JOURNAL OF GEOPHYSICALRESEARCH,VOL. 98, NO. A1, PAGES165-177,JANUARY 1, 1993
Parameterization
of ChaoticParticleDynamics
in a SimpleTime-Dependent
Field Reversal
SANDRA C. CHAPMAN AND NICHOLAS W. WATKINS
SpacePlasmaGroup,Schoolo! Matheraatical
andPhysicalSciences,Universityof Sussex,
Falraer,Brighton,UnitedKingdora
We investigatesinglechargedparticledynamicsin the Earth'smagnetotailusinga simple,
scalefreemagneticfieldmodelwhichis explicitlytime dependent,with a corresponding
induction
electricfield. The time-dependent
Haxniltonianof particlemotionin the simplemodeldescribes
a systemof two coupledoscillators
whichis ch-iven.Whenthe (time-dependent)
ratio of the
oscillationfrequenciesis differentfrom unity, the motion is regular, but if it approachesunity
at some point on the trajectory, the motion becomeschaotic. A parameter a is found which
characterizesthe adiabaticity of the system,and the transitiontime for behaviorin the systemis
given by c•t --. 1. The conditionat • I is equivalentto n m 1 in the static parabolicfield model
discussed
by previousauthors(BuchnerandZelenyi,1989).The explicittimedependence
produces
two possibleclassesof motion, orderedby a. If the reversalis thick and is folding slowly,so that
c• << 1, the motion is a transitionin behavior,from regular t• conservingto chaotic "cucumberlike"
trajectories, when t •, 1/a. If on the other ha•xd,the reversalis thin and folds quickly, so that
c• > 1, the particles execute regular "ring type" trajectories once t > 1/c•. Simple estimates
of presubstormmagnetotailparametersindicatethat electronsin a slowlythinning (15-rain time
scale),thick (1RE) sheethavea • 1, whereasprotonsin a thin (severalhundredsof kilometers)
sheet which thins on a 5-rain time scalehave a • 1. Hence the behavior of the particles, and by
implication, the field reversal which they support, will depend upon the adiabaticity of the system
c•, as well as the "chaotization" parameter c•t; this is shown only to be the case in a model which
is explicitly time dependent and which includes the induction electric field.
1. INTRODUCTION
Sincethe early 1960s[e.g.,Speiser,1965;Sonnerup,1971;
Lyonsand$peiser,1982; Chenand Pa!madesso,
1986]single
chargedparticledynamicshavebeenstudiedin simplestatic
modelsof the magnetic field reversalin the Earth's rnagnetotMi. As well •s giving an insight into the local plasm•
populations,the dynamics of these particles,essentiallythe
currentcarrierssupportingthe tail field reversal,shouldto
someextent dictate the globalstructure and stability of the
magnetotail.Changesin the trajectoriesmay be relatedto
disruptions
of the field geometry,eitherindirectly(e.g.,via
tearingmodeinstability[Buchnerand Zelenyi,1987]),or
directly(by scatteringthe adiabaticcurrentcarriers,caus-
ingthe plasmasheet
to thin [e.g,Mitchellet al., 1990,and
references
therein]).Alternativelychanges
in the dynamics
of someparticlesmay be regardedas providingpopulations
whichdestabilise
the fieldplasmaconfiguration
closeto substormonset[Galeev,1982]and references
therein.
In particular,a staticfieldreversalmodel(with parabolic
magneticfield lines and no inductionelectricfield) whichis
scalefree has been used to characterizethe particle dynam-
ics[Buchner,1986;BuchnerandZelenyi,1989].As will be
demonstrated
here,modelswith no intrinsicscalecanallow
theparticledynamicsto be parameterized
analytically;in
the caseof the parabolicstatic field reversala singlecon-
trolling
parameter,
• [Buchner
andZelenyi,
1986,1989],a
of trajectory are possible:adiabatic trajectorieswhich are
alwaysregular(integrable)and whichconserve
an adiabatic
invariant(suchas/•), and chaoticor stochastictrajectories
which are not regularand do not conservea singleinvariant,
the two types being confinedto separateregionsin phase
space.The steadystate parameter n hasbeenusedin timedependentproblems,e.g., as a substormindicator in the
WKB limit [BuchnerandZelenyi,1987],andto discuss
substorm associated
phenomena[Pulkkinenet al. 1991]. However, the parameter n, being derived from a steady state
model, doesnot containany information about the time dependenceof the field. Temporal changesin the field are
accommodatedby treating the time-dependentsystem as a
seriesof steadystate magneticfield models,each characterized by a slightly different value of n.
This paper is an attempt to introduceexplicit time dependenceinto the systemby examininga scalefree model
which is both time dependentand parameterizable.We examine particle dynamicsin a fully time dependentscalefree
model for the magneticfield reversalwhich "foIdsup" with
increasingtime and which includes the induction electric
field associated with this time dependence, as well as the
constant cross-tail convective electric field by frame trans-
formation. A new parameter a emergeswhich, as well as
being a function of the field strength and thicknessof the
reversal and the particle gyroradius,is now also a function
of the time scale over which the field changesand the p•r-
function
of the field strengthand thickness
of the reversal tide gyroperiod.We showthat the conditionn • 1, which
andtheparticlegyroradius,
determines
whether
theparticle identifies a steady state system which is chaotic for all t, is
motion
is(forall•) adiabatic,
orchaotic.In thissteadystate equivalentto the singletime c•t • i in the time-dependent
system[seeChenandPalmadesso,
1986]twodistinctclasses model. Hence in this model we find that, unlike in a steady
state field reversal, there are no trajectories which are reg-
ular (i.e., adiabatic)for all •. In other words,there are
no integrabletrajectorieswhich can be confinedto a regionin phasesp•cefor all t in the time-dependent
system.
The explicittime dependence
producestwopossibleclasses
Copyright1993by the AmericanGeophysical
Union.
Papernumber92JA01548.
0148-0227/93/925A_01548505.00
165
166
CHAPMAN AND WATKINS:
CHAOTIC DYNAMICS IN NONSTATIC R,EVEI•SAL
of motion, depending on whether the reversal "folds" on
a normalized time scale which is slow, or fast, compared
with the (normalized)chaotizationtimescalet • 1/a. The
rate at which the reversal cha•nges,and its spatial scale, as
compared with the spatial and temporal scales of the particle motion in the reversal, are specifiedby the parameter
c•. If a << 1, representinga thick reversalwhich is folding slowly, the particle experiencesa transition in behavior;
when • < 1/c• it is regular, conservingan invariant tt.,once
t m 1/c•, the particleis pitch anglescattered,and the motion
paper should thereforebe regardedas examining the underlying nature of single particle dynamics in time dependent
field reversals, rather than an attempt to accurately model
the presubstorm magnetotail.
The paper is organized as follows. In the next section
the field model will be presented, with a discussionof the
constraints required for parameterization.
In section 3 the
Hamiltonian equation of motion will be parameterizedto
obtain the control parameters a and at. We will then show
that in the time-dependent model, a transition in behav-
becomeschaoticand "cucumberlike"
[Buchnerand Zelenyi, ior will occuras at .• 1. The role of a as the adiabaticity
!989]. If on the other hand, the reversalis thin and folds parameter of the system will be established, and the be.
quickly, so that c• >> 1, the motion becomesregular within havior for different values of a will be discussed. Numeria Larmor period, exhibiting no extendedinterval of chaotic cally integrated trajectoriesare presentedin section4, and
motion. The parameter c•, which essentially specifies the a summaryof the analyticaland numericalresultsis given
adiabaticityof the system(rather than of the particle), is
in section5. In section6 we will establishthe relationship
shown to have no counterpart in the steady state model.
Explicit time dependenceand the inclusionof the induction
electric field are required to allow the parameterization of
the time scale over which the systemevolvesas compared
with the characteristicfrequenciesof the particle motion.
betweenthe role played by • in the static model, and that
playedby al in the time-dependentmodel, and showthat a
has no analogue in the static model. Estimates of a for the
Earth's magnetotail are given in section7.
Hence we show that the condition • m 1 which has been
2. THE SCALE FREE, PARAMETERIZABLEFIELD MODEL
implied to indicate the onsetof chaoticbehaviorin initially
2.1. Scale Free Models
t•-conserving
particlesin time dependentglobalmodelsof
the magnetotail[Buchnerand Zelenyi, 1987, Pu!kkinenet
We shall use a simple scale free model as opposedto a
a!., 1991]will only do so if the systemchangessufficiently phenomenologicMgeotail model as in general the latter inslowlyandis sufficientlyslowlyspatiallyvarying. In general, troduces intrinsic length and time scaleswhich obscurethe
both the time scMe, and the nature of the transition are parametersthat characterizethe trajectory [Chapmanand
always given by the c• of the system.
Cowley,1985]. The scalefree model is such that the parIn principle our approachcan be applied to folding of ticle equationsof motion,with suitable(scalefree)normalthin current sheets on either short time scales (of the order ization, hereafter denotedby an asterisk,have the following
of a few minutes),impliedby field and plasmain-situ ob- property:
servations
[e.g. Ohtaniet al., 1991;Mitchellet al., 1990]in
dr*
which the inductive electric field will play a substantial role
in particledynamics,or the "global"time scale(of the order
of 15-30min) over whichthe magnetotailas a wholegains
=
energy(i.e., the "growthphase"[Hones,1977]). In prac-
dr*
tice, the constraintsrequiredfor parameterizationresult in
a simple time-dependent model with application restricted
to a "local"regionin the centralplasmasheet. The spatial
validity of the time dependentmodel will be just that of the
steady state parabolic field model which is appropriate for
spatial regionssufficientlycloseto the tail center plane. In
addition, the time dependentmodel will only be appropriate to the magnetotail for a finite range of time. Since we
dt*
will show that
the chaotization
condition
c•t ,•, 1 is satisfied
within a finite time, and as the particle crossesthe tail centre plane, this simple model is sufficientfor a determination
of the "local" conditionsin the plasma sheet required for
chaotization.
(1)
with
A particle trajectory in this systemis uniquely specifiedby
three independentparameters,namely the velocity componentsat somepoint on the trajectory r*(t*) (or the speed,
pitch angle,and õyrophase)at time t* [Chapmanand Cowley, 1985].
The transition from one type of particle behavior to another, and the violation of an adiabatic invariant associated
with a particular type of behavior as the transition takes
placeare thereforealsocharacterizedby no more than three
independent parameters. A scalefree field model, in which
the suitably normalized Lorentz force law has the form of
equation(1), is requiredif we are to determineany of the
By examining a model which is scale free and hence pathree independentparametersanalytically. If a field model
rameterizable, we are also able in principle to obtain the
with some intrinsic spatial or temporal scale is used, the
time scale for chaotization at ,,• 1, and the adiabaticity paright hand sideof (1) will alsobe a functionof that intrinrameter c• for particles of any mass,charge, and gyroscales,
sic scale,and more than three parameterswill be required
in the model field reversalwith any given radius of curvato specifyeach trajectory in the field model, althoughthe
ture at the center plane and time scale for variation of the
additional parameter will not be independentof the first
field. This is generally not the casefor phenomenological three.
field models which more accurately represent the magnetotail where the details of the model obscurethe underlying
œ.œ.Time Dependent Field Model
trends in particle behavior. In this case the dependenceof
the particle behavior on each of the model parameters can
With a:,y and z corresponding
to GSE coordinates
(Geo-
only be determined by numericalintegration of the parti- centtic Sun Earth, with a:toward the Earth, z northward),
cle equations of motion in the model. The analysis in this the scalefree field model to be usedhere,
CHAPMAN
ANDWATKINS:
CHAOTIC
DYNAMICS
tNNONSTATIC
REVEI•SAL
Bx zt
n=( ...•.•-,
0,•)
(a)
167
tiMly trapped,#-conservingtrajectories.In additionthe x
componentof the magneticfield varieslinearly in t, giving
parabolicfield lines for arbitrarily small t > 0 and a current
density which increaseslinearly in t. This linear variation
in
t in generalrepresentsa small time interval of any flucrepresents
parabolicmagneticfieldlinesWhichfoldtoward
2-hV0)
(4)
in thecurrent
density
(e.g.,sint/r or I -e--•) for
thecenterplaneof the magnetotail
(z = 0) withincreasingtuation
time t on time scale r. The field reversal has half thickness t << r, or in particular a system which has approximately
h suchthat at this scale height and at time t = r the field linear time variationin J for arbitrary t. As it will be sitown
z component
hascharacteristicstrengthB=. The inductive that in the time-dependentfield, a transitionoccursin the
dectric
fieldB=z2/2hr
ischosen
tosatisfy
VAE= -OB/Ot
with the additional constraint that there should be no z de-
pendence
in themodelsothat a constant
cross-tail,
in (•r),
particle dynamicson a finite time scale,it is appropriateto
consider a model which is valid for some finite time interval. Our conclusions will then be valid if the variation in the
convection
electricfield E½= vTBz•rcanbe introducedby currentdensityat the centerplane is not stronglynonlinear
in t over the time taken for the transition in behavior, i.e.,
moving
in the• directionwith constant
velocityvT• [Chap- for c•t • 1.
manand Cowley,1985],to representplasmac6nvect.
ion in
the Earth's magnetotail. This requirementfor the inclusion
of a convectionelectric field by frame transformation effectively Constrainsthe linking field Bz to be constantfor all z
andt, so that the model is appropriateprovidedthat on the.
3. ANALYTICAL
DESCRIPTION
DYNAMICS
OF THE PARTICLE
time scalesto be discussedbelow the linking field doesnot
It has been demonstratedthat sttrajectory is uniquely
changesubstantially(i.e., providedthe particleis not con- specified by three independent parameters. Here we will
vectedalonga substantiallengthof the tail). In additionto showthat most, but not all, featuresof the particle behavbeingconstrainedby X7A E -----oqB/0t, the inductionelec- ior are determinedby two parameters: c•t which indicates
tric field is also chosento satisfy V.E = 0 consistentwith the transition time of the particle behavior, and a, which
quasi-neutrality.
yields the "adiabaticity" of the systemwith respectto this
By normalizingtime to 1/fiz (the inverseof the gyrofrequencyin the field at z - O,•% -- eBz/rn) andlengthscMes
transition
time.
toBzr•h/B• it canbeverified
thatthisfieldmodel
isscale 3.1. The Controlling Parameters c• and crt
free,that is, the Lorentz force law becomes(for positively
charged
particles)
dv*
= v' ^
+
(s)
The controlling parametersare revealedby renormalizing the particle and Hamiltonian equationsof motion in the
scalefree model. Normalizingtime to the inverseof the gyrofrequencyat z = 0, 1/f•z, and length to p•, the gyroradius
In thisscalefree normalizationthe vectorpotentialmay be at z = 0, the equations of motion are
written
z2
Z*2• *
J•= crt
T -- x = ϥ--x
2 ,0)
z2
with qb= 0 in the frame where Ec = 0. Field lines (i.e.,
• = -at(at5-- a:)z
= -•zZ
contours
of constantA•) are parabohc,foldingtowardthe
z = 0 plane with increasingt.
(9)
(10)
where we have used the first integral of the y equation of
This simple scale free model representsthe more phe- motion,
nomenologically
accurate model of the tail current sheet
B= (Ctanh(•),
0,Bz)
withintrinsicscaleheighth, overthe restrictedregionz < h.
Herewe alsointroducetime dependence
with
C=B-
t
• = c•t-¾
-- x
(7)
The Hamiltoni•
dH
d •
•=• x
Previous
steadystatestudies[e.g.,ChenandPalmadesso,
1986andkeferences
therein]havea constantC = B•. Here
it will be shownthat stochasticbehavior,that is, jumps in
equation of motion can be written •
•2
z2
•' = •(• +T+•)=•(• ••
(s)
(11)
Z2
-
.•2
- •) = v.E (•2)
Z2
=-•- +•*T-•F•I
(13)
where the controlparameter c•is given by
theparticleactionp, occursasthe particlecrosses
thecen-
B•p,fly
ter planez = 0, so that in order to examinethe conditions
(14)
fortheonsetof chaosweneedonlyto accurately
modelthe
and f•! = I/r, the fold-up "frequency"of the field.
The role of a and at • controllingparametersis apparent
above
B fieldreduces
to equation
(3) andisscalefreeasrefrom the aboveHamiltonian equation of motion.
spatialregioncloseto the centerplane z < h, for whichthe
quired.This approximation
doesimplythat thesimplefield
raodddoesnot have a losscone,that is, all particleswill
ultimately
mirrorand returnto the centerplane[Chenand
First, at any instant in time, we have written the Hamflto_•an
Palmadesso,
1986];
this,however,
willnotaffect
conclusions
concerning
the pitchanglescattering
(onsetofchaos)in ini-
•
•2
•2
168
CHAPMANAND WATKINS: CHAOTICDYNAMICSIN NONSTATICI:{E;VER.
SAL
where
thez = 0 planewhichareapproximately
I, = fv•dz con-
x
(]6)
and • hasbeenwritten as the pseudopotential
for two coupled oscillatorsin x and z. The parameterA = at scales
the Hamiltonian by scalingthe pseudopotential
• within
whichthe particlesmoveat any time t, sothat the ratio of
oscillationfrequencieswill alsodependon c•t.
Second,dH/dt -7/:0 as the v.E term is nonzero,so that
we can write this Ilamiltonian equationof motion as
dH
dt
•A
the motion is /• conserving,result in a drift in the •c^B
direction,i.e., in •r consistent
with guidingcenterdrift, and
if the motion is I, conservingwill producea drift in the•
direction.
A third classof motion will ariseif at somepoint on the
trajectoryWz• f•r•. This weaklychaotic(stochastic
[Chen
and Palraadesso,
1986]or cucumberlike[BuchnerandZe.
ments over which w• • f•n. When the particle is far from
dt
thecenterplane,themotionis approximately
t• conserving.
where
When the particle crossesthe centerplane, it, executesfast
dA
=
The parameter
•:
threadingthe centerplane. The forceterm F•c should,
if
lenyi,1989])motionconsists
of approximately
regular
seg-
OH dA
=
serving,accompaniedby a slow z gyration about tl•e field
(]8)
oscillations
in z which
areapproximately
I• ( = jfv•dz)conserving. The particle completesa half turn about the center
c•scales
the rateat whichthe Ilamilto- planefield B• duringthis sectionof the trajectory.
This type of interaction,wherethe particlemotionswitches
from /• conservingto I• conservingas it interactswith the
OH
field radiusof curvatureat the centerpl,•ne,is
OX= "•A
(19) maximum
reminiscent
of bothSpeiser[Speiser,1965]andcucumberlike
here, a alsoscalesthe rate at whichthe pseudopotential[BuchnerandZelenyi,1986]trajectories.Thesetrajectories
changes.In this sense,c• indicatesthe "adiabaticity"of aredistinguished
by [ChenandPalmadesso,
1986]whether
thesystem.The limit of aninfinitesimMly
slowlychanging they are transient(Speiser)particleswhichcan escapethe
system is
system,or stochastic(cucumberlike)particleswhichmirror
and return to interact with the center plane. In the scale
dtt
free
parabolicfield all particlesof this type will ultimately
a-h- 0
(20)
nian changes,•nd since
implying
mirror and return to the centerplane, so that this discussion
concernscucumberlike,or stochastic,chaotic particles.
We will now establish that the time scale for the transition
betweendifferentclasses
of motionis t •
1/c•. Sincewe
whichcorresponds
to neglectingthe inductionelectricfield. have shown that a yields the adiabaticity of the system,
We will nowdiscuss
the particlebehaviorin titissystem we expect different behavior for different a'. Titere are two
in detail.
classes
of behavior,for the two regimesc•< 1 a.ndc•>_1.
The aboveequations(9)-(13) describethe motionof two
3.œ.a<1
nonlinearcoupledoscillators,
in a:andz. The z equationof
motioncontains
anoscillator
withfrequency
1, the (normMIn this casethe time intervalt <' 1/c• (wheret is norized) Larmorfrequency
at the centerplanez = 0. Sincethe realizedto •;'•) corresponds
to manyLarmorperiods.At
magneticfieldminimizesat the centerp!•ne, this is ,xlsothe early t, that is, for sufficientlysmall at, •:• << 1 and hence
minimumvalueof the Larmorfrequency
fin at,anypointon cv•<< •ql;overthe entiremotion(sincefir. > 1). Duringthis
thetrajectory.In addition,the drivingtermF•(z, at) rep- intervalthe trajectoriesare regular,characterized
by fastosresentsa forcein the a:direction. The z equationof motion cillation about the field at the local Larmor frequency9r.
contains
an oscillator
withfrequency
w=(a•),whichit is im- and slowbouncemotion(at frequency
w,) betweenmirror
portant to note is explicitly time dependent.In section6 we points,that is, they are g conserving.As w• • 1 (with
will showthat in thesteadystatemodelthisfrequency
only increasingaZ), it will approachFtr•in magnitude,the two
hasimplicittime dependence
throughthe particleposition. frequenciesfirst becomingequal at the center plane where
This explicit time dependence
in w• will now be shownto Or. minimizes(i.e., where•qr•= 1). As the particlecrosses
imply that no trajectories
in the time-dependent
modelcan the z = 0 plane(wherethefieldcurvaturemaximises),
# will
haveregular(adiabatic)behaviorfor all t, a.ndwill yieldthe jump. The particlewill undergostrongpitch anglescattertransition time c•t • 1.
ing beforeevolvinginto weaklychaoticcucumberlikebehav-
Particletrajectories
will be regular(adiabatic)for some ior. Forsufficiently
largec• wemayhavew, >>1 (wheno0,
finitetimeintervaloverwhichthetwooscillation
frequenciesis real) overalmostthe entiretrajectorysothat the motion
remaindissimilar.There are in generaltwo possiblemodes becomesregular, that is, w• >> Or. over the entire motion.
ofregular
behavior
forwhichthistimeintervalislonger
than This corresponds
to the regularI• conserving
("ringtype"
the periodof both thefastandslowoscillation.
If f•n > •o• orbits discussedabove. On the other hand, if w• remainsof
overboth the fast and slowoscillatorymotion,and hence the order of •qr. closeto the center plane, the motion will
I >> w•, the motionis adiabatic,conserving
an invariant remain cucumberlike. These two classes of behavior are dis-
tt = v•/B associated
with the fast oscillation
about.the tinguishable by a third parameter, which is related to the
fieldat the Larmorfrequency
f]L, the frequency
w• corre- particleenergyassociatedwith the z componentof the mospondingto the frequencyof the slow bounce motion be-
tion aswill be demonstrated
in section3.4 (seealsoBuchner
tweenmirrorpoints.If •L <<wz, themotion,if completely and Zelenyi [1989]).
regular,is "ring-type"(in the definitionof BuchnerandZeBy inspectionof the ttamilt.onian equationof motionwe
lenyi[1989]),the particleexecuting
fast,z oscillations
across have however established the conditions for "chaotization",
CHAPMANAND WATKINS:CHAOTICDYNAMICSIN NONSTATICREVERSAL
169
origin, intersectingthe center plane at x > -p0 and the trajectory at least twicein the regioncloseto the centerplane.
The motionwill now be cucumberlike(or stochastic):far
from the centerplane the trajectory is still located in the
co•/i2r•<< 1 or # conservingregion, but closeto the center
plane a segmentwill be in the coz/i2r•> 1 or _/..-conserving
region. As the trajectory crossesthe co..- I line, a jump
in the action occurs.
The motion
will remain
cucumber-
like if segmentsof the trajectory remain wholly within the
"
• 03z--1
p-conserving,•-, < I region.
In summary then, we expect that as t -+ !/c• trajectories will exhibit jumps in /• as they crossthe center plane,
S
and that the motion will evolve from y conserving to cu-
cumberlike. Sincejumps in the action /• occurjust as the
particle crosses
the centerplane,this discussion
only applies
to trajectoriesthat crossthe center plane on time scales
shorter than, or of the order of, the chaotizationtimescale
Fig. 1. Sketchof the pseudopotential
•(z,z). The innersolid t -- 1/c•. This is appropriatefor particlesinteractingwith
line marks the minimum ß = 0 correspondingto the location
of the field line or contour of constant vector potential A• = 0. the Earth's magnetotailas in the frame containinga zero
The outer solid line is the locus of points where wz -- I where convectionelectric field most particles will have a signifithe x and z oscillation frequencies become approximately equal. cant field-alignedvelocitydirectedinto the field reversal.
!d line
Both of these lines collapse toward the center plane z = 0 with
increasingt. The dotted lines indicate the energiesassociated
with the z motion of particles with two classesof behavior, S is
either/• conservingor cucumberlikeand I is Iz conserving.
In this casethe intervalt < 1/c• corresponds
to lessthan
a Larmor period, and the particle cannot execute l• conserv-
that is, the conditions for the transition from l• conserving
to cucumberlike
motion.
This is illustrated in Figure 1, which showsa sketchof
thefunction•(x, z) at somet. Markedon the sketches
are
the lines •-- 0, and coz-- 1. The inner line •0 correspondsto the field line, that is, the contourof constantvectorpotentialAy -- 0 (seeequation(6)) aboutwhicha particledescribedby the aboveHamiltoninnequationof motion
gyratesduring the /•-conservingsegmentsof its trajectory.
Thisline, that is, the field linesof the model,movestoward
the center plane z - 0 with increasingt.
When a trajectory approaches
the outer coz-
ing motionfor whichi2r•>>co•.Oncet --+1/c•,providedthe
particlecrosses
the centerplane,the trajectorywill intersect
the co_.= 1 line as above, and the motion must become ei-
ther cucumberlike
or I_.conserving("ring-type"),depending
upon the subsequent
motion, that is, whethersegmentsof
the trajectory remain within the co-,< I region. SinceaJ•
increaseswith (•, we might expect that for sufficientlylarge
t• the motion will becomeregular f_. conserving.In section4
we will numericallydemonstratethat for c• > 1.55, particles
executeregular,f• conserving
(or "ringtype") trajectories,
the motion instead being cucumberlikefor smaller valuesof
1 line,
jumpsin the action(lt or I-,) are expected
to occur.Pro- 3.J. The Third ControllingParameter
videdthat i2r•.• 1, the wz - i line dividesthe x- z plane
intotworegionsof particlebehavior:the regionextending Both classesof particle, thosetitat remaincucumberlike
for c•t > 1 discussedabove, and those that exhibit. regular
intoq-• corresponding
to (co•(1) /,-conserving,
co••(
behavior,movein the pseudopotential
•(x,z).
motion,
andthe regionextending
into -x corresponding
to /•-conserving
These
two
classes
of
behavior
can
be
distinguished
by
their
• • f•r• or I•-conserving
motion.Motionon the q-xside
with the z component
of the motionat
ofthefieldline • -- 0 yieldsan imaginarycoz(seeequation energiesassociated
7 of Buchner
andge!enyi
[1989]).
(10))corresponding
to exponential
growthin z. Thissimply a givenx (seealsoFigure
results
in a net z oscillatorymotionthat is not sinusoidal. From the Hamiltoninn equation of motion the total energy
The w• - I line is closest to the ß -- 0 line at the center
at any time • is
planez -- 0, and it intersects
this planeat x -- -1/crt in
theabovenormalization.At earlytimes,whenat,<• 1, this
H(c•t)
= T + T + •(x,z,at)
(22)
intersection
is at large negativex, and mostof the x- z
whichin terms of the energyassociatedwith the z motion
plane,andthe trajectories
withinit, arein a regionwhere can be written as
•o•/l'•L• 1. Thesetrajectoriesare /• conserving,
and
er•ging
overthefastLarmoroscillation
followthefieldline
z(t) = +
•I/- 0. As we increasea•, the coz- 1 boundarybetween
•/i•
•< i (t•-conserving)
and•o•/Ft•.• ! (Iz conserv-so
ing)behaviormovescloserto the trajectoryuntilat c•t• 1
theintersection
is at ß --1 corresponding
in unnormalizedunitsto x ---p•, the particlegyroradius
in the center
plane. Hence at c•t -• I this chaotic boundaryjust inter-
that
•2
At somefixed as,notionalvaluesof h• are markedS for a
cucumberlike
andI for an I•-conservingtrajectoryon Figure
sects
thetrajectoryonceat thecenterplane,andjumpsin
in riteti•nedependent
field,thevalue
thefastactiong will occurastheparticlecrosses
thecenter 1. Fora givenparticle
plane.As t•t • 1 the co-,: ! boundarymovescloserto the
ofh-,(c•t)
willnotbeconstant
asimplied
bythestraigltt
lines
170
CHAPMAN AND WATKINS:
CHAOTIC DYNAMICS IN NONSTATIC R•V•,RSAL
on the figure, and particles may migrate between the two
classesof behavior. The limiting casefor particle motion
to be regular Iz conservingis just that [cf. Buchner and
(the equivalenttrajectory for a negativelychargedpartick
is obtainedby reflectionin the x - z plane).
Zelenyi,1989,equations(21a)-(2!c)]
.i.1. a < 1 Behavior
=
>
2=0)
....
= 0)=
The overall behavior for a trajectory started at z• = 0.00t
(25)is shown
in Figures 2a and 2b. Figure 2a spansthe period
This implies that the third parameter that characterizesthe
particlemotionwill dependupon h,(ai).
4. NUMERICALLY
INTEGRATED
t = 0 to t > z__
andshows
thetransition
fromtt conserving
t(
chaoticbehavior. As suggestedby the HamiltonJanequati0•
of motion, the particle tt is reasonably well conservedfor
t < l/a = 125, the oscillationsin tt about the averageare
TRAJECTORIES
The two closes of behavior, correspondingto a < 1 and
a _> 1, have been examined numerically. Trajectories with
different values of a have been integrated using an adaptive
time step, adaptive order Adams method [Shampineand
Gordon,1975]. We usethe scalefree normalizationto length
L = pz/a and time T = 1/•tz; the value of a for each
trajectory is then specifiedby the starting position z• =
z/L = az/pz. All trajectorieshaveinitial conditionst = 0
(B constant), and initial velocitieswhich are arbitrary (a
constantv, or vu may be added by frame transformation)
except that the motion is initially toward the center plane
and that the particle crossesthe center plane at least once
every t = 1/a. Trajectoriesare independentof the initial
x and y position, and are for positively charged particles
a result of tt being calculated at the particle, rather than
the guiding center location. The first crossingof the center
plane, at t '" 50, is adiabatic, the second, at t • 80, is
approximately so. After t = 125, however, the z motion
is clearly not a slow oscillation between mirror points, and
tt is no longer well conserved. Three-dimensional plotsof
the trajectory over this period are shown in Figures 3 and
4.
The approximately adiabatic part of the trajectory,
for t = 0- 100, is shown in Figure 3; this is composedof
fast oscillations about the parabolic field, with a slow bounce
motion betweenmirror points. Figure 4 showsthe trajectory
for t = 100 - 190, spanning the chaotization time at --, 1.
During the first centerplane crossingat • = 150 the particle
is strongly pitch angle scattered, from Figure 2a the plot 0t
z versus t shows that just after t = 150, fl• • w•, and on
z
0.05-
0.0
t
-0.05-
-0.1 -
0.021
Ii
0.01-
0.0
-0.01
50
'
100
150
200
250
t
mu
0,10 o
[
I
I
50
100
150
I
I
I
2øø 250 t
Fig. 2a Plot of tt, the perpendicularand parallel velocity components,and the particle z coordinate
versustime for a trajectory with c• = 0.008, that is, with z* = 0.008 at t = 0. The full initial
conditionswerer* = (0,0,0.008), v* = (0,0.008,-0.005). The plot spansthe intervalt = 0 to
t > Io,.
CHAPMANANDWATKINS:CHAOTICDYNAMICSIN NONSTATIC
R,EVERSAL
o.oI
•'
?, ',11,' 1
i/ 200
., ,
,
4-00 •, 600
•..
/800
171
I"-'"'""•2•••'. ;:'-".'.'-'
1000 1200
-0.1:
-,
v,,
O.O2-'
•.01
0.0
-•
200 4.00 600 800 10001200
-0.01-
4,10-'
mu
t
0,10
"
0
200
4.00
I
I
I
I
600
8OO
IOO0
1200
Fig. 2b Plot of •, the perpendicularand parallel velocity components,and the particle z coordinate
versus time for a trajectory with c• - 0.008, spazming the interval from t = 0 to c,t • 1.
z
z
y
y
x
x
x
Fig.3. Threedimensional
plot of the trajectorywith c•= 0.008, Fig. 4. Three dimensionalplot of the trajectorywith a = 0.0 )8,
fortheintervalt = 0 - 100,that is, for t < 1/c•. The motionis for the interval • = 100- 190, that is, spanningt -• 1/c•. The
approximately/•conserving.
particle is stronglypitch angle scattered.
172
CHAPMAN AND WATKINS: CHAOTIC DYNAMICS IN NONSTATIC R,BVI•RSAL
subsequentcrossingsthe motion is more cucumberlike,in
that the particle reachestlxe center plane, executesa single
half turn, and then exits in the same direction from which
it entered.
Figures5-8 showthe sametrajectory for times t >> l/a,
selectedfrom the time period spanned by Figure 2b. Up
to t m 900 the motion is cucumberhke,an example(for
t = 760-825) is shownin Figure 5. From t, m 900 to
1200 the motion is interspersedwith segmentsof regular,
/z-conservingbehavior. Figure 6 showsthe evolutionfrom
cucumberliketo a period of/z-conserving behavior,during
this interval(t = 945- 980) the particlefirst interactstwice
X
with the center plane with cucumberlikemotion before exe-
cuting regular, Iz conservingmotion. The motion continues
to be Iz conservingfor several ß oscillations as shown in
Figure 7, but ultimately reverts to cucumberlikebehavior
as shownin Figure 8.
A number of trajectories have been examined over the
range of c• < 1, and in all cases,chaotization, that is, the
commencementof pitch angle scatteringand the transition
of behavior from /• conservingto cucumberlikehas been
found to occur at at m 1. Some trajectoriessubsequently
exhibit intervMs of/z-conserving behavior as shownhere,
but in all casesthe motiondoesnot continueto be regular, Fig. 6. Three dimensionalplot of the trajectorywith a - 0.008,
reverting to cucumberlikemotion at later t.
for the interval t -- 945- 980, that is, for t > 1/•. The motionis
evolvingfrom cucumberliketo Iz conserving.
4.2. a •_ 1 behavior
For thesetrajectoriesthe time t < 1/a is shorterthan
a Larmor period, so that the particlesdo not execute#.conservingmotion. The overall behaviorof an cr-- 1 trajectory is shownin Figure9. Fromthe time of the first crossing
of the centerplane at t = 6 this particle executescucumberlike motion.The trajectory,for t -- 0- 40 sitownin Figure
10 is similarto the cucumberlikesegmentsof the cr<• 1 trajectory discussedabove. This trajectory again is found to
remaincucumberlikefor large t >> 1/c•, with no evolution
into /z-conservingbehavior. Integration of numex'ous
tra-
greater valuesof • the behaviorevolvesdirectly into regular, I•-conservingmotion once• > l/a,, with no intervalin
whichthe motion is cucumberlike.An exampleof this type
of trajectory is shownin Figures 11 and 12, for an a = 5
particle. The behavior of/z shownin Figure 11 clearlydiffers from that of a cucumberliketrajectory. As discussed
in
jectories reveals the same behavior for a < 1.55. For much
Fig.5. Threedimensional
plotof thetrajectory
withc•= 0.008, Fig. 7. Three dimensionalplot of the trajectory with a' = 0.008,
fortheintervalt = 760- 825,thatis,fort > 1/•. Themotionis for the interval • = 975 - 1000, that is, for t • 1/•. The motion
now cucumberlike.
is now completelyregular, i.e., Iz conserving.
CHAPMAN
ANDWATKINS:
CHAOTIC
DYNAMICS
INNONSTATIC
REVERSAL
z
173
(enclosed
by the 9 = 0 line). The trajectoryitself(Figure
12) shownfor t = 0-15 is plottedfor twoperiodsof the slow
motion, which can be seenin the z-•
about the linking field Bz• in the z-•
y
plot to be a rotation
plane, accompanied
by a drift in the +z direction. The fast motion is an oscilla-
tion in the z direction.Thesenumericalresultssuggestthat
the c• _• I behavior is separatedinto two distinct classesby
x
the value of c• and that particlesdo not migrate between
the two. Sincenumericallyit is difficult to ensurethat all of
phase spacehas been adequatelyexplored,this conclusion
needsto be confirmed analytically.
5. SUMMARY OF ANALYTICAL
x
Fig. 8. Three dimensionalplot of the trajectory with c• = 0.005,
for the intervalt = 1250- 1320, that is, for t • 1/c•. The motion
is now cucumberlike.
section3, the particle slow z motion is sinusoidalwith a net
drift in +z, whereas the fast z motion is not sinusoidal,behugcomposedof exponentialgrowthin z for that part of the
trajectory closestto the center plane wherecozis imaginary
2
/
..//'
AND NUMERICAL
The analytical and numerical results are summarized in
Figure 13, which is a plot of the particle behavior ordered
with the two controllingparameters c•t and c•. The parameter at is plotted along the ordinate, so that the trajectory
of a particle of a given c• is a horizontal line on the plot,
the paxtide coordinatemovingto the right with increasing
.t. Lines denoting the numerically integrated examplesare
marked on the plot. The diagonal t-- I line indicates how
far a particle of a given c• must move acrossthe plot to have
executedof the order of a single Larmor orbit in the center
plane field (i.e., a time t = 1/flz in un-normalizedunits).
The plot showsthe following:
k.\ /'•"•'•, ,/'-• X ,.'-•
\ / \
1.5-
ii
1.0--
0.5--
)
5
10
15
20
25
30
35
40
45
50
55
60
t
-0.50.8-
0.6--
mu
0.20.0
5
10
15
20
25
RESULTS
30
35
40
45
50
55
6O
t
Fig.9. Plotof/•, theperpendicular
andparallel
velocity
components,
andthepaxticle
z andz
coordinateversustime for a trajectorywith c•-.- 1, that is, with z* -- I at •: 0. The full initial
conditions
werer*: (0,0,1),v* ---(0,0,0). Theplotspans
theinterval
•
174
CHAPMAN AND WATKINS: CHAOTIC DYNAMICS IN NONSTATIC P•I•VlglR.
SAL
2. Fora • 1, the particleexecutes
of theorderof a single
Larmor orbit beforet --+ 1/c•, oncet > 1/c• the motionis
cucumberlike. This type of behavior is numericallyfound
for c• < 1.55 approximately.
3. For c• > 1.55 the time t --+ 1/c• is muchshorterthant
Larmorperiod. Oncet > !/c• the particleexecutes
regular,
/z-conserving("ring-type") trajectories.
X
6. RELATIONSHIP WITH THE STEADY STATE MODEL
PARAMETER
,;
We will now establish the relationship between the time-
dependent system discussedabove, and the steady state
parabolicfield model[e.g.,Buchnerand Zelenyi,1989].Ap-
X
Fig. 10. Three dimensioualplot of the trajectory witIx c• - 1, for
the interval • = 0- 40, that is, spanning the interval • = 0 to
•>•. 1 Fort > 1/c• themotionis cucumberlike.
1. For an a << 1 system, the particle executesmany
plication of the steady state model to time dependentfields,
by treating the time-dependentsystemas a seriesof steady
state magneticfield models,eachcharacterizedby a slightly
different vMue of ,•, has been used to imply that regular,
g-conservingmotion becomeschaotic (cucumberlike)when
• • 1 [e.g., Pulkkinen,1991]. We haveseenthat this correspondsto a particular caseof motionin the time-dependent
model, that is, the limit of a slowly changingsystem a • 1.
We will now estabhsh the exact relationship between • and
at, and show that the system adiabaticity c• doesnot have
Larmor orbits beforereachingthe time t --* 1/a. For t <
1/a the motionis regular,andthe particleconserves
t*;once an anMoguein the static model.
In the steady state model:
t > 1/a the motionis in generalcucumberlike.
2O
,"/......
-,
,.
,,.
/
,/
',\,
./'
'x,. /'
'""....
.....
'N, ...,
Z
0 '••AAAAAAAAAAAAAAAAAAA
\/ IN/VVVV•VVVVVVV[VVVVVVVVV•VVvvvvvvvv•vvvvvvvvvv;'v•
-'10
30-
VII
20-
10-
5
2O
10
-10 '
300-
200mu
100-
0 ,
I
I
5
10
15
20
25 t 30
Fig. 11. Plot of t•, the perpendicular
andparalh;1
velocitycomponents,
andthe particlex andz
coordinateversustime for a trajectory with c• = 5, that is, with z* = 5 at t = 0. The full initial
conditions
werer* = (0,0,5), v* = (0,0,0). The plot spansthe intervalt - 0 to t > 'l/c• For
• > 1/c• the motion is Iz conserving.
CHAPMAN
ANDWATKINS:
CHAOTIC
DYNAMICS
INNONSTATIC
r{EVEB.
SAL
175
is now a constant.The parameter.• = C•oscalesthe Hamiltonian by scalingthe pseudopotentialT0 within which the
particlesmove at any time i, so that the ratio of the oscil-
lationfrequencies
will alsodependon a0 andwill nolonger
haveexplicittime dependence.
The parameterao (or
in the steadystatemodelhenceplaysan analogous
roleto
theparameter
at in thetimedependent
model(seeequation
(16)). Asa consequence,
andasshownby previous
authors
y
[Buchner
andZelenyi,1989;ChenandPalmadesso,
1986]
particlesin a givenregimeof behavior(dictatedby a0, or
•) nowrema/nin that regimefor all t. If a0 ½ 1, thenthe
motioncan be regular(integrable)for all t, conserving
an
adiabaticinvariantassociated
with the fast oscillatorymotion. The trajectorywill be confinedto a specificregion'in
phasespace[ChenandPa!madesso,
1986].If (•0 << 1, then
for all t, :Oz0<< 1, and the trajectory will be characterized
by fastoscillations
aboutthe fieldat the Larmorfrequency,
and slow oscillationsbetweenmirror points at frequency
and will conserve p. If a0 >> 1, and if, for all t, •zo > 1,
the trajectory will be ring type (as definedby Buchnerand
x
Zelenyi [1989])and will conservean invariantIz. If, on the
other hand, co..0,• I at some point on the trajectory, the
particle will be cucumberlike for all t. The condition for
chaotic behavior is satisfiedfor all t where the particle in-
y
Fig. 12. Three-dimensionalplot of the trajectory with c• = 5, for
theintervalt = 0 - 15, that is, includingt > 1/c•. For t > 1/a
tersectsthe centerplane (z • 0) at x - -1 (i.e., a distance
z = -pz from the fieldline in unnormalizedunits) if a0 --• 1
x).
In this steadystate model(and in any modelin whichthe
inductionelectricfieldis neglected)we alsohave
dHo
the motion is Iz conserving.
=
o
(35)
so that
Bx z
B = (•--, 0,Bz)
E=0
dA
(26)
(27)
dao
•-7= ,• = o
(3•)
sothat thereis no counterpart
of the parameter
c• = •,
which in the explicitly time-dependent model indicated the
"adiabaticity" of the system. In this sense,the steady state
the Hamiltonian may be written
Ho=•-+ • +To
(28)
system,or any magneticfield model with no induction elec-
where
z2+ •ø-5z2- S
1F=o
2
-52
o:--5
with
2
",l•con••ing
(29)
z2
=
5- -
(30)
z2
ϥ0
=•0 T
Bx pz
1
(32)
a0= B• h = •2
where
n is defined
in equation
(5) of [Buchner
andZelenyi,
p,-co
1989].As in the time dependent
case,the pseudopotential
90 describes
twocoupledoscillators
in z andz. The constant Hamiltonian
o:=0.008
...............
i
.......
has been written as
H0= H0(i,•,z,z,A)= T + T + to(z,z,A)
I
at
(33)
Fig. 13. Particle behavior in the time-dependentmodel ordered
by the two controlling parameters c• and c•t. The diagonal line
t = ! denotes the time taken for a particle of a given c• to execute
a Larmor orbit. The paths of the numericallyintegratedexamples
(34)
are marked on the plot.
where
176
CHAPMAN AND WATKINS: CHAOTIC DYNAMICS IN NONSTATIC REVEP•SAL
tric field (with Hamiltonianof the form H(q,•,)•)), correspondsto the explicitly time dependentsystemin the limit
in which the systemchangesinfinitesimallyslowly,that is,
If a >> 1, representing a reversal ;vhich is thin and folds
quickly,the motion becomesregular ("ring-type") ;vithina
Larmor period, exhibiting no extended interval of chaotic
motion.
3. The parameter c•t plays an analogousrole to a constant
parameter a0 (or in a different normalization,n [B,chner
and Zelenyi, 1989]) which scalesthe steady state system.
Since
= a0--
(38) Hencewhilst at •
this correspondsto a.finite a0 only if the fold up timescaleof
the field r • oo (F•! --• 0), so that the time dependentfield
effectivelybecomessteadystate. Alternatively, if a0 --, 0 so
that a -+ 0, the modelsare identical,both simplydescribing
a particle that gyratesin the constantfield B - (0, 0, 1).
conservingparticles become chaotic in the time dependent
system(with a << 1), particlesin the steadystate system
with a0 m 1 (or n m 1) a.rechaoticfor all t.
4. The parameter a in the time-dependent system, and
as a consequencethe ordering of behavior with a,, do not
have a counterpart in any field model which is static or does
To summarize,the parameterc•0(or •;), whichdetermines not include the inductive electric field associated with time
(for all t) whether a trajectoryis regularor chaoticin the dependence.
steady state system, plays an anMogousrole to the parameter at in the time dependent system. Hence the condition
c•0m 1 (• m 1) dictatesthat a trajectoryis chaoticin the
steady state system for all t, and the condition at •
1 gives
the time at whicha (a << 1) trajectorybecomeschaoticin
the time dependent system. These two systemsare however
generally not identical, that is, the equations of motion
the above steady state system cannot be transformed into
those describingthe time dependentsystemwith the substitution c•0--, at. Hence the regimes of behavior traversedby
a single trajectory in the time dependentmodel as at << 1 ,
at m 1, at >> 1 cannot be identified in a single trajectory in
the steady state model, instead being given by the behavior
of a collection of distinct trajectories specified by a0 << 1,
Multispacecraft in-situ measurements, such as thoseof
the proposedCluster mission[Rolfe, 1990, and references
therein] are neededto unambiguouslydetermineboth the
characteristic length scMe and time scaleof the field in terms
of the particle Larmor scales, to obtain the a of the timedependent model.
Taking both spaceand time derivativesof the normalized
time-dependent field model
OzOt
(c•zt,
O,1)= (c•,0,0)
(39)
yields the a of the system, and will distinguish between
system which is time-dependent, and one which is static,
since
c•0• 1 and a0 > 1 in the steady state model [Bv.chnerand
O
Zelenyi, 1989].
0, =
0)
(40)
Furthermore, trajectories in the time dependent model
We can only crudely estimate c• from single, or two spaceexhibit different behavior that dependson the c• of the syscraft in-situ data, which cannot unambiguouslydistinguish
tem. This ordering of behavior witIt the "adiabaticity" of
between spatial and temporal changesin the field. We will
the systemdoes not have a counterpartin any field model
considertwo cases: a "weak" reversalwhich is thick (half
which is static or does not include the inductive electric field
associated with the time dependence.
7.
CONCLUSIONS
thickness
h = 5000km) andfoldsslowly(r = 20 rains)[e.g.,
Hones,1977],and a "strong"reversalwhichis thin (h = 200
krn) andfoldsquickly(r = 5 min) [Mitchellet al. 1990].In
both casesB• = 20 nT, and the linking field B.. is 5 nT.
Singlechargedparticle dynamicsin a simplescalefree, Electrons then have c• << 1 in both cases, for example, 50
time dependentmodelfor a magneticfield reversalhavebeen keV electronshave c• • 5.3 x 10-? ill the weak reversaland
examined analytically and numerically. Two constraints c•½m 6.4 x 10-s in the strongreversal.Protons,however,
have been imposed: the magneticfield and the associated can be in the c• > 1.55 regime. In the weak reversal,1 keV
induction electric field are independent of as(to allow the
protonshavec•p• 6.0 x 10-a, and in the strongreversal
they havec•p• 0.7. Higherenergyprotons,a.t50 keV,have
transformation),
andthe modelis scalefree(to allowp•ra.m- c•p• 0.04 in the weak reversal and c•r m 5 in the strongreeterization of the particle behavior). The principal results versal.The orderingof particlebehaviorwith c•is therefore
are as follows:
significantfor the evolvingproton distributionin the time
inclusion of a cross-tail convection electric field by frame
1. Unlike the steady state system, behavior at, any given t
in the time-dependentsystemis orderedby two parameters;
c•t, which at any t scalesthe pseudopotential
in whicltthe
particlesmove, and c•, whichscalesthe rate of changeof the
pseudopotential.The time at which a transitionoccursin
the dynamicsis t • 1/c•, and the nature of the transition
dependsupon c•.
2. The explicit time dependenceproducestwo possible
classes of motion:
dependent, presubstorm magnetotail.
The transitiontime t • 1/c• shouldprovidea prediction
for the evolution
in time
of both
the local electron
and protonparticledistributionfunctions.If the change
in
the particle distribution as the dynamics evolvesproduces
substorm-relatedinstabilities,or thermalizesthe distribu-
tionsufficiently
to leadultimatelyto a reconfiguration
ofrite
magnetotail
[Cowley,1991].Mitchellet al, 1990,andreferencestherein],then a characteristic
time scalefor substorm
If c• << 1 representinga thick reversalwhich is folding "evolution",will be t
slowly,the motionis a transitionin behavior;whilet < 1/a
it is regular,conserving
an invariantg; oncet > l/a, the
Acknowledgments.
The authorsareindebted
to S. W. Iq.
particle is pitch angle scatteredand the motion generally Cowley
for illuminating
discussions,
andto D. Willis,R. Hat&
becomes chaotic and "cucumberlike".
arid T. A. Johnson for invaluable input to the 3D graphicspro-
CHAPMANANDWATKINS:CHAOTICDYNAMICS1NNONSTATICREVERSAL
duction.This work wasperformedwhileNWW wassupported
byan SERCgxant.
The Editor thanks L. Lyons and two other refereesfor their
assistance
in evaluatingthis paper.
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