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JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 94, NO. A1, PAGES 299-308, JANUARY 1, 1989 Magnetospheric Interchange Motions DAVID J. SOUTHWOOD AND MARGARET G. KIVELSON1 Institute of Geophysicsand Planetary Physics, University of California, Los Angeles We describe flux tube interchange motion in a corotating magnetosphere,adopting a Hamiltonian formulation that yields a very general criterion for instability. We derive expressionsfor field-aligned currents that reveal the effect of ionospheric conductivity on interchange motion and we calculate instability growth rates. The absenceof net current into the ionospheredemonstratesthe bipolar nature of interchangeflow patterns. We point out that the convection shielding phenomenon in the terrestrial magnetosphereis a direct consequenceof the interchangestability of the system.We concludethe paper with an extended analysis of the nature of interchange motion in the Jovian system. We argue that centrifugally driven interchangedrives convectionand does not give rise to diffusion of Io torus plasma. Neither a large-scaleconvectionnor other forms of unstableinterchangeoverturning appear adequate to explain the plasma distributionsdetectedat Jupiter. INTRODUCTION Steady interchangemotion (convectionor circulation) of the plasma explains much of the known morphology of the terrestrial magnetosphere (see, for example, Cowley [1980]) and is believed to govern transport in other magnetospheres.Interchange motions can be split into two classes,driven and spontaneous. In the latter case, the system is interchange unstable (see, for example, Southwood and Kivelson [1987]). Gold [1959] first suggestedthat the terrestrial magnetospherewas interchange unstable, but Nakada et al. [1965] and many subsequent studies showed that the energeticparticle population is stable to interchange instability. The cold (plasmaspheric) population can be unstable in the outer magnetosphere[Lemaire, 1974]. However, global terrestrial interchange motions have to be driven by momentum transfer from the solar wind. In Jupiter's magnetosphere,the moon, Io, is a substantial source of plasma deep within the magnetospherewhich, by continuity, must be transported away, probably outward, ultimately into the solar wind. Transport has been proposedto be through large-scale convection I-cf. Hill and Liu, 1987] or a small-scaleturbulent interchangecausingdiffusion [e.g., Siscoe et al., 1981]. In the regions where the magnetic field energy greatly exceeds the internal plasma energy, the centrifugal energy associatedwith corotation, or the gravitational energy of the plasma, any such motions will be of interchange form and likely to be spontaneouslydriven. In this paper we derive general formulae associated with interchangemotions and point out some unifying featuresof interchange processes.We give general expressions for the field-aligned current associatedwith interchange motions. We use these to examine the effect of the ionosphere on interchange instability, to illustrate the relation between stable interchangeconfigurations and the occurrenceof the convection shielding phenomenon, to discussthe evolution of interchange instability and to analyze the structure and development of spontaneousinterchange motions. We close with a critique of work on the Jovian problem. •Also at DepartmentoI earth and SpaceSciences,Universityof California, Los Angeles. The dominant forcesin interchange motions vary from one case to another. The plasma pressure gradient forces are dominant in interchange in the terrestrial ring current. In the inner terrestrial magnetosphere, within the plasmasphere, gravity and centrifugal force associatedwith the Earth's rotation can be important (see, for example, Lernaire [1974]). In the Jovian magnetosphere,gravitational and centrifugal forces are also important [Southwood and Kivelson, 1987]. Our approach subsumesall these forces. We shall, however, ignore the effects associated with the Coriolis force in a rotating system (see, for example, Hill [1984]). Coriolis forces can arise in interchange in a rotating systembut are negligible when the ambient magnetic field and planetary ionospheric conductivity are sufficiently strong to impose close to rigid corotation. Often work on interchange instability has ignored the ionosphere [e.g., Southwoodand Kivelson, 1987]. As we shall show, the ionosphere has no influence on the onset of interchange motion; thus stability criteria derived in works such as that of• Southwood and Kivelson are unmodified. The ionosphere acts as a frictional drag on motion in the magnetosphere [Dungey, 1968], and it follows that one expectsit to control the speedof interchange motions. Here we show that both the growth/ damping rate of unstable/stable interchange flow perturbations and the equilibrium velocity of a steady flow are inversely proportional to the ionospheric conductivity. In cases where the system is unstable and a turbulent spectrum of interchange motions develops, the form of the spectrum depends strongly on whether the free energy goes into plasma inertia or into frictional dissipation [cf. Hassam et al., 1986]. FIELD-ALIGNED CURRENT AND INTERCHANGE MOTION Continuity of current governsinterchange flow patterns in a magnetospheric system [Fejer, 1964; Swift, 1967a, b; Vasyiiunas, 1970]. In the ionosphere,current flow is directly related to the electric field [Wolf, 1974], which is quasi-static and also present in the magnetosphere.The two regions are coupled by the field-aligned current fed into the ionosphere, div I = jll sinZ (1) where I is the height-integrated horizontal ionospheric current and ;• is the inclination of the magnetic field to the horizontal. Copyright 1989 by the American Geophysical Union. The field-aligned current,Jll, actsas a sourcefor the iono- Paper number 88JA03629. 0148-0227/89/88JA-03629 $05.00 spheric (and thus also magnetospheric) electric field [e.g., 299 300 SOUTHWOOD AND KIVELSON: MAGNETOSPHERIC INTERCHANGE MOTIONS • = (Sm) = --(1/q)(r3Km/r3fl)= --(1/q)r3K/r3fl Vasyliunas, 1970, 1972]. If the ionosphere is locally uniform and B is vertical, the current and electric field are related in fl = (fl,,) = (1/q)(r3K,,/r3y) = (1/q)r3K/r3y terms of the height-integratedPedersenconductivity•;e by div E = jli/Z e (2) The ionospherecan be treated as a two-dimensionalsystem. We shall show that field-aligned currents associatedwith interchangemotions occur in balanced pairs. The most primitive source is a two-dimensional dipole (i.e., oppositely direc- tedcurrents Iii flowingintoandoutof theionosphere a horizontal distanced apart). The electric potential on the field line at (p, 0) (in locally horizontal ionosphericcoordinatesaligned with and centeredon the current dipole) is then (6) Here the angle brackets denote a bounce average and K is the bounce averaged Hamiltonian. It is convenient to expressf as a functionof theadiabatic invariants/•andJ, withJ = 2• ds viialongthebounceorbit.(Note that our definitions of/• and J differ by factors proportional to rn from the customarydefinitions). In the absenceof sourcesor losses,the particle distribution function for particlesof speciesi, fDt, J, •,/9, t) satisfies the Liouville equation in the form r3f•/r3t + •(r3f•/c3y) + ]J(r3f•/r3fl)= 0 or I d cos 0 (I)--I1_•_.• 2• E v p (3) By analogy one can seehow more complexdistributionsof current "generate" electric fields. Inclusion of geometrical effects and inhomogeneity does not change the principles behind this result. )f•/•t -- q- '(r3fdr3y)(r3K/r3fl) + q- '(r3f•/r3fl)(r3K/r3y) =0 (7) Our normalization convention for integration over velocity (d3v)or overthe tt, J, •, fi coordinates is d•z dfl• d3v fi=d•z dflds d•dJ• v II •b THE HAMILTONIAN APPROACH = dyaft • ni(s, t)= dyaft•i We shall follow Northrop and Tellet's [1960] treatment of particle motion in a magnetospherelikefield configuration ['cf. Swift, 1967a]. We use the magnetic moment and longitudinal adiabatic invariants, t• and J, as velocity space coordinates [Schulz and Lanzerotti, 1974]. The phases (gyro and bounce) associated with each invariant are also regarded as coordinates. (In this paper all distributions are assumedto remain independent of these coordinates.) Like Northrop and Teller [1960], we do not use the third invariant as a coordinate but introduce variables • and fl, the magnetic Euler potentials [Stern, 1967], which are constant on field lines and provide a two-dimensional spatial coordinate system transverse to B. They are related to the magnetic field by B = V• x Vfi (4) The coordinates are a canonical set and the system can be describedby a Hamiltonian as long as drifts of second order in the electric field are ignored [Northrop and Teller, 1960]. The Hamiltonian of a particle of mass rn and charge q can be where n• is the number densityof particlesof speciesi and % is the bounce period; •i is the correspondingflux tube content [Hill, 1984]. Calculation of Field-Aliqned Current In the •, fl coordinate systemthe current density flowing in the y direction at any given point is given by j•,• =Ziq• f d3v h•• proportional to the local value of the rate of change of the coordinate • at a position s on the flux tube. Evidently, the velocity may vary along the flux tube even though • is independent of s. The current in the fl direction is defined analogously. The divergenceof the perpendicular current density is V.jz.• =F-• •• Fj"• +F-• •fi• Fja'• h• ha Km-- mltB(a m,tim,s)+ (1/2)m(v II)2+ qq)(O•m, tim) + mW(o• m, tim, S) where s is the coordinate along the field line, the canonical coordinatesat s are %, and tim,epand W are the electricand gravitational potentials, respectively, t• is the magnetic momentperunitmass,andvii = v ßBIB.In a rotatingsystem, the potential,W, can includea centrifugalpotentialr2f•2/2, but, as noted above, we must assumethat the system is rigid enough that corotation is well maintained. Note that (5) excludes the possibility of parallel electric fields, as the electrostatic potential q>is independentof s. Field-aligned currents associatedwith interchangemotion could themselvesgive rise to parallel potential drops in regions where electron mobility is inhibited. We ignore this possibility in this paper. Interchange motions take place slowly compared to particle bounce motion. Bounce-averagedequations of particle motion are given by (9) In (9),wehaveintroduced thescalefactorsh• andha,where h,ha= B-•. The subscript s is needed to indicatethatJa,m is written (5) (8) (10) where F is the Jacobian,B-•. The total parallel current per unit area into the ionospheres at the two ends of the flux tube is the integral of (10) along the flux tube JH= --Bion • •-• •• F--+ h• F-• ••fi F (11) where Bion is the magnetic field of the ionosphereand we neglectinclination. Using (8) and (9) in (11), we obtain ) where(•)• and(fl•)•arebounce-averaged values. The bounce-averageddrifts include the electric field drift, which is independent of energy and mass and charge and cannot lead to charge accumulation in a charge-neutral SOUTHWOOD AND KIVELSON' MAGNETOSPHERICINTERCHANGE MOTIONS plasma. We exclude thesedrifts by introducing a partial Hamiltonian, H, such that H = K -- q(1) (13) If the integration over # and J in (16) is replaced by a similar integrationover velocityspaceand the field line length and with somealgebraicmanipulation,one recoversthe familiar expressiongiven by Vasyliunas[1970] Henceforth, we shall take the summation over speciesas understood and drop the subscriptsi and b. By use of (6), the form above 301 Vp. (VV x Be)=j, (19) where Be is the field at the equator. is Application to Stability of InterchangeMotions Let us now use our result to examine the effect of the iono- sphereon small perturbationsin interchangemotions.Consider the identity V. 00) =j- or, if we convert to ordinary spatial gradients, Jll =Biof d# dJ [¾f x¾n]. •/B (16) terms in the electric field. Further simplification of the expressionis possibleif H has particular properties. In a fast-rotating system like Jupiter's, the dominant term in H for cold plasma is the energyindependent centrifugal potential T. Plasma is confined to a thin near-equatorial sheet where the potential reachesits low point on the field line. T may be taken outside the integral and the integral simplifies to the expression for the current per unit area of the flux tube used by Hill and coworkers [Hill et al., 1981; Pontius et al., 1986] j, = m[Vr/x V•] ßB (20) Integrating this expressionover the ionosphericsection(dS ds) of the flux tube and noting V ßj = O, we find where B is the local field. The vector form of the integrand in (16) is important. A vector of the form ¾f x VH is divergence free [¾ ß(¾f x ¾H) = 0, identically]. As f and H are bounceaveraged quantities, the flux of ¾f x ¾H must be along B. If the systemis symmetric about the magnetic equator, the fieldaligned flux of ¾f x ¾H into either ionosphere must vanish. From (16), this guarantees that any current into the ionosphere is balanced by a return flow elsewhere.It follows that the simplest electric field source is a current dipole and any more complex sourcesmay be constructedfrom distributions of dipoles. Equation (16) can be related to other forms (see,for example, appendix to Hasegawa and Sato [1979]) by recognizing that the derivatives on H (see equations (5) and (6)) correspondto grad B, curvature,centrifugal,and gravitational drift. The expressioncontains no term correspondingto the parallel vorticity term given by Hasegawa and Sato because of the quasi-static assumption and the requirement to neglect second-order VO + OV.j where we have ignored the inclination of the ionosphericfield. The last integral in (21) is taken at the top of the ionosphere.I is the height-integratedcurrent density. Substituting the expressionfor field-aligned current into (21), fodSIøE---•m dS•)aiønf d•ldJ ß n ag Of OH Of 0n where the integral on the left (right) side is over the ionospheric(magnetospheric)section of the flux tube. As the integral on the magnetosphericside of the ionosphereis expressed in terms of quantities that are independent of s, it may be evaluated at any point along the magnetosphericportion of the flux tube. Now let us introduce a small perturbation in electric potential, (5(I).For convenience, we shall assume that the equilibrium distribution is a function of the coordinate, y, alone and that the time variation is representedby exp (7t). In the linear approximation, the perturbation produces a dependenceon •. The perturbed distribution, 6f, is given by a linearization of equation (7): q _• OH 7(sf-•f 0(500(5f (23) (17) For the moment, we shall assume that the convective time Jaggi and Wolf [1973] considered a monoenergetic 90ø pitch angle distribution convecting under grad B and electric field drift alone. The dominant term in H is/•B, and Jaggi and scaleof the perturbation of the • coordinate is suchthat Wolfs expression forj, (similarin formto (17))is obtainedby where/Ca is the bounceaverageof thenonelectric drift.Recalling that ]•a• 0H/0•, we may drop the secondterm on the taking the Hamiltonian outside the integral. One is not restricted to the use of the particular invariants, /• and J. An important alternative is where the distribution is maintained isotropic in pitch angle by some high-frequency scattering process. The isotropic velocity distribution is a function of energy, W, alone. The adiabatic invariants,/• and J, are not conservedbut there is a single equivalent invariant given by WV 2/3= const where V is the flux tube volume [cfi Harel et al., 1981]. (18) 7 >>fla(O/Ofl) (24) right-hand side of (23): 7 (sf-- (25) The effect of magnetosphere-ionospherecoupling is found by using (22) to secondorder' fondS(5''(sE--'Bion •magdS(sfI) •d•tldJk•-ff]k-•-o• / (26) 302 SOUTHWOODAND KIVELSON.'MAGNETOSPHERIC INTERCHANGEMOTIONS We integrate by parts and substitutefor gffrom (25): n ß = • c•f c•H dit dJ • • dS ag •-•I where Z v is the height-integratedPedersenconductivity of the ionosphere. Then (27) The sign of y determinesthe stability of the system.A necessary and sufficientcondition for stability is fd# dJ Of OH --• < o (28) (31) Observational tests of stability are complicated by the fact that detectorsdo not provide distribution directly as functions of It and J. Nonetheless,the condition has been testedfor the hot plasma (ring current and radiation belts) at Earth [e.g., Nakada et al., 1965] and Jupiter (e.g., Thomsenet al. [1977] who publishedthe relevant resultsin a study of particle diffusion).(The dominanceof gravitational and centrifugaltermsin the cold plasma Hamiltonian can lead to instability in the outer magnetosphere[Lemaire, 1987, and referencestherein], as we remarked earlier. Richmond [1973] discussesthe relationship between the hot and cold plasmas when one distribution is unstable and the other stable, as we mention later.) Condition (28) can be rewritten in special cases to yield more familiar results such as those discussed in Southwood The growth rate is largest for perturbations with the variations g• in the ot direction much smaller than variations in the • direction, and for suchperturbationsthe first term in the bracketsin the denominatorof (31) can be dropped. We shall assumethat the scaleof the unstableperturbation is short compared with the scalesof variations of the background system, i.e., f ho• do• << K (32) (h•)- and f h•do• << (h•)- •(&5•/3•) f ht•d•<< and (33) Kivelson [1987]. For example, where centrifugal forcesdominate, the Hamiltonian H is independent of particle energy and is a monotonically decreasing function of radial distance. Then, the condition for stability (27) is that the integral over It where the integrals are taken over the flux tube. Now and J (i.e., over velocity spaceand flux tube length) increases in the outward direction, or, equivalently, an inward gradient in total flux tube content is required for instability. The expression(28) can be brought into the same form as Taylor's [1964] form for containment devices, but we shall not do that here. Taylor points out that there is a particularly simplesufficientconditionfor stability,equivalentin our nota- (34) lion •ion do• dp (35) tion to The quantitiesd•xdfi and c980/Ofiare independentof position c•f c•H ••<0 c• alongthefluxtube,andh•hts = B- •, sothegrowthrateis c•- 7=[fd. dJ t3ft•H Bio n ho (36, c•'• 0'-•' Z•,-X](B•)(•) or, regardingf as a function of otthrough H, ion As expected,the growth rate, y, is inversely proportional to af <0 -• IIt,./ (29) thelocalionospheric Pedersen conductivity, 2;•,.Now the de- (Taylor also gives necessaryand sufficientconditions for stability for the case where the equilibrium distribution depends on both 0tand fl.) Taylor's calculations were performed for a plasma contained in a toroidal device and no obvious counterpart of the ionospherewas includedin his calculations.Although our calculation specificallytakes accountof the ionosphere,the ionosphericparameters do not enter the conditions (28) and (29). The ionosphere has no effect on the stability condition but does control the speed with which a perturbation evolves in either stable or unstable systems,as we show next. INSTABILITY GROWTH rivative c•H/c• is proportional to the nonelectricdrift in the fi direction (from equation (6)). The sign of the expressionfor V depends on the sign of this drift, which will be made up in general of magnetic components(grad B and curvature) and gravitational and centrifugal drifts. As drifts may be energy dependent,the contribution from the term, c•H/c•, may differ in different parts of the distribution. For example, in the outer Io torus, where ring current impoundment of outward interchange diffusion of torus has been proposed [Siscoe et al., 1981], the Hamiltonian, H, increasesinward for the ring current (hot) particles and outward for the torus plasma. One can rewriteOf/t3• IIt,s c•f/&x lIt,,= c•f/&x Iv+ q- • c•H/&x lIt,,c•f/c•H Ir RATE Equation (27) can be used to derive an estimate of the growthrateforinstability. SetdS= do•d• h•ht•, andnotethat (37) and introduce a critical spatial gradient,c•f/&xIt, for which the systemis marginally stable, namely, t•f/00• Icq-q- I •H/•O•lu,st3f/t3H Ir= 0 6I.tSE =Zv(tSE) 2=Z•, c30t tS• + c3fl Thus (30) c•f/3o• IIt.s= c•f /3o•Iv- c•f/3o• Ic (38) SOUTHWOOD AND KIVELSON:MAGNETOSPHERIC INTERCHANGE MOTIONS 303 Instability proceedsat a rate proportional to the amount by which the actual gradient differs from the local critical gradient. For example, for a case mentioned earlier, the critical gradientin a distributionmaintainedisotropiccorrespondsto the gradient of the "adiabatic" distribution in which the pres- pens in the jovian system (see, for example, Pontius et al. [1986]; Hill and Liu [1987]; Richardsonand McNutt [1987]). The central problem is the outward transport of iogenic plasma. The high rotation rate of the Jovian system means sure satisfies energy. V(p V 5/3)= 0 (39) However, in general one must recall that the critical gradient may be in a different direction at differentenergiesin a distri- that the dominant term in the Hamiltonian is the centrifugal Convective transport in the absence of significant losses conservesflux tube content yet the flux tube content in the region from 6 to 12 Rj falls nearly monotonically[Bagenal and Sullivan, 1981; Richardson and Siscoe, 1981]. Thus it seemsunlikely that a simple two cell convectionpattern [Hill et al., 1981] is present. Convection patterns with short azimuthal wavelengthsalso appear inconsistentwith data. The SHIELDING IN DRIVEN INTERCHANGE MOTIONS magnetic flux must be conservedin the process,so density Interchange motions appear spontaneouslyin a plasma depletedtubes must move in to replace the outward going where the spatial gradient exceeds the critical gradient, enhanceddensity tubes.Thus in this type of convectiveflow, c•f/c•lc. In a stable system,where the appropriate spatial the flux tube content, or, equivalently, the density, might be gradientsare small enough, perturbationsdamp rather than expectedto vary greatly betweensuccessive measurementsdegrow. This is the basisof the convectionshieldingeffectin the pending on whether the measurementwas made in an outterrestrial magnetosphere[daggi and Wolf, 1973; Southwood, ward or inward moving element. Yet Richardsonand McNutt 1977]. [1987] find that density variations between successive Distributions set up in driven interchange motions are natmeasurementstaken about 0.24 s apart (effectivelyseparated urally stable against interchangeinstability. Consider a driven by • 20 km) vary by lessthan 10%. More complicatedmodels convectionsystemin a magnetospherewhere plasma is trans- [e.g.,Hill and Liu, 1987] introducestreamlinesthat reducethe ported inward from a boundary we take to be at large •. radial convectionvelocity,thus increasingthe time over which bution. Wherelosses occurthedistribution mustsatisfy(•f/c•)•,s> 0; otherwise, (•,f/t3•)•,s = 0 in regions wherethereareno sinksor sources. Consider now an enhancement of interchange flow due to an enhancement of the driving source.Throughout the system, the convection field immediately increases.The increase is an interchange perturbation of the system, and wherever (•f/•)•,,s > 0 the perturbation will damp.Hencewe havean alternative way of understanding the convection shielding effect discussedby Vasyliunas[1972], Jaggi and Wolf[1973], Southwood[1977], Harel et al. [1981], Wolf et al. [1982], first pointed out by Wolf [1983]. These authors showedthat currents set up near the inner edge of a ring current type distribution reduce substantially the penetration of solar wind induced convection to the inner magnetosphere. The critical shielding time estimates, given by Jaggi and Wolf [1973] and Southwood[1977] for the onset of screening from low latitudes at the (sharp) inner edge of a hot plasma distribution are of order 1/7. Here, we have shown explicitly that any local enhancement of earthward convection is damped proportional to the difference between the distribution gradient and the gradientlocally required for marginal interchange stability. The damping is exponential and the damping time is proportional to the local ionosphericPedersen conductivity. The shieldingeffect results when the enhanced flow continues to be driven from higher latitude tubes; the radial flow is damped out on the shielding time scale and an azimuthal component developsto maintain continuity. Characteristic shielding times for the dayside of the mag- netosphereare of order of hours. If mean ring current ion energiesare near 10 keV at a characteristicdistanceof 6 Re, n is approximately0.1 cm-3, and with 5;--5 Mho, and • ds/B- 0.05RE/nT,onefinds7- • oforder2 hours. SPONTANEOUS INTERCHANGE IN THE JOVIAN MOTIONS MAGNETOSPHERE The form of interchange motion in the Earth's magnetosphere as discussedabove is generally accepted [Wolf, 1983], but there remains controversy concerning what hap- loss can reduce the flux tube content as a function of radial distance. Loss mechanisms are not identified in this work. The alternative diffusive transport model is consistentwith the presenceof negative radial gradients of flux tube content and lack of variability between successivedensity measurements. A turbulent uncorrelated overturning motion is envisaged with a short scale length such that particles executea random walk. Interchange instability driven by the dense torus plasma has been proposedas the source of such motions. We point out below that the instability of torus plasmadoes not directly lead to random fields. For diffusion to drive the outward flow, contributions of first order in the perturbation velocity (proportional to ,Srl,Sv)must average out for random displacements.However, for the conditionsof the Io torus, a correlationexistsbetween6r/ and 6v • E,/B • 6rl (seeequation (40), below), which produces systematic outflow terms linear in the perturbation velocity.In the simplestconceivable instability conditions the outflow speed of a single tube is linear in time t and such outflow would be expected to dominate over the contributions of second order in 6v which pro- ducethe t•/2-dependentdiffusiveoutflow. We establishedearlier that the simplestmagnetosphericcurrent source distribution corresponds to an electric dipole sourcefor the ionosphere.The idealized particle distribution correspondingto such sourceis a very localized (point) increasein densityabove the background.In Figure 1, we illustrate the field produced by an isolated enhancementand an isolateddepressionof plasma density.On the flanks of each region flow a balanced pair of field-alignedcurrents. In the ionospherethe currentsact as a dipolar sourceof electricfield. If the alignmentof the contoursof K or H are independentof energy, the dipole is simply aligned with the contours. As illustratedin the figure, a depressionin densitywould produce a dipole orientedin the oppositesense.The flow lines associated with the enhancementand depressionare shown.There is flow outside the enhancement and within the enhancement itselfi The plasma in the enhancementmoves toward lower valuesof H and losesenergy. Field patterns are reversedfor a 304 SOUTHWOOD ANDKIVELSON: MAGNETOSPHERIC INTERCHANGE MOTIONS VK Fig.1. Anisolated enhancement ofdensity (darkshading) andanisolated depression ofdensity (lightshading) ina uniform plasma in thepresence ofa gradient oftheHamiltonian K or H. Thedipolelike nature ofthecurrent system linkingtheperturbed fluxtubeswiththeionosphere isillustrated bycharges ontheirboundaries. Darkcurves showflow patterns, anddirections are shownby arrows.Note that the surrounding uniformplasmais setintomotionas the perturbedfluxtubesmovetowardequilibriumpositions. depression and particlesin a depression move to higher return flow in all other sectors. The motion is reminiscent of the convection systemin the Earth'smagnetosphere but with theplasmacorotating withtheplanetto a firstapproximation. Theinwardmotionserves to returnfluxandwouldalsoinject energy.In both casesa net amount of energyis released.In the caseof the depression, energyis lost by the fuller tubesof particlesdisplacedby the motion of the depression. These examplesare closeto the idealizeddescriptionof the interchangewhereinthe energychangeassociated with tubesinter- hot plasma.The energyto sustainthe inward motion is fed from the regionswhereenergyis releasedin the regionof outward flow. A global current systemis set up in which changing positioniscalculated withoutattempting to specify a motionthat wouldaccomplish the change(see,for example, field-alignedcurrentsflowing on the flanks of the outward Cheng [1985]). The electricfield acrossthe tube is directlyproportionalto the enhancement of densityand can be calculatedby balancing the excessmagnetosphericcurrent with the associated Pedersen currentin the ionosphere. For a coldcorotatingdisk with W = r2•2/2, the relation betweenthe azimuthal electric moving plasma feed the ionospheric Pedersen currents throughoutthe convectionsystem.There would be clear re- gionswhereplasmais movingtowardthe planetandregions wherethe plasmais movingawaywith verydifferentpopulationsin the two regions. Hill et al. [1981] (seealsoHill andLiu [1987])proposethat field,E,, and a densityenhancement followsfrom (17) with theflow actuallypenetrates insidethe Io torussourceregion. Jll = -V ß•EpE.If gradientsin/• (or •b)aredominant, Our sketchdoesnot showthis; the flow is likelyto be conE•,= •rI r•2/•, (40) finedto the torussourceand outside.The steepoutward [cf. Hill et al., 1981], where •r/is the enhancementof the flux gradientof plasmawould produceeffectiveconvectionshield- ing as the innerboundaryis very interchange stable.A conse- tubecontentoverthe background tubecontent,r/c,r is radial quence is thatlarge-scale convective flowcouldnotberesponsiblefor thelongitudinal asymmetry in atmospheric hydrogen density [Sandel et al., 1980]despite proposals to thecontrary distance,fl is the angularvelocity.Outsidethe tube thereis a returnflow movingthe surroundingplasmaout of the path of the enhancement; the fieldis that of a dipolewith strength,•/ Arfl2/Y•,whereA istheareaofthetubein theionosphere. Now let us considerthe globalpatternof interchange flow. The simplestis one wherethe field has the two cell pattern exhibitedby our first example.Hill et al. [1981] have describeda corotatingtwo cell convectionsystemlike that illustrated in Figure 2. There is a preferredlongitudefor plasma production and outward flow lines originate there. There is [Dessleret al., 1981]. Hill e• al. [1981] use(40) as an estimateof the outward flow speedin theoutwardmovingflowsector,wherer/cis themean flux tube densityon outwardand inwardmovingtubes.It clearlyservesas a guidebut, as the figureitselfindicates, purely radial flow is unlikely. Any azimuthal variation in sourcerate will giveriseto azimuthalcomponents of flow. One of the problems associatedwith the two-cell model of SOUTHWOOD AND KIVELSON' MAGNETOSPHERIC INTERCHANGE MOTIONS 305 Fig. 2. This diagram representsa possibleflow pattern for the Io plasma torus. It showsa two-cell convectionpattern that could result from a limited region of enhanced plasma production from which outflow originates. The enhanced sourceregion has dark shading.Return flow must be driven in other sectorsto conservemagnetic flux. The inward motion injects hot plasma. An external sink (light stippled region) in which cold plasma is lost from the flux tubes surrounds the entire system.Note that we show the return flow moving no closer to the center of the systemthan the inner edge of the plasma torus, as the outward gradient of plasma at that distanceis interchangestable. Hill et al. [1981] is that in the regions between sink and source regions flux tube content should not vary along streamlinesin outward and inward flow regimes. Steep radial gradients detected at the outer edge of the Io torus suggest that simple radial flow cannot be occurring there. Siscoe and coworkers have favored a diffusive transport model where interchangeinstability causesmulticell convection patterns. Interchange instability could take place on a scale short compared with the region over which plasma is injected and this is likely where production rates vary little with longitude. In drifting fast enoughthat they move more than a wavelengthin a growth time is reduced becausethey have spent part of the time accelerating and part decelerating. However, this situation does not apply for the Io torus as there is a more energetic population of ring current ions whose outward pressure gradient is stabilizing [Siscoe eta!., 1981]. When this is so, perturbations with wavelengths short enough that the fast particles are unaffectedby the perturbation are more unstable [Richmond, 1973] and thus dominate. The shortest wavelength excited will be of order the Larmor radius of the unstable particles and there will be some intermediate wavelength where the growth rate peaks. Such a situation pertains in the outer Io torus where an unstable inward gradient of cold plasma coexistswith a steep stabilizing outward gradient of For the pattern envisaged in Figure 3 to be maintained, hot plasma. There may indeed be particular perpendicular wavelengths perturbationswith a particular wavelengthshould be energetically favored. Although the illustrated steady flow pattern is favored for jovian interchange motions and further investithe result of nonlinear evolution of instability, one may com- gation seemsmerited but the coherenceover the large scalein pare linear growth rate calculations for perturbations of a pattern like that in Figure 3 seemsunlikely. Sourcedistrivarious forms. The growth rate calculated in this paper is butions very uniformly distributed with longitude would be independent of the perpendicular scale length of the pertur- required. The convection may break up into smaller overbation, but we dropped various terms involving spatial struc- turning convection cells but also, if the scale lengths favored ture in our derivation of the growth rate (e.g., seeinequalities for outward and inward motion are short there may be actu(32)-34)). ally a multiplicity of scalesin the convection pattern. Inserting the spacedependentterms in the instability calcuConsider the following alternative to Figure 3. Say the lation usually leads to a reduced growth rate. The reason is source fluctuates fairly randomly. Random azimuthal fluctueasily understood; in a short-scale perturbation, the contri- ations in the density near the source can be expected. The bution to the energy releasefrom that fraction of the particles hamiltonian is roughly azimuthally symmetric and thus any Figure 3 we illustrate one possibilitywhere a short wavelength instability takes place. A similar, more sophisticatedmodel in which departures from corotation are allowed has been presentedby Summersand Siscoe[1986]. 306 SOUTHWOOD ANDKIVELSON: MAGNETOSPHERIC INTERCHANGE MOTIONS SINK Fig.3. Thisflowpattern isalso possible fortheIo torus andcorresponds tothepresence ofa favored wavelength for growth ofaninterchange instability. Shadings correspond tothose ofFigure 2,andasinkregion isalso present. azimuthalvariationsin distributiongiveriseto parallelcur- movingplasmaintermingle.Scalesizescorrespond to the rentsand radial motions.The resultinginterchange motion randomfluctuations in source.Whethera givensectionof will be filamented andtheconvection patternwill be like that plasmamovesoutwardor inwarddependson whetherit con- illustratedin Figure 4, wherepatchesof inward and outward tainsmoreor lessplasma thanthelocalmean.Theflowpat- density ••:*:?'"'"'•: ......... "'"'":'•':':"?:'" "'""" •.,,t • density ......................... ß •..:., .... a.. ...... ... .... •,.,.-.,i; Fig.4. Detailviewofanother flowpattern thatcould beproduced bya temporally varying plasma source intheIo torus. A grayscale represents levels offluxtubecontent in a uniform background plasma. Thepatterns andscales are meant toberandom to reflect variability ofthesource. Overextended timeintervals, regions ofhighfluxtubecontent moveoutandregions of depressed fluxtubecontent movein, butlocallythemotions reflect thelocalvariations of flux tube content. Streamlines evolve with time. SOUTHWOODAND KIVELSON.'MAGNETOSPHERIC INTERCHANGEMOTIONS 307 Acknowledgments. This work was supported by the Division of tern shown in Figure 4 is a snapshot.The spatial structure of the flow pattern correspondsto a fairly random outward and Atmospheric Sciences of the National Science Foundation under grant ATM 86-10858. UCLA Institute of Geophysicsand Planetary inward motion with scale lengths perhaps determined solely PhysicsPublication 3046. by minor fluctuations in production rate. The streamlines The Editor thanks two refereesfor their assistancein evaluating would continually evolve. this paper. Individual parcels of plasma may move inward or outward depending on their immediate surroundings, but a tube with REFERENCES more than average density will move outward overall, and an Bagenal, F., and J. D. Sullivan, Direct plasma measurementsin the Io underpopulated tube will move in. Becausewe have specified torus and inner magnetosphere of Jupiter, J. Geophys. Res., 86, a random source the motion of the fluid at any fixed point in 8480, 1981. space over a period of time is described by a diffusion coef- Cheng, A. F., Magnetospheric interchange instability, J. Geophys. Res., 90, 9900, 1985. ficient. 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