Plunfr. .S;offcfSci.. Vol. 41. No. 3, pp. 245-155. 1993 ~032~633~93 $6.00+ 0.00 Printed in Great Britain. ff 1993 Pergamon Press Ltd Magnetospheric interchange instability in anisotropic plasma Andrew Fazakerley and David Southwood Space and Atmospheric Physics Group. The Blackett Laboratory, Imperial College. London, U.K. Rcceivcd 11 June, 1992 ; revised 22 October. 1992 ; accepred 22 December, 1992 introduction Abstract. We study magnetospheric interchange insta- bility under the assumption that the plasma pressure The idea that the plasma distribution within a planetary distribution is anisotropic. Previous studies of magnetosphere might be maintained in such a way that magnetosphe~c interchange instabihty have only con- plasma motions arise spontaneously was originally pro- sidered the case of isotropic pressure. We also argue posed by Gold (1959) in the context of the terrestrial that, under certain circumstances, substantial particle magnetosphere. Gold recognized that magnetospheric energization can accompany outward interchange plasma. which can usual!y. be described in terms of motions in rapidly rotating magnetic fields. Our studies an MHD fluid. might exhtbtt a form of the well-known of instability treat the plasma as an MHD Ruid and Rayleigh-Taylor fluid instability. in which an adiabatic deal with two special cases in which the plasma pressure exchange of fluid elements results in a reduction of stored evolves anisotropically as the interchange motion pro- energy and the newly released energy is able to fill the ceeds. The first case is that of “fast” interchange role of the kinetic energy associated with the motion. motions, where interchange motions can take place All analyses of magnetospheric interchange instability are rapidly compared with particle bounce times. Our based on this principle. The dominant cause of plasma analysis uses a small perturbation approach and takes circulation in the terrestrial magnetosphere is now known into account the curved magnetic field, and external to be solar-wind-driven convection. though interchange forces such as gravity or an effective gravity arising instability still finds an application in studies of the ter- from rotation of the system. We contrast this with a restrial plasmapause (e.g. Richmond. 1973 : Huang rf al., second case in which the plasma motion conserves the 1990). adiabatic invariants p and J. and in both cases consider The discovery that IO is a strong source of plasma, and the implications for a plasma generated by a satellite that a sheet of logenic plasma exists throughout most of source in the equatorial plane of a rapidly rotating. the rapidly rotating Jovian magnetosphere beyond the lo spin-aligned magnetic field. A consequence of fast inter- torus. led to a number of papers arguing that centrifugally change motions in a corotation-dominated magneto- driven interchange motions are responsible for the radial sphere is that the rapid motions will be accompanied motion of logenic plasma in the Jovian magnetosphere. by motion of ionized material away from (toward) A number of scenarios have been put forward, all involv- the equatorial plane as the material moves outward ing an MHD description of the plasma, ranging from (inward). If an outward (inward) interchange motion large-scale convection patterns (e.g. Hill CI al., 1981) to should be slowed such that it is no longer rapid com- “interchange diffusion” involving ‘-turbulent eddies” (e.g. pared with particle bounce times. particles will resume Siscoe and Summers. 1981). However. there have been bounce motion, but with increased (reduced) parallel difficulties reconciling these models and their later modi- energy. In practice, it is likely that lower energy par- fications (e.g. Pontius ~‘1 ~11.. 1986; Hill and Liu, 1987; ticles in a distribution will violate the longitudinal Summers ct N/., 19X8) with the observed plasma dis- invariant, J, during interchange motion, whereas par- tribution (e.g. Richardson and McNutt. 1987: Vasyliunas, ticles of higher energy wit1 conserve J. Thus our work 1989; Mci and Thorne, 1991). leading to a proposal that implies that the lower the energy of a plasma, the less a pure MHD approach may not be sufbcient to describe likely it is to remain equatorially confined during out- the interchange motions (Southwood and Kivelson, 1989 ; ward interchange motion, whether it is a diffusive or Fazakerley and Southwood. I993 1. steady process. We discuss our results in the context of Despite ail this activity, there have been reiatively few the Jovian magnetosphere. papers in which interch~ln~e stability criteria are studied. The majority of these papers was reviewed by Southwood and Kivelson (1987) who produced a general stability 246 A. Fazakerley and D. Southwood : Magnetospheric interchange instability condition, valid for a plasma with isotropic pressure in a and that the energy, and hence pressure of this plasma curved magnetic field in the presence of gravitational and would become increasingly anisotropic in travelling flux centrifugal forces. The Southwood and Kivelson (1987) tubes. result embodies earlier results (which were usually derived On the basis of these examples and our more general for more specific situations) and. unlike previous results, remarks. we argue that the assumption of isotropic pres- is valid for arbitrary plasma pressure. sure in treatments of interchange instability may often The assumption that the plasma pressure is everywhere result in a poor description of the behaviour of real space isotropic is not generally valid in a collisionless space plasmas. In this paper we do not attempt a general treat- plasma. An isotropic pressure distribution is unlikely to ment for anisotropic plasmas, but concentrate on a special arise unless some non-MHD process (such as scattering) case which we term “fast” interchange, in which the inter- can play the part of collisions and bring about equi- changing plasma moves on a time scale that is short com- partition of energy between parallel and perpendicular pared with a typical bounce period so that J is not con- degrees of freedom, otherwise the parallel and per- served in the plasma. We will show that double adiabatic pendicular pressure (p,, and p_) are free to evolve indepen- theory describes the behaviour of the plasma during a dently. fast interchange motion, which is unsurprising since our The simplest anisotropic pressure scenario assumes definition of “fast” implies that plasma particles do not fluid behaviour both perpendicular and parallel to the have time to travel far along the field during the inter- magnetic field. Localization. the basis for fluid models. is change and that the plasma is thus effectively localized always ensured perpendicular to the magnetic field by the parallel to the field as well as perpendicular to it. There is v x B force, but fluid behaviour parallel to the field is only no possibility of isotropization by scattering effects during possible if the plasma distribution changes sufficiently an interchange motion which occurs on so short a time slowly along the field, and if there is no strong field-aligned scale. current or heat flow. Given localization in both degrees We will also consider the behaviour of individual par- of freedom, the fluid will obey the double adiabatic Chew, ticles associated with a plasma undergoing fast inter- Goldberger and Low (CCL) equations (e.g. see Clemmow change motion in the context of both the MHD picture and Dougherty, 1969) in which separate equations of state of interchange and the recent kinetic picture of Fazakerley apply top,i andp,. Other anisotropic plasma distributions and Southwood (1993). Finally we will consider whether exist in which the plasma behaviour parallel to the field the ideas of fast interchange may be useful in under- cannot be described in terms of a fluid model. Theoretical standing the behaviour of Iogenic plasma in the Jovian studies of the Jovian magnetosphere provide two inter- magnetosphere. esting examples. The Jovian ionosphere is thought to be a source of Jovian magnetospheric plasma, with most of the escaping ionospheric particles entering the magneto- sphere on field lines within the L-shell of IO (90% of the Calculation of stability criteria planet’s surface maps to these field lines). Ions which escape from the ionosphere are expected to have parallel velocities which are at least equal to the local corotation The derivation of instability criteria given here is based velocity, as a consequence of being constrained to move on the small perturbation approach used in Southwood along corotating field lines (Hill et ~1.. 1974). Thus. ions and Kivelson (1987). Additionally. we use a kinetic treat- from opposing hemispheres would form field-aligned ment to describe the independent evolution of parallel beams which are likely to interact. thereby redistributing and perpendicular pressure during a fast interchange some energy into perpendicular degrees of freedom. and motion. giving rise to a trapped. but probably still anisotropic distribution. On L-shells beyond that of the lo torus (L = 6) the magnetospheric plasma is dominated by heavy Unpcrturhed equilibrium ions which originate in the 10 plasma torus and which are thought to be distributed quite differently from magneto- We shall assume in our treatment that if the magneto- spheric ions of ionospheric origin. Observations give sphere is rotating, the planetary--magnetosphere coupling qualitative support to the suggestion of Hill and Michel is strong enough to maintain the magnetospheric plasma (1976) that ions originating at the Jovian satellites would in rigid corotation. In such circumstances Coriolis forces tend to form a sheet of plasma, confined near the spin can be ignored and the sum of the centrifugal and gravi- equatorial plane due to the action of centrifugal force. tational forces can be treated as an “effective” gravity, The idea was expanded upon by Siscoe (1977) who cal- directed away from the planetary rotation axis when the culated how the phase-space distribution of ions picked former force is dominant (e.g.