JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 100, NO. A9, PAGES 17,189-17,203,SEPTEMBER 1, 1995
One- and two-dimensional
simulations
of whistler mode waves in
an anisotropic plasma
P. E. Devine and S.C. Chapman
SpaceScienceCentre,Schoolof MathematicalandPhysicalSciences,
Universityof Sussex,Brighton,England
J. W. Eastwood
AtomicEnergyAuthority,CulhamLaboratory,Abingdon,Oxfordshire,England
Abstract. We presentresultsfrom self-consistent,
one- andtwo-dimensional,
electromagneticsimulationsof the electronwhistlermodeinstabilityrelevantto the near-Earthnightside
plasmasheetregionduringgeomagnetically
disturbedtimes.Specifically,we studythe evolution of energetic,anisotropic(T• > T#) electrondistributions
that areinjectedinto the nightsidering currentregionat geomagnetically
disturbedtimes,the resultinggrowthof electron
whistlermodewaves,andsubsequent
electronpitch anglediffusionvia electronwhistler
wave-particleinteractions.Growthof whistlermodewavesfrom an initial pitch angleanisotropy (T•. -- 4 T#) is studiedin the strongpitch anglediffusionregime(definedashavingscatteringtimesmuchshorterthana typicalelectronbouncetime in the nearEarth'sdipolaffield).
The quasi-linearandsubsequent
nonlinearevolutionof wavesandthe corresponding
migration of electronsin velocityspaceis followedovertimescalessuchthation motionmay be
neglected.Our simulationscontainwave frequenciesandgrowthratesthat are a significant
fractionof the electrongyrofrequency(co- 0.5COc•,
¾- 0.1(l)cebeingtypical) andthe simultaneousevolutionof wavespropagatingbothparallelandnonparallelto the ambientmagnetic
field direction.Effectsdueto thesearenot usuallyaccountedfor in applications
of quasi-linear theoryto the problemof electronwhistlerwave-particleinteractions,sothat our self-consistentsimulationsof the electronwhistlerinstabilityprovidean importantinsightinto the
applicabilityof quasi-lineartheoryto the velocity spacediffusionof electronsdue to electron
whistlerwave-particleinteractions.
We examinethe dependence
of whistlermodewave
growthrates,nonlinearwave modesaturation,andpitchanglediffusionrateson the [5value
of the hot electronspecieswhichcontainsthe resonantpopulation,andwe comparethe differencesbetweenresultsof one- andtwo-dimensional
simulations.
In general,we havefound
significantlylarger averagegrowthratesin a one-dimensionalthanin a two-dimensional
geometry,by up to a factorof-2-3, with the differencebetweensuchgrowthratesbecoming
largeras [5increases.As a result,we find pitch anglediffusionratesare significantlylarger
(by up to a factorof- 10) in a one-dimensional
geometry,andpitch anglediffusionrates
increaseas [• increases.During the early wave growthperiodof the instability,pitch anglediffusionratesD s havebeenfoundto scaleto magneticwaveenergyB2w,
approximately
Ds o•B2w
' We alsoshowthatin the self-consistently
evolvedsystem(containingseveralwave
modes),thepitch anglediffusionis still consistentwith a usualquasi-linearapproachin which
resonantparticlesare takento diffusealongsurfacesof constantwave frame energy.
Introduction
Magnetosphericwhistler mode wave-particle interactions (WPIs) have received much attention since Dungey
[1963] suggestedthat the pitch angle diffusion driven by
lightning-generated
whistlerwavescould occurfor radiation
belt electrons.Kenneland Petschek[ 1966] proposedthat the
radiationbelt'strappedparticlelossconedistributions,
which
are anisotropicin velocity space,may themselvesgenerate
whistlermode waves,which then interactwith the particles
via WPIs.
Copyright1995by theAmericanGeophysical
Union.
Papernumber95JA00842.
0148-0227/95/95JA-00842505.00
Considerablework has addressedthe pitch angle diffusionof electronsdue to whistlermodewave-particleinteractions. Quasi-lineartheory has been applied to predict the
growth of whistler waves from and the effect of whistler
mode wave-particle interactionson anisotropicmagnetosphericelectrondistributions(havinga characteristic
temperatureperpendicularto the magneticfield (T_•) exceeding
thatparallelto the magneticfield (T//)) [e.g.,Rouxand Solomon, 1971]. Applicationof quasi-lineartheory to the pitch
anglediffusionof electronshas often beenrestrictedby the
followingassumptions:
(1) smallwaveamplitudes(i.e., in the
limit of growthrates ¾-• 0), (2) usuallywith wave frequen-
cies tO<<tOce
(electrongyrofrequencytOce),(3) usually
neglectingenergydiffusionof electrons(purepitchanglediffusion), and (4) a wave spectrumusually containingwaves
propagating
parallelto the magneticfieldonly.Suchapplica17,189
17,190
DEVINE ET AL.: ONE- AND TWO-DIMENSIONAL
WHISTLER SIMULATIONS
tionhasfoundsuccess
in describing
thepitchanglediffusion electronwhistlerinstabilityas is the focusin thispaper(e.g.
(andprecipitationrates)of electronsinsidetheplasmasphere, Matsumotoet al. [1984] investigatedthe generationof elecwhereassumptions
(1)-(3) are reasonablywell satisfied[e.g., trostaticwavesin the presenceof an energeticelectronbeam
electronwhistlerwave).We address
Lyonset al., 1972]. Althoughmostquasi-linear
applications anda monochromatic
haveretainedrestriction(4) above,the importanceof includ- the quasi-linearand subsequentlynonlineargenerationof
ing a full (nonmonochromatic)
wave spectrumcontaining whistlermode waves due to anisotropicenergeticelectron
in an initially unperturbed
background
magnetic
bothparallelandnon-parallel
propagating
waveshasbeen distributions
field,thesubsequent
e•01ution
of waves
andthedistribution
stressed
[Lyons
etal., 1972].
andthe differences
introduced
by a 2-D geomeOutside the plasmasphere,the validity of these usual of electrons,
assumptions
is lessclear,particularlyduringgeomagneticallytry. This simulationstudy principally concernsthe electron
disturbedtimes. During a substorm,populationsof electrons whistler instability in two dimensionsfor the first time. A
of pitchanglediffusionratesin sepaaretransported
fromthemagnetotail
intothenear-Earth
ring combinedinvestigation
currentregion,gainingin energyandanisotropy(a character- rateresonantregionsof velocityspaceandvelocityspacedifistictemperature
perpendicular
to the magneticfield exceed- fusionsurfacesis also presentedto addressthe questionof
ing thatparallel, Tñ > T//) [e.g.,Ashour-Abdalla
and Cowley, the applicabilityof quasi-lineartheoryto the velocityspace
of electrons
duetowhistler
WPIs.A dependence
of
1974]. Whistler mode wave growth shouldbe enhancedat diffusion
suchtimes, leadingto a largerrate of pitch angle scattering electronpitch anglediffusionrateson the wave energydeninto the lossconeand thereforean increasedauroralprecipi- sityin the simulationsystemis obtained,andthe pitchangle
tation. Krernser et al. [ 1986] observedan increasein electron dependence
of the pitch anglediffusionrate in eachof three
precipitation
ratecorresponding
to a periodof enhanced
elec- separateresonantregionsof velocity spaceis investigated,
tron whistler mode wave intensity during a magneticsub- both of which allow for a self-consistentinvestigationof
storm, the enhanced precipitation therefore seemingly quasi-lineartheory predictionswith regard to the electron
consistentwith pitch anglediffusiondue to electronwhistler velocityspacediffusion.
We first describe the simulation method and our series of
waves (at enhancedlevels due to growth from anisotropic
electrondistributions).
Whistlermodewaveenergypeaksat simulationstudies.Resultsfrom thesestudiesare then pre-
muchhigherfrequencies
(relativeto thelocalelectrongyrofrequency)thaninsidetheplasmasphere,
frequenciesco~0.25-
0.5COce
beingtypicaloutside
theplasmapause.
Thevalidityof
the usualassumptions
suchas purepitch anglediffusionand
parallelpropagating
wavesneedsto be investigated,
a quantitativeunderstanding
of the degreeto whicha usualquasi-linear applicationexplainsthe electronpitchanglediffusionin
thisregimeis of primeimportance
to thequestion
of thedisturbed-timeaurora,sincethe electronwhistlerinstabilityis
believedto be an importantmechanismby which energetic
electronsareprecipitatedto form thedisturbed-time
nightside
sented and discussed.
Simulation
Details
SimulationConfiguration
A two-dimensional,relativistic, electromagnetic,selfconsistentparticle-in-cellsimulationcode has been developed using the virtual particle (VP) method of Eastwood
[ 1991].All threevectorcomponents
of electromagnetic
fields
E, B, and velocityv are retainedas variablesof two spatial
aurora.
dimensionsand time. Electronsare representedby a set of
by a
Computersimulationrepresents
a usefultool with which computationalparticles,and here ions are represented
to examinethe quasi-lineardiffusionand subsequent
nonlin- fixed,chargeneutralizingbackground,althoughthisneednot
ear evolution of unstable distributions. Previous whistler
be so for otherapplications.Effectively,our implementation
instabilitysimulationshave been performedin one-dimen- of the VP scheme differs from the more traditional 2-D elecsion(I-D) [e.g., Ossakowet al., 1972; Cuperrnanand Salu, tromagneticparticle-in-cellcodes [Birdsall and Langdon,
from parti1973; Pritchett et al., 1991]. Usually, wave propagationin 1985] only in the methodof currentassignment
suchsimulationsis restrictedto be parallel to the magnetic cles to the grid, so that an appropriategrid cell size is Ax
field, andthereforeresonantWPIs to the n = -1 principal equaltotheDebyelength3•De.
Thisavoids
possible
nonphyscyclotronresonance
[Stix, 1962].It is possibleto study ical instabilities[Birdsall and Langdon, 1985], while still
obliquelypropagatingwaves (nonparallelto the magnetic permittingall physicalwaveswith scalelengthsof the order
field) in a 1-D simulation [e.g., Cuperrnanand Sternlieb, or greaterthana Debyelength(e.g.,Langmuirmodes,elec1974; Zhang et al., 1993]. This allows the effects of a tron whistler waves).
restricted set of resonances that include n •
-1
to be inves-
Ourchoiceof doublyperiodicboundary
conditions
effec-
tigated;for example, Zhang et at. [1993] investigatedthe tivelyquantizes
the wavevectorsk = (kx,ky) allowedin
growthfrom an anisotropic
electronbeamof variouselectro- the system, resulting in a discrete set of "wave modes":
staticandelectromagnetic
wavemodesat variouspropaga- kx = _+(2•nx)/L x, where nx = O,1, 2..... Nx/2, with Nx
tion anglesusinga seriesof 1-D simulations.However,wave thenumberof grid cellsin thex direction(a similarsetresults
propagation
is still restrictedto a singlefixeddirection(rela- for ky).We choose
system
lengths
(Lx,Ly) to ensure
the
tiveto themagnetic
field)in suchschemes.
Theperiodic
two- allowedwavemodesadequately
covertheregionof k space
dimensional
(2-D)simulations
presented
hereallowwave thattheproblem
of interest
demands
(seeFigure1, k =
propagation
angles
bothparallelandoblique
to theambient (kx,ky)= (k/l,kñ;seeFigures
2 and3 fora typicalresolution
magnetic
fielddirectionto beincludedin thesamesimulation of thelinearlypredicted
growthregion).Theevolution
of ini-
andtherefore
allowall otherpossible
resonances
(n • -1)
tiallyanisotropic
distributions
canberepresented,
although
in additionto the principalresonance.
Previous
two-dimen- theperiodicboundaryconditions
u•ed allowno sources
or
sionalelectromagnetic
simulations
have not dealt with the sinksof particles.
However,in oursimulations
thenumberof
DEVINE ET AL.: ONE- AND TWO-DIMENSIONAL
WHISTLER SIMULATIONS
17,191
pressure).Parameters
are listedin Table 1. Eachof the four
studieswas performedin both 1-D and 2-D. The quantityA
represents
the electronspeciesanisotropy(A = T.L/T#- 1),
k = (kx,ky) = (k.,
and•// = n,kaT///(B02/2g0)
isjusttheparallel
• value
of
the electronspecies.In all studies,the densityof the simulation electron specieswas taken to representan electron
plasma
frequency
of(o•,=105
rads-1.
In all runs,the numberof simulationparticleswaschosen
to ensurethat the velocity distributionf(v) was represented
adequately.
A temperature
ratioof T.L/T# = 4 is somewhat
large in magnetospheric
terms.This was chosento ensure
that whistlermode growth rates exceednonwhistlernoise
level growthrates.For the purposeof this study,thisnoise
consistsof statisticalfluctuationsin computationalparticle
populations,electronplasma oscillations,and the bremsstrahlungradiationpresentin the system(and all of which are
enhancedin the simulation,relative to a real system,due to
Figure 1. The simulationconfiguration.
usingfar fewer computationalthan there are real particles).
Nonwhistlernoisewas kept to a reasonablelevel using 144
computational
particleslying withina typicalmagnetospheric computational
particlesper cell on a 128 x 128 cell grid in 2losscone(of size •0 = 5ø) is foundto be sufficiently
small D runs,and 256 per cell on a 128 x 1 cell grid in 1-D runs.
(-0.1%) suchthat our resultsare not foundto be affectedby Eachof the studieswasrepeated(in both 1-D and2-D) using
includinga typicalmagnetospheric
lossconesink.
an isotropicelectronf(v) of the sameoverall temperature.
The XWHAMP code, a graphicallyenhancedversionof Theseisotropicrunsare stableto the whistlerinstability,and
the originalwaves in homogeneous,
anisotropic,multicom- thereforereproducethe abovenoisefieldsonly andat similar
ponent plasmas (WHAMP) code [Ronnmark, 1982], was levelsthat are presentin the relevantstudy.This allowsus to
usedto allow comparisonof simulationresultsandlinearthe- makeanestimateof noiseeffectswith regardto ourresults.A
ory. XWHAMP was usedto solvefor the linearly predicted lower numberof 144 particles/cellin 2-D nevertheless
results
complexfrequencyco- cor + iT (c%is real frequency,
¾is in a larger total numberof particlesthan in 1-D. The higher
growthrate)over a desiredrangeof k space,giventhe fol- numberof 256 particlesper cell in 1-D gives an improved
lowing physical parameters:electroncyclotronfrequency representation
in velocity spaceover the entire population
COce,
electronplasmafrequencycoe, the parallelelectron (i.e., over the whole system)than would be obtainedin 1-D
velocity
distribution
temperature
kaY//
= 1/2mV,2h#,
and
the with 144 particlesper cell, andthereforeimproves(relativeto
perpendicular/parallel
temperatureratio T•/T# (a bi-Max- 1-D 144/ce11)
the statisticsof any quantitycalculatedoverthe
wellian electronvelocity distributionfunction was used, so whole 1-D system(e.g., our pitch angle diffusion calculathat T•_/T# is the anisotropyof the electrondistributionf(v)). tions).Suchcalculationscan then be more readily compared
An exampleof XWHAMP's resultsis shownin Figures2 and
•
r
between
1-Dand2-D.System
lengths
(Lx,Ly)
werechosen
to
3 respectively,
whichshowthedependence
of (c%,¾)versus ensurethat an acceptableminimumnumberof allowed(disk//(at k•_= 0) andcontourlevelsof thegrowthrate ¾(k) in
wave vectork = (k#, k•) space,for the parameters
of our
study1 (to be describednext). Both real and imaginaryfre•
quenciesin Figures2 and 3 are normalisedto the electron
Kx--4
i
-
5
i
-
i
6
7
8
-
:1
-
i
gyrofrequency
00c•;thisnormalization
will be usedthrough-
0.9
out.Also,for thetemperature
ratioused(T_L/T#--4), growth
ratesare a significantfractionof the real wave frequencies
0.8
(thepeak¾,n,xof ¾(k) is approximately
20% of the real
0.7
wavefrequencyat the peakin Figure2).
0.6
,
'
ßwavemodesallowedin system
Simulation Studies
0.4
A set of studieshave beenperformedto investigatethe
electronwhistlermodeinstability.The simulationconfigura-
tionis sketched
in Figure1, with an initiallyuniformmag-
neticfieldB0 in thex direction.
In eachsimulation,
therun
timewaschosento be 10xceelectrongyroperiods,
thatis, run
timemuchlessthananiongyroperiod.
The setof four studiescontainsa single,hot (tensof keV)
resonant electron species with an initially anisotropic
(T.L/T# -- 4) bi-Maxwellian velocity distributionf(v). The
difference between the four studies lies in the choice of elec-
0.3
0.2
0.1
0.2
k• v,h•/ face
Figure2. Lineartheory
parallel
dispersion
(k• = 0, Ky= 1)
for our study 1. Dotted lines show the growth modesreprevalue of the resonant electron species (where sentedin the simulation (only parallel modes are shown).
Calculationswere madeusingthe XWHAMP code(seetext).
I] = n•kaT/(B•/2g0)istheusual
ratioofkinetic/magnetic
tronspecies
temperature
andthevalueof lB01;
thatis,in the •
17,192
DEVINEET AL.' ONE-ANDTWO-DIMENSIONAL
WHISTLERSIMULATIONS
Ky=l 2
0.9'
i'
3
4
5
6
7
8
' '
'
' '
"
•
"
...
0.005% over the entire run (as expectedfor our boundary
'
... Kx=11
0.8
... Kx=10
0.7
0.6
.
0.5
ß
0.4
conditions
suchthatno lossof energyoccurs),andby 10Zc•
themagneticwaveenergydensityhasreached-3.1% (l-D)
and-1.9% (2-D) of the initial electronspecieskineticenergy
density(althoughin higher [• studies,the 1-D saturation
waveenergyis not alwayslargerthanit is in 2-D). Notethat
theaveragegrowthrateis largerin a 1-D run thanin a 2-D
runwiththesame[•//value(seeFigure4b).FromFigure4b,
takingthe spatiallyaveraged
magnetic
waveenergydensity
2
in the systemto vary as Bwave
•: exp(2•t), the average
growthrateof themagnetic
waveamplitude
in 1-D study1 is
estimated
to be •o = (0.076 +0.001)O,)ce,
whilein the2-D
study1 it is •2o = (0.054+0.016)O•ce.
Forlinearcompari-
0.3
sonpurposes,
theXWHAMP-predicted
meanof the growth
ratesof all simulationwave modeslying within the linearly
0.2
predictedgrowthregion(seeFigures2 and 3) is approxii
i
t
i
,
mately0.074{0ce
(l-D) and 0.05300ce
(2-D), so that the
i
0.1• 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
kñvth//
/ O}ce
growthin themagnetic
waveenergydensityat earlytimesis
to a goodapproximation
the meangrowthrate from linear
theory.This meansthereis negligiblewavemodecoupling
duringtheearlytimestages
of theinstability(in both1-D and
Figure3. Contour
levelsof thelinearly
predicted
growth 2-D), and hencethe differencein 1-D/2-D growthratesis
rate¾(k) /O•ce
in thewavevector
planek forstudy1.Dotted consistentwith linear theory also. In a 1-D geometry,all
lations
weremadeusingtheXWHAMPcode(seetext).
wave energyis constrained
to one of the allowedparallel
propagating
wave modes,which have relativelyhigher
growthrates(for the samevalueof k//) thanthenonparallel
crete) wave modesfell within the linearly predictedgrowth
wave modes(which sufferfrom Landaudamping[e.g., Kennel & Petschek,1966]) that are presentin a 2-D geometry,
linesshowwavemodesrepresented
in thesimulation.
Calcu-
regionin k space(typically>5 modesin I-D, >10 modesin
hencethe higheraverage1-D growthratesandwavelevels.
Theanisotropy
consequently
fallsfasterin theearlystages
of
the k spacegrowthregion(¾> 0) to be inferred(seeFigures the 1-D runthanin the 2-D run (seeFigure4c). This is dueto
2 and 3).
the higherwave levelsin 1-D than in 2-D over that time
period.
2-D); use of the XWHAMP code allowed the resolutionof
Results
Growth of Dominant
We will first discussin detail resultsfrom study1 to com-
pareI-D and2-D resultsat a single[•//.
Evolution of Magnetic Wave Energy in the System
Initially, the magneticwave energyin the systemgrows
exponentially(seeFigure 4b for the magneticwave energy
densityin the simulationsystem),that is, accordingto linear
theory.The growth of wave energyis accompaniedby a
decreasein both the electronspecieskinetic energyand the
electronspeciesanisotropy(seeFigure4); that is the system
movestowardisotropyand hencea decreasing
amountof
"free" energy available for wave growth. Eventually (by
t= 5- 6•ce(l-D) and8-9•ce(2-D)), thewaveenergysaturates at an almost constantlevel, see Figure 4b. The total
energy in the system is conservedto within better than
Wave Mode
Lookingfirstat thegrowthof wavemodeenergies
in the
1-D study1. Earlyin therun,thewavespectrum
is quiteturbulentin thatmanywavemodesgrowin energy(seeFigure
5). The convenient
dimensionless
Kx valuesassigned
to the
wavemodesin Figure5 arerelatedto the unnormalized
parallelwavenumber
k// by Kx = 1 + (Lx Ik//I/ 2 n). As no electronbeamis presentin theelectronvelocitydistribution
f(v),
growthratesof wavesareexpected
to be symmetric
parallel/
antiparallel
to the background
magneticfield;thatis, wave
modeshavingthe samek = Ikl, but oppositelydirectedk,
shouldgrow at the samerate (within noiseuncertainties).
Eachof the modeenergiesshownin Figure5 containsboth
theparallel/antiparallel
energyin thetwooppositely
directed
wavemodeshavingthe sameIkl (eachhas k•_= 0 (dimen-
sionless
Ky= 1+ (LyIk•.l/2•:)= 1),since
weareconsidering
Table 1, PhysicalParametersfor the Studies
Parallel Electron
Study
Temperature
Anisotropy
lB0I,nT
•//
O•ce
(l)pe
k•tr//, keV
A
8.82 keV
2.89
284
0.14
0.5
3.56 keV
2.95
100
0.45
0.176
6.91 keV
2.91
70
1.79
0.123
8.81 keV
2.89
56.9
3.45
0.1
DEVINE ET AL.: ONE- AND TWO-DIMENSIONAL
o.ox
3.8
WHISTLER
SIMULATIONS
17,193
linear viewpoint(thereforenot entirely correctwhen wave
amplitudeshave grown significantly),there is a maximum
(a)
wavemodekma
x allowedto grow,depending
on the current
anisotropy,
with kma
x givenby Ossakow
et al. [1972]
-- ;t)
= %'[
10'8
c
-
10-lo
Hence, as kma
x decreaseswith decreasinganisotropy
(A = Tilt # -1), then as the simulationprogresses(A
0
2
4
6
8
10
decreasesin time) the rangeof availableresonantk modes
(c)
decreases.
As kmaxfalls belowthe initially availablehigher
Ikl modes,there is a shift of magneticwhistlermodewave
energytowardslower Ikl modes.This canbe seenfrom Figure 5, with magneticenergyshiftingin time from modeKx =
2
4
6
8
10
7 • 6 • 5 • 4. Clearly, when the anisotropydecreases
markedlyfrom its initial level, the quasi-linearwave spect/Xc,
trumwill not agreeexactlywith the linearlypredictedspecFigure 4. Variationwith time of (a) kineticenergydensity trum, with the maximumof the spectrumbeing displaced
(KE), (b) averagemagneticwave energydensity(ME), and toward a lower frequencythan predictedby linear theory.
(by ~ 8'c•.,)in l-D, all
(c) electronspeciesanisotropy
A during1-D and2-D study1. AlsoseenfromFigure5, eventually
10'•
Notethatunits
ofenergy
densities
areJm'3.
modesbut one seem to have saturated,and we are left with a
a 1-D geometry).The resultsshow different wave modes
growing at different rates. Numericallyobtainedreal and
imaginary(growth) wave mode frequenciesfor individual
wavemodesagreefairly well with lineartheory(XWHAMP)
predictions
duringthe early stagesof the run. For example,
this dominant wave mode is not the lowest wave mode
single,dominantwavemode(Kx = 4) in thesystemthatpropagatesparallelto the backgroundmagneticfield. Note that
resolvedby the simulation;that is, we are not observinga
system-related
shift toward ever lower Ikl modesuntil we
reachthelowestpossiblesimulationmode.Sucha final spectrum, dominatedby a singlewave mode,couldfavor saturatheapproximate
growthrates(normalised
to 0•c,)of wave tionof magneticwaveenergyby nonlinearmagnetictrapping
by Ossakowet al. [1972].
modesKx = 5, 6 are0.1 +10% and0.11 +20%, respectively, effects,assuggested
Figure6 showsthe magneticwaveprofileBy (x,t) during
whilethe linearlypredictedgrowthratesfor thesemodesare
0.11 and0.12, respectively.
theearlytime stagesof 1-D study1. Electronwhistlerwaves
As wave energyincreasesand anisotropydecreases,
the propagatingparallelto the backgroundmagneticfield (B0
electromagnetic
wavesright circugeneraltrendis for higherIkl modesto saturatein energyand along•) are transverse
for magneticenergyto shifttowardlower Ikl modes.From a larlypolarized
about
B0, sothatBy(x,t)
isoneof thepairof
10?
,
,
!
,
,
,
,
,
,
•tal
K•4 .............
K•5 .......
K•6 ...................
K•7 xxxxxx
K•8oooooo
'• 10
6
',.•10
s
• 10
4
10•
o
I
I
I
I
I
I
I
I
I
1
2
3
4
5
6
7
8
9
10
t/'Cc,
Figure5. Magneticenergyin resonant
wavemodesKx=4-8 duringthe 1-D study1. Othermodesin thesystemcontainnonwhistlernoiseonly anddo notgrownoticeablyin time.
17,194
DEVINE
ET AL.' ONE- AND TWO-DIMENSIONAL
:i!"?
•
, ..
i ';:.•
:,.4•:,:'
.
WHISTLER
SIMULATIONS
'..
.... %c•,
,,,.....
,.....
't....
'35'
,,,.'..,'
'. :.•:.
', ß' -.,.
",•
";.a('
.,.
.........
...... •':':.,
.
,,'",:.,'
,:::':'"'•
','"->'"
"2'•'
'3.•
.:.•
,-•?•
', ,.•\.'•::':'•
x'•'::•,.
,
',+.
:•,, "• d,.'
.v•:,
'
.•.,',.
'•.',.
,:'.,
20....... t.•.:',
':•::.
','•..%•it•....,,,,
,',•,.,4,
.:,./"::'.
...... ß
:•,:,
:..../.,•,,
• ,,,
•,•..'•
.........
..•:•
•"•:•.
''.'."
.....
•..,..'
"•,•
0,'.'•
•'
•:•'"'.
,.'..,.c':•,...•,.:
",,>
'..•.
',.......
,...1/'
:•' '"..
•' .,.• •'•.•.:•..:!-.:•v•
.,:•i:•:;.':
::•:.';::
."
......
,',,,'•,,,'•,.".•
,.,
'..,"
:":-%3.:
,¾;'":
•.:,'<•'
c..•'..
•.:::,,::•*•:•:t•
.c'v
•:;•
:,,,,,.';l
.x•:,:,:::•:•:•
.:--,
. ....
ß,•,.'
.........
,..,,'..v...:•>,,•'
,.,
?,•'.?•,>,.
v',...•:../i.
......"(•"'"'.
Z• :":.•-o
":*::•.•
.......i:':'--.:;
R.•:•:
::•.'•
•>,".',
' ",..'.•Z"',,'•,'.,
'." \":'.%
,'•"'"'
'<""
' •';,....
'" '.':.:'"•'
,.•'•"".-:",
'•
'-;"';Z."
"'•:/":x..',.".
::::::::::::::::::::::::::
ß
-. •' .. •.....• :,,.........•"-.',:•--'
'..,:..¾,
.....'•.6¾..•.,•.,..•.."?,•::
',,"•.....•"•i''•./,,:•.•
.......
:•.•','. •?,,:•..,•."-'
:•:•':;'•.:;.:•::•:
.
ß............
v:".,..•3.,•:.•
::•:'5'::.:
,.'.'•
:.'t.'::b:•¾?,?•,;?'•
;•:'",.'">
•?*".•"::•:'2':'3:::'"'"..,.
.....
:'",.'-.h•:.•:•!•
i•!•
•
...................
:::"'t"•":':".."-•.
......
'.....',,.•'
.....'"..•
.•:-....?':•
':';4•
.•,,..,x.
I:•:;
,':i•!!:!;
•00
•20
0
x!Ax
Figure6. Magnetic
waveprofile
By(x,t)
during
theearlytimestages
of 1-Dstudy
1.Distance
x ismeasured
inunitsofgridcells(Ax);themagnetic
wavefieldamplitude
ismultiplied
by 3x108
.
circularlypolarizedcomponentsof the magneticwhistler in growingnonparallel(ki • 0) wave modes(early in the
wavefield.Henceboththe growthof magneticwaveampli- run,up to 90% asmuchenergyresidescollectivelyin nonpartude and the shift in time of wave energytowardlonger allel growthwave modesas in parallel growthwave modes
wavelengths
areseenin By(x,t)of Figure6. Theresultantcollectively),thoughof two wavemodeshavingthesamek//
(Kx)butdifferentk• (Ky),theonewithlowerk• (i.e.,the
wave
fieldBy(x,t)
inthesimulation,
isessentially
(excluding
noise
contributions)
thesum
•Bkoe'
(•'r-ø•t+•k)
ofallthedis- moreparallel(lessoblique))tendsto growfaster.This canbe
crete linearly predictedgrowth whistler wave mode waves seenfrom Figure 7, which showsthe growthin magnetic
(eachof amplitude
Bk0,wavevectork, frequency
o)k= wave energy for linearly predicted growth wave modes
{l}rk-['
iYk' initial phase•)k) propagating
both paralleland (Kx,Ky),
eachof thesamedimensionless
parallel
wavenumantiparallel
to B0 (Kx = 4 to 8 in Figure5). Fromsucha setof berKx= 5,anddifferent
perpendicular
wavenumbers
Ky= 1oppositelypropagatingwaves,togetherwith the periodic 6. A similartrendof magneticwave energyshiftingtoward
boundaryconditions
employedin the simulations,
onemay
expecta form of standingwave to eventuallydevelopin
104o
By(x,t),
provided
a fixedphase
relationship
between
theindividual wave modes exists (see Zhang et al. [1993] for the
caseof beatingbetweenlongitudinalelectronplasmawaves).
However,individualwave modesare expectedto be randomlyphased,
withtherebeingnoobvious
phaserelationship
betweenthedifferentmode•)ks. Henceearlyin thesimula- ß•
tion (approximately
t < 3- 41;ce),
whenthe wavespectrum
containsseveralwavemodesof comparable
energy(seeFig-
ure5), theresultant
wavefieldBy(x,t)
in Figure6 doesnot
ß•
appearto be representative
of a stationarywave.Later in the
run, as the wave spectrumbecomesdominatedby fewer
modes,the resultantwave field approaches
the sumof pairs •
of oppositely
propagating
waveshavingroughlyequalamplitude(withinnoiseuncertainties)
andhavinga clearerphase
relationship.
In 2-D runs the wave spectrumis found to be far more
turbulent,sincea far greaternumberof growthwavemodes
arepresent.
Thereisnolonger
therestriction
thatki = 0 (Ky
10e
107
106
10s
i
1040 1
= 1). Instead,a full setof valuesfor the dimensionless
per-
i
3
i
4
i
5
i
6
i
7
i
8
i
9
10
t/xce
pendicular
wavenumber
Kynowresults
asdidforKx.Earlyin
the run, it is found that the region of k-spacethat grows
appreciablyagreeswell with thatpredictedby lineartheory,
with individualwavemodegrowthratesagreeingfairly well
with linearpredictions(as in I-D). Significantenergyresides
i
2
Figure 7.
Magnetic energy in resonantgrowth modes
(Kx,Ky)
having
thesamedimensionless
parallel
wavenumber
Kx = 5 anddifferent
perpendicular
wavenumbers
Ky,during
the 2-D study1.
DEVINE
ET AL.: ONE- AND TWO-DIMENSIONAL
WHISTLER
SIMULATIONS
17,195
lower Ikl modesastime progresses
is seenin 2-D aswasseen
in l-D, with the orderof suchshiftingnow beingfrom domi-
nantwavemodeshavingKx = 6 -->5 -->4. This is bestseen
from Figure 8, which showsthe growthin collectivemag-
netic
waveenergy
ateachKx,summed
overallKy(k.•);that
ß
ß
2
is, we areplottingthe sum•]B•:x•:y overK,.Y' for eachKx. Dur-
ingthe10Xce
runtimein 2-D,wedonotobserve
a finalspectrumdominatedby a single,parallelpropagatingwavemode
in thesystem(aswasseenin l-D). The energyspectrum
late
in therun(after~ 9xc,) is dominated
collectively
by those
107
wave modeshavingKx = 4 (seeFigure8). However,this
energyis sharedroughly equally betweenthe two wave
105
modes
(Kx,Ky)
= (4,1)and(4,2)(seeFigure
9 between
9 and
10x•,).Resultsobtained
by extending
the 2-D run timeto
20Xce(seeFigure9 after10x•) showthateventually
(by 161;ce
) the2-D wavespectrum
becomes
dominated
by a sin-
1050 2
i
4
6
gle, largeamplitude,parallelpropagating
wavemodehaving
i
8
i
10 1•2 1•4 1•6 1•8 20
fiX,.,
(Kx,Ky)
= (4,1);thatis,thesame
eventual
dominant
mode
of
the 1-D studydominatesthe 2-D studybut at a latertime (at
Figure 9.
Magnetic energy in resonantgrowth modes
approximately
8x•.• in I-D, 16x• in 2-D). In both (Kx,Ky)
having
thesamedimensionless
parallel
wavenumber
geometries,
however,thisoccursat a timeoutsidetheperiod Kx = 4 anddifferent
perpendicular
wavenumbers
Ky,during
for whichwe expectquasi-linear
diffusionto occur,aswill be
the 2-D study1.
discussed next.
resonances
with wavemodesof the samek/!(Kx) butdifferent
Pitch Angle Diffusion: Study 1
In orderto investigatethe pitch anglediffusionof electronsdue to the whistlermodewavespresentin the system,
we havecalculated"snapshots"
of the localpitchanglediffusionrate in threeseparateregionsof velocityspace,at varioustimesduringthe simulations.
Theseregionswerechosen
k.• (Ky)arepredicted
(by lineartheory)to havevirtually
identical n = -1 resonant particle parallel velocities
V#res
= (tOr--tOce)/k//,sincethe real frequency(1)
r shows
very little variation(constantwithin -2%) with k.• at a fixed
k!! (for the parameters
relevantto our studies).Hencein our
2-D studies,our threestripsare expectedto containcontributobevelocity
space
"strips"
ofwidth+10• ms'l,centred
at tionsto the local pitch anglediffusionof particlesfrom resov#= -7.84x 107ms'l, -5.15x 107ms'l, and-3.69x 107 nanceswith a whole set of parallel and nonparallelmodes
m s'1.Eachof these
strips
lieswithinthelinearly
predictedwhichhavedimensionless
parallelwavenumbers
Kx = 5, 6,
regionof velocityspacein whichwe expectthereto ben = -1 and7, respectively.Only regionsof velocityspacein which
electron whistler mode resonances with wave modes for
are expectedelectronwhistlerwave n = -1 resonances
were
whichKx = 5, 6, and7 (thesemodesareincludedin Figure5). selectedbecauseonly suchresonances
occurin a 1-D geomeIn l-D, sincewavespropagateparallelto the magneticfield
try for whichwhistlerwavespropagateparallel/antiparallel
to
only, theseresonances
are limited to the threewave modes
the backgroundmagneticfield [e.g.,Stix, 1962], andour aim
Kx=5,6, and7, all withKy= 1 (ork.•= 0). In 2-D,n = -1 is to make 1-D/2-D comparisonsof pitch angle diffusion
1011
•
•
,
,
,
,
,
,
,
rates.
Thestripwidthof +10a m s'1 waschosen
asnarrow
,
Kx=4
101ø
-- Kx--5
-'- Kx--'6
•
..... Kx--7
•'
.-...... .... "
"" -
"'
10g
f o Kx--8 Total
10s
10?
enoughto allow us to distinguishseparateresonances
with
wavemodeshavingdifferentKx, whilecontaininga sufficient
numberof particlesto producemeaningfulstatisticsin our
calculations(typically severalhundredparticleslay within a
1-D strip and more than this in 2-D). Henceforth,we shall
refer to thesethree stripsas mode 5, mode 6, and mode 7,
respectively.
To calculatea snapshotof the local pitch angle
diffusionratefor a particularmode(henceforth
kx = Kx=5, 6,
or 7), overa time interval(t, t + x), we first selectall particles
lying within the modeat time t. At time (t + x), we calculate
the changein pitch angle (Ao• in rad) over the period x for
each particlethat was in the mode at t. A normalizedlocal
pitchangle
(p.a.d.)
rateD• fora mode
kxisthencalculated
I 0s
as:
10s
0
,
I
i
2
,
3
i
4
,
5
i
6
i
7
i
8
,
9
(2)
10
wheretheaveraging{...) is performedoverall theparticles
Figure 8. Collectivemagneticenergyat eachKx, summed in the velocityspacestripcorresponding
to the modeof inter2
totheelectron
gyrofreoverallKy,forlinearly
predicted
resonant
growth
modes
dur- est,andD•n•(inrad), isnormalized
ing the2-D study1.
quencyfce. Our choiceof x = 20At simulationtime steps
17,196
DEVINE ET AL.' ONE- AND TWO-DIMENSIONAL WHISTLER SIMULATIONS
(At) representsa small fraction of an electrongyroperiod phasedependentas well as v// dependent.Mean velocity
(typically
1:<1:c,/8),
andwecalculate
•)ekx
foreach
ofthe changesare small, hence individual energy exchanges
threemodes,
ro•ughly
3 times
pergyroperiod.
Thusaseries
of (betweenwave andparticle)are small(basedon the average
snapshots
of Dekx is obtainedby choosingx to be much kineticenergyper particleof the entire electronspecies,a
shorterthanthetimeintervaloverwhichwaveenergyis seen typicalelectronexperiencesa net lossin its kineticenergyof
to grow (i.e., muchshorterthanan electrongyroperiod-see -1% for the chosentime period, which accountsfor the
Figures4 and5 for wavegrowth).Notethatuseof a standard growthof waveenergyoverthe sameperiod).As notedearquasi-linearapproach[Kenneland Engelrnann,1966] leads lier, the wave spectrumis turbulentduringthis time period,
to a diffusionratethatappliesto time periodsof the orderof and wave-particleinteractionswith sucha spectrumof ranasa stochastic
an electrongyroperiod
or more(overwhichthe waveenergy domlyphasedturbulencecanbe characterized
is assumednot to grow significantly),sincethe equation processdescribedby a diffusionequation(see,for example,
describingthe time evolutionof f(v) is Larmor-phase-aver- [Kennel& Petschek,1966] for a treatmentof pitchangledifaged.In our studies,waveenergydoesnotgrowsignificantly fusion).In our simulations,the turbulentspectrumis effecover the period 1: for which our snapshots
of the p.a.d.are tively representedby a collectionof discretewave modes.
calculated.Hence, althoughour p.a.d. rates are calculated However,ourresolutionof the turbulentspectrum(-5 modes
over muchshortertime scalesthan wouldbe doneusinga in I-D, >10 in 2-D) is sufficientlylarge that the quasi-stoquasi-linearapproach,we shallseethatcertainfeaturesof the chasticprocess
occursin ourchosenquasi-linear
regime.
p.a.d.arewell-described
usingsuchan approach.
After t = 4.5Xc•,the particledynamics
becomestrongly
Calculated
p.a.d.rates•)akxat various
timesduring
the nonlinear,with much largerchangesin particlevelocityand
earlygrowthperiodof waveenergyin boththe 1-D and2-D
energy occurring,and our estimate of p.a.d. in localized
runs of study 1 are shownin Figure 10. Figure 11 shows regionsof phasespaceis no longerapplicable.In thisnonlinvelocityspacetrajectories,in the simulationrestframe,of 30 ear regime,the 1-D wave spectrumis essentiallydominated
of the particleslying within mode7 at t = 0, duringthe 1-D by one/twolargeamplitudewave modes(seeFigure5), and
study1. FromFigure11we seethataftert = 4.5Xce
individ- the stochasticdescriptionno longerapplies.Instead,some
ual particletrajectories
in velocityspaceindicatea morepro- particlesnear resonancemay become "trapped"for some
nouncednonlinearbehaviour.We identify the end of the time aroundresonance,seenas oscillationsin parallelvelocperiodof quasi-lineardiffusionby inspectionof individual ity v// at a "trapping"frequencyandcenteredaboutthe resoparticlephasespacetrajectories,shownas a verticaldashed nant parallel velocity [e.g., Sudan and Ott, 1971].
line in Figure 11, andonly presentp.a.d.ratesup to thistime Oscillations
in v// areevidentin someof the particletrajectoin Figure10.In ourchosen
quasi-linear
regime(t < 4.5x•.e), ries of Figure 11a, and may indicatethat trappingeffects
particletrajectoriesare characterizedby slow variationsin becomeimportantlater in the run. At a given time, someof
velocityand energy,thoughsomeparticlesmove in v space theseparticlesappearto be trapped,while othersdo not.
on a muchfastertimescalethan the "bulk" of particlesover Althoughthis chosengroup of particlesare all initialized
the sametime period,therebysuggesting
that resonanceis with the sameparallel velocity v/l, they have differentper(a) Mode5
0.04
o.o• o •DJ
o
o•.•
1
1.5
2
2.5
3
35
4
4.5
3
3.5
4
4.5
(b) Mode6
0
0.5
1
1.5
2
2.5
(c) Mode7
0.08
/
0'06
l-
........
x
0.04•
• • X
o.o•
0
•
0.5
1
1.5
2
2.5
3
3.5
4
4.5
t/'•ce
Figure 10. Calculatedpitch anglediffusionratesfor (a) mode5, (b) mode6, and (c) mode7 regionsof
velocityspacein 1-D (crosses)
and2-D (opencircles)study1.
DEVINE
ET AL.' ONE- AND TWO-DIMENSIONAL
x 10•
i
i
i
,
!
SIMULATIONS
17,197
2x•.
e, nonwhistler
noisein b• renders
the1-Dand2-D
(a)
i
WHISTLER
i
resultsindistinguishable.
3. Nonwhistlernoiseeffectspreventusfrom distinguishing the whistler-produced
p.a.d. rates in the three different
regionsof v space.Sincewe haveusedmorethan 100 parti-
I
I
I
cles/cell in our simulations (almost 2.5 million total number
in 2-D), andwith thenetnonwhistler
noisein thecodehaving
-2
0
I
I
I
I
I
2
3
4
I
I
I
I
I
5
6
7
8
beenobserved
toscale
to 1/Npcel
1(Npcel
1= number
ofparti9
10
cles/cell), a further reduction in this effect can not be realisti-
cally achieved with this or similar particle-in-cell
6
•
(b)
x 10•6
,
i
i
•
'l '
i
i
i
computationaltechniqueswithoutthe useof a noisereduction
technique(s)that significantlylowersthe noiselevel.
In additionto the p.a.d. calculationsdescribedabove,we
havemadean additionalsetof study1 calculationsin which
i
4
thep.a.d.
rateD•x isinvestigated
asafunction
ofpitchangle
• = atan(vz/Iv,l), that is, b•(c•) is investigated,
1
2
3
4
5
6
7
8
9
10
t/%e
Figure 11. Individual v spacetrajectoriesof 30 particlesin
the simulationrestframe,eachof whichlay within mode7 at
withineachof the threemodeskx = 5, 6, and7. Becauseof
thepitchangleanisotropyof the simulationelectronspecies,
thenumberof simulationparticlesat smallpitchanglesis relativelysmall.This wouldleadto largestatisticaluncertainties
inb• (ct) atsmallctif weattempted
tocalculate
b•k•(ct)
usingthe simulationelectronspecies.Hencewe havechosen
tocalculate
D• (Ix) byusing
a setof 10,000
"tracer"
parti-
clespermode,eachof whichis movedaccordingto the wave
fieldspresentin the systembut doesnot actuallycontributeto
thewavefields.Suchparticlesallowusto uniformlypopulate
velocityspaceregionsof interestin large enoughnumbers
pendicular velocities and positions. Hence the particles that statistical problems can be overcome. To calculate
whosetrajectoriesare shown in Figure 11 all start out with b• (c•) for a particular
modekx overthetimeinterval
differentinitial velocitiesand positionsand thereforediffer- ( t, t + z), we firstloadthe 10,000tracerparticlesuniformlyin
ent Larmorphaseswith respectto the wave fields.As a result, perpendicular
velocity v•_at the centerof the mode(% =
theirsubsequent
phase.
space
motion
ismarkedly
different. const.),at thetime t. Theseparticlesaretagged,accordingto
Pairsof valuesof D•kxarepresented
in Figure10 to show theirpitchangle,intooneof a number
Of5øbinscovering
the
thestartof thesimulation,
in the1-D study1, (a) velocity%
2 v2•)versus
and(b) labframeenergy (v//+
time.
estimates
of thepossible
uncertainty
in thecalculated
dueto noisewave fields,noiseasdiscussed
previouslymeaning any nonwhistlerwave modenot expectedto grow from a
linear viewpoint.Estimatingsuchuncertaintiesallows us to
identifythepitch anglescatteringdue solelyto whistlermode
range{x= 0 to 90ø.At time(t + z), thenormalized
p.a.d.rate
is calculatedin eachbin, asin (2) but now with theaveraging
beingperformedover all particleslying withinthebin at time
t. To completethis set of calculations,the nonwhistlernoiseproducedp.a.d. in eachbin was estimatedin the relevantisowaves.
Toestimate
uncertainties
inD• duetonoise
inapar- tropic noise level run, therebyallowing an estimateto be
ticular study,we repeatedthe simulationswith an initially madeof the p.a.d. as a functionof pitch anglein eachmode
isotropicelectronf(v) to generatesimilar noise level growth kx duesolelyto electronwhistlerWPIs.
rates. The normalized p.a.d. rates due to nonwhistlernoise
Beforelookingat the resultsof suchcalculations,we shall
alonewere estimatedfrom eachisotropicrun, and uncertain- considerthe p.a.d. processwithin one of our modesin some
tiesin D• arerepresented
bytherange
ofvalues
spanned
by detailandmakesomepredictions
regardingtheexpectedvarthe centersof the pairsof symbolsin Figure 10. Inspectionof iation
inD• (Ix). Weshallfirstconcentrate
onthep.a.d.
ina
1-D geometry.
Modecalculations
showtheaveragechangein
p.a.d.ratesin Figure10thenreveals
the•
following:
1. The pitch anglediffusionrate D• growsin time as a particle'swave frame energyto be conservedwithin -4%
the magneticwave energyin the systemgrows.To a reasona- duringthe periodß over whichour p.a.d.ratesare calculated
bleapproximation,
wehavefoundD• •: B2•,whereB2•is for the 1-D study 1 (and conservedwithin lessthan this in 2the magneticwave energydensityin the system.For exam- D). Hencewe considerparticlesto move alonga diffusion
ple,overthetimeperiod
t = (2.5-4) Xce,
thep.a.d.
rate
surfaceof constantenergyin therelevantwaveframeduring
increases
by a factorof- 1.7-2.7,whileB2•growsbya factor theperiodz overwhichp.a.d.ratesarecalculated.
Denoting
of-2.2. Such a result is predicted from a quasi-linear the parallelcomponent
of the wavephase-velocity
as v//w,
approach(see, for example, Kennel and Petschek [1966], thensucha surface
is described
by (% - V//w)2+ v2•= const,
equation(3.10)).
andis a usualfeatureof a quasi-lineartreatment[e.g.,Lyons,
2. The pitch angle diffusionrate due to whistler mode 1974]. Figure 12 showsthe geometryof a mode in phase
wavesis largerin 1-D thanin 2-D for t > 2Xce.By 4Xcethe space.With referenceto Figure 12, we may write the disp.a.d. rate in 1-D is roughly 10 times that in the 2-D study placementA of a resonantparticlealongits diffusionsurface
(the wave energydensityin the 1-D studyis approximately as
10.6 timesthat in the 2-D studyat this time, seeFigure 4b).
We have alreadyseenhow the magneticwave energygrows
more rapidly in I-D, hencethis resultis not so surprising,as
the p.a.d. rate scalesto the magneticwave energy.Before where (vL,c•L) are the particle'slab frame (rest frame)
17,198
R-----1
DEVINE ET AL.: ONE- AND TWO-DIMENSIONAL WHISTLER SIMULATIONS
:esonance
]
V-I-L
......--.........
V.l.w
:
ß
!
ßoo
ßo
ß
.-
lab
o
tame
wavqframe
:
diffusionsurface
o.O
.
:
wherethevelocityVkx//R
= V/ire
• is takento betheresonant
parallelvelocityfor the mode(kx)of interest.
Often,thelimit Iv//r•I Iv. wl (notea n = - 1 resonant
particle and the correspondingwave have oppositelydirected
velocitycomponents
alongthe backgroundmagneticfield) is
takenin a quasi-lineartheoreticalapproach[e.g.,Kenneland
Petschek,1966]. In thiscase,we canimaginethe originof the
waveframe O' in Figure 12 beingtakenclosetowardthe origin O of the lab frame. As a result, the diffusion surfaces
approachsurfacesof constantlab frame energy,the changein
lab framevelocityAv is very smallcomparedto the displace-
mentA along
thediffusion
surface
((Av)2<<
A2 in(3)),and
• V/ires
: VLCOSO•
• --•
Vllw
•'•
Figure 12. A simplisticsketchshowingthe diffusionof a
resonantparticle along its diffusion surface.The displacement of the particle along this surfaceis representedby A.
Dashedlines and subscriptsw indicatethe wave frame, subscriptsL indicatethe lab frame (simulationrestframe).
velocity and pitch angle, (AvL, Actr) the changesin these
quantities,and (f•v,•) are unit vectorsin the directionof
increasingvelocity and pitch angle, respectively.Note a
ratherexaggerateddisplacementA has been usedin Figure
12 in orderthai the geometricaleffectsof the p.a.d.process
may be seenclearly.Sinceall particleswithin the samemode
havethe sameparallelvelocityv//= V//re
s, thenwe may substitutefor a particle'svelocityvL withina mode
cos
b•k = ( •
'•
A
:
) ( cosct)2(
¾kxl/
R
1
)
"•--•c
e
(4)
Comparing(3) and (4), we seethat Av representsa correctionto the pure pitch angle diffusionapproach,so as to
take into accountthe geometricaleffect of a displacement
along a diffusionsurfaceof constantenergyin the resonant
wave frame, that doesnot in generalcoincidewith a surface
of constantlab frame energy. Such a geometricaleffect is
allowedfor in the quasi-linearcalculationsof Lyons[1974],
sinceno assumptionis made about the ratio Iv.r•s?v.wl.
Hence we use the pitch angle diffusioncoefficientof Lyons
[1974] to predict the variation in the normalizedp.a.d. rate
D•nx(ct) withpitchanglect,withina givenmodekx, and
¾//res
¾L --
pure pitch angle diffusionoccurs.In our simulations,a ratio
Iv.=Xv. wl• 1 is typical (as depictedin Figure 12). However,evenwith sucha ratio, at low pitch anglesthe diffusion
surfaces
will approachsurfacesof constantlab frameenergy,
and pure pitch angle diffusionshouldbe a good approximation, that is, at low particle pitch angleswithin one of our
modes(kx) we expect
ct
This is a geometricaleffectdueto the stripnatureof a resonantmode in v space.We wish to solvefor the variationin
Actt with oc
t withina givenmode.Approximately
(seeFigure 12)
over the full pitch angle range. This was done by using
--nka_
Lyons'equation(6) for the p.a.d. coefficientD• applicable
to a resonanceof a given order n (n = -1 for our interests),
with a resonantwave modeof perpendicularwavenumberki
(andat a givenk//). SinceLyons'definitionof thep.a.d.coef-
ficient
is Dn•x'-(vACt)
2/(2At) (At being
anappropriate
scattering
time),useoftherelation
v - V•x
#•/cosctwithina
(vAc•)• = A•- (Av):,
where we are henceforthdroppingsubscriptsL for convenience.
Now assuminga fixed displacementA for all particle
given mode allowed the variation in our normalizedp.a.d.
rateD•kx(ct) = ((Act):)/At tobepredicted
foreach
ofour
modes. The result is
b'•x
•:(cocel
coax
- (sin
ct)
2)2,
(5)
pitch angleswithin the mode (reasonable,
sincein a 1-D
geometryall particles within the same mode are resonant whereconis the real frequencyof the relevantmodekx of
ahdtheconstant
of proportionality
in (5) contains
the
with the same wave mode) and substitutingfor v = vL = interest,
energy
of
the
resonant
wave
mode.
¾//res
/ COS
ct, thenwe have
In order to see whether the p.a.d. occurringwithin our
simulationcan be describedusing the simple geometrical
approachdescribedabove,we have calculatedthe predicted
2
(ac)
2= [(a)2_(av)
2](coscO
2
V//res
variation
in ba•x(ct) withineachof ourthreechosen
reso-
nantmodes,usingboththepurepitchanglediffusion
approxFrom this expression,the variation in the normalized imation(see(4)) andthe full-diffusionLyons'basedequation
t) using(5), therealfrep.a.d.
rateD• x(ct) (see(2))withpitchangle,
within
agiven (5).Tobeableto predictb•kx(c
modekx, canbe predicted
quencyco•of eachmodekx was.taken
fromthelinearlypre-
dicted
disl•ersion
(co,k//)
foreachmode(seeFigure2).
Tracer-obtained
resultsare presentedin Figure 13 for the
p.a.d.b• x(c
t) at t = 2.5Xc•.
Nonwhistler
noiseerrorbars
have been usedrather than pairs of valuesand representthe
DEVINE
ET AL.: ONE- AND TWO-DIMENSIONAL
estimated
uncertainty
in D•k.(at) duetononwhistler
noise,
WHISTLER
SIMULATIONS
17,199
In attempting to apply these descriptionsto tracer-
calculatedfrom a set of tracerswithin the relevantisotropic
obtained
results
forthep.a.d.•
(at) inthe2-Dstudy
1,we
noiselevelrun.Noteagainthatweareconsidering
the1-D found that an unacceptablylarge level of noise uncertainty
stud
y 1.Clearly,
within
aparticular
mode,
thep.a.d.
rateD•k. (~+25%)in D• (at) prevented
usfromdistinguishing
the
decreaseswith increasingat, as we would expectbasedon
either (4) or (5). For each mode shown in Figure 13, two
curves have been drawn to represent the variation in
qualityof the fit of eachof the two predictioncurvesto simulation-obtainedresults.A higherrelativenoiseuncertaintyin
D•k.(at) is pres.
entin 2-D thanin 1-Dduringourchosen
periodbecausethereis a largerlevel of nonwhisD• (at) as predicted
by the purepitchanglediffusion quasi.-linear
tler noise in a 2-D simulation(e.g., 2-D allows electron
plasmaoscillationsin more than just one fixed direction).
thecurves
((A/v•,//n)
2/(Xfce)
in(4))waschosen
bytrial Also, ashasalreadybeenseen,the 2-D whistlerwave energy
and errorto give a reasonable-looking
fit to the Simulation growsmore slowly than in 1-D. Hence a longer period of
results(for a particularmode,the amplitudeof the two curves time is necessaryfor 2-D whistlerwave levelsto grow suffiwas taken to be the same). From Figure 13, it can be seen cientlyover nonwhistlernoiselevelsfor this comparison,
by
that,in general(thoughmostnoticeablyin mode5, whichhas which time we are outsidethe chosenquasi-linearperiod of
approximation(4) and the full-diffusionLyons' basedequa-
tion(5). The unknown"amplitud
e" of variations
of eachof
the largestratio Iv.r•/v. wl= 0.88 of the threemodes),at
4.5Xce(thechosenquasi-linear
periodbasedon 1-D particle
pitchangleslessthanaboutat< 50¸ - 60¸ , thereare not significantdifferences(within noiseuncertainties)in the agreementwith the simulation-obtained
resultsof the two separate
predictioncurvesbasedon (4) and (5). Recallingthat derivation of (4) useda pure pitch anglediffusionapproximation,it
can be seen (Figure 13) that for pitch angles below
at< 50¸ - 60 ¸ , the pitch anglediffusionwithin our threeresonantmodesis well-described
by a purepitchanglediffusion
approach.The pure pitch angle diffusion approximation
dynamics).Nevertheless,2-D p.a.d. resultsshow the same
general
trendof a decreasing
p.a.d.rateD• asthepitch
angle at increases within a mode, consistent with
D• (at)~ ( cos
at)2(see(4)).
Effectof Increasing15//:Studies2 - 4
As we increasethe [5//valueof the electronspecies,
we
observelarger averagemagneticwave energygrowth rates
becomes less accurate, within noise uncertainties, in all
(seeFigures14a and 15a), as predictedby linear theory.The
modesabove at> 65¸ . The agreementwith Lyons' predic- averagegrowthratespresentedin Figure 15a were calculated
tions(our (5)) is suchthat the predictedcurvespassthrough from the growth of averagemagneticwave energy (Figure
all simulation-obtainedresults (within noise uncertainties)in
14a) in the samemanneras describedearlierfor study1
mode 5 and all but two/one in modes 6 and 7.
resultsand representan averagegrowth rate of magnetic
go
(½)
o.o'7'
.
Mode
7
['__i•Yuør
e•_•';
.CI•
.ffc•
'i•'n
' ]_
o.o•
4
O.Ol
øo
1o
a,o
=•o
4o
•o
•o
•'0
8o
¸]
Figure13. Variation
inthetracer-obtained
p.a.d.rates•.
(a) withpitchangleatin(a)mode
5, (b)mode
6,and(c)mode
7 regions
ofvelocity
space
att = 2.5•Xc•
eduring
the1,Dstudy
1.Foreach
mod
e,thedashed
andsolidcurvesrepresent
thepredicted
v.ariation
in Da• (at) basedonour(4) and(5), respectively.
17,200
DEVINE ET AL.: ONE- AND TWO-DIMENSIONAL WHISTLER SIMULATIONS
netic wave energy saturates,as we would expect when
(a)
b•/•x•: B2
W'
Strength of Pitch Angle Diffusion
[=•
. I
ll
Toinferwhether
a givenvalueof O•kximplies
strong
or
10-•
!
I
i
I
0
2
4
6
8
10
weak pitch anglediffusionis occurring,we will estimatethe
time taken for a magnetosphericelectron to be scattered
acrossa realisticloss cone, given that it movesin a region
where
thepitchangle
diffusion
rateisthef)•k•obtained
from
(b)
our simulations.
We will then comparethis time to a typical
magnetospheric
lossconeelectron'sescapetime, whichis the
time taken for a loss cone electron
to be lost to the atmos-
phere in the absenceof scattering(typically a 1/4 bounce
1
period).Takinga typicallossconesizeat L- 6 as3ø [Parks,
1991], thenthetimetakenfor aninitially0øpitchangleelec-
ß x
tron to be scattered out of the loss cone is roughly
T/.c.... (3)2/D•., whereD• = b(xk/c
e (andis thencon-
c
)
verted
tounitsoi• [deg2s
-1]). A typicalelectron
(tensof keV)
on L shell(L- 6) has a bounceperiod TB of a few seconds
Figure 14. Variationwith time of (a) magneticwave energy [Parks,1991]. We will take a typicalnormalisedp.a.d.rateof
-3tad2forloss
cone
particles,
taken
frommode
density
(ME)inJm-3and(b)electron
species
anisotropy
(A) D•, = 5x10
5 calculations
at t -- 2.5xcein the2-D study1 (sincethep.a.d.
in the 2-D studies.
rate within a modeincreaseswith decreasingpitch angle,the
p.a.d.ratefor lossconeparticlesin a particularmodeis a facwaveamplitude
in thesystem.
As we increase
[•//,thetypical tor of-3-4 timeslargerthan that obtainedby averagingover
parallelelectronenergyrequiredfor cyclotron
resonance
with all its pitch angles,and hencethis chosenvalue is somewhat
one of the growth wave modesin the systemdecreases(see largerthanthe pitchangleaveragedvalue(Figure 10) for this
stronger
afterour chosen
the magnetic
energyper particleof KennelandPetschektime).Sincethe•).a.d.becomes
-3 represents
a lowerlimitto
[1966]); that is, the magnitudeIv//I of the parallelelectron time,a valueof D•, = 5x10
the
p.a.d.
occurring
in
these
studies.
A
scattering
time of
velocityv// requiredfor resonance
decreases
into a regionof
T/.c= 7x10-5 siscalculated
bythemethod
described
above.
increasingdistributionfunctionf(v).
As 13//increases,
magnetic
waveenergysaturates
earlier Assumingthat a typical loss cone electronexperiencedthis
(seeFigures14a and 15b), as a resultof the increasinggrowth p.a.d. rate for more than about 51.tsec,it would have been
rates.The differencebetween 1-D and 2-D averagegrowth scatteredout of the loss cone well before completinga
rates,for thesame13//value,
becomes
largeras 13//increases.bounce(i.e. T/.c <<TB/4), hencein this sensewe observe
pitchangle
diffusion.
Since
thevalue
of D•kxonwhich
Again,thesimUlation-obtained
rateof growthof magneticstrong
waveenergydensityat earlytimesarecloseto themeanpre- this calculationwas basedrepresentsa lower limit to the
dictedby linear theory.For example, in study2, it is found p.a.d.occurringin our simulations,it appearsthat,in magnet-
that •lO = (0.172 +_0.01)
(.t}ce
, •2D•--•(0.092+_0.025)
(t}c.e,
whilst the linearly-predictedmeanof the growthratesof all
simulation modes lying within the growth region are
0.6
0.158(0ce
(l-D) and0.123(Oce
(2-D).As 13//increases,
more
0.4
cyclotronresonances(n) becomepossiblein 2-D (our !-D
geometryallowsn = -1 resonance
only) sincetheIv//Irequired
for such resonancesdecreases(into a region of increasing
f(v)), and so we might reasonablyexpectmore differences
0.2
between
1-D
and
2-D
results.
Note
that
the
difference
between1-D and2-D averagegrowthratescanbe as largeas
by a factor- 2.6. Accompanyingthe increasedwave growth
!
o 2D[
x
o
øo
i
i
:'is
I
(b)
rate(andhence
waveamplitude
as•//increases)
area more
.rapiddecreasein the electronspeciesanisotropy,
with a lower
end-of-runanisotropylevel (seeFigure 14b).
Along with an increasing growth rate, the normalized
p.a.d.ratesincrease
as 13//increases
(seeFigure16).This
explainsthefasterrateof decrease
in theelectronspecies
ani-
sotropy
as •// increases.
Duringthegrowthperiodof wave
energy,thep.a.d.ratein 1-D is largerthanin 2-D for the same
13//value.As [•//increases,
theperiodduring
whicha quasilinear approachis expectedto hold shortens,sincewave
energygrowsmorerapidly(andthereforewavemodesreach Figure15, Variationwith 13//of(a) averagegrowthrate¾
nonlinearamplitudes/saturation
sooner).In all studies,p.a.d. and(b) saturation
timestsat of averagemagneticwaveenergy
ratesareseento saturate
at roughly
thesametimeasmag- in the 1-D (crosses)and2-D (opencircles)studies.
DEVINE
ET AL.: ONE- AND TWO-DIMENSIONAL
WHISTLER
SIMULATIONS
17,201
(a) 1D, Mode6
0.5
0.4
0.3
Study1
Study2
Study3
Study4
0.2
0.1
•
o
I
o
o
I
o
o
o
I
(a) 2D, Mode6
0.4
0.3
b•kx
0.2
0.1
Nm
0
m
m
xo
m
I
I
1
2
x
0
0
x
x
3
t/ '•ce
Figure16. Normalized
mode
6 pitchangle
diffusion
rates
bakx
in(a)1-Dand(b)2-Dstudies.
ospheric
terms,strongpitchanglediffusionoccursthrough- not exactlycorrespond
to the linearlypredicted
wavespectrum.This hasbeenpreviouslynotedin a quasi-linear
theoreticaltreatment
by RouxandSolomon
[ 1971], andearlyonedimensional
electronwhistler-instability
simulations[e.g.,
Discussion
Ossakow
et al., 1972].The dependence
of pitchanglediffusionratesonwaveenergyseems
reasonable
froma quasi-linWehave
examined
howaninitially
anisotropic
velocity
distribution
(T, > T/l) may lead to the growthof whistler ear viewpoint[e.g., Kenneland Petschek,1966]. The usual
theoryof resonant
diffusion[KennelandEngelmodewaves,and the subsequent
pitch anglediffusionof quasi-linear
negligiblewave growth-rates/amplielectrons
via electronwhistlermodewave-particle
interac- mann,1966] assumes
tions.Our key resultswere:(1) As waveenergygrows,the tudes(¾-• 0). Often, real wave frequenciesare assumed
anisotropy
of the resonantelectronspeciesdecreases,
and muchlessthanthe electron
cyclotron
frequency
(co<<coce
waveenergyshiftstowardslowerfrequencymodes(gener- [e.g., Kennel and Petschek, 1966]), which allows for an
of purepitchanglediffusion.Neitherof these
ally withoutreachingthe lowestresolvedwave modein the approximation
arecharacteristic
of thesimulations
presented
here
simulation).
(2) Wavegrowthratesbecomelargerasthe [5 conditions
arebotha significant
valueof the resonant
electronspecies
increases.
(3) Larger (growthratesandreal wavefrequencies
average
growthratesareobtained
in a 1-D geometry
thanin a fractionof coc•,seeFigures2 and3). Nevertheless,
at early
2-D geometry,
thedifference
increasing
as[5increases,
being stagesin the simulations,
resonant
particlewaveframeenerlargerbyupto a factorof- 2.6forthestudies
weperformed.giesandthevariationof thep.a.d.rateasa functionof pitch
anglewithina resonant
regionof v spacesuggests
thatresoas magnetic wave energy Bw grows, approximatelynantparticlesdiffusealongsurfaces
of constant
energymeasD,kx
o•B2•.(5) During
theearlywavegrowth
stages
of the uredin the wave frame,with a pure pitch anglediffusion
studies,
pitchanglediffusion
ratesarelargerin a 1-D geome- descriptionsufficingbelow pitch anglesabout at< 50ø. It
try,by upto a factorof-10. (6) For thechosen
parameters,thusappears
thata quasi-linear
description
of thep.a.d.procstrongpitch angle diffusion of electronsvia WPIs occurs essseemsreasonableduringthe early stagesof the simulathroughout.
(7) Duringthe earlywavegrowthstagesof the tions,duringwhichwave energieshavenot grownto large
studies,
resonant
particlesmoveon velocityspacediffusion nonlinearamplitudes.Later in the simulations,when the
surfaces
of constant
waveframeenergy,
withthepitchangle wave spectrumhas becomedominatedby a single largediffusion
for suchparticles
beingconsistent
with thequasi- amplitudewavemode,particletrajectories
indicatethatnontheory of Lyons [1974]. (8) Pitch-anglediffusionrates lineartrapping
of particles
maybeoccurring.
Analysisof parincrease
asthe [5valueof theresonant
species
increases.
(9) ticle trajectoriesand a studyof trappingin the nonlinear
Laterin the run,particletrajectories
suggest
thatnonlinear regimeof the simulations
havebeenperformedand will be
trappingof resonantparticlesby large amplitudeelectron reported
in a separate
paper.Sucheffectsmayplayan imporwhistlermodewave(s)maybe occurring.
tantrolein the overallsaturation
of waveenergyseenin the
The effectof the shiftof magneticwaveenergytoward simulationstudies.We do notexpectsucha saturation
mechlowerfrequency
modesis thatthefinalwavespectrum
does anismto be too importantin the magnetospheric
problemof
out our simulations.
(4)Pitch-angle
diffusion
rates/5,•
have
been
found
togrow
17,202
DEVINE ET AL.' ONE- AND TWO-DIMENSIONAL
pitch anglediffusiondue to whistlermodeturbulence,since
wave lossfrom the wave-growthregionbecomesan important factor in determining saturation,and our simulations
havenotmodeledsucha loss.However,a simulationstudyof
trappingeffectsmay prove to be of someuse in situations
wherethe wave-particleinteractionsare coherentandnonlinear (as opposedto the quasi-linearturbulentcasemore usually treated),which may play somerole in the generationof
magnetospheric
VLF triggeredemissions,as reviewedby
Matsumoto [ 1979].
WHISTLER SIMULATIONS
conesink consideration
is purely to allow the robustness
of
our resultswith respectto particle loss to be evaluated).
Effectsof waveenergylossat the simulationboundaries
on
the evolutionof the instabilityare less clear and are suggestedasfuturework at the endof the discussion.
Nevertheless,sincethe quasi-linearaspectsof our simulationresults
aretakenfromtheearlypartof the simulation
(first4-5xce
electrongyroperiods),
duringwhichtime we do notbelievea
greatdeal of wave lossfrom the simulationregionwould
occur(basedon the linearlypredictedwavephase-velocities,
Our simulationparameterswere chosento be of interest the transit time of a whistler wave across our simulation box
to theproblemof whistlermodewavegrowthfrom andpitch being-8-9 xce),wedonotfeelthequasi-linear
aspects
of our
anglediffusionof anisotropicenergeticelectrondistributions resultswould be significantlyaffectedby a boundarywave
usedin thesestudies).
that are injectedinto the nightsidering currentregionunder loss(for theparameters
an enhancedcross-tailelectricfield duringmagneticallydisAn understanding
of the magnetospheric
problemoutturbedtimes[e.g.,Ashour-Abdalla& Cowley,1974;Krernser lined abovehas relevanceto an understanding
of the diset al., 1986]. Our study1 parametersarerepresentative
of the turbed time aurora. Recent interest in the pitch angle
plasmapause
regionon the nightsideduringgeomagnetically diffusion due to electron whistler WPIs has also been in the
disturbedtimes.Note a rather large anisotropy(temperature questionof which wave modesare responsiblefor the elecformingthe diffuseaurora.It is nowgenerratio T.•/T# --4) was usedto overcomenon-whistlernoise tronprecipitation
levelsinherentin 2-D electromagnetic
particle-in-cellsimula- ally acknowledged
that the precipitationof electronsforming
tions(T.•/T# -- 1.5- 2 appearing
to be moremagnetospheri-thediffuseaurorais unlikelyto be a resultof pitchangledifcallyreasonable
[e.g.,Krernseret al., 1986]).Higher[5value fusion due to electron-cyclotronharmonic wave-particle
studies([5 from approximately
4 to 30) containan electron interactions,sinceobservedelectricfield strengthsare too
speciesof unrealistically
large [5 valuein magnetospheric
i lowto account
fortheobserved
intensities
of lowenergy(less
termsand were madeto investigatethe effectsof increasing than10 keV energies)diffuseauroralelectrons[e.g.,Belmont
[5onthephysicsof theinstability.Sincethe 1-D/2-D simulationspresented
herecovera wide rangeof parameters
(electron temperature,backgroundmagneticfield strength),they
are alsoof generalinterestto a linear and nonlinearunderstandingof the whistlermodeinstabilityandelectronwhistler
wave-particleinteractions.
In all simulations
we found,for a
given initial electrondistributionand backgroundmagnetic
field(given[5 value),the timescalefor pitchanglediffusion
in a 2-D geometryto be greaterthan that in a 1-D geometry
(lower p.a.d. ratesin 2-D), therebyindicatingthe needfor
inclusionof both parallel and obliquepropagatingwhistler
wavesin a 2-D geometryfor a propertreatmentof the wave
growthandelectronpitchanglediffusionassociated
with the
electronwhistlermodeinstability.Our resultsalsoshowthat,
duringthe wave growthperiod of the instability,the quasilinear pictureof velocity spacediffusionalong surfacesof
constantwave frame energy[e.g., Lyons,1974] to be valid,
even when growth-ratesare not negligible,severalwave
modes(both parallel and oblique) are growing simultaneously(2-D wavepropagation)andrelativelylargereal wavefrequenciesexist.In addition,even whenthe ratio of wave-
et al., 1983; Roeder and Koons, 1989]. Johnstoneet al.
[1993] (hereinafterreferredto as JWLH) haverecentlyproposedelectronwhistlerwavepitchanglediffusionasa candidatefor the precipitationmechanismof low energyelectrons
(<10 keV) into the auroral loss cone. In JWLH, a "diffusive
equilibrium"modelwas employedto representthe continuousprecipitationof electrons,in whichwave growthfrom an
unstable loss cone distribution
of electrons balances wave
losses,and electroninjectionat largepitch anglesbalances
electronprecipitationthrough the loss cone. This model
requiresboth a lossof wave energyand particlesfrom the
interactionregion.Our simulationsweredesignedto investigate nonequilibriumsubstorm-injectedelectron distributions,ratherthana lossconedistribution,haveno absorption
of wavesat the boundaries,and are thereforenot directly
applicableto JWLH's paper.However,we believethereto be
two resultsfrom our simulationsof interestto JWLH's proposal.First,our simulations
showthatstrongpitchanglediffusion (in magnetosphericterms) of resonant electrons
havingenergiesbelow 10 keV canoccurvia electronwhistler
wave-particleinteractions.
The electronsresonantwith mode
7 in our study1 (seeFigures10 and13 for thepitchangledifphasevelocity(%) to resonant
electron
parallel-velocity
(v#res)
is notnegligible
<lv,
I 1 typicalhere),a pure fusionratesof mode7 resonantelectrons)have minimum resonantenergies
1/2 meV/Ires
2 = 3.9 keV, beingresonant
witha
whistler
wavemodeat a frequency
0•= 0.670•c•.Wavemode
pitch anglediffusiondescriptionof velocityspacediffusion
appearsto be appropriatefor pitch angles below about
c•< 50ø. The simplifications
presentin our simulations,
relative to the magnetospheric
problem,need to be borne in
mind.Our simulationswereessentiallyperformedin a closed
box (periodicboundaryconditions)from whichno particleor
waveenergylossoccurs,and in whicha discretesetof simulationwave modesare allowed.As outlinedat the beginning
of the paper, our resultsare not found to be affectedby
includinga realisticlossconesinkfor particlesat theboundaries of the simulationbox, sinceso few simulationparticles
fall within a realisticlosscone(typically-5 o for the plasma
sheet;note our simulations are not loss cone driven, the loss
7 is a linearly predictedgrowth mode in our study 1, and
althoughit is not the dominantmodein the wavespectrum,
it
containssignificantmagneticenergyover the chosenquasilinear time period (see Figures5 and 8). Significantwave
energy at such relatively high wave-frequencies(say
0•> 0.50•c•)allowsthepitchanglediffusion
of electrons
having energiesmuchlower than consideredin the past(where
0•<<0•c•wasthe usualassumption
[e.g.,Kenneland Petschek,1966]. This is heavily stressedin JWLH, in fact, it is
vital for the pitch anglediffusionof low energyelectronsto
occur.Our simulations
showthatit is possibleto causestrong
DEVINE
ET AL.: ONE- AND TWO-DIMENSIONAL
WHISTLER
SIMULATIONS
17,203
pitchanglediffusionof low-energy(<10 keV) electronsvia Hardy, D.A., D.M. Walton, A.D. Johnstone,M.E Smith, M.P.
Gough, A. Huber, J.Pantazis,and R. Burkhardt,Low-energy
suchWPIs, providedthereis significantwave-energypresent
plasma analyzer,IEEE Trans. Nucl. Sci., 40(2), 246-251,
overtheresonantfrequencyrange,andhencea consideration
1993.
of the wave data is necessaryin proposingelectronwhistler
Johnstone,A.D., D.M. Walton, R. Liu, and D.A. Hardy, Pitch
WPIs as a low-energy electron precipitationmechanism.
angle diffusion of low energy electronsby whistler mod
Sincethe wave data were not presentedin JWLH, the interwaves,J. Geophys.Res.,98, 5959-5967, 1993.
estingquestionof which wave modeis responsiblefor con- Kennel,C.F., and E Engelmann,Velocityspacediffusionfrom
tinuousauroralprecipitationremainsopen. Second,JWLH
weak plasmaturbulencein a magneticfield, Phys. Fluids
relied on electronvelocity spacediffusion along diffusion
9(12), 2377-2388, 1966.
curvesof constantwaveframeenergy,assuminga wave spec- Kennel,C.F., andH.E. Petschek,Limit on stablytrappedparticle
fluxes,J. Geophys.Res., 71, 1-28, 1966.
trum containingparallel propagatingwhistler-wavesonly,
and(JWLH, p. 5963) questioned
"whetherthe electronsinter- Kremser,G., et al., Energeticelectronprecipitationduring a
magnetospheric
substorm
andits relationship
to waveparticle
actwith whistlernoiseanddiffusealongthesecurvesor not"
whena full wave spectrumincludingobliquelypropagating interaction,J. Geophys.Res.,91, 5711-5718, 1986.
waves is actually present (usually such curves are derived Lyons,L.R., Generalrelationsfor resonantparticlediffusionin
pitchangleandenergy,J. PlasmaPhys.,12, 1, 45-49, 1974.
only for a singlewave modeat a singlefrequency[e.g.,Ken- Lyons,L.R., R.M. Thorne, and C.E Kennel,Pitch anglediffunel & Engelmann,1966]. Both our 1-D and 2-D resultswere
sion of radiationbelt electronswithin the plasmasphere,
J.
consistent
with electrondiffusionalongthe surfacesof conGeophys.Res. 77, 3455-3474, 1972.
stant wave frame energy and may therefore provide an Matsumoto,H., Nonlinear whistler-modeinteractionand triganswerto sucha question.
gered emissionsin the magnetosphere:
a review, in Wave
Instabilitiesin SpacePlasmas,editedby P.J.Palmadesso
and
Future simulationwork regardingthe electronwhistlerK. Papadopoulos,p.p. 163-190, D. Reidel, Norwell, Mass.,
instabilitymay thereforeinvolve:(1) openingthe boundaries
1979.
of thesimulationregionto allowbothwaveandparticleloss,
(2) reductionof noise levels to allow simulationswith more Matsumoto,H., M. Ohashi,and Y. Omura,A computersimulation studyof hook-inducedelectrostaticburstsobservedin the
realisticanisotropies,
(3) use of typicallyobservedmagnetmagnetosphere
by the ISEE satellite,J. Geophys.Res., 89,
osphericenergeticelectrondistributions,
suchascontainedin
3873-3881, 1984.
the CRRES low energyplasmaanalyzer(LEPA) satellitedata Ossakow,S.L., E. Ott, and I. Haber, Nonlinear evolution of whisset [Hardy et al., 1993], and (4) inclusionof a varying
tler instabilities,Phys. Fluids, 15, 2314-2326, 1972.
amountof cold plasmaandits effectson the electronwhistler Parks,G.K., Physicsof SpacePlasmas:An Introduction,Addiinstability,all of which are expectedto make a simulationson-Wesley,Reading,Mass., 1991.
based investigationof the magnetosphericelectron pitch Pritchett, P.L., D. Schriver, and M. Ashour-Abdalla, Simulation
of whistlerwavesexcitedin the presenceof a cold plasma
anglediffusionproblemmorerealistic.
cloud:Implicationsfor the CRRES mission,J. Geophys.Res.,
96, 19,507-19,512, 1991.
Acknowledgments. P. Devine was supportedby a PPARC Roeder,J.L., and H.C. Koons,A surveyof electroncyclotron
researchstudentship.
The EuropeanSimulationNetwork funded
wavesin the magnetosphere
and the diffuseauroralelectron
in partthedevelopmentof the simulationcode.We wouldlike to
precipitation,J. Geophys.Res.,94, 2529-2541, 1989.
acknowledgeI. Roth of Berkeleyfor providingthe XWHAMP
Ronnmark,K., WHAMP- Waves in homogeneous,
anisotropic,
code,developedat DartmouthCollege,New Hampshire,under
multi-component
plasmas,KGI Rep. 179, Kiruna Geophysithe Northstarproject. XWHAMP is basedon K. Ronnmark's
cal Institute, Kiruna, Sweden, June 1982.
originalWHAMP code.
Roux,
A., andJ. Solomon,Self-consistent
solutionof the quasiThe Editor thanks Hiroshi Matsumoto and David Schriver
lineartheory:Applicationto the spectralshapeand intensity
for theirassistance
in evaluatingthispaper.
of VLF wavesin themagnetosphere,
J. Atrnos.Terr.Phys.,33,
1457-1471, 1971.
References
Stix, T., The Theory of Wavesin Plasma, McGraw-Hill, New
York, 1962.
Ashour-Abdalla,
M., andS.W.H. Cowley,Wave-particleinterac- Sudan,R.N., and E. Ott, Theory of triggeredVLF emissions,J.
Geophys.Res., 76, 4463-4476, 1971.
tionsnearthe geostationary
orbit,in Magnetospheric
Physics,
editedby B.M. McCormac,pp. 241-270, D. Reidel, Norwell, Zhang,Y.L., H. Matsumoto,and Y. Omura,Linear andnonlinear
interactionsof an electronbeam with oblique whistler and
Mass., Dordrecht-Holland, 1974.
electrostaticwaves in the magnetosphere,
J. Geophys.Res.,
Belmont,G., D. Fontaine,and P. Canu, Are equatorialelectron
98, 21,353-21,363, 1993.
cyclotronwavesresponsible
for diffuseauroralelectronprecipitation?,J. Geophys.Res.,88, 9163-9170, 1983.
Birdsall,C.K., andA.B. Langdon,PlasmaPhysicsvia Computer
P. E. Devine and S.C. Chapman,Physicsand Astronomy
Simulation, McGraw-Hill, New York, 1985.
SubjectGroup, SpaceScienceCentre,Schoolof Mathematical
Cuperman,S., and Y. Salu, Magnetospheric
implicationsof the
and PhysicalSciences,Universityof Sussex,Falmer, Brighton
nonlinearwhistlerinstabilityobtainedin a computerexperiBN1 9QH, England.(e-mail: pauldv@sussex.ac.uk;
S.C.Chapment,J. Geophys.Res.,78, 4792-4796, 1973.
man@sussex.ac.uk)
Cuperman,S., andA. Sternlieb,Obliquelypropagatingunstable
J. W. Eastwood, AEA Technology,Culham Laboratory,
whistlerwaves: a computersimulation,J. Plasma Phys., 11,
Abingdon,
OxfordshireOX14 3DB, England.(e-mail: jim.east175-188, 1974.
wood@aea.orgn.uk)
Dungey,J., The loss of Van Allen electronsdue to whistlers,
Planet. SpaceSci., 11, 591-595, 1963.
Eastwood,J.W., The virtual particle electromagneticparticle- (ReceivedAugust30, 1994; revisedMarch 8, 1995;
meshmethod,Cornput.Phys.Commune,
64, 252-266, 1991.
acceptedMarch 8, 1995.)