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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
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