Ideal MHD Instabilities, with a Focus on the Rayleigh–Taylor and Kelvin–Helmholtz Instabilities

  • Andrew HillierEmail author
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 591)


In this chapter we focus on the magnetohydrodynamic (MHD) versions of the Rayleigh–Taylor and Kelvin–Helmholtz instabilities, taking the reader beyond the commonly presented situations to include how extra physics influences the stability of the models. After a discussion of the physical processes behind each instability we look at the general framework behind the study of ideal MHD instabilities, providing a detailed look at the derivation of the dispersion relation for a simple model. Extensions to this model are presented, including an investigation into how stability changes in the presence of a time-varying flow. Finally, we take a look at how nonlinearities develop and the role of the MHD in terms of the development of these nonlinearities.



Andrew Hillier is supported by his STFC Ernest Rutherford Fellowship grant number ST/L00397X/2 and STFC research grant ST/R000891/1. This work used the COSMA Data Centric system at Durham University, operated by the Institute for Computational Cosmology on behalf of the STFC DiRAC HPC Facility ( This equipment was funded by a BIS National E-infrastructure capital grant ST/K00042X/1, DiRAC Operations grant ST/K003267/1 and Durham University. DiRAC is part of the UK National E-Infrastructure.


  1. P. Antolin, T. Yokoyama, T. Van Doorsselaere, Fine strand-like structure in the solar corona from magnetohydrodynamic transverse oscillations. Astrophys. J. Lett. 787, L22 (2014)CrossRefGoogle Scholar
  2. C.M. Bender, S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw-Hill, New York, 1978)zbMATHGoogle Scholar
  3. T.E. Berger, G. Slater, N. Hurlburt, R. Shine, T. Tarbell, A. Title, B.W. Lites, T.J. Okamoto, K. Ichimoto, Y. Katsukawa, T. Magara, Y. Suematsu, T. Shimizu, Quiescent prominence dynamics observed with the hinode solar optical telescope. I. Turbulent upflow plumes. Astrophys. J. 716, 1288–1307 (2010)CrossRefGoogle Scholar
  4. J. Carlyle, A. Hillier, The non-linear growth of the magnetic Rayleigh–Taylor instability. Astron. Astrophys. 605, A101 (2017)CrossRefGoogle Scholar
  5. S. Chandrasekhar, Hydrodynamic and Hydromagnetic Stability (Clarendon, Oxford, 1961)zbMATHGoogle Scholar
  6. G. Dimonte, D.L. Youngs, A. Dimits, S. Weber, M. Marinak, S. Wunsch, C. Garasi, A. Robinson, M.J. Andrews, P. Ramaprabhu, A.C. Calder, B. Fryxell, J. Biello, L. Dursi, P. MacNeice, K. Olson, P. Ricker, R. Rosner, F. Timmes, H. Tufo, Y.N. Young, M. Zingale, A comparative study of the turbulent Rayleigh–Taylor instability using high-resolution three-dimensional numerical simulations: the Alpha-Group collaboration. Phys. Fluids 16, 1668–1693 (2004)CrossRefGoogle Scholar
  7. C. Foullon, E. Verwichte, V.M. Nakariakov, K. Nykyri, C. J. Farrugia, Magnetic Kelvin–Helmholtz Instability at the Sun. Astrophys. J. 729, L8 (2011)CrossRefGoogle Scholar
  8. J.P.H. Goedbloed, S. Poedts, Principles of Magnetohydrodynamics (Cambridge University Press, Cambridge, 2004)CrossRefGoogle Scholar
  9. H. Helmholtz, Über discontinuierliche Flüssigkeits-Bewegungen. Monatsberichte der Königlichen Preussische Akademie der Wissenschaften zu Berlin 23, 215–228 (1868)Google Scholar
  10. A. Hillier, On the nature of the magnetic Rayleigh–Taylor instability in astrophysical plasma: the case of uniform magnetic field strength. Mon. Not. R. Astron. Soc. 462, 2256–2265 (2016)CrossRefGoogle Scholar
  11. A. Hillier, The magnetic Rayleigh–Taylor instability in solar prominences. Rev. Mod. Plasma Phys. 2, 1 (2018)CrossRefGoogle Scholar
  12. A. Hillier, V. Polito, Observations of the Kelvin–Helmholtz instability driven by dynamic motions in a solar prominence. Astrophys. J. Lett. 864, L10 (2018)CrossRefGoogle Scholar
  13. A. Hillier, A. Barker, I. Arregui, H. Latter, On Kelvin–Helmholtz and parametric instabilities driven by coronal waves. Mon. Not. R. Astron. Soc. 482, 1143–1153 (2019)CrossRefGoogle Scholar
  14. A.W. Hood, E.R. Priest, Kink instability of solar coronal loops as the cause of solar flares. Solar Phys. 64, 303–321 (1979)CrossRefGoogle Scholar
  15. L.N. Howard, Note on a paper of John W. Miles. J. Fluid Mech. 10, 509–512 (1961)MathSciNetCrossRefGoogle Scholar
  16. D.W. Hughes, S.M. Tobias, On the instability of magnetohydrodynamic shear flows. Proc. R. Soc. Lond. Ser. A 457, 1365 (2001)MathSciNetCrossRefGoogle Scholar
  17. R.E. Kelly, The stability of unsteady Kelvin–Helmholtz flow. J. Fluid Mech. 22, 547–560 (1965)MathSciNetCrossRefGoogle Scholar
  18. L. Kelvin, Hydrokinetic solutions and observations. Philos. Mag. 42, 362–377 (1871)CrossRefGoogle Scholar
  19. M. Kruskal, M. Schwarzschild, Some instabilities of a completely ionized plasma. Proc. R. Soc. Lond. Ser. A 223, 348–360 (1954)MathSciNetCrossRefGoogle Scholar
  20. A. Miura, P.L. Pritchett, Nonlocal stability analysis of the MHD Kelvin–Helmholtz instability in a compressible plasma. J. Geophys. Res. 87, 7431–7444 (1982)CrossRefGoogle Scholar
  21. L. Rayleigh, Investigation of the character of the equilibrium of an incompressible heavy fluid of variable density. Proc. Lond. Math. Soc. 14, 170–177 (1883)MathSciNetzbMATHGoogle Scholar
  22. J.R. Ristorcelli, T.T. Clark, Rayleigh Taylor turbulence: self-similar analysis and direct numerical simulations. J. Fluid Mech. 507, 213–253 (2004)MathSciNetCrossRefGoogle Scholar
  23. B. Roberts, On the hydromagnetic stability of an unsteady Kelvin–Helmholtz flow. J. Fluid Mech. 59, 65–76 (1973)MathSciNetCrossRefGoogle Scholar
  24. M.S. Ruderman, Compressibility effect on the Rayleigh–Taylor instability with sheared magnetic fields. Solar Phys. 292, 47 (2017)CrossRefGoogle Scholar
  25. M.S. Ruderman, J. Terradas, J.L. Ballester, Rayleigh–Taylor instabilities with sheared magnetic fields. Astrophys. J. 785, 110 (2014)CrossRefGoogle Scholar
  26. J.M. Stone, T. Gardiner, Nonlinear evolution of the magnetohydrodynamic Rayleigh–Taylor instability. Phys. Fluids 19(9), 094104 (2007a)CrossRefGoogle Scholar
  27. J.M. Stone, T. Gardiner, The magnetic Rayleigh–Taylor instability in three dimensions. Astrophys. J. 671, 1726–1735 (2007b)CrossRefGoogle Scholar
  28. G. Taylor, The Instability of liquid surfaces when accelerated in a direction perpendicular to their planes. I. Proc. R. Soc. Lond. Ser. A 201, 192–196 (1950)MathSciNetCrossRefGoogle Scholar
  29. J. Terradas, J. Andries, M. Goossens, I. Arregui, R. Oliver, J.L. Ballester, Nonlinear instability of kink oscillations due to shear motions. Astrophys. J. Lett. 687, L115 (2008)CrossRefGoogle Scholar

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© CISM International Centre for Mechanical Sciences 2020

Authors and Affiliations

  1. 1.College of Engineering, Mathematics and Physical SciencesUniversity of ExeterExeterUK

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