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Helical mode interactions and spectral transfer processes in magnetohydrodynamic turbulence

Published online by Cambridge University Press:  15 February 2016

Moritz Linkmann ,
Arjun Berera ,
Mairi McKay  and
Julia Jäger
Moritz Linkmann*
Affiliation:
School of Physics and Astronomy, University of Edinburgh, EdinburghEH9 3FD, UK
Arjun Berera
Affiliation:
School of Physics and Astronomy, University of Edinburgh, EdinburghEH9 3FD, UK
Mairi McKay
Affiliation:
School of Physics and Astronomy, University of Edinburgh, EdinburghEH9 3FD, UK
Julia Jäger
Affiliation:
School of Physics and Astronomy, University of Edinburgh, EdinburghEH9 3FD, UK
*
†Email address for correspondence: m.linkmann@ed.ac.uk
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Abstract

Spectral transfer processes in homogeneous magnetohydrodynamic (MHD) turbulence are investigated analytically by decomposition of the velocity and magnetic fields in Fourier space into helical modes. Steady solutions of the dynamical system which governs the evolution of the helical modes are determined, and a stability analysis of these solutions is carried out. The interpretation of the analysis is that unstable solutions lead to energy transfer between the interacting modes while stable solutions do not. From this, a dependence of possible interscale energy and helicity transfers on the helicities of the interacting modes is derived. As expected from the inverse cascade of magnetic helicity in 3-D MHD turbulence, mode interactions with like helicities lead to transfer of energy and magnetic helicity to smaller wavenumbers. However, some interactions of modes with unlike helicities also contribute to an inverse energy transfer. As such, an inverse energy cascade for non-helical magnetic fields is shown to be possible. Furthermore, it is found that high values of the cross-helicity may have an asymmetric effect on forward and reverse transfer of energy, where forward transfer is more quenched in regions of high cross-helicity than reverse transfer. This conforms with recent observations of solar wind turbulence. For specific helical interactions the relation to dynamo action is established. The present analysis provides new theoretical insights into physical processes where inverse cascade and dynamo action are involved, such as the evolution of cosmological and astrophysical magnetic fields and laboratory plasmas.

JFM classification

Turbulent Flows: Homogeneous turbulence MHD and Electrohydrodynamics: MHD turbulence Turbulent Flows: Turbulence theory
Type
Papers
Information
Journal of Fluid Mechanics , Volume 791 , 25 March 2016 , pp. 61 - 96
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Copyright
© 2016 Cambridge University Press 

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References

Alexakis, A., Mininni, P. D. & Pouquet, A. 2005 Shell-to-shell energy transfer in magnetohydrodynamics. I. Steady state turbulence. Phys. Rev. E 72, 046301.CrossRefGoogle ScholarPubMed
Alexakis, A., Mininni, P. D. & Pouquet, A. 2006 On the inverse cascade of magnetic helicity. Astrophys. J. 640, 335–343.CrossRefGoogle Scholar
André, J. C. & Lesieur, M. 1977 Influence of helicity on the evolution of isotropic turbulence at high Reynolds number. J. Fluid Mech. 81, 187–207.CrossRefGoogle Scholar
Balsara, D. & Pouquet, A. 1999 The formation of large-scale structures in supersonic magnetohydrodynamic flows. Phys. Plasmas 6, 89–99.CrossRefGoogle Scholar
Berera, A. & Linkmann, M. F. 2014 Magnetic helicity and the evolution of decaying magnetohydrodynamic turbulence. Phys. Rev. E 90, 041003(R).CrossRefGoogle ScholarPubMed
Biferale, L., Musacchio, S. & Toschi, F. 2012 Inverse energy cascade in three-dimensional isotropic turbulence. Phys. Rev. Lett. 108, 164501.CrossRefGoogle ScholarPubMed
Biferale, L., Musacchio, S. & Toschi, F. 2013 Split energy-helicity cascades in three dimensional homogeneous and isotropic turbulence. J. Fluid Mech. 730, 309–327.CrossRefGoogle Scholar
Biferale, L. & Titi, E. S. 2013 On the global regularity of a helical-decimated version of the 3D Navier–Stokes equation. J. Stat. Phys. 151, 1089.CrossRefGoogle Scholar
Biskamp, D. 1993 Nonlinear Magnetohydrodynamics, 1st edn. Cambridge University Press.CrossRefGoogle Scholar
Boffetta, G. & Musacchio, S. 2010 Evidence for the double cascade scenario in two-dimensional turbulence. Phys. Rev. E 82, 016307.CrossRefGoogle ScholarPubMed
Brandenburg, A. 2001 The inverse cascade and nonlinear alpha-effect in simulations of isotropic helical magnetohydrodynamic turbulence. Astrophys. J. 550, 824–840.CrossRefGoogle Scholar
Brandenburg, A. 2003 The helicity issue in large scale dynamos. In Turbulence and Magnetic Fields in Astrophysics (ed. Falgarone, E. & Passot, T.), Lecture Notes in Physics, vol. 614, pp. 402–413. Springer.CrossRefGoogle Scholar
Brandenburg, A., Kahniashvili, T. & Tevzadze, A. G. 2015 Nonhelical inverse transfer of a decaying turbulent magnetic field. Phys. Rev. Lett. 114, 075001.CrossRefGoogle ScholarPubMed
Brissaud, A., Frisch, U., Léorat, J., Lesieur, M. & Mazure, A. 1973 Helicity cascades in fully developed isotropic turbulence. Phys. Fluids 16, 1366.CrossRefGoogle Scholar
Carati, D., Debliquy, O., Knaepen, B., Teaca, B. & Verma, M. 2006 Energy transfers in forced MHD turbulence. J. Turbul. 7, 1–12.CrossRefGoogle Scholar
Chen, Q., Chen, S. & Eyink, G. L. 2003a The joint cascade of energy and helicity in three-dimensional turbulence. Phys. Fluids 15, 361–374.CrossRefGoogle Scholar
Chen, Q., Chen, S., Eyink, G. L. & Holm, D. D. 2003b Intermittency in the joint cascade of energy and helicity. Phys. Rev. Lett. 90, 214503.CrossRefGoogle ScholarPubMed
Childress, S. & Gilbert, A. D. 1995 Stretch, Twist, Fold: The Fast Dynamo. Springer.Google Scholar
Cho, J. 2010 Non-locality of hydrodynamic and magnetohydrodynamic turbulence. Astrophys. J. 725, 1786–1791.CrossRefGoogle Scholar
Cho, J. 2011 Magnetic helicity conservation and inverse energy cascade in electron magnetohydrodynamic wave packets. Phys. Rev. Lett. 106, 191104.CrossRefGoogle ScholarPubMed
Christensson, M., Hindmarsh, M. & Brandenburg, A. 2001 Inverse cascade in decaying 3D magnetohydrodynamic turbulence. Phys. Rev. E 64, 056405.CrossRefGoogle Scholar
Coburn, J. T., Smith, C. W., Vasquez, B. J., Forman, M. A. & Stawarz, J. E. 2014 Variable cascade dynamics and intermittency in the solar wind at 1 AU. Astrophys. J. 713, 920–934.Google Scholar
Constantin, P. & Majda, A. 1988 The Beltrami spectrum for incompressible flows. Commun. Math. Phys. 115, 435–456.CrossRefGoogle Scholar
Debliquy, O., Verma, M. K. & Carati, D. 2005 Energy fluxes and shell-to-shell transfers in three-dimensional decaying magnetohydrodynamic turbulence. Phys. Plasmas 12, 042309.CrossRefGoogle Scholar
Dubief, Y., Terrapon, V. E. & Soria, J. 2013 On the mechanism of elasto-inertial turbulence. Phys. Fluids 25, 110817.CrossRefGoogle ScholarPubMed
Frisch, U. 1995 Turbulence: The Legacy of Kolmogorov. Cambridge University Press.CrossRefGoogle Scholar
Frisch, U., Pouquet, A., Léorat, J. & Mazure, A. 1975 Possibility of an inverse cascade of magnetic helicity in magnetohydrodynamic turbulence. J. Fluid Mech. 68, 769–778.CrossRefGoogle Scholar
Karimabadi, H., Roytershteyn, V., Wan, M., Matthaeus, W. H., Daughton, W., Wu, P., Shay, M., Loring, B., Borovsky, J., Leonardis, E., Chapman, S. C. & Nakamura, T. K. M. 2013 Coherent structures, intermittent turbulence, and dissipation in high-temperature plasmas. Phys. Plasmas 20, 012303.CrossRefGoogle Scholar
Kraichnan, R. H. 1967 Inertial ranges in two-dimensional turbulence. Phys. Fluids 10 (10), 1417.CrossRefGoogle Scholar
Krause, F. & Rädler, K. 1980 Mean-Field Magnetohydrodynamics and Dynamo Theory. Pergamon Press, Ltd.Google Scholar
Lessinnes, T., Plunian, F. & Carati, D. 2009 Helical shell models for MHD. Theor. Comput. Fluid Dyn. 23, 439–450.CrossRefGoogle Scholar
McComb, W. D. 2014 Homogeneous, Isotropic Turbulence: Phenomenology, Renormalization and Statistical Closures. Oxford University Press.CrossRefGoogle Scholar
Mininni, P. D. 2011 Scale interactions in magnetohydrodynamic turbulence. Annu. Rev. Fluid Mech. 43, 377–397.CrossRefGoogle Scholar
Mininni, P. D., Alexakis, A. & Pouquet, A. 2005 Shell-to-shell energy transfer in magnetohydrodynamics. II. Kinematic dynamo. Phys. Rev. E 72, 046302.CrossRefGoogle ScholarPubMed
Mininni, P. D., Alexakis, A. & Pouquet, A. G. 2009 Scale interactions and scaling laws in rotating flows at moderate rossby numbers and large Reynolds numbers. Phys. Fluids 21, 015108.CrossRefGoogle Scholar
Mininni, P. D. & Pouquet, A. 2013 Inverse cascade behavior in freely decaying two-dimensional fluid turbulence. Phys. Rev. E 87, 033002.CrossRefGoogle Scholar
Moffatt, H. K. 1969 The degree of knottedness of tangled vortex lines. J. Fluid Mech. 35, 117–129.CrossRefGoogle Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Müller, W. C., Malapaka, S. K. & Busse, A. 2012 Inverse cascade of magnetic helicity in magnetohydrodynamic turbulence. Phys. Rev. E 85, 015302.CrossRefGoogle ScholarPubMed
Parker, E. N. 1979 Cosmical Magnetic Fields: their Origin and their Activity. Clarendon, Oxford University Press.Google Scholar
Pelz, R. B., Shtilman, B. & Tsinober, A. 1986 The helical nature of unforced turbulent flows. Phys. Fluids 29, 3506–3508.CrossRefGoogle Scholar
Polifke, W. 1991 Statistics of helicity fluctuations in homogeneous turbulence. Phys. Fluids A 3, 115.CrossRefGoogle Scholar
Polifke, W. & Shtilman, L. 1989 The dynamics of helical decaying turbulence. Phys. Fluids A 1, 2025.CrossRefGoogle Scholar
Politano, H. & Pouquet, A. 1995 Model of intermittency in magnetohydrodynamic turbulence. Phys. Rev. E 52, 636.CrossRefGoogle ScholarPubMed
Pouquet, A., Frisch, U. & Léorat, J. 1976 Strong MHD helical turbulence and the nonlinear dynamo effect. J. Fluid Mech. 77, 321–354.CrossRefGoogle Scholar
Pouquet, A. & Patterson, G. S. 1978 Numerical simulation of helical magnetohydrodynamic turbulence. J. Fluid Mech. 85, 305–323.CrossRefGoogle Scholar
Sahoo, G., Bonaccorso, F. & Biferale, L. 2015 On the role of helicity for large- and small-scales turbulent fluctuations. Phys. Rev. E 92, 051002.CrossRefGoogle Scholar
Son, D. T. 1999 Magnetohydrodynamics of the early universe and the evolution of primordial magnetic fields. Phys. Rev. D 59, 063008.CrossRefGoogle Scholar
Stawarz, J. E., Smith, C. W., Vasquez, B. J., Forman, M. A. & MacBride, B. T. 2010 The turbulent cascade for high cross-helicity states at 1 AU. Astrophys. J. 713, 920–934.CrossRefGoogle Scholar
Stepanov, R., Frick, P. & Mizeva, I. 2015 Joint inverse cascade of magnetic energy and magnetic helicity in MHD turbulence. Astrophys. J. 798, L35.CrossRefGoogle Scholar
Titchmarsh, E. C. 1939 The Theory of Functions, 2nd edn. Oxford University Press.Google Scholar
Vainshtein, S. I. & Zeldovich, Y. B. 1972 Origin of magnetic fields in astrophysics. Sov. Phys. Uspekhi 15, 159–172.CrossRefGoogle Scholar
Waleffe, F. 1992 The nature of triad interactions in homogeneous turbulence. Phys. Fluids A 4, 350–363.CrossRefGoogle Scholar
Zhu, J.-Z., Yang, W. & Zhu, G.-Y. 2014 Purely helical absolute equilibria and chirality of (magneto)fluid turbulence. J. Fluid Mech. 739, 479–501.CrossRefGoogle Scholar
Zrake, J. 2014 Inverse cascade of nonhelical magnetic turbulence in a relativistic fluid. Astrophys. J. 794, L26.CrossRefGoogle Scholar