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Non-stationary resonant Alfvén surface waves in one-dimensional magnetic plasmas

Published online by Cambridge University Press:  13 March 2009

M. S. Ruderman ,
W. Tirry  and
M. Goossens
M. S. Ruderman
Affiliation:
Centre for Plasma Astrophysics, Katholiek Universiteit Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium
W. Tirry
Affiliation:
Centre for Plasma Astrophysics, Katholiek Universiteit Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium
M. Goossens
Affiliation:
Centre for Plasma Astrophysics, Katholiek Universiteit Leuven, Celestijnenlaan 200B, B-3001 Heverlee, Belgium
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Abstract

This paper uses incompressible visco-resistive MHD to study the propagation of linear resonant waves in an inhomogeneous plasma. The background density and magnetic field are assumed to depend only on one spartial Cartesian coordinate, and the magnetic field is taken to be unidirectional and perpendicular to the direction of inhomogeneity. The equation that governs the component of the velocity normal to the plane formed by the direction of the inhomogeneity and the magnetic field is derived under the assumption that the coefficients of viscosity and resistivity are sufficiently small that dissipation of energy is confined to a narrow dissipative layer. The solutions to this equation are obtained in the form of decaying normal surface modes with wavelengths much larger than the characteristic scale of the inhomogeneity. The effect of non-stationarity inside the dissipative layer is taken into account, and valid solutions are found even when the ratio of the thickness of the dissipative layer to the inhomogeneity length scale is of the order of or smaller than the ratio of the inhoinogeneity length scale to the wavelength. These solutions are the generalization of the solutions obtained by Mok and Einaudi, which are only valid when the first ratio is much larger than the second. The rate of wave damping is shown to be independent of the values of the viscosity and the resistivity. However, the behaviour of the solutions in the dissipative layer depends strongly on the viscosity and the resistivity. In the case that the effect of dissipation dominates the effect of non-stationarity, the solutions behave in the dissipative layer as found by Mok and Einaudi. When the effect of dissipation is steadily decreased in comparison with the effect of nonstationarity, the solutions become more and more oscillatory, and their amplitudes grow very rapidly in the dissipative layer. Eventually, when nonstationarity dominates dissipation, the amplitudes of the solutions become so large in the dissipative layer in comparison with those outside the dissipative layer that practically all the energy of the perturbations is concentrated in the dissipative layer.

Type
Research Article
Information
Journal of Plasma Physics , Volume 54 , Issue 2 , October 1995 , pp. 129 - 148
Copyright
Copyright © Cambridge University Press 1995

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References

REFERENCES

Chen, L. & Hasegawa, A. 1974 Plasma heating by spatial resonance of Alfvén waves. Phys. Fluids 17, 1399–1403.CrossRefGoogle Scholar
Davila, J. M. 1987 Heating of the solar corona by the resonant absorption of Alfvén waves. Astrophys. J. 317, 514–521.CrossRefGoogle Scholar
Einaudi, G. & Mok, J. 1985 Resistive Alfvén normal modes in a non-uniform plasma. J. Plasma Phys. 34, 259–270.CrossRefGoogle Scholar
Einaudi, G. & Mok, J. 1987 Alfvén wave dissipation in the solar atmosphere. Astrophys. J. 319, 520–530.CrossRefGoogle Scholar
Goedbloed, J. P. 1983 Lecture Notes on Ideal Magnetohydrodynamics. Rijnhuizen Report 83–1453.Google Scholar
Goossens, M. 1991 MHD waves and wave heating in non-uniform plasmas. Advances in Solar System Magnetohydroclynamics (ed. Priest, E. R. & Hood, A. W.), p. 135. Cambridge University Press.Google Scholar
Goossens, M., Ruderman, M. S. & Hollweg, J. V. 1995 Dissipative MHD solutions for resonant Alfvén waves in 1-dimensional magnetic tubes. Solar Phys. 157, 75–102.CrossRefGoogle Scholar
Grossmann, W. & Smith, R. A. 1988 Heating of solar coronal loops by resonant absorption of Alfvén waves. Astrophys. J. 332, 476–498.CrossRefGoogle Scholar
Grossmann, W. & Tataronis, J. 1973 Decay of MHD waves by phase mixing. II. Theta pinch in cylindrical geometry. Z. Phys. 261, 217–236.CrossRefGoogle Scholar
Hasegawa, A. & Chen, L. 1976 Kinetic processes in plasma heating by resonant mode conversion of Alfvén wave. Phys. Fluids 19, 1924–1934.CrossRefGoogle Scholar
Hollweg, J. V. 1987a Resonant absorption of magnetohydrodynamic waves: physical discussion. Astrophys. J. 312, 880–885.CrossRefGoogle Scholar
Hollweg, J. V. 1987b Resonant absorption of magnetohydrodynamic waves: viscous effects. Astrophys. J. 320, 875–883.CrossRefGoogle Scholar
Hollweg, J. V. 1990 Heating of the solar corona. Comp. Phys. Rep. 12, 205–232.CrossRefGoogle Scholar
Hollweg, J. V. 1991 Alfvén waves. Mechanism of Chromospheric and Coronal Heating (ed. Ulmschneider, P., Priest, E. R. & Rosner, R.), p. 423. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Hollweg, J. V. & Yang, G. 1988 Resonant absorption of compressible magnetohydrodynamic waves at thin ‘surface’. J. Geophys. Res. 93, 5423–5436.CrossRefGoogle Scholar
Ionson, J. A. 1978 Resonant absorption of Alfvénic surface waves and the heating of solar coronal loops. Astrophys. J. 226, 650–673.CrossRefGoogle Scholar
Ionson, J. A. 1985 The heating of coronae. Solar Phys. 100, 289–308.CrossRefGoogle Scholar
Kuperus, M., Inson, J. A. & Spiser, D. 1981 On the theory of coronal heating mechanisms. Ann. Rev. Astron. Astrophys. 19, 7–40.CrossRefGoogle Scholar
Mok, Y. & Einaudi, G. 1985 Resistive decay of Alfvén waves in a non-uniform plasma. J. Plasma Phys. 33, 199–208.CrossRefGoogle Scholar
Nayfeh, A. H. 1981 Introduction to Perturbation Techniques, p. 123. Wiley-Interscience, New York.Google Scholar
Poedts, S., Goossens, G. & Kerner, W. 1989 Numerical simulation of coronal heating by resonant absorption of Alfvén waves. Solar Phys. 123, 83–115.CrossRefGoogle Scholar
Poedts, S., goossens, M. & Kerner, W. 1990 On the efficiency of coronal heating by resonant absorption. Astrophys. J. 360, 279–287.CrossRefGoogle Scholar
Roberts, B. 1981 Wave propagation in a magnetically structured atmosphere. I: Surface waves at a magnetic interface. Solar Phys. 69, 27–38.CrossRefGoogle Scholar
Ruderman, M. S. & Gossens, M. 1993 Nonlinearity effects on resonant absorption of surface Alfvén waves in incompressible plasmas. Solar Phys. 143, 69–88.CrossRefGoogle Scholar
Sakurai, T., Goossens, M. & Hollweg, J. V. 1991 Resonant behaviour of MHD waves on magnetic flux tubes. I. Connection formulae at the resonant surface. Solar Phys. 133, 227–245.CrossRefGoogle Scholar
Sedláček, Z. 1971 a Electrostatic oscillations in cold inhomogeneous plasma. Part 2. Integral equation approach. J. Plasma Phys. 6, 187–199.CrossRefGoogle Scholar
Sedáček, Z. 1971 b Electrostatic oscillations in cold inhomogeneous plasma. I. Differential equation approach. J. Plasma Phys. 5, 239–263.CrossRefGoogle Scholar
Tataronis, J. & Grossmann, W. 1973 Decay of MHD waves by phase mixing. I. The sheet pinch in plane geometry. Z. Phys. 261, 203–216.CrossRefGoogle Scholar