JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 99, NO. A4, PAGES 5977-5985, APRIL
1, 1994
Properties of single-particle dynamics in a parabolic magnetic
reversal with general time dependence
SandraC. Chapman
SpaceScienceCentre, Universityof Sussex,Falmer, England
We investigate single charged particle dynamics in the earth's magnetotail by examining a
reversalwith simplespatial dependence(the field linesare parabolic)but with generaltime dependence, which includes the associatedinduction electric field. The parabolic spatial dependence
has, in static models, been shown by previous authors to imply that various regular and stochastic
regimes of behavior exist, with particles remaining in a given regime for all time t. Here we show
that, in general, time dependenceyields transitions in behavior between the various regimes of
regular and stochasticbehavior. We identify three independent parametric coordinates which are
functions of the field spatial and temporal scales, and the particle gyroscales, and time, which
will indicate the regime of particle behavior at any given t. For specific time dependent field
models these parametric coordinates can be inverted to directly give the timescalesfor transitions
between one regime of behavior and another in terms of the field and particle spatial and temporal scales. These parmetric coordinates have no direct analogue in static models. For a general
thinning and foldingsheet(representingthe magnetotailmagneticfield closeto the centerplane
just beforesubstom onset)this characterizationimpliesthat stochasticdynamicsmay only occur
over a finite time period and may not occur under certain circumstances. This may have implications for our understanding of the role played by single particle dynamics in the destabilization
or recomSgurationof the magnetotail current sheet associated with substorms.
INTRODUCTION
function of the characteristic field strength and thicknessof
A considerablebody of work exists [e.g., $peiser,1965;
Sonnerup, 1971; Chen and Palmadesso, 1986; Buchner and
Zelenyi, 1989; Chen, 1992; Wagner et al., 1979] on single
charged particle dynamics in simple static models of the
magnetic field reversal in the earth's magnetotail. These
have led to studiesof particle populationsin the quasi-static
reversaland the particle gyroradius[Buchner,1986; Buchher and Zelenyi, 1986, 1989]. In the static parabolicmodel,
this parameter plays the same role as the normalized energy
//definedby ChenandPalmadesso
[1986].Theparameter
• (or //) will notcompletely
specify
thenatureof thedy-
namics(i.e., whetherit is regularor stochastic)[Chenand
magnetotail[e.g., West et al., 1978; Chenet al., 1990; Chen Palmadesso,1986; Chen, 1992] there being ordering with
respectto a secondparameter(e.g., Figure 19 of Buchner
and Burkhart, 1990; Burkhart and Chen, 1991; Ashour- Abdalla et al., 1991] and of conditionsin the pre-substorm and Zelengi[1989];seealso Chapmanand Watkins[1993]).
plasmasheet[e.g., Pulkkinenet al., 1991] and its stability The precisenature of some regimesof behavior in the static
parabolic model have also yet to be completely elucidated
[Buchnerand Zelenyi,1987].
[Chen, 1992]. However,it has been suggestedthat • will
Generally, the characterization of particle dynamics for
characterize
the limiting behaviorof particles(Buchnerand
a given static model requires the numerical integration of
Zelengi
[1989];
seealso Chen[1992]for a discussion
of the
large numbersof trajectories(sufficientto sampleall possivalidity
of
these
limits)
and
in
this
sense
is
a
controlling
pable regionsof phasespace)for differentvaluesof the model
scaling[Gray and Lee,1982, Chenand Palmadesso,
1986].
A given particle will map out the surfaceof a static torus in
phasespaceand the nature of this torus (that is, whether
it is regularor not) will determinethe natureof the motion
(that is, whetherit is regularor stochastic).
Special cases in which the field model is scale free can
be used to analytically characterize the particle dynamics.
A well-studied example is the model where the field lines
are parabolaswhich are static. In this static model, regular
and stochasticregimesof particle behaviorwere identified,
the particle remaining within a given regime for all time
t. The dynamics were shown analytically to be specified
largely (but not completely)by a constantparametern, a
Copyright 1994 by the American GeophysicalUnion.
rameter. In the static model, the value of the constant •
then "characterizes"the regime of behavior of the particles
for all t.
More recently, dynamics in a parabolic field model with
simpletime dependence
(and corresponding
inductionelectric field) wasinvestigated[Chapmanand Watkins,1993;
Chapman,1993], and it was shownthat althoughthe same
regimesof behavior existed as in the static parabolic model,
the particle dynamics migrated from one regime to another.
In the time dependent system, the torus in phase space
which the particle maps out changeswith time. Specifically,
for a single trajectory, the torus may at early t be regular
(sothat the motionis adiabatic,conserving
an approximate
invariant, suchas /•) in the sensethat the systemis close
to an integrablelimit, but at later t be irregular (so that
the motion is stochastic).For the time dependentsystem
in general we cannot therefore determine the regime of par-
Paper number 93JA03036.
ticle behavior(i.e. the nature of the torus) at somet by
0148-0227/94/93JA-03036505.00
integrating the particle trajectory over large t.
5977
5978
CHAPMAN:DYNAMICSIN A REVERSAL
WIqlt G•
General time dependenceis thereforeaddresedanalyticaJlywithin the frameworkof the scalefree spatial dependenceof the static modelwith parabolicfield lines. In principle,thisfieldmodelcouldbe usedto describethe thinning
plasmasheetcloseto the magnetotailcenterplanejust prior
to substormonset. The spatialdependenceof the parabolic
field linesagainproducesthe sameregimesof behaviorfound
in the static model, the time dependenceallowsthe system
Tm• D•P•D•'•C•
coordinates of the particle. Given the form of the Hamiltonjan in an electromagneticfield
1
H= •--•m
(p- eA)
2+ eO
(2)
we have the constraint that in an appropriate coordinate
p. A(5) =
the
mgeotU
is readily achievableby choosinginvariancein the direction
of the cross-tailcurrent (i.e. the GSE (GeocentricSolar
In this paperwe will analyticallyderivethree (time de- Equatorial) • direction)so that A -- Au• , with indepenpendent)parametriccoordinates
in the systemwith gen- dent coodinates qa -- x, q2 -- z. The canonical momenta
to migrate betweenone regime and another.
eral time dependenceby examining the particle Hamiltonjan equation of motion. These parametric coordinateshave
no direct analogue in any static model. The values of the
parametric coordinates at any instant will characterize the
regime of particle behavior of the system at that t. This
are then p• -- ink, p• -- m:• with pa -- m• + eAu as a con-
stantof the motion(whichis zeroin an appropriate
frame).
The magnetic field is then confined to the x, z plane and is
given by contoursof Ay. In this Hamiltonian description
the system therefore has two degreesof freedom and four
"characterization"doesnot correspondto an exact specifi- time dependent coordinates i, •, x and z, which describe
cation of the dynamics of a given particle. Instead, we will the trajectory in four-dimensionalphasespace.
Second, we seek to obtain parametric coordinates which
showthat the parametric coordinateswill indicate at any t
whetherthe systemis regular(i.e., closeto the limitingcase vary with time and the (constant)field and particlescales
only. If the parametric coordinates are a function of one
of integrablebehavior)or stochastic.
We establishthe conditionson these parametric coordi- variable(t) only it is, in principle,possibleto then invert
nates which indicate
a transition
from one class of behavior
them to obtain the times at which transitions in particle
behavior will occur for a given field model. As we will see
next, the choiceof a field model that is scalefree in spaceeffectively allows the spatial dependenceto be separatedfrom
the field and particle scales. For static field models, this
to another. These conditions then allow the time spent by
the particles in any one regime of behavior to be determined
for specificfield models. For somemodels,including examples which describethe thinning presubstormplasma sheet,
to the determination
particles will only spend a finite time executing stochastic leads(via appropriatenormalization)
of
parameters
such
as
n
[Buchner
and
Zelenyi,1989]which
behavior. If the reversalis sufficientlythin and changing
sufficiently quickly, intervals of stochasticbehavior may not characterizethe motion for all t. For time dependentmodels this yields time dependent parametric coordinateswhich
occur. The transitions in behavior are illustrated with nucharacterize
the motion at any given t.
merically integrated trajectoriesin a simple "thinning"mag-
neticreversal(with time-varyinglinkingfield Bz(t)).
The organization of the paper is as follows. We first dis- REGIMES OF BEHAVIOR IN THE STATIC PARABOLIC MODEL
cuss the geometrical constraints which allow the HamiltoThe particle behavior in the static parabolic model has
njan to be written in a sufficiently simplified form to allow
analytical parameterization. We then review the normal- been analyzedin detail by Buchnerand Zelenyi[1989]and
ization of the static model required for parameterization, Chen and Palmadesso[1986]. Here we briefly reiterate the
properties of the system to establishthe regimesof particle
and establish the extent to which behavior can be "charbehavior
that arise as a consequence
of the geometryof this
acterized" analytically. This normalization is then used to
model.
In
this
static
field,
particles
will remain within a
parameterize the general time dependent system. We derive
given
regime
for
all
t;
with
the
addition
of time dependence
the three parametric coordinates,and indicate the possible
these
regimes
are
preserved,
but
particles
can migrate from
regimes of particle behavior as regionsin parametric coorone
regime
to
another
with
time.
We
will
then identify the
dinate space. The consequencesof possibletrajectories in
extent to which parameterssuchas n characterize particle
this parametric coordinatespacefor motion in a thinning
reversalare then discussed,and numericallyintegrated trajectories in the simple thinning model are presented.
behavior.
With X, Y and Z correspondingto GSE coordinates,the
static
model
is
CONSTRAINTSREQUIREDTO OBTAINPARAM•TmC
h0 '
COORDINATES
E---- 0
(4)
Two significantconstraintsare required to analytically
obtain a simplified Hamiltonian equation of motion and
The Larmor scalesassociatedwith the constantlinking field
henceparametric coordinateswhich characterizethe dynamBz define the spatial and temporal scaleswith which we can
ics.
"measure" all other spatial and temporal scales. NormalizFirst, we shall analytically examinea systemof coupled
ing the magnetic field to the linking field Bz, temporal scales
oscillatorsby essentiallyobtainingappropriatescalingof the
particle pseudopotential• so that we require a systemwith
HamiltonJan
of the form
to the inverseof the gyrofrequency
in this field • - eB•/m
and spatial scalesto the particle gyroradiusp• = v/•, the
static
field model
becomes:
2
=
+
,
(
where pj and qj are the generalizedmomenta and position
B = (aoz,0, 1)
r=o
(5)
CHAPMAN:DYNAMICSIN A REVF_•AL WI•rI G•
The parameter
•
D•a•F•NC•
5979
as Ho = h•o + hzo with
•0
• _ ñ_ v•.•)
Bz ho- •2 -
(7)
•0 -
i2
•-
I 2
+-•
2
(•5)
partially determinesthe particle dynamics [Buchner and
Zelenyi, 1989; Chen and Palmadesso, 1986; Chapman and
a•0- ••2
+ •]Z2
(.0•)•'/• - 2•(.0•)
-•/
Z2
Watkins, 1993]. In addition, it can be demonstratedanalytically that at least one other parameter is required to
completelyspecifythe regime of behaviorof a given particle
or
in the parabolicmodel [Buchnerand Zelenyi, 1989; Chapman and Watkins,1993].
•o=5-+•
Since the static parabolic model has no intrinsic scale
height, this is effectively specifiedby the range of z values
as follows:
if we renormalize
the above
equationsto give new variablesz* = a0z, t* = t (that is,
normalizing(3) to length p•/ao insteadof length p•), the
(•)
(•s)
In the limit of a "strong" reversal
2
l ao-- I>>1
equationsof motion describingthe systembecomefunctions
of z* (and z*, y*) and t* only (notethat htis alsorenormalizesvelocities,i.e. v*(t*) = a0v(t)). The singletrajectory
givenby z*(t*) is eitherthat of a particlewith smalla0 and
large z(t) or large a0 and small z(t). Thesecorrespond
to
•- 2•-•ø
;/2 1z2(.o•z)2
•o=T+•
of a giventrajectory [seeChapmanand Cowley,1985].This
can be illustrated
(•)
and in the limit
of a "weak"
(19)
reversal
z2
I-o--I<<
•
(20)
a trajectory with mirror points far from the center plane in the oscillatorsare weaklycoupled,and (15)-(18) imply sega "weak"(e.g., large-scaleheighth) reversalor a trajectory ments of motion with distinct • and z frequenciesw•0 m 1
with mirror points close to the center plane in a "strong" and wzo• aoz/2. One fimi• in which•he motioncompletely
corresponds
•o a zeroreversing
fieldB• = 0 (or an
(e.g., small-scaleheighth) reversal,respectively.Hencethe decouples
choiceof c•0 does not uniquely distinguishdynamics in the infinite shee••hicknesscompared•o •he gyroradiush/p•
static model but does, as we shall see next, characterize the
dynamics by scaling the coupling between the x and z motion.
In the static parabolic system the equations of motion are
2
Z
• = .05
ß
(8)
(9)
(•0)
has w•0 •
•
and w•0 = 1 and does no• decouple
represen• a neu•rM shee• as in •he formalism used here •he
sc•ng (7) canno•distinguish•he physic•y meaningfulc•e
•he time dependen• parametric coordinate •ha• wffi be ob-
•ned from a gener•zafion of •he sc•ng (7) wffi be meaningful for re•sfic time dependen•field modelsappropriate
for •he pre-subs•ormgeo• in which•he magneticreversM
where Ho is a cons•an• of •he motion and
•o=•(-oT• • • ):= •Ag
•
vanishingshee••hickness(seeMso Ghen[1992]). However,
•2
•o - 5- + T + •o(-o,•, •)
a0 •
of a vanishingfining field from •he non-physicMc•e of
The Hamiltonian may be written
i2
o•hertimid,of vanishingfinhng fieldB• = 0 (or equivMenfly
a vanishingshee••hickness•o gyroradiush/p•
•he oscma•orsas written in (15)-(16). This fimi• doesno•
z2
• ----a0(a0
T - a:)z
•), so •ha• a0 = 0 and from (17) and (18) (for finite z)
•he z oscffia•orvanishesleaving gyromofion abou• •he cons•an• fining field B• a• •he unnorm•zed gyrofrequency
• = eBb/m, which has been norm•zed •o w•0 = 1. The
(•)
These equations describe [he behavior of [wo nonfinear
•hins wi•h increasingt (i.e., •hose•ha• do no• includean
infinitely•hin currentshee•a• finite t).
Two regimesof in•eres•here can •hen be identified[see
coupledoscffia•orsin z and z. The s•reng•hof •he coupting Mso Buchner and Zelenyi, 1989; Ghen and Palmadesso,1986;
be[ween [he [wo oscffia[ors wffi essentity dic[a[e [he haGhen,1992]. In a reversMwhichis sufficiently"weak"wi•h
lure of [he particle dynamics.To parame[erize[he coupting respec••o •he particlemotion(i.e., if a0 is sufficientlysm•
s•reng•h, we can a•emp• •o write •he Hamfi•onian in •he
form for •wo decoupled oscffia•ors. If •he z and z oscffia•ors
were decoupled, [hen [he Hamfi[onian
• = •.(-o, i, •) + •(-o, i, •)
(•2)
•ha• I "0z•/ß I<<• ) •heoscma•ors
areweary coupled.
The
motionwffi be regular(in •he sense•ha• i• approaches
•he in•egrabledecoupled
fimi• a0 = 0) conserving
an approfima•e
adiabaticinvarian• • = v•/B. The motionis characterized
by fas• • oscffiafionsabou• •he field, wi•h slow z bounce
motionbetweenmirror points(i.e. w•0 > w•0) .
where
ForlargervMuesof a0, such•ha• I doze/• I• 1 •hemotion
a. = T + •.(-o,•)
•2
h•= •- + •P•(ao,
z)
wffi become s•rongly coupled and wffi be s•och•fic. This
regime of behavior, characterized by a0 > 1 or a0 • 1,
corresponds•o motion which is composedof segmentsof
conservingmotion when •he particle is far from •he cen•er
plane, and fas• oscffiafionsin z which are approfima•ely
(14) I•(= • v,dz)conserving
as•heparticle
crosses
•hecen•er
The Hamiltonianfor the coupledsystem(10) canbe written
plane.
5980
CHAPS:
DYNAMICSIN A REVERSALWITH G•
The simple scalefree parabohc model can representsome
aspectsof more phenomenologicallyaccurate models of the
tail current sheet in a region closeto the center plane. For
tt-conservingand stochastictrajectories, stochasticbehavior
resulting, for example, in jumps in the particle action tt, occurs as the particle crossesthe center plane z = 0, so that in
order to examine the conditionsfor stochasticityin singleparticle dynamics we accurately model the spatial region
close to the center plane. The model should therefore be
relevant to trapped particles in more complex systemsprovided that the segmentof the particle trajectory closeto the
center plane of the reversalwhich resultsin its stochasticity
Tm4E DEPENDENCE
E=• -]• +La0T +E•(•)
(26)
where ao is as defined for the static model.
Note that since the spatial dependenceof the model is
still scalefree, we can again rewrite the equationsthat specify the system as independent of a0 if we renormalize such
that the new position coordinatesz* = a0z; the choiceof
z*(t* = 0) thenspecifies
the a0 of thesystem.Sincethe time
dependenceis in general not scale free this does not ehminate the additional dependenceon the intrinsic timescales
containedin f•(t) and f•(t).
The equations of motion are
(e.g.,wherethe pitchangleof tt conserving
particlesis being
scattered)is containedwithin a regionwherethe fieldhnes
are approximately parabohc. Far from the center plane, the
particle motion will be regular in both the parabohc model
and more phenomenologicallyaccurate models. An additional feature of more phenomenologicallyaccurate models
will be the inclusionof losscones,so that particlesexiting
the parabohcregion with sufficientlysmall pitch angle will
(27)
z2
• = -L• + I•0 •
be lost from the system(e.g., transientorbits [Chen and
Palrnadesso,
1986]).
a, = -a.
(
z2 )
• = -•aof•z= -aof•Z -f•z + f=ao•-- At
(2s)
(29)
The time dependent HamiltonJan equation of motion is
/•- •(
d •(•
1 .•+/•)+•)
THE PAltABOLIC MODEL WITH GENERAL TIM•
DEPENDENCE
(30)
where
We will
now demonstrate
that
the introduction
of time
dependenceallows particle behavior to migrate betweenthe
different regimes summarized above and will obtain convenient parametric coordinateswhich characterizethis migration. We will consider a magnetic field of the form
B = f=(T)B=•-,
O,fz(T)Bz
(21)
so that time dependenceresultsin a pseudopotentialwhich,
as well as having the samespatial dependenceasin the static
model, is a function of time dependent parameters
t
with vector potential
t a0
•,(t)= f*(•i)
A = O,fzB•X- f=Bx
•-• + At(T),
0
(32)
(22)
Aa(t)
=f•(•--•)
with correspondinginduction electric field
(33)
and the frame dependentAt(t). We can transformto the
OT= • -]•B•X+]•B•5• +E•(T) (23) (primed)framein whichA•(t) =
where, in generM, the magnetic field model does not con-
m•,
0 with •' = ß + At/Aa,
z' = z, and t' = t, providedwe do not attempt to take the
hmit of vanishing
hnkingfield (i.e., providedAa• 0). The
= -A,(r), whichdepends
uponthechosenpseudopotential then becomes
frame of reference. If, for example, the } component of the
fieldis constant(f• constant)this "convection"
electricfield
E• can be removed by a de Hoffman Teller frame transfor-
(
9' = •A•
1 ,2_•1A•x'-A,
mation [Chapmanand Watkins,1993]. Gener•y, and for
examplesof specificinterest such• a thinning model where
f• decre,es with time, this w• not be the c•e.
We can ag•n use the characteristicnorma•zation of the
static model, that is, to norm•ze the magneticfield to the
characteristic z field B•, and distances and times to the
Larmor scMesin this field. This yields
' O,/z(•)
B= (/.(•-•)aoz,
=
and
-
+
(34)
Once again these equations describethe behavior of two
nonlinear coupled oscillators in x and z. If in the accelerating frame, we attempt to write them in alecoupledform in
analogyto (16)-(19), then
(35)
(24)
(36)
(2s)
or
CHAPMAN:
DYNAMICS
IN A REVF2•AL
WIqI-IGENEa• Tnv• DEPENOENC•
h'•=•-+•. 2
1-2 hi
5981
(41)
w•
then the parameter
•,21( z')2
Zt2
- IX:I _ ILl
(3s)
(42)
mustindicate"particle"adiabaticitywith respectto Larmor
oscillations,so that if dh2 << 1 segments
of/z-conserving
quencyof the z oscillatorwz • h•h2z'/2 is alsoexplicitly motionmay exist, and if dh2•_ 1 or dh2 > 1, therecan be
no/z-conservingmotion.
time dependent,and the ratio of the two scaleswith hi.
The nature of the system(i.e. that it is two nonhnear
In analogyto the staticmodel,and as wasdemonstrated
with time dependence)
hasintroduced
a
for the simpletime dependent
model[Chapmanand Watkins, coupledoscillators
so that now in the weakly coupled limit the frequency of
the z oscillator w, • h2 = fz is time dependent, the fre-
1993],the valueof the parametriccoordinate
hi (t) scales secondtimescale: the transition timescalegiven by hi •- 1.
the strengthof the couplingbetweenthe two oscillators.If This timescalegivesa slowchangein the pseudopotential
hi = 0, then the motiondecouples.If hi is sufficientlysmall with respectto the Larmorperiod,that is, the systemis
then the couplingwill be weak,andthe characteristic
motion adiabatic if
can be regular/z-conserving.Howeverif hi becomessuffiI• I<<w•
(43)
cientlylarge(again"characterized"
by hi -• 1 or hi > 1) at
somelater t, there will be transitionto stochasticmotion.
so that the parameter
The behavior identified in the static model for different
I I
a0 (constant)for all t by a0 <<1, a0 - 1 anda0 >>1 will be
just the behaviorexhibitedby the time dependentsystem
at differenttimes,that is for hi (t) for timeswhenhi << 1,
hi - 1, and hi >> 1. Hencefor example,for hi (t) increasing
with time, particlesat initially smallhi (t) will be/z conserving and, as hi (t) increases
will exhibitstochastic
behavior.
We can estimate the time at which the transition between
indicatessystem•di•b•ticity. If d• • 1 (the "slowp•s•ge•mit"), thenthe transitionis slow;that is, it occurs
overm•ny gyroperiods,
•nd segments
of •-conserving
motion m•y e•st, where• ffd• • 1 or d• • 1 (the
p•s•ge •mit") therec•n be no •-conserving
motion.
particlesbeing/z conserving
to beingstochastic
takesplace
as whenhi -• 1, providedthe particlescrossthe centerplane
We c•n •ustr•te this by estimating the time t•ken for
to changefrom 0 to 1. Since
on that
timescale.
d•
At• A•
dt
In addition, since
12(:-•T z2+•At)
•'(•,•:,:,:,A,)=•:
(39)
the ch•ge •
• 1 w• be slow (i.e., occurover m•ny
gyroperiods)
if the corresponding
•t • 1/• sothat
d•
then
dt •
= )• OH
OH
•
O•
-
OH
•
OAt
(45)
1
<<
•bove.
In order to characterizetrajectories we only need to iden-
(4o)
sinceEc--•i.t andusing(28).
tify the v•rious regimesof behaviorin •, d•, d•2 sp•ce.
The loc•tion of ß p•rticle •t •y • is known in this sp•ce
• •, d•, •nd d•2 •re just functionsof t •d •re specified
by the m•gnetic field model in whichthe p•rticles move.
Possiblecloses of trajectory •re shownon Figure 1, which
is ß plot of the possibleregimesof behaviorin this p•r•-
The changein the particleHamiltonianwith timeis, aswe
wouldexpect,givenby the changein the pseudopotential, metric coordinate sp•ce. The notion• trajectories shown
beginin ß regionof regular "•di•b•tic" motion,•t ß time
with an additional frame dependentcontribution to v. E
from the y-directed convectionelectricfield Ec. The lat- when • • 1, d• • 1 •nd d•2 • 1 so that the motion
ter will contributeto the (framedependent)energyof the is •-conserving.This initi• conditioncouldcorrespondto
particleat any given[ but shouldnot affectthe characteri- motion in ß revers• •t the e•rly stagesof thinning, so that
zation of the overall behavior, that is, whether the particle the sp•ti• sc•e of the revers• is l•rge comparedto the gymotion is regular or stochastic.
Equation
(40)suggests
that•a andN2scale
therateof
changeof the pseudopotential
and thereforealsomust be
parametriccoordinates
whichcharacterize
the particlemotion. Thesetwo parametriccoordinatesrefer to two classes
of adiabaticity,whichappearsincethereare two timescales
on which the pseudopotential
changes. These timescales
are just the characteristic
particleLarmorperiodgivenby
ha = w• = f• and the transitiontimescalegivenby hi. If
the pseudopotential
changes
slowlywith respectto the Larmot periodthen the particlemotionis "adiabatic"in the
commonlyreferredto sense,that is,
ror•dius, •nd the field is ch•ging slowly with respectto
both the gyroperiod•nd the tr•sition timesc•e.
Two possiblep•ths •re shownfor the p•rticles • • incre•es
•nd the revers•
thins. If the transition
•
•
1 oc-
curswh•e d• • 1 •nd d•2 • 1 (• slowp•s•ge throughthe
transition),the p•rticlesc• continue
to executesegments
•- conservingmotiononce• • 1. The tr•sition is therefore into stoch•tic behavior, which includes segments
both •-conservingmotion•d f•t (I•-conserving)z osc•tions •crossthe centerpl•e. The motion c• only continue
to executesegmentsof •-conservingmotion wh•e d• • 1
ßnd d•2 • 1, if •t some l•ter t this is no longer the c•e
5982
OD. PMAN: DYNAMICSIN A PdgVERSAL
WITH Gzmm•
Tnv• DZPZN•ZNC•
Unnormalized, the magnetic field is
n=
lconserving
'
'
which representsa "thinning" field; that is, the linking field
decreaseswith t. The correspondingelectric field
1
-
i
(48)
has E•(T) = 0, this corresponds
to a frame of reference
in
which •he field fine p•sing through X = 0, Z = 0 is at rest.
We may moveinto an E•(T) • 0 frame by acceleration
in
stochastic
the •
direction; in the moving frame the particle inifiM ve-
locity and subsequent
energisafion
wffidiffer(fromequation
(40)), bu• the z(t)wm be unchanged.
g-con]ervih,g
Again, normMizing the magnetic field to a characteristic
fining field B,, time to the inverse of the gyrofrequency
in field B, (that is, to 1/•,, and lengthto a characteristic
gyroradiusp, = v/•, yields
Figure 1. Notional trajectories for slow and fast passage transitions for a thinning reversal are marked on a plot in parametric
coordinate d)•l, d)•2 versus )•l space. The different regimes of par-
ticle motion are marked on the plot (dashedlinesindicate their
notional boundaries).
a-
0, +
= -0
cannot exist after the transition
•
_• 1.
+
(s0)
1
(as shown),the motion will be entirely composedof fast
(/z-conserving)z oscillations.If on the otherhandthe transition A• • 1 occursafter dA• ___
1 or dA2----1 (a fast passagethroughthe transition)then segmentsof tt-conserving
motion
(4,
The parametric coordinates for this system are
(to+ t
In this
case, stochastic motion composed of segmentsof both
conservingmotion and fast (/z-conserving)motioncannot
occur, and the transition will be from a single period of
regular tt-conservingmotion to a single period of regular
/z-conserving motion.
It should be emphasisedthat the condition for a slow or
fast transition is not simply whether the system is changing
slowor fast with respectto the particlegyroperiod(indicated by d•X2)but also whetherthe changeis slowor fast
with respectto the transitiontimescale(indicatedby dA•).
1
d• = f•zr
(53)
Examples of a fast and a slow passagein this model are
sketched in parametric coordinate space in Figures 23 and
2b, whichshow
givenby (52) and (53), respectively.
In order for a particle to undergo a slow passage,we require that as t increasesA• << 1 to A1 > 1, while both dA• <<
1 and dA•. << 1. From Figure 23 we see that this particle
(represented
by the line markedslow)will, if A•(t = 0) << 1,
NUMERICALLY
INTEGRATED
TRAJECTORIES
Previously,trajectories were numericallyintegrated in the
simpletime dependentmodelof Chapmanand Watkins[19-
initially, execute tt-conservingbehavior. Once • > 1 the
motion will becomestochastic, exhibiting segmentsof both
tt-conservingbehavior when far from the center plane, and
fast z oscillations
across the reversal when close to the center
93] usinga modelfield wheref• = T/r = t/(f•zr) (where plane. Once dA• > 1 the motion will be entirely composed
r is the timescaleon whichthe field changes)and fz = 1 of fast z oscillations across the reversal. This scenario reso that A• = r•ot/(f•zr) = c•t , A• = 1, then dA• = c• was quires dad << 1, as shown by the horizontal hue marked
a constant and dA•. = O. Trajectories were then found to slowon Figure 2b. From (52) and (53)this trajectoryreexhibit a transitionin behaviorwhent _• 1/c•, that is, when quires f•zr >> 1 so that both d•. << 1 and the gradient of
A1 _• 1, and to be ordered in behavior by the constant dA• =
(52), whichis 1/(f•zr) << 1. In addition,to/•z must be
sufficiently small that all three parametric coordinates are
much less than one at the start of the trajectory, so that the
(increasing
A•) with increasing
t. The parametriccoordinate motion is initially tt-conserving.
A numerical solution of such a trajectory is shown in
dA•. could not be identified in this model as it was zero for
Figures
33 and 3b. The equations of motion were inteall trajectories.
In order to further illustrate the above results, a simple grated usingan adaptiveorder and adaptivestepsizescheme
[Shampineand Gordon,1975]. The normahzationis the
field model has been selected which has the property that
c•. On the plot in parametric coordinate space, trajectories
would be horizontal lines, the particle moving to the right
A• = A•(t), dA1= dA•(t), and dA• is constant;trajectories same as usedin the abovediscussion,(i.e. to length pz
and time 1/f•z). The field model then has c•0= 0.01 (a
weak reversal), f•zr = 25 (a slowlychangingreversal),
in this model then exhibit transitions with respect to both
A• and dA• and are ordered with respect to dA2.
CHAPMAN:DYN••
IN A Pdg•AL
WHIt G•
Tnv• DgP•ND•
5983
3a and 3b, whichare plots of z(t) and z(t). The periodin
time spanning from t = 0, so that ;• < 1, to t = 10000, so
that ;• > 1, is shown in Figure 3a. The z motion appears
-conserving
almost linear
due to acceleration
in the induction
electric
field, and the z motion is oscillatory.From the z motion
we see that before t = 2500 the motion is composed of fast
oscillations combined with a slow bounce motion, the par-
ticle is not stronglypitch angle scatteredon crossingthe
centerplane(the mirror pointsat t • 750 andt • 2000are
at approximatelythe samez). However,oncet > 2500,an
interval of stochasticmotion begins;on each crossingof the
centerplane the particle executesfast z oscillationsbefore
beingejectedto executea segmentof it-conserving
motion
awayfromthe centerplane. In Figure3b the sametrajectory
a
b
!
FAST
l•conserving
ß
2000
4
0
6000
8000
10000
12000
t
,,
stochastic
X 104
/
conserving
:
5-
SLOW
4
3
x
2
1
Figure 2. Notional trajectories for slow and fast passagetransitions in the simple thinning model given in the text marked on
i
0
2000
i
i
4000
i
6000
i
8000
10000
12000
t
plots in parametric coordinated,Xl (a) and d,X2(b) versus,Xl
space. The different regimes of particle motion are marked on
b
the plot (dottedlinesindicatetheir notionalboundaries).
and t0/(•z)
: 0.01. The trajectory has initial position
r = (5 x 10-2,0,1), with the z positionchosen
suchthat
the renofinalizedsystemwouldhavez*(t* = 0) = aoz(t =
0) = a0 and the z positionchosensuchthat initially the
•0
particle lies on the rest field line defined by the choice of
E•(T):
O. The initial v = (0,0.5,-0.01) at t = 0, with
-4o
the velocity components chosento ensure that the particle
crossesthe center plane z = 0 on a shorter timescale than
the transition timescalesof the system.
2
Oncet/(flzr) >> to/fl•, that is, oncet >> 0.25 (approximatelya quarterof a gyroperiod)the parametriccoordinates
i
i
'5
i
2
3
3.5
x 105
x 106
1.5
are just
x
,• '"'•-•rt
(54)
1
0.5
I00/ 0.5I
I
1
I
I
1.5
2
t
I
2.5
I
3
3.5
x 105
Figure 3.A numericaJlyintegrated trajectory undergoesslow pas-
1
sagein the simplethinningmodel. The panelsshowz(t) (top)
•nd •(t) (bottom). Th• Not •igu• 3(•) •t•t• •t th• initi•
condition(t = 0) and showsa time period spanning,X•(t) < 1
which yield the transition times, that is, •z _• 1 at • =
,X•(t) > 1. The plot Figure 3(b) starts at the initial condi•,r/ao = 2500,and dA1___
1 at t = (•,r)a/ao = 62500 to
tion (t = 0) and showsa longer time period which also spans
(dAa= 1/25 << 1 for all t). We can comparethesetimes d,X•(t) < 1 to d,X•(t) > 1. In this model d,Xais constantand
with the numerically integrated trajectory shownin Figures d,Xa < 1 for this trajectory.
-d;•2
-- fl,r
(56)
5984
CHAPS: DYNAMICS
IN A POg•AL WI•I-IG•
2
!
•
•
TIMg DEP••CE
tially, thesearegivenby the (time dependent)
scaling
of the strength of the coupling between the x and z
oscillatory motion and the particle and system adia-
•0
baricity.
-2
0
100
200
300
i
i
i
400
500
600
2. For the caseof a thinning reversal, two types of transition are possible: a slow transition from •-conserving
I
7•
800
to stochasticmotion, then subsequentlyto regular
t
motion entirely composedof z oscillationsacrossthe
center plane, and a fast transition into z oscillatory
2O00
motion
1500
with no interval
of stochastic
motion.
3. For specificmodelsfor the field the parametric coordinates can be inverted to give the approximatetimes at
which transitions between the different types of mo-
x 1000
500
/
i
i
i
i
i
i
i
1O0
200
300
400
500
600
700
tion occur.
8OO
t
Figure 4.A numerically integrated trajectory undergoesfast pas-
Multispacecraft in situ measurements,such as those of the
proposed
Glustermission[Rolj%,
1990andreferences
therein]
sagein the simplethinningmodel. The panelsshowz(t) (top) are neededto unambiguouslydetermine both the characterand z(t) (bottom). The time periodstartsat the initial condition istic length scale and timescale of the field in terms of the
(t = 0). The initial conditionsand modelparametersareidentical
particle Larmor scales. This will allow the parametric coto the plotted trajectory in Figure 3, except that now dA2 > 1.
ordinatesto be determinedand hencegive estimatesof the
times at which transitions
between one class of motion and
another should occur. In particular, this should allow the
duration of the period of stochasticmotion to be determined.
is plottedfrom t = 0 to t = 3 x 105,that is, until dA• > 1.
The particle continuesto executestochasticbehavior;at
It has been suggestedthat the changein the particle diseach subsequent
crossingof the centerplane the segment tribution as the dynamicsevolvesin the presubstormthinof fast z oscillationsis longer in time until t • 1.2 x 105
ning plasma sheet may produce substorm-relatedinstabili-
(dA• ___
2) whenthe motionbecomes
entirelycomposed
of
ties [Buchnerand Zelenyi,1987]or may thermalizethe disfast z oscillations.Henceas impliedby the aboveanalysis,
tribution sufficientlyto lead ultimately to a reconfiguration
the conditions
A1 --4 1 and dA1--4 1 (with dA2<< 1) give of the magnetotail[Cowley,1991;Mitchell et al., 1990,and
the approximatetimes after whicha transitionin dynamics
referencestherein]. The implication of these results,that
will occur, the exact times at which the dynamicscan be
stochasticmotion may persist for a finite time or under ceridentifiedto changebeingdependentuponinitial conditions
tain circumstancesmay not occur, may therefore have im(beingsensitiveto the times at whichthe particlecrosses
plicationsfor our understanding
of the roleplayedby singlethe centerplane,for example).
The fast trajectory sketchedin parametric coordinate
space
in Figure2 corresponds
to thegradient
of(52),
particle dynamics in the substorm cycle.
Acknowledgments.The author is indebtedto N. W. Watkins
> 1. Howeverthis impliesd)[2 > 1, so that segmentsof
and J. Chen for illuminating discussions.
#-conserving motion cannot exist. A numerical solution of
such a trajectory is shownin Figure 4, in the sameformat
their assistancein evaluating this paper.
The Editor thanks M. Ashour-Abdalla
and a second referee for
as Figure 3. The field modelin this casehas identicalpaREFERENCES
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sition and velocity of the particle is also identical to the
previouscase. From Figure 4 the motion can be seento be
entirely composedof fast z oscillatorymotion and appears
to be regular.
CONCLUSIONS
Under certain constraintswe can obtain a systemwhich
yields parametric coordinates which are functions of the field
and particle spatial and temporal scales,and time, that
characterizeparticle dynamicsof a time dependentreversal with parabohc field hnes.
Provided that the particlescrossthe centerplane on the
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(ReceivedFebruary22, 1993;
revised August 19, 1993;
acceptedOctober1, 1993)