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Pressure-Driven and Ionosphere-Driven Modes Of View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by UTokyo Repository JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 114, A02224, doi:10.1029/2008JA013663, 2009 Click Here for Full Article Pressure-driven and ionosphere-driven modes of magnetospheric interchange instability Akira Miura1 Received 3 August 2008; revised 9 December 2008; accepted 18 December 2008; published 28 February 2009. [1] A general stability criterion for magnetospheric interchange instability, which includes an ionospheric destabilizing contribution, is derived for an arbitrary finite-b magnetospheric model satisfying the magnetohydrostatic force balance. The derivation is based on the magnetospheric energy principle. Unperturbed field-aligned currents in finite-b nonaxisymmetric magnetospheric models are assumed to close via diamagnetic currents in the magnetosphere or in the ionosphere. By exploiting the limit of a very large perpendicular wave number and the eikonal representation for the perpendicular plasma displacement, the magnetospheric interchange mode is shown to be compressible. In this limit the kink mode makes no contribution to the change in the magnetospheric potential energy. By using magnetospheric flux coordinates, the explicit form of the magnetospheric potential energy change is calculated for interchange perturbations, which do not bend magnetospheric magnetic fields. For a nonaxisymmetric finite-b magnetospheric model, a combined effect of the pressure gradient and field line curvature, not only in the meridional plane but also in the plane parallel to the longitudinal direction, is responsible for pressure-driven interchange instability. For an axisymmetric, north-south symmetric and low-b magnetospheric model, in which the magnetic field is approximated by a dipole field, the m =1orm = 2 ionosphere-driven mode, where m is the azimuthal mode number, has an upper critical equatorial b value for instability in the order of 1. Thus a substantial region of the inner magnetosphere or the near-Earth magnetosphere may be unstable against ionosphere-driven interchange instability caused by a horizontal plasma displacement on the spherical ionospheric surface. Citation: Miura, A. (2009), Pressure-driven and ionosphere-driven modes of magnetospheric interchange instability, J. Geophys. Res., 114, A02224, doi:10.1029/2008JA013663. 1. Introduction corresponds to the adiabatic gradient for the magnetohy- drostatics of a plasma in a dipole field with R being the [2] The possibility of spontaneous large-scale interchange distance from the earth’s center in the equatorial plane, the motion of magnetic flux tubes and the plasma contained in magnetospheric plasma is unstable against fast adiabatic them by pressure-driven interchange instability in the mag- convection. Gold [1959] suggested that a better understand- netosphere has been discussed by Gold [1959]. Such an ing of the processes of magnetic storms and auroras and of interchange perturbation does not bend magnetic field lines the Van Allen radiation belts would all require better in the magnetosphere and allows a class of motions to occur estimates of the interchange motions. The interchange without the need of overcoming magnetic tension force. instability has also been widely discussed as a potentially Since then, many studies have been devoted to clarify important mechanism for the redistribution of mass in characteristics of magnetospheric interchange instability planetary magnetospheres. Therefore, to know accurately [e.g., Sonnerup and Laird, 1963; Cheng, 1985; Rogers the conditions necessary for interchange instability is very and Sonnerup, 1986; Southwood and Kivelson, 1987; important in clarifying the dynamics of the magnetosphere. Ferrie`re et al., 2001]. In particular, stability criteria for [3] The primary objective of the present study is to derive magnetospheric interchange instability have been discussed a general stability criterion for interchange instability, which widely for several limited magnetospheric models. Gold includes an ionospheric destabilizing contribution, for an [1959] discussed the stability of low-b magnetospheric arbitrary finite-b and nonaxisymmetric three-dimensional configuration assuming a dipole magnetic field. He showed magnetospheric plasma, on the basis of a magnetospheric that for pressure gradients steeper than RÀ20/3,which energy principle [Miura, 2007]. The unperturbed magneto- spheric plasma is assumed to satisfy the magnetohydrostatic force balance J Â B = rp and is bounded by ideal 1Department of Earth and Planetary Science, University of Tokyo, Tokyo, Japan. ionospheric boundaries. The magnetospheric energy princi- ple is an extension of the energy principle [Bernstein et al., Copyright 2009 by the American Geophysical Union. 1958] valid for any finite-b magnetohydrostatic equilibrium 0148-0227/09/2008JA013663$09.00 A02224 1of18 A02224 MIURA: MAGNETOSPHERIC INTERCHANGE INSTABILITY A02224 configuration used in fusion plasmas to a magnetospheric system with ideal ionospheric boundary conditions, which satisfy the self-adjointness of the force operator. [4] A stability criterion for the interchange instability has been discussed by Bernstein et al. [1958] for a finite-b axisymmetric system with periodicity in the direction of the axis of symmetry. Hameiri et al. [1991] derived a stability criterion for finite-b axisymmetric magnetosphere with closed field lines, which was originally derived by Spies [1971]. They assumed that field lines are closed loops without any boundaries such as ionospheres and thus Figure 1. A three-dimensional view of the plasma volume avoided complicated problems arising from taking into P surrounded by an ideal ionospheric boundary and by the account ionospheric boundary conditions in the real mag- outer flux surface Sout and the inner flux surface Sin in the netospheric plasma. In these criteria the plasma pressure and magnetohydrostatic magnetospheric equilibrium, which are the specific volume of a magnetic flux tube U are consid- shown by the light grey and the dark grey surfaces, ered to be a function of only one variable representing the respectively. radial coordinate. Here, I d‘ U ¼ internal energy is conserved, a magnetospheric energy prin- B ciple is applicable. The present approach based on the variational magnetospheric energy principle is applicable and the integration is taken for one period or for a closed for any magnetospheric equilibium model satisfiying mag- field line loop. netohydrostatic force balance with and without axisymmetry. [5] These studies assume an axisymmetric three- [7] Since magnetospheric interchange instability is not dimensional system. Therefore, there is no unperturbed field- driven by the solar wind-magnetosphere interaction, one is aligned current. When the plasma pressure p or the specific interested in the stability of a plasma with a finite volume P flux tube volume U is nonaxisymmetric, however, an unper- in the magnetosphere. The plasma volume P is surrounded turbed field-aligned current apppears and stability is different by an ideal ionospheric boundary and two flux surfaces in a from that without a field-aligned current. When there is a field- magnetohydrostatic equilibrium. Shown in Figure 1 are two aligned current in the magnetosphere, the current must be flux surfaces Sout and Sin in the magnetohydrostatic mag- closed three-dimensionally either in the ionosphere or in the netospheric equilibrium, which are the light grey surface magnetosphere to satisfiy current continuity. When there is and the dark grey surface, respectively. Those two flux an unperturbed field-aligned current and when this is closed by surfaces Sout and Sin are virtual boundaries located far from Pedersen current in the ionosphere, Volkov and Mal’tsev disturbed field lines in the plasma volume P. The outer flux [1986] calculated the growth rate of the interchange instability surface Sout is taken inside the magnetopause. Therefore, for a finite-b plasma by using a perturbation stability energy conservation holds in the plasma volume P and one analysis in a local Cartesian coordinate system. can thus apply the magnetospheric energy principle to study [6] However, when the field-aligned current is closed by the magnetospheric interchange instability in the plasma ionospheric Pedersen current, there arises Joule dissipation volume P. in the ionosphere and the zeroth-order state assumed for [8] The energy conservation necessary for the magneto- instability is no longer a steady state but decays with a time spheric energy principle requires that the ideal MHD force constant larger than the Alfve´n transit time [Miura, 1996]. operator must be self-adjoint. Miura [2007] obtained four Hence, the sum of the magnetic energy, kinetic energy and ideal ionospheric boundary conditions, which are compat- the internal energy in ideal magnetohydrodynamics (MHD) ible with the self-adjoint property of the force operator and is not conserved. Therefore, any stability analysis assuming satisfying J Â B = rp in the magnetosphere. Since the a current closure by Pedersen current in the ionosphere interchange of flux tubes involves the motion of a whole cannot be a stability analysis of the steady state in the flux tube, the interchange of magnetospheric flux tubes strictest sense, because the zeroth-order state is not a steady requires horizontal displacement of magnetic field lines in state. In other words, when the field-aligned current is the ionosphere. According to the magnetospheric
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