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On vertical spinning Alfvén waves in a magnetic flux tube

Published online by Cambridge University Press:  13 March 2009

L. M. B. C. Campos  and
N. L. Isaeva
L. M. B. C. Campos
Affiliation:
Max-Planck-Institut für Aeronomie, 3411 Katlenburg-Lindau, Germany
N. L. Isaeva
Affiliation:
Max-Planck-Institut für Aeronomie, 3411 Katlenburg-Lindau, Germany
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Abstract

We derive the Alfvén-wave equation for an atmosphere in the presence of a non-uniform vertical magnetic field and the Hall effect, allowing for Alfvén speed and ion gyrofrequency that may vary with altitude; the pair of coupled second-order differential equations for the horizontal wave variables, namely magnetic field or velocity perturbations, is reduced to a single complex, second-order differential equation. The latter is applied to spinning Alfvén waves in a magnetic flux tube, in magnetohydrostatic equilibrium, in an isothermal atmosphere. The exact solution is found in terms of hypergeometric functions, from which it is shown that at ‘high altitude’the magnetic field perturbation tends to grow to a non-small fraction of the background magnetic field. By ‘high-altitude’ is meant far above the critical level, which acts as a reflecting layer for left-polarized waves incident from below, i.e. from the ‘low-altitude’ range. We also obtain the exact solution near the critical level, where the left-polarized wave has a logarithmic singularity, and the right-polarized wave is finite. The latter is plotted in this region of wave frequency comparable to ion gyrofrequency, and it is shown that the Hall effect can cause oscillations of wave amplitude and non-monotonic phases with slope of alternating sign. The latter corresponds to ‘tunnelling’, i.e. waves propagating in opposite directions or trapped in adjoining atmospheric layers; this could explain the appearance of inward- and outward-propagating waves, with almost random phases, in the solar wind beyond the earth, for which the Hall effect on Alfvén waves should be significant.

Type
Research Article
Information
Journal of Plasma Physics , Volume 48 , Issue 3 , December 1992 , pp. 415 - 434
Copyright
Copyright © Cambridge University Press 1992

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References

Alfvén, H. 1942 Ark. Mat. Astron. Fys. B 29, 1.Google Scholar
Alfvén, H. 1947 Mon. Not. R. Astron. Soc. 107, 211.CrossRefGoogle Scholar
Alfvén, H. 1948 Cosmical Electrodynamics. Oxford University Press.Google Scholar
Alfvén, H. & Falthammer, C. G. 1962 Cosmical Electrodynamics. Oxford University Press.Google Scholar
Banos, A. 1955 Proc. R. Soc. Lond. A 233, 350.Google Scholar
Buelaga, L. F. 1984 Space Sci. Rev. 39, 255.Google Scholar
Bromwich, J. T. A. 1926 Infinite series. Macmillan.Google Scholar
Cabannes, H. 1970 Theoretical Magnetofluid-Dynamics. Academic Press.Google Scholar
Campos, L. M. B. C. 1977 J. Fluid Mech. 81, 529.CrossRefGoogle Scholar
Campos, L. M. B. C. 1982 Port. Maths 41, 13.Google Scholar
Campos, L. M. B. C. 1983 a J. Phys. A 16, 21.Google Scholar
Campos, L. M. B. C. 1883 b Solar Phys. 82, 355.CrossRefGoogle Scholar
Campos, L. M. B. C. 1983 c J. Mech. Theor. Appl. 2, 861.Google Scholar
Campos, L. M. B. C. 1985 Geophys. Astrophys. Fluid Dyn. 40, 217.CrossRefGoogle Scholar
Campos, L. M. B. C. 1987 Rev. Mod. Phys. 69, 363.CrossRefGoogle Scholar
Campos, L. M. B. C. 1988 J. Phys. A 21, 2911.Google Scholar
Campos, L. M. B. C. 1989 Geophys. Astrophys. Fluid Dyn. 48, 193.CrossRefGoogle Scholar
Campos, L. M. B. C. 1992 Phys. Fluids B 4, 2975.CrossRefGoogle Scholar
Campos, L. M. B. C. & Saldanha, R. 1991 Geophys. Astrophys. Fluid Dyn. 56, 237.CrossRefGoogle Scholar
Caratheodory, C. 1953 Theory of Functions. Birkhauser.Google Scholar
Copson, E. T. 1935 Functions of complex variable. Oxford University Press.Google Scholar
Cowling, T. G. 1960 Magnetohydrodynamics. Wiley-Interscience.Google Scholar
Erdelyi, A. (ed.) 1953 Higher Transcendental Functions. McGraw-Hill.Google Scholar
Ferraro, V. C. A. & Plumpton, E. 1958 Astrophys. J. 129, 459.CrossRefGoogle Scholar
Ferraro, V. C. A. & Plumpton, C. 1966 An Introduction to Magnetic-Fluid Dynamics, 2nd edn. Oxford University Press.Google Scholar
Herlofson, N. 1950 Nature 165, 1020.CrossRefGoogle Scholar
Heyvaerts, J. & Priest, E. R. 1983 Astron. Astrophys. 117, 120.Google Scholar
Hollweg, J. V. 1972 Cosmic Electrodyn. 2, 423.Google Scholar
Hollweg, J. V. 1978 Solar Phys. 62, 307.Google Scholar
Knopp, K. 1949 Theory and Application of Infinite Series. Haffner.Google Scholar
Landau, L. D. & Lieshitz, E. M. 1956 Electrodynamics of Continuous Media. ergamon.Google Scholar
Leroy, B. 1980 Astron. Astrophys. 91, 136.Google Scholar
Leroy, B. 1983 Astron. Astrophys. 125, 371.Google Scholar
Lighthill, M. J. 1959 Phil. Trans. R. Soc. Lond. 252, 397.Google Scholar
Lundquist, S. 1949 Phys. Rev. 76, 1805.CrossRefGoogle Scholar
Mckenzie, J. F. 1979 J. Plasma Phys. 22, 361.CrossRefGoogle Scholar
Marsch, E. & Tu, C. Y. 1990 J. Geophys. Res. 95, 8211.CrossRefGoogle Scholar
Moffatt, H. K. 1978 Magnetic Field Generation in Electrically Conducting Fluids. Cambridge University Press.Google Scholar
Nocera, L., Leroy, B. & Priest, E. T. 1984 Astron. Astrophys. 133, 387.Google Scholar
Parker, N. 1963 Interplanetary Dynamical Processes.Google Scholar
Parker, N. 1984 Geophys. Astrophys. Fluid Dyn. 29, 1.CrossRefGoogle Scholar
Roberts, B. 1980 Solar Phys. 97, 77.Google Scholar
Schwartz, S. T., Cally, P. S. & Bel, B. 1984 Solar Phys. 92, 81.CrossRefGoogle Scholar
Steinholfson, R. S. 1985 Astrophys. J. 295, 213.CrossRefGoogle Scholar
Whittaker, E. T. & Watson, G. N. 1927 Course of Modern Analysis. Cambridge University Press.Google Scholar
Yanowitch, M. 1967 Can. J. Phys. 45, 2003.CrossRefGoogle Scholar
Zugzda, Y. A. 1972 Cosmic Electrodyn. 2, 267.Google Scholar
Zugzda, Y. D. & Dzalilov, N. S. 1984 Astron. Astrophys. 132, 45.Google Scholar