, F., Parashar, T. N., Matthaeus, W. H. & Veltri, P. 2017c Turbulence generation during the head-on collision of Alfvénic wave packets. Phys. Rev. E 96, 023201.Google Scholar
Pezzi, O., Servidio, S., Perrone, D., Valentini, F., Sorriso-Valvo, L., Greco, A., Matthaeus, W. H. & Veltri, P. 2018 Velocity–space cascade in magnetized plasmas: numerical simulations. Phys. Plasmas 25, 060704.CrossRefGoogle Scholar
Podesta, J. J. & TenBarge, J. M. 2012 Scale dependence of the variance anisotropy near the proton gyroradius scale: additional evidence for kinetic Alfvén waves in the solar wind at 1 AU. J. Geophys. Res. 117 (1), A10106.CrossRefGoogle Scholar
Pucci, F., Onofri, M. & Malara, F. 2014 Evolution of magnetohydrodynamic waves in low layers of a coronal hole. Astrophys. J. 796 (1), 43.CrossRefGoogle Scholar
Pucci, F., Vásconez, C. L., Pezzi, O., Servisio, S., Valentini, F., Matthaeus, W. H. & Malara, F. 2016 From Alfvén waves to kinetic Alfvén waves in an inhomogeneous equilibrium structure. J. Geophys. Res. Space Phys. 121, 10241045.CrossRefGoogle Scholar
Roberts, D. A., Ghosh, S., Goldstein, M. L. & Matthaeus, W. H. 1991 MHD simulation of the radial evolution and stream structure of the solar wind turbulence. Phys. Rev. Lett. 67, 37413744.CrossRefGoogle Scholar
Roberts, D. A., Goldstein, M. L., Matthaeus, W. H. & Ghosh, S. 1992 Velocity shear generation of solar wind turbulence. J. Geophys. Res. 97, 1711517130.CrossRefGoogle Scholar
Roytershteyn, V. & Daughton, W. 2008 Collisionless instability of thin current sheets in the presence of sheared parallel flows. Phys. Plasmas 15, 082901.CrossRefGoogle Scholar
Sahraoui, F., Goldstein, M. L., Robert, P. & Khotyaintsev, Y. V. 2009 Evidence of a cascade and dissipation of solar-wind turbulence at the electron gyroscale. Phys. Rev. Lett. 102 (23), 231102.CrossRefGoogle ScholarPubMed
Sahraoui, F., Belmont, G. & Goldstein, M. 2012 New insight into short-wavelength solar wind fluctuations from Vlasov theory. Astrophys. J. 748 (2), 100.CrossRefGoogle Scholar
Salem, C., Howes, G., Sundkvist, D., Bale, S., Chaston, C., Chen, C. & Mozer, F. 2012 Identification of kinetic Alfvén wave turbulence in the solar wind. Astrophys. J. Lett. 745 (1), L9.CrossRefGoogle Scholar
Servidio, S., Chasapis, A., Matthaeus, W. H., Perroned, D., Valentini, F., Parashar, T. N. et al. 2017 Magnetospheric multiscale observation of plasma velocity–space cascade: Hermite representation and theory. Phys. Rev. Lett. 119, 205101.CrossRefGoogle ScholarPubMed
Schekochihin, A., Cowley, S., Dorland, W., Hammett, G. H. G., Quataert, E. & Tatsuno, T. 2009 Astrophysical gyrokinetics: kinetic and fluid turbulent cascades in magnetized weakly collisional plasmas. Astrophys. J. Suppl. Ser. 182 (1), 310377.CrossRefGoogle Scholar
Shebalin, J. V., Matthaeus, W. H. & Montgomery, D. 1983 Anisotropy in MHD turbulence due to a mean magnetic field. J. Plasma Phys. 29 (03), 525547.CrossRefGoogle Scholar
Similon, P. L. & Sudan, R. 1989 Energy dissipation of Alfvén wave packets deformed by irregular magnetic fields in solar-coronal arches. Astrophys. J. 336, 442453.CrossRefGoogle Scholar
Sorriso-Valvo, L., Perrone, D., Pezzi, O., Valentini, F., Servidio, S., Zouganelis, I. & Veltri, P. 2018 Local energy transfer rate and kinetic processes: the fate of turbulent energy in two-dimensional hybrid Vlasov-Maxwell numerical simulations. J. Plasma Phys. 84 (2), 725840201.CrossRefGoogle Scholar
Sorriso-Valvo, L., Catapano, F., Retinó, A, Le Contel, O., Perrone, D., Roberts, O. W. et al. 2019 Turbulence–driven ion beams in the magnetospheric Kelvin–Helmholtz instability. Phys. Rev. Lett. 122, 035102.CrossRefGoogle ScholarPubMed
Steinolfson, R. 1985 Resistive wave dissipation on magnetic inhomogeneities normal modes and phase mixing. Astrophys. J. 295, 213219.CrossRefGoogle Scholar
TenBarge, J. & Howes, G. 2012 Evidence of critical balance in kinetic Alfvén wave turbulence simulations. Phys. Plasmas 19 (5), 55901.CrossRefGoogle Scholar
Tomczyk, S., McIntosh, S. W., Keil, S., Judge, P., Schad, T., Seeley, D. & Edmondson, J. 2007 Alfvén waves in the solar corona. Science 317 (5842), 11921196.CrossRefGoogle ScholarPubMed
Tomczyk, S. & McIntosh, S. W. 2009 Time-distance seismology of the solar corona with coMP. Astrophys. J. 697 (2), 13841391.CrossRefGoogle Scholar
Tsiklauri, D., Nakariakov, V. & Rowlands, G. 2002 A three dimensional magnetohydrodynamic pulse in a transversely inhomogeneous medium. Astron. Astrophys. 393 (1), 321329.CrossRefGoogle Scholar
Tsiklauri, D. & Nakariakov, V. 2003 Phase mixing of a three dimensional magnetohydrodynamic pulse. Astron. Astrophys. 400 (3), 10511055.CrossRefGoogle Scholar
Tsiklauri, D., Sakai, J.-I. & Saito, S. 2005 Particle-in-cell simulations of circularly polarised Alfvén wave phase mixing: a new mechanism for electron acceleration in collisionless plasmas. Astron. Astrophys. 435 (3), 11051113.CrossRefGoogle Scholar
Tsiklauri, D. 2011 Particle acceleration by circularly and elliptically polarised dispersive Alfvén waves in a transversely inhomogeneous plasma in the inertial and kinetic regimes. Phys. Plasmas 18 (9), 92903.CrossRefGoogle Scholar
Tsiklauri, D. 2012 Three dimensional particle-in-cell simulation of particle acceleration by circularly polarised inertial Alfvén waves in a transversely inhomogeneous plasma. Phys. Plasmas 19 (8), 82903.CrossRefGoogle Scholar
Tu, C.-Y., Marsch, E. & Qin, Z.-R. 2004 Dependence of the proton beam drift velocity on the proton core plasma beta in the solar wind. J. Geophys. Res. 109, A05101.CrossRefGoogle Scholar
Valentini, F., Trávníček, P., Califano, F., Hellinger, P. & Mangeney, A. 2007 A hybrid-Vlasov model based on the current advance method for the simulation of collisionless magnetized plasma. J. Comput. Phys. 225, 753.CrossRefGoogle Scholar
Valentini, F., Veltri, P., Califano, F. & Mangeney, A. 2008 Cross-scale effects in solar-wind turbulence. Phys. Rev. Lett. 101 (2), 25006.CrossRefGoogle ScholarPubMed
Valentini, F., Perrone, D. & Veltri, P. 2011a Short-wavelength electrostatic fluctuations in the solar wind. Astrophys. J. 739 (1), 54.CrossRefGoogle Scholar
Valentini, F., Califano, F., Perrone, D., Pegoraro, F. & Veltri, P. 2011b New ion-wave path in the energy cascade. Phys. Rev. Lett. 106 (16), 165002.CrossRefGoogle Scholar
Valentini, F., Vásconez, C. L., Pezzi, O., Servidio, S., Malara, F. & Pucci, F. 2017 Transition to kinetic turbulence at proton scales driven by large-amplitude kinetic Alfvén fluctuations. Astron. Astrophys. 599, A8.CrossRefGoogle Scholar
Vásconez, C. L., Valentini, F., Camporeale, E. & Veltri, P. 2014 Vlasov simulations of kinetic Alfvén waves at proton kinetic scales. Phys. Plasmas 21 (11), 112107.CrossRefGoogle Scholar
Vásconez, C. L., Pucci, F., Valentini, F., Servidio, S., Matthaeus, W. H. & Malara, F. 2015 Kinetic Alfvén wave generation by large-scale phase mixing. Astrophys. J. 815, 7.CrossRefGoogle Scholar
Voitenko, Y. & Goossens, M. 2004 Cross-field heating of coronal ions by low-frequency kinetic Alfvén waves. Astrophys. J. Lett. 605 (2), L149.CrossRefGoogle Scholar
Wu, D. & Chen, L. 2013 Excitation of kinetic Alfvén waves by density striation in magneto-plasmas. Astrophys. J. 771 (1), 3.CrossRefGoogle Scholar