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Sufficient conditions of Rayleigh-Taylor stability and instability in equatorial ionosphere

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Abstract

Rayleigh-Taylor (R-T) instability is known as the fundamental mechanism of equatorial plasma bubbles (EPBs). However, the sufficient conditions of R-T instability and stability have not yet been derived. In the present paper, the sufficient conditions of R-T stability and instability are preliminarily derived. Linear equations for small perturbation are first obtained from the electron/ion continuity equations, momentum equations, and the current continuity equation in the equatorial ionosphere. The linear equations can be casted as an eigenvalue equation using a normal mode method. The eigenvalue equation is a variable coefficient linear equation that can be solved using a variational approach. With this approach, the sufficient conditions can be obtained as follows: if the minimum systematic eigenvalue is greater than one, the ionosphere is R-T unstable; while if the maximum systematic eigenvalue is less than one, the ionosphere is R-T stable. An approximate numerical method for obtaining the systematic eigenvalues is introduced, and the R-T stable/unstable areas are calculated. Numerical experiments are designed to validate the sufficient conditions. The results agree with the derived sufficient conditions.

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References

  1. Woodman, R. F. Spread F—an old equatorial aeronomy problem finally resolved? Annales Geophysicae, 27, 1915–1934 (2009)

    Article  Google Scholar 

  2. Kelley, M. C. The Earth’s Ionosphere: Plasma Physics and Electrodynamics, 2nd ed., Elsevier, London (2009)

    Google Scholar 

  3. Huang, C. S., Le, G., de La Beaujardiere, O., Roddy, P. A., Hunton, D. E., Pfaff, R. F., and Hairston, M. R. Relationship between plasma bubbles and density enhancements: observations and interpretation. Journal of Geophysical Research: Space Physics, 119, 1325–1336 (2014)

    Google Scholar 

  4. Carter, B. A., Yizengaw, E., Retterer, J. M., Francis, M., Terkildsen, M., Marshall, R., Norman, R., and Zhang, K. An analysis of the quiet time day-to-day variability in the formation of postsunset equatorial plasma bubbles in the Southeast Asian region. Journal of Geophysical Research: Space Physics, 119, 3206–3223 (2014)

    Google Scholar 

  5. Rayleigh, L. On the stability or instability of certain fluid motions. Proceedings of London Mathematical Society, 11, 57–70 (1880)

    MathSciNet  MATH  Google Scholar 

  6. Fjortoft, R. Application of integral theorems in deriving criteria of stability for laminar flows and for the baroclinic circular vortex. Geofys. Publ. Norske Vid.- Akad. Oslo, 17, 1–52 (1950)

    MathSciNet  Google Scholar 

  7. Howard, L. N. Note on a paper of John W. Miles. Journal of Fluid Mechanics, 10, 509–512 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  8. Drazin, P. D. and Reid, W. H. Hydrodynamics Stability, Cambridge University Press, Cambridge (1981)

    MATH  Google Scholar 

  9. Yi, J. X. Fluid Dynamics (in Chinese), Higher Education Press, Beijing (1982)

    Google Scholar 

  10. Zhao, G. F. and Zhou, H. Weakly nonlinear theory versus theory of secondary instability for plane poiseuille flow. Applied Mathematics and Mechanics (English Edition), 9(7), 617–623 (1988) DOI 10.1007/BF02465691

    Article  MATH  Google Scholar 

  11. Xiong, J. G., Yi, F., and Li, J. The influence of topography on the nonlinear interaction of Rossby waves in the barotropic atmosphere. Applied Mathematics and Mechanics (English Edition), 15(6), 585–594 (1994) DOI 10.1007/BF02450772

    Article  MathSciNet  MATH  Google Scholar 

  12. Zhang, G., Xiang, J., and Li, D. H. Nonlinear saturation of baraclinic instability in the generalized Phillips model I: the upper bound on the evolution of disturbance to the nonlinearly. Applied Mathematics and Mechanics (English Edition), 23(1), 79–88 (2002) DOI 10.1007/BF02437733

    Article  MathSciNet  MATH  Google Scholar 

  13. Zeytounian, R. K. Theory and Applications of Nonviscous Fluid Flows, Springer, Berlin (2002)

    Book  MATH  Google Scholar 

  14. Huang, S. X. and Wu, R. S. Methods of Mathematical Physics in Atmospheric Science (in Chinese), 3rd ed., China Meteorological Press, Beijing (2011)

    Google Scholar 

  15. Ossakow, S. L., Zalesak, S. T., McDonald, B. E., and Chaturvedi, P. K. Nonlinear equatorial spread F: dependence on altitude of the F peak and bottomside background electron density gradient scale length. Journal of Geophysical Research, 84, 17–29 (1979)

    Article  Google Scholar 

  16. Mendillo, M., Baumgardner, J., Pi, X., Sultan, P. J., and Tsunoda R. Onset conditions for equatorial spread F. Journal of Geophysical Research, 97, 13865–13876 (1992)

    Article  Google Scholar 

  17. Huang, C. S., Kelley, M. C., and Hysell, D. L. Nonlinear Rayleigh-Taylor instabilities, atmospheric gravity waves and equatorial spread F. Journal of Geophysical Research, 98, 15631–15642 (1993)

    Article  Google Scholar 

  18. Sultan, P. J. Linear theory and modeling of Rayleigh-Taylor instability leading to the occurrence of equatorial spread F. Journal of Geophysical Research, 101, 26875–26891 (1996)

    Article  Google Scholar 

  19. Rappaport, H. L. Field line integration and localized modes in the equatorial spread F. Journal of Geophysical Research, 101, 24545–24551 (1996)

    Article  Google Scholar 

  20. Migliuolo, S. Nonlocal dynamics of the collisional Rayleigh-Taylor instability: application to the equatorial spread F. Journal of Geophysical Research, 101, 10975–10984 (1996)

    Article  Google Scholar 

  21. Basu, B. On the linear theory of equatorial plasma instability: comparison of different descriptions. Journal of Geophysical Research, 107, 18-1–18-10 (2002)

  22. Chakrabarti, N. and Lakhina, G. S. Collisional Rayleigh-Taylor instability and shear-flow in equatorial spread-F plasma. Annales Geophysicae, 21, 1153–1157 (2003)

    Article  Google Scholar 

  23. Sekar, R. Plasma instabilities and their simulations in the equatorial F region: recent results. Space Science Reviews, 107(1-2), 251–262 (2003)

    Article  Google Scholar 

  24. Lee, C. C. Examine the local linear growth rate of collisional Rayleigh-Taylor instability during solar maximum. Journal of Geophysical Research, 111, A11313 (2006)

    Article  Google Scholar 

  25. Aveiro, H. C. and Huba, J. D. Equatorial spread F studies using SAMI3 with two-dimensional and three-dimensional electrostatics. Annales Geophysicae, 31, 2157–2162 (2013)

    Article  Google Scholar 

  26. Huba, J. D., Bernhardt, P. A., Ossakow, S. L., and Zalesak, S. T. The Rayleigh-Taylor instability is not damped by recombination in the F region. Journal of Geophysical Research, 101, 24553–24556 (1996)

    Article  Google Scholar 

  27. Keskinen, M. J., Ossakow, S. L., and Fejer, B. G. Three-dimensional nonlinear evolution of equatorial ionospheric spread-F bubbles. Geophysical Research Letters, 30, 1855–1858 (2003)

    Article  Google Scholar 

  28. Scannapieco, A. J. and Ossakow, S. L. Nonlinear equatorial spread F. Geophysical Research Letters, 3, 451–454 (1976)

    Article  Google Scholar 

  29. Zalesak, S. T. and Ossakow, S. L. Nonlinear equatorial spread F: spatially large bubbles resulting from large horizontal scale initial perturbations. Journal of Geophysical Research, 85, 2131–2142 (1980)

    Article  Google Scholar 

  30. Basu, B. Nonlinear saturation of Rayleigh-Taylor instability in the presence of time-dependent equilibrium. Journal of Geophysical Research, 104(A4), 6859–6866 (1999)

    Article  Google Scholar 

  31. Xie, H. and Xiao, Z. Numerical simulation of spread-F in low and mid-latitudes (in Chinese). Chinese Journal of Geophysics, 36(1), 18–26 (1993)

    Google Scholar 

  32. Picone, J. M., Hedin, A. E., Drob, D. P., and Aikin, A. C. NRLMSISE-00 empirical model of the atmosphere: statistical comparisons and scientific issues. Journal of Geophysical Research, 107, 15-1–15-16 (2002)

  33. Bilitza, D. and Reinisch, B. W. International Reference Ionosphere 2007: improvements and new parameters. Advances in Space Research, 42, 599–609 (2008)

    Article  Google Scholar 

  34. Zalesak, S. T. and Ossakow, S. L. Nonlinear equatorial spread F: the effect of neutral winds and background Pedersen conductivity. Journal of Geophysical Research, 87, 151–166 (1982)

    Article  Google Scholar 

  35. Chapagain, N. P., Fisher, D. J., Meriwether, J. W., Chau, J. L., and Makela, J. J. Comparison of zonal neutral winds with equatorial plasma bubble and plasma drift velocities. Journal of Geophysical Research: Space Physics, 118, 1802–1812 (2013)

    Google Scholar 

  36. Kudeki, E., Akgiray, A., Milla, M., Chau, J. L., and Hysell, D. L. Equatorial spread-F initiation: posts-sunset vortex, thermospheric winds, gravity waves. Journal of Atmospheric and Solar- Terrestrial Physics, 69, 2416–2427 (2007)

    Article  Google Scholar 

  37. Sekar, R. and Raghavarao, R. Role of vertical winds on the Rayleigh-Taylor mode instabilities of the night time equatorial ionosphere. Journal of Atmospheric and Terrestrial Physics, 49, 981–985 (1987)

    Article  Google Scholar 

  38. Raghavarao, R., Suhasini, R., Mayr, H. G., Hoegy, W. R., and Wharton, L. E. Equatorial spread-F (ESF) and vertical winds. Journal of Atmospheric and Solar-Terrestrial Physics, 61, 607–617 (1999)

    Article  Google Scholar 

  39. Sekar, R., Suhasini, R., and Raghavarao, R. Effects of vertical winds and electric fields in the nonlinear evolution of equatorial spread-F. Journal of Geophysical Research, 99, 2205–2213 (1994)

    Article  Google Scholar 

  40. Raghavarao, R., Sekar, R., and Suhasini, R. Non-linear numerical simulation of equatorial spread-F—effects of winds and electric fields. Advances in Space Research, 12, 227–230 (1992)

    Article  Google Scholar 

  41. McClure, J. P., Hanson, W. B., and Hoffman, J. H. Plasma bubbles and irregularities in the equatorial ionosphere. Journal of Geophysical Research, 82, 2650–2656 (1977)

    Article  Google Scholar 

Download references

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Correspondence to Sixun Huang.

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Project supportedbythe National Natural Science Foundation of China (Nos. 41575026 and 41175025)

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Wang, S., Huang, S. Sufficient conditions of Rayleigh-Taylor stability and instability in equatorial ionosphere. Appl. Math. Mech.-Engl. Ed. 37, 181–192 (2016). https://doi.org/10.1007/s10483-016-2022-8

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  • DOI: https://doi.org/10.1007/s10483-016-2022-8

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Chinese Library Classification

2010 Mathematics Subject Classification

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