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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 interchange motion in a corotating ,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 ionospheredemonstrates the 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 convection and does not give rise to diffusion of Io torus . Neither a large-scaleconvection nor other forms of unstableinterchange overturning appear adequate to explain the plasma distributions detectedat .

INTRODUCTION The dominant forces in interchange motions vary from one Steady interchange motion (convection or circulation) of the case to another. The plasma pressure gradient forces are plasma explains much of the known morphology of the terres- dominant in interchange in the terrestrial ring current. In the trial magnetosphere (see, for example, Cowley [1980]) and is inner terrestrial magnetosphere, within the , believed to govern transport in other .Inter- and associatedwith the Earth's rota- change motions can be split into two classes,driven and spon- tion can be important (see, for example, Lernaire [1974]). In taneous. In the latter case, the system is interchange unstable the Jovian magnetosphere,gravitational and centrifugal forces (see, for example, Southwood and Kivelson [1987]). Gold are also important [Southwood and Kivelson, 1987]. Our ap- [1959] first suggestedthat the terrestrial magnetospherewas proach subsumesall these forces. interchange unstable, but Nakada et al. [1965] and many sub- We shall, however, ignore the effects associated with the sequent studies showed that the energetic particle population Coriolis force in a rotating system (see, for example, Hill is stable to interchange instability. The cold (plasmaspheric) [1984]). Coriolis forces can arise in interchange in a rotating population can be unstable in the outer magnetosphere[Le- system but are negligible when the ambient and maire, 1974]. However, global terrestrial interchange motions planetary ionospheric conductivity are sufficiently strong to have to be driven by transfer from the . impose close to rigid corotation. In Jupiter's magnetosphere,the moon, Io, is a substantial Often work on interchange instability has ignored the iono- source of plasma deep within the magnetospherewhich, by sphere [e.g., Southwoodand Kivelson, 1987]. As we shall show, continuity, must be transported away, probably outward, ulti- the has no influence on the onset of interchange mately into the solar wind. Transport has been proposed to be motion; thus stability criteria derived in works such as that of• through large-scale convection I-cf. Hill and Liu, 1987] or a Southwood and Kivelson are unmodified. The ionosphere acts small-scaleturbulent interchangecausing diffusion [e.g., Siscoe as a frictional drag on motion in the magnetosphere [Dungey, et al., 1981]. In the regions where the magnetic field energy 1968], and it follows that one expectsit to control the speedof greatly exceeds the internal plasma energy, the centrifugal interchange motions. Here we show that both the growth/ energy associatedwith corotation, or the gravitational energy damping rate of unstable/stable interchange flow pertur- of the plasma, any such motions will be of interchange form bations and the equilibrium velocity of a steady flow are in- and likely to be spontaneouslydriven. versely proportional to the ionospheric conductivity. In cases In this paper we derive general formulae associated with where the system is unstable and a turbulent spectrum of interchangemotions and point out some unifying featuresof interchange motions develops, the form of the spectrum de- interchange processes.We give general expressions for the pends strongly on whether the free energy goes into plasma field-aligned current associatedwith interchange motions. We inertia or into frictional dissipation [cf. Hassam et al., 1986]. use these to examine the effect of the ionosphere on inter- FIELD-ALIGNED CURRENT AND INTERCHANGE MOTION change instability, to illustrate the relation between stable in- terchange configurations and the occurrenceof the convection Continuity of current governsinterchange flow patterns in a shielding phenomenon, to discussthe evolution of interchange magnetospheric system [Fejer, 1964; Swift, 1967a, b; Vasy- instability and to analyze the structure and development of iiunas, 1970]. In the ionosphere,current flow is directly related spontaneousinterchange motions. We close with a critique of to the [Wolf, 1974], which is quasi-static and also work on the Jovian problem. present in the magnetosphere.The two regions are coupled by the field-aligned current fed into the ionosphere,

•Also at DepartmentoI earth and SpaceSciences, University of div I = jll sinZ (1) California, Los Angeles. where I is the height-integrated horizontal ionospheric current Copyright 1989 by the American Geophysical Union. and ;• is the inclination of the magnetic field to the horizontal. Paper number 88JA03629. The field-alignedcurrent, J ll, acts as a sourcefor the iono- 0148-0227/89/88JA-03629 $05.00 spheric (and thus also magnetospheric) electric field [e.g.,

299 300 SOUTHWOOD AND KIVELSON: MAGNETOSPHERIC INTERCHANGE MOTIONS

Vasyliunas, 1970, 1972]. If the ionosphere is locally uniform • = (Sm) = --(1/q)(r3Km/r3fl)= --(1/q)r3K/r3fl and B is vertical, the current and electric field are related in (6) fl = (fl,,) = (1/q)(r3K,,/r3y) = (1/q)r3K/r3y terms of the height-integratedPedersen conductivity •;e by Here the angle brackets denote a bounce average and K is the div E = jli/Z e (2) bounce averaged Hamiltonian. It is convenient to expressf as The ionospherecan be treated as a two-dimensionalsystem. a functionof theadiabatic invariants/• and J, withJ = 2• ds We shall show that field-aligned currents associatedwith in- vii alongthe bounceorbit. (Note that our definitionsof/• and terchangemotions occur in balanced pairs. The most primi- J differ by factors proportional to rn from the customary defi- tive source is a two-dimensional dipole (i.e., oppositely direc- nitions). In the absenceof sourcesor losses,the particle distri- tedcurrents Iii flowinginto and out of theionosphere a hori- bution function for particlesof speciesi, fDt, J, •,/9, t) satisfies zontal distanced apart). The electric potential on the the Liouville equation in the form at (p, 0) (in locally horizontal ionosphericcoordinates aligned r3f•/r3t+ •(r3f•/c3y)+ ]J(r3f•/r3fl)= 0 with and centered on the current dipole) is then or I d cos 0 (I)--I1_•_.• (3) 2• E v p )f•/•t -- q- '(r3fdr3y)(r3K/r3fl)+ q- '(r3f•/r3fl)(r3K/r3y)= 0 (7) By analogy one can see how more complex distributionsof Our normalization convention for integration over velocity current "generate" electric fields. Inclusion of geometrical ef- (d3v)or overthe tt, J, •, fi coordinatesis fects and inhomogeneity does not change the principles behind this result. d•zdfl • d3vfi= d•z dfl ds vd• II •b dJ•

THE HAMILTONIAN APPROACH = dyaft • ni(s,t)= dyaft •i (8) We shall follow Northrop and Tellet's [1960] treatment of particle motion in a magnetospherelikefield configuration ['cf. where n• is the number densityof particlesof speciesi and % is Swift, 1967a]. We use the magnetic moment and longitudinal the bounce period; •i is the correspondingflux tube content adiabatic invariants, t• and J, as velocity space coordinates [Hill, 1984]. [Schulz and Lanzerotti, 1974]. The phases (gyro and bounce) Calculation of Field-Aliqned Current associated with each invariant are also regarded as coordi- nates. (In this paper all distributions are assumedto remain In the •, fl coordinate system the current density flowing in independent of these coordinates.) Like Northrop and Teller the y direction at any given point is given by [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 j•,•=Zi q• f d3v h•• (9) two-dimensional spatial coordinate system transverse to B. In (9),we have introduced the scale factors h• andh a, where They are related to the magnetic field by h,ha= B-•. The subscripts is neededto indicatethat Ja,m is proportional to the local value of the rate of change of the B = V• x Vfi (4) coordinate • at a position s on the flux tube. Evidently, the The coordinates are a canonical set and the system can be velocity may vary along the flux tube even though • is inde- described by a Hamiltonian as long as drifts of second order pendent of s. The current in the fl direction is defined analo- in the electric field are ignored [Northrop and Teller, 1960]. gously. The Hamiltonian of a particle of mass rn and charge q can be The divergenceof the perpendicular current density is written V.jz.•= F -• • Fj"• + F-• • Fja'• (10) Km -- mltB(am,tim, s) + (1/2 )m(v II)2 + qq)(O•m,tim)+ mW(o•m, tim, S) • h• •fi ha (5) where F is the Jacobian,B-•. The total parallel current per unit area into the where s is the coordinate along the field line, the canonical at the two ends of the flux tube is the integral of (10) along the coordinatesat s are %, and tim,ep and W are the electricand flux tube gravitational potentials, respectively, t• is the magnetic momentper unit mass, and vii = v ßBIB. In a rotatingsystem, the potential,W, can includea centrifugalpotential r2f•2/2, JH= --Bion• •-• • F--+h• F-• ••fi F (11) but, as noted above, we must assume that the system is rigid enough that corotation is well maintained. Note that (5) ex- where Bion is the magnetic field of the ionosphere and we cludes the possibility of parallel electric fields, as the elec- neglect inclination. Using (8) and (9) in (11), we obtain trostatic potential q>is independentof s. Field-aligned currents associatedwith interchangemotion could themselvesgive rise to parallel potential drops in regions where mobility ) is inhibited. We ignore this possibility in this paper. where(•)• and(fl•)• arebounce-averaged values. Interchange motions take place slowly compared to particle The bounce-averageddrifts include the electric field drift, bounce motion. Bounce-averagedequations of particle motion which is independent of energy and mass and charge and are given by cannot lead to charge accumulation in a charge-neutral SOUTHWOOD AND KIVELSON' MAGNETOSPHERICINTERCHANGE MOTIONS 301 plasma. We exclude thesedrifts by introducing a partial Ham- If the integration over # and J in (16) is replaced by a iltonian, H, such that similar integration over velocityspace and the field line length and with somealgebraic manipulation, one recoversthe famil- H = K -- q(1) (13) iar expressiongiven by Vasyliunas[1970] Henceforth, we shall take the summation over speciesas un- Vp. (VV x Be)=j, (19) derstood and drop the subscripts i and b. By use of (6), the form above is where Be is the field at the equator.

Application to Stability of Interchange Motions Let us now use our result to examine the effect of the iono- sphereon small perturbationsin interchangemotions. Consid- er the identity

V. 00) =j- VO + OV.j (20) or, if we convert to ordinary spatial gradients, Integrating this expressionover the ionosphericsection (dS ds) Jll=Bio- f d#dJ [ ¾ x¾f n].•/B (16)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 bounce- where we have ignored the inclination of the ionosphericfield. averaged quantities, the flux of ¾f x ¾H must be along B. If The last integral in (21) is taken at the top of the ionosphere.I the systemis symmetric about the magnetic equator, the field- is the height-integratedcurrent density. Substituting the ex- aligned flux of ¾f x ¾H into either ionosphere must vanish. pressionfor field-aligned current into (21), From (16), this guarantees that any current into the iono- sphere is balanced by a return flow elsewhere.It follows that the simplest electric field source is a current dipole and any fon dSIøE-- --•mag dS•)aiønf d•ldJ more complex sourcesmay be constructedfrom distributions Of OH Of 0n of dipoles. Equation (16) can be related to other forms (see,for exam- ß ple, appendix to Hasegawa and Sato [1979]) by recognizing where the integral on the left (right) side is over the iono- that the derivatives on H (see equations (5) and (6)) corre- spheric (magnetospheric)section of the flux tube. As the inte- spond to grad B, curvature,centrifugal, and gravitational drift. gral on the magnetosphericside of the ionosphereis expressed The expressioncontains no term correspondingto the parallel in terms of quantities that are independent of s, it may be vorticity term given by Hasegawa and Sato because of the evaluated at any point along the magnetosphericportion of quasi-static assumption and the requirement to neglect the flux tube. second-order terms in the electric field. Now let us introduce a small perturbation in electric poten- Further simplification of the expressionis possibleif H has tial, (5(I).For convenience, we shall assume that the equilibri- particular properties. In a fast-rotating system like Jupiter's, um distribution is a function of the coordinate, y, alone and the dominant term in H for cold plasma is the energy indepen- that the time variation is representedby exp (7t). In the linear dent centrifugal potential T. Plasma is confined to a thin approximation, the perturbation produces a dependenceon •. near-equatorial sheet where the potential reachesits low point The perturbed distribution, 6f, is given by a linearization of on the field line. T may be taken outside the integral and the equation (7): integral simplifies to the expression for the current per unit area of the flux tube used by Hill and coworkers [Hill et al., 7(sf-- •f 0(500(5f q _• OH (23) 1981; Pontius et al., 1986]

j, = m[Vr/x V•] ßB (17) For the moment, we shall assume that the convective time Jaggi and Wolf [1973] considered a monoenergetic 90 ø scaleof the perturbation of the • coordinate is such that pitch angle distribution convecting under grad B and electric 7 >>fla(O/Ofl) (24) field drift alone. The dominant term in H is/•B, and Jaggi and Wolfs expressionfor j, (similarin formto (17))is obtainedby where/Cais the bounceaverage of the nonelectricdrift. Recall- taking the Hamiltonian outside the integral. ing that ]•a• 0H/0•, we may drop the secondterm on the One is not restricted to the use of the particular invariants, right-hand side of (23): /• 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 7 (sf-- (25) function of energy, W, alone. The adiabatic invariants,/• and J, are not conservedbut there is a single equivalent invariant The effect of magnetosphere-ionospherecoupling is found given by by using (22) to secondorder' WV 2/3= const (18) (26) where V is the flux tube volume [cfi Harel et al., 1981]. fondS(5''(sE--'Bion•magdS(sfI)•d•tldJk•-ff]k-•-o•/ 302 SOUTHWOODAND KIVELSON.'MAGNETOSPHERIC INTERCHANGE MOTIONS

We integrate by parts and substitutefor gffrom (25): where Z v is the height-integratedPedersen conductivity of the ionosphere. Then c•f c•H ß = dS dit dJ • • (27) n • ag •-•I The sign of y determinesthe stability of the system.A neces- sary and sufficient condition for stability is

--• < o (28) f d#dJOf OH (31) Observational tests of stability are complicated by the fact The growth rate is largest for perturbations with the vari- that detectorsdo not provide distribution directly as functions ations g• in the ot direction much smaller than variations in of It and J. Nonetheless,the condition has been testedfor the the • direction, and for suchperturbations the first term in the hot plasma (ring current and radiation belts) at Earth [e.g., bracketsin the denominatorof (31) can be dropped. Nakada et al., 1965] and Jupiter (e.g., Thomsenet al. [1977] We shall assumethat the scaleof the unstable perturbation who publishedthe relevant resultsin a study of particle diffu- is short compared with the scales of variations of the back- sion).(The dominanceof gravitational and centrifugalterms in ground system, i.e., the cold plasma Hamiltonian can lead to instability in the outer magnetosphere[Lemaire, 1987, and referencestherein], K (32) as we remarked earlier. Richmond [1973] discussesthe re- f ho•do•<< (h•)- lationship between the hot and cold plasmas when one distri- and bution 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 and (33) (h•)- •(&5•/3•) Kivelson [1987]. For example, where centrifugal forces domi- f h•do• << nate, the Hamiltonian H is independent of particle energy and (34) is a monotonically decreasing function of radial distance. f ht•d• << 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 lion•iondo• dp simplesufficient condition for stability, equivalentin our nota- (35) tion to The quantities d•xdfi and c980/Ofiare independentof position c•f c•H ••<0 alongthe flux tube, and h•hts = B- •, sothe growth rate is c• c•- or, regardingf as a function of otthrough H, 7=[fd.dJc•'•t3ft•H 0'-•' Z•,-X](B•)(•) Bio n ho (36, As expected,the growth rate, y, is inversely proportional to af-• IIt,./ < 0 (29)the localionospheric Pedersen conductivity, 2;•,. Now the de- rivative c•H/c• is proportional to the nonelectricdrift in the fi (Taylor also gives necessaryand sufficient conditions for sta- direction (from equation (6)). The sign of the expressionfor V bility for the case where the equilibrium distribution depends depends on the sign of this drift, which will be made up in on both 0tand fl.) general of magnetic components(grad B and curvature) and Taylor's calculations were performed for a plasma con- gravitational and centrifugal drifts. As drifts may be energy tained in a toroidal device and no obvious counterpart of the dependent,the contribution from the term, c•H/c•, may differ ionospherewas included in his calculations.Although our cal- in different parts of the distribution. For example, in the outer culation specificallytakes account of the ionosphere,the iono- Io torus, where ring current impoundment of outward inter- spheric parameters do not enter the conditions (28) and (29). change diffusion of torus has been proposed [Siscoe et al., The ionosphere has no effect on the stability condition but 1981], the Hamiltonian, H, increasesinward for the ring cur- does control the speed with which a perturbation evolves in rent (hot) particles and outward for the torus plasma. One can either stable or unstable systems,as we show next. rewriteOf/t3• IIt,s INSTABILITY GROWTH RATE c•f/&xlIt,, = c•f/&xIv+ q- • c•H/&xlIt,, c•f/c•H Ir (37) Equation (27) can be used to derive an estimate of the and introduce a critical spatial gradient, c•f/&xIt, for which the growthrate for instability.Set dS = do•d• h•ht•,and note that systemis marginally stable, namely, t•f/00•Ic q- q- I •H/•O•lu,s t3f/t3H Ir = 0 6I.tSE = Zv(tSE) 2= Z•, c30ttS• + c3fl Thus

(30) c•f/3o•IIt.s = c•f/3o• Iv - c•f/3o•Ic (38) SOUTHWOODAND KIVELSON:MAGNETOSPHERIC INTERCHANGE MOTIONS 303

Instability proceedsat a rate proportional to the amount by pens in the jovian system (see, for example, Pontius et al. which the actual gradient differs from the local critical gradi- [1986]; Hill and Liu [1987]; Richardsonand McNutt [1987]). ent. For example, for a case mentioned earlier, the critical The central problem is the outward transport of iogenic gradientin a distributionmaintained isotropic corresponds to plasma. The high rotation rate of the Jovian system means the gradient of the "adiabatic" distribution in which the pres- that the dominant term in the Hamiltonian is the centrifugal sure satisfies energy. Convective transport in the absence of significant losses V(p V 5/3)= 0 (39) conservesflux tube content yet the flux tube content in the However, in general one must recall that the critical gradient region from 6 to 12 Rj falls nearly monotonically[Bagenal may be in a different direction at different energiesin a distri- and Sullivan, 1981; Richardson and Siscoe, 1981]. Thus it bution. seemsunlikely that a simple two cell convectionpattern [Hill et al., 1981] is present. Convection patterns with short azi-

SHIELDING IN DRIVEN INTERCHANGE MOTIONS muthal wavelengthsalso appear inconsistentwith data. The magnetic flux must be conservedin the process,so density Interchange motions appear spontaneously in a plasma depleted tubes 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 gradients are small enough, perturbationsdamp rather than expectedto vary greatly betweensuccessive measurements de- grow. This is the basisof the convectionshielding effect in the pending on whether the measurementwas made in an out- terrestrial 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 nat- measurementstaken about 0.24 s apart (effectivelyseparated urally stable against interchangeinstability. Consider a driven by • 20 km) vary by lessthan 10%. More complicatedmodels convectionsystem in a magnetospherewhere plasma is trans- [e.g., Hill and Liu, 1987] introducestreamlines that reducethe ported inward from a boundary we take to be at large •. radial convectionvelocity, thus increasingthe time over which Wherelosses occur the distribution must satisfy (•f/c•)•,s > 0; loss can reduce the flux tube content as a function of radial otherwise,(•,f/t3•)•,s = 0 in regionswhere there are no sinksor distance. Loss mechanisms are not identified in this work. sources. The alternative diffusive transport model is consistentwith Consider now an enhancement of interchange flow due to the presenceof negative radial gradients of flux tube content an enhancement of the driving source.Throughout the system, and lack of variability between successivedensity measure- the convection field immediately increases.The increase is an ments. A turbulent uncorrelated overturning motion is envis- interchange perturbation of the system, and wherever aged with a short scale length such that particles execute a (•f/•)•,,s > 0 the perturbationwill damp.Hence we havean random walk. Interchange instability driven by the dense alternative way of understanding the convection shielding torus plasma has been proposed as the source of such mo- effect discussedby Vasyliunas[1972], Jaggi and Wolf[1973], tions. Southwood[1977], Harel et al. [1981], Wolf et al. [1982], first We point out below that the instability of torus plasmadoes pointed out by Wolf [1983]. These authors showedthat cur- not directly lead to random fields. For diffusion to drive the rents set up near the inner edge of a ring current type distri- outward flow, contributions of first order in the perturbation bution reduce substantially the penetration of solar wind in- velocity (proportional to ,Srl,Sv) must average out for random duced convection to the inner magnetosphere. displacements.However, for the conditionsof the Io torus, a The critical shielding time estimates, given by Jaggi and correlationexists between 6r/ and 6v • E,/B • 6rl (seeequa- Wolf [1973] and Southwood[1977] for the onset of screening tion (40), below), which produces systematic outflow terms from low latitudes at the (sharp) inner edge of a hot plasma linear in the perturbation velocity.In the simplestconceivable distribution are of order 1/7. Here, we have shown explicitly instability conditions the outflow speed of a single tube is that any local enhancement of earthward convection is linear in time t and such outflow would be expected to domi- damped proportional to the difference between the distri- nate over the contributions of second order in 6v which pro- bution gradient and the gradient locally required for marginal ducethe t•/2-dependentdiffusive outflow. interchange stability. The damping is exponential and the We establishedearlier that the simplestmagnetospheric cur- damping time is proportional to the local ionosphericPed- rent source distribution corresponds to an electric dipole ersen conductivity. The shieldingeffect results when the en- sourcefor the ionosphere.The idealized particle distribution hanced flow continues to be driven from higher latitude tubes; correspondingto such source is a very localized (point) in- the radial flow is damped out on the shielding time scale and creasein density above the background.In Figure 1, we illus- an azimuthal component developsto maintain continuity. trate the field produced by an isolated enhancementand an Characteristic shielding times for the dayside of the mag- isolated depressionof plasma density. On the flanks of each netosphereare of order of hours. If mean ring current ion region flow a balanced pair of field-aligned currents. In the energiesare near 10 keV at a characteristicdistance of 6 R e, n ionospherethe currentsact as a dipolar sourceof electricfield. is approximately0.1 cm-3, and with 5;--5 Mho, and If the alignmentof the contoursof K or H are independentof • ds/B- 0.05RE/nT, one finds 7- • of order2 hours. energy, the dipole is simply aligned with the contours. As illustrated in the figure, a depressionin density would produce SPONTANEOUS INTERCHANGE MOTIONS a dipole oriented in the oppositesense. The flow lines associ- IN THE JOVIAN MAGNETOSPHERE ated with the enhancementand depressionare shown.There is The form of interchange motion in the Earth's mag- flow outside the enhancement and within the enhancement netosphere as discussedabove is generally accepted [Wolf, itselfi The plasma in the enhancementmoves toward lower 1983], but there remains controversy concerning what hap- values of H and losesenergy. Field patterns are reversedfor a 304 SOUTHWOODAND KIVELSON: MAGNETOSPHERIC INTERCHANGE MOTIONS

VK

Fig.1. Anisolated enhancement ofdensity (dark shading) and an isolated depression ofdensity (light shading) in a uniformplasma in thepresence ofa gradientof theHamiltonian K or H. Thedipolelike nature of the current system linkingthe perturbed flux tubes with the ionosphere is illustrated by charges on their boundaries. Dark curves show flow patterns,and directionsare shownby arrows.Note that the surroundinguniform plasma is setinto motionas the perturbedflux tubesmove toward equilibrium positions.

depressionand particlesin a depressionmove to higher return flow in all other sectors. The motion is reminiscent of energy.In both casesa net amount of energyis released.In the convectionsystem in the Earth'smagnetosphere but with the caseof the depression,energy is lost by the fuller tubesof theplasma corotating with the planet to a firstapproximation. particlesdisplaced by the motion of the depression.These The inwardmotion serves to returnflux and would also inject examplesare close to the idealizeddescription of the inter- hot plasma.The energy to sustainthe inward motion is fed changewherein the energychange associated with tubesinter- from the regionswhere energy is releasedin the regionof changingposition is calculatedwithout attempting to specifya outward flow. A global current systemis set up in which motionthat wouldaccomplish the change(see, for example, field-alignedcurrents flowing on the flanks of the outward Cheng [1985]). moving plasma feed the ionospheric Pedersen currents The electricfield acrossthe tube is directlyproportional to throughoutthe convectionsystem. There would be clear re- the enhancementof densityand can be calculatedby balanc- gionswhere plasma is movingtoward the planetand regions ing the excessmagnetospheric current with the associated wherethe plasmais movingaway with very differentpopu- Pedersencurrent in the ionosphere.For a coldcorotating disk lationsin the two regions. with W = r2•2/2, the relation betweenthe azimuthal electric Hill et al. [1981] (seealso Hill andLiu [1987])propose that field,E,, and a densityenhancement follows from (17) with the flow actuallypenetrates inside the Io torussource region. Jll = -V ß•EpE. If gradientsin/• (or •b)are dominant, Our sketchdoes not showthis; the flow is likely to be con- E•,= •rI r•2/•, (40) fined to the torus sourceand outside.The steepoutward gradientof plasmawould produceeffective convection shield- [cf. Hill et al., 1981], where •r/is the enhancementof the flux ing as the innerboundary is very interchangestable. A conse- tubecontent over the backgroundtube content, r/c, r is radial quenceis thatlarge-scale convective flow could not berespon- distance,fl is the angularvelocity. Outside the tube there is a siblefor thelongitudinal asymmetry in atmospherichydrogen returnflow movingthe surroundingplasma out of the path of density[Sandel et al., 1980]despite proposals to thecontrary the enhancement;the fieldis that of a dipolewith strength,•/ [Dessleret al., 1981]. Arfl2/Y•,where A isthe area of thetube in theionosphere. Hill e• al. [1981] use(40) as an estimateof the outward flow Now let us considerthe globalpattern of interchangeflow. speedin theoutward moving flow sector, where r/c is themean The simplestis one wherethe field has the two cell pattern flux tube densityon outwardand inward movingtubes. It exhibitedby our first example.Hill et al. [1981] have de- clearlyserves as a guide but, as the figure itselfindicates, scribeda corotatingtwo cell convectionsystem like that illus- purely radial flow is unlikely. Any azimuthal variation in trated in Figure 2. There is a preferredlongitude for plasma sourcerate will giverise to azimuthalcomponents of flow. production and outward flow lines originate there. There is 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 drifting fast enough that they move more than a wavelengthin source regions flux tube content should not vary along a growth time is reduced becausethey have spent part of the streamlinesin outward and inward flow regimes. Steep radial time accelerating and part decelerating. However, this situ- gradients detected at the outer edge of the Io torus suggest ation does not apply for the Io torus as there is a more ener- that simple radial flow cannot be occurring there. Siscoe and getic population of ring current whose outward pressure coworkers have favored a diffusive transport model where in- gradient is stabilizing [Siscoe eta!., 1981]. When this is so, terchange instability causesmulticell convection patterns. In- perturbations with wavelengths short enough that the fast terchange instability could take place on a scale short com- particles are unaffectedby the perturbation are more unstable pared with the region over which plasma is injected and this is [Richmond, 1973] and thus dominate. The shortest wavelength likely where production rates vary little with longitude. In excited will be of order the Larmor radius of the unstable Figure 3 we illustrate one possibilitywhere a short wavelength particles and there will be some intermediate wavelength instability takes place. A similar, more sophisticatedmodel in where the growth rate peaks. Such a situation pertains in the which departures from corotation are allowed has been pre- outer Io torus where an unstable inward gradient of cold sentedby Summersand Siscoe[1986]. plasma coexistswith a steep stabilizing outward gradient of For the pattern envisaged in Figure 3 to be maintained, hot plasma. perturbationswith a particular wavelengthshould be energet- There may indeed be particular perpendicular wavelengths ically favored. Although the illustrated steady flow pattern is favored for jovian interchange motions and further investi- the 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. Sourcedistri- various 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 over- bation, 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., see inequalities for outward and inward motion are short there may be actu- (32)-34)). ally a multiplicity of scalesin the convection pattern. Inserting the spacedependent terms in the instability calcu- Consider 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 fluctu- easily 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 306 SOUTHWOODANDKIVELSON: MAGNETOSPHERIC INTERCHANGE MOTIONS

SINK Fig.3. Thisflow pattern isalso possible forthe Io torusand corresponds tothe presence ofa favored wavelength for growthofan interchange instability. Shadings correspond tothose of Figure 2,and a sinkregion isalso present. azimuthalvariations in distributiongive rise to parallelcur- movingplasma intermingle. Scale sizes correspond to the rentsand radial motions.The resultinginterchange motion randomfluctuations in source.Whether a givensection of will be filamentedand the convectionpattern will be like that plasmamoves outward or inwarddepends on whetherit con- illustratedin Figure 4, wherepatches of inward and outward tainsmore or lessplasma than the local mean. The flow pat-

density ...... ••:*:?'"'"'•:...... ß•..:., ....a.."'"'":'•':':"?:'" ...... •,.,.-.,i; "'""" •.,,t • density Fig.4. Detailview of another flow pattern that could be produced bya temporallyvarying plasma source in theIo torus.A grayscale represents levels of flux tube content in a uniformbackground plasma. The patterns and scales are meantto berandom to reflectvariability ofthe source. Over extended time intervals, regions ofhigh flux tube content moveout and regions of depressedflux tube content move in, butlocally the motions reflect the local variations of flux tube content. Streamlines evolve with time. SOUTHWOODAND KIVELSON.'MAGNETOSPHERIC INTERCHANGE MOTIONS 307 tern shown in Figure 4 is a snapshot.The spatial structure of Acknowledgments. This work was supported by the Division 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 referees for 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. ficient. 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