Inertial Effects on Tearing Instability


  • F. E. M. Silveira Centro de Ciências Naturais e Humanas, Universidade Federal do ABC, Rua Santa Adélia, 166, Bairro Bangu, CEP 09.210-170, Santo André, SP



resistive instabilities, Ohm’s law, current relaxation


In this work, we explore inertial effects, due to charged species in a resistive plasma, on the tearing instability. The standard theory of tearing modes assumes a long wavelength limit. At shorter wave lengths, inertial effects can become important and the current density flowing in the fluid can acquire a finite relaxation time. The introduction of such a correction into the problem leads to an extension of the standard dispersion relation. In the long wave length limit, we recover the standard scaling of the growth rate γ with the plasma resistivity η, namely γ ≈ η3/5. However, in the short wavelength limit, we find that the scaling of γ with the relevant plasma parameters changes significantly due to the influence of inertia. Notwithstanding, the dependence of γ on the relaxation time of the current density is not determined. In order to achieve such a description, we propose to further rediscuss the problem in the framework of the boundary layer technique.


Killeen J. Rosenbluth M. N. Furth, H. P. Finite resistivity instabilities of a sheet pinch. Phys. Fluids, 6:459, 1963.

Johnson J L Coppi B., Greene J M. The non-linear evolution of resistive interchange modes in a reversed-field pinch. Nucl. Fusion, 6:101, 1961.

Johnson J L Glasser A. H., Greene J M. Resistive instabilities in general toroidal plasma configurations. Phys. Fluids, 18:875, 1975.

Pellat R Rosenbluth M N Rutherford P H Sov. J. Coppi B., Galvão R M O. Sov. J. Plasma Phys., 2, 1976.

Dependence of ideal-mhd kink and ballooning modes on plasma shape and profiles in tokamaks. Phys. Rev. Lett., 38:826, 1977.

J. Mondt J. P., Weiland J. Nonlinear theory of large-mode-number ballooning modes in fully toroidal geometry. Plasma Phys., 34:143, 1985.

Rosenbluth M N Waddell B V White R. W., Monticello D A. Saturation of the tearing mode. Phys. Fluids, 20:800, 1977.

Wesson J Turner M. F. Nucl. Fusion, 22:1069, 1982.

Sov. J. Smolyakov A. I. Plasma Phys., 15:667, 1989.

Matsumoto H McGuire K Peebles W A Ritz Ch P Terry P W Zweben S J Wootton A. J., Carreras B A. Fluctuations and anomalous transport in tokamaks. Phys. Fluids B, 2:2879, 1990.

Kadomtsev B. Plasma transport in tokamaks. Nucl. Fusion, 31:1301, 1991.

Parker E. N. Sweet’s mechanism for merging magnetic fields in conducting fluids. J. Geophys. Res., 62:509, 1957.

Biskamp D. Magnetic reconnection via current sheets. Phys. Fluids, 29:1520, 1986.

Sridhar S Goldreich P. Toward a theory of interstellar turbulence. 2: Strong alfvenic turbulence. Astrophys. J., 438:763, 1995.

Vishniac E Lazarian A. Reconnection in a weakly stochastic field. Astrophys. J., 517:700, 1999.

Vishniac E Otmianowska-Mazur K Kowal G., Lazarian A. Numerical tests of fast reconnection in weakly stochastic magnetic fields. Astrophys. J., 700:63, 2009.

Vishniac E Otmianowska-Mazur K Kowal G., Lazarian A. Reconnection studies under different types of turbulence driving. Nonlinear Proc. Geoph., 19:297, 2012.

Spitzer L. Physics of Fully Ionized Gases. Second edition. Dover Publications, New York, 1969.

Silveira F. E. M. Alfven waves and current relaxation: attenuation at high frequencies and large resistivity. J. Phys.: Conf. Ser., 370:012005, 2012.

Silveira F. E. M. Magnetosonic waves and current relaxation. J. Plasma Phys. and Technol., 79:45, 2013.

Galvão R M O Silveira F. E. M. Magnetorotational instability, current relaxation, and current-vortex sheet. Phys. Plasmas, 20:082126, 2013.

Harris E. G. On a plasma sheath separating regions of oppositely directed magnetic field. Nuovo Cimento, 23:115, 1962.

Bertotti B. Fine structure in current sheaths. Ann. Phys., 22:271, 1963.

Haines M. G. Plasma containment in cusp-shaped magnetic fields. Nucl. Fusion, 17:811, 1977.

Dawson J M Leboeuf J. N., Tajima T. Dynamic magnetic x points. Phys. Fluids, 25:784, 1982.

Ion acceleration and direct ion heating in three-component magnetic reconnection. Phys. Rev. Lett., 76:3328, 1996.

Kosugi T Somov B. V. Collisionless reconnection and high-energy particle acceleration in solar flares. Astrophys. J., 485:859, 1997.

Rutherford P. H. Goldston R. J. Introduction to plasma physics. Institute of Physics, Bristol, 2000.

Jeffreys B S Jeffreys H. Methods of Mathematical Physics. Third edition. Cambridge University Press, Cambridge, 1999.

Orszag S A Bender C. M. Advanced Mathematical Methods for Scientists and Engineers: Asymptotic Methods and Perturbation Theory. Vol. 1. Springer-Verlag, New York, 1999.