GEOPHYSICAl, RI:œSI:,AI<CI
I 1.}œrl"l'll!RS,
V()l,. 28, NO. 2, I•Aœ;11!S
219-222, JANUARY
15, 2001
Particle-in-cell simulations of the lunar wake with high
phase space resolution
Paul C. Birch and Sandra C. Chapman
Space and AstrophysicsGroup, PhysicsDepartment, University of Warwick, UK
Abstract.
The processby which the wake infills was consid-
The evolution of the lunar wake in simpli1
fled geometrycanbe simulatedvia a I•D electromag- eredanalyticallyby Ogilvieet al., [1996].The observed
netic particle-in-cell code. By using a sufficientnumber
of particles per cell, we are able, for the first time, to
resolvethe full phase space dynamics of both electrons
and ions. This simulation begins immediately downstream of the moon, before the solar wind has infilled
the wake region, then evolvesin the solar wind rest
frame. The electrons immediately begin to move into
the void but are trapped by two potential wells, thus
generating vortices in phase spaceon both sidesof the
wake, between which counter-streamingelectron beams
interact. Ion beams are generated after the lighter electrons have moved into the void, creating a two-stream
distribution which mixes in phase space due to the potentials created by the electron two-stream instability.
Other
structures
are also evident.
consistent with both WIND
The simulations
observations
counter-streamingion beams set up a two-stream distribution in the centre of the wake, thus creating an
instability. Once the instability has reachedits saturated state, the beamswill begin folding in phasespace
and eventually produce a monotone-decreasing
distri-
bution. A simplesimulationof a plasmavoid [Farrell
et al., 1998]produceda descriptionof the lunar wake,
consistentin some respectswith the observationsmade
with WIND at 6.5 RL. Due to the number of particles
per cell usedby Farrell et al., [1998],only the ion dynamics were studied in detail.
Ion beams were formed
and subsequentlydisruptedby the two-streaminstability. A density rarefaction wave was also produced,as
predictedand observed
by Ogilvieet al., [1996].
We presentresultsfrom a similarlygeometricallysim-
are
and the results
of earlier electrostatic simulations which focus only on
the ion dynamics.
ple 1•1D kinetic simulationof the lunar wake, but with
significantlymoreparticlesper cell,therebyfully resolving both ion and electronphasespacestructureand dynamics for the first time. This enables a more complete
1.
Introduction
The solar wind flows past the moon, resulting in
a wake structure
behind
it.
Since the moon
comparisonwith the WIND observations,in particular includingthe full electrondynamics.Resolvingthe
electron dynamics also allows a more completeunderstanding of the ion dynamics.
has no
magnetic field or ionosphere,the particles that collide
with the moon will be absorbed
2.
and removed from the
distribution. Directly behind the moon there will be
a void in which no solar wind particles are present.
Following the WIND spacecraft's encounter with the
lunar plasma wake on December 27, 1994, a number
of descriptions of the structure and dynamics of the
wake have been produced. Observations by WIND
showedmany characteristicsof the wake at about 6.5
The
Particle-in-Cell
Simulation
The simulation was produced using a thoroughly
1
tested,selfconsistent,
fullyrelativistic,collisionless
lid
particle-in-cell(PIC) code.In suchsimulations,
the ion
and electron distribution functions are representedby a
collectionof superparticles. These "superparticles",
alongwith the electricand magneticfields,can exhibit
both full ion and electron kinetics. The velocities of the
lunar radii (RL) downstream. These include significant ion and' electron depletions,increasedelectron "superparticles"and the fieldshave vectorcomponents
temperature, counter-streamingion beams and rarefac- in three dimensions,but are a function of one configu1
and time. Knownas I•D simtion waves travelling away from the wake. Also de- ration spacedimension,
tected were a number of electromagneticand electro- ulations, since V. B__- 0, the component of B along
static waves [Ogilvieet al., 1996;Bosquedet al, 1996;
Owen et al., 1996; Kellogg et al., 1996; Farrell et al.,
1996;Farrell et al., 1997;Bale et al., 1997;Bale, 1997].
Copyright
2001by theAmericanGeophysical
Union.
the simulation is constant. The number of superpar-
ticles directly determinesthe phasespaceresolutionof
the simulation, here we use - 2500 per cell compared
with - 40 per cell in Farrell et al., [1998].
Figure 1 showsthe geometryof the simulationwhich
is similar to that usedby Farrell et al., [1998]. The
simulation box begins directly behind the moon, perpendicularto the solar wind flow (-X direction)and
parallelto the IMF (Y direction).A higherdimensional
Papernumber2000GL011958.
0094-8276/01/2000GL011958505.00
219
220
BIRCH
AND
CHAPMAN:
PIC SIMULATIONS
Vconvect
B
•/SW
-X
OF THE
LUNAR
WAKE
hind the moon, along the X axis. The initial void has
begunto infill by 7 RL. A rarefactionwavetravelling
awayfrom the wakeis evidentfrom t = 0 movingat the
ion soundspeed,as predictedby Ogilvieet al., [1996].
Two high densityregionscan be seento form on both
edgesof the wake. By examiningthe electricpotentials
calculated from the electric field data, it is apparent
that two potentialwellsform on either sideof the wake,
-2 ISimulation
slice
trappingthe majority of the electrons(seeFigure3).
Figure 1. Simulation geometry.
The potential wellsare dynamicand quasi-periodically
allow electronsto escape,resultingin the increasedelectron densitiesmovingaway from the wake or towards
the centre that can be seenin figure 2. These clumpsof
electrons travel faster than the electron thermal veloc-
simulation would be required to describethe lunar wake
when the IMF is at oblique angles,and bulk flowsother
than
in the solar wind flow direction
as the simulation
is performed in the rest frame of the solar wind with
zero convection electric field. The IMF will pass almost
unaltered through the moon since the moon can be assumed to be a perfect insulator and therefore no currents can flow to changethe field. The simulation then
begins with a void in an otherwise Maxwellian distribution
of ions and electrons
which mimics the removal
of solar wind particles by the moon. Full shadowingis
ity sinceit will be the most energeticelectronswhich
escapewhen an edge of the potential well is lowered.
These perturbations in electron density also appear in
the electricfield (not shownhere). The phasespace
of the electrons showshow the potential wells capture
the electrons and confine them to the two regions in
the wake (Figure 4). Initially, the electronsmovetowards the centre of the wake, setting up the two potential wells. Note that these potential wells are selfconsistent:trapping the electronswhich set them up.
Thesewellstrap all but the mostenergeticelectrons;the
appropriatesincethe electronand ion gyroradii(95m latter form two streams at the wake centre. This twoand 417m) are significantlysmallerthan one lunar ra- stream distribution is unstable and subsequentlybegins
dius (•1738km). Possiblecrustalremnantfieldswhich to mix in phase space, generating phase space vortices
are of the order of the IMF
at the lunar
surface are ne-
glected in this simple 1D simulation. As we move with
the solar wind, simulation run time can be equated to a
distance behind the moon, effectivelybuilding up a 2D
picture of the wake, provided the IMF and solar wind
flow remain
constant.
We choose an ion to electron
mass ratio of 20 in order
to evolve both ion and electron dynamics. This alters
the simulation's ion plasma frequency and ion thermal
velocity. Since the actual solar wind velocity is 25 times
the ion thermal velocity vthi, the simulation also requires this relationship. Thus, the distance behind the
moon is given by, X = Vconvt,where Vconv= 25vthi and
and creating more potentials which trap the electrons.
These vortices grows as more electrons are captured,
eventually coalescingwith each other. The asymmetry
is due to the nonlinearity of the fully evolved electron
two-stream instability, resulting in one of the vortices
dominating the others.
The characteristic ion time scales are significantly
longerthan the electrontime scales,due to their higher
mass,so the ion dynamicsevolvein the presenceof the
electric potentials created by the electrons. The potentials exclude all but the high energy tail of the ion
distributionfrom the wake;theseform ion beams(Fig-
ure4). After160•;e1 or 7 RL,theionbeams
meetin
t is measured
in plasma
periods,
a•e1. Thisisequiva-the centre of the wake and flow past one another. This
lent to X - O.0437RLt where t is in plasma periods and
RL is the radius of the moon, i.e. half the width of the
void. In reality, vtne> I/cony> Vt,,i; this may affect
the magnitude of the potential and effectively rescales
X = Vco•vt.
With periodic boundary conditions, the simulation
run time is limited to the time for the most energetic
electrons
toreach
theboundaries
andis103•;e1witha
createsa counter-streamingion beam distribution as de-
tectedby WIND [Ogilvieet al., 1996].The beamsare
subsequentlydistorted, howeverhere this is not due to
the ion two-stream instability, but is controlled by the
electric potentials created by the electron two-stream
instability. The timescalefor phase mixing due to ion
two-stream
instability
wouldbe • 6 x 103a3•e
1 in our
simulation. This processdoes not have su•cient time
2048 grid cell simulationbox (grid cell = Debyelength, to thermalize the ions in our simulation. Fluctuations in
AD) and a void of 256ADin the centre(corresponding ion density with correspondingfluctuationsin electric
to a boxlength
of 214.5electron
gyroradii,
8.4•). field (not shownhere) are alsofound travellingaway
Largervoids(up to 512AD)havealsobeenconsidered. from the wake structure close to the ion sound speed
and the ion thermal velocity. These coincidewith times
3.
Results
of the
Simulations
Figure 2 showshow the electron density evolves.The
age of the simulation slice is equated to a distance be-
when the ion beams exit the wake.
Further simulations have been carried out using a
larger initial void. Rather than a 256ADvoid as shown
here, a 512At• void was used. With this alteration, the
BIRCH
AND
CHAPMAN:
PIC SIMULATIONS
OF THE
LUNAR
WAKE
221
0.5
-2
-4
.......
0
0
5
10
15
20
25
-X (RL)= V
cony
30
35
40
45
xt
Figure 2. Electrondensityevolutionshowsdistancealongsimulationbox Y(Rz,) againsttime (distancebehind
moon)X(Rz,) with densitynormalised
to the ambientsolarwind density.The simulationbox extendsto -F8Rz,.
electronsstill moveinto the wake,then set up two potentials at the edges,the samesizeas previously.Then
the samecounter-streaming
electronbeamsare set up,
howeverthere is more spacebetweenthe edgepotentials, allowingmorephasespacevorticesto be generated
by the electrontwo-streaminstability. These vortices
grow and coalesceto fill the wake. The ion beam evolution againis controlledby the potentialgeneratedby
the electrons.Giventhat the moonis about 3.5 x 10sAD
in diameter, one would expect many phasespacevortices to be generatedbetweenthe two edgepotentials,
which grow and coalesceto fill the wake.
150w• in thesimulation.
Figure5 shows
thesimulationafter150w• whichcanbe compared
withthe
data from WIND; in particular, figure I in Ogilvie et
al., [1996],whichshowsplasmaand magneticfield parameters observedduring the lunar wake crossing.
The simulated ion density is in good agreement with
the WIND data given the caveat that the solar wind ion
density dropped while WIND was in the wake, whereas
the
simulation
stant
assumes
ion and electron
that
the solar wind
densities.
WIND
Electrons
4. Comparison
has con-
also detected
Ions
of Results with WIND
Observations
The WIND spacecraftpassedthroughthe wakeabout
6.5 RL behind the moon, which can be equatedto about
-5
5
•
•
0
E-5
5
o
'-'
."=---5
2
-•-
0
o
5
>
0
0
-2
4O
2
Y(R
L)
-2 0
-X(RL)
-1
0
1
Figure 3. A stackplot of electricpotentials(in unitsof
Y(RL)
eV) calculatedfrom electricfield data, plottedagainst
distancealongsimulationbox Y(RL) and time transformedto a distancebehind the moon X(RL), begin- Figure 4. Cut in electron and ion phasespace;plotted
are positionsalong simulationbox Y(R•) againstve-
ningat 40u•t - 1.75Rz,,
thenevery50w• - 2.19Rz,.locitiesalongsimulationbox(vy, in unitsof 10dms
-•),
The simulated obstacle is •1400
times smaller than the
moon, so that the total potential drop is substantially
less in the simulation.
beginning
withthetopplotat 0• • - 0R•, thenevery
100w•- 4.38R•.Ionvelocities
multiplied
byV•.
222
BIRCH AND CHAPMAN:
PIC SIMULATIONS
OF THE LUNAR WAKE
100
ß Dynamics of the ions are governedby electric potentials created by the electrons.
10-2
ß Only the most energeticions enter the wake, creating counter-streamingion beams.
._
z
• "-'"
1 02
In comparison,WIND detected:i) ion and electron
rarefactionwaves,ii) exponentialdensitydecreases
from
the wake edgeto the centre,iii) counter-streaming
ion
beamsin the wake, iv) increasedelectrontemperature
in the wake and v) electrostaticand electromagnetic
10•
'7
•
E
ß Increased electron temperature in the wake.
Te
1
waves in and near the wake.
0
Variations in solar wind parameters while WIND was
passingthrough the wake highlight a limitation of these
i
$$1
$
i
i
simulations. The anglebetweenthe IMF and solarwind
-2
-1
0
1
2
flow varied between5ø and 60ø, with the magnitudeof
Y(RL)
the IMF also changing. This comparesto the simulaFigure 5. Simulationresultsat 6.5 RL showing:Ion tions where the angle is constrainedto be 90ø and magdensity(normalisedto the ambientsolarwind ion den- nitude constant. In order to simulate more closelythe
sity), ion / electrontemperatures
(unitsof 103K)and solar wind parameters during WlND's encounter with
ion speed(unitsof 106ms-l), plottedagainstdistance the lunar wake, and other conditions, two dimensional
along simulation box Y(RL). For comparisonwith simulationsare being undertaken.
WIND data, in particularfig. 1 in Ogilvieet al., [1996].
Acknowledgments.
This work was funded by PPARC
and HEFCE and included usageof the GRAND and CSAR
the rarefactionwaves,in agreement
with the analytical T3E
approachby Ogilvieet al., [1996]andwith simulations.
facilities.
The observedand simulatedtemperaturesare in rea- References
sonableagreement. The electrontemperaturerisesin
the wake accordingto both WIND and the simulations. Bale, S. D., et al., Evidence of currents and unstable particle distributions in an extended region around the lunar
WIND foundthe electrontemperaturepeaksnear the
plasma wake, Geophys. Res. Left., 2•, 1427-1430, 1997.
centrewhereasthe simulationscan resolvepeakswhere Bale, S. D., Shadowedparticle distributions near the Moon,
J. Geophys. Res., 102, 19,773-19,778, 1997.
the phase space vortices are created. The observedion
temperaturesare difficult to comparesincethe ion dis- Bosqued, J. M., et al., Moon-solar wind interaction: First
resultsfrom the WIND/3DP experiment,Geophys.Res.
tribution at this time is essentiallytwo beams.The ion
Left., 23, 1259-1262, 1996.
distributionsampledby WIND clearlyshowscounter- Farrell, W. M., et al., Upstream ULF waves and energetic
streaming beams, which we also find in the simulation.
electrons associated with the lunar wake: Detection
of
precursor activity, Geophys. Res. Left., 23, 1271-1274,
5.
1996.
Conclusion
By usinga highresolutionsimulation,weareable,for
the first time, to fully resolvethe electronphasespace
dynamicsand thus obtain a morecompleteunderstanding of the ion dynamicsin the lunar wake. This yields
Farrell, W. M., et al., Electrostatic instability in the central
lunar wake: A processfor replenishingthe plasma void?,
Geophys. Res. Left., 2•, 1135-1138, 1997.
Farrell, W. M., et al., A simple simulation of a plasma void:
Applications to Wind observationsof the lunar wake, J.
Geophys. Res., 103, 23,653-23,660, 1998.
more complete comparisionswith WIND observations.
In the high resolution simulationswe find:
Kellogg, P. J., et al., Observationsof plasmawavesduring
a traversal of the moon's wake, Geophys.Res. Left., 23,
1267-1270, 1996.
ß Ion and electronrarefactionwaves(alsosimulated Ogilvie, K. W., et al., Observationsof the lunar plasma
wake from the WIND spacecraft on December 27, 1994,
by Farrell et al., [1998]).
Geophys.Res. Left., 23, 1255-1258, 1996.
C. J., et al., The lunar wake at 6.8 RL: WIND magß Two potential wells on both edgesof the wake, Owen,
netic field observations, Geophys. Res. Left., 23, 1263trapping the majority of the electrons.
ß Time fluctuations in the potential correlatedwith
the release of bursts of electrons from the wake
with associated
E fluctuations.
ß Electrontwo-streaminstabilitytrappingelectrons
in the centre of the wake.
1266, 1996.
P. C. Birch and S.C. Chapman, Spaceand Astrophysics
Group, Physics Dept., University of Warwick, Coventry,
CV4 7AL, UK. (e-maih birch@astro.warwick.ac.uk)
(ReceivedJune 27, 2000; revisedSeptember6, 2000;
acceptedSeptember20, 2000.)