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Solar Physics

, 292:148 | Cite as

Optimal Energy Growth in Current Sheets

  • David MacTaggartEmail author
  • Peter Stewart
Article

Abstract

In this article, we investigate the possibility of transient growth in the linear perturbation of current sheets. The resistive magnetohydrodynamics operator for a background field consisting of a current sheet is non-normal, meaning that associated eigenvalues and eigenmodes can be very sensitive to perturbation. In a linear stability analysis of a tearing current sheet, we show that modes that are damped as \(t\rightarrow \infty \) can produce transient energy growth, contributing faster growth rates and higher energy attainment (within a fixed finite time) than the unstable tearing mode found from normal-mode analysis. We determine the transient growth for tearing-stable and tearing-unstable regimes and discuss the consequences of our results for processes in the solar atmosphere, such as flares and coronal heating. Our results have significant potential impact on how fast current sheets can be disrupted. In particular, transient energy growth due to (asymptotically) damped modes may lead to accelerated current sheet thinning and, hence, a faster onset of the plasmoid instability, compared to the rate determined by the tearing mode alone.

Keywords

Magnetohydrodynamics Instabilities Magnetic reconnection, theory 

Notes

Disclosure of Potential Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

  1. Abramowitz, M., Stegun, I.A. (eds.): 1964, Handbook of Mathematical Functions, Dover, New York. zbMATHGoogle Scholar
  2. Bobra, D., Riedel, K.S., Kerner, W., Huysmans, G.T.A., Ottaviani, M., Schmid, P.J.: 1994, Phys. Plasmas 1, 3151. ADSCrossRefMathSciNetGoogle Scholar
  3. Camporeale, E.: 2010, Space Sci. Rev. 172, 397. ADSCrossRefGoogle Scholar
  4. Camporeale, E., Burgess, D., Passot, T.: 2009, Phys. Plasmas 16, 030703. ADSCrossRefGoogle Scholar
  5. Chandrasekhar, S.: 1961, Hydrodynamic and Hydromagnetic Stability, Clarendon, Oxford. zbMATHGoogle Scholar
  6. Comisso, L., Lingam, M., Huang, Y.-M., Bhattacharjee, A.: 2016, Phys. Plasmas 23, 100702. ADSCrossRefGoogle Scholar
  7. Farrell, B.F., Ioannou, P.J.: 1999a, Astrophys. J. 522, 1079. ADSCrossRefGoogle Scholar
  8. Farrell, B.F., Ioannou, P.J.: 1999b, Astrophys. J. 522, 1088. ADSCrossRefGoogle Scholar
  9. Furth, H.P., Kileen, J., Rosenbluth, M.N.: 1963, Phys. Fluids 6, 459. ADSCrossRefGoogle Scholar
  10. Goedbloed, J.P., Keppens, R., Poedts, S.: 2010, Advanced Magnetohydrodynamics, Cambridge University Press, Cambridge. CrossRefGoogle Scholar
  11. Hanifi, A., Schmid, P.J., Henningson, D.S.: 1996, Phys. Fluids 8, 826. ADSCrossRefMathSciNetGoogle Scholar
  12. Hood, A.W., Hughes, D.W.: 2011, Phys. Earth Planet. Inter. 187, 78. ADSCrossRefGoogle Scholar
  13. Hood, A.W., Cargill, P.J., Browning, P.K., Tam, K.V.: 2016, Astrophys. J. Lett. 817, 5. ADSCrossRefGoogle Scholar
  14. Lapenta, G.: 2008, Phys. Rev. Lett. 100, 235001. ADSCrossRefGoogle Scholar
  15. Livermore, P.W., Jackson, A.: 2006, Proc. Roy. Soc. A 462, 2457. ADSCrossRefGoogle Scholar
  16. Lourerio, N.F., Schekochihin, A.A., Cowley, S.C.: 2007, Phys. Plasmas 14, 100703. ADSCrossRefGoogle Scholar
  17. MacNeice, P., Antiochos, S.K., Phillips, S., Spicer, D.S., DeVore, C.R., Olson, K.: 2004, Astrophys. J. 614, 1028. ADSCrossRefGoogle Scholar
  18. MacTaggart, D., Haynes, A.L.: 2014, Mon. Not. Roy. Astron. Soc. 438, 1500. ADSCrossRefGoogle Scholar
  19. MacTaggart, D., Guglielmino, S.L., Haynes, A.L., Simitev, R.D., Zuccarello, F.: 2015, Astron. Astrophys. 576, A4. ADSCrossRefGoogle Scholar
  20. Parker, E.N.: 1988, Astrophys. J. 330, 474. ADSCrossRefGoogle Scholar
  21. Priest, E.R.: 2014, Magnetohydrodynamics of the Sun, Cambridge University Press, Cambridge. Google Scholar
  22. Pritchett, P.L., Lee, Y.C., Drake, J.F.: 1980, Phys. Fluids 23, 1368. ADSCrossRefGoogle Scholar
  23. Pucci, F., Velli, M.: 2014, Astrophys. J. Lett. 780, L19. ADSCrossRefGoogle Scholar
  24. Reddy, S.C., Henningson, D.S.: 1993, J. Fluid Mech. 252, 209. ADSCrossRefMathSciNetGoogle Scholar
  25. Reddy, S.C., Schmid, P.J., Henningson, D.S.: 1993, SIAM J. Appl. Math. 53, 15. CrossRefMathSciNetGoogle Scholar
  26. Schindler, K.: 2006, Physics of Space Plasma Activity, Cambridge University Press, Cambridge. CrossRefGoogle Scholar
  27. Schmid, P.J., Henningson, D.S.: 1994, J. Fluid Mech. 277, 197. ADSCrossRefMathSciNetGoogle Scholar
  28. Tassi, E., Hastie, R.J., Porcelli, F.: 2007, Phys. Plasmas 14, 9. CrossRefGoogle Scholar
  29. Tenerani, A., Rappazzo, A.F., Velli, M., Pucci, F.: 2015, Astrophys. J. 801, 145. ADSCrossRefGoogle Scholar
  30. Tenerani, A., Velli, M., Pucci, F., Landi, S., Rappazzo, A.F.: 2016, J. Plasma Phys. 82, 535820501. CrossRefGoogle Scholar
  31. Terasawa, T.: 1983, Geophys. Res. Lett. 10, 475. ADSCrossRefGoogle Scholar
  32. Tichmarsh, E.C.: 1948, Introduction to the Theory of Fourier Integrals, Oxford University Press, London. Google Scholar
  33. Trefethen, L.N., Bau, D.: 1997, Numerical Linear Algebra, SIAM, Philadelphia. CrossRefzbMATHGoogle Scholar
  34. Trefethen, L.N., Embree, M.: 2005, Spectra and Pseudospectra: The Behaviour of Nonnormal Matrices and Operators, Princeton University Press, Princeton. zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2017

Authors and Affiliations

  1. 1.School of Mathematics & StatisticsUniversity of GlasgowGlasgowUK

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