JOURNALOF GEOPHYSICALRESEARCH,VOL. 94, NO. A1, PAGES227-240,
ON
THE
BULK
MOTION
OF THE
ION CLOUDS
SOLAR
WIND/MAGNETOSHEATH
FORMED
BY
RELEASES
JANUARY1, 1989
THE
AMPTE
Sandra C. Chapman
School of Mathematical Sciences, Queen Mary College, London
Abstract.
A principal objective of the Active
Magnetospheric
Particle Tracer Explorersmissionwas to
release lithium and barium ion clouds that were initially
sufficiently
massdenseto stronglyperturbthe ambientsolar
wind or magnetosheath
flow. A key propertyof these
releasecloudswas that their spatial and temporalscales
were smaller than, or of the order of, the Larmor scales
This was achievedby releasingslowlyexpandingcloudsof
photoionizing
neutral lithium or barium from the German
Ion ReleaseModule (IRM) when it was locatedin the solar
wind, upstreamof the Earth's bow shock, or in the
magnetosheath. Releases were also made in the
magnetotail,but due essentially
to the much lower ambient
plasmaflow speedthey exhibit a different classof behavior
and hence will not be discussed
here (they have been
of all the ion speciesinvolved.A one-dimensional
hybrid
simulationstudy conductedby Chapmanand Schwartz addressed
elsewhere,
[e.g., Bernhardt
et al., 1987]). All of
(1987) showedthat momentum
couldbe transferred
locally the solarwind and magnetosheath
releases
were observed
to
from the oncomingprotonsto the majorityof the release producequalitativelysimilar perturbations
in the ambient
ions, which are associatedwith the diamagneticcavity
field and flow, althoughthe heavier barium, with a much
producedby the release,via the quasi-steady
boundary shorter photoionization
time ('• 30 s comparedwith
layer that forms at the cavity edge. The directionof
'• 1 hour for lithium),gave rise to a muchhigherinitial
motionof these"snowploughed"
releaseions and the field
ion massdensityand hencelonger-liveddisturbance.
structurethat accelerates
them is just that of the oncoming
protons.
The remainder of the release cloud is
photoionizedoutsideof the diamagneticcavity, and as a
consequence
of the "pickup"interactionof theseions in the
oncomingproton flow the protonsare deflectedfrom their
A second spacecraft,the UKS (United Kingdom
Subsatellite)was located close to the IRM during the
AMPTE mission.For the lithium solar wind releases,on
September11, and September20, 1984, this separationwas
ambient flow direction in the release cloud vicinity. This
deflected direction of the oncoming proton flow, just
December 20, 1984, it was '• 170 km (the UKS was
nonoperational for the magnetosheathbarium release, on
upstream of the snowploughfields, then defines the
directionof motion of the snowploughed
releaseions, which
eventuallycomprisethe releaseion cloud that is observed
from the ground. This transversecomponentto the
direction of the releaseion cloud motion suggested
by the
above simple picture is qualitatively consistent with
ground-based
observations
of the bariumreleases.Here this
simple model for the snowploughdynamicsis discussed
quantitativelyfor the particular conditionsof the various
lithium and barium releases. In particular, the model is
found to accurately predict the time taken for the
snowplough
to return to the Ion ReleaseModule(IRM) (the
releasespacecraft)
locationafter the initial expansionof the
cloud, an event which, as suggestedfrom both the
Chapmanand Schwartzsimulationresultsand the IRM in
situ data, corresponds
to the return of the magneticfield at
the IRM. The magnitudeof the transverse
displacement
of
the December 27, 1984 barium release ion cloud that was
seen from ground-basedoptical observationsis also
predictedby resultsfrom the simplemodel. Finally, the
modelsuggests
that for the bariumreleases,',- 10-15% of
the total number of ions released were photoionized
July 18, 1985). For all of theseevents,the releaseplasma
density was sufficientlyhigh at the IRM that the magnetic
field there was seen to be supressed
to zero [Ltihr et al.,
1986a,b] for '• 6 s for lithiumand for '• 80 s for barium.
This diamagnetic cavity did not extend to the UKS,
however,giving an upper limit on the scale size of the
interactionregion. These spatial and temporalscalesare
smallerthan, or of the order of, the Larmor scalesof all
the ion speciesinvolvedin the interaction(releaseions and
oncomingprotons) but are much larger than the Larmor
scalesof the electrons. A hybriddescription,in which the
ions are treatedkineticallyand the electronsare treatedas
a massless,charge neutralizing fluid, was therefore
appropriate for a study of the interaction between the
releaseions and the oncomingproton flow. In order to
understandthe way in which momentumis transferred
locally between the two ion species,a one-dimensional
hybrid simulationwas conductedby Chapman and Schwartz
[1987]. The approachof this studydiffered from that of
previous work, which had concentratedon the global
three-dimensionalaspects of the release.
"Global"
descriptionshave been attemptedfrom both an analytical
'•
30-35 km, and for the first barium release, on
upstreamof the snowplough
field structureand were
[Haerendelet al., 1986; Papadopoulos
et al., 1986; Cheng,
therefore available to balance the transverse momentum of
1987]
and a
numerical [Lui
et
a1.,1986; Brecht and
the snowploughed
release ion cloud, via the deflected Thomas,1987]
point of view and have in most casesbeen
proton flow. Simple considerations suggestthat this principallyaimed at rationalizingthe initial motion of the
fraction should be adequate to balance the observed barium release cloud, which was deduced from
transverse momentum of the release ion cloud.
1. Introduction
A principalobjectiveof the activeexperiments
in space
thatwerepartof theActiveMagnetospheric
Particle
Tracer
Explorers
(AMPTE)mission
[Krimigis
et al., 1982]wasto
releaseion cloudsthat were sufficiently
massdenseto
constitute
a strong
perturbation
to theambient
plasma
flow.
ground-based observations to be transverse to the direction
of the ambient flow, that is, approximately in the opposite
direction of the ambient convection electric field [e.g.,
Valensuela et al., 1986].
The study by Chapman and
Schwartzrevealeddetailsof the interactionwhich had not
beendiscussed
in anyof theseprevious
descriptions
andas
a resultgavethepossibility
of a moredetailed
quantitative
investigation
of the release
cloudmotion. It is justsuch
an investigation
thatwillbe addressed
in thispaper.
The results of the Chapman and Schwartz [1987]
Copyright1989 by the AmericanGeophysical
Union.
simulation showed that a
well-defined, quasi-steady
boundary layer evolves between the release ion population
Paper number88JA03570.
associatedwith the diamagneticcavity and the oncoming
0148-0227/89/88JA-03570505.00
proton flow.
227
Details of the boundary layer structure are
228
Chapman'Bulk Motion of AMPTE ReleaseIon Clouds
found in the IRM in situ data, suggestingthat this
momentum transfer mechanismis in operation during the
releases[Dunlop et al., 1987]. Essentially,the bulk of the
releaseions are moved en masseby a "snowplough"
type
process, gaining momentum from the oncoming protons
which are in turn slowedby the boundarylayer structure.
The directionof motion of the "snowploughed"
releaseions
is just that of the oncomingprotons. It is important to
note that the oncomingprotonsin the releasevicinity have
been deflected from their ambient (undisturbed)flow
(S)
_8
0
TGSE
(epprox.)
¾
,•-VI-
-
direction as they interact with the release ions that were
created outside of the diamagneticcavity and boundary
layer region (see also Chapman and Dunlop [1986]). On
the basis of the simulationresults,Chapmanand Schwartz
[1987] developedan approximateequationof motion, which
for the simple release ion density profile used in the
one-dimensionalsimulationsgave quantitative estimates of
the snowploughdynamicswhich were in good agreement
withthesimulation
results.
Fig.1. Thisfigure
shows
a simplified
representation
of the
We would
nowliketo compare
thebehavior
thatis one-dimensional
snowplough
mechanism
for thetransfer
of
found
by integrating
thisapproximate
snowplough
equationmomentum
between
the oncoming
proton
flowandthe
of motion
withtheobserved
global
behavior
of theactual release
ionsassociated
withthediamagnetic
cavity
(a more
releases,
in orderto establish
whether
or notthe"local" detailed
version
maybe found
in theworkof Chapman
momentum
transfer
process
thatwasfoundto operate
in andSchwartz,
[1987]).Theplane
of thediagram
hasbeen
theone-dimensional
simulations,
andin thein situdata, chosen
to beperpendicular
to theambient
magnetic
field
contributes
substantially
to the observed
release
cloudB_B_o
andto contain
theIRMlocation
(thecenter
of the
introduced
in thenextsection,
along
withdetails
of the ambient
anddisturbed
solar
wind
flowvelocities
V__sw
o and
numerical method required for their integration. The
snowplough
motion
depends
crucially
uponthenumberV_sw
'
the
snowplough
velocity
Vp
and
the
velocity
of
the release ions that are created upstream ofVr the
motion.
Theapproximate
equations
of motion
willfirstbe release
cloud
at t = 0), andhence
it alsocontains
the
density
of release
ionswhich
it encounters.
In section
3 snowplough.
The• axis(ofthecoordinate
system
used
in
wetherefore
derive
a simple
expression
fortherelease
ion thecalculation
of t-hesnowplough
motion)
is aligned
with
densityprofile,normalizing
it to the single-point
thedirection
of motion
of thedeflected
oncoming
proton
measurement
of theionnumber
density
asa function
of flowandconsequently
alsowiththedirection
of motion
of
timethatwasobtained
at theIRMforeach
release.
The the snowplough.
The_•',•', and•' axescorrespond
results
of integrating
thesnowplough
equations
withthis approximately
to x, X, andz GSEfor theDecember
27
model
for the release
iondensity
will be discussed
in barium
release
geometry,
wi•hthe ambient
solarwind
section
4 andcompared
withquantitative
features
of the moving
in the-•' direction
andtheambient
magnetic
field
global
release
ioncloud
behavior.
These
maybeobtainedbeing
directed
approximately
in the•' direction.
from the IRM in situ data, which determine the time taken
for the snowploughedrelease ions to return to the IRM
locationafter the initial expansionof the releasecloud, and
in the case of the December27 barium release, from
one-dimensionalsimulations. Figure 1 is drawn in the
plane which contains the direction of the ambient
ground-based
opticalobservations
of the releaseion cloud (undisturbed)
protonflow V__sw
o and the center of the
motion. A summaryis given in section5.
release cloud at
x -- y = z = 0).
2. TheSnowplough
Description
t
=
0
(i.e.,
the IRM
location,
For simplicity, the ambient magnetic
fieldB_B_o
is takento be perpendicular
to the planeof the
diagram, so that the ambient convection electric field also
lies in the plane.
Since the simple two-dimensional
Beforeintroducing
the snowplough
equations
of motion, description
usedhere doesnot includemechanisms
which
we will beginby settingthe essentially
one-dimensional
can,on average,
change
the direction
of the magnetic
field
snowploughdescriptioninto the context of a (i.e., by "draping"aroundthe obstacleformedby the
three-dimensional
release.An attemptto do thisis shown releasecloud), B alwaysremainsperpendicular
to the
in Figure1, whichis a simplified
versionof Figure7 of plane. A largeproportion
of the release
ionsare assumed
Chapmanand Schwartz[1987]. For the purposes
of future
to be createdwithin, and thereforeassociated
with, the
reference,the _•', •', _•' coordinate
systemof this figure diamagnetic
cavityformedduringthe early stagesof the
has been chosento correspondapproximatelyto x, y, z
release,which is represented
here as boundedby the solid
GSE (x directed toward the Sun, z northward perpendicular
line.
to the ecliptic plane, and y completingthe orthogonalset)
for the particular geometry of the December 27 barium
solar wind release,althoughthe discussion
to be presented
here appliesfor all of the releases. The coordinatesystem
in which we shall later discuss the motion of the
snowplough
is orientedon this diagramby the directionof
created outsideof this field free region and will begin to
be picked up by the oncomingproton flow, extracting
momentumfrom it. The macroscopic
forces which act on
both these releaseions, and the oncomingprotonsin this
"mass loading" region, can be shown from a multifluid
description[Chapman and Dunlop, 1986' Chapman and
motionof the snowplough
itself,5[, that is, the direction
of
the oncoming
disturbed
protonflow V__sw
just upstream
of
Schwartz,
1987] to act in the localU__trAB_
direction,where
U_• is just the relativevelocitybetween
a particularath ion
The remainder of the release ions will, however, be
the snowplough
boundarystructure,the • and _• coordinates speciesand the velocity of the center of charge of the
simply completing the orthogonal set without further
plasma as a whole.
Release ions, which are created
constraint. Figure1 is drawnto representonly thoseion
populations
relevantto the simplesnowplough
model under
discussion
here, so that it must be remembered
that this
doesnot includeall of the detailedbehaviorfound in the
approximately
at rest, i.e., with V__
a = 0 in the oni:oming
solar wind flow V__= V__sw
, will have a velocityrelativeto
the net plasmacenterof chargein the -Y-swdirectionand
will hence move in the -V__swAB
-- direction, i.e.,
Chapman: Bulk Motion of AMPTE Release Ion Clouds
229
approximatelyin +•_', or upward on Figure 1. In the "test
ion" limit, i.e., far upstream, where the release ion mass
density is low in comparisonwith that of the protons, the
oncoming flow will be essentially undisturbed, with
ground-basedobservations[Valensuelaet al., 1986' Rees et
al., 1986].
Finally, the integration of the snowplough
equationsyields the fraction of the total number of release
ions which are expected to be created upstream of the
V_.sw= V_.sw
o, and the release ions just move in the
directionof the ambientconvection
electricfield -V_.swoAl
•
snowplough
field structure. This shouldindicatewhetheror
not it is reasonable
to expectthis populationto balancethe
as defined in the rest frame of the release (i.e., exactly in
transverse momentum of the snowploughedrelease ion
+•_'). However,as the protons approachthe cavity, they
cloud.
move into a region of dense, relatively slowly moving
Before moving on to this detailed quantitative
release pickup ions just outside. The addition of the
investigationit is perhapsworth outlining the differences
slowlymovingreleaseions to the flow effectivelyreduces betweenthe model describedhere (see also Chapmanand
the net centerof chargevelocityof the plasma,so that the
Schwartz[1987]) and those suggested
by Haerendelet al.
protons,
movingwitha velocity',- Vsw
o relativeto the net
[1986]and Cheng[1987]. In thosemodels,the strong
centerof charge,experience
a U_QctAB_
• V__swoAB
• forcethat
"U__aAB_"
interactionbetweenthe oncoming
protonsand the
is directed approximatelyin -_•', or downwardon the
figure, that is, in the oppositesenseto that of the pickup
release ions, momentumbeing conservedbetween the two
species. A deflection and slowing of the oncomingion
flow that is consistentwith this description has been
observedin the in situ ion data for the releases[e.g.,
Haerendel et al., 1986], althoughthe UKS and IRM ion
instruments,which measure energy per charge, cannot be
used to uniquely distinguishion populationsof different
masses. Since B alwaysremainsperpendicularto the plane
release ions located outsideof the cavity, which acts to
deflect the oncomingproton population,is not taken into
account.
Asymmetry in the enhanced magnetic field
surroundingthe cavity and deflection of the oncoming
protonsin the same senseas is expectedto occur from the
U_U_•B__
interactionwere consideredby Cheng [1987] to result
insteadfrom an asymmetricspacechargethat builds up on
the release cloud surface. The amount of transverse
momentumthat may be impartedto the bulk of the release
ions via asymmetricchargingis limited by the quantity of
of the diagram,the •
chargethat may accumulate
and hence is relatedto the
force alwaysactsin the plane,
so that the plane also containsthe velocitiesof the release
ions and oncomingprotons just upstream,or outside, of
the snowplough
field structure.
Understanding the ion behavior upstream of the
flux of oncomingprotons that impinge upon the release
cloud surface, directly inducingthe charge, and the ability
of the ambient plasmato neutralizethe charge, rather than
being ultimately limited by the transversemomentum that
snowplough
field structures,
in terms of the U_tgXB_
force,
can be carried by the release ion populationthat is
has important consequences
for the global motion of the
ion cloud that is acted on by the snowploughfield
photoionized"upstream"of the snowplough(see Figure 1),
which will be estimatedhere. Also, the U_tgXB_
interaction,
structure. In effect, release ions created upstream locally
along with the snowplough boundary structure itself as
exchange momentum with the oncoming proton flow,
envisaged here, arises under the assumption of
slowingand deflectingit as shown. The deflected protons
quasi-neutrality, so that these two descriptionsof events
then interact with the quasi-steady snowplough field
cannot be regardedas equivalent.
structure that forms at the cavity boundary, transferring
Alternatively, the snowplough type mechanism for
momentum to the snowploughedrelease ion population momentum exchangehas been assumedto simply act along
associatedwith it, such that the direction of motion of the
the unperturbed ambient proton flow direction, ultimately
snowplough_
_• is just that of the deflectedproton flow. If
resulting in the collapseof the diamagneticcavity as seen
no other forces are present, the release ions created inside
at the IRM.
The subsequentmotion of this snowplough
the field free region of the diamagneticcavity only gain
structure does not then account for the observedtransverse
momentum by interacting with the snowplough field
motion of the release ion cloud as a whole. Instead, the
structure, so that the entire release ion cloud is eventually transversemomentum of the ion cloud is balanced by that
accelerated by being "gathered up" by the propagating of release ions as they exit the -V_.swoAB_
oriented face of
snowploughfields, and becoming part of the snowploughed the cloud. The momentum imparted to the release ion
ion population.
As the direction of motion of the
cloud by these "extracted" ions directly in this way must
snowploughedrelease ion cloud is just that of the deflected
depend crucially upon the extracted ion momentum in the
solar wind flow, we would expect it to be at some angle to
immediate vicinity of the boundaryof the release ion cloud
the ambient solar wind flow direction. Whether or not this
itself, as subsequently these ions can only exchange
angle is sufficiently large to account for the observed momentum with the cloud via the U_U_ctAB_
coupling process
transversemotion of the barium releaseion cloudsdepends with the oncoming protons as describedabove (and by
upon the fraction of release ions that are created upstream Chapman and Schwartz [1987]). Details of the mechanism
of the snowploughfield structure. It is the momentum of
for the extraction of these ions are not well known, indeed
this population which ultimately must balance the transverse the one-dimensionalfield structuresfound by the Chapman
momentum of the snowploughedrelease ion cloud, with the
and Schwartz simulation study act to prevent release ion
deflected solar wind protons just acting as an intermediary,
migration acrossthe boundarylayer. An added mechanism
transferring momentum between the two release ion
not represented in the one-dimensional hybrid description,
populations.
due for example to the two- to three-dimensional nature
Unfortunately, as the UKS and IRM ion experiments of the problem, or the presenceof microinstabilities,would
cannot distinguishbetween ion species of different masses,
be required to allow ion extraction to take place. The
no exact measurementof the deflected solar wind velocity
momentum imparted by the extracted ions is hence not
is available. Other quantitative features of the simple
known explicitly. In general then, the number of extracted
model of Figure 1 are directly accessiblein the data
ions required to account for the transversemotion of the
however, and will be calculated by integrating the
releaseion cloud directly will differ from the number found
snowplough equations of motion. The time taken for the
to be photoionied upstream from integrating the snowplough
snowploughed release ions to return to the IRM location
equations as is done here, the latter population achieving
after the initial expansionof the release ion cloud can
momentumtransfer via •
accurately be
determined, as the
arrival
of
the
snowploughed ions is clearly evident in the IRM in situ
protons. Calculating the number that are expected to be
photoionized upstream will, in later sections of this paper,
couplingwith the oncoming
data [Chapman and Schwartz, 1987]. The global motion of
the release ion cloud is also approximately known from
allow us to show that this release ion population is
adequate to balance the transverse momentum of the
230
Chapman: Bulk Motion of AMPTE Release Ion Clouds
remaining population that forms the release ion cloud.
Since different fractions of the total release ion population
moving faster than the release ions in its vicinity that have
yet to interact with it.
will generallybe requiredto achievetransversemomentum
balance by these two different mechanisms,the time
dependentmass of the release ion cloud will also differ,
along with its expectedtrajectory.
To a good approximation,the release ions are just
createdwith the expansionvelocityof their sourceneutrals,
and for the purposesof this simple model we assumethat
the ions are not subsequentlyenergized, except by their
In conclusion, we cannot eliminate the possibilitythat
either (or all three) mechanismsare in operation during the
release events, however it will be shown here that the
snowploughdescription(as detailed in Figure 1) can alone
account for the observed quantitative features of the bulk
motion of the various release ion clouds seen both in the
IRM in situ data and from ground-basedmeasurements.
interaction with the snowploughor moving boundary layer
field structure itself. While this is perfectly valid for those
ions created within the diamagnetic cavity, where the
electromagnetic fields have presumably been locally
suppressedby the responseof the release electron cloud, it
is strictly speaking not the case for ions created in the
upstream solar wind flow. For the form of the release ion
cloud employed here, however these release ions, created in
2.1. The Snowplough
Equationsof Motion
the oncomingsolar wind flow upstreamof the snowplough
fields do
not
contribute significantly to
the
mass or
Theone-dimensional
hybridsimulation
study
conductedmomentum
of the snowplough
by directly
interacting
with
by Chapman
andSchwartz
[1987]showed
thatmomentumit, instead
contributing
primarily
to the overallglobal
transfer betweenthe release ions associated
with the
accelerationof the release ion cloud by effectively
diamagnetic
cavityandtheoncoming
proton
flowcouldbe deflecting
and slowing
the oncoming
protonflow as
achieved
viaan interaction
witha propagating,
quasi-steady
described
in the previous
section
(theslowed
proton
flow
fieldstructure.
A simple
model
wasproposed
in which
the speed
thenbeingrequired
asa constant
in theintegration
various
ion populations
encounter
the propagating
field of the snowplough
equations
of motion).We therefore
structure
(thesnowplough),
suffering
an effective
collisionassume
thatbeforeinteracting
withthe snowplough,
the
with it.
The net resultis a lossof momentum
to the
releaseions all initiallymove with the speedsof the
oncoming
protons,
momentum
whichis just ultimatelyneutrals
thatactastheirsource,
Vr(X,t
) = x/t (thesignof
transferredto the release ions that encounterthe
vr in (1) is hence just dependent
upon whetherthe
snowplough
fieldstructure
asit propagates
intotherelease snowplough
is located
upstream
or downstream
of the
cloud. Nowprovided
thattherelease
ionnumber
density release
point).
is suchthatthesnowplough
continually
accelerates,
all the
An expression
fortherateof change
of momentum
of
releaseions that have encountered
it after its formation the snowplough
can be obtainedas follows. An ion
shortly
aftertherelease
time(t = 0) willbe continuallypopulation,
suchastheoncoming
solarwindprotons
with
accelerated
also. The release
ionswill, afterhaving speed
Vswandconstant
number
density
nswis assumed
to
encountered
the snowplough,
simplymovewith the interact
withthesnowplough
fieldstructure
sothattheions
snowplough
speed
Vp,effectively
constituting
theentireeffectively
suffer
collisions
with
coefficient
ofrestitution
e.
snowplough
mass
whichcontinually
gains
momentum
from In the framemoving
withthesnowplough
speed
V.. the
theoncoming
solar
wind.
ions
have
a velocity
-e(Vsw
- Vp)after
theinteractio•n,
so
In thissimple
model,
the snowplough
fieldsact to thattheirchange
of momentum
in therestframe(inwhich
transfer
momentum
along
thedirection
of motion
of the the solar wind protonspeedis Vsw
) is just
snowplough,
_•only
(that
is,all
inthe
the
direction
ofsuch
the AP
x = -(Vsw+ •)mp.isAs
theproton
flux
oncoming
proton
flow)
sothat
variables
here,
interacting
with t•Pe)(1
snowplough
nsw(Vsw
- Vp),
as the snowplough
massper unit area Mp and speedV
along
withtherelease
ionexpansion
speed
vr (i.e.t•
speed of release ions which are still moving within the field
momentum is delivered to the snowplough at the rate
(1 + momentum
e)mpnsw(VswVp)2.rateAnanalogous
gives
transt-er
from
the expression
release
ions
the
freediamagnetic
cavity
andhavenotyetencountered
the encountered
bythesnowplough.
Thesumof these
two
snowplough)
arealljusttreated
asfunctions
of distance
x represents
a netforce
acting
upon
theexisting
snowplough
andtimefromtherelease
t only.Thesnowplough
itself
is ofmass
Mpperunitarea.Assuming
thattherelease
ions
hence
a planar
structure
of infinite
extent
in they andz that haveinteracted
with the snowplough
always
directions,
although
hereweessentially
consider
themotionsubsequently
move
withit at thesnowplough
speed,
sothat
of surface
element
in thisplane
dydz
sufficiently
small
that theyinteract
witha coefficient
of restitution
e = 0,
therelease
ionnumber
density
nr canbesimply
taken
asa conservation
ofmomentum
between
thesnowplough
andthe
function
of x andt, andwewillexamine
theparticularinteracting
protons
andrelease
ions
gives
element which at some time will pass through the IRM
location(i.e., y = 0, z = 0) in order to facilitate
comparison
withthe in situdata. Therateof change
of
the mass of release ions moving with the snowplough at
speed
Vp perunitareais thengivenby
dt
which,
Mp(x,t) - mrnr(X,t) (Vp(x,t)-Vr(X,t))
Vp•Vr
d Vp(X,t)
dt
- (1 + •)nswmp[VswVp(X,t)]2
-mrnr(X,t)[Vr(X,t) - Vp(X,t)]2
d
•
Mp(X,t)
along
(2)
with
dx(t )
(1)
dt
- Vp(x,t)
(3)
dMp(X,t)
dt
-0
Vp•V r
and equation (1),
completely specifies the snowplough
motion. In the results to be discussedlater in this paper
we will assume that e = 0 for the oncoming protons, as
wheremr is the releaseion mass. Implicitin (1) is that
well as the releaseions. This valueis suggested
by the
the snowplough
can of courseonly gain mass if it is
boundarylayer simulationresults[Chapmanand Schwartz,
Chapman: Bulk Motion of AMPTE Release Ion Clouds
1987], althoughin principlea differentor varyingvalue of
e for the protonscould be chosen.
The contribution of the proton mass to the total mass
231
to of 0.05-1 s yields a variationof '• 20% in the time
(approximately a few seconds) that the snowploughis
calculated to take to return to the IRM, with a smaller
per unit area Mp of the snowplough
has also been variation
in all otherparameters
(in thiscaseto = 0.5 is
neglected in comparisonwith that of the release ions.
This assumptionis reasonableif we considerthat for a
used). On varying the initial snowploughmass from 0.01
to 1.0 nswmrL
o it is found that all snowplough
parameters
barium release, for example, the region on which the
diamagnetic cavity extends is known to be limited by the
UKS-IRM separationand is • 100 km. Since the release
event is seen from optical observations
to be r • 400 s in
duration, the mass of solar wind protons that have
impinged on this region of area A • 100 km 2 is
vary by < 3% at all later times of interest here, for all
of the releases. The initial location and speed of the
snowplough can also be varied over several orders of
magnitude to yield negligibly small differences in the
calculatedsnowploughmotion after an initial transientphase
of less than 1 s duration. In particular, values of the
p •- 102s mp.
approximately
vsw ranswTM
This mustbe
initialsnowplough
speedwerevariedfromthatof the local
comparedwith the mass of releasedpt•otoionizedbarium,
• 137 times (a few) times 10•4-10•s mp, which is
releaseion speedvr (a few kilometersper second)to the
oncomingsolar wind speedVsw (a few hundredkilometers
expected to be mostly gathered up by the snowplough. A
similar calculation for the lithium releases shows that the
solar wind mass that would have been expected to
accumulate at the snowplough from our simple picture
would represent an even smaller fraction of the total
snowploughmass.
per second). In summary then, the results to be presented
here may, as far as the integration procedure is concerned,
be considered to be accurate to within a few percent for
the barium releases and to within • 20% for the lithium
releases.
In order to integrate equations (1)-(3) we finally require
2.2. Integration
of the Snowplough
Equations
of Motion
solarwindand releaseion populations.In the modelused
values
Equations
(1) to (3) havebeenintegrated
usinga
fourth-orderRungeKutte schemewith variabletime step,
for
the
number
densities
and
initial
velocities
of
the
here, release ions that are created outside the cavity, in
the oncoming proton flow, are expected to be accelerated,
causing
theprotons
to deflect
anddecelerate
in orderto
conserve
momentum. As no direct measurement
of the
the timestepbeingsufficiently
smallthatthe numericaldisturbed
solarwindvelocity
andnumber
density
in the
errorsaccumulated
overan entireintegration
(i.e., overa vicinity
of the release
ion cloudis available
as required,
timegreater
thanthetimescale
of theevent)
are• 0.3% thesnowplough
equations
of motion
willbe integrated
using
(confirmed
by progressively
reducing
the time step for constant
valuesfor the solarwindspeedand number
several
sample
snowplough
integrations
performed
overthe density,
andin section
4 we willexamine
thesignificance
rangeof parameter
space).
of choosing
different
values
for theseparameters.
It just
Theinitialconditions
for theintegration
arechosen
to remains
to specify
therelease
ionnumber
density
profileto
reflectfeatures
of thesnowplough
structure
formation
seen be usedin theintegration
in thefollowing
section.
in the boundary layer simulationresults of Chapman and
Schwartz [1987]. These and further simulationsover a
range
of different
release
ion masses
suggest
thatthe
formation time
of
the
snowplough field
3. TheIonNumber
Density
Function
structure is
• 0.5-1 s for lithiumandup to a fewseconds
for barium,
whichfor the latteris shortcompared
with the • 80 s
overwhichthe diamagnetic
cavitypersists
at the IRM for
the bariumrelease.Thistimescaledoesrepresent
a larger
fraction
of the • 6 s for whichthe magnetic
fieldis zero
at the IRM in the caseof the lithiumreleases,
so that we
We will nowobtaina simple
expression
for the release
ion number
density
distribution
function
nr(r,v,t
) withinthe
diamagnetic
cavity. First,a functional
formof the density
distribution
function
will be derivedanalytically
fromsimple
considerations
concerning
the properties
of the release
clouds. This functional
form will of coursedependupona
mightexpectquantitative
results
obtained
fromintegratingnumber
of parameters
which
characterize
eachrelease,
such
the snowplough
equations
to compare
lessfavorably
with asthemeanexpansion
velocity
of thereleased
neutrals
that
the corresponding
observations
in thesecases. The act as a source
for the ions,the totalnumber
released,
snowplough
itselfis takento be initially
at restandat a
and their photoionization
time scale. In situ IRM
distancefrom the releasecloud center where the release
measurements
of the ion densityas a functionof time at a
ion numberdensityis just sufficiently
large to be singlepointin spacewill thenbe usedto normalize
the
represented
computationally.
an arbitrarily
smallbut simpleexpression
for the densitydistribution
function,
nonzeroinitial snowplough
massper unit area is also partially
constraining
theseparameters.
requiredas an initial condition,in order to allow
The release
clouds
of bariumor lithiumvaporbecome
integration
of (2). In the sample
results
to be discussedcollision
free almostimmediately
afterexitingthe release
herethe initialmassMo -- 104 nswm
r kg m-• (nswand canisters
(i.e., within•- « s, the totaltimetakenfor the
mr in SI units),or 0.1 nswmrL
o whereLo -- 100km, a releasereaction(G. Haerendel,
privatecommunication,
typicalestimate
of the scalesizeof the bariumrelease 1986)). The neutraldensityis sufficiently
high that
cloud. Thisvaluefor Mo is foundto be reasonable,
as it
photoionization
produces
electron
number
densities
thatare
represents
a negligibly
smallfractionof the typicalmass large enoughto locally supporta diamagnetic
cavity,
per unitareaof the snowplough
at latertimes.
suppressing
the ambientelectricand magneticfields to
To ensure that the above initial conditions do not
zero. Also, the photoionization process is such that the
dictate the propertiesof the subsequent
snowploughinitialion velocities
are just thoseof theirsourceneutrals,
behavior,the integration
was repeatedover a substantial to a goodapproximation.
Withinthe cavitythen,bothions
rangeof initialconditions.
For a bariumrelease,
it is then andneutrals
moveon straight
line trajectories,
theirphase
foundthatif thestarttimeof theintegration
to wasvaried spacecoordinates
at the "release
time",-•o, Eo, to -- 0
from0.1 to 10 s, a change
of only3% is produced
in the (i.e., whenthe cloudbecomes
collision
free),beingrelated
snowplough
parameters
at the timewhenthe snowploughto thoseat sometime later, r, v, t, by -Yo-- v and
returns
to the IRM, for example,
•- 80 s later(to = 1 s -•o= r - _vt. The distribution
function
of the particles
at
wasusedto produce
the results
to be presented
here). In to = 0, ftot(ro,•o,0)
determines
it alongthesephasespace
the caseof a lithium releasehowever,a similarvariationin
trajectoriesat any later time. Here, at to = 0 the particle
232
Chapman: Bulk Motion of AMPTE ReleaseIon Clouds
distribution function is taken for simplicity to have the
following form:
l(
available inside the field free region of the diamagnetic
cavity are of course those of the IRM, giving a single-point
measurement of the ion number density as a function of
time.
ftot(_ro,Vo,0) =
No
distinction
can
therefore
be
made
between
the
different expansionspeedsin different directions,so we
Vtx• Vty.• Vtz = V. The single-point
measurement
(4) take
also
does
not
determine
thedirection
ofthedrift
velocity
VtxVtyVtzRxRyRz
zø2
Vxo2
Vyo2
Vzo2 ofthecloud
asa whole.
Forsimplicity
werestrict
our
.exp
-[ Rx2+RY2 +Rz2 exp
- Vtx•+Vty2+Vtz•J snowplough
attention
here
toacenter,
single
case
where
the
trajectory
ofthe
element
under
intersects
release
ioncloud
i.e.,consideration
where
thecloud
driftisthe
in
The releaseparticle cloud distributionis just Gaussian
the _• direction only, consistent with the observations of
Rees et al. [1986].
It will be shown that no further
details
in
the
model
for
the
release
cloud
need
to
be
in formin bothrealandvelocity
space,
withdifferentintroduced
in order
to obtain
a good
agreement
between
meanvelocities
Vtx,V_ty,
Vtz,andmeanspatial
extentRx,
R.,Rz inthethree
½Cartesian
directions.
Thedenominator
theresults
ofso
integrating
theare
snowplough
the
observations,
that we
justifiedequations
in the and
above
of
•
(4)
represents
the
mean
volume
in
phase
space
occupied
simplifying
assumptions
concerning
the
release
cloud
by the release cloud.
By Liouville's theorem, the
distribution function at some later time t along the phase
spacetrajectoriesftot(r,v,t) = ftot(•,Xo,0) and normalizing
geometry.
finally
The release ion number density function is
to determine the arbitrary constantK finally yields
NO (1 - e
nr(x,y,z,t)
X2
=
•r/•r
•/•
P)
(R2 + V2t2)a/2
(?)
X2 + Vtx2t
ftot (x,y,z,t)
-
x,
y2
R2 + •2t2'
(Rx2+ Vtx2t2)«
(5)
R2 + •2t2
with the IRM location at x = y = z = 0.
y2+Vty2
t
z2+Vtz2t
2'
The IRM ion
densitymeasurements
to be discussed
here are inferred
from
measurements of
[Gurnett et
(Ry2+ Vty2
t2)«
[R2 +z,•2t2]
al.,
the
electron plasma frequency
1986a, b];(R. Treumann, private
communication, 1987).
This
allows an
accurate
determination of the density of a singly ionized ion
(Rz2+ Vtz2t2)«
This does not, of course, represent a unique choice for
ftot, the only constrainton the chosenfunctionbeing its
integrabilityover phase space in order to normalize for K.
population
whichis at sufficiently
low energies
to fall below
the rangeof the ion instrumenton boardthe IRM.
This model for the expandingion cloud then is a
. vd, and
. allofwhich
Thetotalnumber
of released
neutrals
justdecreases
function
ofparameters
No, rp, V,
are only
known approximately,
for R,
example
from
exponentially
by photoionization,
with a time constant
considerations
made prior to release [Haerendelet al.,
density
distribution
becomes
the model by allowing for a nonzero cloud drift speed
If the total numberof releaseparticlesis NO, then the ion
1985].Also,
since
wehave
introduced
anasymmetry
in
relative to the IRM, the single-point measurementof the
nr(x,y,z,t
) =No(1-•t/Vp) ftot(X,y,z,t
) (6) ionnumber
density
attheIRMasa function
oftime
is
Now, since the release is made from two canisters,
insufficientto uniquelydeterminea singleset of parameters
that constitutea best fit to the IRM data. The simplest
whichprior to ignitionhavedriftedsomedistancefrom the
possible
approach,employedhere, is to fit the ion number
IRM in some unknowndirection, we can take the initial
density at the IRM location as a function of time, as
spatialscaleof the cloudto be dominatedby this canister determinedfrom (7), to the IRM data by minimizingwith
separation,
R, i.e., Rx • R•.• Rz • R. These
initial respect
to a single
parameter,
thetotalnumber
of neutrals
release cloud dimensions
wall be significantfor times
t • Rx/vtx,etc., the cloudat later timesbehaving
as if it
NO released,keepingthe other parameters
fixed. We can
then directlycomparedifferentdensityversustime curves
originatedfrom a single point in space (R 4 0).
generatedin this way, each curve corresponding
to a
One
furtherpossible
featureof the behavior
of the release different
choice
of Vp, •, vd, andR, withtheIRM data
particlecloudis that it driftsas a wholewith respect
to directly. This will then determine
whetheror not the
the releasepoint. A drift of the order of the mean chosenmodel (7) is adequateto describethe slow
expansion
speeds
of the particles
wasobserved
in the case expansion
phaseof the release
cloud,as seenby the IRM,
of the first bariumrelease[Reeset al., 1986]; this might
given the expectations
for the values of the various
be expected
to be a nonnegligible
factorin the particle parameters
involved,as well as effectively
allowing
us to
distributionfunction.Assuming
for simplicitythat the center
calculateNO as a functionof theseparameters.
of the Gaussiandistribution
drifts with respectto the
locationof its centerat t = 0 (i.e., the IRM location)in
The deviation
of the fittedcurvenr(t) (i.e., equation
(7))
from the set of ion densitymeasurements
ni(ti) is just
the +• direction at constantspeed vd, we need only
replacex by x - Vdt in (5). In other words,the ion
densitymeasured
at the IRM locationx = 0 is just that of
theionsmoving
withvelocity
-VdR.
A = • [ni(ti)
- nr(ti)]2
(8)
i
We now wish to comparethis model for the ion density
distributionwith in situ measurements,in order to
where ti is the time when the ith measurement
is made.
determinereasonable
valuesfor the parameters
by which Putting•}A/•No = 0 then yieldsan expression
for the value
the release cloud is scaled. The only measurements of NO requiredfor the fitted curve
Chapman'Bulk Motion of AMPTE ReleaseIon Clouds
233
before this time, subsequent measurements being discarded
Log1onr
from the data set ni(ti) used to fit the curves and
determineNo. The curvesshown in Figure 2 have been
drawn
R,
R
% /n
A
1
0.174
0.025
10
0.015
0.329
C
I
0.079
0.165
two
different
for
typical
values of the initial
values of
the
cloud
other
R = 1 km, but using the same data set (i.e.,
40
80
scale size
parameters
(V = 1.0 km sTM, vd = 0.4 km s-l). On the right-hand
side of the plot the appropriate values for N O and •n/n
are given for each curve. Curves A and B correspond to
values of R = 1 and 10 km, respectively, each curve being
producedfrom the interval of IRM data ni(ti) that starts at
the earliest possibletime t• = ti > R/V, as we would not
expect (7) to provide a good description at earlier times
(the earliest data points in time t• used to producethese
fits are marked by triangles).
For the purposes of
comparison, a third curve, C, has been plotted for
NO
B
for
and
the same
value for t•) as for the R = 10 km curve, B.
120
In general the fitted curves found for reasonable ranges
t(s)
Fig. 2. This figureshowsa comparison
betweenthe release of the drift and expansion
speedsdo not vary stronglyin
ion numberdensityseenat the IRM as a functionof time shapeor accuracyof fit. For example,over a rangeof
markedby the dots [from Gurnettet al., 1986b],with
fitted curvesof the form givenby equation(10) in the
text.
The data is for the December27 barium release,
V = 0.5-1.5 km sTMthe deviation
an/n variesby •- 5%,
and over a rangeof vd = 0-1.0 km sTM an/n variesby
less than •- 3% for the example shown. However, these
and the first point used in each fit is marked by a
triangle.The fittedcurvesA andB correspond
to valuesof
fitted curves require values of No which do vary
significantly
with V and vd, and this mustthen be taken
R of I
into accountin the calculationof the snowploughmotion.
and 10 km, respectively, both having been
calculated
usingdifferentsubsets
of the data consistent
with
equation(10), (see text), the curve C corresponding
to
R = I km, but usingthe samesubsetof the data as the
R = 10 km curve B for the purposeof comparison. All
curves have been calculatedfor typical values of the
remaining
parameters
• = I km s-• and vd = 0.4 km sTM.
The valuesof No and an/n appropriate
to eachcurveare
For example, for curves such as B, No varies by
approximately
a factorof 5 in total over the aboveranges
in V andvd. Thisrangeof vd hasbeenchosen
to include
the optical measurement
of •- 0.7 km s-• for this event
[Reeset al., 1986], and the range of expansionspeeds
includesthe expectedvalue of •- I km sTM (the thermal
speedof the neutrals)and valuesimpliedby the optical
given on the right hand side of the plot.
measurementsof the expansionof the release cloud of
•- 1.35 km sTM [Valensuelaet al., 1986] and •- 1.5 km s-•
[Rees et al., 1986].
It is only the choice of the
• ni(ti) fi(ti)
No • i
• f i2(ti)
i
photoionization
timescale
rp thatis notexpected
to
strongly
affectthe calculation-of
the snowplough
motion.
(9) Both
•rn/n
andtheappropriate
value
ofNo vary
byless
than •- 18% over the possible
rangezp = 28 -+ 6 s
[Carlsten, 1975] in the case of a barium release. For the
wherefi(ti) = nr(ti)/No and from (7) is independent
of No.
It is easilyshownthat •2A/•No2 > 0 in general,so that
(9) yieldsa valueof No thatwill always
correspond
to a
lithium releasesthe effect of varyingthe photoionization
time will clearly be negligiblein (7), since the cavity
lifetime
t •- 10 s << Zp •- I hour,andthisis alsofound
minimum in A betweenthe fitted nr(t) curve and the set
to be the case.
of data points. For the purposesof comparison,we will
calculatea normalized#root mean square
# deviationfor
neglectthe uncertaintyin the photoionization
time scales
and simplytake • P = 28 s and I hour for the bariumand
each curve
lithium releases, respectively.
All
• • [ni(ti) - nr(ti)]2 }•
trn
_ __
n
that
Throughoutthis discussionwe therefore
remains
is
to
examine
the
behavior
of
the
fittedcurves
nr(t) astheinitialscale
sizeR is varied.On
i
comparing
the twocurves
A andB plottedin Figure2 that
refer to values of R = 1 and 10 km (and the earliest
• ni(ti)
(10) possible
initialtimes
t• consistent
withtheassumptions
implicitin (7)), it is immediately
clearthat a good
i
agreement with
R
An exampleof the variationof the ion densityat the
IRM is shownin Figure2 for the caseof the first barium
release. Here, the dots denote the discreteset of density
=
10
km
the
curve.
IRM
This
data
is
obtained for
choice
of
R
=
10
km
the
then
represents
the closestfit to the data, yieldingthe smallest
value of the normalizeddeviationan/n < 2% that can be
obtainedfor any R.
measurements
ni(ti) (in cm-•) plottedversustime (in
As the fitted curvesare obtainedby minimizing
the
seconds),
and the solidlinesrepresent
curvesnr(t) (given lineardeviation
A (8), achieved
by simplyallowinga factor
by (7) and (9)) fittedto this dataset. The releasetime in (7), i.e., NO, to vary, it is clear that data pointsat
corresponds
to t = 0 on the plot. After the release,the
measureddensityfirst falls away due to the expansionof
earlier timeson the plot, wherethe ion numberdensityis
largest, will dominatethe propertiesof the fitted curve.
the release cloud within the diamagneticcavity.
Hence by treating t• as a free parameter, different curves
This
behavior continues until t • 77 s, when the density rises
can be obtained for any given value of R.
sharply;this coincides
with the returnof the magneticfield
thereforeensurethat the MbestfitM properties
of the release
We must
at the IRM and hence corresponds
to the arrival of the
ion clouds that are deduced by the fitting procedure
snowplough
field structure
and associated
releaseionsat the
described
aboveare not sensitive
to any restrictions
that are
IRM location, as describedby Chapman and Schwartz
imposedupon the selectionof the time t• at which the
[1987]. Our modelfor the numberdensityfunctionof the
expandingion cloud only appliesto measurements
made
dataset usedto fit the curvesbegins. To this end, curve
C hasbeenplottedin Figure2, whichwascalculated
for R
23q
Chapman:Bulk Motion of AMPTE ReleaseIon Clouds
= I km but with the samedata set as the R = 10 km
curveB, that is, with the samevalue for t•. This choice
corresponding
valuesfor No, are consistent
with prerelease
expectations.However,it must be noted that due to the
of t• for the R = I km curve C now yields a closer fit to
shorter time scalesof the lithium releaseevents, fewer data
the data than the R = I km curveA, with •rn/n ',- 8%.
points, i.e.',- 5-8, are availablefor fitting these curves,
The best possiblefit that can be obtained for the R = I
leading to larger errors in the best fit parameters.For the
km curverequiresan even later time t• than for curve C,
secondbarium release,parametervaluesfor R = 10 km,
and althoughit has a deviation•rn/n ',- 5% approaching rather than the best fit valueshave been given. In order
that of the R = 10 km best fit shown on the figure, it
representsoverall a less satisfactoryfit in that it applies to
a smaller subsetof the data points. The above suggestion
that the R = 10 km curve B providesthe best fit to the
data set is thereforenot affectedby the sensitivityof the
fitted curvesto the earliest data points in time. The best
to fit a function which is only valid for times after
R .,- Vt, the first few points of the IRM data set that in
reality starts close to t = 0 have been discarded. The
fitting procedure is therefore only meaningful if this
representsa small fraction of the data set and hence
becomesless appropriateas R increases.In the case of the
fit curvesfor the remainingreleases
to be discussed
below,
secondbarium release,no distinctminimumin •rn/n is
all calculatedusingthe earliestt• = ti > R/V, have also
beenfoundto be appropriate
in this respect,yielding•rn/n
values that are either smaller than, or indistinguishable
from, thoseof bestfit curvesthat are produced
usingother
arbitrary(but in all caseslater) valuesof t•.
We are
therefore justified in restrictingour attention to fitted
curvescalculatedusingthe earliestpossiblet• = ti > R/V,
found with changingR, even up to R = 20 km, requiring
',- 15% of the data set to be discarded. However,from
the table it is clear that both R = 10 km and R = 20 km
correspond
to valuesof NO consistent
with expectations,
whereasR = I km does not, so that given expectations
also concerning
R, it wouldseemthat R = 10 km probably
represents
the mostreasonable
choicefor this parameter.
i.e., which apply to the largest possiblefraction of the data
setandwhich
areconsistent
withthederived
expression
for
4. Integrating
theSnowplough
Equations:
Results
nr (equation (7)).
Before release it was expected that the separation of
the two release
canisters,
whichthe scalelengthR
presumably
represents,
would
be
smaller,
that
is,
Havingestablished
a function
for the release
ion
number densitynr consistentwith the IRM in situ data, the
approximately
a fewkilometers
[Haerendel
et al., 1985], snowplough
equations
of motion
derived
in section
2 canbe
although
theseparation
is notwellknown.Asmightbe integrated
for theparticular
conditions
appropriate
for the
expected from the plot, using R = I km yields a poorer
agreement between the model and the data, i.e., for curve
various AMPTE solar wind and magnetosheath releases.
This calculation then yields quantitative predictions for the
A, trn/n•- 17%. Thevaluefor thetotalnumber
of global
release
behavior
thatcanthenbe compared
with
released
neutrals
No thatiscalculated
fortheR = 10km observations.
In thiswayweshall
establish
in thissection
whether or not the simple model for the release ion cloud
curve,
•- 3.3
24compares
favorably
with
prerelease
motion
thatwasproposed
in section
2 is quantitatively
expectations
of x',-10
3.4
x 10 24 [Haerendel
et al.,
1985],
whereas
thevalue
of No fortheR = I kmcurve
A is consistent
withwhatcanbe inferred
about
thisglobal
smaller
thanthisbyovera factor
of 10. Thisdifferencemeasurements.
motion
fromboththein situIRMandground-based
optical
in No would be expected to significantly affect any
quantitative properties of the snowplough motion deduced
fromthefittedmodel
for theionnumber
density.If, as
4.1. Comparison
WithIn SituIRMMeasurements
appears here the model for nr using larger values of R
providesa more accurate representation
of the initial
In the precedingdiscussion
it has been demonstrated
releaseion cloudexpansion,
it wouldalsobe expected
to
that the chosen
functionfor nr (equation
(7)) is consistent
ultimately
leadto qualitative
predictions
of the snowploughwith the in situobservations
of the IRM for a rangeof
behavior
whichagreemoreclosely
with observations.
In values
of thedriftandmeanexpansion
speeds
vd andV of
the next section,the snowplough
motionwill thereforebe
calculated
usingboth valuesof R, i.e., the "expected"
valueof I km and the bestfit valueof 10 km, to specify
the appropriate
valueof No.
The best fit R, corresponding
to the smallestvalue of
the ion cloud,giventhe corresponding
valuesfor the total
numberof neutralsNo (from equation(9)). Quantitative
aspects
of the snowplough
dynamics,
deduced
by integrating
the snowplough
equations
of motion,are thereforeexpected
to vary with the expansionand drift speedsof the ion
trn/n, has been deducedfor both the lithium and barium
cloud, so that we will present results calculatedover a
releases
and is givenin Table I for reference. In this
tablethe bestfit and R = I curvesare compared
by listing
the valuesof the percentage
normalized
deviationtrn/n and
the appropriate
valuesfor No. Fromthe tablewe cansee
that for the lithiumreleases,
the bestfit valuesof R are
rangeof thesevalues. It wasalsofoundthat the implied
initial spatialdimensions
of the cloudR, whichgive the
best agreement
betweenthe model (equation(7)) and the
IRM in situ data, are somewhat
largerthan the spatial
separation
of the releasecanisters
that wasexpected
at the
approximately a few kilometers, which, along with the
time of release, at least in the case of the barium releases.
TABLE 1. Comparison of Parameters for Best Fit and R = 1 Curves
Release
Lithium,
Lithium,
Barium,
Barium,
*Not
Event
September 11, 1984
September 20, 1984
December 27, 1984
July 18, 1984
the best
fit
values
Typical
Fit
V,
Vd,
R,
trn/n ,
No,
Crn/n,
NO,
km s -•
km s -•
km
%
x10 TM
%
xlO TM
2.5
2.5
1.
1.
(see text).
0
0
0.4
0
Best Fit
4
1
10
1020*
R = 1 km Fit
1.7
7.2
1.5
4.6
2.6
3.3
2.2
3.8
5.3
8.9
17.4
4.3
4
6.7
0.24
0.49
Chapman: Bulk Motion of AMPTE ReleaseIon Clouds
R = 10
Cavitylifetime
235
km
•' = 1.25 km/s
Cavitylifetime
t(s)
•
V = 350 km/s
sw
t(s)
]
1.5
100 ••
50 500 ]
0
1.25
50
................................................
.2
.4
.6
.8
1.0
0.75
0
1.
-........:• .......14
........•6........:8.......1.
Vd
Vd
Fig. 3. (Left) A plot of the cavity lifetime at the IRM location versusthe drift speed of the ion cloud
vd. The three curves show the results of integratingthe snowploughequationsof motion for the
December 27 barium release for three different values of the oncoming proton flow speed. The plot is
drawn for the best fit value R = 10 km and the typical expansionspeed V = 1.25 km sTM, (Right) This
plot has the same axes as the left hand plot, but the four curves are instead drawn for four different
values of the expansionspeed and for a typical value of the oncoming proton flow speed of 350 km
sTM. The observedcavity lifetime is marked by the thick horizontal line in both plots.
Results will therefore be presented here for the barium
releasestaking both the best fit value, R '• 10 km, and the
value expectedprior to release,R •- 1 km, in turn. One
further parameterwhich is not well known is the velocity
of the oncomingsolar wind protonswhich act as a source
of momentum for the snowploughedrelease ions. It may
be remembered that due to the creation of release ions
outside of the diamagnetic cavity boundary, the ambient
interpreted as a result of the cavity-associatedions being
gathered up by the moving snowploughfield structure, as
detailed in Figure 1. The in situ IRM observationof the
snowploughmotion will be a temporal one, in that the
lifetime of the diamagneticcavity as determined by the
IRM will just correspond to the time taken for the
snowploughedrelease ions to return to the IRM location
after the initial expansion of the cavity.
This initial
flow is expectedto be disturbedbefore reachingthe cavity
boundary where the snowploughfield structures have
formed. The appropriatevalue for the oncomingproton
flow velocity here is hence that just upstream of the
snowplough
field structures,rather than that of the ambient
flow. We will proceed here by consideringa range of
values for the oncoming proton flow velocities that are
suggestedby (but not directly deduciblefrom) in situ
expansionis modeled by the snowploughequationsof
motion here as simply the phasein the snowplough
motion
when the momentumflux of the releaseions striking the
snowploughis sufficiently large that the snowploughis
forced to move against the direction of the oncoming
proton flow (i.e., before the release ion cloud has
expanded substantially). We will
begin by comparing
predicted cavity lifetimes determined by integrating the
measurementsfrom the ion instruments on board the UKS
and IRM for the various release events, the maximum in
this range being just that of the ambient (undisturbed)
proton flow.
In comparingthe snowploughmotion that is determined
snowploughequations of motion for the particular conditions
appropriate for the various releases with the cavity lifetimes
seen at the IRM.
An attempt at such a comparisonis shown in Figure 3,
which displays the results of a series of integrations
by integrationof the approximateequationsgiven in section
performed over a range of the ion cloud drift and
2 with the in situ observations of the IRM
(and with
expansion speeds, and the speed of the oncoming protons,
ground-basedoptical observations),we simply assumehere
that the motion of the entire release ion cloud may be
for the December 27 barium release. The cavity lifetime,
that is, the time taken for the snowploughto return to the
R = 10
Cavitylifetime
km
• = 1.25 km/s
t(s) :..................................................
Cavitylifetime
Vsw
= 250km/s
t(s)'
ß
ß
100
V
370••
ß
100
km/s
1.5
1.25
1.0
0
......................................
-.8
-.4
. ....
0
.4
.8
0.75
0
-.8
-.4
0
.4
.8
Vd
Vd
Fig. 4. This figure has the same format as Figure 3 but is drawn for the July 18 barium release with
the
best fit
value
R
= 10
kin.
236
Chapman:Bulk Motion of AMPTE ReleaseIon Clouds
R=I
Cavitylifetime
t(s)
80
km
•'- 1.25 knds
Cavitylifetime
....
'....
'....
'....
'....
'....
'....
•....
'....
'....
1
t(s)
80
60
60
40
40
Vsw= 350 km/s
....
'....
'....
'....
'....
'....
'....
'....
'....
""
• km/s
1.5
20
20
0
0
1.25
1.0
.2
.4
.6
.8
0.75
1.
.2
.4
.6
.8
Vd
1.
Vd
Fig. 5. This figure is in the same format as Figure 3 and refers to the December 27 barium releasefor
R=I
km.
IRM location, is plotted on the ordinates of the two panels,
release in the magnetosheath,and the results are shown in
with the range of values of the ion cloud drift speed vd
Figure 4.
along the abscissae. In the left-hand panel, the three
curves have been produced for three values of the
oncoming solar wind proton speed and a single typical
the previous figure, except that since no optical
observations are as yet available in a form that will allow
the ion cloud drift (if any) to be determined, values of the
value of the mean expansion speed V = 1.25 km sTM
cavity lifetime have been plotted for both positive and
These
values
for thesolarwindspeed
thenrange
fromth•
negative
values
of vd, themagnitude
of thisrange
being
of
in situ ion data for the event [e.g., Haerendel et al.,
1986]. The four curvesshown in the right-hand panel
have been producedfor a typical value of the disturbed
solar wind flow speed Vsw = 350 km sTM (which is also
would be expectedto be similar, being producedby what
shouldbe an identical releaseprocess. Once again, the
best fit value of R = 10 km has been usedto generatethe
plot. Given that the mean expansionspeedin this case is
featured in the left-hand plot) and correspond to four
values for the mean expansionspeed that include both
prereleaseexpectationsfor V and values suggested
by the
also similar to that of the first barium release, which is
reasonableif, as assumedhere, the principal acceleration
mechanismis that of the snowploughfield structure,we
optical observations. All of the curvesare determined
also find that the predictedcavitylifetimeis consistent
with
using the
that seen at the IRM.
ambientvalueof • 500 km s-1 to valuessuggested
by the
best fit
value
for
the
initial
cloud
size
The format of Figure 4 is identical to that of
the orderof that of the first bariumrelease,sincethese
R = 10 kin. Given someexpectations
or measurements
of
the range in which •, Vd, and Vswlie, the figure provides
The effect of varyingthe initial spatialextent of the
ion cloud on theseresults,all other parametersbeing kept
a prediction of the cavity lifetime as seen at the IRM
location.
The actual cavity lifetime for this event is
denotedby the thick horizontalline on both panels. Given
then that the drift speed is approximately a few times
0.1 km sTM for this event (i.e., .,- 0.7 km s-• [Rees et
al., 1985]) and that the expansion speed lies within the
given range (i.e., from ',-I km s TM [Haerendel et al.,
1985] to ',- 1.25 km s -• [Valensuelaet al., 1986]), the
predictionsobtained from the simple snowploughmodel are
in excellent agreement with the IRM measurement. The
same procedure has been repeated for the July 18 barium
constant, is illustrated in Figures 5 and 6, where the
calculated cavity lifetimes have been plotted in the same
format as in the previousfigures and refer once again to
the December 27 and July 18 barium releases,respectively.
The ion release cloud distributionfunctionsused to produce
these plots are taken to have an initial spatial extent
R = I kin, which would be consistent with prerelease
expectationsbut which does not provide the best agreement
between the model for the release ion cloud number
densityfunction nr (equation(7)) and the in situ IRM data,
as shown in the previous section. We might therefore
R=
Cavitylifetime
I
km
'• = 1.25 km/s
t(s)
Cavitylifetime
V•w= 250 km/s
t(s)
120
100
80
V
km/s
5O
1.5
4o
1.25
1.0
0.75
-.8
-.4
0
.4
.8
Vd
-.8
-.4
0
.4
.8
Vd
Fig. 6. This figure is in the same format as Figure 3 and refers to the July 18 barium release with
R
=1
km.
Chapman:Bulk Motion of AMPTE ReleaseIon Clouds
R=4
Cavity lifetime
237
km
'• = 2.5
Cavitylifetime
km/s
Vsw= 450 km/s
12
12
t(s)
t(s)
V
km/s
3.5
"
450
ß
500
••'
2.5
1.5
0.5
-.8
-.4
0
.4
.8
-.8
-.4
0
.4
Vd
.8
Vd
Fig.7. Thisfigureis in thesameformatasFigure
3 andrefersto theSeptember
11 lithiumrelease.
expectthatthe R = 1 km curves
will not be in as good that the calculation
of the snowplough
motionfor the
agreement
withtheIRM in situmeasurements
of thecavity lithium
releases
is subject
to muchlargererrors
thanin the
lifetimesas the bestfit curvesshownin Figures3 and 4,
and for the firstbariumreleasethisis clearlythe case(the
caseof the bariumreleases,
the principalsources
of error
beingthe sensitivity
of the resultsto the initial conditions
predicted
cavitylifetimes
nowbeingtooshortby abouta
factorof 2). For theJuly18 release,
however,
it is still
possible
thatthecavitylifetime
measured
at theIRM lies
withintheexpected
rangeof theparameters
V, vd, andthe
oncoming
flowspeed,
principally
because
in thiscasevd is
as described
in section
2 andthelimitednumber
of IRM
ionnumber
density
datapoints
ni(ti)thatare available
for
the fittingprocedure
described
in the lastsection,
leading
essentially
to an errorin the totalnumber
of neutrals
No.
It is therefore
not surprising
that the cavitylifetimes
larger. Our conclusion
hereis thenthat although
the
are not in as goodagreement
with the IRM in situ
lesswell known,so that the rangeof this parameter
is
predicted
by integrating
the snowplough
equations
of motion
different
values
for the expected
andthe bestfit values
of
measurements
for the lithiumreleases,
beingin general
too
R implyvalues
for the totalnumber
of neutrals
thatrange smallby a factorof 3.
over an order of magnitudefor the December 27 release,
andabout
halfan orderof magnitude
for theJuly18
4.2. Comparison
WithGround-Based
Optical
Observations
release, the corresponding
values for the cavity lifetimes
vary much less. Therefore,despitethis discrepancy
The otherset of measurements
that are available
for
between
the expected
and the bestfit valuesof R, we are comparison
withthe predictions
of the snowplough
equations
still able to providepredictions
for the cavitylifetimes are the ground-based
opticalobservations
of the December
which are reasonably consistent with the IRM
27 bariumrelease,whichshowthe motionof the entire
measurements,
to within the uncertainties
of the various releaseion cloudin the plane of the night sky. As this
parameters
whichcharacterize
eachrelease.
releasewasperformed
at the flankof the magnetosphere,
September
11 and 20 lithiumreleases,
bothof whichtook
placein the solarwind. Sincethe bestfit values
of R for
these releasesare quite closeto that expectedprior to
release(seeTable1), the cavitylifetimes
that are foundin
convection
electricfield, to a goodapproximation
(i.e., to
within',- 30ø),andtherefore
the observed
ion cloudmotion
alsojust lies approximately
in the planein whichFigure1
is drawn. The _•', •', and_•' axesof Figure1 therefore
We can now turn to consider the case of the
this plane contains the ambient solar wind flow and
both cases for each release are rather similar.
We
therefore confine our attention here to the results obtained
are equivalentto x, X, and z GSE to within ',- 30 o also.
The release ion cloud produced during the July 18 event
usingthe bestfit valuesof R, and theseare shownin
wasalsoobserved
to behavein a mannersimilarto the
Figures7 and 8 in the sameformatas the previous December
27 release,
thatis, to moveinitiallytransverse
to
figures.In interpreting
theseplots,it mustbe remembered the ambientflowdirection
V, i.e. in • VAB_
direction
(M.
R=
Cavity lifetime
1
km
Cavitylifetime
'• = 2.5 km/s
Vsw= 450 knds
10
t(s)
t(s)
V
km/s
3.5
2.5
1.5
0.5
-.8
-.4
0
.4
.8
Vd
Fig.8. Thisfigureis in thesameformatasFigure
3 andrefersto theSeptember
20 lithium
release.
238
Chapman:Bulk Motionof AMPTE ReleaseIon Clouds
Finally, we have only consideredhere the release ion cloud
acceleration that can be achieved directly from the ram
pressure of the oncoming proton flow (via the
enhancements in the electron and magnetic pressures that
constitute the snowploughmoving field structure). The net
acceleration that could be produced by considering the
x10 2
curvature in the magnetic field that must, in some sense,
become draped around the release ion cloud has not been
discussed, as a consequence of the one-dimensionality of
the simple model used here.
The initial justification for
these simplifications in the description presented here is
based upon the uncertainties in the various parameters that
are required to determine the snowplough motion.
The
extent to which the predictions of the simple model have
been found to agree with the observationsthen supports the
approach employed here.
0
5
1o
•5
20
25
30
35
4o
4.3.
Transverse Momentum Transfer
x•o •
T {$>
Fig. 9. This figureis a plot of the predictedsnowplough
displacement
(from the IRM at x = 0) versustime, which
has been calculatedfor the December27 bariumrelease,
for typical parameters Vsw = 350 km s-• ,
One furtherpredictionof the simplesnowplough
model
is requiredin order for the aboveinterpretation
of the
optical observations
to be consistent. It remainsto be
shownwhetheror not a sufficientproportion
of the release
population
is photoionized
upstream
of the snowplough
field
V -- 1.25 km sTM, vd = 0.53 km s-• , and R = 10 kin,
structureto cause a substantialtransversedeflectionin the
chosento alsoyielda cavitylifetimethat is consistent
with
observations
(this point in parameterspaceis markedby
oncomingprotonflow. This deflectionin the oncoming
flow is of courserequiredin order to accountfor the
the asterisk in Figure 3). The observed release ion cloud
displacementsfor this event are also marked on the plot
transverse motion that is observed in the barium release ion
clouds which we have assumed here are accelerated
by crosses
[Reeset al., 1986] and error bars [Valensuela principally by the snowploughprocess. Since the
et al., 1986].
integration
of the snowplough
equations
of motionyieldsthe
snowplough location with respect to that of the release
cloud at all times, it is straightforward to calculate the
Mendillo, private communication,1988), although these
results are as yet not sufficiently processedto allow the
detailed quantitativecomparisonwith the predictionsof the
proportion of release neutrals which are photoionized
upstream of the infinite planar membrane that is used to
represent the snowplough. This calculation is three
snowplough equations that is to be attempted here for the
dimensional in the sense that the function for the neutral
December27 release.
number densityftot(X,y,z,t) (equation(5)) (along with the
In order to compare the motion of the release ion
cloud directly with the predicted snowploughtrajectory, the
displacementof the snowploughas a function of time has
assumptions implicit in (7) concerning the drift and
expansion speeds of the cloud) that was derived in the
previous section will be employed.
been plottedin Figure 9.
This particularsnowplough
trajectory has been calculatedby integratingthe snowplough
equations for typical values of the drift and expansion
speeds
of therelease
ioncloudandthedisturbed
oncoming
protonflow,vd = 0.53km sTM,• = 1.25 km sTM,and
Vsw
= 350kmsTM,which
have
also
been
chosen
togive
a
cavitylifetimethat is consistent
with the IRM measurement
The photoionisation
rate Rp per unit volumeat any
point in space and time is simpl•
Rp- No ftot(X,y,z,t)
-t/rp
e
(11)
rp
fortheevent.Thepointin parameter
space
corresponding
so thatthe fraction
F of the totalnumber
of neutrals
tothischoice
ismarked
bytheasterisk
inFigure
3. The released
thatphotoionize
upstream
of the snowplough
assumptionthat is inherent in the comparison to be made
here is that the snowploughdisplacementthat is shown in
the figure correspondsto the transversedisplacementthat is
observed,
requiring
a suitable
deflection
in the oncoming
solarwindflow. Giventhat thisis the case,we can then
location
Xp between
t - t + /it is
mark the observed release ion cloud displacementsseen by
Rees et al. [1986] (crosses)and Valensuelaet al. [1986]
(error bars) directly on the plot. The bounds of the error
bars correspondto the locations of the edges of release ion
cloud images which were subject to artificial enlargement
due to overexposure and hence must presumably enclose
the location of the snowploughitself.
It is then clear
from the plot that for most times there is good agreement
between the predicted and observed deflections. There are
severalpossiblereasonsfor the discrepancy
betweenthe
observations
and predictionsat later times. First, in
e
F(t ) -
/it
w
P
exp-
•r/•r(R
2 + V2t2)3/2rp -•o
(x- Vdt)2
-
dx
+ V2t 2
(12)
oo
y2
2 q. V2t
oo
Z2
R 2 + V2t
The integrals
over y and z are just standard
in form, so
that (12) may be rewrittenas:
integrating the snowplough equations of motion, we have
assumed a constant value for the oncoming proton flow
snowplough
field
structure
which
isrequired
toaccelerate
e-t/rP/it
the
release
ion
cloud
will
ultimately
be
expected
to
F
speed,
which
will
clearly
not
be
the
case.
Also,
the
i(XeP)
collapse,
other
forces
then
becoming
more
significant.rp•r '• +0
Chapman: Bulk Motion of AMPTE Release Ion Clouds
where
(x-
239
oncoming solar wind or magnetosheath flow is transferred
to the dense ion clouds that were produced during the
early phases of the AMPTE lithium and barium releases.
Vdt)
u(x)-
(R2 + •2 t •) «
This process
was suggested
by the properties
of the
quasi-steady boundary layer structure that was found to
so that whenXp < Vdt, the snowplough
is located formlocally
between
thebulkof therelease
ionsandthe
upstream
of the centerof the drifting
Gaussian
neutral oncoming
protonflow in the one-dimensional
hybrid
cloud.
Equation
(13)
was
evaluated
numerically
at
each
time
simulations
of
Chapman
and
Schwartz
[1987]
allowing
step during the integration of the snowploughequations of
momentum
transfer
to
take
place
between
the
two
motion.Thesumof thefraction
F photoionized
upstreampopulations.
Herewehavemodeled
thisboundary
layerby
at eachtimestepthenyields
thetotalfraction
photoionized
a simple
planar
moving
membrane
(a snowplough),
which
upstream
during
theentireevent.It wasthenfound
for absorbs
momentum
fromtheoncoming
protons,
transferring
bothbarium
release
events
considered
herethat'• 10-15% it to the release
ionswhichit encounters.
As a result,
of the total numberof neutrals
released
NO are thisrelease
ion population
becomes
gathered
up by the
photoionized
upstream
of thesnowplough.
It is difficult
to snowplough
fieldsas theypropagate
intothe release
ion
estimate
the effectthat thismighthavedirectly
on the cloud,so that ultimately
the bulk of the release
ions
oncoming
proton
flowfromsimple
considerations,
principallybecomes
associated
with the snowplough
itself. The
dueto thestrong
timedependence
andtwo-dimensionality
direction
of motion
of the snowplough
is justthatof the
of the problem. However,
sincethe protonpopulation oncoming
protonflow, and sincethe latteris deflected
onlyactsas an intermediary
in the transfer
of transversefromthe direction
of the ambient
flowin the vicinityof
momentum
between
the snowploughed
release
ionsandthe the snowplough
by the presence
of release
ionsthatwere
release
ionscreated
upstream,
we canstillassess
whether created
outside
of the diamagnetic
cavityandupstream
of
or not the observed
transverse
motion
of the release
ion the snowplough
fields,thissimplemodelis qualitatively
cloud is predictedin a consistent
way by our simple
consistentwith the observedtransversemotion of the
snowplough
model.
bariumrelease
ion clouds. In thispaperequations
of
Knowing
the fraction
of the totalmass
of release
ions motionof the snowplough
havebeenintegrated,
using
created
upstream
of the sncgplough
relates
the (transverse)parameters
appropriate
for the particular
conditions
of the
speedof thispopulation
vu withthat of the snowploughvarious
AMPTEreleases,
in orderto establish
quantitatively
itself,i.e., Vp = Vu/9by conservation
of momentum.
To whether
or not the simplesnowplough
modelis consistent
makean estimate
of the speed
of the release
ionsin the with both the IRM in situ data and ground-based
upstream
regionvu we can simplyassume
that they are observations.
The following
properties
havebeenshown:
accelerated
in the ambientconvection
electricfield
1. The lifetimeof the diamagnetic
cavityas seenby
Ec -• 5 mV m-•, whichwill represent
an overestimatethe IRM, whichin the simpledescription
discussed
here
sincethe localconvection
electricfieldin the regionof corresponds
to the timetakenfor thesnowploughed
release
disturbed oncoming proton flow just upstream of the
ions to return to the IRM location after the initial
snowplough
will be reduced
due to the presence
of the expansion
of therelease
cloud,is predicted
fromtheresults
releaseions themselves,
but mightbe expected
to be of integratingthe snowplough
equations,given the
reasonable
to within an order of magnitude. The uncertainties
in the various
parameters
whichare needed
to
momentum
gainedby the upstream
release
ionsappropriate represent
the initialrelease
cloud.
to thiscalculation
is justthat whichthey"pickup"
while
2. The displacement
of the snowplough
calculated
for
interacting
withtheoncoming
protons
thatwillsubsequently
typical
values
of the parameters
appropriate
for the first
transfer
momentum
to thesnowploughed
release
ioncloud. bariumrelease
is consistent
withthe magnitude
of the
This oncoming
protonpopulation
musttherefore
be transverse
displacement
of the release
ion cloudthatwas
expected
to be deflected
over a regionwhichhas actually
found
fromground-based
observations.
dimensions
onthescale
of therelease
cloud
itself,i.e.,on
3. Thesnowplough
motion
calculated
for thebarium
a scalezXL-• 100 km. Giventhatthisis the case,the
releases
hasbeenusedto determine
the totalnumber
of
second
foundfromintegrating
the snowplough
equations
of
the transverse
momentum
of the remainder
of the release
"pickup"
speed
of therelease
ions
upstream
aftertraversing
neutrals
thatphotoionize
upstream
of thesnowplough,
and
thisdistance
is -• 27kmsTM
, implying
a snowplough
speed which
arerequired
to deflect
theoncoming
proton
flow.
of Vp -• 3 kmsTM
. Thisestimate
is in good
agreementSimple
estimates
suggest
thatthe -• 10-15%whichare
withtypical
values
of aproximately
a fewkilometers
per found
to beionized
upstream
would
besufficient
to balance
motion,
andgiventhe approximations
involved,
it also ions,which
formtheobserved
snowploughed
release
ion
compares
favorably
withtheobserved
transverse
velocity
of cloud,
thedeflected
protons
acting
asan intermediary
in
therelease
ioncloud
of -• 5-6 kms-• [Rees
et al., 1986]. the transfer
of momentum
between
the two release
ion
After
At '• 250s, thetimetaken
forsignificant
transverse
populations.
motion to be observedin the release ion cloud (see Figure
9), thesnowplough
displacement
would
beexpected
to be
Acknowledgments.
Theauthor
would
liketo thank
R.
-• VpAt
-• 750km,which
isin good
agreement
withthe Treumann,
R. Anderson,
andO. Bauer
forproviding
the
-• 600-700
kmthatis actually
observed
at thistime. TEe AMPTE/IRM
plasma
wave
instrument
dataused
in section
2
order
of
magnitude
estimate
above
therefore
suggests
that
and
S.
J.
Schwartz
for
invaluable
discussions
concerning
the fractionof releaseionsthat are foundto be created thiswork.
upstream
of the snowplough
by integrating
the simple
Theeditorthanks
R. Roussel-Dupre
andanother
snowplough
equations
of motion
willbeadequate
tobalancereferee
fortheirassistance
inevaluating
thispaper.
the observed transverse momentum of the release ion cloud,
so that we are justifiedin equatingthe predicted
displacements of
the
snowplough with
the
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(RecievedJanuary20, 1988;
revisedAugust29, 1988'
acceptedAugust29, 1988.)