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A compact exact law for compressible isothermal Hall magnetohydrodynamic turbulence

Published online by Cambridge University Press:  19 April 2021

Renaud Ferrand Open the ORCID record for Renaud Ferrand [Opens in a new window] ,
Sébastien Galtier Open the ORCID record for Sébastien Galtier [Opens in a new window]  and
Fouad Sahraoui
Renaud Ferrand*
Affiliation:
Laboratoire de Physique des Plasmas, École polytechnique, CNRS, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, F-91128Palaiseau Cedex, France
Sébastien Galtier
Affiliation:
Laboratoire de Physique des Plasmas, École polytechnique, CNRS, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, F-91128Palaiseau Cedex, France Institut Universitaire de France, 75005Paris, France
Fouad Sahraoui
Affiliation:
Laboratoire de Physique des Plasmas, École polytechnique, CNRS, Sorbonne Université, Université Paris-Saclay, Observatoire de Paris, F-91128Palaiseau Cedex, France
*
†Email address for correspondence: [email protected]
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Abstract

Using mixed second-order structure functions, a compact exact law is derived for isothermal compressible Hall magnetohydrodynamic turbulence with the assumptions of statistical homogeneity, time stationarity and infinite kinetic/magnetic Reynolds numbers. The resulting law is written as the sum of a Yaglom-like flux term, with an overall expression strongly reminiscent of the incompressible law, and a pure compressible source. Being mainly a function of the increments, the compact law is Galilean invariant but is dependent on the background magnetic field if one is present. Only the magnetohydrodynamic source term requires multi-spacecraft data to be estimated whereas the other components, which include those introduced by the Hall term, can be fully computed with single-spacecraft data using the Taylor hypothesis. These properties make this compact law more appropriate for analysing both numerical simulations and in situ data gathered in space plasmas, in particular when only single-spacecraft data are available.

Keywords

plasma nonlinear phenomenaspace plasma physics
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 2 , April 2021 , 905870220
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Andrés, N., Galtier, S. & Sahraoui, F. 2018 Exact law for homogeneous compressible Hall magnetohydrodynamics turbulence. Phys. Rev. E 97, 013204.CrossRefGoogle ScholarPubMed
Andrés, N. & Sahraoui, F. 2017 Alternative derivation of exact law for compressible and isothermal magnetohydrodynamics turbulence. Phys. Rev. E 96 (5), 053205.CrossRefGoogle ScholarPubMed
Andrés, N., Sahraoui, F., Galtier, S., Hadid, L. Z., Dmitruk, P. & Mininni, P. D. 2018 Energy cascade rate in isothermal compressible magnetohydrodynamic turbulence. J. Plasma Phys. 84 (4), 905840404.CrossRefGoogle Scholar
Andrés, N., Sahraoui, F., Galtier, S., Hadid, L. Z., Ferrand, R. & Huang, S. Y. 2019 Energy cascade rate measured in a collisionless space plasma with mms data and compressible hall magnetohydrodynamic turbulence theory. Phys. Rev. Lett. 123, 245101.CrossRefGoogle Scholar
Antonia, R. A., Ould-Rouis, M., Anselmet, F. & Zhu, Y. 1997 Analogy between predictions of Kolmogorov and Yaglom. J. Fluid Mech. 332, 395–409.CrossRefGoogle Scholar
Arzoumanian, D., André, P., Didelon, P., Könyves, V., Schneider, N., Men´shchikov, A., Sousbie, T., Zavagno, A., Bontemps, S., Di Francesco, J., et al. 2011 Characterizing interstellar filaments with herschel in ic 5146. Astron. Astrophys. 529, L6.CrossRefGoogle Scholar
Bandyopadhyay, R., Goldstein, M. L., Maruca, B. A., Matthaeus, W. H., Parashar, T. N., Ruffolo, D., Chhiber, R., Usmanov, A., Chasapis, A., Qudsi, R., et al. 2020 Enhanced energy transfer rate in solar wind turbulence observed near the sun from Parker solar probe. Astrophys. J. Suppl. 246 (2), 48.CrossRefGoogle Scholar
Bandyopadhyay, R., Sorriso-Valvo, L., Chasapis, A., Hellinger, P., Matthaeus, W. H., Verdini, A., Landi, S., Franci, L., Matteini, L., Giles, B. L., et al. 2020 In situ observation of hall magnetohydrodynamic cascade in space plasma. Phys. Rev. Lett. 124 (22), 225101.CrossRefGoogle ScholarPubMed
Banerjee, S. & Galtier, S. 2013 Exact relation with two-point correlation functions and phenomenological approach for compressible magnetohydrodynamic turbulence. Phys. Rev. E 87 (1), 013019.CrossRefGoogle ScholarPubMed
Banerjee, S. & Galtier, S. 2014 A kolmogorov-like exact relation for compressible polytropic turbulence. J. Fluid Mech. 742, 230–242.CrossRefGoogle Scholar
Banerjee, S. & Galtier, S. 2017 An alternative formulation for exact scaling relations in hydrodynamic and magnetohydrodynamic turbulence. J. Phys. A 50, 015501.CrossRefGoogle Scholar
Banerjee, S., Hadid, L., Sahraoui, F. & Galtier, S. 2016 Scaling of compressible magnetohydrodynamic turbulence in the fast solar wind. Astrophys. J. Lett. 829 (2), L27.Google Scholar
Banerjee, S. & Kritsuk, A. G. 2017 Exact relations for energy transfer in self-gravitating isothermal turbulence. Phys. Rev. E 96 (5), 053116.CrossRefGoogle ScholarPubMed
Banerjee, S. & Kritsuk, A. G. 2018 Energy transfer in compressible magnetohydrodynamic turbulence for isothermal self-gravitating fluids. Phys. Rev. E 97 (2), 023107.CrossRefGoogle ScholarPubMed
Belmont, G. & Rezeau, L. 2001 Magnetopause reconnection induced by magnetosheath hall-MHD fluctuations. J. Geophys. Res. 106.CrossRefGoogle Scholar
Bhattacharjee, A. 2004 Impulsive magnetic reconnection in the earth's magnetotail and the solar corona. Annu. Rev. Astron. Astrophys. 42, 365–384.CrossRefGoogle Scholar
Cooper, C. M., Wallace, J., Brookhart, M., Clark, M., Collins, C., Ding, W. X., Flanagan, K., Khalzov, I., Li, Y., Milhone, J., et al. 2014 The Madison plasma dynamo experiment: a facility for studying laboratory plasma astrophysics. Phys. Plasmas 21.CrossRefGoogle Scholar
Federrath, C. 2016 On the universality of interstellar filaments: theory meets simulations and observations. MNRAS 457 (1), 375–388.CrossRefGoogle Scholar
Ferrand, R., Galtier, S., Sahraoui, F. & Federrath, C. 2020 Compressible turbulence in the interstellar medium: new insights from a high-resolution supersonic turbulence simulation. Astrophys. J. 904 (2), 160.CrossRefGoogle Scholar
Ferrand, R., Galtier, S., Sahraoui, F., Meyrand, R., Andrés, N. & Banerjee, S. 2019 On exact laws in incompressible hall magnetohydrodynamic turbulence. Astrophys. J. 881 (1), 50.Google Scholar
Forest, C. B., Flanagan, K., Brookhart, M., Clark, M., Cooper, C. M., Désangles, V., Egedal, J., Endrizzi, D., Khalzov, I. V., Li, H., et al. 2015 The Wisconsin plasma astrophysics laboratory. J. Plasma Phys. 81, 345810501.CrossRefGoogle Scholar
Galtier, S. 2006 Multi-scale turbulence in the inner solar wind. J. Low Temp. Phys. 145, 59–74.Google Scholar
Galtier, S. 2008 von Kármán-Howarth equations for Hall magnetohydrodynamic flows. Phys. Rev. E 77 (1), 015302.CrossRefGoogle ScholarPubMed
Galtier, S. 2012 Kolmogorov vectorial law for solar wind turbulence. Astrophys. J. 746, 184.CrossRefGoogle Scholar
Galtier, S. 2016 Introduction to Modern Magnetohydrodynamics. Cambridge University Press.CrossRefGoogle Scholar
Galtier, S. & Banerjee, S. 2011 Exact relation for correlation functions in compressible isothermal turbulence. Phys. Rev. Lett. 107 (13), 134501.Google ScholarPubMed
Gourgouliatos, K. N. & Cumming, A. 2014 Hall attractor in axially symmetric magnetic fields in neutron star crusts. Phys. Rev. Lett. 112, 171101.CrossRefGoogle ScholarPubMed
Hadid, L., Sahraoui, F. & Galtier, S. 2017 Energy cascade rate in compressible fast and slow solar wind turbulence. Astrophys. J. 838, 11.CrossRefGoogle Scholar
Hadid, L. Z., Sahraoui, F., Galtier, S. & Huang, S. Y. 2018 Compressible magnetohydrodynamic turbulence in the earth's magnetosheath: estimation of the energy cascade rate using in situ spacecraft data. Phys. Rev. Lett. 120, 055102.CrossRefGoogle ScholarPubMed
Hellinger, P., Verdini, A., Landi, S., Franci, L. & Matteini, L. 2018 von Kármán-Howarth equation for hall magnetohydrodynamics: hybrid simulations. Astrophys. J. 857, L19.CrossRefGoogle Scholar
Howes, G. 2009 Limitations of Hall MHD as a model for turbulence in weakly collisional plasmas. Nonlinear Process. Geophys. 16, 219–232.CrossRefGoogle Scholar
Kolmogorov, A. N. 1941 Dissipation of energy in locally isotropic turbulence. Dokl. Akad. Nauk SSSR 32, 16–18.Google Scholar
Kritsuk, A. G., Norman, M. L., Padoan, P. & Wagner, R. 2007 The statistics of supersonic isothermal turbulence. Astrophys. J. 665, 416–431.CrossRefGoogle Scholar
Kritsuk, A. G., Wagner, R. & Norman, M. L. 2013 Energy cascade and scaling in supersonic isothermal turbulence. J. Fluid Mech. 729, 1.CrossRefGoogle Scholar
Kunz, M. W. & Lesur, G. 2013 Magnetic self-organisation in Hall-dominated magnetorotational turbulence. Mon. Not. R. Astron. Soc. 2295, 434.Google Scholar
MacBride, B. T., Smith, C. W. & Forman, M. A. 2008 The turbulent cascade at 1 au: energy transfer and the third-order scaling for MHD. Astrophys. J. 679 (2), 1644.CrossRefGoogle Scholar
Marino, R., Sorriso-Valvo, L., Carbone, V., Noullez, A., Bruno, R. & Bavassano, B. 2008 Heating the solar wind by a magnetohydrodynamic turbulent energy cascade. Astrophys. J. 677, L71.CrossRefGoogle Scholar
Meyrand, R. & Galtier, S. 2013 Anomalous $k_{\perp }^-8/3$ spectrum in electron magnetohydrodynamic turbulence. Phys. Rev. Lett. 111, 264501.CrossRefGoogle ScholarPubMed
Monin, A. S. 1959 On the theory of locally isotropic turbulence. Dokl. Akad. Nauk SSSR 125, 515–518.Google Scholar
Osman, K. T., Wan, M., Matthaeus, W. H., Weygand, J. M. & Dasso, S. 2011 Anisotropic third-moment estimates of the energy cascade in solar wind turbulence using multispacecraft data. Phys. Rev. Lett. 107 (16), 165001.Google ScholarPubMed
Oughton, S., Wan, M., Servidio, S. & Matthaeus, W. H. 2013 On the origin of anisotropy in magnetohydrodynamic turbulence: the role of higher-order correlations. Astrophys. J. 768, 10.CrossRefGoogle Scholar
Politano, H. & Pouquet, A. 1998 Dynamical length scales for turbulent magnetized flows. Geophys. Res. Lett. 25, 273–276.CrossRefGoogle Scholar
Sahraoui, F., Galtier, S. & Belmont, G. 2007 On waves in incompressible Hall magnetohydro-dynamics. J. Plasma Phys. 73, 723–730.Google Scholar
Sahraoui, F., Pincon, J. L., Belmont, G., Rezeau, L., Cornilleau-Wehrlin, N., Robert, P., Mellul, L., Bosqued, J. M., Balogh, A., Canu, P., et al. 2003 ULF wave identification in the magnetosheath: the k-filtering technique applied to Cluster II data. J. Geophys. Res. 108 (A9).CrossRefGoogle Scholar
Sorriso-Valvo, L., Marino, R., Carbone, V., Noullez, A., Lepreti, F., Veltri, P., Bruno, R., Bavassano, B. & Pietropaolo, E. 2007 Observation of inertial energy cascade in interplanetary space plasma. Phys. Rev. Lett. 99 (11), 115001.CrossRefGoogle ScholarPubMed
Stawarz, J. E., Smith, C. W., Vasquez, B. J., Forman, M. A. & MacBride, B. T. 2009 The turbulent cascade and proton heating in the solar wind at 1 AU. Astrophys. J. 697, 1119–1127.CrossRefGoogle Scholar
Wan, M., Oughton, S., Servidio, S. & Matthaeus, W. H. 2012 von kármán self-preservation hypothesis for magnetohydrodynamic turbulence and its consequences for universality. J. Fluid Mech. 697, 296–315.CrossRefGoogle Scholar
Yoshimatsu, K. 2012 Examination of the four-fifths law for longitudinal third-order moments in incompressible magnetohydrodynamic turbulence in a periodic box. Phys. Rev. E 85, 066313.Google Scholar