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. see Southwood and Kivelson, up at Jovian satellites would evolve in the absence of 1987). scattering if the ions were transported away from the Within a magnetosphere in steady equilibrium, balance planetary rotation axis by interchange motion which con- between the plasma pressure, magnetic field pressure, the serves the adiabatic invariants p and J (i.e. the interchange magnetic tension, centrifugal force and gravity must be time scale being large compared with the ion bounce maintained. Note that the gravitational and centrifugal period). Siscoe showed that outward-traveliing plasma forces will usually both have components along the field. would tend to remain equatorially confined in a disc of The general expression for three-dimensional stress constant thickness (in agreement with Hill and Michel) balance in a magnetosphere is : A. Fazakerley and D. Southwood : Magnetospheric interchange instability 247

motion along the field may occur in the interchange time interval due to finite uiI and due to accelerations resulting from the change in local field-aligned forces, we regard where j is the magnetospheric current, P is the pressure u * B = 0 as a good first-order description of particle tensor, g,,is the effective gravitational force, the subscript motion, We treat the motion by assuming that u is so i refers to the perpendicular component, and c is the large that the interchange is effectively instantaneous and curvature vector, b *Vb (where b is the unit vector along parallel particle motions can be ignored. In order to the background magnetic field B) which is perpendicular describe the growth or decay of motions. we take u to be to B. proportional to en’. Thus, instability is represented by a In a collisionless plasma, there is equipartition of kinetic positive c and the displacement of the plasma element is energy between the two dimensions perpendicular to B, given by u/a. To demonstrate stability it will be necessary but not in general involving the dimension parallel to B. to demonstrate that there are no motions possible for Thus pI is not necessarily equal to pii and the pressure which aI1 is real and positive. tensor is termed “gyrotropic”. The stress balance equation We shall calculate the changes in energy and position can be divided into components perpendicular and par- produced by the interchange motion, u, in the linear allel to the magnetic held. The perpendicular component approximation. The derivation is similar to that given can be written (e.g. Ferraro and Plumpton, 1966) by Southwood and Kivelson (1987) for slow. isotropic interchange, The main complication is introduced by the V1PT = nnzg,ni -+ $e-(p;; -Iti)c, (2) requirement that energy cannot be exchanged between perpendicular and parallel degrees of freedom. This will where we have introduced pT the total pressure : be true if scattering processes do not operate, i.e. for both 7 fast and slow anisotropic interchange. As it happens. the PT = pi + E (31 assumption that we can ignore particle motions along the Go. field which characterizes fast interchange motion (i.e. the A similar expression can be written for the parallel com- growth/damping rate D is large enough that /@I z+ ]r>,!b- VI) ponent of the stress balance equation : dramatically simplifies the analysis, and an analytical description of the evolution of perpendicular and parallel pressure is readily obtained using the Liouville theorem. Let us write the total kinetic energy of a particle as B’. indicating that parallel forces cause a variation of pressure Its rate of change in the adiabatic (low frequency) limit is along the length of a flux tube. Note that a plasma can given by (Northrop, 1963) : exhibit a gradient in pressure along the magnetic field and dW yet be isotropic [only the second term on the right-hand -- = ,~~1.6B+yE*v,fr, Fl,. side of equation (4) vanishes for an isotropic plasma]. dt .As in Southwood and Kivelson (1987) we shall assume where 68 is the first-order perturbation in B. F,, contains that interchange motions are displacements of entire flux the force due to a parallel electric field and the effective tubes or field Iines that conserve the direction of the mag- gravitational force. and the first adiabatic invariant, ,u, is netic field (although. if the plasma pressure is comparable given by : with the field pressure, these may not be field strength conserving). The plasma motion is assumed to be strictly perpendicuiar to the ambient field. Of course, the per- pendicular motion may give rise to t~onequilibrium con- ditions along the magnetic field. As we discuss later. the The term vR contains only the magnetic (gradient and result may be secondary acoustic flows taking place on curvature) drift. External force (e.g. gravitational) adia- the time scale of particle motions along the field to restore batic drifts do not enter this expression. because they do equilibrium. The parallel flows would be expected to take no net work; the work done by drift motion due to an place on a time scale comparable with the particle bounce external force along the electric field is exactly equal and time and we shah not attempt to treat them here. opposite to the work done by the electric field drift in the We shall assume that we have an axially symmetric direction of the external force. (or at least approximately: locally axially symmetric) The total kinetic energy can be considered to be the magnetospheric magnetic held in which a hot plasma sum of a gyration energy W,, and a parallel energy W,,, population is trapped. We shall label held lines by their provided that we neglect the kinetic energy due to the equatorial radial distance. L. and furthermore assume electric field and first-order adiabatic drifts. FolIowing that the field curvature is confined to meridian planes. In Northrop and assuming a steady background magnetic interchange motions there must be a component of field field such that iYb/r?f= 0. and dropping terms above first line motion in the L-direction. order in E since this is a linear treatment :

We represent the motion by a velocity field u. where d W,, -- = 2w~u.c-~~,,liiiR+(mg,,,+qE),,t:,,. (8) u-B = 0. Although we acknowledge that some particle dt as 248 A. Fazakerley and D. Southwood : Magnetospheric interchange instability where the field-Iine velocity and the electric field are related defined (but see e.g. Southwood, 1973) : byu = ExB,‘B’. (V*u), = v-u+u*c; (18) We shall assume henceforth that E * b is zero (which follows from our definition of an interchange motion). If similarly : we assume negligible parallel motion whilst the inter- (V-u),, = -U’C. (19) change takes place, there is no possibility of exchanging parallel and gyrational kinetic energy through the action Applying the Liouville theorem we can calculate the of the mirror effect (the z~,,~&B/as term) during the motion, change in the distribution function. Often one regards or of altering W,, through the action of g,,,, . Accordingly, the distribution function as a function f(p, W, L) of the the linear rate of change of W,, the gyration energy, is : invariants or constants of the background adiabatic motion, p, W and .L. However in the special case where d W; -_= &B+qE*+, = /I(DGB+-u*VB), (9) we assume that there is negligible motion along the field dr direction, we may treat the distribution as a function of W,,, W, L where vvB is the magnetic VB drift. Provided that u is very and (despite the two former quantities not being large, the curvature drift term survives the assumption of constants of the long-term motion) and it is convenient to negligible v;: and the linear rate of change of W,;, is : do so: 3” dw,, af d WL ?f dL af __^_____---~- -- (20) (10) 5" dt SW,, dt 63W, dt iYL’ and thus, after integration, substituting for the changes in where v, is the magnetic curvature drift [so that W,, is energy and noting, furthermore, that : changed by the (un)bending of the field line] and the rate of change of W is : dL 56L y=u’oL, (21) dW = psbB+qE*vB. (11) z- (22) The perturbation 6B must be a change in parallel field strength alone, since a property of an interchange motion we have : is that the field is not distorted from the curvature appro- af priate to mechanical equilibrium (such a bending would o~f=2w,,(V-u),aw+w~(P.U)I~-u*Vf. (23) consume much energy). One may straightforwardly derive II 1 an expression relating 6B and u from the frozen-in field Using the general definitions of pXand p,. , we have : condition :

&B = V x (u x B). (12) pll = d3c2Wi,f, (241 s With a little algebraic manipulation. one can show that :

pI = d-‘l,W,f. (25) &?B= -B V.u+u~;+wc . (13) s and with K representing a constant scaling term ( = It follows that : m3f2) in the velocity space Jacobian, we can write : dW1 - = -~B(V*u+u*c). (14) dt d”v = Kd W, ___y$_ (26) We find the changes 6 W,. 6 W,, and 6 W produced by the Using the expressions above for 6 Wl and 6 W,,, and per- disturbance by integrating our expressions for d WJdt, forming the integration, one finds that : d~‘,~~dr and d W]dt. In a linear theory, we use the un- perturbed orbit of a particle to describe the path of the -06P,, = ” *VP,, +I+ (V * “1, + 3Pi, (V. U),,’ (271 particle during the integration time. In this work the inte- - a6p, = u *vp, + 2p, (V *u), +p.l (V *u),, . gration is greatly simplified by our choice of a growth time (28) which is rapid compared with the bounce time, and the For comparison, we remind the reader of the expression expressions for the change in perpendicular and parallel for the pressure perturbation in an isotropic plasma (e.g. energy are : Southwood and Kivelson, 1987) : -G&3 = ypv*usu-vp. The continuity equation for n takes a similar form : so that : -a&2 = u*Vn+n[(V*u), +(V*u),,], (29) crSct/, = --,uB(V-U)~, (16) where 6n is the perturbed density and : a6 U’,! = - 2 w, (V *u) ,/) (17) v-u = (V-u),, -t-(V-u),. (30) where, for reasons that will become clear later, we have Consider the second term on the right-hand side of the A. Fazakerley and D. Southwood : Magnetospheric interchange instability 249 density equation. Its first part, n(V * II),, is inversely pro- If we choose boundaries which lie outside the region in portional to the variation of flux tube area available to which the plasma is displaced, the surface integral the plasma, while the second part, R(V*U),~*is inversely becomes zero. By substituting the expressions for Sp, d^B proportional to the variation in the local length scale along and 6n and also using the stress balance relationship, we the flux tube (see Appendix). The forms of the pressure find that the volume integrand can be rewritten to contain, equations are identical to those derived in the two-fluid among others, a term [dl f a] * incorporating ail di terms {CCL) approximation (see e.g. Clemmow and Dougherty, (which also has the useful property of being positive 1969) and thus show that, in the limit we have chosen, the definite) : plasma is behaving as a two-dimensional gas (ratio of specific heats, y = 2) in respect of the dimensions per- pendicufar to the field and as a one-dimensional gas (1’ c 3) along the field. Thus, our fast interchange criteria will be identical to the criteria for the case of slow inter- +(u*Vn+tld,,)mu*g+ U-VP, -+-3p,!dv-u-E;+ d,, , change in CGL plasma, just as we suggested in the Intro- I) duction. (36) The first-order momentum equation takes the form : where c(, the quantity chosen to make the first term on the nff7mI = -VvI&+ ~;+c+hq&.~ +6p+-bpdc (31) right-hand side a perfect square is given by : I where we have used, from equation (3) : (37)

sp, = sp, t EB. Note also, by virtue of the equilibrium condition, we can PO also write : In the following we shall abbreviate (V - u),, as d,,, (V * u)~ as n,, and geff as g. Also, the I subscripts on g and V will 2~pr = u*Vp,i-pid,,. (38) be dropped and considered to be implicit, wherever these terms appear in a dot product with u (as u * B = 0). Conditions forstability

Encrg_~ equation If rr’ is negative for all possible perturbations, then the system is stable. We can determine the contribution to the Following Southwood and Kivelson’s (1987) approach, stability of a flux tube from a volume element of the tube we develop an expression for the total energy associated by considering the overall sign of the integrand on the with the perturbation, by taking the scalar product of the right-hand side of equation (36) and also discover the momentum equation with u : relative contributions to stability of individual terms. It is necessary to take the sum of all the volume elements composing the Rux tube to determine the stability of the flux tube itself. (33) Let us examine the right-hand side of equation (36) The left-hand side is proportional to the rate of change of more closely. The first term is never positive and so always kinetic energy at a particular point. while the right-hand acts to increase the stability of the system. Its minimum side is the rate of work done per unit volume by the body contribution to stability is when it equals zero, so that: force within the plasma. To obtain a global expression, (d, +cLj = 0, (39) we integrate throughout the volume in which plasma is perturbed : or, using equation (38) :

u - VP, +p.Ld,, + 2p-,d, = 0. (40) It follows, from comparison with the expressions for Sp, and CTB and the perpendicular equilibrium condition, that the condition (above) is exactly equivalent to the require- ment that the total perturbation pressure is balanced, i.e. : Following Southwood and Kivelson (1987) in employing the standard expansions for V(u6p1) and V{u~~~), re- (41) expressing V * u in terms of its parallel and perpendicular components, and using Gauss’ theorem we find : A similar result was found by Southwood and Kivelson (1987) who pointed out that it follows that any unstable d’rnmrru’ = - d’r*u@, interchange motion would be expected to maintain total s pressure balance normal to the field as it is energetically favoured. The requirement specifies d, in terms of the magnetic field properties and the perpendicular plasma pressure; a Less simple relation than that describing d,, 150 A. Fazakerley and D. Southwood : Magnetospheric interchange instability

(which is given by -u .c) reflecting the decoupling of implies that d,! will be positive, and the outward-directed perpendicular and parallel pressures. effective gravity vector implies that u ‘g will be positive. Given that the total pressure is conserved during inter- An outgoing flux tube usually encounters declining mag- change motion, we can further develop the integrand on netic fields and pressures, in which case we expect it to the right-hand side of equation (36), pausing only to note expand to maintain pressure balance, so that V* u will be that in this special case we can use equations (37) and (39) positive. In most cases we would expect a perpendicular to show that: component to the expansion, i.e. positive dL (our third I,& type term), but this need not be the case as we see in - z (dl -4,) = 2pp,d, +p,,d +nmu*g, (42) equation (30). Given that all $-type terms are positive, the condition for stability of a flux tube element against which we shall use shortly. outward motion in the three $u*(,B-&)-type terms of Setting the leading term in the integrand to zero, one is equation (44) is thus that the actual gradients are less left with the following terms : steeply negative than the critical gradients (which are negative, since both VB/B and c are negative). The reverse -d; 2pT + (u - Vn + rid,,)))) . g applies in discussions of stability against inward motions. BVB Note that this statement applies to an infinitesimal element + u.Vp,, +3pid,, -u*---- d\ . of a flux tube, and a full integral over the volume within PO which interchange motions occur is needed to determine Substituting for 2p,d, using equation (38), the integrand the stability of the plasma as a whole. becomes : These criteria for stability are binding in a rigidly rotat- ing magnetosphere. Southwood and Kivelson (1987) remark that in the case of a departure from rigid rotation, the work of Rogers and Sonnerup (1986) implies that a magnetosphere will be more stable to interchange than in VBB’ the rigid case and that their criteria are sufficient, rather + u . VP,, + 3~; d,; - u * - - d’ + (u * Vrz + nd, )mu - g. B ~0 than necessary conditions for stability. This is also the case for the conditions presented here. Substituting for B’/p,(d, - d,l) from equation (42) we find, after selective substitution for d, = -u *c the following form for the energy integral : Discussion CT’ u*Vpi-2pLu.~-pL~*c s Interpretation oj’stability criteria

VB + u*Vp,,;-p~*~~ -3pu.c d The three major terms in the integrand of equation (44) are easily interpreted. Each term compares the change of a parameter (e.g. n) in a travelling flux tube element with VB + u~vn-?lu~ ~~ --I?U’C (43) the change of the local value of the background value of \ B that parameter. The first is associated with the change in gyration energy (i.e. the internal energy stored in the which we can rewrite as : perpendicular degrees of freedom), the second term is associated with the change in energy of the parallel o2 d3rnmu’ = - d’y[u * (VP_ -VP_~.~)(/ (bounce) motion. and the last is associated with the gravi- s s tational or centrifugal energy change. Equally, we can +u. (VP, -Vp,,,)c/, +u * (VII- Vu,, )rnu *g]. (44) regard the stability criteria as expressing the conditions [as can be verified by rewriting the right-hand side of where : equation (43) in terms of 6B, Sp,!, 6p, and 6n] that the number per unit flux (n/B) and the entropy parallel (p,,/B) vpir =p . and perpendicular (pL/Bz) to the field are constant in the background field. for they are inevitably constant in a flux tube undergoing adiabatic motion in the CCL approxi- (1. +d = v-u = --r-U’C. mation. The stability criteria appropriate to the confined plasma (45) disc described by Siscoe (1977) take a similar form, again A system is stable if g’ is negati\,e for all possible per- requiring that n/B, p,,/B, and pl!B’ are constant in the turbations. In order to interpret our expressions. note background field. To see this, recall that in the Siscoe that for positive 4. a term of the form $u * (/i-/3,,) will model, the plasma forms a constant-thickness disc (except contribute to stability if/j is more positive than /& (the near the source) lying in the equatorial plane, the parallel reverse applies for negative 4). At least two of the three energy of ions becomes essentially constant as they move $-type terms will be positive (negative) for an outward away from the source and their perpendicular energy varies (inward) flux tube motion u in a centrifugally dominated inversely with B (conserving p). There must be a re- system; the change in curvature of an outgotng flux tube distribution of plasma along the field. as it travels outward A. Fazakerley and D. Southwood : Magnetospheric interchange instability 251

(or inward) in order to maintain the disc thickness at a form the integration over theentire relevant volume before constant value. Thus. the change in volume of the plasma- we attempt to compare the stability criteria of systems in filled region of a travelling flux tube is a result only of the which the distributions of plasma along the field difier. For changing cross-sectional area of the plasma-filled region. example, the critical density gradient for Siscoe’s confined The usual curvature-related contribution to volume case differs from the gradients in the other two cases (due, change is effectively zero. In this case, the number density as explained above, to the difference in the evoiution of n is more correctty regarded as a number per unit area the plasma volume elements). Despite the difference in than per unit volume, but nevertheless it varies as 1/Band. critical gradients, integration of all relevant volume given the conditions on W, and WI [see equations (24) elements yields the same conclusion in all three cases; that and (X)], pI varies as B’ and pi varies as B, confirming the marginal stability condition is that the mass per unit that we are essentially applying the same constraints to II, flux is the same throughout the magnetosphere. pI and p j as in the fast case, except in a quasi-straight field geometry. Thus, we infer that the stability criteria for the Siscoe case and those for fast interchange in the special case where c = 0 [see equation (43)] are identical. even though Siscoe uses a dipole magnetic field model. Hence According to the assumptions we have made in our treat- the result will be : ment of fast interchange. the trajectories of individual particles during a fast interchange motion are specified by the relation u* B = 0. To illustrate these trajectories. we consider a dipole magnetic field aligned with the planetary spin axis. The field lines are defined by the curves r = L sin’ q, where (r, q) are the radial distance and co- latitude in a polar coordinate system and L is the Mcrlwain parameter. The curves which are everywhere normal to where the condition on pi: naturally disappears in the the field lines are described by r2 = k cos q. where k is a absence of curvature, since I+;! cannot vary without cur- constant. and correspond to the trajectory of particles vature in our fast model [see equation (IO)]. The condition during fast interchange. These curves are directly anal- on the variation of pI is a repetition of the condition on n, ogous to the lines of electric force and of constant electric since W,, does not vary with L, and is thus redundant and potential associated with an electric dipole (Lorrain and absent. The condition on pI incorporates the variation of Corson, 1970). In Fig. 1 we show dipole magnetic field W, with L and is not redundant. tines and precise particle trajectories for the case of par- We can use the fact that VBIB = c for a dipole field to ticles which originate at latitudes of 3 , 6’ and 9” on reexpress our fast interchange stabiIity criteria [equation the L = 6 field line. These points of origin are chosen to (43)J in the form : represent the spread of magnetic latitudes at which ion pickup occurs in the IO torus (at L = 6) allowing direct comparison with the results of Siscoe (1977). Figure 1 shows that particles which are not in the equatorial plane

+ i (47) It is clear that the critical gradients of II and p1 are both steeper in the fast interchange stability criteria than the slow. anisotropic. confined model. It is also interesting to examine the criteria for a slow, isotropic interchange (as described by Southwood and Kivelson. 1987) in a dipole field. This takes the form shown in equation (48). below :

0 5 10 I5 20 Ostonce from dipole 0x1s (R-1

Fig. 1. The.soMcxme~ rcprescnt nupnc~ic ticld lines (with L = 6, The critical gradients for 81are the same in the fast and 12 and 18) for a dipolar magnetic field. The ~&fed lines are slow. isotropic cases. but are different for the Siscoe case. curves which are at all points parallel to the local normal to a The critical gradient for p is steeper than that for pI in the field line and as such represent the trajectories of particles during fast interchange motions. The three curves rcprescnt lrajectories fast case which in turn is steeper than pI in the Siscoe starting at the L-shell of lo (L = 6, roughly) at latitudes above case, while the critical gradient forp is shallower than that the equatorial plant of 3 . 6 and 9 . corresponding to paths for p,; in the fast case. which might be followed by particles originating in the IO torus. The apparent differences between some of these critical Note how the mutual sepxation of the particles and their height gradients reinforces the point that it is important to per- above the equatorial plane incrcasc with increasing L 252 A. Fazakerley and D. Southwood : Magnetospheric interchange instability at the beginning of a fast interchange motion will be dis- express their altered potential energy. The MHD picture placed further from the equator as the flux tube travels of interchange can only accommodate a slowing of the outward (the converse holds for inward transport) and motion of an outward moving flux tube in terms of vari- that outward motion causes the particles to spread apart ations in ionospheric conductivity, i.e. the drag on a flux as the field line ‘stretches’ (compare this with the constant- tube, or by a reduction in the driving force on the flux thickness disc which is expected when J-conserving inter- tube, e.g. when adverse pressure gradients modify the change motions occur). Figure I also shows that as out- instability criteria (e.g. for studies in the Jovian context ward motion proceeds, there is an increase in the distance. see Summers et al.. 1989 and references therein). A flux projected onto the equatorial plane. between a particle tube which is traveiiing rapidly enough to be considered and the point where its field line intersects the equator. “fast” might be slowed sufficiently that its motion This corresponds to an increase in the centrifugal potential becomes “slow” compared with bounce times, at which energy of the particle. In effect the particles would be point particles on the flux tube would resume their bounce raised above the equatorial plane during a fast outward motion but with enhanced parallel energies. interchange motion or lowered toward it during an inward An alternative to the MHD model treats interchange motion. Should the interchange motion then change from at a kinetic level (Fazakerley and Southwood, 1993) in fast to slow (or cease altogether) then the ions will resume which particles interact with drift waves raised by the inter- their bounce motion, falling along the field line towards change unstable plasma. The particles undergo significant the equatorial plane under the influence of the (dominant) radial displacements only when (briefly. but frequently) centrifugal force, thereby converting their centrifugal in resonance with such waves. The radial steps may potential energy into parallel kinetic energy [in precisely be in or out, but the overall motion of the plasma is a the same way as Hill er (I/. (1974) have proposed for diffusive motion away from the source region. The model ionospheric ions]. The increase in potential energy with L- of Fazakeriey and Southwood was constructed in two shell for a particle traveiling outwards along the trajectory dimensions and the inherently three-dimensional argument originating at L = 6 and a latitude of 9 (one of the cases made here was not considered. However, we speculate illustrated in Fig. 1) is presented in Fig. 1. The difference that the individual radial steps taken by particles could between the potential energy at L = 6 and at some larger be sufficiently rapid to be considered as J-violating. If that L-value represents the energy available for conversion were the case, then an outward-moving particle would to parallel kinetic energy should fast interchange motion gain potential energy while in resonance and. after failing cease at that larger L-value. out of resonance, the parallel kinetic energy of the particle The particle energization described here can only occur would be greater than it was prior to entering the res- if an interchange motion transports the particle radially onance (and vice versa for an inward moving particle). during a time interval short compared with the bounce As there is an overall outward diffusion of particles, the period. and then ceases to do so. Particles undergoing fast plasma would make a net gain in energy. interchange all the way to a m~~gnetospheric boundary Paratiel particle acceleration, whether through a J-vio- would have no opportunity to resume bounce motion and lating interchange at the MHD or kinetic level, may be accompanied by perpendicular heating of the plasma if a pitch-angle scattering mechanism comes into play.

It is generally accepted that centrifugal interchange motions transport plasma from the IO plasma torus toward the outer Jovian magnetosphere and that a pres- sure gradient related to the inner boundary of “ring cur- rent” plasma impedes these motions in the region 7 < L < 8 (Siscoe ct d.. 19XI ). Thus. contributions to interchange instability due to plasma pressure are believed to be much less important than the mass-related cen- Fig. 2. The curve rcprcsents lhc LI.OI-Adone on :f particle by centrifugal force as the pnrticlc tr-;l\cls along ii Ii&i hnc from trifugal contribution (except perhaps around L = 7,s). an off-equ~ltori~~l posltion pi, the cqu;~tori;ti piano (so that the Therefore, with the possible exception of studies of regions distance of the particlc from the spin xxi5 pt-OW). If the particlc where plasma pressure has an important role, such as the is initially at rest in the otY-cquatorl;il pos111on (e.g. sifter a fast torus outer boundary, we expect that the differences in interchange motion placed II thcrc) the work done corresponds stability criteria predicted by models using isotropic or to the parallel kinetic energy of the partxle as it crosses the some form of anisotropic plasma pressure will not be very equatorial plane. The figure shon,< the potcntral cquatoritll par- important in studies of the transport of logenic plasma. allel kinetic energy of a particle as ;I I‘uncm~n of the L-shell it However, observational evidence makes it clear that an lies on. The particle is chosen to ttri~inatc at I. = 6 and at 9 above the equatorial plane. and the pax-ticlc IS assumed to follow anisotropic pressure description would be more appro- a Fast interchange trajectory. The cncrpy would bc realized only priate. e.g. Paranicas et al. ( 199 1) have reported VO>~U+Y if the fast interchansc motion ccaxd and the particle rcsumcd LECP observations of pressure anisotropy (with p,, domi- normal bounce motion nant. and most dominant in lower energy channels) in A. Fazakerley and D. Southwood : Magnetospheric interchange instability 253

the Jovian nightside neutral sheet. Whether or not these respectively, if their bounce times differ dramatically. We observations are evidence of fast interchange, our ideas are also interested in estimating the MHD interchange about particle behaviour associated with fast interchange time scale in the Jovian magnetosphere. In a study of motion seem more suitable for testing against data from interchange instability at the 10 torus, Summers and Siscoe the Jovian magnetosphere than our new stability criteria ( 1985) present estimates of the growth time scale TIMas a results. function of the mass per unit flux pl (their Fig. 1). The We have defined fast interchange motions in terms of most rapid growth time appears to be about 1.S h (slightly motion which is rapid compared with particle bounce shorter than a bounce time) for v = 1.08, but Summers times, thus we must investigate whether such rapid and Siscoe (1985) note that an appropriate value of vl motions appear plausible in Jupiter’s magnetosphere. We could lie between 0.01 and 10 (the main cause of uncer- will use the model of Siscoe (19’77) to estimate bounce tainty being poor knowledge of the ionospheric con- times. As with most theoretical arguments regarding the ductivity C,) giving rise to a wide range of possible time Jovian magnetosphere the model assumes a dipole mag- scales. However, we note that riM G rB, the bounce time netic field geometry although this is an increasingly poor for values of $1in the range 0.1 to 30. which encompasses approximation in the region beyond the IO torus due to a significant part of the range of possible values of Q. currents associated with the plasma sheet. Accounting for The lower values of q which would be inconsistent with magnetic and centrifugal forces (gravity can be neglected) s,~ d ‘5Bcorrespond to high values of Cp. Turning to the and using a small-angle approximation for magnetic lati- case of a flux tube further from Jupiter, the expression for tude i, we can express the parallel equation of motion as : fully developed steady-state MHD interchange motion given by Siscoe and Summers (1981) indicates that the d’s characteristic time scale falls with distance as L4 (assuming -= -32 (49) dr’ a shallow gradient in background mass per unit flux), whereas we have remarked above that the bounce time where s is the distance along a field line measured from appears to be independent of distance (at least in a dipole the equatorial plane. This expression reduces to the simple model magnetosphere). These results suggest that even if harmonic motion form : an interchange motion were “SIOW” near IO, it might be “fast” farther from Jupiter. While we remain so uncertain about suitable values for some of the parameters needed to make these estimates, For the case of ions picked up at IO. the bounce period rB and while we use rather idealized (dipole) magnetic field is given by : models, it seems premature to draw firm conclusions regarding the likelihood that MHD interchange motions in the Jovian magnetosphere would violate J. However, assuming that our estimated characteristic times are reasonably valid. it is hard to argue that many bounce where, following Siscoe (1977), s = L/L, the ratio of the cycles occur on the time scale characteristic of MHD current value of L to the value of L at the source and interchange models. and hence it is hard to argue that J <\ = ( I;,, - r’,)i C;,,where vC, is the corotation speed at the is clearly conserved by interchange motions in the torus source and I: is the orbital speed of a neutral particle at region. It follows from the above that even ifJis conserved the source. On their creation at IO, particles have a bounce near the torus, it follows from the steady acceleration of period of 4.7 h. The bounce period increases with increas- an outbound interchanging flux tube and the lack of L- ing Jovicentric distance (as the centrifugal term comes to dependence of bounce time. that the case for “fast” dominate the magnetic term) reaching 5.7 h at only IOR, motions is stronger at larger L-values. We suggest that in and asymptotically approaching a value of 5.8 h at larger reality neither the extreme case of fast MHD interchange distances. The calculation of the bounce period neglects presented here (with virtually no parallel motion during any parallel thermal motion the particle may have had interchange) nor the extreme case of slow MHD inter- prior to pickup. Such an omission is not unreasonable, change (with very many bounces during interchange) con- since the parallel thermal motion of a particle of energy sidered in most earlier work, but some intermediate case 100 eV is an order of magnitude below the equatorial (in which bounce and interchange times are similar and value of the parallei particle velocity at IO predicted by J is again not conserved) may best represent the true Siscoe. and very few particles in the neutral torus are likely situation. to have energies as high as 100 eV. It is also interesting to No attempt has yet been made to compare the time note that bounce motion is dominated by the effects of scales for radial and parallel motion during a resonant centrifugal force. so that the bounce period of a particle interaction of the sort envisaged by Fazakerley and depends only on L. and is virtually independent of the Southwood (1993). but there seems no obvious objection particle energy (for relevant plasma populations) on L- to the idea that a resonant particle could undergo “fast” shells beyond the source L-shell. Particles with much radial motion while in resonance. greater energies than the pickup ions considered here According to the MHD picture, parallel particle ener- will have much shorter bounce times according to the gization will only occur if a fast interchanging flux tube foregoing expressions. Thus energetic particles and pick- were to encounter a change in conditions (e.g. ionospheric up particles on the same flux tube might regard the motion conductivity or a local plasma pressure gradient) which of the interchanging flux tube as “slow” and “fast”, had a signi~cant braking effect. In the light of our current 254 A. Fazakerley and D. Southwood : Magnetospheric interchange instability knowledge, the plasma pressure gradient at the inner edge sheet region. Consequently, the bounce motion of a par- of the ring current would seem the only place in the inner/- ticle found at a particular magnetic latitude and distance middle magnetosphere where interchange is clearly from Jupiter will take it further from the planet than a retarded and parallel particle energization might be dipole model would predict so that its centrifugal potential expected in the fast interchange picture. If that is the case, energy at the aforementioned point would be larger than and MHD interchange motions are not “fast” within the that predicted by a dipole model. The accompanying heat- inner edge of the ring current, then there are no grounds ing of plasma, should it occur, would therefore be greater in the MHD picture for arguing that such particle ener- than suggested by Fig. 2. gization occurs in the Jovian magnetosphere (although particle cooling may occur on inbound flux tubes if their Conclusion motion changes from “fast” to “slow” as they approach the torus). We have begun to address the question of interchange On the other hand, it is a feature of the kinetic model motions in anisotropic plasmas by examining stability of interchange that the interval of fast interchange for a criteria for fast interchange in a plasma of arbitrary /?. particular particle will end when resonance ceases. Thus, We have also inferred the stability criteria for the slow, all resonant particles whose radial motion violates J, wher- confined interchange motions discussed by Siscoe (1977), ever they are, are able to gain parallel energy as they travel a second example of an anisotropic plasma. outwards. As noted above, the plasma behaviour during The special case of fast MHD interchange motions slow (J-conserving), confined interchange has been de- differs from the other interchange motions mentioned scribed by Siscoe (1977) who found that the parallel energy of herein (isotropic and slow, confined) in that the J invar- ions is independent off. while their perpendicular energy iant of the interchanging plasma is not conserved. varies inversely with B. as a consequence of conservation Comparisons between rough estimates of MHD inter- of p. The gyration energy might decline less rapidly than change timescales, which are independent of whether J is this, with increasing distance from Jupiter, if some scat- violated or not, and bounce times suggest that J-violating tering process operates redistributing parallel and per- interchange motions may occur in the Jovian magneto- pendicular energy evenly between the three degrees of sphere (assuming MHD models are appropriate). We note freedom. However, the predictions of Siscoe’s model are however that improved models of the Jovian magneto- contradicted by Vq”ager observations which indicate an sphere, e.g. incorporating the effects of plasma sheet cur- increase in the temperature of the Jovian plasma sheet rents on the magnetic field geometry, are needed to with Jovicentric distance. We suggest that if a resonant improve our confidence in such estimates. interaction between particles and waves lies at the heart Two consequences of fast interchange motions are out- of the plasma transport mechanism. plasma may naturally lined. These are the spreading out of particles along field be heated in the process. The radial motion of a resonant lines and the increase in potential energy of such particles particle will be accompanied by a change in its parallel which occur during outward motion (the reverse will occur energy, with greater amounts of energy being liberated during inward motion). With the additional assumptions at larger distances from Jupiter. On average a diffusing that fast interchanging plasma slows or stops before reach- particle will travel outwards, away from the source at ing the edge of the magnetosphere and that a scattering L = 6, and will on average gain energy as it goes. Pitch- mechanism acts in the magnetosphere, parallel and per- angle scattering could then pass part of this energy into pendicular heating may accompany outward plasma the perpendicular degrees of freedom. This process would transport. We regard such a scenario as unlikely in the therefore seem able to contribute to the heating required MHD interchange picture, but plausible in the kinetic to explain the observed increase of temperature with dis- picture. In the Jovian context, we propose that this heating tance. mechanism may contribute to the maintenance of the Finally, we note that the Jovian magnetic field structure anomalous temperature gradient in the Jovian plasma is not dipolar, unlike the models considered above, since sheet. the field is inflated by plasma pressure and also distorted If, or when, suitable observations of ions on field lines by the centrifugal force. These effects are manifested threading the lo torus and plasma sheet exist. the dis- through adiabatic drifts as ring currents distributed tribution of Iogenic ions in phase space should be exam- throughout the magnetosphere. which locally alter the ined to determine whether it is consistent with the fast curvature according to : interchange hypothesis presented here, or the constant thickness plasma sheet model of Siscoe (I 977). or whether it supports neither proposal.

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