To the theory of transversal small-scale modes in a cylindrical plasma column
1Cheremnykh, OK 1Space Research Institute under NAS and National Space Agency of Ukraine, Kyiv, Ukraine |
Kinemat. fiz. nebesnyh tel (Online) 2015, 31(5):3-19 |
Start Page: Space Physics |
Language: Russian |
Abstract: The equation of small oscillations for a circular cross-section plasma column with a helical magnetic field is received. It is shown that this equation is equivalent to Hain — Lust equation, however has simpler appearance. The given equation allows to receive a number of earlier known results, and to analyse also transversal small-scale MHD perturbations: the criterion of stability of these modes, expression for the maximum increment are obtained, areas of propagation of both stable and unstable modes are defined. The obtained results can be used for interpretation of behavior of solar magnetic tubes. |
Keywords: equation, Hayn-Lust's equation, helical magnetic field |
1.G. Bateman, MHD Instabilities (MIT Press, Cambridge, 1978; Energoizdat, Moscow, 1982).
2.A. Bernstein, “The variational principle for problems of ideal magnetohydrodynamic stability,” in Basic Plasma Physics, Vol. 1, Ed. by A. A. Galeev and R. Sudan (North-Holland, Amsterdam, 1983; Energoatomizdat, Moscow, 1984), pp. 421–451.
3.M. Goossens, “MHD waves and wave heating in nonuniform plasmas,” in Advances in Solar System Magnetohydrodynamics, Ed. by E. R. Priest and A. W. Hood (Cambridge Univ. Press, Cambridge, 1991; Mir, Moscow, 1995), p. 137.
4.A. G. Zagorodnii and O. K. Cheremnykh, Introduction to Plasma Physics (Naukova Dumka, Kiev, 2014) [in Russian].
5.B. B. Kadomtsev, “Hydrodynamic plasma stability,” in Problems of Plasma Theory, Vol. 2, Ed. by M. A. Leontovich (Gosatomizdat, Moscow, 1963), pp. 132–176 [in Russian].
6.K. Miyamoto, Plasma Physics and Controlled Nuclear Fusion (Springer-Verlag, 2005; Fizmatlit, Moscow, 2007).
7.Yu. P. Ladikov-Roev and O. K. Cheremnykh, Mathematical Models of Continuous Media (Naukova Dumka, Kiev, 2010) [in Russian].
8.N. G. Mazur, E. N. Fedorov, and V. A. Pilipenko, “Dispersion relation for ballooning modes and condition of their stability in the near-earth plasma,” Geomagn. Aeron. 52, 603–612 (2012).
https://doi.org/10.1134/S0016793212050118
9.A. B. Mikhailovskii, Theory of Plasma Instabilities, Vol. 2: Instabilities of an Inhomogeneous Plasma in Ser. Studies in Soviet Science (Atomizdat, Moscow, 1971; Consultants Bureau, New York, 1974).
https://doi.org/10.1007/978-1-4899-4785-7
10.E. Parker, Cosmic Magnetic Fields: Their Origin and Their Activity (Clarenton, Oxford, 1979; Mir, Moscow, 1982).
11.E. R. Priest, Solar Magnetohydrodynamics (Reidel, Dordrecht, 1982; Mir, Moscow, 1985).
12.B. Roberts, “MHD waves in the Sun,” in Advances in Solar System Magnetohydrodynamics, Ed. by E. R. Priest and A. W. Hood (Cambridge Univ. Press, Cambridge, 1991; Mir, Moscow, 1995), pp. 105–136.
13.B. R. Suydam, “Stability in a linear pinch,” in Proc. 2nd Int. Conf. on the Peaceful Use of Atomic Energy, Geneva, Sep. 1–13, 1958 (Am. Chem. Soc., 1959; Atomizdat, Moscow, 1959), pp. 157–159.
14.B. A. Trubnikov, Plasma Theory (Energoatomizdat, Moscow, 1996) [in Russian].
15.B. P. Filippov, Eruptive Processes on the Sun (Fizmatlit, Moscow, 2007) [in Russian].
16.Yu. T. Tsap, Yu. G. Kopylova, and A. V. Stepanov, “Ballooning instability and oscillations of coronal loops,” Astron. Rep. 50, 1026–1035 (2006).
https://doi.org/10.1134/S1063772906120079
17.O. K. Cheremnykh, D. Yu. Klimushkin, and D. V. Kostorev, “On the structure of azimuthally small-scale ULF oscillations of hot space plasma in a curved magnetic field. Modes with continuous spectrum,” Kinematics Phys. Celestial Bodies 30, 209–222 (2014).
https://doi.org/10.3103/S088459131405002X
18.V. D. Shafranov, “On the question of hydromagnetic stability of the plasma filament with current in a strong magnetic field,” Zh. Eksp. Teor. Fiz. 40, 241–253 (1970).
19.A. V. Agapitov and O. K. Cheremnykh, “Natural oscillations of the Earth magnetosphere associated with solar wind sudden impulses,” Ukr. J. Phys. 53, 598–512 (2008).
20.A. V. Agapitov and O. K. Cheremnykh, “Polarization of ULF waves in the Earth’s magnetosphere,” Kinematics Phys. Celestial Bodies 27, 117–123 (2011).
https://doi.org/10.3103/S0884591311030020
21.O. S. Burdo, O. K. Cheremnykh, S. M. Revenchuk, and V. D. Pustovitov, “General geometric dispersion relations for toroidal plasma configuration,” Plasma Phys. Controlled Fusion 36, 641–656 (1994).
https://doi.org/10.1088/0741-3335/36/4/006
22.O. S. Burdo, O. K. Cheremnykh, and O. P. Verkhoglyadova, “Study of ballooning modes in the inner magnetosphere of the Earth,” Izv. Ross. Akad. Nauk, Ser. Fiz. 69, 1896–1900 (2000).
23.O. K. Cheremnykh, “Transversally small-scale perturbation in arbitrary plasma configurations with magnetic surfaces,” Plasma Phys. Controlled Fusion 52, 095006 (2010).
https://doi.org/10.1088/0741-3335/52/9/095006
24.O. K. Cheremnykh, Z. M. Andrushchenko, J. W. Edenstrasser, and V. B. Taranov, “Relaxation of non-ideal magnetohydrodynamic plasma in cylindrical column,” Phys. Plasmas 1, 2525–2530 (1994).
https://doi.org/10.1063/1.870580
25.O. K. Cheremnykh and V. V. Danilova, “Transversal small-scale MHD perturbations in space plasma with magnetic surfaces,” Kinematics Phys. Celestial Bodies 27, 98–108 (2011).
https://doi.org/10.3103/S0884591311020036
26.O. K. Cheremnykh and A. S. Parnovski, “The theory of ballooning perturbations in the inner magnetosphere of the Earth,” Adv. Space Res. 33, 769–773 (2004).
https://doi.org/10.1016/S0273-1177(03)00642-2
27.O. K. Cheremnykh and A. S. Parnovski, “Flute and ballooning modes in the inner magnetosphere of the Earth: Stability and influence of the ionospheric confuctivity”, in Space Science: New Research, Ed. by N. S. Maravell, (Nova Sci., New York, 2006), pp. 71–108.
28.O. K. Cheremnykh and A. S. Parnovski, “Influence of ionospheric conductivity of the ballooning modes in the inner magnetosphere of the Earth,” Adv. Space Res. 37, 599–603 (2006). doi 10.1016/j.asr.2005.01.073.
https://doi.org/10.1016/j.asr.2005.01.073
29.R. L. Dewar and A. H. Glasser, “Ballooning mode spectrum in general toroidal systems,” Phys. Fluids 26, 3038–3052 (1983).
https://doi.org/10.1063/1.864028
30.K. Hain and R. Z. Lust, “Zur stabilität zylinder symmetrischer plasma konfigurationen mit volumenströmen,” Naturforsh 13a, 936–940 (1957).
31.D. Yu. Klimushkin, “Spatial structure of transversally small-scale hydromagnetic waves in plane finite-beta model magnetosphere,” Planet. Space Sci. 45, 269–279 (1997). doi 10.1016/S0032-0633(96)00078-5.
https://doi.org/10.1016/S0032-0633(96)00078-5
32.D. Yu. Klimushkin, A. S. Leonovich, and V. A. Mazur, “On the propagation of transversally small-scale standing Alfven waves in a three-dimensionally inhomogeneous magnetosphere,” J. Geophys. Res.: Space Phys. 100, 9527–9534 (1995).
https://doi.org/10.1029/94JA03233
33.M. D. Kruskal, J. L. Johnson, M. B. Gottlieb, and L. M. Goldman, “Hydromagnetic instability in stellarator,” Phys. Fluids 1, 217–224 (1958).
34.A. S. Leonovich and V. A. Mazur, “A theory of transverse small-scale standing Alfven waves in an axially symmetric magnetosphere,” Planet. Space Sci. 41, 697–717 (1993).
https://doi.org/10.1016/0032-0633(93)90055-7
35.C. Mercier, “Critere de stabilite d’un system toroidal hydromagnetique en pressure scalaire,” Nucl. Fus. Suppl. 2, 801–808 (1962).
36.A. V. Stepanov, K. Shibasaki, Yu. G. Kopylova, and Yu. T. Tsap, “MHD-oscillatios of coronal loops and diagnostics of flare plasma,” in Proc. Nobeyama Symp. 2004 on Solar Physics with Nobeyama Radioheliograph, Kiyosato, Oct. 26–29, 2004 (Nobeyama Solar Radio Observatory, Kiyosato, 2006), pp. 23–31.