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Magnetospheric Interchange Motions 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 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 plasma. Neither a large-scaleconvection nor other forms of unstableinterchange overturning appear adequate to explain the plasma distributions detectedat Jupiter. 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 plasmasphere, believed to govern transport in other magnetospheres.Inter- gravity and centrifugal force 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 magnetic field and maire, 1974]. However, global terrestrial interchange motions planetary ionospheric conductivity are sufficiently strong to have to be driven by momentum transfer from the solar wind. 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 ionosphere 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 electric field [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 field line 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
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