JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 99, NO. A9, PAGES 17,391-17,404, SEPTEMBER 1, 1994
Self consistent one-dimensional hybrid code
simulations of a relaxing field reversal
Alan Richardson
• and SandraC. Chapman
SpaceScienceCentre (MAPS), Universityof Sussex,Brighton,United Kingdom
Abstract.
We present the results of self-consistentone-dimensionalhybrid code
(kineticionsand fluid electrons)simulations
of a dipolarizingfield reversal.Local
ion energization processesare studied in terms of both the ion kinetics and bulk
plasmabehavior.We demonstrate
evolutionof the systemthroughtwophases:(1) a
quasi-static"trapping" phase,during whichinfiowingionsare trapped in the reversal
structureat early timesand (2) a dynamic"escape"phase,duringwhichtrapped
ionsescape
fromthe reversal
afteranincrease
in/•1' Weperformsimulations
with
differentlinkingfieldcomponent
(GSE B•) to reversingcomponent
(GSE B•) ratios
and show that this parameter controlsthe details of the ion energization. This
studysuggests
a characteristic
signaturein the magneticfield (a bipolarB•) and
bulkplasma
parameters
(anincrease
in /•l) associated
withthisbehavior
which
should be observablein in situ data. Our results are applicable for times earlier
than the midtail to ionosphereAlfv6n travel time.
mechanisms which occur are all local to the dipolariz-
1. Introduction
ing field line, enablingus to deducewhat energization
can occur locally and by implication what must occur
At the onsetof a magneticsubstorm,magneticfield
(e.g., at a reconnection
site).
linesin the near-Earth geotailundergorapid reconfigu- elsewhere
We
present
the
results
of
two
simulations,
one with a
ration from a taillike geometryto a more dipolarstate
weak
linking
field
component
and
one
with
a stronger
("dipolarization").Characteristic
featuresof dipolarlinking
field.
The
values
we
use
span
the
range
of obization includerapid changesin the local field strength,
theobservation
of high-energy
(• 1MeV) burstsof par- servedlinking field component. We demonstratethat
ticles[e.g.,Honesel al., 1976],andthe arrivalof lower- the ratio of linking field to reversingfield components
energyplasmaseenat around6RE (so called "injec- determinesthe details of ion energization,rather than
tion events")[e.g.,Lennarlson
el al., 1981].To date, the overall evolution of the relaxing reversal.
We find that the reversingmagneticfield structureis
only non-self-consistent
studieshavebeenmade of ion
capable
of trappinginfiowingions. As the trappedions
dynamicsduring dipolarization[e.g., Chapman,1993,
oscillate
acrossthe reversal,they gain net field-aligned
1994;Delcourt,1991;DelcourtandMoore,1992].
In this paper we presentone-dimensional
hybrid code accelerationvia a mechanismapproximately modelled
simulationsof a dipolarizingfield reversal.In order to as first-order Fermi acceleration. The parallel pressure
study the consequences
of dynamicreconfiguration
of of the trapped population increasesdue to individual
the magnetotail magneticfield at substormonset we ion acceleration, and once it has become sufficiently
choose as our initial condition an out-of-equilibrium large to strongly perturb the bulk plasmaconfigurafield reversal. This work is significantin that unlike tion, the trapped population escapes. The escaping
previousstudiesof ion energizationduringdipolariza- population may be in pressurebalanceor may have an
dependingon the sizeof the linking field.
tion [e.g.,Delcourt,1991;DelcourtandMoore,1992], overpressure,
After
the
escape
of the initially trapped population, a
we can self-consistentlyexamine the evolution of local
field structure and ion dynamics. The ion energization continualprocessevolvesin whichionsinitially located
outside of the reversal convectin, are trapped, acceler-
ate, and escapeen masse.A characteristicbipolar By
1Now at TessellaSupport Services,Abingdon,United
Kingdom.
Copyright 1994 by the American Geophysical Union.
Paper number 93JA02929.
0148-0227/ 94/93J A-02929505.00
field componentevolvesacrossthe reversal.
The paper will be structured as follows. We first give
an overview of the simulation, provide details of the
simulation method, and describethe initial conditions
in more depth. We then presentand discussthe results
of two typical simulations. Finally, we summarizeour
key results.
17,391
17,392
RICHARDSON
2. Simulation
2.1.
AND CHAPMAN:
SIMULATIONS
Details
OF RELAXING
REVERSAL
d =
+ v? x n)
(2)
0B
ot
(a)
(4)
(s)
1
VxE-
Overview
•7 x B =/•oJ
An overview of the simulation geometry is given in
Figuresla (initial configuration)and lb (subsequent
evolution).The one-dimensional
hybridcodeemployed
hereallowsvariationin onedirectiononly (GSE z) but
V.J=0
E--V ixB-• JxB vP•+r/j
retains all three vector components. The field reversal where the electron inertial term has been neglected,
is initially out of equilibrium, in that the J x B force
and quasineutrality
impliesnV = niV i + neVe and
is not balanced by the initially isotropic ion distribuJ
ne(V
i
-Ve).
The
number
density
n, andthebulk
tion. The field lines therefore relax as time increases,
ionvelocityV i areobtaineddirectlyasmoments
from
acceleratingthe plasma. The systemis thereforeanalogousto the evolution of a reversalwhen the associated
plasmapopulationhas (by someunstatedmechanism) tional particles that represent the ions. In this one- dibeen newly isotropized. The relaxation of the reversal mensionalgeometrythe assumptionof quasi-neutrality,
thepositions
andvelocities
(thev•) ofthecomputa-
in the simulation occurs by some combination of fast,
slow, and Alfvdn waves which can propagate in GSE
+z as the field lines in principle contract in GSE+x as
shownin Figure lb. This generatesa strongfield region
with infiowing isotropicplasma outsideof the reversal,
and a weak field region in the reversal within which
plasma is accelerated, and within which also particles
can become trapped.
2.2.
Simulation
Model
and continuityof charge,imply that Jz -0 (equation
5), while V.B = 0 constrainsBz to be constant.Full
details of the numerical schemeare given by Terasawa
et al. [1986],with the exceptionthat in orderto obtain
the electron pressurePe, Terasawa et al. assumea polytropic equationof state for the electronfluid whereaswe
obtain Pe from the masslessfluid energyequation:
Ot
The simulations are performed using a one- dimensional hybrid code, in which the electron population
constitutes a massless,charge-neutralizingfluid back-
+ Ve. VPe= -7P•V. Ve + (7 - 1)r/J2
(7)
where7 is the ratio of specificheats. Equations(1)(7) form a closedset, with sevenunknowns(E, B,
n, Vi, Ve, J, and P•) andsevenequations.We se-
groundand the ions(protons)are represented
by com- lected the code originally developedby Terasawaet al.
putational particles. The simulation grid size and time [1986]becauseit is in principlenonresistive,
that is, in
step are such that the ion gyromotionis fully resolved. the region of the evolving field reversal the resistivity
In this limit the full range of "low-frequency"
(i.e., r/= 0 (although,of course,thereis numericalresistivity
w <<cope,
cote)phenomena
areresolved.Quantitiesvary present).This givesthe codegoodresolution,
appropriin one direction only, which we orient in i GSE; however ate for studying the fine structure at the edgesof the
reversal.
all three vector componentsare retained.
The governingequationsare
Equation(6) canbe rewrittenusingequation(q)for
J and a vector identity to give
o•'+ V. [nV]- 0
(1)
B2
- -vi xn+ (n.v)n 1V( +P,)+r/J
' (8)
(a) t=0
The five termsin equation(8) representthe forcesact-
inflow
ing on ionsin the simulation.The V i x B termcan
play a role in the coupling of different speciesin a
multispeciesplasma, where the speciesdiffer in mass
x
[Chapmanand Dunlop,1986; Chapmanand Schwartz,
1987]or representpopulationsof the sameparticletype
with differentdistributionfunctions[Leroy,1983].The
(B.V)B term represents
the curvatureforcedueto the
?
inflow
reversingfield geometryillustrated in Figure la. Since
(b)
z
waves
B = B(z,t) here, the components
of this expression
are B,O/Oz(i:B•+ OBy). At leastinitiallyin the simulation there is no field componentin •, and so this
curvature force simply acts to acceleratethe plasma in
/: (i.e., the directionin whichthe fieldrelaxes).Sub-
x-
sequentlya By componentdevelopsand the dynamics
becomemore complex. The third term on the rightwaves
handsideof equation(8), involvingVB 2, is the field
Figure 1. The magneticfield configurationenvisaged pressure term, and this acts toward the center of the
field reversalalong i to initially trap particlesin the
(a) at time zeroand (b) at somelater time.
RICHARDSON
AND CHAPMAN'
SIMULATIONS
reversal. The fact that the curvature and field pressure
terms act in different directionsshowsthat the physics
of the simulation should not be thought of as being
purely one-dimensional;the propagationof wavesalong
• allows the reversal to collapseand liberate field en-
OF RELAXING
REVERSAL
17,393
here(Figure1); weshallseethat thisexpanding
fieldreversaldrivesthe overallion dynamics.In addition, the
spatial resolutionemployedin the abovepaperswould
not be sufficient to reveal the detailed structure
of the
reversalfound in this study. The resultsfound here and
ergy to the ionsas a bulk plasmaacceleration
in • (as thosein the abovepapersare thereforecomplementary,
as they are designedto examine different features of
illustratedin Figure lb).
Solution of the above equationsrequires the specifi- plasma sheet behavior.
cation of suitable boundary conditions: in these simu- 2.3. Initial
Conditions
lations we have open "inflow" at both boundaries, so
that at the boundariesthe bulk plasma parametersand
We represent the predipolarization conditions as an
ion distribution functions are held constant. To absorb
out-of- equilibrium field reversal. The field component
waves approachingthe boundaries,we increaser/from B• is allowed to reversesmoothly at the center of the
zero at the center
of the box to a finite
value at the
edgesof the box.
A one-dimensional
(i.e. dependence
only on z and
t) simulationhas been chosenfor this study to allow
good spatial and temporal resolution. This is appropriate as we are examining the local dynamics of an
evolvingsystemwhich doesnot include couplingwith
the magnetosphere as a whole, that is, the boundary
conditionssimply damp wavesand absorb particles incident upon them and do not attempt to mimic the
ionosphericplasma which in a global model the field
lines would ultimately encounter. We therefore do not
expect to obtain results of a "global" nature, such as
the width of the current sheet in •Oor the implications
for the time evolution
of current
structures
in the iono-
sphere.
Our choice of boundary conditions also implies that
we cannot study the evolution of this configurationto
equilibrium, since again, this would require coupling
between the region of plasma sheet we are numerically modeling and the external system composedof
the ionosphereand the Earth's dipolar magnetic field.
The couplingwould involveappropriate boundary conditions and would require a simulation code allowing
simulationbox, its variation with positionz along the
simulation axis being given by
B•- B•0
tanh(
z-A0
L/2)
where L is the simulation box length and A0 is the
initial half width of the reversalregion. A smallerlinking field component parallel to the simulation axis Bz
threads the reversal. This componentis constrainedto
be a constantby V ßB - 0 in a onedimensional
system.
The third field component,
By, is initially zerq. These
initial conditionsdescribesharplybent field lineswhose
only directionof variationis alongi (Figurela). Note
that the constant linking field allows us to transform
to differentEy convection
framesby movingwith constant speedin the & direction(the de HoffmanTeller
transformation
[deHoffmanand Teller,1950]).
We find that the parameter which determinesthe de-
tails of ion energizationis the ratio of the linkingfield
componentBz to the reversingcomponentB•. Ion accelerationin x occurson the scaleof the Alfvdn speed
VA,•, whilst the reversalstructureexpandsin z at ap-
proximatelyVA,z. Consequently,
we presenttwo simulationsin whichB•0 is the same(20nT), but B, issmall
(8nT).
spatial variationin morethan one dimension(suchas in the first case(2nT) andlargein the second
the two-dimensional hybrid code used by Thomas et Both thesevaluesare consistent
with observations
[e.g.,
al. [1990]). Instead,the study is designedto explore Wagneret al., 1979].All otherbulk parametersare the
the implications of dynamic changesin the magnetotail field geometry at substorm onset, where by some
unstated mechanism the plasma distribution function
is isotropized and no longer balancesthe J x B force
of the extended magnetotail field. This phenomenon,
localized in both space and time, will transfer field
energy to the plasma and result in ion energization.
The question addressedhere is whether this local phenomenon can completely accountfor the energeticion
signaturesseenduring ion injection events,or whether
energizationfrom somesourceremotefrom the simula-
same in both simulations.
In the simulation rest frame, plasma flows into the
reversalregionfrom both sideswith an inflowspeedof
0.1V•,•. The total plasmadensityis initially constant
acrossthe box, at no - 0.2cm-3. The plasmahasan
ion beta of approximately0.1, whilst the electrontem-
peratureis taken to be ~ 0.1• [e.g., Baumjohannet
al., 1989].
The simulation axis is divided into 512 grid cells.
There are approximately8 grid cells to an ion gyroorbit, giving good spatial resolution. The time step
tion region(i.e., remotefromthe dipolarizing
fieldline) intervalis about 1/400 of the ion gyroperiod.The stais also required.
tistical fluctuationassociated
with representing
the ion
Previousstudies[SwiftandAllen, 1987;Swift,1992] population by N• particles per cell is expectedto be
per cell,reduchave employed two-dimensionalparticle-in-cell simula- ~ x/•-•. We typicallytake100particles
tions to examine "global"propertiesof a plasmasheet ing fluctuationsto better than 10%.
by using boundary conditionsthat attempt to mimic 2.4. Electrostatic
Potential
the ionosphere.The system modelled was however,esWhen discussingthe simulationresultswe make fresentially in equilibrium and hencedid not evolveas an
expanding(i.e. dipolarizing)field reversalas expected quent reference to the electric potential •b. This is de-
17,394
RICHARDSON
AND CHAPMAN: SIMULATIONS
fined in generalby
OF RELAXING
REVERSAL
(a) 8 seconds
(2.5 rqi)
•0•.00•..................
042.50 -
0A
Vx,l
E- -V• Ot
(10)
2085.00
try(B=B(z,t)•naE=E(z,t))ao•snotuniquely
specify
Ji,x
_
Je,x
."
- 1042.50 2.00 -
whereA is the vectorpotential.The simulation
geome- PII-P.L,PB
• andA, andhereweusefor convenience
-
0.00 -
Vx,r
,
1.00- ...................................... "
'......
"....................................
0.00 50.00
-
25.00
0.00
32.50
Z•
By
Zz
-
=
_
OA•
(z' t)
Ot
0.00
•
-32.50
-65.00
15.00
(11)
0.00
•
Ez
-15.00
OAy
Ot(z,t)
04(zt)
2.50
(12)Ey
(13)
We find that the generalcharacterof the potential•
'
1.25
0.00
0.00
-- ' - ' '- ---•k..__•- - .....
-3102.50
-6205.00
21.30
0.00
Bx,By
-21.30
.....................................
0.24
isthatofa potential
well,dueto theeffect
oftheVB2 nl, nr
(fieldpressure)
termin equation
(8) discussed
earlier.
0.12
'""?"•""•'";'•'•"•"•'•'"'•'•';'"""""'•'""
0.00
•'
0.0
Thispotentialwelliscapable
oftrappinginfiowing
ions.
5745.6
(0.0)
11491.2
(31.7)
(63.4)
z/km(rgi)
(b) 32 seconds
(9.8 7'qi
)
1267.50'••.'"" '•"•
-''"'
'_,',.'••:•'•":
3. Results
2535.00
Vx,l
-
0.00
-1267.50
.....
2585.00- ,..........
.•••,•
Wenowpresent
theresults
oftwosimulation
experi- Vx,r
1292.500.00 -
-1292.50 -
.
'" ................... "'"'"•""•'
ments,whichdifferonlyin the ratioof constant
linking
- ---'--•----'"-'•field componentBz to the reversingfield componentPII-P.L,PB 1.50
3.00 -
-•................
0.00 -
Bz.
55.00
Ji,x
-
27.50
o.oo-..........
3.1. Overview
Je,x 0.00•
Wefirstobserve
a weak
burst
offastmagnetohydroEz
0.00•
dynamic
waves
fromthereversal
region.Thesearisedue
-40.00 -80.00 15.00 -
-15.00 2.50 -
to the lackof pressure
balanceinherentin ourchoice
of Ey
1.25 0.00 -
initial conditions. The fast wavesare evident in Figure
2, a snapshot
of magnetic
fieldpressure,
potential,
and *
o.oo-
-5190.00 -10380.00
25.00 -
magnetic
fieldcomponents
takensoon
aftertimezero. Sx,
Sy
o.oo
...................................
•
-25.00 0.40 -
.•.•.•:.z
..............................
Figures
3 and4 showvarious
plasma
parameters
plotted
againstGSEz, thedirection
of variation
in thesimula-nl,nr
0.00
....
,
....
tion. Figure3 is for the weaklinkingfieldcase,Figure
(0.0)
(31.7)
(63.4)
4 for a stronglinkingfield.In bothcases,
quantities
are
z/km(rgi)
givenat earlyandlatetimesin thesimulation.
Figure 3. Plasmaparame[ers
•aken(a) before(b)
0.0
popula[ions
(in kms-•). Thenex[panelshows
PII- P•(oia)
nda
x
0.00.
We [hen give ion and elec[roncurren[sin [he/: direc-
lion (A x 10-•). Theelec[ric
fieldcomponen[s
E, and
Ev (Vm-• x 10-a) andpo•en[ial
• (V) follow.Finally,
-2187.50.
-4375.00.
we givethe magneticfield components
B• (solid)and
Bv (dashed)(nT), and[he densi[yof par[iclesorigina[ingfrom[he lef[- hand(solid)andrighi-hand(dashed)
25.00.
Bx,
By0.00------•
-25.00-
0.0
'14•4.e'
'2889.6
(0.0)
(e.o)
z/kmifS)
(15.9)
Figure 2. A snapshotof magneticfield pressurePB
-= x
po[e[i]
(V)
11491.2
ter the escapeof ionstrapped in the reversalstructure.
The linking field is 2nT. The quan[i[iesshownin each
figure are as follows. The •op [wo panelsshow[he V•
phase spaceconfigurationof the left- and right- hand
Pit242.50.
•
0.00.
•••.•
q)
5745.6
(o]ia)
(ahea)
ompoe[
f[e=
0.6s (0.2r=i,wherer=i istheiongyroperiod)
withB, -
sidesof [he box(in cm-a).
Figures 3 and 4 show the potential well associated
with the reversingfield structure. At early times the
well is approximately20 ion gyroradii across,and it expands in z as the reversalrelaxes. The boundariesto
2nT. A short train of weak f•[ magne[ohydrodynamic the well showa combination
of Alfvdnic(e.g., constant
waves is evident.
field pressure)and slowwave(e.g.,densitydepression)
RICHARDSON AND CHAPMAN: SIMULATIONS OF RELAXING
0.00
17,395
0.9-
,,,o.oo(a)2seconds
(0.6rqi)
Vx,l
REVERSAL
4.0(1.2)
0.0
..
2495.00-
1247.50-
8.0
(ZS) 0.0
"•
'••
,=.o<,.?)
o.o.
0.9
.....
PII-P•,PB 1.ooo.oo
-
65.oo-
Ji'x
32.50o.oo
Je,x -37.50-
......
•
-:••
-75.00-
]6.0
(4.9)0.9
0.0•.0(6.1)0.0-
..............
•2.5o-
0.9
-25.00
•5.oo-
0.9 --
o.oo-- .-Ey
7.50
½ qo•2.5o0.00
' -•
-•
•.0(8.6)0.00.9-
•
-so25.oo32.0
(9.8)
0.0Bx,
By o.oo
.......................................
•
...................................
0.9
............
-20.94""
-"'
;,,, .•e'••&•,'.•
0.9
.........
nl,
nr 0.13•.?
••,•4•
• r,•,•,-•,
20.94 -
o.25-
36.0 (11.1) 0.0
0.00
0.0
5745.6
(0.0)
•
2880.00
v•,]
11491.2
(3• .D
(rgi)
(b) 10seconds
(3.1rqi)
-
.....ß?• .:.•: ":"' '
o.oo
2905.00---
""' ' :•-..4.'
..:•.-
.
(,,.s)
' .. •
........................
•
(3,.•)
(r•)
(0.5)
Figure 5. Evolution
of the electricfieldcomponent
•4•2.•o. .:.
...........................
?.:
.......
:•. "•• '•••:;:•:•32
Vx,r •.oo
0.oo
•.0 (12.3) 0.0
(63.4)
parallel
tothemagnetic
field
lineEll,inunits
ofVmzx
.....................
.•,
........
............. 10-3.
Pii-PA,
PB2.5o
.....................................
55.oo
Ji,x 27.50
•
Je,x
-•.oo
0.00
•
Ez
lO.OO
o.oo•
The linking field is 2nT. The numbersdown
theleft-handsideofeachfigure
indicate
realtimein
seconds
(ion gyroperiods).Only the centralportionof
thesimulation
axisisshown.
0.00
0.00
•o.oo
20.00 -
bulk•1 ofthetrapped
population
incre,es
toapprox-
2o.oo,
Ey ,o.oo- •
ß -•v50•••••
,•ss.000.000.00-
imatelytwiceV•,•, the Alfv•n speedin z. An increase
--
-
2•.00 -
s•.Sy
0.00•
-2•.00:
....................................
•
.................................
capes,that is, they leavethe fieldreversalregion.The
0.•0-
escapingions distort the magneticfield.
0.00• ......
0.•
(0.0)
inPIIaccompanies
theincrede
in •1 Once
PIIhas
become
sufficiently
large
tostrongly
perturb
thebulk
plasmaconfigurationthe trapped populationof ionses.
•
"'
5745.6
(a• .3
• (rg•)
Aftertheinitialescape
a continual
process
of ionsbe-
11491.2
comingtrapped in the well, acceleratingand escaping
(•.4)
from the well is set up. In the stronglinkingfield c•e
Figure4. AsforFigure
3 butwitha stronger
linkingthisprocess
hasa "bursty"
nature,
withbunching
evifield (8nT). Note that escapefrom the reversalregion dentin the escaping
ions.Alsoin this case,ionsremain
occursearlierthan with a weakerlinkingfield. It has trappedin the reversalfor shorterdurations
thanwith
beenverified
by examining
individual
iontrajectories
a weak
linking
field.
(see
e.g.,Figure
12)thatthenoise
generated
inE atthe InFigure
5weplotthecomponent
oftheelectric
field
reversal
boundaries
does
notperturb
theionmotion.parallel
tothemagnetic
field
direction,
Ell, • itevolves
duringthe simulation(linkingfield2nT). It canbe seen
in thevicinityofthefieldreversal
and
propertiesand are approximately4 ion gyroradiiwide. that Ellisnonzero
With a linking field B• - 2nT the boundarieshave ve-
is consistentin direction with the trapping of the pop-
locity in •z of • 5Okras
-•, comparedwith slowwave ulationwithin the reversal.When the initially trapped
andAlfv•n
SPeeds
along
z of3Okras
-• and98kms
-• population
ofions
escapes
from
thereversal
structure
respectively.
With
alinking
field
B•- 8nT
the
bound(atabout
4.9
rgi),
the
region
occupied
by
nonzero
and
incomparison
with
Figure
13,
it will
ary velocityis • 12Okras
-• , compared
with slowwave contracts,
and
Alfv•n
speeds
along
z of11Okras
-• and
40Okras
-• seen
tobe•sociated
withtheregion
within
which
ions
Abipolar
Bycomponent
develops
across
thereversal
are
subsequently
trapped
rather
than
the
distorted
field
'
region •sociated with the escapingions.
A population of trapped ions builds up in the reversal. Someof theseionslay within the weak-fieldregion
ofthereversal
whenthesimulation
began;
others
were 4. Ion Dynamics
initially located outside and have convectedinto the
weak-fieldregion. •apped ionsacceleratein GSE x •
We will now discussthe detailedion dynamicsin both
they oscillateacrossthe reversalin z; consequently,
the the x and z directions.Motions alongz are oscillatory,
17,396
RICHARDSON AND CHAPMAN: SIMULATIONS OF RELAXING REVERSAL
(a)
with no overall gain in velocity and possiblya net loss
as the reversal expands. In the x direction there is net
accelerationto approximately twice the Alfvdn speed.
Acceleration in x can be approximately modeled as a
•vi•
first-orderFermiacceleration[e.g.,Cowley,1980].Mo-
VT
tion in z is understoodin terms of the electric potential
VT
qb(discussed
in section2.4) whichhas the form of a
potential well.
To simplify the discussion,we first consideran ideal
field reversal, where Ba, changesdiscontinuouslyas a
step function, treating motion in x and z as if they were
decoupled. We then consider the more realistic case of
a smooth reversal over a finite distance,retaining the
decoupledapproximation.
4.1. "Zero-order"
]/viz2+(vix+
VT)
2
/
vix+2
VT(
•
viz
Description
) vix+V T
viz
As a "zero order" approximation we suppose that
at time zero, Ba, reversesdiscontinuouslyfrom q-Ba,to
(b ')
-Ba, (in fact, it variessmoothlyon a scalelargerthan
the resolutionAz of the simulation).The potentialwell
in this caseis a squarewell (Figure6a). Evolutionof
the field line between two times to and tX is illustrated
/ viz•(vix+
VT)
2
in Figure 6b. As the field line relaxes,two wave structures bounding a region of zero Ba,moveapart in the z
direction with some speed Vz. Field lines between the
boundaries relax in x with a speed Vt. Our simulation
results
show the boundaries
to have both
Alfv6n
(c ')
vix+Vw•
viz
and
(b)
slow-wave characteristics, and so Vz is determined by
the Alfv6n and slow wave speedsin z. By making the Figure 7. Frame transformationargumentsfor firstapproximation that Vz is just the Alfv•n speedalong z, orderFermiacceleration.
(a) Simulationrestframe.(b)
we now showthat Va,is approximately the Alfvdn speed Field line rest frame.
along x.
Referringto Figure 6b, point A on the relaxingfield
line at time to moves to point B at a later time fl.
by geometry. Clearly, the line AB definesa section of
DefiningAt = tx--tO, we havela,= V•At and lz = VzAt
field line, so la,/lz = Ba,/Bz. HenceV•/Vz = B•/Bz,
and if Vz = VA,zthen Va,= VA,a,.
Given that the field line moves in & with velocity
VT5:= VA,a,5:
in the simulationrestframe,wenowgivea
frame transformationargumentleadingto field-aligned
ion accelerationin this frame. Refer to Figure 7. Under
the decoupledassumptionwe ignore the expansionof
the reversal,that is, motionof the fieldline alongz. (If
we include this, then the ions will losefield-alignedvelocity, whilst at positions b, b' due to the expansionof
t=tl>t0
t=t0
the reversal).Consideran ionenteringthe reversalwith
velocity components-via,i: and -viz ;• in the simulation
restframe(positiona). We transforminto the fieldline
rest frame, where the constant convectionelectric field
Ey is zero, by movingwith velocityVTi:. The ion's x
velocitycomponent
becomes
-(via,+ VT)i: (positiona').
The ion then travels acrossthe zero-B• region of field
line (positionsb, b') and exitsthe reversal.In the field
line frame the ion's energy is constant,so its x compo-
nentsimplyreverses,
becoming
(via,+ VT)i:(positionc').
When
transformed
back into the simulation
rest frame
this becomes(vi• + 2VT)& (positionc). Thus the ion
has gained twice the field line speedin the simulation
(b)
rest frame,i.e., 2VA,a,,in the x direction.
Figure 6. (a) Potentialwelland(b) fieldlineenvisaged We should also consider the effect of the expansion
for a discontinuous reversal.
of the reversal structure
in z.
The boundaries
of the
RICHARDSON
AND CHAPMAN:
SIMULATIONS
OF RELAXING
REVERSAL
17,397
regionof zeroB• eachhavevelocityV•t,zalongz. When
an ion interacts with one of these boundaries, it will
lose energy. A similar frame transformation argument
to that given previously shows that the ion will lose
twice the velocity of the boundary with eachinteraction.
Hence whilst the ion approximately gains field-aligned
velocity(i.e., x directedvelocity)dueto relaxationof
the reversalin x, it will losefield-aligned
velocity(z
directedvelocity)whilstwithinthe reversalat positions
b, b'.
4.2. "First-order"
Description
-11250.00
We improve on the above zero-order description by
consideringthe more realistic case where B• reverses
smoothly over a finite distance at time zero. The po-
.............................
0.0
5745.6
114•1.2
(0.0)
(31.7)
(63.4)
z&m(r•)
Figure •. Evolutionof the potentialwell in •he case
tentiM well in this caseis harmonic(Figure8a). The B• - 2nT. The well widens and deepensin time.
potential wells observed in the simulations have this
form (Figures3, 4 and 9). Evolutionof the field line
between two times to and t x is sketchedin Figure 8b.
The potential well in this model is harmonic, and as
Two wave structures again bound the reversal region,
but these are no longer confinedto an infinitely thin the field line relaxes, the potential well expands. As
an ion moves into the reversal, it interacts with the
region as in the zero-orderapproximation.
In this first-order model, the electric field component expandingwell and losesenergye at a rate given by
Ey acrossthe reversalis not constantbut varieswith
de
dq3
z (this is mostpronounced
in the simulations
at early
d-7
dz
•o•,.
(14)
times).SinceBz isconstantin thisgeometry,
thereisno
singlede Hoffman-Tellerframetransformation
velocity For suitably small initial velocitiesvi,, this lossof enVT = V• that wouldsatisfyEy(z) =-V•B, = 0, and ergy will be sufficientto renderthe ion in a boundstate.
so no singleframe transformation will take us into the Thus the ion becomes
tra.ppedin the reversalstructure
field line rest frame. Instead, VT is a function of z
and oscillates acrossit. Under the approximation that
and the abovefirst-orderFermiacceleration
arguments motion in the x and z directions are decoupled, the
cannot be employedexactly. We would expect net x- trapped ion obeysthe equation of motion
directedion acceleration
to be of similarmagnitudeto
that deducedfrom the zero-ordermodel,that is, gains
d2z E,q
of lessthanor of order2V•t,•,but cannotpredictdetails
= --,
(15)
dt 2
of the trajectories.
m
whereE, = -Oc)/Oz. Suchan equationof motion can
be solvedby taking a suitablemodelfor the potential qb.
A relatively simple model consistsof a simple harmonic
potential well of constant depth qb0expandingfrom an
initial half width A0 with constant velocity v; that is,
•
(a)
t=tl>t0
t=t0
z
t)-
z2
+vt)-1].
(16)
This potential arisesfrom a B• which reverseslinearly
with z; in the simulation,B• ~ tanh(z/Ao),sothispotential results when z << A0. In practice, the depth of
the potential well is observedto increasewith time as
the field relaxes(Figure9); this is not includedin the
simple model. The equation of motion using equation
(16) for qbis solvedin the appendix.It is interestingto
considerthe motion of a population of trapped ions, as
describedby vz versusz phasespaceplots. The analysis
givenin the appendixpredictsthat theseplotsshould
havea markedspiralappearance
(Figure10a). This is
(b)
borne out by simulationresultsfor times prior to exces-
Figure 8. (a) Potentialwelland(b) fieldlineenvisaged sivephasemixing, after whichthe spiral appearanceis
for a gradual reversal.
obscured
by thermaleffects(Figure10b).
17,398
RICHARDSON
AND CHAPMAN:
SIMULATIONS
Vz/kms1
750-
ee
eeee
' "'i'
'.--"-..--'
-220
OF RELAXING
REVERSAL
z versust trajectories of several ions. The two plots
for the weak linking field casereveal that ions remain
trapped for longerperiodsthan with a stronglinking
field. It is alsoevident that after the escapeof the initially trapped populationstarted where B• << Bz, ions
whichwereoriginallylocatedoutsidethe reversalwhere
Br >> Bz continueto fall into the well, accelerate,and
escape.
It is significantthat the initial escapehappensen
massefor the trapped population, indicatinga bulk
plasmamechanismrather than a processsolelydictated
by singleparticle dynamics.
-750
5. Bulk
(•)
Fluid
Behavior
Taking the pressuretensorfor ionsin the vicinity of
thefieldreversal
to begyrotropic,
wecalculate
PIIand
Pñ for the trapped population[Krall and Trivelpiece,
1986](this assumption
has beenfoundto be goodto
within 10%). We find that asthe ionsgainfield-aligned
velocity,PII increases
on the timescale
of trappedion
oscillation.Whenthe difference
between
PII and Pñ
0.00
is sufficientlylarge, the quasi-staticequilibriumof the
trappingphasebreaksdown(Figure13). The trapped
5734'(31.6)
6205 (34.2)
z/km(rgi)
population escapesfrom the reversalregion,traveling
toward either boundaryof the simulationbox. As they
do so, they distort the magneticfield and bulk plasma
properties.
2.5-
(•)
Figure 10. phasespaceplotsof vz versusz as (a) de-
2.0 ÷
ducedby solvingthe equationsof motionof many particlesplacedinitially uniformlyacrossan an expanding
harmonicpotentialwell of constantdepth(b) observed
1.5.
in simulation
results.
4.3. Effect of Linking Field Strength
0.0
0.(
Typical ion v• versusz trajectoriesare givenin Figure 11 for both simulations.
(rgi)
11468'.8
(63)
In both cases the ion starts
with a small thermal v•, and as it oscillatesback and
forth acrossthe reversal,it gainsin v•. The maximum
net field aligned velocity an individual ion can gain be-
fore it escapesfrom the potential well is ~ 2V•,• as
shownin sections4.1 and 4.2. In the weak linking field
casethe ion'snet v• jumps by smallfractionsof VA,•
ß
(o)
•
over each oscillation, whilst with a strong linking field
the ion's net v• increasesmuch more quickly. As we
would expect, ions which have the least number of interactionswith the wallsof the expandingpotential well
before escapingfrom it will ultimately have greater net
field-aligned velocity. We will show that the bulk parallel pressureof the trapped population dictates when
the trapped ions will initially escape, rather than the
velocity of individual ions.
After performing a number of oscillationsin the potential well, trapped ions escapefrom the well and travel
away from the reversal to either end of the simulation
box. This is illustrated in Figure 1:2,which showsthe
2.5
2.0
•1.5
,
,>•g1.0
1468'.8
(63)
Figure 11. Typical z versusv• ion trajectoriesin the
simulationframefor (a) Bz = 2nT and (b) Bz = 8nT.
The dashed line indicates the center of the reversal. Af-
ter exiting the reversal,the ions (movingto the left
here)executegyromotion.
RICHARDSON AND CHAPMAN' SIMULATIONS OF RELAXING REVERSAL
40(12.3)
•
40 (12.3)-
20 (6.1).
•
o (o.o)
0 (0.0)
20 (6.1)-
o (o.o)
o (o.o)
12ooo(66.2)
12000 (66.2)
z/km(rgi)
z/km(rgi)
(a)
(b)
16 (4.9)
•
17,399
16 (4.9)
8(2.5)
•
o (o.o)
0(0.0)
12000'(66.2)
8(2.5)
o (o.o) I
0(0.0)
z/km(rgi)
•"J
12000'
(66.2)
z/km(rgi)
(c)
(d)
Figure 12. The trajectories
of several
ionsinitiated(a) and(c) closeto the reversal
(b) and(d)
somewayfromthe reversal.The top twofigures(a,b) arefor Bz - 2nT, the bottomtwo (c,d)
for Bz - 8nT.
of Plland
The increase
in PII is consistent
with the combinedcapefromthereversalregion.The anisotropy
effectsof Fermiacceleration
andthe trappingforcediscussedearlier. These result in an increasedspreadin
the parallel velocitiesof the trapped ions. This can be
seenin Figure14,whichgivesshaded
viiversus
z phase
space plots.
As we seein Figure 13 (and Figures3 and 4), the
trapped populationinitially escapesen massefrom the
Pñ has beendiscussed
by other authors[e.g.,Pritchett
and Coroniti,1992]as an equilibriumconditionarising
from doubleadiabatictheory [e.g., Bittencourt,1986].
Here however, it appears in a dynamic context as an
indicator of particle escapefrom the reversal.
Figure 13 showsthat in the weaklinkingfield casethe
trapped ions achievepressurebalanceat initial escape,
ø,whilstin thestrong
linking
reversal
regionwhenlocallyPIIis sufficiently
largethat thatis,PII- Pñ-• B2/2P
field case the trapped population produce an overpresB2
Pll-P.L
""2yo
sure,PII- Pñ > B2/2•o'In theformer
case,
individual
vii valuesincrease
in smalljumpsastheyoscillate
(17)ion
acrossthe reversaland the bulk Pll canapproach
the
escapeconditiongradually. In the latter case,individ-
This conditionis just that which double adiabatic the-
ualionviivalues
jumpby muchlargeramounts
during
oscillation
across
the reversal,andconsequently,
PII exreconfiguration
in the bulk plasmaproperties[Krall and ceedsthe escaperequirement.
The applicabilityof equation(17) can be testedby
Trivelpiece,1986]. When this happens,the quasi-static
ory wouldindicateas whenthe excess
PII will forcea
trapping phase breaks down and the trapped ions es-
rearranging it to give
17,400
RICHARDSON
AND CHAPMAN: SIMULATIONS
1.5
0.0
bution function of the ions. For simplicity, we assume
that the trapped populationsare just two beams with
bulk parallel velocities•vcri•, as shownin Figure 15.
3.0
4.0 (1.2)
1.5
0.0
3.0
8.0 (2.5)
REVERSAL
isfy(17) locally.This PII canthenbe relatedto the
bulkviiof the plasmaby assuming
a formforthe distri-
3.0
0.0 (0.0)
OF RELAXING
1.5
In this case,PII is givenby
0.0
3.0
12.0(3.7)
1.5
PII-- PVcrit.
2
0.0
3.0
16.0(4.9)
1.5
0.0
Thus
3.0-
20.0(6.1)
1.50.0-
Vcrit
--
3.0-
24.0 (7.4)
1.50.0
3.0-
28.0 (8.6)
0.0-
1.5
0.0
3.0-
40.0 (12.3)
(20)
Using the simulation bulk parameters, q-vc,.ithas
beenplotted on the phasespaceplots of Figure 14. In
the pre-escapeplots of both simulations,althoughsome
trapped ions have individualparallel velocitiesgreater
1.5-
3.0
36.0(11.1)
+?z
wherePB is themagnetic
fieldpressure
B2/(2/•0).
1.50.0-
3.0-
32.0(9.8)
(19)
1.5
0.0
0.0
57,•5.6
11491.2
than the boundary v•it escapedoes not occur as the
bulk parallel velocity is lessthan this v•it. In the post
escapeplots it is clear that the bulk parallel velocity
5.0-
of the escapingpopulation is greater than vomit,thus
satisfyingthe escapecriterion.
The post escapeplot for the stronglinkingfield case
in Figure 14 also showsclearly the bunchingof the escaping ions. This does not arise in the weak linking
1.0 (0.3)
2.5-
field case.
2.0 (0.6)
2.5-
(a)
(0.0)
(31.7)
(63.4)
z/km(rgi)
0.0 (0.0)
5.1. Magnetic Fields
•:8:
3.0 (0.9)
2.5-
4.0 (1.2)
2.5
5.0 (1.5)
2.5
6.0 (1.8)
2.5-
AlthoughBy is zeroin our initial conditions,
a characteristicbipolarBy signatureis generatedacrossthe
reversalfrom early times(seeFigures3 and4). Hodogramsof the Bz, By field components
are givenin Figure 16 (seealso[Goodrich
and Cargill,1991]). As we
movethroughthe structurewith increasingz, the field
rotatesin a clockwise(right-handed)sensethroughan
anglelessthan •r/2, andthenbackagainin an anticlockwise(left-handed)sense.In the regionsat the edgesof
the reversal,whereBy increases
from zero, the total
field strengthB is constantand the densityreduces.
To supportBy, the electronscarrycurrentin the -&
•:8:
7.0 (2.1)
8.0 (2.5)
2.52.5-
•:8:
9.0 (2.8)
2.5-
direction and the ions in +&. In both simulations,the
electroncurrentis approximately25% largerthan the
•:8:
10.0 (3.1)
2.50.0-
0.0
(b)
(o.o)
57,•5.6
(31.7)
z/km(rgi)
ion current, and so there is a net currentin-•. Prior to
the reversal, the electron current reverses
(63.4)
directionat the edgesof the reversalto thussupportthe
fine structure. After escapethe field becomesdistorted,
11491.2 escapefrom
Figure 13. Evolution
of PII- Pñ (solidline)andmag-
and fine structure
is not as evident.
neticfield pressure
Pls (dashedline) with (a) a weak Whenthe linkingfieldis weak,the magnitudeof By
linkingfield(b) a stronglinkingfield. The unitsof pres- is approximatelytwice that observedwhen the linking
sure are Nm -2 x 10-•ø. The numbers down the left-
fieldis strong.This may be dueto the muchlongertime
hand side of eachfigure indicatereal time in seconds for which ions are trapped in the reversalstructure in
(iongyroperiods).
theformercase,givingthe By component
moretimeto
develop.
B2
The post escapehodograms,Figures 16b and 16d,
showmorestructurein the stronglinkingfieldcasethan
in the weak case. This corresponds
to fluctuationsin Bz
Calculatingthe right-handsidefrom simulationplasma and By at locationsof particlebunchingin the escaping
-
PII•-P•-4 2/•o
parameters
wouldyieldthevalueof PIIrequired
to sat-
ions.
RICHARDSON
AND CHAPMAN:
SIMULATIONS
OF RELAXING
1-I
ß
• •:
-21
9229 15o.9)
17,401
"•'•'•'>'
' • '"
.....-•,.,..
2240(12.4)
REVERSAL
"•:•,
.::.•d•'•..'.,..
•-.....'I,:•'"42.,.,-,. : ' :.:
•-• •'-½."•"
.I•;": ' .;•- *r-"
¾;½,4"• ': :,
2240 (12.4)
9229 (50.9)
z/km(rgi)
(•)
2-
-3
2240 (12.4)
9229 (50.9)
2240 (12.4)
z/km(rgi)
9229 (50.9)
z/km(rgi)
(c)
(d)
Figure 14. Shadedvii versus
z phasesp•ceplots. The top twofigures(•,D) •re for • we•k
linkingfield (2nT), the bottom two (c,d) for a strongerlinkingfield (8nT). In eachcasethe
leftmostplot (a,c) is for a time prior to escape,the rightmostplot (b,d) is postescape(the plots
are for the sametimes as the snapshotsin Figures3 and 4). Note that only the central 60%
of the simulation axis is shown, to clarify structure in the escapingpopulation. The increased
spread in parallel velocitiesin the reversalstructure can clearly be seenin both pre-escapeplots.
In the weak linking field case, the ions escapefrom the reversal with velocitiesless than twice
the z direction Alfvdn speed. With a stronger linking field, they escapewith twice the Alfvdn
speed.Thesolidlinesindicate
thecriticalvelocity
deduced
fromPII- P•- = Ba/2Iuo' In both
simulations, escapefrom the reversal only occurs when the bulk parallel velocity of the trapped
ions exceedsthis critical velocity.
6. Conclusions
which describeswhen the trapped plasma can no longer
be contained and will escape.
We have presented the results of one-dimensional,
4. Following this initial escapea continual processof
self-consistentsimulationsof a dipolarizing field reverinflowing ions becomingtrapped in the reversal,accelsal with varying linking field strengths. We have dis- erating and subsequentlyescapingoccurs.
cussedboth ion dynamics and bulk plasma behavior.
Our principal results are as follows:
1. The largeVB 2 in the reversalstructurecausesthe
trapping of inflowing ions.
2. ThePllofthetrapped
population
increases
in a man-
ner consistentwith net field alignedaccelerationdue to
the Fermi acceleration of ions as they oscillate across
the reversal and decelerationas they interact with the
reversaledge.This ultimatelyresultsin their escape.
3. We havefound a generalcondition,
,• vparallel
-vcrit
S2
PII-P.L
• 2po
+vcrit
Figure 15. Distribution function assumedfor calcu-
(21)latingv•it, f(Vll)-
f0(5(Vll- v•it)q-5(vii+ v•it))
17,402
RICHARDSON
AND CHAPMAN:
SIMULATIONS
(a)
OF RELAXING
REVERSAL
(b)
-11
-11
-21
Bx/nT
,
,
0
21
Bx/nT
(d)
(c)
-6-20
-60
20
-20
Bx/nT
6
Bx/nT
Figure 16. Hodogramsshowingthe bipolarB• component
whichis generatedfrom early times.
The top two graphs(a,b) are for a weak(2nT)linking field,the lowertwo (c,d) for a stronger
linkingfield (8nT). In eachcase,the left figure(a,c) is prior to escapeof the trappedionsand
the figureon the right (b,d) is postescape.
5. We have identifieda characteristic
signaturein By
Appendix: Expanding Potential Well
accompanyingthis behavior.
6. We have shownthat the time an ion remainstrapped
To representan expandingpotential well of constant
is dictated by the ratio of linking field componentto
depth in one dimension,we considera potential of the
reversingcomponent,and as a consequence,
this deter- form
mines whether the post escape phase is characterized
by pressure
balance(PII- P•- -• PB) or overpressure
r)(z,
t)- r)o(Ao
+vt)
•- 1,
(A1)
>
These resultsdefinea signaturewhichshouldbe evwhereA0 is the initialhalfwidthandv is thespeedat
ident in observationsof dipolarizationcloseto the tail
whichthe edgesof the well moveapart. The electric
centerplane. If PII - P•- -> PB outsideof the rever- fieldassociated
with thispotentialis givenby
sal, then field alignedbeamsmovingat Vii ~ 2VA,r
(~ 1- 10keVfromthesimulation
results
shown
here)
will be observed
which,if PII- Pl >>P•, will be bursty
in nature. Insidethe weakfieldregion(identifiableby
the bipolar By signaturein a coordinatesystemanalogousto the simulationconfiguration)
counterstreaming
240z
(Ao+ vt)2
(A2)
beamsof Vii< 2VA,rshouldbe observable.
A particle of massrn and chargeq in this potentialhas
Our simulationsreveala localionenergization
mecha- the equation of motion
nism whichcan be approximatelymodeledasfirst-order
Fermi acceleration.
Ions are accelerated to low-medium
energies(1- 10keV), indicatingthat the high-energy
( ~ 1MeV) burstsof particlesseenduringsubstorms
d2z (2q•0q
•
z
-'-•dt + k,mAo2,]
(1+vt/Ao)
• =0.
(A3)
must be acceleratedelsewhereor by someother mecha- To solve this, we make two substitutions. First, we
nism not representedin the self-consistent
hybrid code. substituteT- 1 + vt/Ao and obtain
RICHARDSON
AND CHAPMAN'
SIMULATIONS
2d2z (2•boq'•
z -0
-o
now make a secondsubstitution,Tdz
ds 2
ds
e• yielding
l- kz - 0
(A7)
has roots
1
D- • 4-V/4
x--k.
7-tan - x[-20 -¬]-
(A13)
(A14)
0
-• xln(•-ø-)
Ao (A15)
C-
V/•i In(
(A16)
sin[
If we now populate the vicinity of the potential well
with a number of equally spacedparticles, we can obtain the expectedphasespacediagram. This involves
calculatingboth the position and the velocity of each
particle at a giventime; the latter quantity is givenby
(A8)
Vz ---- d--•'
• •)•t
AoV(I+
c
case is
k-•+7]
{••sin[qk
_xln(1
+•oo
)+7]
+qk-ixcos[v/k•ln(1
+•0)+71}(a17)
(A9)
where C and 7 are constantsdeterminedby the boundary conditions. Substituting back for s and T gives
A typical phasespaceplot obtained by this method is
givenin Figure 10a, to be comparedwith an actualplot
obtainedfrom a simulationgivenin Figure 10b.
As a final test of this solution, taking the limit of
v -• 0 shouldgive us the trajectory of a particle in a
potential well of fixed width. To do this, we note that
- •)ln(l+•o)+7]
(A10)
From this we seethat the amplitudeof oscillationin-
creases
with timeas v/(1 + vt/Ao). The periodof the
as V•0•
vt) -• A--•'
vt
In(1
+ •0
motion alsoincreasesdue to the logarithmicdependence
on t in the oscillatoryterm (Figure17).
The boundaryconditions,
andhencethe constants
C
)
z-C
sin[•(
2q5øq
t + 7]
time t = 0. For the two caseswe have
z-zo,•- &-O
t-0
(All)
(A19)
Comparingthis with the equationof SHM,
z - C sin[wt+ 7]
.
(A18)
and with somemanipulation,equation(A10) becomes
and7, aredependent
uponwhethertheparticle's
initial
positionz0 liesinsideor outsidethe potentialwellat
Ifzo<Ao,
(A1,2)
and if zo > Ao
The case k < 1/4 would correspondto overdamped
motion. This is outsidethe regimeof our simulations,
wherewe have k > 1/4. The standardsolutionin this
+ •o
v
C - si•-•
(A6)
O2 - O + k - 0
z-Ce'sin[s
t_<
7- tan-X[-2q
k- ¬l
Equationsof this form are associatedwith dampedsimple harmonic motion. To solveit we considerthe auxiliary equation,
which
zo- Ao
Ifz0>A0,z-z0,•-0
wherek is the dimensionless
constant
(2qboq/mv').
We
d•'z
17,403
Usingtheseboundaryconditions,we canobtainexpressionsfor C and 7, giving if z0 < A0
z
+
REVERSAL
&
(A4)
or
d2z
OF RELAXING
(A20)
we identify
/ 2c•oq
co-V•--•0
•
(A21)
as being the frequencyof oscillation. In the caseof a
slowly expandingpotential well, the frequencywould
have this value at time t - 0, and it would slowly de-4
-2
0
x/km
2
4
crease thereafter.
Figure 17. A typical trajectory obtained by solving
Acknowledgments.
The authors are indebted to T.
the equation of motion of a particle in an expanding Terasawa for providing the forerunner to the hybrid code
usedin this study, and to V. Semenovfor illuminating disharmonic potential well of constant depth.
17,404
RICHARDSON
AND CHAPMAN:
SIMULATIONS
cussions.One of the authors (A.R.) was supportedby a
SERC scholarship.
The Editor
thanks two referees for their assistance in eval-
uating this paper.
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(ReceivedMarch 26, 1993;revisedAugust31, 1993;
acceptedOctober14, 1993.)