Hostname: page-component-cd9895bd7-jkksz Total loading time: 0 Render date: 2024-12-27T22:05:21.188Z Has data issue: false hasContentIssue false
  • English
  • Français

Elasticity of tangled magnetic fields

Part of: Focus on Plasma Astrophysics

Published online by Cambridge University Press:  15 October 2020

D. N. Hosking Open the ORCID record for D. N. Hosking [Opens in a new window] ,
A. A. Schekochihin Open the ORCID record for A. A. Schekochihin [Opens in a new window]  and
S. A. Balbus Open the ORCID record for S. A. Balbus [Opens in a new window]
D. N. Hosking*
Affiliation:
Oxford Astrophysics, Denys Wilkinson Building, Keble Road, OxfordOX1 3RH, UK Merton College, Merton Street, OxfordOX1 4JD, UK
A. A. Schekochihin
Affiliation:
Merton College, Merton Street, OxfordOX1 4JD, UK Rudolf Peierls Centre for Theoretical Physics, Clarendon Laboratory, Parks Road, OxfordOX1 3PU, UK
S. A. Balbus
Affiliation:
Oxford Astrophysics, Denys Wilkinson Building, Keble Road, OxfordOX1 3RH, UK New College, Holywell Street, OxfordOX1 3BN, UK
*
†Email address for correspondence: david.hosking@physics.ox.ac.uk
Get access Rights & Permissions [Opens in a new window]

Abstract

The fundamental difference between incompressible ideal magnetohydrodynamics and the dynamics of a non-conducting fluid is that magnetic fields exert a tension force that opposes their bending; magnetic fields behave like elastic strings threading the fluid. It is natural, therefore, to expect that a magnetic field tangled at small length scales should resist a large-scale shear in an elastic way, much as a ball of tangled elastic strings responds elastically to an impulse. Furthermore, a tangled field should support the propagation of ‘magnetoelastic waves’, the isotropic analogue of Alfvén waves on a straight magnetic field. Here, we study magnetoelasticity in the idealised context of an equilibrium tangled field configuration. In contrast to previous treatments, we explicitly account for intermittency of the Maxwell stress, and show that this intermittency necessarily decreases the frequency of magnetoelastic waves in a stable field configuration. We develop a mean-field formalism to describe magnetoelastic behaviour, retaining leading-order corrections due to the coupling of large- and small-scale motions, and solve the initial-value problem for viscous fluids subjected to a large-scale shear, showing that the development of small-scale motions results in anomalous viscous damping of large-scale waves. Finally, we test these analytic predictions using numerical simulations of standing waves on tangled, linear force-free magnetic-field equilibria.

Keywords

plasma wavesastrophysical plasmasplasma dynamics
Type
Research Article
Information
Journal of Plasma Physics , Volume 86 , Issue 5 , October 2020 , 905860511
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

Bernstein, I. B., Frieman, E. A., Kruskal, M. D. & Kulsrud, R. M. 1958 An energy principle for hydromagnetic stability problems. Proc. R. Soc. Lond. 244, 17.Google Scholar
Blandford, R., Yuan, Y., Hoshino, M. & Sironi, L. 2017 Magnetoluminescence. Space Sci. Rev. 207, 291.Google Scholar
Brandenburg, A. & Subramanian, K. 2005 Astrophysical magnetic fields and nonlinear dynamo theory. Phys. Rep. 417, 1.Google Scholar
Burns, K. J., Vasil, G. M., Oishi, J. S., Lecoanet, D. & Brown, B. P. 2020 Dedalus: a flexible framework for numerical simulations with spectral methods. Phys. Rev. Res. 2, 023068.Google Scholar
Chandrasekhar, S. & Woltjer, L. 1958 On force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 285.Google ScholarPubMed
Chen, C. & Diamond, P. H. 2020 Potential vorticity mixing in a tangled magnetic field. Astrophys. J. 892, 24.Google Scholar
East, W. E., Zrake, J., Yuan, Y. & Blandford, R. D. 2015 Spontaneous decay of periodic magnetostatic equilibria. Phys. Rev. Lett. 115, 095002.Google ScholarPubMed
Er-Riani, M., Naji, A. & El Jarroudi, M. 2014 A note on the stability of Beltrami fields for compressible fluid flows. Intl J. Non-Linear Mech. 67, 231.Google Scholar
Gruzinov, A. V. & Diamond, P. H. 1996 Nonlinear mean field electrodynamics of turbulent dynamos. Phys. Plasmas 3, 1853.Google Scholar
Krause, F. & Raedler, K. H. 1980 Mean-field Magnetohydrodynamics and Dynamo Theory. Elsevier.Google Scholar
Kulsrud, R. M. 2005 Plasma Physics for Astrophysics. Princeton University Press.Google Scholar
Lüst, R. & Schlüter, A. 1954 Kraftfreie magnetfelder. Z. Astrophys. 34, 263.Google Scholar
Lyutikov, M., Sironi, L., Komissarov, S. S. & Porth, O. 2017 Particle acceleration in relativistic magnetic flux-merging events. J. Plasma Phys. 83, 635830602.Google Scholar
Maron, J., Cowley, S. & McWilliams, J. 2004 The nonlinear magnetic cascade. Astrophys. J. 603, 569.Google Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Moffatt, H. K. 1986 Magnetostatic equilibria and analogous Euler flows of arbitrarily complex topology. II – stability considerations. J. Fluid Mech. 166, 359.Google Scholar
Molodensky, M. M. 1974 Equilibrium and stability of force-free magnetic field. Sol. Phys. 39, 393.Google Scholar
Müller, W. C. & Grappin, R. 2004 The residual energy in freely decaying magnetohydrodynamic turbulence. Plasma Phys. Control. Fusion 46, B91.Google Scholar
Müller, W. C. & Grappin, R. 2005 Spectral energy dynamics in magnetohydrodynamic turbulence. Phys. Rev. Lett. 95, 114502.Google Scholar
Nalewajko, K. 2018 Three-dimensional kinetic simulations of relativistic magnetostatic equilibria. Mon. Not. R. Astron. Soc. 481, 4342.Google Scholar
Nalewajko, K., Zrake, J., Yuan, Y., East, W. E. & Blandford, R. D. 2016 Kinetic simulations of the lowest-order unstable mode of relativistic magnetostatic equilibria. Astrophys. J. 826, 115.Google Scholar
Ogilvie, G. I. & Proctor, M. R. E. 2003 On the relation between viscoelastic and magnetohydrodynamic flows and their instabilities. J. Fluid Mech. 476, 389.Google Scholar
Qin, B., Salipante, P. F., Hudson, S. D. & Arratia, P. E. 2019 Upstream vortex and elastic wave in the viscoelastic flow around a confined cylinder. J. Fluid Mech. 864, R2.Google ScholarPubMed
Rincon, F. 2019 Dynamo theories. J. Plasma Phys. 85, 205850401.Google Scholar
Schekochihin, A. A., Cowley, S. C., Hammett, G. W., Maron, J. L. & McWilliams, J. C. 2002 A model of nonlinear evolution and saturation of the turbulent MHD dynamo. New J. Phys. 4, 84.Google Scholar
Schekochihin, A. A., Cowley, S. C., Taylor, S. F., Maron, J. L. & McWilliams, J. C. 2004 Simulations of the small-scale turbulent dynamo. Astrophys. J. 612, 276.Google Scholar
Taylor, J. B. 1974 Relaxation of toroidal plasma and generation of reverse magnetic fields. Phys. Rev. Lett. 33, 1139.Google Scholar
Vekshtein, G. E. 1989 Magnetohydrodynamic stability of force-free magnetic fields in a rarefied plasma. J. Expl Theor. Phys. 96, 1263.Google Scholar
Voslamber, D. & Callebaut, D. K. 1962 Stability of force-free magnetic fields. Phys. Rev. 128, 2016.Google Scholar
Williams, P. T. 2004 Turbulent magnetohydrodynamic elasticity: Boussinesq-like approximations for steady shear. New Astron. 10, 133.Google Scholar
Woltjer, L. 1958 A theorem on force-free magnetic fields. Proc. Natl Acad. Sci. USA 44, 489.Google ScholarPubMed
Woltjer, L. 1959 Hydromagnetic equilibrium. II. Stability in the variational formulation. Proc. Natl Acad. Sci. USA 45, 769.Google Scholar
Yuan, Y., Nalewajko, K., Zrake, J., East, W. E. & Blandford, R. D. 2016 Kinetic study of radiation-reaction-limited particle acceleration during the relaxation of unstable force-free equilibria. Astrophys. J. 828, 92.Google Scholar
Zel'dovich, Y. B., Ruzmaikin, A. A. & Sokoloff, D. D. 1983 Magnetic Fields in Astrophysics. Gordon and Breach.Google Scholar
Zrake, J. & East, W. E. 2016 Freely decaying turbulence in force-free electrodynamics. Astrophys. J. 817, 89.Google Scholar