JOURNAL
OF GEOPHYSICAL
RESEARCH,
VOL. 94, NO. A9, PAGES 11,821-11,842, SEPTEMBER
1, 1989
Regular and Chaotic Charged Particle Motion
in Magnetotaillike Field Reversals
1. Basic Theory of Trapped Motion
J6RG BOCHNER
Central Institute for Astrophysics, Academy of Sciences of the German Democratic Republic, Potsdam
LEV
M. ZELENYI
Space ResearchInstitute, Academy of Sciences of the USSR, Moscow
We give a systematic theoretical analysis of trapped nonadiabatic charged particle
motion in two-dimensional taillike magnetic field reversals. Particle dynamics is shown to
becontrolled
bythecurvature
parameter
•, i.e.,theratio•2 __Rmin/Pma:•
between
the
minimum radius of curvature of the magnetic field and the maximum Larmor radius in
it for a particle of given energy. • >> 1 correspondsto the usual adiabatic casewith the
magnetic moment/t as a first-order invariant of motion. As • decreasestoward unity, the
particle motion becomesstochasticdue to deterministic chaos,causedby the overlapping
of nonlinear resonancesbetween the bounce- and the gyro-motion. We determine the
threshold of deterministic chaos and derive the related pitch angle diffusion coefficient
which describes statistically the particle behavior in the limit • -• 1. Such behavior,
which for • --• 1 becomesstrongly chaotic, applies, e.g., to thermal electrons in Earth's
magnetotail and makes its collisionlesstearing mode instability possible. We also show
that in sharply curved field reversals,i.e., for • < 1, both a new kind of adiabaticity and
a partially adiabatic but weakly chaotic type of motion appear. The latter is strongly
effected by separatrix-crossingsin the phase space,which lead to a qualitatively different
chaotic behavior compared with the case • > 1. Both types of trapped particle motion in
sharply curved magnetic field reversals• < 1 are closelyconnectedwith fast oscillations
perpendicular to the reversal plane. However, the trajectories are adiabatic only in the
case that they permanently remain crossingthe reversal plane. The adiabatic are of a
ring type, i.e., they resemblerings in phase space and also in real physical space. For
ring-type orbits the action integral over the fast oscillationsis an adiabatic invariant in
the usual sense. On the other hand, the most common particle trajectories in a sharply
curved field reversal with • • 1 are essentially of a cucumberlike quasi-adiabatic type.
For quasi-adiabatic cucumberlikeorbits the action integral over the fast oscillationsis an
adiabatic invariant only in a piecemeal way between successivetraversals in the phase
space of the fast motion of a separatrix between orbits which do and those, which do not
cross the reversal plane. Due to the effect of scparatrix traversals the slow motion shifts
between different cucumber orbits with a conservationof the action integral on average
but with its chaotic phase space diffusion even for very small perturbation parameters •.
The case • • 1 is applicable, e.g., to thermal ions and high-energy electronsin Earth's
magnetotail. Our findings lead to a systematic interpretation of particle observations
in Earth's magnetotail and of numerous numerical calculations, carried out in the past.
They also explain rather well, e.g., the pitch angle diffusionof plasma sheet particles, the
isotropization of the plasma sheet electron distribution immediately before a substorm
and provide with the transition to chaosa mechanismfor the onset of a large-scale tail
instability and the explosionof isolated substorms. Further implications for magnetotail
physics, such as acceleration processesand the influence of the particle escape from the
field reversal will be discussedin a secondrelated paper.
The topology of such tails is characterizedby a reversal
of the magneticfield directionwith a region of strong
The solar wind interaction with magnetized celestial
field
inhomogeneity. In Earth's magnetotail the field
bodiessuchas Earth leads to the formation of magnetotails, which store a huge amount of energy. This energy line curvature may be so strong that the Larmor radii
(• 102- 103eV) become
compacan be released both permanently and explosively, ac- of thermalelectrons
rable
with
the
curvature
radius,
while
the
Larmor
radii
celeratingparticles and heating plasma to high energies.
of
thermal
ions
(•
110
keV)
exceed
it
many
times.
Magnetotails therefore play an important role in the energy, mass, and momentum transfer between the solar Hence, particles can no longer be consideredadiabatic
1. INTRODUCTION
wind and the innerpart of magnetospheres
(see,for ex- in the usualsense(see,for example,Speiser[1965]aand
Gray and Lee [1982]). This nonadiabaticitycreatesa
ample,the proceedings
editedby Lui [1987]).
Copyright 1989 by the American GeophysicalUnion.
Paper number 89JA00988.
0148-0227]89]89JA-00988505.00
number of difficultiesfor the kinetic theory of magnetotails, of the processesof formation and the balance of
magneticand mechanicalforces,of stability and dynamics of energyconversion.It alsorelatesto the problem of
the stationary velocity spacedistribution and of plasma
11,821
11,822
BOCHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
turbulencein the reversalregion. But what is the proper
In sections4, 5, and 6 we study charged particle moway to treat nonadiabatic trajectories?
tion in a sharply curved field reversal with g ( 1, for
In the past a number of direct numericalintegrations which we previously have shown that two principally
of the equationsof motion have been carried out in the differentsolutionsexist [B•chnev,1984, 1986,B•chnev
nonadiabaticlimit to study this problem[Speiser,1965 and Zelenyi,1986],ring-type adiabaticand cucumbera, b, 1967; Cowley, 1971; Eastwood, 1972, 1974; Purlorkin and Tsyganenko, 1973; Swift, 1977; Wagner et
type, quasi-adiabatic ones. In section 4 we derive the
basic equationsand the governingparameter for a deal., 1979; Tsyganenko,
1982;Lyonsand Speiser,1982]. scription of the particle motion in sharply curvedfields,
Eastwood[1972]alreadypointedout that a strongfield the actionintegralI.
Its normalizedvalue
determines
whetherthe motionis adiabatic(I t ) 1) or
On the basisof their numericalcalculations,Swift[1977] quasi-adiabatic
(I t ( 1 ), wherec•0,fl0 are two specifi-
line curvature seemsto scatter particles at the reversal.
and Wagneret al. [1979]foundthat for certainparam- cally defined velocity pitch anglesat the reversalplane.
eters a very complicatedparticle motion arises,wherein
even small changesin initial conditionslead to a strong
In section 5 we discussthe adiabatic case I t ) I which is
characterizedby ring-type orbits. But most particlesin
modificationof the trajectories.Tsyganenko
[1982]and sharplycurvedmagneticfieldreversals(g (1) moveon
Lyonsand Speiser[1982]analyzedsomeaspectsof this cucumber-typequasi-adiabaticorbits with I t ( 1. In
process
numerically,while Birmingham[1984]evaluated section 6 we show that the motion on "cucumber" orthe related diffusioncoefficientcalculatingin the process bits differsfrom the adiabatic ring-type onesprincipally
the nonadiabaticchangesof the magneticmomentin the because of two separatrix traversals on each "cucumweak field region. Recently, nonadiabatic particle mo- ber" cycle. These traversals correspond to qualitative
tion in two-dimensional
field reversals has been numerichangesof the topologyof fast oscillationsfrom crossing
cally reinvestigatedby meansof Poincar• surfaceof sec- to noncrossingof the neutral plane. In this casethere
tion plots [B•chner, 1984, 1986; Chenand Palmadesso, exist separateinvariants IA and IB valid only away from
1986]and alsoby analyticalmappings[B•chnerand Zelenyi, 1986]. Thesestudiesled to the conclusion
that
the separatrices
(section6.1). They maychangeconsiderablyin the courseof separatrixtraversals(section6.2).
particle motion in magnetotaillikefield reversalsmay be-
In Section 6.3 we derive a general approach to the "cu-
comestochasticin the senseof deterministicchaos[ for cumber" motion. We show that it is weakly chaotic and
example,Lichtenberg
and Liebarman,1983]. Of course, diffusiveas long as g (( 1. We derive the correspondnumerical studies are useful, but for further progressof ing diffusioncoefficient,which is valid for g (( 1. g • 1
the kinetic description of plasma in field reversalsit is leadsto a strongand global chaosof the particle motion.
necessaryto developa generaltheory of particle motion In section 7 we summarize the derived laws of the trapafter the breakdown of adiabaticity.
ped particle motion in magnetotaillikefield reversalsfor
Dungay[1968],Speiser[1968,1970]and Orazberdyevall parts of the phase spaceas they depend on the curand Trakhtengertz
[1973]were the first to discusspos- vature parameterg and the action integrals/•t and I t,
sible consequences
by using the apparent decouplingof
motion parallel and perpendicular to a sharply curved
field reversal to describethe current sheet particle mo-
both of which depend on the particles' equatorial velocity directions. In section 8 we discussthe relation of our
results to previous numerical calculations and several
tion analytically. Speiser[1970]and Sonnerup[1971] applicationsof our results to the physicsof the magsuggestedadiabatic invariance but incorrectly matched netotail. More applications of this theory, the role of
topologicallydifferent solutionsthrough the phasespace trajectories escapingfrom the field reversal,the laws of
current sheet acceleration etc. will be discussed in a reseparatrices.
In this paper we carry out a systematicinvestigation lated secondpaper ( J. Biichnerand Zelenyi,Regular
of trapped particle motion in magnetic field reversals and chaotic chargedparticle motion in magnetotaillike
with arbitrary curvature radii. We show how the type field reversals,2. Physical implications, manuscript in
of trajectory is determinedfirst by the curvatureparam- preparation,henceforth
citedas (BZ2)).
eter •
2
- Rmin/Pmax, the ratio betweenthe minimum
curvature radius Rmi n and the maximum particle Larmot radius Pmax, which depends on the total kinetic
energy of the particle and on the field geometry and
secondlyon the particle adiabatic invariants, which are
determined, for example, by the particle's velocity direction at the moment of crossingthe neutral plane of the
reversal. In section2 we introduce the model of a general two-dimensional magnetic field reversal and derive
the curvature parameter •. In section 3 we investigate
the breakdown of regularity, i.e., the breakdownof the
usual adiabaticity of the magnetic moment /• and the
transition
to deterministic
chaos as • -•
1. To do this
we reduce the original equationsof motion at the border of adiabaticity to a discrete mapping. This allows
a theoretical analysisof the breakdown of regularity as
• -• 1, the chaotization, and chaotic pitch angle diffusion.
2. A MODEL
THE
FIELD
CURVATURE
REVERSAL
PARAMETER
AND
n;
It is appropriateto describetwo-dimensionalmagnetotaillike field reversalsby means of a tangential field,
which corresponds
to the Harris equilibrium[Harris,
1962], and by a small field componentperpendicular
to the neutral plane, which changesweakly along the
reversal(cf. Figure1}:
B-B0.tanh
(•--).
In our analysis we keep the component perpendicular
to the equatorial plane Bn - const., with a ratio bn =
Bn/Bo smallby comparison
with unity (in Earth'smagnetotail, usuallybn < 0.1). As long as Bn - const.,
BOCHNERAND ZELENYhREGULARAND CHAOTICCHARGEDPARTICLEMOTION
11,823
Z=O
(4")
Earth
wherethe equationof a fieldline X(Z) canbe obtained
from expression
(2) settingA --const. From equations
(4) onefindsthe followingconditionfor adiabaticity:
X
Pmaz _
PO
Rmln
-- I,bn
2<<I
Fig. 1. The two-dimensional parabolic magnetic field model
usedin thispaper(cf. Equation
(1')).
a constantelectricfield Ey - const. can be removed
by the de Hoffman and Teller [1950]transformation.
Hence,we will assumewe are in the frame where E - 0.
or, introducingthe curvatureparameter
(5')
with
To simplifyour considerations
we approximate(1) by
Bo' •_ •L.
pn.Bo
L Bn
Bn
the following expression:
Z
Z< L
B -- B0 ßsgnZ 'ez -F Bn 'ez
Z -->L
(1')
(1")
Boll providesa measureof the spatialvariationof the
In this paper we considerthe motion of particlestrap-
(5")
It can easilybe seenthat the curvaturecondition(5) is
muchstrongerthan that for adiabaticityin a straightinhomogeneous
field with Bn -- 0 and a gradientdB/dZ •
BolL, wherepoll << 1 is the conditionfor adiabaticity.
tangential
fieldcomponent
(EquationIt). Dueto equation (1") L alsolimitsthe valueof the tangentialfield.
pedintheinner
partofthefieldreversal
(Z <-L),where
(5)
3. BREAKDOWN
TRANSITION
To
OF ADIABATICITY
CHAOS OF TRAPPED
IN A Two-DIMENSIONAL
the field lines have a parabolic shape. In a secondpaper
FIELD
AND
MOTION
REVERSAL
>
(BZ2) we will discuss
effectsrelatedto the limitationof
Adiabaticity usually occurswhen particle motion may
the field strength at Z -- L. In an appropriate gauge be consideredas a superpositionof fast Larmor rotations
theparabolic
field(1t) corresponds
to a vectorpotential and a sufficiently slower motion of its guiding center.
A:
Provided p -- const. particles may drift up to a distance Z M away from the neutral plane, where the field
A- B..x -
(2)
For the adiabatic conservationof the magnetic moment
strength
reaches
thevalueIBm]= Bn'sin-2 50(50isthe
pitch anglein the equatorialplane Z = 0). For the re-
versal(1•) Zm maybedetermined
fromBn.sin-2(50)=
1 m.vñ 2
/•- •. lBI
•/(B0'
Zm/L)
2+B•
2.Particles
with
"equatorial"
pitch
(3) angles
5ocorresponding
toZm< L stopat Zmandturn
in inhomogeneousmagnetic fields the Larmor radius
must everywhere be small by comparisonwith the radius of curvature. Hence one may derive a necessary
conditionfor the applicability of the adiabatic approach
by comparing the largest possibleLarmor radius Pmaz
to the minimum curvature radius Rmin. In the field
reversal(1) both of thesequantitiesare reachedat the
equatorial plane Z - 0, where a particle rotating with
its total velocity v0 directed perpendicular to the minimum field Bmin - Bn'ez has the maximum Larmor
radius
Pmaz-- Pn-- vo' fin-1
_ m.vo _ Po
- e.B.- b.
(4t)
wherefin - eBn/m is the Larmor frequencyin the field
B = Bn 'ez and P0 = vo/fio with fi0 = eBo/m The
curvature radius at the reversal plane equals
back toward the neutral plane; otherwise, they escape
the reversal(1). We will treat the escapemechanism
separatelyin (BZ2). Here in this paper we consider
particlestrappedin the fieldreversal(It), i.e.,fulfilling
a bouncemotion between the turning points Z-
:EZra,
Zm <L.
Now we will
show how the interaction
between
the
Larmor rotations and the periodic guiding centerbounce
motion destroysthe regularity and adiabaticity of motion. Generally, the bounce frequency dependson the
bounce amplitude Zm, i.e., the bounce motion must be
consideredas a nonlinear oscillationwhose higher harmonics may fall in resonancewith the Larmor rotation.
The resonancecondition may be written as
(6)
where n is a positive integer and < fiL > is the Lar-
11,824
BOCHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
planeandPs-- PIIthefield-aligned
component
ofthe
mechanical
momentum
P -m.
v. We will further use
dimensionless
variables(lowercaseletters)with a time
,½a• ½ - a;• - m/(•. s,) •na • l•n•h ,½a• • 10]
10]
Bn' L/Bo. Then the Hamiltonian(8t) maybe rewritten
as
1 2+
'
102
231'102
(8")
2
where#t _ #/#ma= -- sin250 is the magnetic
moment
normalized
by#,•a=-mVo2/B,, s - S. B,/(BoL) is
the dimensionless
distancefrom the Z - 0 planealong
a fieldline,ps- P•/rnbn2Llothedimensionless
fieldalignedmomentumand b(s) - IB(•)I/B.
and 60 the
equatorial velocity pitch angle.
One can obtain a rough estimate of the ratio of characteristic frequenciesin the system by comparingthe
smallestLarmorfrequencyIL,min -- In with the largestpossible
bounce
frequency
W•,ma=,
viz,thatofsmall
amplitude linear guiding center oscillationsaround the
equatorial plane. The last is easily obtained by ex-
(8")
Fig. 2. Resonances
of ordern, = A•b/(2•r) betweenfrequen- pandingthe potentialfunctionin the Hamiltonian
s - 0. Tofirstorder,onegetswS,ma
==
ciesof the slow guiding center bouncemotion wb and the #t. b(s)around
fastLarmoroscillations
flL asfunctions
of pt, thenormal- Polo/bnL. Henceone obtainsthe followingexpression
ized magneticmomentfor • -- 2, 3, 10 (cf. equations(7),
(•0)).
mot frequency averagedover the bounce period Tb =
2•r/wb. The resonanceconditionmay also be formulated in terms of the Larmor phasegain during half of
the bounce period, that is the time between two subsequent crossingsof the equatorial plane g- 0
z/•.• T•
a• - •
•Ldt- -•-.<•L >- •r.
<•L>
Wb
= •r. •
It is a well-known property of nonlinear parametric
resonancesthat around them phaseoscillationsappear.
If the amplitudeof the phaseoscillationsbecomeslarger
then the distance betweenadjacent resonancefrequen-
for the ratio between the extreme frequencies:
Wb,rna•
IL,min
=
PO
-2
Lbn2 --
•;
(9)
Equation(9) thereforeshowsthat •2 represents
also
the minimum possibleratio of the extreme Larmor to
bounce frequencies of motion. For n >> 1 these two
characteristic frequenciesin the system are well separated from each other.
We will now quantitatively analyze the effect of the
bouncephaseoscillationson the regularity of the whole
motion. First we calculate the Larmor phasegain be-
tweentwo equatorial-plane
crossings,
whichdepends
on
the equatorialpitch angle•0, i.e., on #t = sinZ60. ExpressingIL, P, andds throughb = IBI/B. oneobtains,
cies (Sw• > < IL >), a randomwalk betweenreso- from (7),
nancesstarts [cf., for example, Chirikov1979],which
is now usually called deterministic chaos. Chaotic here
meansstochasticin the sensethat without taking into
account any interaction of the particles, single phase
spacetrajectories launchedwith arbitrarily closeinitial
A•b
- 2fo
'"•ill,
=•. f(•')-•
v/b2 - 1 ßV/1- #t. b
ds
P•
b3
conditionsdivergeexponentiallyfast [cf. , for example, œichtenber9
and œieberman
1983].Of course,if this
happensthe regularity of solutionsis destroyed.
For further discussionwe are going to make use of
an approximate Hamiltonian H0, which describes the
undisturbedguiding center bouncemotion in the mag-
J1
wheresm is the locationof the turning point corresponding to g - gin, at which ps - 0 and the particle is
mirrored back. With Figure 2 we demonstratethe ap-
pearanceof resonances
to differentorders
numerically
calculatingA•b(#t) asa functionof #t. Figexact conservation
of the magneticmoment(compare, ure 2 clearly showsthat for large n > 1 resonancesoccur
for example,Chirikov,[1984]):
only at rather high harmonicswith n )) 1. They are re-
neticmirror (It). It can be obtained,considering
an
strictedto particleswith small#t = sin25o,i.e., those
Ho(#,S,P)-2 m + #. I•(s)l- •l mvo• (8')
Here S is the field-aligned distance to the equatorial
mirroring far from the equatorial plane. We remind the
reader that this limit correspondsto particles which in
bounded field reversalswould escapeinto a loss cone.
That
is the reason for the lower cutoff of the curves.
BOCHNERAND ZELENYI:REGULARAND CHAOTICCHARGEDPARTICLEMOTION
We will discuss the loss mechanism in more detail in
11,825
ping{/•tN;•bN}--• {/•'N+i;•bN+i},whichequivalently
(BZ2). For/•t _• I the approximation
of nonequatorialdescribes the interaction between the fast and slow mooscillations
becomes
invalid(comparethe discussion
of tions, i.e., betweenthe Larmor rotations and the bounce
f(•t, b,) belowandin Birmingham
[1984]).That is the oscillations.
Wewillusethevariable
P- X/•- sin50
reasonfor the upper cutoffof the curvesin Figure 2. As insteadof/•t, because
it is convenient
to analyzemap-
•cdecreases
towardunity lowerorderresonances
alsoap- pingsin an "area preserving"form, i.e., with a Jacobian
pearevenfor larger/•t values.For •c-+ I the distances ID(PN+I, q•N+I)/D(PN, •bN)l= 1. Finally,oneobtains
betweenadjacent harmonicsalso decrease.Obviously, the mapping
the separation between resonancesfirst becomessmaller then the amplitude of the phaseoscillationsfor small
•c. The phaseoscillations
themselves
are causedby small
changesof the magneticmoment5/•t in the weak field
regionnear the equatorialplane Z - 0, which are not
compensatedfor over the Larmor period, in contrast
with the usual adiabatic invariant
which oscillates with
PN+I ----PN -•-AP . sin•bN
=
+
(12t)
whereA•bis givenby expression
(10)and zxP =
ß
F-1(9/8)ß•2.exp {_•2} does
notdepend
onPN. A•
hasto be determinedwith the argumentPN+I because
the motion after the Nth encounter of the reversal is
eter•c[cf.,for example,Chirikov,1978].
already
determinedby the jumped magneticmoment.
The small nonadiabaticchangesAp t at the equatoamplitudesto first order in the small adiabaticparam-
rim plane crossingscan be determinedby integrating
To analyze the phaseoscillationsit is appropriate to
thetimedependent
magnetic
moment/•t(t)throughthe linearize A•b around the resonancepositionsPn deterreversalplane encounterincludingcorrectionsto second minedfrom(7), i.e., aroundA•b(P,) - 2•rn. Introducorderin perturbationtheory(see,for example,Chirikov ing for near-resonance oscillations the new variable J
[1979,1984]and ½ohe•eZal. [19?s]).Fora linearfield defined by
dependence,
as in (lt), relatedcalculations
werecar-
dA•b(P) ß(P-Pn)
fled out by Birmingkam[1984]and BasuandRowlands
[1986]and oneobtains
A•'=r(9/s)
' v/•' •.exp{_•2.
f(/•t,
•)}.sin
sothat A•b• Aq•(Pn)+ •, oneobtainsfor linearoscillationsnearthe nth resonance
the mapping:
JN+• = JN + K,•. sin
•N+• = •N + •N+•
where•0 is the Larmor phaseat the Z = O-planecross-
ingandp• ontheright-hand
sideof (11)isthemagnetic
momentbeforethecrossing.
Thefunctionf(p',
pro•mately constantand of order unity until •' •
1.
(13)
(14t)
with the one and only remainingparameter
Fora graphical
representation
of f(p•, •) wereferyouto
Figure2 in Birmin9ham[1984]or Figure1 in Basuand
Rowlands
[1986].We will furtherdropthe f(p',•) de-
K.- 2.r(9/s)
' •' A•b(P)lp=r
. -exp
pendence,which meansthat we neglecta small number
(14m)
of particleswith large equatoriMpitch angles50 mirror- The mapping(14) is calledthe standardmappingbeing closeto the reversalplane. After eachcrossingof causeit arisesin a numberof problemsof nonlineardythe equatorialplanethejump Ap• changes
the bounce- namics. It was analyzed extensively,for example, by
oscillation
periodand the phaseg•n A•(•') (cf. (10))
by a sm•l amount and that is what causesthe phase
oscillations
A•(t).
For a further quantitative analysisof the phase oscillations,of the transition to chaos,and the laws governing the weakly chaotic motion it is appropriate to
change from the differenti• description of motion to
mappings. In a systemwith two oseillatorydegreesof
freedoma mapping expressesthe relationshipbetween
a p•r of conjugate variablestaken at two subsequent
erossingsof a speei•ly chosensectional plane in the
four-dimension• ph•e space. For the conjugatevariables it is appropriate to choose the Larmor rotation
ph•e •, which changesm•nly during the bouncemotion outside2 • 0, and the norm•ized magneticmo-
mentpt = sin2•0. In the ease•2 )) I thebestchoicefor
a seetion• planeis 2 - 0, becausethe irreversiblejumps
of the magneticmomentApt occurduringtravers• of
Chirikov[1979].Investigations
of the standardmapping
carriedout in the past(see,for example,œichtenberg
and
œieberman
[1983];½hirikov[1984]),revealed
that phase
oscillationsaround resonancesoverlapbeginningwith
K > 1. Hence,
fromtheexpression
forK wecanderive
the exact condition for the breakdownof the stability of
motion, of regularity and adiabatieity and thereforethe
threshold
for the onset of chaos.
The closest packing of resonancesand widest resonance layers occur at high order harmonies n )) 1,
hich IP-
<
<< 1.
Hence chaosis expected to appear first at higher order
resonances
and for small/•t, i.e., for particlesmirroring
farfromZ - 0 (ef. Figure2). ForsmallP - V/-•
7sin•0 the asymptoticbehaviorof the phaseintegral(10),
A•b(P)----(16/15).•2. p-e, canbeusedto evaluate
the
K-parameter of the standard mapping
this plane. Using the Larmor phaseg•n A• between
the Nth and the (N • 1)st erossings
of the 2 = 0plane(seeequation(10)), onethereforeobt•ns a map-
K""5. F(9/s)
16•r
.•;9/4
,P-?ßexp
{_•2} (15)
ß•/•
11,826
BOCHNER AND ZELENYh REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
largepartsof the Hi-spaceup to HI = 0.5 showchaotic
1
behavior,and the diffusionbecomesstronger. Finally,
for • = 2.0 (Figure 6) the resonances
overlapfor most
of phase space, except for some islands of stability for
K<I
adiabaticify
0.5
H• > 0.9 whereresonances
canbe seendirectly.
Our analysis shows that with K - 1 weak diffusion
in phase spacebegins and becomesglobal for K > 4.
In the parameter range between K = 1 and K = 4 the
diffusion rate, related to one iteration of the standard
mapping
nearK •, 1 is givenby D - 0.38.(K- 1)2
Chiriko•[1979]. Making useof this expression
onecan
now determinethe pitch an[•lediffusioncoefficientde-
finedby (2 < (AH')2>). rb-•:
2
Fig. 3. Regionsof adiabaticmotion(K < 1) and determin-
isticchaos
(K > 4, hatched
in thefigure)in /•l _ • space,
where K - Kr• is the parameter of the standard mapping
(14), derivednear the resonanceof nth order for n - 510,000.
The diffusioncoefficient(18) is similarto the estimate
evaluatedby Birmingham[1984],but addsa K-dependent term, which showsthat there is a clear distinction
andsimilarly
theresonance
values
P,•-- X/•,•:
betweenthe adiabaticity(for K • 1), whendiffusion
does not occur and the weakly chaotic regime K • 1
wherethe pitch angle diffusionmay be describedby the
diffusioncoefficient(18). Finally,with • • 1 the term
From these two expressionsone finally obtains the as-
ymptoticestimatefor K at the n-th resonance
(n)) 1):
exp{•
2}isnolonger
small
and
•< (A•)2>
• •. In
this casethe motion becomesstronglychaotic. Particle
dynamicsfor • • 1 will be discussedin the following
sections 4-6.
3 •i-• ••2
•
Kl,<x
-•r(9/8).
(2/3)
7/3'•/•'exp{-•2}
(17)
Figure3 shows
for whichvalues
pl = p2 and• the Kparameter is less than unity and the motion is regular
and adiabatic. The line K - 1 in Figure 3 indicates
the border of adiabaticity. For • less than the value
4. PARTICLE
MOTION
IN SHARPLY CUURVED
FIELDREVERSALS
(• • 1)
In the previous section we showed how the decrease
of • toward unity leads to a chaotization of motion and
pitch anglescatteringafter breakdownof the adiabaticity of the magnetic moment H- On the other hand even
corresponding
to K -- 4 (the hatchedarea),the particle early numerical calculations of particle trajectories in
motionbecomeschaoticand diffusive[cf.,for example, fieldreversals,
carriedout by Speiser[1965a,b, 1967],reChiriko•, 1979]. As alreadydiscussed,
the resonance yealeda quite regular behaviorfor medium energyprooverlap starts at high orders n and for partic]eswith tons in sharply curved two-dimensional field reversals,
smallH•. In fact, K = 1 for • •-, 4 (cf. Figure 3)
is reachedonly at a resonanceof the order n •-, 10,000.
With decreasing
•, however,resonances
beginto overlap
•-' 3.0
alsofor largerHI . For particlesmirroringfar from the
2•
equatorialplanewith H• - 0.2-0.3 the thresholdof weak
chaosis reachedapproximately at • - 3 for n of about
50-150. Between • - 2 and • - 3 the resonance]ayers
overlapfor most of the remaining phasespace,evenfor
larger
P- X/• andat lowerorders
n.
To check these asymptotic predictionswe calculated
directly100iterationsof the mapping(12) for 90 different initial conditions,computingexactly the phaseinte-
grals(10). The resultsare shownin Figures4-6 for • =
3.0, 2.5, and 2.0, respectively. In agreementwith the
predictionsfrom the asymptoticexpression
for K (cf.
Figure3) at •- 3 first contactsbetweenthe resonance
0.1
0.5
layers appear and phase spaceor pitch angle diffusion
startsonlyfor Ht < 0.25 (seeFigure4). However,
the Fig.4. Themapping
{•bN,
p• }
diffusion is so weak that even after 100 iterations, i.e.,
lated
for 90 different
initial
-*
{•bN+
1•/•N+I
' }, calcu-
conditions
over 100 iterations
after50 bounceperiods,onlysmallchanges
of Ht occur. for the case• = 3.0 (cf. equation(12)). Only small and
deviations
fromthe initial/•l occur.
At • - 2.5 ( Figure 5), in agreementwith Figure 3, reversible
BOCHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
11,827
model(1t) andits spatialhomogeneity
in the horizontal
directionsX and Y lead to three trivial integralsof motion: the total energy,a generalizedX momentum and
the canonical
Y momentum:
1
H - • ßmßv02
- const
(19a)
Pz -- m' vz - e . Bn ' Y = const
Py = m.vy + e. Ay - const
(19b)
(19c)
Here v0 is again the particle's total velocityß Unfortu-
0 0.1
0.$
•"
1
nately, the momentaPz and Py are not independent:
their roissonbracket{P=,Py} = -mn. -•- 0 doesnot
vanish.
Hence in order
to describe
the motion
com-
ß
,
,/ZN+1}, calcu- pletely it is necessaryto find a third independentintegral or at least an approximate adiabatic invariant
lated for 90 initial conditions over 100 iterations for the case
s = 2.5 (el. equation(12)). The nonlinearresonance
lay- of motion. The schemefor constructingadiabatic iners for trajectories mirroring far from the equator overlap, variants with nearly periodic solutionson different time
andfor smallpt chaosappears.For largerpt the motion scalesis to split off the rapid time variation into a sinstill remains regular, but resonancestructures can already
glevariable[c/ Kruskal,1962; Whippleet al., 1986],as
be seen.
was done for fast Larmor
rotations
in the case of the
adiabaticityof the magneticmoment/• (cf. section3).
Due to the KAM theorem sucha procedurecan be carried out only if regular solutionsreally exist, i.e., if the
perturbation of regular and integrablesolutionsis small
enough.In our problemof a two-dimensionalmagnetic
field reversalwith a strong curvature due to a small Bn
componentthe Iz action integral is a natural candidate
for being adiabatically invariant, with t• << 1 as the
findingsSpeiser[1970]and So,,erup [1971]suggested small perturbation parameter. In sections 5 and 6 we
will analyze to what extent Iz may be really considered
the existence of a current-sheet
adiabatic invariant related to the Z- motion perpendicular to the reversal. a third adiabatic invariant of motion in sharply curved
However,recentanalysescarriedout by B•chner [1984, magnetotaillikefield reversals.
In order to simplify the further analysisit is appro1986],B•chner and Zelenyi[1986],and Chenand Palpriate
to reducethe original three-dimensionalproblem
madesso
[1986]for g < 1 verifiedadiabaticityonlyfor a
rather limited set of initial conditionsand showedappre- to two oscillatorydegreesof freedomby using the exof the canonicalmomentumPy (19c).
ciable deviations from regularity for most of them. In act conservation
the followingsectionswe will give a theoreticalanalysis Choosingas in section 3 a distance scale X - bn' L
r - (•,)-I - (b,•0)-1 weobtain
of this problem. We will show that in parabolic field re- anda timescale
the
following
expressions
for a two-dimensionalHamilversalswith g < 1 in addition to the Speiser-typeescapvariables)
equivalent
to the origiing trajectories there exist alsotwo types of trapped tra- tonian(in normalized
jectories,ring-type adiabaticand cucumber-typequasi- nal three-dimensional problem:
correspondingto n less than unity. These trajectories
are characterized by fast oscillationsacrossthe reversal
plane g - 0 and slow X- and Y- motion. Speiser-type
trajectories originate from X-infinity, move toward the
field reversal, where they meander across the reversal
in the Z-direction, drifting mainly along the Y-axis,
and leave the reversal to infinity. On the basis of these
adiabaticones.Beforedoingso (in sections5 and 6) we
will
discuss in section
4 some basic ideas about
adia-
baticity in sharply curved field reversals.
In the limit • = 0 due to B, = 0 the equations of mo-
tion in a magneticfieldreversal(1t) maybe separated
__
, ,,,.;
;.;':•, ,-
,II
....
,v >...,:,{•,•,g•
...........
. • ,.,,...:
]
.:.....
......
.,...:
......
.:.,..,
,,•..•:,.,.•
•. r•.:;:j•
....... ;:;...,
:•.:'.:'•'•.•½•,2•.
'?::':
::7•':•.:•1'•'•
.....
'..•:.': ;.•,•. •.•::..'0.
• •. "::•."•4•,"":•'::q...... • ............
• ':;'?:;"•,•'ii'e•t•2a2:•:•:q:;½'.',n:s:.
'W' "•:,•
gralIz - (2•r)-1 j• • dzisanexactintegral
ofmotion.
/ ;....,
e,•**
.,•,.,;•.:, •:,'.'..'•'•:."•'('•'
For finite • due to Bn • 0, however,the motion in the
3 degreesof freedom of the system becomesnonlinearly
I
coupled. Due to the KAM theorem[cf. Lichtenberg
and Lieherman,1983]smallperturbationsof integrable
To solve this problem we first carry out some appropriate transformations. The static nature of the field
,•..•.•.;.;'•f
,'..:,•v
'J•'
•c'(;;•
'.se•.i•.
r•.w.,
.,.•.
•:'._•,•,
.',..
• [, ':•"
- 4.
y•
.'..n..;
}.'•)•.
&•,'rv•2;.
.;:..:;.¾'
F:.•
,.'•'.'.,
i•.•: ..•q
. •' •.....
' '..:,.•.:.•,
. ••C'
completely. The Z-oscillations are nonlinear and can be
describedin terms of elliptical integrals;the action inte-
equations may not destroy the regularity of the solutions. The question arises, whether and in which parts
of the phase-or the velocity-spacethe chargedparticle
motion in sharply curvedfield reversalsremainsregular
for 0 < • < 1,and where possiblyit becomeschaotic.
•:2.0
2•[ .;e.
:...',.".•.s•:....•:'•i•,::'..t'
•.:}';;:f.5•:.Zfi4
.•'•gf
,;[
/
..,•.,. • ß.- . •.1 • '.•' •
.:.'•.'.'::•:,>•>,".e•v;
• • :•t,F3
•:
.,•...>.,.
.....:.. • ....:;,........:,•,:•.••.r'"
C:...•',*.?z.
'::.[;J
....
.
• -.•. i';•Y;.ta
• ..•,[•'•".
. r,•x:•'cx•.v,•,
r]
0 0.1
0.5
Fig.6. Themapping
{•bN,/zk}
-• {•bN+l,/•9+1}'
calcu-
lated for 90 initial
conditions
over 100 iterations
for the case
• = 2.0 (cf. (•2)). Most of the equatorialpitch angleslead
to chaoswith only small islands of regularity remaining for
large/•t.
11,828
BOOHNERAND ZELENYI:REGULARAND CHAOTICCHARGEDPARTICLEMOTION
ing trajectories(a•sh•a lines (B1) •na (B2) in Figure
7). Again the oscillations
are nonlinearbecausethe po-
u Ix,z)
tential is not parabolic. The mathematical conditions
determining whether solutionsfor ß > 0 belongto type
10
(A) or (B) may be expressed
as follows:
X
with
(A) ß
z
<
(B)'
z
>
the transition
V/2hz(z) (21a)
V/2hz(x) (21b)
situation:
(C)'
z
-
V/2hz(z)
(21c)
On the otherhand,lookingat (20c) oneseesthat hz(•)
is the effectivepotential for the z motion. Henceone can
usea Hamiltonianin the form (20c) to investigatethe
slow motion, neglectingto lowest order approximation
the changesof z on the fast time scaleand treating their
Fig. 7. The two-dimensional
potentialu(z,z)for threefixed influence as a small perturbation of the z motion.
To check whether a small perturbation of motion in
valuesz = -3, 0, 3. The dashedhorizontallines correspond
to differentvaluesof the constanteffectiveenergyhz, (A)i
a two-dimensional
magneticfieldreversal(n << 1) leads
to
adiabaticity
or
not,
we will investigatethe actioninteforhz> «z2,z < 0and(A)2forhz> «z2butz > 0,(B1)
gral Iz, which is an exact invariant of motion for n = 0.
and(B2)forhz< •2 and(C)forhz= •2.
Provided the z oscillations really proceed on a faster
time scale than the z motion, one may consider • and
.
.
h•(z) asslowlyvaryingfunctionsof time, whichchange
significantly only on a time scale long by comparison
with the z oscillation period rz. Treating therefore z
and hz as parameters, one may calculate the action in-
•-• lk2+•1•2+ u(•' z)--1•
2 -4 (20a)
.(•,z)-•
•-2
hz(;r
1 4
a-• 15:2+
,z,i)- •1
a,(•,,, z)- • •a+ •(•,,)
(20•)
(20c)
(20d)
tegralIA -- Iz(z, h•) for trajectoriescrossing
thez plane
asfollows(cf. lines(A• and (A)2 in Figure7):
z•(•,a•)
- (•)-•f •d•
2f•"•• 2hz-(•lz2_
•)2dz (22)
---
Here the normalized velocities k and • scale with bnLf•n
= b2nL12o,
Po- vo'f•0-1, z - Z. (b,L)-1 andx Ix - P•,0/(ma.)l-(•.•)-1is•hift•dbyanamount
determined by the initial momentum
• •=• - •. • +da.a•(•)i•ta••mp•it.d•
of
oscillations.It can be determinedby (20d) usingthe
condition• - 0. The integralin (22) can be evaluated
by somestraightforwardcalculationsto be
Py,0
-- Py(to)
- tory(to)
q-eBnX(to)
- eBo
(•0•)
Fromexpressions
(20a)and(20•)it follows
that t•--4
may be considered
as a measureof the dimensionless
energyof a two-dimensional
nonlinearoscillatormovingin
a potentialu(z, z) whoseformis shownin Figure7 for
1
three fixed values ß = const In the case of fixed ß the
fA(k
/ fs(k)
potentialu(•,z) determinesthe z-oscillations,unperturbed by the ß motion. To analyze the z motion, one
may usea Hamiltonianin the form (20d) with hz(x) =
I .-4 1_.2 wherek alsois fixed. For ß < 0 the potential functionu(•,z) of suchHamiltonianhasoneminimum (cf. Figure7). The upperdashedline in Figure
7 denoted(A)i corresponds
to oneparticularlychosen
levelofhz(•)- const
> «•2. Thez oscillations
in this
o.s
/
1
2
k
3
potentialare nonlinear,i.e., their frequencyflz depends
on the amplitudezma=.In contrast,for x > 0 two prin- rig. s. The functionsfA(k) for k < 1 and IB(k) fo• k > 1
cipallydifferenttypesof solutions
arepossible:(A) with separatedby the hatchedregionnear k = 1. The horizontal
to differentI • = const
crossings
of the reversalplanez - 0 (illustratedby the linesdenoteda, b, andc correspond
second
dashed
line (A)2 in Figure7 ) and(B) noncross-(a to I • > 1, b to I • < 1, andc to I • • 1 ).
BOCHNERAND ZELENYI:REGULARAND CHAOTICCHARGEDPARTICLElV[OTION
11,829
with I / < 1 approachthe rightsemiellipse
corresponding
tok0.2
0.3
-1
•2. X
1.
In sections 5 and 6 we will analyze the interaction
between the slow and the fast componentsof motion.
First, in section 5 we are going to discussa type of
motion, which remainsrestricted to the crossingregime
A and then, in section6 those trajectories, which reach
the boundary k- 1.
5. RINGTYPEADIABATIC
MOTION(I t > 1)
IN SHARPLY CURVED FIELD REVERSALS
In this sectionwe investigatethe caseI t - const• 1,
i.e., trajectories which always crossthe neutral plane
z - 0 (k < 1). If onesolvesthe relatedequations
of moFig. 9. Linesl/-
constin the z versus• phasediagramof
tion one finds that the correspondingphasetrajectories
theunperturbed
slow
motion.
ForI / (--0.8thecontours
are in the spacez, •, z lie on the surfaceof a deformedtorus,
labeledby thecorresponding
valuesofI/. Thelastcurve,intersectingthe line of uncertainty k - 1, i.e., the right semiel2
lipsez +•
2
1
=-•-
4
/
,corresponds
toI =09, that forI t--
1.0 is tangential to the ellipse. Inside the ellipse contours
shownin Figure 10. It is worthwhile mentioningthat
Pz = const(19b) leadsto y = •+ const(in normalized
variables).Hencein real spacethe trajectory{•,y, z}
also lies on the surface of a deformed
torus such as that
I t ----constaredrawnfor I t = 1.02,1.04,1.06,1.08,1.10and
shownin Figure10,if onereplaces• by (y-const). The
1.12, if one counts from the outer contour toward the center.
fast oscillationsspiral around the torus' small perimeter while the slow motion occursalong the torus' body.
The nearly circular symmetry of the slow motion is the
The
center of the contours
is related
with
the stable fixed
pointof the unperturbed
slowmotiondetermined
by I / -1.16.
reasonwhy we calledthe orbitscorresponding
to I t • 1
ting-type trajectories.
In our analysiswe assumedfrom the beginningthat
the time scales of the fast oscillations
8 . 2V/•.•-•3
4 ' fA(k)
and of the slow
(23') (•) motionare well separatedfrom eachother. We will
now determine the condition for the period of the fast
rA - rz(k • 1) to be muchsmallerthan the
fA(k)-- (1-- k2)ßK(k) -F(2k2 - 1). E(k) (23") oscillations
period of the slow motion TAA along the torus' body.
One can obtain rA by again treating •, hz as constant
k- • 1+2•/y•
(23'")parameters over the period of the z oscillations:
K(k) and E(k) arecompleteellipticintegralsof the first
and secondkind, respectively. For k •
I the function
fA behaves
approximately
as(3/4)•rk2;fA reaches
its
/dz
¾-4fz•x
dz_
-(-
=4K(k)
(24)
maximum of 1.16 at k - 0.909 and tends to unity as
k --* 1. The exact dependency of fA on k is shown in
Figure 8, to the left of the hatched regionaround k - 1
(cf. alsoSonnerup[1971],Figure2 ), sincethe crossing
condition(21a) can be expressed
alsoin termsof the
argument of the elliptic functions as k < 1.
The undisturbed slow z motion decoupled from the
fast z oscillationsis determinedby IA(z, hz) = const
Takinginto account(20c),onecanvisualizegraphically
cOnst.
the undisturbed slow • motion by meansof a phasedia-
gram (•,i).
In Figure 9 contoursi versusß are shown
for several
valuesI' ----IA/I 0 ----const,I0 = 8/(3• 3)
being a normalization value for IA. All contourscorrespondingto k • 1 are located inside an ellipse drawn
around the center of the coordinate system in Figure
9. The left semiellipse
•2 -3-z2 - •-4 (forz -< 0) is
the boundary of energeticallyallowedmotion, while the
Fig. 10. The invariant torus of the unperturbed ring-type
slowmotionin the regularphasespacedomainwith I t > 1.
Any point on this torus surface represents an actual phase
rightsemiellipse
•2 -3-z2 - •-4 (forz -> 0)corresponds
spacecoordinate•z,
to k -
1. The continuation
of the contours I t -
const
•, z}, the fourth phasespacecoordinate
(•) may be determinedfrom {z, z, •) by usingenergyconservation.Due to (19b) • = y+ const. The torustherefore
to the right of the ellipsewill be discussedlater in section 6. Figure 9 implies that trajectoriescorresponding showsalso the manifold of the ring-type trajectories in real
to I t > 1 never reachthe boundaryk - 1, while those physical space.
11,830
BOCHNER AND ZELENYh REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
From(23) and(24)it follows
that aslongas•'= const Hencefor ring-type trajectorieswith I' > 1 the time
> 1 the period •'z reachesa local maximum at a point
scales of the fast z oscillations and of the slow z motion
k = ki = 0.909,where
are well separatedand the motion may be thereforeadiabatic if • << 1, allowinga splitting off the rapid time
variation in a perturbation approach.
For trajectorieswith I' -• 1, however,whichturn at
- 4.•(•). •/fA
(•). •• • o.•.•• (24•)
•,•
ForI • • 1, i.e., for k+ • 1 (wherek+ corresponds
to k+ valuesverycloseto unity(cf. Figure9), onehasto
TAA,rni
n with the expression
for 7'A,rnaz
given
the positionof m•mum •), the period•A growsin an compare
unlimitedway. Usingan expansion
of K(k) neark - 1, by (24") andoneobtainsan additional
condition
for the
i.e.,fork• = •1 - k2 • 0 (cf.,forexample,
Yankeet al. turningpointk+ value,whichmustfulfilltheinequality
[1960]),onefindsfor I' • 1:
4-g
•A,ma==
•'ln [4]
•l_k 2
exp
{_2_•_•
}
(24") (k+)-•/1-(k+)2))4.
On the other hand, providedI t = constthe period of
the slow motion TAA can be found by an integration on
the slowtime scMe,using(24):
(26")
which may be fulfilled for • <( 1 but only for turning
pointsk+ (andtherefore
(I')) nottoocloseto unity.
Of coursethe frequencyseparationis only a necessary
conditionfor regularityin the phasespacedomainwhere
I' > 1 and the adiabatic conservation of the action inte-
•)
(•)
gral IA therein. As we havedemonstratedin section3,
evenfor clearly distinguishedfrequenciesan overlapping
of higher order nonlinearresonancelayerscan occur,destroyingthe KAM surfacesand thereforedisruptingthe
regularity of motion. As in the case of field reversals
with
a weak curvature, the regularity of motion and the
wherek] andk] arethek values
corresponding
to
the turning points of the slow motion, where i - 0, adiabaticity of IA may also break down in this case if
phase oscillationsnear the resonancesbetween the two
=4(•')•/••
(•')i?(•).
•- •'/i•(•)•
•/•
i.e.,2h•(z•) - g-4 (cf. Equation
(20)) andthereforecharacteristic cyclic motions in the system overlap. For
fA(k]) -- fA(k])- I' • 1 (cf.(23'))corresponding
to ring-typemotion(• (( 1 and I' > 1) the resonance
thelinea andabovein Figure8. The potentiMh•(z, I t)
condition
is
determiningthe z motion, has a minimum at the stable
fixed
point
oftheslow
oscillations
zf - (2.k•- 1)-g-2,
a•n•a•y [dh•(•)/d•][=•
- o.Th•.•uekf • O.9O9
is
,•. < •,• > = •,•,•(•')
(27')
where n is a positive integer and < •'A > is the z oscillation kequency averagedover the period TAA of the slow
z motion. The resonancecondition may be rewritten
k at the stablefixedpoint, (cf. B•chner [1986]). Near
the fixed point, h•(z) h• a nearlyparabolicshapeand in anMogywith (7) in terms of the phasegain of the
smallamplitudemotionsaroundzf are appro•mately fast oscillationsduring a full period of the slow motion,
linear. •om expression(25) one obtainsthe period which can be determinedto lowestorder, integrating on
of sraM1amplitude oscillationsaround the stable fixed
I
•
point,i.e., for I • • I•a • -- 1.16'
the slow time scale using the laws governingthe undisturbed
motion.
limTAA
--4•. kfß•1- k•• 6.74
) 2•
•o• 1 < •' < ••
•h• .au• •(•')a••,
(•')
•ut only
•
slightly,asI t decreases
andreaches
a minimumvaluefor
It •
1:
TAA,min
= TAA[I'-•l = k+--,1
lim TAA----2•r
i](•).• -[z•/i•(•)]
4/•
•A = 2 •/•A wastakenfrom (24).
(25") where
The action integral IA is not an exact constantof the
motion. In secondorder its time rate of changecan be
The period(25") givesa goodestimateof the time a•i•a
f•om (2z) usingthe exactHamiltonian(20a) (in
norm•ized variables).This rate of changecanbe scaled
ingthatTAA= 2•rcorresponds
to 2•r-f•l in realtime. relativeto the phaseangleof the fast oscillations(which
Combining
(24) and(25") oneobtainsfrom•'A,maz< is canonicMlyconjugateto IA) usingd•A/dt - •A -scale of the slow z motion.
It is worthwhile
mention-
TAA,min< TAA,f the inequality
- •). •:(•)
.•/•
•/l•(•f)•(•f)
•r
ß•
2•/i•(•f)•(•f)
•. 0.644•f• (26')
(28)
In the limit • << 1, i.e., for a weaklydisturbedintegrable
BOCHNER AND ZELENYI:
REGULAR
AND CHAOTIC
CHARGED
PARTICLE
MOTION
11,831
systemwhere time scalesof the fast z oscillationsand
also the surfaceof the undisturbed motion in real physi-
the slower z motion are widely separated, one can use
the solution for z oscillations,decoupledfrom the z mo-
cal space{z, y, z}. To obtainPoincar•surfaceof section
plots for ring-type orbits it is appropriate to fix either
or (2) crossings
of
tionz(e) = 2 •f• hz. k. cn[(2/•r)ßK(k). el, wherecnis (1) the reversalplanez = 0 crossings
Jacobi's elliptic cosine.
Adiabaticity of the ring-type trajectoriesmeansthat
althoughIA oscillateswith an amplitudeof the orderof
the small parameter •½,its averageover periods of the
fast oscillationis constant. Using as in the case of p
adiabaticity(for deriving(11)) th• methodproposed
by
the surface• - 0, where ]zI - Zma•. To draw the correspondingPoincar• plots we integrated the equations
of motion in the parabolic field reversal numerically by
means of an extrapolation scheme Stoer and Bulirsch
[1973],usingenergyconservation
for checkingthe accuracy(formoredetailsseeB•chner[1986]).At anycrossHastieet al. [1969](cf. Birmingham[1984];Chirikov ing of the phasetrajectory with the surfacez - 0 (first
phase
[1984]),one can extract a finite nonadiabaticportion type) or • - 0 (secondtype) the corresponding
AIr= f(dlt/d•)d•. Thesecalculations
werecarriedout spacecoordinates{z•,k•} are fixedand plotted. Again
by deformingthe integralinto the complex• plane. The it is worth mentioningthat from (19b) y- k+ const.
resultingnonadiabaticchangeof IA over a period TAA This allows us to interpret the surface of section plots
or the •o• motionbea•ve• • e•p{•} [cr.,ro• e•mp•,
alsoin termsof the spatialcoordinates{z•, y• + const}
Chirikov[1984]). It stemsmainlyfrom the slowz os- of the surfaceof sectioncrossings,i.e., as a trace of the
cillations
neartheturning
pointk+, where
•z - r• •
is minimum. It Mso depends on •0, the value of the
slow component of motion.
Poincardplotsof the first kind (z - 0 crossings)
can
phase
at theturningpointwherek = k+ (fordetails,
cf. only show crossingtrajectories. For the case of twodimensional field reversals they were first obtained by
(aZ2)). Hence,bothdependencies,
a•'(•0)•nd a•(•')
together describethe coupling between the two oscillatory degreesof freedom, the fast z oscillationsand the
slowz motion. This can be usedfor derivinga mapping
Bgchner[1984,1986]and ChenandPalmadesso
[1986],
resonances
definedby (27). Similarto the caseof the p
Figures11 and 12) for severalinitial conditions.Figures
who numerically solved the complete equations of motion for a number of initial conditions, displaying the
crossing
points graphically. The secondkind of Poincard
similarto that givenby (12) for thep adiabaticity,which
can be used to anMyze the phase oscillationsaround the plotsarefirst shownherein thispaper(see,for example,
adiabaticity the stochasticlayers around the nonlinear
9 allows us to interpret the results of these calculations
term adiabatic conservationof IA and to a deterministic chaosof the motion beginningwith a thresholdvalue
slowmotion. By comparingwith Figures9 the first kind
on quMitative aspectsof the break down of the IA adiabaticity, which can also be demonstratedby means of
the powerfulgraphicMmethod of surfaceof sectionplot-
jectories exist and they are adiabatic of the ring type.
Together with Figures 9 the Poincar• plots also clearly
demonstratethat the break down of regularity and adia-
resonances
(27) overlapbeginningwith somefinite per- by comparingthe plottedpoints{z, k} with the related
turbation n > 0. This leads to a breakdownof the long levelsof I • - const,whichcharacterizethe undisturbed
of Poincar•plotsof B•chner [1984,1986]and Chenand
n•it. Leavingout the detailedcMculationsand referring Palmadesso[1986]and also the secondkind of plots,
to (BZ2) andto their analogywith the breakdown
of the Figures11 and 12 (for •½- 0.1 and 0.316,respectively),
p adiabaticity(cf. section3), we will concentrate
here herein this paper,we verifythat for I • > I regulartra-
baricitysetsin with increasing
•½for I • • I andcontinues
to largerand largerI • as •½grows.•½-• I leadsfinally
effectivetool providinga vivid proof of regularity(see, to globalchaosfor all valuesI • > 1. Additionallyin
[1986],one
for example,Hdnonand Heilis [1964],Lichtenberg
and Figures2, 3, and 5 of Chenand Palmadesso
Lieherman[1983]). The idea is the following:In sys- can also identify the appearance of the nonlinear resoting. This methodw• proposed
by Poincard[1892]and
after the introduction of powerful computersbecamean
tems of two oscillatorydegreesof freedomsolutionsare
completely describedby a set of four coordinatesin a
nancesprecedingthe onset of chaos. These resonances
motion acts as a constrMnt, which diminishes by one
the number of dimensions of the manifold of the phase
trajectories. In a two-dimensional case two indepen-
2, 3, and 5 correspond to •½= 0.18, 0.316 and 0.517,
respectively.
On the other hand, our Figures 9 here in this paper
together with the Poincar• plots imply that only a small
leadto pearlsand islandsof stabilityof ordern • •½-•
alongthe linesI • - constof the ring-typetrajectories.
four-dimensionM
phasespace,in our caseby {z, •, z, •}
[1986]parameterH
Any independent
(i.e., beingin invoCa- Note that Chenand Palmadesso's
tion with eachother) integralor adiabaticinvariantof is related
to our•½by •½- 1/•r• H; i.e.,theirFigures
dent integrMsor invariantsof motion lead to phasetrajectories which are located on a two-dimensionalmanifold. Its intersection with an appropriate plane is a
one-dimensionalline. Hence, crossingswith the surface
of sectionlying Mong a line confirmregularity of trajectories.
For ring-type adiabatic trajectories the deformedtorus shownin Figure 10, which is determined by energy
conservation h - const and IA -- const, is that twodimensional
manifold
of the undisturbed
motion
in the
We epet that a,e to (9b)it
fraction of all possibletrajectories are ring-type orbits
with I • > 1. To understandtheir relativeimportance
we will evaluate their specificweight among all possible trajectories, which in a two-dimensional parabolic
field reversal necessarilycross the neutral plane. It is
appropriateto calculatethe relative weightof the phase
spacedomainI • > 1, expressing
I • in termsof the equatorial velocity direction. We choosea spherical velocity
space coordinate system with the polar axis directed
along k, azimuthal angleft0 and polar angle ct0. In this
11,832
BOCHNERAND ZELENYI:REGULARAND CHAOTICCHARGEDPARTICLEMOTION
2.X
Fig. 11. A surface
of section
or Poincar•
plot{z,, k,} of thesecond
kind,i.e.,forsubsequent
crossings
of
thesurface
• = 0 plane
forthecase
n = 0.1.Theleftlimiting
semicircle
z2+ k2_ n-4 isduetoenergy
conservation,
the right is the line of uncertaintyk = 1, wherethe topologyof the trajectorieschanges
fromcrossing
thez- O-plane
(region
(A)) tononcrossing
(region
(B)).ForI' < 1 chaotic
solutions
can
be foundandfor I t • 1 regularones.
frame the dimensionless
velocity componentsare given
and noncrossing
(k > 1) orbits. The resultsof numeri-
by • - n-2.cosa0; • - n-2.sina0.sinri0. Hence, cal computationsof the equationsof motion for several
l'(c•0,f•0)
-- V'sinc•03'
fA[sin(f•0/2)].
If weassume
an
different initial
conditions
are shown for •-
0.316 and
isotropicvelocityspacedistributionat z - 0, the por-
• - 0.1 in Figures 11 and 12, respectively. The loci
integratingover s0 by
(right)semiellipse
in Figures12. Onecannowcompare
the Poincar•plotsFigures11 and 12 with thelinesI t =
constin Figures9 and seesthat for I t • 1 any trajec-
k - 1, i.e.,wherez2+ &2_ •-4, areindicated
in
tionofthephase
space
AF withI t >-1 isgiven
after where
the Figuresas a (right) semicircle
in Figures11 and a
1
4•r
0.1045
(29)
tory reachesthe right semicirclein Figures11 and 12,
crosses
it, and continuesto movein the segmentB with
noncrossing
orbits. There is no energeticlimitation on
motion in the positive x-direction; parabolicfield lines
are "open". Nevertheless,particles are mirrored back
after reachingsome turning point zB which is larger
wherekl = 0.8 is definedby fA(kl) = 1. Hencein the
caseof an isotropicvelocityspacedistributionat z - 0
onlyaboutonetenthof the particlesundergo
ring-type the smallerI t is (cf. Figures9, 11 and 12). Thismeans
motionirrespective
of energysolongas n is small.
that, as in the adiabatic case • ) 1, a parabolicfield
Summingup the resultsof this sectiononemay say reversalseemsto act as a magnetic trap alsofor • • 1.
thatring-typeorbitsin sharplycurvedtwo-dimensionalAfter being mirrored in the negativex-direction, the
field reversalsare adiabaticas long as • •
1 with IA
trajectoriesmovebackwardsand againreachthe bound-
beingthe firstorderadiabaticinvariantof motion.As ary k - 1. There their z oscillationsturn to the crossing
n -• 1 thering-typetrajectories,
firstthosewithI t • 1, type A. The particlesmovefor sometime in the crossing
becomechaoticbeginningwith a criticalvaluenc,.it< 1. regimeA, reachagain the separatrixk - 1, changeover
<
For n • 1 all trajectorieswith I t > 1 becomechaotic.
After the onset of chaos and loss of the correlation be-
tweenthephases
•+ at consecutive
turnings
at k+ the
motion can be describedby a diffusionlaw similar to
1
(18). Moredetailsaboutthephysical
consequences
of
ß ..... ............ ........
z = 0.316
..
thering-typemotionwill be givenin (BZ2).
In section 6 we consider the behavior of the major
portionof the trappedparticlepopulation
in sharply
curvedtaillike two-dimensionalmagneticfieldreversals.
30 •2 x
6. CUCUMBER-TYPE QUASI- ADIABATICMOTION
(I' < 1)
IN SHARPLY
CURVED
FIELDREVERSALS
(•; < 1)
Lookingat Figures9 oneseesthat the contoursI t =
const• 1 reachthe boundaryk- 1 (the right semiellipsein Figures9) and continuein the area B of non- Fig. 12. A Poincar•plot J:a:,•,
•} of the secondkind for the
crossingfast z oscillationswith k • 1. Poincar• plots
case• = 0.316. The left and right semiellipseshave the same
of the secondkind (• - 0 crossings)
allowus to ana- meaning as the correspondingsemicirclesin Figure 11. Only
remainfor I t =~// maz= 1.16
lyze simultaneously
in one plot both crossing
(k • 1) a fewregularsolutions
BOCHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
z
11,833
where
zmin
-- V•' •z --V/2hz
(z)and
Zmaa•
has
al-
(B)2
readybeendefinedin section5. The integral(30) can
be evaluatedin termsof completeelliptic integrals:
•--y+const
x
4.
IB -- 3-•
(B}1
-
(31t)
.
+
- 1)k. E(1/k)
with k the sameas introduced
in section5 but > 1 for
non-crossing
trajectories.The functionlB(k) is shown
in Figures 8, to the right of the hatched area around
k = 1. It decreases
for largek (largerthanthemaximum
k = 3 shown
in Figures
8) as3/16•rk
-1. ForIB • const
Fig. 13. Threeinvariant
halftoriof theunperturbed
quasi- from the form of the functionlB(k) it followsthat the
adiabatic
cucumber-type
motion(ff < 1) in segments
(A), motion is limited in the positivex-direction. In Figure
(B)I and(B)2, whereIt-- const
Thetoriareseparated
by
8 horizontal
the near-separatrixregions(C), wherethe normalizedaction
theslowmotionforI t = const• 1. SettingI t = 2'IB/I0
integralI t changes
nonadiabatically
(equation
(42)). As in
the caseof the invariant torus in Figure 10 the tori of the cucumber motion show the manifold of both phase trajectories
in the {z, i, z} spaceaswellasrealspace{z, St,z}.
lines b and c demonstrate
the limitation
of
one finds the related turning points of the slow motion
kA and kB as
=
= t' = 2.
= .fe(r)
to the noncrossing
regimeB) and repeat thesecycles The physical backgroundfor this matching will be exagain and again. Looking at the Poincar• plots Figures plainedbelow(cf. (40)).
11 and 12, whichdue to (19) alsorepresentthe global
As in the caseof the crossingmotion A, we determine
propertiesof the z-averagedmotion projected onto the
z-y-plane, trajectoriesare seenelongatedin the positive
x-direction. With someimagination one can agree that
they look like cucumbers. That is the reasonwe called
them, for brevity, cucumber-trajectories,in contrast to
the number-eight-like guiding center drift in the adia-
baticcase• ) 1 (cf. section3.) and to the ring-type
adiabaticmotionfor • <<1 but I t > 1 (cf. section5.).
For a better understanding of the meaning of the
boundary k - 1, let us take another look at Figures
7 and the line (C), which corresponds
to k -
1 or
hz(Z)-- «z2. Onesees
.thatline(C) represents
the
the dependenceof the z oscillation period rB on the
parameters x and hz:
dz_ 2.
7'B
-- T
,•Zm
in
2v/-j
4
(33)
Poincar•plots of the secondkind in Figure 11 and 12
andtrajectorycalculations
imply that the phasetrajectories of "cucumber" orbits lie on invariant tori schemat-
boundary
between
thetopologically
different
crossing
A icallyshownin Figure13. Reachingthe edgesof the tori
and noncrossingB motions. Before turning to an in-
vestigationof this transition(in section6.2.), we will
first, in section 6.1., take a closerlook at the motion in
the noncrossingregime B. In section 6.3. we will then
investigatethe regularity of motion as a whole on "cucumber"orbits.
6.1.
with k -• 1, the z oscillationperiod increasesgreatly.
An expansion
of K(1/k) near k - 1 leadsto the followingasymptoticexpression
for rB (cf., for example,
•,•e eZ•l. [19•0]):
2.• [4]
Motion in the Noncrossing Regime B
(34)
Figures11 and 12 demonstratethat orbits in the nonHenceboth z oscillation
periods,rA (cf. (24")) and
crossing
regimeB (for k > 1) againseemto be rather rB diverge logarithmically near k - 1. A phase curve
regular. As in segment A, for • <• 1 the z oscillations proceed considerably faster then the slow z motion. Henceto lowestorder the action integral IB would
appear to be an invariant of the weakly disturbed slow
motion, at least in the noncrossingregion B. Treating
thereforex and hz again as parameters,lB is givenby
along which the oscillationperiod tends to infinity and
which separatesphase spaceregionsof different topology is called a separatrix. C is therefore the separatrix
betweenregionsA and B, i.e., betweenthosetrajectories
which crossand those which do not crossthe symmetry
plane z = 0.
To check under what circumstances rz and the time
IB(z,hz)- •1/ idz
_2_
•r ,•Zmi n
(30)
scalesof the slowmotionare well separatedin segment
B outside the close vicinity of the separatrix, we determine the time interval which a particle spendsin the
noncrossing
segmentB of a cucumber-typeorbit, setting
I t - const(theundisturbed
slowcomponent
ofmotion):
11,834
BOCHNERAND ZELENYI: REGULARAND CHAOTIC CHARGEDPARTICLE MOTION
Near the separatrix(C), however,the interactionbetweenthe z motionand the z oscillationsbecomesstrong
independently
of the valuesof n and I t. The resulting
"nonadiabatic
layer" aroundthe separatrix(C) is indicated in Figure 13 by separating the schematicallyinvariant tori belongingto the crossingand the noncrossing regimesfrom each other and hatching the edgesof
B1 and B2. In the next section we will address the mo-
tion near the separatrix specifically.
6.2.
Fig. 14. The schematicphasetrajectory{z, •} (solidline) of
a separatrixtraversal.For three momentsof time (•2, •3, •4)
the instantaneous solutions of the near-separatrix equation
=
< 0,
for Az = const• 0, and the dashedline denoted(C) the
scparatrixitself (Az = const= 0).
Motion Near the Separatrix C
Fromexpression
(23t) andIA = constonecanasymptoticallyestimatethe positionof the separatrix(k - 1)
as•,ep- (It)2/3'n-2. Theresulting
locations
ofdifferent separatrixtraversalsare shownin Figure 9 as the
right semiellipsearound the origin of the coordinatesystem. This line can be called a line of uncertainty by rea-
sons
wewillexplain
now.Makinguseofh•,,ep--- 1 2
which follows from the condition k - 1, one obtains an
approximate equation of motion near the separatrix:
TB -- 2
-Of••B
,sepdz
•
(35)
1•
•/4Zsep
Zselo
Z2
;•]/•--•1
•"-]2,
- z2ß2(4z,,p
--32)
ßz2'
Within the rangeI t • 1 the followingapproximateexpressionsfor TB may be found:
(39)
whereA• - •- •sep representssmalldeviationsfrom
the separatrixposition(A• • •sep). To illustratethe
motion near the separatrixgiven by (39) we showin
Figure 14 the phasetrajectories of three topologically
different
3 7r2
solutions obtained
for three fixed values of
for A• - const< 0, a crossing
motion(A) shownby the
=
const
=
0)
illustrated
by
the
dashed
line
denoted
C
So, for I t > 1 the x-motion in segmentB is obviously
and
two
noncrossing
trajectories
with
A•
const
•
0
slowerthan the z oscillationperiod,TB, givenby (33).
TB"'8(i,)2
(36)dotted line denotedA; the separatrixitself (for A•
indicated by the dot-dash lines denoted B1 and B2.
Near the separatrix,z oscillationsare no longerfaster
crossingpart of cucumberorbits. The time particles
The same conclusioncan be reachedconcerningthe
on the same
spendin the crossingregimeA betweentwo separatrix than the ß motion:A•(t) thereforechanges
timescaleasz(t) or evenfaster.Hencenoneof thesolu-
traversesis given by
TA -- 2
-a•,sep
dz
•
(37)
•x A
=4(I1)2/3
fkl •.K(k)
dk
tions for A• - const will be fully accomplishedduring
a period •z. In the phase diagram Figure 14 we try to
indicate this by means of a schematicnear-separatrix
phase trajectory, the solid line. This trajectory starts
at a time t• in the crossingregimeA. The dottedline is
the crossing
solutionof (39) at the timet2 corresponding
to A•(t2) • 0. The trajectoryevolves
until it crosses
at
t3 the separatrixgivenby (39) for A•(t3) = 0. •rther,
Againonecanfindan asymptotic
expression
for I t • 1 it continuesin the noncrossing
regime(B2). A particof the maximum time interval, which a particle spends ular solutionof equation(39) in this regionis shown
between two separatrix traverses:
for the momentt4 (the right dot-dashline obtainedfor
TA----z'- 2•,¾-•
j
•om the definitions(22) and (30) of the actionin-
tegrals IA and IB it follows that they are equal to the
(38)areas
encircledby the correspondingphasetrajectories
Fromequations
(24), (33), (34), (36)and (38) onefinds {z, J}, in Figure 14. Hence,if one neglectsthe interthat outside the immediate vicinity of the separatrix
the time scale of the motion in segmentA as well as
action of the z motion with the z oscillations,any sep-
aratrix traversalleadsat least to a doublingB • A]
that in segment
B remains(at leastfor I • •- 1) much or halving A • B of the numerical value of the action
larger than the z oscillationperiod only if • • 1. Hence integral. This is the reasonfor matchingIA and IB to
within regime A as well as within B1 and B2 phase- lowest order m
and spatial-trajectoriesof the unperturbedslowmotion
for • • 1 really lie on invariant half-tori like those
schematicallyshownin Figure 13.
o, 2.lB I'-_ IA
(40)
BOCHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
1
11,835
O.03
'ii
......................
Fig. 15 A Poincar•plot •zs,•s• of the secondkind for severalparticleswith • = 0.03. The limiting
(solid)
semicircle
isforz < 0theenergetic
boundary
z24-•2 _ «•-4 asinFigure
11andforz > 0 the
line of uncertainty where the separatrix is traversed.
as $onnerup[1971]did. Equation(40) indicatesthat
the relationship(32) is valid to lowestorder.
wherethe uppersign(minus)is validfor traversals
from
A to B and the lowersign(plus)for traversalsfromB to
On the other hand a separatrix traversal is an essentiallynonadiabaticprocesscharacterizedby a strong
interaction between the two degreesof freedom. After
our experiencewith nonadiabaticjumps of the magnetic
moment due to the the weak field region near z - 0 in
A. One seesthat in contrast to the nonadiabaticchanges
section 3 and of I t due to the slow z oscillations near
logarithmicallyon the phaseangle at the separatrixin
A/zpcexp•--•2• in the case• >> 1 andin contrast
to
AIt in the case• (( 1 for I t > 1 thejumpsAIt during
a separatrixtraversaldependonly linearly on the small
parameter
•. On the otherhand,AIt(t•ep) depends
theturningpointassociated
with k+ in section4 wecan contrastto the dependency
A/z(•b0). The logarithmic
expectthat I t alsowill changenonadiabatically
during dependency
allowsratherlarge[A/t[ for •sep--•
separatrix traversals. The related jump AI must de- and •ep -• 3•r/2. Thesephases
are relatedwith sepapendon the velocityof separatrixtraversal•zepand on ratfix traversalscloseto {z,i} = {0, 0}. Consequently,
the site of traversing the separatrix curve. The closer separatrix traversescausea strong effect on the regularity of "cucumber"trajectories rather than the small
z motion becomes.It stopscompletelyat the singular- nonadiabatic
changes
A/t and/ki t in theadiabaticcases.
ity {0, 0}. Making useof the near-separatrixequation
The Poincarl plots of the secondkind shownin Figure
of motion(39) and settingA•(t) • •,cp' t or evalu- 11, 12, 15 and 16, which were numerically calculated for
atingAI'= f(dI'/d6)d6 (cf. (28)) directlyoverthe severalinitial conditionsand •½- 0.1, 0.3, 0.03 and 0.01,
motion acrossthe separatrix, one finds the followingex- respectivelyconfirmthis theoreticalfinding. They show
pression
for the jump AIt duringa separatrixtraversal how with increasing•½the shift along the line of uncer(cf. •so Timofee•[1978],Neishtadt[1986],Car• a,d tainty of the z averagedmotion at any separatrixtraver-
it is to the saddlepoint {z,•} - {0, 0}, the slowerthe
Skodje[1988]):
sal (the right semicircles
and semiellipses
in Figures11,
12, 15, 16) becomes
larger. The shiftfrom radial along
the k = 1 curves, the so called line of uncertainty, is a
2
whichdependson the phaseangle8•epat the moment
of separatrixtraversal(8 - 0 refersto the upperturning
point z,•a• and • - •r/2, 3•r/2 to the z=0-plane crossing) and the velocityof the slowmotionat the moment
of separatrix traversal. The z velocity at the separatrix
can be obtainedasymptoticallyfrom the equationI t =
const determining the undisturbedmotion:
V/z_
;•zep
--q•/•-4_zzep
2--q-
measureof AI t, as onecan seeby comparingthe lines
I t - constin Figure9. The Poincarlplots11, 12, 15and
16alsoshowthat, aspredictedby expressions
(42), both
positiveand negativejumpsof AIt occur,controlledby
the phase•zep of the z oscillations
at the momentof
separatrixtraversal.In anotherpaper(J. Biichnerand
L. M. Zelenyi, Separatrix tentacle effect and the formation of the hot ion flow in the plasma sheet boundary
layer,submittedto Geoph!ls.Res. œett.,1989)and also
in (BZ2) we discussthe physicalconsequences
of separatfix traversalson trapping in the regionnear the field
(41") reversal.Herein the next sectionwe investigatetheir in-
where the sign depends on whether the separatrix is
fluenceon the trapped cucumber-typemotion and give
a complete picture of cucumber orbits.
6.3.
crossed
from
A
to
B
Iplus)
or
vice
vets,
(minus).
Putfinally,
ting expressions
(41t and (41") togetheroneobtains
Complete Cucumber-TypeMotion A + B + C
ThePoincarlplotsforsmaller
n • 0.1 (Figures
11,
15,16)demonstrate
that trajectories
withI t < 1, which
AIt•--q:•ß•. 1- (it)4/3.
In2lcosopl(42)
crossthe separatrix, behaverather regularlyinsidesegments A and B, where IA and IB act as adiabatic in-
11,836
B•ICHNERAND ZELENYhREGULARAND CHAOTIC CHARGEDPARTICLEMOTION
•z..• 1
in the segments
A andB and thejumpsof It(8•ep)at
ae-O.Ol
separatrixtraversesone finally obtainsa mappingdescribingthe whole "cucumber"motion. One cucumber
cycle containstwo separatrix traversals, one from A to
B and one from B to A. We will advancein the cy-
clecounting
by « afteranyseparatrix
traversal,
i.e.,
by 1 after a complete cucumber cycle. We start the
mapping
with{O,elo,N,
I•v}, validat the(N + 1)stseparatrix traversal from A to B. As a first step one has
•o½•lcul•• v•u•of•v+(•/•)= •v +
using
expression
(42)withthenegative
sign.
I•+(1/2)
Fig. 16. A Poincar• plot •zs,•s} of the secondkind for
governsthe motion in one of the segmentsB after the
separatrixtraversMand determinesthereforethe phase
&O(I•+(•/•)).
This
canbeobtMned
byevaluating
several
initialI t and•½= 0.01.Theleft-limiting
semiellipsegain
(for z < 0) is due to energyconservation;
the right is the theintegral(44"). The second
jump of I t, whichoccurs
line of uncertainty.
duringthe sepalaffixt•ave•salf•om B backto segment
A canbe determined
using(42) (positivesign)with the
variants of motion. Hence the irregularity of the whole argument
8sev,N+(•/2
) - 8sev,N
+ A8(I•+(•/2)).
This
trajectory seemsto stem from the nonadiabaticsepara- yields for the action integrM governingthe motion in
trix traversals,whereI t undergoes
finitejumps. Pur-
segment
AI•+•
N+(i/2)
+ AIt(OseP,N+
•)' Thecon-
suing this idea we will describethe whole "cucumber" sequent
phasegMnin the segment
A, AON+i, calculated
motion by a mapping of the action angle coordinates
fromexpression
(44t) setting
I t - I•+•, completes
the
betweenseparatrixtraversals(cf. B•chner and gelchili (N + 1)st cucumbercycleof slowmotion. Noticethat
[1986]),i.e., following
the changeof I t on theslowtime the normalizedactionintegralI t w• determinedto take
scaleof the z motion instead of the mappingsobtained
automaticallyinto accountthe halvingand doublingof
before on the fast time scale between subsequenttra- I during separatrix traversMs. Hence one obtains the
versesof the z - 0 or • - 0 planes. The analysis of
following
mapping(cf. B•chnerand2elen•i[1986]):
suchmappingswould give in fact a rather good means
of investigatingthe cucumber-likeparticle motion over
many "cucumber"-periodsand enable the further analytical investigationof the laws governingits irregular,
i.e., chaotic,dynamics. To effect the mapping it is nec-
essary
to determine
in additionto thechanges
AIt(8sep)
•+i - • + •'(0,•,•)
+
- ' +•'(0,•,•+•)
(4s•)
(4s)
(4s•)
the phasegain in segmentsA and B. We do this, taking
into accountthat the phase changesconsiderablyonly
outside the closevicinity of the separatrix, i.e., we con- To study the problem of whether the cucumber-type
siderthe undisturbed
motionwith x andhz(z) changing motion is regular or chaotic we have numericMly an-
on the slowtime scaleTA q-TB. Hence,as in (35) and Myzedthe mapping(45), usingthe exactintegr• ex(37), we integrateon the slowtime scaleandfind
v•e•ion• (44',44")fo• •0(•')•nd (42)fo• •'(•',0,•).
Figures
17and18showtheprojections
of
AOA -- 2
ß dz
•'• •A(Z)
• •A
(43') into the intervMof m•n values0 <
-- Osep,
N • 2• for
•SB -- 2
•sep •(z)
Figure17 for • = 0.03andFigure18 for
(43") {Osep,o,I•),
-- dz
100 iterations, starting from 90 different initiM vMues
• = 0.003. One seesthat even for • << 1 and over many
where ZA and zB are the coordinatesof the turning
points in segmentsA and B, respectively. Using ex-
pressions
(24) and (33) and taking into accountthat
(here100) "cucumber"
cyclesthe motiondoesnot becomeregular,• onewronglycouldsuggestafter looking
at Figure 16, whichshowed,however,only singlecucum-
•j: 2•/•j (j: A, B) in segments
A andB onemay ber cycles.
Recall that in the adiabatic c•e • > 1 with the derewriteA•j • follows:
cre•e of the smMlparameter•-• chaosdisappeared
>
•OA -- 2•-(44') approfimatelyfor • - 3 (cf. section3). Only the ap-
It••
A•s - 4•-It••
k dk
k dk
pearanceof regularityMlowedan adiabatictreatmentof
the whole motion. Cucumber-type trajectories, on the
regularandrem•n chaotic
. (44") otherhand,do not become
overa broad
parameter
range{•, It), because
duetoits
logarithmic
dependency
onOIAI t] maybelargeevenfor
wherekA and kB are the k valuesat the turning points small •. This leadsto a fast lossof the ph•e correlation
givenby equation(32) in the lowestorderappro•ma- betweensubsequentseparatrix traversMs.
tion of an unperturbed and independent motion in
This is an important point for further simplificationof
the anMysisbecauseit Mlowsa "randomphase"approxandz. Combining
thephaseg•ns A•A (I') and
B0CHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
onemay evaluatethe mean value < ZXI• > and the dis-
8e:•03
2•
11,837
persion
< (ZXI•)
2 > byintegrating
over8•ep.Assuming
equalprobabilityfor all 8•epin the interval[0,2•r] one
,..-..,.....
:,..•....,.:..,-.
•.•.'•.:..::::,'?.
]
obtains the phase-averagedvalues
ß :.•..... %,...... ;
ß ?.. v':..I .......
..
ß:'. '.':'..•.,•' ,": ".'..::•
'::.'.'4;•'
.)"'"?:.':.
: $•'......:',
.. o.•...: ': :.'.:...":..'
:..,• '...• :'•
....
.' ". '
.'. ß" ß ". :.. I'
' '
'.•,'.
ß..... v...:.-...:..".:f..
,'.,:'..'
' .' ,..."'....
i..,
,...
.•. •._.• . ..
. ...,
0sep
.
.
I
:.l..... ..::,::..
I
--•:•ß•- 1--(I,)4/3.
•1
:.:•,:•;.".':•.-...:':.
:•';':'•::'::..:.'::,.,:'...,
t ,:..... .
ß ...•'•.
.,,•.•;':'.%'::
, ß :':: ß
',:.
. . ..:..,"
ß' ;::?'.
:.":'.'::'..<
:'7,i:?:"..:'{...':.::..'•
':: I
2•r
.... ß..x.'.-.,?...:' ..: ß..'•.'•-:...'• ::.-•"•.?.'...'..'
ß:"':'::"
"'•'"'"'::
'•'""•'
:"•'
:"I
ß .: ß .%'•:
•.'...:,..
•
...../
ß • ..• ..,'
= 0
ß
ß..::.• ;½:.....:P:.":
'5.:-";'.'
::'i ;/.A:.".::::
...•' '..,•'.:fi....'.'..•",..'
.':.:...:.:.::..
'! '..'•."..;3.:...'.:.:
.;":'.'..?'
." 5
'.•.'?:,,%...
o
1:
0.15 0.3
In2[cosdo•pldd
ß:.".• "' ß '.•' ".2: .... ß
I
0.5
0.7
0.9
(48)
< (AI')
= •.
ß[1-(I')4/3].
•I f02'
[In2[cos•ep[]
2d•
9•c2
I'
Fig. 17. The mapping(45) calculatedfor 90 differentini-
=
tial action angle coordinates of separatrix traversal values
3 •r2
16
ß•:. [I- (I,)4/a]
(49)
{I•,Ose,.,O}
for 100iterations
andfor • - 0.03. Themappingin•catesdeterministic
chaos.
To verify these results, i.e., to verify the ergodic hyv
.•
pothesisfor the cucumber-type motion, it is appropriimation. The Poincar• plots in Figures 11 and 12 show ate to calculatenumericallythe meanvalues( AIt )N
andto check
that the tendencyto chaosis maximum for small values and< (AI') >N for N (>>l) iterations
whether
they
tend
for
N
-•
oo
to
the
phase-averaged
I •. We will use this point to illustrate the fast lossof
correlationbetweensubsequent
valuesof 8•ep,evaluat- valuesgivenby (48) and (49). Henceonehasto test the
ing A• in thelimit of smallI • • 1. In thislimit onecan validity of
use the asymptotic expressionsfor fA and fB discussed
def
in sections5 and 6.1 to obtain from (32) approximate
valueskA and kB for I • • 1:
J4.
-__/'
kAI•,•:•,,,
-- ¾
3•r
N
1
< Aft>/V'-- /V•oo
lim • ' Z AI}
(50)
•=1
(46')
3•r
kBl•,<l
---16.I'
(46")
For smallI t (((1), i.e. for smallkA(((1) andlarge
kB()) 1), one thereforecan estimatethe integralsin
equations(44) and obtainsthe followingapproximate
expressionsfor the phase gains in regionsA and B'
where
ZXI•hastobedetermined
forsubsequent
steps
of
the mapping(45). We checked
the validityof (50) and
A•
A--•
• F(1/4)
F(3/4)
16I']~2.39
(47')
•1 Iv/9•
(51) for severalvaluesofn < 1, includingthe runsshown
in Figure 17 and 18 and foundfor n << 1 goodagreement
8o.'
betweencycle-, or time-, averagedand phase-averaged
The asymptotic
expression
(47")clearlyshows
a strong
dependencyof A•B on I t with the derivativeincreas-
ingas (It)-4 for I t -• 0. ThephasegainASA,onthe
other hand, dependsonly weaklyon I'. Togetherwith
the finite jumps aI t this mixes the phasesof subsequent separatix traversalsvery quickly,evenduring one
cucumber cycle. Hence the separatrix traversalscause
deterministic chaosand ergodicity of the cucumbertrajectories.
On the other hand Figure 18 showsthat for small •
2•
8sep
the randomwalk of I t is limited in phasespace. This
means that its diffusionin phasespacehas a finite characteristic time. We will now estimate quantitative mea-
suresfor the diffusionprocess
in I t space.Deterministic
chaosimplies ergodicity. Time averagingtherefore can
be replacedby averagingover phases,for example,over
8,ep assuminga uniform statisticaldistributionof all
0.3
0.5
0.7
Fig. 18. The mapping(45) calculatedfor 90 differentinitial
values
of{I•, 0•ep,0}
for100iterations
andfor• = 0.003.
A
limited random walk of I t can be seen.
possible
values
0 < 8,ep• 2•r.Withthisassumption
11,838
BOCHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
l':sin•/•Jdo
ira[sin'?•]
1.0
,
[/•) J'-odioboticity
go
ß ill)o.1
(54)
0.7
V/
-
If inequality(54) is violated,the diffusionapproachcan
I __--'
i
J'-quasi-adiQbaficify
not be applied and the chaosis strong. The criterion
(54) indicatesthat a diffusionapproachis valid for a
ß
I•eak
_•
0.5
broad parameter range of interest, including ions and
energeticelectronsin Earth's magnetotail.
chaos)
{S} Ol'I'
ß
0.1
0.5
o
o
In the limit (54), the cucumber-typeorbitsmay be
considered
partiallyregular,andadiabaticin regions
A
)adiaba-
and B outside the separatrix but undergoinga diffusion between invariant half tori of the types shown in
/13) • ticity
/ •tron_•
Figure 13 upon traversalsof the separatrixin •z,•
ChQOS =
1
;•
phasespace. We proposeto call such behavior "quasiadiabatic". Due to the chaotic long term behavior of
lO
Fig. 19. The different types of trapped chargedparticle motion in a parabolic field reversal as they depend on the • parameter and on the complementary phase space coordinates,
the quasi-adiabaticinvariantI I evenfor • •
I quasi-
adiabaticity differsfrom usualadiabaticity. It is closeto
the conceptof an "adiabaticityon average"suggested
by
Janicke[1975].
givenin termsof I I and/•l for • • 1 and• • 1 respectively.
I
/• = sin2
•0 •ndI'(•0,/30)= sinS/2(•0)f• (sin(fi0/2))
,
where c•0,/30and 60 are velocity pitch anglesas definedin
the paper at the moment of the z = 0 reversal plane crossings.
7. SUMMARY
We have shown that the trapped motion of charged
particlesin a sharply curved magneticfield reversalafter the breakdown of adiabaticity of the magnetic moment becomes first stochastic
in the sense of determin-
meanvalues< A/' • and • (A/•) 2 •. Thisconfirms istic chaos, while a further increase of the curvature
leads in one part of velocity space to the appearance
of a new type of adiabaticity and in another part to a
worthwhileto mentionthat the proof of ergodicityneeds weakly chaotic, quasi-adiabatic type of motion. Partiin considerationof the evolution of the trajectory over cle dynamics is shown to be controlled mainly by two
many cucumber cycles. The reasonis the regular be- parameters which determine also the laws of the chaotic
havior of the trajectory outside the near vicinity of the particle motion. The first parameter, •, is determined
line of uncertainty and outsidethe separatrix, while the by the particle's total velocity, i.e., its maximum possichaosis created due to the nonadiabatic effect of sepa- ble Larmor radius Prna=and by the minimum curvature
the correctnessof applying the ergodichypothesisand
the dispersion(49) obtainedby phaseaveraging.It is
ratfix traversals.Usingthe mapping(45) any cucum- radiusof themagnetic
fieldlines:• - V/Rrnin/Prna=
=
ber cycle correspondsonly to two stepsin calculation.
That is why we confirmed ergodicity while Chet• and
(Euion (S)).
(1')
,
B,/Bo
' V/L/po- 'v/B,/BoßLip, (Equation
Palmadesso,
[1986]stressedmorethe regularcharacter
of motion, which they found over the limited number
of cycleswhich they were able to follow, integrating the
complete differential equation of motion. For practical
purposesit is appropriate to derive from the dispersion
per cycle(49) the diffusioncoefficient
-
5"). The secondparameteris the valueof the adia-
batic invariant of the slow motion weakly perturbed by
interaction
with
the fast oscillations.
Its definition
de-
pends on the symmetriesof the problem in the limits of
very large and very small •, i.e., for weakly and sharply
curvedfieldlines. For • )) 1 (weaklycurvedfieldlines)
this invariantis the normalizedmagneticmomentp• =
sin250, where50 is the pitchangleat z - 0. For
-
• (( 1 (sharplycurvedfield lines) particledynamicsis
where TAB -- TA + TB is the whole "cucumber"period,
determined
byI'(ao,fio)- sin(Z/2)(a0)
ßfA(sin
(fi0/2)),
the normalized action integral over the fast oscillations
givenby (35) and (37). It is appropriateto estimate acrossthe reversal, where c•0 and fi0 are the equatothe diffusioncoefficient
(52) for smallI • by usingthe rial velocity-direction anglesin a spherical coordinate
analyticalestimates(36) and (38) for TA and TB:
frame,introducedbefore(29).
Di,
I,• l•.•:z(I')
z
Figure 19 summarizesthe different types of particle
behavior in terms of these parameters. Notice that the
(53)oppositelimits of weak (•))
1) and strong(•c •
1)
curvature, or smaller and larger particle velocities,reLet us finally discussthe applicability of the diffusion spectively,are both related with rather regular trajecapproach.The diffusioncoefficient(52) determines
a tories, which, however, differ qualita•tivelyfrom each
characteristic
diffusion
time7'D • (it)2/Dit,it' From other.
the necessarycondition for diffusivity 'rD • TAB one
thereforederivesthe followingcriterion for applicability
of the diffusion approach to the cucumber motion:
The
combination
of both
limits
with
their
dif-
ferent symmetriesin one figure is therefore quite artificial, i.e., the region around •- I separatesdomainsof
qualitativelydifferentdependencies
on pl and I •. Never-
BOCHNERAND ZELENYI:REGULARAND CHAOTIC CHARGEDPARTICLE MOTION
theless,we choosethis form of presentation to combine
for both limits the more field-aligned equatorial velocities on the top and thesenearly parallel to the equatorial
plane z - 0 on the bottom of the graph. This is the rea-
11,839
thesetrajectoriesin (BZ2). The diffusionapproachto
the quasi-adiabaticcucumber-typemotionbreaksdown
as•
<• 1, firstforsmall
I t corresponding
toorbits
mir-
toting far from the reversalplane z - 0 and closeto the
sonwhy in Figure 19 the/•t axisis directeddownward. lossconein naturalfieldreversals
(cf. (BZ2)). In Figure
Aslongas• ->5 thetrajectories
remain
regular
for 19 the transitionto strongchaos(region3) for • < 1
all but very small pitch angles(region (1) in Figure (cf. equation(50))is indicatedb• a ah-ao line. Near
19). Theseorbits are regularand adiabatic;they can • - 1 the motionbecomes
globallychaoticwith a strong
be describedcompletelyusinga perturbation approach. scatteringof the order A/• •/• and AIt ,• I t.
This is the case of lower energy thermal electrons in
For both cases• > 1 and • < 1, I t < 1, we deEarth's magnetotail. First the nearly field-aligned tra- rived and analy•ed appropriatemappingson the slow
jectories,which correspondto small 50 and/•t, become time scale.Theseallow an effectiveinvestigationof the
chaotic. But under natural conditionsthey belong to long term particle dynamics,its transition to determinthe loss cone and leave the reversal after one encounter
istic chaos, and the laws governingchaotic and quasi-
(cf. (BZ2)). With decreasing
• the pitchangledomain adiabatic motion. We verified them by means of numercorrespondingto chaotic trajectoriesincreasesconsider- ical calculationof the mappings(cf. Figures4-6, 17 and
ably. However, near the threshold of chaos,stochastic- 18), aswellasby meansof a numericalintegrationof the
ity is weak and the particle motion can be describedby
a "deterministic" pitch angle scattering and velocity-
original equationsof motion on the fast time scale,and
representedthe results in the form of Poincar• surface
spacediffusion.The relateddiffusion
coefficient
D•,•,
of sectionplots(cf. Figures11, 12, 15, 16).
is given by (18). The regionof weaklychaoticorbits
wherethe diffusionapproachis validis denoted(2) and
8. DISCUSSION
AND APPLICATIONS
MAGNETOTAIL
hatched in Figure 19. For • -• 1 the chaos becomes
strong,/• changesconsiderablyduring traversalsof the
Our findings explain in a natural way, without any
reversal plane, and the weak-diffusion approximation assumption of wave-particle interaction, rather different propertiesof particle dynamicsin a two-dimensional
breaksdown(region(3)in Figure19).
With • < 1 there arise qualitatively different orbits, magnetotaillikefield reversaland of dynamicsof a colwhich are characterized by fast z oscillationsand a slow lisionlessplasma in it.
motion
in x in contrast
with
Larmor
rotations
around
the field line and the slow guiding center drift in the
limit • > 1. But theseorbits exist only for I t > 1, i.e.
in a small portion of about one tenth of the phasespace
TO THE EARTH'S
First, we providethe reason(overlappingof resonances between the bounce and Larmor frequenciesand
the appearance
of deterministic
chaos)for the threshold
intndtx
oenonadiabatic
pit½
(of. equation(29), region4 in Figure 19). For these angle scatteringin curvedweak field •egions. Ou• •ering-type orbits the adiabatic invariant 1.4 holds until sultsexplainwell the scatteringp•ocessnume•icMlyin•-•1.
vestigatedby Tsyganenko
[1982], Gray and Lee [1982]
Most of velocity space for • < 1 is filled, however,
with weakly chaotic but partially adiabatic, cucumber-
and the transition to strong stochasticityas • • 1 observedby Wagneret al. [1979]and others. Sergeevet
typetrajectorieswith I t < 1, whicharestronglyaffected al. [1983]havecarriedout a tracingof particletrajecby separatrix traversals in the phase space of the fast toriesin a model magnetotailcurrent sheetconfiguracomponent of motion. For • << 1 they correspondto tion. In terms of our •-parameter they found that at
an adiabaticity of the action integrals 1.4 and IB in the • • 3.3 nonadiabaticmotion beginsfor small equatosegmentsA and B of the motion which crossand do not rial pitch anglesand at • • 2.4 they found an overall
crossthe neutral plane. This adiabaticity is interrupted
diffusion, which in fact is the result of a transition to de-
by a scatteringof I t = 1.4/lo • 2IB/Io duringtwo terministic chaos. Our findingsexplain well the results
separatrix traversals which occur in the courseof every obtainedby Imhof et al. [1977,1978,1979],Pytte and
cucumbercycle. This scattering is causedby a quasi[1978],
[1983],
opia,i
periodicchangeof the topologyof the fast z oscili'ations [1981,1985],whoinvestigated
the transitionto isotropy
by traversinga separatrixin the {z, i} phaseplanebe- in the pitch angledistributionof energeticprotonsand
tween orbits which cross z -
0 and those which do not.
electronsby meansof groundbasedand satelliteobser-
We call this a cucumber-type motion becausethe mo- vations. The •-parameter may be usedfor a quantitation on the slow time scalehas cucumberliketopology. tive interpretation of in situ particle measurementsin
For small • << 1 it is characterizedby a partially adia- the magnetosphere,which thereforeprovide a powerful
batic but a weaklychaoticdynamicswith < AIt >= 0.
diagnostictool for an investigationof the distantmagnetotal. If the particleenergyis high enough,the chaotic
adiabatic(•egion(5)in Figure 19). The diffusioncoef- pitch angle diffusioncan take place also in the dipoleficient,whichdescribesthe long term behaviorof I t in like, strongerfield regionnear the Earth's trappingrethe quasi-adiabatic regime with the cucumber-type of gions.This explainsthe findingsof Hjin et al. [1986],
motionis givenby (52) and (53). We mentionthat the who found that ionswith energiesof someMeV become
untrapped(Wagner et al. [1979]or Speiser[1965a,b] scatterednear the trapping regions. Notice that indeed
type) trajectoriesbelongto a lossregion,whicharisesin very high ion energiesare necessaryto reach • • I in
realisticmagneticfieldreversals,whereBa•(z)saturates the caseof large curvatureradii becauseenergyenters
We propose that this type of motion be called quasi-
at some distance from the z = 0 plane. We will discuss the •-parameter via its fourth root.
11,840
BOCHNER AND ZELENYI: REGULAR AND CHAOTIC CHARGED PARTICLE MOTION
The •c--valuesof typical thermal ions in the plasma
sheetof Earth's magnetotailare smallerthan unity. Our
theoreticalfindingsconcerningadiabatic ring-type and
quasi-adiabatic cucumber-type orbits predict therefore
increasingelectron energy and prestorm plasma sheet
thinning,however,g decreases
towardunity and the onset of considerableelectronpitch anglescatteringcan be
expected.This leadsto isotropicPAD and SDF. In fact,
of a transifor ionsalsoa pitchanglescattering(in termsof ct0,•30 Westet al. [1978a,b]reportedobservations
determiningI'(ct0,•30)). This predictionis well con- tion from a butterflylike electron PAD with an empty
firmed by the numerical calculations carried of Propp
lossconefor quiet time dipolelike tail configurationsto
andBeard[1984]for protonswith energies
from 2 eV to an isotropic PAD in more taillike fields characterizing
20 keV (i.e., for 7 • • • 0.7, accordingto the param- the prestorm situation. Hence isotropic PAD functions
etersgivenin their paper) at a geocentricdistanceof are not necessarilyrelated to particle trapping in closed
magnetic structures as is usually thought but may also
be causedby chaotic pitch angle diffusionof particles
(• = 3.3-7) can be considered
adiabatic,while higher trapped in open structures.
Concerningion SDFs in Earth's plasmasheet we reenergyprotons(with • • 1.5) are nonadiabatic
and, as
we now know, chaotic.
call that typical g values for thermal ions are considerThe adiabaticand quasi-adiabaticregimes(• < 1) ably less than unity, a situation permitting both adiaseem to be important for an understandingof ion ac- batic ring-type and quasi-adiabaticcucumber-typetraceleration in the plasma sheet of Earth's magnetotail. pped trajectories with both populations well separated
Ions which escape from the reversal immediately after from each other in velocity space. The plasma sheet
its first traversal(Speisertrajectories)are accelerated
in ion SDF therefore forms differently in the adiabatic and
the dawn-duskelectricfield Ey. This process
wasstud- in the quasi-adiabatic velocity space domains. For the
about 40 RE in a realistic tail-field model. They found
that only protonswith energiesbetween2 eV and 40 eV
fills the lossregion
ied by œyonsand Speiser[1982]and Speiserand œyons latter, diffusionof I t systematically
[1984].But this mechanism
appliesonly to oneclassof (the bottomof the graphin Figure19, (cf. BZ2). The
ions, namely, those belongingto the loss region of the intensity of this mechanism,determinedby the diffusion
(52) and by the separatrixtraversalshasto
distribution function. We discussthe influence of sep- coefficient
aratrix traversals on these trajectories in Biichner and
Zelenyi,submitted,1989;(BZ2). Herein this paperwe
analyzed trapped cucumber-andring-type motions,relevant for ions in Earth's magnetotail. In the slightly in-
be comparedwith the loss to the plasma sheet boundary layer as the main sink of particles. Hence quasiadiabatic motion and separatrix traversalsconsiderably
influence
the formation
of the SDF
in addition
to the
homogeneous
normalfield component
Bn(z) theseions role which sourcesand sinksof particlesplay. Using the
also becomeacceleratedby a quasi-adiabatic compres- quasi-adiabaticinvariant I t and the lawsof its weakly
sion of cucumbersand rings in the courseof their aver-
chaoticdiffusionderivedin section6, or usingmappings
age [E x B] earthwardmotion. We haveaddressed
this instead of integrating many ion trajectories,it now betopicespeciallyin anotherpaperB•chner et al. [1988], comeseasy to derive the ion SDF for given boundary
(cf. (SZ2)). But evenwithoutadditionalcalseealso(BZ2)) and foundthat the acceleration
of ions conditions
on cucumberorbits can explain their energizationalong
the tail and their tendency to escapethe reversal at its
edgewith an enhancedflow energydirected toward the
Earth.
Both effects lead to a contribution
of this mech-
anism to the formation of the plasma sheet boundary
layerPSBL (cf. B6chneret al. [1988]).
Another problem closelyassociatedwith the dynamics of charged particles in a magnetic field reversal is
that of the equilibrium velocity-spacedistribution function. For many years the question has been discussed
of whether electron and ion pitch angle distributions
culations one can expect that the formation of the SDF
of ions with • • 1 is strongly influencedby chaotic diffusion. In the case of a strong sourceof isotropic ions
this would lead to an isotropic SDF. Hence no turbulent mechanism seems to be necessaryto maintain an
isotropicion SDF, as considered
by N•zel e• al. [1985].
This is what $•iles e• al. [1978]observed.On the other
hand, in the case of a weak plasma source,diffusion of
I t can create an anisotropicSDF with a maximumof
the distribution function near an equatorial velocity di-
rectionwhich corresponds
to I t-
1.16, the fixed point
of the ring-type motion.
Our findings concerningthe transition to chaos of
characterizingtaillike plasma equilibria are isotropicor
not and if they are anisotropic,to what extent (cf., for particle motion in two-dimensionalmagnetic field reexample,Wright[1985]). The • parametersof plasma versaIsalso relate to the problem of magnetotail stasheetthermal electronsis larger than unity and usually bility and of the unstable releaseof storedenergy durdo not reach the threshold for deterministic
chaos in
ing magnetosphericsubstorms.In the courseof an encontrast with thermal ions and energeticelectrons.Ac- hanced energy accumulation from the solar wind into
cordingto the ideas developedin this paper, low energy the magnetotail the g parameter of plasma sheet therelectronsin the more dipolelike quiet time magnetotail mal electrons decreases due to the increase of the total
have • considerablylarger than unity and must have a field strength B0 and the thinning of the field reverSDF mainlydepending
on sources
and sinks(losscone) sal, i.e., the enhancedfield line curvature in the equatobecausethey are not pitch angle scatteredwithout col- rial plane. If the tail remains metastable, the threshold
lisions or noise. Hence a loss-cone electron distribution
for deterministic chaos of the electron motion may be
can be expectedin this limit. Baker e• al. [1986]ac- reached. Recently, we have found that the consequent
tually observedsuch an electron PAD outside the cen- chaoticpitch angle diffusiondisruptsthe metastableenergy storageand leads to a large scalemagnetotail intral plasma sheet even in the far magnetotail. With
(PAD) or the stationarydistributionfunctions(SDF)
BOCHNER AND ZELENYI:
REGULAR AND CHAOTIC
CHARGED PARTICLE
MOTION
11,841
stability, i.e., the onsetof isolatedmagnetosphericsub- Bfichner, J., About the third integral of charged particle
motion in strongly curved magnetic fields, Astron. Nachr.,
stormsB•chner and2elen$1i
[1987].Thereweprovideda
307, 191, 1986.
WKB stability analysistaking into accountfinite Bn as Bfichner, J., and L. M. Zelenyi, Deterministic chaos in the
a m•in factor, one which usually stabilizes the collisiondynamics of chargedparticles near a magnetic field rever-
lesstearingmode(cf. Galeevand 2elen$1i
[1976],Coro[9s0],
.,d
[19s2]).Chaoticeecton
motion enablesa spontaneouscollisionless
tearing mode
instability,withoutany assumption
concerning
noise(cf.
½oo,m [9s0])o .nomo.
(½f.
[1985]). In this sensethe chaotization
of singleparticle trajectories due to their nonlinear equations of motion may make up for the lack of turbulent dissipation
sal, Phys. Left. A, 118, 395, 1986.
Bfichner, J., and L. M. Zelenyi, Chaotization of the electron
motion as the cause of an internal magnetotail instability
and substorm onset, J. Geophys. Res., 9•, 13,456, 1987.
Bfichner, J., L. M. Zelenyi, and D. V. Zogin, Quasi-adiabatic
plasmasheetion acceleration and formation of the hot boundary layer flows, Adv. Space Res., 1988, in press.
Cary, J. R. and R. T. Skodje, Reaction probability for sequential separatrix crossings,Phys. Rev. Left., 61, 1795,
1988.
pointedout by Coroniti[1985].Hencethe chaotization Chen, J. and P. J. Palrnadesso,Chaos and nonlinear dynamof electron motion yieldsa reasonablemechanismfor the
macroscopicinstability of Earth's magnetot•il and the
onset of isolated substorms. This agreeswell with the
observationof a transition of the magnetot•il electron
PAD to an isotropic one prior to the onset of isolated
substorms
Baker et al. [1988].Henceto our mind it is
also not necessaryto assumethe generationof free energy due a non-Maxwellian distribution function, which
might be caused by chaotic particle motion as Chen
ics of single-particle orbits in a magnetotaillike magnetic
field, J. Geophys. Res., 91, 1499, 1986.
Chirikov, B. V., The problem of stability of chargedparticle's
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slavskyfor discussionsof the problem of deterministic chaos,
tail, Planet. Space Sci., •,
1641, 1974.
to A. I. Neishtadt for his numerous discussionsof the probGaleev, A. A., and L. M. Zelenyi, Tearing instability in
lem of scparatrix traversalsand to V. Sergeevfor discussions
plasma configurations, Soy. Phys. JETP, 43, 1113, 1976.
of the results of ground-based and satellite observationsof Gray, P. C., and L. C. Lee, Particle pitch angle diffusiondue
PAD functions.
We thank the Academies of Sciences of the
to nonadiabatic effects in the plasma sheet, J. Geophys.
German Democratic Republic and USSR as well as the InterRes., 87, 7445, 1982.
cosmosCouncil for making possible this cooperative study. Harris, E.G., On a plasma sheath separating regions of opThe authors are deeply indebted to T. Birmingham for his
positely directed magnetic fields, Nuovo Cimento, •3,115,
great support in editing and publishing this paper.
1962.
The Editor thanks J. Birn, R.F. Martin, and a third refHastie, R. J., G. D. Hobbs, and J. B. Taylor, Nonadiabatic
eree for their assistancein evaluating this paper.
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(ReceivedMarch 28, 1988;
revisedApril 18, 1989;
acceptedMay 16, 1989.)