Particle dynamics in the plasma sheet

VOL. 84, NO. A3

JOURNAL

OF GEOPHYSICAL

RESEARCH

MARCH

1, 1979

Particle Dynamics in the PlasmaSheet J. S. WAGNER, J. R. KAN, AND S.-I. AKASOFU Geophysical Institute, Universityof Alaska, Fairbanks,Alaska 99701 Trajectoriesof chargedparticlesin the trail regionof the earth'smagnetosphereare studiednumerically usinga model magneticfield. It is shownthat both trapped and untrappedtrajectoriescan be categorized into three groupsusingtwo dimensionless parameters.One of the parametersis the ratio of the x and z componentsof the modelmagneticfield. The otheris the ratio of the plasmasheetthicknessto the particle gyroradius in the midplane. Previoustrajectory studiesare incorporatedinto our classificationscheme which resolvesa numberof apparentcontradictoryconclusionsamongthem. The untrappedparticlesare analyzedin terms of the cross-taildisplacement,the displacementtime, the scatteringof the final pitch angle, and the coefficientof reflectionoff the plasma sheet.The resultsare usedto predict the upper limit on particle energizationduring one neutral sheetinteraction.

1. INTRODUCTION

An understandingof the motions of individual charged particlesin the magnetotailplasmasheetis an important first stepin understandingthe collectivedynamicsof the magnetotail plasma as a whole. A number of authors have attempted analytical solutions of the equationsof motion by various approximations. Among them Speiser [1965], Alexeev and Kropotkin [ 1970],and Sonnerup[ 1971] have obtainedapproximate solutionsin the highly nonadiabaticlimit. On the other hand, Stern and Palmadesso[1975] and Stern [1977] have obtained approximate solutionsin the adiabatic limit. Other authorshave attemptedto numericallyintegratethe equations of motion, for example, Speiser [1967], Cowley [1971], Eastwood [1972], Pudovkin and Tsyganenko[1973], and Swift [1977]. However, due to the assumptionsmade in their studies, each author dealt only with a limited variety of trajectories, and therefore their conclusionscannot be generalizedto all conditionsthat occur in the magentotail. In this paper, a numericalanalysisof all possiblemagnetotail particle orbits is presented.The previousworks are incorporated into a new classificationscheme,which resolvesthe apparent disagreementsamong a number of previousstudies. The new classificationmakes use of the dimensionlessequations of motion suggestedby Swift [1977]. The overall morphologyof the orbits is shownto be determinedby two dimensionlessparametersin the equationsof motion.

The magnetic field in (1) is not a self-consistentsolution of the Vlasov-Maxwell equations,but the field configurationdescribed by (1) does resemblethe two-dimensionalanalytic solutionfor the plasmasheetby Kan [1973] and the numerical solutionby Toichi [ 1972].Thus the field model in (1) shouldbe

adequate for examining thetrajectories of testparticles..• The equationof motionfor a chargedparticlein a magnetic field is given by dv

tn•-]-= q(vX B)

(2)

where v is the particle velocity, rn and q are, respectively,the massand chargeof the particle. To numericallyanalyze (2) it is convenient to rewrite (2) in dimensionlessform. For this purposewe chooseto scalethe distanceby a specialgyroradius Oo,and the time by the related inverseangulargyrofrequency ro, based on the normal field componentB,o and the initial speedVoof the test particle, i.e., 0o = mvo/qB•o

(3)

ro = m/qB•o

(4)

Using (3) and (4), the equationof motion (2) can be rewritten in componentform as dv•*/dt* = vy*

(5)

dvy*/dt* = Bx* vz* - vx*

(6)

2. THE MODEL AND THE EQUATIONSOF MOTION

dv**/dt* = -B•* vy*

, (7)

To analyzethe motion of a chargedparticle in the plasma sheet,we chosea magneticfield which is given by

wherex* = X/0o, y* = Y/0o, z* = Z/0o, t* = t/ro, v* = V/Vo,

B = Bx0tanh (z/L)• + Bzoœ

(1)

where the x axis is assumedto lie along the earth-sun line (positivetowardsthe sun), and the y axis lies in the midplane of the resulting plasma sheetand is parallel to the cross-tail current. The z axis is normal to the midplane to completethe right-handedcoordinate system.Note that the magneticfield dependsonly on the z coordinate,and the normal component, Bz0, is constant. The length L can be regarded as the half thicknessof the plasma sheet,sinceBx approachesB•0 as z/L approachesunity. The electricfield is arbitrarily chosento be zero. The effects of any electric field in the cross-tail direction may still be investigatedthrough the use of a specialLorentz transformation. This will be discussed more later.

Copyright ¸ 1979by theAmerican Geophysical Union. Paper number 8A1039. 0148-0227/79/008A- 1039501.00

891

and

B•* = B* tanh (z*/L*)

(8)

B* = B,,o/B,o

(9)

L* = L/po

(10)

Clearly the only constantsin the equationsof motion (5)-(7) are B* and L* definedin (9) and (10). It will be shown that the major characteristics of a particleorbit are determinedby B*, and more importantly by L*. The parameterB*, which is the ratio B•o/B,o, determinesthe limiting angle of the magnetic field as z*/L* becomesgreaterthan unity. Small valuesof B*, around 0.1, are appropriatefor the distantplasmasheet,while valuesof B* near 1.0 more closelysimulatethe conditionsnear the earth. The parameter L* is the ratio of the plasma sheet thicknessto the scalinggyroradius.L* is the parameter which determines if the motion is adiabatic or not. In fact, adiabatic

892

WAGNER ETAL.:PARTICLES IN THEPLASMA SHEET

motion occurs wheneverL* is much greater than one, while nonadiabaticeffectsare obvious when L* approachesor becomes less than one.

Sincethe initial speedVoof a particle hasbeenchosenasthe scalespeed,the initial velocityof a particle is thereforea unit vector whosedirectionis to be specifiedas an initial condition. The pitch angleis definedas0ø whenthe velocityvectoris field aligned,and 180ø when the velocityvectoris antiparallelto the magnetic field. The phaseangle is defined to be 0 ø when the perpendicularvelocity componentlies in the secondquadrant of the x*-z* plane, and increasesin the direction a proton would gyrate. The energy of the particle is always equal to unity in units of initial particle kinetic energy, and does not change since the electric field is transformed away. Hence variations in both initial particle energy and plasma sheet thicknessare studied together through the singleparameter L*, while the effectsof the limiting magneticfield angle are studiedthrough B*. The actual solutionof the orbit is found by numericallyintegratingthe set of equations(6)-(8) using Bulirschand Stoer [ 1966]rational functionextrapolation,with a maximum percenterror per time stepset at 10-8 for both the position and the velocity components.The number of time stepsvaried between 20 and 1000, dependingon trajectory type.

limited rangeof the parameterL*, or in handlingof the crosstail electricfield(whichwill be discussed in section5), andthus his resultscould not be comparedwith others. Case1 Orbits(L* 1. Actually case I orbits were observedwheneverL* < 0.5, case2 orbits were observed when 0.5 < L* < 50, and case 3 orbits were found when L* > 50, but owing to the wide range of possible L* values and the very gradual transition between cases,the somewhat coarseL* > 1 scheme will sufficein the following discussions. Thesecasesare characterized by nonadiabatic, transitional, and adiabatic motions, respectively.Within eachof thesecasesthere are both trapped particles confined to a region parallel to the midplane and untrappedparticleswhich are not confinedand can enter or escape the midplane region. Most of the earlier trajectory papersrestrictedthemselvesto the untrappedcasesonly. Discussionsof the trapped orbitscan be found in Sonnerup[ 1971], and more recentlyin Stern [ 1977],who suggested that a significant number of plasma sheet particles may be in trapped orbits. A typical case 3 trapped orbit is shown in Figure 1. Case 2 and 3 trapped orbits are not shown becausethey are simply circular orbits confined to the midplane. Representative untrappedorbits for eachof the three casesare shownin Figures 2-4. The remaining portions of this sectionare devoted to discussingthe properties of orbits in each of the above three cases.A usefulempiricalrelationshipis givenwhich relatesthe initial and final pitch angle to the cross-taildisplacementfor the untrappedorbits. A comprehensive summaryof the major characteristicsof the three casesis presentedin Table l, which also lists the previous studiesto show how they fit into the overall picture of particle behaviorin the plasmasheet.It will be shown that the confusingvariety of previouslypublished

trajectoriesis the resultof eachauthorrestrictinghisstudyto a

Untrapped case 1 orbits undergo a net cross-taildis-

placement as they passthroughthe plasmasheetand get ejectedearthward.They can be ejectedon eithersideof the

midplane,but the cross-tail displacement is alwaysalongthe +y* axis.The reflection coefficient for case1 trajectories dependsonB*. The reflection coefficient R wascomputed for 52 testorbitswith L* = 0.1 and L* = 0.01, with pitchangles rangingfrom 180ø to 120ø (every20ø), andaveraged overfour phaseangles90ø apart.For thosetrajectories withB* = 1 the coefficientwas R = 0.35, and for those with B* = 10 the

coefficientwasR = 0.85. In general,the reflectioncoefficient decreases as L* gets larger, until the adiabaticcase3 orbits occur with R = 0.

In many examplesthe initial and final pitch angleswere almost equal. For the test orbits the mean differencebetween

the initial and final pitch angle, in degrees,was 2.4 with a standard deviation of 3.1 when B* = 10, and was 19 with a

standarddeviationof 17 for B* = 1.0.This indicatesat • af for large B*. From the numerical results,we are able to deducea relation-

shipbetweenthe initial pitch angle,the final pitch angle,and the cross-taildisplacement as givenby

doo = (&dBo)([ cos., I + I cos I )

(11)

where Bo = (B,,o 2 + Bzo2) •/•, d is the displacement,Oois the scalinggyroradiusdefinedby (3), and ottand ar refer to the initial and final pitch angles,respectively.Equation (11) is plotted in Figure 5, for B,,o/Bo- 1, showingthat the maximum cross-taildisplacement of 200may occuronly for field-aligned particles.Other particleshavedisplacements between0.00 and 200,dependingon their pitchangles.The formulain (11) is not usefulin determiningthe cross-taildisplacementfor eachtrajectory becausethe final pitch angleis an unknown.However, the formula does put an upper limit on the cross-taildisplacementfor one neutral sheetinteraction which can be used to estimatethe maximumenergygain by a particleundergoing one suchinteraction.During the final typing of this paper, Cowley's [1978] paper came to our attention in which he derived our (11) basedon theoretical considerations. Actually, (11) holds for all untrapped trajectoriesin this

WAGNERET AL.: PARTICLESIN THE PLASMASHEET

I y'"

DIMENSIONLESS

(a)

L• =65.0

y•

PARAMETERS

(a)

B• =t0,0

INITIAL

893

DIMENSIONLESS

VALUES

INITIAL

v;:o,4 PITCH :25.0 PHASE =0.0 FINAL

8'--t0.0 VALUES

v':o.o PITCH=t60,O

VALUES

FINAL

v,":o.se X"

90.0

PITCH=t71,2 PHASE=-t33,8 TOTAL TIME = 296.8 DISPLACEMENT TIME =2tt.95

X#

2.0

JZ* (c)

PHASE=VALUES

Vx*=-0.97 V;=-O. tt V•'=-0.24

PITCH =L:XJ. 8 PHASE=78,5 TOTAL TIME= 4?4. ?

(b)

PARAMETERS

L'=tO,O

t0.0 -

.Z*

t*=o(c)

Z*

(b)

0,0

0.0-

X#

-20.0,

0,0

40,0

Fig. I.

y#

t.O

0.0

-•-•.0

-t0.0•• •

0.0

I

•

-20.0

A trappedcase3 orbitß

I

X* •

-40.0

.0

Y* t.0

0.0

Fig. 3. An untrapped case2 orbitß

study.To check(11) the displacements from 130 orbits were

measuredgraphicallyfrom the computerplotscoveringthe entire rangeinitial conditionsand parametervalues.The uncertaintyof the graphicallymeasureddisplacements wasestimated at roughly 2%. The standard deviation between the

Case 1 orbits can be expectedin the distantmagnetotail (>60 RE). Particleswith 7-keV energyin the distanttail (>60 RE) whereBz = 0.5'7 andL = 1 RE would haveL* = 0.2. Higher particle energies,a smaller Bz, or a thinner plasma sheetwould allow case1 proton orbits closerto the earth. It is unlikely that magnetosphericelectronsfollow case 1 orbits anywhere in the tail.

measuredand computeddisplacements (from equation(11)) for all trajectorieswas only 1%, demonstrating that (11) shouldhold overall conditionslikely to occurin the tail. The time the particlespendcrossingthe plasmasheetwas Case2 Orbits(L* • 1) measuredby markingthe timesfor thefirstandlastmidplane In generalthe motion of case2 trajectoriesincludesa combicrossing andthentakingthe difference. The averagemeasured nation of featuresfound in both the case1 (nonadiabatic)and time for 52 case 1 trajectorieswas 3.5 r0 with a standard

case 3 (adiabatic) orbits which will be described in the next

deviationof 0.6 r0. This is in agreement with Speiser[1965] and also agreeswith the intuitiveargumentthat the time

section. For this reason case 2 orbits will be called transitional.

A typical case2 orbit with L* = l0 and B* = l0 is shown in

shouldbe aroundone half of a gyroperiod,or around•rr0.It

Figure 3. The particlewasstartedjust outsidethe boundaryof will be seenthat othertrajectorytypesspendmuchlonger the plasmasheetat z* = l0 with a pitch angle160ø and phase times crossingthe sheet. angle -90.0 ø. The particlemoveswith guidingcentermotion y# (a) DIMENSIONLESS

PARAMETERS

L*=0.t INITIAL

X#

2.0

INITIAL

PHASE =-t80.0

-/J.o

'

VALUES

V•=-0,8•Vy*: 0.t6 Vz"--0.49

PITCH = 36.0 PHASE =-t65.0 TOTAL TIME = 24.85 OISPLACEMENT TIME=4,31

X*

-t4.0

Fig. 2. An untrappedcase 1 orbit.

PHASE'O.O

FINAL VALUES

t%o (c)

'

TOTAL TIME: 28t.0 DISPLACEMENT TIME = 28t.0

X,11.

•to.o

$0,0

I

PARAMETERS

a' =t0,0

PITCH't60,O

VALUES

0.O'

0.0

L* = t00.0

v.".-o.oe

(b) J

-'1.0

DIMENSIONLESS

VALUES

PITCH=440.O FINAL

(a)

0.0

B*=tO.O

*

Z*

Z*

(b)

t•.•.

(c)

t X* ' -t60.0

•

•/

t.0

Fig. 4. An untrappedcase 3 orbit.

t'-o Y* 0.0

894

WAGNER ET AL.: PARTICLESIN THE PLASMASHEET

TABLE 1. Trajectory Summary

Range of L* Typicalregionof occurance(protons) Illustrations

Trajectory type

Case I

Case 2

L* >

1

near earth

Figures 1,4

adiabaticor if trapped,folded figureeight

d'

d'

moving frame

Typicaltime spentcrossingplasma

71'T o

severalbounceperiods

_LfLV -1ds

Speiser[ 1965] Alexeev[ 1970] Sonnerup[ 1971] Speiser[1967] Cowley[ 1971] Eastwood[1972]

Pudovkinetal. [1973]

Stern [1977]

Swift [ 1977]

None

sheet

Previous authors, without electric field

Previous authors, with electric field

until it nearsthe midplane.The guidingcenterthen cannotbe with a standard deviation of 13.96 ro for those with B* = 1.0. defined as the particle oscillatesaround the midplane and By comparisonwith the case I examplesit is clear that case2 crossesit many times, similar to the motion of the previously trajectorieshave enhancedvariations in their displacement discussed caseI orbitsexceptfor the fact that ejectiondoesnot times. The differencebetween the averagedinitial and final occurafter the first cross-taildisplacement.Transitionalcase2 pitch angleswas20ø with a standarddeviationof 19 for B* = trajectoriesoften leavethe immediatevicinity of the midplane 10, and was 17.9 ø with a standard deviation of 15 for B* = 1.0. only temporarily, suchas in the exampleof Figure 3. Consid- This indicatesa poor relationshipbetweeninitial and final erable variation in the motions of transitional orbits occurs, pitch angle.The considerablevariation of final pitch angleasa even with orbits with nearly identical initial conditions. In function of initial pitch angle is shownin Figure 6, for the 52 somecasesa 1ø changein initial phaseangle was enoughto case2 examples.In addition, the adiabaticcase3 examplesare changethe displacement time by an orderof magnitude.Some plotted showing at = at, a useful check on the numerical trajectorieswere much more complicatedthan the example computations. shown, while others were very similar to case I or case 3 Protonsin case2 orbits can be expectedto predominatein untrappedexamples. the midtail region,at a geocentricdistanceof (20-60 Re). For Trapped case2 orbits are confinedto circular orbits in the example,a 5-keV particlein a field with Bz = 2 3•and a plasma midplane like the case I orbits and therefore will not be sheet thickness of I Re would have L* = 2.0. discussed.Untrapped case 2 orbits undergo a net cross-tail displacementas they passthroughthe plasmasheet.They can be ejected on either side of the plasma sheet. For 52 test Case3 Orbits(L* >> 1) examplesthe reflection coefficientis 0.08 for B* = 1.0 and 0.2 Orbits of case3 can be either trapped, as in Figure I or for B* = 10, indicating higher reflectivityfor high B* fields. untrappedas in Figure 4. Unlike the trappedorbitsof casesl The cross-taildisplacementsobeyed(11) in every example, and 2, the trapped orbits in case3 are not confinedto a thin regardlessof the complexity of the orbit. There was consid- regioncenteredabout the midplaneof the plasmasheet.The erable variation in the time spentcrossingthe plasma sheet, case3 trappedorbits appearas foldedfigureeights.Note that however.The displacementtime was measuredby differentiat- the midplanecrossingpoint doesnot shift acrossthe tail with bounces,in agreementwith Stern's[1977]predicing the times when the particle entered and left the plasma successive sheet(z* = L*) for the first and last time. For 52 testexamples tion that the orbit should be drift free. As we will refer to the the average displacementtime was 96.6 ro with a standard shortly,case3 orbits are fundamentalin understanding structureof the near-earthplasmasheetand probablyexplain deviation of 90.2 ro for those with B* = 10.0 and it was 18.9 ro the formation

of the thin current sheet centered on the mid-

plane.

Sincecase3 trajectoriesare nearlyadiabatic,the magnetic momentis conservedand givesan expression specifyingwhich orbits are trapped and which untrapped. Let at be the loss cone angle on the equatorial plane, then at is related to the magneticfield, throughthe conservation of the magneticmoment, by Bzo

sin (at)= (Bzo: + Bxo:) •/:

Fig. 5. Three-dimensional plot of the cross-taildisplacement versusinitial and final pitch angle.

(12)

where Bxois the asymtotic value of B• for large z. Those particleswith an equatorial pitch angle lessthan that of the losscone(So< at) follow untrappedorbitsasshownin Figure I. Those trajectorieswith ao > at are trapped as shown in Figure 4. The trapped orbits are drift free and do not shift, sinceB•ois assumedconstantin this paper. For the observed

WAGNER ET AL.: PARTICLES IN THE PLASMA SHEET

O Case 3 Adiabatic

variations of Bzo in the tail the net cross-tail shift remains

A

small, as noted by Stern [1977]. Let us examinefirst the representativecase3 trapped orbit in Figure 1. The particleis startedin the equatorialmidplane z* - 0 with pitch angle25ø. The field parametersusedare L* = 65 and B* = 10. The particle'sguidingcenterspendsmostof its time in the mirroring processas illustrated.In a thin layer surroundingthe midplane the guiding center always crosses the midplanein the samedirection and in a short period of

with B* - 0.1 and was 153.5 with a standard deviation of 36.2

for the B* = 1 orbits. The difference between initial and final

pitch angleswas0.12ø with a standarddeviationof 0.1 for all case3 orbitstested,indicatingthat at • ar is a good approximation. The relationship between a• and a r is shown in

t80

--

•60

--

Case

2

Transd•onal

_

t40

-_

time, around 1/100 of the total bounce time. The thicknessof this particular region is roughly 1/10 of the total plasmasheet thickness.It is clear that the gradient drift is responsiblefor

the displacementof nearly -200 during the mirroring process, and the curvaturedrift is responsiblefor the displacementof nearly+ 200as the particlecrosses the thin layer surrounding the midplane.The two drifts averageout so that there is no cross-taildrift. Although the figure-eightorbitsproduceno net current acrossthe tail, they do produce a thin current layer centeredon the midplane.Particlesnear the outer edgeof the plasmasheetare mirroring and their guidingcenteris moving slowly against the electric field. Particles near the midplane move rapidly along the +y* axis. Finally, considerthe typical untrappedorbit as in Figure 3. For illustrativepurposes,and to saveon computertime, the particleis startedat z* - 100in a fieldwith L* = 100andB* = 10. The initial pitch angleis 160ø and the initial phaseangleis 0. Onceagainthe cross-taildisplacement obeys(11). Sincecase 3 orbits are nearly adiabatic, the final pitch angle is always closeto the initial value. Case 3 untrappedorbits alwayspass through the plasmasheetso the reflectioncoefficientis 0. The cross-taildisplacementtime, averagedover pitch and phase angles,was 315.4 with a standarddeviationof 35.7 for those

895

t20

--

t00

--

I 1oo

INITIAL

140 PITCH

i

I t80

ANGLE

Fig.6. Pitchangle dataforcase2(transitional) andcase) (adiabatic) orbits.

L*


1, and this is the type of guidingcentermotion discussedby Stern and Palmadesso[1975]. They pointed out that the symmetricfield of the magnetotailshouldbe drift free for adiabatic particles and that the guiding center curvature and gradient drifts cancelwhen averagedover a trapped orbit. Stern [1977] also showeda net cross-tail drift should occur if Bz varieswith x. In addition, he computedthe averagebounce period of drift free trapped orbits; however, he did not consideruntrappedadiabaticorbits, or nonadiabaticorbits. Thus his orbits belong to trapped class3. Speiser [1967] was the first to numericallyintegrate the equationsof motionof particlesin the magnetotail.In termsof the parameterL* his trajectoriesfall into case1 (untrapped). This is becauseSpeiserusedB•o - 10-: 7, making L* very small (•0.01); in addition, he usedlow-energy(200 ev) particles and an electric field of 3 X 10-4 V/km.

Probably the most extensivenumerical treatment of the orbit problem to date is that of Eastwood[1972, 1975]. The majority of the trajectories Eastwood studied were of the plasmasheet.For example,a 2-keV protonin a fieldwith Bz = Speisertype, case 1. Eastwooduseda B• around 0.4 7, initial 10 7 and a plasmasheetthicknessof 10 Re would have L* = velocity 750 km/s, and an electricfield of 0.3 V/km. This gives 100, and shouldlook like either Figure 1 or Figure 4. L* - 10-4. He did notice that B• plays a fundamental role in determiningorbit morphology and listed the types of orbits 4. PREVIOUS ORBIT STUDIES accordingto whetherthey wereadiabatic,trapped,or coupled. AlexeevandKropotkin[1970]considered trajectoriesin an In their numerical approach, Pudovkin and Tsyganenko infinitelythin plasmasheet,with L* - 0. They did not con- [1973]usedvaluesof 1-5 7 for B•. They usedinitial energiesof siderthe effectof an electricfield, and furthermore assumedB• 10 keV and electricfields around 0.3 V/km. This givesL* • 1 Figure 6. Protons in case 3 orbits can be expectedin the near earth