GEOPHYSICAL
RESEARCH
LETTERS, VOL. 16, NO. 7, PAGES763-766,
THE
EFFECT
OF MASS
LOADING
ON THE
TEMPERATURE
JULY 1989
OF A FLOWING
PLASMA
JonA. Linker1'2, MargaretG. Kivelson
1 and RaymondJ. Walker
Institute of Geophysicsand Planetary Physics,University of California, Los Angeles
1987; MeGruth and Johnson1987]. The torus plasma
flowsat 57 km/s relative to Io, so on average,the neu-
Abstract. We have investigated how the addition of
ions at rest (massloading) affectsthe temperatureof a
flowingplasmain a magnetohydrodynamic
(MHD) approximation,using analytic theory and time dependent,
trals can be assumed to be at rest relative
to Io.
With
that assumption,the normalized MHD equations can be
three-dimensionalMHD simulations of plasma flow past
Io. The MHD equations show that the temperature can
increaseor decreaserelative to the background,depend-
written
in the form:
Op
ing on the local sonicMuchnumber(Ms) of the flow. For
0t
+ V-(pS') - •.
(1)
flows
with
Ms
>ma%/•
(when
7=However,
5/3),
mass
increases
the
plus
temperature.
the loading
simulaP+-5-
tions show a non-linear responseto the addition of mass.
If the massloading rate is large enough,the temperature
increasemay be smaller than expected, or the temperature may actually decrease,becausea large massloading
rate slowsthe flow and decreasesthe thermal energy of
the newly created plasma.
-0
(2)
(3)
d-•
pv(riff+-- d(pp-V)_
7-1
OB
Ot Vx(¾x
•) - -Vx(•TJ) (4)
Introduction
wherep is the density,•' the velocity,P the plasmapressure,B the magneticfield, r/the resistivity,and J - V x B
the current density. /3sis the massadded per unit volume
per unit time. We take 7 - 5/3. (Viscousterms are
The term "massloading" refersto the addition of mass
to a flowingplasmaby ionizationof neutrals. Io, Jupiter's
innermost Galilean satelhte, is embedded in a flowing
plasma(the Io torus) and is alsosurroundedby a cloud
of neutral atoms and molecules[Brownet al., 1983, and
references
therein]. In this paper we presentresultsfrom
addedto (2) and (3)in the simulationrunsfor numerical
stability purposes,but they are irrelevant to the present
discussion.)
Equation(3) canbe derivedin a straightforwardmanner,using(1) and an equationfor the conservation of energy(notethat d/dr is a convective
derivative)
[Linker, 1987]. Becausewe considerthat ionizationof
neutrals results from electron-impact,there should also
be a heat lossterm in (3) representingthe energygiven
up by the electrons.However,the flow energygainedby
an oxygenion createdin the backgroundflow (280 eV)
is largecomparedto the energylost by the electron(1012 eV). The inclusionof the electronenergylossterm did
not alter the simulationresultssignificantly;therefore,we
three-dimensionalMHD simulations of plasma flow past
Io, including the effect of ionization of neutrals in the
equations. Ionization may modify the plasma temperature significantly.We demonstratethese effectsby examining the MHD equation for temperature and the results
of simulation runs for three different values of Ms.
Mass loading plays an important role not only in Io's
interaction with the plasma torus, but also in the solar
wind interaction with comets and with Venus, in plasma
flow past Saturn's satellite Titan, and possibly in other
plasma-satellite interactions. Although the simulation
parameters we use in this paper correspondmost closely
to the Io torus parameter regime, our remarks regard-
have ignored the term.
ing the plasmatemperature(in an MHD approximation)
apply to the plasma in these other systemsas well.
From(1-3) we can alsoderiveequationsfor the plasma
velocity and plasma pressuresimilar to those used by
Schmidtand Wegmann[1982]and Oginoet al. [1988]'
p•-- -V-((P+
•) I • -
The MHD Equations and the Plasma Temperature
We use the time-dependent MHD equations,including
the effect of neutrals ionized near Io. When Jupiter's
gravity is accountedfor, the escapevelocityfor neutrals
from the surfaceof Io is about 2.3 km/s [Linkeret al.,
1985]and the neutral densityin Io's exosphereis dom-
OP
Ot
(5)
+V.(P¾)
- (7- 1)(- P(V.5')
+
inated by lower velocity neutrals on non-escapetrajectories [Watson,1981; Sievekaand Johnson,1984; Linker
(6)
We seefrom (1), (5), and (6) that massloadingcauses
increasesin the plasma density and pressure, and de•Also at Departmentof Earth and SpaceSciences, creasesin the plasma velocity. In the context of the Io
torus, it has been noted that mass loading can increase
University of California, Los Angeles.
the plasmatemperature[e.g.,Goertz, 1980;Brownet al.,
1983; Shemhusky,1988]. However.if we assumean ideal
gaslaw (P = pRT), (6) impliesthat the plasmatemper-
2Nowat PhysicsDepartment,Universityof California, Irvine.
Copyright
1989 by the American Geophysical
ature, T, satisfies
Union.
dt: (7- 1) p.v
pR
Paper number 89GL01160.
0094-8276/89/89GL-01160
$03.00
763
-•'
p
(7)
76•4
Linkeret al.' The Effectof MassLoadingon Temperature
If we neglect effectson the plasma velocity, then, rela-
tive to caseswith no massloading,ionizationaddsto (7)
oneterm that decreases
the temperature()sT/p) andone
term
thatincreases
it ((v•)b,v
paa)' Theformer
termarises
becausewhen massis addedto a fluid element,the thermal energyinitially presentmust be sharedby a larger
numberof ions, and this decreasesthe local temperature.
The latter
term arises because when an ion is added to
the fluid elementand is acceleratedup to the background
flowspeed,someflowenergyof the plasmais convertedto
thermalenergy.This increasesthe thermalenergyin the
fluid element.If the flow energyof the plasmais greater
than the thermal energy,the plasmawill be heated by
massloading. Conversely,if the flow energyis lessthan
the thermal energy, the plasma cools. The ratio of the
_
5 )2
heating
term
tothecooling
term
is•(•-•(Ms)•
- •(Ms
sult of ionization. We note that assumptionsabout the
thermaleffectsof ionizationare an importantcomponent
of studiesof the energybalanceof the Io torus[e.g.,Barbosaet al., 1983; Smith and Strobel, 1985; Shemansky,
1988].
Simulation
Results
Fromour analysisof (7), we wouldexpectthat when
theflowis subsonic
or transonic,
massloadingwouldcool
theplasma,
whereas
whenMs > X/r•-/5,
mass
loading
would heat the plasma. In this section we examine the
resultsfor simulations
run in thesedifferentparameter
regimes.
Oursimulation
codesolves
equations
(1-4)asan
initialvalueproblem
in spherical
coordinates
usinga twostepLax-Wendroff
finitedifference
scheme
[Linkeret al.
1988].Wenotethat fortheresults
shown
in thispaper,
thefluidReynolds
number
Re= 50andRm,themagnetic
pect heating by massloading to occur only if this ratio
Reynolds
number(V•tL/rl, whereV•tis the Alfvdnspeed
is 100. All
isgreater
than1, orMs > X/•-/5.Aswehaveassumed andthe lengthscaleL is an Io diameter),
where Ms is the sonic Much number.
One would ex-
minimal effectsof massloading on the flow velocity,the
scaling for thermal effects that we have derived here is
only approximate. We note that the thermal effects are
not directlydependent
on/3 - 2P/B 2. /3 describes
the
relativeimportanceof thermal energyand magneticenergy,whereasthe effectsof massloadingon the plasma
temperature depend on the relative importance of thermal and flow energieslocally in the plasma.
resultsare shownin Io's restframe,after the simulation
hasreacheda quasi-steady
state.
For electronimpact ionization and a constantelectron
temperature,the massloadingrate )s is proportionalto
the neutral density. We assumethat the neutral density is sphericallysymmetricaboutIo, and we consider
twotypesof radialvariationsfor the neutraldensity.For
mostof the runspresented,
the neutraldensityfallsoff
The Ms in the Io torus is uncertain; Ms - 1.8 for oxygen ions and 2.5 for sulfur ions in the thermal plasma, as-
like1/r 2. We alsoconsider
a radialvariationbasedon a
modelof Io's exosphere
[Linker,1987].We parameterize
suminga backgroundtemperatureof 100 eV [Bagenalet
al., 1985]. However,the flux of ionswith energies> 6 keV
ionizations
per second
occurring
within6 Io radii(R_•o)
has not been measured,and their contributionto the pressureis probably significant. Thus, Ms is probably smaller
than the abovevalues. For example, assumingan average
massnumber of 20, if all of the ions > 6 keV gavethe same
contributionto the plasmapressureas a 1% populationof
30 keV ions, Ms would be 1. Even a 1% population of 10
keV ions would make Ms - 1.4. In this paper we consider
resultsfor simulationswith Ms - 0.55, 1, and 1.5. These
values allow us to examine the effects of mass loading on
plasma temperature for parameter regimesranging from
subsonicto supersonic,and the values are closeenough
to the torus parameter regime to permit useful comparison. Simulationsat higher/• presentedfewer numerical
difiqcultiesand were lessexpensiveto perform, so we used
valuesof the Alfvdn Much number(M•t) of 0.5 and 0.7,
the massloadingmodelsin terms of the total numberof
of Io (roughlythe dimension
of Io's exosphere).
Voyager
observations
limit the ionizationrate of sulfurand oxygen
atomsnearIo to < 1027s
-1 [Shemansky,
1980],butthere
is no limit on the ionizationrate of molecularspecies.Accordingly we consider models with ionization rates both
lessthan and greaterthan 1027s
-1. We note that becausethe electronimpact ionization processreducesthe
electrontemperature,it is not clearthat the torusplasma
couldactually supportthe highestionizationrate that we
consider
here(6 x 1027s-1).A dropin theelectron
temperature near Io that would reduce ionization efficiency
hasbeenreported[Sittlerand Strobel,1987].
Figure 1 showsthe plasmatemperaturein the xz plane,
representedas a surface,for two different simulation runs.
higher than those that actually occur in the Io torus.
The MHD equationsobviouslydo not describeall of the
important phenomena associatedwith the creation of new
ions in a plasma. However, they do give a good description of the global effectsof ionization on the flow. We note
that by using the MHD equations,we make the implicit
assumptionthat when ionization occurs,the ion is introducedinto a volumeelementin thermal equilibriumwith
the fluid containedin it. The assumptionseemsreasonable for the Io torus, where the velocity of the plasmais
nearly perpendicularto the magneticfield, and the ¾x B
electric field acceleratesnewly created ions to the local
flow velocity.Even for flow parallel to the magneticfield,
Omidi and Winske [1987]have shownthat instabilities
driven in the plasma can causethe ions to be pickedup
by the backgroundflow (althoughthe processis not so
efiqcientas in the caseof flow perpendicularto the magnetic field). Plasmainstabilitiescan alsohelp to bring
about thermal equilibrium,but in generalthe plasmadistribution is likely to be more complicatedthan can be
describedby a singletemperaturefor all plasmaspecies.
Nevertheless,a nominal plasma temperature is usefulfor
describingthe net thermal energygainedor lost as a re-
54•
Ca)
....................
s•Cb)
I-3t
-20
-10
4
0
10
Z Axis Cflow-'-)
20
-20
-10
0
10
20
Z Axis Cflow-,-)
Fig. 1. (a) A surfacerepresentationof the plasmatemperaturein the xz plane for a simulationwith M•t = 0.5,
Ms = 0.55 and no massloading. The xz plane is perpendicularto the initial magneticfield. The flow directionis
from left to right. Io is in the centerof the plot and has a
radiusof 2 in the unitsshown.(b) The sameas (a) but
for a simulation
with 6 x 1026ionizations
s-• occurring
within6 R.•oof Io, theionizationfallingoffas1/r 2. Mass
loadingdecreasesthe plasmatemperaturein the wake.
Linkeret al.' The Effect of MassLoadingon Temperature
765
The xz plane cuts through Io's equatorial plane, perpendicular to the initial y-aligned magneticfield and parallel
to the initial z-alignedflow in the simulation(Figure 1
of Linker et al., [1988]depictsthe simulationcoordinate
system). A temperaturevalue of 1 in Figures1, 2, and
4 corresponds
to 670 eV (an averagemassnumberof 20
is assumed).Both runs shownin Figure 1 initially had
MA -- 0.5, and Ms - 0.55 and fl - 1. Figure la is for a
casewith no massloading, and lb is for a casewith a total
of 6 x 1026s
-1 (ionizations
persecond)
anda 1/r 2 fall off
of the ionizationrate. The plasmatemperatureincreases
slightly in Io's wake for the case with no mass loading;
this resultsfrom the slowmode compressiondescribedby
Linker et al. [1988]. Figure lb showspreciselythe be-
havioranticipatedfrom our examinationof (7). When a
smallmassloading rate is introducedinto a subsonicflow,
the plasmatemperature decreasesslightlyin the wake relative to the case with no massloading. Figure 2a shows
the results for a simulation with the same parameters as
the cases
shownin Figure1, but with a total of 6 x 1027
ionizations
s-1. Forthiscasethe decrease
of plasmatemperature in the wake is much larger.
Figure 2b showsthe plasma temperature for a simulation with the same mass loading rate as for the case in
Figure 2a, but with MA -- 0.5, Ms - 1; and fl - 0.3.
This parameter choice for the plasma is probably closer
to the actual torus plasma than the parameters used in
Figures la, lb, and 2a. We find that as in the case of
subsonicflow, massloading in a transonicflow decreases
the plasma temperature in the wake.
The simulationsresults shown in Figures 1 and 2 are
qualitatively consistentwith our expectations based on
(7). However,the decreasein the plasmatemperature
becomesespeciallydramatic as the massloading rate increases. We can interpret the dramatic decreaseby examining the effect of massloading on the plasma velocity.
Figure 3a showsthe plasma velocity in the xz plane for the
samesimulationasFigurela (no massloading).The flow
velocity is representedwith vectors, for a portion of the
total simulationregion. Figure 3b showsthe samedata as
Figure3a, but for the parametersof Figure2a (highmass
loadingrate). Comparisonof the two figuresshowsthat
the plasma velocity in Io's wake diminishes in the presenceof a large massloadingrate. The result suggeststhat
the decreaseof plasma temperature can be a non-linear
process.Initially massis added to the plasma which slows
the flow and reducesMs locally. Slowly flowingfluid elementsremain longer in regionsof high massloading rates,
becomingnearly stagnant. As massis added in the stag-
0
s•
4•Ca)
"-•l•[••••
• ....
2•
-4
...... ,•'.'.•.•
'•
4
6
8
0
Z Axis (flow-.-)
2
4
6
8
Z Axis (flow-*-)
Fig. 3. (a) Plasmavelocityvectorsin the xz planeof the
MA -- 0.5, Ms - 0.55 simulationwith no massloading.
A portion of the simulation downstream of Io is shown.
The length of each vector is proportional to the speedof
the flow at a given point. Io is marked as a circle and has
a radiusof 2 in the unitsshown.(b) The sameas(a) but
for the simulation with 6 x 1027 ionizations s-•. The flow
velocity is greatly decreased,relative to the casewith no
massloading, particularly in the wake.
ducesthe local value of the soundspeed,and this acts to
increase Ms.
Thus far we have examinedcaseswheremassloading
led to coolingof the plasma. We now examinewhat hap-
penswhenweruncases
withMs > Vrff/5. Figure4
showsthe plasmatemperaturein the sameformat as Fig-
ures I and 2, but for simulations with MA -- 0.7 and
Ms - 1.5;fl - 0.26. For Figure4a, 6 x 1026ionizations
s-• wasassumed.
In thissimulation,
mildheatingof the
plasmaoccursin Io's wake region,as we would expectfor
Ms > V/•/5. Figure4bshows
results
froma simulation
with 5 x 1027ionizations
s-• andotherparameters
the
sameas in Figure 4a. Even though the backgroundMs
is large enoughfor massloading to heat the plasma, the
plasmahas cooledin the wake region. Once again, the
explanationof the coolingis apparent from examination
of the flowvelocity(similarto the resultsshownin Figure
3b). The highionizationrate greatlyreducesthe plasma
flowvelocityin the wake,sothat the plasmaflowenergy
in the wake dropsbelow the thermal energy. As more
massis addedto the regionsof reducedflow(primarilyin
the wakeregion),the plasmacools.
nated(lowMs) regionsof the flow,the plasmacoolseven
further. Eventually the plasma temperature reaches a
steady state, because reduction of the temperature re-
2
•
1-•
Ca)
......
2:-•---•'].:•.'-:.:..•
....
-20
-10
0
10
Z Axis Cflow-"-)
20
0
'1•)
20
Z Axis Cflow--)
Fig. 4. (a) The sameasFigurel(a) but for a simulation
z••
/,,'////i//i///iiiilTi,'
.............
0•....'• •....•....• • 0•
....•....•.....
-20
-10
0
10
Z Axis (]flow--.-)
20
-20
-10
0
1•
20
Z Axis Cflow---)
Fig. 2. (a) The sameasFigurel(b), but for a simulation
with 6 x 1027ionizations
s-•. The highermassloading
with MA -- 0.7, Ms - 1.5. For this small massloading
rate, the plasmain the wakeis heatedslightly.(b) The
sameas (a) but for a simulationwith 5 x 1027ionizationss-1. An exospheric
modelwasusedfor the radial
variation of the neutral density, so the ionization falls off
moresteeplythan 1/r 2. The largeionizationrate causes
rate causesa large decreasein the plasma temperature.
the plasmavelocityto decreasein Io's wake, so cooling
of the plasmain the wake occurseven though initially
(b) The sameas (a), but for a simulation
with Ms - 1.
Ms > x/•.
766
Linkeret al.: The Effectof MassLoadingon Temperature
Discussion
Equation(7) predictsthat the valueof Ms determines
whether massloading heats or coolsthe plasma, and our
simulationshave confirmed this prediction qualitatively.
However, Figure 4 shows that the plasma temperature
dependson the local value of Ms. The processis nonlinear becausemassloadingaffectsthe plasmavelocityas
well as the temperature. If the massloadingrate is high
enough, cooling of the plasma can occur even when the
initialvalue
ofMs > X/•_/5.
As we noted earlier, values of Ms in the torus could be
larger than the valueswe have used. It is possiblethat if
the backgroundflow were more stronglysupersonic,the
plasmain the wake would still be heated even though the
flow speedwas reduced in the wake. However, our results
suggestthat evenif Ms in the torus is > 1.5, it is difficult
to heat the torus plasma strongly by heavy massloading
near Io. Increasing•s addsmore massand thermal energy
to plasma in the wake, but becausethe velocity of the
plasma decreasesas )s increases,the thermal energy of
the pickup ions also decreases.In the Io torus at locations
remote from Io, the mass loading rate is expected to be
smallenoughthat the non-lineareffectswe havedescribed
will not be important.
Our simulationsassumeionization is producedby electron impact. Large rates of chargeexchangenear Io have
alsobeeninvokedas a meansof providingthermal energy
to the torus plasma[Shemansky,
1988]. We havenot yet
investigatedthe effectsof chargeexchangenear Io in our
simulation,but we note that chargeexchangecan affect
the plasma velocity in much the sameway as impact ionization. Therefore, strong heating of the torus plasma by
charge exhange near Io may have difficulties similar to
what
we have shown for ionization.
The valuesof Ms pertinent to other plasma regimes,
suchas the solar wind near Venus and comets,are much
higher than the values we have consideredhere, so far
away from these bodies mass loading will only increase
the plasmatemperature. However,closeto the comaof a
comet, the plasma velocity decreasesconsiderably,so the
pickupions couldhavecoolertemperaturesthan the background plasma. Cooling of the plasma near cometsby
chargeexchangehas been noted previously[e.g., Galeev
et al., 1985].
Conclusions
We have used three-dimensional
MHD
simulations
of
plasmaflow past Io to investigatethe effectsof massloading on the plasmatemperature. A straightforwardanaly-
sisof equation(7) revealsthat massloadingcanincrease
Acknowledgements.
This work was supportedby the
Jet PropulsionLaboratoryundercontract955232,NASA
GrantNAGW-93 andthe SanDiegoSupercomputer
Center.
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than expected, and in some regionscoolingmay occur
becausethe addition of mass also decreasesthe plasma
M. G. Kivelsonand R. J. Walker, IGPP, University of
flow velocity.The coolingof the plasmaby massloading
California, Los Angeles,CA 90024-1567.
when the flow is initially supersonicis most likely to be
J. A. Linker, Department of Physics,Universityof Calof interestfor the Io torus parameterregimewhere Ms is
ifornia, Irvine, CA 92717.
probablyin the range of 1-2. Our simulationsshowthat
the effectsof mass loading on plasma temperature and
(ReceivedMay 3, 1989;
velocityare most pronouncedin Io's wake region.
acceptedMay 16, 1989.)
or decreasethe plasma temperature, dependingon the